Properties

Label 731.2.s.a.515.1
Level 731
Weight 2
Character 731.515
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 515.1
Root \(-1.71665 - 2.83780i\)
Character \(\chi\) = 731.515
Dual form 731.2.s.a.687.1

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

\(n\) \(173\) \(562\)
\(\chi(n)\) \(e\left(\frac{5}{16}\right)\) \(-1\)

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 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(4\) −1.41421 + 1.41421i −0.707107 + 0.707107i
\(5\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(6\) 0 0
\(7\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(8\) 0 0
\(9\) −1.14805 + 2.77164i −0.382683 + 0.923880i
\(10\) 0 0
\(11\) −5.18533 3.46473i −1.56344 1.04466i −0.971012 0.239030i \(-0.923171\pi\)
−0.592425 0.805626i \(-0.701829\pi\)
\(12\) 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
\(23\) 5.18975 7.76701i 1.08214 1.61953i 0.352337 0.935873i \(-0.385387\pi\)
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 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(30\) 0 0
\(31\) −6.27575 + 4.19332i −1.12716 + 0.753142i −0.972054 0.234756i \(-0.924571\pi\)
−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 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.49507 + 12.5435i 0.389664 + 1.95897i 0.245299 + 0.969447i \(0.421114\pi\)
0.144364 + 0.989525i \(0.453886\pi\)
\(42\) 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 −0.772051 0.635560i \(-0.780769\pi\)
−0.635560 0.772051i \(-0.719231\pi\)
\(48\) 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 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(62\) 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 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(72\) 0 0
\(73\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 2.23782 + 1.49526i 0.251774 + 0.168230i 0.675053 0.737769i \(-0.264121\pi\)
−0.423279 + 0.906000i \(0.639121\pi\)
\(80\) 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 \(-0.742184\pi\)
0.0245507 + 0.999699i \(0.492184\pi\)
\(84\) 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
\(92\) 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 −0.335287 0.942116i \(-0.608833\pi\)
−0.670297 + 0.742093i \(0.733833\pi\)
\(98\) 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 \(0.230783\pi\)
−0.748481 + 0.663156i \(0.769217\pi\)
\(102\) 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 0.289713 + 0.957114i \(0.406440\pi\)
−0.633932 + 0.773389i \(0.718560\pi\)
\(108\) 0 0
\(109\) 2.08812 + 10.4977i 0.200006 + 1.00550i 0.942133 + 0.335239i \(0.108817\pi\)
−0.742127 + 0.670259i \(0.766183\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(114\) 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 −0.272075 0.962276i \(-0.587710\pi\)
−0.872818 + 0.488046i \(0.837710\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(132\) 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 0.981767 0.190086i \(-0.0608767\pi\)
−0.551323 + 0.834292i \(0.685877\pi\)
\(140\) 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 \(-0.750000\pi\)
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 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(164\) −21.2678 14.2107i −1.66074 1.10967i
\(165\) 0 0
\(166\) 0 0
\(167\) 21.4510 14.3331i 1.65993 1.10913i 0.796557 0.604564i \(-0.206653\pi\)
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
\(173\) −1.75316 2.62379i −0.133290 0.199483i 0.758820 0.651300i \(-0.225776\pi\)
−0.892111 + 0.451817i \(0.850776\pi\)
\(174\) 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 \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(180\) 0 0
\(181\) −15.3621 10.2646i −1.14185 0.762963i −0.167034 0.985951i \(-0.553419\pi\)
−0.974821 + 0.222988i \(0.928419\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −8.33943 + 24.3232i −0.609840 + 1.77869i
\(188\) 27.2946 1.99066
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(192\) 0 0
\(193\) −5.30463 + 7.93894i −0.381836 + 0.571458i −0.971751 0.236007i \(-0.924161\pi\)
0.589915 + 0.807465i \(0.299161\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 5.35757 12.9343i 0.382683 0.923880i
\(197\) 22.4474 4.46506i 1.59931 0.318123i 0.686696 0.726944i \(-0.259060\pi\)
0.912614 + 0.408822i \(0.134060\pi\)
\(198\) 0 0
\(199\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(200\) 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 \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(212\) 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 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(228\) 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 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(234\) 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 \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(242\) 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
\(253\) −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 \(-0.625000\pi\)
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
\(269\) −23.1291 + 15.4544i −1.41021 + 0.942269i −0.410671 + 0.911784i \(0.634705\pi\)
−0.999535 + 0.0304855i \(0.990295\pi\)
\(270\) 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
\(275\) 17.3236 + 25.9267i 1.04466 + 1.56344i
\(276\) 0 0
\(277\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(278\) 0 0
\(279\) −4.41749 22.2082i −0.264468 1.32957i
\(280\) 0 0
\(281\) 10.5669 25.5107i 0.630368 1.52184i −0.208792 0.977960i \(-0.566953\pi\)
0.839161 0.543884i \(-0.183047\pi\)
\(282\) 0 0
\(283\) −10.7005 7.14987i −0.636081 0.425015i 0.195281 0.980747i \(-0.437438\pi\)
−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
\(299\) 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 −0.917950 0.396697i \(-0.870156\pi\)
0.999884 0.0152162i \(-0.00484366\pi\)
\(312\) 0 0
\(313\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −5.27937 + 1.05013i −0.296988 + 0.0590746i
\(317\) 0.329862 + 0.220407i 0.0185269 + 0.0123793i 0.564799 0.825228i \(-0.308954\pi\)
−0.546272 + 0.837608i \(0.683954\pi\)
\(318\) 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
\(337\) 21.0712 14.0793i 1.14782 0.766950i 0.171909 0.985113i \(-0.445006\pi\)
0.975912 + 0.218163i \(0.0700065\pi\)
\(338\) 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 \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(348\) 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 \(-0.707822\pi\)
−0.132119 0.991234i \(-0.542178\pi\)
\(360\) 0 0
\(361\) −13.4350 + 13.4350i −0.707107 + 0.707107i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 5.72742 1.13925i 0.298969 0.0594686i −0.0433266 0.999061i \(-0.513796\pi\)
0.342296 + 0.939592i \(0.388796\pi\)
\(368\) −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 −0.214065 0.976819i \(-0.568670\pi\)
0.571583 0.820544i \(-0.306330\pi\)
\(380\) 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
\(388\) 16.7922 11.2202i 0.852493 0.569618i
\(389\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(390\) 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 0.0669005 + 0.997760i \(0.478689\pi\)
−0.947411 + 0.320018i \(0.896311\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −7.65367 + 18.4776i −0.382683 + 0.923880i
\(401\) −28.3982 + 5.64875i −1.41814 + 0.282085i −0.843854 0.536572i \(-0.819719\pi\)
−0.574283 + 0.818657i \(0.694719\pi\)
\(402\) 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 \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(420\) 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
\(428\) −20.2796 30.3506i −0.980251 1.46705i
\(429\) 0 0
\(430\) 0 0
\(431\) 15.6841 23.4730i 0.755478 1.13065i −0.231972 0.972722i \(-0.574518\pi\)
0.987450 0.157930i \(-0.0504821\pi\)
\(432\) 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
\(436\) −17.7991 11.8930i −0.852421 0.569569i
\(437\) 0 0
\(438\) 0 0
\(439\) 10.0408 6.70905i 0.479222 0.320206i −0.292391 0.956299i \(-0.594451\pi\)
0.771612 + 0.636093i \(0.219451\pi\)
\(440\) 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 \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(450\) 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 \(-0.125000\pi\)
−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 \(0.0841604\pi\)
−0.497748 + 0.867322i \(0.665840\pi\)
\(462\) 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
\(473\) −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 0.898737 + 0.438489i \(0.144486\pi\)
0.0611793 + 0.998127i \(0.480514\pi\)
\(480\) 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 0.759788 0.650171i \(-0.225303\pi\)
−0.309921 + 0.950762i \(0.600303\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(492\) 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 \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(504\) 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
\(517\) 16.6040 + 83.4740i 0.730243 + 3.67118i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(522\) 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
\(527\) 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 −0.704816 0.709390i \(-0.748970\pi\)
0.385670 + 0.922637i \(0.373970\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −18.3255 27.4261i −0.783542 1.17265i −0.981315 0.192406i \(-0.938371\pi\)
0.197773 0.980248i \(-0.436629\pi\)
\(548\) 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 0.809260 0.587450i \(-0.199868\pi\)
−0.587450 + 0.809260i \(0.699868\pi\)
\(558\) 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 −0.619953 0.784639i \(-0.712849\pi\)
−0.993197 + 0.116450i \(0.962849\pi\)
\(570\) 0 0
\(571\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(572\) 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 \(-0.375000\pi\)
−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 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(602\) 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 \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(608\) 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 0.706932 + 0.707281i \(0.250078\pi\)
−0.923785 + 0.382911i \(0.874922\pi\)
\(618\) 0 0
\(619\) 7.32993 + 36.8501i 0.294615 + 1.48113i 0.790350 + 0.612655i \(0.209899\pi\)
−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 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(642\) 0 0
\(643\) 40.2978 26.9261i 1.58919 1.06186i 0.630978 0.775800i \(-0.282654\pi\)
0.958211 0.286062i \(-0.0923464\pi\)
\(644\) 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 \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(654\) 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 0.997423 + 0.0717399i \(0.0228551\pi\)
0.0717399 + 0.997423i \(0.477145\pi\)
\(660\) 0 0
\(661\) −6.05828 + 2.50942i −0.235640 + 0.0976052i −0.497379 0.867533i \(-0.665704\pi\)
0.261739 + 0.965139i \(0.415704\pi\)
\(662\) 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 \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 9.43608 + 9.43608i 0.362926 + 0.362926i
\(677\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −2.61956 3.92045i −0.100235 0.150012i 0.777962 0.628311i \(-0.216253\pi\)
−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 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(692\) 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 −0.868698 0.495342i \(-0.835043\pi\)
0.495342 + 0.868698i \(0.335043\pi\)
\(702\) 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 0.990354 0.138558i \(-0.0442468\pi\)
0.861944 + 0.507003i \(0.169247\pi\)
\(710\) 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 −0.244551 0.969636i \(-0.578641\pi\)
0.597000 0.802242i \(-0.296359\pi\)
\(720\) 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 \(-0.750000\pi\)
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
\(737\) 1.10189 1.64910i 0.0405888 0.0607454i
\(738\) 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 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(744\) 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 \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(752\) −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 \(0.519788\pi\)
−0.998068 + 0.0621249i \(0.980212\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −3.72548 18.7292i −0.134083 0.674080i
\(773\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(774\) 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 0.712923 + 0.701242i \(0.247371\pi\)
−0.927009 + 0.375040i \(0.877629\pi\)
\(788\) −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 0.297687 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
\(802\)