Properties

Label 731.2.s.a.214.1
Level 731
Weight 2
Character 731.214
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 214.1
Root \(-0.792772 + 3.22048i\)
Character \(\chi\) = 731.214
Dual form 731.2.s.a.386.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.41421 - 1.41421i) q^{4} +(1.14805 + 2.77164i) q^{9} +O(q^{10})\) \(q+(-1.41421 - 1.41421i) q^{4} +(1.14805 + 2.77164i) q^{9} +(-0.165842 - 0.248200i) q^{11} +(-4.77872 + 4.77872i) q^{13} +4.00000i q^{16} +(3.56441 + 2.07243i) q^{17} +(7.86853 - 5.25759i) q^{23} +(4.61940 - 1.91342i) q^{25} +(-2.59764 + 3.88764i) q^{31} +(2.29610 - 5.54328i) q^{36} +(11.3414 + 2.25594i) q^{41} +(2.50942 + 6.05828i) q^{43} +(-0.116472 + 0.585544i) q^{44} +(-7.48531 + 7.48531i) q^{47} +(6.46716 + 2.67878i) q^{49} +13.5163 q^{52} +(-3.63588 + 8.77779i) q^{53} +(-11.4652 + 4.74906i) q^{59} +(5.65685 - 5.65685i) q^{64} -11.7985i q^{67} +(-2.10997 - 7.97170i) q^{68} +(-2.35438 - 3.52358i) q^{79} +(-6.36396 + 6.36396i) q^{81} +(-6.05828 - 2.50942i) q^{83} +(-18.5631 - 3.69244i) q^{92} +(3.08269 + 15.4977i) q^{97} +(0.497526 - 0.744600i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 24q^{13} + 56q^{23} - 64q^{59} + 96q^{79} + O(q^{100}) \)

Character values

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

\(n\) \(173\) \(562\)
\(\chi(n)\) \(e\left(\frac{3}{16}\right)\) \(-1\)

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 \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(3\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(4\) −1.41421 1.41421i −0.707107 0.707107i
\(5\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(6\) 0 0
\(7\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(8\) 0 0
\(9\) 1.14805 + 2.77164i 0.382683 + 0.923880i
\(10\) 0 0
\(11\) −0.165842 0.248200i −0.0500032 0.0748351i 0.805626 0.592425i \(-0.201829\pi\)
−0.855629 + 0.517590i \(0.826829\pi\)
\(12\) 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
\(17\) 3.56441 + 2.07243i 0.864496 + 0.502639i
\(18\) 0 0
\(19\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.86853 5.25759i 1.64070 1.09628i 0.729800 0.683660i \(-0.239613\pi\)
0.910902 0.412622i \(-0.135387\pi\)
\(24\) 0 0
\(25\) 4.61940 1.91342i 0.923880 0.382683i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(30\) 0 0
\(31\) −2.59764 + 3.88764i −0.466550 + 0.698241i −0.987898 0.155103i \(-0.950429\pi\)
0.521349 + 0.853344i \(0.325429\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 2.29610 5.54328i 0.382683 0.923880i
\(37\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 11.3414 + 2.25594i 1.77123 + 0.352319i 0.969447 0.245299i \(-0.0788863\pi\)
0.801781 + 0.597619i \(0.203886\pi\)
\(42\) 0 0
\(43\) 2.50942 + 6.05828i 0.382683 + 0.923880i
\(44\) −0.116472 + 0.585544i −0.0175588 + 0.0882740i
\(45\) 0 0
\(46\) 0 0
\(47\) −7.48531 + 7.48531i −1.09185 + 1.09185i −0.0965136 + 0.995332i \(0.530769\pi\)
−0.995332 + 0.0965136i \(0.969231\pi\)
\(48\) 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 0.450367 + 0.892844i \(0.351293\pi\)
−0.949793 + 0.312878i \(0.898707\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −11.4652 + 4.74906i −1.49265 + 0.618274i −0.971891 0.235431i \(-0.924350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(62\) 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 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(72\) 0 0
\(73\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −2.35438 3.52358i −0.264889 0.396434i 0.675053 0.737769i \(-0.264121\pi\)
−0.939942 + 0.341335i \(0.889121\pi\)
\(80\) 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 \(-0.492184\pi\)
−0.689534 + 0.724254i \(0.742184\pi\)
\(84\) 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
\(92\) −18.5631 3.69244i −1.93534 0.384963i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 3.08269 + 15.4977i 0.313000 + 1.57356i 0.742093 + 0.670297i \(0.233833\pi\)
−0.429093 + 0.903260i \(0.641167\pi\)
\(98\) 0 0
\(99\) 0.497526 0.744600i 0.0500032 0.0748351i
\(100\) −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 \(-0.454225\pi\)
0.143310 + 0.989678i \(0.454225\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 15.6772 3.11839i 1.51557 0.301466i 0.633932 0.773389i \(-0.281440\pi\)
0.881640 + 0.471923i \(0.156440\pi\)
\(108\) 0 0
\(109\) −20.2630 4.03056i −1.94084 0.386057i −0.998708 0.0508181i \(-0.983817\pi\)
−0.942133 0.335239i \(-0.891183\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(114\) 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
\(121\) 4.17542 10.0804i 0.379583 0.916396i
\(122\) 0 0
\(123\) 0 0
\(124\) 9.17157 1.82434i 0.823631 0.163830i
\(125\) 0 0
\(126\) 0 0
\(127\) 20.6805 8.56613i 1.83509 0.760121i 0.872818 0.488046i \(-0.162290\pi\)
0.962276 0.272075i \(-0.0877098\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(132\) 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
\(139\) 0.0999825 + 0.0668062i 0.00848041 + 0.00566643i 0.559803 0.828626i \(-0.310877\pi\)
−0.551323 + 0.834292i \(0.685877\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.97859 + 0.393566i 0.165458 + 0.0329116i
\(144\) −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 \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(150\) 0 0
\(151\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(152\) 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 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(164\) −12.8488 19.2295i −1.00332 1.50157i
\(165\) 0 0
\(166\) 0 0
\(167\) −6.28278 + 9.40284i −0.486176 + 0.727613i −0.990742 0.135760i \(-0.956653\pi\)
0.504566 + 0.863373i \(0.331653\pi\)
\(168\) 0 0
\(169\) 32.6723i 2.51326i
\(170\) 0 0
\(171\) 0 0
\(172\) 5.01885 12.1166i 0.382683 0.923880i
\(173\) −21.0658 14.0757i −1.60160 1.07016i −0.950300 0.311335i \(-0.899224\pi\)
−0.651300 0.758820i \(-0.725776\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0.992800 0.663368i 0.0748351 0.0500032i
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(180\) 0 0
\(181\) 5.32439 + 7.96852i 0.395759 + 0.592295i 0.974821 0.222988i \(-0.0715812\pi\)
−0.579062 + 0.815283i \(0.696581\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −0.0767508 1.22838i −0.00561258 0.0898282i
\(188\) 21.1717 1.54410
\(189\) 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
\(193\) −15.6371 + 10.4484i −1.12558 + 0.752090i −0.971751 0.236007i \(-0.924161\pi\)
−0.153831 + 0.988097i \(0.549161\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −5.35757 12.9343i −0.382683 0.923880i
\(197\) −5.33864 + 26.8391i −0.380362 + 1.91221i 0.0284595 + 0.999595i \(0.490940\pi\)
−0.408822 + 0.912614i \(0.634060\pi\)
\(198\) 0 0
\(199\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 23.6056 + 15.7728i 1.64070 + 1.09628i
\(208\) −19.1149 19.1149i −1.32538 1.32538i
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(212\) 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 \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(224\) 0 0
\(225\) 10.6066 + 10.6066i 0.707107 + 0.707107i
\(226\) 0 0
\(227\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(228\) 0 0
\(229\) 8.65964 3.58694i 0.572245 0.237032i −0.0777462 0.996973i \(-0.524772\pi\)
0.649992 + 0.759941i \(0.274772\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(234\) 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 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(242\) 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
\(253\) −2.60986 1.08104i −0.164081 0.0679645i
\(254\) 0 0
\(255\) 0 0
\(256\) −16.0000 −1.00000
\(257\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −16.6856 + 16.6856i −1.01924 + 1.01924i
\(269\) 10.5820 15.8370i 0.645195 0.965602i −0.354341 0.935116i \(-0.615295\pi\)
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
\(275\) −1.24100 0.829209i −0.0748351 0.0500032i
\(276\) 0 0
\(277\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(278\) 0 0
\(279\) −13.7574 2.73651i −0.823631 0.163830i
\(280\) 0 0
\(281\) −12.6171 30.4605i −0.752676 1.81712i −0.543884 0.839161i \(-0.683047\pi\)
−0.208792 0.977960i \(-0.566953\pi\)
\(282\) 0 0
\(283\) −18.6923 27.9750i −1.11114 1.66294i −0.555732 0.831362i \(-0.687562\pi\)
−0.555409 0.831578i \(-0.687438\pi\)
\(284\) 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
\(299\) −12.4770 + 62.7260i −0.721563 + 3.62754i
\(300\) 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 \(-0.0647291\pi\)
0.979395 + 0.201954i \(0.0647291\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 19.6541 3.90944i 1.11448 0.221684i 0.396697 0.917950i \(-0.370156\pi\)
0.717784 + 0.696265i \(0.245156\pi\)
\(312\) 0 0
\(313\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −1.65350 + 8.31270i −0.0930166 + 0.467626i
\(317\) 18.3606 + 27.4786i 1.03123 + 1.54335i 0.825228 + 0.564799i \(0.191046\pi\)
0.206005 + 0.978551i \(0.433954\pi\)
\(318\) 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 \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(332\) 5.01885 + 12.1166i 0.275445 + 0.664983i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 19.0240 28.4714i 1.03630 1.55093i 0.218163 0.975912i \(-0.429994\pi\)
0.818138 0.575022i \(-0.195006\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 1.39571 0.0755819
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(348\) 0 0
\(349\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(350\) 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
\(359\) −3.73079 + 9.00692i −0.196903 + 0.475367i −0.991234 0.132119i \(-0.957822\pi\)
0.794330 + 0.607486i \(0.207822\pi\)
\(360\) 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
\(367\) 6.38915 32.1204i 0.333511 1.67667i −0.342296 0.939592i \(-0.611204\pi\)
0.675806 0.737079i \(-0.263796\pi\)
\(368\) 21.0303 + 31.4741i 1.09628 + 1.64070i
\(369\) 6.76783 + 34.0242i 0.352319 + 1.77123i
\(370\) 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
\(379\) 38.1805 7.59458i 1.96120 0.390107i 0.976819 0.214065i \(-0.0686705\pi\)
0.984383 0.176042i \(-0.0563295\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −13.9104 + 13.9104i −0.707107 + 0.707107i
\(388\) 17.5575 26.2767i 0.891349 1.33400i
\(389\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(390\) 0 0
\(391\) 38.9427 2.43319i 1.96942 0.123051i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) −1.75663 + 0.349416i −0.0882740 + 0.0175588i
\(397\) 28.7196 19.1898i 1.44139 0.963109i 0.443634 0.896208i \(-0.353689\pi\)
0.997760 0.0669005i \(-0.0213110\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 7.65367 + 18.4776i 0.382683 + 0.923880i
\(401\) −7.14894 + 35.9402i −0.357001 + 1.79477i 0.217281 + 0.976109i \(0.430281\pi\)
−0.574283 + 0.818657i \(0.694719\pi\)
\(402\) 0 0
\(403\) −6.16456 30.9913i −0.307079 1.54379i
\(404\) 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 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(420\) 0 0
\(421\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(422\) 0 0
\(423\) −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
\(428\) −26.5810 17.7608i −1.28484 0.858502i
\(429\) 0 0
\(430\) 0 0
\(431\) 31.3742 20.9635i 1.51124 1.00978i 0.523789 0.851848i \(-0.324518\pi\)
0.987450 0.157930i \(-0.0504821\pi\)
\(432\) 0 0
\(433\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 22.9561 + 34.3562i 1.09940 + 1.64537i
\(437\) 0 0
\(438\) 0 0
\(439\) −18.0289 + 26.9822i −0.860473 + 1.28779i 0.0958262 + 0.995398i \(0.469451\pi\)
−0.956299 + 0.292391i \(0.905549\pi\)
\(440\) 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 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(450\) 0 0
\(451\) −1.32095 3.18906i −0.0622012 0.150167i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 10.0377 24.2331i 0.467502 1.12865i −0.497748 0.867322i \(-0.665840\pi\)
0.965250 0.261328i \(-0.0841604\pi\)
\(462\) 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 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(468\) 15.5173 + 37.4622i 0.717290 + 1.73169i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.08750 1.62756i 0.0500032 0.0748351i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −28.5030 −1.30506
\(478\) 0 0
\(479\) −28.9678 19.3556i −1.32357 0.884382i −0.325444 0.945561i \(-0.605514\pi\)
−0.998127 + 0.0611793i \(0.980514\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −20.1607 + 8.35084i −0.916396 + 0.379583i
\(485\) 0 0
\(486\) 0 0
\(487\) 19.2710 + 28.8410i 0.873251 + 1.30691i 0.950762 + 0.309921i \(0.100303\pi\)
−0.0775113 + 0.996991i \(0.524697\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −15.5506 10.3906i −0.698241 0.466550i
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(504\) 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
\(517\) 3.09923 + 0.616476i 0.136304 + 0.0271126i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(522\) 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
\(527\) −17.3159 + 8.47372i −0.754294 + 0.369121i
\(528\) 0 0
\(529\) 25.4699 61.4898i 1.10739 2.67347i
\(530\) 0 0
\(531\) −26.3253 26.3253i −1.14242 1.14242i
\(532\) 0 0
\(533\) −64.9778 + 43.4168i −2.81450 + 1.88059i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −0.407651 2.04940i −0.0175588 0.0882740i
\(540\) 0 0
\(541\) −5.12397 + 7.66857i −0.220297 + 0.329698i −0.925113 0.379693i \(-0.876030\pi\)
0.704816 + 0.709390i \(0.251030\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 35.8915 + 23.9819i 1.53461 + 1.02539i 0.981315 + 0.192406i \(0.0616291\pi\)
0.553293 + 0.832987i \(0.313371\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −0.0469184 0.235875i −0.00198978 0.0100033i
\(557\) 19.6071 19.6071i 0.830780 0.830780i −0.156844 0.987623i \(-0.550132\pi\)
0.987623 + 0.156844i \(0.0501319\pi\)
\(558\) 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 0.138182 + 0.990407i \(0.455874\pi\)
−0.798033 + 0.602614i \(0.794126\pi\)
\(564\) 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 \(-0.787151\pi\)
−0.116450 + 0.993197i \(0.537151\pi\)
\(570\) 0 0
\(571\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(572\) −2.24156 3.35473i −0.0937244 0.140268i
\(573\) 0 0
\(574\) 0 0
\(575\) 26.2879 39.3427i 1.09628 1.64070i
\(576\) 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
\(583\) 2.78163 0.553300i 0.115203 0.0229154i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(594\) 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 \(-0.862902\pi\)
−0.417514 0.908671i \(-0.637098\pi\)
\(600\) 0 0
\(601\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(602\) 0 0
\(603\) 32.7013 13.5453i 1.33170 0.551608i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(608\) 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 \(-0.447303\pi\)
0.164798 + 0.986327i \(0.447303\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 9.51744 1.89314i 0.383158 0.0762148i 0.000246592 1.00000i \(-0.499922\pi\)
0.382911 + 0.923785i \(0.374922\pi\)
\(618\) 0 0
\(619\) −46.2899 9.20763i −1.86055 0.370086i −0.868474 0.495735i \(-0.834899\pi\)
−0.992075 + 0.125649i \(0.959899\pi\)
\(620\) 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 \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(632\) 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 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(642\) 0 0
\(643\) 28.0519 41.9827i 1.10626 1.65563i 0.475281 0.879834i \(-0.342346\pi\)
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
\(649\) 3.08013 + 2.05808i 0.120906 + 0.0807866i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −9.02377 + 45.3656i −0.352319 + 1.77123i
\(657\) 0 0
\(658\) 0 0
\(659\) −2.60446 + 2.60446i −0.101455 + 0.101455i −0.756013 0.654557i \(-0.772855\pi\)
0.654557 + 0.756013i \(0.272855\pi\)
\(660\) 0 0
\(661\) −6.05828 2.50942i −0.235640 0.0976052i 0.261739 0.965139i \(-0.415704\pi\)
−0.497379 + 0.867533i \(0.665704\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 22.1828 4.41243i 0.858279 0.170722i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −46.2056 + 46.2056i −1.77714 + 1.77714i
\(677\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 20.1855 + 13.4875i 0.772378 + 0.516086i 0.878197 0.478299i \(-0.158747\pi\)
−0.105819 + 0.994385i \(0.533747\pi\)
\(684\) 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 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(692\) 9.88545 + 49.6975i 0.375788 + 1.88922i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 35.7501 + 31.5454i 1.35413 + 1.19487i
\(698\) 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
\(704\) −2.34217 0.465887i −0.0882740 0.0175588i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −6.91327 34.7554i −0.259633 1.30526i −0.861944 0.507003i \(-0.830753\pi\)
0.602311 0.798262i \(-0.294247\pi\)
\(710\) 0 0
\(711\) 7.06315 10.5708i 0.264889 0.396434i
\(712\) 0 0
\(713\) 44.2473i 1.65708i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 2.66595 0.530291i 0.0994233 0.0197765i −0.145128 0.989413i \(-0.546359\pi\)
0.244551 + 0.969636i \(0.421359\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 3.73936 18.7990i 0.138972 0.698660i
\(725\) 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
\(729\) −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 \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −2.92840 + 1.95669i −0.107869 + 0.0720756i
\(738\) 0 0
\(739\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 19.6723i 0.719772i
\(748\) −1.62865 + 1.84574i −0.0595495 + 0.0674868i
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(752\) −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 \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(758\) 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
\(767\) 32.0947 77.4836i 1.15887 2.79777i
\(768\) 0 0
\(769\) −39.1416 39.1416i −1.41148 1.41148i −0.749670 0.661812i \(-0.769788\pi\)
−0.661812 0.749670i \(-0.730212\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 36.8904 + 7.33796i 1.32771 + 0.264099i
\(773\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(774\) 0 0
\(775\) −4.56085 + 22.9289i −0.163830 + 0.823631i
\(776\) 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
\(787\) 30.9493 6.15620i 1.10322 0.219445i 0.390301 0.920687i \(-0.372371\pi\)
0.712923 + 0.701242i \(0.247371\pi\)
\(788\) 45.5062 30.4063i 1.62109 1.08318i
\(789\) 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 0.954664 + 0.297687i \(0.0962151\pi\)
0.885545 + 0.464553i \(0.153785\pi\)
\(798\) 0 0
\(799\) −42.1935 + 11.1679i −1.49270 + 0.395093i
\(800\) 0 0
\(801\) 0 0
\(80