Properties

Label 2160.4.a.bo.1.1
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: 3.3.985.1
Defining polynomial: \( x^{3} - x^{2} - 6x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 1080)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.162962\) of defining polynomial
Character \(\chi\) \(=\) 2160.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.00000 q^{5} -26.8184 q^{7} +O(q^{10})\) \(q+5.00000 q^{5} -26.8184 q^{7} -30.8629 q^{11} +32.8629 q^{13} +71.5442 q^{17} +49.3219 q^{19} -72.5035 q^{23} +25.0000 q^{25} +54.4628 q^{29} -146.736 q^{31} -134.092 q^{35} -65.6294 q^{37} +148.626 q^{41} +453.009 q^{43} +171.139 q^{47} +376.228 q^{49} -440.506 q^{53} -154.314 q^{55} +128.143 q^{59} +395.450 q^{61} +164.314 q^{65} +380.925 q^{67} -490.158 q^{71} -288.396 q^{73} +827.694 q^{77} +395.847 q^{79} -845.940 q^{83} +357.721 q^{85} -743.631 q^{89} -881.331 q^{91} +246.610 q^{95} -1840.93 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 15 q^{5} - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 15 q^{5} - 6 q^{7} - 12 q^{11} + 18 q^{13} - 21 q^{17} - 57 q^{19} - 87 q^{23} + 75 q^{25} + 138 q^{29} - 117 q^{31} - 30 q^{35} + 150 q^{37} - 180 q^{43} - 684 q^{47} - 81 q^{49} + 87 q^{53} - 60 q^{55} - 714 q^{59} - 513 q^{61} + 90 q^{65} + 174 q^{67} - 768 q^{71} - 252 q^{73} + 888 q^{77} - 207 q^{79} - 1689 q^{83} - 105 q^{85} + 312 q^{89} - 900 q^{91} - 285 q^{95} - 1080 q^{97}+O(q^{100}) \) 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
\(7\) −26.8184 −1.44806 −0.724030 0.689769i \(-0.757712\pi\)
−0.724030 + 0.689769i \(0.757712\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −30.8629 −0.845956 −0.422978 0.906140i \(-0.639015\pi\)
−0.422978 + 0.906140i \(0.639015\pi\)
\(12\) 0 0
\(13\) 32.8629 0.701117 0.350559 0.936541i \(-0.385992\pi\)
0.350559 + 0.936541i \(0.385992\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 71.5442 1.02071 0.510354 0.859965i \(-0.329515\pi\)
0.510354 + 0.859965i \(0.329515\pi\)
\(18\) 0 0
\(19\) 49.3219 0.595538 0.297769 0.954638i \(-0.403757\pi\)
0.297769 + 0.954638i \(0.403757\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −72.5035 −0.657305 −0.328653 0.944451i \(-0.606595\pi\)
−0.328653 + 0.944451i \(0.606595\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 54.4628 0.348741 0.174370 0.984680i \(-0.444211\pi\)
0.174370 + 0.984680i \(0.444211\pi\)
\(30\) 0 0
\(31\) −146.736 −0.850150 −0.425075 0.905158i \(-0.639752\pi\)
−0.425075 + 0.905158i \(0.639752\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −134.092 −0.647592
\(36\) 0 0
\(37\) −65.6294 −0.291606 −0.145803 0.989314i \(-0.546576\pi\)
−0.145803 + 0.989314i \(0.546576\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 148.626 0.566132 0.283066 0.959100i \(-0.408648\pi\)
0.283066 + 0.959100i \(0.408648\pi\)
\(42\) 0 0
\(43\) 453.009 1.60659 0.803294 0.595583i \(-0.203079\pi\)
0.803294 + 0.595583i \(0.203079\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 171.139 0.531133 0.265566 0.964093i \(-0.414441\pi\)
0.265566 + 0.964093i \(0.414441\pi\)
\(48\) 0 0
\(49\) 376.228 1.09688
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −440.506 −1.14166 −0.570831 0.821067i \(-0.693379\pi\)
−0.570831 + 0.821067i \(0.693379\pi\)
\(54\) 0 0
\(55\) −154.314 −0.378323
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 128.143 0.282760 0.141380 0.989955i \(-0.454846\pi\)
0.141380 + 0.989955i \(0.454846\pi\)
\(60\) 0 0
\(61\) 395.450 0.830036 0.415018 0.909813i \(-0.363775\pi\)
0.415018 + 0.909813i \(0.363775\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 164.314 0.313549
\(66\) 0 0
\(67\) 380.925 0.694587 0.347294 0.937756i \(-0.387101\pi\)
0.347294 + 0.937756i \(0.387101\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −490.158 −0.819311 −0.409655 0.912240i \(-0.634351\pi\)
−0.409655 + 0.912240i \(0.634351\pi\)
\(72\) 0 0
\(73\) −288.396 −0.462386 −0.231193 0.972908i \(-0.574263\pi\)
−0.231193 + 0.972908i \(0.574263\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 827.694 1.22499
\(78\) 0 0
\(79\) 395.847 0.563750 0.281875 0.959451i \(-0.409044\pi\)
0.281875 + 0.959451i \(0.409044\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −845.940 −1.11872 −0.559361 0.828924i \(-0.688954\pi\)
−0.559361 + 0.828924i \(0.688954\pi\)
\(84\) 0 0
\(85\) 357.721 0.456474
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −743.631 −0.885671 −0.442835 0.896603i \(-0.646027\pi\)
−0.442835 + 0.896603i \(0.646027\pi\)
\(90\) 0 0
\(91\) −881.331 −1.01526
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 246.610 0.266333
\(96\) 0 0
\(97\) −1840.93 −1.92699 −0.963494 0.267728i \(-0.913727\pi\)
−0.963494 + 0.267728i \(0.913727\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −9.00402 −0.00887062 −0.00443531 0.999990i \(-0.501412\pi\)
−0.00443531 + 0.999990i \(0.501412\pi\)
\(102\) 0 0
\(103\) −1181.11 −1.12988 −0.564941 0.825131i \(-0.691101\pi\)
−0.564941 + 0.825131i \(0.691101\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1198.10 −1.08247 −0.541235 0.840872i \(-0.682043\pi\)
−0.541235 + 0.840872i \(0.682043\pi\)
\(108\) 0 0
\(109\) −674.102 −0.592360 −0.296180 0.955132i \(-0.595713\pi\)
−0.296180 + 0.955132i \(0.595713\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 779.998 0.649346 0.324673 0.945826i \(-0.394746\pi\)
0.324673 + 0.945826i \(0.394746\pi\)
\(114\) 0 0
\(115\) −362.517 −0.293956
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −1918.70 −1.47804
\(120\) 0 0
\(121\) −378.482 −0.284359
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) −1180.26 −0.824654 −0.412327 0.911036i \(-0.635284\pi\)
−0.412327 + 0.911036i \(0.635284\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −721.126 −0.480955 −0.240477 0.970655i \(-0.577304\pi\)
−0.240477 + 0.970655i \(0.577304\pi\)
\(132\) 0 0
\(133\) −1322.74 −0.862375
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2988.13 1.86345 0.931727 0.363159i \(-0.118302\pi\)
0.931727 + 0.363159i \(0.118302\pi\)
\(138\) 0 0
\(139\) 2231.99 1.36198 0.680990 0.732293i \(-0.261550\pi\)
0.680990 + 0.732293i \(0.261550\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −1014.24 −0.593114
\(144\) 0 0
\(145\) 272.314 0.155962
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1683.57 0.925659 0.462829 0.886447i \(-0.346834\pi\)
0.462829 + 0.886447i \(0.346834\pi\)
\(150\) 0 0
\(151\) −3430.80 −1.84897 −0.924485 0.381218i \(-0.875505\pi\)
−0.924485 + 0.381218i \(0.875505\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −733.682 −0.380199
\(156\) 0 0
\(157\) −1729.76 −0.879296 −0.439648 0.898170i \(-0.644897\pi\)
−0.439648 + 0.898170i \(0.644897\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1944.43 0.951817
\(162\) 0 0
\(163\) −1101.14 −0.529130 −0.264565 0.964368i \(-0.585228\pi\)
−0.264565 + 0.964368i \(0.585228\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1871.07 0.866991 0.433495 0.901156i \(-0.357280\pi\)
0.433495 + 0.901156i \(0.357280\pi\)
\(168\) 0 0
\(169\) −1117.03 −0.508434
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2686.45 1.18062 0.590308 0.807178i \(-0.299006\pi\)
0.590308 + 0.807178i \(0.299006\pi\)
\(174\) 0 0
\(175\) −670.461 −0.289612
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −3620.13 −1.51163 −0.755814 0.654786i \(-0.772759\pi\)
−0.755814 + 0.654786i \(0.772759\pi\)
\(180\) 0 0
\(181\) −444.917 −0.182710 −0.0913548 0.995818i \(-0.529120\pi\)
−0.0913548 + 0.995818i \(0.529120\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −328.147 −0.130410
\(186\) 0 0
\(187\) −2208.06 −0.863473
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 2086.82 0.790561 0.395280 0.918561i \(-0.370647\pi\)
0.395280 + 0.918561i \(0.370647\pi\)
\(192\) 0 0
\(193\) −2750.05 −1.02566 −0.512831 0.858489i \(-0.671403\pi\)
−0.512831 + 0.858489i \(0.671403\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2617.99 0.946824 0.473412 0.880841i \(-0.343022\pi\)
0.473412 + 0.880841i \(0.343022\pi\)
\(198\) 0 0
\(199\) −2697.80 −0.961014 −0.480507 0.876991i \(-0.659547\pi\)
−0.480507 + 0.876991i \(0.659547\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1460.61 −0.504997
\(204\) 0 0
\(205\) 743.128 0.253182
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −1522.22 −0.503799
\(210\) 0 0
\(211\) 1577.75 0.514773 0.257386 0.966309i \(-0.417139\pi\)
0.257386 + 0.966309i \(0.417139\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 2265.05 0.718488
\(216\) 0 0
\(217\) 3935.24 1.23107
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2351.15 0.715635
\(222\) 0 0
\(223\) −3316.84 −0.996018 −0.498009 0.867172i \(-0.665935\pi\)
−0.498009 + 0.867172i \(0.665935\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 6667.43 1.94948 0.974742 0.223332i \(-0.0716935\pi\)
0.974742 + 0.223332i \(0.0716935\pi\)
\(228\) 0 0
\(229\) −4170.75 −1.20354 −0.601770 0.798669i \(-0.705538\pi\)
−0.601770 + 0.798669i \(0.705538\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −5684.72 −1.59836 −0.799181 0.601090i \(-0.794733\pi\)
−0.799181 + 0.601090i \(0.794733\pi\)
\(234\) 0 0
\(235\) 855.697 0.237530
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −2980.17 −0.806573 −0.403286 0.915074i \(-0.632132\pi\)
−0.403286 + 0.915074i \(0.632132\pi\)
\(240\) 0 0
\(241\) 2392.44 0.639462 0.319731 0.947508i \(-0.396407\pi\)
0.319731 + 0.947508i \(0.396407\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1881.14 0.490538
\(246\) 0 0
\(247\) 1620.86 0.417542
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −2328.26 −0.585491 −0.292746 0.956190i \(-0.594569\pi\)
−0.292746 + 0.956190i \(0.594569\pi\)
\(252\) 0 0
\(253\) 2237.67 0.556051
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −136.227 −0.0330646 −0.0165323 0.999863i \(-0.505263\pi\)
−0.0165323 + 0.999863i \(0.505263\pi\)
\(258\) 0 0
\(259\) 1760.08 0.422262
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −4436.05 −1.04007 −0.520036 0.854145i \(-0.674081\pi\)
−0.520036 + 0.854145i \(0.674081\pi\)
\(264\) 0 0
\(265\) −2202.53 −0.510567
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −1679.91 −0.380766 −0.190383 0.981710i \(-0.560973\pi\)
−0.190383 + 0.981710i \(0.560973\pi\)
\(270\) 0 0
\(271\) −846.654 −0.189781 −0.0948904 0.995488i \(-0.530250\pi\)
−0.0948904 + 0.995488i \(0.530250\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −771.572 −0.169191
\(276\) 0 0
\(277\) 554.305 0.120235 0.0601173 0.998191i \(-0.480853\pi\)
0.0601173 + 0.998191i \(0.480853\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −1715.63 −0.364221 −0.182110 0.983278i \(-0.558293\pi\)
−0.182110 + 0.983278i \(0.558293\pi\)
\(282\) 0 0
\(283\) 3327.36 0.698907 0.349454 0.936954i \(-0.386367\pi\)
0.349454 + 0.936954i \(0.386367\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −3985.91 −0.819793
\(288\) 0 0
\(289\) 205.574 0.0418429
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1291.74 0.257557 0.128778 0.991673i \(-0.458894\pi\)
0.128778 + 0.991673i \(0.458894\pi\)
\(294\) 0 0
\(295\) 640.716 0.126454
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −2382.67 −0.460848
\(300\) 0 0
\(301\) −12149.0 −2.32643
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1977.25 0.371203
\(306\) 0 0
\(307\) −10514.4 −1.95468 −0.977339 0.211678i \(-0.932107\pi\)
−0.977339 + 0.211678i \(0.932107\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −8245.53 −1.50341 −0.751706 0.659498i \(-0.770769\pi\)
−0.751706 + 0.659498i \(0.770769\pi\)
\(312\) 0 0
\(313\) −9926.89 −1.79265 −0.896327 0.443393i \(-0.853775\pi\)
−0.896327 + 0.443393i \(0.853775\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −2204.92 −0.390665 −0.195333 0.980737i \(-0.562579\pi\)
−0.195333 + 0.980737i \(0.562579\pi\)
\(318\) 0 0
\(319\) −1680.88 −0.295019
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3528.70 0.607870
\(324\) 0 0
\(325\) 821.572 0.140223
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −4589.69 −0.769112
\(330\) 0 0
\(331\) 8359.73 1.38819 0.694097 0.719882i \(-0.255804\pi\)
0.694097 + 0.719882i \(0.255804\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1904.62 0.310629
\(336\) 0 0
\(337\) 12063.6 1.95000 0.974998 0.222213i \(-0.0713282\pi\)
0.974998 + 0.222213i \(0.0713282\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 4528.71 0.719189
\(342\) 0 0
\(343\) −891.130 −0.140281
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 151.823 0.0234878 0.0117439 0.999931i \(-0.496262\pi\)
0.0117439 + 0.999931i \(0.496262\pi\)
\(348\) 0 0
\(349\) −5990.20 −0.918763 −0.459381 0.888239i \(-0.651929\pi\)
−0.459381 + 0.888239i \(0.651929\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 9917.82 1.49539 0.747695 0.664043i \(-0.231161\pi\)
0.747695 + 0.664043i \(0.231161\pi\)
\(354\) 0 0
\(355\) −2450.79 −0.366407
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 852.720 0.125362 0.0626809 0.998034i \(-0.480035\pi\)
0.0626809 + 0.998034i \(0.480035\pi\)
\(360\) 0 0
\(361\) −4426.35 −0.645334
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −1441.98 −0.206785
\(366\) 0 0
\(367\) −6306.86 −0.897045 −0.448522 0.893772i \(-0.648050\pi\)
−0.448522 + 0.893772i \(0.648050\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 11813.7 1.65320
\(372\) 0 0
\(373\) −5201.75 −0.722082 −0.361041 0.932550i \(-0.617579\pi\)
−0.361041 + 0.932550i \(0.617579\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1789.80 0.244508
\(378\) 0 0
\(379\) −13112.8 −1.77720 −0.888599 0.458684i \(-0.848321\pi\)
−0.888599 + 0.458684i \(0.848321\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 12502.2 1.66797 0.833987 0.551785i \(-0.186053\pi\)
0.833987 + 0.551785i \(0.186053\pi\)
\(384\) 0 0
\(385\) 4138.47 0.547834
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −7814.63 −1.01855 −0.509277 0.860603i \(-0.670087\pi\)
−0.509277 + 0.860603i \(0.670087\pi\)
\(390\) 0 0
\(391\) −5187.20 −0.670916
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1979.23 0.252117
\(396\) 0 0
\(397\) 2748.91 0.347516 0.173758 0.984788i \(-0.444409\pi\)
0.173758 + 0.984788i \(0.444409\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −12676.6 −1.57865 −0.789325 0.613975i \(-0.789570\pi\)
−0.789325 + 0.613975i \(0.789570\pi\)
\(402\) 0 0
\(403\) −4822.19 −0.596055
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2025.51 0.246685
\(408\) 0 0
\(409\) −9710.08 −1.17392 −0.586959 0.809616i \(-0.699675\pi\)
−0.586959 + 0.809616i \(0.699675\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −3436.60 −0.409453
\(414\) 0 0
\(415\) −4229.70 −0.500308
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 3292.09 0.383841 0.191920 0.981411i \(-0.438528\pi\)
0.191920 + 0.981411i \(0.438528\pi\)
\(420\) 0 0
\(421\) −14438.3 −1.67144 −0.835722 0.549152i \(-0.814951\pi\)
−0.835722 + 0.549152i \(0.814951\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1788.61 0.204141
\(426\) 0 0
\(427\) −10605.4 −1.20194
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −1304.43 −0.145783 −0.0728914 0.997340i \(-0.523223\pi\)
−0.0728914 + 0.997340i \(0.523223\pi\)
\(432\) 0 0
\(433\) 9495.26 1.05384 0.526921 0.849914i \(-0.323346\pi\)
0.526921 + 0.849914i \(0.323346\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −3576.01 −0.391450
\(438\) 0 0
\(439\) 2076.89 0.225796 0.112898 0.993607i \(-0.463987\pi\)
0.112898 + 0.993607i \(0.463987\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 406.522 0.0435992 0.0217996 0.999762i \(-0.493060\pi\)
0.0217996 + 0.999762i \(0.493060\pi\)
\(444\) 0 0
\(445\) −3718.15 −0.396084
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −9231.25 −0.970266 −0.485133 0.874440i \(-0.661229\pi\)
−0.485133 + 0.874440i \(0.661229\pi\)
\(450\) 0 0
\(451\) −4587.02 −0.478923
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −4406.66 −0.454038
\(456\) 0 0
\(457\) −5411.09 −0.553873 −0.276936 0.960888i \(-0.589319\pi\)
−0.276936 + 0.960888i \(0.589319\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −16604.5 −1.67755 −0.838774 0.544480i \(-0.816727\pi\)
−0.838774 + 0.544480i \(0.816727\pi\)
\(462\) 0 0
\(463\) 13732.8 1.37843 0.689217 0.724555i \(-0.257955\pi\)
0.689217 + 0.724555i \(0.257955\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 6504.30 0.644503 0.322252 0.946654i \(-0.395560\pi\)
0.322252 + 0.946654i \(0.395560\pi\)
\(468\) 0 0
\(469\) −10215.8 −1.00580
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −13981.2 −1.35910
\(474\) 0 0
\(475\) 1233.05 0.119108
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 8405.21 0.801762 0.400881 0.916130i \(-0.368704\pi\)
0.400881 + 0.916130i \(0.368704\pi\)
\(480\) 0 0
\(481\) −2156.77 −0.204450
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −9204.64 −0.861776
\(486\) 0 0
\(487\) −17940.9 −1.66936 −0.834682 0.550732i \(-0.814349\pi\)
−0.834682 + 0.550732i \(0.814349\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 11905.2 1.09424 0.547122 0.837053i \(-0.315723\pi\)
0.547122 + 0.837053i \(0.315723\pi\)
\(492\) 0 0
\(493\) 3896.50 0.355962
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 13145.3 1.18641
\(498\) 0 0
\(499\) 12372.2 1.10993 0.554964 0.831874i \(-0.312732\pi\)
0.554964 + 0.831874i \(0.312732\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 19878.8 1.76213 0.881065 0.472994i \(-0.156827\pi\)
0.881065 + 0.472994i \(0.156827\pi\)
\(504\) 0 0
\(505\) −45.0201 −0.00396706
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 1624.65 0.141476 0.0707379 0.997495i \(-0.477465\pi\)
0.0707379 + 0.997495i \(0.477465\pi\)
\(510\) 0 0
\(511\) 7734.33 0.669562
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −5905.53 −0.505298
\(516\) 0 0
\(517\) −5281.86 −0.449315
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1131.62 0.0951574 0.0475787 0.998867i \(-0.484850\pi\)
0.0475787 + 0.998867i \(0.484850\pi\)
\(522\) 0 0
\(523\) −4750.46 −0.397176 −0.198588 0.980083i \(-0.563636\pi\)
−0.198588 + 0.980083i \(0.563636\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −10498.1 −0.867754
\(528\) 0 0
\(529\) −6910.24 −0.567950
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 4884.27 0.396925
\(534\) 0 0
\(535\) −5990.48 −0.484095
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −11611.5 −0.927908
\(540\) 0 0
\(541\) 22335.8 1.77503 0.887514 0.460781i \(-0.152431\pi\)
0.887514 + 0.460781i \(0.152431\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −3370.51 −0.264911
\(546\) 0 0
\(547\) −20641.1 −1.61343 −0.806717 0.590937i \(-0.798758\pi\)
−0.806717 + 0.590937i \(0.798758\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 2686.21 0.207688
\(552\) 0 0
\(553\) −10616.0 −0.816344
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −19431.2 −1.47814 −0.739071 0.673628i \(-0.764735\pi\)
−0.739071 + 0.673628i \(0.764735\pi\)
\(558\) 0 0
\(559\) 14887.2 1.12641
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −8084.58 −0.605194 −0.302597 0.953119i \(-0.597854\pi\)
−0.302597 + 0.953119i \(0.597854\pi\)
\(564\) 0 0
\(565\) 3899.99 0.290396
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 5122.63 0.377420 0.188710 0.982033i \(-0.439569\pi\)
0.188710 + 0.982033i \(0.439569\pi\)
\(570\) 0 0
\(571\) −6658.07 −0.487971 −0.243986 0.969779i \(-0.578455\pi\)
−0.243986 + 0.969779i \(0.578455\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1812.59 −0.131461
\(576\) 0 0
\(577\) 18436.5 1.33019 0.665097 0.746757i \(-0.268390\pi\)
0.665097 + 0.746757i \(0.268390\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 22686.8 1.61998
\(582\) 0 0
\(583\) 13595.3 0.965796
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 7573.74 0.532541 0.266271 0.963898i \(-0.414208\pi\)
0.266271 + 0.963898i \(0.414208\pi\)
\(588\) 0 0
\(589\) −7237.33 −0.506297
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 7594.14 0.525892 0.262946 0.964811i \(-0.415306\pi\)
0.262946 + 0.964811i \(0.415306\pi\)
\(594\) 0 0
\(595\) −9593.52 −0.661001
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 12198.7 0.832094 0.416047 0.909343i \(-0.363415\pi\)
0.416047 + 0.909343i \(0.363415\pi\)
\(600\) 0 0
\(601\) 26299.6 1.78500 0.892498 0.451051i \(-0.148951\pi\)
0.892498 + 0.451051i \(0.148951\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −1892.41 −0.127169
\(606\) 0 0
\(607\) 27293.4 1.82505 0.912523 0.409025i \(-0.134131\pi\)
0.912523 + 0.409025i \(0.134131\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 5624.13 0.372386
\(612\) 0 0
\(613\) −15607.9 −1.02838 −0.514191 0.857676i \(-0.671908\pi\)
−0.514191 + 0.857676i \(0.671908\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −22973.4 −1.49898 −0.749492 0.662013i \(-0.769702\pi\)
−0.749492 + 0.662013i \(0.769702\pi\)
\(618\) 0 0
\(619\) 4542.72 0.294971 0.147486 0.989064i \(-0.452882\pi\)
0.147486 + 0.989064i \(0.452882\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 19943.0 1.28250
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −4695.40 −0.297644
\(630\) 0 0
\(631\) −6102.70 −0.385015 −0.192507 0.981296i \(-0.561662\pi\)
−0.192507 + 0.981296i \(0.561662\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −5901.30 −0.368797
\(636\) 0 0
\(637\) 12363.9 0.769038
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 19133.6 1.17899 0.589493 0.807773i \(-0.299328\pi\)
0.589493 + 0.807773i \(0.299328\pi\)
\(642\) 0 0
\(643\) 4281.30 0.262579 0.131289 0.991344i \(-0.458088\pi\)
0.131289 + 0.991344i \(0.458088\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −3061.84 −0.186048 −0.0930242 0.995664i \(-0.529653\pi\)
−0.0930242 + 0.995664i \(0.529653\pi\)
\(648\) 0 0
\(649\) −3954.87 −0.239202
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 19647.5 1.17744 0.588718 0.808338i \(-0.299633\pi\)
0.588718 + 0.808338i \(0.299633\pi\)
\(654\) 0 0
\(655\) −3605.63 −0.215089
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 4733.23 0.279788 0.139894 0.990166i \(-0.455324\pi\)
0.139894 + 0.990166i \(0.455324\pi\)
\(660\) 0 0
\(661\) −26923.2 −1.58425 −0.792127 0.610356i \(-0.791026\pi\)
−0.792127 + 0.610356i \(0.791026\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −6613.68 −0.385666
\(666\) 0 0
\(667\) −3948.74 −0.229229
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −12204.7 −0.702174
\(672\) 0 0
\(673\) 5071.60 0.290484 0.145242 0.989396i \(-0.453604\pi\)
0.145242 + 0.989396i \(0.453604\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 218.456 0.0124017 0.00620086 0.999981i \(-0.498026\pi\)
0.00620086 + 0.999981i \(0.498026\pi\)
\(678\) 0 0
\(679\) 49370.8 2.79039
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −20868.6 −1.16913 −0.584565 0.811347i \(-0.698735\pi\)
−0.584565 + 0.811347i \(0.698735\pi\)
\(684\) 0 0
\(685\) 14940.7 0.833362
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −14476.3 −0.800439
\(690\) 0 0
\(691\) 14869.6 0.818618 0.409309 0.912396i \(-0.365770\pi\)
0.409309 + 0.912396i \(0.365770\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 11160.0 0.609096
\(696\) 0 0
\(697\) 10633.3 0.577855
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −20898.6 −1.12600 −0.563002 0.826455i \(-0.690354\pi\)
−0.563002 + 0.826455i \(0.690354\pi\)
\(702\) 0 0
\(703\) −3236.97 −0.173662
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 241.474 0.0128452
\(708\) 0 0
\(709\) −27866.7 −1.47610 −0.738051 0.674745i \(-0.764253\pi\)
−0.738051 + 0.674745i \(0.764253\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 10638.9 0.558808
\(714\) 0 0
\(715\) −5071.22 −0.265249
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −19210.2 −0.996412 −0.498206 0.867059i \(-0.666008\pi\)
−0.498206 + 0.867059i \(0.666008\pi\)
\(720\) 0 0
\(721\) 31675.4 1.63614
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1361.57 0.0697482
\(726\) 0 0
\(727\) −15121.7 −0.771433 −0.385716 0.922617i \(-0.626046\pi\)
−0.385716 + 0.922617i \(0.626046\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 32410.2 1.63986
\(732\) 0 0
\(733\) −6552.13 −0.330162 −0.165081 0.986280i \(-0.552788\pi\)
−0.165081 + 0.986280i \(0.552788\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −11756.4 −0.587590
\(738\) 0 0
\(739\) 13514.5 0.672719 0.336359 0.941734i \(-0.390804\pi\)
0.336359 + 0.941734i \(0.390804\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 25986.4 1.28311 0.641553 0.767079i \(-0.278290\pi\)
0.641553 + 0.767079i \(0.278290\pi\)
\(744\) 0 0
\(745\) 8417.83 0.413967
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 32131.0 1.56748
\(750\) 0 0
\(751\) −21586.1 −1.04885 −0.524425 0.851457i \(-0.675720\pi\)
−0.524425 + 0.851457i \(0.675720\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −17154.0 −0.826885
\(756\) 0 0
\(757\) 8240.02 0.395626 0.197813 0.980240i \(-0.436616\pi\)
0.197813 + 0.980240i \(0.436616\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 21903.1 1.04335 0.521674 0.853145i \(-0.325308\pi\)
0.521674 + 0.853145i \(0.325308\pi\)
\(762\) 0 0
\(763\) 18078.4 0.857772
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 4211.15 0.198248
\(768\) 0 0
\(769\) 8344.70 0.391310 0.195655 0.980673i \(-0.437317\pi\)
0.195655 + 0.980673i \(0.437317\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −10226.3 −0.475826 −0.237913 0.971286i \(-0.576463\pi\)
−0.237913 + 0.971286i \(0.576463\pi\)
\(774\) 0 0
\(775\) −3668.41 −0.170030
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 7330.50 0.337153
\(780\) 0 0
\(781\) 15127.7 0.693100
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −8648.78 −0.393233
\(786\) 0 0
\(787\) −5355.21 −0.242557 −0.121279 0.992619i \(-0.538699\pi\)
−0.121279 + 0.992619i \(0.538699\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −20918.3 −0.940291
\(792\) 0 0
\(793\) 12995.6 0.581953
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −24006.7 −1.06695 −0.533476 0.845815i \(-0.679115\pi\)
−0.533476 + 0.845815i \(0.679115\pi\)
\(798\) 0 0
\(799\) 12244.0 0.542131
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 8900.73 0.391158
\(804\) 0 0
\(805\) 9722.15 0.425666
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −845.632 −0.0367501 −0.0183751 0.999831i \(-0.505849\pi\)
−0.0183751 + 0.999831i \(0.505849\pi\)
\(810\) 0 0
\(811\) −3745.89 −0.162190 −0.0810950 0.996706i \(-0.525842\pi\)
−0.0810950 + 0.996706i \(0.525842\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −5505.72 −0.236634
\(816\) 0 0
\(817\) 22343.3 0.956784
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 27575.2 1.17221 0.586104 0.810236i \(-0.300661\pi\)
0.586104 + 0.810236i \(0.300661\pi\)
\(822\) 0 0
\(823\) 5541.15 0.234693 0.117347 0.993091i \(-0.462561\pi\)
0.117347 + 0.993091i \(0.462561\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −38373.8 −1.61353 −0.806765 0.590873i \(-0.798784\pi\)
−0.806765 + 0.590873i \(0.798784\pi\)
\(828\) 0 0
\(829\) −1413.64 −0.0592252 −0.0296126 0.999561i \(-0.509427\pi\)
−0.0296126 + 0.999561i \(0.509427\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 26917.0 1.11959
\(834\) 0 0
\(835\) 9355.33 0.387730
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −15354.0 −0.631798 −0.315899 0.948793i \(-0.602306\pi\)
−0.315899 + 0.948793i \(0.602306\pi\)
\(840\) 0 0
\(841\) −21422.8 −0.878380
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −5585.15 −0.227379
\(846\) 0 0
\(847\) 10150.3 0.411769
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 4758.36 0.191674
\(852\) 0 0
\(853\) −29717.0 −1.19284 −0.596419 0.802674i \(-0.703410\pi\)
−0.596419 + 0.802674i \(0.703410\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 31742.7 1.26524 0.632619 0.774463i \(-0.281980\pi\)
0.632619 + 0.774463i \(0.281980\pi\)
\(858\) 0 0
\(859\) 5520.83 0.219288 0.109644 0.993971i \(-0.465029\pi\)
0.109644 + 0.993971i \(0.465029\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −19947.1 −0.786799 −0.393399 0.919368i \(-0.628701\pi\)
−0.393399 + 0.919368i \(0.628701\pi\)
\(864\) 0 0
\(865\) 13432.2 0.527988
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −12217.0 −0.476908
\(870\) 0 0
\(871\) 12518.3 0.486987
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −3352.30 −0.129518
\(876\) 0 0
\(877\) 24074.1 0.926937 0.463469 0.886113i \(-0.346605\pi\)
0.463469 + 0.886113i \(0.346605\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −29201.8 −1.11672 −0.558362 0.829598i \(-0.688570\pi\)
−0.558362 + 0.829598i \(0.688570\pi\)
\(882\) 0 0
\(883\) 7705.46 0.293669 0.146834 0.989161i \(-0.453092\pi\)
0.146834 + 0.989161i \(0.453092\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −4225.85 −0.159966 −0.0799832 0.996796i \(-0.525487\pi\)
−0.0799832 + 0.996796i \(0.525487\pi\)
\(888\) 0 0
\(889\) 31652.7 1.19415
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 8440.92 0.316310
\(894\) 0 0
\(895\) −18100.7 −0.676021
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −7991.68 −0.296482
\(900\) 0 0
\(901\) −31515.6 −1.16530
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −2224.59 −0.0817102
\(906\) 0 0
\(907\) −16052.5 −0.587667 −0.293833 0.955857i \(-0.594931\pi\)
−0.293833 + 0.955857i \(0.594931\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −911.735 −0.0331582 −0.0165791 0.999863i \(-0.505278\pi\)
−0.0165791 + 0.999863i \(0.505278\pi\)
\(912\) 0 0
\(913\) 26108.2 0.946390
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 19339.5 0.696451
\(918\) 0 0
\(919\) −2367.26 −0.0849713 −0.0424856 0.999097i \(-0.513528\pi\)
−0.0424856 + 0.999097i \(0.513528\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −16108.0 −0.574433
\(924\) 0 0
\(925\) −1640.73 −0.0583211
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 14903.0 0.526319 0.263159 0.964752i \(-0.415235\pi\)
0.263159 + 0.964752i \(0.415235\pi\)
\(930\) 0 0
\(931\) 18556.3 0.653231
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −11040.3 −0.386157
\(936\) 0 0
\(937\) 44983.7 1.56836 0.784180 0.620534i \(-0.213084\pi\)
0.784180 + 0.620534i \(0.213084\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 35749.8 1.23848 0.619240 0.785202i \(-0.287441\pi\)
0.619240 + 0.785202i \(0.287441\pi\)
\(942\) 0 0
\(943\) −10775.9 −0.372122
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −46315.7 −1.58929 −0.794646 0.607074i \(-0.792343\pi\)
−0.794646 + 0.607074i \(0.792343\pi\)
\(948\) 0 0
\(949\) −9477.52 −0.324187
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 39448.6 1.34089 0.670444 0.741960i \(-0.266104\pi\)
0.670444 + 0.741960i \(0.266104\pi\)
\(954\) 0 0
\(955\) 10434.1 0.353549
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −80137.0 −2.69839
\(960\) 0 0
\(961\) −8259.40 −0.277245
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −13750.3 −0.458690
\(966\) 0 0
\(967\) −43691.9 −1.45299 −0.726493 0.687174i \(-0.758851\pi\)
−0.726493 + 0.687174i \(0.758851\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 5070.26 0.167572 0.0837860 0.996484i \(-0.473299\pi\)
0.0837860 + 0.996484i \(0.473299\pi\)
\(972\) 0 0
\(973\) −59858.6 −1.97223
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 34289.3 1.12284 0.561419 0.827532i \(-0.310256\pi\)
0.561419 + 0.827532i \(0.310256\pi\)
\(978\) 0 0
\(979\) 22950.6 0.749238
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 57986.0 1.88145 0.940725 0.339170i \(-0.110146\pi\)
0.940725 + 0.339170i \(0.110146\pi\)
\(984\) 0 0
\(985\) 13090.0 0.423432
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −32844.8 −1.05602
\(990\) 0 0
\(991\) 24526.7 0.786193 0.393096 0.919497i \(-0.371404\pi\)
0.393096 + 0.919497i \(0.371404\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −13489.0 −0.429778
\(996\) 0 0
\(997\) −16061.9 −0.510215 −0.255108 0.966913i \(-0.582111\pi\)
−0.255108 + 0.966913i \(0.582111\pi\)
\(998\) 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.bo.1.1 3
3.2 odd 2 2160.4.a.bg.1.1 3
4.3 odd 2 1080.4.a.m.1.3 yes 3
12.11 even 2 1080.4.a.g.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1080.4.a.g.1.3 3 12.11 even 2
1080.4.a.m.1.3 yes 3 4.3 odd 2
2160.4.a.bg.1.1 3 3.2 odd 2
2160.4.a.bo.1.1 3 1.1 even 1 trivial