Properties

Label 2240.2.a.x.1.1
Level $2240$
Weight $2$
Character 2240.1
Self dual yes
Analytic conductor $17.886$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2240,2,Mod(1,2240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2240.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.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: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1120)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2240.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} +1.00000 q^{15} +1.00000 q^{17} +1.00000 q^{21} +4.00000 q^{23} +1.00000 q^{25} -5.00000 q^{27} +1.00000 q^{29} +10.0000 q^{31} -1.00000 q^{33} +1.00000 q^{35} +8.00000 q^{37} +1.00000 q^{39} +2.00000 q^{41} +4.00000 q^{43} -2.00000 q^{45} +11.0000 q^{47} +1.00000 q^{49} +1.00000 q^{51} -4.00000 q^{53} -1.00000 q^{55} -8.00000 q^{59} -10.0000 q^{61} -2.00000 q^{63} +1.00000 q^{65} +4.00000 q^{67} +4.00000 q^{69} +12.0000 q^{71} +6.00000 q^{73} +1.00000 q^{75} -1.00000 q^{77} +7.00000 q^{79} +1.00000 q^{81} +8.00000 q^{83} +1.00000 q^{85} +1.00000 q^{87} -6.00000 q^{89} +1.00000 q^{91} +10.0000 q^{93} +1.00000 q^{97} +2.00000 q^{99} +O(q^{100})\)

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
\(3\) 1.00000 0.577350 0.288675 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 1.00000 0.242536 0.121268 0.992620i \(-0.461304\pi\)
0.121268 + 0.992620i \(0.461304\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) 1.00000 0.185695 0.0928477 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 0 0
\(31\) 10.0000 1.79605 0.898027 0.439941i \(-0.145001\pi\)
0.898027 + 0.439941i \(0.145001\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) −2.00000 −0.298142
\(46\) 0 0
\(47\) 11.0000 1.60451 0.802257 0.596978i \(-0.203632\pi\)
0.802257 + 0.596978i \(0.203632\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 1.00000 0.140028
\(52\) 0 0
\(53\) −4.00000 −0.549442 −0.274721 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) −2.00000 −0.251976
\(64\) 0 0
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) 7.00000 0.787562 0.393781 0.919204i \(-0.371167\pi\)
0.393781 + 0.919204i \(0.371167\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 8.00000 0.878114 0.439057 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) 0 0
\(85\) 1.00000 0.108465
\(86\) 0 0
\(87\) 1.00000 0.107211
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) 10.0000 1.03695
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.00000 0.101535 0.0507673 0.998711i \(-0.483833\pi\)
0.0507673 + 0.998711i \(0.483833\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 0 0
\(103\) −5.00000 −0.492665 −0.246332 0.969185i \(-0.579225\pi\)
−0.246332 + 0.969185i \(0.579225\pi\)
\(104\) 0 0
\(105\) 1.00000 0.0975900
\(106\) 0 0
\(107\) −10.0000 −0.966736 −0.483368 0.875417i \(-0.660587\pi\)
−0.483368 + 0.875417i \(0.660587\pi\)
\(108\) 0 0
\(109\) −13.0000 −1.24517 −0.622587 0.782551i \(-0.713918\pi\)
−0.622587 + 0.782551i \(0.713918\pi\)
\(110\) 0 0
\(111\) 8.00000 0.759326
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) 1.00000 0.0916698
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 0 0
\(123\) 2.00000 0.180334
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −5.00000 −0.430331
\(136\) 0 0
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 0 0
\(139\) 2.00000 0.169638 0.0848189 0.996396i \(-0.472969\pi\)
0.0848189 + 0.996396i \(0.472969\pi\)
\(140\) 0 0
\(141\) 11.0000 0.926367
\(142\) 0 0
\(143\) −1.00000 −0.0836242
\(144\) 0 0
\(145\) 1.00000 0.0830455
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 7.00000 0.569652 0.284826 0.958579i \(-0.408064\pi\)
0.284826 + 0.958579i \(0.408064\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) 10.0000 0.803219
\(156\) 0 0
\(157\) 6.00000 0.478852 0.239426 0.970915i \(-0.423041\pi\)
0.239426 + 0.970915i \(0.423041\pi\)
\(158\) 0 0
\(159\) −4.00000 −0.317221
\(160\) 0 0
\(161\) 4.00000 0.315244
\(162\) 0 0
\(163\) 22.0000 1.72317 0.861586 0.507611i \(-0.169471\pi\)
0.861586 + 0.507611i \(0.169471\pi\)
\(164\) 0 0
\(165\) −1.00000 −0.0778499
\(166\) 0 0
\(167\) −5.00000 −0.386912 −0.193456 0.981109i \(-0.561970\pi\)
−0.193456 + 0.981109i \(0.561970\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −21.0000 −1.59660 −0.798300 0.602260i \(-0.794267\pi\)
−0.798300 + 0.602260i \(0.794267\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −8.00000 −0.601317
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 12.0000 0.891953 0.445976 0.895045i \(-0.352856\pi\)
0.445976 + 0.895045i \(0.352856\pi\)
\(182\) 0 0
\(183\) −10.0000 −0.739221
\(184\) 0 0
\(185\) 8.00000 0.588172
\(186\) 0 0
\(187\) −1.00000 −0.0731272
\(188\) 0 0
\(189\) −5.00000 −0.363696
\(190\) 0 0
\(191\) 21.0000 1.51951 0.759753 0.650211i \(-0.225320\pi\)
0.759753 + 0.650211i \(0.225320\pi\)
\(192\) 0 0
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) 0 0
\(195\) 1.00000 0.0716115
\(196\) 0 0
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) 1.00000 0.0701862
\(204\) 0 0
\(205\) 2.00000 0.139686
\(206\) 0 0
\(207\) −8.00000 −0.556038
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 17.0000 1.17033 0.585164 0.810915i \(-0.301030\pi\)
0.585164 + 0.810915i \(0.301030\pi\)
\(212\) 0 0
\(213\) 12.0000 0.822226
\(214\) 0 0
\(215\) 4.00000 0.272798
\(216\) 0 0
\(217\) 10.0000 0.678844
\(218\) 0 0
\(219\) 6.00000 0.405442
\(220\) 0 0
\(221\) 1.00000 0.0672673
\(222\) 0 0
\(223\) −21.0000 −1.40626 −0.703132 0.711059i \(-0.748216\pi\)
−0.703132 + 0.711059i \(0.748216\pi\)
\(224\) 0 0
\(225\) −2.00000 −0.133333
\(226\) 0 0
\(227\) −7.00000 −0.464606 −0.232303 0.972643i \(-0.574626\pi\)
−0.232303 + 0.972643i \(0.574626\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) −1.00000 −0.0657952
\(232\) 0 0
\(233\) −24.0000 −1.57229 −0.786146 0.618041i \(-0.787927\pi\)
−0.786146 + 0.618041i \(0.787927\pi\)
\(234\) 0 0
\(235\) 11.0000 0.717561
\(236\) 0 0
\(237\) 7.00000 0.454699
\(238\) 0 0
\(239\) 15.0000 0.970269 0.485135 0.874439i \(-0.338771\pi\)
0.485135 + 0.874439i \(0.338771\pi\)
\(240\) 0 0
\(241\) 20.0000 1.28831 0.644157 0.764894i \(-0.277208\pi\)
0.644157 + 0.764894i \(0.277208\pi\)
\(242\) 0 0
\(243\) 16.0000 1.02640
\(244\) 0 0
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 8.00000 0.506979
\(250\) 0 0
\(251\) −10.0000 −0.631194 −0.315597 0.948893i \(-0.602205\pi\)
−0.315597 + 0.948893i \(0.602205\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) 0 0
\(255\) 1.00000 0.0626224
\(256\) 0 0
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 0 0
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 0 0
\(263\) −26.0000 −1.60323 −0.801614 0.597841i \(-0.796025\pi\)
−0.801614 + 0.597841i \(0.796025\pi\)
\(264\) 0 0
\(265\) −4.00000 −0.245718
\(266\) 0 0
\(267\) −6.00000 −0.367194
\(268\) 0 0
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) 22.0000 1.33640 0.668202 0.743980i \(-0.267064\pi\)
0.668202 + 0.743980i \(0.267064\pi\)
\(272\) 0 0
\(273\) 1.00000 0.0605228
\(274\) 0 0
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) −26.0000 −1.56219 −0.781094 0.624413i \(-0.785338\pi\)
−0.781094 + 0.624413i \(0.785338\pi\)
\(278\) 0 0
\(279\) −20.0000 −1.19737
\(280\) 0 0
\(281\) −9.00000 −0.536895 −0.268447 0.963294i \(-0.586511\pi\)
−0.268447 + 0.963294i \(0.586511\pi\)
\(282\) 0 0
\(283\) 7.00000 0.416107 0.208053 0.978117i \(-0.433287\pi\)
0.208053 + 0.978117i \(0.433287\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2.00000 0.118056
\(288\) 0 0
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) 1.00000 0.0586210
\(292\) 0 0
\(293\) 7.00000 0.408944 0.204472 0.978872i \(-0.434452\pi\)
0.204472 + 0.978872i \(0.434452\pi\)
\(294\) 0 0
\(295\) −8.00000 −0.465778
\(296\) 0 0
\(297\) 5.00000 0.290129
\(298\) 0 0
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 0 0
\(303\) −12.0000 −0.689382
\(304\) 0 0
\(305\) −10.0000 −0.572598
\(306\) 0 0
\(307\) −5.00000 −0.285365 −0.142683 0.989769i \(-0.545573\pi\)
−0.142683 + 0.989769i \(0.545573\pi\)
\(308\) 0 0
\(309\) −5.00000 −0.284440
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 19.0000 1.07394 0.536972 0.843600i \(-0.319568\pi\)
0.536972 + 0.843600i \(0.319568\pi\)
\(314\) 0 0
\(315\) −2.00000 −0.112687
\(316\) 0 0
\(317\) 22.0000 1.23564 0.617822 0.786318i \(-0.288015\pi\)
0.617822 + 0.786318i \(0.288015\pi\)
\(318\) 0 0
\(319\) −1.00000 −0.0559893
\(320\) 0 0
\(321\) −10.0000 −0.558146
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) 0 0
\(327\) −13.0000 −0.718902
\(328\) 0 0
\(329\) 11.0000 0.606450
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) 0 0
\(333\) −16.0000 −0.876795
\(334\) 0 0
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) 30.0000 1.63420 0.817102 0.576493i \(-0.195579\pi\)
0.817102 + 0.576493i \(0.195579\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −10.0000 −0.541530
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 4.00000 0.215353
\(346\) 0 0
\(347\) −36.0000 −1.93258 −0.966291 0.257454i \(-0.917117\pi\)
−0.966291 + 0.257454i \(0.917117\pi\)
\(348\) 0 0
\(349\) 28.0000 1.49881 0.749403 0.662114i \(-0.230341\pi\)
0.749403 + 0.662114i \(0.230341\pi\)
\(350\) 0 0
\(351\) −5.00000 −0.266880
\(352\) 0 0
\(353\) 21.0000 1.11772 0.558859 0.829263i \(-0.311239\pi\)
0.558859 + 0.829263i \(0.311239\pi\)
\(354\) 0 0
\(355\) 12.0000 0.636894
\(356\) 0 0
\(357\) 1.00000 0.0529256
\(358\) 0 0
\(359\) −32.0000 −1.68890 −0.844448 0.535638i \(-0.820071\pi\)
−0.844448 + 0.535638i \(0.820071\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) −10.0000 −0.524864
\(364\) 0 0
\(365\) 6.00000 0.314054
\(366\) 0 0
\(367\) −17.0000 −0.887393 −0.443696 0.896177i \(-0.646333\pi\)
−0.443696 + 0.896177i \(0.646333\pi\)
\(368\) 0 0
\(369\) −4.00000 −0.208232
\(370\) 0 0
\(371\) −4.00000 −0.207670
\(372\) 0 0
\(373\) 6.00000 0.310668 0.155334 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 1.00000 0.0515026
\(378\) 0 0
\(379\) −32.0000 −1.64373 −0.821865 0.569683i \(-0.807066\pi\)
−0.821865 + 0.569683i \(0.807066\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 0 0
\(383\) −20.0000 −1.02195 −0.510976 0.859595i \(-0.670716\pi\)
−0.510976 + 0.859595i \(0.670716\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) 0 0
\(387\) −8.00000 −0.406663
\(388\) 0 0
\(389\) −9.00000 −0.456318 −0.228159 0.973624i \(-0.573271\pi\)
−0.228159 + 0.973624i \(0.573271\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 7.00000 0.352208
\(396\) 0 0
\(397\) −17.0000 −0.853206 −0.426603 0.904439i \(-0.640290\pi\)
−0.426603 + 0.904439i \(0.640290\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −11.0000 −0.549314 −0.274657 0.961542i \(-0.588564\pi\)
−0.274657 + 0.961542i \(0.588564\pi\)
\(402\) 0 0
\(403\) 10.0000 0.498135
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −8.00000 −0.396545
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 0 0
\(411\) −2.00000 −0.0986527
\(412\) 0 0
\(413\) −8.00000 −0.393654
\(414\) 0 0
\(415\) 8.00000 0.392705
\(416\) 0 0
\(417\) 2.00000 0.0979404
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 15.0000 0.731055 0.365528 0.930800i \(-0.380889\pi\)
0.365528 + 0.930800i \(0.380889\pi\)
\(422\) 0 0
\(423\) −22.0000 −1.06968
\(424\) 0 0
\(425\) 1.00000 0.0485071
\(426\) 0 0
\(427\) −10.0000 −0.483934
\(428\) 0 0
\(429\) −1.00000 −0.0482805
\(430\) 0 0
\(431\) 25.0000 1.20421 0.602104 0.798418i \(-0.294329\pi\)
0.602104 + 0.798418i \(0.294329\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) 1.00000 0.0479463
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −6.00000 −0.286364 −0.143182 0.989696i \(-0.545733\pi\)
−0.143182 + 0.989696i \(0.545733\pi\)
\(440\) 0 0
\(441\) −2.00000 −0.0952381
\(442\) 0 0
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 0 0
\(445\) −6.00000 −0.284427
\(446\) 0 0
\(447\) 6.00000 0.283790
\(448\) 0 0
\(449\) 15.0000 0.707894 0.353947 0.935266i \(-0.384839\pi\)
0.353947 + 0.935266i \(0.384839\pi\)
\(450\) 0 0
\(451\) −2.00000 −0.0941763
\(452\) 0 0
\(453\) 7.00000 0.328889
\(454\) 0 0
\(455\) 1.00000 0.0468807
\(456\) 0 0
\(457\) −26.0000 −1.21623 −0.608114 0.793849i \(-0.708074\pi\)
−0.608114 + 0.793849i \(0.708074\pi\)
\(458\) 0 0
\(459\) −5.00000 −0.233380
\(460\) 0 0
\(461\) 32.0000 1.49039 0.745194 0.666847i \(-0.232357\pi\)
0.745194 + 0.666847i \(0.232357\pi\)
\(462\) 0 0
\(463\) 18.0000 0.836531 0.418265 0.908325i \(-0.362638\pi\)
0.418265 + 0.908325i \(0.362638\pi\)
\(464\) 0 0
\(465\) 10.0000 0.463739
\(466\) 0 0
\(467\) −33.0000 −1.52706 −0.763529 0.645774i \(-0.776535\pi\)
−0.763529 + 0.645774i \(0.776535\pi\)
\(468\) 0 0
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) 6.00000 0.276465
\(472\) 0 0
\(473\) −4.00000 −0.183920
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 8.00000 0.366295
\(478\) 0 0
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) 8.00000 0.364769
\(482\) 0 0
\(483\) 4.00000 0.182006
\(484\) 0 0
\(485\) 1.00000 0.0454077
\(486\) 0 0
\(487\) −18.0000 −0.815658 −0.407829 0.913058i \(-0.633714\pi\)
−0.407829 + 0.913058i \(0.633714\pi\)
\(488\) 0 0
\(489\) 22.0000 0.994874
\(490\) 0 0
\(491\) −31.0000 −1.39901 −0.699505 0.714628i \(-0.746596\pi\)
−0.699505 + 0.714628i \(0.746596\pi\)
\(492\) 0 0
\(493\) 1.00000 0.0450377
\(494\) 0 0
\(495\) 2.00000 0.0898933
\(496\) 0 0
\(497\) 12.0000 0.538274
\(498\) 0 0
\(499\) −41.0000 −1.83541 −0.917706 0.397260i \(-0.869961\pi\)
−0.917706 + 0.397260i \(0.869961\pi\)
\(500\) 0 0
\(501\) −5.00000 −0.223384
\(502\) 0 0
\(503\) 13.0000 0.579641 0.289821 0.957081i \(-0.406404\pi\)
0.289821 + 0.957081i \(0.406404\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) 0 0
\(507\) −12.0000 −0.532939
\(508\) 0 0
\(509\) −2.00000 −0.0886484 −0.0443242 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) 0 0
\(511\) 6.00000 0.265424
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −5.00000 −0.220326
\(516\) 0 0
\(517\) −11.0000 −0.483779
\(518\) 0 0
\(519\) −21.0000 −0.921798
\(520\) 0 0
\(521\) 44.0000 1.92767 0.963837 0.266491i \(-0.0858642\pi\)
0.963837 + 0.266491i \(0.0858642\pi\)
\(522\) 0 0
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 0 0
\(525\) 1.00000 0.0436436
\(526\) 0 0
\(527\) 10.0000 0.435607
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 16.0000 0.694341
\(532\) 0 0
\(533\) 2.00000 0.0866296
\(534\) 0 0
\(535\) −10.0000 −0.432338
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) 9.00000 0.386940 0.193470 0.981106i \(-0.438026\pi\)
0.193470 + 0.981106i \(0.438026\pi\)
\(542\) 0 0
\(543\) 12.0000 0.514969
\(544\) 0 0
\(545\) −13.0000 −0.556859
\(546\) 0 0
\(547\) 34.0000 1.45374 0.726868 0.686778i \(-0.240975\pi\)
0.726868 + 0.686778i \(0.240975\pi\)
\(548\) 0 0
\(549\) 20.0000 0.853579
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 7.00000 0.297670
\(554\) 0 0
\(555\) 8.00000 0.339581
\(556\) 0 0
\(557\) −24.0000 −1.01691 −0.508456 0.861088i \(-0.669784\pi\)
−0.508456 + 0.861088i \(0.669784\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) −1.00000 −0.0422200
\(562\) 0 0
\(563\) −20.0000 −0.842900 −0.421450 0.906852i \(-0.638479\pi\)
−0.421450 + 0.906852i \(0.638479\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −42.0000 −1.76073 −0.880366 0.474295i \(-0.842703\pi\)
−0.880366 + 0.474295i \(0.842703\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 0 0
\(573\) 21.0000 0.877288
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) −41.0000 −1.70685 −0.853426 0.521214i \(-0.825479\pi\)
−0.853426 + 0.521214i \(0.825479\pi\)
\(578\) 0 0
\(579\) −10.0000 −0.415586
\(580\) 0 0
\(581\) 8.00000 0.331896
\(582\) 0 0
\(583\) 4.00000 0.165663
\(584\) 0 0
\(585\) −2.00000 −0.0826898
\(586\) 0 0
\(587\) −36.0000 −1.48588 −0.742940 0.669359i \(-0.766569\pi\)
−0.742940 + 0.669359i \(0.766569\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) 0 0
\(593\) −33.0000 −1.35515 −0.677574 0.735455i \(-0.736969\pi\)
−0.677574 + 0.735455i \(0.736969\pi\)
\(594\) 0 0
\(595\) 1.00000 0.0409960
\(596\) 0 0
\(597\) −4.00000 −0.163709
\(598\) 0 0
\(599\) 9.00000 0.367730 0.183865 0.982952i \(-0.441139\pi\)
0.183865 + 0.982952i \(0.441139\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 0 0
\(603\) −8.00000 −0.325785
\(604\) 0 0
\(605\) −10.0000 −0.406558
\(606\) 0 0
\(607\) −3.00000 −0.121766 −0.0608831 0.998145i \(-0.519392\pi\)
−0.0608831 + 0.998145i \(0.519392\pi\)
\(608\) 0 0
\(609\) 1.00000 0.0405220
\(610\) 0 0
\(611\) 11.0000 0.445012
\(612\) 0 0
\(613\) −26.0000 −1.05013 −0.525065 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) 0 0
\(615\) 2.00000 0.0806478
\(616\) 0 0
\(617\) 22.0000 0.885687 0.442843 0.896599i \(-0.353970\pi\)
0.442843 + 0.896599i \(0.353970\pi\)
\(618\) 0 0
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 0 0
\(621\) −20.0000 −0.802572
\(622\) 0 0
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 8.00000 0.318981
\(630\) 0 0
\(631\) 49.0000 1.95066 0.975330 0.220754i \(-0.0708517\pi\)
0.975330 + 0.220754i \(0.0708517\pi\)
\(632\) 0 0
\(633\) 17.0000 0.675689
\(634\) 0 0
\(635\) −8.00000 −0.317470
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 0 0
\(639\) −24.0000 −0.949425
\(640\) 0 0
\(641\) −34.0000 −1.34292 −0.671460 0.741041i \(-0.734332\pi\)
−0.671460 + 0.741041i \(0.734332\pi\)
\(642\) 0 0
\(643\) 17.0000 0.670415 0.335207 0.942144i \(-0.391194\pi\)
0.335207 + 0.942144i \(0.391194\pi\)
\(644\) 0 0
\(645\) 4.00000 0.157500
\(646\) 0 0
\(647\) −36.0000 −1.41531 −0.707653 0.706560i \(-0.750246\pi\)
−0.707653 + 0.706560i \(0.750246\pi\)
\(648\) 0 0
\(649\) 8.00000 0.314027
\(650\) 0 0
\(651\) 10.0000 0.391931
\(652\) 0 0
\(653\) −4.00000 −0.156532 −0.0782660 0.996933i \(-0.524938\pi\)
−0.0782660 + 0.996933i \(0.524938\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −12.0000 −0.468165
\(658\) 0 0
\(659\) −21.0000 −0.818044 −0.409022 0.912525i \(-0.634130\pi\)
−0.409022 + 0.912525i \(0.634130\pi\)
\(660\) 0 0
\(661\) −36.0000 −1.40024 −0.700119 0.714026i \(-0.746870\pi\)
−0.700119 + 0.714026i \(0.746870\pi\)
\(662\) 0 0
\(663\) 1.00000 0.0388368
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 4.00000 0.154881
\(668\) 0 0
\(669\) −21.0000 −0.811907
\(670\) 0 0
\(671\) 10.0000 0.386046
\(672\) 0 0
\(673\) −18.0000 −0.693849 −0.346925 0.937893i \(-0.612774\pi\)
−0.346925 + 0.937893i \(0.612774\pi\)
\(674\) 0 0
\(675\) −5.00000 −0.192450
\(676\) 0 0
\(677\) 21.0000 0.807096 0.403548 0.914959i \(-0.367777\pi\)
0.403548 + 0.914959i \(0.367777\pi\)
\(678\) 0 0
\(679\) 1.00000 0.0383765
\(680\) 0 0
\(681\) −7.00000 −0.268241
\(682\) 0 0
\(683\) −42.0000 −1.60709 −0.803543 0.595247i \(-0.797054\pi\)
−0.803543 + 0.595247i \(0.797054\pi\)
\(684\) 0 0
\(685\) −2.00000 −0.0764161
\(686\) 0 0
\(687\) −10.0000 −0.381524
\(688\) 0 0
\(689\) −4.00000 −0.152388
\(690\) 0 0
\(691\) 32.0000 1.21734 0.608669 0.793424i \(-0.291704\pi\)
0.608669 + 0.793424i \(0.291704\pi\)
\(692\) 0 0
\(693\) 2.00000 0.0759737
\(694\) 0 0
\(695\) 2.00000 0.0758643
\(696\) 0 0
\(697\) 2.00000 0.0757554
\(698\) 0 0
\(699\) −24.0000 −0.907763
\(700\) 0 0
\(701\) 33.0000 1.24639 0.623196 0.782065i \(-0.285834\pi\)
0.623196 + 0.782065i \(0.285834\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 11.0000 0.414284
\(706\) 0 0
\(707\) −12.0000 −0.451306
\(708\) 0 0
\(709\) 1.00000 0.0375558 0.0187779 0.999824i \(-0.494022\pi\)
0.0187779 + 0.999824i \(0.494022\pi\)
\(710\) 0 0
\(711\) −14.0000 −0.525041
\(712\) 0 0
\(713\) 40.0000 1.49801
\(714\) 0 0
\(715\) −1.00000 −0.0373979
\(716\) 0 0
\(717\) 15.0000 0.560185
\(718\) 0 0
\(719\) 12.0000 0.447524 0.223762 0.974644i \(-0.428166\pi\)
0.223762 + 0.974644i \(0.428166\pi\)
\(720\) 0 0
\(721\) −5.00000 −0.186210
\(722\) 0 0
\(723\) 20.0000 0.743808
\(724\) 0 0
\(725\) 1.00000 0.0371391
\(726\) 0 0
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 4.00000 0.147945
\(732\) 0 0
\(733\) −11.0000 −0.406294 −0.203147 0.979148i \(-0.565117\pi\)
−0.203147 + 0.979148i \(0.565117\pi\)
\(734\) 0 0
\(735\) 1.00000 0.0368856
\(736\) 0 0
\(737\) −4.00000 −0.147342
\(738\) 0 0
\(739\) −17.0000 −0.625355 −0.312678 0.949859i \(-0.601226\pi\)
−0.312678 + 0.949859i \(0.601226\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 8.00000 0.293492 0.146746 0.989174i \(-0.453120\pi\)
0.146746 + 0.989174i \(0.453120\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) 0 0
\(747\) −16.0000 −0.585409
\(748\) 0 0
\(749\) −10.0000 −0.365392
\(750\) 0 0
\(751\) 3.00000 0.109472 0.0547358 0.998501i \(-0.482568\pi\)
0.0547358 + 0.998501i \(0.482568\pi\)
\(752\) 0 0
\(753\) −10.0000 −0.364420
\(754\) 0 0
\(755\) 7.00000 0.254756
\(756\) 0 0
\(757\) −4.00000 −0.145382 −0.0726912 0.997354i \(-0.523159\pi\)
−0.0726912 + 0.997354i \(0.523159\pi\)
\(758\) 0 0
\(759\) −4.00000 −0.145191
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) −13.0000 −0.470632
\(764\) 0 0
\(765\) −2.00000 −0.0723102
\(766\) 0 0
\(767\) −8.00000 −0.288863
\(768\) 0 0
\(769\) 54.0000 1.94729 0.973645 0.228069i \(-0.0732413\pi\)
0.973645 + 0.228069i \(0.0732413\pi\)
\(770\) 0 0
\(771\) 2.00000 0.0720282
\(772\) 0 0
\(773\) 21.0000 0.755318 0.377659 0.925945i \(-0.376729\pi\)
0.377659 + 0.925945i \(0.376729\pi\)
\(774\) 0 0
\(775\) 10.0000 0.359211
\(776\) 0 0
\(777\) 8.00000 0.286998
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −12.0000 −0.429394
\(782\) 0 0
\(783\) −5.00000 −0.178685
\(784\) 0 0
\(785\) 6.00000 0.214149
\(786\) 0 0
\(787\) 17.0000 0.605985 0.302992 0.952993i \(-0.402014\pi\)
0.302992 + 0.952993i \(0.402014\pi\)
\(788\) 0 0
\(789\) −26.0000 −0.925625
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −10.0000 −0.355110
\(794\) 0 0
\(795\) −4.00000 −0.141865
\(796\) 0 0
\(797\) −29.0000 −1.02723 −0.513616 0.858020i \(-0.671695\pi\)
−0.513616 + 0.858020i \(0.671695\pi\)
\(798\) 0 0
\(799\) 11.0000 0.389152
\(800\) 0 0
\(801\) 12.0000 0.423999
\(802\) 0 0
\(803\) −6.00000 −0.211735
\(804\) 0 0
\(805\) 4.00000 0.140981
\(806\) 0 0
\(807\) −6.00000 −0.211210
\(808\) 0 0
\(809\) −23.0000 −0.808637 −0.404318 0.914618i \(-0.632491\pi\)
−0.404318 + 0.914618i \(0.632491\pi\)
\(810\) 0 0
\(811\) 34.0000 1.19390 0.596951 0.802278i \(-0.296379\pi\)
0.596951 + 0.802278i \(0.296379\pi\)
\(812\) 0 0
\(813\) 22.0000 0.771574
\(814\) 0 0
\(815\) 22.0000 0.770626
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −2.00000 −0.0698857
\(820\) 0 0
\(821\) −49.0000 −1.71011 −0.855056 0.518536i \(-0.826477\pi\)
−0.855056 + 0.518536i \(0.826477\pi\)
\(822\) 0 0
\(823\) 46.0000 1.60346 0.801730 0.597687i \(-0.203913\pi\)
0.801730 + 0.597687i \(0.203913\pi\)
\(824\) 0 0
\(825\) −1.00000 −0.0348155
\(826\) 0 0
\(827\) −6.00000 −0.208640 −0.104320 0.994544i \(-0.533267\pi\)
−0.104320 + 0.994544i \(0.533267\pi\)
\(828\) 0 0
\(829\) 30.0000 1.04194 0.520972 0.853574i \(-0.325570\pi\)
0.520972 + 0.853574i \(0.325570\pi\)
\(830\) 0 0
\(831\) −26.0000 −0.901930
\(832\) 0 0
\(833\) 1.00000 0.0346479
\(834\) 0 0
\(835\) −5.00000 −0.173032
\(836\) 0 0
\(837\) −50.0000 −1.72825
\(838\) 0 0
\(839\) −44.0000 −1.51905 −0.759524 0.650479i \(-0.774568\pi\)
−0.759524 + 0.650479i \(0.774568\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 0 0
\(843\) −9.00000 −0.309976
\(844\) 0 0
\(845\) −12.0000 −0.412813
\(846\) 0 0
\(847\) −10.0000 −0.343604
\(848\) 0 0
\(849\) 7.00000 0.240239
\(850\) 0 0
\(851\) 32.0000 1.09695
\(852\) 0 0
\(853\) 18.0000 0.616308 0.308154 0.951336i \(-0.400289\pi\)
0.308154 + 0.951336i \(0.400289\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) 52.0000 1.77422 0.887109 0.461561i \(-0.152710\pi\)
0.887109 + 0.461561i \(0.152710\pi\)
\(860\) 0 0
\(861\) 2.00000 0.0681598
\(862\) 0 0
\(863\) 18.0000 0.612727 0.306364 0.951915i \(-0.400888\pi\)
0.306364 + 0.951915i \(0.400888\pi\)
\(864\) 0 0
\(865\) −21.0000 −0.714021
\(866\) 0 0
\(867\) −16.0000 −0.543388
\(868\) 0 0
\(869\) −7.00000 −0.237459
\(870\) 0 0
\(871\) 4.00000 0.135535
\(872\) 0 0
\(873\) −2.00000 −0.0676897
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) 44.0000 1.48577 0.742887 0.669417i \(-0.233456\pi\)
0.742887 + 0.669417i \(0.233456\pi\)
\(878\) 0 0
\(879\) 7.00000 0.236104
\(880\) 0 0
\(881\) 40.0000 1.34763 0.673817 0.738898i \(-0.264654\pi\)
0.673817 + 0.738898i \(0.264654\pi\)
\(882\) 0 0
\(883\) 2.00000 0.0673054 0.0336527 0.999434i \(-0.489286\pi\)
0.0336527 + 0.999434i \(0.489286\pi\)
\(884\) 0 0
\(885\) −8.00000 −0.268917
\(886\) 0 0
\(887\) 8.00000 0.268614 0.134307 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 4.00000 0.133556
\(898\) 0 0
\(899\) 10.0000 0.333519
\(900\) 0 0
\(901\) −4.00000 −0.133259
\(902\) 0 0
\(903\) 4.00000 0.133112
\(904\) 0 0
\(905\) 12.0000 0.398893
\(906\) 0 0
\(907\) 44.0000 1.46100 0.730498 0.682915i \(-0.239288\pi\)
0.730498 + 0.682915i \(0.239288\pi\)
\(908\) 0 0
\(909\) 24.0000 0.796030
\(910\) 0 0
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) 0 0
\(913\) −8.00000 −0.264761
\(914\) 0 0
\(915\) −10.0000 −0.330590
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −5.00000 −0.164935 −0.0824674 0.996594i \(-0.526280\pi\)
−0.0824674 + 0.996594i \(0.526280\pi\)
\(920\) 0 0
\(921\) −5.00000 −0.164756
\(922\) 0 0
\(923\) 12.0000 0.394985
\(924\) 0 0
\(925\) 8.00000 0.263038
\(926\) 0 0
\(927\) 10.0000 0.328443
\(928\) 0 0
\(929\) −24.0000 −0.787414 −0.393707 0.919236i \(-0.628808\pi\)
−0.393707 + 0.919236i \(0.628808\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −1.00000 −0.0327035
\(936\) 0 0
\(937\) −19.0000 −0.620703 −0.310351 0.950622i \(-0.600447\pi\)
−0.310351 + 0.950622i \(0.600447\pi\)
\(938\) 0 0
\(939\) 19.0000 0.620042
\(940\) 0 0
\(941\) −50.0000 −1.62995 −0.814977 0.579494i \(-0.803250\pi\)
−0.814977 + 0.579494i \(0.803250\pi\)
\(942\) 0 0
\(943\) 8.00000 0.260516
\(944\) 0 0
\(945\) −5.00000 −0.162650
\(946\) 0 0
\(947\) −10.0000 −0.324956 −0.162478 0.986712i \(-0.551949\pi\)
−0.162478 + 0.986712i \(0.551949\pi\)
\(948\) 0 0
\(949\) 6.00000 0.194768
\(950\) 0 0
\(951\) 22.0000 0.713399
\(952\) 0 0
\(953\) 26.0000 0.842223 0.421111 0.907009i \(-0.361640\pi\)
0.421111 + 0.907009i \(0.361640\pi\)
\(954\) 0 0
\(955\) 21.0000 0.679544
\(956\) 0 0
\(957\) −1.00000 −0.0323254
\(958\) 0 0
\(959\) −2.00000 −0.0645834
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) 0 0
\(963\) 20.0000 0.644491
\(964\) 0 0
\(965\) −10.0000 −0.321911
\(966\) 0 0
\(967\) −34.0000 −1.09337 −0.546683 0.837340i \(-0.684110\pi\)
−0.546683 + 0.837340i \(0.684110\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −6.00000 −0.192549 −0.0962746 0.995355i \(-0.530693\pi\)
−0.0962746 + 0.995355i \(0.530693\pi\)
\(972\) 0 0
\(973\) 2.00000 0.0641171
\(974\) 0 0
\(975\) 1.00000 0.0320256
\(976\) 0 0
\(977\) −40.0000 −1.27971 −0.639857 0.768494i \(-0.721006\pi\)
−0.639857 + 0.768494i \(0.721006\pi\)
\(978\) 0 0
\(979\) 6.00000 0.191761
\(980\) 0 0
\(981\) 26.0000 0.830116
\(982\) 0 0
\(983\) 9.00000 0.287055 0.143528 0.989646i \(-0.454155\pi\)
0.143528 + 0.989646i \(0.454155\pi\)
\(984\) 0 0
\(985\) 6.00000 0.191176
\(986\) 0 0
\(987\) 11.0000 0.350134
\(988\) 0 0
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) −32.0000 −1.01651 −0.508257 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) 0 0
\(993\) 12.0000 0.380808
\(994\) 0 0
\(995\) −4.00000 −0.126809
\(996\) 0 0
\(997\) 55.0000 1.74187 0.870934 0.491400i \(-0.163515\pi\)
0.870934 + 0.491400i \(0.163515\pi\)
\(998\) 0 0
\(999\) −40.0000 −1.26554
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 2240.2.a.x.1.1 1
4.3 odd 2 2240.2.a.i.1.1 1
8.3 odd 2 1120.2.a.k.1.1 yes 1
8.5 even 2 1120.2.a.c.1.1 1
40.19 odd 2 5600.2.a.g.1.1 1
40.29 even 2 5600.2.a.q.1.1 1
56.13 odd 2 7840.2.a.u.1.1 1
56.27 even 2 7840.2.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1120.2.a.c.1.1 1 8.5 even 2
1120.2.a.k.1.1 yes 1 8.3 odd 2
2240.2.a.i.1.1 1 4.3 odd 2
2240.2.a.x.1.1 1 1.1 even 1 trivial
5600.2.a.g.1.1 1 40.19 odd 2
5600.2.a.q.1.1 1 40.29 even 2
7840.2.a.j.1.1 1 56.27 even 2
7840.2.a.u.1.1 1 56.13 odd 2