Properties

Label 731.2.s.a.386.2
Level 731
Weight 2
Character 731.386
Analytic conductor 5.837
Analytic rank 0
Dimension 16
CM discriminant -43
Inner twists 4

Related objects

Downloads

Learn more about

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 386.2
Root \(1.71665 + 2.83780i\)
Character \(\chi\) = 731.386
Dual form 731.2.s.a.214.2

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.41421 + 1.41421i) q^{4} +(1.14805 - 2.77164i) q^{9} +O(q^{10})\) \(q+(-1.41421 + 1.41421i) q^{4} +(1.14805 - 2.77164i) q^{9} +(1.25563 - 1.87919i) q^{11} +(1.77872 + 1.77872i) q^{13} -4.00000i q^{16} +(1.05499 + 3.98585i) q^{17} +(1.81025 + 1.20957i) q^{23} +(4.61940 + 1.91342i) q^{25} +(4.54859 + 6.80745i) q^{31} +(2.29610 + 5.54328i) q^{36} +(0.646300 - 0.128557i) q^{41} +(-2.50942 + 6.05828i) q^{43} +(0.881839 + 4.43330i) q^{44} +(9.65010 + 9.65010i) q^{47} +(6.46716 - 2.67878i) q^{49} -5.03098 q^{52} +(-5.55651 - 13.4146i) q^{53} +(-2.19162 - 0.907798i) q^{59} +(5.65685 + 5.65685i) q^{64} -0.318032i q^{67} +(-7.12882 - 4.14487i) q^{68} +(9.76218 - 14.6101i) q^{79} +(-6.36396 - 6.36396i) q^{81} +(6.05828 - 2.50942i) q^{83} +(-4.27067 + 0.849489i) q^{92} +(-3.29947 + 16.5875i) q^{97} +(-3.76690 - 5.63756i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 24q^{13} + 56q^{23} - 64q^{59} + 96q^{79} + O(q^{100}) \)

Character values

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

\(n\) \(173\) \(562\)
\(\chi(n)\) \(e\left(\frac{13}{16}\right)\) \(-1\)

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 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(4\) −1.41421 + 1.41421i −0.707107 + 0.707107i
\(5\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(6\) 0 0
\(7\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(8\) 0 0
\(9\) 1.14805 2.77164i 0.382683 0.923880i
\(10\) 0 0
\(11\) 1.25563 1.87919i 0.378587 0.566596i −0.592425 0.805626i \(-0.701829\pi\)
0.971012 + 0.239030i \(0.0768293\pi\)
\(12\) 0 0
\(13\) 1.77872 + 1.77872i 0.493328 + 0.493328i 0.909353 0.416025i \(-0.136577\pi\)
−0.416025 + 0.909353i \(0.636577\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000i 1.00000i
\(17\) 1.05499 + 3.98585i 0.255872 + 0.966711i
\(18\) 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
\(23\) 1.81025 + 1.20957i 0.377463 + 0.252213i 0.729800 0.683660i \(-0.239613\pi\)
−0.352337 + 0.935873i \(0.614613\pi\)
\(24\) 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 \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(30\) 0 0
\(31\) 4.54859 + 6.80745i 0.816952 + 1.22265i 0.972054 + 0.234756i \(0.0754291\pi\)
−0.155103 + 0.987898i \(0.549571\pi\)
\(32\) 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 \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.646300 0.128557i 0.100935 0.0200772i −0.144364 0.989525i \(-0.546114\pi\)
0.245299 + 0.969447i \(0.421114\pi\)
\(42\) 0 0
\(43\) −2.50942 + 6.05828i −0.382683 + 0.923880i
\(44\) 0.881839 + 4.43330i 0.132942 + 0.668346i
\(45\) 0 0
\(46\) 0 0
\(47\) 9.65010 + 9.65010i 1.40761 + 1.40761i 0.772051 + 0.635560i \(0.219231\pi\)
0.635560 + 0.772051i \(0.280769\pi\)
\(48\) 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 −0.450367 0.892844i \(-0.648707\pi\)
−0.312878 0.949793i \(-0.601293\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.19162 0.907798i −0.285324 0.118185i 0.235431 0.971891i \(-0.424350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(62\) 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 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(72\) 0 0
\(73\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 9.76218 14.6101i 1.09833 1.64377i 0.423279 0.906000i \(-0.360879\pi\)
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
\(92\) −4.27067 + 0.849489i −0.445248 + 0.0885654i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −3.29947 + 16.5875i −0.335010 + 1.68421i 0.335287 + 0.942116i \(0.391167\pi\)
−0.670297 + 0.742093i \(0.733833\pi\)
\(98\) 0 0
\(99\) −3.76690 5.63756i −0.378587 0.566596i
\(100\) −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 \(-0.704225\pi\)
−0.598473 + 0.801143i \(0.704225\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −9.55425 1.90046i −0.923645 0.183724i −0.289713 0.957114i \(-0.593560\pi\)
−0.633932 + 0.773389i \(0.718560\pi\)
\(108\) 0 0
\(109\) 17.5842 3.49771i 1.68426 0.335020i 0.742127 0.670259i \(-0.233817\pi\)
0.942133 + 0.335239i \(0.108817\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(114\) 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
\(121\) 2.25479 + 5.44354i 0.204981 + 0.494867i
\(122\) 0 0
\(123\) 0 0
\(124\) −16.0599 3.19451i −1.44222 0.286875i
\(125\) 0 0
\(126\) 0 0
\(127\) −12.9023 5.34430i −1.14489 0.474230i −0.272075 0.962276i \(-0.587710\pi\)
−0.872818 + 0.488046i \(0.837710\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(132\) 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
\(139\) −18.0749 + 12.0772i −1.53309 + 1.02438i −0.551323 + 0.834292i \(0.685877\pi\)
−0.981767 + 0.190086i \(0.939123\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 5.57596 1.10913i 0.466285 0.0927499i
\(144\) −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
\(153\) 12.2585 + 1.65191i 0.991042 + 0.133549i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(164\) −0.732199 + 1.09581i −0.0571751 + 0.0855686i
\(165\) 0 0
\(166\) 0 0
\(167\) 0.863457 + 1.29225i 0.0668163 + 0.0999976i 0.863373 0.504566i \(-0.168347\pi\)
−0.796557 + 0.604564i \(0.793347\pi\)
\(168\) 0 0
\(169\) 6.67232i 0.513255i
\(170\) 0 0
\(171\) 0 0
\(172\) −5.01885 12.1166i −0.382683 0.923880i
\(173\) 21.7146 14.5092i 1.65093 1.10312i 0.758820 0.651300i \(-0.225776\pi\)
0.892111 0.451817i \(-0.149224\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −7.51674 5.02253i −0.566596 0.378587i
\(177\) 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
\(181\) −10.8677 + 16.2646i −0.807788 + 1.20894i 0.167034 + 0.985951i \(0.446581\pi\)
−0.974821 + 0.222988i \(0.928419\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 8.81483 + 3.02224i 0.644604 + 0.221008i
\(188\) −27.2946 −1.99066
\(189\) 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
\(193\) −21.6954 14.4964i −1.56167 1.04347i −0.971751 0.236007i \(-0.924161\pi\)
−0.589915 0.807465i \(-0.700839\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −5.35757 + 12.9343i −0.382683 + 0.923880i
\(197\) 3.17091 + 15.9412i 0.225918 + 1.13577i 0.912614 + 0.408822i \(0.134060\pi\)
−0.686696 + 0.726944i \(0.740940\pi\)
\(198\) 0 0
\(199\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 5.43075 3.62871i 0.377463 0.252213i
\(208\) 7.11488 7.11488i 0.493328 0.493328i
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(212\) 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 \(-0.625000\pi\)
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 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(228\) 0 0
\(229\) −24.9231 10.3235i −1.64696 0.682195i −0.649992 0.759941i \(-0.725228\pi\)
−0.996973 + 0.0777462i \(0.975228\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(234\) 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 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 18.7787 + 18.7787i 1.18530 + 1.18530i 0.978351 + 0.206951i \(0.0663540\pi\)
0.206951 + 0.978351i \(0.433646\pi\)
\(252\) 0 0
\(253\) 4.54602 1.88302i 0.285806 0.118385i
\(254\) 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
\(268\) 0.449765 + 0.449765i 0.0274738 + 0.0274738i
\(269\) −9.65810 14.4544i −0.588865 0.881298i 0.410671 0.911784i \(-0.365295\pi\)
−0.999535 + 0.0304855i \(0.990295\pi\)
\(270\) 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
\(275\) 9.39593 6.27816i 0.566596 0.378587i
\(276\) 0 0
\(277\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(278\) 0 0
\(279\) 24.0898 4.79176i 1.44222 0.286875i
\(280\) 0 0
\(281\) 10.5669 25.5107i 0.630368 1.52184i −0.208792 0.977960i \(-0.566953\pi\)
0.839161 0.543884i \(-0.183047\pi\)
\(282\) 0 0
\(283\) −17.2708 + 25.8476i −1.02664 + 1.53648i −0.195281 + 0.980747i \(0.562562\pi\)
−0.831362 + 0.555732i \(0.812438\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −14.7740 + 8.41004i −0.869059 + 0.494708i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 24.1149 24.1149i 1.40881 1.40881i 0.642627 0.766179i \(-0.277845\pi\)
0.766179 0.642627i \(-0.222155\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.06844 + 5.37141i 0.0617895 + 0.310637i
\(300\) 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 \(-0.314729\pi\)
0.549734 + 0.835340i \(0.314729\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −33.8214 6.72749i −1.91783 0.381481i −0.917950 0.396697i \(-0.870156\pi\)
−0.999884 + 0.0152162i \(0.995156\pi\)
\(312\) 0 0
\(313\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 6.85605 + 34.4677i 0.385683 + 1.93896i
\(317\) 19.7821 29.6060i 1.11107 1.66284i 0.546272 0.837608i \(-0.316046\pi\)
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 \(-0.875000\pi\)
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
\(337\) 14.7595 + 22.0892i 0.804003 + 1.20328i 0.975912 + 0.218163i \(0.0700065\pi\)
−0.171909 + 0.985113i \(0.554994\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 18.5038 1.00204
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(348\) 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
\(353\) −15.2213 15.2213i −0.810147 0.810147i 0.174509 0.984656i \(-0.444166\pi\)
−0.984656 + 0.174509i \(0.944166\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 14.0135 + 33.8316i 0.739605 + 1.78556i 0.607486 + 0.794330i \(0.292178\pi\)
0.132119 + 0.991234i \(0.457822\pi\)
\(360\) 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
\(367\) 7.38746 + 37.1393i 0.385622 + 1.93865i 0.342296 + 0.939592i \(0.388796\pi\)
0.0433266 + 0.999061i \(0.486204\pi\)
\(368\) 4.83828 7.24100i 0.252213 0.377463i
\(369\) 0.385671 1.93890i 0.0200772 0.100935i
\(370\) 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
\(379\) −15.2949 3.04235i −0.785648 0.156275i −0.214065 0.976819i \(-0.568670\pi\)
−0.571583 + 0.820544i \(0.693670\pi\)
\(380\) 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
\(388\) −18.7922 28.1245i −0.954028 1.42780i
\(389\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(390\) 0 0
\(391\) −2.91138 + 8.49147i −0.147235 + 0.429432i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 13.2999 + 2.64552i 0.668346 + 0.132942i
\(397\) 20.2100 + 13.5039i 1.01431 + 0.677742i 0.947411 0.320018i \(-0.103689\pi\)
0.0669005 + 0.997760i \(0.478689\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 7.65367 18.4776i 0.382683 0.923880i
\(401\) 5.39817 + 27.1384i 0.269572 + 1.35523i 0.843854 + 0.536572i \(0.180281\pi\)
−0.574283 + 0.818657i \(0.694719\pi\)
\(402\) 0 0
\(403\) −4.01787 + 20.1992i −0.200145 + 1.00619i
\(404\) 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 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(420\) 0 0
\(421\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(422\) 0 0
\(423\) 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
\(428\) 16.1994 10.8241i 0.783028 0.523203i
\(429\) 0 0
\(430\) 0 0
\(431\) 25.3159 + 16.9155i 1.21942 + 0.814792i 0.987450 0.157930i \(-0.0504821\pi\)
0.231972 + 0.972722i \(0.425482\pi\)
\(432\) 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
\(436\) −19.9213 + 29.8143i −0.954057 + 1.42785i
\(437\) 0 0
\(438\) 0 0
\(439\) −22.2933 33.3643i −1.06400 1.59239i −0.771612 0.636093i \(-0.780549\pi\)
−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 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(450\) 0 0
\(451\) 0.569932 1.37594i 0.0268371 0.0647904i
\(452\) 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 \(0.125000\pi\)
\(468\) −5.77582 + 13.9441i −0.266987 + 0.644564i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 8.23373 + 12.3226i 0.378587 + 0.566596i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −43.5596 −1.99446
\(478\) 0 0
\(479\) −18.3308 + 12.2483i −0.837557 + 0.559638i −0.898737 0.438489i \(-0.855514\pi\)
0.0611793 + 0.998127i \(0.480514\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −10.8871 4.50958i −0.494867 0.204981i
\(485\) 0 0
\(486\) 0 0
\(487\) −23.6064 + 35.3295i −1.06971 + 1.60093i −0.309921 + 0.950762i \(0.600303\pi\)
−0.759788 + 0.650171i \(0.774697\pi\)
\(488\) 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
\(496\) 27.2298 18.1944i 1.22265 0.816952i
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(504\) 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
\(517\) 30.2513 6.01736i 1.33045 0.264643i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(522\) 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
\(527\) −22.3348 + 25.3118i −0.972918 + 1.10260i
\(528\) 0 0
\(529\) −6.98777 16.8700i −0.303816 0.733477i
\(530\) 0 0
\(531\) −5.03218 + 5.03218i −0.218378 + 0.218378i
\(532\) 0 0
\(533\) 1.37825 + 0.920919i 0.0596988 + 0.0398894i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 3.08644 15.5166i 0.132942 0.668346i
\(540\) 0 0
\(541\) −25.3641 37.9600i −1.09049 1.63203i −0.704816 0.709390i \(-0.748970\pi\)
−0.385670 0.922637i \(-0.626030\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −27.5766 + 18.4261i −1.17909 + 0.787842i −0.981315 0.192406i \(-0.938371\pi\)
−0.197773 + 0.980248i \(0.563371\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 8.48193 42.6415i 0.359714 1.80840i
\(557\) −5.23491 5.23491i −0.221810 0.221810i 0.587450 0.809260i \(-0.300132\pi\)
−0.809260 + 0.587450i \(0.800132\pi\)
\(558\) 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 −0.602614 0.798033i \(-0.705874\pi\)
−0.138182 0.990407i \(-0.544126\pi\)
\(564\) 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 \(-0.0371515\pi\)
0.619953 + 0.784639i \(0.287151\pi\)
\(570\) 0 0
\(571\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(572\) −6.31706 + 9.45414i −0.264129 + 0.395298i
\(573\) 0 0
\(574\) 0 0
\(575\) 6.04785 + 9.05125i 0.252213 + 0.377463i
\(576\) 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
\(583\) −32.1855 6.40209i −1.33299 0.265147i
\(584\) 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 \(-0.362902\pi\)
0.908671 0.417514i \(-0.137098\pi\)
\(600\) 0 0
\(601\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(602\) 0 0
\(603\) −0.881469 0.365116i −0.0358962 0.0148687i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(608\) 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 \(-0.802697\pi\)
−0.813969 + 0.580909i \(0.802697\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −40.5062 8.05718i −1.63072 0.324370i −0.706932 0.707281i \(-0.749922\pi\)
−0.923785 + 0.382911i \(0.874922\pi\)
\(618\) 0 0
\(619\) −31.9974 + 6.36469i −1.28609 + 0.255818i −0.790350 0.612655i \(-0.790101\pi\)
−0.495735 + 0.868474i \(0.665101\pi\)
\(620\) 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
\(637\) 16.2681 + 6.73845i 0.644564 + 0.266987i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(642\) 0 0
\(643\) −8.29778 12.4185i −0.327233 0.489738i 0.630978 0.775800i \(-0.282654\pi\)
−0.958211 + 0.286062i \(0.907654\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) −4.45779 + 2.97860i −0.174983 + 0.116920i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −0.514228 2.58520i −0.0200772 0.100935i
\(657\) 0 0
\(658\) 0 0
\(659\) −27.4465 27.4465i −1.06916 1.06916i −0.997423 0.0717399i \(-0.977145\pi\)
−0.0717399 0.997423i \(-0.522855\pi\)
\(660\) 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
\(668\) −3.04864 0.606411i −0.117955 0.0234628i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 9.43608 + 9.43608i 0.362926 + 0.362926i
\(677\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −43.2825 + 28.9204i −1.65616 + 1.10661i −0.777962 + 0.628311i \(0.783747\pi\)
−0.878197 + 0.478299i \(0.841253\pi\)
\(684\) 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 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(692\) −10.1899 + 51.2283i −0.387363 + 1.94741i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1.19425 + 2.44043i 0.0452353 + 0.0924378i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −9.88512 + 9.88512i −0.373356 + 0.373356i −0.868698 0.495342i \(-0.835043\pi\)
0.495342 + 0.868698i \(0.335043\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 17.7332 3.52735i 0.668346 0.132942i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −3.41918 + 17.1894i −0.128410 + 0.645561i 0.861944 + 0.507003i \(0.169247\pi\)
−0.990354 + 0.138558i \(0.955753\pi\)
\(710\) 0 0
\(711\) −29.2865 43.8304i −1.09833 1.64377i
\(712\) 0 0
\(713\) 17.8250i 0.667553i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −22.5655 4.48856i −0.841551 0.167395i −0.244551 0.969636i \(-0.578641\pi\)
−0.597000 + 0.802242i \(0.703641\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) −7.63244 38.3708i −0.283657 1.42604i
\(725\) 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
\(731\) −26.7948 3.61077i −0.991042 0.133549i
\(732\) 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
\(737\) −0.597641 0.399331i −0.0220144 0.0147095i
\(738\) 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 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 19.6723i 0.719772i
\(748\) −16.7402 + 8.19195i −0.612081 + 0.299528i
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(752\) 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 \(-0.375000\pi\)
−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
\(767\) −2.28356 5.51299i −0.0824544 0.199063i
\(768\) 0 0
\(769\) 29.4001 29.4001i 1.06019 1.06019i 0.0621249 0.998068i \(-0.480212\pi\)
0.998068 0.0621249i \(-0.0197877\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 51.1829 10.1809i 1.84211 0.366419i
\(773\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(774\) 0 0
\(775\) 7.98627 + 40.1497i 0.286875 + 1.44222i
\(776\) 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
\(787\) 46.0059 + 9.15113i 1.63993 + 0.326203i 0.927009 0.375040i \(-0.122371\pi\)
0.712923 + 0.701242i \(0.247371\pi\)
\(788\) −27.0287 18.0600i −0.962856 0.643360i
\(789\) 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 0.297687 0.954664i \(-0.403785\pi\)
0.885545 + 0.464553i \(0.153785\pi\)
\(798\) 0 0
\(799\) −28.2831 + 48.6446i −1.00058 + 1.72092i
\(800\) 0 0
\(801\) 0 0