Properties

Label 2671.1.b.a.2670.7
Level $2671$
Weight $1$
Character 2671.2670
Self dual yes
Analytic conductor $1.333$
Analytic rank $0$
Dimension $11$
Projective image $D_{23}$
CM discriminant -2671
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2671,1,Mod(2670,2671)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2671, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2671.2670");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2671 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2671.b (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: \(1.33300264874\)
Analytic rank: \(0\)
Dimension: \(11\)
Coefficient field: \(\Q(\zeta_{46})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - x^{10} - 10x^{9} + 9x^{8} + 36x^{7} - 28x^{6} - 56x^{5} + 35x^{4} + 35x^{3} - 15x^{2} - 6x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{23}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{23} - \cdots)\)

Embedding invariants

Embedding label 2670.7
Root \(0.136485\) of defining polynomial
Character \(\chi\) \(=\) 2671.2670

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.406912 q^{2} -0.834423 q^{4} +1.92583 q^{5} -0.746449 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.406912 q^{2} -0.834423 q^{4} +1.92583 q^{5} -0.746449 q^{8} +1.00000 q^{9} +0.783645 q^{10} +0.530684 q^{16} -0.669759 q^{17} +0.406912 q^{18} -1.60696 q^{20} +2.70884 q^{25} +0.962390 q^{32} -0.272533 q^{34} -0.834423 q^{36} -1.55142 q^{37} -1.43754 q^{40} +1.36511 q^{43} +1.92583 q^{45} -1.98137 q^{47} +1.00000 q^{49} +1.10226 q^{50} -1.98137 q^{61} -0.139076 q^{64} +0.920130 q^{67} +0.558862 q^{68} -0.136485 q^{71} -0.746449 q^{72} -0.631293 q^{74} +1.02201 q^{80} +1.00000 q^{81} -1.55142 q^{83} -1.28985 q^{85} +0.555478 q^{86} +1.36511 q^{89} +0.783645 q^{90} -0.806244 q^{94} +0.406912 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q - q^{2} + 10 q^{4} - q^{5} - 2 q^{8} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 11 q - q^{2} + 10 q^{4} - q^{5} - 2 q^{8} + 11 q^{9} - 2 q^{10} + 9 q^{16} - q^{17} - q^{18} - 3 q^{20} + 10 q^{25} - 3 q^{32} - 2 q^{34} + 10 q^{36} - q^{37} - 4 q^{40} - q^{43} - q^{45} - q^{47} + 11 q^{49} - 3 q^{50} - q^{61} + 8 q^{64} - q^{67} - 3 q^{68} - q^{71} - 2 q^{72} - 2 q^{74} - 5 q^{80} + 11 q^{81} - q^{83} - 2 q^{85} - 2 q^{86} - q^{89} - 2 q^{90} - 2 q^{94} - q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2671\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.406912 0.406912 0.203456 0.979084i \(-0.434783\pi\)
0.203456 + 0.979084i \(0.434783\pi\)
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) −0.834423 −0.834423
\(5\) 1.92583 1.92583 0.962917 0.269797i \(-0.0869565\pi\)
0.962917 + 0.269797i \(0.0869565\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) −0.746449 −0.746449
\(9\) 1.00000 1.00000
\(10\) 0.783645 0.783645
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.530684 0.530684
\(17\) −0.669759 −0.669759 −0.334880 0.942261i \(-0.608696\pi\)
−0.334880 + 0.942261i \(0.608696\pi\)
\(18\) 0.406912 0.406912
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) −1.60696 −1.60696
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 2.70884 2.70884
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0.962390 0.962390
\(33\) 0 0
\(34\) −0.272533 −0.272533
\(35\) 0 0
\(36\) −0.834423 −0.834423
\(37\) −1.55142 −1.55142 −0.775711 0.631088i \(-0.782609\pi\)
−0.775711 + 0.631088i \(0.782609\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −1.43754 −1.43754
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 1.36511 1.36511 0.682553 0.730836i \(-0.260870\pi\)
0.682553 + 0.730836i \(0.260870\pi\)
\(44\) 0 0
\(45\) 1.92583 1.92583
\(46\) 0 0
\(47\) −1.98137 −1.98137 −0.990686 0.136167i \(-0.956522\pi\)
−0.990686 + 0.136167i \(0.956522\pi\)
\(48\) 0 0
\(49\) 1.00000 1.00000
\(50\) 1.10226 1.10226
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −1.98137 −1.98137 −0.990686 0.136167i \(-0.956522\pi\)
−0.990686 + 0.136167i \(0.956522\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −0.139076 −0.139076
\(65\) 0 0
\(66\) 0 0
\(67\) 0.920130 0.920130 0.460065 0.887885i \(-0.347826\pi\)
0.460065 + 0.887885i \(0.347826\pi\)
\(68\) 0.558862 0.558862
\(69\) 0 0
\(70\) 0 0
\(71\) −0.136485 −0.136485 −0.0682424 0.997669i \(-0.521739\pi\)
−0.0682424 + 0.997669i \(0.521739\pi\)
\(72\) −0.746449 −0.746449
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) −0.631293 −0.631293
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 1.02201 1.02201
\(81\) 1.00000 1.00000
\(82\) 0 0
\(83\) −1.55142 −1.55142 −0.775711 0.631088i \(-0.782609\pi\)
−0.775711 + 0.631088i \(0.782609\pi\)
\(84\) 0 0
\(85\) −1.28985 −1.28985
\(86\) 0.555478 0.555478
\(87\) 0 0
\(88\) 0 0
\(89\) 1.36511 1.36511 0.682553 0.730836i \(-0.260870\pi\)
0.682553 + 0.730836i \(0.260870\pi\)
\(90\) 0.783645 0.783645
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) −0.806244 −0.806244
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0.406912 0.406912
\(99\) 0 0
\(100\) −2.26032 −2.26032
\(101\) 0.920130 0.920130 0.460065 0.887885i \(-0.347826\pi\)
0.460065 + 0.887885i \(0.347826\pi\)
\(102\) 0 0
\(103\) −1.15336 −1.15336 −0.576680 0.816970i \(-0.695652\pi\)
−0.576680 + 0.816970i \(0.695652\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.70884 1.70884 0.854419 0.519584i \(-0.173913\pi\)
0.854419 + 0.519584i \(0.173913\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0.920130 0.920130 0.460065 0.887885i \(-0.347826\pi\)
0.460065 + 0.887885i \(0.347826\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 1.00000
\(122\) −0.806244 −0.806244
\(123\) 0 0
\(124\) 0 0
\(125\) 3.29094 3.29094
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) −1.01898 −1.01898
\(129\) 0 0
\(130\) 0 0
\(131\) −1.15336 −1.15336 −0.576680 0.816970i \(-0.695652\pi\)
−0.576680 + 0.816970i \(0.695652\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0.374412 0.374412
\(135\) 0 0
\(136\) 0.499941 0.499941
\(137\) −0.136485 −0.136485 −0.0682424 0.997669i \(-0.521739\pi\)
−0.0682424 + 0.997669i \(0.521739\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −0.0555373 −0.0555373
\(143\) 0 0
\(144\) 0.530684 0.530684
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 1.29454 1.29454
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) −1.55142 −1.55142 −0.775711 0.631088i \(-0.782609\pi\)
−0.775711 + 0.631088i \(0.782609\pi\)
\(152\) 0 0
\(153\) −0.669759 −0.669759
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 1.85340 1.85340
\(161\) 0 0
\(162\) 0.406912 0.406912
\(163\) −0.669759 −0.669759 −0.334880 0.942261i \(-0.608696\pi\)
−0.334880 + 0.942261i \(0.608696\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −0.631293 −0.631293
\(167\) −1.83442 −1.83442 −0.917211 0.398401i \(-0.869565\pi\)
−0.917211 + 0.398401i \(0.869565\pi\)
\(168\) 0 0
\(169\) 1.00000 1.00000
\(170\) −0.524854 −0.524854
\(171\) 0 0
\(172\) −1.13908 −1.13908
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0.555478 0.555478
\(179\) −0.136485 −0.136485 −0.0682424 0.997669i \(-0.521739\pi\)
−0.0682424 + 0.997669i \(0.521739\pi\)
\(180\) −1.60696 −1.60696
\(181\) −1.83442 −1.83442 −0.917211 0.398401i \(-0.869565\pi\)
−0.917211 + 0.398401i \(0.869565\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −2.98778 −2.98778
\(186\) 0 0
\(187\) 0 0
\(188\) 1.65330 1.65330
\(189\) 0 0
\(190\) 0 0
\(191\) −0.669759 −0.669759 −0.334880 0.942261i \(-0.608696\pi\)
−0.334880 + 0.942261i \(0.608696\pi\)
\(192\) 0 0
\(193\) −1.98137 −1.98137 −0.990686 0.136167i \(-0.956522\pi\)
−0.990686 + 0.136167i \(0.956522\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −0.834423 −0.834423
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 1.70884 1.70884 0.854419 0.519584i \(-0.173913\pi\)
0.854419 + 0.519584i \(0.173913\pi\)
\(200\) −2.02201 −2.02201
\(201\) 0 0
\(202\) 0.374412 0.374412
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) −0.469316 −0.469316
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0.695347 0.695347
\(215\) 2.62897 2.62897
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 2.70884 2.70884
\(226\) 0.374412 0.374412
\(227\) −1.83442 −1.83442 −0.917211 0.398401i \(-0.869565\pi\)
−0.917211 + 0.398401i \(0.869565\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) −3.81579 −3.81579
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0.920130 0.920130 0.460065 0.887885i \(-0.347826\pi\)
0.460065 + 0.887885i \(0.347826\pi\)
\(240\) 0 0
\(241\) −1.15336 −1.15336 −0.576680 0.816970i \(-0.695652\pi\)
−0.576680 + 0.816970i \(0.695652\pi\)
\(242\) 0.406912 0.406912
\(243\) 0 0
\(244\) 1.65330 1.65330
\(245\) 1.92583 1.92583
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 1.33912 1.33912
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −0.275560 −0.275560
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) −0.469316 −0.469316
\(263\) 1.36511 1.36511 0.682553 0.730836i \(-0.260870\pi\)
0.682553 + 0.730836i \(0.260870\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −0.767777 −0.767777
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) −0.669759 −0.669759 −0.334880 0.942261i \(-0.608696\pi\)
−0.334880 + 0.942261i \(0.608696\pi\)
\(272\) −0.355430 −0.355430
\(273\) 0 0
\(274\) −0.0555373 −0.0555373
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −0.136485 −0.136485 −0.0682424 0.997669i \(-0.521739\pi\)
−0.0682424 + 0.997669i \(0.521739\pi\)
\(284\) 0.113886 0.113886
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0.962390 0.962390
\(289\) −0.551423 −0.551423
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 1.15806 1.15806
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) −0.631293 −0.631293
\(303\) 0 0
\(304\) 0 0
\(305\) −3.81579 −3.81579
\(306\) −0.272533 −0.272533
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −1.55142 −1.55142 −0.775711 0.631088i \(-0.782609\pi\)
−0.775711 + 0.631088i \(0.782609\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.83442 −1.83442 −0.917211 0.398401i \(-0.869565\pi\)
−0.917211 + 0.398401i \(0.869565\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −0.267836 −0.267836
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −0.834423 −0.834423
\(325\) 0 0
\(326\) −0.272533 −0.272533
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1.15336 −1.15336 −0.576680 0.816970i \(-0.695652\pi\)
−0.576680 + 0.816970i \(0.695652\pi\)
\(332\) 1.29454 1.29454
\(333\) −1.55142 −1.55142
\(334\) −0.746449 −0.746449
\(335\) 1.77202 1.77202
\(336\) 0 0
\(337\) 0.406912 0.406912 0.203456 0.979084i \(-0.434783\pi\)
0.203456 + 0.979084i \(0.434783\pi\)
\(338\) 0.406912 0.406912
\(339\) 0 0
\(340\) 1.07628 1.07628
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) −1.01898 −1.01898
\(345\) 0 0
\(346\) 0 0
\(347\) −1.98137 −1.98137 −0.990686 0.136167i \(-0.956522\pi\)
−0.990686 + 0.136167i \(0.956522\pi\)
\(348\) 0 0
\(349\) 0.406912 0.406912 0.203456 0.979084i \(-0.434783\pi\)
0.203456 + 0.979084i \(0.434783\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −0.136485 −0.136485 −0.0682424 0.997669i \(-0.521739\pi\)
−0.0682424 + 0.997669i \(0.521739\pi\)
\(354\) 0 0
\(355\) −0.262847 −0.262847
\(356\) −1.13908 −1.13908
\(357\) 0 0
\(358\) −0.0555373 −0.0555373
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) −1.43754 −1.43754
\(361\) 1.00000 1.00000
\(362\) −0.746449 −0.746449
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) −1.21576 −1.21576
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 1.47899 1.47899
\(377\) 0 0
\(378\) 0 0
\(379\) 0.406912 0.406912 0.203456 0.979084i \(-0.434783\pi\)
0.203456 + 0.979084i \(0.434783\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −0.272533 −0.272533
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −0.806244 −0.806244
\(387\) 1.36511 1.36511
\(388\) 0 0
\(389\) 0.406912 0.406912 0.203456 0.979084i \(-0.434783\pi\)
0.203456 + 0.979084i \(0.434783\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −0.746449 −0.746449
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 1.36511 1.36511 0.682553 0.730836i \(-0.260870\pi\)
0.682553 + 0.730836i \(0.260870\pi\)
\(398\) 0.695347 0.695347
\(399\) 0 0
\(400\) 1.43754 1.43754
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −0.767777 −0.767777
\(405\) 1.92583 1.92583
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0.920130 0.920130 0.460065 0.887885i \(-0.347826\pi\)
0.460065 + 0.887885i \(0.347826\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0.962390 0.962390
\(413\) 0 0
\(414\) 0 0
\(415\) −2.98778 −2.98778
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) −1.98137 −1.98137
\(424\) 0 0
\(425\) −1.81427 −1.81427
\(426\) 0 0
\(427\) 0 0
\(428\) −1.42589 −1.42589
\(429\) 0 0
\(430\) 1.06976 1.06976
\(431\) −1.83442 −1.83442 −0.917211 0.398401i \(-0.869565\pi\)
−0.917211 + 0.398401i \(0.869565\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0.406912 0.406912 0.203456 0.979084i \(-0.434783\pi\)
0.203456 + 0.979084i \(0.434783\pi\)
\(440\) 0 0
\(441\) 1.00000 1.00000
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 2.62897 2.62897
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −1.55142 −1.55142 −0.775711 0.631088i \(-0.782609\pi\)
−0.775711 + 0.631088i \(0.782609\pi\)
\(450\) 1.10226 1.10226
\(451\) 0 0
\(452\) −0.767777 −0.767777
\(453\) 0 0
\(454\) −0.746449 −0.746449
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 1.70884 1.70884 0.854419 0.519584i \(-0.173913\pi\)
0.854419 + 0.519584i \(0.173913\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −1.55269 −1.55269
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0.374412 0.374412
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −0.469316 −0.469316
\(483\) 0 0
\(484\) −0.834423 −0.834423
\(485\) 0 0
\(486\) 0 0
\(487\) 0.406912 0.406912 0.203456 0.979084i \(-0.434783\pi\)
0.203456 + 0.979084i \(0.434783\pi\)
\(488\) 1.47899 1.47899
\(489\) 0 0
\(490\) 0.783645 0.783645
\(491\) 1.70884 1.70884 0.854419 0.519584i \(-0.173913\pi\)
0.854419 + 0.519584i \(0.173913\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −1.98137 −1.98137 −0.990686 0.136167i \(-0.956522\pi\)
−0.990686 + 0.136167i \(0.956522\pi\)
\(500\) −2.74604 −2.74604
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 1.77202 1.77202
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −0.136485 −0.136485 −0.0682424 0.997669i \(-0.521739\pi\)
−0.0682424 + 0.997669i \(0.521739\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0.906853 0.906853
\(513\) 0 0
\(514\) 0 0
\(515\) −2.22118 −2.22118
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 1.70884 1.70884 0.854419 0.519584i \(-0.173913\pi\)
0.854419 + 0.519584i \(0.173913\pi\)
\(524\) 0.962390 0.962390
\(525\) 0 0
\(526\) 0.555478 0.555478
\(527\) 0 0
\(528\) 0 0
\(529\) 1.00000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 3.29094 3.29094
\(536\) −0.686830 −0.686830
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −1.15336 −1.15336 −0.576680 0.816970i \(-0.695652\pi\)
−0.576680 + 0.816970i \(0.695652\pi\)
\(542\) −0.272533 −0.272533
\(543\) 0 0
\(544\) −0.644570 −0.644570
\(545\) 0 0
\(546\) 0 0
\(547\) 1.92583 1.92583 0.962917 0.269797i \(-0.0869565\pi\)
0.962917 + 0.269797i \(0.0869565\pi\)
\(548\) 0.113886 0.113886
\(549\) −1.98137 −1.98137
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −1.83442 −1.83442 −0.917211 0.398401i \(-0.869565\pi\)
−0.917211 + 0.398401i \(0.869565\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 1.92583 1.92583 0.962917 0.269797i \(-0.0869565\pi\)
0.962917 + 0.269797i \(0.0869565\pi\)
\(564\) 0 0
\(565\) 1.77202 1.77202
\(566\) −0.0555373 −0.0555373
\(567\) 0 0
\(568\) 0.101879 0.101879
\(569\) −1.98137 −1.98137 −0.990686 0.136167i \(-0.956522\pi\)
−0.990686 + 0.136167i \(0.956522\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −0.139076 −0.139076
\(577\) −1.83442 −1.83442 −0.917211 0.398401i \(-0.869565\pi\)
−0.917211 + 0.398401i \(0.869565\pi\)
\(578\) −0.224380 −0.224380
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0.406912 0.406912 0.203456 0.979084i \(-0.434783\pi\)
0.203456 + 0.979084i \(0.434783\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −0.823315 −0.823315
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 1.92583 1.92583 0.962917 0.269797i \(-0.0869565\pi\)
0.962917 + 0.269797i \(0.0869565\pi\)
\(600\) 0 0
\(601\) −0.136485 −0.136485 −0.0682424 0.997669i \(-0.521739\pi\)
−0.0682424 + 0.997669i \(0.521739\pi\)
\(602\) 0 0
\(603\) 0.920130 0.920130
\(604\) 1.29454 1.29454
\(605\) 1.92583 1.92583
\(606\) 0 0
\(607\) −0.136485 −0.136485 −0.0682424 0.997669i \(-0.521739\pi\)
−0.0682424 + 0.997669i \(0.521739\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −1.55269 −1.55269
\(611\) 0 0
\(612\) 0.558862 0.558862
\(613\) 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
\(622\) −0.631293 −0.631293
\(623\) 0 0
\(624\) 0 0
\(625\) 3.62897 3.62897
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1.03908 1.03908
\(630\) 0 0
\(631\) 1.92583 1.92583 0.962917 0.269797i \(-0.0869565\pi\)
0.962917 + 0.269797i \(0.0869565\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −0.746449 −0.746449
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −0.136485 −0.136485
\(640\) −1.96239 −1.96239
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) −0.136485 −0.136485 −0.0682424 0.997669i \(-0.521739\pi\)
−0.0682424 + 0.997669i \(0.521739\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) −0.746449 −0.746449
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0.558862 0.558862
\(653\) −1.15336 −1.15336 −0.576680 0.816970i \(-0.695652\pi\)
−0.576680 + 0.816970i \(0.695652\pi\)
\(654\) 0 0
\(655\) −2.22118 −2.22118
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0.406912 0.406912 0.203456 0.979084i \(-0.434783\pi\)
0.203456 + 0.979084i \(0.434783\pi\)
\(660\) 0 0
\(661\) 0.920130 0.920130 0.460065 0.887885i \(-0.347826\pi\)
0.460065 + 0.887885i \(0.347826\pi\)
\(662\) −0.469316 −0.469316
\(663\) 0 0
\(664\) 1.15806 1.15806
\(665\) 0 0
\(666\) −0.631293 −0.631293
\(667\) 0 0
\(668\) 1.53068 1.53068
\(669\) 0 0
\(670\) 0.721056 0.721056
\(671\) 0 0
\(672\) 0 0
\(673\) 1.36511 1.36511 0.682553 0.730836i \(-0.260870\pi\)
0.682553 + 0.730836i \(0.260870\pi\)
\(674\) 0.165577 0.165577
\(675\) 0 0
\(676\) −0.834423 −0.834423
\(677\) 1.70884 1.70884 0.854419 0.519584i \(-0.173913\pi\)
0.854419 + 0.519584i \(0.173913\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0.962803 0.962803
\(681\) 0 0
\(682\) 0 0
\(683\) 0.920130 0.920130 0.460065 0.887885i \(-0.347826\pi\)
0.460065 + 0.887885i \(0.347826\pi\)
\(684\) 0 0
\(685\) −0.262847 −0.262847
\(686\) 0 0
\(687\) 0 0
\(688\) 0.724440 0.724440
\(689\) 0 0
\(690\) 0 0
\(691\) 1.92583 1.92583 0.962917 0.269797i \(-0.0869565\pi\)
0.962917 + 0.269797i \(0.0869565\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −0.806244 −0.806244
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0.165577 0.165577
\(699\) 0 0
\(700\) 0 0
\(701\) −1.98137 −1.98137 −0.990686 0.136167i \(-0.956522\pi\)
−0.990686 + 0.136167i \(0.956522\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) −0.0555373 −0.0555373
\(707\) 0 0
\(708\) 0 0
\(709\) −0.669759 −0.669759 −0.334880 0.942261i \(-0.608696\pi\)
−0.334880 + 0.942261i \(0.608696\pi\)
\(710\) −0.106956 −0.106956
\(711\) 0 0
\(712\) −1.01898 −1.01898
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0.113886 0.113886
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 1.02201 1.02201
\(721\) 0 0
\(722\) 0.406912 0.406912
\(723\) 0 0
\(724\) 1.53068 1.53068
\(725\) 0 0
\(726\) 0 0
\(727\) 0.920130 0.920130 0.460065 0.887885i \(-0.347826\pi\)
0.460065 + 0.887885i \(0.347826\pi\)
\(728\) 0 0
\(729\) 1.00000 1.00000
\(730\) 0 0
\(731\) −0.914293 −0.914293
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 1.92583 1.92583 0.962917 0.269797i \(-0.0869565\pi\)
0.962917 + 0.269797i \(0.0869565\pi\)
\(740\) 2.49307 2.49307
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −1.55142 −1.55142
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 1.36511 1.36511 0.682553 0.730836i \(-0.260870\pi\)
0.682553 + 0.730836i \(0.260870\pi\)
\(752\) −1.05148 −1.05148
\(753\) 0 0
\(754\) 0 0
\(755\) −2.98778 −2.98778
\(756\) 0 0
\(757\) −0.136485 −0.136485 −0.0682424 0.997669i \(-0.521739\pi\)
−0.0682424 + 0.997669i \(0.521739\pi\)
\(758\) 0.165577 0.165577
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0.558862 0.558862
\(765\) −1.28985 −1.28985
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0.406912 0.406912 0.203456 0.979084i \(-0.434783\pi\)
0.203456 + 0.979084i \(0.434783\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.65330 1.65330
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0.555478 0.555478
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0.165577 0.165577
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0.530684 0.530684
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0.555478 0.555478
\(795\) 0 0
\(796\) −1.42589 −1.42589
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 1.32704 1.32704
\(800\) 2.60696 2.60696
\(801\) 1.36511 1.36511
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) −0.686830 −0.686830
\(809\) −1.15336 −1.15336 −0.576680 0.816970i \(-0.695652\pi\)
−0.576680 + 0.816970i \(0.695652\pi\)
\(810\) 0.783645 0.783645
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −1.28985 −1.28985
\(816\) 0 0
\(817\) 0 0
\(818\) 0.374412 0.374412
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 1.92583 1.92583 0.962917 0.269797i \(-0.0869565\pi\)
0.962917 + 0.269797i \(0.0869565\pi\)
\(824\) 0.860924 0.860924
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) −1.21576 −1.21576
\(831\) 0 0
\(832\) 0 0
\(833\) −0.669759 −0.669759
\(834\) 0 0
\(835\) −3.53279 −3.53279
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −1.15336 −1.15336 −0.576680 0.816970i \(-0.695652\pi\)
−0.576680 + 0.816970i \(0.695652\pi\)
\(840\) 0 0
\(841\) 1.00000 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.92583 1.92583
\(846\) −0.806244 −0.806244
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) −0.738248 −0.738248
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −1.27556 −1.27556
\(857\) −1.55142 −1.55142 −0.775711 0.631088i \(-0.782609\pi\)
−0.775711 + 0.631088i \(0.782609\pi\)
\(858\) 0 0
\(859\) −1.55142 −1.55142 −0.775711 0.631088i \(-0.782609\pi\)
−0.775711 + 0.631088i \(0.782609\pi\)
\(860\) −2.19367 −2.19367
\(861\) 0 0
\(862\) −0.746449 −0.746449
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −0.669759 −0.669759 −0.334880 0.942261i \(-0.608696\pi\)
−0.334880 + 0.942261i \(0.608696\pi\)
\(878\) 0.165577 0.165577
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0.406912 0.406912
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1.92583 1.92583 0.962917 0.269797i \(-0.0869565\pi\)
0.962917 + 0.269797i \(0.0869565\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 1.06976 1.06976
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −0.262847 −0.262847
\(896\) 0 0
\(897\) 0 0
\(898\) −0.631293 −0.631293
\(899\) 0 0
\(900\) −2.26032 −2.26032
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) −0.686830 −0.686830
\(905\) −3.53279 −3.53279
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 1.53068 1.53068
\(909\) 0.920130 0.920130
\(910\) 0 0
\(911\) −1.98137 −1.98137 −0.990686 0.136167i \(-0.956522\pi\)
−0.990686 + 0.136167i \(0.956522\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0.695347 0.695347
\(923\) 0 0
\(924\) 0 0
\(925\) −4.20255 −4.20255
\(926\) 0 0
\(927\) −1.15336 −1.15336
\(928\) 0 0
\(929\) 0.406912 0.406912 0.203456 0.979084i \(-0.434783\pi\)
0.203456 + 0.979084i \(0.434783\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −1.83442 −1.83442 −0.917211 0.398401i \(-0.869565\pi\)
−0.917211 + 0.398401i \(0.869565\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 3.18399 3.18399
\(941\) −0.136485 −0.136485 −0.0682424 0.997669i \(-0.521739\pi\)
−0.0682424 + 0.997669i \(0.521739\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) −1.28985 −1.28985
\(956\) −0.767777 −0.767777
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 1.00000 1.00000
\(962\) 0 0
\(963\) 1.70884 1.70884
\(964\) 0.962390 0.962390
\(965\) −3.81579 −3.81579
\(966\) 0 0
\(967\) −0.136485 −0.136485 −0.0682424 0.997669i \(-0.521739\pi\)
−0.0682424 + 0.997669i \(0.521739\pi\)
\(968\) −0.746449 −0.746449
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0.165577 0.165577
\(975\) 0 0
\(976\) −1.05148 −1.05148
\(977\) −1.15336 −1.15336 −0.576680 0.816970i \(-0.695652\pi\)
−0.576680 + 0.816970i \(0.695652\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −1.60696 −1.60696
\(981\) 0 0
\(982\) 0.695347 0.695347
\(983\) 1.70884 1.70884 0.854419 0.519584i \(-0.173913\pi\)
0.854419 + 0.519584i \(0.173913\pi\)
\(984\) 0 0
\(985\) 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
\(993\) 0 0
\(994\) 0 0
\(995\) 3.29094 3.29094
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) −0.806244 −0.806244
\(999\) 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 2671.1.b.a.2670.7 11
2671.2670 odd 2 CM 2671.1.b.a.2670.7 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2671.1.b.a.2670.7 11 1.1 even 1 trivial
2671.1.b.a.2670.7 11 2671.2670 odd 2 CM