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Properties

Label 35.11.c.b.34.1
Level $35$
Weight $11$
Character 35.34
Self dual yes
Analytic conductor $22.238$
Analytic rank $0$
Dimension $1$
CM discriminant -35
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] = [35,11,Mod(34,35)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(35, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("35.34");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 35.c (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: \(22.2375038436\)
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 34.1
Character \(\chi\) \(=\) 35.34

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+361.000 q^{3} +1024.00 q^{4} -3125.00 q^{5} +16807.0 q^{7} +71272.0 q^{9} +6227.00 q^{11} +369664. q^{12} +606461. q^{13} -1.12812e6 q^{15} +1.04858e6 q^{16} -2.62041e6 q^{17} -3.20000e6 q^{20} +6.06733e6 q^{21} +9.76562e6 q^{25} +4.41250e6 q^{27} +1.72104e7 q^{28} +3.66114e7 q^{29} +2.24795e6 q^{33} -5.25219e7 q^{35} +7.29825e7 q^{36} +2.18932e8 q^{39} +6.37645e6 q^{44} -2.22725e8 q^{45} -4.55938e8 q^{47} +3.78536e8 q^{48} +2.82475e8 q^{49} -9.45968e8 q^{51} +6.21016e8 q^{52} -1.94594e7 q^{55} -1.15520e9 q^{60} +1.19787e9 q^{63} +1.07374e9 q^{64} -1.89519e9 q^{65} -2.68330e9 q^{68} +2.53915e8 q^{71} -3.82588e9 q^{73} +3.52539e9 q^{75} +1.04657e8 q^{77} -5.20480e9 q^{79} -3.27680e9 q^{80} -2.61563e9 q^{81} +3.20240e9 q^{83} +6.21294e9 q^{84} +8.18878e9 q^{85} +1.32167e10 q^{87} +1.01928e10 q^{91} -1.62284e10 q^{97} +4.43811e8 q^{99} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) 361.000 1.48560 0.742798 − 0.669515i \(-0.233498\pi\)
0.742798 + 0.669515i \(0.233498\pi\)
\(4\) 1024.00 1.00000
\(5\) −3125.00 −1.00000
\(6\) 0 0
\(7\) 16807.0 1.00000
\(8\) 0 0
\(9\) 71272.0 1.20700
\(10\) 0 0
\(11\) 6227.00 0.0386648 0.0193324 − 0.999813i \(-0.493846\pi\)
0.0193324 + 0.999813i \(0.493846\pi\)
\(12\) 369664. 1.48560
\(13\) 606461. 1.63338 0.816688 − 0.577080i \(-0.195808\pi\)
0.816688 + 0.577080i \(0.195808\pi\)
\(14\) 0 0
\(15\) −1.12812e6 −1.48560
\(16\) 1.04858e6 1.00000
\(17\) −2.62041e6 −1.84555 −0.922773 − 0.385344i \(-0.874083\pi\)
−0.922773 + 0.385344i \(0.874083\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) −3.20000e6 −1.00000
\(21\) 6.06733e6 1.48560
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 9.76562e6 1.00000
\(26\) 0 0
\(27\) 4.41250e6 0.307515
\(28\) 1.72104e7 1.00000
\(29\) 3.66114e7 1.78495 0.892476 − 0.451095i \(-0.148966\pi\)
0.892476 + 0.451095i \(0.148966\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 2.24795e6 0.0574403
\(34\) 0 0
\(35\) −5.25219e7 −1.00000
\(36\) 7.29825e7 1.20700
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 2.18932e8 2.42654
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 6.37645e6 0.0386648
\(45\) −2.22725e8 −1.20700
\(46\) 0 0
\(47\) −4.55938e8 −1.98800 −0.994001 − 0.109375i \(-0.965115\pi\)
−0.994001 + 0.109375i \(0.965115\pi\)
\(48\) 3.78536e8 1.48560
\(49\) 2.82475e8 1.00000
\(50\) 0 0
\(51\) −9.45968e8 −2.74174
\(52\) 6.21016e8 1.63338
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) −1.94594e7 −0.0386648
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) −1.15520e9 −1.48560
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 1.19787e9 1.20700
\(64\) 1.07374e9 1.00000
\(65\) −1.89519e9 −1.63338
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) −2.68330e9 −1.84555
\(69\) 0 0
\(70\) 0 0
\(71\) 2.53915e8 0.140733 0.0703667 − 0.997521i \(-0.477583\pi\)
0.0703667 + 0.997521i \(0.477583\pi\)
\(72\) 0 0
\(73\) −3.82588e9 −1.84551 −0.922757 − 0.385383i \(-0.874069\pi\)
−0.922757 + 0.385383i \(0.874069\pi\)
\(74\) 0 0
\(75\) 3.52539e9 1.48560
\(76\) 0 0
\(77\) 1.04657e8 0.0386648
\(78\) 0 0
\(79\) −5.20480e9 −1.69149 −0.845743 − 0.533591i \(-0.820842\pi\)
−0.845743 + 0.533591i \(0.820842\pi\)
\(80\) −3.27680e9 −1.00000
\(81\) −2.61563e9 −0.750154
\(82\) 0 0
\(83\) 3.20240e9 0.812990 0.406495 − 0.913653i \(-0.366751\pi\)
0.406495 + 0.913653i \(0.366751\pi\)
\(84\) 6.21294e9 1.48560
\(85\) 8.18878e9 1.84555
\(86\) 0 0
\(87\) 1.32167e10 2.65172
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 1.01928e10 1.63338
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.62284e10 −1.88980 −0.944901 − 0.327357i \(-0.893842\pi\)
−0.944901 + 0.327357i \(0.893842\pi\)
\(98\) 0 0
\(99\) 4.43811e8 0.0466683
\(100\) 1.00000e10 1.00000
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −6.04135e9 −0.521132 −0.260566 − 0.965456i \(-0.583909\pi\)
−0.260566 + 0.965456i \(0.583909\pi\)
\(104\) 0 0
\(105\) −1.89604e10 −1.48560
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 4.51840e9 0.307515
\(109\) −2.28317e10 −1.48390 −0.741952 − 0.670453i \(-0.766100\pi\)
−0.741952 + 0.670453i \(0.766100\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.76234e10 1.00000
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 3.74901e10 1.78495
\(117\) 4.32237e10 1.97148
\(118\) 0 0
\(119\) −4.40412e10 −1.84555
\(120\) 0 0
\(121\) −2.58986e10 −0.998505
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −3.05176e10 −1.00000
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 2.30190e9 0.0574403
\(133\) 0 0
\(134\) 0 0
\(135\) −1.37891e10 −0.307515
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) −5.37824e10 −1.00000
\(141\) −1.64594e11 −2.95337
\(142\) 0 0
\(143\) 3.77643e9 0.0631541
\(144\) 7.47341e10 1.20700
\(145\) −1.14411e11 −1.78495
\(146\) 0 0
\(147\) 1.01974e11 1.48560
\(148\) 0 0
\(149\) 1.16172e11 1.58187 0.790937 − 0.611897i \(-0.209593\pi\)
0.790937 + 0.611897i \(0.209593\pi\)
\(150\) 0 0
\(151\) −3.35425e10 −0.427278 −0.213639 − 0.976913i \(-0.568532\pi\)
−0.213639 + 0.976913i \(0.568532\pi\)
\(152\) 0 0
\(153\) −1.86762e11 −2.22757
\(154\) 0 0
\(155\) 0 0
\(156\) 2.24187e11 2.42654
\(157\) 1.53738e11 1.61169 0.805846 − 0.592125i \(-0.201711\pi\)
0.805846 + 0.592125i \(0.201711\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) −7.02483e9 −0.0574403
\(166\) 0 0
\(167\) 2.59731e11 1.99959 0.999797 − 0.0201320i \(-0.00640864\pi\)
0.999797 + 0.0201320i \(0.00640864\pi\)
\(168\) 0 0
\(169\) 2.29936e11 1.66792
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −9.82661e10 −0.634123 −0.317061 − 0.948405i \(-0.602696\pi\)
−0.317061 + 0.948405i \(0.602696\pi\)
\(174\) 0 0
\(175\) 1.64131e11 1.00000
\(176\) 6.52948e9 0.0386648
\(177\) 0 0
\(178\) 0 0
\(179\) −4.83793e10 −0.263266 −0.131633 − 0.991299i \(-0.542022\pi\)
−0.131633 + 0.991299i \(0.542022\pi\)
\(180\) −2.28070e11 −1.20700
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −1.63173e10 −0.0713576
\(188\) −4.66881e11 −1.98800
\(189\) 7.41609e10 0.307515
\(190\) 0 0
\(191\) −2.81264e11 −1.10649 −0.553245 − 0.833019i \(-0.686611\pi\)
−0.553245 + 0.833019i \(0.686611\pi\)
\(192\) 3.87621e11 1.48560
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) −6.84164e11 −2.42654
\(196\) 2.89255e11 1.00000
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 6.15328e11 1.78495
\(204\) −9.68672e11 −2.74174
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 6.35920e11 1.63338
\(209\) 0 0
\(210\) 0 0
\(211\) 8.01759e11 1.91704 0.958521 − 0.285023i \(-0.0920014\pi\)
0.958521 + 0.285023i \(0.0920014\pi\)
\(212\) 0 0
\(213\) 9.16634e10 0.209073
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −1.38114e12 −2.74169
\(220\) −1.99264e10 −0.0386648
\(221\) −1.58918e12 −3.01447
\(222\) 0 0
\(223\) −7.05975e11 −1.28016 −0.640081 − 0.768307i \(-0.721099\pi\)
−0.640081 + 0.768307i \(0.721099\pi\)
\(224\) 0 0
\(225\) 6.96016e11 1.20700
\(226\) 0 0
\(227\) 1.07143e12 1.77761 0.888803 − 0.458290i \(-0.151538\pi\)
0.888803 + 0.458290i \(0.151538\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 3.77812e10 0.0574403
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 1.42481e12 1.98800
\(236\) 0 0
\(237\) −1.87893e12 −2.51286
\(238\) 0 0
\(239\) 1.53213e12 1.96475 0.982375 − 0.186923i \(-0.0598515\pi\)
0.982375 + 0.186923i \(0.0598515\pi\)
\(240\) −1.18292e12 −1.48560
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) −1.20480e12 −1.42194
\(244\) 0 0
\(245\) −8.82735e11 −1.00000
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 1.15607e12 1.20777
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 1.22662e12 1.20700
\(253\) 0 0
\(254\) 0 0
\(255\) 2.95615e12 2.74174
\(256\) 1.09951e12 1.00000
\(257\) 3.82481e11 0.341149 0.170574 − 0.985345i \(-0.445438\pi\)
0.170574 + 0.985345i \(0.445438\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −1.94068e12 −1.63338
\(261\) 2.60937e12 2.15443
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) −2.74770e12 −1.84555
\(273\) 3.67960e12 2.42654
\(274\) 0 0
\(275\) 6.08105e10 0.0386648
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −7.06814e11 −0.403435 −0.201717 − 0.979444i \(-0.564652\pi\)
−0.201717 + 0.979444i \(0.564652\pi\)
\(282\) 0 0
\(283\) −2.56255e12 −1.41169 −0.705845 − 0.708366i \(-0.749433\pi\)
−0.705845 + 0.708366i \(0.749433\pi\)
\(284\) 2.60009e11 0.140733
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 4.85056e12 2.40604
\(290\) 0 0
\(291\) −5.85844e12 −2.80748
\(292\) −3.91770e12 −1.84551
\(293\) −6.97213e11 −0.322870 −0.161435 − 0.986883i \(-0.551612\pi\)
−0.161435 + 0.986883i \(0.551612\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 2.74767e10 0.0118900
\(298\) 0 0
\(299\) 0 0
\(300\) 3.61000e12 1.48560
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −1.89773e12 −0.695892 −0.347946 − 0.937515i \(-0.613121\pi\)
−0.347946 + 0.937515i \(0.613121\pi\)
\(308\) 1.07169e11 0.0386648
\(309\) −2.18093e12 −0.774193
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 5.84557e12 1.94583 0.972916 − 0.231160i \(-0.0742520\pi\)
0.972916 + 0.231160i \(0.0742520\pi\)
\(314\) 0 0
\(315\) −3.74334e12 −1.20700
\(316\) −5.32971e12 −1.69149
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 2.27979e11 0.0690148
\(320\) −3.35544e12 −1.00000
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −2.67840e12 −0.750154
\(325\) 5.92247e12 1.63338
\(326\) 0 0
\(327\) −8.24225e12 −2.20448
\(328\) 0 0
\(329\) −7.66295e12 −1.98800
\(330\) 0 0
\(331\) 4.78315e12 1.20385 0.601927 − 0.798551i \(-0.294400\pi\)
0.601927 + 0.798551i \(0.294400\pi\)
\(332\) 3.27926e12 0.812990
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 6.36205e12 1.48560
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 8.38532e12 1.84555
\(341\) 0 0
\(342\) 0 0
\(343\) 4.74756e12 1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 1.35339e13 2.65172
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 2.67601e12 0.502287
\(352\) 0 0
\(353\) −9.17017e12 −1.67303 −0.836515 − 0.547943i \(-0.815411\pi\)
−0.836515 + 0.547943i \(0.815411\pi\)
\(354\) 0 0
\(355\) −7.93485e11 −0.140733
\(356\) 0 0
\(357\) −1.58989e13 −2.74174
\(358\) 0 0
\(359\) −1.19194e13 −1.99885 −0.999426 − 0.0338878i \(-0.989211\pi\)
−0.999426 + 0.0338878i \(0.989211\pi\)
\(360\) 0 0
\(361\) 6.13107e12 1.00000
\(362\) 0 0
\(363\) −9.34941e12 −1.48338
\(364\) 1.04374e13 1.63338
\(365\) 1.19559e13 1.84551
\(366\) 0 0
\(367\) −4.34842e12 −0.653133 −0.326566 − 0.945174i \(-0.605892\pi\)
−0.326566 + 0.945174i \(0.605892\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) −1.10168e13 −1.48560
\(376\) 0 0
\(377\) 2.22034e13 2.91550
\(378\) 0 0
\(379\) 7.18906e12 0.919340 0.459670 − 0.888090i \(-0.347968\pi\)
0.459670 + 0.888090i \(0.347968\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 8.42209e12 1.02194 0.510971 − 0.859598i \(-0.329286\pi\)
0.510971 + 0.859598i \(0.329286\pi\)
\(384\) 0 0
\(385\) −3.27054e11 −0.0386648
\(386\) 0 0
\(387\) 0 0
\(388\) −1.66179e13 −1.88980
\(389\) 1.16301e12 0.130568 0.0652838 − 0.997867i \(-0.479205\pi\)
0.0652838 + 0.997867i \(0.479205\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1.62650e13 1.69149
\(396\) 4.54462e11 0.0466683
\(397\) 1.08349e13 1.09869 0.549343 − 0.835597i \(-0.314878\pi\)
0.549343 + 0.835597i \(0.314878\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.02400e13 1.00000
\(401\) −9.12709e12 −0.880259 −0.440130 − 0.897934i \(-0.645068\pi\)
−0.440130 + 0.897934i \(0.645068\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 8.17383e12 0.750154
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −6.18635e12 −0.521132
\(413\) 0 0
\(414\) 0 0
\(415\) −1.00075e13 −0.812990
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) −1.94154e13 −1.48560
\(421\) −6.28161e12 −0.474964 −0.237482 − 0.971392i \(-0.576322\pi\)
−0.237482 + 0.971392i \(0.576322\pi\)
\(422\) 0 0
\(423\) −3.24956e13 −2.39951
\(424\) 0 0
\(425\) −2.55900e13 −1.84555
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 1.36329e12 0.0938215
\(430\) 0 0
\(431\) −2.23006e13 −1.49944 −0.749722 − 0.661753i \(-0.769813\pi\)
−0.749722 + 0.661753i \(0.769813\pi\)
\(432\) 4.62684e12 0.307515
\(433\) 2.56220e13 1.68335 0.841673 − 0.539988i \(-0.181571\pi\)
0.841673 + 0.539988i \(0.181571\pi\)
\(434\) 0 0
\(435\) −4.13023e13 −2.65172
\(436\) −2.33797e13 −1.48390
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 2.01326e13 1.20700
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 4.19383e13 2.35003
\(448\) 1.80464e13 1.00000
\(449\) 1.96700e13 1.07788 0.538942 − 0.842343i \(-0.318824\pi\)
0.538942 + 0.842343i \(0.318824\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −1.21088e13 −0.634762
\(454\) 0 0
\(455\) −3.18525e13 −1.63338
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) −1.15626e13 −0.567533
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 3.83899e13 1.78495
\(465\) 0 0
\(466\) 0 0
\(467\) 1.21122e13 0.545303 0.272651 − 0.962113i \(-0.412099\pi\)
0.272651 + 0.962113i \(0.412099\pi\)
\(468\) 4.42611e13 1.97148
\(469\) 0 0
\(470\) 0 0
\(471\) 5.54993e13 2.39433
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) −4.50982e13 −1.84555
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −2.65202e13 −0.998505
\(485\) 5.07137e13 1.88980
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 2.96316e13 1.03836 0.519179 − 0.854665i \(-0.326238\pi\)
0.519179 + 0.854665i \(0.326238\pi\)
\(492\) 0 0
\(493\) −9.59370e13 −3.29421
\(494\) 0 0
\(495\) −1.38691e12 −0.0466683
\(496\) 0 0
\(497\) 4.26755e12 0.140733
\(498\) 0 0
\(499\) −3.63094e13 −1.17359 −0.586795 − 0.809735i \(-0.699611\pi\)
−0.586795 + 0.809735i \(0.699611\pi\)
\(500\) −3.12500e13 −1.00000
\(501\) 9.37630e13 2.97059
\(502\) 0 0
\(503\) −4.97863e13 −1.54622 −0.773108 − 0.634275i \(-0.781299\pi\)
−0.773108 + 0.634275i \(0.781299\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 8.30071e13 2.47785
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) −6.43016e13 −1.84551
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1.88792e13 0.521132
\(516\) 0 0
\(517\) −2.83913e12 −0.0768656
\(518\) 0 0
\(519\) −3.54741e13 −0.942050
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 7.82522e13 1.99981 0.999903 − 0.0139193i \(-0.00443078\pi\)
0.999903 + 0.0139193i \(0.00443078\pi\)
\(524\) 0 0
\(525\) 5.92512e13 1.48560
\(526\) 0 0
\(527\) 0 0
\(528\) 2.35714e12 0.0574403
\(529\) 4.14265e13 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −1.74649e13 −0.391107
\(538\) 0 0
\(539\) 1.75897e12 0.0386648
\(540\) −1.41200e13 −0.307515
\(541\) −6.63534e13 −1.43178 −0.715891 − 0.698212i \(-0.753979\pi\)
−0.715891 + 0.698212i \(0.753979\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 7.13491e13 1.48390
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −8.74770e13 −1.69149
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −5.50732e13 −1.00000
\(561\) −5.89055e12 −0.106009
\(562\) 0 0
\(563\) 1.09051e14 1.92791 0.963953 − 0.266073i \(-0.0857261\pi\)
0.963953 + 0.266073i \(0.0857261\pi\)
\(564\) −1.68544e14 −2.95337
\(565\) 0 0
\(566\) 0 0
\(567\) −4.39608e13 −0.750154
\(568\) 0 0
\(569\) −3.63657e13 −0.609720 −0.304860 − 0.952397i \(-0.598610\pi\)
−0.304860 + 0.952397i \(0.598610\pi\)
\(570\) 0 0
\(571\) −6.00630e13 −0.989525 −0.494762 − 0.869028i \(-0.664745\pi\)
−0.494762 + 0.869028i \(0.664745\pi\)
\(572\) 3.86707e12 0.0631541
\(573\) −1.01536e14 −1.64380
\(574\) 0 0
\(575\) 0 0
\(576\) 7.65277e13 1.20700
\(577\) 1.60315e13 0.250665 0.125333 − 0.992115i \(-0.460000\pi\)
0.125333 + 0.992115i \(0.460000\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) −1.17157e14 −1.78495
\(581\) 5.38227e13 0.812990
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −1.35074e14 −1.97148
\(586\) 0 0
\(587\) −3.92828e13 −0.563654 −0.281827 − 0.959465i \(-0.590940\pi\)
−0.281827 + 0.959465i \(0.590940\pi\)
\(588\) 1.04421e14 1.48560
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −5.65806e13 −0.771604 −0.385802 − 0.922582i \(-0.626075\pi\)
−0.385802 + 0.922582i \(0.626075\pi\)
\(594\) 0 0
\(595\) 1.37629e14 1.84555
\(596\) 1.18961e14 1.58187
\(597\) 0 0
\(598\) 0 0
\(599\) 1.50973e14 1.95779 0.978893 − 0.204375i \(-0.0655161\pi\)
0.978893 + 0.204375i \(0.0655161\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −3.43475e13 −0.427278
\(605\) 8.09333e13 0.998505
\(606\) 0 0
\(607\) −7.97639e13 −0.967972 −0.483986 − 0.875076i \(-0.660811\pi\)
−0.483986 + 0.875076i \(0.660811\pi\)
\(608\) 0 0
\(609\) 2.22133e14 2.65172
\(610\) 0 0
\(611\) −2.76509e14 −3.24715
\(612\) −1.91244e14 −2.22757
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 2.29567e14 2.42654
\(625\) 9.53674e13 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 1.57427e14 1.61169
\(629\) 0 0
\(630\) 0 0
\(631\) −1.78457e14 −1.78397 −0.891983 − 0.452068i \(-0.850686\pi\)
−0.891983 + 0.452068i \(0.850686\pi\)
\(632\) 0 0
\(633\) 2.89435e14 2.84795
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 1.71310e14 1.63338
\(638\) 0 0
\(639\) 1.80970e13 0.169865
\(640\) 0 0
\(641\) 1.90692e14 1.76215 0.881074 − 0.472979i \(-0.156821\pi\)
0.881074 + 0.472979i \(0.156821\pi\)
\(642\) 0 0
\(643\) 5.31460e13 0.483521 0.241761 − 0.970336i \(-0.422275\pi\)
0.241761 + 0.970336i \(0.422275\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −2.04022e14 −1.79951 −0.899757 − 0.436392i \(-0.856256\pi\)
−0.899757 + 0.436392i \(0.856256\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −2.72678e14 −2.22753
\(658\) 0 0
\(659\) 1.00437e14 0.808104 0.404052 − 0.914736i \(-0.367601\pi\)
0.404052 + 0.914736i \(0.367601\pi\)
\(660\) −7.19343e12 −0.0574403
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) −5.73693e14 −4.47829
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 2.65965e14 1.99959
\(669\) −2.54857e14 −1.90180
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 4.30908e13 0.307515
\(676\) 2.35455e14 1.66792
\(677\) −1.12469e13 −0.0790838 −0.0395419 − 0.999218i \(-0.512590\pi\)
−0.0395419 + 0.999218i \(0.512590\pi\)
\(678\) 0 0
\(679\) −2.72750e14 −1.88980
\(680\) 0 0
\(681\) 3.86787e14 2.64081
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) −1.00624e14 −0.634123
\(693\) 7.45913e12 0.0466683
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 1.68070e14 1.00000
\(701\) −3.05570e14 −1.80518 −0.902591 − 0.430500i \(-0.858337\pi\)
−0.902591 + 0.430500i \(0.858337\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 6.68619e12 0.0386648
\(705\) 5.14355e14 2.95337
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −3.52012e14 −1.96484 −0.982419 − 0.186688i \(-0.940225\pi\)
−0.982419 + 0.186688i \(0.940225\pi\)
\(710\) 0 0
\(711\) −3.70956e14 −2.04162
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −1.18014e13 −0.0631541
\(716\) −4.95404e13 −0.263266
\(717\) 5.53100e14 2.91882
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) −2.33544e14 −1.20700
\(721\) −1.01537e14 −0.521132
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 3.57533e14 1.78495
\(726\) 0 0
\(727\) −2.25285e14 −1.10933 −0.554665 − 0.832074i \(-0.687153\pi\)
−0.554665 + 0.832074i \(0.687153\pi\)
\(728\) 0 0
\(729\) −2.80481e14 −1.36228
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −3.32669e14 −1.57214 −0.786071 − 0.618136i \(-0.787888\pi\)
−0.786071 + 0.618136i \(0.787888\pi\)
\(734\) 0 0
\(735\) −3.18667e14 −1.48560
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 1.98250e14 0.899481 0.449741 − 0.893159i \(-0.351516\pi\)
0.449741 + 0.893159i \(0.351516\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) −3.63039e14 −1.58187
\(746\) 0 0
\(747\) 2.28241e14 0.981277
\(748\) −1.67089e13 −0.0713576
\(749\) 0 0
\(750\) 0 0
\(751\) 3.50600e14 1.46762 0.733808 − 0.679357i \(-0.237741\pi\)
0.733808 + 0.679357i \(0.237741\pi\)
\(752\) −4.78086e14 −1.98800
\(753\) 0 0
\(754\) 0 0
\(755\) 1.04820e14 0.427278
\(756\) 7.59408e13 0.307515
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) −3.83733e14 −1.48390
\(764\) −2.88015e14 −1.10649
\(765\) 5.83631e14 2.22757
\(766\) 0 0
\(767\) 0 0
\(768\) 3.96924e14 1.48560
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 1.38076e14 0.506810
\(772\) 0 0
\(773\) 5.07930e14 1.84037 0.920187 − 0.391478i \(-0.128036\pi\)
0.920187 + 0.391478i \(0.128036\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) −7.00584e14 −2.42654
\(781\) 1.58113e12 0.00544142
\(782\) 0 0
\(783\) 1.61548e14 0.548899
\(784\) 2.96197e14 1.00000
\(785\) −4.80431e14 −1.61169
\(786\) 0 0
\(787\) 5.99148e14 1.98454 0.992272 − 0.124085i \(-0.0395995\pi\)
0.992272 + 0.124085i \(0.0395995\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.02699e14 −0.319357 −0.159678 − 0.987169i \(-0.551046\pi\)
−0.159678 + 0.987169i \(0.551046\pi\)
\(798\) 0 0
\(799\) 1.19475e15 3.66895
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −2.38238e13 −0.0713564
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −2.04769e14 −0.590912 −0.295456 − 0.955356i \(-0.595472\pi\)
−0.295456 + 0.955356i \(0.595472\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 6.30096e14 1.78495
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) −9.91920e14 −2.74174
\(817\) 0 0
\(818\) 0 0
\(819\) 7.26461e14 1.97148
\(820\) 0 0
\(821\) 3.81368e14 1.02242 0.511209 − 0.859457i \(-0.329198\pi\)
0.511209 + 0.859457i \(0.329198\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 2.19526e13 0.0574403
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 6.51183e14 1.63338
\(833\) −7.40201e14 −1.84555
\(834\) 0 0
\(835\) −8.11660e14 −1.99959
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 9.19689e14 2.18605
\(842\) 0 0
\(843\) −2.55160e14 −0.599342
\(844\) 8.21001e14 1.91704
\(845\) −7.18551e14 −1.66792
\(846\) 0 0
\(847\) −4.35279e14 −0.998505
\(848\) 0 0
\(849\) −9.25079e14 −2.09720
\(850\) 0 0
\(851\) 0 0
\(852\) 9.38633e13 0.209073
\(853\) 2.25339e14 0.498989 0.249495 − 0.968376i \(-0.419735\pi\)
0.249495 + 0.968376i \(0.419735\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −7.87286e14 −1.70305 −0.851527 − 0.524311i \(-0.824323\pi\)
−0.851527 + 0.524311i \(0.824323\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 3.07082e14 0.634123
\(866\) 0 0
\(867\) 1.75105e15 3.57440
\(868\) 0 0
\(869\) −3.24103e13 −0.0654009
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −1.15663e15 −2.28099
\(874\) 0 0
\(875\) −5.12909e14 −1.00000
\(876\) −1.41429e15 −2.74169
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) −2.51694e14 −0.479654
\(880\) −2.04046e13 −0.0386648
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) −1.62732e15 −3.01447
\(885\) 0 0
\(886\) 0 0
\(887\) 1.08412e15 1.97452 0.987258 − 0.159130i \(-0.0508690\pi\)
0.987258 + 0.159130i \(0.0508690\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −1.62875e13 −0.0290045
\(892\) −7.22918e14 −1.28016
\(893\) 0 0
\(894\) 0 0
\(895\) 1.51185e14 0.263266
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 7.12720e14 1.20700
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 1.09715e15 1.77761
\(909\) 0 0
\(910\) 0 0
\(911\) 5.23629e14 0.834511 0.417256 − 0.908789i \(-0.362992\pi\)
0.417256 + 0.908789i \(0.362992\pi\)
\(912\) 0 0
\(913\) 1.99413e13 0.0314341
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −3.55893e14 −0.542928 −0.271464 − 0.962449i \(-0.587508\pi\)
−0.271464 + 0.962449i \(0.587508\pi\)
\(920\) 0 0
\(921\) −6.85080e14 −1.03382
\(922\) 0 0
\(923\) 1.53990e14 0.229870
\(924\) 3.86880e13 0.0574403
\(925\) 0 0
\(926\) 0 0
\(927\) −4.30579e14 −0.629006
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 5.09916e13 0.0713576
\(936\) 0 0
\(937\) 6.57883e14 0.910859 0.455429 − 0.890272i \(-0.349486\pi\)
0.455429 + 0.890272i \(0.349486\pi\)
\(938\) 0 0
\(939\) 2.11025e15 2.89072
\(940\) 1.45900e15 1.98800
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −2.31753e14 −0.307515
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) −1.92403e15 −2.51286
\(949\) −2.32025e15 −3.01442
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 8.78951e14 1.10649
\(956\) 1.56890e15 1.96475
\(957\) 8.23005e13 0.102528
\(958\) 0 0
\(959\) 0 0
\(960\) −1.21131e15 −1.48560
\(961\) 8.19628e14 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) −1.23371e15 −1.42194
\(973\) 0 0
\(974\) 0 0
\(975\) 2.13801e15 2.42654
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −9.03921e14 −1.00000
\(981\) −1.62726e15 −1.79107
\(982\) 0 0
\(983\) 1.97662e14 0.215356 0.107678 − 0.994186i \(-0.465658\pi\)
0.107678 + 0.994186i \(0.465658\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −2.76633e15 −2.95337
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 1.82986e15 1.91448 0.957239 − 0.289298i \(-0.0934219\pi\)
0.957239 + 0.289298i \(0.0934219\pi\)
\(992\) 0 0
\(993\) 1.72672e15 1.78844
\(994\) 0 0
\(995\) 0 0
\(996\) 1.18381e15 1.20777
\(997\) 1.94338e15 1.97280 0.986398 − 0.164376i \(-0.0525610\pi\)
0.986398 + 0.164376i \(0.0525610\pi\)
\(998\) 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 35.11.c.b.34.1 yes 1
5.4 even 2 35.11.c.a.34.1 ✓ 1
7.6 odd 2 35.11.c.a.34.1 ✓ 1
35.34 odd 2 CM 35.11.c.b.34.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.11.c.a.34.1 ✓ 1 5.4 even 2
35.11.c.a.34.1 ✓ 1 7.6 odd 2
35.11.c.b.34.1 yes 1 1.1 even 1 trivial
35.11.c.b.34.1 yes 1 35.34 odd 2 CM