Properties

Label 1728.4.a.bj.1.2
Level $1728$
Weight $4$
Character 1728.1
Self dual yes
Analytic conductor $101.955$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1728.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(101.955300490\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 216)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 1728.1

$q$-expansion

\(f(q)\) \(=\) \(q+11.4164 q^{5} +29.8328 q^{7} +O(q^{10})\) \(q+11.4164 q^{5} +29.8328 q^{7} +66.2492 q^{11} -39.8328 q^{13} +107.416 q^{17} +70.3313 q^{19} -6.91796 q^{23} +5.33437 q^{25} -36.6687 q^{29} +231.331 q^{31} +340.584 q^{35} -36.8359 q^{37} +429.325 q^{41} -74.3344 q^{43} -52.5836 q^{47} +546.997 q^{49} -288.170 q^{53} +756.328 q^{55} -783.745 q^{59} -439.158 q^{61} -454.748 q^{65} -218.337 q^{67} -790.492 q^{71} +1098.00 q^{73} +1976.40 q^{77} -439.827 q^{79} -50.8452 q^{83} +1226.31 q^{85} +719.745 q^{89} -1188.33 q^{91} +802.930 q^{95} -1199.64 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{5} + 6q^{7} + O(q^{10}) \) \( 2q - 4q^{5} + 6q^{7} + 52q^{11} - 26q^{13} + 188q^{17} - 74q^{19} - 148q^{23} + 118q^{25} - 288q^{29} + 248q^{31} + 708q^{35} - 342q^{37} - 256q^{43} - 132q^{47} + 772q^{49} - 952q^{53} + 976q^{55} - 1004q^{59} + 34q^{61} - 668q^{65} - 866q^{67} - 776q^{71} + 1874q^{73} + 2316q^{77} - 182q^{79} - 1336q^{83} - 16q^{85} + 876q^{89} - 1518q^{91} + 3028q^{95} - 38q^{97} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 11.4164 1.02111 0.510557 0.859844i \(-0.329439\pi\)
0.510557 + 0.859844i \(0.329439\pi\)
\(6\) 0 0
\(7\) 29.8328 1.61082 0.805410 0.592718i \(-0.201945\pi\)
0.805410 + 0.592718i \(0.201945\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 66.2492 1.81590 0.907950 0.419079i \(-0.137647\pi\)
0.907950 + 0.419079i \(0.137647\pi\)
\(12\) 0 0
\(13\) −39.8328 −0.849818 −0.424909 0.905236i \(-0.639694\pi\)
−0.424909 + 0.905236i \(0.639694\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 107.416 1.53249 0.766244 0.642549i \(-0.222123\pi\)
0.766244 + 0.642549i \(0.222123\pi\)
\(18\) 0 0
\(19\) 70.3313 0.849216 0.424608 0.905377i \(-0.360412\pi\)
0.424608 + 0.905377i \(0.360412\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −6.91796 −0.0627172 −0.0313586 0.999508i \(-0.509983\pi\)
−0.0313586 + 0.999508i \(0.509983\pi\)
\(24\) 0 0
\(25\) 5.33437 0.0426749
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −36.6687 −0.234800 −0.117400 0.993085i \(-0.537456\pi\)
−0.117400 + 0.993085i \(0.537456\pi\)
\(30\) 0 0
\(31\) 231.331 1.34027 0.670134 0.742240i \(-0.266237\pi\)
0.670134 + 0.742240i \(0.266237\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 340.584 1.64483
\(36\) 0 0
\(37\) −36.8359 −0.163670 −0.0818350 0.996646i \(-0.526078\pi\)
−0.0818350 + 0.996646i \(0.526078\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 429.325 1.63535 0.817674 0.575681i \(-0.195263\pi\)
0.817674 + 0.575681i \(0.195263\pi\)
\(42\) 0 0
\(43\) −74.3344 −0.263625 −0.131813 0.991275i \(-0.542080\pi\)
−0.131813 + 0.991275i \(0.542080\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −52.5836 −0.163194 −0.0815969 0.996665i \(-0.526002\pi\)
−0.0815969 + 0.996665i \(0.526002\pi\)
\(48\) 0 0
\(49\) 546.997 1.59474
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −288.170 −0.746853 −0.373427 0.927660i \(-0.621817\pi\)
−0.373427 + 0.927660i \(0.621817\pi\)
\(54\) 0 0
\(55\) 756.328 1.85424
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −783.745 −1.72940 −0.864702 0.502285i \(-0.832493\pi\)
−0.864702 + 0.502285i \(0.832493\pi\)
\(60\) 0 0
\(61\) −439.158 −0.921777 −0.460889 0.887458i \(-0.652469\pi\)
−0.460889 + 0.887458i \(0.652469\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −454.748 −0.867762
\(66\) 0 0
\(67\) −218.337 −0.398122 −0.199061 0.979987i \(-0.563789\pi\)
−0.199061 + 0.979987i \(0.563789\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −790.492 −1.32133 −0.660663 0.750682i \(-0.729725\pi\)
−0.660663 + 0.750682i \(0.729725\pi\)
\(72\) 0 0
\(73\) 1098.00 1.76042 0.880211 0.474582i \(-0.157401\pi\)
0.880211 + 0.474582i \(0.157401\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1976.40 2.92509
\(78\) 0 0
\(79\) −439.827 −0.626384 −0.313192 0.949690i \(-0.601398\pi\)
−0.313192 + 0.949690i \(0.601398\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −50.8452 −0.0672408 −0.0336204 0.999435i \(-0.510704\pi\)
−0.0336204 + 0.999435i \(0.510704\pi\)
\(84\) 0 0
\(85\) 1226.31 1.56485
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 719.745 0.857222 0.428611 0.903489i \(-0.359003\pi\)
0.428611 + 0.903489i \(0.359003\pi\)
\(90\) 0 0
\(91\) −1188.33 −1.36890
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 802.930 0.867147
\(96\) 0 0
\(97\) −1199.64 −1.25573 −0.627863 0.778324i \(-0.716070\pi\)
−0.627863 + 0.778324i \(0.716070\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −245.981 −0.242337 −0.121169 0.992632i \(-0.538664\pi\)
−0.121169 + 0.992632i \(0.538664\pi\)
\(102\) 0 0
\(103\) −575.820 −0.550847 −0.275424 0.961323i \(-0.588818\pi\)
−0.275424 + 0.961323i \(0.588818\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1015.89 −0.917849 −0.458924 0.888475i \(-0.651765\pi\)
−0.458924 + 0.888475i \(0.651765\pi\)
\(108\) 0 0
\(109\) −1471.64 −1.29319 −0.646595 0.762834i \(-0.723807\pi\)
−0.646595 + 0.762834i \(0.723807\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1903.57 −1.58472 −0.792359 0.610055i \(-0.791147\pi\)
−0.792359 + 0.610055i \(0.791147\pi\)
\(114\) 0 0
\(115\) −78.9783 −0.0640414
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3204.53 2.46856
\(120\) 0 0
\(121\) 3057.96 2.29749
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1366.15 −0.977539
\(126\) 0 0
\(127\) −230.310 −0.160919 −0.0804593 0.996758i \(-0.525639\pi\)
−0.0804593 + 0.996758i \(0.525639\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 794.310 0.529764 0.264882 0.964281i \(-0.414667\pi\)
0.264882 + 0.964281i \(0.414667\pi\)
\(132\) 0 0
\(133\) 2098.18 1.36793
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −683.076 −0.425979 −0.212989 0.977055i \(-0.568320\pi\)
−0.212989 + 0.977055i \(0.568320\pi\)
\(138\) 0 0
\(139\) −1512.64 −0.923025 −0.461513 0.887134i \(-0.652693\pi\)
−0.461513 + 0.887134i \(0.652693\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −2638.89 −1.54318
\(144\) 0 0
\(145\) −418.625 −0.239758
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2307.14 −1.26851 −0.634255 0.773124i \(-0.718693\pi\)
−0.634255 + 0.773124i \(0.718693\pi\)
\(150\) 0 0
\(151\) −2523.17 −1.35982 −0.679910 0.733296i \(-0.737981\pi\)
−0.679910 + 0.733296i \(0.737981\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2640.97 1.36857
\(156\) 0 0
\(157\) 2918.98 1.48382 0.741912 0.670497i \(-0.233919\pi\)
0.741912 + 0.670497i \(0.233919\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −206.382 −0.101026
\(162\) 0 0
\(163\) 1096.99 0.527133 0.263567 0.964641i \(-0.415101\pi\)
0.263567 + 0.964641i \(0.415101\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 732.729 0.339523 0.169761 0.985485i \(-0.445700\pi\)
0.169761 + 0.985485i \(0.445700\pi\)
\(168\) 0 0
\(169\) −610.347 −0.277809
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −722.675 −0.317595 −0.158798 0.987311i \(-0.550762\pi\)
−0.158798 + 0.987311i \(0.550762\pi\)
\(174\) 0 0
\(175\) 159.139 0.0687417
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1010.16 0.421806 0.210903 0.977507i \(-0.432360\pi\)
0.210903 + 0.977507i \(0.432360\pi\)
\(180\) 0 0
\(181\) 4256.79 1.74809 0.874045 0.485844i \(-0.161488\pi\)
0.874045 + 0.485844i \(0.161488\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −420.534 −0.167126
\(186\) 0 0
\(187\) 7116.25 2.78284
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −3429.39 −1.29917 −0.649585 0.760289i \(-0.725057\pi\)
−0.649585 + 0.760289i \(0.725057\pi\)
\(192\) 0 0
\(193\) 967.341 0.360781 0.180390 0.983595i \(-0.442264\pi\)
0.180390 + 0.983595i \(0.442264\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 468.413 0.169406 0.0847032 0.996406i \(-0.473006\pi\)
0.0847032 + 0.996406i \(0.473006\pi\)
\(198\) 0 0
\(199\) 2673.47 0.952347 0.476174 0.879351i \(-0.342023\pi\)
0.476174 + 0.879351i \(0.342023\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1093.93 −0.378221
\(204\) 0 0
\(205\) 4901.35 1.66988
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 4659.39 1.54209
\(210\) 0 0
\(211\) 2870.62 0.936594 0.468297 0.883571i \(-0.344868\pi\)
0.468297 + 0.883571i \(0.344868\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −848.631 −0.269192
\(216\) 0 0
\(217\) 6901.26 2.15893
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −4278.70 −1.30234
\(222\) 0 0
\(223\) 1246.60 0.374343 0.187172 0.982327i \(-0.440068\pi\)
0.187172 + 0.982327i \(0.440068\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 560.461 0.163873 0.0819364 0.996638i \(-0.473890\pi\)
0.0819364 + 0.996638i \(0.473890\pi\)
\(228\) 0 0
\(229\) 1145.38 0.330519 0.165260 0.986250i \(-0.447154\pi\)
0.165260 + 0.986250i \(0.447154\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −623.709 −0.175367 −0.0876836 0.996148i \(-0.527946\pi\)
−0.0876836 + 0.996148i \(0.527946\pi\)
\(234\) 0 0
\(235\) −600.316 −0.166639
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −6265.80 −1.69582 −0.847910 0.530141i \(-0.822139\pi\)
−0.847910 + 0.530141i \(0.822139\pi\)
\(240\) 0 0
\(241\) −640.653 −0.171237 −0.0856185 0.996328i \(-0.527287\pi\)
−0.0856185 + 0.996328i \(0.527287\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 6244.74 1.62842
\(246\) 0 0
\(247\) −2801.49 −0.721679
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −7748.91 −1.94863 −0.974316 0.225183i \(-0.927702\pi\)
−0.974316 + 0.225183i \(0.927702\pi\)
\(252\) 0 0
\(253\) −458.310 −0.113888
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3191.76 0.774693 0.387347 0.921934i \(-0.373392\pi\)
0.387347 + 0.921934i \(0.373392\pi\)
\(258\) 0 0
\(259\) −1098.92 −0.263643
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 3776.41 0.885413 0.442706 0.896667i \(-0.354018\pi\)
0.442706 + 0.896667i \(0.354018\pi\)
\(264\) 0 0
\(265\) −3289.87 −0.762623
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 4065.91 0.921572 0.460786 0.887511i \(-0.347568\pi\)
0.460786 + 0.887511i \(0.347568\pi\)
\(270\) 0 0
\(271\) 5508.17 1.23468 0.617339 0.786697i \(-0.288211\pi\)
0.617339 + 0.786697i \(0.288211\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 353.398 0.0774934
\(276\) 0 0
\(277\) 8306.91 1.80186 0.900928 0.433970i \(-0.142887\pi\)
0.900928 + 0.433970i \(0.142887\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −2962.67 −0.628962 −0.314481 0.949264i \(-0.601831\pi\)
−0.314481 + 0.949264i \(0.601831\pi\)
\(282\) 0 0
\(283\) −4509.67 −0.947250 −0.473625 0.880727i \(-0.657055\pi\)
−0.473625 + 0.880727i \(0.657055\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 12808.0 2.63425
\(288\) 0 0
\(289\) 6625.28 1.34852
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1814.46 −0.361781 −0.180890 0.983503i \(-0.557898\pi\)
−0.180890 + 0.983503i \(0.557898\pi\)
\(294\) 0 0
\(295\) −8947.55 −1.76592
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 275.562 0.0532982
\(300\) 0 0
\(301\) −2217.60 −0.424653
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −5013.61 −0.941240
\(306\) 0 0
\(307\) 4395.62 0.817171 0.408585 0.912720i \(-0.366022\pi\)
0.408585 + 0.912720i \(0.366022\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 1377.23 0.251111 0.125555 0.992087i \(-0.459929\pi\)
0.125555 + 0.992087i \(0.459929\pi\)
\(312\) 0 0
\(313\) 7347.95 1.32693 0.663467 0.748205i \(-0.269084\pi\)
0.663467 + 0.748205i \(0.269084\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −2607.78 −0.462043 −0.231021 0.972949i \(-0.574207\pi\)
−0.231021 + 0.972949i \(0.574207\pi\)
\(318\) 0 0
\(319\) −2429.28 −0.426374
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 7554.73 1.30141
\(324\) 0 0
\(325\) −212.483 −0.0362659
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −1568.72 −0.262876
\(330\) 0 0
\(331\) −6541.33 −1.08624 −0.543118 0.839656i \(-0.682756\pi\)
−0.543118 + 0.839656i \(0.682756\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −2492.63 −0.406528
\(336\) 0 0
\(337\) 4315.66 0.697594 0.348797 0.937198i \(-0.386590\pi\)
0.348797 + 0.937198i \(0.386590\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 15325.5 2.43379
\(342\) 0 0
\(343\) 6085.80 0.958025
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 3895.76 0.602695 0.301347 0.953514i \(-0.402564\pi\)
0.301347 + 0.953514i \(0.402564\pi\)
\(348\) 0 0
\(349\) 4877.42 0.748087 0.374044 0.927411i \(-0.377971\pi\)
0.374044 + 0.927411i \(0.377971\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 1739.79 0.262322 0.131161 0.991361i \(-0.458129\pi\)
0.131161 + 0.991361i \(0.458129\pi\)
\(354\) 0 0
\(355\) −9024.58 −1.34923
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −188.828 −0.0277604 −0.0138802 0.999904i \(-0.504418\pi\)
−0.0138802 + 0.999904i \(0.504418\pi\)
\(360\) 0 0
\(361\) −1912.51 −0.278833
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 12535.2 1.79759
\(366\) 0 0
\(367\) 6338.15 0.901495 0.450747 0.892651i \(-0.351157\pi\)
0.450747 + 0.892651i \(0.351157\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −8596.93 −1.20305
\(372\) 0 0
\(373\) −8641.41 −1.19956 −0.599779 0.800166i \(-0.704745\pi\)
−0.599779 + 0.800166i \(0.704745\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1460.62 0.199538
\(378\) 0 0
\(379\) −4143.88 −0.561627 −0.280814 0.959762i \(-0.590604\pi\)
−0.280814 + 0.959762i \(0.590604\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 4193.94 0.559531 0.279766 0.960068i \(-0.409743\pi\)
0.279766 + 0.960068i \(0.409743\pi\)
\(384\) 0 0
\(385\) 22563.4 2.98685
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −10870.4 −1.41684 −0.708422 0.705789i \(-0.750593\pi\)
−0.708422 + 0.705789i \(0.750593\pi\)
\(390\) 0 0
\(391\) −743.102 −0.0961133
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −5021.24 −0.639610
\(396\) 0 0
\(397\) 8890.87 1.12398 0.561990 0.827144i \(-0.310036\pi\)
0.561990 + 0.827144i \(0.310036\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 15095.4 1.87987 0.939936 0.341349i \(-0.110884\pi\)
0.939936 + 0.341349i \(0.110884\pi\)
\(402\) 0 0
\(403\) −9214.58 −1.13898
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −2440.35 −0.297208
\(408\) 0 0
\(409\) 1618.34 0.195652 0.0978260 0.995204i \(-0.468811\pi\)
0.0978260 + 0.995204i \(0.468811\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −23381.3 −2.78576
\(414\) 0 0
\(415\) −580.470 −0.0686606
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 10011.9 1.16733 0.583666 0.811994i \(-0.301618\pi\)
0.583666 + 0.811994i \(0.301618\pi\)
\(420\) 0 0
\(421\) −7234.43 −0.837493 −0.418747 0.908103i \(-0.637530\pi\)
−0.418747 + 0.908103i \(0.637530\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 572.999 0.0653989
\(426\) 0 0
\(427\) −13101.3 −1.48482
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −5379.03 −0.601157 −0.300579 0.953757i \(-0.597180\pi\)
−0.300579 + 0.953757i \(0.597180\pi\)
\(432\) 0 0
\(433\) −1602.96 −0.177906 −0.0889530 0.996036i \(-0.528352\pi\)
−0.0889530 + 0.996036i \(0.528352\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −486.549 −0.0532604
\(438\) 0 0
\(439\) −6725.87 −0.731226 −0.365613 0.930767i \(-0.619141\pi\)
−0.365613 + 0.930767i \(0.619141\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 1169.92 0.125473 0.0627367 0.998030i \(-0.480017\pi\)
0.0627367 + 0.998030i \(0.480017\pi\)
\(444\) 0 0
\(445\) 8216.90 0.875322
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −2996.56 −0.314958 −0.157479 0.987522i \(-0.550337\pi\)
−0.157479 + 0.987522i \(0.550337\pi\)
\(450\) 0 0
\(451\) 28442.5 2.96963
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −13566.4 −1.39781
\(456\) 0 0
\(457\) −11152.1 −1.14151 −0.570757 0.821119i \(-0.693350\pi\)
−0.570757 + 0.821119i \(0.693350\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 2947.17 0.297752 0.148876 0.988856i \(-0.452435\pi\)
0.148876 + 0.988856i \(0.452435\pi\)
\(462\) 0 0
\(463\) −6563.38 −0.658804 −0.329402 0.944190i \(-0.606847\pi\)
−0.329402 + 0.944190i \(0.606847\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −2727.33 −0.270248 −0.135124 0.990829i \(-0.543143\pi\)
−0.135124 + 0.990829i \(0.543143\pi\)
\(468\) 0 0
\(469\) −6513.62 −0.641303
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −4924.59 −0.478717
\(474\) 0 0
\(475\) 375.173 0.0362402
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −14512.1 −1.38429 −0.692146 0.721758i \(-0.743335\pi\)
−0.692146 + 0.721758i \(0.743335\pi\)
\(480\) 0 0
\(481\) 1467.28 0.139090
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −13695.6 −1.28224
\(486\) 0 0
\(487\) 14189.9 1.32034 0.660170 0.751116i \(-0.270484\pi\)
0.660170 + 0.751116i \(0.270484\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −7215.91 −0.663238 −0.331619 0.943413i \(-0.607595\pi\)
−0.331619 + 0.943413i \(0.607595\pi\)
\(492\) 0 0
\(493\) −3938.82 −0.359829
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −23582.6 −2.12842
\(498\) 0 0
\(499\) 13032.6 1.16917 0.584587 0.811331i \(-0.301257\pi\)
0.584587 + 0.811331i \(0.301257\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −9502.84 −0.842367 −0.421184 0.906975i \(-0.638385\pi\)
−0.421184 + 0.906975i \(0.638385\pi\)
\(504\) 0 0
\(505\) −2808.22 −0.247454
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −3794.30 −0.330411 −0.165206 0.986259i \(-0.552829\pi\)
−0.165206 + 0.986259i \(0.552829\pi\)
\(510\) 0 0
\(511\) 32756.3 2.83572
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −6573.80 −0.562478
\(516\) 0 0
\(517\) −3483.62 −0.296343
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 14811.2 1.24547 0.622735 0.782432i \(-0.286021\pi\)
0.622735 + 0.782432i \(0.286021\pi\)
\(522\) 0 0
\(523\) −345.532 −0.0288892 −0.0144446 0.999896i \(-0.504598\pi\)
−0.0144446 + 0.999896i \(0.504598\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 24848.8 2.05395
\(528\) 0 0
\(529\) −12119.1 −0.996067
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −17101.2 −1.38975
\(534\) 0 0
\(535\) −11597.8 −0.937229
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 36238.1 2.89589
\(540\) 0 0
\(541\) 5474.45 0.435056 0.217528 0.976054i \(-0.430201\pi\)
0.217528 + 0.976054i \(0.430201\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −16800.9 −1.32049
\(546\) 0 0
\(547\) 977.278 0.0763901 0.0381951 0.999270i \(-0.487839\pi\)
0.0381951 + 0.999270i \(0.487839\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −2578.96 −0.199396
\(552\) 0 0
\(553\) −13121.3 −1.00899
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 18833.3 1.43266 0.716330 0.697762i \(-0.245821\pi\)
0.716330 + 0.697762i \(0.245821\pi\)
\(558\) 0 0
\(559\) 2960.95 0.224033
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −20967.0 −1.56954 −0.784772 0.619784i \(-0.787220\pi\)
−0.784772 + 0.619784i \(0.787220\pi\)
\(564\) 0 0
\(565\) −21732.0 −1.61818
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 13488.7 0.993804 0.496902 0.867807i \(-0.334471\pi\)
0.496902 + 0.867807i \(0.334471\pi\)
\(570\) 0 0
\(571\) −2248.90 −0.164822 −0.0824110 0.996598i \(-0.526262\pi\)
−0.0824110 + 0.996598i \(0.526262\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −36.9030 −0.00267645
\(576\) 0 0
\(577\) −1968.87 −0.142054 −0.0710270 0.997474i \(-0.522628\pi\)
−0.0710270 + 0.997474i \(0.522628\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −1516.86 −0.108313
\(582\) 0 0
\(583\) −19091.1 −1.35621
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 4107.66 0.288827 0.144413 0.989517i \(-0.453871\pi\)
0.144413 + 0.989517i \(0.453871\pi\)
\(588\) 0 0
\(589\) 16269.8 1.13818
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −7497.21 −0.519180 −0.259590 0.965719i \(-0.583587\pi\)
−0.259590 + 0.965719i \(0.583587\pi\)
\(594\) 0 0
\(595\) 36584.3 2.52069
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 27107.1 1.84903 0.924513 0.381151i \(-0.124472\pi\)
0.924513 + 0.381151i \(0.124472\pi\)
\(600\) 0 0
\(601\) 6802.17 0.461674 0.230837 0.972992i \(-0.425854\pi\)
0.230837 + 0.972992i \(0.425854\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 34910.9 2.34600
\(606\) 0 0
\(607\) −16992.4 −1.13625 −0.568123 0.822943i \(-0.692330\pi\)
−0.568123 + 0.822943i \(0.692330\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 2094.55 0.138685
\(612\) 0 0
\(613\) 13766.6 0.907062 0.453531 0.891241i \(-0.350164\pi\)
0.453531 + 0.891241i \(0.350164\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −17356.0 −1.13246 −0.566229 0.824248i \(-0.691598\pi\)
−0.566229 + 0.824248i \(0.691598\pi\)
\(618\) 0 0
\(619\) 2924.89 0.189922 0.0949608 0.995481i \(-0.469727\pi\)
0.0949608 + 0.995481i \(0.469727\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 21472.0 1.38083
\(624\) 0 0
\(625\) −16263.3 −1.04085
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −3956.78 −0.250822
\(630\) 0 0
\(631\) −6158.40 −0.388530 −0.194265 0.980949i \(-0.562232\pi\)
−0.194265 + 0.980949i \(0.562232\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −2629.31 −0.164316
\(636\) 0 0
\(637\) −21788.4 −1.35524
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 11814.2 0.727979 0.363990 0.931403i \(-0.381414\pi\)
0.363990 + 0.931403i \(0.381414\pi\)
\(642\) 0 0
\(643\) −16914.7 −1.03741 −0.518703 0.854954i \(-0.673585\pi\)
−0.518703 + 0.854954i \(0.673585\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 28652.4 1.74103 0.870513 0.492145i \(-0.163787\pi\)
0.870513 + 0.492145i \(0.163787\pi\)
\(648\) 0 0
\(649\) −51922.5 −3.14042
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 5498.48 0.329513 0.164757 0.986334i \(-0.447316\pi\)
0.164757 + 0.986334i \(0.447316\pi\)
\(654\) 0 0
\(655\) 9068.16 0.540950
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −18785.1 −1.11042 −0.555209 0.831711i \(-0.687362\pi\)
−0.555209 + 0.831711i \(0.687362\pi\)
\(660\) 0 0
\(661\) 17304.5 1.01825 0.509127 0.860691i \(-0.329968\pi\)
0.509127 + 0.860691i \(0.329968\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 23953.7 1.39682
\(666\) 0 0
\(667\) 253.673 0.0147260
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −29093.9 −1.67385
\(672\) 0 0
\(673\) −13711.5 −0.785346 −0.392673 0.919678i \(-0.628450\pi\)
−0.392673 + 0.919678i \(0.628450\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 30566.9 1.73527 0.867637 0.497198i \(-0.165638\pi\)
0.867637 + 0.497198i \(0.165638\pi\)
\(678\) 0 0
\(679\) −35788.8 −2.02275
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 4445.72 0.249064 0.124532 0.992216i \(-0.460257\pi\)
0.124532 + 0.992216i \(0.460257\pi\)
\(684\) 0 0
\(685\) −7798.27 −0.434973
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 11478.6 0.634690
\(690\) 0 0
\(691\) −12198.7 −0.671580 −0.335790 0.941937i \(-0.609003\pi\)
−0.335790 + 0.941937i \(0.609003\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −17268.9 −0.942515
\(696\) 0 0
\(697\) 46116.6 2.50615
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −21613.4 −1.16452 −0.582258 0.813004i \(-0.697831\pi\)
−0.582258 + 0.813004i \(0.697831\pi\)
\(702\) 0 0
\(703\) −2590.72 −0.138991
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −7338.32 −0.390362
\(708\) 0 0
\(709\) 4924.85 0.260870 0.130435 0.991457i \(-0.458363\pi\)
0.130435 + 0.991457i \(0.458363\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −1600.34 −0.0840578
\(714\) 0 0
\(715\) −30126.7 −1.57577
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 205.354 0.0106515 0.00532573 0.999986i \(-0.498305\pi\)
0.00532573 + 0.999986i \(0.498305\pi\)
\(720\) 0 0
\(721\) −17178.3 −0.887316
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −195.605 −0.0100201
\(726\) 0 0
\(727\) −15796.0 −0.805836 −0.402918 0.915236i \(-0.632004\pi\)
−0.402918 + 0.915236i \(0.632004\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −7984.73 −0.404003
\(732\) 0 0
\(733\) −32125.3 −1.61880 −0.809398 0.587261i \(-0.800206\pi\)
−0.809398 + 0.587261i \(0.800206\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −14464.7 −0.722949
\(738\) 0 0
\(739\) 28938.9 1.44051 0.720255 0.693710i \(-0.244025\pi\)
0.720255 + 0.693710i \(0.244025\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −36108.5 −1.78289 −0.891447 0.453124i \(-0.850309\pi\)
−0.891447 + 0.453124i \(0.850309\pi\)
\(744\) 0 0
\(745\) −26339.2 −1.29529
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −30306.9 −1.47849
\(750\) 0 0
\(751\) 23231.7 1.12881 0.564405 0.825498i \(-0.309106\pi\)
0.564405 + 0.825498i \(0.309106\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −28805.5 −1.38853
\(756\) 0 0
\(757\) −22762.6 −1.09289 −0.546447 0.837494i \(-0.684020\pi\)
−0.546447 + 0.837494i \(0.684020\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −40518.6 −1.93009 −0.965044 0.262088i \(-0.915589\pi\)
−0.965044 + 0.262088i \(0.915589\pi\)
\(762\) 0 0
\(763\) −43903.2 −2.08310
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 31218.8 1.46968
\(768\) 0 0
\(769\) −27401.1 −1.28493 −0.642464 0.766316i \(-0.722088\pi\)
−0.642464 + 0.766316i \(0.722088\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 35698.4 1.66104 0.830520 0.556989i \(-0.188043\pi\)
0.830520 + 0.556989i \(0.188043\pi\)
\(774\) 0 0
\(775\) 1234.01 0.0571959
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 30195.0 1.38876
\(780\) 0 0
\(781\) −52369.5 −2.39940
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 33324.3 1.51515
\(786\) 0 0
\(787\) −9566.36 −0.433296 −0.216648 0.976250i \(-0.569512\pi\)
−0.216648 + 0.976250i \(0.569512\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −56789.0 −2.55270
\(792\) 0 0
\(793\) 17492.9 0.783343
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 5306.01 0.235820 0.117910 0.993024i \(-0.462381\pi\)
0.117910 + 0.993024i \(0.462381\pi\)
\(798\) 0 0
\(799\) −5648.34 −0.250093
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 72741.4 3.19675
\(804\) 0 0
\(805\) −2356.14 −0.103159
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 9362.75 0.406893 0.203447 0.979086i \(-0.434786\pi\)
0.203447 + 0.979086i \(0.434786\pi\)
\(810\) 0 0
\(811\) 32610.2 1.41196 0.705981 0.708231i \(-0.250507\pi\)
0.705981 + 0.708231i \(0.250507\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 12523.7 0.538263
\(816\) 0 0
\(817\) −5228.03 −0.223875
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −18930.7 −0.804732 −0.402366 0.915479i \(-0.631812\pi\)
−0.402366 + 0.915479i \(0.631812\pi\)
\(822\) 0 0
\(823\) −40673.1 −1.72269 −0.861346 0.508018i \(-0.830378\pi\)
−0.861346 + 0.508018i \(0.830378\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −30770.2 −1.29382 −0.646908 0.762568i \(-0.723938\pi\)
−0.646908 + 0.762568i \(0.723938\pi\)
\(828\) 0 0
\(829\) −10464.9 −0.438434 −0.219217 0.975676i \(-0.570350\pi\)
−0.219217 + 0.975676i \(0.570350\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 58756.4 2.44393
\(834\) 0 0
\(835\) 8365.13 0.346691
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −22716.4 −0.934752 −0.467376 0.884059i \(-0.654800\pi\)
−0.467376 + 0.884059i \(0.654800\pi\)
\(840\) 0 0
\(841\) −23044.4 −0.944869
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −6967.97 −0.283675
\(846\) 0 0
\(847\) 91227.5 3.70084
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 254.829 0.0102649
\(852\) 0 0
\(853\) 10181.2 0.408671 0.204335 0.978901i \(-0.434497\pi\)
0.204335 + 0.978901i \(0.434497\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 3885.59 0.154877 0.0774384 0.996997i \(-0.475326\pi\)
0.0774384 + 0.996997i \(0.475326\pi\)
\(858\) 0 0
\(859\) −17734.9 −0.704433 −0.352217 0.935919i \(-0.614572\pi\)
−0.352217 + 0.935919i \(0.614572\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −19658.7 −0.775421 −0.387711 0.921781i \(-0.626734\pi\)
−0.387711 + 0.921781i \(0.626734\pi\)
\(864\) 0 0
\(865\) −8250.35 −0.324301
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −29138.2 −1.13745
\(870\) 0 0
\(871\) 8697.00 0.338331
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −40756.2 −1.57464
\(876\) 0 0
\(877\) −5586.82 −0.215112 −0.107556 0.994199i \(-0.534303\pi\)
−0.107556 + 0.994199i \(0.534303\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −8050.03 −0.307846 −0.153923 0.988083i \(-0.549191\pi\)
−0.153923 + 0.988083i \(0.549191\pi\)
\(882\) 0 0
\(883\) −2326.88 −0.0886815 −0.0443408 0.999016i \(-0.514119\pi\)
−0.0443408 + 0.999016i \(0.514119\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 19343.5 0.732233 0.366116 0.930569i \(-0.380687\pi\)
0.366116 + 0.930569i \(0.380687\pi\)
\(888\) 0 0
\(889\) −6870.78 −0.259211
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −3698.27 −0.138587
\(894\) 0 0
\(895\) 11532.4 0.430712
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −8482.63 −0.314696
\(900\) 0 0
\(901\) −30954.2 −1.14454
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 48597.2 1.78500
\(906\) 0 0
\(907\) −299.317 −0.0109577 −0.00547886 0.999985i \(-0.501744\pi\)
−0.00547886 + 0.999985i \(0.501744\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 21022.5 0.764551 0.382275 0.924048i \(-0.375141\pi\)
0.382275 + 0.924048i \(0.375141\pi\)
\(912\) 0 0
\(913\) −3368.46 −0.122103
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 23696.5 0.853356
\(918\) 0 0
\(919\) 18375.8 0.659587 0.329794 0.944053i \(-0.393021\pi\)
0.329794 + 0.944053i \(0.393021\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 31487.5 1.12289
\(924\) 0 0
\(925\) −196.496 −0.00698461
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −29548.0 −1.04353 −0.521764 0.853090i \(-0.674726\pi\)
−0.521764 + 0.853090i \(0.674726\pi\)
\(930\) 0 0
\(931\) 38471.0 1.35428
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 81242.1 2.84160
\(936\) 0 0
\(937\) 36751.2 1.28133 0.640667 0.767819i \(-0.278658\pi\)
0.640667 + 0.767819i \(0.278658\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −26024.7 −0.901575 −0.450788 0.892631i \(-0.648857\pi\)
−0.450788 + 0.892631i \(0.648857\pi\)
\(942\) 0 0
\(943\) −2970.05 −0.102564
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 40238.6 1.38076 0.690379 0.723448i \(-0.257444\pi\)
0.690379 + 0.723448i \(0.257444\pi\)
\(948\) 0 0
\(949\) −43736.3 −1.49604
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −52396.4 −1.78099 −0.890496 0.454991i \(-0.849643\pi\)
−0.890496 + 0.454991i \(0.849643\pi\)
\(954\) 0 0
\(955\) −39151.3 −1.32660
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −20378.1 −0.686176
\(960\) 0 0
\(961\) 23723.2 0.796319
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 11043.6 0.368399
\(966\) 0 0
\(967\) 9007.03 0.299531 0.149765 0.988722i \(-0.452148\pi\)
0.149765 + 0.988722i \(0.452148\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −1110.63 −0.0367062 −0.0183531 0.999832i \(-0.505842\pi\)
−0.0183531 + 0.999832i \(0.505842\pi\)
\(972\) 0 0
\(973\) −45126.3 −1.48683
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −27544.4 −0.901969 −0.450984 0.892532i \(-0.648927\pi\)
−0.450984 + 0.892532i \(0.648927\pi\)
\(978\) 0 0
\(979\) 47682.5 1.55663
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 8657.48 0.280906 0.140453 0.990087i \(-0.455144\pi\)
0.140453 + 0.990087i \(0.455144\pi\)
\(984\) 0 0
\(985\) 5347.60 0.172983
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 514.242 0.0165338
\(990\) 0 0
\(991\) 54045.9 1.73242 0.866208 0.499683i \(-0.166550\pi\)
0.866208 + 0.499683i \(0.166550\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 30521.4 0.972456
\(996\) 0 0
\(997\) 32591.1 1.03528 0.517638 0.855600i \(-0.326811\pi\)
0.517638 + 0.855600i \(0.326811\pi\)
\(998\) 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 1728.4.a.bj.1.2 2
3.2 odd 2 1728.4.a.br.1.1 2
4.3 odd 2 1728.4.a.bi.1.2 2
8.3 odd 2 216.4.a.g.1.1 yes 2
8.5 even 2 432.4.a.r.1.1 2
12.11 even 2 1728.4.a.bq.1.1 2
24.5 odd 2 432.4.a.p.1.2 2
24.11 even 2 216.4.a.f.1.2 2
72.11 even 6 648.4.i.r.433.1 4
72.43 odd 6 648.4.i.o.433.2 4
72.59 even 6 648.4.i.r.217.1 4
72.67 odd 6 648.4.i.o.217.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
216.4.a.f.1.2 2 24.11 even 2
216.4.a.g.1.1 yes 2 8.3 odd 2
432.4.a.p.1.2 2 24.5 odd 2
432.4.a.r.1.1 2 8.5 even 2
648.4.i.o.217.2 4 72.67 odd 6
648.4.i.o.433.2 4 72.43 odd 6
648.4.i.r.217.1 4 72.59 even 6
648.4.i.r.433.1 4 72.11 even 6
1728.4.a.bi.1.2 2 4.3 odd 2
1728.4.a.bj.1.2 2 1.1 even 1 trivial
1728.4.a.bq.1.1 2 12.11 even 2
1728.4.a.br.1.1 2 3.2 odd 2