Properties

Label 1080.4.a.p.1.3
Level $1080$
Weight $4$
Character 1080.1
Self dual yes
Analytic conductor $63.722$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1080.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.7220628062\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \( x^{4} - x^{3} - 141x^{2} + 200x + 3500 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-4.73555\) of defining polynomial
Character \(\chi\) \(=\) 1080.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.00000 q^{5} +11.2995 q^{7} +O(q^{10})\) \(q+5.00000 q^{5} +11.2995 q^{7} +61.3933 q^{11} -75.7226 q^{13} +11.1657 q^{17} +71.5271 q^{19} +126.156 q^{23} +25.0000 q^{25} -235.502 q^{29} +110.923 q^{31} +56.4973 q^{35} +434.358 q^{37} -1.15792 q^{41} -77.6254 q^{43} -231.442 q^{47} -215.322 q^{49} +500.296 q^{53} +306.967 q^{55} -334.718 q^{59} +147.171 q^{61} -378.613 q^{65} +84.9722 q^{67} -101.308 q^{71} -50.4773 q^{73} +693.712 q^{77} -818.225 q^{79} +206.044 q^{83} +55.8283 q^{85} -648.508 q^{89} -855.625 q^{91} +357.636 q^{95} +1885.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 20 q^{5} + 14 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 20 q^{5} + 14 q^{7} - 4 q^{11} + 30 q^{13} - 28 q^{17} + 78 q^{19} + 182 q^{23} + 100 q^{25} + 202 q^{29} - 76 q^{31} + 70 q^{35} + 302 q^{37} + 380 q^{41} + 178 q^{43} + 114 q^{47} + 958 q^{49} - 256 q^{53} - 20 q^{55} - 204 q^{59} + 766 q^{61} + 150 q^{65} + 330 q^{67} - 1060 q^{71} + 1442 q^{73} + 216 q^{77} + 742 q^{79} - 768 q^{83} - 140 q^{85} - 400 q^{89} + 3066 q^{91} + 390 q^{95} + 3338 q^{97}+O(q^{100}) \) 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
\(7\) 11.2995 0.610114 0.305057 0.952334i \(-0.401325\pi\)
0.305057 + 0.952334i \(0.401325\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 61.3933 1.68280 0.841399 0.540414i \(-0.181732\pi\)
0.841399 + 0.540414i \(0.181732\pi\)
\(12\) 0 0
\(13\) −75.7226 −1.61551 −0.807757 0.589516i \(-0.799319\pi\)
−0.807757 + 0.589516i \(0.799319\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 11.1657 0.159298 0.0796491 0.996823i \(-0.474620\pi\)
0.0796491 + 0.996823i \(0.474620\pi\)
\(18\) 0 0
\(19\) 71.5271 0.863655 0.431828 0.901956i \(-0.357869\pi\)
0.431828 + 0.901956i \(0.357869\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 126.156 1.14371 0.571855 0.820355i \(-0.306224\pi\)
0.571855 + 0.820355i \(0.306224\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −235.502 −1.50798 −0.753992 0.656883i \(-0.771874\pi\)
−0.753992 + 0.656883i \(0.771874\pi\)
\(30\) 0 0
\(31\) 110.923 0.642654 0.321327 0.946968i \(-0.395871\pi\)
0.321327 + 0.946968i \(0.395871\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 56.4973 0.272851
\(36\) 0 0
\(37\) 434.358 1.92995 0.964973 0.262349i \(-0.0844971\pi\)
0.964973 + 0.262349i \(0.0844971\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.15792 −0.00441066 −0.00220533 0.999998i \(-0.500702\pi\)
−0.00220533 + 0.999998i \(0.500702\pi\)
\(42\) 0 0
\(43\) −77.6254 −0.275297 −0.137648 0.990481i \(-0.543954\pi\)
−0.137648 + 0.990481i \(0.543954\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −231.442 −0.718282 −0.359141 0.933283i \(-0.616930\pi\)
−0.359141 + 0.933283i \(0.616930\pi\)
\(48\) 0 0
\(49\) −215.322 −0.627761
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 500.296 1.29662 0.648310 0.761376i \(-0.275476\pi\)
0.648310 + 0.761376i \(0.275476\pi\)
\(54\) 0 0
\(55\) 306.967 0.752571
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −334.718 −0.738587 −0.369293 0.929313i \(-0.620400\pi\)
−0.369293 + 0.929313i \(0.620400\pi\)
\(60\) 0 0
\(61\) 147.171 0.308906 0.154453 0.988000i \(-0.450638\pi\)
0.154453 + 0.988000i \(0.450638\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −378.613 −0.722480
\(66\) 0 0
\(67\) 84.9722 0.154940 0.0774702 0.996995i \(-0.475316\pi\)
0.0774702 + 0.996995i \(0.475316\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −101.308 −0.169339 −0.0846695 0.996409i \(-0.526983\pi\)
−0.0846695 + 0.996409i \(0.526983\pi\)
\(72\) 0 0
\(73\) −50.4773 −0.0809304 −0.0404652 0.999181i \(-0.512884\pi\)
−0.0404652 + 0.999181i \(0.512884\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 693.712 1.02670
\(78\) 0 0
\(79\) −818.225 −1.16528 −0.582642 0.812729i \(-0.697981\pi\)
−0.582642 + 0.812729i \(0.697981\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 206.044 0.272486 0.136243 0.990675i \(-0.456497\pi\)
0.136243 + 0.990675i \(0.456497\pi\)
\(84\) 0 0
\(85\) 55.8283 0.0712403
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −648.508 −0.772379 −0.386189 0.922420i \(-0.626209\pi\)
−0.386189 + 0.922420i \(0.626209\pi\)
\(90\) 0 0
\(91\) −855.625 −0.985647
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 357.636 0.386238
\(96\) 0 0
\(97\) 1885.00 1.97313 0.986563 0.163381i \(-0.0522399\pi\)
0.986563 + 0.163381i \(0.0522399\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1923.79 1.89529 0.947647 0.319321i \(-0.103455\pi\)
0.947647 + 0.319321i \(0.103455\pi\)
\(102\) 0 0
\(103\) 1808.72 1.73028 0.865139 0.501531i \(-0.167230\pi\)
0.865139 + 0.501531i \(0.167230\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1342.31 −1.21277 −0.606385 0.795171i \(-0.707381\pi\)
−0.606385 + 0.795171i \(0.707381\pi\)
\(108\) 0 0
\(109\) 1173.14 1.03088 0.515442 0.856924i \(-0.327628\pi\)
0.515442 + 0.856924i \(0.327628\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −284.667 −0.236984 −0.118492 0.992955i \(-0.537806\pi\)
−0.118492 + 0.992955i \(0.537806\pi\)
\(114\) 0 0
\(115\) 630.779 0.511483
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 126.166 0.0971900
\(120\) 0 0
\(121\) 2438.14 1.83181
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) 2413.23 1.68614 0.843068 0.537807i \(-0.180747\pi\)
0.843068 + 0.537807i \(0.180747\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −118.912 −0.0793085 −0.0396543 0.999213i \(-0.512626\pi\)
−0.0396543 + 0.999213i \(0.512626\pi\)
\(132\) 0 0
\(133\) 808.218 0.526928
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1097.16 −0.684212 −0.342106 0.939661i \(-0.611140\pi\)
−0.342106 + 0.939661i \(0.611140\pi\)
\(138\) 0 0
\(139\) −355.233 −0.216766 −0.108383 0.994109i \(-0.534567\pi\)
−0.108383 + 0.994109i \(0.534567\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −4648.86 −2.71858
\(144\) 0 0
\(145\) −1177.51 −0.674391
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2001.55 −1.10049 −0.550246 0.835002i \(-0.685466\pi\)
−0.550246 + 0.835002i \(0.685466\pi\)
\(150\) 0 0
\(151\) 2652.70 1.42963 0.714813 0.699316i \(-0.246512\pi\)
0.714813 + 0.699316i \(0.246512\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 554.613 0.287404
\(156\) 0 0
\(157\) −3087.09 −1.56928 −0.784640 0.619952i \(-0.787152\pi\)
−0.784640 + 0.619952i \(0.787152\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1425.49 0.697793
\(162\) 0 0
\(163\) 1139.42 0.547523 0.273761 0.961798i \(-0.411732\pi\)
0.273761 + 0.961798i \(0.411732\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2531.03 1.17280 0.586398 0.810023i \(-0.300545\pi\)
0.586398 + 0.810023i \(0.300545\pi\)
\(168\) 0 0
\(169\) 3536.92 1.60988
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2872.89 1.26255 0.631277 0.775557i \(-0.282531\pi\)
0.631277 + 0.775557i \(0.282531\pi\)
\(174\) 0 0
\(175\) 282.487 0.122023
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 3827.16 1.59807 0.799036 0.601283i \(-0.205343\pi\)
0.799036 + 0.601283i \(0.205343\pi\)
\(180\) 0 0
\(181\) −659.781 −0.270946 −0.135473 0.990781i \(-0.543255\pi\)
−0.135473 + 0.990781i \(0.543255\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 2171.79 0.863098
\(186\) 0 0
\(187\) 685.497 0.268067
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −4376.11 −1.65782 −0.828911 0.559380i \(-0.811039\pi\)
−0.828911 + 0.559380i \(0.811039\pi\)
\(192\) 0 0
\(193\) −3802.02 −1.41801 −0.709004 0.705204i \(-0.750855\pi\)
−0.709004 + 0.705204i \(0.750855\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3340.27 1.20804 0.604021 0.796968i \(-0.293564\pi\)
0.604021 + 0.796968i \(0.293564\pi\)
\(198\) 0 0
\(199\) 4274.83 1.52279 0.761393 0.648290i \(-0.224516\pi\)
0.761393 + 0.648290i \(0.224516\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −2661.04 −0.920042
\(204\) 0 0
\(205\) −5.78960 −0.00197251
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 4391.29 1.45336
\(210\) 0 0
\(211\) −126.348 −0.0412233 −0.0206117 0.999788i \(-0.506561\pi\)
−0.0206117 + 0.999788i \(0.506561\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −388.127 −0.123117
\(216\) 0 0
\(217\) 1253.36 0.392092
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −845.493 −0.257348
\(222\) 0 0
\(223\) 3423.31 1.02799 0.513996 0.857793i \(-0.328165\pi\)
0.513996 + 0.857793i \(0.328165\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −2145.09 −0.627200 −0.313600 0.949555i \(-0.601535\pi\)
−0.313600 + 0.949555i \(0.601535\pi\)
\(228\) 0 0
\(229\) 3524.09 1.01694 0.508468 0.861081i \(-0.330212\pi\)
0.508468 + 0.861081i \(0.330212\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1479.58 0.416011 0.208005 0.978128i \(-0.433303\pi\)
0.208005 + 0.978128i \(0.433303\pi\)
\(234\) 0 0
\(235\) −1157.21 −0.321226
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −3075.33 −0.832328 −0.416164 0.909290i \(-0.636626\pi\)
−0.416164 + 0.909290i \(0.636626\pi\)
\(240\) 0 0
\(241\) 2600.22 0.694998 0.347499 0.937680i \(-0.387031\pi\)
0.347499 + 0.937680i \(0.387031\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1076.61 −0.280743
\(246\) 0 0
\(247\) −5416.22 −1.39525
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1186.65 0.298410 0.149205 0.988806i \(-0.452329\pi\)
0.149205 + 0.988806i \(0.452329\pi\)
\(252\) 0 0
\(253\) 7745.13 1.92463
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3364.26 0.816562 0.408281 0.912856i \(-0.366128\pi\)
0.408281 + 0.912856i \(0.366128\pi\)
\(258\) 0 0
\(259\) 4908.01 1.17749
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −1655.19 −0.388074 −0.194037 0.980994i \(-0.562158\pi\)
−0.194037 + 0.980994i \(0.562158\pi\)
\(264\) 0 0
\(265\) 2501.48 0.579867
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −2646.65 −0.599886 −0.299943 0.953957i \(-0.596968\pi\)
−0.299943 + 0.953957i \(0.596968\pi\)
\(270\) 0 0
\(271\) −3798.31 −0.851405 −0.425703 0.904863i \(-0.639973\pi\)
−0.425703 + 0.904863i \(0.639973\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1534.83 0.336560
\(276\) 0 0
\(277\) −5471.39 −1.18680 −0.593400 0.804908i \(-0.702215\pi\)
−0.593400 + 0.804908i \(0.702215\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −3611.68 −0.766743 −0.383371 0.923594i \(-0.625237\pi\)
−0.383371 + 0.923594i \(0.625237\pi\)
\(282\) 0 0
\(283\) −1914.83 −0.402208 −0.201104 0.979570i \(-0.564453\pi\)
−0.201104 + 0.979570i \(0.564453\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −13.0839 −0.00269100
\(288\) 0 0
\(289\) −4788.33 −0.974624
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1706.82 0.340319 0.170159 0.985417i \(-0.445572\pi\)
0.170159 + 0.985417i \(0.445572\pi\)
\(294\) 0 0
\(295\) −1673.59 −0.330306
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −9552.85 −1.84768
\(300\) 0 0
\(301\) −877.126 −0.167962
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 735.853 0.138147
\(306\) 0 0
\(307\) 5888.56 1.09472 0.547358 0.836898i \(-0.315634\pi\)
0.547358 + 0.836898i \(0.315634\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −4902.98 −0.893962 −0.446981 0.894543i \(-0.647501\pi\)
−0.446981 + 0.894543i \(0.647501\pi\)
\(312\) 0 0
\(313\) 6209.11 1.12128 0.560638 0.828061i \(-0.310556\pi\)
0.560638 + 0.828061i \(0.310556\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2427.25 0.430056 0.215028 0.976608i \(-0.431016\pi\)
0.215028 + 0.976608i \(0.431016\pi\)
\(318\) 0 0
\(319\) −14458.2 −2.53763
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 798.648 0.137579
\(324\) 0 0
\(325\) −1893.07 −0.323103
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −2615.17 −0.438234
\(330\) 0 0
\(331\) 6058.23 1.00601 0.503006 0.864283i \(-0.332227\pi\)
0.503006 + 0.864283i \(0.332227\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 424.861 0.0692915
\(336\) 0 0
\(337\) 117.584 0.0190065 0.00950326 0.999955i \(-0.496975\pi\)
0.00950326 + 0.999955i \(0.496975\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 6809.90 1.08146
\(342\) 0 0
\(343\) −6308.74 −0.993119
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −1596.08 −0.246922 −0.123461 0.992349i \(-0.539399\pi\)
−0.123461 + 0.992349i \(0.539399\pi\)
\(348\) 0 0
\(349\) 3555.89 0.545394 0.272697 0.962100i \(-0.412084\pi\)
0.272697 + 0.962100i \(0.412084\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −2973.17 −0.448289 −0.224145 0.974556i \(-0.571959\pi\)
−0.224145 + 0.974556i \(0.571959\pi\)
\(354\) 0 0
\(355\) −506.541 −0.0757307
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −12834.7 −1.88688 −0.943442 0.331537i \(-0.892433\pi\)
−0.943442 + 0.331537i \(0.892433\pi\)
\(360\) 0 0
\(361\) −1742.87 −0.254099
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −252.386 −0.0361932
\(366\) 0 0
\(367\) 7503.89 1.06730 0.533651 0.845705i \(-0.320819\pi\)
0.533651 + 0.845705i \(0.320819\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 5653.07 0.791086
\(372\) 0 0
\(373\) −10834.1 −1.50394 −0.751970 0.659197i \(-0.770896\pi\)
−0.751970 + 0.659197i \(0.770896\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 17832.8 2.43617
\(378\) 0 0
\(379\) 1526.64 0.206909 0.103454 0.994634i \(-0.467010\pi\)
0.103454 + 0.994634i \(0.467010\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 8794.30 1.17328 0.586642 0.809846i \(-0.300450\pi\)
0.586642 + 0.809846i \(0.300450\pi\)
\(384\) 0 0
\(385\) 3468.56 0.459154
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 7955.47 1.03691 0.518456 0.855104i \(-0.326507\pi\)
0.518456 + 0.855104i \(0.326507\pi\)
\(390\) 0 0
\(391\) 1408.61 0.182191
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −4091.12 −0.521131
\(396\) 0 0
\(397\) −3359.28 −0.424679 −0.212339 0.977196i \(-0.568108\pi\)
−0.212339 + 0.977196i \(0.568108\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −3387.77 −0.421888 −0.210944 0.977498i \(-0.567654\pi\)
−0.210944 + 0.977498i \(0.567654\pi\)
\(402\) 0 0
\(403\) −8399.34 −1.03822
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 26666.7 3.24771
\(408\) 0 0
\(409\) −7919.60 −0.957454 −0.478727 0.877964i \(-0.658902\pi\)
−0.478727 + 0.877964i \(0.658902\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −3782.14 −0.450622
\(414\) 0 0
\(415\) 1030.22 0.121859
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −10671.6 −1.24425 −0.622127 0.782917i \(-0.713731\pi\)
−0.622127 + 0.782917i \(0.713731\pi\)
\(420\) 0 0
\(421\) −8445.59 −0.977703 −0.488852 0.872367i \(-0.662584\pi\)
−0.488852 + 0.872367i \(0.662584\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 279.141 0.0318596
\(426\) 0 0
\(427\) 1662.95 0.188468
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −11206.9 −1.25248 −0.626241 0.779630i \(-0.715407\pi\)
−0.626241 + 0.779630i \(0.715407\pi\)
\(432\) 0 0
\(433\) −5119.10 −0.568148 −0.284074 0.958802i \(-0.591686\pi\)
−0.284074 + 0.958802i \(0.591686\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 9023.57 0.987771
\(438\) 0 0
\(439\) −14018.6 −1.52408 −0.762039 0.647531i \(-0.775802\pi\)
−0.762039 + 0.647531i \(0.775802\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −10393.3 −1.11468 −0.557338 0.830286i \(-0.688177\pi\)
−0.557338 + 0.830286i \(0.688177\pi\)
\(444\) 0 0
\(445\) −3242.54 −0.345418
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −2522.28 −0.265108 −0.132554 0.991176i \(-0.542318\pi\)
−0.132554 + 0.991176i \(0.542318\pi\)
\(450\) 0 0
\(451\) −71.0886 −0.00742225
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −4278.13 −0.440795
\(456\) 0 0
\(457\) 10879.1 1.11357 0.556785 0.830657i \(-0.312035\pi\)
0.556785 + 0.830657i \(0.312035\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −288.174 −0.0291141 −0.0145570 0.999894i \(-0.504634\pi\)
−0.0145570 + 0.999894i \(0.504634\pi\)
\(462\) 0 0
\(463\) −3397.84 −0.341060 −0.170530 0.985352i \(-0.554548\pi\)
−0.170530 + 0.985352i \(0.554548\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −16484.3 −1.63341 −0.816703 0.577058i \(-0.804201\pi\)
−0.816703 + 0.577058i \(0.804201\pi\)
\(468\) 0 0
\(469\) 960.140 0.0945313
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −4765.68 −0.463269
\(474\) 0 0
\(475\) 1788.18 0.172731
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −8583.18 −0.818738 −0.409369 0.912369i \(-0.634251\pi\)
−0.409369 + 0.912369i \(0.634251\pi\)
\(480\) 0 0
\(481\) −32890.7 −3.11785
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 9425.02 0.882409
\(486\) 0 0
\(487\) −6076.49 −0.565404 −0.282702 0.959208i \(-0.591231\pi\)
−0.282702 + 0.959208i \(0.591231\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 1652.84 0.151918 0.0759589 0.997111i \(-0.475798\pi\)
0.0759589 + 0.997111i \(0.475798\pi\)
\(492\) 0 0
\(493\) −2629.53 −0.240219
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1144.73 −0.103316
\(498\) 0 0
\(499\) −18468.0 −1.65680 −0.828400 0.560137i \(-0.810748\pi\)
−0.828400 + 0.560137i \(0.810748\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −3044.06 −0.269837 −0.134918 0.990857i \(-0.543077\pi\)
−0.134918 + 0.990857i \(0.543077\pi\)
\(504\) 0 0
\(505\) 9618.97 0.847601
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 8238.76 0.717440 0.358720 0.933445i \(-0.383213\pi\)
0.358720 + 0.933445i \(0.383213\pi\)
\(510\) 0 0
\(511\) −570.366 −0.0493767
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 9043.61 0.773804
\(516\) 0 0
\(517\) −14209.0 −1.20872
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 11609.7 0.976261 0.488131 0.872771i \(-0.337679\pi\)
0.488131 + 0.872771i \(0.337679\pi\)
\(522\) 0 0
\(523\) −10413.6 −0.870662 −0.435331 0.900270i \(-0.643369\pi\)
−0.435331 + 0.900270i \(0.643369\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1238.52 0.102374
\(528\) 0 0
\(529\) 3748.31 0.308072
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 87.6808 0.00712547
\(534\) 0 0
\(535\) −6711.57 −0.542367
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −13219.3 −1.05640
\(540\) 0 0
\(541\) 15045.2 1.19565 0.597823 0.801628i \(-0.296032\pi\)
0.597823 + 0.801628i \(0.296032\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 5865.70 0.461026
\(546\) 0 0
\(547\) −8132.94 −0.635721 −0.317860 0.948138i \(-0.602964\pi\)
−0.317860 + 0.948138i \(0.602964\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −16844.8 −1.30238
\(552\) 0 0
\(553\) −9245.50 −0.710956
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 19419.4 1.47725 0.738623 0.674118i \(-0.235476\pi\)
0.738623 + 0.674118i \(0.235476\pi\)
\(558\) 0 0
\(559\) 5878.00 0.444746
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 5696.48 0.426426 0.213213 0.977006i \(-0.431607\pi\)
0.213213 + 0.977006i \(0.431607\pi\)
\(564\) 0 0
\(565\) −1423.33 −0.105982
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −18615.2 −1.37151 −0.685754 0.727833i \(-0.740527\pi\)
−0.685754 + 0.727833i \(0.740527\pi\)
\(570\) 0 0
\(571\) −11250.1 −0.824521 −0.412260 0.911066i \(-0.635261\pi\)
−0.412260 + 0.911066i \(0.635261\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 3153.90 0.228742
\(576\) 0 0
\(577\) 2336.40 0.168571 0.0842856 0.996442i \(-0.473139\pi\)
0.0842856 + 0.996442i \(0.473139\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 2328.19 0.166247
\(582\) 0 0
\(583\) 30714.8 2.18195
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 20277.2 1.42578 0.712889 0.701277i \(-0.247386\pi\)
0.712889 + 0.701277i \(0.247386\pi\)
\(588\) 0 0
\(589\) 7933.97 0.555031
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −25051.8 −1.73483 −0.867414 0.497587i \(-0.834220\pi\)
−0.867414 + 0.497587i \(0.834220\pi\)
\(594\) 0 0
\(595\) 630.830 0.0434647
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 26572.3 1.81254 0.906271 0.422697i \(-0.138916\pi\)
0.906271 + 0.422697i \(0.138916\pi\)
\(600\) 0 0
\(601\) 4224.74 0.286740 0.143370 0.989669i \(-0.454206\pi\)
0.143370 + 0.989669i \(0.454206\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 12190.7 0.819211
\(606\) 0 0
\(607\) −17428.2 −1.16539 −0.582693 0.812692i \(-0.698001\pi\)
−0.582693 + 0.812692i \(0.698001\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 17525.4 1.16039
\(612\) 0 0
\(613\) −7100.27 −0.467826 −0.233913 0.972258i \(-0.575153\pi\)
−0.233913 + 0.972258i \(0.575153\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −26045.2 −1.69941 −0.849707 0.527255i \(-0.823221\pi\)
−0.849707 + 0.527255i \(0.823221\pi\)
\(618\) 0 0
\(619\) 8865.32 0.575650 0.287825 0.957683i \(-0.407068\pi\)
0.287825 + 0.957683i \(0.407068\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −7327.79 −0.471239
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 4849.89 0.307437
\(630\) 0 0
\(631\) 177.883 0.0112225 0.00561126 0.999984i \(-0.498214\pi\)
0.00561126 + 0.999984i \(0.498214\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 12066.1 0.754063
\(636\) 0 0
\(637\) 16304.8 1.01416
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −27742.7 −1.70947 −0.854736 0.519064i \(-0.826281\pi\)
−0.854736 + 0.519064i \(0.826281\pi\)
\(642\) 0 0
\(643\) 9182.34 0.563166 0.281583 0.959537i \(-0.409140\pi\)
0.281583 + 0.959537i \(0.409140\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 16429.9 0.998341 0.499171 0.866504i \(-0.333638\pi\)
0.499171 + 0.866504i \(0.333638\pi\)
\(648\) 0 0
\(649\) −20549.5 −1.24289
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −10965.8 −0.657158 −0.328579 0.944477i \(-0.606570\pi\)
−0.328579 + 0.944477i \(0.606570\pi\)
\(654\) 0 0
\(655\) −594.561 −0.0354678
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −23252.7 −1.37450 −0.687250 0.726421i \(-0.741182\pi\)
−0.687250 + 0.726421i \(0.741182\pi\)
\(660\) 0 0
\(661\) −2215.22 −0.130351 −0.0651755 0.997874i \(-0.520761\pi\)
−0.0651755 + 0.997874i \(0.520761\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 4041.09 0.235649
\(666\) 0 0
\(667\) −29709.9 −1.72470
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 9035.30 0.519827
\(672\) 0 0
\(673\) −20009.1 −1.14605 −0.573026 0.819537i \(-0.694230\pi\)
−0.573026 + 0.819537i \(0.694230\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 18842.2 1.06967 0.534834 0.844957i \(-0.320374\pi\)
0.534834 + 0.844957i \(0.320374\pi\)
\(678\) 0 0
\(679\) 21299.5 1.20383
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −20258.1 −1.13493 −0.567463 0.823399i \(-0.692075\pi\)
−0.567463 + 0.823399i \(0.692075\pi\)
\(684\) 0 0
\(685\) −5485.82 −0.305989
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −37883.7 −2.09471
\(690\) 0 0
\(691\) 22738.2 1.25181 0.625905 0.779899i \(-0.284730\pi\)
0.625905 + 0.779899i \(0.284730\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −1776.17 −0.0969407
\(696\) 0 0
\(697\) −12.9289 −0.000702610 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −8306.55 −0.447552 −0.223776 0.974641i \(-0.571838\pi\)
−0.223776 + 0.974641i \(0.571838\pi\)
\(702\) 0 0
\(703\) 31068.4 1.66681
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 21737.8 1.15634
\(708\) 0 0
\(709\) 24716.6 1.30924 0.654620 0.755958i \(-0.272829\pi\)
0.654620 + 0.755958i \(0.272829\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 13993.5 0.735009
\(714\) 0 0
\(715\) −23244.3 −1.21579
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −9888.23 −0.512891 −0.256446 0.966559i \(-0.582551\pi\)
−0.256446 + 0.966559i \(0.582551\pi\)
\(720\) 0 0
\(721\) 20437.6 1.05567
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −5887.54 −0.301597
\(726\) 0 0
\(727\) 6335.29 0.323195 0.161598 0.986857i \(-0.448335\pi\)
0.161598 + 0.986857i \(0.448335\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −866.739 −0.0438543
\(732\) 0 0
\(733\) −13170.9 −0.663680 −0.331840 0.943336i \(-0.607669\pi\)
−0.331840 + 0.943336i \(0.607669\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 5216.73 0.260734
\(738\) 0 0
\(739\) −31361.7 −1.56111 −0.780554 0.625088i \(-0.785063\pi\)
−0.780554 + 0.625088i \(0.785063\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −20002.1 −0.987624 −0.493812 0.869569i \(-0.664397\pi\)
−0.493812 + 0.869569i \(0.664397\pi\)
\(744\) 0 0
\(745\) −10007.8 −0.492155
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −15167.4 −0.739927
\(750\) 0 0
\(751\) −25698.3 −1.24866 −0.624331 0.781160i \(-0.714628\pi\)
−0.624331 + 0.781160i \(0.714628\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 13263.5 0.639348
\(756\) 0 0
\(757\) −33958.5 −1.63044 −0.815219 0.579153i \(-0.803384\pi\)
−0.815219 + 0.579153i \(0.803384\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −13216.4 −0.629559 −0.314780 0.949165i \(-0.601931\pi\)
−0.314780 + 0.949165i \(0.601931\pi\)
\(762\) 0 0
\(763\) 13255.9 0.628957
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 25345.8 1.19320
\(768\) 0 0
\(769\) 3107.27 0.145710 0.0728551 0.997343i \(-0.476789\pi\)
0.0728551 + 0.997343i \(0.476789\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 20568.2 0.957034 0.478517 0.878078i \(-0.341175\pi\)
0.478517 + 0.878078i \(0.341175\pi\)
\(774\) 0 0
\(775\) 2773.06 0.128531
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −82.8228 −0.00380929
\(780\) 0 0
\(781\) −6219.65 −0.284963
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −15435.5 −0.701803
\(786\) 0 0
\(787\) 19353.9 0.876609 0.438304 0.898827i \(-0.355579\pi\)
0.438304 + 0.898827i \(0.355579\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −3216.58 −0.144587
\(792\) 0 0
\(793\) −11144.1 −0.499042
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 29243.9 1.29971 0.649856 0.760057i \(-0.274829\pi\)
0.649856 + 0.760057i \(0.274829\pi\)
\(798\) 0 0
\(799\) −2584.20 −0.114421
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −3098.97 −0.136190
\(804\) 0 0
\(805\) 7127.47 0.312063
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −31460.5 −1.36723 −0.683617 0.729841i \(-0.739594\pi\)
−0.683617 + 0.729841i \(0.739594\pi\)
\(810\) 0 0
\(811\) 22991.8 0.995503 0.497751 0.867320i \(-0.334159\pi\)
0.497751 + 0.867320i \(0.334159\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 5697.10 0.244860
\(816\) 0 0
\(817\) −5552.32 −0.237762
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 32144.1 1.36643 0.683214 0.730218i \(-0.260581\pi\)
0.683214 + 0.730218i \(0.260581\pi\)
\(822\) 0 0
\(823\) 908.840 0.0384935 0.0192468 0.999815i \(-0.493873\pi\)
0.0192468 + 0.999815i \(0.493873\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −23098.9 −0.971255 −0.485628 0.874166i \(-0.661409\pi\)
−0.485628 + 0.874166i \(0.661409\pi\)
\(828\) 0 0
\(829\) 42491.6 1.78021 0.890104 0.455757i \(-0.150631\pi\)
0.890104 + 0.455757i \(0.150631\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −2404.21 −0.100001
\(834\) 0 0
\(835\) 12655.2 0.524491
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 12239.5 0.503642 0.251821 0.967774i \(-0.418971\pi\)
0.251821 + 0.967774i \(0.418971\pi\)
\(840\) 0 0
\(841\) 31072.0 1.27402
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 17684.6 0.719962
\(846\) 0 0
\(847\) 27549.7 1.11761
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 54796.8 2.20730
\(852\) 0 0
\(853\) 9020.23 0.362071 0.181036 0.983477i \(-0.442055\pi\)
0.181036 + 0.983477i \(0.442055\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 39913.4 1.59092 0.795459 0.606008i \(-0.207230\pi\)
0.795459 + 0.606008i \(0.207230\pi\)
\(858\) 0 0
\(859\) 26038.5 1.03425 0.517126 0.855910i \(-0.327002\pi\)
0.517126 + 0.855910i \(0.327002\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 6044.84 0.238434 0.119217 0.992868i \(-0.461962\pi\)
0.119217 + 0.992868i \(0.461962\pi\)
\(864\) 0 0
\(865\) 14364.5 0.564631
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −50233.5 −1.96094
\(870\) 0 0
\(871\) −6434.32 −0.250308
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1412.43 0.0545702
\(876\) 0 0
\(877\) 28918.4 1.11346 0.556731 0.830693i \(-0.312055\pi\)
0.556731 + 0.830693i \(0.312055\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 25949.7 0.992359 0.496179 0.868220i \(-0.334736\pi\)
0.496179 + 0.868220i \(0.334736\pi\)
\(882\) 0 0
\(883\) 39784.1 1.51624 0.758121 0.652114i \(-0.226118\pi\)
0.758121 + 0.652114i \(0.226118\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 16227.6 0.614284 0.307142 0.951664i \(-0.400627\pi\)
0.307142 + 0.951664i \(0.400627\pi\)
\(888\) 0 0
\(889\) 27268.2 1.02873
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −16554.4 −0.620348
\(894\) 0 0
\(895\) 19135.8 0.714680
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −26122.4 −0.969112
\(900\) 0 0
\(901\) 5586.13 0.206549
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −3298.91 −0.121171
\(906\) 0 0
\(907\) −2107.66 −0.0771596 −0.0385798 0.999256i \(-0.512283\pi\)
−0.0385798 + 0.999256i \(0.512283\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 3471.16 0.126240 0.0631199 0.998006i \(-0.479895\pi\)
0.0631199 + 0.998006i \(0.479895\pi\)
\(912\) 0 0
\(913\) 12649.8 0.458538
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −1343.65 −0.0483872
\(918\) 0 0
\(919\) −7620.88 −0.273547 −0.136773 0.990602i \(-0.543673\pi\)
−0.136773 + 0.990602i \(0.543673\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 7671.32 0.273569
\(924\) 0 0
\(925\) 10858.9 0.385989
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −23396.4 −0.826276 −0.413138 0.910669i \(-0.635567\pi\)
−0.413138 + 0.910669i \(0.635567\pi\)
\(930\) 0 0
\(931\) −15401.4 −0.542169
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 3427.48 0.119883
\(936\) 0 0
\(937\) 23669.1 0.825226 0.412613 0.910906i \(-0.364616\pi\)
0.412613 + 0.910906i \(0.364616\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −15682.0 −0.543272 −0.271636 0.962400i \(-0.587565\pi\)
−0.271636 + 0.962400i \(0.587565\pi\)
\(942\) 0 0
\(943\) −146.079 −0.00504451
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 28683.3 0.984247 0.492123 0.870526i \(-0.336221\pi\)
0.492123 + 0.870526i \(0.336221\pi\)
\(948\) 0 0
\(949\) 3822.27 0.130744
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 20850.6 0.708728 0.354364 0.935108i \(-0.384697\pi\)
0.354364 + 0.935108i \(0.384697\pi\)
\(954\) 0 0
\(955\) −21880.5 −0.741401
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −12397.4 −0.417447
\(960\) 0 0
\(961\) −17487.2 −0.586996
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −19010.1 −0.634152
\(966\) 0 0
\(967\) −5002.87 −0.166372 −0.0831858 0.996534i \(-0.526509\pi\)
−0.0831858 + 0.996534i \(0.526509\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −39413.3 −1.30261 −0.651304 0.758817i \(-0.725778\pi\)
−0.651304 + 0.758817i \(0.725778\pi\)
\(972\) 0 0
\(973\) −4013.94 −0.132252
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −11631.3 −0.380879 −0.190439 0.981699i \(-0.560991\pi\)
−0.190439 + 0.981699i \(0.560991\pi\)
\(978\) 0 0
\(979\) −39814.1 −1.29976
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 38605.2 1.25261 0.626304 0.779579i \(-0.284567\pi\)
0.626304 + 0.779579i \(0.284567\pi\)
\(984\) 0 0
\(985\) 16701.3 0.540253
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −9792.90 −0.314860
\(990\) 0 0
\(991\) 382.345 0.0122559 0.00612795 0.999981i \(-0.498049\pi\)
0.00612795 + 0.999981i \(0.498049\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 21374.1 0.681011
\(996\) 0 0
\(997\) 36072.1 1.14585 0.572926 0.819607i \(-0.305808\pi\)
0.572926 + 0.819607i \(0.305808\pi\)
\(998\) 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 1080.4.a.p.1.3 yes 4
3.2 odd 2 1080.4.a.o.1.3 4
4.3 odd 2 2160.4.a.bv.1.2 4
12.11 even 2 2160.4.a.bu.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1080.4.a.o.1.3 4 3.2 odd 2
1080.4.a.p.1.3 yes 4 1.1 even 1 trivial
2160.4.a.bu.1.2 4 12.11 even 2
2160.4.a.bv.1.2 4 4.3 odd 2