Properties

Label 1520.2.g.a.1519.1
Level $1520$
Weight $2$
Character 1520.1519
Analytic conductor $12.137$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-19}) \)
Defining polynomial: \(x^{2} - x + 5\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 1519.1
Root \(0.500000 + 2.17945i\) of defining polynomial
Character \(\chi\) \(=\) 1520.1519
Dual form 1520.2.g.a.1519.2

$q$-expansion

\(f(q)\) \(=\) \(q+(-0.500000 - 2.17945i) q^{5} -3.00000 q^{7} +3.00000 q^{9} +O(q^{10})\) \(q+(-0.500000 - 2.17945i) q^{5} -3.00000 q^{7} +3.00000 q^{9} -4.35890i q^{11} +4.35890i q^{17} -4.35890i q^{19} -4.00000 q^{23} +(-4.50000 + 2.17945i) q^{25} +(1.50000 + 6.53835i) q^{35} +1.00000 q^{43} +(-1.50000 - 6.53835i) q^{45} -13.0000 q^{47} +2.00000 q^{49} +(-9.50000 + 2.17945i) q^{55} -15.0000 q^{61} -9.00000 q^{63} -13.0767i q^{73} +13.0767i q^{77} +9.00000 q^{81} -16.0000 q^{83} +(9.50000 - 2.17945i) q^{85} +(-9.50000 + 2.17945i) q^{95} -13.0767i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{5} - 6 q^{7} + 6 q^{9} + O(q^{10}) \) \( 2 q - q^{5} - 6 q^{7} + 6 q^{9} - 8 q^{23} - 9 q^{25} + 3 q^{35} + 2 q^{43} - 3 q^{45} - 26 q^{47} + 4 q^{49} - 19 q^{55} - 30 q^{61} - 18 q^{63} + 18 q^{81} - 32 q^{83} + 19 q^{85} - 19 q^{95} + O(q^{100}) \)

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
\(5\) −0.500000 2.17945i −0.223607 0.974679i
\(6\) 0 0
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) 0 0
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) 4.35890i 1.31426i −0.753778 0.657129i \(-0.771771\pi\)
0.753778 0.657129i \(-0.228229\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.35890i 1.05719i 0.848875 + 0.528594i \(0.177281\pi\)
−0.848875 + 0.528594i \(0.822719\pi\)
\(18\) 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
\(25\) −4.50000 + 2.17945i −0.900000 + 0.435890i
\(26\) 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
\(35\) 1.50000 + 6.53835i 0.253546 + 1.10518i
\(36\) 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
\(43\) 1.00000 0.152499 0.0762493 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) 0 0
\(45\) −1.50000 6.53835i −0.223607 0.974679i
\(46\) 0 0
\(47\) −13.0000 −1.89624 −0.948122 0.317905i \(-0.897021\pi\)
−0.948122 + 0.317905i \(0.897021\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) −9.50000 + 2.17945i −1.28098 + 0.293877i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −15.0000 −1.92055 −0.960277 0.279050i \(-0.909981\pi\)
−0.960277 + 0.279050i \(0.909981\pi\)
\(62\) 0 0
\(63\) −9.00000 −1.13389
\(64\) 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
\(73\) 13.0767i 1.53051i −0.643726 0.765256i \(-0.722612\pi\)
0.643726 0.765256i \(-0.277388\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 13.0767i 1.49023i
\(78\) 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
\(85\) 9.50000 2.17945i 1.03042 0.236394i
\(86\) 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
\(95\) −9.50000 + 2.17945i −0.974679 + 0.223607i
\(96\) 0 0
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) 13.0767i 1.31426i
\(100\) 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
\(115\) 2.00000 + 8.71780i 0.186501 + 0.812939i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 13.0767i 1.19874i
\(120\) 0 0
\(121\) −8.00000 −0.727273
\(122\) 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
\(131\) 21.7945i 1.90419i 0.305796 + 0.952097i \(0.401077\pi\)
−0.305796 + 0.952097i \(0.598923\pi\)
\(132\) 0 0
\(133\) 13.0767i 1.13389i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.35890i 0.372406i −0.982511 0.186203i \(-0.940382\pi\)
0.982511 0.186203i \(-0.0596182\pi\)
\(138\) 0 0
\(139\) 21.7945i 1.84858i −0.381685 0.924292i \(-0.624656\pi\)
0.381685 0.924292i \(-0.375344\pi\)
\(140\) 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
\(149\) −11.0000 −0.901155 −0.450578 0.892737i \(-0.648782\pi\)
−0.450578 + 0.892737i \(0.648782\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 13.0767i 1.05719i
\(154\) 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
\(161\) 12.0000 0.945732
\(162\) 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
\(175\) 13.5000 6.53835i 1.02050 0.494253i
\(176\) 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
\(187\) 19.0000 1.38942
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 21.7945i 1.57699i 0.615038 + 0.788497i \(0.289140\pi\)
−0.615038 + 0.788497i \(0.710860\pi\)
\(192\) 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
\(199\) 13.0767i 0.926982i −0.886102 0.463491i \(-0.846597\pi\)
0.886102 0.463491i \(-0.153403\pi\)
\(200\) 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
\(209\) −19.0000 −1.31426
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −0.500000 2.17945i −0.0340997 0.148637i
\(216\) 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
\(225\) −13.5000 + 6.53835i −0.900000 + 0.435890i
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 21.0000 1.38772 0.693860 0.720110i \(-0.255909\pi\)
0.693860 + 0.720110i \(0.255909\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 30.5123i 1.99893i −0.0327561 0.999463i \(-0.510428\pi\)
0.0327561 0.999463i \(-0.489572\pi\)
\(234\) 0 0
\(235\) 6.50000 + 28.3328i 0.424013 + 1.84823i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 30.5123i 1.97368i −0.161712 0.986838i \(-0.551701\pi\)
0.161712 0.986838i \(-0.448299\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.00000 4.35890i −0.0638877 0.278480i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 21.7945i 1.37566i 0.725874 + 0.687828i \(0.241436\pi\)
−0.725874 + 0.687828i \(0.758564\pi\)
\(252\) 0 0
\(253\) 17.4356i 1.09617i
\(254\) 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
\(263\) −11.0000 −0.678289 −0.339145 0.940734i \(-0.610138\pi\)
−0.339145 + 0.940734i \(0.610138\pi\)
\(264\) 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
\(275\) 9.50000 + 19.6150i 0.572872 + 1.18283i
\(276\) 0 0
\(277\) 4.35890i 0.261901i 0.991389 + 0.130950i \(0.0418029\pi\)
−0.991389 + 0.130950i \(0.958197\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 31.0000 1.84276 0.921379 0.388664i \(-0.127063\pi\)
0.921379 + 0.388664i \(0.127063\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −2.00000 −0.117647
\(290\) 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
\(301\) −3.00000 −0.172917
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 7.50000 + 32.6917i 0.429449 + 1.87192i
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 4.35890i 0.247170i −0.992334 0.123585i \(-0.960561\pi\)
0.992334 0.123585i \(-0.0394392\pi\)
\(312\) 0 0
\(313\) 34.8712i 1.97104i 0.169570 + 0.985518i \(0.445762\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 0 0
\(315\) 4.50000 + 19.6150i 0.253546 + 1.10518i
\(316\) 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
\(323\) 19.0000 1.05719
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 39.0000 2.15014
\(330\) 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
\(343\) 15.0000 0.809924
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 37.0000 1.98626 0.993132 0.116999i \(-0.0373274\pi\)
0.993132 + 0.116999i \(0.0373274\pi\)
\(348\) 0 0
\(349\) 35.0000 1.87351 0.936754 0.349990i \(-0.113815\pi\)
0.936754 + 0.349990i \(0.113815\pi\)
\(350\) 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
\(359\) 21.7945i 1.15027i −0.818059 0.575135i \(-0.804950\pi\)
0.818059 0.575135i \(-0.195050\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −28.5000 + 6.53835i −1.49176 + 0.342233i
\(366\) 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
\(385\) 28.5000 6.53835i 1.45249 0.333225i
\(386\) 0 0
\(387\) 3.00000 0.152499
\(388\) 0 0
\(389\) −25.0000 −1.26755 −0.633775 0.773517i \(-0.718496\pi\)
−0.633775 + 0.773517i \(0.718496\pi\)
\(390\) 0 0
\(391\) 17.4356i 0.881756i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 39.2301i 1.96890i −0.175660 0.984451i \(-0.556206\pi\)
0.175660 0.984451i \(-0.443794\pi\)
\(398\) 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
\(405\) −4.50000 19.6150i −0.223607 0.974679i
\(406\) 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
\(415\) 8.00000 + 34.8712i 0.392705 + 1.71176i
\(416\) 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
\(423\) −39.0000 −1.89624
\(424\) 0 0
\(425\) −9.50000 19.6150i −0.460818 0.951469i
\(426\) 0 0
\(427\) 45.0000 2.17770
\(428\) 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
\(441\) 6.00000 0.285714
\(442\) 0 0
\(443\) 29.0000 1.37783 0.688916 0.724841i \(-0.258087\pi\)
0.688916 + 0.724841i \(0.258087\pi\)
\(444\) 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
\(457\) 39.2301i 1.83511i 0.397613 + 0.917553i \(0.369839\pi\)
−0.397613 + 0.917553i \(0.630161\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −37.0000 −1.72326 −0.861631 0.507535i \(-0.830557\pi\)
−0.861631 + 0.507535i \(0.830557\pi\)
\(462\) 0 0
\(463\) 41.0000 1.90543 0.952716 0.303863i \(-0.0982765\pi\)
0.952716 + 0.303863i \(0.0982765\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −43.0000 −1.98980 −0.994901 0.100853i \(-0.967843\pi\)
−0.994901 + 0.100853i \(0.967843\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 4.35890i 0.200422i
\(474\) 0 0
\(475\) 9.50000 + 19.6150i 0.435890 + 0.900000i
\(476\) 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
\(495\) −28.5000 + 6.53835i −1.28098 + 0.293877i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 21.7945i 0.975656i −0.872940 0.487828i \(-0.837789\pi\)
0.872940 0.487828i \(-0.162211\pi\)
\(500\) 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
\(505\) −5.00000 21.7945i −0.222497 0.969842i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 39.2301i 1.73544i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 56.6657i 2.49215i
\(518\) 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
\(539\) 8.71780i 0.375502i
\(540\) 0 0
\(541\) 25.0000 1.07483 0.537417 0.843317i \(-0.319400\pi\)
0.537417 + 0.843317i \(0.319400\pi\)
\(542\) 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
\(549\) −45.0000 −1.92055
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 4.35890i 0.184692i 0.995727 + 0.0923462i \(0.0294367\pi\)
−0.995727 + 0.0923462i \(0.970563\pi\)
\(558\) 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
\(567\) −27.0000 −1.13389
\(568\) 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
\(575\) 18.0000 8.71780i 0.750652 0.363557i
\(576\) 0 0
\(577\) 47.9479i 1.99610i −0.0624458 0.998048i \(-0.519890\pi\)
0.0624458 0.998048i \(-0.480110\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 48.0000 1.99138
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −7.00000 −0.288921 −0.144460 0.989511i \(-0.546145\pi\)
−0.144460 + 0.989511i \(0.546145\pi\)
\(588\) 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
\(595\) −28.5000 + 6.53835i −1.16839 + 0.268046i
\(596\) 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
\(605\) 4.00000 + 17.4356i 0.162623 + 0.708858i
\(606\) 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
\(613\) 30.5123i 1.23238i −0.787598 0.616190i \(-0.788675\pi\)
0.787598 0.616190i \(-0.211325\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 47.9479i 1.93031i 0.261680 + 0.965155i \(0.415723\pi\)
−0.261680 + 0.965155i \(0.584277\pi\)
\(618\) 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
\(625\) 15.5000 19.6150i 0.620000 0.784602i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 47.9479i 1.90878i 0.298570 + 0.954388i \(0.403490\pi\)
−0.298570 + 0.954388i \(0.596510\pi\)
\(632\) 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
\(643\) −49.0000 −1.93237 −0.966186 0.257847i \(-0.916987\pi\)
−0.966186 + 0.257847i \(0.916987\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −17.0000 −0.668339 −0.334169 0.942513i \(-0.608456\pi\)
−0.334169 + 0.942513i \(0.608456\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 30.5123i 1.19404i −0.802227 0.597019i \(-0.796352\pi\)
0.802227 0.597019i \(-0.203648\pi\)
\(654\) 0 0
\(655\) 47.5000 10.8972i 1.85598 0.425791i
\(656\) 0 0
\(657\) 39.2301i 1.53051i
\(658\) 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
\(665\) 28.5000 6.53835i 1.10518 0.253546i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 65.3835i 2.52410i
\(672\) 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
\(685\) −9.50000 + 2.17945i −0.362976 + 0.0832725i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 39.2301i 1.49238i −0.665731 0.746191i \(-0.731880\pi\)
0.665731 0.746191i \(-0.268120\pi\)
\(692\) 0 0
\(693\) 39.2301i 1.49023i
\(694\) 0 0
\(695\) −47.5000 + 10.8972i −1.80178 + 0.413356i
\(696\) 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
\(707\) −30.0000 −1.12827
\(708\) 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
\(719\) 21.7945i 0.812798i −0.913696 0.406399i \(-0.866784\pi\)
0.913696 0.406399i \(-0.133216\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −37.0000 −1.37225 −0.686127 0.727482i \(-0.740691\pi\)
−0.686127 + 0.727482i \(0.740691\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 4.35890i 0.161220i
\(732\) 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
\(739\) 30.5123i 1.12241i −0.827676 0.561206i \(-0.810337\pi\)
0.827676 0.561206i \(-0.189663\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 5.50000 + 23.9739i 0.201504 + 0.878337i
\(746\) 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
\(757\) 47.9479i 1.74270i 0.490666 + 0.871348i \(0.336754\pi\)
−0.490666 + 0.871348i \(0.663246\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −55.0000 −1.99375 −0.996874 0.0790050i \(-0.974826\pi\)
−0.996874 + 0.0790050i \(0.974826\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 28.5000 6.53835i 1.03042 0.236394i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −51.0000 −1.83911 −0.919554 0.392965i \(-0.871449\pi\)
−0.919554 + 0.392965i \(0.871449\pi\)
\(770\) 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
\(785\) −38.0000 + 8.71780i −1.35628 + 0.311152i
\(786\) 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
\(799\) 56.6657i 2.00469i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −57.0000 −2.01149
\(804\) 0 0
\(805\) −6.00000 26.1534i −0.211472 0.921786i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −5.00000 −0.175791 −0.0878953 0.996130i \(-0.528014\pi\)
−0.0878953 + 0.996130i \(0.528014\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)