Properties

Label 2736.2.d.b.2015.14
Level $2736$
Weight $2$
Character 2736.2015
Analytic conductor $21.847$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2015.14
Character \(\chi\) \(=\) 2736.2015
Dual form 2736.2.d.b.2015.11

$q$-expansion

\(f(q)\) \(=\) \(q+1.11119i q^{5} +1.19380i q^{7} +O(q^{10})\) \(q+1.11119i q^{5} +1.19380i q^{7} +3.72045 q^{11} +0.113448 q^{13} -2.89258i q^{17} -1.00000i q^{19} +1.35549 q^{23} +3.76526 q^{25} -0.860408i q^{29} -9.01283i q^{31} -1.32654 q^{35} +5.64396 q^{37} +10.5323i q^{41} -2.47195i q^{43} -5.05623 q^{47} +5.57485 q^{49} +9.51146i q^{53} +4.13413i q^{55} +8.73964 q^{59} -8.52172 q^{61} +0.126062i q^{65} -0.857080i q^{67} -1.91011 q^{71} +3.06213 q^{73} +4.44146i q^{77} -7.62732i q^{79} +10.0020 q^{83} +3.21420 q^{85} +13.1659i q^{89} +0.135434i q^{91} +1.11119 q^{95} +11.6659 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24q + O(q^{10}) \) \( 24q + 8q^{25} - 32q^{37} - 32q^{49} + 8q^{73} + 40q^{85} + 16q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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
\(5\) 1.11119i 0.496939i 0.968640 + 0.248470i \(0.0799276\pi\)
−0.968640 + 0.248470i \(0.920072\pi\)
\(6\) 0 0
\(7\) 1.19380i 0.451213i 0.974219 + 0.225606i \(0.0724363\pi\)
−0.974219 + 0.225606i \(0.927564\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.72045 1.12176 0.560879 0.827898i \(-0.310463\pi\)
0.560879 + 0.827898i \(0.310463\pi\)
\(12\) 0 0
\(13\) 0.113448 0.0314648 0.0157324 0.999876i \(-0.494992\pi\)
0.0157324 + 0.999876i \(0.494992\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 2.89258i − 0.701553i −0.936459 0.350776i \(-0.885918\pi\)
0.936459 0.350776i \(-0.114082\pi\)
\(18\) 0 0
\(19\) − 1.00000i − 0.229416i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.35549 0.282639 0.141319 0.989964i \(-0.454866\pi\)
0.141319 + 0.989964i \(0.454866\pi\)
\(24\) 0 0
\(25\) 3.76526 0.753051
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 0.860408i − 0.159774i −0.996804 0.0798869i \(-0.974544\pi\)
0.996804 0.0798869i \(-0.0254559\pi\)
\(30\) 0 0
\(31\) − 9.01283i − 1.61875i −0.587290 0.809376i \(-0.699805\pi\)
0.587290 0.809376i \(-0.300195\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.32654 −0.224225
\(36\) 0 0
\(37\) 5.64396 0.927862 0.463931 0.885871i \(-0.346439\pi\)
0.463931 + 0.885871i \(0.346439\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 10.5323i 1.64487i 0.568859 + 0.822435i \(0.307385\pi\)
−0.568859 + 0.822435i \(0.692615\pi\)
\(42\) 0 0
\(43\) − 2.47195i − 0.376969i −0.982076 0.188484i \(-0.939643\pi\)
0.982076 0.188484i \(-0.0603575\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.05623 −0.737527 −0.368764 0.929523i \(-0.620219\pi\)
−0.368764 + 0.929523i \(0.620219\pi\)
\(48\) 0 0
\(49\) 5.57485 0.796407
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 9.51146i 1.30650i 0.757143 + 0.653250i \(0.226595\pi\)
−0.757143 + 0.653250i \(0.773405\pi\)
\(54\) 0 0
\(55\) 4.13413i 0.557446i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 8.73964 1.13780 0.568902 0.822405i \(-0.307368\pi\)
0.568902 + 0.822405i \(0.307368\pi\)
\(60\) 0 0
\(61\) −8.52172 −1.09109 −0.545547 0.838080i \(-0.683678\pi\)
−0.545547 + 0.838080i \(0.683678\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.126062i 0.0156361i
\(66\) 0 0
\(67\) − 0.857080i − 0.104709i −0.998629 0.0523545i \(-0.983327\pi\)
0.998629 0.0523545i \(-0.0166726\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −1.91011 −0.226689 −0.113344 0.993556i \(-0.536156\pi\)
−0.113344 + 0.993556i \(0.536156\pi\)
\(72\) 0 0
\(73\) 3.06213 0.358395 0.179198 0.983813i \(-0.442650\pi\)
0.179198 + 0.983813i \(0.442650\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.44146i 0.506152i
\(78\) 0 0
\(79\) − 7.62732i − 0.858141i −0.903271 0.429071i \(-0.858841\pi\)
0.903271 0.429071i \(-0.141159\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 10.0020 1.09787 0.548933 0.835867i \(-0.315034\pi\)
0.548933 + 0.835867i \(0.315034\pi\)
\(84\) 0 0
\(85\) 3.21420 0.348629
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 13.1659i 1.39558i 0.716300 + 0.697792i \(0.245834\pi\)
−0.716300 + 0.697792i \(0.754166\pi\)
\(90\) 0 0
\(91\) 0.135434i 0.0141973i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.11119 0.114006
\(96\) 0 0
\(97\) 11.6659 1.18450 0.592249 0.805755i \(-0.298240\pi\)
0.592249 + 0.805755i \(0.298240\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1.64199i 0.163384i 0.996658 + 0.0816921i \(0.0260324\pi\)
−0.996658 + 0.0816921i \(0.973968\pi\)
\(102\) 0 0
\(103\) 2.09681i 0.206605i 0.994650 + 0.103302i \(0.0329410\pi\)
−0.994650 + 0.103302i \(0.967059\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −2.03841 −0.197061 −0.0985303 0.995134i \(-0.531414\pi\)
−0.0985303 + 0.995134i \(0.531414\pi\)
\(108\) 0 0
\(109\) −2.32276 −0.222480 −0.111240 0.993794i \(-0.535482\pi\)
−0.111240 + 0.993794i \(0.535482\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 5.91983i 0.556891i 0.960452 + 0.278445i \(0.0898191\pi\)
−0.960452 + 0.278445i \(0.910181\pi\)
\(114\) 0 0
\(115\) 1.50621i 0.140454i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.45315 0.316550
\(120\) 0 0
\(121\) 2.84176 0.258342
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 9.73987i 0.871160i
\(126\) 0 0
\(127\) 12.4051i 1.10078i 0.834909 + 0.550388i \(0.185520\pi\)
−0.834909 + 0.550388i \(0.814480\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 11.9820 1.04687 0.523434 0.852066i \(-0.324651\pi\)
0.523434 + 0.852066i \(0.324651\pi\)
\(132\) 0 0
\(133\) 1.19380 0.103515
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 3.69469i 0.315658i 0.987466 + 0.157829i \(0.0504495\pi\)
−0.987466 + 0.157829i \(0.949550\pi\)
\(138\) 0 0
\(139\) − 0.965743i − 0.0819133i −0.999161 0.0409566i \(-0.986959\pi\)
0.999161 0.0409566i \(-0.0130406\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0.422078 0.0352960
\(144\) 0 0
\(145\) 0.956077 0.0793978
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 3.59705i − 0.294682i −0.989086 0.147341i \(-0.952929\pi\)
0.989086 0.147341i \(-0.0470715\pi\)
\(150\) 0 0
\(151\) 2.17609i 0.177088i 0.996072 + 0.0885440i \(0.0282214\pi\)
−0.996072 + 0.0885440i \(0.971779\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 10.0150 0.804422
\(156\) 0 0
\(157\) 24.4292 1.94966 0.974832 0.222943i \(-0.0715663\pi\)
0.974832 + 0.222943i \(0.0715663\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1.61818i 0.127530i
\(162\) 0 0
\(163\) − 18.6476i − 1.46059i −0.683129 0.730297i \(-0.739381\pi\)
0.683129 0.730297i \(-0.260619\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 9.46847 0.732692 0.366346 0.930479i \(-0.380609\pi\)
0.366346 + 0.930479i \(0.380609\pi\)
\(168\) 0 0
\(169\) −12.9871 −0.999010
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 8.52499i 0.648143i 0.946033 + 0.324071i \(0.105052\pi\)
−0.946033 + 0.324071i \(0.894948\pi\)
\(174\) 0 0
\(175\) 4.49495i 0.339786i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −5.16062 −0.385723 −0.192861 0.981226i \(-0.561777\pi\)
−0.192861 + 0.981226i \(0.561777\pi\)
\(180\) 0 0
\(181\) 17.3864 1.29232 0.646160 0.763202i \(-0.276374\pi\)
0.646160 + 0.763202i \(0.276374\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 6.27151i 0.461091i
\(186\) 0 0
\(187\) − 10.7617i − 0.786973i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0.768626 0.0556159 0.0278079 0.999613i \(-0.491147\pi\)
0.0278079 + 0.999613i \(0.491147\pi\)
\(192\) 0 0
\(193\) −12.1274 −0.872951 −0.436475 0.899716i \(-0.643773\pi\)
−0.436475 + 0.899716i \(0.643773\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.95333i 0.210416i 0.994450 + 0.105208i \(0.0335509\pi\)
−0.994450 + 0.105208i \(0.966449\pi\)
\(198\) 0 0
\(199\) − 17.3465i − 1.22966i −0.788660 0.614829i \(-0.789225\pi\)
0.788660 0.614829i \(-0.210775\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.02715 0.0720920
\(204\) 0 0
\(205\) −11.7034 −0.817400
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 3.72045i − 0.257349i
\(210\) 0 0
\(211\) − 2.32941i − 0.160363i −0.996780 0.0801815i \(-0.974450\pi\)
0.996780 0.0801815i \(-0.0255500\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 2.74680 0.187331
\(216\) 0 0
\(217\) 10.7595 0.730402
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 0.328157i − 0.0220742i
\(222\) 0 0
\(223\) 10.1883i 0.682257i 0.940017 + 0.341129i \(0.110809\pi\)
−0.940017 + 0.341129i \(0.889191\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 14.2877 0.948305 0.474153 0.880443i \(-0.342754\pi\)
0.474153 + 0.880443i \(0.342754\pi\)
\(228\) 0 0
\(229\) 4.84008 0.319842 0.159921 0.987130i \(-0.448876\pi\)
0.159921 + 0.987130i \(0.448876\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 14.5288i − 0.951813i −0.879496 0.475907i \(-0.842120\pi\)
0.879496 0.475907i \(-0.157880\pi\)
\(234\) 0 0
\(235\) − 5.61843i − 0.366506i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −29.9420 −1.93679 −0.968395 0.249422i \(-0.919759\pi\)
−0.968395 + 0.249422i \(0.919759\pi\)
\(240\) 0 0
\(241\) −10.0657 −0.648390 −0.324195 0.945990i \(-0.605093\pi\)
−0.324195 + 0.945990i \(0.605093\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 6.19472i 0.395766i
\(246\) 0 0
\(247\) − 0.113448i − 0.00721853i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 16.6664 1.05197 0.525986 0.850493i \(-0.323696\pi\)
0.525986 + 0.850493i \(0.323696\pi\)
\(252\) 0 0
\(253\) 5.04303 0.317053
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 21.7438i 1.35634i 0.734906 + 0.678169i \(0.237226\pi\)
−0.734906 + 0.678169i \(0.762774\pi\)
\(258\) 0 0
\(259\) 6.73774i 0.418663i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −30.8883 −1.90466 −0.952328 0.305075i \(-0.901319\pi\)
−0.952328 + 0.305075i \(0.901319\pi\)
\(264\) 0 0
\(265\) −10.5690 −0.649251
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 4.71830i − 0.287680i −0.989601 0.143840i \(-0.954055\pi\)
0.989601 0.143840i \(-0.0459450\pi\)
\(270\) 0 0
\(271\) − 4.77033i − 0.289777i −0.989448 0.144888i \(-0.953718\pi\)
0.989448 0.144888i \(-0.0462823\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 14.0085 0.844742
\(276\) 0 0
\(277\) −5.73084 −0.344333 −0.172166 0.985068i \(-0.555077\pi\)
−0.172166 + 0.985068i \(0.555077\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 12.7428i 0.760171i 0.924951 + 0.380085i \(0.124105\pi\)
−0.924951 + 0.380085i \(0.875895\pi\)
\(282\) 0 0
\(283\) − 24.1940i − 1.43818i −0.694914 0.719092i \(-0.744558\pi\)
0.694914 0.719092i \(-0.255442\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −12.5734 −0.742186
\(288\) 0 0
\(289\) 8.63300 0.507824
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 8.35727i − 0.488237i −0.969745 0.244118i \(-0.921501\pi\)
0.969745 0.244118i \(-0.0784986\pi\)
\(294\) 0 0
\(295\) 9.71140i 0.565420i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0.153778 0.00889319
\(300\) 0 0
\(301\) 2.95100 0.170093
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 9.46925i − 0.542208i
\(306\) 0 0
\(307\) 13.0461i 0.744578i 0.928117 + 0.372289i \(0.121427\pi\)
−0.928117 + 0.372289i \(0.878573\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 3.71584 0.210706 0.105353 0.994435i \(-0.466403\pi\)
0.105353 + 0.994435i \(0.466403\pi\)
\(312\) 0 0
\(313\) 7.32643 0.414114 0.207057 0.978329i \(-0.433611\pi\)
0.207057 + 0.978329i \(0.433611\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 4.42123i − 0.248321i −0.992262 0.124161i \(-0.960376\pi\)
0.992262 0.124161i \(-0.0396238\pi\)
\(318\) 0 0
\(319\) − 3.20111i − 0.179228i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −2.89258 −0.160947
\(324\) 0 0
\(325\) 0.427161 0.0236946
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 6.03611i − 0.332782i
\(330\) 0 0
\(331\) − 1.97474i − 0.108541i −0.998526 0.0542707i \(-0.982717\pi\)
0.998526 0.0542707i \(-0.0172834\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0.952379 0.0520340
\(336\) 0 0
\(337\) 24.4795 1.33348 0.666741 0.745290i \(-0.267689\pi\)
0.666741 + 0.745290i \(0.267689\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 33.5318i − 1.81585i
\(342\) 0 0
\(343\) 15.0118i 0.810562i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 18.8988 1.01454 0.507271 0.861787i \(-0.330654\pi\)
0.507271 + 0.861787i \(0.330654\pi\)
\(348\) 0 0
\(349\) −10.1508 −0.543361 −0.271680 0.962388i \(-0.587579\pi\)
−0.271680 + 0.962388i \(0.587579\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 20.6591i − 1.09957i −0.835306 0.549785i \(-0.814709\pi\)
0.835306 0.549785i \(-0.185291\pi\)
\(354\) 0 0
\(355\) − 2.12250i − 0.112650i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1.15543 0.0609810 0.0304905 0.999535i \(-0.490293\pi\)
0.0304905 + 0.999535i \(0.490293\pi\)
\(360\) 0 0
\(361\) −1.00000 −0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 3.40261i 0.178101i
\(366\) 0 0
\(367\) 25.9993i 1.35715i 0.734530 + 0.678576i \(0.237403\pi\)
−0.734530 + 0.678576i \(0.762597\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −11.3547 −0.589509
\(372\) 0 0
\(373\) 18.6006 0.963102 0.481551 0.876418i \(-0.340074\pi\)
0.481551 + 0.876418i \(0.340074\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 0.0976116i − 0.00502726i
\(378\) 0 0
\(379\) − 6.01294i − 0.308864i −0.988003 0.154432i \(-0.950645\pi\)
0.988003 0.154432i \(-0.0493547\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −10.6395 −0.543653 −0.271827 0.962346i \(-0.587628\pi\)
−0.271827 + 0.962346i \(0.587628\pi\)
\(384\) 0 0
\(385\) −4.93531 −0.251527
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 15.2848i 0.774971i 0.921876 + 0.387485i \(0.126656\pi\)
−0.921876 + 0.387485i \(0.873344\pi\)
\(390\) 0 0
\(391\) − 3.92085i − 0.198286i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 8.47541 0.426444
\(396\) 0 0
\(397\) −11.7056 −0.587487 −0.293743 0.955884i \(-0.594901\pi\)
−0.293743 + 0.955884i \(0.594901\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 3.79270i − 0.189399i −0.995506 0.0946993i \(-0.969811\pi\)
0.995506 0.0946993i \(-0.0301890\pi\)
\(402\) 0 0
\(403\) − 1.02249i − 0.0509338i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 20.9981 1.04084
\(408\) 0 0
\(409\) 0.556303 0.0275074 0.0137537 0.999905i \(-0.495622\pi\)
0.0137537 + 0.999905i \(0.495622\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 10.4334i 0.513392i
\(414\) 0 0
\(415\) 11.1142i 0.545572i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −26.4184 −1.29063 −0.645313 0.763918i \(-0.723273\pi\)
−0.645313 + 0.763918i \(0.723273\pi\)
\(420\) 0 0
\(421\) −24.0288 −1.17109 −0.585546 0.810639i \(-0.699120\pi\)
−0.585546 + 0.810639i \(0.699120\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 10.8913i − 0.528305i
\(426\) 0 0
\(427\) − 10.1732i − 0.492316i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −30.3393 −1.46139 −0.730697 0.682702i \(-0.760805\pi\)
−0.730697 + 0.682702i \(0.760805\pi\)
\(432\) 0 0
\(433\) −15.1246 −0.726843 −0.363421 0.931625i \(-0.618391\pi\)
−0.363421 + 0.931625i \(0.618391\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 1.35549i − 0.0648418i
\(438\) 0 0
\(439\) 28.6582i 1.36778i 0.729585 + 0.683890i \(0.239713\pi\)
−0.729585 + 0.683890i \(0.760287\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −36.9951 −1.75769 −0.878845 0.477108i \(-0.841685\pi\)
−0.878845 + 0.477108i \(0.841685\pi\)
\(444\) 0 0
\(445\) −14.6298 −0.693521
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 17.4824i − 0.825044i −0.910948 0.412522i \(-0.864648\pi\)
0.910948 0.412522i \(-0.135352\pi\)
\(450\) 0 0
\(451\) 39.1849i 1.84515i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −0.150493 −0.00705522
\(456\) 0 0
\(457\) −33.2001 −1.55303 −0.776517 0.630096i \(-0.783015\pi\)
−0.776517 + 0.630096i \(0.783015\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 13.2270i − 0.616043i −0.951379 0.308022i \(-0.900333\pi\)
0.951379 0.308022i \(-0.0996669\pi\)
\(462\) 0 0
\(463\) 19.7916i 0.919794i 0.887972 + 0.459897i \(0.152114\pi\)
−0.887972 + 0.459897i \(0.847886\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.7076 0.588038 0.294019 0.955800i \(-0.405007\pi\)
0.294019 + 0.955800i \(0.405007\pi\)
\(468\) 0 0
\(469\) 1.02318 0.0472460
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 9.19677i − 0.422868i
\(474\) 0 0
\(475\) − 3.76526i − 0.172762i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −6.28722 −0.287271 −0.143635 0.989631i \(-0.545879\pi\)
−0.143635 + 0.989631i \(0.545879\pi\)
\(480\) 0 0
\(481\) 0.640297 0.0291950
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 12.9631i 0.588623i
\(486\) 0 0
\(487\) − 18.6477i − 0.845008i −0.906361 0.422504i \(-0.861151\pi\)
0.906361 0.422504i \(-0.138849\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 15.8097 0.713480 0.356740 0.934204i \(-0.383888\pi\)
0.356740 + 0.934204i \(0.383888\pi\)
\(492\) 0 0
\(493\) −2.48880 −0.112090
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 2.28029i − 0.102285i
\(498\) 0 0
\(499\) − 13.1361i − 0.588052i −0.955797 0.294026i \(-0.905005\pi\)
0.955797 0.294026i \(-0.0949952\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 3.07107 0.136932 0.0684661 0.997653i \(-0.478189\pi\)
0.0684661 + 0.997653i \(0.478189\pi\)
\(504\) 0 0
\(505\) −1.82456 −0.0811920
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 21.1047i − 0.935449i −0.883874 0.467725i \(-0.845074\pi\)
0.883874 0.467725i \(-0.154926\pi\)
\(510\) 0 0
\(511\) 3.65556i 0.161713i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −2.32995 −0.102670
\(516\) 0 0
\(517\) −18.8115 −0.827327
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 19.1133i − 0.837367i −0.908132 0.418684i \(-0.862492\pi\)
0.908132 0.418684i \(-0.137508\pi\)
\(522\) 0 0
\(523\) − 29.0588i − 1.27065i −0.772244 0.635326i \(-0.780866\pi\)
0.772244 0.635326i \(-0.219134\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −26.0703 −1.13564
\(528\) 0 0
\(529\) −21.1627 −0.920115
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 1.19487i 0.0517556i
\(534\) 0 0
\(535\) − 2.26506i − 0.0979272i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 20.7410 0.893376
\(540\) 0 0
\(541\) −5.90913 −0.254053 −0.127027 0.991899i \(-0.540543\pi\)
−0.127027 + 0.991899i \(0.540543\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 2.58103i − 0.110559i
\(546\) 0 0
\(547\) − 13.7241i − 0.586802i −0.955990 0.293401i \(-0.905213\pi\)
0.955990 0.293401i \(-0.0947870\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −0.860408 −0.0366546
\(552\) 0 0
\(553\) 9.10548 0.387204
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 7.35820i 0.311777i 0.987775 + 0.155889i \(0.0498241\pi\)
−0.987775 + 0.155889i \(0.950176\pi\)
\(558\) 0 0
\(559\) − 0.280438i − 0.0118613i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 9.41425 0.396763 0.198382 0.980125i \(-0.436431\pi\)
0.198382 + 0.980125i \(0.436431\pi\)
\(564\) 0 0
\(565\) −6.57805 −0.276741
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 14.6116i − 0.612551i −0.951943 0.306276i \(-0.900917\pi\)
0.951943 0.306276i \(-0.0990829\pi\)
\(570\) 0 0
\(571\) − 39.1353i − 1.63776i −0.573963 0.818881i \(-0.694595\pi\)
0.573963 0.818881i \(-0.305405\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 5.10376 0.212842
\(576\) 0 0
\(577\) −1.36547 −0.0568451 −0.0284226 0.999596i \(-0.509048\pi\)
−0.0284226 + 0.999596i \(0.509048\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 11.9404i 0.495371i
\(582\) 0 0
\(583\) 35.3869i 1.46558i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 22.0683 0.910855 0.455427 0.890273i \(-0.349486\pi\)
0.455427 + 0.890273i \(0.349486\pi\)
\(588\) 0 0
\(589\) −9.01283 −0.371367
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 12.0401i − 0.494429i −0.968961 0.247215i \(-0.920485\pi\)
0.968961 0.247215i \(-0.0795153\pi\)
\(594\) 0 0
\(595\) 3.83710i 0.157306i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −40.0405 −1.63601 −0.818006 0.575210i \(-0.804920\pi\)
−0.818006 + 0.575210i \(0.804920\pi\)
\(600\) 0 0
\(601\) −32.8278 −1.33907 −0.669537 0.742778i \(-0.733508\pi\)
−0.669537 + 0.742778i \(0.733508\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 3.15774i 0.128380i
\(606\) 0 0
\(607\) 4.55151i 0.184740i 0.995725 + 0.0923700i \(0.0294443\pi\)
−0.995725 + 0.0923700i \(0.970556\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −0.573620 −0.0232062
\(612\) 0 0
\(613\) −42.4038 −1.71267 −0.856336 0.516419i \(-0.827265\pi\)
−0.856336 + 0.516419i \(0.827265\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 16.9498i − 0.682373i −0.939996 0.341187i \(-0.889171\pi\)
0.939996 0.341187i \(-0.110829\pi\)
\(618\) 0 0
\(619\) 22.7120i 0.912872i 0.889756 + 0.456436i \(0.150874\pi\)
−0.889756 + 0.456436i \(0.849126\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −15.7174 −0.629705
\(624\) 0 0
\(625\) 8.00344 0.320138
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 16.3256i − 0.650944i
\(630\) 0 0
\(631\) − 0.732322i − 0.0291533i −0.999894 0.0145766i \(-0.995360\pi\)
0.999894 0.0145766i \(-0.00464005\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −13.7844 −0.547019
\(636\) 0 0
\(637\) 0.632456 0.0250588
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 7.90566i − 0.312255i −0.987737 0.156127i \(-0.950099\pi\)
0.987737 0.156127i \(-0.0499010\pi\)
\(642\) 0 0
\(643\) 36.9435i 1.45691i 0.685095 + 0.728454i \(0.259761\pi\)
−0.685095 + 0.728454i \(0.740239\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −3.47909 −0.136777 −0.0683886 0.997659i \(-0.521786\pi\)
−0.0683886 + 0.997659i \(0.521786\pi\)
\(648\) 0 0
\(649\) 32.5154 1.27634
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 37.0587i − 1.45022i −0.688634 0.725109i \(-0.741790\pi\)
0.688634 0.725109i \(-0.258210\pi\)
\(654\) 0 0
\(655\) 13.3142i 0.520230i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −38.8706 −1.51418 −0.757091 0.653309i \(-0.773380\pi\)
−0.757091 + 0.653309i \(0.773380\pi\)
\(660\) 0 0
\(661\) 17.6229 0.685450 0.342725 0.939436i \(-0.388650\pi\)
0.342725 + 0.939436i \(0.388650\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.32654i 0.0514408i
\(666\) 0 0
\(667\) − 1.16627i − 0.0451583i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −31.7047 −1.22394
\(672\) 0 0
\(673\) −20.3151 −0.783090 −0.391545 0.920159i \(-0.628059\pi\)
−0.391545 + 0.920159i \(0.628059\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 11.8396i − 0.455032i −0.973774 0.227516i \(-0.926940\pi\)
0.973774 0.227516i \(-0.0730604\pi\)
\(678\) 0 0
\(679\) 13.9268i 0.534460i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 50.6817 1.93928 0.969640 0.244535i \(-0.0786355\pi\)
0.969640 + 0.244535i \(0.0786355\pi\)
\(684\) 0 0
\(685\) −4.10550 −0.156863
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1.07906i 0.0411088i
\(690\) 0 0
\(691\) − 32.6962i − 1.24382i −0.783088 0.621910i \(-0.786357\pi\)
0.783088 0.621910i \(-0.213643\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1.07312 0.0407059
\(696\) 0 0
\(697\) 30.4655 1.15396
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 50.1158i − 1.89285i −0.322924 0.946425i \(-0.604666\pi\)
0.322924 0.946425i \(-0.395334\pi\)
\(702\) 0 0
\(703\) − 5.64396i − 0.212866i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −1.96020 −0.0737210
\(708\) 0 0
\(709\) −12.4838 −0.468840 −0.234420 0.972135i \(-0.575319\pi\)
−0.234420 + 0.972135i \(0.575319\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 12.2168i − 0.457522i
\(714\) 0 0
\(715\) 0.469009i 0.0175399i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −20.4623 −0.763116 −0.381558 0.924345i \(-0.624612\pi\)
−0.381558 + 0.924345i \(0.624612\pi\)
\(720\) 0 0
\(721\) −2.50317 −0.0932228
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 3.23966i − 0.120318i
\(726\) 0 0
\(727\) 20.7963i 0.771292i 0.922647 + 0.385646i \(0.126021\pi\)
−0.922647 + 0.385646i \(0.873979\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −7.15030 −0.264463
\(732\) 0 0
\(733\) 13.3029 0.491352 0.245676 0.969352i \(-0.420990\pi\)
0.245676 + 0.969352i \(0.420990\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 3.18873i − 0.117458i
\(738\) 0 0
\(739\) − 6.39272i − 0.235160i −0.993063 0.117580i \(-0.962486\pi\)
0.993063 0.117580i \(-0.0375136\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 34.2605 1.25689 0.628447 0.777852i \(-0.283691\pi\)
0.628447 + 0.777852i \(0.283691\pi\)
\(744\) 0 0
\(745\) 3.99701 0.146439
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 2.43345i − 0.0889163i
\(750\) 0 0
\(751\) 45.5988i 1.66392i 0.554833 + 0.831962i \(0.312782\pi\)
−0.554833 + 0.831962i \(0.687218\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −2.41805 −0.0880020
\(756\) 0 0
\(757\) 25.1493 0.914066 0.457033 0.889450i \(-0.348912\pi\)
0.457033 + 0.889450i \(0.348912\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 12.3805i 0.448793i 0.974498 + 0.224396i \(0.0720411\pi\)
−0.974498 + 0.224396i \(0.927959\pi\)
\(762\) 0 0
\(763\) − 2.77291i − 0.100386i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0.991496 0.0358008
\(768\) 0 0
\(769\) −37.9127 −1.36717 −0.683584 0.729872i \(-0.739580\pi\)
−0.683584 + 0.729872i \(0.739580\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 41.1007i 1.47829i 0.673546 + 0.739145i \(0.264770\pi\)
−0.673546 + 0.739145i \(0.735230\pi\)
\(774\) 0 0
\(775\) − 33.9356i − 1.21900i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 10.5323 0.377359
\(780\) 0 0
\(781\) −7.10648 −0.254290
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 27.1455i 0.968864i
\(786\) 0 0
\(787\) 0.408338i 0.0145557i 0.999974 + 0.00727783i \(0.00231663\pi\)
−0.999974 + 0.00727783i \(0.997683\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −7.06707 −0.251276
\(792\) 0 0
\(793\) −0.966773 −0.0343311
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 14.1914i − 0.502686i −0.967898 0.251343i \(-0.919128\pi\)
0.967898 0.251343i \(-0.0808723\pi\)
\(798\) 0 0
\(799\) 14.6255i 0.517414i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 11.3925 0.402033
\(804\) 0 0
\(805\) −1.79810 −0.0633748
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 15.2844i 0.537370i 0.963228 + 0.268685i \(0.0865890\pi\)
−0.963228 + 0.268685i \(0.913411\pi\)
\(810\) 0 0
\(811\) 1.35096i 0.0474386i 0.999719 + 0.0237193i \(0.00755080\pi\)
−0.999719 + 0.0237193i \(0.992449\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 20.7211 0.725827
\(816\) 0 0
\(817\) −2.47195 −0.0864825
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 45.3093i − 1.58131i −0.612264 0.790653i \(-0.709741\pi\)
0.612264 0.790653i \(-0.290259\pi\)
\(822\) 0 0
\(823\) 2.98709i 0.104123i 0.998644 + 0.0520617i \(0.0165793\pi\)
−0.998644 + 0.0520617i \(0.983421\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −20.0319 −0.696579 −0.348289 0.937387i \(-0.613237\pi\)
−0.348289 + 0.937387i \(0.613237\pi\)
\(828\) 0 0
\(829\) −17.6585 −0.613304 −0.306652 0.951822i \(-0.599209\pi\)
−0.306652 + 0.951822i \(0.599209\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 16.1257i − 0.558722i
\(834\) 0 0
\(835\) 10.5213i 0.364103i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 9.22764 0.318574 0.159287 0.987232i \(-0.449081\pi\)
0.159287 + 0.987232i \(0.449081\pi\)
\(840\) 0 0
\(841\) 28.2597 0.974472
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 14.4312i − 0.496447i
\(846\) 0 0
\(847\) 3.39249i 0.116567i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 7.65033 0.262250
\(852\) 0 0
\(853\) −34.7289 −1.18909 −0.594547 0.804061i \(-0.702669\pi\)
−0.594547 + 0.804061i \(0.702669\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 28.3620i 0.968828i 0.874839 + 0.484414i \(0.160967\pi\)
−0.874839 + 0.484414i \(0.839033\pi\)
\(858\) 0 0
\(859\) − 13.7817i − 0.470226i −0.971968 0.235113i \(-0.924454\pi\)
0.971968 0.235113i \(-0.0755460\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −36.1403 −1.23023 −0.615114 0.788438i \(-0.710890\pi\)
−0.615114 + 0.788438i \(0.710890\pi\)
\(864\) 0 0
\(865\) −9.47288 −0.322088
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 28.3771i − 0.962627i
\(870\) 0 0
\(871\) − 0.0972341i − 0.00329465i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −11.6274 −0.393079
\(876\) 0 0
\(877\) 34.2580 1.15681 0.578405 0.815750i \(-0.303675\pi\)
0.578405 + 0.815750i \(0.303675\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 21.4623i 0.723085i 0.932356 + 0.361542i \(0.117750\pi\)
−0.932356 + 0.361542i \(0.882250\pi\)
\(882\) 0 0
\(883\) − 5.60029i − 0.188465i −0.995550 0.0942323i \(-0.969960\pi\)
0.995550 0.0942323i \(-0.0300396\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −38.9684 −1.30843 −0.654216 0.756308i \(-0.727001\pi\)
−0.654216 + 0.756308i \(0.727001\pi\)
\(888\) 0 0
\(889\) −14.8092 −0.496685
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 5.05623i 0.169200i
\(894\) 0 0
\(895\) − 5.73443i − 0.191681i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −7.75471 −0.258634
\(900\) 0 0
\(901\) 27.5126 0.916578
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 19.3196i 0.642205i
\(906\) 0 0
\(907\) 23.8500i 0.791927i 0.918266 + 0.395964i \(0.129589\pi\)
−0.918266 + 0.395964i \(0.870411\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 12.5268 0.415032 0.207516 0.978232i \(-0.433462\pi\)
0.207516 + 0.978232i \(0.433462\pi\)
\(912\) 0 0
\(913\) 37.2121 1.23154
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 14.3040i 0.472360i
\(918\) 0 0
\(919\) − 22.4838i − 0.741671i −0.928699 0.370836i \(-0.879071\pi\)
0.928699 0.370836i \(-0.120929\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −0.216699 −0.00713272
\(924\) 0 0
\(925\) 21.2510 0.698727
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 34.9715i − 1.14738i −0.819073 0.573689i \(-0.805512\pi\)
0.819073 0.573689i \(-0.194488\pi\)
\(930\) 0 0
\(931\) − 5.57485i − 0.182708i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 11.9583 0.391078
\(936\) 0 0
\(937\) 19.0472 0.622245 0.311122 0.950370i \(-0.399295\pi\)
0.311122 + 0.950370i \(0.399295\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 21.1518i 0.689527i 0.938690 + 0.344764i \(0.112041\pi\)
−0.938690 + 0.344764i \(0.887959\pi\)
\(942\) 0 0
\(943\) 14.2764i 0.464904i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −20.6426 −0.670793 −0.335396 0.942077i \(-0.608870\pi\)
−0.335396 + 0.942077i \(0.608870\pi\)
\(948\) 0 0
\(949\) 0.347393 0.0112768
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 48.3617i − 1.56659i −0.621650 0.783295i \(-0.713537\pi\)
0.621650 0.783295i \(-0.286463\pi\)
\(954\) 0 0
\(955\) 0.854090i 0.0276377i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −4.41071 −0.142429
\(960\) 0 0
\(961\) −50.2312 −1.62036
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 13.4759i − 0.433804i
\(966\) 0 0
\(967\) − 62.1680i − 1.99919i −0.0285042 0.999594i \(-0.509074\pi\)
0.0285042 0.999594i \(-0.490926\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −39.5108 −1.26796 −0.633981 0.773348i \(-0.718580\pi\)
−0.633981 + 0.773348i \(0.718580\pi\)
\(972\) 0 0
\(973\) 1.15290 0.0369603
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 36.3792i 1.16387i 0.813234 + 0.581937i \(0.197705\pi\)
−0.813234 + 0.581937i \(0.802295\pi\)
\(978\) 0 0
\(979\) 48.9832i 1.56551i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −51.8584 −1.65403 −0.827013 0.562182i \(-0.809962\pi\)
−0.827013 + 0.562182i \(0.809962\pi\)
\(984\) 0 0
\(985\) −3.28171 −0.104564
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 3.35070i − 0.106546i
\(990\) 0 0
\(991\) − 51.5075i − 1.63619i −0.575083 0.818095i \(-0.695030\pi\)
0.575083 0.818095i \(-0.304970\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 19.2752 0.611066
\(996\) 0 0
\(997\) −19.9320 −0.631252 −0.315626 0.948884i \(-0.602214\pi\)
−0.315626 + 0.948884i \(0.602214\pi\)
\(998\) 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 2736.2.d.b.2015.14 yes 24
3.2 odd 2 inner 2736.2.d.b.2015.12 yes 24
4.3 odd 2 inner 2736.2.d.b.2015.13 yes 24
12.11 even 2 inner 2736.2.d.b.2015.11 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2736.2.d.b.2015.11 24 12.11 even 2 inner
2736.2.d.b.2015.12 yes 24 3.2 odd 2 inner
2736.2.d.b.2015.13 yes 24 4.3 odd 2 inner
2736.2.d.b.2015.14 yes 24 1.1 even 1 trivial