Properties

Label 5184.2.a.by.1.2
Level $5184$
Weight $2$
Character 5184.1
Self dual yes
Analytic conductor $41.394$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5184 = 2^{6} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5184.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(41.3944484078\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \(x^{2} - 6\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 288)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.44949\) of defining polynomial
Character \(\chi\) \(=\) 5184.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{5} +3.44949 q^{7} +O(q^{10})\) \(q+1.00000 q^{5} +3.44949 q^{7} -1.44949 q^{11} -5.89898 q^{13} +4.89898 q^{17} -4.00000 q^{19} -5.44949 q^{23} -4.00000 q^{25} -0.101021 q^{29} +2.55051 q^{31} +3.44949 q^{35} +0.898979 q^{37} -11.8990 q^{41} +2.34847 q^{43} -6.34847 q^{47} +4.89898 q^{49} -8.89898 q^{53} -1.44949 q^{55} -14.3485 q^{59} +7.89898 q^{61} -5.89898 q^{65} -12.3485 q^{67} +7.79796 q^{71} -4.89898 q^{73} -5.00000 q^{77} +13.4495 q^{79} +0.550510 q^{83} +4.89898 q^{85} -12.8990 q^{89} -20.3485 q^{91} -4.00000 q^{95} -3.89898 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} + 2 q^{7} + O(q^{10}) \) \( 2 q + 2 q^{5} + 2 q^{7} + 2 q^{11} - 2 q^{13} - 8 q^{19} - 6 q^{23} - 8 q^{25} - 10 q^{29} + 10 q^{31} + 2 q^{35} - 8 q^{37} - 14 q^{41} - 10 q^{43} + 2 q^{47} - 8 q^{53} + 2 q^{55} - 14 q^{59} + 6 q^{61} - 2 q^{65} - 10 q^{67} - 4 q^{71} - 10 q^{77} + 22 q^{79} + 6 q^{83} - 16 q^{89} - 26 q^{91} - 8 q^{95} + 2 q^{97} + O(q^{100}) \)

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
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0 0
\(7\) 3.44949 1.30378 0.651892 0.758312i \(-0.273975\pi\)
0.651892 + 0.758312i \(0.273975\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.44949 −0.437038 −0.218519 0.975833i \(-0.570122\pi\)
−0.218519 + 0.975833i \(0.570122\pi\)
\(12\) 0 0
\(13\) −5.89898 −1.63608 −0.818041 0.575160i \(-0.804940\pi\)
−0.818041 + 0.575160i \(0.804940\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.89898 1.18818 0.594089 0.804400i \(-0.297513\pi\)
0.594089 + 0.804400i \(0.297513\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −5.44949 −1.13630 −0.568149 0.822926i \(-0.692340\pi\)
−0.568149 + 0.822926i \(0.692340\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −0.101021 −0.0187590 −0.00937952 0.999956i \(-0.502986\pi\)
−0.00937952 + 0.999956i \(0.502986\pi\)
\(30\) 0 0
\(31\) 2.55051 0.458085 0.229043 0.973416i \(-0.426440\pi\)
0.229043 + 0.973416i \(0.426440\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.44949 0.583070
\(36\) 0 0
\(37\) 0.898979 0.147791 0.0738957 0.997266i \(-0.476457\pi\)
0.0738957 + 0.997266i \(0.476457\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −11.8990 −1.85831 −0.929154 0.369692i \(-0.879463\pi\)
−0.929154 + 0.369692i \(0.879463\pi\)
\(42\) 0 0
\(43\) 2.34847 0.358138 0.179069 0.983836i \(-0.442691\pi\)
0.179069 + 0.983836i \(0.442691\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.34847 −0.926019 −0.463010 0.886353i \(-0.653230\pi\)
−0.463010 + 0.886353i \(0.653230\pi\)
\(48\) 0 0
\(49\) 4.89898 0.699854
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −8.89898 −1.22237 −0.611184 0.791488i \(-0.709307\pi\)
−0.611184 + 0.791488i \(0.709307\pi\)
\(54\) 0 0
\(55\) −1.44949 −0.195449
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −14.3485 −1.86801 −0.934006 0.357258i \(-0.883712\pi\)
−0.934006 + 0.357258i \(0.883712\pi\)
\(60\) 0 0
\(61\) 7.89898 1.01136 0.505680 0.862721i \(-0.331242\pi\)
0.505680 + 0.862721i \(0.331242\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −5.89898 −0.731678
\(66\) 0 0
\(67\) −12.3485 −1.50861 −0.754303 0.656527i \(-0.772025\pi\)
−0.754303 + 0.656527i \(0.772025\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 7.79796 0.925447 0.462724 0.886503i \(-0.346872\pi\)
0.462724 + 0.886503i \(0.346872\pi\)
\(72\) 0 0
\(73\) −4.89898 −0.573382 −0.286691 0.958023i \(-0.592555\pi\)
−0.286691 + 0.958023i \(0.592555\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −5.00000 −0.569803
\(78\) 0 0
\(79\) 13.4495 1.51319 0.756593 0.653886i \(-0.226863\pi\)
0.756593 + 0.653886i \(0.226863\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0.550510 0.0604264 0.0302132 0.999543i \(-0.490381\pi\)
0.0302132 + 0.999543i \(0.490381\pi\)
\(84\) 0 0
\(85\) 4.89898 0.531369
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −12.8990 −1.36729 −0.683645 0.729815i \(-0.739606\pi\)
−0.683645 + 0.729815i \(0.739606\pi\)
\(90\) 0 0
\(91\) −20.3485 −2.13310
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.00000 −0.410391
\(96\) 0 0
\(97\) −3.89898 −0.395881 −0.197941 0.980214i \(-0.563425\pi\)
−0.197941 + 0.980214i \(0.563425\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −12.7980 −1.27344 −0.636722 0.771093i \(-0.719710\pi\)
−0.636722 + 0.771093i \(0.719710\pi\)
\(102\) 0 0
\(103\) 12.5505 1.23664 0.618319 0.785927i \(-0.287814\pi\)
0.618319 + 0.785927i \(0.287814\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 5.79796 0.560510 0.280255 0.959926i \(-0.409581\pi\)
0.280255 + 0.959926i \(0.409581\pi\)
\(108\) 0 0
\(109\) −0.898979 −0.0861066 −0.0430533 0.999073i \(-0.513709\pi\)
−0.0430533 + 0.999073i \(0.513709\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 8.79796 0.827642 0.413821 0.910358i \(-0.364194\pi\)
0.413821 + 0.910358i \(0.364194\pi\)
\(114\) 0 0
\(115\) −5.44949 −0.508168
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 16.8990 1.54913
\(120\) 0 0
\(121\) −8.89898 −0.808998
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 13.2474 1.15743 0.578717 0.815528i \(-0.303553\pi\)
0.578717 + 0.815528i \(0.303553\pi\)
\(132\) 0 0
\(133\) −13.7980 −1.19643
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −10.1010 −0.862988 −0.431494 0.902116i \(-0.642013\pi\)
−0.431494 + 0.902116i \(0.642013\pi\)
\(138\) 0 0
\(139\) 3.44949 0.292582 0.146291 0.989242i \(-0.453266\pi\)
0.146291 + 0.989242i \(0.453266\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 8.55051 0.715030
\(144\) 0 0
\(145\) −0.101021 −0.00838930
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −4.10102 −0.335969 −0.167984 0.985790i \(-0.553726\pi\)
−0.167984 + 0.985790i \(0.553726\pi\)
\(150\) 0 0
\(151\) 15.2474 1.24082 0.620410 0.784278i \(-0.286966\pi\)
0.620410 + 0.784278i \(0.286966\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2.55051 0.204862
\(156\) 0 0
\(157\) 10.7980 0.861771 0.430885 0.902407i \(-0.358201\pi\)
0.430885 + 0.902407i \(0.358201\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −18.7980 −1.48149
\(162\) 0 0
\(163\) 5.79796 0.454131 0.227066 0.973879i \(-0.427087\pi\)
0.227066 + 0.973879i \(0.427087\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 4.55051 0.352129 0.176065 0.984379i \(-0.443663\pi\)
0.176065 + 0.984379i \(0.443663\pi\)
\(168\) 0 0
\(169\) 21.7980 1.67677
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −3.00000 −0.228086 −0.114043 0.993476i \(-0.536380\pi\)
−0.114043 + 0.993476i \(0.536380\pi\)
\(174\) 0 0
\(175\) −13.7980 −1.04303
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −10.6969 −0.795097 −0.397549 0.917581i \(-0.630139\pi\)
−0.397549 + 0.917581i \(0.630139\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.898979 0.0660943
\(186\) 0 0
\(187\) −7.10102 −0.519278
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −12.5505 −0.908123 −0.454062 0.890970i \(-0.650025\pi\)
−0.454062 + 0.890970i \(0.650025\pi\)
\(192\) 0 0
\(193\) 4.10102 0.295198 0.147599 0.989047i \(-0.452846\pi\)
0.147599 + 0.989047i \(0.452846\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 17.5959 1.25366 0.626829 0.779157i \(-0.284353\pi\)
0.626829 + 0.779157i \(0.284353\pi\)
\(198\) 0 0
\(199\) −7.79796 −0.552783 −0.276391 0.961045i \(-0.589139\pi\)
−0.276391 + 0.961045i \(0.589139\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −0.348469 −0.0244577
\(204\) 0 0
\(205\) −11.8990 −0.831061
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 5.79796 0.401053
\(210\) 0 0
\(211\) −10.5505 −0.726327 −0.363164 0.931725i \(-0.618303\pi\)
−0.363164 + 0.931725i \(0.618303\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 2.34847 0.160164
\(216\) 0 0
\(217\) 8.79796 0.597244
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −28.8990 −1.94396
\(222\) 0 0
\(223\) −16.1464 −1.08124 −0.540622 0.841265i \(-0.681811\pi\)
−0.540622 + 0.841265i \(0.681811\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −7.65153 −0.507850 −0.253925 0.967224i \(-0.581722\pi\)
−0.253925 + 0.967224i \(0.581722\pi\)
\(228\) 0 0
\(229\) 1.00000 0.0660819 0.0330409 0.999454i \(-0.489481\pi\)
0.0330409 + 0.999454i \(0.489481\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) −6.34847 −0.414128
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 14.1464 0.915056 0.457528 0.889195i \(-0.348735\pi\)
0.457528 + 0.889195i \(0.348735\pi\)
\(240\) 0 0
\(241\) 7.00000 0.450910 0.225455 0.974254i \(-0.427613\pi\)
0.225455 + 0.974254i \(0.427613\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 4.89898 0.312984
\(246\) 0 0
\(247\) 23.5959 1.50137
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −21.7980 −1.37587 −0.687937 0.725770i \(-0.741484\pi\)
−0.687937 + 0.725770i \(0.741484\pi\)
\(252\) 0 0
\(253\) 7.89898 0.496605
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −10.7980 −0.673558 −0.336779 0.941584i \(-0.609338\pi\)
−0.336779 + 0.941584i \(0.609338\pi\)
\(258\) 0 0
\(259\) 3.10102 0.192688
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 5.44949 0.336030 0.168015 0.985784i \(-0.446264\pi\)
0.168015 + 0.985784i \(0.446264\pi\)
\(264\) 0 0
\(265\) −8.89898 −0.546660
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −7.10102 −0.432957 −0.216478 0.976287i \(-0.569457\pi\)
−0.216478 + 0.976287i \(0.569457\pi\)
\(270\) 0 0
\(271\) 29.3939 1.78555 0.892775 0.450502i \(-0.148755\pi\)
0.892775 + 0.450502i \(0.148755\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 5.79796 0.349630
\(276\) 0 0
\(277\) 14.7980 0.889123 0.444562 0.895748i \(-0.353359\pi\)
0.444562 + 0.895748i \(0.353359\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −10.1010 −0.602576 −0.301288 0.953533i \(-0.597417\pi\)
−0.301288 + 0.953533i \(0.597417\pi\)
\(282\) 0 0
\(283\) 3.44949 0.205051 0.102525 0.994730i \(-0.467308\pi\)
0.102525 + 0.994730i \(0.467308\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −41.0454 −2.42283
\(288\) 0 0
\(289\) 7.00000 0.411765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −0.797959 −0.0466173 −0.0233086 0.999728i \(-0.507420\pi\)
−0.0233086 + 0.999728i \(0.507420\pi\)
\(294\) 0 0
\(295\) −14.3485 −0.835400
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 32.1464 1.85908
\(300\) 0 0
\(301\) 8.10102 0.466935
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 7.89898 0.452294
\(306\) 0 0
\(307\) 21.7980 1.24408 0.622038 0.782987i \(-0.286305\pi\)
0.622038 + 0.782987i \(0.286305\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −15.2474 −0.864603 −0.432302 0.901729i \(-0.642298\pi\)
−0.432302 + 0.901729i \(0.642298\pi\)
\(312\) 0 0
\(313\) −17.0000 −0.960897 −0.480448 0.877023i \(-0.659526\pi\)
−0.480448 + 0.877023i \(0.659526\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −25.8990 −1.45463 −0.727316 0.686303i \(-0.759232\pi\)
−0.727316 + 0.686303i \(0.759232\pi\)
\(318\) 0 0
\(319\) 0.146428 0.00819841
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −19.5959 −1.09035
\(324\) 0 0
\(325\) 23.5959 1.30887
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −21.8990 −1.20733
\(330\) 0 0
\(331\) −11.2474 −0.618216 −0.309108 0.951027i \(-0.600030\pi\)
−0.309108 + 0.951027i \(0.600030\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −12.3485 −0.674669
\(336\) 0 0
\(337\) −10.7980 −0.588202 −0.294101 0.955774i \(-0.595020\pi\)
−0.294101 + 0.955774i \(0.595020\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −3.69694 −0.200200
\(342\) 0 0
\(343\) −7.24745 −0.391325
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −6.14643 −0.329958 −0.164979 0.986297i \(-0.552756\pi\)
−0.164979 + 0.986297i \(0.552756\pi\)
\(348\) 0 0
\(349\) 14.7980 0.792117 0.396058 0.918225i \(-0.370378\pi\)
0.396058 + 0.918225i \(0.370378\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −13.6969 −0.729014 −0.364507 0.931201i \(-0.618763\pi\)
−0.364507 + 0.931201i \(0.618763\pi\)
\(354\) 0 0
\(355\) 7.79796 0.413873
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1.79796 −0.0948926 −0.0474463 0.998874i \(-0.515108\pi\)
−0.0474463 + 0.998874i \(0.515108\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −4.89898 −0.256424
\(366\) 0 0
\(367\) −4.34847 −0.226988 −0.113494 0.993539i \(-0.536204\pi\)
−0.113494 + 0.993539i \(0.536204\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −30.6969 −1.59371
\(372\) 0 0
\(373\) −31.6969 −1.64121 −0.820603 0.571499i \(-0.806362\pi\)
−0.820603 + 0.571499i \(0.806362\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0.595918 0.0306913
\(378\) 0 0
\(379\) 26.0000 1.33553 0.667765 0.744372i \(-0.267251\pi\)
0.667765 + 0.744372i \(0.267251\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 28.5505 1.45886 0.729431 0.684054i \(-0.239785\pi\)
0.729431 + 0.684054i \(0.239785\pi\)
\(384\) 0 0
\(385\) −5.00000 −0.254824
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 6.79796 0.344670 0.172335 0.985038i \(-0.444869\pi\)
0.172335 + 0.985038i \(0.444869\pi\)
\(390\) 0 0
\(391\) −26.6969 −1.35012
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 13.4495 0.676717
\(396\) 0 0
\(397\) 10.6969 0.536864 0.268432 0.963299i \(-0.413495\pi\)
0.268432 + 0.963299i \(0.413495\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −13.6969 −0.683992 −0.341996 0.939701i \(-0.611103\pi\)
−0.341996 + 0.939701i \(0.611103\pi\)
\(402\) 0 0
\(403\) −15.0454 −0.749465
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1.30306 −0.0645904
\(408\) 0 0
\(409\) 26.5959 1.31508 0.657542 0.753418i \(-0.271596\pi\)
0.657542 + 0.753418i \(0.271596\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −49.4949 −2.43548
\(414\) 0 0
\(415\) 0.550510 0.0270235
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 31.2474 1.52654 0.763269 0.646081i \(-0.223593\pi\)
0.763269 + 0.646081i \(0.223593\pi\)
\(420\) 0 0
\(421\) −0.101021 −0.00492344 −0.00246172 0.999997i \(-0.500784\pi\)
−0.00246172 + 0.999997i \(0.500784\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −19.5959 −0.950542
\(426\) 0 0
\(427\) 27.2474 1.31860
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 14.2020 0.684088 0.342044 0.939684i \(-0.388881\pi\)
0.342044 + 0.939684i \(0.388881\pi\)
\(432\) 0 0
\(433\) −8.49490 −0.408239 −0.204119 0.978946i \(-0.565433\pi\)
−0.204119 + 0.978946i \(0.565433\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 21.7980 1.04274
\(438\) 0 0
\(439\) −36.3485 −1.73482 −0.867409 0.497596i \(-0.834216\pi\)
−0.867409 + 0.497596i \(0.834216\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −23.4495 −1.11412 −0.557059 0.830473i \(-0.688070\pi\)
−0.557059 + 0.830473i \(0.688070\pi\)
\(444\) 0 0
\(445\) −12.8990 −0.611470
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 11.1010 0.523890 0.261945 0.965083i \(-0.415636\pi\)
0.261945 + 0.965083i \(0.415636\pi\)
\(450\) 0 0
\(451\) 17.2474 0.812151
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −20.3485 −0.953951
\(456\) 0 0
\(457\) −31.4949 −1.47327 −0.736635 0.676291i \(-0.763586\pi\)
−0.736635 + 0.676291i \(0.763586\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −35.6969 −1.66257 −0.831286 0.555845i \(-0.812395\pi\)
−0.831286 + 0.555845i \(0.812395\pi\)
\(462\) 0 0
\(463\) −31.2474 −1.45219 −0.726096 0.687593i \(-0.758667\pi\)
−0.726096 + 0.687593i \(0.758667\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −39.5959 −1.83228 −0.916140 0.400858i \(-0.868712\pi\)
−0.916140 + 0.400858i \(0.868712\pi\)
\(468\) 0 0
\(469\) −42.5959 −1.96690
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −3.40408 −0.156520
\(474\) 0 0
\(475\) 16.0000 0.734130
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 25.0454 1.14435 0.572177 0.820130i \(-0.306099\pi\)
0.572177 + 0.820130i \(0.306099\pi\)
\(480\) 0 0
\(481\) −5.30306 −0.241799
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −3.89898 −0.177044
\(486\) 0 0
\(487\) −17.7980 −0.806503 −0.403251 0.915089i \(-0.632120\pi\)
−0.403251 + 0.915089i \(0.632120\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −10.7526 −0.485256 −0.242628 0.970119i \(-0.578009\pi\)
−0.242628 + 0.970119i \(0.578009\pi\)
\(492\) 0 0
\(493\) −0.494897 −0.0222891
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 26.8990 1.20658
\(498\) 0 0
\(499\) 1.65153 0.0739327 0.0369663 0.999317i \(-0.488231\pi\)
0.0369663 + 0.999317i \(0.488231\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −39.7980 −1.77450 −0.887252 0.461286i \(-0.847388\pi\)
−0.887252 + 0.461286i \(0.847388\pi\)
\(504\) 0 0
\(505\) −12.7980 −0.569502
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 15.4949 0.686799 0.343400 0.939189i \(-0.388421\pi\)
0.343400 + 0.939189i \(0.388421\pi\)
\(510\) 0 0
\(511\) −16.8990 −0.747567
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 12.5505 0.553042
\(516\) 0 0
\(517\) 9.20204 0.404705
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) −13.5959 −0.594508 −0.297254 0.954798i \(-0.596071\pi\)
−0.297254 + 0.954798i \(0.596071\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 12.4949 0.544286
\(528\) 0 0
\(529\) 6.69694 0.291171
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 70.1918 3.04035
\(534\) 0 0
\(535\) 5.79796 0.250668
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −7.10102 −0.305863
\(540\) 0 0
\(541\) −29.5959 −1.27243 −0.636214 0.771513i \(-0.719500\pi\)
−0.636214 + 0.771513i \(0.719500\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −0.898979 −0.0385081
\(546\) 0 0
\(547\) 6.55051 0.280080 0.140040 0.990146i \(-0.455277\pi\)
0.140040 + 0.990146i \(0.455277\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0.404082 0.0172145
\(552\) 0 0
\(553\) 46.3939 1.97287
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 18.6969 0.792215 0.396107 0.918204i \(-0.370361\pi\)
0.396107 + 0.918204i \(0.370361\pi\)
\(558\) 0 0
\(559\) −13.8536 −0.585944
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −4.34847 −0.183266 −0.0916331 0.995793i \(-0.529209\pi\)
−0.0916331 + 0.995793i \(0.529209\pi\)
\(564\) 0 0
\(565\) 8.79796 0.370133
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −1.00000 −0.0419222 −0.0209611 0.999780i \(-0.506673\pi\)
−0.0209611 + 0.999780i \(0.506673\pi\)
\(570\) 0 0
\(571\) 3.65153 0.152812 0.0764059 0.997077i \(-0.475656\pi\)
0.0764059 + 0.997077i \(0.475656\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 21.7980 0.909038
\(576\) 0 0
\(577\) −19.1010 −0.795186 −0.397593 0.917562i \(-0.630154\pi\)
−0.397593 + 0.917562i \(0.630154\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1.89898 0.0787829
\(582\) 0 0
\(583\) 12.8990 0.534221
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 29.9444 1.23594 0.617969 0.786203i \(-0.287956\pi\)
0.617969 + 0.786203i \(0.287956\pi\)
\(588\) 0 0
\(589\) −10.2020 −0.420368
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 28.8990 1.18674 0.593369 0.804930i \(-0.297797\pi\)
0.593369 + 0.804930i \(0.297797\pi\)
\(594\) 0 0
\(595\) 16.8990 0.692791
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 28.5505 1.16654 0.583271 0.812278i \(-0.301773\pi\)
0.583271 + 0.812278i \(0.301773\pi\)
\(600\) 0 0
\(601\) 10.3031 0.420271 0.210135 0.977672i \(-0.432610\pi\)
0.210135 + 0.977672i \(0.432610\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −8.89898 −0.361795
\(606\) 0 0
\(607\) 25.9444 1.05305 0.526525 0.850160i \(-0.323495\pi\)
0.526525 + 0.850160i \(0.323495\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 37.4495 1.51504
\(612\) 0 0
\(613\) 26.6969 1.07828 0.539140 0.842216i \(-0.318750\pi\)
0.539140 + 0.842216i \(0.318750\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 23.6969 0.954003 0.477001 0.878903i \(-0.341724\pi\)
0.477001 + 0.878903i \(0.341724\pi\)
\(618\) 0 0
\(619\) 26.1464 1.05091 0.525457 0.850820i \(-0.323894\pi\)
0.525457 + 0.850820i \(0.323894\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −44.4949 −1.78265
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 4.40408 0.175602
\(630\) 0 0
\(631\) −6.20204 −0.246899 −0.123450 0.992351i \(-0.539396\pi\)
−0.123450 + 0.992351i \(0.539396\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 8.00000 0.317470
\(636\) 0 0
\(637\) −28.8990 −1.14502
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 26.5959 1.05048 0.525238 0.850955i \(-0.323976\pi\)
0.525238 + 0.850955i \(0.323976\pi\)
\(642\) 0 0
\(643\) −4.34847 −0.171487 −0.0857434 0.996317i \(-0.527327\pi\)
−0.0857434 + 0.996317i \(0.527327\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 9.79796 0.385198 0.192599 0.981278i \(-0.438308\pi\)
0.192599 + 0.981278i \(0.438308\pi\)
\(648\) 0 0
\(649\) 20.7980 0.816391
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 17.0000 0.665261 0.332631 0.943057i \(-0.392064\pi\)
0.332631 + 0.943057i \(0.392064\pi\)
\(654\) 0 0
\(655\) 13.2474 0.517621
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −7.65153 −0.298061 −0.149031 0.988833i \(-0.547615\pi\)
−0.149031 + 0.988833i \(0.547615\pi\)
\(660\) 0 0
\(661\) 30.3939 1.18218 0.591092 0.806604i \(-0.298697\pi\)
0.591092 + 0.806604i \(0.298697\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −13.7980 −0.535062
\(666\) 0 0
\(667\) 0.550510 0.0213158
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −11.4495 −0.442003
\(672\) 0 0
\(673\) 27.2929 1.05206 0.526031 0.850465i \(-0.323680\pi\)
0.526031 + 0.850465i \(0.323680\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −24.3939 −0.937533 −0.468766 0.883322i \(-0.655301\pi\)
−0.468766 + 0.883322i \(0.655301\pi\)
\(678\) 0 0
\(679\) −13.4495 −0.516144
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 17.3939 0.665558 0.332779 0.943005i \(-0.392014\pi\)
0.332779 + 0.943005i \(0.392014\pi\)
\(684\) 0 0
\(685\) −10.1010 −0.385940
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 52.4949 1.99990
\(690\) 0 0
\(691\) 37.9444 1.44347 0.721736 0.692168i \(-0.243344\pi\)
0.721736 + 0.692168i \(0.243344\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 3.44949 0.130847
\(696\) 0 0
\(697\) −58.2929 −2.20800
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 12.4949 0.471926 0.235963 0.971762i \(-0.424176\pi\)
0.235963 + 0.971762i \(0.424176\pi\)
\(702\) 0 0
\(703\) −3.59592 −0.135623
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −44.1464 −1.66030
\(708\) 0 0
\(709\) 5.00000 0.187779 0.0938895 0.995583i \(-0.470070\pi\)
0.0938895 + 0.995583i \(0.470070\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −13.8990 −0.520521
\(714\) 0 0
\(715\) 8.55051 0.319771
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 31.1918 1.16326 0.581630 0.813454i \(-0.302415\pi\)
0.581630 + 0.813454i \(0.302415\pi\)
\(720\) 0 0
\(721\) 43.2929 1.61231
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0.404082 0.0150072
\(726\) 0 0
\(727\) −18.5505 −0.688000 −0.344000 0.938970i \(-0.611782\pi\)
−0.344000 + 0.938970i \(0.611782\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 11.5051 0.425532
\(732\) 0 0
\(733\) 11.8990 0.439499 0.219749 0.975556i \(-0.429476\pi\)
0.219749 + 0.975556i \(0.429476\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 17.8990 0.659317
\(738\) 0 0
\(739\) 14.0000 0.514998 0.257499 0.966279i \(-0.417102\pi\)
0.257499 + 0.966279i \(0.417102\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −35.4495 −1.30052 −0.650258 0.759714i \(-0.725339\pi\)
−0.650258 + 0.759714i \(0.725339\pi\)
\(744\) 0 0
\(745\) −4.10102 −0.150250
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 20.0000 0.730784
\(750\) 0 0
\(751\) −42.6413 −1.55600 −0.778002 0.628262i \(-0.783767\pi\)
−0.778002 + 0.628262i \(0.783767\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 15.2474 0.554911
\(756\) 0 0
\(757\) 42.0000 1.52652 0.763258 0.646094i \(-0.223599\pi\)
0.763258 + 0.646094i \(0.223599\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −25.2929 −0.916865 −0.458433 0.888729i \(-0.651589\pi\)
−0.458433 + 0.888729i \(0.651589\pi\)
\(762\) 0 0
\(763\) −3.10102 −0.112264
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 84.6413 3.05622
\(768\) 0 0
\(769\) −36.5959 −1.31968 −0.659841 0.751405i \(-0.729376\pi\)
−0.659841 + 0.751405i \(0.729376\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 36.4949 1.31263 0.656315 0.754487i \(-0.272114\pi\)
0.656315 + 0.754487i \(0.272114\pi\)
\(774\) 0 0
\(775\) −10.2020 −0.366468
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 47.5959 1.70530
\(780\) 0 0
\(781\) −11.3031 −0.404455
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 10.7980 0.385396
\(786\) 0 0
\(787\) 21.2474 0.757390 0.378695 0.925522i \(-0.376373\pi\)
0.378695 + 0.925522i \(0.376373\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 30.3485 1.07907
\(792\) 0 0
\(793\) −46.5959 −1.65467
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −47.6969 −1.68951 −0.844756 0.535151i \(-0.820255\pi\)
−0.844756 + 0.535151i \(0.820255\pi\)
\(798\) 0 0
\(799\) −31.1010 −1.10028
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 7.10102 0.250590
\(804\) 0 0
\(805\) −18.7980 −0.662541
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −32.4949 −1.14246 −0.571230 0.820790i \(-0.693534\pi\)
−0.571230 + 0.820790i \(0.693534\pi\)
\(810\) 0 0
\(811\) −8.40408 −0.295107 −0.147554 0.989054i \(-0.547140\pi\)
−0.147554 + 0.989054i \(0.547140\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 5.79796 0.203094
\(816\) 0 0
\(817\) −9.39388 −0.328650
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −15.2929 −0.533724 −0.266862 0.963735i \(-0.585987\pi\)
−0.266862 + 0.963735i \(0.585987\pi\)
\(822\) 0 0
\(823\) 10.5505 0.367768 0.183884 0.982948i \(-0.441133\pi\)
0.183884 + 0.982948i \(0.441133\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 28.0000 0.973655 0.486828 0.873498i \(-0.338154\pi\)
0.486828 + 0.873498i \(0.338154\pi\)
\(828\) 0 0
\(829\) 23.1010 0.802332 0.401166 0.916005i \(-0.368605\pi\)
0.401166 + 0.916005i \(0.368605\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 24.0000 0.831551
\(834\) 0 0
\(835\) 4.55051 0.157477
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −30.3485 −1.04775 −0.523873 0.851797i \(-0.675513\pi\)
−0.523873 + 0.851797i \(0.675513\pi\)
\(840\) 0 0
\(841\) −28.9898 −0.999648
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 21.7980 0.749873
\(846\) 0 0
\(847\) −30.6969 −1.05476
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −4.89898 −0.167935
\(852\) 0 0
\(853\) 7.89898 0.270456 0.135228 0.990815i \(-0.456823\pi\)
0.135228 + 0.990815i \(0.456823\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −27.8990 −0.953011 −0.476505 0.879171i \(-0.658097\pi\)
−0.476505 + 0.879171i \(0.658097\pi\)
\(858\) 0 0
\(859\) 21.4495 0.731847 0.365924 0.930645i \(-0.380753\pi\)
0.365924 + 0.930645i \(0.380753\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −17.7980 −0.605850 −0.302925 0.953014i \(-0.597963\pi\)
−0.302925 + 0.953014i \(0.597963\pi\)
\(864\) 0 0
\(865\) −3.00000 −0.102003
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −19.4949 −0.661319
\(870\) 0 0
\(871\) 72.8434 2.46820
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −31.0454 −1.04953
\(876\) 0 0
\(877\) 27.8990 0.942082 0.471041 0.882111i \(-0.343878\pi\)
0.471041 + 0.882111i \(0.343878\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −24.4949 −0.825254 −0.412627 0.910900i \(-0.635389\pi\)
−0.412627 + 0.910900i \(0.635389\pi\)
\(882\) 0 0
\(883\) −39.5959 −1.33251 −0.666254 0.745725i \(-0.732103\pi\)
−0.666254 + 0.745725i \(0.732103\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −34.8434 −1.16993 −0.584963 0.811060i \(-0.698891\pi\)
−0.584963 + 0.811060i \(0.698891\pi\)
\(888\) 0 0
\(889\) 27.5959 0.925537
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 25.3939 0.849774
\(894\) 0 0
\(895\) 12.0000 0.401116
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −0.257654 −0.00859324
\(900\) 0 0
\(901\) −43.5959 −1.45239
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −10.6969 −0.355578
\(906\) 0 0
\(907\) −25.2474 −0.838328 −0.419164 0.907911i \(-0.637677\pi\)
−0.419164 + 0.907911i \(0.637677\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −43.5403 −1.44255 −0.721277 0.692646i \(-0.756445\pi\)
−0.721277 + 0.692646i \(0.756445\pi\)
\(912\) 0 0
\(913\) −0.797959 −0.0264086
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 45.6969 1.50905
\(918\) 0 0
\(919\) −9.79796 −0.323205 −0.161602 0.986856i \(-0.551666\pi\)
−0.161602 + 0.986856i \(0.551666\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −46.0000 −1.51411
\(924\) 0 0
\(925\) −3.59592 −0.118233
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 47.6969 1.56489 0.782443 0.622722i \(-0.213973\pi\)
0.782443 + 0.622722i \(0.213973\pi\)
\(930\) 0 0
\(931\) −19.5959 −0.642230
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −7.10102 −0.232228
\(936\) 0 0
\(937\) 56.4949 1.84561 0.922804 0.385270i \(-0.125892\pi\)
0.922804 + 0.385270i \(0.125892\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −7.00000 −0.228193 −0.114097 0.993470i \(-0.536397\pi\)
−0.114097 + 0.993470i \(0.536397\pi\)
\(942\) 0 0
\(943\) 64.8434 2.11159
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 20.7526 0.674367 0.337184 0.941439i \(-0.390526\pi\)
0.337184 + 0.941439i \(0.390526\pi\)
\(948\) 0 0
\(949\) 28.8990 0.938101
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 43.1010 1.39618 0.698090 0.716011i \(-0.254034\pi\)
0.698090 + 0.716011i \(0.254034\pi\)
\(954\) 0 0
\(955\) −12.5505 −0.406125
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −34.8434 −1.12515
\(960\) 0 0
\(961\) −24.4949 −0.790158
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 4.10102 0.132017
\(966\) 0 0
\(967\) −15.2474 −0.490325 −0.245162 0.969482i \(-0.578841\pi\)
−0.245162 + 0.969482i \(0.578841\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 10.0000 0.320915 0.160458 0.987043i \(-0.448703\pi\)
0.160458 + 0.987043i \(0.448703\pi\)
\(972\) 0 0
\(973\) 11.8990 0.381464
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 13.2020 0.422371 0.211185 0.977446i \(-0.432268\pi\)
0.211185 + 0.977446i \(0.432268\pi\)
\(978\) 0 0
\(979\) 18.6969 0.597557
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 19.6515 0.626786 0.313393 0.949623i \(-0.398534\pi\)
0.313393 + 0.949623i \(0.398534\pi\)
\(984\) 0 0
\(985\) 17.5959 0.560653
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −12.7980 −0.406951
\(990\) 0 0
\(991\) −21.3939 −0.679599 −0.339799 0.940498i \(-0.610359\pi\)
−0.339799 + 0.940498i \(0.610359\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −7.79796 −0.247212
\(996\) 0 0
\(997\) −32.3939 −1.02592 −0.512962 0.858411i \(-0.671452\pi\)
−0.512962 + 0.858411i \(0.671452\pi\)
\(998\) 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 5184.2.a.by.1.2 2
3.2 odd 2 5184.2.a.bn.1.2 2
4.3 odd 2 5184.2.a.bu.1.1 2
8.3 odd 2 2592.2.a.j.1.1 2
8.5 even 2 2592.2.a.n.1.2 2
9.2 odd 6 1728.2.i.k.577.1 4
9.4 even 3 576.2.i.i.385.1 4
9.5 odd 6 1728.2.i.k.1153.1 4
9.7 even 3 576.2.i.i.193.2 4
12.11 even 2 5184.2.a.bj.1.1 2
24.5 odd 2 2592.2.a.s.1.2 2
24.11 even 2 2592.2.a.o.1.1 2
36.7 odd 6 576.2.i.m.193.1 4
36.11 even 6 1728.2.i.m.577.2 4
36.23 even 6 1728.2.i.m.1153.2 4
36.31 odd 6 576.2.i.m.385.2 4
72.5 odd 6 864.2.i.c.289.1 4
72.11 even 6 864.2.i.e.577.2 4
72.13 even 6 288.2.i.e.97.2 yes 4
72.29 odd 6 864.2.i.c.577.1 4
72.43 odd 6 288.2.i.c.193.2 yes 4
72.59 even 6 864.2.i.e.289.2 4
72.61 even 6 288.2.i.e.193.1 yes 4
72.67 odd 6 288.2.i.c.97.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
288.2.i.c.97.1 4 72.67 odd 6
288.2.i.c.193.2 yes 4 72.43 odd 6
288.2.i.e.97.2 yes 4 72.13 even 6
288.2.i.e.193.1 yes 4 72.61 even 6
576.2.i.i.193.2 4 9.7 even 3
576.2.i.i.385.1 4 9.4 even 3
576.2.i.m.193.1 4 36.7 odd 6
576.2.i.m.385.2 4 36.31 odd 6
864.2.i.c.289.1 4 72.5 odd 6
864.2.i.c.577.1 4 72.29 odd 6
864.2.i.e.289.2 4 72.59 even 6
864.2.i.e.577.2 4 72.11 even 6
1728.2.i.k.577.1 4 9.2 odd 6
1728.2.i.k.1153.1 4 9.5 odd 6
1728.2.i.m.577.2 4 36.11 even 6
1728.2.i.m.1153.2 4 36.23 even 6
2592.2.a.j.1.1 2 8.3 odd 2
2592.2.a.n.1.2 2 8.5 even 2
2592.2.a.o.1.1 2 24.11 even 2
2592.2.a.s.1.2 2 24.5 odd 2
5184.2.a.bj.1.1 2 12.11 even 2
5184.2.a.bn.1.2 2 3.2 odd 2
5184.2.a.bu.1.1 2 4.3 odd 2
5184.2.a.by.1.2 2 1.1 even 1 trivial