Properties

Label 432.9.e.g.161.1
Level $432$
Weight $9$
Character 432.161
Analytic conductor $175.988$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [432,9,Mod(161,432)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(432, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("432.161");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 432.e (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(175.987559546\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5}\cdot 3 \)
Twist minimal: no (minimal twist has level 54)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 161.1
Root \(-1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 432.161
Dual form 432.9.e.g.161.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-678.823i q^{5} +2065.00 q^{7} +O(q^{10})\) \(q-678.823i q^{5} +2065.00 q^{7} +6652.46i q^{11} +8063.00 q^{13} +21586.6i q^{17} +226609. q^{19} -368329. i q^{23} -70175.0 q^{25} +937047. i q^{29} -826370. q^{31} -1.40177e6i q^{35} +1.34458e6 q^{37} +5.19191e6i q^{41} +6.14774e6 q^{43} +5.91078e6i q^{47} -1.50058e6 q^{49} +768156. i q^{53} +4.51584e6 q^{55} +473954. i q^{59} -1.49857e7 q^{61} -5.47335e6i q^{65} +1.00237e7 q^{67} +4.54849e7i q^{71} -2.32616e7 q^{73} +1.37373e7i q^{77} -1.42672e7 q^{79} +3.61918e7i q^{83} +1.46534e7 q^{85} -1.15088e8i q^{89} +1.66501e7 q^{91} -1.53827e8i q^{95} -4.05716e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4130 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4130 q^{7} + 16126 q^{13} + 453218 q^{19} - 140350 q^{25} - 1652740 q^{31} + 2689150 q^{37} + 12295484 q^{43} - 3001152 q^{49} + 9031680 q^{55} - 29971394 q^{61} + 20047394 q^{67} - 46523138 q^{73} - 28534366 q^{79} + 29306880 q^{85} + 33300190 q^{91} - 81143234 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/432\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(5\) − 678.823i − 1.08612i −0.839695 0.543058i \(-0.817266\pi\)
0.839695 0.543058i \(-0.182734\pi\)
\(6\) 0 0
\(7\) 2065.00 0.860058 0.430029 0.902815i \(-0.358503\pi\)
0.430029 + 0.902815i \(0.358503\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 6652.46i 0.454372i 0.973851 + 0.227186i \(0.0729525\pi\)
−0.973851 + 0.227186i \(0.927047\pi\)
\(12\) 0 0
\(13\) 8063.00 0.282308 0.141154 0.989988i \(-0.454919\pi\)
0.141154 + 0.989988i \(0.454919\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 21586.6i 0.258457i 0.991615 + 0.129228i \(0.0412500\pi\)
−0.991615 + 0.129228i \(0.958750\pi\)
\(18\) 0 0
\(19\) 226609. 1.73885 0.869426 0.494063i \(-0.164489\pi\)
0.869426 + 0.494063i \(0.164489\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 368329.i − 1.31621i −0.752927 0.658104i \(-0.771359\pi\)
0.752927 0.658104i \(-0.228641\pi\)
\(24\) 0 0
\(25\) −70175.0 −0.179648
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 937047.i 1.32486i 0.749125 + 0.662429i \(0.230474\pi\)
−0.749125 + 0.662429i \(0.769526\pi\)
\(30\) 0 0
\(31\) −826370. −0.894804 −0.447402 0.894333i \(-0.647651\pi\)
−0.447402 + 0.894333i \(0.647651\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 1.40177e6i − 0.934123i
\(36\) 0 0
\(37\) 1.34458e6 0.717428 0.358714 0.933448i \(-0.383215\pi\)
0.358714 + 0.933448i \(0.383215\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.19191e6i 1.83735i 0.395017 + 0.918674i \(0.370739\pi\)
−0.395017 + 0.918674i \(0.629261\pi\)
\(42\) 0 0
\(43\) 6.14774e6 1.79822 0.899108 0.437727i \(-0.144216\pi\)
0.899108 + 0.437727i \(0.144216\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.91078e6i 1.21130i 0.795729 + 0.605652i \(0.207088\pi\)
−0.795729 + 0.605652i \(0.792912\pi\)
\(48\) 0 0
\(49\) −1.50058e6 −0.260300
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 768156.i 0.0973522i 0.998815 + 0.0486761i \(0.0155002\pi\)
−0.998815 + 0.0486761i \(0.984500\pi\)
\(54\) 0 0
\(55\) 4.51584e6 0.493501
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 473954.i 0.0391136i 0.999809 + 0.0195568i \(0.00622552\pi\)
−0.999809 + 0.0195568i \(0.993774\pi\)
\(60\) 0 0
\(61\) −1.49857e7 −1.08232 −0.541162 0.840918i \(-0.682016\pi\)
−0.541162 + 0.840918i \(0.682016\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 5.47335e6i − 0.306619i
\(66\) 0 0
\(67\) 1.00237e7 0.497426 0.248713 0.968577i \(-0.419992\pi\)
0.248713 + 0.968577i \(0.419992\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.54849e7i 1.78992i 0.446145 + 0.894961i \(0.352797\pi\)
−0.446145 + 0.894961i \(0.647203\pi\)
\(72\) 0 0
\(73\) −2.32616e7 −0.819120 −0.409560 0.912283i \(-0.634318\pi\)
−0.409560 + 0.912283i \(0.634318\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.37373e7i 0.390786i
\(78\) 0 0
\(79\) −1.42672e7 −0.366294 −0.183147 0.983086i \(-0.558628\pi\)
−0.183147 + 0.983086i \(0.558628\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.61918e7i 0.762602i 0.924451 + 0.381301i \(0.124524\pi\)
−0.924451 + 0.381301i \(0.875476\pi\)
\(84\) 0 0
\(85\) 1.46534e7 0.280714
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 1.15088e8i − 1.83429i −0.398549 0.917147i \(-0.630486\pi\)
0.398549 0.917147i \(-0.369514\pi\)
\(90\) 0 0
\(91\) 1.66501e7 0.242801
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 1.53827e8i − 1.88860i
\(96\) 0 0
\(97\) −4.05716e7 −0.458285 −0.229142 0.973393i \(-0.573592\pi\)
−0.229142 + 0.973393i \(0.573592\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 6.49465e7i 0.624123i 0.950062 + 0.312061i \(0.101019\pi\)
−0.950062 + 0.312061i \(0.898981\pi\)
\(102\) 0 0
\(103\) 1.37263e8 1.21956 0.609782 0.792569i \(-0.291257\pi\)
0.609782 + 0.792569i \(0.291257\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 1.39108e8i − 1.06125i −0.847607 0.530625i \(-0.821957\pi\)
0.847607 0.530625i \(-0.178043\pi\)
\(108\) 0 0
\(109\) −4.29417e7 −0.304210 −0.152105 0.988364i \(-0.548605\pi\)
−0.152105 + 0.988364i \(0.548605\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2.91649e8i 1.78874i 0.447329 + 0.894369i \(0.352375\pi\)
−0.447329 + 0.894369i \(0.647625\pi\)
\(114\) 0 0
\(115\) −2.50030e8 −1.42956
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 4.45762e7i 0.222288i
\(120\) 0 0
\(121\) 1.70104e8 0.793546
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 2.17529e8i − 0.890997i
\(126\) 0 0
\(127\) 3.39515e8 1.30510 0.652551 0.757745i \(-0.273699\pi\)
0.652551 + 0.757745i \(0.273699\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 2.24537e8i − 0.762436i −0.924485 0.381218i \(-0.875505\pi\)
0.924485 0.381218i \(-0.124495\pi\)
\(132\) 0 0
\(133\) 4.67948e8 1.49551
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 4.01263e8i − 1.13906i −0.821970 0.569531i \(-0.807125\pi\)
0.821970 0.569531i \(-0.192875\pi\)
\(138\) 0 0
\(139\) −2.69764e8 −0.722645 −0.361322 0.932441i \(-0.617675\pi\)
−0.361322 + 0.932441i \(0.617675\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 5.36388e7i 0.128273i
\(144\) 0 0
\(145\) 6.36088e8 1.43895
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1.94198e8i 0.394003i 0.980403 + 0.197002i \(0.0631204\pi\)
−0.980403 + 0.197002i \(0.936880\pi\)
\(150\) 0 0
\(151\) 8.75100e7 0.168325 0.0841627 0.996452i \(-0.473178\pi\)
0.0841627 + 0.996452i \(0.473178\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 5.60959e8i 0.971861i
\(156\) 0 0
\(157\) 2.84655e8 0.468512 0.234256 0.972175i \(-0.424735\pi\)
0.234256 + 0.972175i \(0.424735\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 7.60600e8i − 1.13202i
\(162\) 0 0
\(163\) 2.63153e8 0.372785 0.186393 0.982475i \(-0.440320\pi\)
0.186393 + 0.982475i \(0.440320\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.16384e9i 1.49633i 0.663511 + 0.748167i \(0.269066\pi\)
−0.663511 + 0.748167i \(0.730934\pi\)
\(168\) 0 0
\(169\) −7.50719e8 −0.920302
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.32417e9i 1.47829i 0.673547 + 0.739145i \(0.264770\pi\)
−0.673547 + 0.739145i \(0.735230\pi\)
\(174\) 0 0
\(175\) −1.44911e8 −0.154508
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 4.06696e8i − 0.396148i −0.980187 0.198074i \(-0.936531\pi\)
0.980187 0.198074i \(-0.0634686\pi\)
\(180\) 0 0
\(181\) −1.29071e9 −1.20258 −0.601289 0.799032i \(-0.705346\pi\)
−0.601289 + 0.799032i \(0.705346\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 9.12728e8i − 0.779210i
\(186\) 0 0
\(187\) −1.43604e8 −0.117435
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 2.97714e8i − 0.223700i −0.993725 0.111850i \(-0.964322\pi\)
0.993725 0.111850i \(-0.0356776\pi\)
\(192\) 0 0
\(193\) 2.22004e9 1.60004 0.800021 0.599972i \(-0.204822\pi\)
0.800021 + 0.599972i \(0.204822\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.91580e9i 1.27199i 0.771693 + 0.635996i \(0.219410\pi\)
−0.771693 + 0.635996i \(0.780590\pi\)
\(198\) 0 0
\(199\) 1.75472e9 1.11891 0.559457 0.828859i \(-0.311010\pi\)
0.559457 + 0.828859i \(0.311010\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.93500e9i 1.13945i
\(204\) 0 0
\(205\) 3.52438e9 1.99557
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1.50751e9i 0.790086i
\(210\) 0 0
\(211\) 2.15389e9 1.08666 0.543331 0.839519i \(-0.317163\pi\)
0.543331 + 0.839519i \(0.317163\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 4.17323e9i − 1.95307i
\(216\) 0 0
\(217\) −1.70645e9 −0.769583
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.74052e8i 0.0729644i
\(222\) 0 0
\(223\) 2.39374e9 0.967959 0.483980 0.875079i \(-0.339191\pi\)
0.483980 + 0.875079i \(0.339191\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 9.92657e8i − 0.373849i −0.982374 0.186924i \(-0.940148\pi\)
0.982374 0.186924i \(-0.0598519\pi\)
\(228\) 0 0
\(229\) −2.22150e9 −0.807800 −0.403900 0.914803i \(-0.632346\pi\)
−0.403900 + 0.914803i \(0.632346\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 1.24042e9i − 0.420867i −0.977608 0.210433i \(-0.932512\pi\)
0.977608 0.210433i \(-0.0674875\pi\)
\(234\) 0 0
\(235\) 4.01237e9 1.31562
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 7.59290e8i − 0.232711i −0.993208 0.116355i \(-0.962879\pi\)
0.993208 0.116355i \(-0.0371211\pi\)
\(240\) 0 0
\(241\) 4.47467e9 1.32646 0.663228 0.748417i \(-0.269186\pi\)
0.663228 + 0.748417i \(0.269186\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.01862e9i 0.282716i
\(246\) 0 0
\(247\) 1.82715e9 0.490892
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 1.07356e9i − 0.270477i −0.990813 0.135239i \(-0.956820\pi\)
0.990813 0.135239i \(-0.0431801\pi\)
\(252\) 0 0
\(253\) 2.45029e9 0.598048
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 7.58666e9i − 1.73907i −0.493867 0.869537i \(-0.664417\pi\)
0.493867 0.869537i \(-0.335583\pi\)
\(258\) 0 0
\(259\) 2.77655e9 0.617030
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2.81146e9i 0.587637i 0.955861 + 0.293819i \(0.0949262\pi\)
−0.955861 + 0.293819i \(0.905074\pi\)
\(264\) 0 0
\(265\) 5.21441e8 0.105736
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 2.96285e9i − 0.565849i −0.959142 0.282924i \(-0.908695\pi\)
0.959142 0.282924i \(-0.0913045\pi\)
\(270\) 0 0
\(271\) 8.40415e9 1.55818 0.779089 0.626914i \(-0.215682\pi\)
0.779089 + 0.626914i \(0.215682\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 4.66836e8i − 0.0816270i
\(276\) 0 0
\(277\) 5.98162e9 1.01601 0.508007 0.861353i \(-0.330382\pi\)
0.508007 + 0.861353i \(0.330382\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 1.08341e10i − 1.73767i −0.495105 0.868833i \(-0.664870\pi\)
0.495105 0.868833i \(-0.335130\pi\)
\(282\) 0 0
\(283\) −3.78670e9 −0.590357 −0.295179 0.955442i \(-0.595379\pi\)
−0.295179 + 0.955442i \(0.595379\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.07213e10i 1.58023i
\(288\) 0 0
\(289\) 6.50978e9 0.933200
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 7.32039e9i 0.993262i 0.867962 + 0.496631i \(0.165430\pi\)
−0.867962 + 0.496631i \(0.834570\pi\)
\(294\) 0 0
\(295\) 3.21731e8 0.0424819
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 2.96984e9i − 0.371576i
\(300\) 0 0
\(301\) 1.26951e10 1.54657
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1.01726e10i 1.17553i
\(306\) 0 0
\(307\) 2.21986e8 0.0249903 0.0124951 0.999922i \(-0.496023\pi\)
0.0124951 + 0.999922i \(0.496023\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 7.19555e9i − 0.769171i −0.923089 0.384585i \(-0.874344\pi\)
0.923089 0.384585i \(-0.125656\pi\)
\(312\) 0 0
\(313\) 4.23341e9 0.441076 0.220538 0.975378i \(-0.429219\pi\)
0.220538 + 0.975378i \(0.429219\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 5.40044e9i 0.534801i 0.963586 + 0.267400i \(0.0861646\pi\)
−0.963586 + 0.267400i \(0.913835\pi\)
\(318\) 0 0
\(319\) −6.23367e9 −0.601978
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 4.89171e9i 0.449418i
\(324\) 0 0
\(325\) −5.65821e8 −0.0507161
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 1.22058e10i 1.04179i
\(330\) 0 0
\(331\) 1.35941e10 1.13250 0.566252 0.824232i \(-0.308393\pi\)
0.566252 + 0.824232i \(0.308393\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) − 6.80431e9i − 0.540263i
\(336\) 0 0
\(337\) −2.35433e9 −0.182536 −0.0912680 0.995826i \(-0.529092\pi\)
−0.0912680 + 0.995826i \(0.529092\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 5.49739e9i − 0.406574i
\(342\) 0 0
\(343\) −1.50030e10 −1.08393
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 8.33815e9i − 0.575111i −0.957764 0.287555i \(-0.907157\pi\)
0.957764 0.287555i \(-0.0928426\pi\)
\(348\) 0 0
\(349\) −1.26463e10 −0.852434 −0.426217 0.904621i \(-0.640154\pi\)
−0.426217 + 0.904621i \(0.640154\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 3.76139e9i 0.242242i 0.992638 + 0.121121i \(0.0386490\pi\)
−0.992638 + 0.121121i \(0.961351\pi\)
\(354\) 0 0
\(355\) 3.08762e10 1.94406
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1.64726e10i 0.991707i 0.868406 + 0.495853i \(0.165145\pi\)
−0.868406 + 0.495853i \(0.834855\pi\)
\(360\) 0 0
\(361\) 3.43681e10 2.02361
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1.57905e10i 0.889659i
\(366\) 0 0
\(367\) −7.38723e9 −0.407209 −0.203605 0.979053i \(-0.565266\pi\)
−0.203605 + 0.979053i \(0.565266\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 1.58624e9i 0.0837286i
\(372\) 0 0
\(373\) 2.30025e10 1.18834 0.594168 0.804341i \(-0.297482\pi\)
0.594168 + 0.804341i \(0.297482\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 7.55541e9i 0.374018i
\(378\) 0 0
\(379\) 2.05891e10 0.997883 0.498942 0.866636i \(-0.333722\pi\)
0.498942 + 0.866636i \(0.333722\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 1.43645e10i − 0.667569i −0.942649 0.333785i \(-0.891674\pi\)
0.942649 0.333785i \(-0.108326\pi\)
\(384\) 0 0
\(385\) 9.32521e9 0.424439
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 3.33296e10i − 1.45557i −0.685808 0.727783i \(-0.740551\pi\)
0.685808 0.727783i \(-0.259449\pi\)
\(390\) 0 0
\(391\) 7.95096e9 0.340183
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 9.68488e9i 0.397838i
\(396\) 0 0
\(397\) −4.32631e9 −0.174163 −0.0870814 0.996201i \(-0.527754\pi\)
−0.0870814 + 0.996201i \(0.527754\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 1.20770e10i − 0.467069i −0.972348 0.233535i \(-0.924971\pi\)
0.972348 0.233535i \(-0.0750293\pi\)
\(402\) 0 0
\(403\) −6.66302e9 −0.252610
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 8.94473e9i 0.325979i
\(408\) 0 0
\(409\) −3.81221e10 −1.36234 −0.681168 0.732127i \(-0.738528\pi\)
−0.681168 + 0.732127i \(0.738528\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 9.78715e8i 0.0336400i
\(414\) 0 0
\(415\) 2.45678e10 0.828275
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 2.52616e10i 0.819605i 0.912174 + 0.409803i \(0.134402\pi\)
−0.912174 + 0.409803i \(0.865598\pi\)
\(420\) 0 0
\(421\) −3.33965e9 −0.106310 −0.0531548 0.998586i \(-0.516928\pi\)
−0.0531548 + 0.998586i \(0.516928\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 1.51484e9i − 0.0464312i
\(426\) 0 0
\(427\) −3.09455e10 −0.930862
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 5.55244e9i − 0.160907i −0.996758 0.0804534i \(-0.974363\pi\)
0.996758 0.0804534i \(-0.0256368\pi\)
\(432\) 0 0
\(433\) 1.14713e10 0.326334 0.163167 0.986598i \(-0.447829\pi\)
0.163167 + 0.986598i \(0.447829\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 8.34667e10i − 2.28869i
\(438\) 0 0
\(439\) 5.98486e10 1.61137 0.805686 0.592343i \(-0.201797\pi\)
0.805686 + 0.592343i \(0.201797\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 3.61609e9i − 0.0938910i −0.998897 0.0469455i \(-0.985051\pi\)
0.998897 0.0469455i \(-0.0149487\pi\)
\(444\) 0 0
\(445\) −7.81241e10 −1.99226
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 2.39980e9i − 0.0590459i −0.999564 0.0295230i \(-0.990601\pi\)
0.999564 0.0295230i \(-0.00939882\pi\)
\(450\) 0 0
\(451\) −3.45390e10 −0.834839
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 1.13025e10i − 0.263710i
\(456\) 0 0
\(457\) −4.11731e10 −0.943948 −0.471974 0.881612i \(-0.656458\pi\)
−0.471974 + 0.881612i \(0.656458\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 2.07400e10i 0.459203i 0.973285 + 0.229601i \(0.0737422\pi\)
−0.973285 + 0.229601i \(0.926258\pi\)
\(462\) 0 0
\(463\) 7.35380e9 0.160025 0.0800125 0.996794i \(-0.474504\pi\)
0.0800125 + 0.996794i \(0.474504\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 9.09281e10i − 1.91175i −0.293776 0.955874i \(-0.594912\pi\)
0.293776 0.955874i \(-0.405088\pi\)
\(468\) 0 0
\(469\) 2.06989e10 0.427816
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 4.08976e10i 0.817059i
\(474\) 0 0
\(475\) −1.59023e10 −0.312381
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 1.91197e10i − 0.363194i −0.983373 0.181597i \(-0.941873\pi\)
0.983373 0.181597i \(-0.0581266\pi\)
\(480\) 0 0
\(481\) 1.08413e10 0.202536
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2.75409e10i 0.497750i
\(486\) 0 0
\(487\) −5.28737e10 −0.939992 −0.469996 0.882669i \(-0.655745\pi\)
−0.469996 + 0.882669i \(0.655745\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 9.78032e10i 1.68278i 0.540429 + 0.841389i \(0.318262\pi\)
−0.540429 + 0.841389i \(0.681738\pi\)
\(492\) 0 0
\(493\) −2.02276e10 −0.342418
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 9.39263e10i 1.53944i
\(498\) 0 0
\(499\) −8.89351e10 −1.43440 −0.717201 0.696866i \(-0.754577\pi\)
−0.717201 + 0.696866i \(0.754577\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 2.97791e10i − 0.465200i −0.972572 0.232600i \(-0.925277\pi\)
0.972572 0.232600i \(-0.0747233\pi\)
\(504\) 0 0
\(505\) 4.40871e10 0.677870
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 2.67186e10i − 0.398054i −0.979994 0.199027i \(-0.936222\pi\)
0.979994 0.199027i \(-0.0637781\pi\)
\(510\) 0 0
\(511\) −4.80351e10 −0.704491
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 9.31772e10i − 1.32459i
\(516\) 0 0
\(517\) −3.93212e10 −0.550383
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 1.06779e10i − 0.144922i −0.997371 0.0724610i \(-0.976915\pi\)
0.997371 0.0724610i \(-0.0230853\pi\)
\(522\) 0 0
\(523\) −4.63928e10 −0.620075 −0.310037 0.950724i \(-0.600342\pi\)
−0.310037 + 0.950724i \(0.600342\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 1.78385e10i − 0.231268i
\(528\) 0 0
\(529\) −5.73553e10 −0.732405
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 4.18623e10i 0.518698i
\(534\) 0 0
\(535\) −9.44298e10 −1.15264
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 9.98252e9i − 0.118273i
\(540\) 0 0
\(541\) −1.22420e11 −1.42911 −0.714553 0.699581i \(-0.753370\pi\)
−0.714553 + 0.699581i \(0.753370\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 2.91498e10i 0.330407i
\(546\) 0 0
\(547\) 5.53975e10 0.618786 0.309393 0.950934i \(-0.399874\pi\)
0.309393 + 0.950934i \(0.399874\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 2.12343e11i 2.30373i
\(552\) 0 0
\(553\) −2.94617e10 −0.315034
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 1.17293e11i − 1.21857i −0.792951 0.609285i \(-0.791456\pi\)
0.792951 0.609285i \(-0.208544\pi\)
\(558\) 0 0
\(559\) 4.95692e10 0.507651
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 1.03752e11i 1.03267i 0.856385 + 0.516337i \(0.172705\pi\)
−0.856385 + 0.516337i \(0.827295\pi\)
\(564\) 0 0
\(565\) 1.97978e11 1.94278
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 2.67515e10i − 0.255211i −0.991825 0.127605i \(-0.959271\pi\)
0.991825 0.127605i \(-0.0407291\pi\)
\(570\) 0 0
\(571\) −1.14161e11 −1.07393 −0.536963 0.843606i \(-0.680428\pi\)
−0.536963 + 0.843606i \(0.680428\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 2.58475e10i 0.236454i
\(576\) 0 0
\(577\) 1.43414e11 1.29386 0.646930 0.762549i \(-0.276052\pi\)
0.646930 + 0.762549i \(0.276052\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 7.47361e10i 0.655883i
\(582\) 0 0
\(583\) −5.11012e9 −0.0442341
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.93256e11i 1.62773i 0.581057 + 0.813863i \(0.302639\pi\)
−0.581057 + 0.813863i \(0.697361\pi\)
\(588\) 0 0
\(589\) −1.87263e11 −1.55593
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 2.21385e10i 0.179031i 0.995985 + 0.0895156i \(0.0285319\pi\)
−0.995985 + 0.0895156i \(0.971468\pi\)
\(594\) 0 0
\(595\) 3.02594e10 0.241430
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 1.32315e11i − 1.02778i −0.857856 0.513891i \(-0.828204\pi\)
0.857856 0.513891i \(-0.171796\pi\)
\(600\) 0 0
\(601\) −2.29562e10 −0.175955 −0.0879775 0.996122i \(-0.528040\pi\)
−0.0879775 + 0.996122i \(0.528040\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 1.15470e11i − 0.861883i
\(606\) 0 0
\(607\) −1.56978e11 −1.15634 −0.578169 0.815917i \(-0.696233\pi\)
−0.578169 + 0.815917i \(0.696233\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 4.76586e10i 0.341961i
\(612\) 0 0
\(613\) −1.72931e11 −1.22470 −0.612352 0.790586i \(-0.709776\pi\)
−0.612352 + 0.790586i \(0.709776\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 1.42425e10i − 0.0982758i −0.998792 0.0491379i \(-0.984353\pi\)
0.998792 0.0491379i \(-0.0156474\pi\)
\(618\) 0 0
\(619\) −1.29983e11 −0.885367 −0.442684 0.896678i \(-0.645973\pi\)
−0.442684 + 0.896678i \(0.645973\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 2.37656e11i − 1.57760i
\(624\) 0 0
\(625\) −1.75075e11 −1.14737
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 2.90247e10i 0.185424i
\(630\) 0 0
\(631\) 5.79110e9 0.0365295 0.0182648 0.999833i \(-0.494186\pi\)
0.0182648 + 0.999833i \(0.494186\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 2.30471e11i − 1.41749i
\(636\) 0 0
\(637\) −1.20991e10 −0.0734847
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 2.18083e11i − 1.29178i −0.763429 0.645892i \(-0.776486\pi\)
0.763429 0.645892i \(-0.223514\pi\)
\(642\) 0 0
\(643\) −1.74278e11 −1.01953 −0.509764 0.860314i \(-0.670267\pi\)
−0.509764 + 0.860314i \(0.670267\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 2.22876e11i 1.27188i 0.771738 + 0.635941i \(0.219388\pi\)
−0.771738 + 0.635941i \(0.780612\pi\)
\(648\) 0 0
\(649\) −3.15296e9 −0.0177721
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 2.56934e11i 1.41308i 0.707671 + 0.706542i \(0.249746\pi\)
−0.707671 + 0.706542i \(0.750254\pi\)
\(654\) 0 0
\(655\) −1.52421e11 −0.828094
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 2.69559e11i − 1.42926i −0.699500 0.714632i \(-0.746594\pi\)
0.699500 0.714632i \(-0.253406\pi\)
\(660\) 0 0
\(661\) −2.56807e11 −1.34525 −0.672623 0.739986i \(-0.734832\pi\)
−0.672623 + 0.739986i \(0.734832\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 3.17653e11i − 1.62430i
\(666\) 0 0
\(667\) 3.45142e11 1.74379
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 9.96918e10i − 0.491778i
\(672\) 0 0
\(673\) 1.67895e11 0.818421 0.409210 0.912440i \(-0.365804\pi\)
0.409210 + 0.912440i \(0.365804\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1.70201e11i 0.810230i 0.914266 + 0.405115i \(0.132769\pi\)
−0.914266 + 0.405115i \(0.867231\pi\)
\(678\) 0 0
\(679\) −8.37804e10 −0.394152
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 3.04016e11i − 1.39706i −0.715582 0.698529i \(-0.753838\pi\)
0.715582 0.698529i \(-0.246162\pi\)
\(684\) 0 0
\(685\) −2.72387e11 −1.23715
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 6.19364e9i 0.0274833i
\(690\) 0 0
\(691\) 4.17969e10 0.183329 0.0916646 0.995790i \(-0.470781\pi\)
0.0916646 + 0.995790i \(0.470781\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1.83122e11i 0.784876i
\(696\) 0 0
\(697\) −1.12075e11 −0.474875
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 2.07821e11i 0.860631i 0.902678 + 0.430316i \(0.141598\pi\)
−0.902678 + 0.430316i \(0.858402\pi\)
\(702\) 0 0
\(703\) 3.04693e11 1.24750
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1.34114e11i 0.536782i
\(708\) 0 0
\(709\) −3.47189e11 −1.37398 −0.686990 0.726667i \(-0.741068\pi\)
−0.686990 + 0.726667i \(0.741068\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 3.04376e11i 1.17775i
\(714\) 0 0
\(715\) 3.64112e10 0.139319
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1.56177e11i 0.584387i 0.956359 + 0.292194i \(0.0943851\pi\)
−0.956359 + 0.292194i \(0.905615\pi\)
\(720\) 0 0
\(721\) 2.83448e11 1.04890
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 6.57572e10i − 0.238008i
\(726\) 0 0
\(727\) 2.70116e11 0.966969 0.483485 0.875353i \(-0.339371\pi\)
0.483485 + 0.875353i \(0.339371\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1.32709e11i 0.464761i
\(732\) 0 0
\(733\) 1.53459e11 0.531589 0.265795 0.964030i \(-0.414366\pi\)
0.265795 + 0.964030i \(0.414366\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 6.66822e10i 0.226017i
\(738\) 0 0
\(739\) −7.60266e10 −0.254911 −0.127455 0.991844i \(-0.540681\pi\)
−0.127455 + 0.991844i \(0.540681\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 4.18589e11i − 1.37351i −0.726888 0.686756i \(-0.759034\pi\)
0.726888 0.686756i \(-0.240966\pi\)
\(744\) 0 0
\(745\) 1.31826e11 0.427933
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 2.87259e11i − 0.912737i
\(750\) 0 0
\(751\) −3.73788e11 −1.17507 −0.587537 0.809197i \(-0.699902\pi\)
−0.587537 + 0.809197i \(0.699902\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 5.94038e10i − 0.182821i
\(756\) 0 0
\(757\) 4.74806e11 1.44588 0.722941 0.690910i \(-0.242790\pi\)
0.722941 + 0.690910i \(0.242790\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 5.91673e11i − 1.76418i −0.471079 0.882091i \(-0.656135\pi\)
0.471079 0.882091i \(-0.343865\pi\)
\(762\) 0 0
\(763\) −8.86745e10 −0.261638
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 3.82149e9i 0.0110421i
\(768\) 0 0
\(769\) 4.29184e11 1.22727 0.613633 0.789592i \(-0.289708\pi\)
0.613633 + 0.789592i \(0.289708\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1.95537e11i 0.547660i 0.961778 + 0.273830i \(0.0882906\pi\)
−0.961778 + 0.273830i \(0.911709\pi\)
\(774\) 0 0
\(775\) 5.79905e10 0.160750
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.17653e12i 3.19488i
\(780\) 0 0
\(781\) −3.02587e11 −0.813290
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 1.93230e11i − 0.508858i
\(786\) 0 0
\(787\) 1.30204e10 0.0339409 0.0169705 0.999856i \(-0.494598\pi\)
0.0169705 + 0.999856i \(0.494598\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 6.02256e11i 1.53842i
\(792\) 0 0
\(793\) −1.20830e11 −0.305549
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 1.33234e11i − 0.330204i −0.986277 0.165102i \(-0.947205\pi\)
0.986277 0.165102i \(-0.0527953\pi\)
\(798\) 0 0
\(799\) −1.27593e11 −0.313070
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 1.54747e11i − 0.372185i
\(804\) 0 0
\(805\) −5.16312e11 −1.22950
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 7.75707e11i 1.81094i 0.424413 + 0.905469i \(0.360481\pi\)
−0.424413 + 0.905469i \(0.639519\pi\)
\(810\) 0 0
\(811\) 7.76807e11 1.79568 0.897842 0.440318i \(-0.145134\pi\)
0.897842 + 0.440318i \(0.145134\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 1.78634e11i − 0.404888i
\(816\) 0 0
\(817\) 1.39313e12 3.12683
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 2.75649e10i 0.0606713i 0.999540 + 0.0303356i \(0.00965761\pi\)
−0.999540 + 0.0303356i \(0.990342\pi\)
\(822\) 0 0
\(823\) −2.84608e11 −0.620366 −0.310183 0.950677i \(-0.600390\pi\)
−0.310183 + 0.950677i \(0.600390\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 2.82378e11i 0.603683i 0.953358 + 0.301842i \(0.0976014\pi\)
−0.953358 + 0.301842i \(0.902399\pi\)
\(828\) 0 0
\(829\) 8.40257e11 1.77907 0.889537 0.456863i \(-0.151027\pi\)
0.889537 + 0.456863i \(0.151027\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 3.23923e10i − 0.0672762i
\(834\) 0 0
\(835\) 7.90043e11 1.62519
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 3.34946e11i − 0.675968i −0.941152 0.337984i \(-0.890255\pi\)
0.941152 0.337984i \(-0.109745\pi\)
\(840\) 0 0
\(841\) −3.77810e11 −0.755248
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 5.09605e11i 0.999555i
\(846\) 0 0
\(847\) 3.51264e11 0.682496
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 4.95246e11i − 0.944284i
\(852\) 0 0
\(853\) 6.09757e10 0.115176 0.0575878 0.998340i \(-0.481659\pi\)
0.0575878 + 0.998340i \(0.481659\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 4.12820e11i − 0.765311i −0.923891 0.382655i \(-0.875010\pi\)
0.923891 0.382655i \(-0.124990\pi\)
\(858\) 0 0
\(859\) −4.28958e11 −0.787847 −0.393924 0.919143i \(-0.628883\pi\)
−0.393924 + 0.919143i \(0.628883\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 9.65828e11i − 1.74123i −0.491963 0.870616i \(-0.663720\pi\)
0.491963 0.870616i \(-0.336280\pi\)
\(864\) 0 0
\(865\) 8.98876e11 1.60559
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 9.49119e10i − 0.166434i
\(870\) 0 0
\(871\) 8.08211e10 0.140427
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 4.49197e11i − 0.766310i
\(876\) 0 0
\(877\) −1.01992e12 −1.72412 −0.862059 0.506808i \(-0.830825\pi\)
−0.862059 + 0.506808i \(0.830825\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 7.44050e11i − 1.23509i −0.786535 0.617545i \(-0.788127\pi\)
0.786535 0.617545i \(-0.211873\pi\)
\(882\) 0 0
\(883\) −1.35286e11 −0.222540 −0.111270 0.993790i \(-0.535492\pi\)
−0.111270 + 0.993790i \(0.535492\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 5.56566e11i 0.899129i 0.893248 + 0.449565i \(0.148421\pi\)
−0.893248 + 0.449565i \(0.851579\pi\)
\(888\) 0 0
\(889\) 7.01099e11 1.12246
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1.33944e12i 2.10628i
\(894\) 0 0
\(895\) −2.76074e11 −0.430263
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 7.74347e11i − 1.18549i
\(900\) 0 0
\(901\) −1.65818e10 −0.0251613
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 8.76161e11i 1.30614i
\(906\) 0 0
\(907\) −3.75761e10 −0.0555243 −0.0277621 0.999615i \(-0.508838\pi\)
−0.0277621 + 0.999615i \(0.508838\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 5.43769e11i 0.789479i 0.918793 + 0.394740i \(0.129165\pi\)
−0.918793 + 0.394740i \(0.870835\pi\)
\(912\) 0 0
\(913\) −2.40765e11 −0.346505
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 4.63670e11i − 0.655740i
\(918\) 0 0
\(919\) −9.08317e10 −0.127343 −0.0636715 0.997971i \(-0.520281\pi\)
−0.0636715 + 0.997971i \(0.520281\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 3.66745e11i 0.505309i
\(924\) 0 0
\(925\) −9.43556e10 −0.128884
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1.38421e11i 0.185841i 0.995674 + 0.0929203i \(0.0296202\pi\)
−0.995674 + 0.0929203i \(0.970380\pi\)
\(930\) 0 0
\(931\) −3.40044e11 −0.452623
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 9.74814e10i 0.127549i
\(936\) 0 0
\(937\) 1.28993e12 1.67343 0.836714 0.547640i \(-0.184474\pi\)
0.836714 + 0.547640i \(0.184474\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 1.44651e12i − 1.84486i −0.386163 0.922431i \(-0.626200\pi\)
0.386163 0.922431i \(-0.373800\pi\)
\(942\) 0 0
\(943\) 1.91233e12 2.41833
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 6.92009e11i − 0.860423i −0.902728 0.430212i \(-0.858439\pi\)
0.902728 0.430212i \(-0.141561\pi\)
\(948\) 0 0
\(949\) −1.87558e11 −0.231244
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 3.33259e11i 0.404027i 0.979383 + 0.202013i \(0.0647484\pi\)
−0.979383 + 0.202013i \(0.935252\pi\)
\(954\) 0 0
\(955\) −2.02095e11 −0.242964
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 8.28609e11i − 0.979659i
\(960\) 0 0
\(961\) −1.70004e11 −0.199326
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 1.50701e12i − 1.73783i
\(966\) 0 0
\(967\) 1.25085e12 1.43054 0.715270 0.698848i \(-0.246304\pi\)
0.715270 + 0.698848i \(0.246304\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 7.48637e11i 0.842160i 0.907024 + 0.421080i \(0.138349\pi\)
−0.907024 + 0.421080i \(0.861651\pi\)
\(972\) 0 0
\(973\) −5.57063e11 −0.621516
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 6.85254e11i 0.752096i 0.926600 + 0.376048i \(0.122717\pi\)
−0.926600 + 0.376048i \(0.877283\pi\)
\(978\) 0 0
\(979\) 7.65616e11 0.833452
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 1.15973e12i 1.24206i 0.783785 + 0.621032i \(0.213286\pi\)
−0.783785 + 0.621032i \(0.786714\pi\)
\(984\) 0 0
\(985\) 1.30048e12 1.38153
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 2.26439e12i − 2.36683i
\(990\) 0 0
\(991\) 1.55112e12 1.60824 0.804121 0.594465i \(-0.202636\pi\)
0.804121 + 0.594465i \(0.202636\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 1.19115e12i − 1.21527i
\(996\) 0 0
\(997\) −3.61247e11 −0.365615 −0.182807 0.983149i \(-0.558518\pi\)
−0.182807 + 0.983149i \(0.558518\pi\)
\(998\) 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 432.9.e.g.161.1 2
3.2 odd 2 inner 432.9.e.g.161.2 2
4.3 odd 2 54.9.b.a.53.1 2
12.11 even 2 54.9.b.a.53.2 yes 2
36.7 odd 6 162.9.d.c.53.1 4
36.11 even 6 162.9.d.c.53.2 4
36.23 even 6 162.9.d.c.107.1 4
36.31 odd 6 162.9.d.c.107.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
54.9.b.a.53.1 2 4.3 odd 2
54.9.b.a.53.2 yes 2 12.11 even 2
162.9.d.c.53.1 4 36.7 odd 6
162.9.d.c.53.2 4 36.11 even 6
162.9.d.c.107.1 4 36.23 even 6
162.9.d.c.107.2 4 36.31 odd 6
432.9.e.g.161.1 2 1.1 even 1 trivial
432.9.e.g.161.2 2 3.2 odd 2 inner