Properties

Label 2144.1.bu.a.1999.1
Level $2144$
Weight $1$
Character 2144.1999
Analytic conductor $1.070$
Analytic rank $0$
Dimension $20$
Projective image $D_{33}$
CM discriminant -8
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2144,1,Mod(47,2144)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2144, base_ring=CyclotomicField(66))
 
chi = DirichletCharacter(H, H._module([33, 33, 50]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2144.47");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2144 = 2^{5} \cdot 67 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2144.bu (of order \(66\), degree \(20\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.06999538709\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\Q(\zeta_{33})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 536)
Projective image: \(D_{33}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{33} - \cdots)\)

Embedding invariants

Embedding label 1999.1
Root \(-0.327068 + 0.945001i\) of defining polynomial
Character \(\chi\) \(=\) 2144.1999
Dual form 2144.1.bu.a.207.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.0395325 - 0.0865641i) q^{3} +(0.648930 - 0.748905i) q^{9} +O(q^{10})\) \(q+(-0.0395325 - 0.0865641i) q^{3} +(0.648930 - 0.748905i) q^{9} +(-1.21769 - 1.16106i) q^{11} +(-1.45949 - 1.14776i) q^{17} +(-0.462997 + 0.0892353i) q^{19} +(0.841254 - 0.540641i) q^{25} +(-0.181791 - 0.0533787i) q^{27} +(-0.0523681 + 0.151308i) q^{33} +(-1.21590 + 0.486774i) q^{41} +(0.205996 - 1.43273i) q^{43} +(0.928368 + 0.371662i) q^{49} +(-0.0416572 + 0.171713i) q^{51} +(0.0260280 + 0.0365512i) q^{57} +(1.49547 + 0.961081i) q^{59} +(0.959493 - 0.281733i) q^{67} +(-1.13779 + 1.08488i) q^{73} +(-0.0800569 - 0.0514495i) q^{75} +(-0.138460 - 0.963011i) q^{81} +(0.452418 - 1.86489i) q^{83} +(-0.271738 + 0.595023i) q^{89} +(0.786053 - 1.36148i) q^{97} +(-1.65972 + 0.158484i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 2 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 2 q^{3} - 2 q^{11} - 12 q^{17} + q^{19} - 2 q^{25} + 2 q^{27} - 2 q^{33} + 2 q^{41} - 2 q^{43} + q^{49} - q^{51} + q^{57} + 9 q^{59} + 2 q^{67} - q^{73} + 9 q^{75} + 2 q^{81} + 9 q^{83} + 2 q^{89} - q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2144\mathbb{Z}\right)^\times\).

\(n\) \(671\) \(805\) \(1409\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{13}{33}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.0395325 0.0865641i −0.0395325 0.0865641i 0.888835 0.458227i \(-0.151515\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(4\) 0 0
\(5\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(6\) 0 0
\(7\) 0 0 −0.981929 0.189251i \(-0.939394\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(8\) 0 0
\(9\) 0.648930 0.748905i 0.648930 0.748905i
\(10\) 0 0
\(11\) −1.21769 1.16106i −1.21769 1.16106i −0.981929 0.189251i \(-0.939394\pi\)
−0.235759 0.971812i \(-0.575758\pi\)
\(12\) 0 0
\(13\) 0 0 −0.0475819 0.998867i \(-0.515152\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.45949 1.14776i −1.45949 1.14776i −0.959493 0.281733i \(-0.909091\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(18\) 0 0
\(19\) −0.462997 + 0.0892353i −0.462997 + 0.0892353i −0.415415 0.909632i \(-0.636364\pi\)
−0.0475819 + 0.998867i \(0.515152\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.995472 0.0950560i \(-0.969697\pi\)
0.995472 + 0.0950560i \(0.0303030\pi\)
\(24\) 0 0
\(25\) 0.841254 0.540641i 0.841254 0.540641i
\(26\) 0 0
\(27\) −0.181791 0.0533787i −0.181791 0.0533787i
\(28\) 0 0
\(29\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(30\) 0 0
\(31\) 0 0 0.0475819 0.998867i \(-0.484848\pi\)
−0.0475819 + 0.998867i \(0.515152\pi\)
\(32\) 0 0
\(33\) −0.0523681 + 0.151308i −0.0523681 + 0.151308i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.21590 + 0.486774i −1.21590 + 0.486774i −0.888835 0.458227i \(-0.848485\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(42\) 0 0
\(43\) 0.205996 1.43273i 0.205996 1.43273i −0.580057 0.814576i \(-0.696970\pi\)
0.786053 0.618159i \(-0.212121\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.580057 0.814576i \(-0.303030\pi\)
−0.580057 + 0.814576i \(0.696970\pi\)
\(48\) 0 0
\(49\) 0.928368 + 0.371662i 0.928368 + 0.371662i
\(50\) 0 0
\(51\) −0.0416572 + 0.171713i −0.0416572 + 0.171713i
\(52\) 0 0
\(53\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0.0260280 + 0.0365512i 0.0260280 + 0.0365512i
\(58\) 0 0
\(59\) 1.49547 + 0.961081i 1.49547 + 0.961081i 0.995472 + 0.0950560i \(0.0303030\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(60\) 0 0
\(61\) 0 0 0.723734 0.690079i \(-0.242424\pi\)
−0.723734 + 0.690079i \(0.757576\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0.959493 0.281733i 0.959493 0.281733i
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.786053 0.618159i \(-0.212121\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(72\) 0 0
\(73\) −1.13779 + 1.08488i −1.13779 + 1.08488i −0.142315 + 0.989821i \(0.545455\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(74\) 0 0
\(75\) −0.0800569 0.0514495i −0.0800569 0.0514495i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.888835 0.458227i \(-0.848485\pi\)
0.888835 + 0.458227i \(0.151515\pi\)
\(80\) 0 0
\(81\) −0.138460 0.963011i −0.138460 0.963011i
\(82\) 0 0
\(83\) 0.452418 1.86489i 0.452418 1.86489i −0.0475819 0.998867i \(-0.515152\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −0.271738 + 0.595023i −0.271738 + 0.595023i −0.995472 0.0950560i \(-0.969697\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.786053 1.36148i 0.786053 1.36148i −0.142315 0.989821i \(-0.545455\pi\)
0.928368 0.371662i \(-0.121212\pi\)
\(98\) 0 0
\(99\) −1.65972 + 0.158484i −1.65972 + 0.158484i
\(100\) 0 0
\(101\) 0 0 0.327068 0.945001i \(-0.393939\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(102\) 0 0
\(103\) 0 0 0.0475819 0.998867i \(-0.484848\pi\)
−0.0475819 + 0.998867i \(0.515152\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0.797176 + 0.234072i 0.797176 + 0.234072i 0.654861 0.755750i \(-0.272727\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(108\) 0 0
\(109\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −0.419102 1.72756i −0.419102 1.72756i −0.654861 0.755750i \(-0.727273\pi\)
0.235759 0.971812i \(-0.424242\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0.0871144 + 1.82876i 0.0871144 + 1.82876i
\(122\) 0 0
\(123\) 0.0902048 + 0.0860101i 0.0902048 + 0.0860101i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 −0.981929 0.189251i \(-0.939394\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(128\) 0 0
\(129\) −0.132167 + 0.0388077i −0.132167 + 0.0388077i
\(130\) 0 0
\(131\) 0.544078 + 1.19136i 0.544078 + 1.19136i 0.959493 + 0.281733i \(0.0909091\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0.481929 + 1.05528i 0.481929 + 1.05528i 0.981929 + 0.189251i \(0.0606061\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(138\) 0 0
\(139\) −0.959493 + 0.281733i −0.959493 + 0.281733i −0.723734 0.690079i \(-0.757576\pi\)
−0.235759 + 0.971812i \(0.575758\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −0.00452808 0.0950560i −0.00452808 0.0950560i
\(148\) 0 0
\(149\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(150\) 0 0
\(151\) 0 0 −0.786053 0.618159i \(-0.787879\pi\)
0.786053 + 0.618159i \(0.212121\pi\)
\(152\) 0 0
\(153\) −1.80667 + 0.348207i −1.80667 + 0.348207i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 −0.995472 0.0950560i \(-0.969697\pi\)
0.995472 + 0.0950560i \(0.0303030\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0.981929 + 1.70075i 0.981929 + 1.70075i 0.654861 + 0.755750i \(0.272727\pi\)
0.327068 + 0.945001i \(0.393939\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.327068 0.945001i \(-0.393939\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(168\) 0 0
\(169\) −0.995472 + 0.0950560i −0.995472 + 0.0950560i
\(170\) 0 0
\(171\) −0.233624 + 0.404648i −0.233624 + 0.404648i
\(172\) 0 0
\(173\) 0 0 0.888835 0.458227i \(-0.151515\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0.0240754 0.167448i 0.0240754 0.167448i
\(178\) 0 0
\(179\) 0.797176 1.74557i 0.797176 1.74557i 0.142315 0.989821i \(-0.454545\pi\)
0.654861 0.755750i \(-0.272727\pi\)
\(180\) 0 0
\(181\) 0 0 0.580057 0.814576i \(-0.303030\pi\)
−0.580057 + 0.814576i \(0.696970\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0.444587 + 3.09217i 0.444587 + 3.09217i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 −0.580057 0.814576i \(-0.696970\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(192\) 0 0
\(193\) −0.239446 0.153882i −0.239446 0.153882i 0.415415 0.909632i \(-0.363636\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 0.786053 0.618159i \(-0.212121\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(198\) 0 0
\(199\) 0 0 −0.327068 0.945001i \(-0.606061\pi\)
0.327068 + 0.945001i \(0.393939\pi\)
\(200\) 0 0
\(201\) −0.0623191 0.0719200i −0.0623191 0.0719200i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0.667393 + 0.428908i 0.667393 + 0.428908i
\(210\) 0 0
\(211\) 0.379436 + 0.532843i 0.379436 + 0.532843i 0.959493 0.281733i \(-0.0909091\pi\)
−0.580057 + 0.814576i \(0.696970\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0.138891 + 0.0556035i 0.138891 + 0.0556035i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(224\) 0 0
\(225\) 0.141026 0.980857i 0.141026 0.980857i
\(226\) 0 0
\(227\) −1.82318 + 0.729892i −1.82318 + 0.729892i −0.841254 + 0.540641i \(0.818182\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(228\) 0 0
\(229\) 0 0 0.888835 0.458227i \(-0.151515\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0.995472 0.0950560i 0.995472 0.0950560i 0.415415 0.909632i \(-0.363636\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(240\) 0 0
\(241\) 1.84125 + 0.540641i 1.84125 + 0.540641i 1.00000 \(0\)
0.841254 + 0.540641i \(0.181818\pi\)
\(242\) 0 0
\(243\) −0.237277 + 0.152489i −0.237277 + 0.152489i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −0.179318 + 0.0345607i −0.179318 + 0.0345607i
\(250\) 0 0
\(251\) 1.54370 + 1.21398i 1.54370 + 1.21398i 0.888835 + 0.458227i \(0.151515\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −0.205996 0.196417i −0.205996 0.196417i 0.580057 0.814576i \(-0.303030\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0.0622501 0.0622501
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.65210 0.318417i −1.65210 0.318417i
\(276\) 0 0
\(277\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0.0883470 + 1.85463i 0.0883470 + 1.85463i 0.415415 + 0.909632i \(0.363636\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(282\) 0 0
\(283\) −0.654861 0.755750i −0.654861 0.755750i 0.327068 0.945001i \(-0.393939\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 0.577012 + 2.37848i 0.577012 + 2.37848i
\(290\) 0 0
\(291\) −0.148930 0.0142211i −0.148930 0.0142211i
\(292\) 0 0
\(293\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0.159389 + 0.276070i 0.159389 + 0.276070i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 1.28656 0.663268i 1.28656 0.663268i 0.327068 0.945001i \(-0.393939\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(312\) 0 0
\(313\) 0.195876 0.428908i 0.195876 0.428908i −0.786053 0.618159i \(-0.787879\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 −0.928368 0.371662i \(-0.878788\pi\)
0.928368 + 0.371662i \(0.121212\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −0.0112521 0.0782602i −0.0112521 0.0782602i
\(322\) 0 0
\(323\) 0.778161 + 0.401170i 0.778161 + 0.401170i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0.370638 0.291473i 0.370638 0.291473i −0.415415 0.909632i \(-0.636364\pi\)
0.786053 + 0.618159i \(0.212121\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0.0930932 + 0.268975i 0.0930932 + 0.268975i 0.981929 0.189251i \(-0.0606061\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(338\) 0 0
\(339\) −0.132977 + 0.104574i −0.132977 + 0.104574i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −0.888835 0.458227i −0.888835 0.458227i −0.0475819 0.998867i \(-0.515152\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(348\) 0 0
\(349\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0.771316 + 0.308788i 0.771316 + 0.308788i 0.723734 0.690079i \(-0.242424\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(360\) 0 0
\(361\) −0.721965 + 0.289031i −0.721965 + 0.289031i
\(362\) 0 0
\(363\) 0.154861 0.0798363i 0.154861 0.0798363i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 0.995472 0.0950560i \(-0.0303030\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(368\) 0 0
\(369\) −0.424489 + 1.22648i −0.424489 + 1.22648i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −1.30379 0.124497i −1.30379 0.124497i −0.580057 0.814576i \(-0.696970\pi\)
−0.723734 + 0.690079i \(0.757576\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.981929 0.189251i \(-0.0606061\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −0.939306 1.08402i −0.939306 1.08402i
\(388\) 0 0
\(389\) 0 0 −0.0475819 0.998867i \(-0.515152\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0.0816206 0.0941952i 0.0816206 0.0941952i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −0.284630 −0.284630 −0.142315 0.989821i \(-0.545455\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0.815816 + 0.157236i 0.815816 + 0.157236i 0.580057 0.814576i \(-0.303030\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(410\) 0 0
\(411\) 0.0722972 0.0834354i 0.0722972 0.0834354i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0.0623191 + 0.0719200i 0.0623191 + 0.0719200i
\(418\) 0 0
\(419\) −1.23576 0.971812i −1.23576 0.971812i −0.235759 0.971812i \(-0.575758\pi\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 0 0 0.981929 0.189251i \(-0.0606061\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1.84833 0.176494i −1.84833 0.176494i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(432\) 0 0
\(433\) 0.0552004 1.15880i 0.0552004 1.15880i −0.786053 0.618159i \(-0.787879\pi\)
0.841254 0.540641i \(-0.181818\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(440\) 0 0
\(441\) 0.880786 0.454077i 0.880786 0.454077i
\(442\) 0 0
\(443\) −1.72373 + 0.690079i −1.72373 + 0.690079i −0.723734 + 0.690079i \(0.757576\pi\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0.975950 1.37053i 0.975950 1.37053i 0.0475819 0.998867i \(-0.484848\pi\)
0.928368 0.371662i \(-0.121212\pi\)
\(450\) 0 0
\(451\) 2.04577 + 0.819001i 2.04577 + 0.819001i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0.581419 + 0.299742i 0.581419 + 0.299742i 0.723734 0.690079i \(-0.242424\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(458\) 0 0
\(459\) 0.204057 + 0.286558i 0.204057 + 0.286558i
\(460\) 0 0
\(461\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(462\) 0 0
\(463\) 0 0 0.723734 0.690079i \(-0.242424\pi\)
−0.723734 + 0.690079i \(0.757576\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0.550294 + 1.58997i 0.550294 + 1.58997i 0.786053 + 0.618159i \(0.212121\pi\)
−0.235759 + 0.971812i \(0.575758\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −1.91433 + 1.50545i −1.91433 + 1.50545i
\(474\) 0 0
\(475\) −0.341254 + 0.325385i −0.341254 + 0.325385i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.580057 0.814576i \(-0.696970\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 −0.928368 0.371662i \(-0.878788\pi\)
0.928368 + 0.371662i \(0.121212\pi\)
\(488\) 0 0
\(489\) 0.108406 0.152235i 0.108406 0.152235i
\(490\) 0 0
\(491\) 0.415415 0.909632i 0.415415 0.909632i −0.580057 0.814576i \(-0.696970\pi\)
0.995472 0.0950560i \(-0.0303030\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −0.995472 + 1.72421i −0.995472 + 1.72421i −0.415415 + 0.909632i \(0.636364\pi\)
−0.580057 + 0.814576i \(0.696970\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.327068 0.945001i \(-0.393939\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0.0475819 + 0.0824143i 0.0475819 + 0.0824143i
\(508\) 0 0
\(509\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0.0889320 + 0.00849198i 0.0889320 + 0.00849198i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.30379 + 1.50465i 1.30379 + 1.50465i 0.723734 + 0.690079i \(0.242424\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(522\) 0 0
\(523\) −0.0552004 1.15880i −0.0552004 1.15880i −0.841254 0.540641i \(-0.818182\pi\)
0.786053 0.618159i \(-0.212121\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 0.981929 + 0.189251i 0.981929 + 0.189251i
\(530\) 0 0
\(531\) 1.69022 0.496292i 1.69022 0.496292i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −0.182618 −0.182618
\(538\) 0 0
\(539\) −0.698939 1.53046i −0.698939 1.53046i
\(540\) 0 0
\(541\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1.38884 + 1.32425i 1.38884 + 1.32425i 0.888835 + 0.458227i \(0.151515\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 −0.235759 0.971812i \(-0.575758\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0.250095 0.160727i 0.250095 0.160727i
\(562\) 0 0
\(563\) −1.91030 0.560914i −1.91030 0.560914i −0.981929 0.189251i \(-0.939394\pi\)
−0.928368 0.371662i \(-0.878788\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0.514186 1.48564i 0.514186 1.48564i −0.327068 0.945001i \(-0.606061\pi\)
0.841254 0.540641i \(-0.181818\pi\)
\(570\) 0 0
\(571\) 1.84833 0.176494i 1.84833 0.176494i 0.888835 0.458227i \(-0.151515\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −0.607279 + 0.243118i −0.607279 + 0.243118i −0.654861 0.755750i \(-0.727273\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(578\) 0 0
\(579\) −0.00385480 + 0.0268107i −0.00385480 + 0.0268107i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0.370638 1.52779i 0.370638 1.52779i −0.415415 0.909632i \(-0.636364\pi\)
0.786053 0.618159i \(-0.212121\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −0.911911 1.28060i −0.911911 1.28060i −0.959493 0.281733i \(-0.909091\pi\)
0.0475819 0.998867i \(-0.484848\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 0.786053 0.618159i \(-0.212121\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(600\) 0 0
\(601\) −0.642315 1.85585i −0.642315 1.85585i −0.500000 0.866025i \(-0.666667\pi\)
−0.142315 0.989821i \(-0.545455\pi\)
\(602\) 0 0
\(603\) 0.411653 0.901394i 0.411653 0.901394i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.786053 0.618159i \(-0.212121\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 −0.580057 0.814576i \(-0.696970\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0.186393 + 1.29639i 0.186393 + 1.29639i 0.841254 + 0.540641i \(0.181818\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(618\) 0 0
\(619\) −0.273507 + 1.12741i −0.273507 + 1.12741i 0.654861 + 0.755750i \(0.272727\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0.415415 0.909632i 0.415415 0.909632i
\(626\) 0 0
\(627\) 0.0107443 0.0747281i 0.0107443 0.0747281i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 0.888835 0.458227i \(-0.151515\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(632\) 0 0
\(633\) 0.0311250 0.0539102i 0.0311250 0.0539102i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −0.981929 1.70075i −0.981929 1.70075i −0.654861 0.755750i \(-0.727273\pi\)
−0.327068 0.945001i \(-0.606061\pi\)
\(642\) 0 0
\(643\) −1.70566 0.500828i −1.70566 0.500828i −0.723734 0.690079i \(-0.757576\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.995472 0.0950560i \(-0.969697\pi\)
0.995472 + 0.0950560i \(0.0303030\pi\)
\(648\) 0 0
\(649\) −0.705142 2.90663i −0.705142 2.90663i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 −0.786053 0.618159i \(-0.787879\pi\)
0.786053 + 0.618159i \(0.212121\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0.0741264 + 1.55610i 0.0741264 + 1.55610i
\(658\) 0 0
\(659\) 0.723734 + 0.690079i 0.723734 + 0.690079i 0.959493 0.281733i \(-0.0909091\pi\)
−0.235759 + 0.971812i \(0.575758\pi\)
\(660\) 0 0
\(661\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −0.827068 1.81103i −0.827068 1.81103i −0.500000 0.866025i \(-0.666667\pi\)
−0.327068 0.945001i \(-0.606061\pi\)
\(674\) 0 0
\(675\) −0.181791 + 0.0533787i −0.181791 + 0.0533787i
\(676\) 0 0
\(677\) 0 0 −0.981929 0.189251i \(-0.939394\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0.135257 + 0.128968i 0.135257 + 0.128968i
\(682\) 0 0
\(683\) −0.0552004 1.15880i −0.0552004 1.15880i −0.841254 0.540641i \(-0.818182\pi\)
0.786053 0.618159i \(-0.212121\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −0.195876 0.807410i −0.195876 0.807410i −0.981929 0.189251i \(-0.939394\pi\)
0.786053 0.618159i \(-0.212121\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 2.33330 + 0.685119i 2.33330 + 0.685119i
\(698\) 0 0
\(699\) −0.0475819 0.0824143i −0.0475819 0.0824143i
\(700\) 0 0
\(701\) 0 0 0.0475819 0.998867i \(-0.484848\pi\)
−0.0475819 + 0.998867i \(0.515152\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 0.888835 0.458227i \(-0.151515\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.928368 0.371662i \(-0.878788\pi\)
0.928368 + 0.371662i \(0.121212\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −0.0259893 0.180759i −0.0259893 0.180759i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 −0.580057 0.814576i \(-0.696970\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(728\) 0 0
\(729\) −0.795887 0.511485i −0.795887 0.511485i
\(730\) 0 0
\(731\) −1.94508 + 1.85463i −1.94508 + 1.85463i
\(732\) 0 0
\(733\) 0 0 0.786053 0.618159i \(-0.212121\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1.49547 0.770969i −1.49547 0.770969i
\(738\) 0 0
\(739\) −0.581419 1.67990i −0.581419 1.67990i −0.723734 0.690079i \(-0.757576\pi\)
0.142315 0.989821i \(-0.454545\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.723734 0.690079i \(-0.242424\pi\)
−0.723734 + 0.690079i \(0.757576\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −1.10304 1.54900i −1.10304 1.54900i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(752\) 0 0
\(753\) 0.0440606 0.181620i 0.0440606 0.181620i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 0.580057 0.814576i \(-0.303030\pi\)
−0.580057 + 0.814576i \(0.696970\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −0.118239 + 0.822373i −0.118239 + 0.822373i 0.841254 + 0.540641i \(0.181818\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −1.95496 + 0.186677i −1.95496 + 0.186677i −0.995472 0.0950560i \(-0.969697\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(770\) 0 0
\(771\) −0.00885911 + 0.0255967i −0.00885911 + 0.0255967i
\(772\) 0 0
\(773\) 0 0 0.0475819 0.998867i \(-0.484848\pi\)
−0.0475819 + 0.998867i \(0.515152\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0.519522 0.333877i 0.519522 0.333877i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0.653077 + 0.513585i 0.653077 + 0.513585i 0.888835 0.458227i \(-0.151515\pi\)
−0.235759 + 0.971812i \(0.575758\pi\)
\(788\) 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
\(797\) 0 0 −0.981929 0.189251i \(-0.939394\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0.269277 + 0.589634i 0.269277 + 0.589634i
\(802\) 0 0
\(803\) 2.64508 2.64508
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.84125 0.540641i 1.84125 0.540641i 0.841254 0.540641i \(-0.181818\pi\)
1.00000 \(0\)
\(810\) 0 0
\(811\) 1.88431 + 0.363170i 1.88431 + 0.363170i 0.995472 0.0950560i \(-0.0303030\pi\)
0.888835 + 0.458227i \(0.151515\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0.0324750 + 0.681734i 0.0324750 + 0.681734i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 −0.786053 0.618159i \(-0.787879\pi\)
0.786053 + 0.618159i \(0.212121\pi\)
\(822\) 0 0
\(823\) 0 0 0.981929 0.189251i \(-0.0606061\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(824\) 0 0
\(825\) 0.0377483 + 0.155600i 0.0377483 + 0.155600i
\(826\) 0 0
\(827\) 1.67489 + 0.159932i 1.67489 + 0.159932i 0.888835 0.458227i \(-0.151515\pi\)
0.786053 + 0.618159i \(0.212121\pi\)
\(828\) 0 0
\(829\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −0.928368 1.60798i −0.928368 1.60798i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 0.995472 0.0950560i \(-0.0303030\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(840\) 0 0
\(841\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(842\) 0 0
\(843\) 0.157052 0.0809659i 0.157052 0.0809659i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −0.0395325 + 0.0865641i −0.0395325 + 0.0865641i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 −0.928368 0.371662i \(-0.878788\pi\)
0.928368 + 0.371662i \(0.121212\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0.0405070 + 0.281733i 0.0405070 + 0.281733i 1.00000 \(0\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(858\) 0 0
\(859\) 1.65033 + 0.850806i 1.65033 + 0.850806i 0.995472 + 0.0950560i \(0.0303030\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0.183080 0.143976i 0.183080 0.143976i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −0.509529 1.47219i −0.509529 1.47219i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 0.723734 0.690079i \(-0.242424\pi\)
−0.723734 + 0.690079i \(0.757576\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0.975950 + 1.37053i 0.975950 + 1.37053i 0.928368 + 0.371662i \(0.121212\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(882\) 0 0
\(883\) −1.16413 0.600149i −1.16413 0.600149i −0.235759 0.971812i \(-0.575758\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.235759 0.971812i \(-0.424242\pi\)
−0.235759 + 0.971812i \(0.575758\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −0.949515 + 1.33341i −0.949515 + 1.33341i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0.0135432 0.284307i 0.0135432 0.284307i −0.981929 0.189251i \(-0.939394\pi\)
0.995472 0.0950560i \(-0.0303030\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(912\) 0 0
\(913\) −2.71616 + 1.74557i −2.71616 + 1.74557i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 0.981929 0.189251i \(-0.0606061\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(920\) 0 0
\(921\) −0.108276 0.0851493i −0.108276 0.0851493i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1.30379 1.50465i 1.30379 1.50465i 0.580057 0.814576i \(-0.303030\pi\)
0.723734 0.690079i \(-0.242424\pi\)
\(930\) 0 0
\(931\) −0.462997 0.0892353i −0.462997 0.0892353i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0.830830 0.830830 0.415415 0.909632i \(-0.363636\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(938\) 0 0
\(939\) −0.0448714 −0.0448714
\(940\) 0 0
\(941\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −0.428368 + 0.494363i −0.428368 + 0.494363i −0.928368 0.371662i \(-0.878788\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −0.947890 1.09392i −0.947890 1.09392i −0.995472 0.0950560i \(-0.969697\pi\)
0.0475819 0.998867i \(-0.484848\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −0.995472 0.0950560i −0.995472 0.0950560i
\(962\) 0 0
\(963\) 0.692609 0.445113i 0.692609 0.445113i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(968\) 0 0
\(969\) 0.00396426 0.0832201i 0.00396426 0.0832201i
\(970\) 0 0
\(971\) 0.473420 1.36786i 0.473420 1.36786i −0.415415 0.909632i \(-0.636364\pi\)
0.888835 0.458227i \(-0.151515\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.70566 0.879330i 1.70566 0.879330i 0.723734 0.690079i \(-0.242424\pi\)
0.981929 0.189251i \(-0.0606061\pi\)
\(978\) 0 0
\(979\) 1.02175 0.409047i 1.02175 0.409047i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(992\) 0 0
\(993\) −0.0398833 0.0205613i −0.0398833 0.0205613i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(998\) 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 2144.1.bu.a.1999.1 20
4.3 odd 2 536.1.ba.a.123.1 20
8.3 odd 2 CM 2144.1.bu.a.1999.1 20
8.5 even 2 536.1.ba.a.123.1 20
67.6 even 33 inner 2144.1.bu.a.207.1 20
268.207 odd 66 536.1.ba.a.475.1 yes 20
536.341 even 66 536.1.ba.a.475.1 yes 20
536.475 odd 66 inner 2144.1.bu.a.207.1 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
536.1.ba.a.123.1 20 4.3 odd 2
536.1.ba.a.123.1 20 8.5 even 2
536.1.ba.a.475.1 yes 20 268.207 odd 66
536.1.ba.a.475.1 yes 20 536.341 even 66
2144.1.bu.a.207.1 20 67.6 even 33 inner
2144.1.bu.a.207.1 20 536.475 odd 66 inner
2144.1.bu.a.1999.1 20 1.1 even 1 trivial
2144.1.bu.a.1999.1 20 8.3 odd 2 CM