Properties

Label 731.2.s.a.687.1
Level 731
Weight 2
Character 731.687
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 687.1
Root \(-1.71665 + 2.83780i\)
Character \(\chi\) = 731.687
Dual form 731.2.s.a.515.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.41421 - 1.41421i) q^{4} +(-1.14805 - 2.77164i) q^{9} +O(q^{10})\) \(q+(-1.41421 - 1.41421i) q^{4} +(-1.14805 - 2.77164i) q^{9} +(-5.18533 + 3.46473i) q^{11} +(1.77872 - 1.77872i) q^{13} +4.00000i q^{16} +(-1.05499 + 3.98585i) q^{17} +(5.18975 + 7.76701i) q^{23} +(-4.61940 + 1.91342i) q^{25} +(-6.27575 - 4.19332i) q^{31} +(-2.29610 + 5.54328i) q^{36} +(2.49507 - 12.5435i) q^{41} +(2.50942 + 6.05828i) q^{43} +(12.2330 + 2.43330i) q^{44} +(-9.65010 + 9.65010i) q^{47} +(-6.46716 - 2.67878i) q^{49} -5.03098 q^{52} +(-5.55651 + 13.4146i) q^{53} +(-2.19162 + 0.907798i) q^{59} +(5.65685 - 5.65685i) q^{64} -0.318032i q^{67} +(7.12882 - 4.14487i) q^{68} +(2.23782 - 1.49526i) q^{79} +(-6.36396 + 6.36396i) q^{81} +(-6.05828 - 2.50942i) q^{83} +(3.64480 - 18.3236i) q^{92} +(-9.90386 + 1.97000i) q^{97} +(15.5560 + 10.3942i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 24q^{13} + 56q^{23} - 64q^{59} + 96q^{79} + O(q^{100}) \)

Character values

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

\(n\) \(173\) \(562\)
\(\chi(n)\) \(e\left(\frac{11}{16}\right)\) \(-1\)

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 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(4\) −1.41421 1.41421i −0.707107 0.707107i
\(5\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(6\) 0 0
\(7\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(8\) 0 0
\(9\) −1.14805 2.77164i −0.382683 0.923880i
\(10\) 0 0
\(11\) −5.18533 + 3.46473i −1.56344 + 1.04466i −0.592425 + 0.805626i \(0.701829\pi\)
−0.971012 + 0.239030i \(0.923171\pi\)
\(12\) 0 0
\(13\) 1.77872 1.77872i 0.493328 0.493328i −0.416025 0.909353i \(-0.636577\pi\)
0.909353 + 0.416025i \(0.136577\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000i 1.00000i
\(17\) −1.05499 + 3.98585i −0.255872 + 0.966711i
\(18\) 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
\(23\) 5.18975 + 7.76701i 1.08214 + 1.61953i 0.729800 + 0.683660i \(0.239613\pi\)
0.352337 + 0.935873i \(0.385387\pi\)
\(24\) 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 \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(30\) 0 0
\(31\) −6.27575 4.19332i −1.12716 0.753142i −0.155103 0.987898i \(-0.549571\pi\)
−0.972054 + 0.234756i \(0.924571\pi\)
\(32\) 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 \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.49507 12.5435i 0.389664 1.95897i 0.144364 0.989525i \(-0.453886\pi\)
0.245299 0.969447i \(-0.421114\pi\)
\(42\) 0 0
\(43\) 2.50942 + 6.05828i 0.382683 + 0.923880i
\(44\) 12.2330 + 2.43330i 1.84420 + 0.366834i
\(45\) 0 0
\(46\) 0 0
\(47\) −9.65010 + 9.65010i −1.40761 + 1.40761i −0.635560 + 0.772051i \(0.719231\pi\)
−0.772051 + 0.635560i \(0.780769\pi\)
\(48\) 0 0
\(49\) −6.46716 2.67878i −0.923880 0.382683i
\(50\) 0 0
\(51\) 0 0
\(52\) −5.03098 −0.697671
\(53\) −5.55651 + 13.4146i −0.763245 + 1.84264i −0.312878 + 0.949793i \(0.601293\pi\)
−0.450367 + 0.892844i \(0.648707\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.19162 + 0.907798i −0.285324 + 0.118185i −0.520756 0.853706i \(-0.674350\pi\)
0.235431 + 0.971891i \(0.424350\pi\)
\(60\) 0 0
\(61\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 5.65685 5.65685i 0.707107 0.707107i
\(65\) 0 0
\(66\) 0 0
\(67\) 0.318032i 0.0388538i −0.999811 0.0194269i \(-0.993816\pi\)
0.999811 0.0194269i \(-0.00618416\pi\)
\(68\) 7.12882 4.14487i 0.864496 0.502639i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(72\) 0 0
\(73\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 2.23782 1.49526i 0.251774 0.168230i −0.423279 0.906000i \(-0.639121\pi\)
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 \(-0.492184\pi\)
−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
\(92\) 3.64480 18.3236i 0.379996 1.91037i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −9.90386 + 1.97000i −1.00558 + 0.200023i −0.670297 0.742093i \(-0.733833\pi\)
−0.335287 + 0.942116i \(0.608833\pi\)
\(98\) 0 0
\(99\) 15.5560 + 10.3942i 1.56344 + 1.04466i
\(100\) 9.23880 + 3.82683i 0.923880 + 0.382683i
\(101\) 13.3293i 1.32631i −0.748481 0.663156i \(-0.769217\pi\)
0.748481 0.663156i \(-0.230783\pi\)
\(102\) 0 0
\(103\) 12.1477 1.19695 0.598473 0.801143i \(-0.295775\pi\)
0.598473 + 0.801143i \(0.295775\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −3.56062 17.9005i −0.344218 1.73050i −0.633932 0.773389i \(-0.718560\pi\)
0.289713 0.957114i \(-0.406440\pi\)
\(108\) 0 0
\(109\) 2.08812 10.4977i 0.200006 1.00550i −0.742127 0.670259i \(-0.766183\pi\)
0.942133 0.335239i \(-0.108817\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −6.97203 2.88791i −0.644564 0.266987i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 10.6738 25.7689i 0.970348 2.34263i
\(122\) 0 0
\(123\) 0 0
\(124\) 2.94500 + 14.8055i 0.264468 + 1.32957i
\(125\) 0 0
\(126\) 0 0
\(127\) −12.9023 + 5.34430i −1.14489 + 0.474230i −0.872818 0.488046i \(-0.837710\pi\)
−0.272075 + 0.962276i \(0.587710\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(132\) 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
\(139\) 5.07487 7.59508i 0.430445 0.644206i −0.551323 0.834292i \(-0.685877\pi\)
0.981767 + 0.190086i \(0.0608767\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −3.06047 + 15.3860i −0.255930 + 1.28664i
\(144\) 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
\(153\) 12.2585 1.65191i 0.991042 0.133549i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
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 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(164\) −21.2678 + 14.2107i −1.66074 + 1.10967i
\(165\) 0 0
\(166\) 0 0
\(167\) 21.4510 + 14.3331i 1.65993 + 1.10913i 0.863373 + 0.504566i \(0.168347\pi\)
0.796557 + 0.604564i \(0.206653\pi\)
\(168\) 0 0
\(169\) 6.67232i 0.513255i
\(170\) 0 0
\(171\) 0 0
\(172\) 5.01885 12.1166i 0.382683 0.923880i
\(173\) −1.75316 + 2.62379i −0.133290 + 0.199483i −0.892111 0.451817i \(-0.850776\pi\)
0.758820 + 0.651300i \(0.225776\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −13.8589 20.7413i −1.04466 1.56344i
\(177\) 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
\(181\) −15.3621 + 10.2646i −1.14185 + 0.762963i −0.974821 0.222988i \(-0.928419\pi\)
−0.167034 + 0.985951i \(0.553419\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −8.33943 24.3232i −0.609840 1.77869i
\(188\) 27.2946 1.99066
\(189\) 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
\(193\) −5.30463 7.93894i −0.381836 0.571458i 0.589915 0.807465i \(-0.299161\pi\)
−0.971751 + 0.236007i \(0.924161\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 5.35757 + 12.9343i 0.382683 + 0.923880i
\(197\) 22.4474 + 4.46506i 1.59931 + 0.318123i 0.912614 0.408822i \(-0.134060\pi\)
0.686696 + 0.726944i \(0.259060\pi\)
\(198\) 0 0
\(199\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 15.5692 23.3010i 1.08214 1.61953i
\(208\) 7.11488 + 7.11488i 0.493328 + 0.493328i
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(212\) 26.8292 11.1130i 1.84264 0.763245i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 5.21318 + 8.96623i 0.350677 + 0.603134i
\(222\) 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 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(228\) 0 0
\(229\) −24.9231 + 10.3235i −1.64696 + 0.682195i −0.996973 0.0777462i \(-0.975228\pi\)
−0.649992 + 0.759941i \(0.725228\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 4.38324 + 1.81560i 0.285324 + 0.118185i
\(237\) 0 0
\(238\) 0 0
\(239\) −29.0726 −1.88055 −0.940275 0.340415i \(-0.889432\pi\)
−0.940275 + 0.340415i \(0.889432\pi\)
\(240\) 0 0
\(241\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(242\) 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 0.206951 0.978351i \(-0.433646\pi\)
0.978351 0.206951i \(-0.0663540\pi\)
\(252\) 0 0
\(253\) −53.8212 22.2935i −3.38371 1.40158i
\(254\) 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
\(268\) −0.449765 + 0.449765i −0.0274738 + 0.0274738i
\(269\) −23.1291 15.4544i −1.41021 0.942269i −0.999535 0.0304855i \(-0.990295\pi\)
−0.410671 0.911784i \(-0.634705\pi\)
\(270\) 0 0
\(271\) 9.77548i 0.593818i −0.954906 0.296909i \(-0.904044\pi\)
0.954906 0.296909i \(-0.0959558\pi\)
\(272\) −15.9434 4.21995i −0.966711 0.255872i
\(273\) 0 0
\(274\) 0 0
\(275\) 17.3236 25.9267i 1.04466 1.56344i
\(276\) 0 0
\(277\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(278\) 0 0
\(279\) −4.41749 + 22.2082i −0.264468 + 1.32957i
\(280\) 0 0
\(281\) 10.5669 + 25.5107i 0.630368 + 1.52184i 0.839161 + 0.543884i \(0.183047\pi\)
−0.208792 + 0.977960i \(0.566953\pi\)
\(282\) 0 0
\(283\) −10.7005 + 7.14987i −0.636081 + 0.425015i −0.831362 0.555732i \(-0.812438\pi\)
0.195281 + 0.980747i \(0.437438\pi\)
\(284\) 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 0.766179 + 0.642627i \(0.222155\pi\)
0.642627 + 0.766179i \(0.277845\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 23.0464 + 4.58422i 1.33281 + 0.265112i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −19.2642 −1.09947 −0.549734 0.835340i \(-0.685271\pi\)
−0.549734 + 0.835340i \(0.685271\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 1.44493 + 7.26417i 0.0819346 + 0.411913i 0.999884 + 0.0152162i \(0.00484366\pi\)
−0.917950 + 0.396697i \(0.870156\pi\)
\(312\) 0 0
\(313\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −5.27937 1.05013i −0.296988 0.0590746i
\(317\) 0.329862 0.220407i 0.0185269 0.0123793i −0.546272 0.837608i \(-0.683954\pi\)
0.564799 + 0.825228i \(0.308954\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 18.0000 1.00000
\(325\) −4.81318 + 11.6200i −0.266987 + 0.644564i
\(326\) 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
\(337\) 21.0712 + 14.0793i 1.14782 + 0.766950i 0.975912 0.218163i \(-0.0700065\pi\)
0.171909 + 0.985113i \(0.445006\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 47.0706 2.54901
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(348\) 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
\(353\) −15.2213 + 15.2213i −0.810147 + 0.810147i −0.984656 0.174509i \(-0.944166\pi\)
0.174509 + 0.984656i \(0.444166\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −14.0135 + 33.8316i −0.739605 + 1.78556i −0.132119 + 0.991234i \(0.542178\pi\)
−0.607486 + 0.794330i \(0.707822\pi\)
\(360\) 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
\(367\) 5.72742 + 1.13925i 0.298969 + 0.0594686i 0.342296 0.939592i \(-0.388796\pi\)
−0.0433266 + 0.999061i \(0.513796\pi\)
\(368\) −31.0680 + 20.7590i −1.61953 + 1.08214i
\(369\) −37.6306 + 7.48520i −1.95897 + 0.389664i
\(370\) 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
\(379\) 6.96013 + 34.9909i 0.357518 + 1.79736i 0.571583 + 0.820544i \(0.306330\pi\)
−0.214065 + 0.976819i \(0.568670\pi\)
\(380\) 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
\(388\) 16.7922 + 11.2202i 0.852493 + 0.569618i
\(389\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(390\) 0 0
\(391\) −36.4333 + 12.4915i −1.84251 + 0.631721i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) −7.29991 36.6991i −0.366834 1.84420i
\(397\) −17.5441 26.2565i −0.880511 1.31778i −0.947411 0.320018i \(-0.896311\pi\)
0.0669005 0.997760i \(-0.478689\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −7.65367 18.4776i −0.382683 0.923880i
\(401\) −28.3982 5.64875i −1.41814 0.282085i −0.574283 0.818657i \(-0.694719\pi\)
−0.843854 + 0.536572i \(0.819719\pi\)
\(402\) 0 0
\(403\) −18.6215 + 3.70405i −0.927604 + 0.184512i
\(404\) −18.8504 + 18.8504i −0.937844 + 0.937844i
\(405\) 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
\(412\) −17.1794 17.1794i −0.846368 0.846368i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(420\) 0 0
\(421\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(422\) 0 0
\(423\) 37.8254 + 15.6678i 1.83913 + 0.761794i
\(424\) 0 0
\(425\) −2.75319 20.4309i −0.133549 0.991042i
\(426\) 0 0
\(427\) 0 0
\(428\) −20.2796 + 30.3506i −0.980251 + 1.46705i
\(429\) 0 0
\(430\) 0 0
\(431\) 15.6841 + 23.4730i 0.755478 + 1.13065i 0.987450 + 0.157930i \(0.0504821\pi\)
−0.231972 + 0.972722i \(0.574518\pi\)
\(432\) 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
\(436\) −17.7991 + 11.8930i −0.852421 + 0.569569i
\(437\) 0 0
\(438\) 0 0
\(439\) 10.0408 + 6.70905i 0.479222 + 0.320206i 0.771612 0.636093i \(-0.219451\pi\)
−0.292391 + 0.956299i \(0.594451\pi\)
\(440\) 0 0
\(441\) 21.0000i 1.00000i
\(442\) 0 0
\(443\) 19.0107 0.903223 0.451612 0.892215i \(-0.350849\pi\)
0.451612 + 0.892215i \(0.350849\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(450\) 0 0
\(451\) 30.5222 + 73.6872i 1.43724 + 3.46979i
\(452\) 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 \(0.875000\pi\)
\(468\) 5.77582 + 13.9441i 0.266987 + 0.644564i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −34.0025 22.7198i −1.56344 1.04466i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 43.5596 1.99446
\(478\) 0 0
\(479\) 21.0088 31.4419i 0.959916 1.43662i 0.0611793 0.998127i \(-0.480514\pi\)
0.898737 0.438489i \(-0.144486\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −51.5378 + 21.3476i −2.34263 + 0.970348i
\(485\) 0 0
\(486\) 0 0
\(487\) 9.92770 6.63348i 0.449867 0.300592i −0.309921 0.950762i \(-0.600303\pi\)
0.759788 + 0.650171i \(0.225303\pi\)
\(488\) 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
\(496\) 16.7733 25.1030i 0.753142 1.12716i
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 25.8046 + 10.6886i 1.14489 + 0.474230i
\(509\) 41.1874i 1.82560i −0.408408 0.912799i \(-0.633916\pi\)
0.408408 0.912799i \(-0.366084\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 16.6040 83.4740i 0.730243 3.67118i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(522\) 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
\(527\) 23.3348 20.5903i 1.01648 0.896927i
\(528\) 0 0
\(529\) −24.5912 + 59.3684i −1.06918 + 2.58124i
\(530\) 0 0
\(531\) 5.03218 + 5.03218i 0.218378 + 0.218378i
\(532\) 0 0
\(533\) −17.8734 26.7495i −0.774184 1.15865i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 42.8156 8.51656i 1.84420 0.366834i
\(540\) 0 0
\(541\) −7.42314 4.95998i −0.319146 0.213246i 0.385670 0.922637i \(-0.373970\pi\)
−0.704816 + 0.709390i \(0.748970\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −18.3255 + 27.4261i −0.783542 + 1.17265i 0.197773 + 0.980248i \(0.436629\pi\)
−0.981315 + 0.192406i \(0.938371\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −17.9180 + 3.56411i −0.759893 + 0.151152i
\(557\) 5.23491 5.23491i 0.221810 0.221810i −0.587450 0.809260i \(-0.699868\pi\)
0.809260 + 0.587450i \(0.199868\pi\)
\(558\) 0 0
\(559\) 15.2395 + 6.31243i 0.644564 + 0.266987i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −17.5773 + 42.4354i −0.740796 + 1.78844i −0.138182 + 0.990407i \(0.544126\pi\)
−0.602614 + 0.798033i \(0.705874\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −38.4796 + 15.9388i −1.61315 + 0.668189i −0.993197 0.116450i \(-0.962849\pi\)
−0.619953 + 0.784639i \(0.712849\pi\)
\(570\) 0 0
\(571\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(572\) 26.0873 17.4310i 1.09076 0.728826i
\(573\) 0 0
\(574\) 0 0
\(575\) −38.8350 25.9487i −1.61953 1.08214i
\(576\) −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
\(583\) −17.6656 88.8110i −0.731635 3.67818i
\(584\) 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 \(0.137098\pi\)
0.417514 + 0.908671i \(0.362902\pi\)
\(600\) 0 0
\(601\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(602\) 0 0
\(603\) −0.881469 + 0.365116i −0.0358962 + 0.0148687i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 34.3296i 1.38883i
\(612\) −19.6723 15.0000i −0.795206 0.606339i
\(613\) 40.3058 1.62794 0.813969 0.580909i \(-0.197303\pi\)
0.813969 + 0.580909i \(0.197303\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −5.38651 27.0798i −0.216853 1.09019i −0.923785 0.382911i \(-0.874922\pi\)
0.706932 0.707281i \(-0.250078\pi\)
\(618\) 0 0
\(619\) 7.32993 36.8501i 0.294615 1.48113i −0.495735 0.868474i \(-0.665101\pi\)
0.790350 0.612655i \(-0.209899\pi\)
\(620\) 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
\(637\) −16.2681 + 6.73845i −0.644564 + 0.266987i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(642\) 0 0
\(643\) 40.2978 + 26.9261i 1.58919 + 1.06186i 0.958211 + 0.286062i \(0.0923464\pi\)
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
\(649\) 8.21900 12.3006i 0.322624 0.482841i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 50.1742 + 9.98026i 1.95897 + 0.389664i
\(657\) 0 0
\(658\) 0 0
\(659\) 27.4465 27.4465i 1.06916 1.06916i 0.0717399 0.997423i \(-0.477145\pi\)
0.997423 0.0717399i \(-0.0228551\pi\)
\(660\) 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
\(668\) −10.0662 50.6064i −0.389475 1.95802i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 9.43608 9.43608i 0.362926 0.362926i
\(677\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −2.61956 + 3.92045i −0.100235 + 0.150012i −0.878197 0.478299i \(-0.841253\pi\)
0.777962 + 0.628311i \(0.216253\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −24.2331 + 10.0377i −0.923880 + 0.382683i
\(689\) 13.9773 + 33.7443i 0.532494 + 1.28555i
\(690\) 0 0
\(691\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(692\) 6.18993 1.23125i 0.235306 0.0468053i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 47.3644 + 23.1782i 1.79406 + 0.877938i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −9.88512 9.88512i −0.373356 0.373356i 0.495342 0.868698i \(-0.335043\pi\)
−0.868698 + 0.495342i \(0.835043\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −9.73321 + 48.9322i −0.366834 + 1.84420i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 49.3213 9.81061i 1.85230 0.368445i 0.861944 0.507003i \(-0.169247\pi\)
0.990354 + 0.138558i \(0.0442468\pi\)
\(710\) 0 0
\(711\) −6.71345 4.48579i −0.251774 0.168230i
\(712\) 0 0
\(713\) 70.5060i 2.64047i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 9.45061 + 47.5114i 0.352448 + 1.77188i 0.597000 + 0.802242i \(0.296359\pi\)
−0.244551 + 0.969636i \(0.578641\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 36.2416 + 7.20891i 1.34691 + 0.267917i
\(725\) 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
\(731\) −26.7948 + 3.61077i −0.991042 + 0.133549i
\(732\) 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
\(737\) 1.10189 + 1.64910i 0.0405888 + 0.0607454i
\(738\) 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 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 19.6723i 0.719772i
\(748\) −22.6045 + 46.1920i −0.826502 + 1.68895i
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(752\) −38.6004 38.6004i −1.40761 1.40761i
\(753\) 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
\(767\) −2.28356 + 5.51299i −0.0824544 + 0.199063i
\(768\) 0 0
\(769\) −29.4001 29.4001i −1.06019 1.06019i −0.998068 0.0621249i \(-0.980212\pi\)
−0.0621249 0.998068i \(-0.519788\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −3.72548 + 18.7292i −0.134083 + 0.674080i
\(773\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(774\) 0 0
\(775\) 37.0137 + 7.36249i 1.32957 + 0.264468i
\(776\) 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
\(787\) −6.00586 30.1935i −0.214086 1.07628i −0.927009 0.375040i \(-0.877629\pi\)
0.712923 0.701242i \(-0.247371\pi\)
\(788\) −25.4309 38.0600i −0.905937 1.35583i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 33.4041 + 13.8364i 1.18323 + 0.490111i 0.885545 0.464553i \(-0.153785\pi\)
0.297687 + 0.954664i \(0.403785\pi\)
\(798\) 0 0
\(799\) −28.2831 48.6446i −1.00058 1.72092i
\(800\) 0 0
\(801\) 0 0
\(802\)