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Properties

Label 144.12.a.f.1.1
Level $144$
Weight $12$
Character 144.1
Self dual yes
Analytic conductor $110.641$
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] = [144,12,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, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 12 \)
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: \(110.641418001\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 12)
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)
 
\(f(q)\) \(=\) \(q-2862.00 q^{5} -9128.00 q^{7} +O(q^{10})\) \(q-2862.00 q^{5} -9128.00 q^{7} +668196. q^{11} +2.05295e6 q^{13} -1.60418e6 q^{17} +230500. q^{19} -4.30127e7 q^{23} -4.06371e7 q^{25} +1.41745e8 q^{29} -2.33222e8 q^{31} +2.61243e7 q^{35} +2.78270e8 q^{37} +1.18158e9 q^{41} -8.56975e8 q^{43} -1.66405e9 q^{47} -1.89401e9 q^{49} +3.85118e9 q^{53} -1.91238e9 q^{55} +1.03390e10 q^{59} +1.85948e8 q^{61} -5.87554e9 q^{65} -2.91501e9 q^{67} +1.26623e10 q^{71} -1.52013e10 q^{73} -6.09929e9 q^{77} +3.66440e10 q^{79} -9.21764e9 q^{83} +4.59116e9 q^{85} -3.05738e10 q^{89} -1.87393e10 q^{91} -6.59691e8 q^{95} +1.45702e11 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
\(5\) −2862.00 −0.409576 −0.204788 − 0.978806i \(-0.565651\pi\)
−0.204788 + 0.978806i \(0.565651\pi\)
\(6\) 0 0
\(7\) −9128.00 −0.205275 −0.102638 − 0.994719i \(-0.532728\pi\)
−0.102638 + 0.994719i \(0.532728\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 668196. 1.25096 0.625481 − 0.780239i \(-0.284903\pi\)
0.625481 + 0.780239i \(0.284903\pi\)
\(12\) 0 0
\(13\) 2.05295e6 1.53352 0.766761 − 0.641933i \(-0.221867\pi\)
0.766761 + 0.641933i \(0.221867\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.60418e6 −0.274021 −0.137010 − 0.990570i \(-0.543749\pi\)
−0.137010 + 0.990570i \(0.543749\pi\)
\(18\) 0 0
\(19\) 230500. 0.0213563 0.0106782 − 0.999943i \(-0.496601\pi\)
0.0106782 + 0.999943i \(0.496601\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.30127e7 −1.39346 −0.696729 − 0.717334i \(-0.745362\pi\)
−0.696729 + 0.717334i \(0.745362\pi\)
\(24\) 0 0
\(25\) −4.06371e7 −0.832247
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.41745e8 1.28327 0.641637 − 0.767008i \(-0.278256\pi\)
0.641637 + 0.767008i \(0.278256\pi\)
\(30\) 0 0
\(31\) −2.33222e8 −1.46312 −0.731560 − 0.681777i \(-0.761207\pi\)
−0.731560 + 0.681777i \(0.761207\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.61243e7 0.0840758
\(36\) 0 0
\(37\) 2.78270e8 0.659715 0.329858 − 0.944031i \(-0.392999\pi\)
0.329858 + 0.944031i \(0.392999\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.18158e9 1.59276 0.796381 − 0.604795i \(-0.206745\pi\)
0.796381 + 0.604795i \(0.206745\pi\)
\(42\) 0 0
\(43\) −8.56975e8 −0.888979 −0.444490 − 0.895784i \(-0.646615\pi\)
−0.444490 + 0.895784i \(0.646615\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.66405e9 −1.05835 −0.529175 − 0.848513i \(-0.677499\pi\)
−0.529175 + 0.848513i \(0.677499\pi\)
\(48\) 0 0
\(49\) −1.89401e9 −0.957862
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.85118e9 1.26496 0.632480 − 0.774577i \(-0.282037\pi\)
0.632480 + 0.774577i \(0.282037\pi\)
\(54\) 0 0
\(55\) −1.91238e9 −0.512364
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.03390e10 1.88275 0.941375 − 0.337363i \(-0.109535\pi\)
0.941375 + 0.337363i \(0.109535\pi\)
\(60\) 0 0
\(61\) 1.85948e8 0.0281889 0.0140944 − 0.999901i \(-0.495513\pi\)
0.0140944 + 0.999901i \(0.495513\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −5.87554e9 −0.628094
\(66\) 0 0
\(67\) −2.91501e9 −0.263772 −0.131886 − 0.991265i \(-0.542103\pi\)
−0.131886 + 0.991265i \(0.542103\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1.26623e10 0.832899 0.416449 − 0.909159i \(-0.363274\pi\)
0.416449 + 0.909159i \(0.363274\pi\)
\(72\) 0 0
\(73\) −1.52013e10 −0.858231 −0.429115 − 0.903250i \(-0.641175\pi\)
−0.429115 + 0.903250i \(0.641175\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.09929e9 −0.256791
\(78\) 0 0
\(79\) 3.66440e10 1.33984 0.669922 − 0.742432i \(-0.266328\pi\)
0.669922 + 0.742432i \(0.266328\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −9.21764e9 −0.256856 −0.128428 − 0.991719i \(-0.540993\pi\)
−0.128428 + 0.991719i \(0.540993\pi\)
\(84\) 0 0
\(85\) 4.59116e9 0.112232
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −3.05738e10 −0.580370 −0.290185 − 0.956971i \(-0.593717\pi\)
−0.290185 + 0.956971i \(0.593717\pi\)
\(90\) 0 0
\(91\) −1.87393e10 −0.314794
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −6.59691e8 −0.00874703
\(96\) 0 0
\(97\) 1.45702e11 1.72274 0.861371 − 0.507976i \(-0.169606\pi\)
0.861371 + 0.507976i \(0.169606\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 5.59904e10 0.530085 0.265043 − 0.964237i \(-0.414614\pi\)
0.265043 + 0.964237i \(0.414614\pi\)
\(102\) 0 0
\(103\) 4.39677e10 0.373705 0.186852 − 0.982388i \(-0.440171\pi\)
0.186852 + 0.982388i \(0.440171\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1.49244e11 −1.02869 −0.514347 − 0.857582i \(-0.671966\pi\)
−0.514347 + 0.857582i \(0.671966\pi\)
\(108\) 0 0
\(109\) 1.20912e9 0.00752704 0.00376352 − 0.999993i \(-0.498802\pi\)
0.00376352 + 0.999993i \(0.498802\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2.75811e11 1.40825 0.704125 − 0.710077i \(-0.251340\pi\)
0.704125 + 0.710077i \(0.251340\pi\)
\(114\) 0 0
\(115\) 1.23102e11 0.570727
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1.46429e10 0.0562497
\(120\) 0 0
\(121\) 1.61174e11 0.564906
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 2.56049e11 0.750445
\(126\) 0 0
\(127\) −5.48292e10 −0.147262 −0.0736312 − 0.997286i \(-0.523459\pi\)
−0.0736312 + 0.997286i \(0.523459\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 7.69885e11 1.74355 0.871774 − 0.489909i \(-0.162970\pi\)
0.871774 + 0.489909i \(0.162970\pi\)
\(132\) 0 0
\(133\) −2.10400e9 −0.00438392
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −3.90519e11 −0.691321 −0.345660 − 0.938360i \(-0.612345\pi\)
−0.345660 + 0.938360i \(0.612345\pi\)
\(138\) 0 0
\(139\) −5.14102e10 −0.0840365 −0.0420183 − 0.999117i \(-0.513379\pi\)
−0.0420183 + 0.999117i \(0.513379\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.37177e12 1.91838
\(144\) 0 0
\(145\) −4.05675e11 −0.525598
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1.39815e12 1.55966 0.779830 − 0.625991i \(-0.215305\pi\)
0.779830 + 0.625991i \(0.215305\pi\)
\(150\) 0 0
\(151\) 4.21187e11 0.436618 0.218309 − 0.975880i \(-0.429946\pi\)
0.218309 + 0.975880i \(0.429946\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 6.67481e11 0.599259
\(156\) 0 0
\(157\) −1.27939e12 −1.07042 −0.535210 − 0.844719i \(-0.679768\pi\)
−0.535210 + 0.844719i \(0.679768\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 3.92620e11 0.286042
\(162\) 0 0
\(163\) 4.72791e11 0.321838 0.160919 − 0.986968i \(-0.448554\pi\)
0.160919 + 0.986968i \(0.448554\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.04628e12 0.623313 0.311656 − 0.950195i \(-0.399116\pi\)
0.311656 + 0.950195i \(0.399116\pi\)
\(168\) 0 0
\(169\) 2.42244e12 1.35169
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.46150e12 0.717044 0.358522 − 0.933521i \(-0.383281\pi\)
0.358522 + 0.933521i \(0.383281\pi\)
\(174\) 0 0
\(175\) 3.70935e11 0.170840
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −4.23853e11 −0.172395 −0.0861974 − 0.996278i \(-0.527472\pi\)
−0.0861974 + 0.996278i \(0.527472\pi\)
\(180\) 0 0
\(181\) 1.83136e12 0.700717 0.350358 − 0.936616i \(-0.386060\pi\)
0.350358 + 0.936616i \(0.386060\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −7.96408e11 −0.270204
\(186\) 0 0
\(187\) −1.07191e12 −0.342790
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2.94498e12 −0.838297 −0.419149 − 0.907918i \(-0.637671\pi\)
−0.419149 + 0.907918i \(0.637671\pi\)
\(192\) 0 0
\(193\) 6.25393e12 1.68108 0.840538 − 0.541752i \(-0.182239\pi\)
0.840538 + 0.541752i \(0.182239\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 6.66278e12 1.59989 0.799947 − 0.600071i \(-0.204861\pi\)
0.799947 + 0.600071i \(0.204861\pi\)
\(198\) 0 0
\(199\) 7.06530e12 1.60487 0.802433 − 0.596742i \(-0.203539\pi\)
0.802433 + 0.596742i \(0.203539\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1.29385e12 −0.263424
\(204\) 0 0
\(205\) −3.38167e12 −0.652357
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1.54019e11 0.0267159
\(210\) 0 0
\(211\) 3.11902e12 0.513411 0.256705 − 0.966490i \(-0.417363\pi\)
0.256705 + 0.966490i \(0.417363\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 2.45266e12 0.364105
\(216\) 0 0
\(217\) 2.12885e12 0.300342
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −3.29330e12 −0.420217
\(222\) 0 0
\(223\) −1.34244e13 −1.63011 −0.815055 − 0.579384i \(-0.803293\pi\)
−0.815055 + 0.579384i \(0.803293\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 9.95920e12 1.09669 0.548343 − 0.836254i \(-0.315259\pi\)
0.548343 + 0.836254i \(0.315259\pi\)
\(228\) 0 0
\(229\) −1.19130e13 −1.25005 −0.625023 − 0.780607i \(-0.714910\pi\)
−0.625023 + 0.780607i \(0.714910\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.96588e13 −1.87542 −0.937710 − 0.347420i \(-0.887058\pi\)
−0.937710 + 0.347420i \(0.887058\pi\)
\(234\) 0 0
\(235\) 4.76253e12 0.433475
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 3.00146e11 0.0248968 0.0124484 − 0.999923i \(-0.496037\pi\)
0.0124484 + 0.999923i \(0.496037\pi\)
\(240\) 0 0
\(241\) 8.82505e12 0.699235 0.349618 − 0.936893i \(-0.386312\pi\)
0.349618 + 0.936893i \(0.386312\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 5.42065e12 0.392317
\(246\) 0 0
\(247\) 4.73205e11 0.0327504
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1.54036e12 0.0975928 0.0487964 − 0.998809i \(-0.484461\pi\)
0.0487964 + 0.998809i \(0.484461\pi\)
\(252\) 0 0
\(253\) −2.87409e13 −1.74316
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.36700e12 0.131694 0.0658470 − 0.997830i \(-0.479025\pi\)
0.0658470 + 0.997830i \(0.479025\pi\)
\(258\) 0 0
\(259\) −2.54005e12 −0.135423
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1.69269e12 0.0829508 0.0414754 − 0.999140i \(-0.486794\pi\)
0.0414754 + 0.999140i \(0.486794\pi\)
\(264\) 0 0
\(265\) −1.10221e13 −0.518097
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 3.24983e13 1.40677 0.703386 − 0.710808i \(-0.251671\pi\)
0.703386 + 0.710808i \(0.251671\pi\)
\(270\) 0 0
\(271\) −8.84500e12 −0.367593 −0.183796 − 0.982964i \(-0.558839\pi\)
−0.183796 + 0.982964i \(0.558839\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −2.71535e13 −1.04111
\(276\) 0 0
\(277\) 4.16847e13 1.53581 0.767906 − 0.640563i \(-0.221299\pi\)
0.767906 + 0.640563i \(0.221299\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −3.13771e13 −1.06838 −0.534192 − 0.845363i \(-0.679384\pi\)
−0.534192 + 0.845363i \(0.679384\pi\)
\(282\) 0 0
\(283\) −3.97011e13 −1.30010 −0.650051 − 0.759891i \(-0.725252\pi\)
−0.650051 + 0.759891i \(0.725252\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −1.07854e13 −0.326955
\(288\) 0 0
\(289\) −3.16985e13 −0.924913
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −3.71473e13 −1.00497 −0.502487 − 0.864584i \(-0.667582\pi\)
−0.502487 + 0.864584i \(0.667582\pi\)
\(294\) 0 0
\(295\) −2.95902e13 −0.771129
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −8.83030e13 −2.13690
\(300\) 0 0
\(301\) 7.82247e12 0.182485
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −5.32183e11 −0.0115455
\(306\) 0 0
\(307\) 5.82939e13 1.22001 0.610003 − 0.792399i \(-0.291168\pi\)
0.610003 + 0.792399i \(0.291168\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −3.38262e13 −0.659281 −0.329641 − 0.944106i \(-0.606928\pi\)
−0.329641 + 0.944106i \(0.606928\pi\)
\(312\) 0 0
\(313\) 7.69062e13 1.44700 0.723498 − 0.690326i \(-0.242533\pi\)
0.723498 + 0.690326i \(0.242533\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −4.59943e13 −0.807008 −0.403504 − 0.914978i \(-0.632208\pi\)
−0.403504 + 0.914978i \(0.632208\pi\)
\(318\) 0 0
\(319\) 9.47136e13 1.60533
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −3.69763e11 −0.00585207
\(324\) 0 0
\(325\) −8.34259e13 −1.27627
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 1.51895e13 0.217253
\(330\) 0 0
\(331\) −1.12182e14 −1.55192 −0.775960 − 0.630782i \(-0.782734\pi\)
−0.775960 + 0.630782i \(0.782734\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 8.34276e12 0.108035
\(336\) 0 0
\(337\) −6.29192e13 −0.788531 −0.394265 − 0.918997i \(-0.629001\pi\)
−0.394265 + 0.918997i \(0.629001\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −1.55838e14 −1.83031
\(342\) 0 0
\(343\) 3.53375e13 0.401900
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.34320e14 1.43327 0.716635 − 0.697448i \(-0.245681\pi\)
0.716635 + 0.697448i \(0.245681\pi\)
\(348\) 0 0
\(349\) 6.49874e13 0.671876 0.335938 − 0.941884i \(-0.390947\pi\)
0.335938 + 0.941884i \(0.390947\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −6.06398e13 −0.588839 −0.294420 − 0.955676i \(-0.595126\pi\)
−0.294420 + 0.955676i \(0.595126\pi\)
\(354\) 0 0
\(355\) −3.62395e13 −0.341135
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1.95954e13 0.173435 0.0867173 − 0.996233i \(-0.472362\pi\)
0.0867173 + 0.996233i \(0.472362\pi\)
\(360\) 0 0
\(361\) −1.16437e14 −0.999544
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 4.35060e13 0.351511
\(366\) 0 0
\(367\) −1.02500e14 −0.803636 −0.401818 − 0.915719i \(-0.631622\pi\)
−0.401818 + 0.915719i \(0.631622\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −3.51536e13 −0.259665
\(372\) 0 0
\(373\) 8.91645e13 0.639431 0.319715 − 0.947514i \(-0.396413\pi\)
0.319715 + 0.947514i \(0.396413\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.90996e14 1.96793
\(378\) 0 0
\(379\) 2.61504e14 1.71776 0.858879 − 0.512179i \(-0.171161\pi\)
0.858879 + 0.512179i \(0.171161\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1.38948e14 0.861508 0.430754 − 0.902469i \(-0.358248\pi\)
0.430754 + 0.902469i \(0.358248\pi\)
\(384\) 0 0
\(385\) 1.74562e13 0.105176
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 1.83753e14 1.04595 0.522976 − 0.852347i \(-0.324822\pi\)
0.522976 + 0.852347i \(0.324822\pi\)
\(390\) 0 0
\(391\) 6.90001e13 0.381836
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −1.04875e14 −0.548768
\(396\) 0 0
\(397\) 1.94932e13 0.0992053 0.0496026 − 0.998769i \(-0.484205\pi\)
0.0496026 + 0.998769i \(0.484205\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2.33384e14 1.12403 0.562013 − 0.827128i \(-0.310027\pi\)
0.562013 + 0.827128i \(0.310027\pi\)
\(402\) 0 0
\(403\) −4.78793e14 −2.24373
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.85939e14 0.825279
\(408\) 0 0
\(409\) −1.58513e14 −0.684837 −0.342419 − 0.939548i \(-0.611246\pi\)
−0.342419 + 0.939548i \(0.611246\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −9.43744e13 −0.386482
\(414\) 0 0
\(415\) 2.63809e13 0.105202
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −2.82880e14 −1.07010 −0.535052 − 0.844819i \(-0.679708\pi\)
−0.535052 + 0.844819i \(0.679708\pi\)
\(420\) 0 0
\(421\) 4.96375e14 1.82919 0.914593 − 0.404375i \(-0.132511\pi\)
0.914593 + 0.404375i \(0.132511\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 6.51891e13 0.228053
\(426\) 0 0
\(427\) −1.69733e12 −0.00578647
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 3.97677e14 1.28797 0.643985 − 0.765038i \(-0.277280\pi\)
0.643985 + 0.765038i \(0.277280\pi\)
\(432\) 0 0
\(433\) −1.49632e14 −0.472433 −0.236217 − 0.971700i \(-0.575908\pi\)
−0.236217 + 0.971700i \(0.575908\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −9.91443e12 −0.0297591
\(438\) 0 0
\(439\) 1.05728e14 0.309481 0.154741 − 0.987955i \(-0.450546\pi\)
0.154741 + 0.987955i \(0.450546\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 3.31088e13 0.0921984 0.0460992 − 0.998937i \(-0.485321\pi\)
0.0460992 + 0.998937i \(0.485321\pi\)
\(444\) 0 0
\(445\) 8.75023e13 0.237706
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.86449e14 0.482176 0.241088 − 0.970503i \(-0.422496\pi\)
0.241088 + 0.970503i \(0.422496\pi\)
\(450\) 0 0
\(451\) 7.89525e14 1.99249
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 5.36320e13 0.128932
\(456\) 0 0
\(457\) 5.52462e13 0.129647 0.0648236 − 0.997897i \(-0.479352\pi\)
0.0648236 + 0.997897i \(0.479352\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −7.95826e12 −0.0178018 −0.00890088 − 0.999960i \(-0.502833\pi\)
−0.00890088 + 0.999960i \(0.502833\pi\)
\(462\) 0 0
\(463\) 5.56637e14 1.21584 0.607920 − 0.793998i \(-0.292004\pi\)
0.607920 + 0.793998i \(0.292004\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 9.04316e13 0.188398 0.0941992 − 0.995553i \(-0.469971\pi\)
0.0941992 + 0.995553i \(0.469971\pi\)
\(468\) 0 0
\(469\) 2.66082e13 0.0541458
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −5.72627e14 −1.11208
\(474\) 0 0
\(475\) −9.36685e12 −0.0177737
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 4.91081e14 0.889832 0.444916 − 0.895572i \(-0.353234\pi\)
0.444916 + 0.895572i \(0.353234\pi\)
\(480\) 0 0
\(481\) 5.71274e14 1.01169
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −4.16999e14 −0.705594
\(486\) 0 0
\(487\) −2.00849e14 −0.332246 −0.166123 − 0.986105i \(-0.553125\pi\)
−0.166123 + 0.986105i \(0.553125\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 7.45240e13 0.117855 0.0589275 − 0.998262i \(-0.481232\pi\)
0.0589275 + 0.998262i \(0.481232\pi\)
\(492\) 0 0
\(493\) −2.27385e14 −0.351644
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1.15582e14 −0.170973
\(498\) 0 0
\(499\) −6.79790e14 −0.983608 −0.491804 − 0.870706i \(-0.663662\pi\)
−0.491804 + 0.870706i \(0.663662\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −3.23785e14 −0.448367 −0.224183 − 0.974547i \(-0.571971\pi\)
−0.224183 + 0.974547i \(0.571971\pi\)
\(504\) 0 0
\(505\) −1.60244e14 −0.217110
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 5.26272e13 0.0682751 0.0341376 − 0.999417i \(-0.489132\pi\)
0.0341376 + 0.999417i \(0.489132\pi\)
\(510\) 0 0
\(511\) 1.38757e14 0.176173
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1.25835e14 −0.153061
\(516\) 0 0
\(517\) −1.11191e15 −1.32396
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 8.60352e14 0.981904 0.490952 − 0.871187i \(-0.336649\pi\)
0.490952 + 0.871187i \(0.336649\pi\)
\(522\) 0 0
\(523\) −8.88719e14 −0.993127 −0.496564 − 0.868000i \(-0.665405\pi\)
−0.496564 + 0.868000i \(0.665405\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3.74129e14 0.400925
\(528\) 0 0
\(529\) 8.97285e14 0.941725
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 2.42572e15 2.44254
\(534\) 0 0
\(535\) 4.27136e14 0.421328
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −1.26557e15 −1.19825
\(540\) 0 0
\(541\) −4.09513e14 −0.379912 −0.189956 − 0.981793i \(-0.560835\pi\)
−0.189956 + 0.981793i \(0.560835\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −3.46051e12 −0.00308290
\(546\) 0 0
\(547\) 1.99316e14 0.174025 0.0870124 − 0.996207i \(-0.472268\pi\)
0.0870124 + 0.996207i \(0.472268\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 3.26723e13 0.0274060
\(552\) 0 0
\(553\) −3.34487e14 −0.275037
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −1.34904e15 −1.06615 −0.533077 − 0.846067i \(-0.678965\pi\)
−0.533077 + 0.846067i \(0.678965\pi\)
\(558\) 0 0
\(559\) −1.75933e15 −1.36327
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −2.54316e15 −1.89486 −0.947432 − 0.319958i \(-0.896331\pi\)
−0.947432 + 0.319958i \(0.896331\pi\)
\(564\) 0 0
\(565\) −7.89370e14 −0.576785
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −2.74174e15 −1.92712 −0.963561 − 0.267490i \(-0.913806\pi\)
−0.963561 + 0.267490i \(0.913806\pi\)
\(570\) 0 0
\(571\) −2.76777e15 −1.90823 −0.954117 − 0.299433i \(-0.903202\pi\)
−0.954117 + 0.299433i \(0.903202\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.74791e15 1.15970
\(576\) 0 0
\(577\) 6.94653e14 0.452169 0.226085 − 0.974108i \(-0.427407\pi\)
0.226085 + 0.974108i \(0.427407\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 8.41386e13 0.0527262
\(582\) 0 0
\(583\) 2.57334e15 1.58242
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −9.04959e14 −0.535944 −0.267972 − 0.963427i \(-0.586353\pi\)
−0.267972 + 0.963427i \(0.586353\pi\)
\(588\) 0 0
\(589\) −5.37576e13 −0.0312468
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1.41540e15 −0.792645 −0.396323 − 0.918111i \(-0.629714\pi\)
−0.396323 + 0.918111i \(0.629714\pi\)
\(594\) 0 0
\(595\) −4.19081e13 −0.0230385
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 2.22145e15 1.17703 0.588516 − 0.808486i \(-0.299712\pi\)
0.588516 + 0.808486i \(0.299712\pi\)
\(600\) 0 0
\(601\) −2.80969e15 −1.46167 −0.730833 − 0.682556i \(-0.760868\pi\)
−0.730833 + 0.682556i \(0.760868\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −4.61281e14 −0.231372
\(606\) 0 0
\(607\) 7.67540e14 0.378062 0.189031 − 0.981971i \(-0.439465\pi\)
0.189031 + 0.981971i \(0.439465\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −3.41622e15 −1.62300
\(612\) 0 0
\(613\) 1.47031e15 0.686083 0.343042 − 0.939320i \(-0.388543\pi\)
0.343042 + 0.939320i \(0.388543\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1.22736e15 −0.552592 −0.276296 − 0.961073i \(-0.589107\pi\)
−0.276296 + 0.961073i \(0.589107\pi\)
\(618\) 0 0
\(619\) 1.00809e15 0.445862 0.222931 − 0.974834i \(-0.428437\pi\)
0.222931 + 0.974834i \(0.428437\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 2.79078e14 0.119136
\(624\) 0 0
\(625\) 1.25142e15 0.524883
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −4.46394e14 −0.180776
\(630\) 0 0
\(631\) 2.98051e15 1.18612 0.593061 − 0.805158i \(-0.297919\pi\)
0.593061 + 0.805158i \(0.297919\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1.56921e14 0.0603151
\(636\) 0 0
\(637\) −3.88830e15 −1.46890
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −3.59154e15 −1.31087 −0.655437 − 0.755249i \(-0.727516\pi\)
−0.655437 + 0.755249i \(0.727516\pi\)
\(642\) 0 0
\(643\) 1.78687e15 0.641110 0.320555 − 0.947230i \(-0.396131\pi\)
0.320555 + 0.947230i \(0.396131\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 7.66508e14 0.265793 0.132896 − 0.991130i \(-0.457572\pi\)
0.132896 + 0.991130i \(0.457572\pi\)
\(648\) 0 0
\(649\) 6.90848e15 2.35525
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 2.56438e14 0.0845201 0.0422600 − 0.999107i \(-0.486544\pi\)
0.0422600 + 0.999107i \(0.486544\pi\)
\(654\) 0 0
\(655\) −2.20341e15 −0.714115
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −3.63270e15 −1.13857 −0.569286 − 0.822140i \(-0.692780\pi\)
−0.569286 + 0.822140i \(0.692780\pi\)
\(660\) 0 0
\(661\) 5.24024e14 0.161526 0.0807632 − 0.996733i \(-0.474264\pi\)
0.0807632 + 0.996733i \(0.474264\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 6.02166e12 0.00179555
\(666\) 0 0
\(667\) −6.09685e15 −1.78819
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1.24250e14 0.0352632
\(672\) 0 0
\(673\) −3.22347e15 −0.899996 −0.449998 − 0.893029i \(-0.648575\pi\)
−0.449998 + 0.893029i \(0.648575\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −6.08105e14 −0.164339 −0.0821695 − 0.996618i \(-0.526185\pi\)
−0.0821695 + 0.996618i \(0.526185\pi\)
\(678\) 0 0
\(679\) −1.32997e15 −0.353636
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1.63384e14 0.0420626 0.0210313 − 0.999779i \(-0.493305\pi\)
0.0210313 + 0.999779i \(0.493305\pi\)
\(684\) 0 0
\(685\) 1.11767e15 0.283148
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 7.90628e15 1.93984
\(690\) 0 0
\(691\) 2.57486e15 0.621763 0.310882 − 0.950449i \(-0.399376\pi\)
0.310882 + 0.950449i \(0.399376\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1.47136e14 0.0344194
\(696\) 0 0
\(697\) −1.89546e15 −0.436450
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 6.95102e15 1.55096 0.775478 − 0.631374i \(-0.217509\pi\)
0.775478 + 0.631374i \(0.217509\pi\)
\(702\) 0 0
\(703\) 6.41412e13 0.0140891
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −5.11080e14 −0.108813
\(708\) 0 0
\(709\) −1.64191e15 −0.344187 −0.172094 − 0.985081i \(-0.555053\pi\)
−0.172094 + 0.985081i \(0.555053\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1.00315e16 2.03880
\(714\) 0 0
\(715\) −3.92601e15 −0.785722
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 7.03155e15 1.36472 0.682358 − 0.731018i \(-0.260955\pi\)
0.682358 + 0.731018i \(0.260955\pi\)
\(720\) 0 0
\(721\) −4.01337e14 −0.0767123
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −5.76011e15 −1.06800
\(726\) 0 0
\(727\) 9.65145e15 1.76260 0.881300 − 0.472558i \(-0.156669\pi\)
0.881300 + 0.472558i \(0.156669\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1.37474e15 0.243599
\(732\) 0 0
\(733\) 5.38706e15 0.940329 0.470164 − 0.882579i \(-0.344195\pi\)
0.470164 + 0.882579i \(0.344195\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1.94780e15 −0.329969
\(738\) 0 0
\(739\) −6.23212e15 −1.04014 −0.520070 − 0.854124i \(-0.674094\pi\)
−0.520070 + 0.854124i \(0.674094\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 3.05132e15 0.494366 0.247183 − 0.968969i \(-0.420495\pi\)
0.247183 + 0.968969i \(0.420495\pi\)
\(744\) 0 0
\(745\) −4.00151e15 −0.638800
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1.36230e15 0.211165
\(750\) 0 0
\(751\) −3.92991e15 −0.600292 −0.300146 − 0.953893i \(-0.597035\pi\)
−0.300146 + 0.953893i \(0.597035\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −1.20544e15 −0.178828
\(756\) 0 0
\(757\) 5.24662e15 0.767101 0.383551 − 0.923520i \(-0.374701\pi\)
0.383551 + 0.923520i \(0.374701\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −4.83177e15 −0.686264 −0.343132 − 0.939287i \(-0.611488\pi\)
−0.343132 + 0.939287i \(0.611488\pi\)
\(762\) 0 0
\(763\) −1.10369e13 −0.00154512
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 2.12255e16 2.88724
\(768\) 0 0
\(769\) −6.55230e15 −0.878615 −0.439308 − 0.898337i \(-0.644776\pi\)
−0.439308 + 0.898337i \(0.644776\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −2.31322e15 −0.301460 −0.150730 − 0.988575i \(-0.548162\pi\)
−0.150730 + 0.988575i \(0.548162\pi\)
\(774\) 0 0
\(775\) 9.47746e15 1.21768
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 2.72354e14 0.0340155
\(780\) 0 0
\(781\) 8.46091e15 1.04192
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 3.66161e15 0.438419
\(786\) 0 0
\(787\) 1.47182e16 1.73778 0.868888 − 0.495008i \(-0.164835\pi\)
0.868888 + 0.495008i \(0.164835\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −2.51760e15 −0.289079
\(792\) 0 0
\(793\) 3.81742e14 0.0432282
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −2.50458e15 −0.275877 −0.137938 − 0.990441i \(-0.544048\pi\)
−0.137938 + 0.990441i \(0.544048\pi\)
\(798\) 0 0
\(799\) 2.66944e15 0.290010
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −1.01574e16 −1.07361
\(804\) 0 0
\(805\) −1.12368e15 −0.117156
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.29400e16 1.31286 0.656431 − 0.754386i \(-0.272066\pi\)
0.656431 + 0.754386i \(0.272066\pi\)
\(810\) 0 0
\(811\) 4.70410e15 0.470828 0.235414 − 0.971895i \(-0.424355\pi\)
0.235414 + 0.971895i \(0.424355\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −1.35313e15 −0.131817
\(816\) 0 0
\(817\) −1.97533e14 −0.0189853
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1.12296e16 −1.05069 −0.525346 − 0.850889i \(-0.676064\pi\)
−0.525346 + 0.850889i \(0.676064\pi\)
\(822\) 0 0
\(823\) −8.53732e15 −0.788175 −0.394087 − 0.919073i \(-0.628939\pi\)
−0.394087 + 0.919073i \(0.628939\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1.02091e16 0.917712 0.458856 − 0.888511i \(-0.348259\pi\)
0.458856 + 0.888511i \(0.348259\pi\)
\(828\) 0 0
\(829\) 2.25785e15 0.200284 0.100142 − 0.994973i \(-0.468070\pi\)
0.100142 + 0.994973i \(0.468070\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 3.03832e15 0.262474
\(834\) 0 0
\(835\) −2.99444e15 −0.255294
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −2.05099e16 −1.70322 −0.851612 − 0.524172i \(-0.824375\pi\)
−0.851612 + 0.524172i \(0.824375\pi\)
\(840\) 0 0
\(841\) 7.89119e15 0.646792
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −6.93303e15 −0.553619
\(846\) 0 0
\(847\) −1.47120e15 −0.115961
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −1.19691e16 −0.919285
\(852\) 0 0
\(853\) 1.36009e16 1.03121 0.515606 − 0.856826i \(-0.327567\pi\)
0.515606 + 0.856826i \(0.327567\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1.03910e16 −0.767828 −0.383914 − 0.923369i \(-0.625424\pi\)
−0.383914 + 0.923369i \(0.625424\pi\)
\(858\) 0 0
\(859\) −1.33142e16 −0.971298 −0.485649 − 0.874154i \(-0.661417\pi\)
−0.485649 + 0.874154i \(0.661417\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 8.38331e15 0.596151 0.298076 − 0.954542i \(-0.403655\pi\)
0.298076 + 0.954542i \(0.403655\pi\)
\(864\) 0 0
\(865\) −4.18282e15 −0.293684
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 2.44854e16 1.67609
\(870\) 0 0
\(871\) −5.98437e15 −0.404500
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −2.33722e15 −0.154048
\(876\) 0 0
\(877\) −1.51493e16 −0.986042 −0.493021 − 0.870017i \(-0.664107\pi\)
−0.493021 + 0.870017i \(0.664107\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 2.86242e15 0.181705 0.0908524 − 0.995864i \(-0.471041\pi\)
0.0908524 + 0.995864i \(0.471041\pi\)
\(882\) 0 0
\(883\) 4.51715e15 0.283192 0.141596 − 0.989925i \(-0.454777\pi\)
0.141596 + 0.989925i \(0.454777\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1.46802e15 0.0897741 0.0448871 − 0.998992i \(-0.485707\pi\)
0.0448871 + 0.998992i \(0.485707\pi\)
\(888\) 0 0
\(889\) 5.00481e14 0.0302293
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −3.83565e14 −0.0226025
\(894\) 0 0
\(895\) 1.21307e15 0.0706088
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −3.30581e16 −1.87758
\(900\) 0 0
\(901\) −6.17798e15 −0.346625
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −5.24136e15 −0.286997
\(906\) 0 0
\(907\) −2.09558e16 −1.13361 −0.566805 − 0.823852i \(-0.691821\pi\)
−0.566805 + 0.823852i \(0.691821\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1.93579e15 0.102213 0.0511065 − 0.998693i \(-0.483725\pi\)
0.0511065 + 0.998693i \(0.483725\pi\)
\(912\) 0 0
\(913\) −6.15919e15 −0.321317
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −7.02751e15 −0.357907
\(918\) 0 0
\(919\) 3.35035e16 1.68599 0.842994 − 0.537923i \(-0.180791\pi\)
0.842994 + 0.537923i \(0.180791\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 2.59951e16 1.27727
\(924\) 0 0
\(925\) −1.13081e16 −0.549046
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1.65748e16 0.785892 0.392946 − 0.919562i \(-0.371456\pi\)
0.392946 + 0.919562i \(0.371456\pi\)
\(930\) 0 0
\(931\) −4.36568e14 −0.0204564
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 3.06779e15 0.140398
\(936\) 0 0
\(937\) −1.11596e16 −0.504755 −0.252378 − 0.967629i \(-0.581212\pi\)
−0.252378 + 0.967629i \(0.581212\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −2.08676e16 −0.921997 −0.460999 − 0.887401i \(-0.652509\pi\)
−0.460999 + 0.887401i \(0.652509\pi\)
\(942\) 0 0
\(943\) −5.08229e16 −2.21945
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 3.24403e16 1.38407 0.692037 − 0.721862i \(-0.256713\pi\)
0.692037 + 0.721862i \(0.256713\pi\)
\(948\) 0 0
\(949\) −3.12074e16 −1.31612
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −3.61199e16 −1.48846 −0.744228 − 0.667926i \(-0.767182\pi\)
−0.744228 + 0.667926i \(0.767182\pi\)
\(954\) 0 0
\(955\) 8.42852e15 0.343347
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 3.56466e15 0.141911
\(960\) 0 0
\(961\) 2.89840e16 1.14072
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −1.78987e16 −0.688529
\(966\) 0 0
\(967\) 4.35846e14 0.0165763 0.00828815 − 0.999966i \(-0.497362\pi\)
0.00828815 + 0.999966i \(0.497362\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −4.01243e15 −0.149177 −0.0745884 − 0.997214i \(-0.523764\pi\)
−0.0745884 + 0.997214i \(0.523764\pi\)
\(972\) 0 0
\(973\) 4.69273e14 0.0172506
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −4.27191e16 −1.53533 −0.767665 − 0.640851i \(-0.778582\pi\)
−0.767665 + 0.640851i \(0.778582\pi\)
\(978\) 0 0
\(979\) −2.04293e16 −0.726021
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −3.72875e16 −1.29574 −0.647871 − 0.761750i \(-0.724340\pi\)
−0.647871 + 0.761750i \(0.724340\pi\)
\(984\) 0 0
\(985\) −1.90689e16 −0.655278
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 3.68608e16 1.23876
\(990\) 0 0
\(991\) 4.47101e16 1.48594 0.742969 − 0.669326i \(-0.233417\pi\)
0.742969 + 0.669326i \(0.233417\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −2.02209e16 −0.657315
\(996\) 0 0
\(997\) −1.16976e16 −0.376073 −0.188037 − 0.982162i \(-0.560212\pi\)
−0.188037 + 0.982162i \(0.560212\pi\)
\(998\) 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 144.12.a.f.1.1 1
3.2 odd 2 48.12.a.c.1.1 1
4.3 odd 2 36.12.a.b.1.1 1
12.11 even 2 12.12.a.b.1.1 ✓ 1
24.5 odd 2 192.12.a.n.1.1 1
24.11 even 2 192.12.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.12.a.b.1.1 ✓ 1 12.11 even 2
36.12.a.b.1.1 1 4.3 odd 2
48.12.a.c.1.1 1 3.2 odd 2
144.12.a.f.1.1 1 1.1 even 1 trivial
192.12.a.d.1.1 1 24.11 even 2
192.12.a.n.1.1 1 24.5 odd 2