Properties

Label 2160.4.a.bn.1.3
Level $2160$
Weight $4$
Character 2160.1
Self dual yes
Analytic conductor $127.444$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2160.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(127.444125612\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.47977.1
Defining polynomial: \( x^{3} - x^{2} - 60x - 44 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 1080)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.749725\) of defining polynomial
Character \(\chi\) \(=\) 2160.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.00000 q^{5} +24.3916 q^{7} +O(q^{10})\) \(q+5.00000 q^{5} +24.3916 q^{7} -26.8932 q^{11} -68.2815 q^{13} +61.8899 q^{17} +75.1747 q^{19} -43.3916 q^{23} +25.0000 q^{25} -174.951 q^{29} -222.666 q^{31} +121.958 q^{35} +67.1813 q^{37} +22.2070 q^{41} +84.7442 q^{43} -585.650 q^{47} +251.948 q^{49} -38.3625 q^{53} -134.466 q^{55} +92.0000 q^{59} -226.300 q^{61} -341.407 q^{65} -858.035 q^{67} +116.912 q^{71} +911.769 q^{73} -655.967 q^{77} +285.129 q^{79} -999.099 q^{83} +309.450 q^{85} +374.848 q^{89} -1665.49 q^{91} +375.873 q^{95} -227.413 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 15 q^{5} - 9 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 15 q^{5} - 9 q^{7} - 18 q^{11} - 21 q^{13} + 84 q^{17} - 21 q^{19} - 48 q^{23} + 75 q^{25} - 36 q^{29} - 324 q^{31} - 45 q^{35} + 33 q^{37} + 114 q^{41} - 282 q^{43} - 282 q^{47} + 228 q^{49} + 222 q^{53} - 90 q^{55} + 276 q^{59} + 303 q^{61} - 105 q^{65} - 1035 q^{67} - 510 q^{71} + 447 q^{73} - 1578 q^{77} - 777 q^{79} - 78 q^{83} + 420 q^{85} - 324 q^{89} - 1995 q^{91} - 105 q^{95} + 1191 q^{97}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 0 0
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) 24.3916 1.31702 0.658510 0.752572i \(-0.271187\pi\)
0.658510 + 0.752572i \(0.271187\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −26.8932 −0.737146 −0.368573 0.929599i \(-0.620154\pi\)
−0.368573 + 0.929599i \(0.620154\pi\)
\(12\) 0 0
\(13\) −68.2815 −1.45676 −0.728380 0.685174i \(-0.759726\pi\)
−0.728380 + 0.685174i \(0.759726\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 61.8899 0.882971 0.441485 0.897268i \(-0.354452\pi\)
0.441485 + 0.897268i \(0.354452\pi\)
\(18\) 0 0
\(19\) 75.1747 0.907697 0.453849 0.891079i \(-0.350051\pi\)
0.453849 + 0.891079i \(0.350051\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −43.3916 −0.393381 −0.196691 0.980466i \(-0.563019\pi\)
−0.196691 + 0.980466i \(0.563019\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −174.951 −1.12026 −0.560131 0.828404i \(-0.689249\pi\)
−0.560131 + 0.828404i \(0.689249\pi\)
\(30\) 0 0
\(31\) −222.666 −1.29007 −0.645033 0.764154i \(-0.723157\pi\)
−0.645033 + 0.764154i \(0.723157\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 121.958 0.588989
\(36\) 0 0
\(37\) 67.1813 0.298501 0.149250 0.988799i \(-0.452314\pi\)
0.149250 + 0.988799i \(0.452314\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 22.2070 0.0845890 0.0422945 0.999105i \(-0.486533\pi\)
0.0422945 + 0.999105i \(0.486533\pi\)
\(42\) 0 0
\(43\) 84.7442 0.300543 0.150272 0.988645i \(-0.451985\pi\)
0.150272 + 0.988645i \(0.451985\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −585.650 −1.81757 −0.908785 0.417264i \(-0.862989\pi\)
−0.908785 + 0.417264i \(0.862989\pi\)
\(48\) 0 0
\(49\) 251.948 0.734542
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −38.3625 −0.0994245 −0.0497122 0.998764i \(-0.515830\pi\)
−0.0497122 + 0.998764i \(0.515830\pi\)
\(54\) 0 0
\(55\) −134.466 −0.329662
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 92.0000 0.203006 0.101503 0.994835i \(-0.467635\pi\)
0.101503 + 0.994835i \(0.467635\pi\)
\(60\) 0 0
\(61\) −226.300 −0.474997 −0.237498 0.971388i \(-0.576327\pi\)
−0.237498 + 0.971388i \(0.576327\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −341.407 −0.651482
\(66\) 0 0
\(67\) −858.035 −1.56456 −0.782281 0.622925i \(-0.785944\pi\)
−0.782281 + 0.622925i \(0.785944\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 116.912 0.195422 0.0977108 0.995215i \(-0.468848\pi\)
0.0977108 + 0.995215i \(0.468848\pi\)
\(72\) 0 0
\(73\) 911.769 1.46184 0.730921 0.682462i \(-0.239091\pi\)
0.730921 + 0.682462i \(0.239091\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −655.967 −0.970836
\(78\) 0 0
\(79\) 285.129 0.406070 0.203035 0.979171i \(-0.434919\pi\)
0.203035 + 0.979171i \(0.434919\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −999.099 −1.32127 −0.660635 0.750707i \(-0.729713\pi\)
−0.660635 + 0.750707i \(0.729713\pi\)
\(84\) 0 0
\(85\) 309.450 0.394877
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 374.848 0.446447 0.223223 0.974767i \(-0.428342\pi\)
0.223223 + 0.974767i \(0.428342\pi\)
\(90\) 0 0
\(91\) −1665.49 −1.91858
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 375.873 0.405935
\(96\) 0 0
\(97\) −227.413 −0.238044 −0.119022 0.992892i \(-0.537976\pi\)
−0.119022 + 0.992892i \(0.537976\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1857.75 −1.83023 −0.915115 0.403193i \(-0.867900\pi\)
−0.915115 + 0.403193i \(0.867900\pi\)
\(102\) 0 0
\(103\) −172.787 −0.165294 −0.0826468 0.996579i \(-0.526337\pi\)
−0.0826468 + 0.996579i \(0.526337\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1806.79 −1.63242 −0.816209 0.577756i \(-0.803928\pi\)
−0.816209 + 0.577756i \(0.803928\pi\)
\(108\) 0 0
\(109\) 287.644 0.252764 0.126382 0.991982i \(-0.459663\pi\)
0.126382 + 0.991982i \(0.459663\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 671.875 0.559333 0.279667 0.960097i \(-0.409776\pi\)
0.279667 + 0.960097i \(0.409776\pi\)
\(114\) 0 0
\(115\) −216.958 −0.175925
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1509.59 1.16289
\(120\) 0 0
\(121\) −607.756 −0.456616
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) −1818.41 −1.27053 −0.635265 0.772294i \(-0.719109\pi\)
−0.635265 + 0.772294i \(0.719109\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1768.99 1.17983 0.589915 0.807465i \(-0.299161\pi\)
0.589915 + 0.807465i \(0.299161\pi\)
\(132\) 0 0
\(133\) 1833.63 1.19546
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1842.85 1.14924 0.574618 0.818422i \(-0.305151\pi\)
0.574618 + 0.818422i \(0.305151\pi\)
\(138\) 0 0
\(139\) −1371.14 −0.836678 −0.418339 0.908291i \(-0.637388\pi\)
−0.418339 + 0.908291i \(0.637388\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1836.31 1.07384
\(144\) 0 0
\(145\) −874.756 −0.500997
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2139.65 1.17642 0.588212 0.808707i \(-0.299832\pi\)
0.588212 + 0.808707i \(0.299832\pi\)
\(150\) 0 0
\(151\) 1199.81 0.646616 0.323308 0.946294i \(-0.395205\pi\)
0.323308 + 0.946294i \(0.395205\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −1113.33 −0.576935
\(156\) 0 0
\(157\) 1847.13 0.938960 0.469480 0.882943i \(-0.344441\pi\)
0.469480 + 0.882943i \(0.344441\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −1058.39 −0.518091
\(162\) 0 0
\(163\) 3203.12 1.53919 0.769595 0.638532i \(-0.220458\pi\)
0.769595 + 0.638532i \(0.220458\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 590.884 0.273796 0.136898 0.990585i \(-0.456287\pi\)
0.136898 + 0.990585i \(0.456287\pi\)
\(168\) 0 0
\(169\) 2465.36 1.12215
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −3110.68 −1.36705 −0.683527 0.729925i \(-0.739555\pi\)
−0.683527 + 0.729925i \(0.739555\pi\)
\(174\) 0 0
\(175\) 609.789 0.263404
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −2448.94 −1.02258 −0.511291 0.859408i \(-0.670832\pi\)
−0.511291 + 0.859408i \(0.670832\pi\)
\(180\) 0 0
\(181\) 2416.13 0.992206 0.496103 0.868264i \(-0.334764\pi\)
0.496103 + 0.868264i \(0.334764\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 335.906 0.133494
\(186\) 0 0
\(187\) −1664.42 −0.650879
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1652.25 −0.625929 −0.312964 0.949765i \(-0.601322\pi\)
−0.312964 + 0.949765i \(0.601322\pi\)
\(192\) 0 0
\(193\) 4736.65 1.76659 0.883295 0.468818i \(-0.155320\pi\)
0.883295 + 0.468818i \(0.155320\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2106.79 0.761944 0.380972 0.924587i \(-0.375589\pi\)
0.380972 + 0.924587i \(0.375589\pi\)
\(198\) 0 0
\(199\) −891.149 −0.317447 −0.158723 0.987323i \(-0.550738\pi\)
−0.158723 + 0.987323i \(0.550738\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −4267.33 −1.47541
\(204\) 0 0
\(205\) 111.035 0.0378294
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −2021.69 −0.669105
\(210\) 0 0
\(211\) −3607.11 −1.17689 −0.588444 0.808538i \(-0.700259\pi\)
−0.588444 + 0.808538i \(0.700259\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 423.721 0.134407
\(216\) 0 0
\(217\) −5431.18 −1.69904
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −4225.93 −1.28628
\(222\) 0 0
\(223\) −2507.12 −0.752865 −0.376432 0.926444i \(-0.622849\pi\)
−0.376432 + 0.926444i \(0.622849\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −3970.82 −1.16102 −0.580512 0.814252i \(-0.697147\pi\)
−0.580512 + 0.814252i \(0.697147\pi\)
\(228\) 0 0
\(229\) 5494.88 1.58564 0.792820 0.609455i \(-0.208612\pi\)
0.792820 + 0.609455i \(0.208612\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6895.27 −1.93873 −0.969365 0.245625i \(-0.921007\pi\)
−0.969365 + 0.245625i \(0.921007\pi\)
\(234\) 0 0
\(235\) −2928.25 −0.812842
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −4308.62 −1.16612 −0.583058 0.812431i \(-0.698144\pi\)
−0.583058 + 0.812431i \(0.698144\pi\)
\(240\) 0 0
\(241\) 3448.92 0.921843 0.460922 0.887441i \(-0.347519\pi\)
0.460922 + 0.887441i \(0.347519\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1259.74 0.328497
\(246\) 0 0
\(247\) −5133.04 −1.32230
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −6184.80 −1.55530 −0.777652 0.628695i \(-0.783589\pi\)
−0.777652 + 0.628695i \(0.783589\pi\)
\(252\) 0 0
\(253\) 1166.94 0.289979
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −4616.98 −1.12062 −0.560310 0.828283i \(-0.689318\pi\)
−0.560310 + 0.828283i \(0.689318\pi\)
\(258\) 0 0
\(259\) 1638.66 0.393132
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −728.557 −0.170816 −0.0854082 0.996346i \(-0.527219\pi\)
−0.0854082 + 0.996346i \(0.527219\pi\)
\(264\) 0 0
\(265\) −191.813 −0.0444640
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −6718.17 −1.52273 −0.761364 0.648325i \(-0.775470\pi\)
−0.761364 + 0.648325i \(0.775470\pi\)
\(270\) 0 0
\(271\) 2435.89 0.546015 0.273007 0.962012i \(-0.411982\pi\)
0.273007 + 0.962012i \(0.411982\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −672.330 −0.147429
\(276\) 0 0
\(277\) −4808.18 −1.04295 −0.521473 0.853268i \(-0.674617\pi\)
−0.521473 + 0.853268i \(0.674617\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 3535.67 0.750606 0.375303 0.926902i \(-0.377539\pi\)
0.375303 + 0.926902i \(0.377539\pi\)
\(282\) 0 0
\(283\) −3070.64 −0.644984 −0.322492 0.946572i \(-0.604521\pi\)
−0.322492 + 0.946572i \(0.604521\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 541.663 0.111405
\(288\) 0 0
\(289\) −1082.64 −0.220362
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1245.18 −0.248273 −0.124136 0.992265i \(-0.539616\pi\)
−0.124136 + 0.992265i \(0.539616\pi\)
\(294\) 0 0
\(295\) 460.000 0.0907872
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 2962.84 0.573061
\(300\) 0 0
\(301\) 2067.04 0.395822
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −1131.50 −0.212425
\(306\) 0 0
\(307\) 4269.79 0.793778 0.396889 0.917867i \(-0.370090\pi\)
0.396889 + 0.917867i \(0.370090\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −2584.39 −0.471213 −0.235607 0.971849i \(-0.575708\pi\)
−0.235607 + 0.971849i \(0.575708\pi\)
\(312\) 0 0
\(313\) −8899.44 −1.60711 −0.803556 0.595229i \(-0.797061\pi\)
−0.803556 + 0.595229i \(0.797061\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −6168.58 −1.09294 −0.546470 0.837479i \(-0.684029\pi\)
−0.546470 + 0.837479i \(0.684029\pi\)
\(318\) 0 0
\(319\) 4705.00 0.825797
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 4652.55 0.801470
\(324\) 0 0
\(325\) −1707.04 −0.291352
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −14284.9 −2.39378
\(330\) 0 0
\(331\) 9996.33 1.65996 0.829982 0.557791i \(-0.188351\pi\)
0.829982 + 0.557791i \(0.188351\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −4290.18 −0.699694
\(336\) 0 0
\(337\) −2497.40 −0.403685 −0.201843 0.979418i \(-0.564693\pi\)
−0.201843 + 0.979418i \(0.564693\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 5988.21 0.950967
\(342\) 0 0
\(343\) −2220.90 −0.349614
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −9001.22 −1.39254 −0.696269 0.717780i \(-0.745158\pi\)
−0.696269 + 0.717780i \(0.745158\pi\)
\(348\) 0 0
\(349\) −10707.4 −1.64227 −0.821135 0.570733i \(-0.806659\pi\)
−0.821135 + 0.570733i \(0.806659\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 2843.35 0.428714 0.214357 0.976755i \(-0.431234\pi\)
0.214357 + 0.976755i \(0.431234\pi\)
\(354\) 0 0
\(355\) 584.561 0.0873952
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 9328.89 1.37148 0.685738 0.727849i \(-0.259480\pi\)
0.685738 + 0.727849i \(0.259480\pi\)
\(360\) 0 0
\(361\) −1207.77 −0.176086
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 4558.85 0.653756
\(366\) 0 0
\(367\) −46.7367 −0.00664751 −0.00332376 0.999994i \(-0.501058\pi\)
−0.00332376 + 0.999994i \(0.501058\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −935.721 −0.130944
\(372\) 0 0
\(373\) −3791.98 −0.526384 −0.263192 0.964743i \(-0.584775\pi\)
−0.263192 + 0.964743i \(0.584775\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 11945.9 1.63195
\(378\) 0 0
\(379\) −1881.74 −0.255035 −0.127518 0.991836i \(-0.540701\pi\)
−0.127518 + 0.991836i \(0.540701\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −12011.9 −1.60256 −0.801280 0.598289i \(-0.795847\pi\)
−0.801280 + 0.598289i \(0.795847\pi\)
\(384\) 0 0
\(385\) −3279.83 −0.434171
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −7758.91 −1.01129 −0.505645 0.862741i \(-0.668746\pi\)
−0.505645 + 0.862741i \(0.668746\pi\)
\(390\) 0 0
\(391\) −2685.50 −0.347344
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1425.65 0.181600
\(396\) 0 0
\(397\) 14841.9 1.87630 0.938152 0.346223i \(-0.112536\pi\)
0.938152 + 0.346223i \(0.112536\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 435.446 0.0542273 0.0271136 0.999632i \(-0.491368\pi\)
0.0271136 + 0.999632i \(0.491368\pi\)
\(402\) 0 0
\(403\) 15204.0 1.87932
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1806.72 −0.220039
\(408\) 0 0
\(409\) 10550.6 1.27554 0.637770 0.770227i \(-0.279857\pi\)
0.637770 + 0.770227i \(0.279857\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 2244.02 0.267364
\(414\) 0 0
\(415\) −4995.50 −0.590890
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 4161.11 0.485164 0.242582 0.970131i \(-0.422006\pi\)
0.242582 + 0.970131i \(0.422006\pi\)
\(420\) 0 0
\(421\) 154.745 0.0179140 0.00895701 0.999960i \(-0.497149\pi\)
0.00895701 + 0.999960i \(0.497149\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1547.25 0.176594
\(426\) 0 0
\(427\) −5519.82 −0.625580
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −4393.72 −0.491040 −0.245520 0.969392i \(-0.578959\pi\)
−0.245520 + 0.969392i \(0.578959\pi\)
\(432\) 0 0
\(433\) 4437.08 0.492454 0.246227 0.969212i \(-0.420809\pi\)
0.246227 + 0.969212i \(0.420809\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −3261.95 −0.357071
\(438\) 0 0
\(439\) 7046.68 0.766104 0.383052 0.923727i \(-0.374873\pi\)
0.383052 + 0.923727i \(0.374873\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 7335.09 0.786683 0.393341 0.919392i \(-0.371319\pi\)
0.393341 + 0.919392i \(0.371319\pi\)
\(444\) 0 0
\(445\) 1874.24 0.199657
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −731.914 −0.0769290 −0.0384645 0.999260i \(-0.512247\pi\)
−0.0384645 + 0.999260i \(0.512247\pi\)
\(450\) 0 0
\(451\) −597.217 −0.0623545
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −8327.45 −0.858016
\(456\) 0 0
\(457\) −12611.7 −1.29092 −0.645462 0.763792i \(-0.723335\pi\)
−0.645462 + 0.763792i \(0.723335\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −12971.9 −1.31055 −0.655275 0.755391i \(-0.727447\pi\)
−0.655275 + 0.755391i \(0.727447\pi\)
\(462\) 0 0
\(463\) −9899.48 −0.993667 −0.496833 0.867846i \(-0.665504\pi\)
−0.496833 + 0.867846i \(0.665504\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −2180.60 −0.216073 −0.108037 0.994147i \(-0.534456\pi\)
−0.108037 + 0.994147i \(0.534456\pi\)
\(468\) 0 0
\(469\) −20928.8 −2.06056
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −2279.04 −0.221544
\(474\) 0 0
\(475\) 1879.37 0.181539
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −4132.54 −0.394197 −0.197099 0.980384i \(-0.563152\pi\)
−0.197099 + 0.980384i \(0.563152\pi\)
\(480\) 0 0
\(481\) −4587.23 −0.434844
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −1137.07 −0.106457
\(486\) 0 0
\(487\) 19277.8 1.79376 0.896880 0.442275i \(-0.145828\pi\)
0.896880 + 0.442275i \(0.145828\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 1315.88 0.120946 0.0604732 0.998170i \(-0.480739\pi\)
0.0604732 + 0.998170i \(0.480739\pi\)
\(492\) 0 0
\(493\) −10827.7 −0.989159
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2851.67 0.257374
\(498\) 0 0
\(499\) −10477.9 −0.939988 −0.469994 0.882670i \(-0.655744\pi\)
−0.469994 + 0.882670i \(0.655744\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 9312.29 0.825476 0.412738 0.910850i \(-0.364572\pi\)
0.412738 + 0.910850i \(0.364572\pi\)
\(504\) 0 0
\(505\) −9288.76 −0.818504
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −21595.3 −1.88054 −0.940270 0.340429i \(-0.889428\pi\)
−0.940270 + 0.340429i \(0.889428\pi\)
\(510\) 0 0
\(511\) 22239.5 1.92528
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −863.936 −0.0739215
\(516\) 0 0
\(517\) 15750.0 1.33981
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 9386.88 0.789341 0.394670 0.918823i \(-0.370859\pi\)
0.394670 + 0.918823i \(0.370859\pi\)
\(522\) 0 0
\(523\) 19068.9 1.59431 0.797156 0.603773i \(-0.206337\pi\)
0.797156 + 0.603773i \(0.206337\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −13780.8 −1.13909
\(528\) 0 0
\(529\) −10284.2 −0.845251
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −1516.33 −0.123226
\(534\) 0 0
\(535\) −9033.94 −0.730040
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −6775.69 −0.541465
\(540\) 0 0
\(541\) −22209.4 −1.76499 −0.882493 0.470325i \(-0.844137\pi\)
−0.882493 + 0.470325i \(0.844137\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1438.22 0.113040
\(546\) 0 0
\(547\) 2233.14 0.174556 0.0872782 0.996184i \(-0.472183\pi\)
0.0872782 + 0.996184i \(0.472183\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −13151.9 −1.01686
\(552\) 0 0
\(553\) 6954.74 0.534802
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 19727.7 1.50070 0.750351 0.661039i \(-0.229884\pi\)
0.750351 + 0.661039i \(0.229884\pi\)
\(558\) 0 0
\(559\) −5786.46 −0.437819
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 12025.7 0.900216 0.450108 0.892974i \(-0.351385\pi\)
0.450108 + 0.892974i \(0.351385\pi\)
\(564\) 0 0
\(565\) 3359.37 0.250142
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 19708.8 1.45209 0.726044 0.687648i \(-0.241357\pi\)
0.726044 + 0.687648i \(0.241357\pi\)
\(570\) 0 0
\(571\) −3965.65 −0.290643 −0.145322 0.989384i \(-0.546422\pi\)
−0.145322 + 0.989384i \(0.546422\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1084.79 −0.0786762
\(576\) 0 0
\(577\) 4134.92 0.298334 0.149167 0.988812i \(-0.452341\pi\)
0.149167 + 0.988812i \(0.452341\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −24369.6 −1.74014
\(582\) 0 0
\(583\) 1031.69 0.0732904
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −10949.0 −0.769873 −0.384937 0.922943i \(-0.625777\pi\)
−0.384937 + 0.922943i \(0.625777\pi\)
\(588\) 0 0
\(589\) −16738.9 −1.17099
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −17834.7 −1.23505 −0.617523 0.786553i \(-0.711864\pi\)
−0.617523 + 0.786553i \(0.711864\pi\)
\(594\) 0 0
\(595\) 7547.95 0.520060
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 6052.79 0.412872 0.206436 0.978460i \(-0.433814\pi\)
0.206436 + 0.978460i \(0.433814\pi\)
\(600\) 0 0
\(601\) 5233.02 0.355174 0.177587 0.984105i \(-0.443171\pi\)
0.177587 + 0.984105i \(0.443171\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −3038.78 −0.204205
\(606\) 0 0
\(607\) 24209.4 1.61883 0.809413 0.587240i \(-0.199785\pi\)
0.809413 + 0.587240i \(0.199785\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 39989.0 2.64776
\(612\) 0 0
\(613\) 7255.45 0.478050 0.239025 0.971013i \(-0.423172\pi\)
0.239025 + 0.971013i \(0.423172\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 29694.7 1.93754 0.968772 0.247951i \(-0.0797573\pi\)
0.968772 + 0.247951i \(0.0797573\pi\)
\(618\) 0 0
\(619\) −10775.2 −0.699665 −0.349833 0.936812i \(-0.613762\pi\)
−0.349833 + 0.936812i \(0.613762\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 9143.11 0.587979
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 4157.84 0.263568
\(630\) 0 0
\(631\) 3661.18 0.230981 0.115491 0.993309i \(-0.463156\pi\)
0.115491 + 0.993309i \(0.463156\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −9092.03 −0.568199
\(636\) 0 0
\(637\) −17203.4 −1.07005
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 5627.43 0.346755 0.173378 0.984855i \(-0.444532\pi\)
0.173378 + 0.984855i \(0.444532\pi\)
\(642\) 0 0
\(643\) 10732.5 0.658243 0.329122 0.944288i \(-0.393247\pi\)
0.329122 + 0.944288i \(0.393247\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −28753.1 −1.74714 −0.873571 0.486697i \(-0.838202\pi\)
−0.873571 + 0.486697i \(0.838202\pi\)
\(648\) 0 0
\(649\) −2474.17 −0.149645
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −27398.6 −1.64195 −0.820974 0.570966i \(-0.806569\pi\)
−0.820974 + 0.570966i \(0.806569\pi\)
\(654\) 0 0
\(655\) 8844.97 0.527636
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 26446.7 1.56331 0.781653 0.623714i \(-0.214377\pi\)
0.781653 + 0.623714i \(0.214377\pi\)
\(660\) 0 0
\(661\) 25643.7 1.50897 0.754483 0.656320i \(-0.227888\pi\)
0.754483 + 0.656320i \(0.227888\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 9168.13 0.534624
\(666\) 0 0
\(667\) 7591.40 0.440690
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 6085.94 0.350142
\(672\) 0 0
\(673\) 10796.0 0.618357 0.309179 0.951004i \(-0.399946\pi\)
0.309179 + 0.951004i \(0.399946\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −13183.2 −0.748406 −0.374203 0.927347i \(-0.622084\pi\)
−0.374203 + 0.927347i \(0.622084\pi\)
\(678\) 0 0
\(679\) −5546.96 −0.313509
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1349.06 0.0755786 0.0377893 0.999286i \(-0.487968\pi\)
0.0377893 + 0.999286i \(0.487968\pi\)
\(684\) 0 0
\(685\) 9214.26 0.513954
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 2619.45 0.144838
\(690\) 0 0
\(691\) −25133.8 −1.38370 −0.691849 0.722042i \(-0.743204\pi\)
−0.691849 + 0.722042i \(0.743204\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −6855.68 −0.374174
\(696\) 0 0
\(697\) 1374.39 0.0746896
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −14808.9 −0.797895 −0.398948 0.916974i \(-0.630625\pi\)
−0.398948 + 0.916974i \(0.630625\pi\)
\(702\) 0 0
\(703\) 5050.33 0.270948
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −45313.5 −2.41045
\(708\) 0 0
\(709\) −13857.2 −0.734019 −0.367009 0.930217i \(-0.619618\pi\)
−0.367009 + 0.930217i \(0.619618\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 9661.84 0.507488
\(714\) 0 0
\(715\) 9181.54 0.480238
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 21853.5 1.13352 0.566759 0.823884i \(-0.308197\pi\)
0.566759 + 0.823884i \(0.308197\pi\)
\(720\) 0 0
\(721\) −4214.55 −0.217695
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −4373.78 −0.224053
\(726\) 0 0
\(727\) 1442.23 0.0735757 0.0367878 0.999323i \(-0.488287\pi\)
0.0367878 + 0.999323i \(0.488287\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 5244.81 0.265371
\(732\) 0 0
\(733\) 2606.50 0.131342 0.0656708 0.997841i \(-0.479081\pi\)
0.0656708 + 0.997841i \(0.479081\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 23075.3 1.15331
\(738\) 0 0
\(739\) −9072.37 −0.451600 −0.225800 0.974174i \(-0.572500\pi\)
−0.225800 + 0.974174i \(0.572500\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 10093.2 0.498364 0.249182 0.968457i \(-0.419838\pi\)
0.249182 + 0.968457i \(0.419838\pi\)
\(744\) 0 0
\(745\) 10698.3 0.526113
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −44070.4 −2.14993
\(750\) 0 0
\(751\) −22033.4 −1.07059 −0.535294 0.844666i \(-0.679799\pi\)
−0.535294 + 0.844666i \(0.679799\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 5999.04 0.289175
\(756\) 0 0
\(757\) 8362.75 0.401518 0.200759 0.979641i \(-0.435659\pi\)
0.200759 + 0.979641i \(0.435659\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −11405.1 −0.543280 −0.271640 0.962399i \(-0.587566\pi\)
−0.271640 + 0.962399i \(0.587566\pi\)
\(762\) 0 0
\(763\) 7016.09 0.332896
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −6281.89 −0.295731
\(768\) 0 0
\(769\) 1447.34 0.0678704 0.0339352 0.999424i \(-0.489196\pi\)
0.0339352 + 0.999424i \(0.489196\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 25835.9 1.20214 0.601070 0.799196i \(-0.294741\pi\)
0.601070 + 0.799196i \(0.294741\pi\)
\(774\) 0 0
\(775\) −5566.66 −0.258013
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1669.40 0.0767812
\(780\) 0 0
\(781\) −3144.14 −0.144054
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 9235.63 0.419916
\(786\) 0 0
\(787\) −15859.9 −0.718355 −0.359178 0.933269i \(-0.616943\pi\)
−0.359178 + 0.933269i \(0.616943\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 16388.1 0.736653
\(792\) 0 0
\(793\) 15452.1 0.691956
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 32805.2 1.45799 0.728997 0.684517i \(-0.239987\pi\)
0.728997 + 0.684517i \(0.239987\pi\)
\(798\) 0 0
\(799\) −36245.8 −1.60486
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −24520.4 −1.07759
\(804\) 0 0
\(805\) −5291.94 −0.231697
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 17068.4 0.741772 0.370886 0.928678i \(-0.379054\pi\)
0.370886 + 0.928678i \(0.379054\pi\)
\(810\) 0 0
\(811\) −12079.5 −0.523018 −0.261509 0.965201i \(-0.584220\pi\)
−0.261509 + 0.965201i \(0.584220\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 16015.6 0.688347
\(816\) 0 0
\(817\) 6370.61 0.272802
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 34364.3 1.46081 0.730403 0.683016i \(-0.239332\pi\)
0.730403 + 0.683016i \(0.239332\pi\)
\(822\) 0 0
\(823\) −12983.0 −0.549891 −0.274946 0.961460i \(-0.588660\pi\)
−0.274946 + 0.961460i \(0.588660\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 25503.7 1.07237 0.536186 0.844100i \(-0.319865\pi\)
0.536186 + 0.844100i \(0.319865\pi\)
\(828\) 0 0
\(829\) 26016.9 1.08999 0.544997 0.838438i \(-0.316531\pi\)
0.544997 + 0.838438i \(0.316531\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 15593.0 0.648579
\(834\) 0 0
\(835\) 2954.42 0.122445
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 11090.3 0.456353 0.228177 0.973620i \(-0.426724\pi\)
0.228177 + 0.973620i \(0.426724\pi\)
\(840\) 0 0
\(841\) 6218.91 0.254988
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 12326.8 0.501839
\(846\) 0 0
\(847\) −14824.1 −0.601372
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −2915.10 −0.117425
\(852\) 0 0
\(853\) 41006.8 1.64601 0.823004 0.568036i \(-0.192296\pi\)
0.823004 + 0.568036i \(0.192296\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 38890.1 1.55013 0.775065 0.631881i \(-0.217717\pi\)
0.775065 + 0.631881i \(0.217717\pi\)
\(858\) 0 0
\(859\) −43062.9 −1.71046 −0.855232 0.518246i \(-0.826585\pi\)
−0.855232 + 0.518246i \(0.826585\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 2294.97 0.0905232 0.0452616 0.998975i \(-0.485588\pi\)
0.0452616 + 0.998975i \(0.485588\pi\)
\(864\) 0 0
\(865\) −15553.4 −0.611365
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −7668.03 −0.299333
\(870\) 0 0
\(871\) 58587.9 2.27919
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 3048.94 0.117798
\(876\) 0 0
\(877\) 33278.9 1.28136 0.640678 0.767810i \(-0.278653\pi\)
0.640678 + 0.767810i \(0.278653\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −9821.12 −0.375575 −0.187788 0.982210i \(-0.560132\pi\)
−0.187788 + 0.982210i \(0.560132\pi\)
\(882\) 0 0
\(883\) 13745.6 0.523869 0.261934 0.965086i \(-0.415640\pi\)
0.261934 + 0.965086i \(0.415640\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 24565.8 0.929922 0.464961 0.885331i \(-0.346068\pi\)
0.464961 + 0.885331i \(0.346068\pi\)
\(888\) 0 0
\(889\) −44353.7 −1.67331
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −44026.0 −1.64980
\(894\) 0 0
\(895\) −12244.7 −0.457312
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 38955.7 1.44521
\(900\) 0 0
\(901\) −2374.25 −0.0877889
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 12080.6 0.443728
\(906\) 0 0
\(907\) −20218.5 −0.740181 −0.370091 0.928996i \(-0.620673\pi\)
−0.370091 + 0.928996i \(0.620673\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 4552.96 0.165583 0.0827915 0.996567i \(-0.473616\pi\)
0.0827915 + 0.996567i \(0.473616\pi\)
\(912\) 0 0
\(913\) 26869.0 0.973969
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 43148.5 1.55386
\(918\) 0 0
\(919\) −19814.0 −0.711211 −0.355605 0.934636i \(-0.615725\pi\)
−0.355605 + 0.934636i \(0.615725\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −7982.94 −0.284682
\(924\) 0 0
\(925\) 1679.53 0.0597002
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −40202.9 −1.41982 −0.709911 0.704291i \(-0.751265\pi\)
−0.709911 + 0.704291i \(0.751265\pi\)
\(930\) 0 0
\(931\) 18940.1 0.666742
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −8322.09 −0.291082
\(936\) 0 0
\(937\) 24587.9 0.857260 0.428630 0.903480i \(-0.358996\pi\)
0.428630 + 0.903480i \(0.358996\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 33462.1 1.15923 0.579614 0.814891i \(-0.303203\pi\)
0.579614 + 0.814891i \(0.303203\pi\)
\(942\) 0 0
\(943\) −963.596 −0.0332757
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −8385.17 −0.287731 −0.143866 0.989597i \(-0.545953\pi\)
−0.143866 + 0.989597i \(0.545953\pi\)
\(948\) 0 0
\(949\) −62256.9 −2.12955
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 55532.0 1.88758 0.943788 0.330552i \(-0.107235\pi\)
0.943788 + 0.330552i \(0.107235\pi\)
\(954\) 0 0
\(955\) −8261.23 −0.279924
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 44950.0 1.51357
\(960\) 0 0
\(961\) 19789.3 0.664272
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 23683.3 0.790043
\(966\) 0 0
\(967\) 44080.8 1.46592 0.732960 0.680272i \(-0.238138\pi\)
0.732960 + 0.680272i \(0.238138\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 29721.4 0.982291 0.491146 0.871077i \(-0.336578\pi\)
0.491146 + 0.871077i \(0.336578\pi\)
\(972\) 0 0
\(973\) −33444.2 −1.10192
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 462.881 0.0151575 0.00757876 0.999971i \(-0.497588\pi\)
0.00757876 + 0.999971i \(0.497588\pi\)
\(978\) 0 0
\(979\) −10080.9 −0.329096
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 27389.7 0.888703 0.444351 0.895853i \(-0.353434\pi\)
0.444351 + 0.895853i \(0.353434\pi\)
\(984\) 0 0
\(985\) 10534.0 0.340752
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −3677.18 −0.118228
\(990\) 0 0
\(991\) −32596.6 −1.04487 −0.522435 0.852679i \(-0.674976\pi\)
−0.522435 + 0.852679i \(0.674976\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −4455.75 −0.141966
\(996\) 0 0
\(997\) −26630.6 −0.845936 −0.422968 0.906145i \(-0.639012\pi\)
−0.422968 + 0.906145i \(0.639012\pi\)
\(998\) 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 2160.4.a.bn.1.3 3
3.2 odd 2 2160.4.a.bf.1.3 3
4.3 odd 2 1080.4.a.n.1.1 yes 3
12.11 even 2 1080.4.a.h.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1080.4.a.h.1.1 3 12.11 even 2
1080.4.a.n.1.1 yes 3 4.3 odd 2
2160.4.a.bf.1.3 3 3.2 odd 2
2160.4.a.bn.1.3 3 1.1 even 1 trivial