Properties

Label 1152.2.k.d.289.4
Level $1152$
Weight $2$
Character 1152.289
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 289.4
Root \(-0.767178 - 1.18804i\) of defining polynomial
Character \(\chi\) \(=\) 1152.289
Dual form 1152.2.k.d.865.4

$q$-expansion

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

Character values

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

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{3}{4}\right)\)

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 0.0647108 0.997904i \(-0.479388\pi\)
0.997904 0.0647108i \(-0.0206125\pi\)
\(6\) 0 0
\(7\) 3.64575i 1.37796i −0.724778 0.688982i \(-0.758058\pi\)
0.724778 0.688982i \(-0.241942\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.841723 + 0.841723i −0.253789 + 0.253789i −0.822522 0.568733i \(-0.807434\pi\)
0.568733 + 0.822522i \(0.307434\pi\)
\(12\) 0 0
\(13\) 2.64575 + 2.64575i 0.733799 + 0.733799i 0.971370 0.237571i \(-0.0763512\pi\)
−0.237571 + 0.971370i \(0.576351\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.06871 0.744272 0.372136 0.928178i \(-0.378625\pi\)
0.372136 + 0.928178i \(0.378625\pi\)
\(18\) 0 0
\(19\) 1.64575 + 1.64575i 0.377561 + 0.377561i 0.870222 0.492660i \(-0.163976\pi\)
−0.492660 + 0.870222i \(0.663976\pi\)
\(20\) 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 \(-0.221011\pi\)
−0.639867 + 0.768486i \(0.721011\pi\)
\(30\) 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 −0.976999 0.213242i \(-0.931598\pi\)
0.213242 + 0.976999i \(0.431598\pi\)
\(38\) 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 −0.991451 0.130482i \(-0.958348\pi\)
0.130482 + 0.991451i \(0.458348\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −11.1878 −1.63190 −0.815951 0.578121i \(-0.803786\pi\)
−0.815951 + 0.578121i \(0.803786\pi\)
\(48\) 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 \(-0.572625\pi\)
0.974084 + 0.226185i \(0.0726253\pi\)
\(54\) 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 0.0183591 0.999831i \(-0.494156\pi\)
0.999831 0.0183591i \(-0.00584422\pi\)
\(60\) 0 0
\(61\) −4.64575 4.64575i −0.594828 0.594828i 0.344104 0.938932i \(-0.388183\pi\)
−0.938932 + 0.344104i \(0.888183\pi\)
\(62\) 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 0.907889 0.419211i \(-0.137693\pi\)
−0.419211 + 0.907889i \(0.637693\pi\)
\(68\) 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 \(0.140325\pi\)
−0.904392 + 0.426703i \(0.859675\pi\)
\(74\) 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 \(-0.229190\pi\)
−0.659401 + 0.751792i \(0.729190\pi\)
\(84\) 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 \(-0.696534\pi\)
0.815370 + 0.578941i \(0.196534\pi\)
\(102\) 0 0
\(103\) 1.06275i 0.104715i −0.998628 0.0523577i \(-0.983326\pi\)
0.998628 0.0523577i \(-0.0166736\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −9.50432 + 9.50432i −0.918817 + 0.918817i −0.996943 0.0781266i \(-0.975106\pi\)
0.0781266 + 0.996943i \(0.475106\pi\)
\(108\) 0 0
\(109\) 1.35425 + 1.35425i 0.129713 + 0.129713i 0.768983 0.639269i \(-0.220763\pi\)
−0.639269 + 0.768983i \(0.720763\pi\)
\(110\) 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 \(-0.283165\pi\)
−0.776814 + 0.629730i \(0.783165\pi\)
\(132\) 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 −0.370478 0.928841i \(-0.620806\pi\)
0.928841 + 0.370478i \(0.120806\pi\)
\(140\) 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 \(-0.819917\pi\)
0.536047 + 0.844188i \(0.319917\pi\)
\(150\) 0 0
\(151\) 15.6458i 1.27323i 0.771180 + 0.636617i \(0.219667\pi\)
−0.771180 + 0.636617i \(0.780333\pi\)
\(152\) 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 0.759079 0.650998i \(-0.225650\pi\)
−0.650998 + 0.759079i \(0.725650\pi\)
\(158\) 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 0.910722 0.413019i \(-0.135526\pi\)
−0.413019 + 0.910722i \(0.635526\pi\)
\(164\) 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 \(-0.344558\pi\)
−0.883115 + 0.469156i \(0.844558\pi\)
\(174\) 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 \(-0.0824804\pi\)
−0.256230 + 0.966616i \(0.582480\pi\)
\(180\) 0 0
\(181\) −4.64575 + 4.64575i −0.345316 + 0.345316i −0.858361 0.513045i \(-0.828517\pi\)
0.513045 + 0.858361i \(0.328517\pi\)
\(182\) 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 \(-0.367355\pi\)
0.404759 + 0.914423i \(0.367355\pi\)
\(192\) 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 \(-0.896415\pi\)
0.319709 + 0.947516i \(0.396415\pi\)
\(198\) 0 0
\(199\) 8.35425i 0.592217i −0.955154 0.296108i \(-0.904311\pi\)
0.955154 0.296108i \(-0.0956890\pi\)
\(200\) 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 0.896413 0.443220i \(-0.146164\pi\)
−0.443220 + 0.896413i \(0.646164\pi\)
\(212\) 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 0.972381 + 0.233399i \(0.0749848\pi\)
0.233399 + 0.972381i \(0.425015\pi\)
\(228\) 0 0
\(229\) −2.06275 + 2.06275i −0.136310 + 0.136310i −0.771970 0.635659i \(-0.780728\pi\)
0.635659 + 0.771970i \(0.280728\pi\)
\(230\) 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 \(-0.225769\pi\)
0.758835 + 0.651283i \(0.225769\pi\)
\(240\) 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 \(-0.949071\pi\)
0.159317 + 0.987228i \(0.449071\pi\)
\(252\) 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 \(-0.330957\pi\)
0.506452 + 0.862268i \(0.330957\pi\)
\(258\) 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 \(-0.369630\pi\)
−0.917292 + 0.398215i \(0.869630\pi\)
\(270\) 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 −0.270115 + 0.962828i \(0.587062\pi\)
−0.962828 + 0.270115i \(0.912938\pi\)
\(278\) 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 0.110803 0.993842i \(-0.464658\pi\)
0.993842 0.110803i \(-0.0353421\pi\)
\(284\) 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 \(-0.961871\pi\)
0.119498 + 0.992834i \(0.461871\pi\)
\(294\) 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 −0.988183 0.153277i \(-0.951017\pi\)
−0.153277 0.988183i \(-0.548983\pi\)
\(308\) 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 \(-0.988379\pi\)
0.999334 0.0365000i \(-0.0116209\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −2.37608 2.37608i −0.133454 0.133454i 0.637224 0.770678i \(-0.280082\pi\)
−0.770678 + 0.637224i \(0.780082\pi\)
\(318\) 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 −0.891918 0.452198i \(-0.850640\pi\)
0.452198 + 0.891918i \(0.350640\pi\)
\(332\) 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 \(-0.760172\pi\)
0.684153 + 0.729339i \(0.260172\pi\)
\(348\) 0 0
\(349\) 23.2288 + 23.2288i 1.24341 + 1.24341i 0.958579 + 0.284828i \(0.0919366\pi\)
0.284828 + 0.958579i \(0.408063\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −28.5129 −1.51759 −0.758796 0.651329i \(-0.774212\pi\)
−0.758796 + 0.651329i \(0.774212\pi\)
\(354\) 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 −0.597790 0.801653i \(-0.703954\pi\)
0.801653 + 0.597790i \(0.203954\pi\)
\(374\) 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 −0.263676 0.964611i \(-0.584935\pi\)
0.964611 + 0.263676i \(0.0849350\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −11.1878 −0.571668 −0.285834 0.958279i \(-0.592271\pi\)
−0.285834 + 0.958279i \(0.592271\pi\)
\(384\) 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 \(-0.664215\pi\)
0.869850 + 0.493316i \(0.164215\pi\)
\(390\) 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 −0.973748 0.227628i \(-0.926903\pi\)
−0.227628 0.973748i \(-0.573097\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 9.20614 0.459733 0.229866 0.973222i \(-0.426171\pi\)
0.229866 + 0.973222i \(0.426171\pi\)
\(402\) 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 \(0.139516\pi\)
−0.905474 + 0.424402i \(0.860484\pi\)
\(410\) 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 \(-0.100316\pi\)
−0.309962 + 0.950749i \(0.600316\pi\)
\(420\) 0 0
\(421\) −16.6458 + 16.6458i −0.811264 + 0.811264i −0.984823 0.173559i \(-0.944473\pi\)
0.173559 + 0.984823i \(0.444473\pi\)
\(422\) 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 \(-0.508335\pi\)
−0.0261816 + 0.999657i \(0.508335\pi\)
\(432\) 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 \(0.353090\pi\)
−0.445319 + 0.895372i \(0.646910\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −23.2172 + 23.2172i −1.10308 + 1.10308i −0.109048 + 0.994036i \(0.534780\pi\)
−0.994036 + 0.109048i \(0.965220\pi\)
\(444\) 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 \(-0.150678\pi\)
0.890037 + 0.455889i \(0.150678\pi\)
\(450\) 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 \(0.213578\pi\)
−0.783216 + 0.621750i \(0.786422\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 8.81168 + 8.81168i 0.410401 + 0.410401i 0.881878 0.471477i \(-0.156279\pi\)
−0.471477 + 0.881878i \(0.656279\pi\)
\(462\) 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 \(-0.372049\pi\)
−0.920293 + 0.391231i \(0.872049\pi\)
\(468\) 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 \(-0.409524\pi\)
0.280426 + 0.959876i \(0.409524\pi\)
\(480\) 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 \(0.135523\pi\)
−0.910726 + 0.413012i \(0.864477\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 13.9583 13.9583i 0.629929 0.629929i −0.318121 0.948050i \(-0.603052\pi\)
0.948050 + 0.318121i \(0.103052\pi\)
\(492\) 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 0.429369 0.903129i \(-0.358736\pi\)
−0.903129 + 0.429369i \(0.858736\pi\)
\(500\) 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 0.972096 + 0.234585i \(0.0753731\pi\)
0.234585 + 0.972096i \(0.424627\pi\)
\(510\) 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 −0.770891 0.636967i \(-0.780189\pi\)
0.636967 + 0.770891i \(0.280189\pi\)
\(524\) 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 0.499817 0.866131i \(-0.333401\pi\)
−0.866131 + 0.499817i \(0.833401\pi\)
\(542\) 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 0.686786 0.726860i \(-0.259021\pi\)
−0.726860 + 0.686786i \(0.759021\pi\)
\(548\) 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 −0.998749 0.0500103i \(-0.984075\pi\)
−0.0500103 0.998749i \(-0.515925\pi\)
\(558\) 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 \(-0.413450\pi\)
−0.963261 + 0.268566i \(0.913450\pi\)
\(564\) 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 −0.618438 0.785833i \(-0.712234\pi\)
0.785833 + 0.618438i \(0.212234\pi\)
\(572\) 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 \(-0.562804\pi\)
0.980598 + 0.196028i \(0.0628044\pi\)
\(588\) 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 \(-0.418901\pi\)
0.252034 + 0.967718i \(0.418901\pi\)
\(594\) 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 \(0.0475132\pi\)
−0.988880 + 0.148713i \(0.952487\pi\)
\(602\) 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 −0.991627 0.129137i \(-0.958779\pi\)
0.129137 + 0.991627i \(0.458779\pi\)
\(614\) 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 −0.139490 0.990224i \(-0.544546\pi\)
0.990224 + 0.139490i \(0.0445462\pi\)
\(620\) 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 \(-0.881808\pi\)
0.931852 0.362838i \(-0.118192\pi\)
\(632\) 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 \(-0.441804\pi\)
0.181810 + 0.983334i \(0.441804\pi\)
\(642\) 0 0
\(643\) 1.64575 + 1.64575i 0.0649021 + 0.0649021i 0.738813 0.673911i \(-0.235387\pi\)
−0.673911 + 0.738813i \(0.735387\pi\)
\(644\) 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 \(-0.315881\pi\)
−0.837324 + 0.546706i \(0.815881\pi\)
\(654\) 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 \(-0.480591\pi\)
−0.998142 + 0.0609372i \(0.980591\pi\)
\(660\) 0 0
\(661\) 29.2288 29.2288i 1.13687 1.13687i 0.147858 0.989009i \(-0.452762\pi\)
0.989009 0.147858i \(-0.0472380\pi\)
\(662\) 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 \(-0.799883\pi\)
0.588081 + 0.808802i \(0.299883\pi\)
\(678\) 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 −0.481769 + 0.876298i \(0.660006\pi\)
−0.876298 + 0.481769i \(0.839994\pi\)
\(684\) 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 0.737717 0.675110i \(-0.235904\pi\)
−0.675110 + 0.737717i \(0.735904\pi\)
\(692\) 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 \(-0.396514\pi\)
−0.947616 + 0.319413i \(0.896514\pi\)
\(702\) 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 0.411744 0.911299i \(-0.364920\pi\)
0.911299 0.411744i \(-0.135080\pi\)
\(710\) 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 \(-0.560313\pi\)
−0.188346 + 0.982103i \(0.560313\pi\)
\(720\) 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 \(-0.772095\pi\)
0.754448 0.656360i \(-0.227905\pi\)
\(728\) 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 0.829758 0.558123i \(-0.188478\pi\)
−0.558123 + 0.829758i \(0.688478\pi\)
\(734\) 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 0.967620 + 0.252411i \(0.0812236\pi\)
0.252411 + 0.967620i \(0.418776\pi\)
\(740\) 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 −0.964169 0.265288i \(-0.914533\pi\)
0.265288 + 0.964169i \(0.414533\pi\)
\(758\) 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 \(-0.628461\pi\)
0.919664 + 0.392707i \(0.128461\pi\)
\(774\) 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 0.247747 0.968825i \(-0.420310\pi\)
−0.968825 + 0.247747i \(0.920310\pi\)
\(788\) 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 \(-0.369602\pi\)
−0.917257 + 0.398295i \(0.869602\pi\)
\(798\) 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 0