Properties

Label 1520.2.g.d.1519.4
Level $1520$
Weight $2$
Character 1520.1519
Analytic conductor $12.137$
Analytic rank $0$
Dimension $4$
CM discriminant -19
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1520.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.1372611072\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-19})\)
Defining polynomial: \(x^{4} - x^{3} - 4 x^{2} - 5 x + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 1519.4
Root \(-1.63746 - 1.52274i\) of defining polynomial
Character \(\chi\) \(=\) 1520.1519
Dual form 1520.2.g.d.1519.3

$q$-expansion

\(f(q)\) \(=\) \(q+(2.13746 + 0.656712i) q^{5} -2.27492 q^{7} +3.00000 q^{9} +O(q^{10})\) \(q+(2.13746 + 0.656712i) q^{5} -2.27492 q^{7} +3.00000 q^{9} -2.15068i q^{11} -8.24163i q^{17} -4.35890i q^{19} -4.00000 q^{23} +(4.13746 + 2.80739i) q^{25} +(-4.86254 - 1.49397i) q^{35} +10.8248 q^{43} +(6.41238 + 1.97014i) q^{45} +10.2749 q^{47} -1.82475 q^{49} +(1.41238 - 4.59698i) q^{55} +3.72508 q^{61} -6.82475 q^{63} -2.98793i q^{73} +4.89261i q^{77} +9.00000 q^{81} -16.0000 q^{83} +(5.41238 - 17.6161i) q^{85} +(2.86254 - 9.31697i) q^{95} -6.45203i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{5} + 6 q^{7} + 12 q^{9} + O(q^{10}) \) \( 4 q + q^{5} + 6 q^{7} + 12 q^{9} - 16 q^{23} + 9 q^{25} - 27 q^{35} - 2 q^{43} + 3 q^{45} + 26 q^{47} + 38 q^{49} - 17 q^{55} + 30 q^{61} + 18 q^{63} + 36 q^{81} - 64 q^{83} - q^{85} + 19 q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(401\) \(1141\) \(1217\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

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
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 0 0
\(5\) 2.13746 + 0.656712i 0.955901 + 0.293691i
\(6\) 0 0
\(7\) −2.27492 −0.859838 −0.429919 0.902867i \(-0.641458\pi\)
−0.429919 + 0.902867i \(0.641458\pi\)
\(8\) 0 0
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) 2.15068i 0.648454i −0.945979 0.324227i \(-0.894896\pi\)
0.945979 0.324227i \(-0.105104\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 8.24163i 1.99889i −0.0333386 0.999444i \(-0.510614\pi\)
0.0333386 0.999444i \(-0.489386\pi\)
\(18\) 0 0
\(19\) 4.35890i 1.00000i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 4.13746 + 2.80739i 0.827492 + 0.561478i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.86254 1.49397i −0.821920 0.252526i
\(36\) 0 0
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 10.8248 1.65076 0.825380 0.564578i \(-0.190961\pi\)
0.825380 + 0.564578i \(0.190961\pi\)
\(44\) 0 0
\(45\) 6.41238 + 1.97014i 0.955901 + 0.293691i
\(46\) 0 0
\(47\) 10.2749 1.49875 0.749375 0.662145i \(-0.230354\pi\)
0.749375 + 0.662145i \(0.230354\pi\)
\(48\) 0 0
\(49\) −1.82475 −0.260679
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 1.41238 4.59698i 0.190445 0.619857i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 3.72508 0.476948 0.238474 0.971149i \(-0.423353\pi\)
0.238474 + 0.971149i \(0.423353\pi\)
\(62\) 0 0
\(63\) −6.82475 −0.859838
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 2.98793i 0.349711i −0.984594 0.174855i \(-0.944054\pi\)
0.984594 0.174855i \(-0.0559458\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.89261i 0.557565i
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) −16.0000 −1.75623 −0.878114 0.478451i \(-0.841198\pi\)
−0.878114 + 0.478451i \(0.841198\pi\)
\(84\) 0 0
\(85\) 5.41238 17.6161i 0.587055 1.91074i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.86254 9.31697i 0.293691 0.955901i
\(96\) 0 0
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) 6.45203i 0.648454i
\(100\) 0 0
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) −8.54983 2.62685i −0.797276 0.244955i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 18.7490i 1.71872i
\(120\) 0 0
\(121\) 6.37459 0.579508
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 7.00000 + 8.71780i 0.626099 + 0.779744i
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 16.9594i 1.48175i −0.671642 0.740876i \(-0.734411\pi\)
0.671642 0.740876i \(-0.265589\pi\)
\(132\) 0 0
\(133\) 9.91613i 0.859838i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 22.0980i 1.88796i 0.329999 + 0.943981i \(0.392952\pi\)
−0.329999 + 0.943981i \(0.607048\pi\)
\(138\) 0 0
\(139\) 3.10302i 0.263195i 0.991303 + 0.131597i \(0.0420106\pi\)
−0.991303 + 0.131597i \(0.957989\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −13.3746 −1.09569 −0.547844 0.836580i \(-0.684551\pi\)
−0.547844 + 0.836580i \(0.684551\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 24.7249i 1.99889i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 17.4356i 1.39151i −0.718278 0.695756i \(-0.755069\pi\)
0.718278 0.695756i \(-0.244931\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 9.09967 0.717154
\(162\) 0 0
\(163\) 24.0000 1.87983 0.939913 0.341415i \(-0.110906\pi\)
0.939913 + 0.341415i \(0.110906\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 13.0767i 1.00000i
\(172\) 0 0
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) −9.41238 6.38658i −0.711509 0.482780i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −17.7251 −1.29619
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3.82518i 0.276781i 0.990378 + 0.138390i \(0.0441928\pi\)
−0.990378 + 0.138390i \(0.955807\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 17.4356i 1.24223i −0.783718 0.621117i \(-0.786679\pi\)
0.783718 0.621117i \(-0.213321\pi\)
\(198\) 0 0
\(199\) 28.1890i 1.99826i 0.0416556 + 0.999132i \(0.486737\pi\)
−0.0416556 + 0.999132i \(0.513263\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −12.0000 −0.834058
\(208\) 0 0
\(209\) −9.37459 −0.648454
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 23.1375 + 7.10874i 1.57796 + 0.484812i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 12.4124 + 8.42217i 0.827492 + 0.561478i
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) −29.3746 −1.94113 −0.970564 0.240845i \(-0.922576\pi\)
−0.970564 + 0.240845i \(0.922576\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 16.1222i 1.05620i 0.849183 + 0.528099i \(0.177095\pi\)
−0.849183 + 0.528099i \(0.822905\pi\)
\(234\) 0 0
\(235\) 21.9622 + 6.74766i 1.43266 + 0.440169i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 19.5863i 1.26693i 0.773771 + 0.633465i \(0.218368\pi\)
−0.773771 + 0.633465i \(0.781632\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −3.90033 1.19834i −0.249183 0.0765589i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 30.8158i 1.94508i −0.232740 0.972539i \(-0.574769\pi\)
0.232740 0.972539i \(-0.425231\pi\)
\(252\) 0 0
\(253\) 8.60271i 0.540848i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 31.9244 1.96854 0.984272 0.176659i \(-0.0565291\pi\)
0.984272 + 0.176659i \(0.0565291\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 26.1534i 1.58871i 0.607457 + 0.794353i \(0.292190\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 6.03779 8.89834i 0.364092 0.536590i
\(276\) 0 0
\(277\) 26.3994i 1.58619i 0.609101 + 0.793093i \(0.291530\pi\)
−0.609101 + 0.793093i \(0.708470\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −26.8248 −1.59457 −0.797283 0.603606i \(-0.793730\pi\)
−0.797283 + 0.603606i \(0.793730\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −50.9244 −2.99555
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −24.6254 −1.41939
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 7.96221 + 2.44631i 0.455915 + 0.140075i
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 32.4903i 1.84236i 0.389139 + 0.921179i \(0.372773\pi\)
−0.389139 + 0.921179i \(0.627227\pi\)
\(312\) 0 0
\(313\) 34.8712i 1.97104i 0.169570 + 0.985518i \(0.445762\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 0 0
\(315\) −14.5876 4.48190i −0.821920 0.252526i
\(316\) 0 0
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −35.9244 −1.99889
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −23.3746 −1.28868
\(330\) 0 0
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 20.0756 1.08398
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −22.2749 −1.19578 −0.597890 0.801578i \(-0.703994\pi\)
−0.597890 + 0.801578i \(0.703994\pi\)
\(348\) 0 0
\(349\) −28.8248 −1.54295 −0.771477 0.636257i \(-0.780482\pi\)
−0.771477 + 0.636257i \(0.780482\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 34.8712i 1.85601i −0.372572 0.928003i \(-0.621524\pi\)
0.372572 0.928003i \(-0.378476\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 37.7440i 1.99205i 0.0890519 + 0.996027i \(0.471616\pi\)
−0.0890519 + 0.996027i \(0.528384\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1.96221 6.38658i 0.102707 0.334289i
\(366\) 0 0
\(367\) 28.0000 1.46159 0.730794 0.682598i \(-0.239150\pi\)
0.730794 + 0.682598i \(0.239150\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 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
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) −3.21304 + 10.4578i −0.163752 + 0.532977i
\(386\) 0 0
\(387\) 32.4743 1.65076
\(388\) 0 0
\(389\) −13.9244 −0.705996 −0.352998 0.935624i \(-0.614838\pi\)
−0.352998 + 0.935624i \(0.614838\pi\)
\(390\) 0 0
\(391\) 32.9665i 1.66719i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 25.6772i 1.28870i 0.764730 + 0.644351i \(0.222873\pi\)
−0.764730 + 0.644351i \(0.777127\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 19.2371 + 5.91041i 0.955901 + 0.293691i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −34.1993 10.5074i −1.67878 0.515788i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 8.71780i 0.425892i −0.977064 0.212946i \(-0.931694\pi\)
0.977064 0.212946i \(-0.0683059\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 30.8248 1.49875
\(424\) 0 0
\(425\) 23.1375 34.0994i 1.12233 1.65406i
\(426\) 0 0
\(427\) −8.47425 −0.410098
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 17.4356i 0.834058i
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) −5.47425 −0.260679
\(442\) 0 0
\(443\) 11.9244 0.566546 0.283273 0.959039i \(-0.408580\pi\)
0.283273 + 0.959039i \(0.408580\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 4.89261i 0.228867i −0.993431 0.114433i \(-0.963495\pi\)
0.993431 0.114433i \(-0.0365053\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 37.3746 1.74071 0.870354 0.492427i \(-0.163890\pi\)
0.870354 + 0.492427i \(0.163890\pi\)
\(462\) 0 0
\(463\) −9.17525 −0.426410 −0.213205 0.977007i \(-0.568390\pi\)
−0.213205 + 0.977007i \(0.568390\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 17.7251 0.820219 0.410110 0.912036i \(-0.365490\pi\)
0.410110 + 0.912036i \(0.365490\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 23.2806i 1.07044i
\(474\) 0 0
\(475\) 12.2371 18.0348i 0.561478 0.827492i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 43.5890i 1.99163i −0.0913823 0.995816i \(-0.529129\pi\)
0.0913823 0.995816i \(-0.470871\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 43.5890i 1.96714i −0.180517 0.983572i \(-0.557777\pi\)
0.180517 0.983572i \(-0.442223\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 4.23713 13.7910i 0.190445 0.619857i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 44.6722i 1.99980i 0.0139987 + 0.999902i \(0.495544\pi\)
−0.0139987 + 0.999902i \(0.504456\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −44.0000 −1.96186 −0.980932 0.194354i \(-0.937739\pi\)
−0.980932 + 0.194354i \(0.937739\pi\)
\(504\) 0 0
\(505\) 21.3746 + 6.56712i 0.951157 + 0.292233i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 6.79730i 0.300695i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 22.0980i 0.971870i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 3.92445i 0.169038i
\(540\) 0 0
\(541\) −46.4743 −1.99808 −0.999042 0.0437584i \(-0.986067\pi\)
−0.999042 + 0.0437584i \(0.986067\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 11.1752 0.476948
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 42.8826i 1.81700i −0.417889 0.908498i \(-0.637230\pi\)
0.417889 0.908498i \(-0.362770\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −20.4743 −0.859838
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 26.1534i 1.09449i −0.836974 0.547243i \(-0.815677\pi\)
0.836974 0.547243i \(-0.184323\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −16.5498 11.2296i −0.690176 0.468305i
\(576\) 0 0
\(577\) 21.3759i 0.889889i 0.895558 + 0.444945i \(0.146777\pi\)
−0.895558 + 0.444945i \(0.853223\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 36.3987 1.51007
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 45.0241 1.85834 0.929172 0.369649i \(-0.120522\pi\)
0.929172 + 0.369649i \(0.120522\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 34.8712i 1.43199i 0.698106 + 0.715994i \(0.254026\pi\)
−0.698106 + 0.715994i \(0.745974\pi\)
\(594\) 0 0
\(595\) −12.3127 + 40.0753i −0.504772 + 1.64293i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 13.6254 + 4.18627i 0.553952 + 0.170196i
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 18.5188i 0.747969i −0.927435 0.373985i \(-0.877991\pi\)
0.927435 0.373985i \(-0.122009\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 35.2323i 1.41840i −0.705008 0.709199i \(-0.749057\pi\)
0.705008 0.709199i \(-0.250943\pi\)
\(618\) 0 0
\(619\) 43.5890i 1.75199i 0.482321 + 0.875995i \(0.339794\pi\)
−0.482321 + 0.875995i \(0.660206\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 9.23713 + 23.2309i 0.369485 + 0.929237i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 10.9836i 0.437249i −0.975809 0.218624i \(-0.929843\pi\)
0.975809 0.218624i \(-0.0701569\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 13.1752 0.519581 0.259791 0.965665i \(-0.416346\pi\)
0.259791 + 0.965665i \(0.416346\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −33.0241 −1.29831 −0.649155 0.760656i \(-0.724878\pi\)
−0.649155 + 0.760656i \(0.724878\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 50.7632i 1.98652i 0.115920 + 0.993259i \(0.463018\pi\)
−0.115920 + 0.993259i \(0.536982\pi\)
\(654\) 0 0
\(655\) 11.1375 36.2501i 0.435177 1.41641i
\(656\) 0 0
\(657\) 8.96379i 0.349711i
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −6.51204 + 21.1953i −0.252526 + 0.821920i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 8.01145i 0.309279i
\(672\) 0 0
\(673\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) −14.5120 + 47.2336i −0.554477 + 1.80470i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 49.9259i 1.89927i 0.313355 + 0.949636i \(0.398547\pi\)
−0.313355 + 0.949636i \(0.601453\pi\)
\(692\) 0 0
\(693\) 14.6778i 0.557565i
\(694\) 0 0
\(695\) −2.03779 + 6.63257i −0.0772978 + 0.251588i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 50.0000 1.88847 0.944237 0.329267i \(-0.106802\pi\)
0.944237 + 0.329267i \(0.106802\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −22.7492 −0.855571
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 31.5380i 1.17617i −0.808800 0.588084i \(-0.799882\pi\)
0.808800 0.588084i \(-0.200118\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 52.4743 1.94616 0.973081 0.230463i \(-0.0740239\pi\)
0.973081 + 0.230463i \(0.0740239\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 89.2136i 3.29968i
\(732\) 0 0
\(733\) 52.3068i 1.93200i −0.258551 0.965998i \(-0.583245\pi\)
0.258551 0.965998i \(-0.416755\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 54.2273i 1.99478i 0.0721811 + 0.997392i \(0.477004\pi\)
−0.0721811 + 0.997392i \(0.522996\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) −28.5876 8.78325i −1.04737 0.321793i
\(746\) 0 0
\(747\) −48.0000 −1.75623
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0.591258i 0.0214896i −0.999942 0.0107448i \(-0.996580\pi\)
0.999942 0.0107448i \(-0.00342025\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 23.7251 0.860034 0.430017 0.902821i \(-0.358508\pi\)
0.430017 + 0.902821i \(0.358508\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 16.2371 52.8484i 0.587055 1.91074i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 6.62541 0.238919 0.119459 0.992839i \(-0.461884\pi\)
0.119459 + 0.992839i \(0.461884\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 11.4502 37.2679i 0.408674 1.33015i
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 84.6820i 2.99584i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −6.42608 −0.226771
\(804\) 0 0
\(805\) 19.4502 + 5.97586i 0.685528 + 0.210621i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 51.5739 1.81324 0.906621 0.421945i \(-0.138653\pi\)
0.906621 + 0.421945i \(0.138653\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0