Properties

Label 731.2.s.a.515.2
Level 731
Weight 2
Character 731.515
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 515.2
Root \(0.792772 + 3.22048i\)
Character \(\chi\) = 731.515
Dual form 731.2.s.a.687.2

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.41421 + 1.41421i) q^{4} +(-1.14805 + 2.77164i) q^{9} +O(q^{10})\) \(q+(-1.41421 + 1.41421i) q^{4} +(-1.14805 + 2.77164i) q^{9} +(5.50976 + 3.68150i) q^{11} +(-4.77872 - 4.77872i) q^{13} -4.00000i q^{16} +(-3.56441 + 2.07243i) q^{17} +(-0.868534 + 1.29985i) q^{23} +(-4.61940 - 1.91342i) q^{25} +(-8.40313 + 5.61479i) q^{31} +(-2.29610 - 5.54328i) q^{36} +(1.07359 + 5.39731i) q^{41} +(-2.50942 + 6.05828i) q^{43} +(-12.9984 + 2.58554i) q^{44} +(7.48531 + 7.48531i) q^{47} +(-6.46716 + 2.67878i) q^{49} +13.5163 q^{52} +(-3.63588 - 8.77779i) q^{53} +(-11.4652 - 4.74906i) q^{59} +(5.65685 + 5.65685i) q^{64} -11.7985i q^{67} +(2.10997 - 7.97170i) q^{68} +(14.3544 + 9.59129i) q^{79} +(-6.36396 - 6.36396i) q^{81} +(6.05828 - 2.50942i) q^{83} +(-0.609977 - 3.06656i) q^{92} +(11.5348 + 2.29442i) q^{97} +(-16.5293 + 11.0445i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 24q^{13} + 56q^{23} - 64q^{59} + 96q^{79} + O(q^{100}) \)

Character values

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

\(n\) \(173\) \(562\)
\(\chi(n)\) \(e\left(\frac{5}{16}\right)\) \(-1\)

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 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(3\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(4\) −1.41421 + 1.41421i −0.707107 + 0.707107i
\(5\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(6\) 0 0
\(7\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(8\) 0 0
\(9\) −1.14805 + 2.77164i −0.382683 + 0.923880i
\(10\) 0 0
\(11\) 5.50976 + 3.68150i 1.66125 + 1.11001i 0.855629 + 0.517590i \(0.173171\pi\)
0.805626 + 0.592425i \(0.201829\pi\)
\(12\) 0 0
\(13\) −4.77872 4.77872i −1.32538 1.32538i −0.909353 0.416025i \(-0.863423\pi\)
−0.416025 0.909353i \(-0.636577\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000i 1.00000i
\(17\) −3.56441 + 2.07243i −0.864496 + 0.502639i
\(18\) 0 0
\(19\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −0.868534 + 1.29985i −0.181102 + 0.271038i −0.910902 0.412622i \(-0.864613\pi\)
0.729800 + 0.683660i \(0.239613\pi\)
\(24\) 0 0
\(25\) −4.61940 1.91342i −0.923880 0.382683i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(30\) 0 0
\(31\) −8.40313 + 5.61479i −1.50925 + 1.00845i −0.521349 + 0.853344i \(0.674571\pi\)
−0.987898 + 0.155103i \(0.950429\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −2.29610 5.54328i −0.382683 0.923880i
\(37\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.07359 + 5.39731i 0.167667 + 0.842918i 0.969447 + 0.245299i \(0.0788863\pi\)
−0.801781 + 0.597619i \(0.796114\pi\)
\(42\) 0 0
\(43\) −2.50942 + 6.05828i −0.382683 + 0.923880i
\(44\) −12.9984 + 2.58554i −1.95958 + 0.389785i
\(45\) 0 0
\(46\) 0 0
\(47\) 7.48531 + 7.48531i 1.09185 + 1.09185i 0.995332 + 0.0965136i \(0.0307691\pi\)
0.0965136 + 0.995332i \(0.469231\pi\)
\(48\) 0 0
\(49\) −6.46716 + 2.67878i −0.923880 + 0.382683i
\(50\) 0 0
\(51\) 0 0
\(52\) 13.5163 1.87437
\(53\) −3.63588 8.77779i −0.499426 1.20572i −0.949793 0.312878i \(-0.898707\pi\)
0.450367 0.892844i \(-0.351293\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −11.4652 4.74906i −1.49265 0.618274i −0.520756 0.853706i \(-0.674350\pi\)
−0.971891 + 0.235431i \(0.924350\pi\)
\(60\) 0 0
\(61\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 5.65685 + 5.65685i 0.707107 + 0.707107i
\(65\) 0 0
\(66\) 0 0
\(67\) 11.7985i 1.44142i −0.693236 0.720710i \(-0.743816\pi\)
0.693236 0.720710i \(-0.256184\pi\)
\(68\) 2.10997 7.97170i 0.255872 0.966711i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(72\) 0 0
\(73\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 14.3544 + 9.59129i 1.61499 + 1.07910i 0.939942 + 0.341335i \(0.110879\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 0 0
\(81\) −6.36396 6.36396i −0.707107 0.707107i
\(82\) 0 0
\(83\) 6.05828 2.50942i 0.664983 0.275445i −0.0245507 0.999699i \(-0.507816\pi\)
0.689534 + 0.724254i \(0.257816\pi\)
\(84\) 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
\(92\) −0.609977 3.06656i −0.0635945 0.319711i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 11.5348 + 2.29442i 1.17119 + 0.232963i 0.742093 0.670297i \(-0.233833\pi\)
0.429093 + 0.903260i \(0.358833\pi\)
\(98\) 0 0
\(99\) −16.5293 + 11.0445i −1.66125 + 1.11001i
\(100\) 9.23880 3.82683i 0.923880 0.382683i
\(101\) 1.21270i 0.120668i 0.998178 + 0.0603342i \(0.0192166\pi\)
−0.998178 + 0.0603342i \(0.980783\pi\)
\(102\) 0 0
\(103\) −2.90887 −0.286620 −0.143310 0.989678i \(-0.545775\pi\)
−0.143310 + 0.989678i \(0.545775\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −2.56231 + 12.8816i −0.247708 + 1.24531i 0.633932 + 0.773389i \(0.281440\pi\)
−0.881640 + 0.471923i \(0.843560\pi\)
\(108\) 0 0
\(109\) 0.590659 + 2.96944i 0.0565749 + 0.284421i 0.998708 0.0508181i \(-0.0161829\pi\)
−0.942133 + 0.335239i \(0.891183\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 18.7311 7.75867i 1.73169 0.717290i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 12.5945 + 30.4057i 1.14495 + 2.76415i
\(122\) 0 0
\(123\) 0 0
\(124\) 3.94331 19.8243i 0.354119 1.78028i
\(125\) 0 0
\(126\) 0 0
\(127\) 20.6805 + 8.56613i 1.83509 + 0.760121i 0.962276 + 0.272075i \(0.0877098\pi\)
0.872818 + 0.488046i \(0.162290\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(132\) 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
\(139\) −13.1000 19.6055i −1.11113 1.66292i −0.559803 0.828626i \(-0.689123\pi\)
−0.551323 0.834292i \(-0.685877\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −8.73672 43.9225i −0.730601 3.67298i
\(144\) 11.0866 + 4.59220i 0.923880 + 0.382683i
\(145\) 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 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(152\) 0 0
\(153\) −1.65191 12.2585i −0.133549 0.991042i
\(154\) 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 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(164\) −9.15123 6.11466i −0.714591 0.477475i
\(165\) 0 0
\(166\) 0 0
\(167\) 19.3236 12.9116i 1.49531 0.999133i 0.504566 0.863373i \(-0.331653\pi\)
0.990742 0.135760i \(-0.0433475\pi\)
\(168\) 0 0
\(169\) 32.6723i 2.51326i
\(170\) 0 0
\(171\) 0 0
\(172\) −5.01885 12.1166i −0.382683 0.923880i
\(173\) 3.93274 + 5.88576i 0.299001 + 0.447486i 0.950300 0.311335i \(-0.100776\pi\)
−0.651300 + 0.758820i \(0.725776\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 14.7260 22.0390i 1.11001 1.66125i
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(180\) 0 0
\(181\) 20.9054 + 13.9685i 1.55388 + 1.03827i 0.974821 + 0.222988i \(0.0715812\pi\)
0.579062 + 0.815283i \(0.303419\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −27.2687 1.70378i −1.99409 0.124593i
\(188\) −21.1717 −1.54410
\(189\) 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
\(193\) −11.3629 + 17.0058i −0.817921 + 1.22410i 0.153831 + 0.988097i \(0.450839\pi\)
−0.971751 + 0.236007i \(0.924161\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 5.35757 12.9343i 0.382683 0.923880i
\(197\) −6.13753 + 1.22083i −0.437281 + 0.0869806i −0.408822 0.912614i \(-0.634060\pi\)
−0.0284595 + 0.999595i \(0.509060\pi\)
\(198\) 0 0
\(199\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −2.60560 3.89956i −0.181102 0.271038i
\(208\) −19.1149 + 19.1149i −1.32538 + 1.32538i
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(212\) 17.5556 + 7.27176i 1.20572 + 0.499426i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 26.9369 + 7.12974i 1.81197 + 0.479598i
\(222\) 0 0
\(223\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(224\) 0 0
\(225\) 10.6066 10.6066i 0.707107 0.707107i
\(226\) 0 0
\(227\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(228\) 0 0
\(229\) 8.65964 + 3.58694i 0.572245 + 0.237032i 0.649992 0.759941i \(-0.274772\pi\)
−0.0777462 + 0.996973i \(0.524772\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 22.9305 9.49811i 1.49265 0.618274i
\(237\) 0 0
\(238\) 0 0
\(239\) −10.5254 −0.680830 −0.340415 0.940275i \(-0.610568\pi\)
−0.340415 + 0.940275i \(0.610568\pi\)
\(240\) 0 0
\(241\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(242\) 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
\(251\) 12.2213 + 12.2213i 0.771400 + 0.771400i 0.978351 0.206951i \(-0.0663540\pi\)
−0.206951 + 0.978351i \(0.566354\pi\)
\(252\) 0 0
\(253\) −9.57082 + 3.96436i −0.601712 + 0.249237i
\(254\) 0 0
\(255\) 0 0
\(256\) −16.0000 −1.00000
\(257\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 16.6856 + 16.6856i 1.01924 + 1.01924i
\(269\) 22.2052 14.8370i 1.35388 0.904631i 0.354341 0.935116i \(-0.384705\pi\)
0.999535 + 0.0304855i \(0.00970535\pi\)
\(270\) 0 0
\(271\) 15.3188i 0.930548i −0.885167 0.465274i \(-0.845956\pi\)
0.885167 0.465274i \(-0.154044\pi\)
\(272\) 8.28973 + 14.2576i 0.502639 + 0.864496i
\(273\) 0 0
\(274\) 0 0
\(275\) −18.4075 27.5488i −1.11001 1.66125i
\(276\) 0 0
\(277\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(278\) 0 0
\(279\) −5.91496 29.7365i −0.354119 1.78028i
\(280\) 0 0
\(281\) −12.6171 + 30.4605i −0.752676 + 1.81712i −0.208792 + 0.977960i \(0.566953\pi\)
−0.543884 + 0.839161i \(0.683047\pi\)
\(282\) 0 0
\(283\) −0.00543901 0.00363423i −0.000323316 0.000216033i 0.555409 0.831578i \(-0.312562\pi\)
−0.555732 + 0.831362i \(0.687562\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 8.41004 14.7740i 0.494708 0.869059i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −2.11488 + 2.11488i −0.123552 + 0.123552i −0.766179 0.642627i \(-0.777845\pi\)
0.642627 + 0.766179i \(0.277845\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 10.3621 2.06115i 0.599256 0.119199i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −34.3208 −1.95879 −0.979395 0.201954i \(-0.935271\pi\)
−0.979395 + 0.201954i \(0.935271\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −5.66244 + 28.4670i −0.321087 + 1.61422i 0.396697 + 0.917950i \(0.370156\pi\)
−0.717784 + 0.696265i \(0.754844\pi\)
\(312\) 0 0
\(313\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −33.8643 + 6.73603i −1.90502 + 0.378931i
\(317\) 11.0250 + 7.36664i 0.619223 + 0.413752i 0.825228 0.564799i \(-0.191046\pi\)
−0.206005 + 0.978551i \(0.566046\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 18.0000 1.00000
\(325\) 12.9311 + 31.2185i 0.717290 + 1.73169i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(332\) −5.01885 + 12.1166i −0.275445 + 0.664983i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −11.0141 + 7.35937i −0.599975 + 0.400890i −0.818138 0.575022i \(-0.804994\pi\)
0.218163 + 0.975912i \(0.429994\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −66.9701 −3.62663
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(348\) 0 0
\(349\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −21.7787 21.7787i −1.15916 1.15916i −0.984656 0.174509i \(-0.944166\pi\)
−0.174509 0.984656i \(-0.555834\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 3.73079 + 9.00692i 0.196903 + 0.475367i 0.991234 0.132119i \(-0.0421781\pi\)
−0.794330 + 0.607486i \(0.792178\pi\)
\(360\) 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
\(367\) −19.5040 + 3.87959i −1.01810 + 0.202513i −0.675806 0.737079i \(-0.736204\pi\)
−0.342296 + 0.939592i \(0.611204\pi\)
\(368\) 5.19941 + 3.47413i 0.271038 + 0.181102i
\(369\) −16.1919 3.22077i −0.842918 0.167667i
\(370\) 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
\(379\) −0.147240 + 0.740225i −0.00756320 + 0.0380228i −0.984383 0.176042i \(-0.943670\pi\)
0.976819 + 0.214065i \(0.0686705\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −13.9104 13.9104i −0.707107 0.707107i
\(388\) −19.5575 + 13.0679i −0.992883 + 0.663423i
\(389\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(390\) 0 0
\(391\) 0.401953 6.43319i 0.0203276 0.325340i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 7.75663 38.9952i 0.389785 1.95958i
\(397\) 11.0409 16.5238i 0.554126 0.829307i −0.443634 0.896208i \(-0.646311\pi\)
0.997760 + 0.0669005i \(0.0213110\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −7.65367 + 18.4776i −0.382683 + 0.923880i
\(401\) −15.8511 + 3.15297i −0.791564 + 0.157452i −0.574283 0.818657i \(-0.694719\pi\)
−0.217281 + 0.976109i \(0.569719\pi\)
\(402\) 0 0
\(403\) 66.9877 + 13.3247i 3.33690 + 0.663750i
\(404\) −1.71502 1.71502i −0.0853254 0.0853254i
\(405\) 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
\(412\) 4.11377 4.11377i 0.202671 0.202671i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(420\) 0 0
\(421\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(422\) 0 0
\(423\) −29.3401 + 12.1531i −1.42656 + 0.590902i
\(424\) 0 0
\(425\) 20.4309 2.75319i 0.991042 0.133549i
\(426\) 0 0
\(427\) 0 0
\(428\) −14.5937 21.8410i −0.705413 1.05573i
\(429\) 0 0
\(430\) 0 0
\(431\) 9.62585 14.4061i 0.463661 0.693917i −0.523789 0.851848i \(-0.675482\pi\)
0.987450 + 0.157930i \(0.0504821\pi\)
\(432\) 0 0
\(433\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −5.03475 3.36411i −0.241121 0.161112i
\(437\) 0 0
\(438\) 0 0
\(439\) −22.0445 + 14.7296i −1.05213 + 0.703008i −0.956299 0.292391i \(-0.905549\pi\)
−0.0958262 + 0.995398i \(0.530549\pi\)
\(440\) 0 0
\(441\) 21.0000i 1.00000i
\(442\) 0 0
\(443\) 37.5579 1.78443 0.892215 0.451612i \(-0.149151\pi\)
0.892215 + 0.451612i \(0.149151\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(450\) 0 0
\(451\) −13.9550 + 33.6903i −0.657114 + 1.58641i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −10.0377 24.2331i −0.467502 1.12865i −0.965250 0.261328i \(-0.915840\pi\)
0.497748 0.867322i \(-0.334160\pi\)
\(462\) 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 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(468\) −15.5173 + 37.4622i −0.717290 + 1.73169i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −36.1299 + 24.1412i −1.66125 + 1.11001i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 28.5030 1.30506
\(478\) 0 0
\(479\) −14.7224 22.0336i −0.672683 1.00674i −0.998127 0.0611793i \(-0.980514\pi\)
0.325444 0.945561i \(-0.394486\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −60.8114 25.1889i −2.76415 1.14495i
\(485\) 0 0
\(486\) 0 0
\(487\) 22.6920 + 15.1623i 1.02827 + 0.687070i 0.950762 0.309921i \(-0.100303\pi\)
0.0775113 + 0.996991i \(0.475303\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 22.4592 + 33.6125i 1.00845 + 1.50925i
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) −41.3609 + 17.1323i −1.83509 + 0.760121i
\(509\) 16.0932i 0.713316i 0.934235 + 0.356658i \(0.116084\pi\)
−0.934235 + 0.356658i \(0.883916\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 13.6851 + 68.7995i 0.601869 + 3.02580i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(522\) 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
\(527\) 18.3159 37.4284i 0.797854 1.63040i
\(528\) 0 0
\(529\) 7.86645 + 18.9913i 0.342020 + 0.825709i
\(530\) 0 0
\(531\) 26.3253 26.3253i 1.14242 1.14242i
\(532\) 0 0
\(533\) 20.6618 30.9226i 0.894963 1.33941i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −45.4944 9.04940i −1.95958 0.389785i
\(540\) 0 0
\(541\) 37.9112 25.3314i 1.62993 1.08908i 0.704816 0.709390i \(-0.251030\pi\)
0.925113 0.379693i \(-0.123970\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 10.0106 + 14.9819i 0.428022 + 0.640580i 0.981315 0.192406i \(-0.0616291\pi\)
−0.553293 + 0.832987i \(0.686629\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 46.2526 + 9.20020i 1.96155 + 0.390176i
\(557\) −19.6071 19.6071i −0.830780 0.830780i 0.156844 0.987623i \(-0.449868\pi\)
−0.987623 + 0.156844i \(0.949868\pi\)
\(558\) 0 0
\(559\) 40.9427 16.9590i 1.73169 0.717290i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −15.6567 37.7986i −0.659851 1.59302i −0.798033 0.602614i \(-0.794126\pi\)
0.138182 0.990407i \(-0.455874\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 21.4943 + 8.90324i 0.901088 + 0.373243i 0.784639 0.619953i \(-0.212849\pi\)
0.116450 + 0.993197i \(0.462849\pi\)
\(570\) 0 0
\(571\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(572\) 74.4713 + 49.7601i 3.11380 + 2.08058i
\(573\) 0 0
\(574\) 0 0
\(575\) 6.49926 4.34267i 0.271038 0.181102i
\(576\) −22.1731 + 9.18440i −0.923880 + 0.382683i
\(577\) 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
\(583\) 12.2826 61.7490i 0.508695 2.55738i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(594\) 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 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(602\) 0 0
\(603\) 32.7013 + 13.5453i 1.33170 + 0.551608i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 71.5404i 2.89422i
\(612\) 19.6723 + 15.0000i 0.795206 + 0.606339i
\(613\) −8.16043 −0.329597 −0.164798 0.986327i \(-0.552697\pi\)
−0.164798 + 0.986327i \(0.552697\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 9.50519 47.7858i 0.382665 1.92379i −0.000246592 1.00000i \(-0.500078\pi\)
0.382911 0.923785i \(-0.374922\pi\)
\(618\) 0 0
\(619\) 3.07516 + 15.4599i 0.123601 + 0.621385i 0.992075 + 0.125649i \(0.0401014\pi\)
−0.868474 + 0.495735i \(0.834899\pi\)
\(620\) 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 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 43.7059 + 18.1036i 1.73169 + 0.717290i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(642\) 0 0
\(643\) 3.94809 2.63803i 0.155697 0.104034i −0.475281 0.879834i \(-0.657654\pi\)
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
\(649\) −45.6870 68.3754i −1.79337 2.68397i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 21.5892 4.29437i 0.842918 0.167667i
\(657\) 0 0
\(658\) 0 0
\(659\) 2.60446 + 2.60446i 0.101455 + 0.101455i 0.756013 0.654557i \(-0.227145\pi\)
−0.654557 + 0.756013i \(0.727145\pi\)
\(660\) 0 0
\(661\) 6.05828 2.50942i 0.235640 0.0976052i −0.261739 0.965139i \(-0.584296\pi\)
0.497379 + 0.867533i \(0.334296\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −9.06793 + 45.5876i −0.350849 + 1.76384i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −46.2056 46.2056i −1.77714 1.77714i
\(677\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 25.7165 + 38.4875i 0.984016 + 1.47268i 0.878197 + 0.478299i \(0.158747\pi\)
0.105819 + 0.994385i \(0.466253\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 24.2331 + 10.0377i 0.923880 + 0.382683i
\(689\) −24.5717 + 59.3214i −0.936109 + 2.25997i
\(690\) 0 0
\(691\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(692\) −13.8855 2.76199i −0.527846 0.104995i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −15.0123 17.0133i −0.568631 0.644424i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −36.1149 + 36.1149i −1.36404 + 1.36404i −0.495342 + 0.868698i \(0.664957\pi\)
−0.868698 + 0.495342i \(0.835043\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 10.3422 + 51.9936i 0.389785 + 1.95958i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −38.9888 7.75535i −1.46426 0.291258i −0.602311 0.798262i \(-0.705753\pi\)
−0.861944 + 0.507003i \(0.830753\pi\)
\(710\) 0 0
\(711\) −43.0632 + 28.7739i −1.61499 + 1.07910i
\(712\) 0 0
\(713\) 15.7995i 0.591695i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 10.4489 52.5303i 0.389679 1.95905i 0.145128 0.989413i \(-0.453641\pi\)
0.244551 0.969636i \(-0.421359\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) −49.3191 + 9.81018i −1.83293 + 0.364593i
\(725\) 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
\(729\) 24.9447 10.3325i 0.923880 0.382683i
\(730\) 0 0
\(731\) −3.61077 26.7948i −0.133549 0.991042i
\(732\) 0 0
\(733\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 43.4363 65.0071i 1.60000 2.39457i
\(738\) 0 0
\(739\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 19.6723i 0.719772i
\(748\) 40.9733 36.1543i 1.49813 1.32193i
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(752\) 29.9413 29.9413i 1.09185 1.09185i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(758\) 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
\(767\) 32.0947 + 77.4836i 1.15887 + 2.79777i
\(768\) 0 0
\(769\) 39.1416 39.1416i 1.41148 1.41148i 0.661812 0.749670i \(-0.269788\pi\)
0.749670 0.661812i \(-0.230212\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −7.98025 40.1194i −0.287215 1.44393i
\(773\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(774\) 0 0
\(775\) 49.5608 9.85827i 1.78028 0.354119i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 10.7151 + 25.8686i 0.382683 + 0.923880i
\(785\) 0 0
\(786\) 0 0
\(787\) 9.05068 45.5008i 0.322622 1.62193i −0.390301 0.920687i \(-0.627629\pi\)
0.712923 0.701242i \(-0.247371\pi\)
\(788\) 6.95327 10.4063i 0.247700 0.370709i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 51.9513 21.5189i 1.84021 0.762240i 0.885545 0.464553i \(-0.153785\pi\)
0.954664 0.297687i \(-0.0962151\pi\)
\(798\) 0 0
\(799\) −42.1935 11.1679i −1.49270 0.395093i
\(800\) 0 0
\(801\) 0 0
\(802\) 0