Properties

Label 731.2.s.a.214.2
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 214.2
Root \(1.71665 - 2.83780i\)
Character \(\chi\) = 731.214
Dual form 731.2.s.a.386.2

$q$-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 \(\left(\mathbb{Z}/731\mathbb{Z}\right)^\times\).

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

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
\(11\) 1.25563 + 1.87919i 0.378587 + 0.566596i 0.971012 0.239030i \(-0.0768293\pi\)
−0.592425 + 0.805626i \(0.701829\pi\)
\(12\) 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
\(23\) 1.81025 1.20957i 0.377463 0.252213i −0.352337 0.935873i \(-0.614613\pi\)
0.729800 + 0.683660i \(0.239613\pi\)
\(24\) 0 0
\(25\) 4.61940 1.91342i 0.923880 0.382683i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(30\) 0 0
\(31\) 4.54859 6.80745i 0.816952 1.22265i −0.155103 0.987898i \(-0.549571\pi\)
0.972054 0.234756i \(-0.0754291\pi\)
\(32\) 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
\(41\) 0.646300 + 0.128557i 0.100935 + 0.0200772i 0.245299 0.969447i \(-0.421114\pi\)
−0.144364 + 0.989525i \(0.546114\pi\)
\(42\) 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 0.635560 0.772051i \(-0.280769\pi\)
0.772051 0.635560i \(-0.219231\pi\)
\(48\) 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 0.195090 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
\(67\) 0.318032i 0.0388538i 0.999811 + 0.0194269i \(0.00618416\pi\)
−0.999811 + 0.0194269i \(0.993816\pi\)
\(68\) −7.12882 + 4.14487i −0.864496 + 0.502639i
\(69\) 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
\(79\) 9.76218 + 14.6101i 1.09833 + 1.64377i 0.675053 + 0.737769i \(0.264121\pi\)
0.423279 + 0.906000i \(0.360879\pi\)
\(80\) 0 0
\(81\) −6.36396 + 6.36396i −0.707107 + 0.707107i
\(82\) 0 0
\(83\) 6.05828 + 2.50942i 0.664983 + 0.275445i 0.689534 0.724254i \(-0.257816\pi\)
−0.0245507 + 0.999699i \(0.507816\pi\)
\(84\) 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
\(92\) −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 −0.670297 0.742093i \(-0.733833\pi\)
0.335287 0.942116i \(-0.391167\pi\)
\(98\) 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 \(0.230783\pi\)
−0.748481 + 0.663156i \(0.769217\pi\)
\(102\) 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 −0.633932 0.773389i \(-0.718560\pi\)
−0.289713 + 0.957114i \(0.593560\pi\)
\(108\) 0 0
\(109\) 17.5842 + 3.49771i 1.68426 + 0.335020i 0.942133 0.335239i \(-0.108817\pi\)
0.742127 + 0.670259i \(0.233817\pi\)
\(110\) 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
\(117\) 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 −0.872818 0.488046i \(-0.837710\pi\)
−0.272075 + 0.962276i \(0.587710\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 −0.195090 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
\(139\) −18.0749 12.0772i −1.53309 1.02438i −0.981767 0.190086i \(-0.939123\pi\)
−0.551323 0.834292i \(-0.685877\pi\)
\(140\) 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 \(-0.250000\pi\)
−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 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(164\) −0.732199 1.09581i −0.0571751 0.0855686i
\(165\) 0 0
\(166\) 0 0
\(167\) 0.863457 1.29225i 0.0668163 0.0999976i −0.796557 0.604564i \(-0.793347\pi\)
0.863373 + 0.504566i \(0.168347\pi\)
\(168\) 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 0.892111 + 0.451817i \(0.149224\pi\)
0.758820 + 0.651300i \(0.225776\pi\)
\(174\) 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 \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(180\) 0 0
\(181\) −10.8677 16.2646i −0.807788 1.20894i −0.974821 0.222988i \(-0.928419\pi\)
0.167034 0.985951i \(-0.446581\pi\)
\(182\) 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 \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(192\) 0 0
\(193\) −21.6954 + 14.4964i −1.56167 + 1.04347i −0.589915 + 0.807465i \(0.700839\pi\)
−0.971751 + 0.236007i \(0.924161\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −5.35757 12.9343i −0.382683 0.923880i
\(197\) 3.17091 15.9412i 0.225918 1.13577i −0.686696 0.726944i \(-0.740940\pi\)
0.912614 0.408822i \(-0.134060\pi\)
\(198\) 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
\(207\) 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 \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(212\) 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 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(228\) 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 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(234\) 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 \(-0.812500\pi\)
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
\(251\) 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
\(253\) 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 \(-0.375000\pi\)
−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
\(269\) −9.65810 + 14.4544i −0.588865 + 0.881298i −0.999535 0.0304855i \(-0.990295\pi\)
0.410671 + 0.911784i \(0.365295\pi\)
\(270\) 0 0
\(271\) 9.77548i 0.593818i 0.954906 + 0.296909i \(0.0959558\pi\)
−0.954906 + 0.296909i \(0.904044\pi\)
\(272\) 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 \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(278\) 0 0
\(279\) 24.0898 + 4.79176i 1.44222 + 0.286875i
\(280\) 0 0
\(281\) 10.5669 + 25.5107i 0.630368 + 1.52184i 0.839161 + 0.543884i \(0.183047\pi\)
−0.208792 + 0.977960i \(0.566953\pi\)
\(282\) 0 0
\(283\) −17.2708 25.8476i −1.02664 1.53648i −0.831362 0.555732i \(-0.812438\pi\)
−0.195281 0.980747i \(-0.562562\pi\)
\(284\) 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
\(299\) 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 −0.999884 0.0152162i \(-0.995156\pi\)
−0.917950 + 0.396697i \(0.870156\pi\)
\(312\) 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
\(316\) 6.85605 34.4677i 0.385683 1.93896i
\(317\) 19.7821 + 29.6060i 1.11107 + 1.66284i 0.564799 + 0.825228i \(0.308954\pi\)
0.546272 + 0.837608i \(0.316046\pi\)
\(318\) 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
\(337\) 14.7595 22.0892i 0.804003 1.20328i −0.171909 0.985113i \(-0.554994\pi\)
0.975912 0.218163i \(-0.0700065\pi\)
\(338\) 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 \(-0.937500\pi\)
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
\(353\) −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 \(-0.457822\pi\)
0.607486 0.794330i \(-0.292178\pi\)
\(360\) 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 0.0433266 0.999061i \(-0.486204\pi\)
0.342296 0.939592i \(-0.388796\pi\)
\(368\) 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 −0.571583 0.820544i \(-0.693670\pi\)
−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
\(388\) −18.7922 + 28.1245i −0.954028 + 1.42780i
\(389\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(390\) 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 0.0669005 0.997760i \(-0.478689\pi\)
0.947411 + 0.320018i \(0.103689\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 7.65367 + 18.4776i 0.382683 + 0.923880i
\(401\) 5.39817 27.1384i 0.269572 1.35523i −0.574283 0.818657i \(-0.694719\pi\)
0.843854 0.536572i \(-0.180281\pi\)
\(402\) 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 \(-0.687500\pi\)
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
\(423\) 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 0.231972 0.972722i \(-0.425482\pi\)
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
\(436\) −19.9213 29.8143i −0.954057 1.42785i
\(437\) 0 0
\(438\) 0 0
\(439\) −22.2933 + 33.3643i −1.06400 + 1.59239i −0.292391 + 0.956299i \(0.594451\pi\)
−0.771612 + 0.636093i \(0.780549\pi\)
\(440\) 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 \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(450\) 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 \(-0.875000\pi\)
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 \(0.334160\pi\)
−0.965250 + 0.261328i \(0.915840\pi\)
\(462\) 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
\(473\) 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 0.0611793 0.998127i \(-0.480514\pi\)
−0.898737 + 0.438489i \(0.855514\pi\)
\(480\) 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 −0.759788 0.650171i \(-0.774697\pi\)
−0.309921 0.950762i \(-0.600303\pi\)
\(488\) 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
\(496\) 27.2298 + 18.1944i 1.22265 + 0.816952i
\(497\) 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
\(508\) 25.8046 + 10.6886i 1.14489 + 0.474230i
\(509\) 41.1874i 1.82560i 0.408408 + 0.912799i \(0.366084\pi\)
−0.408408 + 0.912799i \(0.633916\pi\)
\(510\) 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 \(-0.687500\pi\)
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
\(527\) −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 −0.385670 + 0.922637i \(0.626030\pi\)
−0.704816 + 0.709390i \(0.748970\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −27.5766 18.4261i −1.17909 0.787842i −0.197773 0.980248i \(-0.563371\pi\)
−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
\(556\) 8.48193 + 42.6415i 0.359714 + 1.80840i
\(557\) −5.23491 + 5.23491i −0.221810 + 0.221810i −0.809260 0.587450i \(-0.800132\pi\)
0.587450 + 0.809260i \(0.300132\pi\)
\(558\) 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 0.619953 0.784639i \(-0.287151\pi\)
0.993197 + 0.116450i \(0.0371515\pi\)
\(570\) 0 0
\(571\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(572\) −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 \(-0.625000\pi\)
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 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(602\) 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 \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(608\) 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 −0.923785 0.382911i \(-0.874922\pi\)
−0.706932 + 0.707281i \(0.749922\pi\)
\(618\) 0 0
\(619\) −31.9974 6.36469i −1.28609 0.255818i −0.495735 0.868474i \(-0.665101\pi\)
−0.790350 + 0.612655i \(0.790101\pi\)
\(620\) 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 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(642\) 0 0
\(643\) −8.29778 + 12.4185i −0.327233 + 0.489738i −0.958211 0.286062i \(-0.907654\pi\)
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
\(649\) −4.45779 2.97860i −0.174983 0.116920i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(654\) 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 −0.0717399 + 0.997423i \(0.522855\pi\)
−0.997423 + 0.0717399i \(0.977145\pi\)
\(660\) 0 0
\(661\) 6.05828 + 2.50942i 0.235640 + 0.0976052i 0.497379 0.867533i \(-0.334296\pi\)
−0.261739 + 0.965139i \(0.584296\pi\)
\(662\) 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 \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 9.43608 9.43608i 0.362926 0.362926i
\(677\) 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
\(683\) −43.2825 28.9204i −1.65616 1.10661i −0.878197 0.478299i \(-0.841253\pi\)
−0.777962 0.628311i \(-0.783747\pi\)
\(684\) 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 \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(692\) −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 0.495342 0.868698i \(-0.335043\pi\)
−0.868698 + 0.495342i \(0.835043\pi\)
\(702\) 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 −0.990354 0.138558i \(-0.955753\pi\)
0.861944 0.507003i \(-0.169247\pi\)
\(710\) 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 −0.597000 0.802242i \(-0.703641\pi\)
−0.244551 + 0.969636i \(0.578641\pi\)
\(720\) 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 \(-0.250000\pi\)
−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
\(737\) −0.597641 + 0.399331i −0.0220144 + 0.0147095i
\(738\) 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
\(748\) −16.7402 8.19195i −0.612081 0.299528i
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(752\) 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 \(0.0197877\pi\)
0.0621249 + 0.998068i \(0.480212\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 51.1829 + 10.1809i 1.84211 + 0.366419i
\(773\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(774\) 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 0.712923 0.701242i \(-0.247371\pi\)
0.927009 + 0.375040i \(0.122371\pi\)
\(788\) −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 0.885545 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
\(802\) 0