Properties

Label 2736.2.k.n.2431.3
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 2431.3
Root \(1.68614 + 0.396143i\) of defining polynomial
Character \(\chi\) \(=\) 2736.2431
Dual form 2736.2.k.n.2431.4

$q$-expansion

\(f(q)\) \(=\) \(q+3.37228 q^{5} -0.792287i q^{7} +O(q^{10})\) \(q+3.37228 q^{5} -0.792287i q^{7} +0.792287i q^{11} -5.04868i q^{13} -5.37228 q^{17} +(-4.00000 + 1.73205i) q^{19} -8.51278i q^{23} +6.37228 q^{25} -10.0974i q^{29} -8.74456 q^{31} -2.67181i q^{35} -5.04868i q^{37} -6.92820i q^{41} +9.30506i q^{43} -4.25639i q^{47} +6.37228 q^{49} +3.16915i q^{53} +2.67181i q^{55} -4.00000 q^{59} +5.37228 q^{61} -17.0256i q^{65} +9.48913 q^{67} -4.00000 q^{71} -4.11684 q^{73} +0.627719 q^{77} -4.74456 q^{79} -3.46410i q^{83} -18.1168 q^{85} +13.2665i q^{89} -4.00000 q^{91} +(-13.4891 + 5.84096i) q^{95} +13.2665i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{5} + O(q^{10}) \) \( 4q + 2q^{5} - 10q^{17} - 16q^{19} + 14q^{25} - 12q^{31} + 14q^{49} - 16q^{59} + 10q^{61} - 8q^{67} - 16q^{71} + 18q^{73} + 14q^{77} + 4q^{79} - 38q^{85} - 16q^{91} - 8q^{95} + O(q^{100}) \)

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
\(5\) 3.37228 1.50813 0.754065 0.656800i \(-0.228090\pi\)
0.754065 + 0.656800i \(0.228090\pi\)
\(6\) 0 0
\(7\) 0.792287i 0.299456i −0.988727 0.149728i \(-0.952160\pi\)
0.988727 0.149728i \(-0.0478399\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.792287i 0.238884i 0.992841 + 0.119442i \(0.0381105\pi\)
−0.992841 + 0.119442i \(0.961890\pi\)
\(12\) 0 0
\(13\) 5.04868i 1.40025i −0.714020 0.700125i \(-0.753127\pi\)
0.714020 0.700125i \(-0.246873\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.37228 −1.30297 −0.651485 0.758662i \(-0.725854\pi\)
−0.651485 + 0.758662i \(0.725854\pi\)
\(18\) 0 0
\(19\) −4.00000 + 1.73205i −0.917663 + 0.397360i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.51278i 1.77504i −0.460772 0.887518i \(-0.652428\pi\)
0.460772 0.887518i \(-0.347572\pi\)
\(24\) 0 0
\(25\) 6.37228 1.27446
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 10.0974i 1.87503i −0.347943 0.937516i \(-0.613120\pi\)
0.347943 0.937516i \(-0.386880\pi\)
\(30\) 0 0
\(31\) −8.74456 −1.57057 −0.785285 0.619135i \(-0.787484\pi\)
−0.785285 + 0.619135i \(0.787484\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.67181i 0.451619i
\(36\) 0 0
\(37\) 5.04868i 0.829997i −0.909822 0.414999i \(-0.863782\pi\)
0.909822 0.414999i \(-0.136218\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.92820i 1.08200i −0.841021 0.541002i \(-0.818045\pi\)
0.841021 0.541002i \(-0.181955\pi\)
\(42\) 0 0
\(43\) 9.30506i 1.41901i 0.704701 + 0.709504i \(0.251081\pi\)
−0.704701 + 0.709504i \(0.748919\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.25639i 0.620858i −0.950597 0.310429i \(-0.899527\pi\)
0.950597 0.310429i \(-0.100473\pi\)
\(48\) 0 0
\(49\) 6.37228 0.910326
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.16915i 0.435316i 0.976025 + 0.217658i \(0.0698417\pi\)
−0.976025 + 0.217658i \(0.930158\pi\)
\(54\) 0 0
\(55\) 2.67181i 0.360267i
\(56\) 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
\(61\) 5.37228 0.687850 0.343925 0.938997i \(-0.388243\pi\)
0.343925 + 0.938997i \(0.388243\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 17.0256i 2.11176i
\(66\) 0 0
\(67\) 9.48913 1.15928 0.579641 0.814872i \(-0.303193\pi\)
0.579641 + 0.814872i \(0.303193\pi\)
\(68\) 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
\(73\) −4.11684 −0.481840 −0.240920 0.970545i \(-0.577449\pi\)
−0.240920 + 0.970545i \(0.577449\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.627719 0.0715352
\(78\) 0 0
\(79\) −4.74456 −0.533805 −0.266903 0.963724i \(-0.586000\pi\)
−0.266903 + 0.963724i \(0.586000\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.46410i 0.380235i −0.981761 0.190117i \(-0.939113\pi\)
0.981761 0.190117i \(-0.0608868\pi\)
\(84\) 0 0
\(85\) −18.1168 −1.96505
\(86\) 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
\(95\) −13.4891 + 5.84096i −1.38396 + 0.599270i
\(96\) 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
\(101\) 14.7446 1.46714 0.733569 0.679615i \(-0.237853\pi\)
0.733569 + 0.679615i \(0.237853\pi\)
\(102\) 0 0
\(103\) 12.7446 1.25576 0.627880 0.778311i \(-0.283923\pi\)
0.627880 + 0.778311i \(0.283923\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 9.48913 0.917348 0.458674 0.888605i \(-0.348325\pi\)
0.458674 + 0.888605i \(0.348325\pi\)
\(108\) 0 0
\(109\) 8.21782i 0.787125i 0.919298 + 0.393562i \(0.128757\pi\)
−0.919298 + 0.393562i \(0.871243\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 3.75906i 0.353622i −0.984245 0.176811i \(-0.943422\pi\)
0.984245 0.176811i \(-0.0565782\pi\)
\(114\) 0 0
\(115\) 28.7075i 2.67699i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 4.25639i 0.390183i
\(120\) 0 0
\(121\) 10.3723 0.942935
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 4.62772 0.413916
\(126\) 0 0
\(127\) −12.7446 −1.13090 −0.565449 0.824784i \(-0.691297\pi\)
−0.565449 + 0.824784i \(0.691297\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 2.37686i 0.207667i −0.994595 0.103834i \(-0.966889\pi\)
0.994595 0.103834i \(-0.0331110\pi\)
\(132\) 0 0
\(133\) 1.37228 + 3.16915i 0.118992 + 0.274800i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.11684 0.351726 0.175863 0.984415i \(-0.443728\pi\)
0.175863 + 0.984415i \(0.443728\pi\)
\(138\) 0 0
\(139\) 5.54601i 0.470406i 0.971946 + 0.235203i \(0.0755756\pi\)
−0.971946 + 0.235203i \(0.924424\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) 34.0511i 2.82779i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2.11684 −0.173419 −0.0867093 0.996234i \(-0.527635\pi\)
−0.0867093 + 0.996234i \(0.527635\pi\)
\(150\) 0 0
\(151\) 8.74456 0.711622 0.355811 0.934558i \(-0.384205\pi\)
0.355811 + 0.934558i \(0.384205\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −29.4891 −2.36862
\(156\) 0 0
\(157\) 7.48913 0.597697 0.298849 0.954301i \(-0.403397\pi\)
0.298849 + 0.954301i \(0.403397\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −6.74456 −0.531546
\(162\) 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
\(167\) 17.4891 1.35335 0.676675 0.736282i \(-0.263420\pi\)
0.676675 + 0.736282i \(0.263420\pi\)
\(168\) 0 0
\(169\) −12.4891 −0.960702
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 20.1947i 1.53537i −0.640825 0.767687i \(-0.721407\pi\)
0.640825 0.767687i \(-0.278593\pi\)
\(174\) 0 0
\(175\) 5.04868i 0.381644i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1.48913 0.111302 0.0556512 0.998450i \(-0.482277\pi\)
0.0556512 + 0.998450i \(0.482277\pi\)
\(180\) 0 0
\(181\) 5.63858i 0.419113i −0.977797 0.209556i \(-0.932798\pi\)
0.977797 0.209556i \(-0.0672019\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 17.0256i 1.25174i
\(186\) 0 0
\(187\) 4.25639i 0.311258i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3.66648i 0.265297i −0.991163 0.132649i \(-0.957652\pi\)
0.991163 0.132649i \(-0.0423482\pi\)
\(192\) 0 0
\(193\) 13.8564i 0.997406i 0.866773 + 0.498703i \(0.166190\pi\)
−0.866773 + 0.498703i \(0.833810\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 5.25544 0.374434 0.187217 0.982319i \(-0.440053\pi\)
0.187217 + 0.982319i \(0.440053\pi\)
\(198\) 0 0
\(199\) 21.5769i 1.52955i −0.644300 0.764773i \(-0.722851\pi\)
0.644300 0.764773i \(-0.277149\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −8.00000 −0.561490
\(204\) 0 0
\(205\) 23.3639i 1.63180i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −1.37228 3.16915i −0.0949227 0.219215i
\(210\) 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
\(215\) 31.3793i 2.14005i
\(216\) 0 0
\(217\) 6.92820i 0.470317i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 27.1229i 1.82448i
\(222\) 0 0
\(223\) −0.744563 −0.0498596 −0.0249298 0.999689i \(-0.507936\pi\)
−0.0249298 + 0.999689i \(0.507936\pi\)
\(224\) 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
\(229\) −24.1168 −1.59369 −0.796843 0.604186i \(-0.793498\pi\)
−0.796843 + 0.604186i \(0.793498\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 4.11684 0.269703 0.134852 0.990866i \(-0.456944\pi\)
0.134852 + 0.990866i \(0.456944\pi\)
\(234\) 0 0
\(235\) 14.3537i 0.936335i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 29.7947i 1.92726i 0.267240 + 0.963630i \(0.413888\pi\)
−0.267240 + 0.963630i \(0.586112\pi\)
\(240\) 0 0
\(241\) 17.0256i 1.09671i −0.836245 0.548356i \(-0.815254\pi\)
0.836245 0.548356i \(-0.184746\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 21.4891 1.37289
\(246\) 0 0
\(247\) 8.74456 + 20.1947i 0.556403 + 1.28496i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 6.13592i 0.387296i −0.981071 0.193648i \(-0.937968\pi\)
0.981071 0.193648i \(-0.0620319\pi\)
\(252\) 0 0
\(253\) 6.74456 0.424027
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3.75906i 0.234483i 0.993103 + 0.117242i \(0.0374052\pi\)
−0.993103 + 0.117242i \(0.962595\pi\)
\(258\) 0 0
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 18.1128i 1.11688i −0.829544 0.558441i \(-0.811400\pi\)
0.829544 0.558441i \(-0.188600\pi\)
\(264\) 0 0
\(265\) 10.6873i 0.656513i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 10.6873i 0.651614i −0.945436 0.325807i \(-0.894364\pi\)
0.945436 0.325807i \(-0.105636\pi\)
\(270\) 0 0
\(271\) 13.5615i 0.823800i −0.911229 0.411900i \(-0.864865\pi\)
0.911229 0.411900i \(-0.135135\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 5.04868i 0.304447i
\(276\) 0 0
\(277\) −2.62772 −0.157884 −0.0789422 0.996879i \(-0.525154\pi\)
−0.0789422 + 0.996879i \(0.525154\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 10.0974i 0.602357i −0.953568 0.301179i \(-0.902620\pi\)
0.953568 0.301179i \(-0.0973801\pi\)
\(282\) 0 0
\(283\) 27.3253i 1.62432i −0.583435 0.812160i \(-0.698292\pi\)
0.583435 0.812160i \(-0.301708\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −5.48913 −0.324013
\(288\) 0 0
\(289\) 11.8614 0.697730
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 13.8564i 0.809500i −0.914427 0.404750i \(-0.867359\pi\)
0.914427 0.404750i \(-0.132641\pi\)
\(294\) 0 0
\(295\) −13.4891 −0.785367
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −42.9783 −2.48550
\(300\) 0 0
\(301\) 7.37228 0.424931
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 18.1168 1.03737
\(306\) 0 0
\(307\) 17.4891 0.998157 0.499079 0.866557i \(-0.333672\pi\)
0.499079 + 0.866557i \(0.333672\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 14.3537i 0.813926i −0.913445 0.406963i \(-0.866588\pi\)
0.913445 0.406963i \(-0.133412\pi\)
\(312\) 0 0
\(313\) 19.4891 1.10159 0.550795 0.834640i \(-0.314325\pi\)
0.550795 + 0.834640i \(0.314325\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 10.6873i 0.600256i 0.953899 + 0.300128i \(0.0970294\pi\)
−0.953899 + 0.300128i \(0.902971\pi\)
\(318\) 0 0
\(319\) 8.00000 0.447914
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 21.4891 9.30506i 1.19569 0.517748i
\(324\) 0 0
\(325\) 32.1716i 1.78456i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −3.37228 −0.185920
\(330\) 0 0
\(331\) −9.48913 −0.521569 −0.260785 0.965397i \(-0.583981\pi\)
−0.260785 + 0.965397i \(0.583981\pi\)
\(332\) 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 \(0.191550\pi\)
−0.824333 + 0.566105i \(0.808450\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 6.92820i 0.375183i
\(342\) 0 0
\(343\) 10.5947i 0.572059i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 13.0641i 0.701319i −0.936503 0.350659i \(-0.885958\pi\)
0.936503 0.350659i \(-0.114042\pi\)
\(348\) 0 0
\(349\) −25.6060 −1.37066 −0.685328 0.728234i \(-0.740341\pi\)
−0.685328 + 0.728234i \(0.740341\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 23.4891 1.25020 0.625100 0.780545i \(-0.285058\pi\)
0.625100 + 0.780545i \(0.285058\pi\)
\(354\) 0 0
\(355\) −13.4891 −0.715928
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 13.7638i 0.726427i −0.931706 0.363214i \(-0.881680\pi\)
0.931706 0.363214i \(-0.118320\pi\)
\(360\) 0 0
\(361\) 13.0000 13.8564i 0.684211 0.729285i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −13.8832 −0.726678
\(366\) 0 0
\(367\) 26.8280i 1.40041i 0.713943 + 0.700204i \(0.246908\pi\)
−0.713943 + 0.700204i \(0.753092\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2.51087 0.130358
\(372\) 0 0
\(373\) 1.87953i 0.0973183i −0.998815 0.0486591i \(-0.984505\pi\)
0.998815 0.0486591i \(-0.0154948\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −50.9783 −2.62551
\(378\) 0 0
\(379\) −26.9783 −1.38578 −0.692890 0.721043i \(-0.743663\pi\)
−0.692890 + 0.721043i \(0.743663\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 30.9783 1.58291 0.791457 0.611224i \(-0.209323\pi\)
0.791457 + 0.611224i \(0.209323\pi\)
\(384\) 0 0
\(385\) 2.11684 0.107884
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −6.11684 −0.310136 −0.155068 0.987904i \(-0.549560\pi\)
−0.155068 + 0.987904i \(0.549560\pi\)
\(390\) 0 0
\(391\) 45.7330i 2.31282i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −16.0000 −0.805047
\(396\) 0 0
\(397\) 11.8832 0.596399 0.298199 0.954504i \(-0.403614\pi\)
0.298199 + 0.954504i \(0.403614\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 23.9538i 1.19619i 0.801424 + 0.598097i \(0.204076\pi\)
−0.801424 + 0.598097i \(0.795924\pi\)
\(402\) 0 0
\(403\) 44.1485i 2.19919i
\(404\) 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 \(-0.945207\pi\)
0.985221 0.171289i \(-0.0547931\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 3.16915i 0.155944i
\(414\) 0 0
\(415\) 11.6819i 0.573443i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0.294954i 0.0144094i 0.999974 + 0.00720471i \(0.00229335\pi\)
−0.999974 + 0.00720471i \(0.997707\pi\)
\(420\) 0 0
\(421\) 32.7615i 1.59670i 0.602196 + 0.798349i \(0.294293\pi\)
−0.602196 + 0.798349i \(0.705707\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −34.2337 −1.66058
\(426\) 0 0
\(427\) 4.25639i 0.205981i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 18.5109 0.891637 0.445819 0.895123i \(-0.352913\pi\)
0.445819 + 0.895123i \(0.352913\pi\)
\(432\) 0 0
\(433\) 16.4356i 0.789847i 0.918714 + 0.394923i \(0.129229\pi\)
−0.918714 + 0.394923i \(0.870771\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 14.7446 + 34.0511i 0.705328 + 1.62889i
\(438\) 0 0
\(439\) 39.7228 1.89587 0.947933 0.318469i \(-0.103169\pi\)
0.947933 + 0.318469i \(0.103169\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 19.4024i 0.921837i −0.887443 0.460918i \(-0.847520\pi\)
0.887443 0.460918i \(-0.152480\pi\)
\(444\) 0 0
\(445\) 44.7384i 2.12080i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 17.6155i 0.831325i 0.909519 + 0.415663i \(0.136450\pi\)
−0.909519 + 0.415663i \(0.863550\pi\)
\(450\) 0 0
\(451\) 5.48913 0.258473
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −13.4891 −0.632380
\(456\) 0 0
\(457\) −8.11684 −0.379690 −0.189845 0.981814i \(-0.560799\pi\)
−0.189845 + 0.981814i \(0.560799\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −32.6277 −1.51962 −0.759812 0.650143i \(-0.774709\pi\)
−0.759812 + 0.650143i \(0.774709\pi\)
\(462\) 0 0
\(463\) 0.202380i 0.00940538i −0.999989 0.00470269i \(-0.998503\pi\)
0.999989 0.00470269i \(-0.00149692\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 31.0843i 1.43841i 0.694797 + 0.719206i \(0.255494\pi\)
−0.694797 + 0.719206i \(0.744506\pi\)
\(468\) 0 0
\(469\) 7.51811i 0.347154i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −7.37228 −0.338978
\(474\) 0 0
\(475\) −25.4891 + 11.0371i −1.16952 + 0.506418i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 21.7793i 0.995121i 0.867429 + 0.497560i \(0.165771\pi\)
−0.867429 + 0.497560i \(0.834229\pi\)
\(480\) 0 0
\(481\) −25.4891 −1.16220
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 44.7384i 2.03146i
\(486\) 0 0
\(487\) −5.76631 −0.261297 −0.130648 0.991429i \(-0.541706\pi\)
−0.130648 + 0.991429i \(0.541706\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 21.0796i 0.951307i −0.879633 0.475654i \(-0.842212\pi\)
0.879633 0.475654i \(-0.157788\pi\)
\(492\) 0 0
\(493\) 54.2458i 2.44311i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 3.16915i 0.142156i
\(498\) 0 0
\(499\) 4.55134i 0.203746i −0.994797 0.101873i \(-0.967516\pi\)
0.994797 0.101873i \(-0.0324835\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 16.0309i 0.714782i −0.933955 0.357391i \(-0.883666\pi\)
0.933955 0.357391i \(-0.116334\pi\)
\(504\) 0 0
\(505\) 49.7228 2.21264
\(506\) 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
\(511\) 3.26172i 0.144290i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 42.9783 1.89385
\(516\) 0 0
\(517\) 3.37228 0.148313
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 3.75906i 0.164687i −0.996604 0.0823436i \(-0.973760\pi\)
0.996604 0.0823436i \(-0.0262405\pi\)
\(522\) 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
\(527\) 46.9783 2.04640
\(528\) 0 0
\(529\) −49.4674 −2.15076
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −34.9783 −1.51508
\(534\) 0 0
\(535\) 32.0000 1.38348
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 5.04868i 0.217462i
\(540\) 0 0
\(541\) −4.11684 −0.176997 −0.0884985 0.996076i \(-0.528207\pi\)
−0.0884985 + 0.996076i \(0.528207\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 27.7128i 1.18709i
\(546\) 0 0
\(547\) 29.4891 1.26086 0.630432 0.776245i \(-0.282878\pi\)
0.630432 + 0.776245i \(0.282878\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 17.4891 + 40.3894i 0.745062 + 1.72065i
\(552\) 0 0
\(553\) 3.75906i 0.159851i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −11.1386 −0.471957 −0.235979 0.971758i \(-0.575830\pi\)
−0.235979 + 0.971758i \(0.575830\pi\)
\(558\) 0 0
\(559\) 46.9783 1.98697
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −22.9783 −0.968418 −0.484209 0.874952i \(-0.660893\pi\)
−0.484209 + 0.874952i \(0.660893\pi\)
\(564\) 0 0
\(565\) 12.6766i 0.533308i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 27.1229i 1.13705i −0.822666 0.568526i \(-0.807514\pi\)
0.822666 0.568526i \(-0.192486\pi\)
\(570\) 0 0
\(571\) 2.87419i 0.120281i −0.998190 0.0601406i \(-0.980845\pi\)
0.998190 0.0601406i \(-0.0191549\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 54.2458i 2.26221i
\(576\) 0 0
\(577\) −37.6060 −1.56556 −0.782778 0.622300i \(-0.786198\pi\)
−0.782778 + 0.622300i \(0.786198\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −2.74456 −0.113864
\(582\) 0 0
\(583\) −2.51087 −0.103990
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.38219i 0.0570493i 0.999593 + 0.0285246i \(0.00908090\pi\)
−0.999593 + 0.0285246i \(0.990919\pi\)
\(588\) 0 0
\(589\) 34.9783 15.1460i 1.44125 0.624081i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 23.4891 0.964583 0.482291 0.876011i \(-0.339805\pi\)
0.482291 + 0.876011i \(0.339805\pi\)
\(594\) 0 0
\(595\) 14.3537i 0.588446i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 44.4674 1.81689 0.908444 0.418007i \(-0.137271\pi\)
0.908444 + 0.418007i \(0.137271\pi\)
\(600\) 0 0
\(601\) 19.6048i 0.799696i 0.916581 + 0.399848i \(0.130937\pi\)
−0.916581 + 0.399848i \(0.869063\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 34.9783 1.42207
\(606\) 0 0
\(607\) 40.7446 1.65377 0.826885 0.562370i \(-0.190111\pi\)
0.826885 + 0.562370i \(0.190111\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −21.4891 −0.869357
\(612\) 0 0
\(613\) 5.37228 0.216984 0.108492 0.994097i \(-0.465398\pi\)
0.108492 + 0.994097i \(0.465398\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 19.0951 0.768740 0.384370 0.923179i \(-0.374419\pi\)
0.384370 + 0.923179i \(0.374419\pi\)
\(618\) 0 0
\(619\) 10.3923i 0.417702i −0.977947 0.208851i \(-0.933028\pi\)
0.977947 0.208851i \(-0.0669724\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 10.5109 0.421109
\(624\) 0 0
\(625\) −16.2554 −0.650217
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 27.1229i 1.08146i
\(630\) 0 0
\(631\) 42.7663i 1.70250i 0.524762 + 0.851249i \(0.324154\pi\)
−0.524762 + 0.851249i \(0.675846\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −42.9783 −1.70554
\(636\) 0 0
\(637\) 32.1716i 1.27468i
\(638\) 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
\(643\) 0.792287i 0.0312447i −0.999878 0.0156224i \(-0.995027\pi\)
0.999878 0.0156224i \(-0.00497296\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 36.7229i 1.44373i 0.692035 + 0.721864i \(0.256714\pi\)
−0.692035 + 0.721864i \(0.743286\pi\)
\(648\) 0 0
\(649\) 3.16915i 0.124400i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −22.1168 −0.865499 −0.432749 0.901514i \(-0.642457\pi\)
−0.432749 + 0.901514i \(0.642457\pi\)
\(654\) 0 0
\(655\) 8.01544i 0.313189i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 9.48913 0.369644 0.184822 0.982772i \(-0.440829\pi\)
0.184822 + 0.982772i \(0.440829\pi\)
\(660\) 0 0
\(661\) 46.0280i 1.79028i 0.445784 + 0.895141i \(0.352925\pi\)
−0.445784 + 0.895141i \(0.647075\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 4.62772 + 10.6873i 0.179455 + 0.414434i
\(666\) 0 0
\(667\) −85.9565 −3.32825
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 4.25639i 0.164316i
\(672\) 0 0
\(673\) 14.4463i 0.556864i 0.960456 + 0.278432i \(0.0898148\pi\)
−0.960456 + 0.278432i \(0.910185\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 21.3745i 0.821489i 0.911750 + 0.410745i \(0.134731\pi\)
−0.911750 + 0.410745i \(0.865269\pi\)
\(678\) 0 0
\(679\) 10.5109 0.403370
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 6.97825 0.267015 0.133508 0.991048i \(-0.457376\pi\)
0.133508 + 0.991048i \(0.457376\pi\)
\(684\) 0 0
\(685\) 13.8832 0.530448
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 16.0000 0.609551
\(690\) 0 0
\(691\) 27.9152i 1.06194i −0.847389 0.530972i \(-0.821827\pi\)
0.847389 0.530972i \(-0.178173\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 18.7027i 0.709434i
\(696\) 0 0
\(697\) 37.2203i 1.40982i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −37.7228 −1.42477 −0.712385 0.701788i \(-0.752385\pi\)
−0.712385 + 0.701788i \(0.752385\pi\)
\(702\) 0 0
\(703\) 8.74456 + 20.1947i 0.329807 + 0.761658i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 11.6819i 0.439344i
\(708\) 0 0
\(709\) 43.9565 1.65082 0.825411 0.564533i \(-0.190944\pi\)
0.825411 + 0.564533i \(0.190944\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 74.4405i 2.78782i
\(714\) 0 0
\(715\) 13.4891 0.504465
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 13.7638i 0.513304i −0.966504 0.256652i \(-0.917381\pi\)
0.966504 0.256652i \(-0.0826195\pi\)
\(720\) 0 0
\(721\) 10.0974i 0.376045i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 64.3432i 2.38965i
\(726\) 0 0
\(727\) 24.7460i 0.917780i −0.888493 0.458890i \(-0.848247\pi\)
0.888493 0.458890i \(-0.151753\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 49.9894i 1.84893i
\(732\) 0 0
\(733\) −20.9783 −0.774849 −0.387425 0.921901i \(-0.626635\pi\)
−0.387425 + 0.921901i \(0.626635\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 7.51811i 0.276933i
\(738\) 0 0
\(739\) 38.0125i 1.39831i −0.714968 0.699157i \(-0.753559\pi\)
0.714968 0.699157i \(-0.246441\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −34.9783 −1.28323 −0.641614 0.767028i \(-0.721735\pi\)
−0.641614 + 0.767028i \(0.721735\pi\)
\(744\) 0 0
\(745\) −7.13859 −0.261538
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 7.51811i 0.274706i
\(750\) 0 0
\(751\) −15.7228 −0.573734 −0.286867 0.957970i \(-0.592614\pi\)
−0.286867 + 0.957970i \(0.592614\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 29.4891 1.07322
\(756\) 0 0
\(757\) 10.8614 0.394765 0.197382 0.980327i \(-0.436756\pi\)
0.197382 + 0.980327i \(0.436756\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −6.86141 −0.248726 −0.124363 0.992237i \(-0.539689\pi\)
−0.124363 + 0.992237i \(0.539689\pi\)
\(762\) 0 0
\(763\) 6.51087 0.235709
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 20.1947i 0.729188i
\(768\) 0 0
\(769\) −9.13859 −0.329546 −0.164773 0.986332i \(-0.552689\pi\)
−0.164773 + 0.986332i \(0.552689\pi\)
\(770\) 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
\(775\) −55.7228 −2.00162
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 12.0000 + 27.7128i 0.429945 + 0.992915i
\(780\) 0 0
\(781\) 3.16915i 0.113401i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 25.2554 0.901405
\(786\) 0 0
\(787\) −10.9783 −0.391332 −0.195666 0.980671i \(-0.562687\pi\)
−0.195666 + 0.980671i \(0.562687\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −2.97825 −0.105894
\(792\) 0 0
\(793\) 27.1229i 0.963163i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 10.0974i 0.357667i −0.983879 0.178833i \(-0.942768\pi\)
0.983879 0.178833i \(-0.0572323\pi\)
\(798\) 0 0
\(799\) 22.8665i 0.808959i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 3.26172i 0.115104i
\(804\) 0 0
\(805\) −22.7446 −0.801640
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 32.1168