Properties

Label 8001.2.a.i.1.2
Level $8001$
Weight $2$
Character 8001.1
Self dual yes
Analytic conductor $63.888$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8001,2,Mod(1,8001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8001, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8001.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.a (trivial)

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: \(63.8883066572\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2667)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 8001.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{4} +3.56155 q^{5} -1.00000 q^{7} +O(q^{10})\) \(q-2.00000 q^{4} +3.56155 q^{5} -1.00000 q^{7} -3.12311 q^{11} -3.56155 q^{13} +4.00000 q^{16} -6.56155 q^{17} +4.00000 q^{19} -7.12311 q^{20} +4.68466 q^{23} +7.68466 q^{25} +2.00000 q^{28} -4.12311 q^{29} +4.68466 q^{31} -3.56155 q^{35} -1.87689 q^{37} +10.8078 q^{41} -4.24621 q^{43} +6.24621 q^{44} +4.87689 q^{47} +1.00000 q^{49} +7.12311 q^{52} +11.0000 q^{53} -11.1231 q^{55} +4.43845 q^{59} -9.56155 q^{61} -8.00000 q^{64} -12.6847 q^{65} -2.00000 q^{67} +13.1231 q^{68} +13.1231 q^{71} -4.43845 q^{73} -8.00000 q^{76} +3.12311 q^{77} -0.315342 q^{79} +14.2462 q^{80} -17.8078 q^{83} -23.3693 q^{85} -7.80776 q^{89} +3.56155 q^{91} -9.36932 q^{92} +14.2462 q^{95} -8.80776 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} + 3 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{4} + 3 q^{5} - 2 q^{7} + 2 q^{11} - 3 q^{13} + 8 q^{16} - 9 q^{17} + 8 q^{19} - 6 q^{20} - 3 q^{23} + 3 q^{25} + 4 q^{28} - 3 q^{31} - 3 q^{35} - 12 q^{37} + q^{41} + 8 q^{43} - 4 q^{44} + 18 q^{47} + 2 q^{49} + 6 q^{52} + 22 q^{53} - 14 q^{55} + 13 q^{59} - 15 q^{61} - 16 q^{64} - 13 q^{65} - 4 q^{67} + 18 q^{68} + 18 q^{71} - 13 q^{73} - 16 q^{76} - 2 q^{77} - 13 q^{79} + 12 q^{80} - 15 q^{83} - 22 q^{85} + 5 q^{89} + 3 q^{91} + 6 q^{92} + 12 q^{95} + 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 0 0
\(4\) −2.00000 −1.00000
\(5\) 3.56155 1.59277 0.796387 0.604787i \(-0.206742\pi\)
0.796387 + 0.604787i \(0.206742\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.12311 −0.941652 −0.470826 0.882226i \(-0.656044\pi\)
−0.470826 + 0.882226i \(0.656044\pi\)
\(12\) 0 0
\(13\) −3.56155 −0.987797 −0.493899 0.869520i \(-0.664429\pi\)
−0.493899 + 0.869520i \(0.664429\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) −6.56155 −1.59141 −0.795705 0.605684i \(-0.792900\pi\)
−0.795705 + 0.605684i \(0.792900\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) −7.12311 −1.59277
\(21\) 0 0
\(22\) 0 0
\(23\) 4.68466 0.976819 0.488409 0.872615i \(-0.337577\pi\)
0.488409 + 0.872615i \(0.337577\pi\)
\(24\) 0 0
\(25\) 7.68466 1.53693
\(26\) 0 0
\(27\) 0 0
\(28\) 2.00000 0.377964
\(29\) −4.12311 −0.765641 −0.382821 0.923823i \(-0.625047\pi\)
−0.382821 + 0.923823i \(0.625047\pi\)
\(30\) 0 0
\(31\) 4.68466 0.841389 0.420695 0.907202i \(-0.361786\pi\)
0.420695 + 0.907202i \(0.361786\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.56155 −0.602012
\(36\) 0 0
\(37\) −1.87689 −0.308560 −0.154280 0.988027i \(-0.549306\pi\)
−0.154280 + 0.988027i \(0.549306\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 10.8078 1.68789 0.843945 0.536430i \(-0.180228\pi\)
0.843945 + 0.536430i \(0.180228\pi\)
\(42\) 0 0
\(43\) −4.24621 −0.647541 −0.323771 0.946136i \(-0.604951\pi\)
−0.323771 + 0.946136i \(0.604951\pi\)
\(44\) 6.24621 0.941652
\(45\) 0 0
\(46\) 0 0
\(47\) 4.87689 0.711368 0.355684 0.934606i \(-0.384248\pi\)
0.355684 + 0.934606i \(0.384248\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 7.12311 0.987797
\(53\) 11.0000 1.51097 0.755483 0.655168i \(-0.227402\pi\)
0.755483 + 0.655168i \(0.227402\pi\)
\(54\) 0 0
\(55\) −11.1231 −1.49984
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.43845 0.577837 0.288918 0.957354i \(-0.406704\pi\)
0.288918 + 0.957354i \(0.406704\pi\)
\(60\) 0 0
\(61\) −9.56155 −1.22423 −0.612116 0.790768i \(-0.709681\pi\)
−0.612116 + 0.790768i \(0.709681\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) −12.6847 −1.57334
\(66\) 0 0
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) 13.1231 1.59141
\(69\) 0 0
\(70\) 0 0
\(71\) 13.1231 1.55743 0.778713 0.627380i \(-0.215873\pi\)
0.778713 + 0.627380i \(0.215873\pi\)
\(72\) 0 0
\(73\) −4.43845 −0.519481 −0.259740 0.965678i \(-0.583637\pi\)
−0.259740 + 0.965678i \(0.583637\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −8.00000 −0.917663
\(77\) 3.12311 0.355911
\(78\) 0 0
\(79\) −0.315342 −0.0354787 −0.0177393 0.999843i \(-0.505647\pi\)
−0.0177393 + 0.999843i \(0.505647\pi\)
\(80\) 14.2462 1.59277
\(81\) 0 0
\(82\) 0 0
\(83\) −17.8078 −1.95466 −0.977328 0.211731i \(-0.932090\pi\)
−0.977328 + 0.211731i \(0.932090\pi\)
\(84\) 0 0
\(85\) −23.3693 −2.53476
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −7.80776 −0.827621 −0.413811 0.910363i \(-0.635802\pi\)
−0.413811 + 0.910363i \(0.635802\pi\)
\(90\) 0 0
\(91\) 3.56155 0.373352
\(92\) −9.36932 −0.976819
\(93\) 0 0
\(94\) 0 0
\(95\) 14.2462 1.46163
\(96\) 0 0
\(97\) −8.80776 −0.894293 −0.447146 0.894461i \(-0.647560\pi\)
−0.447146 + 0.894461i \(0.647560\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −15.3693 −1.53693
\(101\) −19.8078 −1.97095 −0.985473 0.169832i \(-0.945678\pi\)
−0.985473 + 0.169832i \(0.945678\pi\)
\(102\) 0 0
\(103\) −12.0000 −1.18240 −0.591198 0.806527i \(-0.701345\pi\)
−0.591198 + 0.806527i \(0.701345\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −4.00000 −0.377964
\(113\) −2.87689 −0.270635 −0.135318 0.990802i \(-0.543206\pi\)
−0.135318 + 0.990802i \(0.543206\pi\)
\(114\) 0 0
\(115\) 16.6847 1.55585
\(116\) 8.24621 0.765641
\(117\) 0 0
\(118\) 0 0
\(119\) 6.56155 0.601497
\(120\) 0 0
\(121\) −1.24621 −0.113292
\(122\) 0 0
\(123\) 0 0
\(124\) −9.36932 −0.841389
\(125\) 9.56155 0.855211
\(126\) 0 0
\(127\) 1.00000 0.0887357
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −13.6847 −1.19563 −0.597817 0.801633i \(-0.703965\pi\)
−0.597817 + 0.801633i \(0.703965\pi\)
\(132\) 0 0
\(133\) −4.00000 −0.346844
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 19.5616 1.67126 0.835628 0.549296i \(-0.185104\pi\)
0.835628 + 0.549296i \(0.185104\pi\)
\(138\) 0 0
\(139\) 12.5616 1.06546 0.532729 0.846286i \(-0.321167\pi\)
0.532729 + 0.846286i \(0.321167\pi\)
\(140\) 7.12311 0.602012
\(141\) 0 0
\(142\) 0 0
\(143\) 11.1231 0.930161
\(144\) 0 0
\(145\) −14.6847 −1.21949
\(146\) 0 0
\(147\) 0 0
\(148\) 3.75379 0.308560
\(149\) −14.0000 −1.14692 −0.573462 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) 0 0
\(151\) −22.4924 −1.83041 −0.915204 0.402992i \(-0.867970\pi\)
−0.915204 + 0.402992i \(0.867970\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 16.6847 1.34014
\(156\) 0 0
\(157\) 0.246211 0.0196498 0.00982490 0.999952i \(-0.496873\pi\)
0.00982490 + 0.999952i \(0.496873\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −4.68466 −0.369203
\(162\) 0 0
\(163\) −22.6155 −1.77138 −0.885692 0.464272i \(-0.846316\pi\)
−0.885692 + 0.464272i \(0.846316\pi\)
\(164\) −21.6155 −1.68789
\(165\) 0 0
\(166\) 0 0
\(167\) −10.2462 −0.792876 −0.396438 0.918062i \(-0.629754\pi\)
−0.396438 + 0.918062i \(0.629754\pi\)
\(168\) 0 0
\(169\) −0.315342 −0.0242570
\(170\) 0 0
\(171\) 0 0
\(172\) 8.49242 0.647541
\(173\) 1.36932 0.104107 0.0520536 0.998644i \(-0.483423\pi\)
0.0520536 + 0.998644i \(0.483423\pi\)
\(174\) 0 0
\(175\) −7.68466 −0.580906
\(176\) −12.4924 −0.941652
\(177\) 0 0
\(178\) 0 0
\(179\) 2.87689 0.215029 0.107515 0.994204i \(-0.465711\pi\)
0.107515 + 0.994204i \(0.465711\pi\)
\(180\) 0 0
\(181\) −3.19224 −0.237277 −0.118639 0.992938i \(-0.537853\pi\)
−0.118639 + 0.992938i \(0.537853\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −6.68466 −0.491466
\(186\) 0 0
\(187\) 20.4924 1.49855
\(188\) −9.75379 −0.711368
\(189\) 0 0
\(190\) 0 0
\(191\) −19.3693 −1.40151 −0.700757 0.713400i \(-0.747154\pi\)
−0.700757 + 0.713400i \(0.747154\pi\)
\(192\) 0 0
\(193\) 7.12311 0.512732 0.256366 0.966580i \(-0.417475\pi\)
0.256366 + 0.966580i \(0.417475\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −2.00000 −0.142857
\(197\) 27.3693 1.94998 0.974992 0.222242i \(-0.0713375\pi\)
0.974992 + 0.222242i \(0.0713375\pi\)
\(198\) 0 0
\(199\) −9.56155 −0.677801 −0.338900 0.940822i \(-0.610055\pi\)
−0.338900 + 0.940822i \(0.610055\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 4.12311 0.289385
\(204\) 0 0
\(205\) 38.4924 2.68843
\(206\) 0 0
\(207\) 0 0
\(208\) −14.2462 −0.987797
\(209\) −12.4924 −0.864119
\(210\) 0 0
\(211\) −28.6847 −1.97473 −0.987367 0.158452i \(-0.949350\pi\)
−0.987367 + 0.158452i \(0.949350\pi\)
\(212\) −22.0000 −1.51097
\(213\) 0 0
\(214\) 0 0
\(215\) −15.1231 −1.03139
\(216\) 0 0
\(217\) −4.68466 −0.318015
\(218\) 0 0
\(219\) 0 0
\(220\) 22.2462 1.49984
\(221\) 23.3693 1.57199
\(222\) 0 0
\(223\) −10.8078 −0.723741 −0.361871 0.932228i \(-0.617862\pi\)
−0.361871 + 0.932228i \(0.617862\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 13.3693 0.887353 0.443676 0.896187i \(-0.353674\pi\)
0.443676 + 0.896187i \(0.353674\pi\)
\(228\) 0 0
\(229\) −9.43845 −0.623710 −0.311855 0.950130i \(-0.600950\pi\)
−0.311855 + 0.950130i \(0.600950\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −24.3693 −1.59649 −0.798244 0.602335i \(-0.794237\pi\)
−0.798244 + 0.602335i \(0.794237\pi\)
\(234\) 0 0
\(235\) 17.3693 1.13305
\(236\) −8.87689 −0.577837
\(237\) 0 0
\(238\) 0 0
\(239\) −3.63068 −0.234849 −0.117425 0.993082i \(-0.537464\pi\)
−0.117425 + 0.993082i \(0.537464\pi\)
\(240\) 0 0
\(241\) −5.43845 −0.350321 −0.175161 0.984540i \(-0.556044\pi\)
−0.175161 + 0.984540i \(0.556044\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 19.1231 1.22423
\(245\) 3.56155 0.227539
\(246\) 0 0
\(247\) −14.2462 −0.906465
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 22.8078 1.43961 0.719807 0.694175i \(-0.244230\pi\)
0.719807 + 0.694175i \(0.244230\pi\)
\(252\) 0 0
\(253\) −14.6307 −0.919823
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 10.0540 0.627150 0.313575 0.949563i \(-0.398473\pi\)
0.313575 + 0.949563i \(0.398473\pi\)
\(258\) 0 0
\(259\) 1.87689 0.116625
\(260\) 25.3693 1.57334
\(261\) 0 0
\(262\) 0 0
\(263\) 13.6155 0.839569 0.419785 0.907624i \(-0.362106\pi\)
0.419785 + 0.907624i \(0.362106\pi\)
\(264\) 0 0
\(265\) 39.1771 2.40663
\(266\) 0 0
\(267\) 0 0
\(268\) 4.00000 0.244339
\(269\) 4.80776 0.293135 0.146567 0.989201i \(-0.453178\pi\)
0.146567 + 0.989201i \(0.453178\pi\)
\(270\) 0 0
\(271\) 12.9309 0.785494 0.392747 0.919646i \(-0.371525\pi\)
0.392747 + 0.919646i \(0.371525\pi\)
\(272\) −26.2462 −1.59141
\(273\) 0 0
\(274\) 0 0
\(275\) −24.0000 −1.44725
\(276\) 0 0
\(277\) −1.75379 −0.105375 −0.0526875 0.998611i \(-0.516779\pi\)
−0.0526875 + 0.998611i \(0.516779\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −8.56155 −0.510739 −0.255370 0.966843i \(-0.582197\pi\)
−0.255370 + 0.966843i \(0.582197\pi\)
\(282\) 0 0
\(283\) −15.6847 −0.932356 −0.466178 0.884691i \(-0.654369\pi\)
−0.466178 + 0.884691i \(0.654369\pi\)
\(284\) −26.2462 −1.55743
\(285\) 0 0
\(286\) 0 0
\(287\) −10.8078 −0.637962
\(288\) 0 0
\(289\) 26.0540 1.53259
\(290\) 0 0
\(291\) 0 0
\(292\) 8.87689 0.519481
\(293\) 10.0000 0.584206 0.292103 0.956387i \(-0.405645\pi\)
0.292103 + 0.956387i \(0.405645\pi\)
\(294\) 0 0
\(295\) 15.8078 0.920364
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −16.6847 −0.964899
\(300\) 0 0
\(301\) 4.24621 0.244748
\(302\) 0 0
\(303\) 0 0
\(304\) 16.0000 0.917663
\(305\) −34.0540 −1.94992
\(306\) 0 0
\(307\) −34.8078 −1.98658 −0.993292 0.115633i \(-0.963110\pi\)
−0.993292 + 0.115633i \(0.963110\pi\)
\(308\) −6.24621 −0.355911
\(309\) 0 0
\(310\) 0 0
\(311\) −20.0540 −1.13716 −0.568578 0.822629i \(-0.692506\pi\)
−0.568578 + 0.822629i \(0.692506\pi\)
\(312\) 0 0
\(313\) 11.9309 0.674373 0.337186 0.941438i \(-0.390525\pi\)
0.337186 + 0.941438i \(0.390525\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0.630683 0.0354787
\(317\) 7.75379 0.435496 0.217748 0.976005i \(-0.430129\pi\)
0.217748 + 0.976005i \(0.430129\pi\)
\(318\) 0 0
\(319\) 12.8769 0.720968
\(320\) −28.4924 −1.59277
\(321\) 0 0
\(322\) 0 0
\(323\) −26.2462 −1.46038
\(324\) 0 0
\(325\) −27.3693 −1.51818
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −4.87689 −0.268872
\(330\) 0 0
\(331\) −6.24621 −0.343323 −0.171661 0.985156i \(-0.554914\pi\)
−0.171661 + 0.985156i \(0.554914\pi\)
\(332\) 35.6155 1.95466
\(333\) 0 0
\(334\) 0 0
\(335\) −7.12311 −0.389177
\(336\) 0 0
\(337\) 9.75379 0.531323 0.265661 0.964066i \(-0.414410\pi\)
0.265661 + 0.964066i \(0.414410\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 46.7386 2.53476
\(341\) −14.6307 −0.792296
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 25.4924 1.36850 0.684252 0.729245i \(-0.260129\pi\)
0.684252 + 0.729245i \(0.260129\pi\)
\(348\) 0 0
\(349\) 21.0540 1.12699 0.563497 0.826118i \(-0.309456\pi\)
0.563497 + 0.826118i \(0.309456\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 2.87689 0.153122 0.0765608 0.997065i \(-0.475606\pi\)
0.0765608 + 0.997065i \(0.475606\pi\)
\(354\) 0 0
\(355\) 46.7386 2.48063
\(356\) 15.6155 0.827621
\(357\) 0 0
\(358\) 0 0
\(359\) −21.9309 −1.15747 −0.578734 0.815517i \(-0.696453\pi\)
−0.578734 + 0.815517i \(0.696453\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 0 0
\(364\) −7.12311 −0.373352
\(365\) −15.8078 −0.827416
\(366\) 0 0
\(367\) −7.17708 −0.374641 −0.187320 0.982299i \(-0.559980\pi\)
−0.187320 + 0.982299i \(0.559980\pi\)
\(368\) 18.7386 0.976819
\(369\) 0 0
\(370\) 0 0
\(371\) −11.0000 −0.571092
\(372\) 0 0
\(373\) 3.12311 0.161708 0.0808541 0.996726i \(-0.474235\pi\)
0.0808541 + 0.996726i \(0.474235\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 14.6847 0.756298
\(378\) 0 0
\(379\) −27.1231 −1.39322 −0.696610 0.717450i \(-0.745309\pi\)
−0.696610 + 0.717450i \(0.745309\pi\)
\(380\) −28.4924 −1.46163
\(381\) 0 0
\(382\) 0 0
\(383\) 33.9309 1.73379 0.866893 0.498494i \(-0.166113\pi\)
0.866893 + 0.498494i \(0.166113\pi\)
\(384\) 0 0
\(385\) 11.1231 0.566886
\(386\) 0 0
\(387\) 0 0
\(388\) 17.6155 0.894293
\(389\) 16.4924 0.836199 0.418100 0.908401i \(-0.362696\pi\)
0.418100 + 0.908401i \(0.362696\pi\)
\(390\) 0 0
\(391\) −30.7386 −1.55452
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −1.12311 −0.0565096
\(396\) 0 0
\(397\) 38.0540 1.90987 0.954937 0.296808i \(-0.0959222\pi\)
0.954937 + 0.296808i \(0.0959222\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 30.7386 1.53693
\(401\) 14.3693 0.717569 0.358785 0.933420i \(-0.383191\pi\)
0.358785 + 0.933420i \(0.383191\pi\)
\(402\) 0 0
\(403\) −16.6847 −0.831122
\(404\) 39.6155 1.97095
\(405\) 0 0
\(406\) 0 0
\(407\) 5.86174 0.290556
\(408\) 0 0
\(409\) −18.0000 −0.890043 −0.445021 0.895520i \(-0.646804\pi\)
−0.445021 + 0.895520i \(0.646804\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 24.0000 1.18240
\(413\) −4.43845 −0.218402
\(414\) 0 0
\(415\) −63.4233 −3.11333
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −0.876894 −0.0428391 −0.0214195 0.999771i \(-0.506819\pi\)
−0.0214195 + 0.999771i \(0.506819\pi\)
\(420\) 0 0
\(421\) −38.2462 −1.86401 −0.932003 0.362450i \(-0.881940\pi\)
−0.932003 + 0.362450i \(0.881940\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −50.4233 −2.44589
\(426\) 0 0
\(427\) 9.56155 0.462716
\(428\) 8.00000 0.386695
\(429\) 0 0
\(430\) 0 0
\(431\) 18.4924 0.890749 0.445374 0.895344i \(-0.353071\pi\)
0.445374 + 0.895344i \(0.353071\pi\)
\(432\) 0 0
\(433\) −30.4924 −1.46537 −0.732686 0.680567i \(-0.761734\pi\)
−0.732686 + 0.680567i \(0.761734\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −4.00000 −0.191565
\(437\) 18.7386 0.896390
\(438\) 0 0
\(439\) −1.19224 −0.0569023 −0.0284512 0.999595i \(-0.509058\pi\)
−0.0284512 + 0.999595i \(0.509058\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 0 0
\(445\) −27.8078 −1.31821
\(446\) 0 0
\(447\) 0 0
\(448\) 8.00000 0.377964
\(449\) −23.6155 −1.11449 −0.557243 0.830350i \(-0.688141\pi\)
−0.557243 + 0.830350i \(0.688141\pi\)
\(450\) 0 0
\(451\) −33.7538 −1.58940
\(452\) 5.75379 0.270635
\(453\) 0 0
\(454\) 0 0
\(455\) 12.6847 0.594666
\(456\) 0 0
\(457\) 3.00000 0.140334 0.0701670 0.997535i \(-0.477647\pi\)
0.0701670 + 0.997535i \(0.477647\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −33.3693 −1.55585
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) 8.68466 0.403610 0.201805 0.979426i \(-0.435319\pi\)
0.201805 + 0.979426i \(0.435319\pi\)
\(464\) −16.4924 −0.765641
\(465\) 0 0
\(466\) 0 0
\(467\) 6.43845 0.297936 0.148968 0.988842i \(-0.452405\pi\)
0.148968 + 0.988842i \(0.452405\pi\)
\(468\) 0 0
\(469\) 2.00000 0.0923514
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 13.2614 0.609758
\(474\) 0 0
\(475\) 30.7386 1.41039
\(476\) −13.1231 −0.601497
\(477\) 0 0
\(478\) 0 0
\(479\) −35.4384 −1.61922 −0.809612 0.586965i \(-0.800322\pi\)
−0.809612 + 0.586965i \(0.800322\pi\)
\(480\) 0 0
\(481\) 6.68466 0.304794
\(482\) 0 0
\(483\) 0 0
\(484\) 2.49242 0.113292
\(485\) −31.3693 −1.42441
\(486\) 0 0
\(487\) 26.2462 1.18933 0.594665 0.803974i \(-0.297285\pi\)
0.594665 + 0.803974i \(0.297285\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 40.4924 1.82740 0.913699 0.406392i \(-0.133213\pi\)
0.913699 + 0.406392i \(0.133213\pi\)
\(492\) 0 0
\(493\) 27.0540 1.21845
\(494\) 0 0
\(495\) 0 0
\(496\) 18.7386 0.841389
\(497\) −13.1231 −0.588652
\(498\) 0 0
\(499\) −32.4924 −1.45456 −0.727280 0.686341i \(-0.759216\pi\)
−0.727280 + 0.686341i \(0.759216\pi\)
\(500\) −19.1231 −0.855211
\(501\) 0 0
\(502\) 0 0
\(503\) 3.68466 0.164291 0.0821454 0.996620i \(-0.473823\pi\)
0.0821454 + 0.996620i \(0.473823\pi\)
\(504\) 0 0
\(505\) −70.5464 −3.13927
\(506\) 0 0
\(507\) 0 0
\(508\) −2.00000 −0.0887357
\(509\) 15.6847 0.695210 0.347605 0.937641i \(-0.386995\pi\)
0.347605 + 0.937641i \(0.386995\pi\)
\(510\) 0 0
\(511\) 4.43845 0.196345
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −42.7386 −1.88329
\(516\) 0 0
\(517\) −15.2311 −0.669861
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 5.12311 0.224447 0.112224 0.993683i \(-0.464203\pi\)
0.112224 + 0.993683i \(0.464203\pi\)
\(522\) 0 0
\(523\) −15.1771 −0.663647 −0.331824 0.943341i \(-0.607664\pi\)
−0.331824 + 0.943341i \(0.607664\pi\)
\(524\) 27.3693 1.19563
\(525\) 0 0
\(526\) 0 0
\(527\) −30.7386 −1.33900
\(528\) 0 0
\(529\) −1.05398 −0.0458250
\(530\) 0 0
\(531\) 0 0
\(532\) 8.00000 0.346844
\(533\) −38.4924 −1.66729
\(534\) 0 0
\(535\) −14.2462 −0.615917
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −3.12311 −0.134522
\(540\) 0 0
\(541\) −2.24621 −0.0965722 −0.0482861 0.998834i \(-0.515376\pi\)
−0.0482861 + 0.998834i \(0.515376\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 7.12311 0.305120
\(546\) 0 0
\(547\) −4.49242 −0.192082 −0.0960411 0.995377i \(-0.530618\pi\)
−0.0960411 + 0.995377i \(0.530618\pi\)
\(548\) −39.1231 −1.67126
\(549\) 0 0
\(550\) 0 0
\(551\) −16.4924 −0.702601
\(552\) 0 0
\(553\) 0.315342 0.0134097
\(554\) 0 0
\(555\) 0 0
\(556\) −25.1231 −1.06546
\(557\) 8.87689 0.376126 0.188063 0.982157i \(-0.439779\pi\)
0.188063 + 0.982157i \(0.439779\pi\)
\(558\) 0 0
\(559\) 15.1231 0.639639
\(560\) −14.2462 −0.602012
\(561\) 0 0
\(562\) 0 0
\(563\) 10.7386 0.452579 0.226290 0.974060i \(-0.427340\pi\)
0.226290 + 0.974060i \(0.427340\pi\)
\(564\) 0 0
\(565\) −10.2462 −0.431061
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −12.6307 −0.529506 −0.264753 0.964316i \(-0.585290\pi\)
−0.264753 + 0.964316i \(0.585290\pi\)
\(570\) 0 0
\(571\) −12.8769 −0.538881 −0.269441 0.963017i \(-0.586839\pi\)
−0.269441 + 0.963017i \(0.586839\pi\)
\(572\) −22.2462 −0.930161
\(573\) 0 0
\(574\) 0 0
\(575\) 36.0000 1.50130
\(576\) 0 0
\(577\) −9.80776 −0.408303 −0.204151 0.978939i \(-0.565443\pi\)
−0.204151 + 0.978939i \(0.565443\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 29.3693 1.21949
\(581\) 17.8078 0.738791
\(582\) 0 0
\(583\) −34.3542 −1.42280
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 31.3002 1.29190 0.645948 0.763381i \(-0.276462\pi\)
0.645948 + 0.763381i \(0.276462\pi\)
\(588\) 0 0
\(589\) 18.7386 0.772112
\(590\) 0 0
\(591\) 0 0
\(592\) −7.50758 −0.308560
\(593\) −24.6847 −1.01368 −0.506839 0.862041i \(-0.669186\pi\)
−0.506839 + 0.862041i \(0.669186\pi\)
\(594\) 0 0
\(595\) 23.3693 0.958049
\(596\) 28.0000 1.14692
\(597\) 0 0
\(598\) 0 0
\(599\) −46.6155 −1.90466 −0.952329 0.305072i \(-0.901320\pi\)
−0.952329 + 0.305072i \(0.901320\pi\)
\(600\) 0 0
\(601\) 23.9309 0.976161 0.488080 0.872799i \(-0.337697\pi\)
0.488080 + 0.872799i \(0.337697\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 44.9848 1.83041
\(605\) −4.43845 −0.180449
\(606\) 0 0
\(607\) −3.80776 −0.154552 −0.0772762 0.997010i \(-0.524622\pi\)
−0.0772762 + 0.997010i \(0.524622\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −17.3693 −0.702687
\(612\) 0 0
\(613\) 31.3693 1.26699 0.633497 0.773745i \(-0.281619\pi\)
0.633497 + 0.773745i \(0.281619\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −33.1080 −1.33288 −0.666438 0.745560i \(-0.732182\pi\)
−0.666438 + 0.745560i \(0.732182\pi\)
\(618\) 0 0
\(619\) 22.3153 0.896929 0.448465 0.893801i \(-0.351971\pi\)
0.448465 + 0.893801i \(0.351971\pi\)
\(620\) −33.3693 −1.34014
\(621\) 0 0
\(622\) 0 0
\(623\) 7.80776 0.312811
\(624\) 0 0
\(625\) −4.36932 −0.174773
\(626\) 0 0
\(627\) 0 0
\(628\) −0.492423 −0.0196498
\(629\) 12.3153 0.491045
\(630\) 0 0
\(631\) −10.4924 −0.417697 −0.208848 0.977948i \(-0.566972\pi\)
−0.208848 + 0.977948i \(0.566972\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 3.56155 0.141336
\(636\) 0 0
\(637\) −3.56155 −0.141114
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 14.8078 0.584872 0.292436 0.956285i \(-0.405534\pi\)
0.292436 + 0.956285i \(0.405534\pi\)
\(642\) 0 0
\(643\) 30.7386 1.21221 0.606107 0.795383i \(-0.292730\pi\)
0.606107 + 0.795383i \(0.292730\pi\)
\(644\) 9.36932 0.369203
\(645\) 0 0
\(646\) 0 0
\(647\) −16.0540 −0.631147 −0.315573 0.948901i \(-0.602197\pi\)
−0.315573 + 0.948901i \(0.602197\pi\)
\(648\) 0 0
\(649\) −13.8617 −0.544121
\(650\) 0 0
\(651\) 0 0
\(652\) 45.2311 1.77138
\(653\) 21.8617 0.855516 0.427758 0.903893i \(-0.359304\pi\)
0.427758 + 0.903893i \(0.359304\pi\)
\(654\) 0 0
\(655\) −48.7386 −1.90438
\(656\) 43.2311 1.68789
\(657\) 0 0
\(658\) 0 0
\(659\) −30.4384 −1.18571 −0.592857 0.805308i \(-0.702000\pi\)
−0.592857 + 0.805308i \(0.702000\pi\)
\(660\) 0 0
\(661\) 19.5616 0.760856 0.380428 0.924810i \(-0.375777\pi\)
0.380428 + 0.924810i \(0.375777\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −14.2462 −0.552444
\(666\) 0 0
\(667\) −19.3153 −0.747893
\(668\) 20.4924 0.792876
\(669\) 0 0
\(670\) 0 0
\(671\) 29.8617 1.15280
\(672\) 0 0
\(673\) 25.1080 0.967840 0.483920 0.875112i \(-0.339212\pi\)
0.483920 + 0.875112i \(0.339212\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0.630683 0.0242570
\(677\) −21.9309 −0.842872 −0.421436 0.906858i \(-0.638474\pi\)
−0.421436 + 0.906858i \(0.638474\pi\)
\(678\) 0 0
\(679\) 8.80776 0.338011
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −30.7386 −1.17618 −0.588091 0.808795i \(-0.700120\pi\)
−0.588091 + 0.808795i \(0.700120\pi\)
\(684\) 0 0
\(685\) 69.6695 2.66193
\(686\) 0 0
\(687\) 0 0
\(688\) −16.9848 −0.647541
\(689\) −39.1771 −1.49253
\(690\) 0 0
\(691\) 17.3002 0.658130 0.329065 0.944307i \(-0.393266\pi\)
0.329065 + 0.944307i \(0.393266\pi\)
\(692\) −2.73863 −0.104107
\(693\) 0 0
\(694\) 0 0
\(695\) 44.7386 1.69703
\(696\) 0 0
\(697\) −70.9157 −2.68612
\(698\) 0 0
\(699\) 0 0
\(700\) 15.3693 0.580906
\(701\) −20.9309 −0.790548 −0.395274 0.918563i \(-0.629350\pi\)
−0.395274 + 0.918563i \(0.629350\pi\)
\(702\) 0 0
\(703\) −7.50758 −0.283154
\(704\) 24.9848 0.941652
\(705\) 0 0
\(706\) 0 0
\(707\) 19.8078 0.744948
\(708\) 0 0
\(709\) −8.06913 −0.303043 −0.151521 0.988454i \(-0.548417\pi\)
−0.151521 + 0.988454i \(0.548417\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 21.9460 0.821885
\(714\) 0 0
\(715\) 39.6155 1.48154
\(716\) −5.75379 −0.215029
\(717\) 0 0
\(718\) 0 0
\(719\) 12.1771 0.454128 0.227064 0.973880i \(-0.427087\pi\)
0.227064 + 0.973880i \(0.427087\pi\)
\(720\) 0 0
\(721\) 12.0000 0.446903
\(722\) 0 0
\(723\) 0 0
\(724\) 6.38447 0.237277
\(725\) −31.6847 −1.17674
\(726\) 0 0
\(727\) −14.1771 −0.525799 −0.262899 0.964823i \(-0.584679\pi\)
−0.262899 + 0.964823i \(0.584679\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 27.8617 1.03050
\(732\) 0 0
\(733\) −8.68466 −0.320775 −0.160388 0.987054i \(-0.551274\pi\)
−0.160388 + 0.987054i \(0.551274\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 6.24621 0.230082
\(738\) 0 0
\(739\) 3.87689 0.142614 0.0713069 0.997454i \(-0.477283\pi\)
0.0713069 + 0.997454i \(0.477283\pi\)
\(740\) 13.3693 0.491466
\(741\) 0 0
\(742\) 0 0
\(743\) −27.1080 −0.994494 −0.497247 0.867609i \(-0.665656\pi\)
−0.497247 + 0.867609i \(0.665656\pi\)
\(744\) 0 0
\(745\) −49.8617 −1.82679
\(746\) 0 0
\(747\) 0 0
\(748\) −40.9848 −1.49855
\(749\) 4.00000 0.146157
\(750\) 0 0
\(751\) −37.1231 −1.35464 −0.677321 0.735688i \(-0.736859\pi\)
−0.677321 + 0.735688i \(0.736859\pi\)
\(752\) 19.5076 0.711368
\(753\) 0 0
\(754\) 0 0
\(755\) −80.1080 −2.91543
\(756\) 0 0
\(757\) −33.1080 −1.20333 −0.601664 0.798749i \(-0.705496\pi\)
−0.601664 + 0.798749i \(0.705496\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −44.5464 −1.61481 −0.807403 0.590001i \(-0.799128\pi\)
−0.807403 + 0.590001i \(0.799128\pi\)
\(762\) 0 0
\(763\) −2.00000 −0.0724049
\(764\) 38.7386 1.40151
\(765\) 0 0
\(766\) 0 0
\(767\) −15.8078 −0.570785
\(768\) 0 0
\(769\) −5.05398 −0.182251 −0.0911255 0.995839i \(-0.529046\pi\)
−0.0911255 + 0.995839i \(0.529046\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −14.2462 −0.512732
\(773\) 23.3002 0.838049 0.419025 0.907975i \(-0.362372\pi\)
0.419025 + 0.907975i \(0.362372\pi\)
\(774\) 0 0
\(775\) 36.0000 1.29316
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 43.2311 1.54891
\(780\) 0 0
\(781\) −40.9848 −1.46655
\(782\) 0 0
\(783\) 0 0
\(784\) 4.00000 0.142857
\(785\) 0.876894 0.0312977
\(786\) 0 0
\(787\) 40.7926 1.45410 0.727050 0.686585i \(-0.240891\pi\)
0.727050 + 0.686585i \(0.240891\pi\)
\(788\) −54.7386 −1.94998
\(789\) 0 0
\(790\) 0 0
\(791\) 2.87689 0.102291
\(792\) 0 0
\(793\) 34.0540 1.20929
\(794\) 0 0
\(795\) 0 0
\(796\) 19.1231 0.677801
\(797\) 11.7538 0.416341 0.208170 0.978093i \(-0.433249\pi\)
0.208170 + 0.978093i \(0.433249\pi\)
\(798\) 0 0
\(799\) −32.0000 −1.13208
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 13.8617 0.489170
\(804\) 0 0
\(805\) −16.6847 −0.588057
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −6.87689 −0.241779 −0.120889 0.992666i \(-0.538575\pi\)
−0.120889 + 0.992666i \(0.538575\pi\)
\(810\) 0 0
\(811\) 1.56155 0.0548335 0.0274168 0.999624i \(-0.491272\pi\)
0.0274168 + 0.999624i \(0.491272\pi\)
\(812\) −8.24621 −0.289385
\(813\) 0 0
\(814\) 0 0
\(815\) −80.5464 −2.82142
\(816\) 0 0
\(817\) −16.9848 −0.594225
\(818\) 0 0
\(819\) 0 0
\(820\) −76.9848 −2.68843
\(821\) 47.0000 1.64031 0.820156 0.572140i \(-0.193887\pi\)
0.820156 + 0.572140i \(0.193887\pi\)
\(822\) 0 0
\(823\) 26.4233 0.921058 0.460529 0.887645i \(-0.347660\pi\)
0.460529 + 0.887645i \(0.347660\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 31.0000 1.07798 0.538988 0.842314i \(-0.318807\pi\)
0.538988 + 0.842314i \(0.318807\pi\)
\(828\) 0 0
\(829\) 29.2311 1.01524 0.507618 0.861582i \(-0.330526\pi\)
0.507618 + 0.861582i \(0.330526\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 28.4924 0.987797
\(833\) −6.56155 −0.227344
\(834\) 0 0
\(835\) −36.4924 −1.26287
\(836\) 24.9848 0.864119
\(837\) 0 0
\(838\) 0 0
\(839\) −51.4233 −1.77533 −0.887665 0.460491i \(-0.847674\pi\)
−0.887665 + 0.460491i \(0.847674\pi\)
\(840\) 0 0
\(841\) −12.0000 −0.413793
\(842\) 0 0
\(843\) 0 0
\(844\) 57.3693 1.97473
\(845\) −1.12311 −0.0386360
\(846\) 0 0
\(847\) 1.24621 0.0428203
\(848\) 44.0000 1.51097
\(849\) 0 0
\(850\) 0 0
\(851\) −8.79261 −0.301407
\(852\) 0 0
\(853\) −7.43845 −0.254688 −0.127344 0.991859i \(-0.540645\pi\)
−0.127344 + 0.991859i \(0.540645\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −56.0540 −1.91477 −0.957384 0.288819i \(-0.906737\pi\)
−0.957384 + 0.288819i \(0.906737\pi\)
\(858\) 0 0
\(859\) 38.9157 1.32779 0.663894 0.747827i \(-0.268903\pi\)
0.663894 + 0.747827i \(0.268903\pi\)
\(860\) 30.2462 1.03139
\(861\) 0 0
\(862\) 0 0
\(863\) −38.1231 −1.29773 −0.648863 0.760905i \(-0.724755\pi\)
−0.648863 + 0.760905i \(0.724755\pi\)
\(864\) 0 0
\(865\) 4.87689 0.165819
\(866\) 0 0
\(867\) 0 0
\(868\) 9.36932 0.318015
\(869\) 0.984845 0.0334086
\(870\) 0 0
\(871\) 7.12311 0.241357
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −9.56155 −0.323239
\(876\) 0 0
\(877\) 49.5464 1.67306 0.836531 0.547919i \(-0.184580\pi\)
0.836531 + 0.547919i \(0.184580\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) −44.4924 −1.49984
\(881\) −46.9309 −1.58114 −0.790571 0.612371i \(-0.790216\pi\)
−0.790571 + 0.612371i \(0.790216\pi\)
\(882\) 0 0
\(883\) 22.9309 0.771685 0.385843 0.922565i \(-0.373911\pi\)
0.385843 + 0.922565i \(0.373911\pi\)
\(884\) −46.7386 −1.57199
\(885\) 0 0
\(886\) 0 0
\(887\) −26.4384 −0.887716 −0.443858 0.896097i \(-0.646391\pi\)
−0.443858 + 0.896097i \(0.646391\pi\)
\(888\) 0 0
\(889\) −1.00000 −0.0335389
\(890\) 0 0
\(891\) 0 0
\(892\) 21.6155 0.723741
\(893\) 19.5076 0.652796
\(894\) 0 0
\(895\) 10.2462 0.342493
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −19.3153 −0.644203
\(900\) 0 0
\(901\) −72.1771 −2.40457
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −11.3693 −0.377929
\(906\) 0 0
\(907\) −37.4924 −1.24492 −0.622458 0.782653i \(-0.713866\pi\)
−0.622458 + 0.782653i \(0.713866\pi\)
\(908\) −26.7386 −0.887353
\(909\) 0 0
\(910\) 0 0
\(911\) 0.630683 0.0208955 0.0104477 0.999945i \(-0.496674\pi\)
0.0104477 + 0.999945i \(0.496674\pi\)
\(912\) 0 0
\(913\) 55.6155 1.84061
\(914\) 0 0
\(915\) 0 0
\(916\) 18.8769 0.623710
\(917\) 13.6847 0.451907
\(918\) 0 0
\(919\) 4.86174 0.160374 0.0801870 0.996780i \(-0.474448\pi\)
0.0801870 + 0.996780i \(0.474448\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −46.7386 −1.53842
\(924\) 0 0
\(925\) −14.4233 −0.474235
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 3.61553 0.118622 0.0593108 0.998240i \(-0.481110\pi\)
0.0593108 + 0.998240i \(0.481110\pi\)
\(930\) 0 0
\(931\) 4.00000 0.131095
\(932\) 48.7386 1.59649
\(933\) 0 0
\(934\) 0 0
\(935\) 72.9848 2.38686
\(936\) 0 0
\(937\) −1.12311 −0.0366903 −0.0183451 0.999832i \(-0.505840\pi\)
−0.0183451 + 0.999832i \(0.505840\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −34.7386 −1.13305
\(941\) 37.0540 1.20793 0.603963 0.797013i \(-0.293588\pi\)
0.603963 + 0.797013i \(0.293588\pi\)
\(942\) 0 0
\(943\) 50.6307 1.64876
\(944\) 17.7538 0.577837
\(945\) 0 0
\(946\) 0 0
\(947\) 38.4233 1.24859 0.624295 0.781189i \(-0.285386\pi\)
0.624295 + 0.781189i \(0.285386\pi\)
\(948\) 0 0
\(949\) 15.8078 0.513142
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −24.7386 −0.801363 −0.400681 0.916217i \(-0.631227\pi\)
−0.400681 + 0.916217i \(0.631227\pi\)
\(954\) 0 0
\(955\) −68.9848 −2.23230
\(956\) 7.26137 0.234849
\(957\) 0 0
\(958\) 0 0
\(959\) −19.5616 −0.631675
\(960\) 0 0
\(961\) −9.05398 −0.292064
\(962\) 0 0
\(963\) 0 0
\(964\) 10.8769 0.350321
\(965\) 25.3693 0.816667
\(966\) 0 0
\(967\) −50.3542 −1.61928 −0.809640 0.586926i \(-0.800338\pi\)
−0.809640 + 0.586926i \(0.800338\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −29.3693 −0.942506 −0.471253 0.881998i \(-0.656198\pi\)
−0.471253 + 0.881998i \(0.656198\pi\)
\(972\) 0 0
\(973\) −12.5616 −0.402705
\(974\) 0 0
\(975\) 0 0
\(976\) −38.2462 −1.22423
\(977\) 52.9848 1.69514 0.847568 0.530687i \(-0.178066\pi\)
0.847568 + 0.530687i \(0.178066\pi\)
\(978\) 0 0
\(979\) 24.3845 0.779331
\(980\) −7.12311 −0.227539
\(981\) 0 0
\(982\) 0 0
\(983\) 10.8078 0.344714 0.172357 0.985035i \(-0.444862\pi\)
0.172357 + 0.985035i \(0.444862\pi\)
\(984\) 0 0
\(985\) 97.4773 3.10588
\(986\) 0 0
\(987\) 0 0
\(988\) 28.4924 0.906465
\(989\) −19.8920 −0.632530
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −34.0540 −1.07958
\(996\) 0 0
\(997\) 26.0000 0.823428 0.411714 0.911313i \(-0.364930\pi\)
0.411714 + 0.911313i \(0.364930\pi\)
\(998\) 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 8001.2.a.i.1.2 2
3.2 odd 2 2667.2.a.f.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2667.2.a.f.1.1 2 3.2 odd 2
8001.2.a.i.1.2 2 1.1 even 1 trivial