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Properties

Label 144.14.a.k.1.1
Level $144$
Weight $14$
Character 144.1
Self dual yes
Analytic conductor $154.413$
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] = [144,14,Mod(1,144)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(144, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 14, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.1");
 
S:= CuspForms(chi, 14);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 144.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: \(154.412537691\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 144.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+30210.0 q^{5} -235088. q^{7} -1.11829e7 q^{11} +8.04961e6 q^{13} +1.17495e8 q^{17} +2.14061e8 q^{19} +8.30556e8 q^{23} -3.08059e8 q^{25} +1.25240e9 q^{29} -6.15935e9 q^{31} -7.10201e9 q^{35} -5.49819e9 q^{37} +4.67869e9 q^{41} -7.11501e9 q^{43} -2.95288e10 q^{47} -4.16226e10 q^{49} +2.04125e11 q^{53} -3.37836e11 q^{55} -2.99098e10 q^{59} -1.34392e11 q^{61} +2.43179e11 q^{65} -3.48519e11 q^{67} +1.31434e12 q^{71} -1.17888e12 q^{73} +2.62897e12 q^{77} +1.07242e12 q^{79} +1.12403e12 q^{83} +3.54951e12 q^{85} -2.23561e12 q^{89} -1.89237e12 q^{91} +6.46679e12 q^{95} -1.42153e13 q^{97} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 30210.0 0.864661 0.432330 − 0.901715i \(-0.357691\pi\)
0.432330 + 0.901715i \(0.357691\pi\)
\(6\) 0 0
\(7\) −235088. −0.755254 −0.377627 − 0.925958i \(-0.623260\pi\)
−0.377627 + 0.925958i \(0.623260\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.11829e7 −1.90328 −0.951639 − 0.307218i \(-0.900602\pi\)
−0.951639 + 0.307218i \(0.900602\pi\)
\(12\) 0 0
\(13\) 8.04961e6 0.462534 0.231267 − 0.972890i \(-0.425713\pi\)
0.231267 + 0.972890i \(0.425713\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.17495e8 1.18059 0.590296 − 0.807187i \(-0.299011\pi\)
0.590296 + 0.807187i \(0.299011\pi\)
\(18\) 0 0
\(19\) 2.14061e8 1.04385 0.521927 − 0.852990i \(-0.325213\pi\)
0.521927 + 0.852990i \(0.325213\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.30556e8 1.16987 0.584935 − 0.811080i \(-0.301120\pi\)
0.584935 + 0.811080i \(0.301120\pi\)
\(24\) 0 0
\(25\) −3.08059e8 −0.252362
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.25240e9 0.390981 0.195491 − 0.980706i \(-0.437370\pi\)
0.195491 + 0.980706i \(0.437370\pi\)
\(30\) 0 0
\(31\) −6.15935e9 −1.24648 −0.623238 − 0.782032i \(-0.714183\pi\)
−0.623238 + 0.782032i \(0.714183\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −7.10201e9 −0.653039
\(36\) 0 0
\(37\) −5.49819e9 −0.352297 −0.176148 − 0.984364i \(-0.556364\pi\)
−0.176148 + 0.984364i \(0.556364\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.67869e9 0.153826 0.0769129 − 0.997038i \(-0.475494\pi\)
0.0769129 + 0.997038i \(0.475494\pi\)
\(42\) 0 0
\(43\) −7.11501e9 −0.171645 −0.0858224 − 0.996310i \(-0.527352\pi\)
−0.0858224 + 0.996310i \(0.527352\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.95288e10 −0.399585 −0.199793 − 0.979838i \(-0.564027\pi\)
−0.199793 + 0.979838i \(0.564027\pi\)
\(48\) 0 0
\(49\) −4.16226e10 −0.429591
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.04125e11 1.26504 0.632518 − 0.774545i \(-0.282021\pi\)
0.632518 + 0.774545i \(0.282021\pi\)
\(54\) 0 0
\(55\) −3.37836e11 −1.64569
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.99098e10 −0.0923157 −0.0461579 − 0.998934i \(-0.514698\pi\)
−0.0461579 + 0.998934i \(0.514698\pi\)
\(60\) 0 0
\(61\) −1.34392e11 −0.333987 −0.166993 − 0.985958i \(-0.553406\pi\)
−0.166993 + 0.985958i \(0.553406\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.43179e11 0.399935
\(66\) 0 0
\(67\) −3.48519e11 −0.470695 −0.235348 − 0.971911i \(-0.575623\pi\)
−0.235348 + 0.971911i \(0.575623\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1.31434e12 1.21766 0.608831 − 0.793300i \(-0.291639\pi\)
0.608831 + 0.793300i \(0.291639\pi\)
\(72\) 0 0
\(73\) −1.17888e12 −0.911737 −0.455868 − 0.890047i \(-0.650671\pi\)
−0.455868 + 0.890047i \(0.650671\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.62897e12 1.43746
\(78\) 0 0
\(79\) 1.07242e12 0.496351 0.248176 − 0.968715i \(-0.420169\pi\)
0.248176 + 0.968715i \(0.420169\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.12403e12 0.377371 0.188685 − 0.982038i \(-0.439577\pi\)
0.188685 + 0.982038i \(0.439577\pi\)
\(84\) 0 0
\(85\) 3.54951e12 1.02081
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −2.23561e12 −0.476827 −0.238414 − 0.971164i \(-0.576627\pi\)
−0.238414 + 0.971164i \(0.576627\pi\)
\(90\) 0 0
\(91\) −1.89237e12 −0.349330
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 6.46679e12 0.902580
\(96\) 0 0
\(97\) −1.42153e13 −1.73276 −0.866380 − 0.499385i \(-0.833559\pi\)
−0.866380 + 0.499385i \(0.833559\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.70194e13 −1.59535 −0.797675 − 0.603088i \(-0.793937\pi\)
−0.797675 + 0.603088i \(0.793937\pi\)
\(102\) 0 0
\(103\) −1.09904e13 −0.906928 −0.453464 − 0.891275i \(-0.649812\pi\)
−0.453464 + 0.891275i \(0.649812\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1.96403e13 −1.26519 −0.632593 − 0.774485i \(-0.718009\pi\)
−0.632593 + 0.774485i \(0.718009\pi\)
\(108\) 0 0
\(109\) −9.82099e12 −0.560897 −0.280448 − 0.959869i \(-0.590483\pi\)
−0.280448 + 0.959869i \(0.590483\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.70267e13 0.769344 0.384672 − 0.923053i \(-0.374315\pi\)
0.384672 + 0.923053i \(0.374315\pi\)
\(114\) 0 0
\(115\) 2.50911e13 1.01154
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −2.76216e13 −0.891648
\(120\) 0 0
\(121\) 9.05347e13 2.62247
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −4.61839e13 −1.08287
\(126\) 0 0
\(127\) 4.49347e13 0.950292 0.475146 − 0.879907i \(-0.342395\pi\)
0.475146 + 0.879907i \(0.342395\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1.20182e12 −0.0207768 −0.0103884 − 0.999946i \(-0.503307\pi\)
−0.0103884 + 0.999946i \(0.503307\pi\)
\(132\) 0 0
\(133\) −5.03233e13 −0.788375
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1.71562e13 −0.221685 −0.110842 − 0.993838i \(-0.535355\pi\)
−0.110842 + 0.993838i \(0.535355\pi\)
\(138\) 0 0
\(139\) −1.05644e14 −1.24236 −0.621182 − 0.783666i \(-0.713347\pi\)
−0.621182 + 0.783666i \(0.713347\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −9.00181e13 −0.880330
\(144\) 0 0
\(145\) 3.78350e13 0.338066
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 8.53533e13 0.639012 0.319506 − 0.947584i \(-0.396483\pi\)
0.319506 + 0.947584i \(0.396483\pi\)
\(150\) 0 0
\(151\) 6.16414e13 0.423177 0.211589 − 0.977359i \(-0.432136\pi\)
0.211589 + 0.977359i \(0.432136\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −1.86074e14 −1.07778
\(156\) 0 0
\(157\) −1.18021e14 −0.628942 −0.314471 − 0.949267i \(-0.601827\pi\)
−0.314471 + 0.949267i \(0.601827\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −1.95254e14 −0.883550
\(162\) 0 0
\(163\) −1.54710e14 −0.646099 −0.323050 − 0.946382i \(-0.604708\pi\)
−0.323050 + 0.946382i \(0.604708\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 3.76012e14 1.34136 0.670679 − 0.741748i \(-0.266003\pi\)
0.670679 + 0.741748i \(0.266003\pi\)
\(168\) 0 0
\(169\) −2.38079e14 −0.786063
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −3.73562e14 −1.05941 −0.529704 − 0.848182i \(-0.677697\pi\)
−0.529704 + 0.848182i \(0.677697\pi\)
\(174\) 0 0
\(175\) 7.24210e13 0.190597
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 4.23349e13 0.0961952 0.0480976 − 0.998843i \(-0.484684\pi\)
0.0480976 + 0.998843i \(0.484684\pi\)
\(180\) 0 0
\(181\) −3.10447e14 −0.656261 −0.328130 − 0.944632i \(-0.606419\pi\)
−0.328130 + 0.944632i \(0.606419\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.66100e14 −0.304617
\(186\) 0 0
\(187\) −1.31393e15 −2.24700
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −8.62273e14 −1.28507 −0.642537 − 0.766255i \(-0.722118\pi\)
−0.642537 + 0.766255i \(0.722118\pi\)
\(192\) 0 0
\(193\) −9.37837e14 −1.30618 −0.653092 − 0.757278i \(-0.726529\pi\)
−0.653092 + 0.757278i \(0.726529\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 6.71715e14 0.818756 0.409378 − 0.912365i \(-0.365746\pi\)
0.409378 + 0.912365i \(0.365746\pi\)
\(198\) 0 0
\(199\) 4.36451e13 0.0498185 0.0249093 − 0.999690i \(-0.492070\pi\)
0.0249093 + 0.999690i \(0.492070\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −2.94424e14 −0.295290
\(204\) 0 0
\(205\) 1.41343e14 0.133007
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −2.39383e15 −1.98675
\(210\) 0 0
\(211\) 1.62162e15 1.26507 0.632534 − 0.774533i \(-0.282015\pi\)
0.632534 + 0.774533i \(0.282015\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −2.14945e14 −0.148415
\(216\) 0 0
\(217\) 1.44799e15 0.941406
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 9.45786e14 0.546064
\(222\) 0 0
\(223\) −1.47333e15 −0.802266 −0.401133 − 0.916020i \(-0.631383\pi\)
−0.401133 + 0.916020i \(0.631383\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −3.74889e15 −1.81859 −0.909294 − 0.416153i \(-0.863378\pi\)
−0.909294 + 0.416153i \(0.863378\pi\)
\(228\) 0 0
\(229\) −1.47993e13 −0.00678126 −0.00339063 − 0.999994i \(-0.501079\pi\)
−0.00339063 + 0.999994i \(0.501079\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −3.63053e15 −1.48647 −0.743236 − 0.669030i \(-0.766710\pi\)
−0.743236 + 0.669030i \(0.766710\pi\)
\(234\) 0 0
\(235\) −8.92064e14 −0.345506
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −4.33900e15 −1.50592 −0.752962 − 0.658063i \(-0.771376\pi\)
−0.752962 + 0.658063i \(0.771376\pi\)
\(240\) 0 0
\(241\) 3.02372e15 0.994103 0.497051 − 0.867721i \(-0.334416\pi\)
0.497051 + 0.867721i \(0.334416\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.25742e15 −0.371450
\(246\) 0 0
\(247\) 1.72311e15 0.482818
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −1.75146e15 −0.442099 −0.221050 − 0.975263i \(-0.570948\pi\)
−0.221050 + 0.975263i \(0.570948\pi\)
\(252\) 0 0
\(253\) −9.28803e15 −2.22659
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −4.87604e15 −1.05561 −0.527803 − 0.849367i \(-0.676984\pi\)
−0.527803 + 0.849367i \(0.676984\pi\)
\(258\) 0 0
\(259\) 1.29256e15 0.266074
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 4.67882e15 0.871815 0.435907 − 0.899992i \(-0.356428\pi\)
0.435907 + 0.899992i \(0.356428\pi\)
\(264\) 0 0
\(265\) 6.16662e15 1.09383
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 1.80262e15 0.290078 0.145039 − 0.989426i \(-0.453669\pi\)
0.145039 + 0.989426i \(0.453669\pi\)
\(270\) 0 0
\(271\) −6.10016e15 −0.935494 −0.467747 − 0.883862i \(-0.654934\pi\)
−0.467747 + 0.883862i \(0.654934\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.44500e15 0.480315
\(276\) 0 0
\(277\) −1.07023e16 −1.42351 −0.711754 − 0.702428i \(-0.752099\pi\)
−0.711754 + 0.702428i \(0.752099\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2.45460e15 0.297433 0.148717 − 0.988880i \(-0.452486\pi\)
0.148717 + 0.988880i \(0.452486\pi\)
\(282\) 0 0
\(283\) −4.01155e15 −0.464195 −0.232098 − 0.972692i \(-0.574559\pi\)
−0.232098 + 0.972692i \(0.574559\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −1.09990e15 −0.116178
\(288\) 0 0
\(289\) 3.90041e15 0.393799
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −2.08187e15 −0.192227 −0.0961133 − 0.995370i \(-0.530641\pi\)
−0.0961133 + 0.995370i \(0.530641\pi\)
\(294\) 0 0
\(295\) −9.03576e14 −0.0798218
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 6.68565e15 0.541104
\(300\) 0 0
\(301\) 1.67265e15 0.129636
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −4.05998e15 −0.288785
\(306\) 0 0
\(307\) 1.32352e16 0.902260 0.451130 − 0.892458i \(-0.351021\pi\)
0.451130 + 0.892458i \(0.351021\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −8.09301e15 −0.507187 −0.253593 − 0.967311i \(-0.581612\pi\)
−0.253593 + 0.967311i \(0.581612\pi\)
\(312\) 0 0
\(313\) −1.48181e16 −0.890748 −0.445374 − 0.895345i \(-0.646929\pi\)
−0.445374 + 0.895345i \(0.646929\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.43171e16 1.34594 0.672970 − 0.739670i \(-0.265018\pi\)
0.672970 + 0.739670i \(0.265018\pi\)
\(318\) 0 0
\(319\) −1.40055e16 −0.744147
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2.51511e16 1.23237
\(324\) 0 0
\(325\) −2.47976e15 −0.116726
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 6.94186e15 0.301788
\(330\) 0 0
\(331\) −1.16232e16 −0.485783 −0.242892 − 0.970053i \(-0.578096\pi\)
−0.242892 + 0.970053i \(0.578096\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −1.05288e16 −0.406992
\(336\) 0 0
\(337\) 4.62652e16 1.72052 0.860262 − 0.509853i \(-0.170300\pi\)
0.860262 + 0.509853i \(0.170300\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 6.88795e16 2.37239
\(342\) 0 0
\(343\) 3.25624e16 1.07970
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 4.79404e15 0.147421 0.0737106 − 0.997280i \(-0.476516\pi\)
0.0737106 + 0.997280i \(0.476516\pi\)
\(348\) 0 0
\(349\) 3.76900e16 1.11651 0.558253 − 0.829671i \(-0.311472\pi\)
0.558253 + 0.829671i \(0.311472\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −4.80179e16 −1.32089 −0.660446 − 0.750873i \(-0.729633\pi\)
−0.660446 + 0.750873i \(0.729633\pi\)
\(354\) 0 0
\(355\) 3.97061e16 1.05286
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 4.06616e16 1.00247 0.501234 − 0.865312i \(-0.332880\pi\)
0.501234 + 0.865312i \(0.332880\pi\)
\(360\) 0 0
\(361\) 3.76929e15 0.0896320
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −3.56138e16 −0.788343
\(366\) 0 0
\(367\) −2.96733e16 −0.633923 −0.316961 − 0.948438i \(-0.602663\pi\)
−0.316961 + 0.948438i \(0.602663\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −4.79873e16 −0.955424
\(372\) 0 0
\(373\) −9.01346e16 −1.73294 −0.866471 − 0.499227i \(-0.833617\pi\)
−0.866471 + 0.499227i \(0.833617\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.00813e16 0.180842
\(378\) 0 0
\(379\) 1.54841e16 0.268369 0.134184 − 0.990956i \(-0.457159\pi\)
0.134184 + 0.990956i \(0.457159\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 9.37088e15 0.151701 0.0758505 − 0.997119i \(-0.475833\pi\)
0.0758505 + 0.997119i \(0.475833\pi\)
\(384\) 0 0
\(385\) 7.94211e16 1.24291
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −2.95806e16 −0.432847 −0.216423 − 0.976300i \(-0.569439\pi\)
−0.216423 + 0.976300i \(0.569439\pi\)
\(390\) 0 0
\(391\) 9.75858e16 1.38114
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 3.23978e16 0.429175
\(396\) 0 0
\(397\) 1.80617e16 0.231538 0.115769 − 0.993276i \(-0.463067\pi\)
0.115769 + 0.993276i \(0.463067\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.20412e17 1.44621 0.723107 − 0.690736i \(-0.242713\pi\)
0.723107 + 0.690736i \(0.242713\pi\)
\(402\) 0 0
\(403\) −4.95804e16 −0.576537
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 6.14858e16 0.670519
\(408\) 0 0
\(409\) −1.77522e16 −0.187521 −0.0937606 − 0.995595i \(-0.529889\pi\)
−0.0937606 + 0.995595i \(0.529889\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 7.03144e15 0.0697219
\(414\) 0 0
\(415\) 3.39568e16 0.326298
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 1.75670e17 1.58602 0.793008 − 0.609212i \(-0.208514\pi\)
0.793008 + 0.609212i \(0.208514\pi\)
\(420\) 0 0
\(421\) 1.84473e17 1.61473 0.807365 − 0.590052i \(-0.200893\pi\)
0.807365 + 0.590052i \(0.200893\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3.61953e16 −0.297937
\(426\) 0 0
\(427\) 3.15939e16 0.252245
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 8.05532e16 0.605314 0.302657 − 0.953100i \(-0.402126\pi\)
0.302657 + 0.953100i \(0.402126\pi\)
\(432\) 0 0
\(433\) −1.97092e17 −1.43714 −0.718568 − 0.695457i \(-0.755202\pi\)
−0.718568 + 0.695457i \(0.755202\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.77790e17 1.22117
\(438\) 0 0
\(439\) −9.89007e16 −0.659447 −0.329724 − 0.944078i \(-0.606956\pi\)
−0.329724 + 0.944078i \(0.606956\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −1.25104e17 −0.786404 −0.393202 − 0.919452i \(-0.628633\pi\)
−0.393202 + 0.919452i \(0.628633\pi\)
\(444\) 0 0
\(445\) −6.75378e16 −0.412294
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.80095e17 1.03729 0.518645 − 0.854990i \(-0.326437\pi\)
0.518645 + 0.854990i \(0.326437\pi\)
\(450\) 0 0
\(451\) −5.23213e16 −0.292773
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −5.71684e16 −0.302052
\(456\) 0 0
\(457\) −9.43597e16 −0.484542 −0.242271 − 0.970209i \(-0.577892\pi\)
−0.242271 + 0.970209i \(0.577892\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −8.00500e16 −0.388423 −0.194212 − 0.980960i \(-0.562215\pi\)
−0.194212 + 0.980960i \(0.562215\pi\)
\(462\) 0 0
\(463\) −2.14174e17 −1.01039 −0.505196 − 0.863004i \(-0.668580\pi\)
−0.505196 + 0.863004i \(0.668580\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −1.80681e17 −0.806031 −0.403015 − 0.915193i \(-0.632038\pi\)
−0.403015 + 0.915193i \(0.632038\pi\)
\(468\) 0 0
\(469\) 8.19326e16 0.355495
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 7.95665e16 0.326688
\(474\) 0 0
\(475\) −6.59435e16 −0.263429
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −2.66712e17 −1.00893 −0.504466 − 0.863431i \(-0.668311\pi\)
−0.504466 + 0.863431i \(0.668311\pi\)
\(480\) 0 0
\(481\) −4.42583e16 −0.162949
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −4.29443e17 −1.49825
\(486\) 0 0
\(487\) 2.63552e17 0.895216 0.447608 − 0.894230i \(-0.352276\pi\)
0.447608 + 0.894230i \(0.352276\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 4.11733e17 1.32613 0.663065 − 0.748562i \(-0.269255\pi\)
0.663065 + 0.748562i \(0.269255\pi\)
\(492\) 0 0
\(493\) 1.47150e17 0.461590
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −3.08984e17 −0.919645
\(498\) 0 0
\(499\) −3.99658e17 −1.15887 −0.579435 − 0.815018i \(-0.696727\pi\)
−0.579435 + 0.815018i \(0.696727\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −2.83581e17 −0.780702 −0.390351 − 0.920666i \(-0.627646\pi\)
−0.390351 + 0.920666i \(0.627646\pi\)
\(504\) 0 0
\(505\) −5.14157e17 −1.37944
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −6.40327e17 −1.63206 −0.816030 − 0.578009i \(-0.803830\pi\)
−0.816030 + 0.578009i \(0.803830\pi\)
\(510\) 0 0
\(511\) 2.77140e17 0.688593
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −3.32021e17 −0.784185
\(516\) 0 0
\(517\) 3.30218e17 0.760522
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 4.01348e17 0.879175 0.439588 − 0.898200i \(-0.355125\pi\)
0.439588 + 0.898200i \(0.355125\pi\)
\(522\) 0 0
\(523\) 5.05985e17 1.08113 0.540564 − 0.841303i \(-0.318211\pi\)
0.540564 + 0.841303i \(0.318211\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −7.23691e17 −1.47158
\(528\) 0 0
\(529\) 1.85786e17 0.368597
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 3.76616e16 0.0711496
\(534\) 0 0
\(535\) −5.93334e17 −1.09396
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 4.65462e17 0.817631
\(540\) 0 0
\(541\) −1.69124e17 −0.290017 −0.145009 − 0.989430i \(-0.546321\pi\)
−0.145009 + 0.989430i \(0.546321\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −2.96692e17 −0.484986
\(546\) 0 0
\(547\) 4.32104e17 0.689717 0.344858 − 0.938655i \(-0.387927\pi\)
0.344858 + 0.938655i \(0.387927\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 2.68091e17 0.408128
\(552\) 0 0
\(553\) −2.52113e17 −0.374871
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −1.36804e18 −1.94107 −0.970534 − 0.240966i \(-0.922536\pi\)
−0.970534 + 0.240966i \(0.922536\pi\)
\(558\) 0 0
\(559\) −5.72731e16 −0.0793915
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −9.52405e17 −1.26043 −0.630213 − 0.776422i \(-0.717032\pi\)
−0.630213 + 0.776422i \(0.717032\pi\)
\(564\) 0 0
\(565\) 5.14377e17 0.665221
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −1.53632e17 −0.189780 −0.0948902 − 0.995488i \(-0.530250\pi\)
−0.0948902 + 0.995488i \(0.530250\pi\)
\(570\) 0 0
\(571\) 1.27956e18 1.54500 0.772498 − 0.635017i \(-0.219007\pi\)
0.772498 + 0.635017i \(0.219007\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −2.55860e17 −0.295231
\(576\) 0 0
\(577\) 3.56770e17 0.402481 0.201241 − 0.979542i \(-0.435503\pi\)
0.201241 + 0.979542i \(0.435503\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −2.64245e17 −0.285011
\(582\) 0 0
\(583\) −2.28271e18 −2.40772
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.28968e18 1.30118 0.650588 − 0.759431i \(-0.274523\pi\)
0.650588 + 0.759431i \(0.274523\pi\)
\(588\) 0 0
\(589\) −1.31848e18 −1.30114
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1.88640e18 1.78147 0.890735 − 0.454523i \(-0.150190\pi\)
0.890735 + 0.454523i \(0.150190\pi\)
\(594\) 0 0
\(595\) −8.34448e17 −0.770973
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −1.44668e18 −1.27967 −0.639834 − 0.768513i \(-0.720997\pi\)
−0.639834 + 0.768513i \(0.720997\pi\)
\(600\) 0 0
\(601\) −4.44358e16 −0.0384635 −0.0192317 − 0.999815i \(-0.506122\pi\)
−0.0192317 + 0.999815i \(0.506122\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 2.73505e18 2.26755
\(606\) 0 0
\(607\) −2.98050e16 −0.0241860 −0.0120930 − 0.999927i \(-0.503849\pi\)
−0.0120930 + 0.999927i \(0.503849\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −2.37695e17 −0.184822
\(612\) 0 0
\(613\) −8.84082e17 −0.672976 −0.336488 − 0.941688i \(-0.609239\pi\)
−0.336488 + 0.941688i \(0.609239\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1.43684e18 −1.04846 −0.524232 − 0.851575i \(-0.675648\pi\)
−0.524232 + 0.851575i \(0.675648\pi\)
\(618\) 0 0
\(619\) −1.68862e18 −1.20654 −0.603272 − 0.797535i \(-0.706137\pi\)
−0.603272 + 0.797535i \(0.706137\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 5.25565e17 0.360126
\(624\) 0 0
\(625\) −1.01917e18 −0.683951
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −6.46008e17 −0.415919
\(630\) 0 0
\(631\) 3.53490e17 0.222939 0.111470 − 0.993768i \(-0.464444\pi\)
0.111470 + 0.993768i \(0.464444\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1.35748e18 0.821680
\(636\) 0 0
\(637\) −3.35046e17 −0.198700
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −1.61802e18 −0.921313 −0.460656 − 0.887579i \(-0.652386\pi\)
−0.460656 + 0.887579i \(0.652386\pi\)
\(642\) 0 0
\(643\) 1.96065e18 1.09403 0.547015 − 0.837123i \(-0.315764\pi\)
0.547015 + 0.837123i \(0.315764\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −5.96114e17 −0.319486 −0.159743 − 0.987159i \(-0.551067\pi\)
−0.159743 + 0.987159i \(0.551067\pi\)
\(648\) 0 0
\(649\) 3.34479e17 0.175703
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 2.58318e18 1.30382 0.651912 − 0.758295i \(-0.273967\pi\)
0.651912 + 0.758295i \(0.273967\pi\)
\(654\) 0 0
\(655\) −3.63071e16 −0.0179648
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 2.64137e18 1.25624 0.628121 − 0.778116i \(-0.283824\pi\)
0.628121 + 0.778116i \(0.283824\pi\)
\(660\) 0 0
\(661\) 4.12451e18 1.92337 0.961685 − 0.274156i \(-0.0883983\pi\)
0.961685 + 0.274156i \(0.0883983\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −1.52027e18 −0.681677
\(666\) 0 0
\(667\) 1.04019e18 0.457398
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1.50289e18 0.635670
\(672\) 0 0
\(673\) 2.79726e18 1.16047 0.580236 − 0.814449i \(-0.302961\pi\)
0.580236 + 0.814449i \(0.302961\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 4.25553e18 1.69874 0.849372 − 0.527795i \(-0.176981\pi\)
0.849372 + 0.527795i \(0.176981\pi\)
\(678\) 0 0
\(679\) 3.34184e18 1.30867
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1.60893e18 0.606461 0.303230 − 0.952917i \(-0.401935\pi\)
0.303230 + 0.952917i \(0.401935\pi\)
\(684\) 0 0
\(685\) −5.18287e17 −0.191682
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1.64313e18 0.585122
\(690\) 0 0
\(691\) 3.06331e18 1.07049 0.535247 − 0.844696i \(-0.320218\pi\)
0.535247 + 0.844696i \(0.320218\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −3.19151e18 −1.07422
\(696\) 0 0
\(697\) 5.49721e17 0.181606
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −2.99144e18 −0.952166 −0.476083 − 0.879400i \(-0.657944\pi\)
−0.476083 + 0.879400i \(0.657944\pi\)
\(702\) 0 0
\(703\) −1.17695e18 −0.367747
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 4.00106e18 1.20489
\(708\) 0 0
\(709\) −5.31694e18 −1.57203 −0.786015 − 0.618207i \(-0.787859\pi\)
−0.786015 + 0.618207i \(0.787859\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −5.11568e18 −1.45822
\(714\) 0 0
\(715\) −2.71945e18 −0.761187
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −4.03153e18 −1.08826 −0.544129 − 0.839001i \(-0.683140\pi\)
−0.544129 + 0.839001i \(0.683140\pi\)
\(720\) 0 0
\(721\) 2.58372e18 0.684961
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −3.85813e17 −0.0986688
\(726\) 0 0
\(727\) 4.77643e18 1.19986 0.599928 − 0.800054i \(-0.295196\pi\)
0.599928 + 0.800054i \(0.295196\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −8.35976e17 −0.202643
\(732\) 0 0
\(733\) 1.71668e18 0.408803 0.204401 − 0.978887i \(-0.434475\pi\)
0.204401 + 0.978887i \(0.434475\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 3.89745e18 0.895864
\(738\) 0 0
\(739\) 8.69723e17 0.196423 0.0982114 − 0.995166i \(-0.468688\pi\)
0.0982114 + 0.995166i \(0.468688\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 2.40272e18 0.523933 0.261966 − 0.965077i \(-0.415629\pi\)
0.261966 + 0.965077i \(0.415629\pi\)
\(744\) 0 0
\(745\) 2.57852e18 0.552529
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 4.61721e18 0.955537
\(750\) 0 0
\(751\) −9.37175e18 −1.90617 −0.953084 − 0.302706i \(-0.902110\pi\)
−0.953084 + 0.302706i \(0.902110\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1.86219e18 0.365905
\(756\) 0 0
\(757\) 3.09120e18 0.597040 0.298520 − 0.954403i \(-0.403507\pi\)
0.298520 + 0.954403i \(0.403507\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 7.97787e18 1.48897 0.744486 − 0.667638i \(-0.232694\pi\)
0.744486 + 0.667638i \(0.232694\pi\)
\(762\) 0 0
\(763\) 2.30880e18 0.423620
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −2.40763e17 −0.0426991
\(768\) 0 0
\(769\) 7.37344e18 1.28573 0.642863 − 0.765981i \(-0.277746\pi\)
0.642863 + 0.765981i \(0.277746\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1.67335e18 0.282111 0.141056 − 0.990002i \(-0.454950\pi\)
0.141056 + 0.990002i \(0.454950\pi\)
\(774\) 0 0
\(775\) 1.89744e18 0.314563
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.00153e18 0.160572
\(780\) 0 0
\(781\) −1.46981e19 −2.31755
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −3.56541e18 −0.543822
\(786\) 0 0
\(787\) 3.75359e18 0.563133 0.281566 − 0.959542i \(-0.409146\pi\)
0.281566 + 0.959542i \(0.409146\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −4.00277e18 −0.581050
\(792\) 0 0
\(793\) −1.08180e18 −0.154480
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −3.38853e18 −0.468309 −0.234154 − 0.972199i \(-0.575232\pi\)
−0.234154 + 0.972199i \(0.575232\pi\)
\(798\) 0 0
\(799\) −3.46947e18 −0.471747
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 1.31833e19 1.73529
\(804\) 0 0
\(805\) −5.89861e18 −0.763971
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 3.13119e18 0.392685 0.196343 − 0.980535i \(-0.437094\pi\)
0.196343 + 0.980535i \(0.437094\pi\)
\(810\) 0 0
\(811\) −1.04731e19 −1.29253 −0.646264 − 0.763114i \(-0.723669\pi\)
−0.646264 + 0.763114i \(0.723669\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −4.67379e18 −0.558657
\(816\) 0 0
\(817\) −1.52305e18 −0.179172
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 6.85162e18 0.780841 0.390421 − 0.920637i \(-0.372330\pi\)
0.390421 + 0.920637i \(0.372330\pi\)
\(822\) 0 0
\(823\) −3.06934e17 −0.0344308 −0.0172154 − 0.999852i \(-0.505480\pi\)
−0.0172154 + 0.999852i \(0.505480\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 7.75365e18 0.842792 0.421396 − 0.906877i \(-0.361540\pi\)
0.421396 + 0.906877i \(0.361540\pi\)
\(828\) 0 0
\(829\) 2.34336e18 0.250747 0.125373 − 0.992110i \(-0.459987\pi\)
0.125373 + 0.992110i \(0.459987\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −4.89044e18 −0.507172
\(834\) 0 0
\(835\) 1.13593e19 1.15982
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −1.63297e19 −1.61632 −0.808158 − 0.588965i \(-0.799535\pi\)
−0.808158 + 0.588965i \(0.799535\pi\)
\(840\) 0 0
\(841\) −8.69212e18 −0.847134
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −7.19236e18 −0.679677
\(846\) 0 0
\(847\) −2.12836e19 −1.98063
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −4.56655e18 −0.412142
\(852\) 0 0
\(853\) 1.93794e19 1.72255 0.861276 − 0.508137i \(-0.169666\pi\)
0.861276 + 0.508137i \(0.169666\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1.20537e19 −1.03931 −0.519656 − 0.854376i \(-0.673940\pi\)
−0.519656 + 0.854376i \(0.673940\pi\)
\(858\) 0 0
\(859\) 1.00612e19 0.854465 0.427232 − 0.904142i \(-0.359489\pi\)
0.427232 + 0.904142i \(0.359489\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 8.01407e18 0.660363 0.330182 − 0.943917i \(-0.392890\pi\)
0.330182 + 0.943917i \(0.392890\pi\)
\(864\) 0 0
\(865\) −1.12853e19 −0.916029
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −1.19928e19 −0.944694
\(870\) 0 0
\(871\) −2.80544e18 −0.217712
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1.08573e19 0.817841
\(876\) 0 0
\(877\) 8.87791e17 0.0658891 0.0329445 − 0.999457i \(-0.489512\pi\)
0.0329445 + 0.999457i \(0.489512\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −3.78774e18 −0.272921 −0.136460 − 0.990646i \(-0.543573\pi\)
−0.136460 + 0.990646i \(0.543573\pi\)
\(882\) 0 0
\(883\) 2.75428e19 1.95553 0.977763 − 0.209712i \(-0.0672526\pi\)
0.977763 + 0.209712i \(0.0672526\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −2.84165e19 −1.95914 −0.979572 − 0.201092i \(-0.935551\pi\)
−0.979572 + 0.201092i \(0.935551\pi\)
\(888\) 0 0
\(889\) −1.05636e19 −0.717712
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −6.32097e18 −0.417109
\(894\) 0 0
\(895\) 1.27894e18 0.0831762
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −7.71397e18 −0.487349
\(900\) 0 0
\(901\) 2.39836e19 1.49349
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −9.37860e18 −0.567443
\(906\) 0 0
\(907\) −7.51023e18 −0.447926 −0.223963 − 0.974598i \(-0.571899\pi\)
−0.223963 + 0.974598i \(0.571899\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −2.34028e19 −1.35643 −0.678216 − 0.734863i \(-0.737247\pi\)
−0.678216 + 0.734863i \(0.737247\pi\)
\(912\) 0 0
\(913\) −1.25699e19 −0.718242
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 2.82535e17 0.0156917
\(918\) 0 0
\(919\) −2.20881e19 −1.20950 −0.604752 − 0.796414i \(-0.706728\pi\)
−0.604752 + 0.796414i \(0.706728\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 1.05799e19 0.563210
\(924\) 0 0
\(925\) 1.69377e18 0.0889063
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1.05632e19 0.539132 0.269566 − 0.962982i \(-0.413120\pi\)
0.269566 + 0.962982i \(0.413120\pi\)
\(930\) 0 0
\(931\) −8.90980e18 −0.448430
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −3.96939e19 −1.94289
\(936\) 0 0
\(937\) −1.72833e19 −0.834294 −0.417147 − 0.908839i \(-0.636970\pi\)
−0.417147 + 0.908839i \(0.636970\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −2.14038e19 −1.00498 −0.502490 − 0.864583i \(-0.667583\pi\)
−0.502490 + 0.864583i \(0.667583\pi\)
\(942\) 0 0
\(943\) 3.88591e18 0.179956
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −5.91308e17 −0.0266403 −0.0133202 − 0.999911i \(-0.504240\pi\)
−0.0133202 + 0.999911i \(0.504240\pi\)
\(948\) 0 0
\(949\) −9.48950e18 −0.421709
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 2.39396e19 1.03517 0.517587 − 0.855631i \(-0.326830\pi\)
0.517587 + 0.855631i \(0.326830\pi\)
\(954\) 0 0
\(955\) −2.60493e19 −1.11115
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 4.03321e18 0.167429
\(960\) 0 0
\(961\) 1.35201e19 0.553702
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −2.83320e19 −1.12941
\(966\) 0 0
\(967\) −1.99045e18 −0.0782853 −0.0391427 − 0.999234i \(-0.512463\pi\)
−0.0391427 + 0.999234i \(0.512463\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 4.12385e19 1.57898 0.789492 − 0.613761i \(-0.210344\pi\)
0.789492 + 0.613761i \(0.210344\pi\)
\(972\) 0 0
\(973\) 2.48357e19 0.938301
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.25374e19 0.461202 0.230601 − 0.973048i \(-0.425931\pi\)
0.230601 + 0.973048i \(0.425931\pi\)
\(978\) 0 0
\(979\) 2.50006e19 0.907535
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −1.83966e19 −0.650340 −0.325170 − 0.945656i \(-0.605422\pi\)
−0.325170 + 0.945656i \(0.605422\pi\)
\(984\) 0 0
\(985\) 2.02925e19 0.707946
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −5.90941e18 −0.200802
\(990\) 0 0
\(991\) 3.43064e19 1.15053 0.575263 − 0.817969i \(-0.304900\pi\)
0.575263 + 0.817969i \(0.304900\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 1.31852e18 0.0430761
\(996\) 0 0
\(997\) −5.72458e19 −1.84597 −0.922985 − 0.384835i \(-0.874258\pi\)
−0.922985 + 0.384835i \(0.874258\pi\)
\(998\) 0 0
\(999\) 0 0
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 144.14.a.k.1.1 1
3.2 odd 2 48.14.a.c.1.1 1
4.3 odd 2 9.14.a.a.1.1 1
12.11 even 2 3.14.a.a.1.1 ✓ 1
24.5 odd 2 192.14.a.e.1.1 1
24.11 even 2 192.14.a.j.1.1 1
60.23 odd 4 75.14.b.b.49.2 2
60.47 odd 4 75.14.b.b.49.1 2
60.59 even 2 75.14.a.a.1.1 1
84.83 odd 2 147.14.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.14.a.a.1.1 ✓ 1 12.11 even 2
9.14.a.a.1.1 1 4.3 odd 2
48.14.a.c.1.1 1 3.2 odd 2
75.14.a.a.1.1 1 60.59 even 2
75.14.b.b.49.1 2 60.47 odd 4
75.14.b.b.49.2 2 60.23 odd 4
144.14.a.k.1.1 1 1.1 even 1 trivial
147.14.a.a.1.1 1 84.83 odd 2
192.14.a.e.1.1 1 24.5 odd 2
192.14.a.j.1.1 1 24.11 even 2