Properties

Label 1520.2.g.d.1519.1
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 1519.1
Root \(2.13746 + 0.656712i\) of defining polynomial
Character \(\chi\) \(=\) 1520.1519
Dual form 1520.2.g.d.1519.2

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.63746 - 1.52274i) q^{5} +5.27492 q^{7} +3.00000 q^{9} +O(q^{10})\) \(q+(-1.63746 - 1.52274i) q^{5} +5.27492 q^{7} +3.00000 q^{9} -6.50958i q^{11} -3.88273i q^{17} +4.35890i q^{19} -4.00000 q^{23} +(0.362541 + 4.98684i) q^{25} +(-8.63746 - 8.03231i) q^{35} -11.8248 q^{43} +(-4.91238 - 4.56821i) q^{45} +2.72508 q^{47} +20.8248 q^{49} +(-9.91238 + 10.6592i) q^{55} +11.2749 q^{61} +15.8248 q^{63} -16.0646i q^{73} -34.3375i q^{77} +9.00000 q^{81} -16.0000 q^{83} +(-5.91238 + 6.35781i) q^{85} +(6.63746 - 7.13752i) q^{95} -19.5287i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{5} + 6 q^{7} + 12 q^{9} + O(q^{10}) \) \( 4 q + q^{5} + 6 q^{7} + 12 q^{9} - 16 q^{23} + 9 q^{25} - 27 q^{35} - 2 q^{43} + 3 q^{45} + 26 q^{47} + 38 q^{49} - 17 q^{55} + 30 q^{61} + 18 q^{63} + 36 q^{81} - 64 q^{83} - q^{85} + 19 q^{95} + O(q^{100}) \)

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
\(5\) −1.63746 1.52274i −0.732294 0.680989i
\(6\) 0 0
\(7\) 5.27492 1.99373 0.996866 0.0791130i \(-0.0252088\pi\)
0.996866 + 0.0791130i \(0.0252088\pi\)
\(8\) 0 0
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) 6.50958i 1.96271i −0.192201 0.981356i \(-0.561563\pi\)
0.192201 0.981356i \(-0.438437\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.88273i 0.941700i −0.882213 0.470850i \(-0.843947\pi\)
0.882213 0.470850i \(-0.156053\pi\)
\(18\) 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
\(25\) 0.362541 + 4.98684i 0.0725083 + 0.997368i
\(26\) 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
\(35\) −8.63746 8.03231i −1.46000 1.35771i
\(36\) 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
\(43\) −11.8248 −1.80326 −0.901629 0.432511i \(-0.857628\pi\)
−0.901629 + 0.432511i \(0.857628\pi\)
\(44\) 0 0
\(45\) −4.91238 4.56821i −0.732294 0.680989i
\(46\) 0 0
\(47\) 2.72508 0.397494 0.198747 0.980051i \(-0.436313\pi\)
0.198747 + 0.980051i \(0.436313\pi\)
\(48\) 0 0
\(49\) 20.8248 2.97496
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) −9.91238 + 10.6592i −1.33658 + 1.43728i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 11.2749 1.44361 0.721803 0.692099i \(-0.243314\pi\)
0.721803 + 0.692099i \(0.243314\pi\)
\(62\) 0 0
\(63\) 15.8248 1.99373
\(64\) 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
\(73\) 16.0646i 1.88022i −0.340868 0.940111i \(-0.610721\pi\)
0.340868 0.940111i \(-0.389279\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 34.3375i 3.91312i
\(78\) 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
\(85\) −5.91238 + 6.35781i −0.641287 + 0.689601i
\(86\) 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
\(95\) 6.63746 7.13752i 0.680989 0.732294i
\(96\) 0 0
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) 19.5287i 1.96271i
\(100\) 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
\(115\) 6.54983 + 6.09095i 0.610775 + 0.567984i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 20.4811i 1.87750i
\(120\) 0 0
\(121\) −31.3746 −2.85224
\(122\) 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
\(131\) 4.83507i 0.422442i 0.977438 + 0.211221i \(0.0677440\pi\)
−0.977438 + 0.211221i \(0.932256\pi\)
\(132\) 0 0
\(133\) 22.9928i 1.99373i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 17.7391i 1.51556i 0.652512 + 0.757778i \(0.273715\pi\)
−0.652512 + 0.757778i \(0.726285\pi\)
\(138\) 0 0
\(139\) 18.6915i 1.58539i −0.609618 0.792695i \(-0.708677\pi\)
0.609618 0.792695i \(-0.291323\pi\)
\(140\) 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
\(149\) 24.3746 1.99684 0.998422 0.0561570i \(-0.0178847\pi\)
0.998422 + 0.0561570i \(0.0178847\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 11.6482i 0.941700i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 17.4356i 1.39151i 0.718278 + 0.695756i \(0.244931\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −21.0997 −1.66289
\(162\) 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
\(175\) 1.91238 + 26.3052i 0.144562 + 1.98848i
\(176\) 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
\(187\) −25.2749 −1.84828
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 25.6197i 1.85377i 0.375339 + 0.926887i \(0.377526\pi\)
−0.375339 + 0.926887i \(0.622474\pi\)
\(192\) 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 \(0.213321\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) 0 0
\(199\) 15.1123i 1.07128i 0.844446 + 0.535641i \(0.179930\pi\)
−0.844446 + 0.535641i \(0.820070\pi\)
\(200\) 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
\(209\) 28.3746 1.96271
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 19.3625 + 18.0060i 1.32051 + 1.22800i
\(216\) 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
\(225\) 1.08762 + 14.9605i 0.0725083 + 0.997368i
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 8.37459 0.553408 0.276704 0.960955i \(-0.410758\pi\)
0.276704 + 0.960955i \(0.410758\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 14.3901i 0.942728i −0.881939 0.471364i \(-0.843762\pi\)
0.881939 0.471364i \(-0.156238\pi\)
\(234\) 0 0
\(235\) −4.46221 4.14959i −0.291083 0.270689i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 10.9260i 0.706745i −0.935483 0.353373i \(-0.885035\pi\)
0.935483 0.353373i \(-0.114965\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −34.0997 31.7106i −2.17855 2.02592i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 9.02134i 0.569422i −0.958613 0.284711i \(-0.908102\pi\)
0.958613 0.284711i \(-0.0918976\pi\)
\(252\) 0 0
\(253\) 26.0383i 1.63701i
\(254\) 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
\(263\) −20.9244 −1.29026 −0.645128 0.764075i \(-0.723196\pi\)
−0.645128 + 0.764075i \(0.723196\pi\)
\(264\) 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 \(-0.707810\pi\)
0.607457 0.794353i \(-0.292190\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 32.4622 2.35999i 1.95754 0.142313i
\(276\) 0 0
\(277\) 30.7583i 1.84809i 0.382288 + 0.924043i \(0.375136\pi\)
−0.382288 + 0.924043i \(0.624864\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −4.17525 −0.248193 −0.124096 0.992270i \(-0.539603\pi\)
−0.124096 + 0.992270i \(0.539603\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1.92442 0.113201
\(290\) 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
\(301\) −62.3746 −3.59521
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −18.4622 17.1687i −1.05714 0.983079i
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 28.1314i 1.59519i 0.603195 + 0.797594i \(0.293894\pi\)
−0.603195 + 0.797594i \(0.706106\pi\)
\(312\) 0 0
\(313\) 34.8712i 1.97104i −0.169570 0.985518i \(-0.554238\pi\)
0.169570 0.985518i \(-0.445762\pi\)
\(314\) 0 0
\(315\) −25.9124 24.0969i −1.46000 1.35771i
\(316\) 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
\(323\) 16.9244 0.941700
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 14.3746 0.792497
\(330\) 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
\(343\) 72.9244 3.93755
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −14.7251 −0.790484 −0.395242 0.918577i \(-0.629339\pi\)
−0.395242 + 0.918577i \(0.629339\pi\)
\(348\) 0 0
\(349\) −6.17525 −0.330553 −0.165277 0.986247i \(-0.552852\pi\)
−0.165277 + 0.986247i \(0.552852\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 34.8712i 1.85601i 0.372572 + 0.928003i \(0.378476\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 15.9495i 0.841785i 0.907111 + 0.420892i \(0.138283\pi\)
−0.907111 + 0.420892i \(0.861717\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −24.4622 + 26.3052i −1.28041 + 1.37688i
\(366\) 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
\(385\) −52.2870 + 56.2262i −2.66479 + 2.86555i
\(386\) 0 0
\(387\) −35.4743 −1.80326
\(388\) 0 0
\(389\) 38.9244 1.97355 0.986773 0.162107i \(-0.0518289\pi\)
0.986773 + 0.162107i \(0.0518289\pi\)
\(390\) 0 0
\(391\) 15.5309i 0.785432i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 13.5529i 0.680199i −0.940389 0.340099i \(-0.889539\pi\)
0.940389 0.340099i \(-0.110461\pi\)
\(398\) 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
\(405\) −14.7371 13.7046i −0.732294 0.680989i
\(406\) 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
\(415\) 26.1993 + 24.3638i 1.28607 + 1.19597i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 8.71780i 0.425892i 0.977064 + 0.212946i \(0.0683059\pi\)
−0.977064 + 0.212946i \(0.931694\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 8.17525 0.397494
\(424\) 0 0
\(425\) 19.3625 1.40765i 0.939221 0.0682810i
\(426\) 0 0
\(427\) 59.4743 2.87816
\(428\) 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
\(441\) 62.4743 2.97496
\(442\) 0 0
\(443\) −40.9244 −1.94438 −0.972189 0.234198i \(-0.924754\pi\)
−0.972189 + 0.234198i \(0.924754\pi\)
\(444\) 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
\(457\) 34.3375i 1.60624i 0.595818 + 0.803120i \(0.296828\pi\)
−0.595818 + 0.803120i \(0.703172\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −0.374586 −0.0174462 −0.00872311 0.999962i \(-0.502777\pi\)
−0.00872311 + 0.999962i \(0.502777\pi\)
\(462\) 0 0
\(463\) −31.8248 −1.47902 −0.739511 0.673145i \(-0.764943\pi\)
−0.739511 + 0.673145i \(0.764943\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 25.2749 1.16958 0.584792 0.811183i \(-0.301176\pi\)
0.584792 + 0.811183i \(0.301176\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 76.9741i 3.53927i
\(474\) 0 0
\(475\) −21.7371 + 1.58028i −0.997368 + 0.0725083i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 43.5890i 1.99163i 0.0913823 + 0.995816i \(0.470871\pi\)
−0.0913823 + 0.995816i \(0.529129\pi\)
\(480\) 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 \(0.442223\pi\)
−0.180517 + 0.983572i \(0.557777\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −29.7371 + 31.9775i −1.33658 + 1.43728i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 22.8777i 1.02415i 0.858941 + 0.512074i \(0.171123\pi\)
−0.858941 + 0.512074i \(0.828877\pi\)
\(500\) 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
\(505\) −16.3746 15.2274i −0.728660 0.677609i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 84.7396i 3.74866i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 17.7391i 0.780166i
\(518\) 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
\(539\) 135.560i 5.83900i
\(540\) 0 0
\(541\) 21.4743 0.923250 0.461625 0.887075i \(-0.347267\pi\)
0.461625 + 0.887075i \(0.347267\pi\)
\(542\) 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
\(549\) 33.8248 1.44361
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 38.5237i 1.63230i −0.577838 0.816152i \(-0.696103\pi\)
0.577838 0.816152i \(-0.303897\pi\)
\(558\) 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
\(567\) 47.4743 1.99373
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 26.1534i 1.09449i 0.836974 + 0.547243i \(0.184323\pi\)
−0.836974 + 0.547243i \(0.815677\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1.45017 19.9474i −0.0604761 0.831862i
\(576\) 0 0
\(577\) 26.5720i 1.10621i −0.833112 0.553104i \(-0.813443\pi\)
0.833112 0.553104i \(-0.186557\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −84.3987 −3.50145
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −38.0241 −1.56942 −0.784711 0.619862i \(-0.787189\pi\)
−0.784711 + 0.619862i \(0.787189\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 34.8712i 1.43199i −0.698106 0.715994i \(-0.745974\pi\)
0.698106 0.715994i \(-0.254026\pi\)
\(594\) 0 0
\(595\) −31.1873 + 33.5369i −1.27855 + 1.37488i
\(596\) 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
\(605\) 51.3746 + 47.7753i 2.08867 + 1.94234i
\(606\) 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
\(613\) 49.0311i 1.98035i −0.139837 0.990174i \(-0.544658\pi\)
0.139837 0.990174i \(-0.455342\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 12.7156i 0.511911i 0.966689 + 0.255956i \(0.0823901\pi\)
−0.966689 + 0.255956i \(0.917610\pi\)
\(618\) 0 0
\(619\) 43.5890i 1.75199i −0.482321 0.875995i \(-0.660206\pi\)
0.482321 0.875995i \(-0.339794\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −24.7371 + 3.61587i −0.989485 + 0.144635i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 36.9643i 1.47153i 0.677239 + 0.735763i \(0.263176\pi\)
−0.677239 + 0.735763i \(0.736824\pi\)
\(632\) 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
\(643\) 35.8248 1.41279 0.706395 0.707818i \(-0.250320\pi\)
0.706395 + 0.707818i \(0.250320\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 50.0241 1.96665 0.983325 0.181857i \(-0.0582109\pi\)
0.983325 + 0.181857i \(0.0582109\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 20.2509i 0.792479i 0.918147 + 0.396239i \(0.129685\pi\)
−0.918147 + 0.396239i \(0.870315\pi\)
\(654\) 0 0
\(655\) 7.36254 7.91723i 0.287678 0.309352i
\(656\) 0 0
\(657\) 48.1939i 1.88022i
\(658\) 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
\(665\) 35.0120 37.6498i 1.35771 1.46000i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 73.3949i 2.83338i
\(672\) 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
\(685\) 27.0120 29.0471i 1.03208 1.10983i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 10.6958i 0.406889i 0.979086 + 0.203445i \(0.0652137\pi\)
−0.979086 + 0.203445i \(0.934786\pi\)
\(692\) 0 0
\(693\) 103.012i 3.91312i
\(694\) 0 0
\(695\) −28.4622 + 30.6065i −1.07963 + 1.16097i
\(696\) 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
\(707\) 52.7492 1.98384
\(708\) 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
\(719\) 53.3325i 1.98897i −0.104896 0.994483i \(-0.533451\pi\)
0.104896 0.994483i \(-0.466549\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −15.4743 −0.573908 −0.286954 0.957944i \(-0.592643\pi\)
−0.286954 + 0.957944i \(0.592643\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 45.9123i 1.69813i
\(732\) 0 0
\(733\) 52.3068i 1.93200i 0.258551 + 0.965998i \(0.416755\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 23.7150i 0.872370i 0.899857 + 0.436185i \(0.143671\pi\)
−0.899857 + 0.436185i \(0.856329\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) −39.9124 37.1161i −1.46228 1.35983i
\(746\) 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
\(757\) 47.3566i 1.72121i 0.509276 + 0.860603i \(0.329913\pi\)
−0.509276 + 0.860603i \(0.670087\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 31.2749 1.13371 0.566857 0.823816i \(-0.308159\pi\)
0.566857 + 0.823816i \(0.308159\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −17.7371 + 19.0734i −0.641287 + 0.689601i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 44.3746 1.60019 0.800094 0.599874i \(-0.204783\pi\)
0.800094 + 0.599874i \(0.204783\pi\)
\(770\) 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
\(785\) 26.5498 28.5501i 0.947604 1.01900i
\(786\) 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
\(799\) 10.5808i 0.374320i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −104.574 −3.69033
\(804\) 0 0
\(805\) 34.5498 + 32.1293i 1.21772 + 1.13241i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −46.5739 −1.63745 −0.818726 0.574184i \(-0.805319\pi\)
−0.818726 + 0.574184i \(0.805319\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0