Properties

Label 8.23.d.a.3.1
Level $8$
Weight $23$
Character 8.3
Self dual yes
Analytic conductor $24.537$
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] = [8,23,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, 23, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8.3");
 
S:= CuspForms(chi, 23);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 23 \)
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: \(24.5365947873\)
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)
 
\(f(q)\) \(=\) \(q-2048.00 q^{2} -199058. q^{3} +4.19430e6 q^{4} +4.07671e8 q^{6} -8.58993e9 q^{8} +8.24303e9 q^{9} +O(q^{10})\) \(q-2048.00 q^{2} -199058. q^{3} +4.19430e6 q^{4} +4.07671e8 q^{6} -8.58993e9 q^{8} +8.24303e9 q^{9} -5.50486e11 q^{11} -8.34910e11 q^{12} +1.75922e13 q^{16} -4.13415e13 q^{17} -1.68817e13 q^{18} -8.64417e13 q^{19} +1.12740e15 q^{22} +1.70990e15 q^{24} +2.38419e15 q^{25} +4.60581e15 q^{27} -3.60288e16 q^{32} +1.09579e17 q^{33} +8.46675e16 q^{34} +3.45738e16 q^{36} +1.77033e17 q^{38} +2.91476e17 q^{41} -1.81053e18 q^{43} -2.30891e18 q^{44} -3.50187e18 q^{48} +3.90982e18 q^{49} -4.88281e18 q^{50} +8.22936e18 q^{51} -9.43270e18 q^{54} +1.72069e19 q^{57} +4.98678e19 q^{59} +7.37870e19 q^{64} -2.24417e20 q^{66} +2.04466e20 q^{67} -1.73399e20 q^{68} -7.08071e19 q^{72} +5.31356e20 q^{73} -4.74591e20 q^{75} -3.62563e20 q^{76} -1.17550e21 q^{81} -5.96943e20 q^{82} -2.47007e21 q^{83} +3.70796e21 q^{86} +4.72864e21 q^{88} +5.07757e21 q^{89} +7.17182e21 q^{96} -9.42268e21 q^{97} -8.00731e21 q^{98} -4.53767e21 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2048.00 −1.00000
\(3\) −199058. −1.12369 −0.561844 0.827243i \(-0.689908\pi\)
−0.561844 + 0.827243i \(0.689908\pi\)
\(4\) 4.19430e6 1.00000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 4.07671e8 1.12369
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) −8.58993e9 −1.00000
\(9\) 8.24303e9 0.262675
\(10\) 0 0
\(11\) −5.50486e11 −1.92942 −0.964710 0.263315i \(-0.915184\pi\)
−0.964710 + 0.263315i \(0.915184\pi\)
\(12\) −8.34910e11 −1.12369
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.75922e13 1.00000
\(17\) −4.13415e13 −1.20628 −0.603141 0.797635i \(-0.706084\pi\)
−0.603141 + 0.797635i \(0.706084\pi\)
\(18\) −1.68817e13 −0.262675
\(19\) −8.64417e13 −0.742051 −0.371025 0.928623i \(-0.620994\pi\)
−0.371025 + 0.928623i \(0.620994\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.12740e15 1.92942
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 1.70990e15 1.12369
\(25\) 2.38419e15 1.00000
\(26\) 0 0
\(27\) 4.60581e15 0.828523
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −3.60288e16 −1.00000
\(33\) 1.09579e17 2.16807
\(34\) 8.46675e16 1.20628
\(35\) 0 0
\(36\) 3.45738e16 0.262675
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 1.77033e17 0.742051
\(39\) 0 0
\(40\) 0 0
\(41\) 2.91476e17 0.529639 0.264820 0.964298i \(-0.414688\pi\)
0.264820 + 0.964298i \(0.414688\pi\)
\(42\) 0 0
\(43\) −1.81053e18 −1.94828 −0.974142 0.225938i \(-0.927455\pi\)
−0.974142 + 0.225938i \(0.927455\pi\)
\(44\) −2.30891e18 −1.92942
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) −3.50187e18 −1.12369
\(49\) 3.90982e18 1.00000
\(50\) −4.88281e18 −1.00000
\(51\) 8.22936e18 1.35548
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) −9.43270e18 −0.828523
\(55\) 0 0
\(56\) 0 0
\(57\) 1.72069e19 0.833834
\(58\) 0 0
\(59\) 4.98678e19 1.65367 0.826834 0.562446i \(-0.190139\pi\)
0.826834 + 0.562446i \(0.190139\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 7.37870e19 1.00000
\(65\) 0 0
\(66\) −2.24417e20 −2.16807
\(67\) 2.04466e20 1.67417 0.837083 0.547076i \(-0.184259\pi\)
0.837083 + 0.547076i \(0.184259\pi\)
\(68\) −1.73399e20 −1.20628
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) −7.08071e19 −0.262675
\(73\) 5.31356e20 1.69369 0.846845 0.531840i \(-0.178499\pi\)
0.846845 + 0.531840i \(0.178499\pi\)
\(74\) 0 0
\(75\) −4.74591e20 −1.12369
\(76\) −3.62563e20 −0.742051
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) −1.17550e21 −1.19368
\(82\) −5.96943e20 −0.529639
\(83\) −2.47007e21 −1.91800 −0.959002 0.283398i \(-0.908538\pi\)
−0.959002 + 0.283398i \(0.908538\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 3.70796e21 1.94828
\(87\) 0 0
\(88\) 4.72864e21 1.92942
\(89\) 5.07757e21 1.82964 0.914821 0.403861i \(-0.132332\pi\)
0.914821 + 0.403861i \(0.132332\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 7.17182e21 1.12369
\(97\) −9.42268e21 −1.31730 −0.658651 0.752448i \(-0.728873\pi\)
−0.658651 + 0.752448i \(0.728873\pi\)
\(98\) −8.00731e21 −1.00000
\(99\) −4.53767e21 −0.506811
\(100\) 1.00000e22 1.00000
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) −1.68537e22 −1.35548
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.13372e22 −1.96390 −0.981950 0.189140i \(-0.939430\pi\)
−0.981950 + 0.189140i \(0.939430\pi\)
\(108\) 1.93182e22 0.828523
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 7.48466e22 1.95123 0.975617 0.219480i \(-0.0704360\pi\)
0.975617 + 0.219480i \(0.0704360\pi\)
\(114\) −3.52398e22 −0.833834
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) −1.02129e23 −1.65367
\(119\) 0 0
\(120\) 0 0
\(121\) 2.21632e23 2.72266
\(122\) 0 0
\(123\) −5.80206e22 −0.595150
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) −1.51116e23 −1.00000
\(129\) 3.60400e23 2.18926
\(130\) 0 0
\(131\) −1.91595e23 −0.982651 −0.491326 0.870976i \(-0.663487\pi\)
−0.491326 + 0.870976i \(0.663487\pi\)
\(132\) 4.59606e23 2.16807
\(133\) 0 0
\(134\) −4.18747e23 −1.67417
\(135\) 0 0
\(136\) 3.55121e23 1.20628
\(137\) −5.33554e23 −1.67206 −0.836031 0.548682i \(-0.815130\pi\)
−0.836031 + 0.548682i \(0.815130\pi\)
\(138\) 0 0
\(139\) −1.81204e23 −0.484178 −0.242089 0.970254i \(-0.577833\pi\)
−0.242089 + 0.970254i \(0.577833\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 1.45013e23 0.262675
\(145\) 0 0
\(146\) −1.08822e24 −1.69369
\(147\) −7.78281e23 −1.12369
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 9.71963e23 1.12369
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 7.42529e23 0.742051
\(153\) −3.40779e23 −0.316860
\(154\) 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
\(162\) 2.40742e24 1.19368
\(163\) 6.62670e23 0.307068 0.153534 0.988143i \(-0.450935\pi\)
0.153534 + 0.988143i \(0.450935\pi\)
\(164\) 1.22254e24 0.529639
\(165\) 0 0
\(166\) 5.05870e24 1.91800
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 3.21184e24 1.00000
\(170\) 0 0
\(171\) −7.12541e23 −0.194918
\(172\) −7.59390e24 −1.94828
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −9.68425e24 −1.92942
\(177\) −9.92659e24 −1.85821
\(178\) −1.03989e25 −1.82964
\(179\) 1.04643e25 1.73112 0.865561 0.500803i \(-0.166962\pi\)
0.865561 + 0.500803i \(0.166962\pi\)
\(180\) 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
\(187\) 2.27579e25 2.32742
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) −1.46879e25 −1.12369
\(193\) −8.65291e24 −0.625219 −0.312609 0.949882i \(-0.601203\pi\)
−0.312609 + 0.949882i \(0.601203\pi\)
\(194\) 1.92977e25 1.31730
\(195\) 0 0
\(196\) 1.63990e25 1.00000
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 9.29315e24 0.506811
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −2.04800e25 −1.00000
\(201\) −4.07006e25 −1.88124
\(202\) 0 0
\(203\) 0 0
\(204\) 3.45165e25 1.35548
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 4.75849e25 1.43173
\(210\) 0 0
\(211\) −4.90517e24 −0.132907 −0.0664534 0.997790i \(-0.521168\pi\)
−0.0664534 + 0.997790i \(0.521168\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 8.46586e25 1.96390
\(215\) 0 0
\(216\) −3.95636e25 −0.828523
\(217\) 0 0
\(218\) 0 0
\(219\) −1.05771e26 −1.90318
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 1.96529e25 0.262675
\(226\) −1.53286e26 −1.95123
\(227\) −7.86790e25 −0.954057 −0.477028 0.878888i \(-0.658286\pi\)
−0.477028 + 0.878888i \(0.658286\pi\)
\(228\) 7.21710e25 0.833834
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.26283e26 −1.14929 −0.574646 0.818402i \(-0.694860\pi\)
−0.574646 + 0.818402i \(0.694860\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 2.09161e26 1.65367
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 3.14720e26 1.97577 0.987887 0.155176i \(-0.0495944\pi\)
0.987887 + 0.155176i \(0.0495944\pi\)
\(242\) −4.53903e26 −2.72266
\(243\) 8.94571e25 0.512798
\(244\) 0 0
\(245\) 0 0
\(246\) 1.18826e26 0.595150
\(247\) 0 0
\(248\) 0 0
\(249\) 4.91687e26 2.15524
\(250\) 0 0
\(251\) 2.48085e26 0.995842 0.497921 0.867222i \(-0.334097\pi\)
0.497921 + 0.867222i \(0.334097\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 3.09485e26 1.00000
\(257\) −5.15705e25 −0.159638 −0.0798191 0.996809i \(-0.525434\pi\)
−0.0798191 + 0.996809i \(0.525434\pi\)
\(258\) −7.38099e26 −2.18926
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 3.92386e26 0.982651
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) −9.41273e26 −2.16807
\(265\) 0 0
\(266\) 0 0
\(267\) −1.01073e27 −2.05595
\(268\) 8.57593e26 1.67417
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) −7.27288e26 −1.20628
\(273\) 0 0
\(274\) 1.09272e27 1.67206
\(275\) −1.31246e27 −1.92942
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 3.71106e26 0.484178
\(279\) 0 0
\(280\) 0 0
\(281\) −1.71679e27 −1.99044 −0.995219 0.0976635i \(-0.968863\pi\)
−0.995219 + 0.0976635i \(0.968863\pi\)
\(282\) 0 0
\(283\) 1.86425e27 1.99919 0.999595 0.0284676i \(-0.00906274\pi\)
0.999595 + 0.0284676i \(0.00906274\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −2.96986e26 −0.262675
\(289\) 5.34560e26 0.455114
\(290\) 0 0
\(291\) 1.87566e27 1.48024
\(292\) 2.22867e27 1.69369
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 1.59392e27 1.12369
\(295\) 0 0
\(296\) 0 0
\(297\) −2.53543e27 −1.59857
\(298\) 0 0
\(299\) 0 0
\(300\) −1.99058e27 −1.12369
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −1.52070e27 −0.742051
\(305\) 0 0
\(306\) 6.97916e26 0.316860
\(307\) 2.82900e27 1.23911 0.619556 0.784952i \(-0.287313\pi\)
0.619556 + 0.784952i \(0.287313\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −5.64844e27 −1.99959 −0.999794 0.0202739i \(-0.993546\pi\)
−0.999794 + 0.0202739i \(0.993546\pi\)
\(314\) 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
\(321\) 8.22850e27 2.20681
\(322\) 0 0
\(323\) 3.57363e27 0.895122
\(324\) −4.93040e27 −1.19368
\(325\) 0 0
\(326\) −1.35715e27 −0.307068
\(327\) 0 0
\(328\) −2.50376e27 −0.529639
\(329\) 0 0
\(330\) 0 0
\(331\) −2.40940e27 −0.461106 −0.230553 0.973060i \(-0.574053\pi\)
−0.230553 + 0.973060i \(0.574053\pi\)
\(332\) −1.03602e28 −1.91800
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 8.70597e27 1.36738 0.683689 0.729774i \(-0.260375\pi\)
0.683689 + 0.729774i \(0.260375\pi\)
\(338\) −6.57785e27 −1.00000
\(339\) −1.48988e28 −2.19258
\(340\) 0 0
\(341\) 0 0
\(342\) 1.45928e27 0.194918
\(343\) 0 0
\(344\) 1.55523e28 1.94828
\(345\) 0 0
\(346\) 0 0
\(347\) 1.60603e28 1.82864 0.914320 0.404992i \(-0.132726\pi\)
0.914320 + 0.404992i \(0.132726\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.98333e28 1.92942
\(353\) −1.68524e27 −0.158906 −0.0794529 0.996839i \(-0.525317\pi\)
−0.0794529 + 0.996839i \(0.525317\pi\)
\(354\) 2.03297e28 1.85821
\(355\) 0 0
\(356\) 2.12969e28 1.82964
\(357\) 0 0
\(358\) −2.14309e28 −1.73112
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) −6.09781e27 −0.449361
\(362\) 0 0
\(363\) −4.41176e28 −3.05942
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 2.40264e27 0.139123
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) −4.66083e28 −2.32742
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 5.58804e27 0.241117 0.120559 0.992706i \(-0.461531\pi\)
0.120559 + 0.992706i \(0.461531\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 3.00808e28 1.12369
\(385\) 0 0
\(386\) 1.77212e28 0.625219
\(387\) −1.49242e28 −0.511766
\(388\) −3.95216e28 −1.31730
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −3.35851e28 −1.00000
\(393\) 3.81385e28 1.10419
\(394\) 0 0
\(395\) 0 0
\(396\) −1.90324e28 −0.506811
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 4.19430e28 1.00000
\(401\) 8.50261e28 1.97226 0.986130 0.165973i \(-0.0530766\pi\)
0.986130 + 0.165973i \(0.0530766\pi\)
\(402\) 8.33549e28 1.88124
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) −7.06897e28 −1.35548
\(409\) −1.06755e29 −1.99265 −0.996327 0.0856271i \(-0.972711\pi\)
−0.996327 + 0.0856271i \(0.972711\pi\)
\(410\) 0 0
\(411\) 1.06208e29 1.87888
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 3.60701e28 0.544065
\(418\) −9.74540e28 −1.43173
\(419\) 1.16092e29 1.66130 0.830649 0.556796i \(-0.187970\pi\)
0.830649 + 0.556796i \(0.187970\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 1.00458e28 0.132907
\(423\) 0 0
\(424\) 0 0
\(425\) −9.85659e28 −1.20628
\(426\) 0 0
\(427\) 0 0
\(428\) −1.73381e29 −1.96390
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 8.10263e28 0.828523
\(433\) −1.97229e29 −1.96610 −0.983048 0.183346i \(-0.941307\pi\)
−0.983048 + 0.183346i \(0.941307\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 2.16618e29 1.90318
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 3.22288e28 0.262675
\(442\) 0 0
\(443\) 2.35590e28 0.182691 0.0913454 0.995819i \(-0.470883\pi\)
0.0913454 + 0.995819i \(0.470883\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −2.93310e29 −1.96163 −0.980814 0.194947i \(-0.937546\pi\)
−0.980814 + 0.194947i \(0.937546\pi\)
\(450\) −4.02492e28 −0.262675
\(451\) −1.60453e29 −1.02190
\(452\) 3.13930e29 1.95123
\(453\) 0 0
\(454\) 1.61135e29 0.954057
\(455\) 0 0
\(456\) −1.47806e29 −0.833834
\(457\) 8.76876e28 0.482903 0.241451 0.970413i \(-0.422377\pi\)
0.241451 + 0.970413i \(0.422377\pi\)
\(458\) 0 0
\(459\) −1.90411e29 −0.999432
\(460\) 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
\(466\) 2.58628e29 1.14929
\(467\) 1.79665e29 0.779789 0.389894 0.920860i \(-0.372512\pi\)
0.389894 + 0.920860i \(0.372512\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −4.28361e29 −1.65367
\(473\) 9.96670e29 3.75906
\(474\) 0 0
\(475\) −2.06093e29 −0.742051
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −6.44547e29 −1.97577
\(483\) 0 0
\(484\) 9.29592e29 2.72266
\(485\) 0 0
\(486\) −1.83208e29 −0.512798
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) −1.31910e29 −0.345048
\(490\) 0 0
\(491\) 5.20379e29 1.30144 0.650719 0.759319i \(-0.274468\pi\)
0.650719 + 0.759319i \(0.274468\pi\)
\(492\) −2.43356e29 −0.595150
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) −1.00697e30 −2.15524
\(499\) −7.12542e29 −1.49178 −0.745890 0.666070i \(-0.767975\pi\)
−0.745890 + 0.666070i \(0.767975\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −5.08079e29 −0.995842
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −6.39342e29 −1.12369
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −6.33825e29 −1.00000
\(513\) −3.98134e29 −0.614806
\(514\) 1.05616e29 0.159638
\(515\) 0 0
\(516\) 1.51163e30 2.18926
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.49085e30 1.94187 0.970935 0.239344i \(-0.0769323\pi\)
0.970935 + 0.239344i \(0.0769323\pi\)
\(522\) 0 0
\(523\) 1.46322e30 1.82722 0.913611 0.406589i \(-0.133282\pi\)
0.913611 + 0.406589i \(0.133282\pi\)
\(524\) −8.03607e29 −0.982651
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 1.92773e30 2.16807
\(529\) 9.07846e29 1.00000
\(530\) 0 0
\(531\) 4.11062e29 0.434378
\(532\) 0 0
\(533\) 0 0
\(534\) 2.06998e30 2.05595
\(535\) 0 0
\(536\) −1.75635e30 −1.67417
\(537\) −2.08301e30 −1.94524
\(538\) 0 0
\(539\) −2.15230e30 −1.92942
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 1.48949e30 1.20628
\(545\) 0 0
\(546\) 0 0
\(547\) −1.68878e30 −1.28740 −0.643699 0.765279i \(-0.722601\pi\)
−0.643699 + 0.765279i \(0.722601\pi\)
\(548\) −2.23789e30 −1.67206
\(549\) 0 0
\(550\) 2.68792e30 1.92942
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −7.60025e29 −0.484178
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −4.53015e30 −2.61530
\(562\) 3.51600e30 1.99044
\(563\) 2.86339e30 1.58960 0.794800 0.606871i \(-0.207576\pi\)
0.794800 + 0.606871i \(0.207576\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −3.81799e30 −1.99919
\(567\) 0 0
\(568\) 0 0
\(569\) −5.00609e29 −0.247323 −0.123661 0.992324i \(-0.539464\pi\)
−0.123661 + 0.992324i \(0.539464\pi\)
\(570\) 0 0
\(571\) 1.91064e30 0.908201 0.454100 0.890951i \(-0.349961\pi\)
0.454100 + 0.890951i \(0.349961\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 6.08228e29 0.262675
\(577\) −8.99818e28 −0.0381260 −0.0190630 0.999818i \(-0.506068\pi\)
−0.0190630 + 0.999818i \(0.506068\pi\)
\(578\) −1.09478e30 −0.455114
\(579\) 1.72243e30 0.702551
\(580\) 0 0
\(581\) 0 0
\(582\) −3.84135e30 −1.48024
\(583\) 0 0
\(584\) −4.56431e30 −1.69369
\(585\) 0 0
\(586\) 0 0
\(587\) 5.22861e30 1.83387 0.916933 0.399042i \(-0.130657\pi\)
0.916933 + 0.399042i \(0.130657\pi\)
\(588\) −3.26435e30 −1.12369
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 2.91319e30 0.913622 0.456811 0.889564i \(-0.348992\pi\)
0.456811 + 0.889564i \(0.348992\pi\)
\(594\) 5.19257e30 1.59857
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 4.07671e30 1.12369
\(601\) 5.78992e29 0.156694 0.0783471 0.996926i \(-0.475036\pi\)
0.0783471 + 0.996926i \(0.475036\pi\)
\(602\) 0 0
\(603\) 1.68542e30 0.439762
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 3.11439e30 0.742051
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −1.42933e30 −0.316860
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) −5.79379e30 −1.23911
\(615\) 0 0
\(616\) 0 0
\(617\) 1.24653e30 0.252677 0.126339 0.991987i \(-0.459677\pi\)
0.126339 + 0.991987i \(0.459677\pi\)
\(618\) 0 0
\(619\) −9.11164e30 −1.78238 −0.891189 0.453633i \(-0.850128\pi\)
−0.891189 + 0.453633i \(0.850128\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 5.68434e30 1.00000
\(626\) 1.15680e31 1.99959
\(627\) −9.47216e30 −1.60882
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 9.76414e29 0.149346
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1.33684e31 1.78091 0.890455 0.455071i \(-0.150386\pi\)
0.890455 + 0.455071i \(0.150386\pi\)
\(642\) −1.68520e31 −2.20681
\(643\) −1.31905e31 −1.69801 −0.849005 0.528384i \(-0.822798\pi\)
−0.849005 + 0.528384i \(0.822798\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −7.31880e30 −0.895122
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 1.00975e31 1.19368
\(649\) −2.74515e31 −3.19062
\(650\) 0 0
\(651\) 0 0
\(652\) 2.77944e30 0.307068
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 5.12770e30 0.529639
\(657\) 4.37998e30 0.444890
\(658\) 0 0
\(659\) 5.38367e29 0.0528858 0.0264429 0.999650i \(-0.491582\pi\)
0.0264429 + 0.999650i \(0.491582\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 4.93444e30 0.461106
\(663\) 0 0
\(664\) 2.12177e31 1.91800
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1.10556e31 0.861818 0.430909 0.902395i \(-0.358193\pi\)
0.430909 + 0.902395i \(0.358193\pi\)
\(674\) −1.78298e31 −1.36738
\(675\) 1.09811e31 0.828523
\(676\) 1.34714e31 1.00000
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 3.05128e31 2.19258
\(679\) 0 0
\(680\) 0 0
\(681\) 1.56617e31 1.07206
\(682\) 0 0
\(683\) 3.32933e30 0.220663 0.110331 0.993895i \(-0.464809\pi\)
0.110331 + 0.993895i \(0.464809\pi\)
\(684\) −2.98861e30 −0.194918
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −3.18511e31 −1.94828
\(689\) 0 0
\(690\) 0 0
\(691\) 4.24628e30 0.247600 0.123800 0.992307i \(-0.460492\pi\)
0.123800 + 0.992307i \(0.460492\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −3.28914e31 −1.82864
\(695\) 0 0
\(696\) 0 0
\(697\) −1.20501e31 −0.638894
\(698\) 0 0
\(699\) 2.51377e31 1.29145
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −4.06187e31 −1.92942
\(705\) 0 0
\(706\) 3.45137e30 0.158906
\(707\) 0 0
\(708\) −4.16351e31 −1.85821
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −4.36160e31 −1.82964
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 4.38905e31 1.73112
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 1.24883e31 0.449361
\(723\) −6.26476e31 −2.22015
\(724\) 0 0
\(725\) 0 0
\(726\) 9.03529e31 3.05942
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 1.90812e31 0.617452
\(730\) 0 0
\(731\) 7.48500e31 2.35018
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1.12556e32 −3.23017
\(738\) −4.92062e30 −0.139123
\(739\) 4.76521e31 1.32737 0.663687 0.748011i \(-0.268991\pi\)
0.663687 + 0.748011i \(0.268991\pi\)
\(740\) 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
\(747\) −2.03608e31 −0.503812
\(748\) 9.54537e31 2.32742
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) −4.93834e31 −1.11902
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) −1.14443e31 −0.241117
\(759\) 0 0
\(760\) 0 0
\(761\) −1.66580e31 −0.336040 −0.168020 0.985784i \(-0.553737\pi\)
−0.168020 + 0.985784i \(0.553737\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −6.16055e31 −1.12369
\(769\) 1.00095e32 1.79978 0.899892 0.436112i \(-0.143645\pi\)
0.899892 + 0.436112i \(0.143645\pi\)
\(770\) 0 0
\(771\) 1.02655e31 0.179384
\(772\) −3.62930e31 −0.625219
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 3.05648e31 0.511766
\(775\) 0 0
\(776\) 8.09402e31 1.31730
\(777\) 0 0
\(778\) 0 0
\(779\) −2.51957e31 −0.393019
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 6.87823e31 1.00000
\(785\) 0 0
\(786\) −7.81076e31 −1.10419
\(787\) −8.51540e31 −1.18709 −0.593544 0.804801i \(-0.702272\pi\)
−0.593544 + 0.804801i \(0.702272\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 3.89783e31 0.506811
\(793\) 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
\(800\) −8.58993e31 −1.00000
\(801\) 4.18546e31 0.480601
\(802\) −1.74133e32 −1.97226
\(803\) −2.92504e32 −3.26784
\(804\) −1.70711e32 −1.88124
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.33002e32 1.36907 0.684534 0.728981i \(-0.260006\pi\)
0.684534 + 0.728981i \(0.260006\pi\)
\(810\) 0 0
\(811\) 1.86762e32 1.87094 0.935470 0.353405i \(-0.114976\pi\)
0.935470 + 0.353405i \(0.114976\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 1.44773e32 1.35548
\(817\) 1.56505e32 1.44573
\(818\) 2.18634e32 1.99265
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) −2.17515e32 −1.87888
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 2.61256e32 2.16807
\(826\) 0 0
\(827\) 3.02585e31 0.244505 0.122252 0.992499i \(-0.460988\pi\)
0.122252 + 0.992499i \(0.460988\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −1.61638e32 −1.20628
\(834\) −7.38716e31 −0.544065
\(835\) 0 0
\(836\) 1.99586e32 1.43173
\(837\) 0 0
\(838\) −2.37756e32 −1.66130
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 1.48852e32 1.00000
\(842\) 0 0
\(843\) 3.41742e32 2.23663
\(844\) −2.05738e31 −0.132907
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −3.71094e32 −2.24647
\(850\) 2.01863e32 1.20628
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 3.55084e32 1.96390
\(857\) −2.81285e32 −1.53588 −0.767939 0.640523i \(-0.778718\pi\)
−0.767939 + 0.640523i \(0.778718\pi\)
\(858\) 0 0
\(859\) −3.75245e32 −1.99705 −0.998527 0.0542638i \(-0.982719\pi\)
−0.998527 + 0.0542638i \(0.982719\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) −1.65942e32 −0.828523
\(865\) 0 0
\(866\) 4.03926e32 1.96610
\(867\) −1.06408e32 −0.511406
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −7.76714e31 −0.346023
\(874\) 0 0
\(875\) 0 0
\(876\) −4.43634e32 −1.90318
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −4.36914e32 −1.76060 −0.880301 0.474416i \(-0.842659\pi\)
−0.880301 + 0.474416i \(0.842659\pi\)
\(882\) −6.60045e31 −0.262675
\(883\) −3.75605e32 −1.47626 −0.738132 0.674656i \(-0.764292\pi\)
−0.738132 + 0.674656i \(0.764292\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −4.82489e31 −0.182691
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 6.47095e32 2.30310
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 6.00698e32 1.96163
\(899\) 0 0
\(900\) 8.24303e31 0.262675
\(901\) 0 0
\(902\) 3.28609e32 1.02190
\(903\) 0 0
\(904\) −6.42928e32 −1.95123
\(905\) 0 0
\(906\) 0 0
\(907\) 5.90418e32 1.72775 0.863873 0.503710i \(-0.168032\pi\)
0.863873 + 0.503710i \(0.168032\pi\)
\(908\) −3.30003e32 −0.954057
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 3.02707e32 0.833834
\(913\) 1.35974e33 3.70064
\(914\) −1.79584e32 −0.482903
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 3.89962e32 0.999432
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) −5.63135e32 −1.39238
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −3.30591e32 −0.743221 −0.371610 0.928389i \(-0.621194\pi\)
−0.371610 + 0.928389i \(0.621194\pi\)
\(930\) 0 0
\(931\) −3.37972e32 −0.742051
\(932\) −5.29670e32 −1.14929
\(933\) 0 0
\(934\) −3.67953e32 −0.779789
\(935\) 0 0
\(936\) 0 0
\(937\) −9.05204e32 −1.85187 −0.925936 0.377681i \(-0.876722\pi\)
−0.925936 + 0.377681i \(0.876722\pi\)
\(938\) 0 0
\(939\) 1.12437e33 2.24691
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 8.77284e32 1.65367
\(945\) 0 0
\(946\) −2.04118e33 −3.75906
\(947\) −2.24420e32 −0.408519 −0.204260 0.978917i \(-0.565479\pi\)
−0.204260 + 0.978917i \(0.565479\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 4.22079e32 0.742051
\(951\) 0 0
\(952\) 0 0
\(953\) 8.61327e32 1.46267 0.731335 0.682018i \(-0.238898\pi\)
0.731335 + 0.682018i \(0.238898\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 6.45591e32 1.00000
\(962\) 0 0
\(963\) −3.40744e32 −0.515868
\(964\) 1.32003e33 1.97577
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) −1.90381e33 −2.72266
\(969\) −7.11360e32 −1.00584
\(970\) 0 0
\(971\) 6.98717e32 0.965806 0.482903 0.875674i \(-0.339582\pi\)
0.482903 + 0.875674i \(0.339582\pi\)
\(972\) 3.75210e32 0.512798
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 8.01352e32 1.03510 0.517551 0.855652i \(-0.326844\pi\)
0.517551 + 0.855652i \(0.326844\pi\)
\(978\) 2.70151e32 0.345048
\(979\) −2.79513e33 −3.53015
\(980\) 0 0
\(981\) 0 0
\(982\) −1.06574e33 −1.30144
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 4.98393e32 0.595150
\(985\) 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
\(993\) 4.79610e32 0.518139
\(994\) 0 0
\(995\) 0 0
\(996\) 2.06228e33 2.15524
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 1.45929e33 1.49178
\(999\) 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 8.23.d.a.3.1 1
4.3 odd 2 32.23.d.a.15.1 1
8.3 odd 2 CM 8.23.d.a.3.1 1
8.5 even 2 32.23.d.a.15.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.23.d.a.3.1 1 1.1 even 1 trivial
8.23.d.a.3.1 1 8.3 odd 2 CM
32.23.d.a.15.1 1 4.3 odd 2
32.23.d.a.15.1 1 8.5 even 2