Properties

Label 731.2.s.a.343.2
Level 731
Weight 2
Character 731.343
Analytic conductor 5.837
Analytic rank 0
Dimension 16
CM discriminant -43
Inner twists 4

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 731 = 17 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 731.s (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.83706438776\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{16})\)
Coefficient field: 16.0.3289935900927224469054816256.1
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{16}]$

Embedding invariants

Embedding label 343.2
Root \(2.83780 - 1.71665i\)
Character \(\chi\) = 731.343
Dual form 731.2.s.a.601.2

$q$-expansion

\(f(q)\) \(=\) \(q+(1.41421 - 1.41421i) q^{4} +(2.77164 + 1.14805i) q^{9} +O(q^{10})\) \(q+(1.41421 - 1.41421i) q^{4} +(2.77164 + 1.14805i) q^{9} +(1.17208 + 5.89244i) q^{11} +(1.77872 + 1.77872i) q^{13} -4.00000i q^{16} +(-3.98585 + 1.05499i) q^{17} +(7.98829 - 1.58897i) q^{23} +(-1.91342 + 4.61940i) q^{25} +(2.17064 - 10.9126i) q^{31} +(5.54328 - 2.29610i) q^{36} +(4.76537 - 7.13187i) q^{41} +(-6.05828 - 2.50942i) q^{43} +(9.99074 + 6.67560i) q^{44} +(0.935734 + 0.935734i) q^{47} +(-2.67878 - 6.46716i) q^{49} +5.03098 q^{52} +(-1.00093 + 0.414599i) q^{53} +(-5.80838 + 14.0227i) q^{59} +(-5.65685 - 5.65685i) q^{64} +16.3676i q^{67} +(-4.14487 + 7.12882i) q^{68} +(-2.05270 - 10.3196i) q^{79} +(6.36396 + 6.36396i) q^{81} +(-2.50942 - 6.05828i) q^{83} +(9.05001 - 13.5443i) q^{92} +(15.8804 - 10.6110i) q^{97} +(-3.51624 + 17.6773i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 24q^{13} + 56q^{23} - 64q^{59} + 96q^{79} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/731\mathbb{Z}\right)^\times\).

\(n\) \(173\) \(562\)
\(\chi(n)\) \(e\left(\frac{1}{16}\right)\) \(-1\)

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 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(3\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(4\) 1.41421 1.41421i 0.707107 0.707107i
\(5\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(6\) 0 0
\(7\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(8\) 0 0
\(9\) 2.77164 + 1.14805i 0.923880 + 0.382683i
\(10\) 0 0
\(11\) 1.17208 + 5.89244i 0.353395 + 1.77664i 0.592425 + 0.805626i \(0.298171\pi\)
−0.239030 + 0.971012i \(0.576829\pi\)
\(12\) 0 0
\(13\) 1.77872 + 1.77872i 0.493328 + 0.493328i 0.909353 0.416025i \(-0.136577\pi\)
−0.416025 + 0.909353i \(0.636577\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000i 1.00000i
\(17\) −3.98585 + 1.05499i −0.966711 + 0.255872i
\(18\) 0 0
\(19\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.98829 1.58897i 1.66567 0.331323i 0.729800 0.683660i \(-0.239613\pi\)
0.935873 + 0.352337i \(0.114613\pi\)
\(24\) 0 0
\(25\) −1.91342 + 4.61940i −0.382683 + 0.923880i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(30\) 0 0
\(31\) 2.17064 10.9126i 0.389859 1.95995i 0.155103 0.987898i \(-0.450429\pi\)
0.234756 0.972054i \(-0.424571\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 5.54328 2.29610i 0.923880 0.382683i
\(37\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.76537 7.13187i 0.744225 1.11381i −0.245299 0.969447i \(-0.578886\pi\)
0.989525 0.144364i \(-0.0461137\pi\)
\(42\) 0 0
\(43\) −6.05828 2.50942i −0.923880 0.382683i
\(44\) 9.99074 + 6.67560i 1.50616 + 1.00638i
\(45\) 0 0
\(46\) 0 0
\(47\) 0.935734 + 0.935734i 0.136491 + 0.136491i 0.772051 0.635560i \(-0.219231\pi\)
−0.635560 + 0.772051i \(0.719231\pi\)
\(48\) 0 0
\(49\) −2.67878 6.46716i −0.382683 0.923880i
\(50\) 0 0
\(51\) 0 0
\(52\) 5.03098 0.697671
\(53\) −1.00093 + 0.414599i −0.137488 + 0.0569495i −0.450367 0.892844i \(-0.648707\pi\)
0.312878 + 0.949793i \(0.398707\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −5.80838 + 14.0227i −0.756187 + 1.82560i −0.235431 + 0.971891i \(0.575650\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −5.65685 5.65685i −0.707107 0.707107i
\(65\) 0 0
\(66\) 0 0
\(67\) 16.3676i 1.99962i 0.0194269 + 0.999811i \(0.493816\pi\)
−0.0194269 + 0.999811i \(0.506184\pi\)
\(68\) −4.14487 + 7.12882i −0.502639 + 0.864496i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(72\) 0 0
\(73\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −2.05270 10.3196i −0.230947 1.16105i −0.906000 0.423279i \(-0.860879\pi\)
0.675053 0.737769i \(-0.264121\pi\)
\(80\) 0 0
\(81\) 6.36396 + 6.36396i 0.707107 + 0.707107i
\(82\) 0 0
\(83\) −2.50942 6.05828i −0.275445 0.664983i 0.724254 0.689534i \(-0.242184\pi\)
−0.999699 + 0.0245507i \(0.992184\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 9.05001 13.5443i 0.943528 1.41209i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 15.8804 10.6110i 1.61241 1.07738i 0.670297 0.742093i \(-0.266167\pi\)
0.942116 0.335287i \(-0.108833\pi\)
\(98\) 0 0
\(99\) −3.51624 + 17.6773i −0.353395 + 1.77664i
\(100\) 3.82683 + 9.23880i 0.382683 + 0.923880i
\(101\) 15.0443i 1.49696i −0.663156 0.748481i \(-0.730783\pi\)
0.663156 0.748481i \(-0.269217\pi\)
\(102\) 0 0
\(103\) −16.2614 −1.60229 −0.801143 0.598473i \(-0.795775\pi\)
−0.801143 + 0.598473i \(0.795775\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3.34302 + 5.00318i 0.323182 + 0.483676i 0.957114 0.289713i \(-0.0935599\pi\)
−0.633932 + 0.773389i \(0.718560\pi\)
\(108\) 0 0
\(109\) 2.83844 4.24803i 0.271874 0.406888i −0.670259 0.742127i \(-0.733817\pi\)
0.942133 + 0.335239i \(0.108817\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 2.88791 + 6.97203i 0.266987 + 0.644564i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −23.1844 + 9.60330i −2.10767 + 0.873027i
\(122\) 0 0
\(123\) 0 0
\(124\) −12.3629 18.5024i −1.11022 1.66157i
\(125\) 0 0
\(126\) 0 0
\(127\) −6.77003 + 16.3443i −0.600743 + 1.45032i 0.272075 + 0.962276i \(0.412290\pi\)
−0.872818 + 0.488046i \(0.837710\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) −8.74108 1.73871i −0.741409 0.147475i −0.190086 0.981767i \(-0.560877\pi\)
−0.551323 + 0.834292i \(0.685877\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −8.39620 + 12.5658i −0.702125 + 1.05080i
\(144\) 4.59220 11.0866i 0.382683 0.923880i
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(150\) 0 0
\(151\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(152\) 0 0
\(153\) −12.2585 1.65191i −0.991042 0.133549i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(164\) −3.34675 16.8252i −0.261337 1.31383i
\(165\) 0 0
\(166\) 0 0
\(167\) −3.34455 + 16.8142i −0.258809 + 1.30112i 0.604564 + 0.796557i \(0.293347\pi\)
−0.863373 + 0.504566i \(0.831653\pi\)
\(168\) 0 0
\(169\) 6.67232i 0.513255i
\(170\) 0 0
\(171\) 0 0
\(172\) −12.1166 + 5.01885i −0.923880 + 0.382683i
\(173\) −15.9235 3.16737i −1.21064 0.240811i −0.451817 0.892111i \(-0.649224\pi\)
−0.758820 + 0.651300i \(0.774224\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 23.5698 4.68832i 1.77664 0.353395i
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(180\) 0 0
\(181\) 0.149740 + 0.752792i 0.0111301 + 0.0559546i 0.985951 0.167034i \(-0.0534188\pi\)
−0.974821 + 0.222988i \(0.928419\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −10.8882 22.2499i −0.796223 1.62707i
\(188\) 2.64666 0.193027
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(192\) 0 0
\(193\) −24.7177 + 4.91665i −1.77922 + 0.353908i −0.971751 0.236007i \(-0.924161\pi\)
−0.807465 + 0.589915i \(0.799161\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −12.9343 5.35757i −0.923880 0.382683i
\(197\) −23.0123 15.3763i −1.63956 1.09552i −0.912614 0.408822i \(-0.865940\pi\)
−0.726944 0.686696i \(-0.759060\pi\)
\(198\) 0 0
\(199\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 23.9649 + 4.76691i 1.66567 + 0.331323i
\(208\) 7.11488 7.11488i 0.493328 0.493328i
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(212\) −0.829197 + 2.00186i −0.0569495 + 0.137488i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −8.96623 5.21318i −0.603134 0.350677i
\(222\) 0 0
\(223\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(224\) 0 0
\(225\) −10.6066 + 10.6066i −0.707107 + 0.707107i
\(226\) 0 0
\(227\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(228\) 0 0
\(229\) 5.25078 12.6765i 0.346982 0.837688i −0.649992 0.759941i \(-0.725228\pi\)
0.996973 0.0777462i \(-0.0247724\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 11.6168 + 28.0453i 0.756187 + 1.82560i
\(237\) 0 0
\(238\) 0 0
\(239\) 29.0726 1.88055 0.940275 0.340415i \(-0.110568\pi\)
0.940275 + 0.340415i \(0.110568\pi\)
\(240\) 0 0
\(241\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 18.7787 + 18.7787i 1.18530 + 1.18530i 0.978351 + 0.206951i \(0.0663540\pi\)
0.206951 + 0.978351i \(0.433646\pi\)
\(252\) 0 0
\(253\) 18.7258 + 45.2081i 1.17728 + 2.84221i
\(254\) 0 0
\(255\) 0 0
\(256\) −16.0000 −1.00000
\(257\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 23.1473 + 23.1473i 1.41395 + 1.41395i
\(269\) −1.43923 + 7.23550i −0.0877514 + 0.441156i 0.911784 + 0.410671i \(0.134705\pi\)
−0.999535 + 0.0304855i \(0.990295\pi\)
\(270\) 0 0
\(271\) 31.4395i 1.90981i −0.296909 0.954906i \(-0.595956\pi\)
0.296909 0.954906i \(-0.404044\pi\)
\(272\) 4.21995 + 15.9434i 0.255872 + 0.966711i
\(273\) 0 0
\(274\) 0 0
\(275\) −29.4622 5.86040i −1.77664 0.353395i
\(276\) 0 0
\(277\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(278\) 0 0
\(279\) 18.5444 27.7537i 1.11022 1.66157i
\(280\) 0 0
\(281\) −17.5669 7.27645i −1.04795 0.434076i −0.208792 0.977960i \(-0.566953\pi\)
−0.839161 + 0.543884i \(0.816953\pi\)
\(282\) 0 0
\(283\) −2.51306 12.6340i −0.149386 0.751013i −0.980747 0.195281i \(-0.937438\pi\)
0.831362 0.555732i \(-0.187562\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 14.7740 8.41004i 0.869059 0.494708i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 24.1149 24.1149i 1.40881 1.40881i 0.642627 0.766179i \(-0.277845\pi\)
0.766179 0.642627i \(-0.222155\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 17.0353 + 11.3826i 0.985174 + 0.658272i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −29.2727 −1.67068 −0.835340 0.549734i \(-0.814729\pi\)
−0.835340 + 0.549734i \(0.814729\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 16.4566 + 24.6290i 0.933166 + 1.39658i 0.917950 + 0.396697i \(0.129844\pi\)
0.0152162 + 0.999884i \(0.495156\pi\)
\(312\) 0 0
\(313\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −17.4971 11.6912i −0.984289 0.657681i
\(317\) 4.85722 + 24.4189i 0.272808 + 1.37150i 0.837608 + 0.546272i \(0.183954\pi\)
−0.564799 + 0.825228i \(0.691046\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 18.0000 1.00000
\(325\) −11.6200 + 4.81318i −0.644564 + 0.266987i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(332\) −12.1166 5.01885i −0.664983 0.275445i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0.168898 0.849110i 0.00920048 0.0462539i −0.975912 0.218163i \(-0.929994\pi\)
0.985113 + 0.171909i \(0.0549935\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 66.8458 3.61990
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(348\) 0 0
\(349\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −15.2213 15.2213i −0.810147 0.810147i 0.174509 0.984656i \(-0.444166\pi\)
−0.984656 + 0.174509i \(0.944166\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −33.8316 + 14.0135i −1.78556 + 0.739605i −0.794330 + 0.607486i \(0.792178\pi\)
−0.991234 + 0.132119i \(0.957822\pi\)
\(360\) 0 0
\(361\) 13.4350 13.4350i 0.707107 0.707107i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 25.6967 + 17.1700i 1.34136 + 0.896266i 0.999061 0.0433266i \(-0.0137956\pi\)
0.342296 + 0.939592i \(0.388796\pi\)
\(368\) −6.35588 31.9532i −0.331323 1.66567i
\(369\) 21.3956 14.2961i 1.11381 0.744225i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 20.1417 + 30.1442i 1.03461 + 1.54840i 0.820544 + 0.571583i \(0.193670\pi\)
0.214065 + 0.976819i \(0.431330\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −13.9104 13.9104i −0.707107 0.707107i
\(388\) 7.45215 37.4645i 0.378326 1.90197i
\(389\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(390\) 0 0
\(391\) −30.1638 + 14.7609i −1.52545 + 0.746493i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 20.0268 + 29.9722i 1.00638 + 1.50616i
\(397\) 5.04333 1.00318i 0.253118 0.0503482i −0.0669005 0.997760i \(-0.521311\pi\)
0.320018 + 0.947411i \(0.396311\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 18.4776 + 7.65367i 0.923880 + 0.382683i
\(401\) −0.755151 0.504576i −0.0377105 0.0251973i 0.536572 0.843854i \(-0.319719\pi\)
−0.574283 + 0.818657i \(0.694719\pi\)
\(402\) 0 0
\(403\) 23.2713 15.5494i 1.15923 0.774571i
\(404\) −21.2758 21.2758i −1.05851 1.05851i
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −22.9971 + 22.9971i −1.13299 + 1.13299i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(420\) 0 0
\(421\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(422\) 0 0
\(423\) 1.51925 + 3.66779i 0.0738683 + 0.178334i
\(424\) 0 0
\(425\) 2.75319 20.4309i 0.133549 0.991042i
\(426\) 0 0
\(427\) 0 0
\(428\) 11.8033 + 2.34783i 0.570535 + 0.113486i
\(429\) 0 0
\(430\) 0 0
\(431\) 40.6942 8.09459i 1.96017 0.389903i 0.972722 0.231972i \(-0.0745179\pi\)
0.987450 0.157930i \(-0.0504821\pi\)
\(432\) 0 0
\(433\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −1.99346 10.0218i −0.0954693 0.479957i
\(437\) 0 0
\(438\) 0 0
\(439\) −7.20138 + 36.2038i −0.343703 + 1.72791i 0.292391 + 0.956299i \(0.405549\pi\)
−0.636093 + 0.771612i \(0.719451\pi\)
\(440\) 0 0
\(441\) 21.0000i 1.00000i
\(442\) 0 0
\(443\) −19.0107 −0.903223 −0.451612 0.892215i \(-0.649151\pi\)
−0.451612 + 0.892215i \(0.649151\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(450\) 0 0
\(451\) 47.6095 + 19.7205i 2.24185 + 0.928603i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −24.2331 + 10.0377i −1.12865 + 0.467502i −0.867322 0.497748i \(-0.834160\pi\)
−0.261328 + 0.965250i \(0.584160\pi\)
\(462\) 0 0
\(463\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(468\) 13.9441 + 5.77582i 0.644564 + 0.266987i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 7.68584 38.6393i 0.353395 1.77664i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −3.25020 −0.148816
\(478\) 0 0
\(479\) −10.9358 2.17526i −0.499668 0.0993902i −0.0611793 0.998127i \(-0.519486\pi\)
−0.438489 + 0.898737i \(0.644486\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −19.2066 + 46.3688i −0.873027 + 2.10767i
\(485\) 0 0
\(486\) 0 0
\(487\) −7.50866 37.7486i −0.340250 1.71055i −0.650171 0.759788i \(-0.725303\pi\)
0.309921 0.950762i \(-0.399697\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −43.6502 8.68257i −1.95995 0.389859i
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 13.5401 + 32.6886i 0.600743 + 1.45032i
\(509\) 18.4282i 0.816817i −0.912799 0.408408i \(-0.866084\pi\)
0.912799 0.408408i \(-0.133916\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −4.41700 + 6.61051i −0.194260 + 0.290730i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(522\) 0 0
\(523\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.86075 + 45.7858i 0.124616 + 1.99446i
\(528\) 0 0
\(529\) 40.0387 16.5846i 1.74081 0.721069i
\(530\) 0 0
\(531\) −32.1975 + 32.1975i −1.39725 + 1.39725i
\(532\) 0 0
\(533\) 21.1619 4.20935i 0.916622 0.182327i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 34.9676 23.3646i 1.50616 1.00638i
\(540\) 0 0
\(541\) 5.06639 25.4705i 0.217821 1.09506i −0.704816 0.709390i \(-0.748970\pi\)
0.922637 0.385670i \(-0.126030\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −45.8771 9.12552i −1.96156 0.390179i −0.981315 0.192406i \(-0.938371\pi\)
−0.980248 0.197773i \(-0.936629\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −14.8207 + 9.90285i −0.628536 + 0.419974i
\(557\) 32.9636 + 32.9636i 1.39671 + 1.39671i 0.809260 + 0.587450i \(0.199868\pi\)
0.587450 + 0.809260i \(0.300132\pi\)
\(558\) 0 0
\(559\) −6.31243 15.2395i −0.266987 0.644564i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 11.0199 4.56459i 0.464433 0.192374i −0.138182 0.990407i \(-0.544126\pi\)
0.602614 + 0.798033i \(0.294126\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 15.9388 38.4796i 0.668189 1.61315i −0.116450 0.993197i \(-0.537151\pi\)
0.784639 0.619953i \(-0.212849\pi\)
\(570\) 0 0
\(571\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(572\) 5.89671 + 29.6447i 0.246554 + 1.23951i
\(573\) 0 0
\(574\) 0 0
\(575\) −7.94485 + 39.9414i −0.331323 + 1.66567i
\(576\) −9.18440 22.1731i −0.382683 0.923880i
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −3.61617 5.41198i −0.149766 0.224141i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −32.4577 + 32.4577i −1.32618 + 1.32618i −0.417514 + 0.908671i \(0.637098\pi\)
−0.908671 + 0.417514i \(0.862902\pi\)
\(600\) 0 0
\(601\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(602\) 0 0
\(603\) −18.7908 + 45.3651i −0.765222 + 1.84741i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 3.32882i 0.134670i
\(612\) −19.6723 + 15.0000i −0.795206 + 0.606339i
\(613\) −28.7652 −1.16182 −0.580909 0.813969i \(-0.697303\pi\)
−0.580909 + 0.813969i \(0.697303\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 5.37785 + 8.04852i 0.216504 + 0.324021i 0.923785 0.382911i \(-0.125078\pi\)
−0.707281 + 0.706932i \(0.750078\pi\)
\(618\) 0 0
\(619\) 27.5764 41.2711i 1.10839 1.65882i 0.495735 0.868474i \(-0.334899\pi\)
0.612655 0.790350i \(-0.290101\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −17.6777 17.6777i −0.707107 0.707107i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 6.73845 16.2681i 0.266987 0.644564i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(642\) 0 0
\(643\) 8.74620 43.9701i 0.344916 1.73401i −0.286062 0.958211i \(-0.592346\pi\)
0.630978 0.775800i \(-0.282654\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) −89.4357 17.7899i −3.51066 0.698313i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −28.5275 19.0615i −1.11381 0.744225i
\(657\) 0 0
\(658\) 0 0
\(659\) 23.7632 + 23.7632i 0.925684 + 0.925684i 0.997423 0.0717399i \(-0.0228551\pi\)
−0.0717399 + 0.997423i \(0.522855\pi\)
\(660\) 0 0
\(661\) −2.50942 6.05828i −0.0976052 0.235640i 0.867533 0.497379i \(-0.165704\pi\)
−0.965139 + 0.261739i \(0.915704\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 19.0490 + 28.5088i 0.737027 + 1.10304i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −9.43608 9.43608i −0.362926 0.362926i
\(677\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −39.3715 7.83147i −1.50651 0.299663i −0.628311 0.777962i \(-0.716253\pi\)
−0.878197 + 0.478299i \(0.841253\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −10.0377 + 24.2331i −0.382683 + 0.923880i
\(689\) −2.51783 1.04292i −0.0959216 0.0397320i
\(690\) 0 0
\(691\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(692\) −26.9985 + 18.0398i −1.02633 + 0.685771i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −11.4700 + 33.4540i −0.434457 + 1.26716i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −9.88512 + 9.88512i −0.373356 + 0.373356i −0.868698 0.495342i \(-0.835043\pi\)
0.495342 + 0.868698i \(0.335043\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 26.7024 39.9630i 1.00638 1.50616i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 19.2616 12.8702i 0.723386 0.483351i −0.138558 0.990354i \(-0.544247\pi\)
0.861944 + 0.507003i \(0.169247\pi\)
\(710\) 0 0
\(711\) 6.15810 30.9589i 0.230947 1.16105i
\(712\) 0 0
\(713\) 90.6217i 3.39381i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −28.0689 42.0081i −1.04679 1.56664i −0.802242 0.597000i \(-0.796359\pi\)
−0.244551 0.969636i \(-0.578641\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 1.27637 + 0.852845i 0.0474360 + 0.0316957i
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(728\) 0 0
\(729\) 10.3325 + 24.9447i 0.382683 + 0.923880i
\(730\) 0 0
\(731\) 26.7948 + 3.61077i 0.991042 + 0.133549i
\(732\) 0 0
\(733\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −96.4452 + 19.1841i −3.55261 + 0.706657i
\(738\) 0 0
\(739\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 19.6723i 0.719772i
\(748\) −46.8643 16.0678i −1.71353 0.587498i
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(752\) 3.74294 3.74294i 0.136491 0.136491i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −35.2739 + 14.6109i −1.27367 + 0.527570i
\(768\) 0 0
\(769\) 25.9545 25.9545i 0.935943 0.935943i −0.0621249 0.998068i \(-0.519788\pi\)
0.998068 + 0.0621249i \(0.0197877\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −28.0029 + 41.9092i −1.00785 + 1.50835i
\(773\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(774\) 0 0
\(775\) 46.2561 + 30.9073i 1.66157 + 1.11022i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −25.8686 + 10.7151i −0.923880 + 0.382683i
\(785\) 0 0
\(786\) 0 0
\(787\) 30.5212 + 45.6782i 1.08796 + 1.62825i 0.712923 + 0.701242i \(0.247371\pi\)
0.375040 + 0.927009i \(0.377629\pi\)
\(788\) −54.2897 + 10.7989i −1.93399 + 0.384695i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 16.5959 + 40.0662i 0.587859 + 1.41922i 0.885545 + 0.464553i \(0.153785\pi\)
−0.297687 + 0.954664i \(0.596215\pi\)
\(798\) 0 0
\(799\) −4.71688 2.74251i −0.166871 0.0970230i
\(800\) 0 0
\(801\) 0 0
\(802\)