Properties

Label 8.19.d.a.3.1
Level $8$
Weight $19$
Character 8.3
Self dual yes
Analytic conductor $16.431$
Analytic rank $0$
Dimension $1$
CM discriminant -8
Inner twists $2$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8,19,Mod(3,8)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 19, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8.3");
 
S:= CuspForms(chi, 19);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 19 \)
Character orbit: \([\chi]\) \(=\) 8.d (of order \(2\), degree \(1\), 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: yes
Analytic conductor: \(16.4308910168\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 3.1
Character \(\chi\) \(=\) 8.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-512.000 q^{2} -3266.00 q^{3} +262144. q^{4} +1.67219e6 q^{6} -1.34218e8 q^{8} -3.76754e8 q^{9} +O(q^{10})\) \(q-512.000 q^{2} -3266.00 q^{3} +262144. q^{4} +1.67219e6 q^{6} -1.34218e8 q^{8} -3.76754e8 q^{9} -3.54350e8 q^{11} -8.56162e8 q^{12} +6.87195e10 q^{16} +1.19842e11 q^{17} +1.92898e11 q^{18} +3.35014e11 q^{19} +1.81427e11 q^{22} +4.38355e11 q^{24} +3.81470e12 q^{25} +2.49579e12 q^{27} -3.51844e13 q^{32} +1.15731e12 q^{33} -6.13593e13 q^{34} -9.87637e13 q^{36} -1.71527e14 q^{38} +5.22163e14 q^{41} +1.00025e15 q^{43} -9.28906e13 q^{44} -2.24438e14 q^{48} +1.62841e15 q^{49} -1.95312e15 q^{50} -3.91405e14 q^{51} -1.27785e15 q^{54} -1.09415e15 q^{57} -1.02290e16 q^{59} +1.80144e16 q^{64} -5.92541e14 q^{66} -5.04671e16 q^{67} +3.14160e16 q^{68} +5.05670e16 q^{72} +6.06152e16 q^{73} -1.24588e16 q^{75} +8.78218e16 q^{76} +1.37811e17 q^{81} -2.67347e17 q^{82} -3.52960e17 q^{83} -5.12128e17 q^{86} +4.75600e16 q^{88} +4.87058e17 q^{89} +1.14912e17 q^{96} +1.50107e18 q^{97} -8.33748e17 q^{98} +1.33503e17 q^{99} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −512.000 −1.00000
\(3\) −3266.00 −0.165930 −0.0829650 0.996552i \(-0.526439\pi\)
−0.0829650 + 0.996552i \(0.526439\pi\)
\(4\) 262144. 1.00000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 1.67219e6 0.165930
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) −1.34218e8 −1.00000
\(9\) −3.76754e8 −0.972467
\(10\) 0 0
\(11\) −3.54350e8 −0.150279 −0.0751394 0.997173i \(-0.523940\pi\)
−0.0751394 + 0.997173i \(0.523940\pi\)
\(12\) −8.56162e8 −0.165930
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 6.87195e10 1.00000
\(17\) 1.19842e11 1.01058 0.505290 0.862950i \(-0.331386\pi\)
0.505290 + 0.862950i \(0.331386\pi\)
\(18\) 1.92898e11 0.972467
\(19\) 3.35014e11 1.03820 0.519099 0.854714i \(-0.326268\pi\)
0.519099 + 0.854714i \(0.326268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.81427e11 0.150279
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 4.38355e11 0.165930
\(25\) 3.81470e12 1.00000
\(26\) 0 0
\(27\) 2.49579e12 0.327291
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −3.51844e13 −1.00000
\(33\) 1.15731e12 0.0249358
\(34\) −6.13593e13 −1.01058
\(35\) 0 0
\(36\) −9.87637e13 −0.972467
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) −1.71527e14 −1.03820
\(39\) 0 0
\(40\) 0 0
\(41\) 5.22163e14 1.59496 0.797482 0.603342i \(-0.206165\pi\)
0.797482 + 0.603342i \(0.206165\pi\)
\(42\) 0 0
\(43\) 1.00025e15 1.99018 0.995091 0.0989682i \(-0.0315542\pi\)
0.995091 + 0.0989682i \(0.0315542\pi\)
\(44\) −9.28906e13 −0.150279
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) −2.24438e14 −0.165930
\(49\) 1.62841e15 1.00000
\(50\) −1.95312e15 −1.00000
\(51\) −3.91405e14 −0.167685
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) −1.27785e15 −0.327291
\(55\) 0 0
\(56\) 0 0
\(57\) −1.09415e15 −0.172268
\(58\) 0 0
\(59\) −1.02290e16 −1.18077 −0.590383 0.807123i \(-0.701023\pi\)
−0.590383 + 0.807123i \(0.701023\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.80144e16 1.00000
\(65\) 0 0
\(66\) −5.92541e14 −0.0249358
\(67\) −5.04671e16 −1.85496 −0.927481 0.373871i \(-0.878030\pi\)
−0.927481 + 0.373871i \(0.878030\pi\)
\(68\) 3.14160e16 1.01058
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 5.05670e16 0.972467
\(73\) 6.06152e16 1.02962 0.514808 0.857305i \(-0.327863\pi\)
0.514808 + 0.857305i \(0.327863\pi\)
\(74\) 0 0
\(75\) −1.24588e16 −0.165930
\(76\) 8.78218e16 1.03820
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 1.37811e17 0.918160
\(82\) −2.67347e17 −1.59496
\(83\) −3.52960e17 −1.88809 −0.944046 0.329813i \(-0.893014\pi\)
−0.944046 + 0.329813i \(0.893014\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −5.12128e17 −1.99018
\(87\) 0 0
\(88\) 4.75600e16 0.150279
\(89\) 4.87058e17 1.39018 0.695089 0.718923i \(-0.255365\pi\)
0.695089 + 0.718923i \(0.255365\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 1.14912e17 0.165930
\(97\) 1.50107e18 1.97449 0.987244 0.159214i \(-0.0508959\pi\)
0.987244 + 0.159214i \(0.0508959\pi\)
\(98\) −8.33748e17 −1.00000
\(99\) 1.33503e17 0.146141
\(100\) 1.00000e18 1.00000
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 2.00400e17 0.167685
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −2.02760e18 −1.10288 −0.551439 0.834215i \(-0.685921\pi\)
−0.551439 + 0.834215i \(0.685921\pi\)
\(108\) 6.54257e17 0.327291
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −4.68774e18 −1.56048 −0.780238 0.625482i \(-0.784902\pi\)
−0.780238 + 0.625482i \(0.784902\pi\)
\(114\) 5.60207e17 0.172268
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 5.23723e18 1.18077
\(119\) 0 0
\(120\) 0 0
\(121\) −5.43435e18 −0.977416
\(122\) 0 0
\(123\) −1.70538e18 −0.264652
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) −9.22337e18 −1.00000
\(129\) −3.26682e18 −0.330231
\(130\) 0 0
\(131\) 1.90150e19 1.67361 0.836807 0.547497i \(-0.184419\pi\)
0.836807 + 0.547497i \(0.184419\pi\)
\(132\) 3.03381e17 0.0249358
\(133\) 0 0
\(134\) 2.58391e19 1.85496
\(135\) 0 0
\(136\) −1.60850e19 −1.01058
\(137\) 1.59042e18 0.0935461 0.0467730 0.998906i \(-0.485106\pi\)
0.0467730 + 0.998906i \(0.485106\pi\)
\(138\) 0 0
\(139\) 3.64855e19 1.88359 0.941797 0.336182i \(-0.109136\pi\)
0.941797 + 0.336182i \(0.109136\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −2.58903e19 −0.972467
\(145\) 0 0
\(146\) −3.10350e19 −1.02962
\(147\) −5.31840e18 −0.165930
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 6.37891e18 0.165930
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) −4.49648e19 −1.03820
\(153\) −4.51511e19 −0.982755
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) −7.05592e19 −0.918160
\(163\) −2.57981e19 −0.317613 −0.158807 0.987310i \(-0.550765\pi\)
−0.158807 + 0.987310i \(0.550765\pi\)
\(164\) 1.36882e20 1.59496
\(165\) 0 0
\(166\) 1.80716e20 1.88809
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 1.12455e20 1.00000
\(170\) 0 0
\(171\) −1.26218e20 −1.00961
\(172\) 2.62210e20 1.99018
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −2.43507e19 −0.150279
\(177\) 3.34078e19 0.195924
\(178\) −2.49374e20 −1.39018
\(179\) −2.84970e20 −1.51050 −0.755252 0.655434i \(-0.772486\pi\)
−0.755252 + 0.655434i \(0.772486\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −4.24661e19 −0.151869
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) −5.88350e19 −0.165930
\(193\) 5.49022e20 1.47766 0.738829 0.673893i \(-0.235379\pi\)
0.738829 + 0.673893i \(0.235379\pi\)
\(194\) −7.68546e20 −1.97449
\(195\) 0 0
\(196\) 4.26879e20 1.00000
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) −6.83533e19 −0.146141
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −5.12000e20 −1.00000
\(201\) 1.64825e20 0.307794
\(202\) 0 0
\(203\) 0 0
\(204\) −1.02605e20 −0.167685
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −1.18712e20 −0.156019
\(210\) 0 0
\(211\) 1.54335e21 1.86176 0.930878 0.365331i \(-0.119044\pi\)
0.930878 + 0.365331i \(0.119044\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 1.03813e21 1.10288
\(215\) 0 0
\(216\) −3.34980e20 −0.327291
\(217\) 0 0
\(218\) 0 0
\(219\) −1.97969e20 −0.170844
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) −1.43720e21 −0.972467
\(226\) 2.40012e21 1.56048
\(227\) −3.87177e20 −0.241923 −0.120961 0.992657i \(-0.538598\pi\)
−0.120961 + 0.992657i \(0.538598\pi\)
\(228\) −2.86826e20 −0.172268
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −3.95634e21 −1.95475 −0.977374 0.211520i \(-0.932159\pi\)
−0.977374 + 0.211520i \(0.932159\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −2.68146e21 −1.18077
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −3.03581e21 −1.10693 −0.553467 0.832871i \(-0.686695\pi\)
−0.553467 + 0.832871i \(0.686695\pi\)
\(242\) 2.78239e21 0.977416
\(243\) −1.41701e21 −0.479642
\(244\) 0 0
\(245\) 0 0
\(246\) 8.73156e20 0.264652
\(247\) 0 0
\(248\) 0 0
\(249\) 1.15277e21 0.313291
\(250\) 0 0
\(251\) 7.58375e21 1.91788 0.958938 0.283615i \(-0.0915339\pi\)
0.958938 + 0.283615i \(0.0915339\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 4.72237e21 1.00000
\(257\) 9.57081e21 1.95682 0.978410 0.206675i \(-0.0662644\pi\)
0.978410 + 0.206675i \(0.0662644\pi\)
\(258\) 1.67261e21 0.330231
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) −9.73570e21 −1.67361
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) −1.55331e20 −0.0249358
\(265\) 0 0
\(266\) 0 0
\(267\) −1.59073e21 −0.230672
\(268\) −1.32296e22 −1.85496
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 8.23551e21 1.01058
\(273\) 0 0
\(274\) −8.14293e20 −0.0935461
\(275\) −1.35174e21 −0.150279
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) −1.86806e22 −1.88359
\(279\) 0 0
\(280\) 0 0
\(281\) 1.55808e22 1.42637 0.713186 0.700975i \(-0.247252\pi\)
0.713186 + 0.700975i \(0.247252\pi\)
\(282\) 0 0
\(283\) −2.24902e22 −1.93159 −0.965795 0.259307i \(-0.916506\pi\)
−0.965795 + 0.259307i \(0.916506\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 1.32558e22 0.972467
\(289\) 2.99128e20 0.0212704
\(290\) 0 0
\(291\) −4.90249e21 −0.327627
\(292\) 1.58899e22 1.02962
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 2.72302e21 0.165930
\(295\) 0 0
\(296\) 0 0
\(297\) −8.84383e20 −0.0491850
\(298\) 0 0
\(299\) 0 0
\(300\) −3.26600e21 −0.165930
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 2.30220e22 1.03820
\(305\) 0 0
\(306\) 2.31174e22 0.982755
\(307\) −1.47844e22 −0.610322 −0.305161 0.952301i \(-0.598710\pi\)
−0.305161 + 0.952301i \(0.598710\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −2.48227e22 −0.860894 −0.430447 0.902616i \(-0.641644\pi\)
−0.430447 + 0.902616i \(0.641644\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 6.62213e21 0.183000
\(322\) 0 0
\(323\) 4.01489e22 1.04918
\(324\) 3.61263e22 0.918160
\(325\) 0 0
\(326\) 1.32086e22 0.317613
\(327\) 0 0
\(328\) −7.00835e22 −1.59496
\(329\) 0 0
\(330\) 0 0
\(331\) 1.80468e22 0.378398 0.189199 0.981939i \(-0.439411\pi\)
0.189199 + 0.981939i \(0.439411\pi\)
\(332\) −9.25265e22 −1.88809
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −8.13695e22 −1.45142 −0.725709 0.688001i \(-0.758488\pi\)
−0.725709 + 0.688001i \(0.758488\pi\)
\(338\) −5.75772e22 −1.00000
\(339\) 1.53101e22 0.258930
\(340\) 0 0
\(341\) 0 0
\(342\) 6.46234e22 1.00961
\(343\) 0 0
\(344\) −1.34251e23 −1.99018
\(345\) 0 0
\(346\) 0 0
\(347\) 1.42037e23 1.94731 0.973657 0.228018i \(-0.0732243\pi\)
0.973657 + 0.228018i \(0.0732243\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.24676e22 0.150279
\(353\) 1.01143e23 1.18841 0.594203 0.804315i \(-0.297468\pi\)
0.594203 + 0.804315i \(0.297468\pi\)
\(354\) −1.71048e22 −0.195924
\(355\) 0 0
\(356\) 1.27679e23 1.39018
\(357\) 0 0
\(358\) 1.45905e23 1.51050
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 8.10683e21 0.0778550
\(362\) 0 0
\(363\) 1.77486e22 0.162183
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) −1.96727e23 −1.55105
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 2.17427e22 0.151869
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −2.21819e23 −1.37482 −0.687411 0.726268i \(-0.741253\pi\)
−0.687411 + 0.726268i \(0.741253\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 3.01235e22 0.165930
\(385\) 0 0
\(386\) −2.81099e23 −1.47766
\(387\) −3.76848e23 −1.93539
\(388\) 3.93496e23 1.97449
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −2.18562e23 −1.00000
\(393\) −6.21031e22 −0.277703
\(394\) 0 0
\(395\) 0 0
\(396\) 3.49969e22 0.146141
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 2.62144e23 1.00000
\(401\) 4.86318e23 1.81393 0.906967 0.421203i \(-0.138392\pi\)
0.906967 + 0.421203i \(0.138392\pi\)
\(402\) −8.43906e22 −0.307794
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 5.25335e22 0.167685
\(409\) 3.84499e23 1.20056 0.600282 0.799788i \(-0.295055\pi\)
0.600282 + 0.799788i \(0.295055\pi\)
\(410\) 0 0
\(411\) −5.19430e21 −0.0155221
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −1.19162e23 −0.312545
\(418\) 6.07805e22 0.156019
\(419\) 2.65749e23 0.667646 0.333823 0.942636i \(-0.391661\pi\)
0.333823 + 0.942636i \(0.391661\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) −7.90196e23 −1.86176
\(423\) 0 0
\(424\) 0 0
\(425\) 4.57163e23 1.01058
\(426\) 0 0
\(427\) 0 0
\(428\) −5.31522e23 −1.10288
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 1.71510e23 0.327291
\(433\) 3.09750e23 0.578921 0.289461 0.957190i \(-0.406524\pi\)
0.289461 + 0.957190i \(0.406524\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 1.01360e23 0.170844
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) −6.13511e23 −0.972467
\(442\) 0 0
\(443\) 4.63503e23 0.705373 0.352687 0.935741i \(-0.385268\pi\)
0.352687 + 0.935741i \(0.385268\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −1.10365e24 −1.48804 −0.744020 0.668157i \(-0.767083\pi\)
−0.744020 + 0.668157i \(0.767083\pi\)
\(450\) 7.35847e23 0.972467
\(451\) −1.85028e23 −0.239689
\(452\) −1.22886e24 −1.56048
\(453\) 0 0
\(454\) 1.98235e23 0.241923
\(455\) 0 0
\(456\) 1.46855e23 0.172268
\(457\) −1.21141e24 −1.39331 −0.696653 0.717408i \(-0.745328\pi\)
−0.696653 + 0.717408i \(0.745328\pi\)
\(458\) 0 0
\(459\) 2.99102e23 0.330754
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 2.02565e24 1.95475
\(467\) −6.80067e23 −0.643724 −0.321862 0.946787i \(-0.604309\pi\)
−0.321862 + 0.946787i \(0.604309\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 1.37291e24 1.18077
\(473\) −3.54438e23 −0.299082
\(474\) 0 0
\(475\) 1.27798e24 1.03820
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 1.55434e24 1.10693
\(483\) 0 0
\(484\) −1.42458e24 −0.977416
\(485\) 0 0
\(486\) 7.25510e23 0.479642
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 8.42565e22 0.0527016
\(490\) 0 0
\(491\) 3.00554e24 1.81213 0.906065 0.423139i \(-0.139072\pi\)
0.906065 + 0.423139i \(0.139072\pi\)
\(492\) −4.47056e23 −0.264652
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) −5.90218e23 −0.313291
\(499\) −3.27870e24 −1.70921 −0.854607 0.519275i \(-0.826202\pi\)
−0.854607 + 0.519275i \(0.826202\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −3.88288e24 −1.91788
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −3.67279e23 −0.165930
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −2.41785e24 −1.00000
\(513\) 8.36125e23 0.339793
\(514\) −4.90026e24 −1.95682
\(515\) 0 0
\(516\) −8.56377e23 −0.330231
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 4.06522e24 1.43729 0.718645 0.695377i \(-0.244763\pi\)
0.718645 + 0.695377i \(0.244763\pi\)
\(522\) 0 0
\(523\) −2.12505e24 −0.725862 −0.362931 0.931816i \(-0.618224\pi\)
−0.362931 + 0.931816i \(0.618224\pi\)
\(524\) 4.98468e24 1.67361
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 7.95295e22 0.0249358
\(529\) 3.24415e24 1.00000
\(530\) 0 0
\(531\) 3.85380e24 1.14826
\(532\) 0 0
\(533\) 0 0
\(534\) 8.14455e23 0.230672
\(535\) 0 0
\(536\) 6.77357e24 1.85496
\(537\) 9.30712e23 0.250638
\(538\) 0 0
\(539\) −5.77028e23 −0.150279
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −4.21658e24 −1.01058
\(545\) 0 0
\(546\) 0 0
\(547\) −8.68027e24 −1.97992 −0.989960 0.141351i \(-0.954855\pi\)
−0.989960 + 0.141351i \(0.954855\pi\)
\(548\) 4.16918e23 0.0935461
\(549\) 0 0
\(550\) 6.92089e23 0.150279
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 9.56446e24 1.88359
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 1.38694e23 0.0251996
\(562\) −7.97736e24 −1.42637
\(563\) −1.10183e25 −1.93883 −0.969417 0.245419i \(-0.921075\pi\)
−0.969417 + 0.245419i \(0.921075\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 1.15150e25 1.93159
\(567\) 0 0
\(568\) 0 0
\(569\) −1.07894e25 −1.72578 −0.862890 0.505392i \(-0.831348\pi\)
−0.862890 + 0.505392i \(0.831348\pi\)
\(570\) 0 0
\(571\) 8.85990e24 1.37311 0.686553 0.727080i \(-0.259123\pi\)
0.686553 + 0.727080i \(0.259123\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −6.78699e24 −0.972467
\(577\) 2.21137e23 0.0311946 0.0155973 0.999878i \(-0.495035\pi\)
0.0155973 + 0.999878i \(0.495035\pi\)
\(578\) −1.53153e23 −0.0212704
\(579\) −1.79310e24 −0.245188
\(580\) 0 0
\(581\) 0 0
\(582\) 2.51007e24 0.327627
\(583\) 0 0
\(584\) −8.13563e24 −1.02962
\(585\) 0 0
\(586\) 0 0
\(587\) 1.01955e25 1.23216 0.616081 0.787683i \(-0.288719\pi\)
0.616081 + 0.787683i \(0.288719\pi\)
\(588\) −1.39419e24 −0.165930
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1.56385e25 −1.72465 −0.862326 0.506353i \(-0.830993\pi\)
−0.862326 + 0.506353i \(0.830993\pi\)
\(594\) 4.52804e23 0.0491850
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 1.67219e24 0.165930
\(601\) 2.03965e25 1.99382 0.996910 0.0785507i \(-0.0250293\pi\)
0.996910 + 0.0785507i \(0.0250293\pi\)
\(602\) 0 0
\(603\) 1.90137e25 1.80389
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) −1.17872e25 −1.03820
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −1.18361e25 −0.982755
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 7.56963e24 0.610322
\(615\) 0 0
\(616\) 0 0
\(617\) −1.61927e25 −1.24955 −0.624775 0.780805i \(-0.714809\pi\)
−0.624775 + 0.780805i \(0.714809\pi\)
\(618\) 0 0
\(619\) 2.37714e25 1.78171 0.890857 0.454283i \(-0.150105\pi\)
0.890857 + 0.454283i \(0.150105\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.45519e25 1.00000
\(626\) 1.27092e25 0.860894
\(627\) 3.87713e23 0.0258883
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) −5.04059e24 −0.308921
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −1.17518e25 −0.643252 −0.321626 0.946867i \(-0.604229\pi\)
−0.321626 + 0.946867i \(0.604229\pi\)
\(642\) −3.39053e24 −0.183000
\(643\) −3.73263e25 −1.98662 −0.993312 0.115462i \(-0.963165\pi\)
−0.993312 + 0.115462i \(0.963165\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −2.05562e25 −1.04918
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) −1.84967e25 −0.918160
\(649\) 3.62463e24 0.177444
\(650\) 0 0
\(651\) 0 0
\(652\) −6.76281e24 −0.317613
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 3.58827e25 1.59496
\(657\) −2.28370e25 −1.00127
\(658\) 0 0
\(659\) −4.62488e25 −1.97302 −0.986511 0.163697i \(-0.947658\pi\)
−0.986511 + 0.163697i \(0.947658\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) −9.23998e24 −0.378398
\(663\) 0 0
\(664\) 4.73736e25 1.88809
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 5.32210e25 1.87909 0.939545 0.342426i \(-0.111249\pi\)
0.939545 + 0.342426i \(0.111249\pi\)
\(674\) 4.16612e25 1.45142
\(675\) 9.52069e24 0.327291
\(676\) 2.94795e25 1.00000
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) −7.83880e24 −0.258930
\(679\) 0 0
\(680\) 0 0
\(681\) 1.26452e24 0.0401422
\(682\) 0 0
\(683\) 2.99137e25 0.924877 0.462438 0.886651i \(-0.346975\pi\)
0.462438 + 0.886651i \(0.346975\pi\)
\(684\) −3.30872e25 −1.00961
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 6.87367e25 1.99018
\(689\) 0 0
\(690\) 0 0
\(691\) −6.80314e25 −1.89412 −0.947059 0.321059i \(-0.895961\pi\)
−0.947059 + 0.321059i \(0.895961\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −7.27229e25 −1.94731
\(695\) 0 0
\(696\) 0 0
\(697\) 6.25772e25 1.61184
\(698\) 0 0
\(699\) 1.29214e25 0.324351
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −6.38340e24 −0.150279
\(705\) 0 0
\(706\) −5.17854e25 −1.18841
\(707\) 0 0
\(708\) 8.75766e24 0.195924
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −6.53718e25 −1.39018
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −7.47032e25 −1.51050
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −4.15070e24 −0.0778550
\(723\) 9.91497e24 0.183674
\(724\) 0 0
\(725\) 0 0
\(726\) −9.08728e24 −0.162183
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) −4.87628e25 −0.838573
\(730\) 0 0
\(731\) 1.19872e26 2.01124
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.78830e25 0.278761
\(738\) 1.00724e26 1.55105
\(739\) −1.20793e26 −1.83756 −0.918782 0.394766i \(-0.870826\pi\)
−0.918782 + 0.394766i \(0.870826\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 1.32979e26 1.83611
\(748\) −1.11322e25 −0.151869
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) −2.47685e25 −0.318233
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 1.13572e26 1.37482
\(759\) 0 0
\(760\) 0 0
\(761\) −1.43585e26 −1.67744 −0.838718 0.544567i \(-0.816694\pi\)
−0.838718 + 0.544567i \(0.816694\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −1.54232e25 −0.165930
\(769\) −9.21536e25 −0.979885 −0.489943 0.871755i \(-0.662982\pi\)
−0.489943 + 0.871755i \(0.662982\pi\)
\(770\) 0 0
\(771\) −3.12583e25 −0.324695
\(772\) 1.43923e26 1.47766
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 1.92946e26 1.93539
\(775\) 0 0
\(776\) −2.01470e26 −1.97449
\(777\) 0 0
\(778\) 0 0
\(779\) 1.74932e26 1.65589
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 1.11904e26 1.00000
\(785\) 0 0
\(786\) 3.17968e25 0.277703
\(787\) −1.35055e26 −1.16610 −0.583051 0.812436i \(-0.698141\pi\)
−0.583051 + 0.812436i \(0.698141\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) −1.79184e25 −0.146141
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −1.34218e26 −1.00000
\(801\) −1.83501e26 −1.35190
\(802\) −2.48995e26 −1.81393
\(803\) −2.14790e25 −0.154730
\(804\) 4.32080e25 0.307794
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 9.65583e25 0.650508 0.325254 0.945627i \(-0.394550\pi\)
0.325254 + 0.945627i \(0.394550\pi\)
\(810\) 0 0
\(811\) −2.53712e26 −1.67168 −0.835841 0.548972i \(-0.815019\pi\)
−0.835841 + 0.548972i \(0.815019\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) −2.68972e25 −0.167685
\(817\) 3.35098e26 2.06620
\(818\) −1.96864e26 −1.20056
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 2.65948e24 0.0155221
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 4.41477e24 0.0249358
\(826\) 0 0
\(827\) 3.61565e26 1.99819 0.999096 0.0425114i \(-0.0135359\pi\)
0.999096 + 0.0425114i \(0.0135359\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1.95153e26 1.01058
\(834\) 6.10108e25 0.312545
\(835\) 0 0
\(836\) −3.11196e25 −0.156019
\(837\) 0 0
\(838\) −1.36064e26 −0.667646
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 2.10457e26 1.00000
\(842\) 0 0
\(843\) −5.08869e25 −0.236678
\(844\) 4.04580e26 1.86176
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 7.34529e25 0.320509
\(850\) −2.34067e26 −1.01058
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 2.72139e26 1.10288
\(857\) 3.25681e26 1.30607 0.653033 0.757329i \(-0.273496\pi\)
0.653033 + 0.757329i \(0.273496\pi\)
\(858\) 0 0
\(859\) −4.40249e26 −1.72886 −0.864431 0.502752i \(-0.832321\pi\)
−0.864431 + 0.502752i \(0.832321\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) −8.78129e25 −0.327291
\(865\) 0 0
\(866\) −1.58592e26 −0.578921
\(867\) −9.76951e23 −0.00352940
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −5.65533e26 −1.92013
\(874\) 0 0
\(875\) 0 0
\(876\) −5.18964e25 −0.170844
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −4.73095e26 −1.47967 −0.739836 0.672787i \(-0.765097\pi\)
−0.739836 + 0.672787i \(0.765097\pi\)
\(882\) 3.14118e26 0.972467
\(883\) −5.36481e26 −1.64402 −0.822011 0.569471i \(-0.807148\pi\)
−0.822011 + 0.569471i \(0.807148\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −2.37314e26 −0.705373
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −4.88332e25 −0.137980
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 5.65069e26 1.48804
\(899\) 0 0
\(900\) −3.76754e26 −0.972467
\(901\) 0 0
\(902\) 9.47344e25 0.239689
\(903\) 0 0
\(904\) 6.29177e26 1.56048
\(905\) 0 0
\(906\) 0 0
\(907\) 8.22752e26 1.98063 0.990315 0.138840i \(-0.0443374\pi\)
0.990315 + 0.138840i \(0.0443374\pi\)
\(908\) −1.01496e26 −0.241923
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) −7.51897e25 −0.172268
\(913\) 1.25071e26 0.283740
\(914\) 6.20244e26 1.39331
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) −1.53140e26 −0.330754
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 4.82860e25 0.101271
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −7.61266e26 −1.47705 −0.738525 0.674226i \(-0.764477\pi\)
−0.738525 + 0.674226i \(0.764477\pi\)
\(930\) 0 0
\(931\) 5.45541e26 1.03820
\(932\) −1.03713e27 −1.95475
\(933\) 0 0
\(934\) 3.48194e26 0.643724
\(935\) 0 0
\(936\) 0 0
\(937\) 4.30663e26 0.773537 0.386768 0.922177i \(-0.373591\pi\)
0.386768 + 0.922177i \(0.373591\pi\)
\(938\) 0 0
\(939\) 8.10709e25 0.142848
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −7.02929e26 −1.18077
\(945\) 0 0
\(946\) 1.81472e26 0.299082
\(947\) 1.18392e27 1.93273 0.966366 0.257170i \(-0.0827902\pi\)
0.966366 + 0.257170i \(0.0827902\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −6.54324e26 −1.03820
\(951\) 0 0
\(952\) 0 0
\(953\) −8.06436e26 −1.24375 −0.621876 0.783116i \(-0.713629\pi\)
−0.621876 + 0.783116i \(0.713629\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 6.99054e26 1.00000
\(962\) 0 0
\(963\) 7.63904e26 1.07251
\(964\) −7.95820e26 −1.10693
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 7.29387e26 0.977416
\(969\) −1.31126e26 −0.174091
\(970\) 0 0
\(971\) −1.53443e27 −1.99974 −0.999871 0.0160616i \(-0.994887\pi\)
−0.999871 + 0.0160616i \(0.994887\pi\)
\(972\) −3.71461e26 −0.479642
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 2.57600e26 0.317610 0.158805 0.987310i \(-0.449236\pi\)
0.158805 + 0.987310i \(0.449236\pi\)
\(978\) −4.31393e25 −0.0527016
\(979\) −1.72589e26 −0.208914
\(980\) 0 0
\(981\) 0 0
\(982\) −1.53884e27 −1.81213
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 2.28893e26 0.264652
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) −5.89410e25 −0.0627876
\(994\) 0 0
\(995\) 0 0
\(996\) 3.02191e26 0.313291
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 1.67869e27 1.70921
\(999\) 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 8.19.d.a.3.1 1
3.2 odd 2 72.19.b.a.19.1 1
4.3 odd 2 32.19.d.a.15.1 1
8.3 odd 2 CM 8.19.d.a.3.1 1
8.5 even 2 32.19.d.a.15.1 1
24.11 even 2 72.19.b.a.19.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.19.d.a.3.1 1 1.1 even 1 trivial
8.19.d.a.3.1 1 8.3 odd 2 CM
32.19.d.a.15.1 1 4.3 odd 2
32.19.d.a.15.1 1 8.5 even 2
72.19.b.a.19.1 1 3.2 odd 2
72.19.b.a.19.1 1 24.11 even 2