Properties

Label 2736.2.k.n.2431.1
Level $2736$
Weight $2$
Character 2736.2431
Analytic conductor $21.847$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Defining polynomial: \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 912)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2431.1
Root \(-1.18614 + 1.26217i\) of defining polynomial
Character \(\chi\) \(=\) 2736.2431
Dual form 2736.2.k.n.2431.2

$q$-expansion

\(f(q)\) \(=\) \(q-2.37228 q^{5} -2.52434i q^{7} +O(q^{10})\) \(q-2.37228 q^{5} -2.52434i q^{7} +2.52434i q^{11} -1.58457i q^{13} +0.372281 q^{17} +(-4.00000 - 1.73205i) q^{19} +1.87953i q^{23} +0.627719 q^{25} -3.16915i q^{29} +2.74456 q^{31} +5.98844i q^{35} -1.58457i q^{37} +6.92820i q^{41} +0.644810i q^{43} +0.939764i q^{47} +0.627719 q^{49} +10.0974i q^{53} -5.98844i q^{55} -4.00000 q^{59} -0.372281 q^{61} +3.75906i q^{65} -13.4891 q^{67} -4.00000 q^{71} +13.1168 q^{73} +6.37228 q^{77} +6.74456 q^{79} +3.46410i q^{83} -0.883156 q^{85} +13.2665i q^{89} -4.00000 q^{91} +(9.48913 + 4.10891i) q^{95} +13.2665i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{5} + O(q^{10}) \) \( 4q + 2q^{5} - 10q^{17} - 16q^{19} + 14q^{25} - 12q^{31} + 14q^{49} - 16q^{59} + 10q^{61} - 8q^{67} - 16q^{71} + 18q^{73} + 14q^{77} + 4q^{79} - 38q^{85} - 16q^{91} - 8q^{95} + O(q^{100}) \)

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
\(5\) −2.37228 −1.06092 −0.530458 0.847711i \(-0.677980\pi\)
−0.530458 + 0.847711i \(0.677980\pi\)
\(6\) 0 0
\(7\) 2.52434i 0.954110i −0.878873 0.477055i \(-0.841704\pi\)
0.878873 0.477055i \(-0.158296\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.52434i 0.761116i 0.924757 + 0.380558i \(0.124268\pi\)
−0.924757 + 0.380558i \(0.875732\pi\)
\(12\) 0 0
\(13\) 1.58457i 0.439482i −0.975558 0.219741i \(-0.929479\pi\)
0.975558 0.219741i \(-0.0705212\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.372281 0.0902915 0.0451457 0.998980i \(-0.485625\pi\)
0.0451457 + 0.998980i \(0.485625\pi\)
\(18\) 0 0
\(19\) −4.00000 1.73205i −0.917663 0.397360i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.87953i 0.391909i 0.980613 + 0.195954i \(0.0627804\pi\)
−0.980613 + 0.195954i \(0.937220\pi\)
\(24\) 0 0
\(25\) 0.627719 0.125544
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.16915i 0.588496i −0.955729 0.294248i \(-0.904931\pi\)
0.955729 0.294248i \(-0.0950692\pi\)
\(30\) 0 0
\(31\) 2.74456 0.492938 0.246469 0.969151i \(-0.420730\pi\)
0.246469 + 0.969151i \(0.420730\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 5.98844i 1.01223i
\(36\) 0 0
\(37\) 1.58457i 0.260502i −0.991481 0.130251i \(-0.958422\pi\)
0.991481 0.130251i \(-0.0415784\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.92820i 1.08200i 0.841021 + 0.541002i \(0.181955\pi\)
−0.841021 + 0.541002i \(0.818045\pi\)
\(42\) 0 0
\(43\) 0.644810i 0.0983326i 0.998791 + 0.0491663i \(0.0156564\pi\)
−0.998791 + 0.0491663i \(0.984344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.939764i 0.137079i 0.997648 + 0.0685393i \(0.0218339\pi\)
−0.997648 + 0.0685393i \(0.978166\pi\)
\(48\) 0 0
\(49\) 0.627719 0.0896741
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 10.0974i 1.38698i 0.720467 + 0.693489i \(0.243927\pi\)
−0.720467 + 0.693489i \(0.756073\pi\)
\(54\) 0 0
\(55\) 5.98844i 0.807481i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −0.372281 −0.0476657 −0.0238329 0.999716i \(-0.507587\pi\)
−0.0238329 + 0.999716i \(0.507587\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.75906i 0.466253i
\(66\) 0 0
\(67\) −13.4891 −1.64796 −0.823979 0.566620i \(-0.808251\pi\)
−0.823979 + 0.566620i \(0.808251\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 0 0
\(73\) 13.1168 1.53521 0.767605 0.640923i \(-0.221448\pi\)
0.767605 + 0.640923i \(0.221448\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.37228 0.726189
\(78\) 0 0
\(79\) 6.74456 0.758823 0.379411 0.925228i \(-0.376127\pi\)
0.379411 + 0.925228i \(0.376127\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.46410i 0.380235i 0.981761 + 0.190117i \(0.0608868\pi\)
−0.981761 + 0.190117i \(0.939113\pi\)
\(84\) 0 0
\(85\) −0.883156 −0.0957917
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 13.2665i 1.40625i 0.711068 + 0.703123i \(0.248212\pi\)
−0.711068 + 0.703123i \(0.751788\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 9.48913 + 4.10891i 0.973564 + 0.421565i
\(96\) 0 0
\(97\) 13.2665i 1.34701i 0.739183 + 0.673504i \(0.235212\pi\)
−0.739183 + 0.673504i \(0.764788\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 3.25544 0.323928 0.161964 0.986797i \(-0.448217\pi\)
0.161964 + 0.986797i \(0.448217\pi\)
\(102\) 0 0
\(103\) 1.25544 0.123702 0.0618510 0.998085i \(-0.480300\pi\)
0.0618510 + 0.998085i \(0.480300\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −13.4891 −1.30404 −0.652021 0.758200i \(-0.726079\pi\)
−0.652021 + 0.758200i \(0.726079\pi\)
\(108\) 0 0
\(109\) 11.6819i 1.11893i 0.828855 + 0.559463i \(0.188993\pi\)
−0.828855 + 0.559463i \(0.811007\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 17.0256i 1.60163i 0.598912 + 0.800815i \(0.295600\pi\)
−0.598912 + 0.800815i \(0.704400\pi\)
\(114\) 0 0
\(115\) 4.45877i 0.415782i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.939764i 0.0861480i
\(120\) 0 0
\(121\) 4.62772 0.420702
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 10.3723 0.927725
\(126\) 0 0
\(127\) −1.25544 −0.111402 −0.0557010 0.998447i \(-0.517739\pi\)
−0.0557010 + 0.998447i \(0.517739\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 7.57301i 0.661657i −0.943691 0.330829i \(-0.892672\pi\)
0.943691 0.330829i \(-0.107328\pi\)
\(132\) 0 0
\(133\) −4.37228 + 10.0974i −0.379125 + 0.875551i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −13.1168 −1.12065 −0.560324 0.828274i \(-0.689323\pi\)
−0.560324 + 0.828274i \(0.689323\pi\)
\(138\) 0 0
\(139\) 17.6704i 1.49878i 0.662128 + 0.749390i \(0.269653\pi\)
−0.662128 + 0.749390i \(0.730347\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) 7.51811i 0.624345i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 15.1168 1.23842 0.619210 0.785225i \(-0.287453\pi\)
0.619210 + 0.785225i \(0.287453\pi\)
\(150\) 0 0
\(151\) −2.74456 −0.223349 −0.111675 0.993745i \(-0.535621\pi\)
−0.111675 + 0.993745i \(0.535621\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −6.51087 −0.522966
\(156\) 0 0
\(157\) −15.4891 −1.23617 −0.618083 0.786113i \(-0.712091\pi\)
−0.618083 + 0.786113i \(0.712091\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 4.74456 0.373924
\(162\) 0 0
\(163\) 19.8997i 1.55867i 0.626608 + 0.779334i \(0.284443\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −5.48913 −0.424761 −0.212381 0.977187i \(-0.568122\pi\)
−0.212381 + 0.977187i \(0.568122\pi\)
\(168\) 0 0
\(169\) 10.4891 0.806856
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 6.33830i 0.481892i −0.970539 0.240946i \(-0.922542\pi\)
0.970539 0.240946i \(-0.0774576\pi\)
\(174\) 0 0
\(175\) 1.58457i 0.119783i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −21.4891 −1.60617 −0.803086 0.595863i \(-0.796810\pi\)
−0.803086 + 0.595863i \(0.796810\pi\)
\(180\) 0 0
\(181\) 25.5383i 1.89825i 0.314901 + 0.949125i \(0.398029\pi\)
−0.314901 + 0.949125i \(0.601971\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 3.75906i 0.276371i
\(186\) 0 0
\(187\) 0.939764i 0.0687223i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 26.1831i 1.89455i −0.320428 0.947273i \(-0.603827\pi\)
0.320428 0.947273i \(-0.396173\pi\)
\(192\) 0 0
\(193\) 13.8564i 0.997406i −0.866773 0.498703i \(-0.833810\pi\)
0.866773 0.498703i \(-0.166190\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 16.7446 1.19300 0.596500 0.802613i \(-0.296557\pi\)
0.596500 + 0.802613i \(0.296557\pi\)
\(198\) 0 0
\(199\) 18.2603i 1.29444i 0.762305 + 0.647218i \(0.224068\pi\)
−0.762305 + 0.647218i \(0.775932\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −8.00000 −0.561490
\(204\) 0 0
\(205\) 16.4356i 1.14792i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 4.37228 10.0974i 0.302437 0.698448i
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1.52967i 0.104323i
\(216\) 0 0
\(217\) 6.92820i 0.470317i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0.589907i 0.0396815i
\(222\) 0 0
\(223\) 10.7446 0.719509 0.359755 0.933047i \(-0.382860\pi\)
0.359755 + 0.933047i \(0.382860\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 0 0
\(229\) −6.88316 −0.454852 −0.227426 0.973795i \(-0.573031\pi\)
−0.227426 + 0.973795i \(0.573031\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −13.1168 −0.859313 −0.429657 0.902992i \(-0.641365\pi\)
−0.429657 + 0.902992i \(0.641365\pi\)
\(234\) 0 0
\(235\) 2.22938i 0.145429i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 6.57835i 0.425518i −0.977105 0.212759i \(-0.931755\pi\)
0.977105 0.212759i \(-0.0682449\pi\)
\(240\) 0 0
\(241\) 3.75906i 0.242142i 0.992644 + 0.121071i \(0.0386329\pi\)
−0.992644 + 0.121071i \(0.961367\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.48913 −0.0951367
\(246\) 0 0
\(247\) −2.74456 + 6.33830i −0.174632 + 0.403296i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 9.45254i 0.596639i 0.954466 + 0.298320i \(0.0964261\pi\)
−0.954466 + 0.298320i \(0.903574\pi\)
\(252\) 0 0
\(253\) −4.74456 −0.298288
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 17.0256i 1.06202i −0.847364 0.531012i \(-0.821812\pi\)
0.847364 0.531012i \(-0.178188\pi\)
\(258\) 0 0
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 14.7962i 0.912371i 0.889885 + 0.456185i \(0.150785\pi\)
−0.889885 + 0.456185i \(0.849215\pi\)
\(264\) 0 0
\(265\) 23.9538i 1.47147i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 23.9538i 1.46049i 0.683187 + 0.730243i \(0.260593\pi\)
−0.683187 + 0.730243i \(0.739407\pi\)
\(270\) 0 0
\(271\) 0.294954i 0.0179172i 0.999960 + 0.00895858i \(0.00285164\pi\)
−0.999960 + 0.00895858i \(0.997148\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.58457i 0.0955534i
\(276\) 0 0
\(277\) −8.37228 −0.503042 −0.251521 0.967852i \(-0.580931\pi\)
−0.251521 + 0.967852i \(0.580931\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 3.16915i 0.189056i −0.995522 0.0945278i \(-0.969866\pi\)
0.995522 0.0945278i \(-0.0301341\pi\)
\(282\) 0 0
\(283\) 29.0573i 1.72728i −0.504110 0.863640i \(-0.668179\pi\)
0.504110 0.863640i \(-0.331821\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 17.4891 1.03235
\(288\) 0 0
\(289\) −16.8614 −0.991847
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 13.8564i 0.809500i 0.914427 + 0.404750i \(0.132641\pi\)
−0.914427 + 0.404750i \(0.867359\pi\)
\(294\) 0 0
\(295\) 9.48913 0.552478
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 2.97825 0.172237
\(300\) 0 0
\(301\) 1.62772 0.0938201
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0.883156 0.0505694
\(306\) 0 0
\(307\) −5.48913 −0.313281 −0.156640 0.987656i \(-0.550066\pi\)
−0.156640 + 0.987656i \(0.550066\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 2.22938i 0.126417i −0.998000 0.0632084i \(-0.979867\pi\)
0.998000 0.0632084i \(-0.0201333\pi\)
\(312\) 0 0
\(313\) −3.48913 −0.197217 −0.0986085 0.995126i \(-0.531439\pi\)
−0.0986085 + 0.995126i \(0.531439\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 23.9538i 1.34538i −0.739926 0.672689i \(-0.765139\pi\)
0.739926 0.672689i \(-0.234861\pi\)
\(318\) 0 0
\(319\) 8.00000 0.447914
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −1.48913 0.644810i −0.0828571 0.0358782i
\(324\) 0 0
\(325\) 0.994667i 0.0551742i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 2.37228 0.130788
\(330\) 0 0
\(331\) 13.4891 0.741429 0.370715 0.928747i \(-0.379113\pi\)
0.370715 + 0.928747i \(0.379113\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 32.0000 1.74835
\(336\) 0 0
\(337\) 20.7846i 1.13221i −0.824333 0.566105i \(-0.808450\pi\)
0.824333 0.566105i \(-0.191550\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 6.92820i 0.375183i
\(342\) 0 0
\(343\) 19.2549i 1.03967i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 16.3807i 0.879364i 0.898153 + 0.439682i \(0.144909\pi\)
−0.898153 + 0.439682i \(0.855091\pi\)
\(348\) 0 0
\(349\) 14.6060 0.781840 0.390920 0.920425i \(-0.372157\pi\)
0.390920 + 0.920425i \(0.372157\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0.510875 0.0271911 0.0135956 0.999908i \(-0.495672\pi\)
0.0135956 + 0.999908i \(0.495672\pi\)
\(354\) 0 0
\(355\) 9.48913 0.503630
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 29.3523i 1.54915i −0.632479 0.774577i \(-0.717963\pi\)
0.632479 0.774577i \(-0.282037\pi\)
\(360\) 0 0
\(361\) 13.0000 + 13.8564i 0.684211 + 0.729285i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −31.1168 −1.62873
\(366\) 0 0
\(367\) 12.9715i 0.677109i 0.940947 + 0.338555i \(0.109938\pi\)
−0.940947 + 0.338555i \(0.890062\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 25.4891 1.32333
\(372\) 0 0
\(373\) 8.51278i 0.440775i 0.975412 + 0.220387i \(0.0707322\pi\)
−0.975412 + 0.220387i \(0.929268\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −5.02175 −0.258633
\(378\) 0 0
\(379\) 18.9783 0.974847 0.487424 0.873166i \(-0.337937\pi\)
0.487424 + 0.873166i \(0.337937\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −14.9783 −0.765353 −0.382676 0.923882i \(-0.624998\pi\)
−0.382676 + 0.923882i \(0.624998\pi\)
\(384\) 0 0
\(385\) −15.1168 −0.770426
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 11.1168 0.563646 0.281823 0.959466i \(-0.409061\pi\)
0.281823 + 0.959466i \(0.409061\pi\)
\(390\) 0 0
\(391\) 0.699713i 0.0353860i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −16.0000 −0.805047
\(396\) 0 0
\(397\) 29.1168 1.46133 0.730666 0.682735i \(-0.239210\pi\)
0.730666 + 0.682735i \(0.239210\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 10.6873i 0.533696i −0.963739 0.266848i \(-0.914018\pi\)
0.963739 0.266848i \(-0.0859822\pi\)
\(402\) 0 0
\(403\) 4.34896i 0.216637i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 4.00000 0.198273
\(408\) 0 0
\(409\) 6.92820i 0.342578i 0.985221 + 0.171289i \(0.0547931\pi\)
−0.985221 + 0.171289i \(0.945207\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 10.0974i 0.496858i
\(414\) 0 0
\(415\) 8.21782i 0.403397i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 13.5615i 0.662520i −0.943539 0.331260i \(-0.892526\pi\)
0.943539 0.331260i \(-0.107474\pi\)
\(420\) 0 0
\(421\) 26.1282i 1.27341i −0.771106 0.636706i \(-0.780296\pi\)
0.771106 0.636706i \(-0.219704\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0.233688 0.0113355
\(426\) 0 0
\(427\) 0.939764i 0.0454784i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 41.4891 1.99846 0.999230 0.0392245i \(-0.0124888\pi\)
0.999230 + 0.0392245i \(0.0124888\pi\)
\(432\) 0 0
\(433\) 23.3639i 1.12279i 0.827546 + 0.561397i \(0.189736\pi\)
−0.827546 + 0.561397i \(0.810264\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.25544 7.51811i 0.155729 0.359640i
\(438\) 0 0
\(439\) −17.7228 −0.845864 −0.422932 0.906161i \(-0.638999\pi\)
−0.422932 + 0.906161i \(0.638999\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 3.81396i 0.181207i −0.995887 0.0906033i \(-0.971120\pi\)
0.995887 0.0906033i \(-0.0288795\pi\)
\(444\) 0 0
\(445\) 31.4719i 1.49191i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 30.8820i 1.45741i −0.684828 0.728705i \(-0.740123\pi\)
0.684828 0.728705i \(-0.259877\pi\)
\(450\) 0 0
\(451\) −17.4891 −0.823531
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 9.48913 0.444857
\(456\) 0 0
\(457\) 9.11684 0.426468 0.213234 0.977001i \(-0.431600\pi\)
0.213234 + 0.977001i \(0.431600\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −38.3723 −1.78718 −0.893588 0.448889i \(-0.851820\pi\)
−0.893588 + 0.448889i \(0.851820\pi\)
\(462\) 0 0
\(463\) 29.6472i 1.37782i −0.724845 0.688912i \(-0.758089\pi\)
0.724845 0.688912i \(-0.241911\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.0318i 0.556764i 0.960470 + 0.278382i \(0.0897982\pi\)
−0.960470 + 0.278382i \(0.910202\pi\)
\(468\) 0 0
\(469\) 34.0511i 1.57233i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −1.62772 −0.0748426
\(474\) 0 0
\(475\) −2.51087 1.08724i −0.115207 0.0498860i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 11.3870i 0.520284i 0.965570 + 0.260142i \(0.0837694\pi\)
−0.965570 + 0.260142i \(0.916231\pi\)
\(480\) 0 0
\(481\) −2.51087 −0.114486
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 31.4719i 1.42906i
\(486\) 0 0
\(487\) −40.2337 −1.82316 −0.911581 0.411120i \(-0.865138\pi\)
−0.911581 + 0.411120i \(0.865138\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 34.3461i 1.55002i 0.631951 + 0.775008i \(0.282254\pi\)
−0.631951 + 0.775008i \(0.717746\pi\)
\(492\) 0 0
\(493\) 1.17981i 0.0531362i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 10.0974i 0.452928i
\(498\) 0 0
\(499\) 14.5012i 0.649164i 0.945858 + 0.324582i \(0.105224\pi\)
−0.945858 + 0.324582i \(0.894776\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 35.9306i 1.60207i 0.598619 + 0.801034i \(0.295716\pi\)
−0.598619 + 0.801034i \(0.704284\pi\)
\(504\) 0 0
\(505\) −7.72281 −0.343661
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 13.2665i 0.588027i −0.955801 0.294014i \(-0.905009\pi\)
0.955801 0.294014i \(-0.0949911\pi\)
\(510\) 0 0
\(511\) 33.1113i 1.46476i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −2.97825 −0.131237
\(516\) 0 0
\(517\) −2.37228 −0.104333
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 17.0256i 0.745903i 0.927851 + 0.372952i \(0.121654\pi\)
−0.927851 + 0.372952i \(0.878346\pi\)
\(522\) 0 0
\(523\) −32.0000 −1.39926 −0.699631 0.714504i \(-0.746652\pi\)
−0.699631 + 0.714504i \(0.746652\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.02175 0.0445081
\(528\) 0 0
\(529\) 19.4674 0.846408
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 10.9783 0.475521
\(534\) 0 0
\(535\) 32.0000 1.38348
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1.58457i 0.0682524i
\(540\) 0 0
\(541\) 13.1168 0.563937 0.281969 0.959424i \(-0.409013\pi\)
0.281969 + 0.959424i \(0.409013\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 27.7128i 1.18709i
\(546\) 0 0
\(547\) 6.51087 0.278385 0.139192 0.990265i \(-0.455549\pi\)
0.139192 + 0.990265i \(0.455549\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −5.48913 + 12.6766i −0.233845 + 0.540041i
\(552\) 0 0
\(553\) 17.0256i 0.724000i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −39.8614 −1.68898 −0.844491 0.535570i \(-0.820097\pi\)
−0.844491 + 0.535570i \(0.820097\pi\)
\(558\) 0 0
\(559\) 1.02175 0.0432154
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 22.9783 0.968418 0.484209 0.874952i \(-0.339107\pi\)
0.484209 + 0.874952i \(0.339107\pi\)
\(564\) 0 0
\(565\) 40.3894i 1.69920i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0.589907i 0.0247302i 0.999924 + 0.0123651i \(0.00393603\pi\)
−0.999924 + 0.0123651i \(0.996064\pi\)
\(570\) 0 0
\(571\) 23.6588i 0.990090i −0.868867 0.495045i \(-0.835152\pi\)
0.868867 0.495045i \(-0.164848\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.17981i 0.0492017i
\(576\) 0 0
\(577\) 2.60597 0.108488 0.0542440 0.998528i \(-0.482725\pi\)
0.0542440 + 0.998528i \(0.482725\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 8.74456 0.362786
\(582\) 0 0
\(583\) −25.4891 −1.05565
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 24.5986i 1.01529i −0.861566 0.507646i \(-0.830516\pi\)
0.861566 0.507646i \(-0.169484\pi\)
\(588\) 0 0
\(589\) −10.9783 4.75372i −0.452351 0.195874i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0.510875 0.0209791 0.0104896 0.999945i \(-0.496661\pi\)
0.0104896 + 0.999945i \(0.496661\pi\)
\(594\) 0 0
\(595\) 2.22938i 0.0913958i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −24.4674 −0.999710 −0.499855 0.866109i \(-0.666613\pi\)
−0.499855 + 0.866109i \(0.666613\pi\)
\(600\) 0 0
\(601\) 33.4612i 1.36491i 0.730927 + 0.682455i \(0.239088\pi\)
−0.730927 + 0.682455i \(0.760912\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −10.9783 −0.446329
\(606\) 0 0
\(607\) 29.2554 1.18744 0.593721 0.804671i \(-0.297658\pi\)
0.593721 + 0.804671i \(0.297658\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.48913 0.0602436
\(612\) 0 0
\(613\) −0.372281 −0.0150363 −0.00751815 0.999972i \(-0.502393\pi\)
−0.00751815 + 0.999972i \(0.502393\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −44.0951 −1.77520 −0.887601 0.460613i \(-0.847629\pi\)
−0.887601 + 0.460613i \(0.847629\pi\)
\(618\) 0 0
\(619\) 10.3923i 0.417702i 0.977947 + 0.208851i \(0.0669724\pi\)
−0.977947 + 0.208851i \(0.933028\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 33.4891 1.34171
\(624\) 0 0
\(625\) −27.7446 −1.10978
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0.589907i 0.0235211i
\(630\) 0 0
\(631\) 20.2496i 0.806124i 0.915173 + 0.403062i \(0.132054\pi\)
−0.915173 + 0.403062i \(0.867946\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 2.97825 0.118188
\(636\) 0 0
\(637\) 0.994667i 0.0394101i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 13.2665i 0.523995i 0.965069 + 0.261998i \(0.0843813\pi\)
−0.965069 + 0.261998i \(0.915619\pi\)
\(642\) 0 0
\(643\) 2.52434i 0.0995502i −0.998760 0.0497751i \(-0.984150\pi\)
0.998760 0.0497751i \(-0.0158505\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 13.5065i 0.530997i −0.964111 0.265499i \(-0.914463\pi\)
0.964111 0.265499i \(-0.0855366\pi\)
\(648\) 0 0
\(649\) 10.0974i 0.396356i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −4.88316 −0.191093 −0.0955463 0.995425i \(-0.530460\pi\)
−0.0955463 + 0.995425i \(0.530460\pi\)
\(654\) 0 0
\(655\) 17.9653i 0.701963i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −13.4891 −0.525462 −0.262731 0.964869i \(-0.584623\pi\)
−0.262731 + 0.964869i \(0.584623\pi\)
\(660\) 0 0
\(661\) 12.8617i 0.500264i −0.968212 0.250132i \(-0.919526\pi\)
0.968212 0.250132i \(-0.0804740\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 10.3723 23.9538i 0.402220 0.928887i
\(666\) 0 0
\(667\) 5.95650 0.230637
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0.939764i 0.0362792i
\(672\) 0 0
\(673\) 40.9793i 1.57964i −0.613341 0.789818i \(-0.710175\pi\)
0.613341 0.789818i \(-0.289825\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 47.9075i 1.84124i −0.390465 0.920618i \(-0.627686\pi\)
0.390465 0.920618i \(-0.372314\pi\)
\(678\) 0 0
\(679\) 33.4891 1.28519
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −38.9783 −1.49146 −0.745731 0.666248i \(-0.767899\pi\)
−0.745731 + 0.666248i \(0.767899\pi\)
\(684\) 0 0
\(685\) 31.1168 1.18891
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 16.0000 0.609551
\(690\) 0 0
\(691\) 1.93443i 0.0735892i −0.999323 0.0367946i \(-0.988285\pi\)
0.999323 0.0367946i \(-0.0117147\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 41.9191i 1.59008i
\(696\) 0 0
\(697\) 2.57924i 0.0976957i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 19.7228 0.744920 0.372460 0.928048i \(-0.378514\pi\)
0.372460 + 0.928048i \(0.378514\pi\)
\(702\) 0 0
\(703\) −2.74456 + 6.33830i −0.103513 + 0.239053i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 8.21782i 0.309063i
\(708\) 0 0
\(709\) −47.9565 −1.80104 −0.900522 0.434810i \(-0.856815\pi\)
−0.900522 + 0.434810i \(0.856815\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 5.15848i 0.193187i
\(714\) 0 0
\(715\) −9.48913 −0.354873
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 29.3523i 1.09466i −0.836918 0.547328i \(-0.815645\pi\)
0.836918 0.547328i \(-0.184355\pi\)
\(720\) 0 0
\(721\) 3.16915i 0.118025i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1.98933i 0.0738820i
\(726\) 0 0
\(727\) 8.16292i 0.302746i 0.988477 + 0.151373i \(0.0483695\pi\)
−0.988477 + 0.151373i \(0.951631\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0.240051i 0.00887860i
\(732\) 0 0
\(733\) 24.9783 0.922593 0.461296 0.887246i \(-0.347384\pi\)
0.461296 + 0.887246i \(0.347384\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 34.0511i 1.25429i
\(738\) 0 0
\(739\) 5.10358i 0.187738i −0.995585 0.0938691i \(-0.970076\pi\)
0.995585 0.0938691i \(-0.0299235\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 10.9783 0.402753 0.201376 0.979514i \(-0.435459\pi\)
0.201376 + 0.979514i \(0.435459\pi\)
\(744\) 0 0
\(745\) −35.8614 −1.31386
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 34.0511i 1.24420i
\(750\) 0 0
\(751\) 41.7228 1.52249 0.761244 0.648466i \(-0.224589\pi\)
0.761244 + 0.648466i \(0.224589\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 6.51087 0.236955
\(756\) 0 0
\(757\) −17.8614 −0.649184 −0.324592 0.945854i \(-0.605227\pi\)
−0.324592 + 0.945854i \(0.605227\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 21.8614 0.792475 0.396238 0.918148i \(-0.370316\pi\)
0.396238 + 0.918148i \(0.370316\pi\)
\(762\) 0 0
\(763\) 29.4891 1.06758
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 6.33830i 0.228863i
\(768\) 0 0
\(769\) −37.8614 −1.36532 −0.682659 0.730737i \(-0.739176\pi\)
−0.682659 + 0.730737i \(0.739176\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 39.7995i 1.43149i −0.698363 0.715744i \(-0.746088\pi\)
0.698363 0.715744i \(-0.253912\pi\)
\(774\) 0 0
\(775\) 1.72281 0.0618853
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 12.0000 27.7128i 0.429945 0.992915i
\(780\) 0 0
\(781\) 10.0974i 0.361312i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 36.7446 1.31147
\(786\) 0 0
\(787\) 34.9783 1.24684 0.623420 0.781887i \(-0.285743\pi\)
0.623420 + 0.781887i \(0.285743\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 42.9783 1.52813
\(792\) 0 0
\(793\) 0.589907i 0.0209482i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 3.16915i 0.112257i −0.998424 0.0561285i \(-0.982124\pi\)
0.998424 0.0561285i \(-0.0178756\pi\)
\(798\) 0 0
\(799\) 0.349857i 0.0123770i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 33.1113i 1.16847i
\(804\) 0 0
\(805\) −11.2554 −0.396702
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 14.8832