Properties

Label 731.2.s.a.601.1
Level 731
Weight 2
Character 731.601
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 601.1
Root \(-3.22048 - 0.792772i\)
Character \(\chi\) = 731.601
Dual form 731.2.s.a.343.1

$q$-expansion

\(f(q)\) \(=\) \(q+(1.41421 + 1.41421i) q^{4} +(2.77164 - 1.14805i) q^{9} +O(q^{10})\) \(q+(1.41421 + 1.41421i) q^{4} +(2.77164 - 1.14805i) q^{9} +(-0.955307 + 4.80265i) q^{11} +(-4.77872 + 4.77872i) q^{13} +4.00000i q^{16} +(2.07243 - 3.56441i) q^{17} +(5.47887 + 1.08981i) q^{23} +(-1.91342 - 4.61940i) q^{25} +(0.749168 + 3.76632i) q^{31} +(5.54328 + 2.29610i) q^{36} +(-2.38087 - 3.56322i) q^{41} +(6.05828 - 2.50942i) q^{43} +(-8.14298 + 5.44097i) q^{44} +(-6.16199 + 6.16199i) q^{47} +(-2.67878 + 6.46716i) q^{49} -13.5163 q^{52} +(10.1933 + 4.22221i) q^{53} +(3.46524 + 8.36582i) q^{59} +(-5.65685 + 5.65685i) q^{64} -11.3488i q^{67} +(7.97170 - 2.10997i) q^{68} +(2.96615 - 14.9118i) q^{79} +(6.36396 - 6.36396i) q^{81} +(2.50942 - 6.05828i) q^{83} +(6.20706 + 9.28952i) q^{92} +(-16.2049 - 10.8277i) q^{97} +(2.86592 + 14.4080i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 24q^{13} + 56q^{23} - 64q^{59} + 96q^{79} + O(q^{100}) \)

Character values

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

\(n\) \(173\) \(562\)
\(\chi(n)\) \(e\left(\frac{15}{16}\right)\) \(-1\)

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 \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(3\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(4\) 1.41421 + 1.41421i 0.707107 + 0.707107i
\(5\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(6\) 0 0
\(7\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(8\) 0 0
\(9\) 2.77164 1.14805i 0.923880 0.382683i
\(10\) 0 0
\(11\) −0.955307 + 4.80265i −0.288036 + 1.44805i 0.517590 + 0.855629i \(0.326829\pi\)
−0.805626 + 0.592425i \(0.798171\pi\)
\(12\) 0 0
\(13\) −4.77872 + 4.77872i −1.32538 + 1.32538i −0.416025 + 0.909353i \(0.636577\pi\)
−0.909353 + 0.416025i \(0.863423\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000i 1.00000i
\(17\) 2.07243 3.56441i 0.502639 0.864496i
\(18\) 0 0
\(19\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.47887 + 1.08981i 1.14242 + 0.227242i 0.729800 0.683660i \(-0.239613\pi\)
0.412622 + 0.910902i \(0.364613\pi\)
\(24\) 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 \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(30\) 0 0
\(31\) 0.749168 + 3.76632i 0.134555 + 0.676451i 0.987898 + 0.155103i \(0.0495709\pi\)
−0.853344 + 0.521349i \(0.825429\pi\)
\(32\) 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 \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.38087 3.56322i −0.371829 0.556481i 0.597619 0.801781i \(-0.296114\pi\)
−0.969447 + 0.245299i \(0.921114\pi\)
\(42\) 0 0
\(43\) 6.05828 2.50942i 0.923880 0.382683i
\(44\) −8.14298 + 5.44097i −1.22760 + 0.820257i
\(45\) 0 0
\(46\) 0 0
\(47\) −6.16199 + 6.16199i −0.898818 + 0.898818i −0.995332 0.0965136i \(-0.969231\pi\)
0.0965136 + 0.995332i \(0.469231\pi\)
\(48\) 0 0
\(49\) −2.67878 + 6.46716i −0.382683 + 0.923880i
\(50\) 0 0
\(51\) 0 0
\(52\) −13.5163 −1.87437
\(53\) 10.1933 + 4.22221i 1.40016 + 0.579965i 0.949793 0.312878i \(-0.101293\pi\)
0.450367 + 0.892844i \(0.351293\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.46524 + 8.36582i 0.451135 + 1.08914i 0.971891 + 0.235431i \(0.0756503\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −5.65685 + 5.65685i −0.707107 + 0.707107i
\(65\) 0 0
\(66\) 0 0
\(67\) 11.3488i 1.38647i −0.720710 0.693236i \(-0.756184\pi\)
0.720710 0.693236i \(-0.243816\pi\)
\(68\) 7.97170 2.10997i 0.966711 0.255872i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(72\) 0 0
\(73\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 2.96615 14.9118i 0.333717 1.67771i −0.341335 0.939942i \(-0.610879\pi\)
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 \(-0.757816\pi\)
0.999699 + 0.0245507i \(0.00781552\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 6.20706 + 9.28952i 0.647130 + 0.968499i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −16.2049 10.8277i −1.64535 1.09939i −0.903260 0.429093i \(-0.858833\pi\)
−0.742093 0.670297i \(-0.766167\pi\)
\(98\) 0 0
\(99\) 2.86592 + 14.4080i 0.288036 + 1.44805i
\(100\) 3.82683 9.23880i 0.382683 0.923880i
\(101\) 20.0631i 1.99636i 0.0603342 + 0.998178i \(0.480783\pi\)
−0.0603342 + 0.998178i \(0.519217\pi\)
\(102\) 0 0
\(103\) 20.0883 1.97936 0.989678 0.143310i \(-0.0457745\pi\)
0.989678 + 0.143310i \(0.0457745\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 11.4391 17.1198i 1.10585 1.65503i 0.471923 0.881640i \(-0.343560\pi\)
0.633932 0.773389i \(-0.281440\pi\)
\(108\) 0 0
\(109\) −9.30560 13.9268i −0.891315 1.33395i −0.942133 0.335239i \(-0.891183\pi\)
0.0508181 0.998708i \(-0.483817\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −7.75867 + 18.7311i −0.717290 + 1.73169i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −11.9902 4.96649i −1.09002 0.451499i
\(122\) 0 0
\(123\) 0 0
\(124\) −4.26690 + 6.38587i −0.383179 + 0.573468i
\(125\) 0 0
\(126\) 0 0
\(127\) −1.00814 2.43387i −0.0894583 0.215971i 0.872818 0.488046i \(-0.162290\pi\)
−0.962276 + 0.272075i \(0.912290\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(132\) 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
\(139\) −16.2694 + 3.23618i −1.37995 + 0.274489i −0.828626 0.559803i \(-0.810877\pi\)
−0.551323 + 0.834292i \(0.685877\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −18.3854 27.5157i −1.53746 2.30098i
\(144\) 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 \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(150\) 0 0
\(151\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(152\) 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 \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(164\) 1.67210 8.40620i 0.130569 0.656414i
\(165\) 0 0
\(166\) 0 0
\(167\) −4.76603 23.9604i −0.368806 1.85411i −0.504566 0.863373i \(-0.668347\pi\)
0.135760 0.990742i \(-0.456653\pi\)
\(168\) 0 0
\(169\) 32.6723i 2.51326i
\(170\) 0 0
\(171\) 0 0
\(172\) 12.1166 + 5.01885i 0.923880 + 0.382683i
\(173\) 12.6615 2.51852i 0.962634 0.191480i 0.311335 0.950300i \(-0.399224\pi\)
0.651300 + 0.758820i \(0.274224\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −19.2106 3.82123i −1.44805 0.288036i
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(180\) 0 0
\(181\) 2.14636 10.7905i 0.159538 0.802050i −0.815283 0.579062i \(-0.803419\pi\)
0.974821 0.222988i \(-0.0715812\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 15.1388 + 13.3583i 1.10706 + 0.976854i
\(188\) −17.4287 −1.27112
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(192\) 0 0
\(193\) −27.2271 5.41580i −1.95985 0.389838i −0.988097 0.153831i \(-0.950839\pi\)
−0.971751 0.236007i \(-0.924161\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −12.9343 + 5.35757i −0.923880 + 0.382683i
\(197\) 19.7681 13.2086i 1.40842 0.941074i 0.408822 0.912614i \(-0.365940\pi\)
0.999595 0.0284595i \(-0.00906017\pi\)
\(198\) 0 0
\(199\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 16.4366 3.26944i 1.14242 0.227242i
\(208\) −19.1149 19.1149i −1.32538 1.32538i
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(212\) 8.44442 + 20.3866i 0.579965 + 1.40016i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 7.12974 + 26.9369i 0.479598 + 1.81197i
\(222\) 0 0
\(223\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(224\) 0 0
\(225\) −10.6066 10.6066i −0.707107 0.707107i
\(226\) 0 0
\(227\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(228\) 0 0
\(229\) 11.0127 + 26.5869i 0.727738 + 1.75691i 0.649992 + 0.759941i \(0.274772\pi\)
0.0777462 + 0.996973i \(0.475228\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −6.93047 + 16.7316i −0.451135 + 1.08914i
\(237\) 0 0
\(238\) 0 0
\(239\) 10.5254 0.680830 0.340415 0.940275i \(-0.389432\pi\)
0.340415 + 0.940275i \(0.389432\pi\)
\(240\) 0 0
\(241\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(242\) 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 −0.206951 0.978351i \(-0.566354\pi\)
0.978351 + 0.206951i \(0.0663540\pi\)
\(252\) 0 0
\(253\) −10.4680 + 25.2720i −0.658117 + 1.58884i
\(254\) 0 0
\(255\) 0 0
\(256\) −16.0000 −1.00000
\(257\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 16.0496 16.0496i 0.980384 0.980384i
\(269\) 1.05655 + 5.31162i 0.0644188 + 0.323855i 0.999535 0.0304855i \(-0.00970535\pi\)
−0.935116 + 0.354341i \(0.884705\pi\)
\(270\) 0 0
\(271\) 29.1434i 1.77033i −0.465274 0.885167i \(-0.654044\pi\)
0.465274 0.885167i \(-0.345956\pi\)
\(272\) 14.2576 + 8.28973i 0.864496 + 0.502639i
\(273\) 0 0
\(274\) 0 0
\(275\) 24.0133 4.77653i 1.44805 0.288036i
\(276\) 0 0
\(277\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(278\) 0 0
\(279\) 6.40035 + 9.57880i 0.383179 + 0.573468i
\(280\) 0 0
\(281\) 5.61715 2.32670i 0.335091 0.138799i −0.208792 0.977960i \(-0.566953\pi\)
0.543884 + 0.839161i \(0.316953\pi\)
\(282\) 0 0
\(283\) −4.64044 + 23.3291i −0.275846 + 1.38677i 0.555732 + 0.831362i \(0.312438\pi\)
−0.831578 + 0.555409i \(0.812562\pi\)
\(284\) 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 0.642627 0.766179i \(-0.277845\pi\)
−0.766179 + 0.642627i \(0.777845\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −31.3899 + 20.9740i −1.81532 + 1.21296i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 7.07703 0.403907 0.201954 0.979395i \(-0.435271\pi\)
0.201954 + 0.979395i \(0.435271\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −19.2746 + 28.8465i −1.09296 + 1.63573i −0.396697 + 0.917950i \(0.629844\pi\)
−0.696265 + 0.717784i \(0.745156\pi\)
\(312\) 0 0
\(313\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 25.2833 16.8937i 1.42229 0.950347i
\(317\) 2.72983 13.7238i 0.153323 0.770804i −0.825228 0.564799i \(-0.808954\pi\)
0.978551 0.206005i \(-0.0660464\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 18.0000 1.00000
\(325\) 31.2185 + 12.9311i 1.73169 + 0.717290i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(332\) 12.1166 5.01885i 0.664983 0.275445i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 6.55106 + 32.9344i 0.356859 + 1.79405i 0.575022 + 0.818138i \(0.304994\pi\)
−0.218163 + 0.975912i \(0.570006\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −18.8040 −1.01829
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(348\) 0 0
\(349\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −21.7787 + 21.7787i −1.15916 + 1.15916i −0.174509 + 0.984656i \(0.555834\pi\)
−0.984656 + 0.174509i \(0.944166\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 9.00692 + 3.73079i 0.475367 + 0.196903i 0.607486 0.794330i \(-0.292178\pi\)
−0.132119 + 0.991234i \(0.542178\pi\)
\(360\) 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
\(367\) 7.56297 5.05341i 0.394784 0.263786i −0.342296 0.939592i \(-0.611204\pi\)
0.737079 + 0.675806i \(0.236204\pi\)
\(368\) −4.35926 + 21.9155i −0.227242 + 1.14242i
\(369\) −10.6897 7.14260i −0.556481 0.371829i
\(370\) 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
\(379\) −15.5895 + 23.3313i −0.800777 + 1.19845i 0.176042 + 0.984383i \(0.443670\pi\)
−0.976819 + 0.214065i \(0.931330\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 13.9104 13.9104i 0.707107 0.707107i
\(388\) −7.60439 38.2298i −0.386054 1.94083i
\(389\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(390\) 0 0
\(391\) 15.2391 17.2704i 0.770676 0.873400i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) −16.3229 + 24.4289i −0.820257 + 1.22760i
\(397\) −37.7370 7.50636i −1.89397 0.376734i −0.896208 0.443634i \(-0.853689\pi\)
−0.997760 + 0.0669005i \(0.978689\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 18.4776 7.65367i 0.923880 0.382683i
\(401\) −31.0466 + 20.7447i −1.55039 + 1.03594i −0.574283 + 0.818657i \(0.694719\pi\)
−0.976109 + 0.217281i \(0.930281\pi\)
\(402\) 0 0
\(403\) −21.5783 14.4181i −1.07489 0.718218i
\(404\) −28.3736 + 28.3736i −1.41164 + 1.41164i
\(405\) 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
\(412\) 28.4091 + 28.4091i 1.39962 + 1.39962i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(420\) 0 0
\(421\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(422\) 0 0
\(423\) −10.0045 + 24.1531i −0.486437 + 1.17436i
\(424\) 0 0
\(425\) −20.4309 2.75319i −0.991042 0.133549i
\(426\) 0 0
\(427\) 0 0
\(428\) 40.3882 8.03372i 1.95224 0.388325i
\(429\) 0 0
\(430\) 0 0
\(431\) 38.1848 + 7.59543i 1.83930 + 0.365859i 0.987450 0.157930i \(-0.0504821\pi\)
0.851848 + 0.523789i \(0.175482\pi\)
\(432\) 0 0
\(433\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 6.53539 32.8556i 0.312988 1.57350i
\(437\) 0 0
\(438\) 0 0
\(439\) −0.819217 4.11848i −0.0390991 0.196564i 0.956299 0.292391i \(-0.0944507\pi\)
−0.995398 + 0.0958262i \(0.969451\pi\)
\(440\) 0 0
\(441\) 21.0000i 1.00000i
\(442\) 0 0
\(443\) −37.5579 −1.78443 −0.892215 0.451612i \(-0.850849\pi\)
−0.892215 + 0.451612i \(0.850849\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(450\) 0 0
\(451\) 19.3873 8.03050i 0.912915 0.378142i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 24.2331 + 10.0377i 1.12865 + 0.467502i 0.867322 0.497748i \(-0.165840\pi\)
0.261328 + 0.965250i \(0.415840\pi\)
\(462\) 0 0
\(463\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(468\) −37.4622 + 15.5173i −1.73169 + 0.717290i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 6.26437 + 31.4931i 0.288036 + 1.44805i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 33.0995 1.51552
\(478\) 0 0
\(479\) 42.5397 8.46167i 1.94369 0.386624i 0.945561 0.325444i \(-0.105514\pi\)
0.998127 0.0611793i \(-0.0194862\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −9.93298 23.9803i −0.451499 1.09002i
\(485\) 0 0
\(486\) 0 0
\(487\) 1.02019 5.12883i 0.0462292 0.232410i −0.950762 0.309921i \(-0.899697\pi\)
0.996991 + 0.0775113i \(0.0246974\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −15.0653 + 2.99667i −0.676451 + 0.134555i
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 2.01629 4.86775i 0.0894583 0.215971i
\(509\) 42.1546i 1.86847i −0.356658 0.934235i \(-0.616084\pi\)
0.356658 0.934235i \(-0.383916\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −23.7073 35.4805i −1.04265 1.56043i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(522\) 0 0
\(523\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 14.9773 + 5.13511i 0.652422 + 0.223689i
\(528\) 0 0
\(529\) 7.58106 + 3.14018i 0.329611 + 0.136529i
\(530\) 0 0
\(531\) 19.2088 + 19.2088i 0.833589 + 0.833589i
\(532\) 0 0
\(533\) 28.4051 + 5.65013i 1.23036 + 0.244734i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −28.5004 19.0434i −1.22760 0.820257i
\(540\) 0 0
\(541\) 7.56216 + 38.0176i 0.325123 + 1.63450i 0.704816 + 0.709390i \(0.251030\pi\)
−0.379693 + 0.925113i \(0.623970\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 42.4330 8.44044i 1.81430 0.360887i 0.832987 0.553293i \(-0.186629\pi\)
0.981315 + 0.192406i \(0.0616291\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −27.5850 18.4317i −1.16986 0.781678i
\(557\) −27.0104 + 27.0104i −1.14447 + 1.14447i −0.156844 + 0.987623i \(0.550132\pi\)
−0.987623 + 0.156844i \(0.949868\pi\)
\(558\) 0 0
\(559\) −16.9590 + 40.9427i −0.717290 + 1.73169i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 22.2141 + 9.20140i 0.936214 + 0.387793i 0.798033 0.602614i \(-0.205874\pi\)
0.138182 + 0.990407i \(0.455874\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −8.90324 21.4943i −0.373243 0.901088i −0.993197 0.116450i \(-0.962849\pi\)
0.619953 0.784639i \(-0.287151\pi\)
\(570\) 0 0
\(571\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(572\) 12.9122 64.9139i 0.539885 2.71419i
\(573\) 0 0
\(574\) 0 0
\(575\) −5.44907 27.3943i −0.227242 1.14242i
\(576\) −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
\(583\) −30.0155 + 44.9214i −1.24312 + 1.86046i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 32.4577 + 32.4577i 1.32618 + 1.32618i 0.908671 + 0.417514i \(0.137098\pi\)
0.417514 + 0.908671i \(0.362902\pi\)
\(600\) 0 0
\(601\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(602\) 0 0
\(603\) −13.0290 31.4547i −0.530580 1.28093i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 58.8928i 2.38255i
\(612\) 19.6723 15.0000i 0.795206 0.606339i
\(613\) −48.8406 −1.97265 −0.986327 0.164798i \(-0.947303\pi\)
−0.986327 + 0.164798i \(0.947303\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 15.3282 22.9402i 0.617089 0.923539i −0.382911 0.923785i \(-0.625078\pi\)
1.00000 0.000246592i \(7.84928e-5\pi\)
\(618\) 0 0
\(619\) 24.7335 + 37.0163i 0.994123 + 1.48781i 0.868474 + 0.495735i \(0.165101\pi\)
0.125649 + 0.992075i \(0.459899\pi\)
\(620\) 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 \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −18.1036 43.7059i −0.717290 1.73169i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(642\) 0 0
\(643\) −6.31034 31.7242i −0.248856 1.25108i −0.879834 0.475281i \(-0.842346\pi\)
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
\(649\) −43.4885 + 8.65040i −1.70707 + 0.339558i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 14.2529 9.52346i 0.556481 0.371829i
\(657\) 0 0
\(658\) 0 0
\(659\) −36.2107 + 36.2107i −1.41057 + 1.41057i −0.654557 + 0.756013i \(0.727145\pi\)
−0.756013 + 0.654557i \(0.772855\pi\)
\(660\) 0 0
\(661\) 2.50942 6.05828i 0.0976052 0.235640i −0.867533 0.497379i \(-0.834296\pi\)
0.965139 + 0.261739i \(0.0842959\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 27.1450 40.6254i 1.05027 1.57184i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 46.2056 46.2056i 1.77714 1.77714i
\(677\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 48.9386 9.73449i 1.87258 0.372480i 0.878197 0.478299i \(-0.158747\pi\)
0.994385 + 0.105819i \(0.0337466\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 10.0377 + 24.2331i 0.382683 + 0.923880i
\(689\) −68.8878 + 28.5342i −2.62441 + 1.08707i
\(690\) 0 0
\(691\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(692\) 21.4678 + 14.3443i 0.816082 + 0.545289i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −17.6350 + 1.10185i −0.667972 + 0.0417357i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −36.1149 36.1149i −1.36404 1.36404i −0.868698 0.495342i \(-0.835043\pi\)
−0.495342 0.868698i \(-0.664957\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −21.7639 32.5719i −0.820257 1.22760i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −44.2064 29.5378i −1.66021 1.10931i −0.861944 0.507003i \(-0.830753\pi\)
−0.798262 0.602311i \(-0.794247\pi\)
\(710\) 0 0
\(711\) −8.89844 44.7355i −0.333717 1.67771i
\(712\) 0 0
\(713\) 21.4516i 0.803370i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −19.9729 + 29.8915i −0.744862 + 1.11476i 0.244551 + 0.969636i \(0.421359\pi\)
−0.989413 + 0.145128i \(0.953641\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 18.2955 12.2246i 0.679946 0.454325i
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(728\) 0 0
\(729\) 10.3325 24.9447i 0.382683 0.923880i
\(730\) 0 0
\(731\) 3.61077 26.7948i 0.133549 0.991042i
\(732\) 0 0
\(733\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 54.5042 + 10.8416i 2.00769 + 0.399354i
\(738\) 0 0
\(739\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 19.6723i 0.719772i
\(748\) 2.51805 + 40.3010i 0.0920691 + 1.47355i
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(752\) −24.6479 24.6479i −0.898818 0.898818i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −56.5373 23.4185i −2.04144 0.845594i
\(768\) 0 0
\(769\) −2.43637 2.43637i −0.0878579 0.0878579i 0.661812 0.749670i \(-0.269788\pi\)
−0.749670 + 0.661812i \(0.769788\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −30.8458 46.1640i −1.11016 1.66148i
\(773\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(774\) 0 0
\(775\) 15.9647 10.6673i 0.573468 0.383179i
\(776\) 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
\(787\) −5.82852 + 8.72299i −0.207764 + 0.310941i −0.920687 0.390301i \(-0.872371\pi\)
0.712923 + 0.701242i \(0.247371\pi\)
\(788\) 46.6361 + 9.27649i 1.66134 + 0.330461i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.95129 + 4.71083i −0.0691182 + 0.166866i −0.954664 0.297687i \(-0.903785\pi\)
0.885545 + 0.464553i \(0.153785\pi\)
\(798\) 0 0
\(799\) 9.19354 + 34.7342i 0.325244 + 1.22881i
\(800\) 0 0
\(801\) 0 0
\(802\)