Properties

Label 240.2.o.a.239.1
Level $240$
Weight $2$
Character 240.239
Analytic conductor $1.916$
Analytic rank $0$
Dimension $4$
CM discriminant -20
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 240.o (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.91640964851\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-5})\)
Defining polynomial: \(x^{4} + 4 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 239.1
Root \(-0.707107 - 1.58114i\) of defining polynomial
Character \(\chi\) \(=\) 240.239
Dual form 240.2.o.a.239.2

$q$-expansion

\(f(q)\) \(=\) \(q+(-0.707107 - 1.58114i) q^{3} -2.23607i q^{5} -4.24264 q^{7} +(-2.00000 + 2.23607i) q^{9} +O(q^{10})\) \(q+(-0.707107 - 1.58114i) q^{3} -2.23607i q^{5} -4.24264 q^{7} +(-2.00000 + 2.23607i) q^{9} +(-3.53553 + 1.58114i) q^{15} +(3.00000 + 6.70820i) q^{21} -9.48683i q^{23} -5.00000 q^{25} +(4.94975 + 1.58114i) q^{27} -8.94427i q^{29} +9.48683i q^{35} -4.47214i q^{41} +12.7279 q^{43} +(5.00000 + 4.47214i) q^{45} +9.48683i q^{47} +11.0000 q^{49} -8.00000 q^{61} +(8.48528 - 9.48683i) q^{63} +4.24264 q^{67} +(-15.0000 + 6.70820i) q^{69} +(3.53553 + 7.90569i) q^{75} +(-1.00000 - 8.94427i) q^{81} -9.48683i q^{83} +(-14.1421 + 6.32456i) q^{87} +17.8885i q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{9} + O(q^{10}) \) \( 4q - 8q^{9} + 12q^{21} - 20q^{25} + 20q^{45} + 44q^{49} - 32q^{61} - 60q^{69} - 4q^{81} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(161\) \(181\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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
\(3\) −0.707107 1.58114i −0.408248 0.912871i
\(4\) 0 0
\(5\) 2.23607i 1.00000i
\(6\) 0 0
\(7\) −4.24264 −1.60357 −0.801784 0.597614i \(-0.796115\pi\)
−0.801784 + 0.597614i \(0.796115\pi\)
\(8\) 0 0
\(9\) −2.00000 + 2.23607i −0.666667 + 0.745356i
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) −3.53553 + 1.58114i −0.912871 + 0.408248i
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 3.00000 + 6.70820i 0.654654 + 1.46385i
\(22\) 0 0
\(23\) 9.48683i 1.97814i −0.147442 0.989071i \(-0.547104\pi\)
0.147442 0.989071i \(-0.452896\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 4.94975 + 1.58114i 0.952579 + 0.304290i
\(28\) 0 0
\(29\) 8.94427i 1.66091i −0.557086 0.830455i \(-0.688081\pi\)
0.557086 0.830455i \(-0.311919\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 9.48683i 1.60357i
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.47214i 0.698430i −0.937043 0.349215i \(-0.886448\pi\)
0.937043 0.349215i \(-0.113552\pi\)
\(42\) 0 0
\(43\) 12.7279 1.94099 0.970495 0.241121i \(-0.0775152\pi\)
0.970495 + 0.241121i \(0.0775152\pi\)
\(44\) 0 0
\(45\) 5.00000 + 4.47214i 0.745356 + 0.666667i
\(46\) 0 0
\(47\) 9.48683i 1.38380i 0.721995 + 0.691898i \(0.243225\pi\)
−0.721995 + 0.691898i \(0.756775\pi\)
\(48\) 0 0
\(49\) 11.0000 1.57143
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 0 0
\(63\) 8.48528 9.48683i 1.06904 1.19523i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.24264 0.518321 0.259161 0.965834i \(-0.416554\pi\)
0.259161 + 0.965834i \(0.416554\pi\)
\(68\) 0 0
\(69\) −15.0000 + 6.70820i −1.80579 + 0.807573i
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 3.53553 + 7.90569i 0.408248 + 0.912871i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) −1.00000 8.94427i −0.111111 0.993808i
\(82\) 0 0
\(83\) 9.48683i 1.04132i −0.853766 0.520658i \(-0.825687\pi\)
0.853766 0.520658i \(-0.174313\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −14.1421 + 6.32456i −1.51620 + 0.678064i
\(88\) 0 0
\(89\) 17.8885i 1.89618i 0.317999 + 0.948091i \(0.396989\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 8.94427i 0.889988i −0.895533 0.444994i \(-0.853206\pi\)
0.895533 0.444994i \(-0.146794\pi\)
\(102\) 0 0
\(103\) −12.7279 −1.25412 −0.627060 0.778971i \(-0.715742\pi\)
−0.627060 + 0.778971i \(0.715742\pi\)
\(104\) 0 0
\(105\) 15.0000 6.70820i 1.46385 0.654654i
\(106\) 0 0
\(107\) 9.48683i 0.917127i −0.888662 0.458563i \(-0.848364\pi\)
0.888662 0.458563i \(-0.151636\pi\)
\(108\) 0 0
\(109\) 16.0000 1.53252 0.766261 0.642529i \(-0.222115\pi\)
0.766261 + 0.642529i \(0.222115\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) −21.2132 −1.97814
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) −7.07107 + 3.16228i −0.637577 + 0.285133i
\(124\) 0 0
\(125\) 11.1803i 1.00000i
\(126\) 0 0
\(127\) 4.24264 0.376473 0.188237 0.982124i \(-0.439723\pi\)
0.188237 + 0.982124i \(0.439723\pi\)
\(128\) 0 0
\(129\) −9.00000 20.1246i −0.792406 1.77187i
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 3.53553 11.0680i 0.304290 0.952579i
\(136\) 0 0
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 15.0000 6.70820i 1.26323 0.564933i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −20.0000 −1.66091
\(146\) 0 0
\(147\) −7.77817 17.3925i −0.641533 1.43451i
\(148\) 0 0
\(149\) 4.47214i 0.366372i −0.983078 0.183186i \(-0.941359\pi\)
0.983078 0.183186i \(-0.0586410\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 40.2492i 3.17208i
\(162\) 0 0
\(163\) −12.7279 −0.996928 −0.498464 0.866910i \(-0.666102\pi\)
−0.498464 + 0.866910i \(0.666102\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 9.48683i 0.734113i 0.930199 + 0.367057i \(0.119634\pi\)
−0.930199 + 0.367057i \(0.880366\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 21.2132 1.60357
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 5.65685 + 12.6491i 0.418167 + 0.935049i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −21.0000 6.70820i −1.52753 0.487950i
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) −3.00000 6.70820i −0.211604 0.473160i
\(202\) 0 0
\(203\) 37.9473i 2.66338i
\(204\) 0 0
\(205\) −10.0000 −0.698430
\(206\) 0 0
\(207\) 21.2132 + 18.9737i 1.47442 + 1.31876i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 28.4605i 1.94099i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 29.6985 1.98876 0.994379 0.105881i \(-0.0337662\pi\)
0.994379 + 0.105881i \(0.0337662\pi\)
\(224\) 0 0
\(225\) 10.0000 11.1803i 0.666667 0.745356i
\(226\) 0 0
\(227\) 28.4605i 1.88899i −0.328526 0.944495i \(-0.606552\pi\)
0.328526 0.944495i \(-0.393448\pi\)
\(228\) 0 0
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 21.2132 1.38380
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 28.0000 1.80364 0.901819 0.432113i \(-0.142232\pi\)
0.901819 + 0.432113i \(0.142232\pi\)
\(242\) 0 0
\(243\) −13.4350 + 7.90569i −0.861858 + 0.507151i
\(244\) 0 0
\(245\) 24.5967i 1.57143i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −15.0000 + 6.70820i −0.950586 + 0.425115i
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 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
\(261\) 20.0000 + 17.8885i 1.23797 + 1.10727i
\(262\) 0 0
\(263\) 28.4605i 1.75495i 0.479623 + 0.877475i \(0.340774\pi\)
−0.479623 + 0.877475i \(0.659226\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 28.2843 12.6491i 1.73097 0.774113i
\(268\) 0 0
\(269\) 22.3607i 1.36335i 0.731653 + 0.681677i \(0.238749\pi\)
−0.731653 + 0.681677i \(0.761251\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 31.3050i 1.86750i −0.357930 0.933748i \(-0.616517\pi\)
0.357930 0.933748i \(-0.383483\pi\)
\(282\) 0 0
\(283\) −29.6985 −1.76539 −0.882696 0.469945i \(-0.844274\pi\)
−0.882696 + 0.469945i \(0.844274\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 18.9737i 1.11998i
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 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
\(301\) −54.0000 −3.11251
\(302\) 0 0
\(303\) −14.1421 + 6.32456i −0.812444 + 0.363336i
\(304\) 0 0
\(305\) 17.8885i 1.02430i
\(306\) 0 0
\(307\) −4.24264 −0.242140 −0.121070 0.992644i \(-0.538633\pi\)
−0.121070 + 0.992644i \(0.538633\pi\)
\(308\) 0 0
\(309\) 9.00000 + 20.1246i 0.511992 + 1.14485i
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) −21.2132 18.9737i −1.19523 1.06904i
\(316\) 0 0
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −15.0000 + 6.70820i −0.837218 + 0.374415i
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −11.3137 25.2982i −0.625650 1.39899i
\(328\) 0 0
\(329\) 40.2492i 2.21901i
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 9.48683i 0.518321i
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −16.9706 −0.916324
\(344\) 0 0
\(345\) 15.0000 + 33.5410i 0.807573 + 1.80579i
\(346\) 0 0
\(347\) 28.4605i 1.52784i −0.645311 0.763920i \(-0.723272\pi\)
0.645311 0.763920i \(-0.276728\pi\)
\(348\) 0 0
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 19.0000 1.00000
\(362\) 0 0
\(363\) 7.77817 + 17.3925i 0.408248 + 0.912871i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −38.1838 −1.99318 −0.996588 0.0825348i \(-0.973698\pi\)
−0.996588 + 0.0825348i \(0.973698\pi\)
\(368\) 0 0
\(369\) 10.0000 + 8.94427i 0.520579 + 0.465620i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 17.6777 7.90569i 0.912871 0.408248i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) −3.00000 6.70820i −0.153695 0.343672i
\(382\) 0 0
\(383\) 28.4605i 1.45426i −0.686498 0.727132i \(-0.740853\pi\)
0.686498 0.727132i \(-0.259147\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −25.4558 + 28.4605i −1.29399 + 1.44673i
\(388\) 0 0
\(389\) 31.3050i 1.58722i −0.608424 0.793612i \(-0.708198\pi\)
0.608424 0.793612i \(-0.291802\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 35.7771i 1.78662i −0.449439 0.893311i \(-0.648376\pi\)
0.449439 0.893311i \(-0.351624\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −20.0000 + 2.23607i −0.993808 + 0.111111i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 4.00000 0.197787 0.0988936 0.995098i \(-0.468470\pi\)
0.0988936 + 0.995098i \(0.468470\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −21.2132 −1.04132
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −8.00000 −0.389896 −0.194948 0.980814i \(-0.562454\pi\)
−0.194948 + 0.980814i \(0.562454\pi\)
\(422\) 0 0
\(423\) −21.2132 18.9737i −1.03142 0.922531i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 33.9411 1.64253
\(428\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 14.1421 + 31.6228i 0.678064 + 1.51620i
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) −22.0000 + 24.5967i −1.04762 + 1.17127i
\(442\) 0 0
\(443\) 9.48683i 0.450733i 0.974274 + 0.225367i \(0.0723580\pi\)
−0.974274 + 0.225367i \(0.927642\pi\)
\(444\) 0 0
\(445\) 40.0000 1.89618
\(446\) 0 0
\(447\) −7.07107 + 3.16228i −0.334450 + 0.149571i
\(448\) 0 0
\(449\) 22.3607i 1.05527i 0.849473 + 0.527633i \(0.176920\pi\)
−0.849473 + 0.527633i \(0.823080\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 8.94427i 0.416576i −0.978068 0.208288i \(-0.933211\pi\)
0.978068 0.208288i \(-0.0667892\pi\)
\(462\) 0 0
\(463\) 12.7279 0.591517 0.295758 0.955263i \(-0.404428\pi\)
0.295758 + 0.955263i \(0.404428\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 28.4605i 1.31699i 0.752583 + 0.658497i \(0.228808\pi\)
−0.752583 + 0.658497i \(0.771192\pi\)
\(468\) 0 0
\(469\) −18.0000 −0.831163
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 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
\(483\) 63.6396 28.4605i 2.89570 1.29500i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 38.1838 1.73027 0.865136 0.501538i \(-0.167232\pi\)
0.865136 + 0.501538i \(0.167232\pi\)
\(488\) 0 0
\(489\) 9.00000 + 20.1246i 0.406994 + 0.910066i
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 0 0
\(501\) 15.0000 6.70820i 0.670151 0.299700i
\(502\) 0 0
\(503\) 9.48683i 0.422997i 0.977378 + 0.211498i \(0.0678343\pi\)
−0.977378 + 0.211498i \(0.932166\pi\)
\(504\) 0 0
\(505\) −20.0000 −0.889988
\(506\) 0 0
\(507\) −9.19239 20.5548i −0.408248 0.912871i
\(508\) 0 0
\(509\) 44.7214i 1.98224i 0.132973 + 0.991120i \(0.457548\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 28.4605i 1.25412i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 17.8885i 0.783711i 0.920027 + 0.391856i \(0.128167\pi\)
−0.920027 + 0.391856i \(0.871833\pi\)
\(522\) 0 0
\(523\) 29.6985 1.29862 0.649312 0.760522i \(-0.275057\pi\)
0.649312 + 0.760522i \(0.275057\pi\)
\(524\) 0 0
\(525\) −15.0000 33.5410i −0.654654 1.46385i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −67.0000 −2.91304
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −21.2132 −0.917127
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) 0 0
\(543\) −1.41421 3.16228i −0.0606897 0.135706i
\(544\) 0 0
\(545\) 35.7771i 1.53252i
\(546\) 0 0
\(547\) 46.6690 1.99542 0.997712 0.0676046i \(-0.0215356\pi\)
0.997712 + 0.0676046i \(0.0215356\pi\)
\(548\) 0 0
\(549\) 16.0000 17.8885i 0.682863 0.763464i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 47.4342i 1.99911i −0.0298010 0.999556i \(-0.509487\pi\)
0.0298010 0.999556i \(-0.490513\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 4.24264 + 37.9473i 0.178174 + 1.59364i
\(568\) 0 0
\(569\) 31.3050i 1.31237i −0.754599 0.656186i \(-0.772169\pi\)
0.754599 0.656186i \(-0.227831\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 47.4342i 1.97814i
\(576\) 0 0
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 40.2492i 1.66982i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 47.4342i 1.95782i 0.204298 + 0.978909i \(0.434509\pi\)
−0.204298 + 0.978909i \(0.565491\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 28.0000 1.14214 0.571072 0.820900i \(-0.306528\pi\)
0.571072 + 0.820900i \(0.306528\pi\)
\(602\) 0 0
\(603\) −8.48528 + 9.48683i −0.345547 + 0.386334i
\(604\) 0 0
\(605\) 24.5967i 1.00000i
\(606\) 0 0
\(607\) −46.6690 −1.89424 −0.947119 0.320882i \(-0.896021\pi\)
−0.947119 + 0.320882i \(0.896021\pi\)
\(608\) 0 0
\(609\) 60.0000 26.8328i 2.43132 1.08732i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 7.07107 + 15.8114i 0.285133 + 0.637577i
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 15.0000 46.9574i 0.601929 1.88434i
\(622\) 0 0
\(623\) 75.8947i 3.04066i
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 9.48683i 0.376473i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 49.1935i 1.94303i 0.236986 + 0.971513i \(0.423841\pi\)
−0.236986 + 0.971513i \(0.576159\pi\)
\(642\) 0 0
\(643\) 29.6985 1.17119 0.585597 0.810602i \(-0.300860\pi\)
0.585597 + 0.810602i \(0.300860\pi\)
\(644\) 0 0
\(645\) −45.0000 + 20.1246i −1.77187 + 0.792406i
\(646\) 0 0
\(647\) 47.4342i 1.86483i −0.361390 0.932415i \(-0.617698\pi\)
0.361390 0.932415i \(-0.382302\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −32.0000 −1.24466 −0.622328 0.782757i \(-0.713813\pi\)
−0.622328 + 0.782757i \(0.713813\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −84.8528 −3.28551
\(668\) 0 0
\(669\) −21.0000 46.9574i −0.811907 1.81548i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) −24.7487 7.90569i −0.952579 0.304290i
\(676\) 0 0
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −45.0000 + 20.1246i −1.72440 + 0.771177i
\(682\) 0 0
\(683\) 28.4605i 1.08901i −0.838757 0.544505i \(-0.816717\pi\)
0.838757 0.544505i \(-0.183283\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −9.89949 22.1359i −0.377689 0.844539i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 22.3607i 0.844551i 0.906467 + 0.422276i \(0.138769\pi\)
−0.906467 + 0.422276i \(0.861231\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −15.0000 33.5410i −0.564933 1.26323i
\(706\) 0 0
\(707\) 37.9473i 1.42716i
\(708\) 0 0
\(709\) −46.0000 −1.72757 −0.863783 0.503864i \(-0.831911\pi\)
−0.863783 + 0.503864i \(0.831911\pi\)
\(710\) 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
\(721\) 54.0000 2.01107
\(722\) 0 0
\(723\) −19.7990 44.2719i −0.736332 1.64649i
\(724\) 0 0
\(725\) 44.7214i 1.66091i
\(726\) 0 0
\(727\) 4.24264 0.157351 0.0786754 0.996900i \(-0.474931\pi\)
0.0786754 + 0.996900i \(0.474931\pi\)
\(728\) 0 0
\(729\) 22.0000 + 15.6525i 0.814815 + 0.579721i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) −38.8909 + 17.3925i −1.43451 + 0.641533i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 47.4342i 1.74019i 0.492883 + 0.870095i \(0.335943\pi\)
−0.492883 + 0.870095i \(0.664057\pi\)
\(744\) 0 0
\(745\) −10.0000 −0.366372
\(746\) 0 0
\(747\) 21.2132 + 18.9737i 0.776151 + 0.694210i
\(748\) 0 0
\(749\) 40.2492i 1.47067i
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 35.7771i 1.29692i −0.761249 0.648459i \(-0.775414\pi\)
0.761249 0.648459i \(-0.224586\pi\)
\(762\) 0 0
\(763\) −67.8823 −2.45750
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 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
\(783\) 14.1421 44.2719i 0.505399 1.58215i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −38.1838 −1.36110 −0.680552 0.732700i \(-0.738260\pi\)
−0.680552 + 0.732700i \(0.738260\pi\)
\(788\) 0 0
\(789\) 45.0000 20.1246i 1.60204 0.716455i
\(790\) 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
\(799\) 0 0
\(800\) 0 0
\(801\) −40.0000 35.7771i −1.41333 1.26412i
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 90.0000 3.17208
\(806\) 0 0
\(807\) 35.3553 15.8114i 1.24457 0.556587i
\(808\) 0 0
\(809\) 17.8885i 0.628928i 0.949269 + 0.314464i \(0.101825\pi\)
−0.949269 + 0.314464i \(0.898175\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\)