Properties

Label 2160.4.a.be.1.3
Level $2160$
Weight $4$
Character 2160.1
Self dual yes
Analytic conductor $127.444$
Analytic rank $0$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.5637.1
Defining polynomial: \( x^{3} - x^{2} - 23x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
Twist minimal: no (minimal twist has level 135)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-4.45938\) of defining polynomial
Character \(\chi\) \(=\) 2160.1

$q$-expansion

\(f(q)\) \(=\) \(q-5.00000 q^{5} -5.08123 q^{7} +O(q^{10})\) \(q-5.00000 q^{5} -5.08123 q^{7} +58.3007 q^{11} +21.2119 q^{13} +68.8451 q^{17} +40.8133 q^{19} -144.318 q^{23} +25.0000 q^{25} +220.058 q^{29} -291.545 q^{31} +25.4062 q^{35} +260.637 q^{37} +169.766 q^{41} +438.596 q^{43} -255.481 q^{47} -317.181 q^{49} -214.714 q^{53} -291.503 q^{55} +331.524 q^{59} +54.9647 q^{61} -106.060 q^{65} -758.179 q^{67} -904.348 q^{71} +866.622 q^{73} -296.239 q^{77} -206.961 q^{79} -463.397 q^{83} -344.225 q^{85} -601.736 q^{89} -107.783 q^{91} -204.066 q^{95} +229.363 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 15 q^{5} - 44 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 15 q^{5} - 44 q^{7} + 38 q^{11} + 28 q^{13} + 19 q^{17} - 187 q^{19} - 81 q^{23} + 75 q^{25} - 160 q^{29} - 227 q^{31} + 220 q^{35} + 78 q^{37} + 338 q^{41} - 22 q^{43} - 472 q^{47} - 197 q^{49} - 521 q^{53} - 190 q^{55} + 140 q^{59} + 595 q^{61} - 140 q^{65} - 878 q^{67} - 602 q^{71} + 1294 q^{73} - 288 q^{77} - 629 q^{79} - 1287 q^{83} - 95 q^{85} - 2154 q^{89} + 440 q^{91} + 935 q^{95} + 1392 q^{97}+O(q^{100}) \) 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
\(7\) −5.08123 −0.274361 −0.137180 0.990546i \(-0.543804\pi\)
−0.137180 + 0.990546i \(0.543804\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 58.3007 1.59803 0.799014 0.601312i \(-0.205355\pi\)
0.799014 + 0.601312i \(0.205355\pi\)
\(12\) 0 0
\(13\) 21.2119 0.452548 0.226274 0.974064i \(-0.427345\pi\)
0.226274 + 0.974064i \(0.427345\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 68.8451 0.982199 0.491099 0.871104i \(-0.336595\pi\)
0.491099 + 0.871104i \(0.336595\pi\)
\(18\) 0 0
\(19\) 40.8133 0.492800 0.246400 0.969168i \(-0.420752\pi\)
0.246400 + 0.969168i \(0.420752\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −144.318 −1.30837 −0.654184 0.756336i \(-0.726988\pi\)
−0.654184 + 0.756336i \(0.726988\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 220.058 1.40909 0.704547 0.709657i \(-0.251150\pi\)
0.704547 + 0.709657i \(0.251150\pi\)
\(30\) 0 0
\(31\) −291.545 −1.68913 −0.844566 0.535452i \(-0.820141\pi\)
−0.844566 + 0.535452i \(0.820141\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 25.4062 0.122698
\(36\) 0 0
\(37\) 260.637 1.15807 0.579033 0.815304i \(-0.303430\pi\)
0.579033 + 0.815304i \(0.303430\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 169.766 0.646660 0.323330 0.946286i \(-0.395198\pi\)
0.323330 + 0.946286i \(0.395198\pi\)
\(42\) 0 0
\(43\) 438.596 1.55547 0.777735 0.628592i \(-0.216369\pi\)
0.777735 + 0.628592i \(0.216369\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −255.481 −0.792887 −0.396444 0.918059i \(-0.629756\pi\)
−0.396444 + 0.918059i \(0.629756\pi\)
\(48\) 0 0
\(49\) −317.181 −0.924726
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −214.714 −0.556477 −0.278239 0.960512i \(-0.589751\pi\)
−0.278239 + 0.960512i \(0.589751\pi\)
\(54\) 0 0
\(55\) −291.503 −0.714660
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 331.524 0.731537 0.365769 0.930706i \(-0.380806\pi\)
0.365769 + 0.930706i \(0.380806\pi\)
\(60\) 0 0
\(61\) 54.9647 0.115369 0.0576845 0.998335i \(-0.481628\pi\)
0.0576845 + 0.998335i \(0.481628\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −106.060 −0.202386
\(66\) 0 0
\(67\) −758.179 −1.38248 −0.691241 0.722624i \(-0.742936\pi\)
−0.691241 + 0.722624i \(0.742936\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −904.348 −1.51164 −0.755819 0.654780i \(-0.772761\pi\)
−0.755819 + 0.654780i \(0.772761\pi\)
\(72\) 0 0
\(73\) 866.622 1.38946 0.694729 0.719271i \(-0.255524\pi\)
0.694729 + 0.719271i \(0.255524\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −296.239 −0.438437
\(78\) 0 0
\(79\) −206.961 −0.294746 −0.147373 0.989081i \(-0.547082\pi\)
−0.147373 + 0.989081i \(0.547082\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −463.397 −0.612825 −0.306412 0.951899i \(-0.599129\pi\)
−0.306412 + 0.951899i \(0.599129\pi\)
\(84\) 0 0
\(85\) −344.225 −0.439253
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −601.736 −0.716673 −0.358337 0.933592i \(-0.616656\pi\)
−0.358337 + 0.933592i \(0.616656\pi\)
\(90\) 0 0
\(91\) −107.783 −0.124162
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −204.066 −0.220387
\(96\) 0 0
\(97\) 229.363 0.240086 0.120043 0.992769i \(-0.461697\pi\)
0.120043 + 0.992769i \(0.461697\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1345.66 −1.32573 −0.662863 0.748740i \(-0.730659\pi\)
−0.662863 + 0.748740i \(0.730659\pi\)
\(102\) 0 0
\(103\) 1596.30 1.52707 0.763534 0.645768i \(-0.223463\pi\)
0.763534 + 0.645768i \(0.223463\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 958.786 0.866256 0.433128 0.901333i \(-0.357410\pi\)
0.433128 + 0.901333i \(0.357410\pi\)
\(108\) 0 0
\(109\) 1690.23 1.48527 0.742635 0.669696i \(-0.233576\pi\)
0.742635 + 0.669696i \(0.233576\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 11.6211 0.00967456 0.00483728 0.999988i \(-0.498460\pi\)
0.00483728 + 0.999988i \(0.498460\pi\)
\(114\) 0 0
\(115\) 721.592 0.585120
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −349.818 −0.269477
\(120\) 0 0
\(121\) 2067.97 1.55370
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) −309.141 −0.215999 −0.107999 0.994151i \(-0.534444\pi\)
−0.107999 + 0.994151i \(0.534444\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 2785.03 1.85747 0.928736 0.370742i \(-0.120897\pi\)
0.928736 + 0.370742i \(0.120897\pi\)
\(132\) 0 0
\(133\) −207.382 −0.135205
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2489.41 1.55244 0.776222 0.630460i \(-0.217134\pi\)
0.776222 + 0.630460i \(0.217134\pi\)
\(138\) 0 0
\(139\) −1786.05 −1.08986 −0.544931 0.838481i \(-0.683444\pi\)
−0.544931 + 0.838481i \(0.683444\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1236.67 0.723185
\(144\) 0 0
\(145\) −1100.29 −0.630166
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1568.83 −0.862575 −0.431288 0.902214i \(-0.641941\pi\)
−0.431288 + 0.902214i \(0.641941\pi\)
\(150\) 0 0
\(151\) 438.327 0.236229 0.118114 0.993000i \(-0.462315\pi\)
0.118114 + 0.993000i \(0.462315\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1457.73 0.755403
\(156\) 0 0
\(157\) −44.7479 −0.0227469 −0.0113735 0.999935i \(-0.503620\pi\)
−0.0113735 + 0.999935i \(0.503620\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 733.315 0.358965
\(162\) 0 0
\(163\) 2611.84 1.25506 0.627531 0.778591i \(-0.284065\pi\)
0.627531 + 0.778591i \(0.284065\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −188.947 −0.0875516 −0.0437758 0.999041i \(-0.513939\pi\)
−0.0437758 + 0.999041i \(0.513939\pi\)
\(168\) 0 0
\(169\) −1747.05 −0.795200
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1505.02 0.661413 0.330707 0.943734i \(-0.392713\pi\)
0.330707 + 0.943734i \(0.392713\pi\)
\(174\) 0 0
\(175\) −127.031 −0.0548722
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −3136.62 −1.30973 −0.654865 0.755746i \(-0.727275\pi\)
−0.654865 + 0.755746i \(0.727275\pi\)
\(180\) 0 0
\(181\) 4512.67 1.85317 0.926586 0.376084i \(-0.122730\pi\)
0.926586 + 0.376084i \(0.122730\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1303.18 −0.517902
\(186\) 0 0
\(187\) 4013.71 1.56958
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1207.43 0.457418 0.228709 0.973495i \(-0.426550\pi\)
0.228709 + 0.973495i \(0.426550\pi\)
\(192\) 0 0
\(193\) 923.164 0.344305 0.172152 0.985070i \(-0.444928\pi\)
0.172152 + 0.985070i \(0.444928\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1180.87 0.427075 0.213537 0.976935i \(-0.431501\pi\)
0.213537 + 0.976935i \(0.431501\pi\)
\(198\) 0 0
\(199\) 839.805 0.299157 0.149578 0.988750i \(-0.452208\pi\)
0.149578 + 0.988750i \(0.452208\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1118.17 −0.386600
\(204\) 0 0
\(205\) −848.832 −0.289195
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 2379.44 0.787509
\(210\) 0 0
\(211\) 2589.65 0.844923 0.422461 0.906381i \(-0.361166\pi\)
0.422461 + 0.906381i \(0.361166\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −2192.98 −0.695627
\(216\) 0 0
\(217\) 1481.41 0.463432
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1460.34 0.444492
\(222\) 0 0
\(223\) 4180.76 1.25544 0.627722 0.778437i \(-0.283987\pi\)
0.627722 + 0.778437i \(0.283987\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −2602.67 −0.760993 −0.380497 0.924782i \(-0.624247\pi\)
−0.380497 + 0.924782i \(0.624247\pi\)
\(228\) 0 0
\(229\) −1845.35 −0.532508 −0.266254 0.963903i \(-0.585786\pi\)
−0.266254 + 0.963903i \(0.585786\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −240.637 −0.0676594 −0.0338297 0.999428i \(-0.510770\pi\)
−0.0338297 + 0.999428i \(0.510770\pi\)
\(234\) 0 0
\(235\) 1277.40 0.354590
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 2567.64 0.694924 0.347462 0.937694i \(-0.387044\pi\)
0.347462 + 0.937694i \(0.387044\pi\)
\(240\) 0 0
\(241\) 3987.99 1.06593 0.532965 0.846137i \(-0.321078\pi\)
0.532965 + 0.846137i \(0.321078\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1585.91 0.413550
\(246\) 0 0
\(247\) 865.728 0.223016
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 967.393 0.243272 0.121636 0.992575i \(-0.461186\pi\)
0.121636 + 0.992575i \(0.461186\pi\)
\(252\) 0 0
\(253\) −8413.86 −2.09081
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −3743.03 −0.908498 −0.454249 0.890875i \(-0.650092\pi\)
−0.454249 + 0.890875i \(0.650092\pi\)
\(258\) 0 0
\(259\) −1324.36 −0.317728
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −2497.47 −0.585554 −0.292777 0.956181i \(-0.594579\pi\)
−0.292777 + 0.956181i \(0.594579\pi\)
\(264\) 0 0
\(265\) 1073.57 0.248864
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 2142.91 0.485709 0.242855 0.970063i \(-0.421916\pi\)
0.242855 + 0.970063i \(0.421916\pi\)
\(270\) 0 0
\(271\) −1540.64 −0.345341 −0.172671 0.984980i \(-0.555240\pi\)
−0.172671 + 0.984980i \(0.555240\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1457.52 0.319606
\(276\) 0 0
\(277\) 6777.80 1.47018 0.735088 0.677972i \(-0.237141\pi\)
0.735088 + 0.677972i \(0.237141\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −827.653 −0.175707 −0.0878535 0.996133i \(-0.528001\pi\)
−0.0878535 + 0.996133i \(0.528001\pi\)
\(282\) 0 0
\(283\) 3171.98 0.666270 0.333135 0.942879i \(-0.391894\pi\)
0.333135 + 0.942879i \(0.391894\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −862.623 −0.177418
\(288\) 0 0
\(289\) −173.358 −0.0352856
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1376.02 −0.274362 −0.137181 0.990546i \(-0.543804\pi\)
−0.137181 + 0.990546i \(0.543804\pi\)
\(294\) 0 0
\(295\) −1657.62 −0.327153
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −3061.27 −0.592099
\(300\) 0 0
\(301\) −2228.61 −0.426760
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −274.823 −0.0515946
\(306\) 0 0
\(307\) 119.504 0.0222165 0.0111083 0.999938i \(-0.496464\pi\)
0.0111083 + 0.999938i \(0.496464\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 2139.35 0.390069 0.195035 0.980796i \(-0.437518\pi\)
0.195035 + 0.980796i \(0.437518\pi\)
\(312\) 0 0
\(313\) 5163.50 0.932455 0.466227 0.884665i \(-0.345613\pi\)
0.466227 + 0.884665i \(0.345613\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8631.69 1.52935 0.764675 0.644416i \(-0.222899\pi\)
0.764675 + 0.644416i \(0.222899\pi\)
\(318\) 0 0
\(319\) 12829.5 2.25177
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2809.79 0.484028
\(324\) 0 0
\(325\) 530.298 0.0905097
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 1298.16 0.217537
\(330\) 0 0
\(331\) 2942.34 0.488597 0.244298 0.969700i \(-0.421442\pi\)
0.244298 + 0.969700i \(0.421442\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 3790.90 0.618265
\(336\) 0 0
\(337\) 9897.46 1.59985 0.799924 0.600101i \(-0.204873\pi\)
0.799924 + 0.600101i \(0.204873\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −16997.3 −2.69928
\(342\) 0 0
\(343\) 3354.53 0.528070
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −12015.7 −1.85890 −0.929451 0.368947i \(-0.879718\pi\)
−0.929451 + 0.368947i \(0.879718\pi\)
\(348\) 0 0
\(349\) 7894.62 1.21086 0.605428 0.795900i \(-0.293002\pi\)
0.605428 + 0.795900i \(0.293002\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −741.154 −0.111750 −0.0558748 0.998438i \(-0.517795\pi\)
−0.0558748 + 0.998438i \(0.517795\pi\)
\(354\) 0 0
\(355\) 4521.74 0.676025
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −11564.4 −1.70012 −0.850060 0.526685i \(-0.823435\pi\)
−0.850060 + 0.526685i \(0.823435\pi\)
\(360\) 0 0
\(361\) −5193.28 −0.757148
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −4333.11 −0.621385
\(366\) 0 0
\(367\) 1148.57 0.163365 0.0816823 0.996658i \(-0.473971\pi\)
0.0816823 + 0.996658i \(0.473971\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 1091.01 0.152676
\(372\) 0 0
\(373\) −4602.98 −0.638963 −0.319481 0.947593i \(-0.603509\pi\)
−0.319481 + 0.947593i \(0.603509\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4667.85 0.637683
\(378\) 0 0
\(379\) 3988.46 0.540563 0.270282 0.962781i \(-0.412883\pi\)
0.270282 + 0.962781i \(0.412883\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 7788.20 1.03906 0.519528 0.854454i \(-0.326108\pi\)
0.519528 + 0.854454i \(0.326108\pi\)
\(384\) 0 0
\(385\) 1481.20 0.196075
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −8524.87 −1.11113 −0.555563 0.831474i \(-0.687497\pi\)
−0.555563 + 0.831474i \(0.687497\pi\)
\(390\) 0 0
\(391\) −9935.60 −1.28508
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1034.80 0.131814
\(396\) 0 0
\(397\) −155.729 −0.0196872 −0.00984361 0.999952i \(-0.503133\pi\)
−0.00984361 + 0.999952i \(0.503133\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −5933.96 −0.738972 −0.369486 0.929236i \(-0.620466\pi\)
−0.369486 + 0.929236i \(0.620466\pi\)
\(402\) 0 0
\(403\) −6184.23 −0.764414
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 15195.3 1.85062
\(408\) 0 0
\(409\) 14161.4 1.71207 0.856035 0.516917i \(-0.172921\pi\)
0.856035 + 0.516917i \(0.172921\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −1684.55 −0.200705
\(414\) 0 0
\(415\) 2316.99 0.274064
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 9624.24 1.12214 0.561068 0.827770i \(-0.310391\pi\)
0.561068 + 0.827770i \(0.310391\pi\)
\(420\) 0 0
\(421\) −1536.26 −0.177845 −0.0889223 0.996039i \(-0.528342\pi\)
−0.0889223 + 0.996039i \(0.528342\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1721.13 0.196440
\(426\) 0 0
\(427\) −279.288 −0.0316527
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 11582.2 1.29441 0.647207 0.762314i \(-0.275937\pi\)
0.647207 + 0.762314i \(0.275937\pi\)
\(432\) 0 0
\(433\) 14892.6 1.65287 0.826437 0.563029i \(-0.190364\pi\)
0.826437 + 0.563029i \(0.190364\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −5890.10 −0.644764
\(438\) 0 0
\(439\) 1642.51 0.178571 0.0892853 0.996006i \(-0.471542\pi\)
0.0892853 + 0.996006i \(0.471542\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 3916.31 0.420021 0.210011 0.977699i \(-0.432650\pi\)
0.210011 + 0.977699i \(0.432650\pi\)
\(444\) 0 0
\(445\) 3008.68 0.320506
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 3985.25 0.418876 0.209438 0.977822i \(-0.432837\pi\)
0.209438 + 0.977822i \(0.432837\pi\)
\(450\) 0 0
\(451\) 9897.50 1.03338
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 538.914 0.0555267
\(456\) 0 0
\(457\) −14177.8 −1.45122 −0.725611 0.688105i \(-0.758443\pi\)
−0.725611 + 0.688105i \(0.758443\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 16394.3 1.65631 0.828154 0.560500i \(-0.189391\pi\)
0.828154 + 0.560500i \(0.189391\pi\)
\(462\) 0 0
\(463\) −3319.60 −0.333207 −0.166603 0.986024i \(-0.553280\pi\)
−0.166603 + 0.986024i \(0.553280\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −2529.79 −0.250674 −0.125337 0.992114i \(-0.540001\pi\)
−0.125337 + 0.992114i \(0.540001\pi\)
\(468\) 0 0
\(469\) 3852.49 0.379299
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 25570.4 2.48569
\(474\) 0 0
\(475\) 1020.33 0.0985601
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −8646.75 −0.824802 −0.412401 0.911002i \(-0.635310\pi\)
−0.412401 + 0.911002i \(0.635310\pi\)
\(480\) 0 0
\(481\) 5528.60 0.524080
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −1146.82 −0.107370
\(486\) 0 0
\(487\) 15251.7 1.41914 0.709569 0.704636i \(-0.248890\pi\)
0.709569 + 0.704636i \(0.248890\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −1766.87 −0.162398 −0.0811992 0.996698i \(-0.525875\pi\)
−0.0811992 + 0.996698i \(0.525875\pi\)
\(492\) 0 0
\(493\) 15149.9 1.38401
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4595.20 0.414734
\(498\) 0 0
\(499\) −11733.8 −1.05266 −0.526332 0.850279i \(-0.676433\pi\)
−0.526332 + 0.850279i \(0.676433\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −977.608 −0.0866589 −0.0433294 0.999061i \(-0.513797\pi\)
−0.0433294 + 0.999061i \(0.513797\pi\)
\(504\) 0 0
\(505\) 6728.31 0.592883
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −9674.72 −0.842484 −0.421242 0.906948i \(-0.638406\pi\)
−0.421242 + 0.906948i \(0.638406\pi\)
\(510\) 0 0
\(511\) −4403.51 −0.381213
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −7981.49 −0.682926
\(516\) 0 0
\(517\) −14894.7 −1.26706
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −5178.95 −0.435497 −0.217748 0.976005i \(-0.569871\pi\)
−0.217748 + 0.976005i \(0.569871\pi\)
\(522\) 0 0
\(523\) −14280.7 −1.19398 −0.596992 0.802248i \(-0.703637\pi\)
−0.596992 + 0.802248i \(0.703637\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −20071.5 −1.65906
\(528\) 0 0
\(529\) 8660.78 0.711826
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 3601.07 0.292645
\(534\) 0 0
\(535\) −4793.93 −0.387401
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −18491.9 −1.47774
\(540\) 0 0
\(541\) 12923.8 1.02706 0.513529 0.858072i \(-0.328338\pi\)
0.513529 + 0.858072i \(0.328338\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −8451.14 −0.664233
\(546\) 0 0
\(547\) −13653.3 −1.06723 −0.533614 0.845728i \(-0.679166\pi\)
−0.533614 + 0.845728i \(0.679166\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 8981.28 0.694402
\(552\) 0 0
\(553\) 1051.62 0.0808666
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 9313.63 0.708494 0.354247 0.935152i \(-0.384737\pi\)
0.354247 + 0.935152i \(0.384737\pi\)
\(558\) 0 0
\(559\) 9303.46 0.703925
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −10625.4 −0.795394 −0.397697 0.917517i \(-0.630190\pi\)
−0.397697 + 0.917517i \(0.630190\pi\)
\(564\) 0 0
\(565\) −58.1057 −0.00432660
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −9060.50 −0.667550 −0.333775 0.942653i \(-0.608323\pi\)
−0.333775 + 0.942653i \(0.608323\pi\)
\(570\) 0 0
\(571\) 21379.1 1.56688 0.783440 0.621467i \(-0.213463\pi\)
0.783440 + 0.621467i \(0.213463\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −3607.96 −0.261674
\(576\) 0 0
\(577\) 6347.76 0.457991 0.228996 0.973427i \(-0.426456\pi\)
0.228996 + 0.973427i \(0.426456\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 2354.63 0.168135
\(582\) 0 0
\(583\) −12518.0 −0.889266
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 14773.7 1.03880 0.519401 0.854531i \(-0.326155\pi\)
0.519401 + 0.854531i \(0.326155\pi\)
\(588\) 0 0
\(589\) −11898.9 −0.832405
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −26868.1 −1.86061 −0.930304 0.366790i \(-0.880457\pi\)
−0.930304 + 0.366790i \(0.880457\pi\)
\(594\) 0 0
\(595\) 1749.09 0.120514
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −1749.83 −0.119359 −0.0596794 0.998218i \(-0.519008\pi\)
−0.0596794 + 0.998218i \(0.519008\pi\)
\(600\) 0 0
\(601\) −17964.0 −1.21924 −0.609622 0.792692i \(-0.708679\pi\)
−0.609622 + 0.792692i \(0.708679\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −10339.8 −0.694834
\(606\) 0 0
\(607\) 4418.22 0.295437 0.147718 0.989029i \(-0.452807\pi\)
0.147718 + 0.989029i \(0.452807\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −5419.24 −0.358820
\(612\) 0 0
\(613\) −20179.7 −1.32961 −0.664804 0.747018i \(-0.731485\pi\)
−0.664804 + 0.747018i \(0.731485\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 4673.01 0.304908 0.152454 0.988311i \(-0.451282\pi\)
0.152454 + 0.988311i \(0.451282\pi\)
\(618\) 0 0
\(619\) 19976.8 1.29715 0.648574 0.761151i \(-0.275366\pi\)
0.648574 + 0.761151i \(0.275366\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 3057.56 0.196627
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 17943.5 1.13745
\(630\) 0 0
\(631\) −10457.5 −0.659757 −0.329879 0.944023i \(-0.607008\pi\)
−0.329879 + 0.944023i \(0.607008\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1545.70 0.0965975
\(636\) 0 0
\(637\) −6728.02 −0.418483
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −4395.86 −0.270867 −0.135434 0.990786i \(-0.543243\pi\)
−0.135434 + 0.990786i \(0.543243\pi\)
\(642\) 0 0
\(643\) −5786.36 −0.354886 −0.177443 0.984131i \(-0.556783\pi\)
−0.177443 + 0.984131i \(0.556783\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −25367.7 −1.54143 −0.770717 0.637178i \(-0.780102\pi\)
−0.770717 + 0.637178i \(0.780102\pi\)
\(648\) 0 0
\(649\) 19328.1 1.16902
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 21633.2 1.29644 0.648218 0.761454i \(-0.275514\pi\)
0.648218 + 0.761454i \(0.275514\pi\)
\(654\) 0 0
\(655\) −13925.1 −0.830687
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 18312.4 1.08248 0.541238 0.840870i \(-0.317956\pi\)
0.541238 + 0.840870i \(0.317956\pi\)
\(660\) 0 0
\(661\) −5526.08 −0.325174 −0.162587 0.986694i \(-0.551984\pi\)
−0.162587 + 0.986694i \(0.551984\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1036.91 0.0604656
\(666\) 0 0
\(667\) −31758.4 −1.84361
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 3204.48 0.184363
\(672\) 0 0
\(673\) 1437.24 0.0823204 0.0411602 0.999153i \(-0.486895\pi\)
0.0411602 + 0.999153i \(0.486895\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 23405.9 1.32875 0.664373 0.747401i \(-0.268699\pi\)
0.664373 + 0.747401i \(0.268699\pi\)
\(678\) 0 0
\(679\) −1165.45 −0.0658701
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 19227.4 1.07719 0.538593 0.842566i \(-0.318956\pi\)
0.538593 + 0.842566i \(0.318956\pi\)
\(684\) 0 0
\(685\) −12447.1 −0.694274
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −4554.50 −0.251833
\(690\) 0 0
\(691\) 35284.8 1.94254 0.971271 0.237975i \(-0.0764835\pi\)
0.971271 + 0.237975i \(0.0764835\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 8930.26 0.487401
\(696\) 0 0
\(697\) 11687.6 0.635149
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −9173.00 −0.494236 −0.247118 0.968985i \(-0.579484\pi\)
−0.247118 + 0.968985i \(0.579484\pi\)
\(702\) 0 0
\(703\) 10637.4 0.570695
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 6837.63 0.363728
\(708\) 0 0
\(709\) −33951.6 −1.79842 −0.899210 0.437517i \(-0.855858\pi\)
−0.899210 + 0.437517i \(0.855858\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 42075.3 2.21001
\(714\) 0 0
\(715\) −6183.35 −0.323418
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −6727.90 −0.348968 −0.174484 0.984660i \(-0.555826\pi\)
−0.174484 + 0.984660i \(0.555826\pi\)
\(720\) 0 0
\(721\) −8111.17 −0.418968
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 5501.45 0.281819
\(726\) 0 0
\(727\) 36726.1 1.87359 0.936793 0.349885i \(-0.113779\pi\)
0.936793 + 0.349885i \(0.113779\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 30195.1 1.52778
\(732\) 0 0
\(733\) 26691.4 1.34498 0.672489 0.740108i \(-0.265225\pi\)
0.672489 + 0.740108i \(0.265225\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −44202.4 −2.20925
\(738\) 0 0
\(739\) −12207.0 −0.607634 −0.303817 0.952730i \(-0.598261\pi\)
−0.303817 + 0.952730i \(0.598261\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −12473.3 −0.615882 −0.307941 0.951405i \(-0.599640\pi\)
−0.307941 + 0.951405i \(0.599640\pi\)
\(744\) 0 0
\(745\) 7844.16 0.385755
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −4871.82 −0.237667
\(750\) 0 0
\(751\) 15102.6 0.733825 0.366913 0.930255i \(-0.380415\pi\)
0.366913 + 0.930255i \(0.380415\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −2191.63 −0.105645
\(756\) 0 0
\(757\) −3418.34 −0.164124 −0.0820618 0.996627i \(-0.526150\pi\)
−0.0820618 + 0.996627i \(0.526150\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 12684.5 0.604224 0.302112 0.953272i \(-0.402308\pi\)
0.302112 + 0.953272i \(0.402308\pi\)
\(762\) 0 0
\(763\) −8588.45 −0.407500
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 7032.25 0.331056
\(768\) 0 0
\(769\) −27580.2 −1.29333 −0.646663 0.762776i \(-0.723836\pi\)
−0.646663 + 0.762776i \(0.723836\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −17386.0 −0.808966 −0.404483 0.914546i \(-0.632548\pi\)
−0.404483 + 0.914546i \(0.632548\pi\)
\(774\) 0 0
\(775\) −7288.63 −0.337826
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 6928.72 0.318674
\(780\) 0 0
\(781\) −52724.1 −2.41564
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 223.739 0.0101727
\(786\) 0 0
\(787\) −4680.29 −0.211988 −0.105994 0.994367i \(-0.533802\pi\)
−0.105994 + 0.994367i \(0.533802\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −59.0498 −0.00265432
\(792\) 0 0
\(793\) 1165.91 0.0522100
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −7278.62 −0.323490 −0.161745 0.986833i \(-0.551712\pi\)
−0.161745 + 0.986833i \(0.551712\pi\)
\(798\) 0 0
\(799\) −17588.6 −0.778773
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 50524.7 2.22039
\(804\) 0 0
\(805\) −3666.58 −0.160534
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −29212.5 −1.26954 −0.634770 0.772701i \(-0.718905\pi\)
−0.634770 + 0.772701i \(0.718905\pi\)
\(810\) 0 0
\(811\) −41992.4 −1.81819 −0.909094 0.416590i \(-0.863225\pi\)
−0.909094 + 0.416590i \(0.863225\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −13059.2 −0.561281
\(816\) 0 0
\(817\) 17900.5 0.766536
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 8722.99 0.370809 0.185405 0.982662i \(-0.440640\pi\)
0.185405 + 0.982662i \(0.440640\pi\)
\(822\) 0 0
\(823\) −13584.8 −0.575379 −0.287690 0.957724i \(-0.592887\pi\)
−0.287690 + 0.957724i \(0.592887\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 26573.6 1.11736 0.558679 0.829384i \(-0.311308\pi\)
0.558679 + 0.829384i \(0.311308\pi\)
\(828\) 0 0
\(829\) −43238.4 −1.81150 −0.905748 0.423816i \(-0.860690\pi\)
−0.905748 + 0.423816i \(0.860690\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −21836.3 −0.908265
\(834\) 0 0
\(835\) 944.733 0.0391543
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −4093.21 −0.168431 −0.0842154 0.996448i \(-0.526838\pi\)
−0.0842154 + 0.996448i \(0.526838\pi\)
\(840\) 0 0
\(841\) 24036.5 0.985546
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 8735.27 0.355624
\(846\) 0 0
\(847\) −10507.8 −0.426273
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −37614.7 −1.51517
\(852\) 0 0
\(853\) −20201.5 −0.810886 −0.405443 0.914120i \(-0.632883\pi\)
−0.405443 + 0.914120i \(0.632883\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −4551.65 −0.181425 −0.0907126 0.995877i \(-0.528914\pi\)
−0.0907126 + 0.995877i \(0.528914\pi\)
\(858\) 0 0
\(859\) −11962.6 −0.475154 −0.237577 0.971369i \(-0.576353\pi\)
−0.237577 + 0.971369i \(0.576353\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −7164.61 −0.282603 −0.141301 0.989967i \(-0.545129\pi\)
−0.141301 + 0.989967i \(0.545129\pi\)
\(864\) 0 0
\(865\) −7525.10 −0.295793
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −12065.9 −0.471012
\(870\) 0 0
\(871\) −16082.4 −0.625640
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 635.154 0.0245396
\(876\) 0 0
\(877\) −19218.7 −0.739987 −0.369994 0.929034i \(-0.620640\pi\)
−0.369994 + 0.929034i \(0.620640\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 45914.5 1.75585 0.877923 0.478802i \(-0.158929\pi\)
0.877923 + 0.478802i \(0.158929\pi\)
\(882\) 0 0
\(883\) 44656.7 1.70194 0.850972 0.525211i \(-0.176014\pi\)
0.850972 + 0.525211i \(0.176014\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 14975.2 0.566873 0.283437 0.958991i \(-0.408525\pi\)
0.283437 + 0.958991i \(0.408525\pi\)
\(888\) 0 0
\(889\) 1570.82 0.0592616
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −10427.0 −0.390735
\(894\) 0 0
\(895\) 15683.1 0.585729
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −64156.8 −2.38015
\(900\) 0 0
\(901\) −14782.0 −0.546571
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −22563.3 −0.828763
\(906\) 0 0
\(907\) 14818.1 0.542479 0.271240 0.962512i \(-0.412566\pi\)
0.271240 + 0.962512i \(0.412566\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 21846.5 0.794518 0.397259 0.917707i \(-0.369961\pi\)
0.397259 + 0.917707i \(0.369961\pi\)
\(912\) 0 0
\(913\) −27016.4 −0.979312
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −14151.4 −0.509618
\(918\) 0 0
\(919\) −28878.9 −1.03659 −0.518296 0.855201i \(-0.673434\pi\)
−0.518296 + 0.855201i \(0.673434\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −19182.9 −0.684089
\(924\) 0 0
\(925\) 6515.92 0.231613
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −4608.79 −0.162766 −0.0813829 0.996683i \(-0.525934\pi\)
−0.0813829 + 0.996683i \(0.525934\pi\)
\(930\) 0 0
\(931\) −12945.2 −0.455705
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −20068.6 −0.701938
\(936\) 0 0
\(937\) −21063.8 −0.734391 −0.367195 0.930144i \(-0.619682\pi\)
−0.367195 + 0.930144i \(0.619682\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 20244.8 0.701339 0.350670 0.936499i \(-0.385954\pi\)
0.350670 + 0.936499i \(0.385954\pi\)
\(942\) 0 0
\(943\) −24500.4 −0.846069
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −29978.3 −1.02868 −0.514342 0.857585i \(-0.671964\pi\)
−0.514342 + 0.857585i \(0.671964\pi\)
\(948\) 0 0
\(949\) 18382.7 0.628797
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −5782.65 −0.196556 −0.0982782 0.995159i \(-0.531334\pi\)
−0.0982782 + 0.995159i \(0.531334\pi\)
\(954\) 0 0
\(955\) −6037.17 −0.204564
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −12649.3 −0.425930
\(960\) 0 0
\(961\) 55207.7 1.85317
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −4615.82 −0.153978
\(966\) 0 0
\(967\) 26119.2 0.868600 0.434300 0.900768i \(-0.356996\pi\)
0.434300 + 0.900768i \(0.356996\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 5101.93 0.168619 0.0843093 0.996440i \(-0.473132\pi\)
0.0843093 + 0.996440i \(0.473132\pi\)
\(972\) 0 0
\(973\) 9075.34 0.299016
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 45902.4 1.50312 0.751559 0.659666i \(-0.229302\pi\)
0.751559 + 0.659666i \(0.229302\pi\)
\(978\) 0 0
\(979\) −35081.6 −1.14526
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 13416.5 0.435321 0.217661 0.976025i \(-0.430157\pi\)
0.217661 + 0.976025i \(0.430157\pi\)
\(984\) 0 0
\(985\) −5904.37 −0.190994
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −63297.4 −2.03513
\(990\) 0 0
\(991\) −3806.66 −0.122021 −0.0610104 0.998137i \(-0.519432\pi\)
−0.0610104 + 0.998137i \(0.519432\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −4199.02 −0.133787
\(996\) 0 0
\(997\) −22523.8 −0.715483 −0.357742 0.933821i \(-0.616453\pi\)
−0.357742 + 0.933821i \(0.616453\pi\)
\(998\) 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.be.1.3 3
3.2 odd 2 2160.4.a.bm.1.3 3
4.3 odd 2 135.4.a.g.1.1 yes 3
12.11 even 2 135.4.a.f.1.3 3
20.3 even 4 675.4.b.k.649.5 6
20.7 even 4 675.4.b.k.649.2 6
20.19 odd 2 675.4.a.q.1.3 3
36.7 odd 6 405.4.e.r.271.3 6
36.11 even 6 405.4.e.t.271.1 6
36.23 even 6 405.4.e.t.136.1 6
36.31 odd 6 405.4.e.r.136.3 6
60.23 odd 4 675.4.b.l.649.2 6
60.47 odd 4 675.4.b.l.649.5 6
60.59 even 2 675.4.a.r.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.4.a.f.1.3 3 12.11 even 2
135.4.a.g.1.1 yes 3 4.3 odd 2
405.4.e.r.136.3 6 36.31 odd 6
405.4.e.r.271.3 6 36.7 odd 6
405.4.e.t.136.1 6 36.23 even 6
405.4.e.t.271.1 6 36.11 even 6
675.4.a.q.1.3 3 20.19 odd 2
675.4.a.r.1.1 3 60.59 even 2
675.4.b.k.649.2 6 20.7 even 4
675.4.b.k.649.5 6 20.3 even 4
675.4.b.l.649.2 6 60.23 odd 4
675.4.b.l.649.5 6 60.47 odd 4
2160.4.a.be.1.3 3 1.1 even 1 trivial
2160.4.a.bm.1.3 3 3.2 odd 2