Properties

Label 960.4.a.y.1.1
Level $960$
Weight $4$
Character 960.1
Self dual yes
Analytic conductor $56.642$
Analytic rank $1$
Dimension $1$
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] = [960,4,Mod(1,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 960.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: \(56.6418336055\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 480)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 960.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} -5.00000 q^{5} +12.0000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -5.00000 q^{5} +12.0000 q^{7} +9.00000 q^{9} -24.0000 q^{11} -38.0000 q^{13} -15.0000 q^{15} -6.00000 q^{17} +104.000 q^{19} +36.0000 q^{21} -100.000 q^{23} +25.0000 q^{25} +27.0000 q^{27} -230.000 q^{29} +56.0000 q^{31} -72.0000 q^{33} -60.0000 q^{35} -190.000 q^{37} -114.000 q^{39} +202.000 q^{41} -148.000 q^{43} -45.0000 q^{45} -124.000 q^{47} -199.000 q^{49} -18.0000 q^{51} -206.000 q^{53} +120.000 q^{55} +312.000 q^{57} -128.000 q^{59} -190.000 q^{61} +108.000 q^{63} +190.000 q^{65} -204.000 q^{67} -300.000 q^{69} +440.000 q^{71} +1210.00 q^{73} +75.0000 q^{75} -288.000 q^{77} -816.000 q^{79} +81.0000 q^{81} -1412.00 q^{83} +30.0000 q^{85} -690.000 q^{87} -214.000 q^{89} -456.000 q^{91} +168.000 q^{93} -520.000 q^{95} +1202.00 q^{97} -216.000 q^{99} +O(q^{100})\)

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
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 12.0000 0.647939 0.323970 0.946068i \(-0.394982\pi\)
0.323970 + 0.946068i \(0.394982\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −24.0000 −0.657843 −0.328921 0.944357i \(-0.606685\pi\)
−0.328921 + 0.944357i \(0.606685\pi\)
\(12\) 0 0
\(13\) −38.0000 −0.810716 −0.405358 0.914158i \(-0.632853\pi\)
−0.405358 + 0.914158i \(0.632853\pi\)
\(14\) 0 0
\(15\) −15.0000 −0.258199
\(16\) 0 0
\(17\) −6.00000 −0.0856008 −0.0428004 0.999084i \(-0.513628\pi\)
−0.0428004 + 0.999084i \(0.513628\pi\)
\(18\) 0 0
\(19\) 104.000 1.25575 0.627875 0.778314i \(-0.283925\pi\)
0.627875 + 0.778314i \(0.283925\pi\)
\(20\) 0 0
\(21\) 36.0000 0.374088
\(22\) 0 0
\(23\) −100.000 −0.906584 −0.453292 0.891362i \(-0.649751\pi\)
−0.453292 + 0.891362i \(0.649751\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −230.000 −1.47276 −0.736378 0.676570i \(-0.763465\pi\)
−0.736378 + 0.676570i \(0.763465\pi\)
\(30\) 0 0
\(31\) 56.0000 0.324448 0.162224 0.986754i \(-0.448133\pi\)
0.162224 + 0.986754i \(0.448133\pi\)
\(32\) 0 0
\(33\) −72.0000 −0.379806
\(34\) 0 0
\(35\) −60.0000 −0.289767
\(36\) 0 0
\(37\) −190.000 −0.844211 −0.422106 0.906547i \(-0.638709\pi\)
−0.422106 + 0.906547i \(0.638709\pi\)
\(38\) 0 0
\(39\) −114.000 −0.468067
\(40\) 0 0
\(41\) 202.000 0.769441 0.384721 0.923033i \(-0.374298\pi\)
0.384721 + 0.923033i \(0.374298\pi\)
\(42\) 0 0
\(43\) −148.000 −0.524879 −0.262439 0.964948i \(-0.584527\pi\)
−0.262439 + 0.964948i \(0.584527\pi\)
\(44\) 0 0
\(45\) −45.0000 −0.149071
\(46\) 0 0
\(47\) −124.000 −0.384835 −0.192418 0.981313i \(-0.561633\pi\)
−0.192418 + 0.981313i \(0.561633\pi\)
\(48\) 0 0
\(49\) −199.000 −0.580175
\(50\) 0 0
\(51\) −18.0000 −0.0494217
\(52\) 0 0
\(53\) −206.000 −0.533892 −0.266946 0.963711i \(-0.586015\pi\)
−0.266946 + 0.963711i \(0.586015\pi\)
\(54\) 0 0
\(55\) 120.000 0.294196
\(56\) 0 0
\(57\) 312.000 0.725007
\(58\) 0 0
\(59\) −128.000 −0.282444 −0.141222 0.989978i \(-0.545103\pi\)
−0.141222 + 0.989978i \(0.545103\pi\)
\(60\) 0 0
\(61\) −190.000 −0.398803 −0.199402 0.979918i \(-0.563900\pi\)
−0.199402 + 0.979918i \(0.563900\pi\)
\(62\) 0 0
\(63\) 108.000 0.215980
\(64\) 0 0
\(65\) 190.000 0.362563
\(66\) 0 0
\(67\) −204.000 −0.371979 −0.185989 0.982552i \(-0.559549\pi\)
−0.185989 + 0.982552i \(0.559549\pi\)
\(68\) 0 0
\(69\) −300.000 −0.523417
\(70\) 0 0
\(71\) 440.000 0.735470 0.367735 0.929931i \(-0.380133\pi\)
0.367735 + 0.929931i \(0.380133\pi\)
\(72\) 0 0
\(73\) 1210.00 1.94000 0.969999 0.243111i \(-0.0781678\pi\)
0.969999 + 0.243111i \(0.0781678\pi\)
\(74\) 0 0
\(75\) 75.0000 0.115470
\(76\) 0 0
\(77\) −288.000 −0.426242
\(78\) 0 0
\(79\) −816.000 −1.16212 −0.581058 0.813862i \(-0.697361\pi\)
−0.581058 + 0.813862i \(0.697361\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −1412.00 −1.86731 −0.933657 0.358167i \(-0.883402\pi\)
−0.933657 + 0.358167i \(0.883402\pi\)
\(84\) 0 0
\(85\) 30.0000 0.0382818
\(86\) 0 0
\(87\) −690.000 −0.850296
\(88\) 0 0
\(89\) −214.000 −0.254876 −0.127438 0.991847i \(-0.540675\pi\)
−0.127438 + 0.991847i \(0.540675\pi\)
\(90\) 0 0
\(91\) −456.000 −0.525294
\(92\) 0 0
\(93\) 168.000 0.187320
\(94\) 0 0
\(95\) −520.000 −0.561588
\(96\) 0 0
\(97\) 1202.00 1.25819 0.629096 0.777328i \(-0.283425\pi\)
0.629096 + 0.777328i \(0.283425\pi\)
\(98\) 0 0
\(99\) −216.000 −0.219281
\(100\) 0 0
\(101\) −1342.00 −1.32212 −0.661059 0.750334i \(-0.729893\pi\)
−0.661059 + 0.750334i \(0.729893\pi\)
\(102\) 0 0
\(103\) −908.000 −0.868620 −0.434310 0.900763i \(-0.643008\pi\)
−0.434310 + 0.900763i \(0.643008\pi\)
\(104\) 0 0
\(105\) −180.000 −0.167297
\(106\) 0 0
\(107\) −876.000 −0.791459 −0.395730 0.918367i \(-0.629508\pi\)
−0.395730 + 0.918367i \(0.629508\pi\)
\(108\) 0 0
\(109\) −302.000 −0.265379 −0.132690 0.991158i \(-0.542361\pi\)
−0.132690 + 0.991158i \(0.542361\pi\)
\(110\) 0 0
\(111\) −570.000 −0.487405
\(112\) 0 0
\(113\) −998.000 −0.830831 −0.415416 0.909632i \(-0.636364\pi\)
−0.415416 + 0.909632i \(0.636364\pi\)
\(114\) 0 0
\(115\) 500.000 0.405437
\(116\) 0 0
\(117\) −342.000 −0.270239
\(118\) 0 0
\(119\) −72.0000 −0.0554641
\(120\) 0 0
\(121\) −755.000 −0.567243
\(122\) 0 0
\(123\) 606.000 0.444237
\(124\) 0 0
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) −836.000 −0.584118 −0.292059 0.956400i \(-0.594340\pi\)
−0.292059 + 0.956400i \(0.594340\pi\)
\(128\) 0 0
\(129\) −444.000 −0.303039
\(130\) 0 0
\(131\) −1480.00 −0.987085 −0.493543 0.869722i \(-0.664298\pi\)
−0.493543 + 0.869722i \(0.664298\pi\)
\(132\) 0 0
\(133\) 1248.00 0.813649
\(134\) 0 0
\(135\) −135.000 −0.0860663
\(136\) 0 0
\(137\) 1346.00 0.839391 0.419695 0.907665i \(-0.362137\pi\)
0.419695 + 0.907665i \(0.362137\pi\)
\(138\) 0 0
\(139\) 824.000 0.502811 0.251406 0.967882i \(-0.419107\pi\)
0.251406 + 0.967882i \(0.419107\pi\)
\(140\) 0 0
\(141\) −372.000 −0.222185
\(142\) 0 0
\(143\) 912.000 0.533324
\(144\) 0 0
\(145\) 1150.00 0.658637
\(146\) 0 0
\(147\) −597.000 −0.334964
\(148\) 0 0
\(149\) 2450.00 1.34706 0.673530 0.739160i \(-0.264777\pi\)
0.673530 + 0.739160i \(0.264777\pi\)
\(150\) 0 0
\(151\) −2696.00 −1.45296 −0.726481 0.687186i \(-0.758846\pi\)
−0.726481 + 0.687186i \(0.758846\pi\)
\(152\) 0 0
\(153\) −54.0000 −0.0285336
\(154\) 0 0
\(155\) −280.000 −0.145098
\(156\) 0 0
\(157\) 554.000 0.281618 0.140809 0.990037i \(-0.455030\pi\)
0.140809 + 0.990037i \(0.455030\pi\)
\(158\) 0 0
\(159\) −618.000 −0.308243
\(160\) 0 0
\(161\) −1200.00 −0.587411
\(162\) 0 0
\(163\) −1364.00 −0.655440 −0.327720 0.944775i \(-0.606280\pi\)
−0.327720 + 0.944775i \(0.606280\pi\)
\(164\) 0 0
\(165\) 360.000 0.169854
\(166\) 0 0
\(167\) 2004.00 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) −753.000 −0.342740
\(170\) 0 0
\(171\) 936.000 0.418583
\(172\) 0 0
\(173\) 1546.00 0.679423 0.339712 0.940530i \(-0.389671\pi\)
0.339712 + 0.940530i \(0.389671\pi\)
\(174\) 0 0
\(175\) 300.000 0.129588
\(176\) 0 0
\(177\) −384.000 −0.163069
\(178\) 0 0
\(179\) −1072.00 −0.447626 −0.223813 0.974632i \(-0.571850\pi\)
−0.223813 + 0.974632i \(0.571850\pi\)
\(180\) 0 0
\(181\) 3754.00 1.54162 0.770808 0.637067i \(-0.219853\pi\)
0.770808 + 0.637067i \(0.219853\pi\)
\(182\) 0 0
\(183\) −570.000 −0.230249
\(184\) 0 0
\(185\) 950.000 0.377543
\(186\) 0 0
\(187\) 144.000 0.0563119
\(188\) 0 0
\(189\) 324.000 0.124696
\(190\) 0 0
\(191\) −1224.00 −0.463694 −0.231847 0.972752i \(-0.574477\pi\)
−0.231847 + 0.972752i \(0.574477\pi\)
\(192\) 0 0
\(193\) −1694.00 −0.631797 −0.315898 0.948793i \(-0.602306\pi\)
−0.315898 + 0.948793i \(0.602306\pi\)
\(194\) 0 0
\(195\) 570.000 0.209326
\(196\) 0 0
\(197\) −3134.00 −1.13344 −0.566721 0.823909i \(-0.691788\pi\)
−0.566721 + 0.823909i \(0.691788\pi\)
\(198\) 0 0
\(199\) −2560.00 −0.911928 −0.455964 0.889998i \(-0.650705\pi\)
−0.455964 + 0.889998i \(0.650705\pi\)
\(200\) 0 0
\(201\) −612.000 −0.214762
\(202\) 0 0
\(203\) −2760.00 −0.954256
\(204\) 0 0
\(205\) −1010.00 −0.344105
\(206\) 0 0
\(207\) −900.000 −0.302195
\(208\) 0 0
\(209\) −2496.00 −0.826086
\(210\) 0 0
\(211\) −1856.00 −0.605556 −0.302778 0.953061i \(-0.597914\pi\)
−0.302778 + 0.953061i \(0.597914\pi\)
\(212\) 0 0
\(213\) 1320.00 0.424624
\(214\) 0 0
\(215\) 740.000 0.234733
\(216\) 0 0
\(217\) 672.000 0.210223
\(218\) 0 0
\(219\) 3630.00 1.12006
\(220\) 0 0
\(221\) 228.000 0.0693979
\(222\) 0 0
\(223\) −1596.00 −0.479265 −0.239632 0.970864i \(-0.577027\pi\)
−0.239632 + 0.970864i \(0.577027\pi\)
\(224\) 0 0
\(225\) 225.000 0.0666667
\(226\) 0 0
\(227\) −3996.00 −1.16839 −0.584193 0.811614i \(-0.698589\pi\)
−0.584193 + 0.811614i \(0.698589\pi\)
\(228\) 0 0
\(229\) 6090.00 1.75737 0.878687 0.477399i \(-0.158420\pi\)
0.878687 + 0.477399i \(0.158420\pi\)
\(230\) 0 0
\(231\) −864.000 −0.246091
\(232\) 0 0
\(233\) −894.000 −0.251364 −0.125682 0.992071i \(-0.540112\pi\)
−0.125682 + 0.992071i \(0.540112\pi\)
\(234\) 0 0
\(235\) 620.000 0.172104
\(236\) 0 0
\(237\) −2448.00 −0.670948
\(238\) 0 0
\(239\) 6336.00 1.71482 0.857410 0.514635i \(-0.172072\pi\)
0.857410 + 0.514635i \(0.172072\pi\)
\(240\) 0 0
\(241\) 338.000 0.0903423 0.0451711 0.998979i \(-0.485617\pi\)
0.0451711 + 0.998979i \(0.485617\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 995.000 0.259462
\(246\) 0 0
\(247\) −3952.00 −1.01806
\(248\) 0 0
\(249\) −4236.00 −1.07809
\(250\) 0 0
\(251\) −4872.00 −1.22517 −0.612585 0.790404i \(-0.709870\pi\)
−0.612585 + 0.790404i \(0.709870\pi\)
\(252\) 0 0
\(253\) 2400.00 0.596390
\(254\) 0 0
\(255\) 90.0000 0.0221020
\(256\) 0 0
\(257\) −1878.00 −0.455823 −0.227911 0.973682i \(-0.573190\pi\)
−0.227911 + 0.973682i \(0.573190\pi\)
\(258\) 0 0
\(259\) −2280.00 −0.546997
\(260\) 0 0
\(261\) −2070.00 −0.490919
\(262\) 0 0
\(263\) −2244.00 −0.526125 −0.263063 0.964779i \(-0.584733\pi\)
−0.263063 + 0.964779i \(0.584733\pi\)
\(264\) 0 0
\(265\) 1030.00 0.238764
\(266\) 0 0
\(267\) −642.000 −0.147153
\(268\) 0 0
\(269\) 4314.00 0.977804 0.488902 0.872339i \(-0.337398\pi\)
0.488902 + 0.872339i \(0.337398\pi\)
\(270\) 0 0
\(271\) 6392.00 1.43279 0.716395 0.697694i \(-0.245791\pi\)
0.716395 + 0.697694i \(0.245791\pi\)
\(272\) 0 0
\(273\) −1368.00 −0.303279
\(274\) 0 0
\(275\) −600.000 −0.131569
\(276\) 0 0
\(277\) −4398.00 −0.953972 −0.476986 0.878911i \(-0.658271\pi\)
−0.476986 + 0.878911i \(0.658271\pi\)
\(278\) 0 0
\(279\) 504.000 0.108149
\(280\) 0 0
\(281\) −7622.00 −1.61812 −0.809058 0.587729i \(-0.800022\pi\)
−0.809058 + 0.587729i \(0.800022\pi\)
\(282\) 0 0
\(283\) 1020.00 0.214250 0.107125 0.994246i \(-0.465836\pi\)
0.107125 + 0.994246i \(0.465836\pi\)
\(284\) 0 0
\(285\) −1560.00 −0.324233
\(286\) 0 0
\(287\) 2424.00 0.498551
\(288\) 0 0
\(289\) −4877.00 −0.992673
\(290\) 0 0
\(291\) 3606.00 0.726417
\(292\) 0 0
\(293\) 3746.00 0.746907 0.373453 0.927649i \(-0.378174\pi\)
0.373453 + 0.927649i \(0.378174\pi\)
\(294\) 0 0
\(295\) 640.000 0.126313
\(296\) 0 0
\(297\) −648.000 −0.126602
\(298\) 0 0
\(299\) 3800.00 0.734982
\(300\) 0 0
\(301\) −1776.00 −0.340089
\(302\) 0 0
\(303\) −4026.00 −0.763326
\(304\) 0 0
\(305\) 950.000 0.178350
\(306\) 0 0
\(307\) 9700.00 1.80328 0.901642 0.432483i \(-0.142362\pi\)
0.901642 + 0.432483i \(0.142362\pi\)
\(308\) 0 0
\(309\) −2724.00 −0.501498
\(310\) 0 0
\(311\) −4152.00 −0.757036 −0.378518 0.925594i \(-0.623566\pi\)
−0.378518 + 0.925594i \(0.623566\pi\)
\(312\) 0 0
\(313\) 6362.00 1.14889 0.574443 0.818544i \(-0.305219\pi\)
0.574443 + 0.818544i \(0.305219\pi\)
\(314\) 0 0
\(315\) −540.000 −0.0965891
\(316\) 0 0
\(317\) −10886.0 −1.92877 −0.964383 0.264511i \(-0.914790\pi\)
−0.964383 + 0.264511i \(0.914790\pi\)
\(318\) 0 0
\(319\) 5520.00 0.968842
\(320\) 0 0
\(321\) −2628.00 −0.456949
\(322\) 0 0
\(323\) −624.000 −0.107493
\(324\) 0 0
\(325\) −950.000 −0.162143
\(326\) 0 0
\(327\) −906.000 −0.153217
\(328\) 0 0
\(329\) −1488.00 −0.249350
\(330\) 0 0
\(331\) 4128.00 0.685485 0.342742 0.939429i \(-0.388644\pi\)
0.342742 + 0.939429i \(0.388644\pi\)
\(332\) 0 0
\(333\) −1710.00 −0.281404
\(334\) 0 0
\(335\) 1020.00 0.166354
\(336\) 0 0
\(337\) 12002.0 1.94003 0.970016 0.243042i \(-0.0781453\pi\)
0.970016 + 0.243042i \(0.0781453\pi\)
\(338\) 0 0
\(339\) −2994.00 −0.479681
\(340\) 0 0
\(341\) −1344.00 −0.213436
\(342\) 0 0
\(343\) −6504.00 −1.02386
\(344\) 0 0
\(345\) 1500.00 0.234079
\(346\) 0 0
\(347\) 6276.00 0.970932 0.485466 0.874256i \(-0.338650\pi\)
0.485466 + 0.874256i \(0.338650\pi\)
\(348\) 0 0
\(349\) 9362.00 1.43592 0.717960 0.696084i \(-0.245076\pi\)
0.717960 + 0.696084i \(0.245076\pi\)
\(350\) 0 0
\(351\) −1026.00 −0.156022
\(352\) 0 0
\(353\) −838.000 −0.126352 −0.0631760 0.998002i \(-0.520123\pi\)
−0.0631760 + 0.998002i \(0.520123\pi\)
\(354\) 0 0
\(355\) −2200.00 −0.328912
\(356\) 0 0
\(357\) −216.000 −0.0320222
\(358\) 0 0
\(359\) 6896.00 1.01381 0.506904 0.862003i \(-0.330790\pi\)
0.506904 + 0.862003i \(0.330790\pi\)
\(360\) 0 0
\(361\) 3957.00 0.576906
\(362\) 0 0
\(363\) −2265.00 −0.327498
\(364\) 0 0
\(365\) −6050.00 −0.867593
\(366\) 0 0
\(367\) 1132.00 0.161008 0.0805040 0.996754i \(-0.474347\pi\)
0.0805040 + 0.996754i \(0.474347\pi\)
\(368\) 0 0
\(369\) 1818.00 0.256480
\(370\) 0 0
\(371\) −2472.00 −0.345930
\(372\) 0 0
\(373\) 12578.0 1.74602 0.873008 0.487705i \(-0.162166\pi\)
0.873008 + 0.487705i \(0.162166\pi\)
\(374\) 0 0
\(375\) −375.000 −0.0516398
\(376\) 0 0
\(377\) 8740.00 1.19399
\(378\) 0 0
\(379\) 11752.0 1.59277 0.796385 0.604790i \(-0.206743\pi\)
0.796385 + 0.604790i \(0.206743\pi\)
\(380\) 0 0
\(381\) −2508.00 −0.337241
\(382\) 0 0
\(383\) 7372.00 0.983529 0.491764 0.870728i \(-0.336352\pi\)
0.491764 + 0.870728i \(0.336352\pi\)
\(384\) 0 0
\(385\) 1440.00 0.190621
\(386\) 0 0
\(387\) −1332.00 −0.174960
\(388\) 0 0
\(389\) −6654.00 −0.867278 −0.433639 0.901087i \(-0.642771\pi\)
−0.433639 + 0.901087i \(0.642771\pi\)
\(390\) 0 0
\(391\) 600.000 0.0776044
\(392\) 0 0
\(393\) −4440.00 −0.569894
\(394\) 0 0
\(395\) 4080.00 0.519714
\(396\) 0 0
\(397\) −4278.00 −0.540823 −0.270411 0.962745i \(-0.587160\pi\)
−0.270411 + 0.962745i \(0.587160\pi\)
\(398\) 0 0
\(399\) 3744.00 0.469761
\(400\) 0 0
\(401\) 9074.00 1.13001 0.565005 0.825088i \(-0.308874\pi\)
0.565005 + 0.825088i \(0.308874\pi\)
\(402\) 0 0
\(403\) −2128.00 −0.263035
\(404\) 0 0
\(405\) −405.000 −0.0496904
\(406\) 0 0
\(407\) 4560.00 0.555358
\(408\) 0 0
\(409\) 6682.00 0.807833 0.403916 0.914796i \(-0.367649\pi\)
0.403916 + 0.914796i \(0.367649\pi\)
\(410\) 0 0
\(411\) 4038.00 0.484623
\(412\) 0 0
\(413\) −1536.00 −0.183006
\(414\) 0 0
\(415\) 7060.00 0.835089
\(416\) 0 0
\(417\) 2472.00 0.290298
\(418\) 0 0
\(419\) 4832.00 0.563386 0.281693 0.959505i \(-0.409104\pi\)
0.281693 + 0.959505i \(0.409104\pi\)
\(420\) 0 0
\(421\) −3974.00 −0.460050 −0.230025 0.973185i \(-0.573881\pi\)
−0.230025 + 0.973185i \(0.573881\pi\)
\(422\) 0 0
\(423\) −1116.00 −0.128278
\(424\) 0 0
\(425\) −150.000 −0.0171202
\(426\) 0 0
\(427\) −2280.00 −0.258400
\(428\) 0 0
\(429\) 2736.00 0.307915
\(430\) 0 0
\(431\) −1112.00 −0.124276 −0.0621382 0.998068i \(-0.519792\pi\)
−0.0621382 + 0.998068i \(0.519792\pi\)
\(432\) 0 0
\(433\) 1106.00 0.122751 0.0613753 0.998115i \(-0.480451\pi\)
0.0613753 + 0.998115i \(0.480451\pi\)
\(434\) 0 0
\(435\) 3450.00 0.380264
\(436\) 0 0
\(437\) −10400.0 −1.13844
\(438\) 0 0
\(439\) 9280.00 1.00891 0.504454 0.863439i \(-0.331694\pi\)
0.504454 + 0.863439i \(0.331694\pi\)
\(440\) 0 0
\(441\) −1791.00 −0.193392
\(442\) 0 0
\(443\) 7004.00 0.751174 0.375587 0.926787i \(-0.377441\pi\)
0.375587 + 0.926787i \(0.377441\pi\)
\(444\) 0 0
\(445\) 1070.00 0.113984
\(446\) 0 0
\(447\) 7350.00 0.777725
\(448\) 0 0
\(449\) −11502.0 −1.20894 −0.604469 0.796629i \(-0.706615\pi\)
−0.604469 + 0.796629i \(0.706615\pi\)
\(450\) 0 0
\(451\) −4848.00 −0.506172
\(452\) 0 0
\(453\) −8088.00 −0.838868
\(454\) 0 0
\(455\) 2280.00 0.234919
\(456\) 0 0
\(457\) 11578.0 1.18511 0.592556 0.805529i \(-0.298119\pi\)
0.592556 + 0.805529i \(0.298119\pi\)
\(458\) 0 0
\(459\) −162.000 −0.0164739
\(460\) 0 0
\(461\) 6362.00 0.642750 0.321375 0.946952i \(-0.395855\pi\)
0.321375 + 0.946952i \(0.395855\pi\)
\(462\) 0 0
\(463\) −2892.00 −0.290286 −0.145143 0.989411i \(-0.546364\pi\)
−0.145143 + 0.989411i \(0.546364\pi\)
\(464\) 0 0
\(465\) −840.000 −0.0837722
\(466\) 0 0
\(467\) −11036.0 −1.09354 −0.546772 0.837281i \(-0.684144\pi\)
−0.546772 + 0.837281i \(0.684144\pi\)
\(468\) 0 0
\(469\) −2448.00 −0.241019
\(470\) 0 0
\(471\) 1662.00 0.162592
\(472\) 0 0
\(473\) 3552.00 0.345288
\(474\) 0 0
\(475\) 2600.00 0.251150
\(476\) 0 0
\(477\) −1854.00 −0.177964
\(478\) 0 0
\(479\) 13664.0 1.30339 0.651695 0.758481i \(-0.274058\pi\)
0.651695 + 0.758481i \(0.274058\pi\)
\(480\) 0 0
\(481\) 7220.00 0.684415
\(482\) 0 0
\(483\) −3600.00 −0.339142
\(484\) 0 0
\(485\) −6010.00 −0.562680
\(486\) 0 0
\(487\) −4820.00 −0.448491 −0.224245 0.974533i \(-0.571992\pi\)
−0.224245 + 0.974533i \(0.571992\pi\)
\(488\) 0 0
\(489\) −4092.00 −0.378418
\(490\) 0 0
\(491\) 17464.0 1.60517 0.802586 0.596537i \(-0.203457\pi\)
0.802586 + 0.596537i \(0.203457\pi\)
\(492\) 0 0
\(493\) 1380.00 0.126069
\(494\) 0 0
\(495\) 1080.00 0.0980654
\(496\) 0 0
\(497\) 5280.00 0.476540
\(498\) 0 0
\(499\) 13960.0 1.25238 0.626188 0.779672i \(-0.284614\pi\)
0.626188 + 0.779672i \(0.284614\pi\)
\(500\) 0 0
\(501\) 6012.00 0.536120
\(502\) 0 0
\(503\) −20388.0 −1.80727 −0.903634 0.428305i \(-0.859111\pi\)
−0.903634 + 0.428305i \(0.859111\pi\)
\(504\) 0 0
\(505\) 6710.00 0.591269
\(506\) 0 0
\(507\) −2259.00 −0.197881
\(508\) 0 0
\(509\) 12954.0 1.12805 0.564024 0.825759i \(-0.309253\pi\)
0.564024 + 0.825759i \(0.309253\pi\)
\(510\) 0 0
\(511\) 14520.0 1.25700
\(512\) 0 0
\(513\) 2808.00 0.241669
\(514\) 0 0
\(515\) 4540.00 0.388459
\(516\) 0 0
\(517\) 2976.00 0.253161
\(518\) 0 0
\(519\) 4638.00 0.392265
\(520\) 0 0
\(521\) −3542.00 −0.297846 −0.148923 0.988849i \(-0.547581\pi\)
−0.148923 + 0.988849i \(0.547581\pi\)
\(522\) 0 0
\(523\) 1532.00 0.128087 0.0640437 0.997947i \(-0.479600\pi\)
0.0640437 + 0.997947i \(0.479600\pi\)
\(524\) 0 0
\(525\) 900.000 0.0748176
\(526\) 0 0
\(527\) −336.000 −0.0277730
\(528\) 0 0
\(529\) −2167.00 −0.178105
\(530\) 0 0
\(531\) −1152.00 −0.0941479
\(532\) 0 0
\(533\) −7676.00 −0.623798
\(534\) 0 0
\(535\) 4380.00 0.353951
\(536\) 0 0
\(537\) −3216.00 −0.258437
\(538\) 0 0
\(539\) 4776.00 0.381664
\(540\) 0 0
\(541\) 11826.0 0.939814 0.469907 0.882716i \(-0.344287\pi\)
0.469907 + 0.882716i \(0.344287\pi\)
\(542\) 0 0
\(543\) 11262.0 0.890053
\(544\) 0 0
\(545\) 1510.00 0.118681
\(546\) 0 0
\(547\) −11260.0 −0.880151 −0.440076 0.897961i \(-0.645048\pi\)
−0.440076 + 0.897961i \(0.645048\pi\)
\(548\) 0 0
\(549\) −1710.00 −0.132934
\(550\) 0 0
\(551\) −23920.0 −1.84941
\(552\) 0 0
\(553\) −9792.00 −0.752980
\(554\) 0 0
\(555\) 2850.00 0.217974
\(556\) 0 0
\(557\) −4374.00 −0.332733 −0.166367 0.986064i \(-0.553203\pi\)
−0.166367 + 0.986064i \(0.553203\pi\)
\(558\) 0 0
\(559\) 5624.00 0.425527
\(560\) 0 0
\(561\) 432.000 0.0325117
\(562\) 0 0
\(563\) −11604.0 −0.868651 −0.434325 0.900756i \(-0.643013\pi\)
−0.434325 + 0.900756i \(0.643013\pi\)
\(564\) 0 0
\(565\) 4990.00 0.371559
\(566\) 0 0
\(567\) 972.000 0.0719932
\(568\) 0 0
\(569\) −7990.00 −0.588679 −0.294339 0.955701i \(-0.595100\pi\)
−0.294339 + 0.955701i \(0.595100\pi\)
\(570\) 0 0
\(571\) −26080.0 −1.91141 −0.955704 0.294329i \(-0.904904\pi\)
−0.955704 + 0.294329i \(0.904904\pi\)
\(572\) 0 0
\(573\) −3672.00 −0.267714
\(574\) 0 0
\(575\) −2500.00 −0.181317
\(576\) 0 0
\(577\) 13922.0 1.00447 0.502236 0.864731i \(-0.332511\pi\)
0.502236 + 0.864731i \(0.332511\pi\)
\(578\) 0 0
\(579\) −5082.00 −0.364768
\(580\) 0 0
\(581\) −16944.0 −1.20991
\(582\) 0 0
\(583\) 4944.00 0.351217
\(584\) 0 0
\(585\) 1710.00 0.120854
\(586\) 0 0
\(587\) 26340.0 1.85208 0.926038 0.377431i \(-0.123193\pi\)
0.926038 + 0.377431i \(0.123193\pi\)
\(588\) 0 0
\(589\) 5824.00 0.407426
\(590\) 0 0
\(591\) −9402.00 −0.654394
\(592\) 0 0
\(593\) −9478.00 −0.656349 −0.328174 0.944617i \(-0.606433\pi\)
−0.328174 + 0.944617i \(0.606433\pi\)
\(594\) 0 0
\(595\) 360.000 0.0248043
\(596\) 0 0
\(597\) −7680.00 −0.526502
\(598\) 0 0
\(599\) −6528.00 −0.445287 −0.222643 0.974900i \(-0.571469\pi\)
−0.222643 + 0.974900i \(0.571469\pi\)
\(600\) 0 0
\(601\) 2090.00 0.141852 0.0709259 0.997482i \(-0.477405\pi\)
0.0709259 + 0.997482i \(0.477405\pi\)
\(602\) 0 0
\(603\) −1836.00 −0.123993
\(604\) 0 0
\(605\) 3775.00 0.253679
\(606\) 0 0
\(607\) −8788.00 −0.587634 −0.293817 0.955862i \(-0.594926\pi\)
−0.293817 + 0.955862i \(0.594926\pi\)
\(608\) 0 0
\(609\) −8280.00 −0.550940
\(610\) 0 0
\(611\) 4712.00 0.311992
\(612\) 0 0
\(613\) 2626.00 0.173023 0.0865115 0.996251i \(-0.472428\pi\)
0.0865115 + 0.996251i \(0.472428\pi\)
\(614\) 0 0
\(615\) −3030.00 −0.198669
\(616\) 0 0
\(617\) −29214.0 −1.90618 −0.953089 0.302691i \(-0.902115\pi\)
−0.953089 + 0.302691i \(0.902115\pi\)
\(618\) 0 0
\(619\) 22984.0 1.49242 0.746208 0.665713i \(-0.231873\pi\)
0.746208 + 0.665713i \(0.231873\pi\)
\(620\) 0 0
\(621\) −2700.00 −0.174472
\(622\) 0 0
\(623\) −2568.00 −0.165144
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) −7488.00 −0.476941
\(628\) 0 0
\(629\) 1140.00 0.0722651
\(630\) 0 0
\(631\) −4472.00 −0.282136 −0.141068 0.990000i \(-0.545054\pi\)
−0.141068 + 0.990000i \(0.545054\pi\)
\(632\) 0 0
\(633\) −5568.00 −0.349618
\(634\) 0 0
\(635\) 4180.00 0.261226
\(636\) 0 0
\(637\) 7562.00 0.470357
\(638\) 0 0
\(639\) 3960.00 0.245157
\(640\) 0 0
\(641\) −8798.00 −0.542122 −0.271061 0.962562i \(-0.587374\pi\)
−0.271061 + 0.962562i \(0.587374\pi\)
\(642\) 0 0
\(643\) −29428.0 −1.80486 −0.902432 0.430833i \(-0.858220\pi\)
−0.902432 + 0.430833i \(0.858220\pi\)
\(644\) 0 0
\(645\) 2220.00 0.135523
\(646\) 0 0
\(647\) −4860.00 −0.295311 −0.147656 0.989039i \(-0.547173\pi\)
−0.147656 + 0.989039i \(0.547173\pi\)
\(648\) 0 0
\(649\) 3072.00 0.185804
\(650\) 0 0
\(651\) 2016.00 0.121372
\(652\) 0 0
\(653\) 6570.00 0.393727 0.196864 0.980431i \(-0.436924\pi\)
0.196864 + 0.980431i \(0.436924\pi\)
\(654\) 0 0
\(655\) 7400.00 0.441438
\(656\) 0 0
\(657\) 10890.0 0.646666
\(658\) 0 0
\(659\) 26496.0 1.56622 0.783109 0.621885i \(-0.213633\pi\)
0.783109 + 0.621885i \(0.213633\pi\)
\(660\) 0 0
\(661\) 19642.0 1.15580 0.577901 0.816107i \(-0.303872\pi\)
0.577901 + 0.816107i \(0.303872\pi\)
\(662\) 0 0
\(663\) 684.000 0.0400669
\(664\) 0 0
\(665\) −6240.00 −0.363875
\(666\) 0 0
\(667\) 23000.0 1.33518
\(668\) 0 0
\(669\) −4788.00 −0.276704
\(670\) 0 0
\(671\) 4560.00 0.262350
\(672\) 0 0
\(673\) −19582.0 −1.12159 −0.560795 0.827954i \(-0.689505\pi\)
−0.560795 + 0.827954i \(0.689505\pi\)
\(674\) 0 0
\(675\) 675.000 0.0384900
\(676\) 0 0
\(677\) 22914.0 1.30082 0.650411 0.759582i \(-0.274597\pi\)
0.650411 + 0.759582i \(0.274597\pi\)
\(678\) 0 0
\(679\) 14424.0 0.815232
\(680\) 0 0
\(681\) −11988.0 −0.674569
\(682\) 0 0
\(683\) −13764.0 −0.771105 −0.385553 0.922686i \(-0.625989\pi\)
−0.385553 + 0.922686i \(0.625989\pi\)
\(684\) 0 0
\(685\) −6730.00 −0.375387
\(686\) 0 0
\(687\) 18270.0 1.01462
\(688\) 0 0
\(689\) 7828.00 0.432835
\(690\) 0 0
\(691\) −34688.0 −1.90969 −0.954843 0.297109i \(-0.903977\pi\)
−0.954843 + 0.297109i \(0.903977\pi\)
\(692\) 0 0
\(693\) −2592.00 −0.142081
\(694\) 0 0
\(695\) −4120.00 −0.224864
\(696\) 0 0
\(697\) −1212.00 −0.0658648
\(698\) 0 0
\(699\) −2682.00 −0.145125
\(700\) 0 0
\(701\) 17226.0 0.928127 0.464064 0.885802i \(-0.346391\pi\)
0.464064 + 0.885802i \(0.346391\pi\)
\(702\) 0 0
\(703\) −19760.0 −1.06012
\(704\) 0 0
\(705\) 1860.00 0.0993640
\(706\) 0 0
\(707\) −16104.0 −0.856652
\(708\) 0 0
\(709\) 12970.0 0.687022 0.343511 0.939149i \(-0.388384\pi\)
0.343511 + 0.939149i \(0.388384\pi\)
\(710\) 0 0
\(711\) −7344.00 −0.387372
\(712\) 0 0
\(713\) −5600.00 −0.294140
\(714\) 0 0
\(715\) −4560.00 −0.238510
\(716\) 0 0
\(717\) 19008.0 0.990051
\(718\) 0 0
\(719\) −10832.0 −0.561843 −0.280922 0.959731i \(-0.590640\pi\)
−0.280922 + 0.959731i \(0.590640\pi\)
\(720\) 0 0
\(721\) −10896.0 −0.562813
\(722\) 0 0
\(723\) 1014.00 0.0521592
\(724\) 0 0
\(725\) −5750.00 −0.294551
\(726\) 0 0
\(727\) −35588.0 −1.81552 −0.907762 0.419486i \(-0.862210\pi\)
−0.907762 + 0.419486i \(0.862210\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 888.000 0.0449300
\(732\) 0 0
\(733\) −19238.0 −0.969402 −0.484701 0.874680i \(-0.661072\pi\)
−0.484701 + 0.874680i \(0.661072\pi\)
\(734\) 0 0
\(735\) 2985.00 0.149801
\(736\) 0 0
\(737\) 4896.00 0.244703
\(738\) 0 0
\(739\) −4072.00 −0.202694 −0.101347 0.994851i \(-0.532315\pi\)
−0.101347 + 0.994851i \(0.532315\pi\)
\(740\) 0 0
\(741\) −11856.0 −0.587775
\(742\) 0 0
\(743\) −31268.0 −1.54389 −0.771946 0.635688i \(-0.780716\pi\)
−0.771946 + 0.635688i \(0.780716\pi\)
\(744\) 0 0
\(745\) −12250.0 −0.602423
\(746\) 0 0
\(747\) −12708.0 −0.622438
\(748\) 0 0
\(749\) −10512.0 −0.512817
\(750\) 0 0
\(751\) 28216.0 1.37099 0.685497 0.728075i \(-0.259585\pi\)
0.685497 + 0.728075i \(0.259585\pi\)
\(752\) 0 0
\(753\) −14616.0 −0.707353
\(754\) 0 0
\(755\) 13480.0 0.649785
\(756\) 0 0
\(757\) 33874.0 1.62638 0.813191 0.581997i \(-0.197728\pi\)
0.813191 + 0.581997i \(0.197728\pi\)
\(758\) 0 0
\(759\) 7200.00 0.344326
\(760\) 0 0
\(761\) 5466.00 0.260371 0.130186 0.991490i \(-0.458443\pi\)
0.130186 + 0.991490i \(0.458443\pi\)
\(762\) 0 0
\(763\) −3624.00 −0.171950
\(764\) 0 0
\(765\) 270.000 0.0127606
\(766\) 0 0
\(767\) 4864.00 0.228982
\(768\) 0 0
\(769\) −8878.00 −0.416318 −0.208159 0.978095i \(-0.566747\pi\)
−0.208159 + 0.978095i \(0.566747\pi\)
\(770\) 0 0
\(771\) −5634.00 −0.263169
\(772\) 0 0
\(773\) 9538.00 0.443801 0.221900 0.975069i \(-0.428774\pi\)
0.221900 + 0.975069i \(0.428774\pi\)
\(774\) 0 0
\(775\) 1400.00 0.0648897
\(776\) 0 0
\(777\) −6840.00 −0.315809
\(778\) 0 0
\(779\) 21008.0 0.966226
\(780\) 0 0
\(781\) −10560.0 −0.483824
\(782\) 0 0
\(783\) −6210.00 −0.283432
\(784\) 0 0
\(785\) −2770.00 −0.125943
\(786\) 0 0
\(787\) −5404.00 −0.244767 −0.122384 0.992483i \(-0.539054\pi\)
−0.122384 + 0.992483i \(0.539054\pi\)
\(788\) 0 0
\(789\) −6732.00 −0.303759
\(790\) 0 0
\(791\) −11976.0 −0.538328
\(792\) 0 0
\(793\) 7220.00 0.323316
\(794\) 0 0
\(795\) 3090.00 0.137850
\(796\) 0 0
\(797\) −16326.0 −0.725592 −0.362796 0.931869i \(-0.618178\pi\)
−0.362796 + 0.931869i \(0.618178\pi\)
\(798\) 0 0
\(799\) 744.000 0.0329422
\(800\) 0 0
\(801\) −1926.00 −0.0849586
\(802\) 0 0
\(803\) −29040.0 −1.27621
\(804\) 0 0
\(805\) 6000.00 0.262698
\(806\) 0 0
\(807\) 12942.0 0.564535
\(808\) 0 0
\(809\) 26202.0 1.13871 0.569353 0.822093i \(-0.307194\pi\)
0.569353 + 0.822093i \(0.307194\pi\)
\(810\) 0 0
\(811\) −26208.0 −1.13476 −0.567378 0.823457i \(-0.692042\pi\)
−0.567378 + 0.823457i \(0.692042\pi\)
\(812\) 0 0
\(813\) 19176.0 0.827222
\(814\) 0 0
\(815\) 6820.00 0.293122
\(816\) 0 0
\(817\) −15392.0 −0.659116
\(818\) 0 0
\(819\) −4104.00 −0.175098
\(820\) 0 0
\(821\) 1986.00 0.0844237 0.0422119 0.999109i \(-0.486560\pi\)
0.0422119 + 0.999109i \(0.486560\pi\)
\(822\) 0 0
\(823\) 5236.00 0.221769 0.110884 0.993833i \(-0.464632\pi\)
0.110884 + 0.993833i \(0.464632\pi\)
\(824\) 0 0
\(825\) −1800.00 −0.0759612
\(826\) 0 0
\(827\) −26044.0 −1.09509 −0.547545 0.836777i \(-0.684437\pi\)
−0.547545 + 0.836777i \(0.684437\pi\)
\(828\) 0 0
\(829\) −21246.0 −0.890113 −0.445057 0.895502i \(-0.646816\pi\)
−0.445057 + 0.895502i \(0.646816\pi\)
\(830\) 0 0
\(831\) −13194.0 −0.550776
\(832\) 0 0
\(833\) 1194.00 0.0496634
\(834\) 0 0
\(835\) −10020.0 −0.415277
\(836\) 0 0
\(837\) 1512.00 0.0624401
\(838\) 0 0
\(839\) 17440.0 0.717635 0.358817 0.933408i \(-0.383180\pi\)
0.358817 + 0.933408i \(0.383180\pi\)
\(840\) 0 0
\(841\) 28511.0 1.16901
\(842\) 0 0
\(843\) −22866.0 −0.934219
\(844\) 0 0
\(845\) 3765.00 0.153278
\(846\) 0 0
\(847\) −9060.00 −0.367539
\(848\) 0 0
\(849\) 3060.00 0.123697
\(850\) 0 0
\(851\) 19000.0 0.765349
\(852\) 0 0
\(853\) −33582.0 −1.34798 −0.673989 0.738741i \(-0.735421\pi\)
−0.673989 + 0.738741i \(0.735421\pi\)
\(854\) 0 0
\(855\) −4680.00 −0.187196
\(856\) 0 0
\(857\) −18942.0 −0.755013 −0.377507 0.926007i \(-0.623218\pi\)
−0.377507 + 0.926007i \(0.623218\pi\)
\(858\) 0 0
\(859\) 25720.0 1.02160 0.510800 0.859699i \(-0.329349\pi\)
0.510800 + 0.859699i \(0.329349\pi\)
\(860\) 0 0
\(861\) 7272.00 0.287839
\(862\) 0 0
\(863\) −32436.0 −1.27941 −0.639707 0.768619i \(-0.720944\pi\)
−0.639707 + 0.768619i \(0.720944\pi\)
\(864\) 0 0
\(865\) −7730.00 −0.303847
\(866\) 0 0
\(867\) −14631.0 −0.573120
\(868\) 0 0
\(869\) 19584.0 0.764490
\(870\) 0 0
\(871\) 7752.00 0.301569
\(872\) 0 0
\(873\) 10818.0 0.419397
\(874\) 0 0
\(875\) −1500.00 −0.0579534
\(876\) 0 0
\(877\) −8646.00 −0.332902 −0.166451 0.986050i \(-0.553231\pi\)
−0.166451 + 0.986050i \(0.553231\pi\)
\(878\) 0 0
\(879\) 11238.0 0.431227
\(880\) 0 0
\(881\) 27442.0 1.04943 0.524713 0.851279i \(-0.324173\pi\)
0.524713 + 0.851279i \(0.324173\pi\)
\(882\) 0 0
\(883\) −14116.0 −0.537986 −0.268993 0.963142i \(-0.586691\pi\)
−0.268993 + 0.963142i \(0.586691\pi\)
\(884\) 0 0
\(885\) 1920.00 0.0729267
\(886\) 0 0
\(887\) −4124.00 −0.156111 −0.0780554 0.996949i \(-0.524871\pi\)
−0.0780554 + 0.996949i \(0.524871\pi\)
\(888\) 0 0
\(889\) −10032.0 −0.378473
\(890\) 0 0
\(891\) −1944.00 −0.0730937
\(892\) 0 0
\(893\) −12896.0 −0.483257
\(894\) 0 0
\(895\) 5360.00 0.200184
\(896\) 0 0
\(897\) 11400.0 0.424342
\(898\) 0 0
\(899\) −12880.0 −0.477833
\(900\) 0 0
\(901\) 1236.00 0.0457016
\(902\) 0 0
\(903\) −5328.00 −0.196351
\(904\) 0 0
\(905\) −18770.0 −0.689432
\(906\) 0 0
\(907\) 42100.0 1.54124 0.770622 0.637293i \(-0.219946\pi\)
0.770622 + 0.637293i \(0.219946\pi\)
\(908\) 0 0
\(909\) −12078.0 −0.440706
\(910\) 0 0
\(911\) −34152.0 −1.24205 −0.621024 0.783791i \(-0.713283\pi\)
−0.621024 + 0.783791i \(0.713283\pi\)
\(912\) 0 0
\(913\) 33888.0 1.22840
\(914\) 0 0
\(915\) 2850.00 0.102971
\(916\) 0 0
\(917\) −17760.0 −0.639571
\(918\) 0 0
\(919\) −41984.0 −1.50699 −0.753495 0.657453i \(-0.771634\pi\)
−0.753495 + 0.657453i \(0.771634\pi\)
\(920\) 0 0
\(921\) 29100.0 1.04113
\(922\) 0 0
\(923\) −16720.0 −0.596257
\(924\) 0 0
\(925\) −4750.00 −0.168842
\(926\) 0 0
\(927\) −8172.00 −0.289540
\(928\) 0 0
\(929\) 48434.0 1.71051 0.855257 0.518204i \(-0.173399\pi\)
0.855257 + 0.518204i \(0.173399\pi\)
\(930\) 0 0
\(931\) −20696.0 −0.728554
\(932\) 0 0
\(933\) −12456.0 −0.437075
\(934\) 0 0
\(935\) −720.000 −0.0251834
\(936\) 0 0
\(937\) −25590.0 −0.892197 −0.446099 0.894984i \(-0.647187\pi\)
−0.446099 + 0.894984i \(0.647187\pi\)
\(938\) 0 0
\(939\) 19086.0 0.663310
\(940\) 0 0
\(941\) 18906.0 0.654961 0.327480 0.944858i \(-0.393800\pi\)
0.327480 + 0.944858i \(0.393800\pi\)
\(942\) 0 0
\(943\) −20200.0 −0.697564
\(944\) 0 0
\(945\) −1620.00 −0.0557657
\(946\) 0 0
\(947\) −36316.0 −1.24616 −0.623079 0.782159i \(-0.714118\pi\)
−0.623079 + 0.782159i \(0.714118\pi\)
\(948\) 0 0
\(949\) −45980.0 −1.57279
\(950\) 0 0
\(951\) −32658.0 −1.11357
\(952\) 0 0
\(953\) 15890.0 0.540113 0.270056 0.962844i \(-0.412958\pi\)
0.270056 + 0.962844i \(0.412958\pi\)
\(954\) 0 0
\(955\) 6120.00 0.207370
\(956\) 0 0
\(957\) 16560.0 0.559361
\(958\) 0 0
\(959\) 16152.0 0.543874
\(960\) 0 0
\(961\) −26655.0 −0.894733
\(962\) 0 0
\(963\) −7884.00 −0.263820
\(964\) 0 0
\(965\) 8470.00 0.282548
\(966\) 0 0
\(967\) −116.000 −0.00385761 −0.00192880 0.999998i \(-0.500614\pi\)
−0.00192880 + 0.999998i \(0.500614\pi\)
\(968\) 0 0
\(969\) −1872.00 −0.0620612
\(970\) 0 0
\(971\) 4744.00 0.156789 0.0783945 0.996922i \(-0.475021\pi\)
0.0783945 + 0.996922i \(0.475021\pi\)
\(972\) 0 0
\(973\) 9888.00 0.325791
\(974\) 0 0
\(975\) −2850.00 −0.0936134
\(976\) 0 0
\(977\) −18374.0 −0.601675 −0.300837 0.953675i \(-0.597266\pi\)
−0.300837 + 0.953675i \(0.597266\pi\)
\(978\) 0 0
\(979\) 5136.00 0.167668
\(980\) 0 0
\(981\) −2718.00 −0.0884598
\(982\) 0 0
\(983\) 13036.0 0.422974 0.211487 0.977381i \(-0.432169\pi\)
0.211487 + 0.977381i \(0.432169\pi\)
\(984\) 0 0
\(985\) 15670.0 0.506891
\(986\) 0 0
\(987\) −4464.00 −0.143962
\(988\) 0 0
\(989\) 14800.0 0.475847
\(990\) 0 0
\(991\) −15224.0 −0.487998 −0.243999 0.969775i \(-0.578459\pi\)
−0.243999 + 0.969775i \(0.578459\pi\)
\(992\) 0 0
\(993\) 12384.0 0.395765
\(994\) 0 0
\(995\) 12800.0 0.407826
\(996\) 0 0
\(997\) 43794.0 1.39114 0.695572 0.718457i \(-0.255151\pi\)
0.695572 + 0.718457i \(0.255151\pi\)
\(998\) 0 0
\(999\) −5130.00 −0.162468
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 960.4.a.y.1.1 1
4.3 odd 2 960.4.a.d.1.1 1
8.3 odd 2 480.4.a.j.1.1 yes 1
8.5 even 2 480.4.a.e.1.1 1
24.5 odd 2 1440.4.a.g.1.1 1
24.11 even 2 1440.4.a.d.1.1 1
40.19 odd 2 2400.4.a.g.1.1 1
40.29 even 2 2400.4.a.p.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
480.4.a.e.1.1 1 8.5 even 2
480.4.a.j.1.1 yes 1 8.3 odd 2
960.4.a.d.1.1 1 4.3 odd 2
960.4.a.y.1.1 1 1.1 even 1 trivial
1440.4.a.d.1.1 1 24.11 even 2
1440.4.a.g.1.1 1 24.5 odd 2
2400.4.a.g.1.1 1 40.19 odd 2
2400.4.a.p.1.1 1 40.29 even 2