Properties

Label 55.2.e.b.43.2
Level 55
Weight 2
Character 55.43
Analytic conductor 0.439
Analytic rank 0
Dimension 4
CM discriminant -11
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 55 = 5 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 55.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.439177211117\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{11})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 43.2
Root \(1.65831 - 0.500000i\)
Character \(\chi\) = 55.43
Dual form 55.2.e.b.32.2

$q$-expansion

\(f(q)\) \(=\) \(q+(1.15831 + 1.15831i) q^{3} -2.00000i q^{4} +(-1.65831 + 1.50000i) q^{5} -0.316625i q^{9} +O(q^{10})\) \(q+(1.15831 + 1.15831i) q^{3} -2.00000i q^{4} +(-1.65831 + 1.50000i) q^{5} -0.316625i q^{9} -3.31662 q^{11} +(2.31662 - 2.31662i) q^{12} +(-3.65831 - 0.183375i) q^{15} -4.00000 q^{16} +(3.00000 + 3.31662i) q^{20} +(6.15831 + 6.15831i) q^{23} +(0.500000 - 4.97494i) q^{25} +(3.84169 - 3.84169i) q^{27} +9.94987 q^{31} +(-3.84169 - 3.84169i) q^{33} -0.633250 q^{36} +(-8.47494 + 8.47494i) q^{37} +6.63325i q^{44} +(0.474937 + 0.525063i) q^{45} +(-2.68338 + 2.68338i) q^{47} +(-4.63325 - 4.63325i) q^{48} -7.00000i q^{49} +(-9.63325 - 9.63325i) q^{53} +(5.50000 - 4.97494i) q^{55} -3.31662i q^{59} +(-0.366750 + 7.31662i) q^{60} +8.00000i q^{64} +(1.52506 - 1.52506i) q^{67} +14.2665i q^{69} -3.00000 q^{71} +(6.34169 - 5.18338i) q^{75} +(6.63325 - 6.00000i) q^{80} +7.94987 q^{81} -9.00000i q^{89} +(12.3166 - 12.3166i) q^{92} +(11.5251 + 11.5251i) q^{93} +(-13.4749 + 13.4749i) q^{97} +1.05013i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{3} + O(q^{10}) \) \( 4q - 2q^{3} - 4q^{12} - 8q^{15} - 16q^{16} + 12q^{20} + 18q^{23} + 2q^{25} + 22q^{27} - 22q^{33} + 24q^{36} - 14q^{37} - 18q^{45} - 24q^{47} + 8q^{48} - 12q^{53} + 22q^{55} - 28q^{60} + 26q^{67} - 12q^{71} + 32q^{75} - 8q^{81} + 36q^{92} + 66q^{93} - 34q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/55\mathbb{Z}\right)^\times\).

\(n\) \(12\) \(46\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(-1\)

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 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 1.15831 + 1.15831i 0.668752 + 0.668752i 0.957427 0.288675i \(-0.0932147\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 2.00000i 1.00000i
\(5\) −1.65831 + 1.50000i −0.741620 + 0.670820i
\(6\) 0 0
\(7\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(8\) 0 0
\(9\) 0.316625i 0.105542i
\(10\) 0 0
\(11\) −3.31662 −1.00000
\(12\) 2.31662 2.31662i 0.668752 0.668752i
\(13\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(14\) 0 0
\(15\) −3.65831 0.183375i −0.944572 0.0473473i
\(16\) −4.00000 −1.00000
\(17\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 3.00000 + 3.31662i 0.670820 + 0.741620i
\(21\) 0 0
\(22\) 0 0
\(23\) 6.15831 + 6.15831i 1.28410 + 1.28410i 0.938315 + 0.345782i \(0.112386\pi\)
0.345782 + 0.938315i \(0.387614\pi\)
\(24\) 0 0
\(25\) 0.500000 4.97494i 0.100000 0.994987i
\(26\) 0 0
\(27\) 3.84169 3.84169i 0.739333 0.739333i
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 9.94987 1.78705 0.893525 0.449013i \(-0.148224\pi\)
0.893525 + 0.449013i \(0.148224\pi\)
\(32\) 0 0
\(33\) −3.84169 3.84169i −0.668752 0.668752i
\(34\) 0 0
\(35\) 0 0
\(36\) −0.633250 −0.105542
\(37\) −8.47494 + 8.47494i −1.39327 + 1.39327i −0.575396 + 0.817875i \(0.695152\pi\)
−0.817875 + 0.575396i \(0.804848\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(44\) 6.63325i 1.00000i
\(45\) 0.474937 + 0.525063i 0.0707995 + 0.0782717i
\(46\) 0 0
\(47\) −2.68338 + 2.68338i −0.391411 + 0.391411i −0.875190 0.483779i \(-0.839264\pi\)
0.483779 + 0.875190i \(0.339264\pi\)
\(48\) −4.63325 4.63325i −0.668752 0.668752i
\(49\) 7.00000i 1.00000i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −9.63325 9.63325i −1.32323 1.32323i −0.911147 0.412082i \(-0.864802\pi\)
−0.412082 0.911147i \(-0.635198\pi\)
\(54\) 0 0
\(55\) 5.50000 4.97494i 0.741620 0.670820i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.31662i 0.431788i −0.976417 0.215894i \(-0.930733\pi\)
0.976417 0.215894i \(-0.0692665\pi\)
\(60\) −0.366750 + 7.31662i −0.0473473 + 0.944572i
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 8.00000i 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) 1.52506 1.52506i 0.186316 0.186316i −0.607785 0.794101i \(-0.707942\pi\)
0.794101 + 0.607785i \(0.207942\pi\)
\(68\) 0 0
\(69\) 14.2665i 1.71748i
\(70\) 0 0
\(71\) −3.00000 −0.356034 −0.178017 0.984027i \(-0.556968\pi\)
−0.178017 + 0.984027i \(0.556968\pi\)
\(72\) 0 0
\(73\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(74\) 0 0
\(75\) 6.34169 5.18338i 0.732275 0.598525i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 6.63325 6.00000i 0.741620 0.670820i
\(81\) 7.94987 0.883319
\(82\) 0 0
\(83\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 9.00000i 0.953998i −0.878904 0.476999i \(-0.841725\pi\)
0.878904 0.476999i \(-0.158275\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 12.3166 12.3166i 1.28410 1.28410i
\(93\) 11.5251 + 11.5251i 1.19509 + 1.19509i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −13.4749 + 13.4749i −1.36817 + 1.36817i −0.505128 + 0.863044i \(0.668555\pi\)
−0.863044 + 0.505128i \(0.831445\pi\)
\(98\) 0 0
\(99\) 1.05013i 0.105542i
\(100\) −9.94987 1.00000i −0.994987 0.100000i
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 11.9499 + 11.9499i 1.17746 + 1.17746i 0.980390 + 0.197066i \(0.0631413\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(108\) −7.68338 7.68338i −0.739333 0.739333i
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) −19.6332 −1.86351
\(112\) 0 0
\(113\) −8.84169 8.84169i −0.831756 0.831756i 0.156001 0.987757i \(-0.450140\pi\)
−0.987757 + 0.156001i \(0.950140\pi\)
\(114\) 0 0
\(115\) −19.4499 0.974937i −1.81371 0.0909134i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 19.8997i 1.78705i
\(125\) 6.63325 + 9.00000i 0.593296 + 0.804984i
\(126\) 0 0
\(127\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) −7.68338 + 7.68338i −0.668752 + 0.668752i
\(133\) 0 0
\(134\) 0 0
\(135\) −0.608187 + 12.1332i −0.0523444 + 1.04426i
\(136\) 0 0
\(137\) 13.1082 13.1082i 1.11991 1.11991i 0.128154 0.991754i \(-0.459095\pi\)
0.991754 0.128154i \(-0.0409051\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) −6.21637 −0.523513
\(142\) 0 0
\(143\) 0 0
\(144\) 1.26650i 0.105542i
\(145\) 0 0
\(146\) 0 0
\(147\) 8.10819 8.10819i 0.668752 0.668752i
\(148\) 16.9499 + 16.9499i 1.39327 + 1.39327i
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −16.5000 + 14.9248i −1.32531 + 1.19879i
\(156\) 0 0
\(157\) 6.52506 6.52506i 0.520757 0.520757i −0.397043 0.917800i \(-0.629964\pi\)
0.917800 + 0.397043i \(0.129964\pi\)
\(158\) 0 0
\(159\) 22.3166i 1.76982i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1.94987 + 1.94987i 0.152726 + 0.152726i 0.779334 0.626608i \(-0.215557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 0 0
\(165\) 12.1332 + 0.608187i 0.944572 + 0.0473473i
\(166\) 0 0
\(167\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(168\) 0 0
\(169\) 13.0000i 1.00000i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 13.2665 1.00000
\(177\) 3.84169 3.84169i 0.288759 0.288759i
\(178\) 0 0
\(179\) 21.0000i 1.56961i 0.619740 + 0.784807i \(0.287238\pi\)
−0.619740 + 0.784807i \(0.712762\pi\)
\(180\) 1.05013 0.949874i 0.0782717 0.0707995i
\(181\) 9.94987 0.739568 0.369784 0.929118i \(-0.379432\pi\)
0.369784 + 0.929118i \(0.379432\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1.34169 26.7665i 0.0986428 1.96791i
\(186\) 0 0
\(187\) 0 0
\(188\) 5.36675 + 5.36675i 0.391411 + 0.391411i
\(189\) 0 0
\(190\) 0 0
\(191\) −23.2164 −1.67988 −0.839939 0.542681i \(-0.817409\pi\)
−0.839939 + 0.542681i \(0.817409\pi\)
\(192\) −9.26650 + 9.26650i −0.668752 + 0.668752i
\(193\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −14.0000 −1.00000
\(197\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(198\) 0 0
\(199\) 19.8997i 1.41066i −0.708881 0.705328i \(-0.750800\pi\)
0.708881 0.705328i \(-0.249200\pi\)
\(200\) 0 0
\(201\) 3.53300 0.249198
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1.94987 1.94987i 0.135526 0.135526i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) −19.2665 + 19.2665i −1.32323 + 1.32323i
\(213\) −3.47494 3.47494i −0.238099 0.238099i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) −9.94987 11.0000i −0.670820 0.741620i
\(221\) 0 0
\(222\) 0 0
\(223\) −15.4248 15.4248i −1.03292 1.03292i −0.999439 0.0334825i \(-0.989340\pi\)
−0.0334825 0.999439i \(-0.510660\pi\)
\(224\) 0 0
\(225\) −1.57519 0.158312i −0.105013 0.0105542i
\(226\) 0 0
\(227\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(228\) 0 0
\(229\) 29.8496i 1.97252i 0.165205 + 0.986259i \(0.447172\pi\)
−0.165205 + 0.986259i \(0.552828\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(234\) 0 0
\(235\) 0.424812 8.47494i 0.0277117 0.552844i
\(236\) −6.63325 −0.431788
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 14.6332 + 0.733501i 0.944572 + 0.0473473i
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) −2.31662 2.31662i −0.148612 0.148612i
\(244\) 0 0
\(245\) 10.5000 + 11.6082i 0.670820 + 0.741620i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 27.0000 1.70422 0.852112 0.523359i \(-0.175321\pi\)
0.852112 + 0.523359i \(0.175321\pi\)
\(252\) 0 0
\(253\) −20.4248 20.4248i −1.28410 1.28410i
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) −4.26650 + 4.26650i −0.266137 + 0.266137i −0.827541 0.561405i \(-0.810261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(264\) 0 0
\(265\) 30.4248 + 1.52506i 1.86898 + 0.0936839i
\(266\) 0 0
\(267\) 10.4248 10.4248i 0.637988 0.637988i
\(268\) −3.05013 3.05013i −0.186316 0.186316i
\(269\) 13.2665i 0.808873i 0.914566 + 0.404436i \(0.132532\pi\)
−0.914566 + 0.404436i \(0.867468\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.65831 + 16.5000i −0.100000 + 0.994987i
\(276\) 28.5330 1.71748
\(277\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(278\) 0 0
\(279\) 3.15038i 0.188608i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(284\) 6.00000i 0.356034i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 17.0000i 1.00000i
\(290\) 0 0
\(291\) −31.2164 −1.82994
\(292\) 0 0
\(293\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(294\) 0 0
\(295\) 4.97494 + 5.50000i 0.289652 + 0.320222i
\(296\) 0 0
\(297\) −12.7414 + 12.7414i −0.739333 + 0.739333i
\(298\) 0 0
\(299\) 0 0
\(300\) −10.3668 12.6834i −0.598525 0.732275i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(308\) 0 0
\(309\) 27.6834i 1.57485i
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) −5.42481 5.42481i −0.306628 0.306628i 0.536972 0.843600i \(-0.319568\pi\)
−0.843600 + 0.536972i \(0.819568\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.89181 + 1.89181i −0.106255 + 0.106255i −0.758236 0.651981i \(-0.773938\pi\)
0.651981 + 0.758236i \(0.273938\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −12.0000 13.2665i −0.670820 0.741620i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 15.8997i 0.883319i
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 9.94987 0.546895 0.273447 0.961887i \(-0.411836\pi\)
0.273447 + 0.961887i \(0.411836\pi\)
\(332\) 0 0
\(333\) 2.68338 + 2.68338i 0.147048 + 0.147048i
\(334\) 0 0
\(335\) −0.241436 + 4.81662i −0.0131911 + 0.263160i
\(336\) 0 0
\(337\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(338\) 0 0
\(339\) 20.4829i 1.11248i
\(340\) 0 0
\(341\) −33.0000 −1.78705
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −21.3997 23.6583i −1.15212 1.27372i
\(346\) 0 0
\(347\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 22.7414 + 22.7414i 1.21040 + 1.21040i 0.970894 + 0.239511i \(0.0769871\pi\)
0.239511 + 0.970894i \(0.423013\pi\)
\(354\) 0 0
\(355\) 4.97494 4.50000i 0.264042 0.238835i
\(356\) −18.0000 −0.953998
\(357\) 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
\(363\) 12.7414 + 12.7414i 0.668752 + 0.668752i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −23.4749 + 23.4749i −1.22538 + 1.22538i −0.259690 + 0.965692i \(0.583620\pi\)
−0.965692 + 0.259690i \(0.916380\pi\)
\(368\) −24.6332 24.6332i −1.28410 1.28410i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 23.0501 23.0501i 1.19509 1.19509i
\(373\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(374\) 0 0
\(375\) −2.74144 + 18.1082i −0.141567 + 0.935103i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 29.8496i 1.53327i 0.642082 + 0.766636i \(0.278071\pi\)
−0.642082 + 0.766636i \(0.721929\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 21.1583 + 21.1583i 1.08114 + 1.08114i 0.996403 + 0.0847358i \(0.0270046\pi\)
0.0847358 + 0.996403i \(0.472995\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 26.9499 + 26.9499i 1.36817 + 1.36817i
\(389\) 36.4829i 1.84976i −0.380265 0.924878i \(-0.624167\pi\)
0.380265 0.924878i \(-0.375833\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 2.10025 0.105542
\(397\) 18.8997 18.8997i 0.948551 0.948551i −0.0501886 0.998740i \(-0.515982\pi\)
0.998740 + 0.0501886i \(0.0159822\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −2.00000 + 19.8997i −0.100000 + 0.994987i
\(401\) 26.5330 1.32499 0.662497 0.749064i \(-0.269497\pi\)
0.662497 + 0.749064i \(0.269497\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −13.1834 + 11.9248i −0.655087 + 0.592549i
\(406\) 0 0
\(407\) 28.1082 28.1082i 1.39327 1.39327i
\(408\) 0 0
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) 30.3668 1.49788
\(412\) 23.8997 23.8997i 1.17746 1.17746i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 24.0000i 1.17248i −0.810139 0.586238i \(-0.800608\pi\)
0.810139 0.586238i \(-0.199392\pi\)
\(420\) 0 0
\(421\) −39.7995 −1.93971 −0.969854 0.243685i \(-0.921644\pi\)
−0.969854 + 0.243685i \(0.921644\pi\)
\(422\) 0 0
\(423\) 0.849623 + 0.849623i 0.0413101 + 0.0413101i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) −15.3668 + 15.3668i −0.739333 + 0.739333i
\(433\) −0.424812 0.424812i −0.0204151 0.0204151i 0.696826 0.717241i \(-0.254595\pi\)
−0.717241 + 0.696826i \(0.754595\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) −2.21637 −0.105542
\(442\) 0 0
\(443\) 7.74144 + 7.74144i 0.367807 + 0.367807i 0.866677 0.498870i \(-0.166252\pi\)
−0.498870 + 0.866677i \(0.666252\pi\)
\(444\) 39.2665i 1.86351i
\(445\) 13.5000 + 14.9248i 0.639961 + 0.707504i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 39.0000i 1.84052i −0.391303 0.920262i \(-0.627976\pi\)
0.391303 0.920262i \(-0.372024\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −17.6834 + 17.6834i −0.831756 + 0.831756i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −1.94987 + 38.8997i −0.0909134 + 1.81371i
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −30.4248 30.4248i −1.41396 1.41396i −0.720346 0.693615i \(-0.756017\pi\)
−0.693615 0.720346i \(-0.743983\pi\)
\(464\) 0 0
\(465\) −36.3997 1.82456i −1.68800 0.0846120i
\(466\) 0 0
\(467\) −20.0581 + 20.0581i −0.928176 + 0.928176i −0.997588 0.0694117i \(-0.977888\pi\)
0.0694117 + 0.997588i \(0.477888\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 15.1161 0.696514
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −3.05013 + 3.05013i −0.139656 + 0.139656i
\(478\) 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
\(484\) 22.0000i 1.00000i
\(485\) 2.13325 42.5581i 0.0968659 1.93246i
\(486\) 0 0
\(487\) 16.5251 16.5251i 0.748822 0.748822i −0.225436 0.974258i \(-0.572381\pi\)
0.974258 + 0.225436i \(0.0723806\pi\)
\(488\) 0 0
\(489\) 4.51713i 0.204272i
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −1.57519 1.74144i −0.0707995 0.0782717i
\(496\) −39.7995 −1.78705
\(497\) 0 0
\(498\) 0 0
\(499\) 19.8997i 0.890835i −0.895323 0.445418i \(-0.853055\pi\)
0.895323 0.445418i \(-0.146945\pi\)
\(500\) 18.0000 13.2665i 0.804984 0.593296i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −15.0581 + 15.0581i −0.668752 + 0.668752i
\(508\) 0 0
\(509\) 3.31662i 0.147007i −0.997295 0.0735034i \(-0.976582\pi\)
0.997295 0.0735034i \(-0.0234180\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −37.7414 1.89181i −1.66309 0.0833633i
\(516\) 0 0
\(517\) 8.89975 8.89975i 0.391411 0.391411i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 43.1161 1.88895 0.944476 0.328581i \(-0.106570\pi\)
0.944476 + 0.328581i \(0.106570\pi\)
\(522\) 0 0
\(523\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 15.3668 + 15.3668i 0.668752 + 0.668752i
\(529\) 52.8496i 2.29781i
\(530\) 0 0
\(531\) −1.05013 −0.0455716
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −24.3246 + 24.3246i −1.04968 + 1.04968i
\(538\) 0 0
\(539\) 23.2164i 1.00000i
\(540\) 24.2665 + 1.21637i 1.04426 + 0.0523444i
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 11.5251 + 11.5251i 0.494588 + 0.494588i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(548\) −26.2164 26.2164i −1.11991 1.11991i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 32.5581 29.4499i 1.38201 1.25008i
\(556\) 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
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(564\) 12.4327i 0.523513i
\(565\) 27.9248 + 1.39975i 1.17481 + 0.0588879i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) −26.8918 26.8918i −1.12342 1.12342i
\(574\) 0 0
\(575\) 33.7164 27.5581i 1.40607 1.14925i
\(576\) 2.53300 0.105542
\(577\) −28.4749 + 28.4749i −1.18543 + 1.18543i −0.207109 + 0.978318i \(0.566406\pi\)
−0.978318 + 0.207109i \(0.933594\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 31.9499 + 31.9499i 1.32323 + 1.32323i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 27.3166 27.3166i 1.12748 1.12748i 0.136892 0.990586i \(-0.456289\pi\)
0.990586 0.136892i \(-0.0437113\pi\)
\(588\) −16.2164 16.2164i −0.668752 0.668752i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 33.8997 33.8997i 1.39327 1.39327i
\(593\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 23.0501 23.0501i 0.943379 0.943379i
\(598\) 0 0
\(599\) 36.0000i 1.47092i 0.677568 + 0.735460i \(0.263034\pi\)
−0.677568 + 0.735460i \(0.736966\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) −0.482873 0.482873i −0.0196641 0.0196641i
\(604\) 0 0
\(605\) −18.2414 + 16.5000i −0.741620 + 0.670820i
\(606\) 0 0
\(607\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(608\) 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
\(617\) −34.2665 + 34.2665i −1.37952 + 1.37952i −0.534089 + 0.845428i \(0.679345\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) 0 0
\(619\) 1.00000i 0.0401934i 0.999798 + 0.0200967i \(0.00639741\pi\)
−0.999798 + 0.0200967i \(0.993603\pi\)
\(620\) 29.8496 + 33.0000i 1.19879 + 1.32531i
\(621\) 47.3166 1.89875
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −24.5000 4.97494i −0.980000 0.198997i
\(626\) 0 0
\(627\) 0 0
\(628\) −13.0501 13.0501i −0.520757 0.520757i
\(629\) 0 0
\(630\) 0 0
\(631\) 7.00000 0.278666 0.139333 0.990246i \(-0.455504\pi\)
0.139333 + 0.990246i \(0.455504\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) −44.6332 −1.76982
\(637\) 0 0
\(638\) 0 0
\(639\) 0.949874i 0.0375764i
\(640\) 0 0
\(641\) −23.2164 −0.916992 −0.458496 0.888697i \(-0.651612\pi\)
−0.458496 + 0.888697i \(0.651612\pi\)
\(642\) 0 0
\(643\) −35.4248 35.4248i −1.39702 1.39702i −0.808441 0.588577i \(-0.799688\pi\)
−0.588577 0.808441i \(-0.700312\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −35.0581 + 35.0581i −1.37827 + 1.37827i −0.530740 + 0.847535i \(0.678086\pi\)
−0.847535 + 0.530740i \(0.821914\pi\)
\(648\) 0 0
\(649\) 11.0000i 0.431788i
\(650\) 0 0
\(651\) 0 0
\(652\) 3.89975 3.89975i 0.152726 0.152726i
\(653\) −23.8417 23.8417i −0.932997 0.932997i 0.0648948 0.997892i \(-0.479329\pi\)
−0.997892 + 0.0648948i \(0.979329\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 1.21637 24.2665i 0.0473473 0.944572i
\(661\) −13.0000 −0.505641 −0.252821 0.967513i \(-0.581358\pi\)
−0.252821 + 0.967513i \(0.581358\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 35.7335i 1.38154i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(674\) 0 0
\(675\) −17.1913 21.0330i −0.661694 0.809560i
\(676\) 26.0000 1.00000
\(677\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −11.2164 11.2164i −0.429183 0.429183i 0.459167 0.888350i \(-0.348148\pi\)
−0.888350 + 0.459167i \(0.848148\pi\)
\(684\) 0 0
\(685\) −2.07519 + 41.3997i −0.0792889 + 1.58180i
\(686\) 0 0
\(687\) −34.5752 + 34.5752i −1.31913 + 1.31913i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 17.0000 0.646710 0.323355 0.946278i \(-0.395189\pi\)
0.323355 + 0.946278i \(0.395189\pi\)
\(692\) 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
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 26.5330i 1.00000i
\(705\) 10.3087 9.32456i 0.388248 0.351183i
\(706\) 0 0
\(707\) 0 0
\(708\) −7.68338 7.68338i −0.288759 0.288759i
\(709\) 19.0000i 0.713560i −0.934188 0.356780i \(-0.883875\pi\)
0.934188 0.356780i \(-0.116125\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 61.2744 + 61.2744i 2.29475 + 2.29475i
\(714\) 0 0
\(715\) 0 0
\(716\) 42.0000 1.56961
\(717\) 0 0
\(718\) 0 0
\(719\) 51.0000i 1.90198i 0.309223 + 0.950990i \(0.399931\pi\)
−0.309223 + 0.950990i \(0.600069\pi\)
\(720\) −1.89975 2.10025i −0.0707995 0.0782717i
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 19.8997i 0.739568i
\(725\) 0 0
\(726\) 0 0
\(727\) 21.5251 21.5251i 0.798320 0.798320i −0.184510 0.982831i \(-0.559070\pi\)
0.982831 + 0.184510i \(0.0590699\pi\)
\(728\) 0 0
\(729\) 29.2164i 1.08209i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(734\) 0 0
\(735\) −1.28363 + 25.6082i −0.0473473 + 0.944572i
\(736\) 0 0
\(737\) −5.05806 + 5.05806i −0.186316 + 0.186316i
\(738\) 0 0
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) −53.5330 2.68338i −1.96791 0.0986428i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −23.0000 −0.839282 −0.419641 0.907690i \(-0.637844\pi\)
−0.419641 + 0.907690i \(0.637844\pi\)
\(752\) 10.7335 10.7335i 0.391411 0.391411i
\(753\) 31.2744 + 31.2744i 1.13970 + 1.13970i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 38.8997 38.8997i 1.41384 1.41384i 0.690567 0.723269i \(-0.257361\pi\)
0.723269 0.690567i \(-0.242639\pi\)
\(758\) 0 0
\(759\) 47.3166i 1.71748i
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 46.4327i 1.67988i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 18.5330 + 18.5330i 0.668752 + 0.668752i
\(769\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) −9.88388 −0.355959
\(772\) 0 0
\(773\) 20.3668 + 20.3668i 0.732541 + 0.732541i 0.971123 0.238581i \(-0.0766824\pi\)
−0.238581 + 0.971123i \(0.576682\pi\)
\(774\) 0 0
\(775\) 4.97494 49.5000i 0.178705 1.77809i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 9.94987 0.356034
\(782\) 0 0
\(783\) 0 0
\(784\) 28.0000i 1.00000i
\(785\) −1.03300 + 20.6082i −0.0368693 + 0.735538i
\(786\) 0 0
\(787\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 33.4749 + 37.0079i 1.18723 + 1.31254i
\(796\) −39.7995 −1.41066
\(797\) 29.6913 29.6913i 1.05172 1.05172i 0.0531327 0.998587i \(-0.483079\pi\)
0.998587 0.0531327i \(-0.0169206\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −2.84962 −0.100686
\(802\) 0 0
\(803\) 0 0
\(804\) 7.06600i 0.249198i
\(805\) 0 0
\(806\) 0 0
\(807\) −15.3668 + 15.3668i −0.540935 + 0.540935i
\(808\) 0 0
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) </