Properties

Label 731.2.s.a.558.2
Level 731
Weight 2
Character 731.558
Analytic conductor 5.837
Analytic rank 0
Dimension 16
CM discriminant -43
Inner twists 4

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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 558.2
Root \(-2.83780 + 1.71665i\)
Character \(\chi\) = 731.558
Dual form 731.2.s.a.300.2

$q$-expansion

\(f(q)\) \(=\) \(q+(1.41421 - 1.41421i) q^{4} +(-2.77164 - 1.14805i) q^{9} +O(q^{10})\) \(q+(1.41421 - 1.41421i) q^{4} +(-2.77164 - 1.14805i) q^{9} +(2.75762 - 0.548525i) q^{11} +(1.77872 + 1.77872i) q^{13} -4.00000i q^{16} +(3.98585 - 1.05499i) q^{17} +(-0.988290 - 4.96847i) q^{23} +(1.91342 - 4.61940i) q^{25} +(-0.443490 - 0.0882157i) q^{31} +(-5.54328 + 2.29610i) q^{36} +(-7.90673 - 5.28311i) q^{41} +(6.05828 + 2.50942i) q^{43} +(3.12414 - 4.67560i) q^{44} +(-0.935734 - 0.935734i) q^{47} +(2.67878 + 6.46716i) q^{49} +5.03098 q^{52} +(-1.00093 + 0.414599i) q^{53} +(-5.80838 + 14.0227i) q^{59} +(-5.65685 - 5.65685i) q^{64} -16.3676i q^{67} +(4.14487 - 7.12882i) q^{68} +(14.0527 - 2.79526i) q^{79} +(6.36396 + 6.36396i) q^{81} +(2.50942 + 6.05828i) q^{83} +(-8.42413 - 5.62882i) q^{92} +(-2.67711 - 4.00657i) q^{97} +(-8.27287 - 1.64558i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 24q^{13} + 56q^{23} - 64q^{59} + 96q^{79} + O(q^{100}) \)

Character values

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

\(n\) \(173\) \(562\)
\(\chi(n)\) \(e\left(\frac{9}{16}\right)\) \(-1\)

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 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(4\) 1.41421 1.41421i 0.707107 0.707107i
\(5\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(6\) 0 0
\(7\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(8\) 0 0
\(9\) −2.77164 1.14805i −0.923880 0.382683i
\(10\) 0 0
\(11\) 2.75762 0.548525i 0.831455 0.165387i 0.239030 0.971012i \(-0.423171\pi\)
0.592425 + 0.805626i \(0.298171\pi\)
\(12\) 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
\(23\) −0.988290 4.96847i −0.206073 1.03600i −0.935873 0.352337i \(-0.885387\pi\)
0.729800 0.683660i \(-0.239613\pi\)
\(24\) 0 0
\(25\) 1.91342 4.61940i 0.382683 0.923880i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(30\) 0 0
\(31\) −0.443490 0.0882157i −0.0796532 0.0158440i 0.155103 0.987898i \(-0.450429\pi\)
−0.234756 + 0.972054i \(0.575429\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −5.54328 + 2.29610i −0.923880 + 0.382683i
\(37\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −7.90673 5.28311i −1.23482 0.825083i −0.245299 0.969447i \(-0.578886\pi\)
−0.989525 + 0.144364i \(0.953886\pi\)
\(42\) 0 0
\(43\) 6.05828 + 2.50942i 0.923880 + 0.382683i
\(44\) 3.12414 4.67560i 0.470981 0.704873i
\(45\) 0 0
\(46\) 0 0
\(47\) −0.935734 0.935734i −0.136491 0.136491i 0.635560 0.772051i \(-0.280769\pi\)
−0.772051 + 0.635560i \(0.780769\pi\)
\(48\) 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 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(62\) 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 \(-0.506184\pi\)
0.0194269 0.999811i \(-0.493816\pi\)
\(68\) 4.14487 7.12882i 0.502639 0.864496i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(72\) 0 0
\(73\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 14.0527 2.79526i 1.58105 0.314491i 0.675053 0.737769i \(-0.264121\pi\)
0.906000 + 0.423279i \(0.139121\pi\)
\(80\) 0 0
\(81\) 6.36396 + 6.36396i 0.707107 + 0.707107i
\(82\) 0 0
\(83\) 2.50942 + 6.05828i 0.275445 + 0.664983i 0.999699 0.0245507i \(-0.00781552\pi\)
−0.724254 + 0.689534i \(0.757816\pi\)
\(84\) 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
\(92\) −8.42413 5.62882i −0.878276 0.586845i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −2.67711 4.00657i −0.271819 0.406806i 0.670297 0.742093i \(-0.266167\pi\)
−0.942116 + 0.335287i \(0.891167\pi\)
\(98\) 0 0
\(99\) −8.27287 1.64558i −0.831455 0.165387i
\(100\) −3.82683 9.23880i −0.382683 0.923880i
\(101\) 15.0443i 1.49696i 0.663156 + 0.748481i \(0.269217\pi\)
−0.663156 + 0.748481i \(0.730783\pi\)
\(102\) 0 0
\(103\) 16.2614 1.60229 0.801143 0.598473i \(-0.204225\pi\)
0.801143 + 0.598473i \(0.204225\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −16.4579 + 10.9968i −1.59105 + 1.06310i −0.633932 + 0.773389i \(0.718560\pi\)
−0.957114 + 0.289713i \(0.906440\pi\)
\(108\) 0 0
\(109\) 16.8339 + 11.2480i 1.61239 + 1.07737i 0.942133 + 0.335239i \(0.108817\pi\)
0.670259 + 0.742127i \(0.266183\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(114\) 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
\(121\) −2.85907 + 1.18427i −0.259915 + 0.107661i
\(122\) 0 0
\(123\) 0 0
\(124\) −0.751946 + 0.502434i −0.0675267 + 0.0451199i
\(125\) 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 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(132\) 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
\(139\) −4.25892 + 21.4110i −0.361237 + 1.81606i 0.190086 + 0.981767i \(0.439123\pi\)
−0.551323 + 0.834292i \(0.685877\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 5.88071 + 3.92936i 0.491770 + 0.328590i
\(144\) −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 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(164\) −18.6533 + 3.71036i −1.45657 + 0.289731i
\(165\) 0 0
\(166\) 0 0
\(167\) −18.9699 3.77335i −1.46794 0.291991i −0.604564 0.796557i \(-0.706653\pi\)
−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
\(173\) −4.03800 + 20.3004i −0.307003 + 1.54341i 0.451817 + 0.892111i \(0.350776\pi\)
−0.758820 + 0.651300i \(0.774224\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −2.19410 11.0305i −0.165387 0.831455i
\(177\) 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
\(181\) −26.3795 + 5.24721i −1.96077 + 0.390022i −0.974821 + 0.222988i \(0.928419\pi\)
−0.985951 + 0.167034i \(0.946581\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 10.4128 5.09560i 0.761458 0.372627i
\(188\) −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
\(193\) −2.28234 11.4741i −0.164286 0.825923i −0.971751 0.236007i \(-0.924161\pi\)
0.807465 0.589915i \(-0.200839\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 12.9343 + 5.35757i 0.923880 + 0.382683i
\(197\) −2.60600 + 3.90016i −0.185670 + 0.277875i −0.912614 0.408822i \(-0.865940\pi\)
0.726944 + 0.686696i \(0.240940\pi\)
\(198\) 0 0
\(199\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −2.96487 + 14.9054i −0.206073 + 1.03600i
\(208\) 7.11488 7.11488i 0.493328 0.493328i
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(212\) −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 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(228\) 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 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(234\) 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 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(242\) 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
\(253\) −5.45066 13.1591i −0.342680 0.827303i
\(254\) 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
\(269\) −31.3480 6.23550i −1.91132 0.380185i −0.911784 0.410671i \(-0.865295\pi\)
−0.999535 + 0.0304855i \(0.990295\pi\)
\(270\) 0 0
\(271\) 31.4395i 1.90981i 0.296909 + 0.954906i \(0.404044\pi\)
−0.296909 + 0.954906i \(0.595956\pi\)
\(272\) −4.21995 15.9434i −0.255872 0.966711i
\(273\) 0 0
\(274\) 0 0
\(275\) 2.74263 13.7881i 0.165387 0.831455i
\(276\) 0 0
\(277\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(278\) 0 0
\(279\) 1.12792 + 0.753651i 0.0675267 + 0.0451199i
\(280\) 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
\(283\) 30.4844 6.06372i 1.81211 0.360451i 0.831362 0.555732i \(-0.187562\pi\)
0.980747 + 0.195281i \(0.0625619\pi\)
\(284\) 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
\(299\) 7.07962 10.5954i 0.409425 0.612748i
\(300\) 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 \(-0.185271\pi\)
0.835340 + 0.549734i \(0.185271\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 15.9199 10.6373i 0.902733 0.603187i −0.0152162 0.999884i \(-0.504844\pi\)
0.917950 + 0.396697i \(0.129844\pi\)
\(312\) 0 0
\(313\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 15.9204 23.8266i 0.895594 1.34035i
\(317\) −24.9691 + 4.96667i −1.40241 + 0.278956i −0.837608 0.546272i \(-0.816046\pi\)
−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
\(337\) −35.9996 7.16077i −1.96103 0.390072i −0.985113 0.171909i \(-0.945006\pi\)
−0.975912 0.218163i \(-0.929994\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −1.27137 −0.0688484
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(348\) 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 \(-0.207822\pi\)
0.991234 0.132119i \(-0.0421781\pi\)
\(360\) 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
\(367\) −12.5818 + 18.8300i −0.656765 + 0.982919i 0.342296 + 0.939592i \(0.388796\pi\)
−0.999061 + 0.0433266i \(0.986204\pi\)
\(368\) −19.8739 + 3.95316i −1.03600 + 0.206073i
\(369\) 15.8493 + 23.7202i 0.825083 + 1.23482i
\(370\) 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
\(379\) −11.8069 + 7.88911i −0.606479 + 0.405236i −0.820544 0.571583i \(-0.806330\pi\)
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
\(388\) −9.45215 1.88015i −0.479860 0.0954501i
\(389\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(390\) 0 0
\(391\) −9.18085 18.7609i −0.464295 0.948781i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) −14.0268 + 9.37241i −0.704873 + 0.470981i
\(397\) −7.70930 38.7573i −0.386919 1.94517i −0.320018 0.947411i \(-0.603689\pi\)
−0.0669005 0.997760i \(-0.521311\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −18.4776 7.65367i −0.923880 0.382683i
\(401\) −22.2448 + 33.2918i −1.11085 + 1.66251i −0.536572 + 0.843854i \(0.680281\pi\)
−0.574283 + 0.818657i \(0.694719\pi\)
\(402\) 0 0
\(403\) −0.631934 0.945756i −0.0314789 0.0471114i
\(404\) 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 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(420\) 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
\(428\) −7.72314 + 38.8268i −0.373312 + 1.87677i
\(429\) 0 0
\(430\) 0 0
\(431\) 0.305758 + 1.53715i 0.0147278 + 0.0740419i 0.987450 0.157930i \(-0.0504821\pi\)
−0.972722 + 0.231972i \(0.925482\pi\)
\(432\) 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
\(436\) 39.7138 7.89957i 1.90195 0.378321i
\(437\) 0 0
\(438\) 0 0
\(439\) 19.4539 + 3.86962i 0.928484 + 0.184687i 0.636093 0.771612i \(-0.280549\pi\)
0.292391 + 0.956299i \(0.405549\pi\)
\(440\) 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 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(450\) 0 0
\(451\) −24.7017 10.2318i −1.16316 0.481796i
\(452\) 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 0.261328 0.965250i \(-0.415840\pi\)
0.867322 + 0.497748i \(0.165840\pi\)
\(462\) 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
\(473\) 18.0829 + 3.59692i 0.831455 + 0.165387i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 3.25020 0.148816
\(478\) 0 0
\(479\) 8.25782 41.5149i 0.377310 1.89686i −0.0611793 0.998127i \(-0.519486\pi\)
0.438489 0.898737i \(-0.355514\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −2.36853 + 5.71814i −0.107661 + 0.259915i
\(485\) 0 0
\(486\) 0 0
\(487\) 21.1874 4.21443i 0.960092 0.190974i 0.309921 0.950762i \(-0.399697\pi\)
0.650171 + 0.759788i \(0.274697\pi\)
\(488\) 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
\(496\) −0.352863 + 1.77396i −0.0158440 + 0.0796532i
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(504\) 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 \(0.133916\pi\)
−0.912799 + 0.408408i \(0.866084\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −3.09368 2.06713i −0.136060 0.0909122i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(522\) 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
\(527\) −1.86075 + 0.116262i −0.0810556 + 0.00506445i
\(528\) 0 0
\(529\) −2.45974 + 1.01886i −0.106945 + 0.0442981i
\(530\) 0 0
\(531\) 32.1975 32.1975i 1.39725 1.39725i
\(532\) 0 0
\(533\) −4.66669 23.4610i −0.202137 1.01621i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 10.9345 + 16.3646i 0.470981 + 0.704873i
\(540\) 0 0
\(541\) −37.8536 7.52955i −1.62745 0.323720i −0.704816 0.709390i \(-0.748970\pi\)
−0.922637 + 0.385670i \(0.873970\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −0.0249681 + 0.125523i −0.00106756 + 0.00536697i −0.981315 0.192406i \(-0.938371\pi\)
0.980248 + 0.197773i \(0.0633709\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 24.2567 + 36.3028i 1.02872 + 1.53958i
\(557\) −32.9636 32.9636i −1.39671 1.39671i −0.809260 0.587450i \(-0.800132\pi\)
−0.587450 0.809260i \(-0.699868\pi\)
\(558\) 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 \(0.462849\pi\)
−0.784639 + 0.619953i \(0.787151\pi\)
\(570\) 0 0
\(571\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(572\) 13.8735 2.75962i 0.580082 0.115385i
\(573\) 0 0
\(574\) 0 0
\(575\) −24.8423 4.94145i −1.03600 0.206073i
\(576\) 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
\(583\) −2.53277 + 1.69234i −0.104897 + 0.0700897i
\(584\) 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 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(602\) 0 0
\(603\) −18.7908 + 45.3651i −0.765222 + 1.84741i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(608\) 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 \(-0.302697\pi\)
0.580909 + 0.813969i \(0.302697\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 40.5148 27.0712i 1.63107 1.08984i 0.707281 0.706932i \(-0.249922\pi\)
0.923785 0.382911i \(-0.125078\pi\)
\(618\) 0 0
\(619\) −2.90893 1.94369i −0.116920 0.0781233i 0.495735 0.868474i \(-0.334899\pi\)
−0.612655 + 0.790350i \(0.709899\pi\)
\(620\) 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 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(642\) 0 0
\(643\) 23.2538 + 4.62547i 0.917040 + 0.182411i 0.630978 0.775800i \(-0.282654\pi\)
0.286062 + 0.958211i \(0.407654\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) −8.32554 + 41.8553i −0.326806 + 1.64296i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −21.1324 + 31.6269i −0.825083 + 1.23482i
\(657\) 0 0
\(658\) 0 0
\(659\) −23.7632 23.7632i −0.925684 0.925684i 0.0717399 0.997423i \(-0.477145\pi\)
−0.997423 + 0.0717399i \(0.977145\pi\)
\(660\) 0 0
\(661\) 2.50942 + 6.05828i 0.0976052 + 0.235640i 0.965139 0.261739i \(-0.0842959\pi\)
−0.867533 + 0.497379i \(0.834296\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −32.1638 + 21.4912i −1.24446 + 0.831519i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −9.43608 9.43608i −0.362926 0.362926i
\(677\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −6.53059 + 32.8315i −0.249886 + 1.25626i 0.628311 + 0.777962i \(0.283747\pi\)
−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 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(692\) 22.9985 + 34.4197i 0.874272 + 1.30844i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −37.0887 12.7162i −1.40483 0.481660i
\(698\) 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
\(704\) −18.7024 12.4965i −0.704873 0.470981i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 26.6404 + 39.8702i 1.00050 + 1.49736i 0.861944 + 0.507003i \(0.169247\pi\)
0.138558 + 0.990354i \(0.455753\pi\)
\(710\) 0 0
\(711\) −42.1581 8.38577i −1.58105 0.314491i
\(712\) 0 0
\(713\) 2.29065i 0.0857855i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 14.9540 9.99195i 0.557690 0.372637i −0.244551 0.969636i \(-0.578641\pi\)
0.802242 + 0.597000i \(0.203641\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) −29.8856 + 44.7269i −1.11069 + 1.66226i
\(725\) 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
\(737\) −8.97805 45.1357i −0.330711 1.66260i
\(738\) 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 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 19.6723i 0.719772i
\(748\) 7.51964 21.9322i 0.274945 0.801919i
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(752\) −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 −0.998068 0.0621249i \(-0.980212\pi\)
0.0621249 + 0.998068i \(0.480212\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −19.4545 12.9991i −0.700184 0.467848i
\(773\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(774\) 0 0
\(775\) −1.25609 + 1.87986i −0.0451199 + 0.0675267i
\(776\) 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
\(787\) 9.47882 6.33354i 0.337883 0.225766i −0.375040 0.927009i \(-0.622371\pi\)
0.712923 + 0.701242i \(0.247371\pi\)
\(788\) 1.83021 + 9.20110i 0.0651986 + 0.327776i
\(789\) 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