Properties

Label 1024.2.e.a.257.1
Level $1024$
Weight $2$
Character 1024.257
Analytic conductor $8.177$
Analytic rank $1$
Dimension $2$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1024 = 2^{10} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1024.e (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.17668116698\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 128)
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 257.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1024.257
Dual form 1024.2.e.a.769.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-2.00000 - 2.00000i) q^{3} +5.00000i q^{9} +O(q^{10})\) \(q+(-2.00000 - 2.00000i) q^{3} +5.00000i q^{9} +(2.00000 - 2.00000i) q^{11} -6.00000 q^{17} +(-6.00000 - 6.00000i) q^{19} +5.00000i q^{25} +(4.00000 - 4.00000i) q^{27} -8.00000 q^{33} +6.00000i q^{41} +(-6.00000 + 6.00000i) q^{43} +7.00000 q^{49} +(12.0000 + 12.0000i) q^{51} +24.0000i q^{57} +(-10.0000 + 10.0000i) q^{59} +(6.00000 + 6.00000i) q^{67} +2.00000i q^{73} +(10.0000 - 10.0000i) q^{75} -1.00000 q^{81} +(2.00000 + 2.00000i) q^{83} -18.0000i q^{89} -10.0000 q^{97} +(10.0000 + 10.0000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{3} + O(q^{10}) \) \( 2q - 4q^{3} + 4q^{11} - 12q^{17} - 12q^{19} + 8q^{27} - 16q^{33} - 12q^{43} + 14q^{49} + 24q^{51} - 20q^{59} + 12q^{67} + 20q^{75} - 2q^{81} + 4q^{83} - 20q^{97} + 20q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(1023\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\)

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
\(3\) −2.00000 2.00000i −1.15470 1.15470i −0.985599 0.169102i \(-0.945913\pi\)
−0.169102 0.985599i \(-0.554087\pi\)
\(4\) 0 0
\(5\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 5.00000i 1.66667i
\(10\) 0 0
\(11\) 2.00000 2.00000i 0.603023 0.603023i −0.338091 0.941113i \(-0.609781\pi\)
0.941113 + 0.338091i \(0.109781\pi\)
\(12\) 0 0
\(13\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) −6.00000 6.00000i −1.37649 1.37649i −0.850469 0.526026i \(-0.823682\pi\)
−0.526026 0.850469i \(-0.676318\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 5.00000i 1.00000i
\(26\) 0 0
\(27\) 4.00000 4.00000i 0.769800 0.769800i
\(28\) 0 0
\(29\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) −8.00000 −1.39262
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000i 0.937043i 0.883452 + 0.468521i \(0.155213\pi\)
−0.883452 + 0.468521i \(0.844787\pi\)
\(42\) 0 0
\(43\) −6.00000 + 6.00000i −0.914991 + 0.914991i −0.996660 0.0816682i \(-0.973975\pi\)
0.0816682 + 0.996660i \(0.473975\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 12.0000 + 12.0000i 1.68034 + 1.68034i
\(52\) 0 0
\(53\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 24.0000i 3.17888i
\(58\) 0 0
\(59\) −10.0000 + 10.0000i −1.30189 + 1.30189i −0.374772 + 0.927117i \(0.622279\pi\)
−0.927117 + 0.374772i \(0.877721\pi\)
\(60\) 0 0
\(61\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 6.00000 + 6.00000i 0.733017 + 0.733017i 0.971216 0.238200i \(-0.0765572\pi\)
−0.238200 + 0.971216i \(0.576557\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 2.00000i 0.234082i 0.993127 + 0.117041i \(0.0373409\pi\)
−0.993127 + 0.117041i \(0.962659\pi\)
\(74\) 0 0
\(75\) 10.0000 10.0000i 1.15470 1.15470i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −1.00000 −0.111111
\(82\) 0 0
\(83\) 2.00000 + 2.00000i 0.219529 + 0.219529i 0.808300 0.588771i \(-0.200388\pi\)
−0.588771 + 0.808300i \(0.700388\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 18.0000i 1.90800i −0.299813 0.953998i \(-0.596924\pi\)
0.299813 0.953998i \(-0.403076\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) 10.0000 + 10.0000i 1.00504 + 1.00504i
\(100\) 0 0
\(101\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −14.0000 + 14.0000i −1.35343 + 1.35343i −0.471640 + 0.881791i \(0.656338\pi\)
−0.881791 + 0.471640i \(0.843662\pi\)
\(108\) 0 0
\(109\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 3.00000i 0.272727i
\(122\) 0 0
\(123\) 12.0000 12.0000i 1.08200 1.08200i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 24.0000 2.11308
\(130\) 0 0
\(131\) −10.0000 10.0000i −0.873704 0.873704i 0.119170 0.992874i \(-0.461977\pi\)
−0.992874 + 0.119170i \(0.961977\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.00000i 0.512615i 0.966595 + 0.256307i \(0.0825059\pi\)
−0.966595 + 0.256307i \(0.917494\pi\)
\(138\) 0 0
\(139\) −6.00000 + 6.00000i −0.508913 + 0.508913i −0.914193 0.405279i \(-0.867174\pi\)
0.405279 + 0.914193i \(0.367174\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −14.0000 14.0000i −1.15470 1.15470i
\(148\) 0 0
\(149\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 30.0000i 2.42536i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −18.0000 18.0000i −1.40987 1.40987i −0.760319 0.649550i \(-0.774958\pi\)
−0.649550 0.760319i \(-0.725042\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 13.0000i 1.00000i
\(170\) 0 0
\(171\) 30.0000 30.0000i 2.29416 2.29416i
\(172\) 0 0
\(173\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 40.0000 3.00658
\(178\) 0 0
\(179\) −14.0000 14.0000i −1.04641 1.04641i −0.998869 0.0475398i \(-0.984862\pi\)
−0.0475398 0.998869i \(-0.515138\pi\)
\(180\) 0 0
\(181\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −12.0000 + 12.0000i −0.877527 + 0.877527i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 22.0000 1.58359 0.791797 0.610784i \(-0.209146\pi\)
0.791797 + 0.610784i \(0.209146\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 24.0000i 1.69283i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −24.0000 −1.66011
\(210\) 0 0
\(211\) 18.0000 + 18.0000i 1.23917 + 1.23917i 0.960340 + 0.278831i \(0.0899469\pi\)
0.278831 + 0.960340i \(0.410053\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 4.00000 4.00000i 0.270295 0.270295i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) −25.0000 −1.66667
\(226\) 0 0
\(227\) −2.00000 2.00000i −0.132745 0.132745i 0.637613 0.770357i \(-0.279922\pi\)
−0.770357 + 0.637613i \(0.779922\pi\)
\(228\) 0 0
\(229\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 30.0000i 1.96537i −0.185296 0.982683i \(-0.559325\pi\)
0.185296 0.982683i \(-0.440675\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 26.0000 1.67481 0.837404 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) 0 0
\(243\) −10.0000 10.0000i −0.641500 0.641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 8.00000i 0.506979i
\(250\) 0 0
\(251\) 22.0000 22.0000i 1.38863 1.38863i 0.560417 0.828210i \(-0.310641\pi\)
0.828210 0.560417i \(-0.189359\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −30.0000 −1.87135 −0.935674 0.352865i \(-0.885208\pi\)
−0.935674 + 0.352865i \(0.885208\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −36.0000 + 36.0000i −2.20316 + 2.20316i
\(268\) 0 0
\(269\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 10.0000 + 10.0000i 0.603023 + 0.603023i
\(276\) 0 0
\(277\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 18.0000i 1.07379i −0.843649 0.536895i \(-0.819597\pi\)
0.843649 0.536895i \(-0.180403\pi\)
\(282\) 0 0
\(283\) −18.0000 + 18.0000i −1.06999 + 1.06999i −0.0726300 + 0.997359i \(0.523139\pi\)
−0.997359 + 0.0726300i \(0.976861\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 20.0000 + 20.0000i 1.17242 + 1.17242i
\(292\) 0 0
\(293\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 16.0000i 0.928414i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −6.00000 6.00000i −0.342438 0.342438i 0.514845 0.857283i \(-0.327849\pi\)
−0.857283 + 0.514845i \(0.827849\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 10.0000i 0.565233i 0.959233 + 0.282617i \(0.0912024\pi\)
−0.959233 + 0.282617i \(0.908798\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 56.0000 3.12562
\(322\) 0 0
\(323\) 36.0000 + 36.0000i 2.00309 + 2.00309i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 18.0000 18.0000i 0.989369 0.989369i −0.0105746 0.999944i \(-0.503366\pi\)
0.999944 + 0.0105746i \(0.00336607\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 0 0
\(339\) 36.0000 + 36.0000i 1.95525 + 1.95525i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −26.0000 + 26.0000i −1.39575 + 1.39575i −0.583998 + 0.811755i \(0.698512\pi\)
−0.811755 + 0.583998i \(0.801488\pi\)
\(348\) 0 0
\(349\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −30.0000 −1.59674 −0.798369 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 53.0000i 2.78947i
\(362\) 0 0
\(363\) 6.00000 6.00000i 0.314918 0.314918i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) −30.0000 −1.56174
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 6.00000 6.00000i 0.308199 0.308199i −0.536011 0.844211i \(-0.680070\pi\)
0.844211 + 0.536011i \(0.180070\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −30.0000 30.0000i −1.52499 1.52499i
\(388\) 0 0
\(389\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 40.0000i 2.01773i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 22.0000i 1.08783i −0.839140 0.543915i \(-0.816941\pi\)
0.839140 0.543915i \(-0.183059\pi\)
\(410\) 0 0
\(411\) 12.0000 12.0000i 0.591916 0.591916i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 24.0000 1.17529
\(418\) 0 0
\(419\) −26.0000 26.0000i −1.27018 1.27018i −0.945991 0.324192i \(-0.894908\pi\)
−0.324192 0.945991i \(-0.605092\pi\)
\(420\) 0 0
\(421\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 30.0000i 1.45521i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −38.0000 −1.82616 −0.913082 0.407777i \(-0.866304\pi\)
−0.913082 + 0.407777i \(0.866304\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 35.0000i 1.66667i
\(442\) 0 0
\(443\) −2.00000 + 2.00000i −0.0950229 + 0.0950229i −0.753020 0.657997i \(-0.771404\pi\)
0.657997 + 0.753020i \(0.271404\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −42.0000 −1.98210 −0.991051 0.133482i \(-0.957384\pi\)
−0.991051 + 0.133482i \(0.957384\pi\)
\(450\) 0 0
\(451\) 12.0000 + 12.0000i 0.565058 + 0.565058i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 26.0000i 1.21623i −0.793849 0.608114i \(-0.791926\pi\)
0.793849 0.608114i \(-0.208074\pi\)
\(458\) 0 0
\(459\) −24.0000 + 24.0000i −1.12022 + 1.12022i
\(460\) 0 0
\(461\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −22.0000 22.0000i −1.01804 1.01804i −0.999834 0.0182043i \(-0.994205\pi\)
−0.0182043 0.999834i \(-0.505795\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 24.0000i 1.10352i
\(474\) 0 0
\(475\) 30.0000 30.0000i 1.37649 1.37649i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 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
\(489\) 72.0000i 3.25595i
\(490\) 0 0
\(491\) 10.0000 10.0000i 0.451294 0.451294i −0.444490 0.895784i \(-0.646615\pi\)
0.895784 + 0.444490i \(0.146615\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −30.0000 30.0000i −1.34298 1.34298i −0.893073 0.449911i \(-0.851456\pi\)
−0.449911 0.893073i \(-0.648544\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −26.0000 + 26.0000i −1.15470 + 1.15470i
\(508\) 0 0
\(509\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −48.0000 −2.11925
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 6.00000i 0.262865i 0.991325 + 0.131432i \(0.0419576\pi\)
−0.991325 + 0.131432i \(0.958042\pi\)
\(522\) 0 0
\(523\) 18.0000 18.0000i 0.787085 0.787085i −0.193930 0.981015i \(-0.562124\pi\)
0.981015 + 0.193930i \(0.0621236\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) −50.0000 50.0000i −2.16982 2.16982i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 56.0000i 2.41658i
\(538\) 0 0
\(539\) 14.0000 14.0000i 0.603023 0.603023i
\(540\) 0 0
\(541\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 6.00000 + 6.00000i 0.256541 + 0.256541i 0.823646 0.567104i \(-0.191936\pi\)
−0.567104 + 0.823646i \(0.691936\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 48.0000 2.02656
\(562\) 0 0
\(563\) 26.0000 + 26.0000i 1.09577 + 1.09577i 0.994900 + 0.100870i \(0.0321625\pi\)
0.100870 + 0.994900i \(0.467837\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 42.0000i 1.76073i 0.474295 + 0.880366i \(0.342703\pi\)
−0.474295 + 0.880366i \(0.657297\pi\)
\(570\) 0 0
\(571\) 30.0000 30.0000i 1.25546 1.25546i 0.302224 0.953237i \(-0.402271\pi\)
0.953237 0.302224i \(-0.0977291\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 34.0000 1.41544 0.707719 0.706494i \(-0.249724\pi\)
0.707719 + 0.706494i \(0.249724\pi\)
\(578\) 0 0
\(579\) −44.0000 44.0000i −1.82858 1.82858i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 34.0000 34.0000i 1.40333 1.40333i 0.614109 0.789221i \(-0.289516\pi\)
0.789221 0.614109i \(-0.210484\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −18.0000 −0.739171 −0.369586 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 46.0000i 1.87638i 0.346122 + 0.938190i \(0.387498\pi\)
−0.346122 + 0.938190i \(0.612502\pi\)
\(602\) 0 0
\(603\) −30.0000 + 30.0000i −1.22169 + 1.22169i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 30.0000i 1.20775i −0.797077 0.603877i \(-0.793622\pi\)
0.797077 0.603877i \(-0.206378\pi\)
\(618\) 0 0
\(619\) −30.0000 + 30.0000i −1.20580 + 1.20580i −0.233428 + 0.972374i \(0.574994\pi\)
−0.972374 + 0.233428i \(0.925006\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −25.0000 −1.00000
\(626\) 0 0
\(627\) 48.0000 + 48.0000i 1.91694 + 1.91694i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 72.0000i 2.86174i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −42.0000 −1.65890 −0.829450 0.558581i \(-0.811346\pi\)
−0.829450 + 0.558581i \(0.811346\pi\)
\(642\) 0 0
\(643\) 6.00000 + 6.00000i 0.236617 + 0.236617i 0.815448 0.578831i \(-0.196491\pi\)
−0.578831 + 0.815448i \(0.696491\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 40.0000i 1.57014i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −10.0000 −0.390137
\(658\) 0 0
\(659\) 34.0000 + 34.0000i 1.32445 + 1.32445i 0.910131 + 0.414321i \(0.135981\pi\)
0.414321 + 0.910131i \(0.364019\pi\)
\(660\) 0 0
\(661\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −10.0000 −0.385472 −0.192736 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) 0 0
\(675\) 20.0000 + 20.0000i 0.769800 + 0.769800i
\(676\) 0 0
\(677\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 8.00000i 0.306561i
\(682\) 0 0
\(683\) −22.0000 + 22.0000i −0.841807 + 0.841807i −0.989094 0.147287i \(-0.952946\pi\)
0.147287 + 0.989094i \(0.452946\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 18.0000 + 18.0000i 0.684752 + 0.684752i 0.961067 0.276315i \(-0.0891133\pi\)
−0.276315 + 0.961067i \(0.589113\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 36.0000i 1.36360i
\(698\) 0 0
\(699\) −60.0000 + 60.0000i −2.26941 + 2.26941i
\(700\) 0 0
\(701\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(710\) 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
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −52.0000 52.0000i −1.93390 1.93390i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 43.0000i 1.59259i
\(730\) 0 0
\(731\) 36.0000 36.0000i 1.33151 1.33151i
\(732\) 0 0
\(733\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 24.0000 0.884051
\(738\) 0 0
\(739\) 30.0000 + 30.0000i 1.10357 + 1.10357i 0.993977 + 0.109591i \(0.0349541\pi\)
0.109591 + 0.993977i \(0.465046\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −10.0000 + 10.0000i −0.365881 + 0.365881i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) −88.0000 −3.20690
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 54.0000i 1.95750i −0.205061 0.978749i \(-0.565739\pi\)
0.205061 0.978749i \(-0.434261\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 22.0000 0.793340 0.396670 0.917961i \(-0.370166\pi\)
0.396670 + 0.917961i \(0.370166\pi\)
\(770\) 0 0
\(771\) 60.0000 + 60.0000i 2.16085 + 2.16085i
\(772\) 0 0
\(773\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 36.0000 36.0000i 1.28983 1.28983i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 18.0000 + 18.0000i 0.641631 + 0.641631i 0.950956 0.309326i \(-0.100103\pi\)
−0.309326 + 0.950956i \(0.600103\pi\)
\(788\) 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 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 90.0000 3.17999
\(802\) 0 0
\(803\) 4.00000 + 4.00000i 0.141157 + 0.141157i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 6.00000i 0.210949i 0.994422 + 0.105474i \(0.0336361\pi\)