Properties

Label 731.2.s.a.300.1
Level 731
Weight 2
Character 731.300
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 300.1
Root \(3.22048 + 0.792772i\)
Character \(\chi\) = 731.300
Dual form 731.2.s.a.558.1

$q$-expansion

\(f(q)\) \(=\) \(q+(1.41421 + 1.41421i) q^{4} +(-2.77164 + 1.14805i) q^{9} +O(q^{10})\) \(q+(1.41421 + 1.41421i) q^{4} +(-2.77164 + 1.14805i) q^{9} +(-4.38861 - 0.872949i) q^{11} +(-4.77872 + 4.77872i) q^{13} +4.00000i q^{16} +(-2.07243 + 3.56441i) q^{17} +(1.52113 - 7.64725i) q^{23} +(1.91342 + 4.61940i) q^{25} +(10.2516 - 2.03917i) q^{31} +(-5.54328 - 2.29610i) q^{36} +(-10.0341 + 6.70458i) q^{41} +(-6.05828 + 2.50942i) q^{43} +(-4.97189 - 7.44097i) q^{44} +(6.16199 - 6.16199i) q^{47} +(2.67878 - 6.46716i) q^{49} -13.5163 q^{52} +(10.1933 + 4.22221i) q^{53} +(3.46524 + 8.36582i) q^{59} +(-5.65685 + 5.65685i) q^{64} +11.3488i q^{67} +(-7.97170 + 2.10997i) q^{68} +(9.03385 + 1.79695i) q^{79} +(6.36396 - 6.36396i) q^{81} +(-2.50942 + 6.05828i) q^{83} +(12.9661 - 8.66364i) q^{92} +(1.58732 - 2.37559i) q^{97} +(13.1658 - 2.61885i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 24q^{13} + 56q^{23} - 64q^{59} + 96q^{79} + O(q^{100}) \)

Character values

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

\(n\) \(173\) \(562\)
\(\chi(n)\) \(e\left(\frac{7}{16}\right)\) \(-1\)

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 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(3\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(4\) 1.41421 + 1.41421i 0.707107 + 0.707107i
\(5\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(6\) 0 0
\(7\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(8\) 0 0
\(9\) −2.77164 + 1.14805i −0.923880 + 0.382683i
\(10\) 0 0
\(11\) −4.38861 0.872949i −1.32322 0.263204i −0.517590 0.855629i \(-0.673171\pi\)
−0.805626 + 0.592425i \(0.798171\pi\)
\(12\) 0 0
\(13\) −4.77872 + 4.77872i −1.32538 + 1.32538i −0.416025 + 0.909353i \(0.636577\pi\)
−0.909353 + 0.416025i \(0.863423\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000i 1.00000i
\(17\) −2.07243 + 3.56441i −0.502639 + 0.864496i
\(18\) 0 0
\(19\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.52113 7.64725i 0.317178 1.59456i −0.412622 0.910902i \(-0.635387\pi\)
0.729800 0.683660i \(-0.239613\pi\)
\(24\) 0 0
\(25\) 1.91342 + 4.61940i 0.382683 + 0.923880i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(30\) 0 0
\(31\) 10.2516 2.03917i 1.84124 0.366246i 0.853344 0.521349i \(-0.174571\pi\)
0.987898 + 0.155103i \(0.0495709\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −5.54328 2.29610i −0.923880 0.382683i
\(37\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −10.0341 + 6.70458i −1.56707 + 1.04708i −0.597619 + 0.801781i \(0.703886\pi\)
−0.969447 + 0.245299i \(0.921114\pi\)
\(42\) 0 0
\(43\) −6.05828 + 2.50942i −0.923880 + 0.382683i
\(44\) −4.97189 7.44097i −0.749541 1.12177i
\(45\) 0 0
\(46\) 0 0
\(47\) 6.16199 6.16199i 0.898818 0.898818i −0.0965136 0.995332i \(-0.530769\pi\)
0.995332 + 0.0965136i \(0.0307691\pi\)
\(48\) 0 0
\(49\) 2.67878 6.46716i 0.382683 0.923880i
\(50\) 0 0
\(51\) 0 0
\(52\) −13.5163 −1.87437
\(53\) 10.1933 + 4.22221i 1.40016 + 0.579965i 0.949793 0.312878i \(-0.101293\pi\)
0.450367 + 0.892844i \(0.351293\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.46524 + 8.36582i 0.451135 + 1.08914i 0.971891 + 0.235431i \(0.0756503\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −5.65685 + 5.65685i −0.707107 + 0.707107i
\(65\) 0 0
\(66\) 0 0
\(67\) 11.3488i 1.38647i 0.720710 + 0.693236i \(0.243816\pi\)
−0.720710 + 0.693236i \(0.756184\pi\)
\(68\) −7.97170 + 2.10997i −0.966711 + 0.255872i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(72\) 0 0
\(73\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 9.03385 + 1.79695i 1.01639 + 0.202172i 0.675053 0.737769i \(-0.264121\pi\)
0.341335 + 0.939942i \(0.389121\pi\)
\(80\) 0 0
\(81\) 6.36396 6.36396i 0.707107 0.707107i
\(82\) 0 0
\(83\) −2.50942 + 6.05828i −0.275445 + 0.664983i −0.999699 0.0245507i \(-0.992184\pi\)
0.724254 + 0.689534i \(0.242184\pi\)
\(84\) 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
\(92\) 12.9661 8.66364i 1.35180 0.903247i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.58732 2.37559i 0.161168 0.241204i −0.742093 0.670297i \(-0.766167\pi\)
0.903260 + 0.429093i \(0.141167\pi\)
\(98\) 0 0
\(99\) 13.1658 2.61885i 1.32322 0.263204i
\(100\) −3.82683 + 9.23880i −0.382683 + 0.923880i
\(101\) 20.0631i 1.99636i −0.0603342 0.998178i \(-0.519217\pi\)
0.0603342 0.998178i \(-0.480783\pi\)
\(102\) 0 0
\(103\) −20.0883 −1.97936 −0.989678 0.143310i \(-0.954225\pi\)
−0.989678 + 0.143310i \(0.954225\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.67583 + 1.11975i 0.162008 + 0.108250i 0.633932 0.773389i \(-0.281440\pi\)
−0.471923 + 0.881640i \(0.656440\pi\)
\(108\) 0 0
\(109\) −10.3667 + 6.92682i −0.992951 + 0.663469i −0.942133 0.335239i \(-0.891183\pi\)
−0.0508181 + 0.998708i \(0.516183\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 7.75867 18.7311i 0.717290 1.73169i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 8.33518 + 3.45254i 0.757743 + 0.313868i
\(122\) 0 0
\(123\) 0 0
\(124\) 17.3818 + 11.6141i 1.56093 + 1.04298i
\(125\) 0 0
\(126\) 0 0
\(127\) −1.00814 2.43387i −0.0894583 0.215971i 0.872818 0.488046i \(-0.162290\pi\)
−0.962276 + 0.272075i \(0.912290\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(132\) 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
\(139\) 3.26935 + 16.4361i 0.277303 + 1.39410i 0.828626 + 0.559803i \(0.189123\pi\)
−0.551323 + 0.834292i \(0.685877\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 25.1435 16.8004i 2.10261 1.40492i
\(144\) −4.59220 11.0866i −0.382683 0.923880i
\(145\) 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 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(152\) 0 0
\(153\) 1.65191 12.2585i 0.133549 0.991042i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(164\) −23.6721 4.70867i −1.84848 0.367686i
\(165\) 0 0
\(166\) 0 0
\(167\) −8.27483 + 1.64597i −0.640326 + 0.127369i −0.504566 0.863373i \(-0.668347\pi\)
−0.135760 + 0.990742i \(0.543347\pi\)
\(168\) 0 0
\(169\) 32.6723i 2.51326i
\(170\) 0 0
\(171\) 0 0
\(172\) −12.1166 5.01885i −0.923880 0.382683i
\(173\) 4.47155 + 22.4800i 0.339965 + 1.70912i 0.651300 + 0.758820i \(0.274224\pi\)
−0.311335 + 0.950300i \(0.600776\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 3.49179 17.5544i 0.263204 1.32322i
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(180\) 0 0
\(181\) 24.0834 + 4.79048i 1.79010 + 0.356074i 0.974821 0.222988i \(-0.0715812\pi\)
0.815283 + 0.579062i \(0.196581\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 12.2066 13.8337i 0.892638 1.01162i
\(188\) 17.4287 1.27112
\(189\) 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
\(193\) 0.227085 1.14163i 0.0163460 0.0821767i −0.971751 0.236007i \(-0.924161\pi\)
0.988097 + 0.153831i \(0.0491610\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 12.9343 5.35757i 0.923880 0.382683i
\(197\) −8.29190 12.4097i −0.590773 0.884155i 0.408822 0.912614i \(-0.365940\pi\)
−0.999595 + 0.0284595i \(0.990940\pi\)
\(198\) 0 0
\(199\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 4.56340 + 22.9418i 0.317178 + 1.59456i
\(208\) −19.1149 19.1149i −1.32538 1.32538i
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(212\) 8.44442 + 20.3866i 0.579965 + 1.40016i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −7.12974 26.9369i −0.479598 1.81197i
\(222\) 0 0
\(223\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(224\) 0 0
\(225\) −10.6066 10.6066i −0.707107 0.707107i
\(226\) 0 0
\(227\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(228\) 0 0
\(229\) 11.0127 + 26.5869i 0.727738 + 1.75691i 0.649992 + 0.759941i \(0.274772\pi\)
0.0777462 + 0.996973i \(0.475228\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −6.93047 + 16.7316i −0.451135 + 1.08914i
\(237\) 0 0
\(238\) 0 0
\(239\) 10.5254 0.680830 0.340415 0.940275i \(-0.389432\pi\)
0.340415 + 0.940275i \(0.389432\pi\)
\(240\) 0 0
\(241\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 12.2213 12.2213i 0.771400 0.771400i −0.206951 0.978351i \(-0.566354\pi\)
0.978351 + 0.206951i \(0.0663540\pi\)
\(252\) 0 0
\(253\) −13.3513 + 32.2329i −0.839390 + 2.02647i
\(254\) 0 0
\(255\) 0 0
\(256\) −16.0000 −1.00000
\(257\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −16.0496 + 16.0496i −0.980384 + 0.980384i
\(269\) 31.7306 6.31162i 1.93465 0.384826i 0.935116 0.354341i \(-0.115295\pi\)
0.999535 0.0304855i \(-0.00970535\pi\)
\(270\) 0 0
\(271\) 29.1434i 1.77033i 0.465274 + 0.885167i \(0.345956\pi\)
−0.465274 + 0.885167i \(0.654044\pi\)
\(272\) −14.2576 8.28973i −0.864496 0.502639i
\(273\) 0 0
\(274\) 0 0
\(275\) −4.36474 21.9430i −0.263204 1.32322i
\(276\) 0 0
\(277\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(278\) 0 0
\(279\) −26.0727 + 17.4212i −1.56093 + 1.04298i
\(280\) 0 0
\(281\) 5.61715 2.32670i 0.335091 0.138799i −0.208792 0.977960i \(-0.566953\pi\)
0.543884 + 0.839161i \(0.316953\pi\)
\(282\) 0 0
\(283\) 23.3382 + 4.64225i 1.38731 + 0.275953i 0.831578 0.555409i \(-0.187438\pi\)
0.555732 + 0.831362i \(0.312438\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −8.41004 14.7740i −0.494708 0.869059i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −2.11488 2.11488i −0.123552 0.123552i 0.642627 0.766179i \(-0.277845\pi\)
−0.766179 + 0.642627i \(0.777845\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 29.2750 + 43.8131i 1.69302 + 2.53378i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −7.07703 −0.403907 −0.201954 0.979395i \(-0.564729\pi\)
−0.201954 + 0.979395i \(0.564729\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 5.28295 + 3.52995i 0.299568 + 0.200165i 0.696265 0.717784i \(-0.254844\pi\)
−0.396697 + 0.917950i \(0.629844\pi\)
\(312\) 0 0
\(313\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 10.2345 + 15.3171i 0.575738 + 0.861652i
\(317\) −32.1154 6.38815i −1.80378 0.358794i −0.825228 0.564799i \(-0.808954\pi\)
−0.978551 + 0.206005i \(0.933954\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 18.0000 1.00000
\(325\) −31.2185 12.9311i −1.73169 0.717290i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(332\) −12.1166 + 5.01885i −0.664983 + 0.275445i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −14.5609 + 2.89635i −0.793185 + 0.157774i −0.575022 0.818138i \(-0.695006\pi\)
−0.218163 + 0.975912i \(0.570006\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −46.7704 −2.53276
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(348\) 0 0
\(349\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −21.7787 + 21.7787i −1.15916 + 1.15916i −0.174509 + 0.984656i \(0.555834\pi\)
−0.984656 + 0.174509i \(0.944166\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −9.00692 3.73079i −0.475367 0.196903i 0.132119 0.991234i \(-0.457822\pi\)
−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
\(367\) −20.6778 30.9466i −1.07937 1.61540i −0.737079 0.675806i \(-0.763796\pi\)
−0.342296 0.939592i \(-0.611204\pi\)
\(368\) 30.5890 + 6.08453i 1.59456 + 0.317178i
\(369\) 20.1137 30.1024i 1.04708 1.56707i
\(370\) 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
\(379\) −22.4438 14.9965i −1.15286 0.770318i −0.176042 0.984383i \(-0.556330\pi\)
−0.976819 + 0.214065i \(0.931330\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 13.9104 13.9104i 0.707107 0.707107i
\(388\) 5.60439 1.11478i 0.284520 0.0565945i
\(389\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(390\) 0 0
\(391\) 24.1055 + 21.2704i 1.21907 + 1.07569i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 22.3229 + 14.9157i 1.12177 + 0.749541i
\(397\) −2.02340 + 10.1723i −0.101552 + 0.510535i 0.896208 + 0.443634i \(0.146311\pi\)
−0.997760 + 0.0669005i \(0.978689\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −18.4776 + 7.65367i −0.923880 + 0.382683i
\(401\) 8.04657 + 12.0425i 0.401826 + 0.601376i 0.976109 0.217281i \(-0.0697189\pi\)
−0.574283 + 0.818657i \(0.694719\pi\)
\(402\) 0 0
\(403\) −39.2449 + 58.7342i −1.95493 + 2.92576i
\(404\) 28.3736 28.3736i 1.41164 1.41164i
\(405\) 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
\(412\) −28.4091 28.4091i −1.39962 1.39962i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(420\) 0 0
\(421\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(422\) 0 0
\(423\) −10.0045 + 24.1531i −0.486437 + 1.17436i
\(424\) 0 0
\(425\) −20.4309 2.75319i −0.991042 0.133549i
\(426\) 0 0
\(427\) 0 0
\(428\) 0.786408 + 3.95354i 0.0380125 + 0.191102i
\(429\) 0 0
\(430\) 0 0
\(431\) 2.81518 14.1529i 0.135603 0.681720i −0.851848 0.523789i \(-0.824518\pi\)
0.987450 0.157930i \(-0.0504821\pi\)
\(432\) 0 0
\(433\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −24.4567 4.86475i −1.17127 0.232979i
\(437\) 0 0
\(438\) 0 0
\(439\) 40.8926 8.13404i 1.95170 0.388217i 0.956299 0.292391i \(-0.0944507\pi\)
0.995398 0.0958262i \(-0.0305493\pi\)
\(440\) 0 0
\(441\) 21.0000i 1.00000i
\(442\) 0 0
\(443\) −37.5579 −1.78443 −0.892215 0.451612i \(-0.850849\pi\)
−0.892215 + 0.451612i \(0.850849\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(450\) 0 0
\(451\) 49.8886 20.6645i 2.34916 0.973055i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −24.2331 10.0377i −1.12865 0.467502i −0.261328 0.965250i \(-0.584160\pi\)
−0.867322 + 0.497748i \(0.834160\pi\)
\(462\) 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 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(468\) 37.4622 15.5173i 1.73169 0.717290i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 28.7780 5.72431i 1.32322 0.263204i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −33.0995 −1.51552
\(478\) 0 0
\(479\) 1.15045 + 5.78372i 0.0525656 + 0.264265i 0.998127 0.0611793i \(-0.0194862\pi\)
−0.945561 + 0.325444i \(0.894486\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 6.90509 + 16.6704i 0.313868 + 0.757743i
\(485\) 0 0
\(486\) 0 0
\(487\) −42.9832 8.54989i −1.94775 0.387432i −0.996991 0.0775113i \(-0.975303\pi\)
−0.950762 0.309921i \(-0.899697\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 8.15668 + 41.0064i 0.366246 + 1.84124i
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 2.01629 4.86775i 0.0894583 0.215971i
\(509\) 42.1546i 1.86847i 0.356658 + 0.934235i \(0.383916\pi\)
−0.356658 + 0.934235i \(0.616084\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −32.4216 + 21.6635i −1.42590 + 0.952758i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(522\) 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
\(527\) −13.9773 + 40.7670i −0.608862 + 1.77584i
\(528\) 0 0
\(529\) −34.9174 14.4633i −1.51815 0.628837i
\(530\) 0 0
\(531\) −19.2088 19.2088i −0.833589 0.833589i
\(532\) 0 0
\(533\) 15.9109 79.9896i 0.689178 3.46473i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −17.4016 + 26.0434i −0.749541 + 1.12177i
\(540\) 0 0
\(541\) 25.2250 5.01757i 1.08451 0.215722i 0.379693 0.925113i \(-0.376030\pi\)
0.704816 + 0.709390i \(0.251030\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 3.46912 + 17.4404i 0.148329 + 0.745699i 0.981315 + 0.192406i \(0.0616291\pi\)
−0.832987 + 0.553293i \(0.813371\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −18.6207 + 27.8678i −0.789692 + 1.18186i
\(557\) 27.0104 27.0104i 1.14447 1.14447i 0.156844 0.987623i \(-0.449868\pi\)
0.987623 0.156844i \(-0.0501319\pi\)
\(558\) 0 0
\(559\) 16.9590 40.9427i 0.717290 1.73169i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 22.2141 + 9.20140i 0.936214 + 0.387793i 0.798033 0.602614i \(-0.205874\pi\)
0.138182 + 0.990407i \(0.455874\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 8.90324 + 21.4943i 0.373243 + 0.901088i 0.993197 + 0.116450i \(0.0371515\pi\)
−0.619953 + 0.784639i \(0.712849\pi\)
\(570\) 0 0
\(571\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(572\) 59.3176 + 11.7990i 2.48019 + 0.493341i
\(573\) 0 0
\(574\) 0 0
\(575\) 38.2363 7.60567i 1.59456 0.317178i
\(576\) 9.18440 22.1731i 0.382683 0.923880i
\(577\) 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
\(583\) −41.0487 27.4279i −1.70006 1.13595i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(594\) 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 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(602\) 0 0
\(603\) −13.0290 31.4547i −0.530580 1.28093i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 58.8928i 2.38255i
\(612\) 19.6723 15.0000i 0.795206 0.606339i
\(613\) 48.8406 1.97265 0.986327 0.164798i \(-0.0526973\pi\)
0.986327 + 0.164798i \(0.0526973\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −34.3508 22.9525i −1.38291 0.924032i −1.00000 0.000246592i \(-0.999922\pi\)
−0.382911 0.923785i \(-0.625078\pi\)
\(618\) 0 0
\(619\) 18.4813 12.3488i 0.742824 0.496339i −0.125649 0.992075i \(-0.540101\pi\)
0.868474 + 0.495735i \(0.165101\pi\)
\(620\) 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 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 18.1036 + 43.7059i 0.717290 + 1.73169i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(642\) 0 0
\(643\) 38.3103 7.62040i 1.51081 0.300519i 0.630978 0.775800i \(-0.282654\pi\)
0.879834 + 0.475281i \(0.157654\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) −7.90464 39.7393i −0.310284 1.55990i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −26.8183 40.1365i −1.04708 1.56707i
\(657\) 0 0
\(658\) 0 0
\(659\) 36.2107 36.2107i 1.41057 1.41057i 0.654557 0.756013i \(-0.272855\pi\)
0.756013 0.654557i \(-0.227145\pi\)
\(660\) 0 0
\(661\) −2.50942 + 6.05828i −0.0976052 + 0.235640i −0.965139 0.261739i \(-0.915704\pi\)
0.867533 + 0.497379i \(0.165704\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −14.0301 9.37463i −0.542842 0.362715i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 46.2056 46.2056i 1.77714 1.77714i
\(677\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −3.03650 15.2655i −0.116188 0.584119i −0.994385 0.105819i \(-0.966253\pi\)
0.878197 0.478299i \(-0.158747\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −10.0377 24.2331i −0.382683 0.923880i
\(689\) −68.8878 + 28.5342i −2.62441 + 1.08707i
\(690\) 0 0
\(691\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(692\) −25.4678 + 38.1152i −0.968139 + 1.44892i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −3.10285 49.6605i −0.117529 1.88103i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −36.1149 36.1149i −1.36404 1.36404i −0.868698 0.495342i \(-0.835043\pi\)
−0.495342 0.868698i \(-0.664957\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 29.7639 19.8876i 1.12177 0.749541i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1.69568 + 2.53776i −0.0636826 + 0.0953078i −0.861944 0.507003i \(-0.830753\pi\)
0.798262 + 0.602311i \(0.205753\pi\)
\(710\) 0 0
\(711\) −27.1016 + 5.39084i −1.01639 + 0.202172i
\(712\) 0 0
\(713\) 81.4984i 3.05214i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 33.0877 + 22.1085i 1.23396 + 0.824508i 0.989413 0.145128i \(-0.0463594\pi\)
0.244551 + 0.969636i \(0.421359\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 27.2843 + 40.8338i 1.01401 + 1.51758i
\(725\) 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
\(729\) −10.3325 + 24.9447i −0.382683 + 0.923880i
\(730\) 0 0
\(731\) 3.61077 26.7948i 0.133549 0.991042i
\(732\) 0 0
\(733\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 9.90689 49.8053i 0.364925 1.83460i
\(738\) 0 0
\(739\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 19.6723i 0.719772i
\(748\) 36.8266 2.30097i 1.34651 0.0841318i
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(752\) 24.6479 + 24.6479i 0.898818 + 0.898818i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(758\) 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
\(767\) −56.5373 23.4185i −2.04144 0.845594i
\(768\) 0 0
\(769\) 2.43637 + 2.43637i 0.0878579 + 0.0878579i 0.749670 0.661812i \(-0.230212\pi\)
−0.661812 + 0.749670i \(0.730212\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.93566 1.29337i 0.0696660 0.0465493i
\(773\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(774\) 0 0
\(775\) 29.0353 + 43.4544i 1.04298 + 1.56093i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 25.8686 + 10.7151i 0.923880 + 0.382683i
\(785\) 0 0
\(786\) 0 0
\(787\) 45.8285 + 30.6216i 1.63361 + 1.09154i 0.920687 + 0.390301i \(0.127629\pi\)
0.712923 + 0.701242i \(0.247371\pi\)
\(788\) 5.82346 29.2765i 0.207452 1.04293i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.95129 + 4.71083i −0.0691182 + 0.166866i −0.954664 0.297687i \(-0.903785\pi\)
0.885545 + 0.464553i \(0.153785\pi\)
\(798\) 0 0
\(799\) 9.19354 + 34.7342i 0.325244 + 1.22881i
\(800\) 0 0
\(801\) 0 0
\(802\) 0