Properties

Label 1152.2.k.d.865.1
Level $1152$
Weight $2$
Character 1152.865
Analytic conductor $9.199$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1152.k (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.19876631285\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.629407744.1
Defining polynomial: \(x^{8} - 2 x^{6} + 2 x^{4} - 8 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 865.1
Root \(0.767178 - 1.18804i\) of defining polynomial
Character \(\chi\) \(=\) 1152.865
Dual form 1152.2.k.d.289.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-2.37608 - 2.37608i) q^{5} +3.64575i q^{7} +O(q^{10})\) \(q+(-2.37608 - 2.37608i) q^{5} +3.64575i q^{7} +(0.841723 + 0.841723i) q^{11} +(2.64575 - 2.64575i) q^{13} -3.06871 q^{17} +(1.64575 - 1.64575i) q^{19} -7.82087i q^{23} +6.29150i q^{25} +(-0.692633 + 0.692633i) q^{29} +0.354249 q^{31} +(8.66259 - 8.66259i) q^{35} +(-4.64575 - 4.64575i) q^{37} -6.43560i q^{41} +(-5.64575 - 5.64575i) q^{43} +11.1878 q^{47} -6.29150 q^{49} +(-5.44479 - 5.44479i) q^{53} -4.00000i q^{55} +(-7.82087 - 7.82087i) q^{59} +(-4.64575 + 4.64575i) q^{61} -12.5730 q^{65} +(4.00000 - 4.00000i) q^{67} +3.36689i q^{71} -7.29150i q^{73} +(-3.06871 + 3.06871i) q^{77} -4.35425 q^{79} +(-0.841723 + 0.841723i) q^{83} +(7.29150 + 7.29150i) q^{85} -9.50432i q^{89} +(9.64575 + 9.64575i) q^{91} -7.82087 q^{95} +10.5830 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + O(q^{10}) \) \( 8 q - 8 q^{19} + 24 q^{31} - 16 q^{37} - 24 q^{43} - 8 q^{49} - 16 q^{61} + 32 q^{67} - 56 q^{79} + 16 q^{85} + 56 q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{4}\right)\)

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
\(4\) 0 0
\(5\) −2.37608 2.37608i −1.06261 1.06261i −0.997904 0.0647108i \(-0.979388\pi\)
−0.0647108 0.997904i \(-0.520612\pi\)
\(6\) 0 0
\(7\) 3.64575i 1.37796i 0.724778 + 0.688982i \(0.241942\pi\)
−0.724778 + 0.688982i \(0.758058\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.841723 + 0.841723i 0.253789 + 0.253789i 0.822522 0.568733i \(-0.192566\pi\)
−0.568733 + 0.822522i \(0.692566\pi\)
\(12\) 0 0
\(13\) 2.64575 2.64575i 0.733799 0.733799i −0.237571 0.971370i \(-0.576351\pi\)
0.971370 + 0.237571i \(0.0763512\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.06871 −0.744272 −0.372136 0.928178i \(-0.621375\pi\)
−0.372136 + 0.928178i \(0.621375\pi\)
\(18\) 0 0
\(19\) 1.64575 1.64575i 0.377561 0.377561i −0.492660 0.870222i \(-0.663976\pi\)
0.870222 + 0.492660i \(0.163976\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.82087i 1.63076i −0.578923 0.815382i \(-0.696527\pi\)
0.578923 0.815382i \(-0.303473\pi\)
\(24\) 0 0
\(25\) 6.29150i 1.25830i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −0.692633 + 0.692633i −0.128619 + 0.128619i −0.768486 0.639867i \(-0.778989\pi\)
0.639867 + 0.768486i \(0.278989\pi\)
\(30\) 0 0
\(31\) 0.354249 0.0636249 0.0318125 0.999494i \(-0.489872\pi\)
0.0318125 + 0.999494i \(0.489872\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 8.66259 8.66259i 1.46425 1.46425i
\(36\) 0 0
\(37\) −4.64575 4.64575i −0.763757 0.763757i 0.213242 0.976999i \(-0.431598\pi\)
−0.976999 + 0.213242i \(0.931598\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.43560i 1.00507i −0.864556 0.502536i \(-0.832400\pi\)
0.864556 0.502536i \(-0.167600\pi\)
\(42\) 0 0
\(43\) −5.64575 5.64575i −0.860969 0.860969i 0.130482 0.991451i \(-0.458348\pi\)
−0.991451 + 0.130482i \(0.958348\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 11.1878 1.63190 0.815951 0.578121i \(-0.196214\pi\)
0.815951 + 0.578121i \(0.196214\pi\)
\(48\) 0 0
\(49\) −6.29150 −0.898786
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −5.44479 5.44479i −0.747900 0.747900i 0.226185 0.974084i \(-0.427375\pi\)
−0.974084 + 0.226185i \(0.927375\pi\)
\(54\) 0 0
\(55\) 4.00000i 0.539360i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −7.82087 7.82087i −1.01819 1.01819i −0.999831 0.0183591i \(-0.994156\pi\)
−0.0183591 0.999831i \(-0.505844\pi\)
\(60\) 0 0
\(61\) −4.64575 + 4.64575i −0.594828 + 0.594828i −0.938932 0.344104i \(-0.888183\pi\)
0.344104 + 0.938932i \(0.388183\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −12.5730 −1.55949
\(66\) 0 0
\(67\) 4.00000 4.00000i 0.488678 0.488678i −0.419211 0.907889i \(-0.637693\pi\)
0.907889 + 0.419211i \(0.137693\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.36689i 0.399577i 0.979839 + 0.199788i \(0.0640254\pi\)
−0.979839 + 0.199788i \(0.935975\pi\)
\(72\) 0 0
\(73\) 7.29150i 0.853406i −0.904392 0.426703i \(-0.859675\pi\)
0.904392 0.426703i \(-0.140325\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.06871 + 3.06871i −0.349712 + 0.349712i
\(78\) 0 0
\(79\) −4.35425 −0.489891 −0.244946 0.969537i \(-0.578770\pi\)
−0.244946 + 0.969537i \(0.578770\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −0.841723 + 0.841723i −0.0923911 + 0.0923911i −0.751792 0.659401i \(-0.770810\pi\)
0.659401 + 0.751792i \(0.270810\pi\)
\(84\) 0 0
\(85\) 7.29150 + 7.29150i 0.790875 + 0.790875i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 9.50432i 1.00746i −0.863862 0.503728i \(-0.831961\pi\)
0.863862 0.503728i \(-0.168039\pi\)
\(90\) 0 0
\(91\) 9.64575 + 9.64575i 1.01115 + 1.01115i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −7.82087 −0.802404
\(96\) 0 0
\(97\) 10.5830 1.07454 0.537271 0.843410i \(-0.319455\pi\)
0.537271 + 0.843410i \(0.319455\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −2.37608 2.37608i −0.236429 0.236429i 0.578941 0.815370i \(-0.303466\pi\)
−0.815370 + 0.578941i \(0.803466\pi\)
\(102\) 0 0
\(103\) 1.06275i 0.104715i 0.998628 + 0.0523577i \(0.0166736\pi\)
−0.998628 + 0.0523577i \(0.983326\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 9.50432 + 9.50432i 0.918817 + 0.918817i 0.996943 0.0781266i \(-0.0248938\pi\)
−0.0781266 + 0.996943i \(0.524894\pi\)
\(108\) 0 0
\(109\) 1.35425 1.35425i 0.129713 0.129713i −0.639269 0.768983i \(-0.720763\pi\)
0.768983 + 0.639269i \(0.220763\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) −18.5830 + 18.5830i −1.73287 + 1.73287i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 11.1878i 1.02558i
\(120\) 0 0
\(121\) 9.58301i 0.871182i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 3.06871 3.06871i 0.274474 0.274474i
\(126\) 0 0
\(127\) 14.9373 1.32547 0.662733 0.748855i \(-0.269396\pi\)
0.662733 + 0.748855i \(0.269396\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1.68345 1.68345i 0.147083 0.147083i −0.629730 0.776814i \(-0.716835\pi\)
0.776814 + 0.629730i \(0.216835\pi\)
\(132\) 0 0
\(133\) 6.00000 + 6.00000i 0.520266 + 0.520266i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 15.9399i 1.36184i 0.732358 + 0.680920i \(0.238420\pi\)
−0.732358 + 0.680920i \(0.761580\pi\)
\(138\) 0 0
\(139\) 6.58301 + 6.58301i 0.558363 + 0.558363i 0.928841 0.370478i \(-0.120806\pi\)
−0.370478 + 0.928841i \(0.620806\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.45398 0.372460
\(144\) 0 0
\(145\) 3.29150 0.273344
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3.76135 + 3.76135i 0.308141 + 0.308141i 0.844188 0.536047i \(-0.180083\pi\)
−0.536047 + 0.844188i \(0.680083\pi\)
\(150\) 0 0
\(151\) 15.6458i 1.27323i −0.771180 0.636617i \(-0.780333\pi\)
0.771180 0.636617i \(-0.219667\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −0.841723 0.841723i −0.0676088 0.0676088i
\(156\) 0 0
\(157\) 1.35425 1.35425i 0.108081 0.108081i −0.650998 0.759079i \(-0.725650\pi\)
0.759079 + 0.650998i \(0.225650\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 28.5129 2.24714
\(162\) 0 0
\(163\) 6.35425 6.35425i 0.497703 0.497703i −0.413019 0.910722i \(-0.635526\pi\)
0.910722 + 0.413019i \(0.135526\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 4.45398i 0.344659i 0.985039 + 0.172330i \(0.0551294\pi\)
−0.985039 + 0.172330i \(0.944871\pi\)
\(168\) 0 0
\(169\) 1.00000i 0.0769231i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 5.44479 5.44479i 0.413960 0.413960i −0.469156 0.883115i \(-0.655442\pi\)
0.883115 + 0.469156i \(0.155442\pi\)
\(174\) 0 0
\(175\) −22.9373 −1.73389
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −9.50432 + 9.50432i −0.710386 + 0.710386i −0.966616 0.256230i \(-0.917520\pi\)
0.256230 + 0.966616i \(0.417520\pi\)
\(180\) 0 0
\(181\) −4.64575 4.64575i −0.345316 0.345316i 0.513045 0.858361i \(-0.328517\pi\)
−0.858361 + 0.513045i \(0.828517\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 22.0773i 1.62316i
\(186\) 0 0
\(187\) −2.58301 2.58301i −0.188888 0.188888i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −11.1878 −0.809518 −0.404759 0.914423i \(-0.632645\pi\)
−0.404759 + 0.914423i \(0.632645\pi\)
\(192\) 0 0
\(193\) −19.8745 −1.43060 −0.715299 0.698818i \(-0.753710\pi\)
−0.715299 + 0.698818i \(0.753710\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 8.81168 + 8.81168i 0.627806 + 0.627806i 0.947516 0.319709i \(-0.103585\pi\)
−0.319709 + 0.947516i \(0.603585\pi\)
\(198\) 0 0
\(199\) 8.35425i 0.592217i 0.955154 + 0.296108i \(0.0956890\pi\)
−0.955154 + 0.296108i \(0.904311\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −2.52517 2.52517i −0.177232 0.177232i
\(204\) 0 0
\(205\) −15.2915 + 15.2915i −1.06800 + 1.06800i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 2.77053 0.191642
\(210\) 0 0
\(211\) 6.58301 6.58301i 0.453193 0.453193i −0.443220 0.896413i \(-0.646164\pi\)
0.896413 + 0.443220i \(0.146164\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 26.8295i 1.82976i
\(216\) 0 0
\(217\) 1.29150i 0.0876729i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −8.11905 + 8.11905i −0.546146 + 0.546146i
\(222\) 0 0
\(223\) 19.6458 1.31558 0.657788 0.753203i \(-0.271492\pi\)
0.657788 + 0.753203i \(0.271492\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −18.1669 + 18.1669i −1.20578 + 1.20578i −0.233399 + 0.972381i \(0.574985\pi\)
−0.972381 + 0.233399i \(0.925015\pi\)
\(228\) 0 0
\(229\) −2.06275 2.06275i −0.136310 0.136310i 0.635659 0.771970i \(-0.280728\pi\)
−0.771970 + 0.635659i \(0.780728\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 21.7792i 1.42680i −0.700757 0.713400i \(-0.747154\pi\)
0.700757 0.713400i \(-0.252846\pi\)
\(234\) 0 0
\(235\) −26.5830 26.5830i −1.73408 1.73408i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −23.4626 −1.51767 −0.758835 0.651283i \(-0.774231\pi\)
−0.758835 + 0.651283i \(0.774231\pi\)
\(240\) 0 0
\(241\) 9.29150 0.598518 0.299259 0.954172i \(-0.403260\pi\)
0.299259 + 0.954172i \(0.403260\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 14.9491 + 14.9491i 0.955063 + 0.955063i
\(246\) 0 0
\(247\) 8.70850i 0.554108i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 13.1166 + 13.1166i 0.827911 + 0.827911i 0.987228 0.159317i \(-0.0509291\pi\)
−0.159317 + 0.987228i \(0.550929\pi\)
\(252\) 0 0
\(253\) 6.58301 6.58301i 0.413870 0.413870i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −16.2381 −1.01290 −0.506452 0.862268i \(-0.669043\pi\)
−0.506452 + 0.862268i \(0.669043\pi\)
\(258\) 0 0
\(259\) 16.9373 16.9373i 1.05243 1.05243i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 15.6417i 0.964511i 0.876031 + 0.482256i \(0.160182\pi\)
−0.876031 + 0.482256i \(0.839818\pi\)
\(264\) 0 0
\(265\) 25.8745i 1.58946i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 8.51350 8.51350i 0.519077 0.519077i −0.398215 0.917292i \(-0.630370\pi\)
0.917292 + 0.398215i \(0.130370\pi\)
\(270\) 0 0
\(271\) −11.6458 −0.707429 −0.353715 0.935353i \(-0.615082\pi\)
−0.353715 + 0.935353i \(0.615082\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −5.29570 + 5.29570i −0.319343 + 0.319343i
\(276\) 0 0
\(277\) −20.5203 20.5203i −1.23294 1.23294i −0.962828 0.270115i \(-0.912938\pi\)
−0.270115 0.962828i \(-0.587062\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 9.50432i 0.566980i −0.958975 0.283490i \(-0.908508\pi\)
0.958975 0.283490i \(-0.0914923\pi\)
\(282\) 0 0
\(283\) 18.5830 + 18.5830i 1.10465 + 1.10465i 0.993842 + 0.110803i \(0.0353421\pi\)
0.110803 + 0.993842i \(0.464658\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 23.4626 1.38495
\(288\) 0 0
\(289\) −7.58301 −0.446059
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 14.9491 + 14.9491i 0.873336 + 0.873336i 0.992834 0.119498i \(-0.0381286\pi\)
−0.119498 + 0.992834i \(0.538129\pi\)
\(294\) 0 0
\(295\) 37.1660i 2.16389i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −20.6921 20.6921i −1.19665 1.19665i
\(300\) 0 0
\(301\) 20.5830 20.5830i 1.18638 1.18638i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 22.0773 1.26415
\(306\) 0 0
\(307\) −20.0000 + 20.0000i −1.14146 + 1.14146i −0.153277 + 0.988183i \(0.548983\pi\)
−0.988183 + 0.153277i \(0.951017\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 15.6417i 0.886962i 0.896284 + 0.443481i \(0.146257\pi\)
−0.896284 + 0.443481i \(0.853743\pi\)
\(312\) 0 0
\(313\) 1.29150i 0.0730000i 0.999334 + 0.0365000i \(0.0116209\pi\)
−0.999334 + 0.0365000i \(0.988379\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.37608 2.37608i 0.133454 0.133454i −0.637224 0.770678i \(-0.719918\pi\)
0.770678 + 0.637224i \(0.219918\pi\)
\(318\) 0 0
\(319\) −1.16601 −0.0652841
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −5.05034 + 5.05034i −0.281008 + 0.281008i
\(324\) 0 0
\(325\) 16.6458 + 16.6458i 0.923340 + 0.923340i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 40.7878i 2.24870i
\(330\) 0 0
\(331\) −8.00000 8.00000i −0.439720 0.439720i 0.452198 0.891918i \(-0.350640\pi\)
−0.891918 + 0.452198i \(0.850640\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −19.0086 −1.03855
\(336\) 0 0
\(337\) 15.2915 0.832981 0.416491 0.909140i \(-0.363260\pi\)
0.416491 + 0.909140i \(0.363260\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0.298179 + 0.298179i 0.0161473 + 0.0161473i
\(342\) 0 0
\(343\) 2.58301i 0.139469i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0.841723 + 0.841723i 0.0451861 + 0.0451861i 0.729339 0.684153i \(-0.239828\pi\)
−0.684153 + 0.729339i \(0.739828\pi\)
\(348\) 0 0
\(349\) 23.2288 23.2288i 1.24341 1.24341i 0.284828 0.958579i \(-0.408063\pi\)
0.958579 0.284828i \(-0.0919366\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 28.5129 1.51759 0.758796 0.651329i \(-0.225788\pi\)
0.758796 + 0.651329i \(0.225788\pi\)
\(354\) 0 0
\(355\) 8.00000 8.00000i 0.424596 0.424596i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 19.0086i 1.00324i −0.865089 0.501619i \(-0.832738\pi\)
0.865089 0.501619i \(-0.167262\pi\)
\(360\) 0 0
\(361\) 13.5830i 0.714895i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −17.3252 + 17.3252i −0.906842 + 0.906842i
\(366\) 0 0
\(367\) 12.8118 0.668769 0.334384 0.942437i \(-0.391472\pi\)
0.334384 + 0.942437i \(0.391472\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 19.8504 19.8504i 1.03058 1.03058i
\(372\) 0 0
\(373\) 3.93725 + 3.93725i 0.203863 + 0.203863i 0.801653 0.597790i \(-0.203954\pi\)
−0.597790 + 0.801653i \(0.703954\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3.66507i 0.188761i
\(378\) 0 0
\(379\) 13.6458 + 13.6458i 0.700935 + 0.700935i 0.964611 0.263676i \(-0.0849350\pi\)
−0.263676 + 0.964611i \(0.584935\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 11.1878 0.571668 0.285834 0.958279i \(-0.407729\pi\)
0.285834 + 0.958279i \(0.407729\pi\)
\(384\) 0 0
\(385\) 14.5830 0.743219
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −7.42642 7.42642i −0.376534 0.376534i 0.493316 0.869850i \(-0.335785\pi\)
−0.869850 + 0.493316i \(0.835785\pi\)
\(390\) 0 0
\(391\) 24.0000i 1.21373i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 10.3460 + 10.3460i 0.520566 + 0.520566i
\(396\) 0 0
\(397\) −23.9373 + 23.9373i −1.20138 + 1.20138i −0.227628 + 0.973748i \(0.573097\pi\)
−0.973748 + 0.227628i \(0.926903\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −9.20614 −0.459733 −0.229866 0.973222i \(-0.573829\pi\)
−0.229866 + 0.973222i \(0.573829\pi\)
\(402\) 0 0
\(403\) 0.937254 0.937254i 0.0466879 0.0466879i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 7.82087i 0.387666i
\(408\) 0 0
\(409\) 17.1660i 0.848805i −0.905474 0.424402i \(-0.860484\pi\)
0.905474 0.424402i \(-0.139516\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 28.5129 28.5129i 1.40303 1.40303i
\(414\) 0 0
\(415\) 4.00000 0.196352
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −13.1166 + 13.1166i −0.640786 + 0.640786i −0.950749 0.309962i \(-0.899684\pi\)
0.309962 + 0.950749i \(0.399684\pi\)
\(420\) 0 0
\(421\) −16.6458 16.6458i −0.811264 0.811264i 0.173559 0.984823i \(-0.444473\pi\)
−0.984823 + 0.173559i \(0.944473\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 19.3068i 0.936518i
\(426\) 0 0
\(427\) −16.9373 16.9373i −0.819651 0.819651i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 1.08709 0.0523632 0.0261816 0.999657i \(-0.491665\pi\)
0.0261816 + 0.999657i \(0.491665\pi\)
\(432\) 0 0
\(433\) −4.00000 −0.192228 −0.0961139 0.995370i \(-0.530641\pi\)
−0.0961139 + 0.995370i \(0.530641\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −12.8712 12.8712i −0.615713 0.615713i
\(438\) 0 0
\(439\) 37.5203i 1.79074i −0.445319 0.895372i \(-0.646910\pi\)
0.445319 0.895372i \(-0.353090\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 23.2172 + 23.2172i 1.10308 + 1.10308i 0.994036 + 0.109048i \(0.0347803\pi\)
0.109048 + 0.994036i \(0.465220\pi\)
\(444\) 0 0
\(445\) −22.5830 + 22.5830i −1.07054 + 1.07054i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −37.7191 −1.78007 −0.890037 0.455889i \(-0.849322\pi\)
−0.890037 + 0.455889i \(0.849322\pi\)
\(450\) 0 0
\(451\) 5.41699 5.41699i 0.255076 0.255076i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 45.8381i 2.14892i
\(456\) 0 0
\(457\) 26.5830i 1.24350i −0.783216 0.621750i \(-0.786422\pi\)
0.783216 0.621750i \(-0.213578\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −8.81168 + 8.81168i −0.410401 + 0.410401i −0.881878 0.471477i \(-0.843721\pi\)
0.471477 + 0.881878i \(0.343721\pi\)
\(462\) 0 0
\(463\) −30.9373 −1.43778 −0.718888 0.695126i \(-0.755349\pi\)
−0.718888 + 0.695126i \(0.755349\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 11.4331 11.4331i 0.529062 0.529062i −0.391231 0.920293i \(-0.627951\pi\)
0.920293 + 0.391231i \(0.127951\pi\)
\(468\) 0 0
\(469\) 14.5830 + 14.5830i 0.673381 + 0.673381i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 9.50432i 0.437009i
\(474\) 0 0
\(475\) 10.3542 + 10.3542i 0.475086 + 0.475086i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −12.2748 −0.560852 −0.280426 0.959876i \(-0.590476\pi\)
−0.280426 + 0.959876i \(0.590476\pi\)
\(480\) 0 0
\(481\) −24.5830 −1.12089
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −25.1461 25.1461i −1.14182 1.14182i
\(486\) 0 0
\(487\) 18.2288i 0.826024i −0.910726 0.413012i \(-0.864477\pi\)
0.910726 0.413012i \(-0.135523\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −13.9583 13.9583i −0.629929 0.629929i 0.318121 0.948050i \(-0.396948\pi\)
−0.948050 + 0.318121i \(0.896948\pi\)
\(492\) 0 0
\(493\) 2.12549 2.12549i 0.0957274 0.0957274i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −12.2748 −0.550602
\(498\) 0 0
\(499\) −10.5830 + 10.5830i −0.473760 + 0.473760i −0.903129 0.429369i \(-0.858736\pi\)
0.429369 + 0.903129i \(0.358736\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 25.7424i 1.14780i 0.818926 + 0.573899i \(0.194570\pi\)
−0.818926 + 0.573899i \(0.805430\pi\)
\(504\) 0 0
\(505\) 11.2915i 0.502465i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −27.2240 + 27.2240i −1.20668 + 1.20668i −0.234585 + 0.972096i \(0.575373\pi\)
−0.972096 + 0.234585i \(0.924627\pi\)
\(510\) 0 0
\(511\) 26.5830 1.17596
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 2.52517 2.52517i 0.111272 0.111272i
\(516\) 0 0
\(517\) 9.41699 + 9.41699i 0.414159 + 0.414159i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 18.7105i 0.819720i −0.912148 0.409860i \(-0.865578\pi\)
0.912148 0.409860i \(-0.134422\pi\)
\(522\) 0 0
\(523\) −3.06275 3.06275i −0.133925 0.133925i 0.636967 0.770891i \(-0.280189\pi\)
−0.770891 + 0.636967i \(0.780189\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −1.08709 −0.0473543
\(528\) 0 0
\(529\) −38.1660 −1.65939
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −17.0270 17.0270i −0.737522 0.737522i
\(534\) 0 0
\(535\) 45.1660i 1.95270i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −5.29570 5.29570i −0.228102 0.228102i
\(540\) 0 0
\(541\) −8.52026 + 8.52026i −0.366315 + 0.366315i −0.866131 0.499817i \(-0.833401\pi\)
0.499817 + 0.866131i \(0.333401\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −6.43560 −0.275671
\(546\) 0 0
\(547\) −0.937254 + 0.937254i −0.0400741 + 0.0400741i −0.726860 0.686786i \(-0.759021\pi\)
0.686786 + 0.726860i \(0.259021\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 2.27980i 0.0971229i
\(552\) 0 0
\(553\) 15.8745i 0.675053i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 24.7516 24.7516i 1.04876 1.04876i 0.0500103 0.998749i \(-0.484075\pi\)
0.998749 0.0500103i \(-0.0159254\pi\)
\(558\) 0 0
\(559\) −29.8745 −1.26356
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 16.4835 16.4835i 0.694695 0.694695i −0.268566 0.963261i \(-0.586550\pi\)
0.963261 + 0.268566i \(0.0865498\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 11.9767i 0.502088i 0.967976 + 0.251044i \(0.0807739\pi\)
−0.967976 + 0.251044i \(0.919226\pi\)
\(570\) 0 0
\(571\) 4.00000 + 4.00000i 0.167395 + 0.167395i 0.785833 0.618438i \(-0.212234\pi\)
−0.618438 + 0.785833i \(0.712234\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 49.2050 2.05199
\(576\) 0 0
\(577\) 35.0405 1.45876 0.729378 0.684111i \(-0.239810\pi\)
0.729378 + 0.684111i \(0.239810\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −3.06871 3.06871i −0.127312 0.127312i
\(582\) 0 0
\(583\) 9.16601i 0.379617i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −19.0086 19.0086i −0.784570 0.784570i 0.196028 0.980598i \(-0.437196\pi\)
−0.980598 + 0.196028i \(0.937196\pi\)
\(588\) 0 0
\(589\) 0.583005 0.583005i 0.0240223 0.0240223i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −12.2748 −0.504068 −0.252034 0.967718i \(-0.581099\pi\)
−0.252034 + 0.967718i \(0.581099\pi\)
\(594\) 0 0
\(595\) −26.5830 + 26.5830i −1.08980 + 1.08980i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 30.1964i 1.23379i −0.787045 0.616896i \(-0.788390\pi\)
0.787045 0.616896i \(-0.211610\pi\)
\(600\) 0 0
\(601\) 7.29150i 0.297427i −0.988880 0.148713i \(-0.952487\pi\)
0.988880 0.148713i \(-0.0475132\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −22.7700 + 22.7700i −0.925731 + 0.925731i
\(606\) 0 0
\(607\) 44.1033 1.79010 0.895048 0.445970i \(-0.147141\pi\)
0.895048 + 0.445970i \(0.147141\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 29.6000 29.6000i 1.19749 1.19749i
\(612\) 0 0
\(613\) −21.3542 21.3542i −0.862490 0.862490i 0.129137 0.991627i \(-0.458779\pi\)
−0.991627 + 0.129137i \(0.958779\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 25.7424i 1.03635i −0.855274 0.518175i \(-0.826611\pi\)
0.855274 0.518175i \(-0.173389\pi\)
\(618\) 0 0
\(619\) 21.1660 + 21.1660i 0.850734 + 0.850734i 0.990224 0.139490i \(-0.0445462\pi\)
−0.139490 + 0.990224i \(0.544546\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 34.6504 1.38824
\(624\) 0 0
\(625\) 16.8745 0.674980
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 14.2565 + 14.2565i 0.568443 + 0.568443i
\(630\) 0 0
\(631\) 18.2288i 0.725675i 0.931852 + 0.362838i \(0.118192\pi\)
−0.931852 + 0.362838i \(0.881808\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −35.4921 35.4921i −1.40846 1.40846i
\(636\) 0 0
\(637\) −16.6458 + 16.6458i −0.659529 + 0.659529i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −9.20614 −0.363621 −0.181810 0.983334i \(-0.558196\pi\)
−0.181810 + 0.983334i \(0.558196\pi\)
\(642\) 0 0
\(643\) 1.64575 1.64575i 0.0649021 0.0649021i −0.673911 0.738813i \(-0.735387\pi\)
0.738813 + 0.673911i \(0.235387\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 7.82087i 0.307470i −0.988112 0.153735i \(-0.950870\pi\)
0.988112 0.153735i \(-0.0491302\pi\)
\(648\) 0 0
\(649\) 13.1660i 0.516811i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 7.42642 7.42642i 0.290618 0.290618i −0.546706 0.837324i \(-0.684119\pi\)
0.837324 + 0.546706i \(0.184119\pi\)
\(654\) 0 0
\(655\) −8.00000 −0.312586
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 24.0590 24.0590i 0.937204 0.937204i −0.0609372 0.998142i \(-0.519409\pi\)
0.998142 + 0.0609372i \(0.0194089\pi\)
\(660\) 0 0
\(661\) 29.2288 + 29.2288i 1.13687 + 1.13687i 0.989009 + 0.147858i \(0.0472380\pi\)
0.147858 + 0.989009i \(0.452762\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 28.5129i 1.10568i
\(666\) 0 0
\(667\) 5.41699 + 5.41699i 0.209747 + 0.209747i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −7.82087 −0.301921
\(672\) 0 0
\(673\) 20.0000 0.770943 0.385472 0.922720i \(-0.374039\pi\)
0.385472 + 0.922720i \(0.374039\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 5.74297 + 5.74297i 0.220720 + 0.220720i 0.808802 0.588081i \(-0.200117\pi\)
−0.588081 + 0.808802i \(0.700117\pi\)
\(678\) 0 0
\(679\) 38.5830i 1.48068i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 35.4921 + 35.4921i 1.35807 + 1.35807i 0.876298 + 0.481769i \(0.160006\pi\)
0.481769 + 0.876298i \(0.339994\pi\)
\(684\) 0 0
\(685\) 37.8745 37.8745i 1.44711 1.44711i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −28.8111 −1.09762
\(690\) 0 0
\(691\) 1.64575 1.64575i 0.0626073 0.0626073i −0.675110 0.737717i \(-0.735904\pi\)
0.737717 + 0.675110i \(0.235904\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 31.2835i 1.18665i
\(696\) 0 0
\(697\) 19.7490i 0.748047i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 16.6326 16.6326i 0.628203 0.628203i −0.319413 0.947616i \(-0.603486\pi\)
0.947616 + 0.319413i \(0.103486\pi\)
\(702\) 0 0
\(703\) −15.2915 −0.576730
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 8.66259 8.66259i 0.325790 0.325790i
\(708\) 0 0
\(709\) 35.2288 + 35.2288i 1.32304 + 1.32304i 0.911299 + 0.411744i \(0.135080\pi\)
0.411744 + 0.911299i \(0.364920\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2.77053i 0.103757i
\(714\) 0 0
\(715\) −10.5830 10.5830i −0.395782 0.395782i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 10.1007 0.376692 0.188346 0.982103i \(-0.439687\pi\)
0.188346 + 0.982103i \(0.439687\pi\)
\(720\) 0 0
\(721\) −3.87451 −0.144294
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −4.35770 4.35770i −0.161841 0.161841i
\(726\) 0 0
\(727\) 35.3948i 1.31272i 0.754448 + 0.656360i \(0.227905\pi\)
−0.754448 + 0.656360i \(0.772095\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 17.3252 + 17.3252i 0.640795 + 0.640795i
\(732\) 0 0
\(733\) 7.35425 7.35425i 0.271635 0.271635i −0.558123 0.829758i \(-0.688478\pi\)
0.829758 + 0.558123i \(0.188478\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 6.73378 0.248042
\(738\) 0 0
\(739\) 33.1660 33.1660i 1.22003 1.22003i 0.252411 0.967620i \(-0.418776\pi\)
0.967620 0.252411i \(-0.0812236\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 14.5547i 0.533958i 0.963702 + 0.266979i \(0.0860255\pi\)
−0.963702 + 0.266979i \(0.913974\pi\)
\(744\) 0 0
\(745\) 17.8745i 0.654871i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −34.6504 + 34.6504i −1.26610 + 1.26610i
\(750\) 0 0
\(751\) 2.93725 0.107182 0.0535910 0.998563i \(-0.482933\pi\)
0.0535910 + 0.998563i \(0.482933\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −37.1755 + 37.1755i −1.35296 + 1.35296i
\(756\) 0 0
\(757\) −19.2288 19.2288i −0.698881 0.698881i 0.265288 0.964169i \(-0.414533\pi\)
−0.964169 + 0.265288i \(0.914533\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 6.43560i 0.233290i −0.993174 0.116645i \(-0.962786\pi\)
0.993174 0.116645i \(-0.0372140\pi\)
\(762\) 0 0
\(763\) 4.93725 + 4.93725i 0.178741 + 0.178741i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −41.3842 −1.49430
\(768\) 0 0
\(769\) 6.70850 0.241915 0.120957 0.992658i \(-0.461404\pi\)
0.120957 + 0.992658i \(0.461404\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −14.6509 14.6509i −0.526957 0.526957i 0.392707 0.919664i \(-0.371539\pi\)
−0.919664 + 0.392707i \(0.871539\pi\)
\(774\) 0 0
\(775\) 2.22876i 0.0800593i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −10.5914 10.5914i −0.379476 0.379476i
\(780\) 0 0
\(781\) −2.83399 + 2.83399i −0.101408 + 0.101408i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −6.43560 −0.229697
\(786\) 0 0
\(787\) −20.2288 + 20.2288i −0.721077 + 0.721077i −0.968825 0.247747i \(-0.920310\pi\)
0.247747 + 0.968825i \(0.420310\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 24.5830i 0.872968i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 14.6509 14.6509i 0.518962 0.518962i −0.398295 0.917257i \(-0.630398\pi\)
0.917257 + 0.398295i \(0.130398\pi\)
\(798\) 0 0
\(799\) −34.3320 −1.21458
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 6.13742 6.13742i 0.216585 0.216585i
\(804\) 0 0
\(805\) −67.7490 67.7490i −2.38784 2.38784i
\(806\) 0