Properties

Label 1089.2.d.g.1088.12
Level $1089$
Weight $2$
Character 1089.1088
Analytic conductor $8.696$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1089.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.69570878012\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 2 x^{14} - 16 x^{12} - 72 x^{10} + 26 x^{8} + 360 x^{6} + 725 x^{4} + 1000 x^{2} + 625\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 99)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1088.12
Root \(-1.90184 - 0.0324487i\) of defining polynomial
Character \(\chi\) \(=\) 1089.1088
Dual form 1089.2.d.g.1088.11

$q$-expansion

\(f(q)\) \(=\) \(q+0.688291 q^{2} -1.52626 q^{4} +0.0401087i q^{5} +0.246848i q^{7} -2.42709 q^{8} +O(q^{10})\) \(q+0.688291 q^{2} -1.52626 q^{4} +0.0401087i q^{5} +0.246848i q^{7} -2.42709 q^{8} +0.0276065i q^{10} -2.30152i q^{13} +0.169903i q^{14} +1.38197 q^{16} +4.27621 q^{17} +6.20004i q^{19} -0.0612162i q^{20} +6.79984i q^{23} +4.99839 q^{25} -1.58412i q^{26} -0.376753i q^{28} +5.59574 q^{29} +4.79738 q^{31} +5.80538 q^{32} +2.94328 q^{34} -0.00990077 q^{35} -4.03084 q^{37} +4.26743i q^{38} -0.0973475i q^{40} -9.60293 q^{41} -1.03166i q^{43} +4.68027i q^{46} +11.1262i q^{47} +6.93907 q^{49} +3.44035 q^{50} +3.51271i q^{52} +8.96776i q^{53} -0.599123i q^{56} +3.85150 q^{58} +2.78832i q^{59} -8.48450i q^{61} +3.30199 q^{62} +1.23186 q^{64} +0.0923111 q^{65} +7.94588 q^{67} -6.52659 q^{68} -0.00681461 q^{70} +3.32850i q^{71} -11.8537i q^{73} -2.77439 q^{74} -9.46284i q^{76} -3.01100i q^{79} +0.0554289i q^{80} -6.60961 q^{82} +5.29380 q^{83} +0.171513i q^{85} -0.710085i q^{86} +8.54422i q^{89} +0.568126 q^{91} -10.3783i q^{92} +7.65807i q^{94} -0.248676 q^{95} -3.02824 q^{97} +4.77610 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{4} + O(q^{10}) \) \( 16 q + 16 q^{4} + 40 q^{16} - 32 q^{25} + 16 q^{31} + 40 q^{34} + 8 q^{37} + 16 q^{49} + 32 q^{58} - 104 q^{64} + 96 q^{67} - 64 q^{70} + 88 q^{82} + 48 q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(848\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.688291 0.486695 0.243348 0.969939i \(-0.421754\pi\)
0.243348 + 0.969939i \(0.421754\pi\)
\(3\) 0 0
\(4\) −1.52626 −0.763128
\(5\) 0.0401087i 0.0179372i 0.999960 + 0.00896859i \(0.00285483\pi\)
−0.999960 + 0.00896859i \(0.997145\pi\)
\(6\) 0 0
\(7\) 0.246848i 0.0932998i 0.998911 + 0.0466499i \(0.0148545\pi\)
−0.998911 + 0.0466499i \(0.985145\pi\)
\(8\) −2.42709 −0.858106
\(9\) 0 0
\(10\) 0.0276065i 0.00872994i
\(11\) 0 0
\(12\) 0 0
\(13\) − 2.30152i − 0.638327i −0.947700 0.319164i \(-0.896598\pi\)
0.947700 0.319164i \(-0.103402\pi\)
\(14\) 0.169903i 0.0454086i
\(15\) 0 0
\(16\) 1.38197 0.345492
\(17\) 4.27621 1.03713 0.518567 0.855037i \(-0.326466\pi\)
0.518567 + 0.855037i \(0.326466\pi\)
\(18\) 0 0
\(19\) 6.20004i 1.42239i 0.702997 + 0.711193i \(0.251845\pi\)
−0.702997 + 0.711193i \(0.748155\pi\)
\(20\) − 0.0612162i − 0.0136884i
\(21\) 0 0
\(22\) 0 0
\(23\) 6.79984i 1.41786i 0.705277 + 0.708932i \(0.250823\pi\)
−0.705277 + 0.708932i \(0.749177\pi\)
\(24\) 0 0
\(25\) 4.99839 0.999678
\(26\) − 1.58412i − 0.310671i
\(27\) 0 0
\(28\) − 0.376753i − 0.0711997i
\(29\) 5.59574 1.03910 0.519552 0.854439i \(-0.326099\pi\)
0.519552 + 0.854439i \(0.326099\pi\)
\(30\) 0 0
\(31\) 4.79738 0.861635 0.430817 0.902439i \(-0.358225\pi\)
0.430817 + 0.902439i \(0.358225\pi\)
\(32\) 5.80538 1.02626
\(33\) 0 0
\(34\) 2.94328 0.504768
\(35\) −0.00990077 −0.00167354
\(36\) 0 0
\(37\) −4.03084 −0.662667 −0.331333 0.943514i \(-0.607498\pi\)
−0.331333 + 0.943514i \(0.607498\pi\)
\(38\) 4.26743i 0.692268i
\(39\) 0 0
\(40\) − 0.0973475i − 0.0153920i
\(41\) −9.60293 −1.49973 −0.749863 0.661593i \(-0.769881\pi\)
−0.749863 + 0.661593i \(0.769881\pi\)
\(42\) 0 0
\(43\) − 1.03166i − 0.157327i −0.996901 0.0786636i \(-0.974935\pi\)
0.996901 0.0786636i \(-0.0250653\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 4.68027i 0.690068i
\(47\) 11.1262i 1.62292i 0.584405 + 0.811462i \(0.301328\pi\)
−0.584405 + 0.811462i \(0.698672\pi\)
\(48\) 0 0
\(49\) 6.93907 0.991295
\(50\) 3.44035 0.486539
\(51\) 0 0
\(52\) 3.51271i 0.487125i
\(53\) 8.96776i 1.23182i 0.787818 + 0.615908i \(0.211211\pi\)
−0.787818 + 0.615908i \(0.788789\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) − 0.599123i − 0.0800611i
\(57\) 0 0
\(58\) 3.85150 0.505727
\(59\) 2.78832i 0.363008i 0.983390 + 0.181504i \(0.0580966\pi\)
−0.983390 + 0.181504i \(0.941903\pi\)
\(60\) 0 0
\(61\) − 8.48450i − 1.08633i −0.839627 0.543164i \(-0.817226\pi\)
0.839627 0.543164i \(-0.182774\pi\)
\(62\) 3.30199 0.419354
\(63\) 0 0
\(64\) 1.23186 0.153982
\(65\) 0.0923111 0.0114498
\(66\) 0 0
\(67\) 7.94588 0.970744 0.485372 0.874308i \(-0.338684\pi\)
0.485372 + 0.874308i \(0.338684\pi\)
\(68\) −6.52659 −0.791465
\(69\) 0 0
\(70\) −0.00681461 −0.000814502 0
\(71\) 3.32850i 0.395020i 0.980301 + 0.197510i \(0.0632855\pi\)
−0.980301 + 0.197510i \(0.936714\pi\)
\(72\) 0 0
\(73\) − 11.8537i − 1.38737i −0.720278 0.693685i \(-0.755986\pi\)
0.720278 0.693685i \(-0.244014\pi\)
\(74\) −2.77439 −0.322517
\(75\) 0 0
\(76\) − 9.46284i − 1.08546i
\(77\) 0 0
\(78\) 0 0
\(79\) − 3.01100i − 0.338764i −0.985550 0.169382i \(-0.945823\pi\)
0.985550 0.169382i \(-0.0541772\pi\)
\(80\) 0.0554289i 0.00619714i
\(81\) 0 0
\(82\) −6.60961 −0.729909
\(83\) 5.29380 0.581070 0.290535 0.956864i \(-0.406167\pi\)
0.290535 + 0.956864i \(0.406167\pi\)
\(84\) 0 0
\(85\) 0.171513i 0.0186032i
\(86\) − 0.710085i − 0.0765704i
\(87\) 0 0
\(88\) 0 0
\(89\) 8.54422i 0.905686i 0.891590 + 0.452843i \(0.149590\pi\)
−0.891590 + 0.452843i \(0.850410\pi\)
\(90\) 0 0
\(91\) 0.568126 0.0595558
\(92\) − 10.3783i − 1.08201i
\(93\) 0 0
\(94\) 7.65807i 0.789870i
\(95\) −0.248676 −0.0255136
\(96\) 0 0
\(97\) −3.02824 −0.307471 −0.153736 0.988112i \(-0.549130\pi\)
−0.153736 + 0.988112i \(0.549130\pi\)
\(98\) 4.77610 0.482459
\(99\) 0 0
\(100\) −7.62882 −0.762882
\(101\) 6.95198 0.691747 0.345874 0.938281i \(-0.387583\pi\)
0.345874 + 0.938281i \(0.387583\pi\)
\(102\) 0 0
\(103\) 13.1194 1.29269 0.646347 0.763044i \(-0.276296\pi\)
0.646347 + 0.763044i \(0.276296\pi\)
\(104\) 5.58600i 0.547752i
\(105\) 0 0
\(106\) 6.17243i 0.599519i
\(107\) −1.36847 −0.132295 −0.0661476 0.997810i \(-0.521071\pi\)
−0.0661476 + 0.997810i \(0.521071\pi\)
\(108\) 0 0
\(109\) 7.34454i 0.703480i 0.936098 + 0.351740i \(0.114410\pi\)
−0.936098 + 0.351740i \(0.885590\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0.341136i 0.0322343i
\(113\) 0.528542i 0.0497211i 0.999691 + 0.0248605i \(0.00791417\pi\)
−0.999691 + 0.0248605i \(0.992086\pi\)
\(114\) 0 0
\(115\) −0.272733 −0.0254325
\(116\) −8.54053 −0.792968
\(117\) 0 0
\(118\) 1.91917i 0.176674i
\(119\) 1.05557i 0.0967644i
\(120\) 0 0
\(121\) 0 0
\(122\) − 5.83980i − 0.528711i
\(123\) 0 0
\(124\) −7.32203 −0.657537
\(125\) 0.401023i 0.0358686i
\(126\) 0 0
\(127\) − 12.7719i − 1.13332i −0.823952 0.566660i \(-0.808235\pi\)
0.823952 0.566660i \(-0.191765\pi\)
\(128\) −10.7629 −0.951313
\(129\) 0 0
\(130\) 0.0635369 0.00557256
\(131\) −18.1534 −1.58607 −0.793033 0.609178i \(-0.791499\pi\)
−0.793033 + 0.609178i \(0.791499\pi\)
\(132\) 0 0
\(133\) −1.53047 −0.132708
\(134\) 5.46908 0.472456
\(135\) 0 0
\(136\) −10.3787 −0.889970
\(137\) − 16.0730i − 1.37321i −0.727032 0.686604i \(-0.759101\pi\)
0.727032 0.686604i \(-0.240899\pi\)
\(138\) 0 0
\(139\) 3.85730i 0.327172i 0.986529 + 0.163586i \(0.0523061\pi\)
−0.986529 + 0.163586i \(0.947694\pi\)
\(140\) 0.0151111 0.00127712
\(141\) 0 0
\(142\) 2.29098i 0.192255i
\(143\) 0 0
\(144\) 0 0
\(145\) 0.224438i 0.0186386i
\(146\) − 8.15879i − 0.675227i
\(147\) 0 0
\(148\) 6.15210 0.505699
\(149\) −10.2700 −0.841350 −0.420675 0.907211i \(-0.638207\pi\)
−0.420675 + 0.907211i \(0.638207\pi\)
\(150\) 0 0
\(151\) − 8.77042i − 0.713726i −0.934157 0.356863i \(-0.883846\pi\)
0.934157 0.356863i \(-0.116154\pi\)
\(152\) − 15.0480i − 1.22056i
\(153\) 0 0
\(154\) 0 0
\(155\) 0.192417i 0.0154553i
\(156\) 0 0
\(157\) −13.0308 −1.03997 −0.519987 0.854174i \(-0.674063\pi\)
−0.519987 + 0.854174i \(0.674063\pi\)
\(158\) − 2.07245i − 0.164875i
\(159\) 0 0
\(160\) 0.232846i 0.0184081i
\(161\) −1.67853 −0.132287
\(162\) 0 0
\(163\) −9.17774 −0.718856 −0.359428 0.933173i \(-0.617028\pi\)
−0.359428 + 0.933173i \(0.617028\pi\)
\(164\) 14.6565 1.14448
\(165\) 0 0
\(166\) 3.64367 0.282804
\(167\) 7.60934 0.588828 0.294414 0.955678i \(-0.404876\pi\)
0.294414 + 0.955678i \(0.404876\pi\)
\(168\) 0 0
\(169\) 7.70300 0.592538
\(170\) 0.118051i 0.00905411i
\(171\) 0 0
\(172\) 1.57458i 0.120061i
\(173\) 16.6204 1.26363 0.631813 0.775121i \(-0.282311\pi\)
0.631813 + 0.775121i \(0.282311\pi\)
\(174\) 0 0
\(175\) 1.23384i 0.0932698i
\(176\) 0 0
\(177\) 0 0
\(178\) 5.88091i 0.440793i
\(179\) − 7.04979i − 0.526926i −0.964669 0.263463i \(-0.915135\pi\)
0.964669 0.263463i \(-0.0848647\pi\)
\(180\) 0 0
\(181\) −21.1236 −1.57011 −0.785053 0.619429i \(-0.787364\pi\)
−0.785053 + 0.619429i \(0.787364\pi\)
\(182\) 0.391036 0.0289855
\(183\) 0 0
\(184\) − 16.5038i − 1.21668i
\(185\) − 0.161672i − 0.0118864i
\(186\) 0 0
\(187\) 0 0
\(188\) − 16.9814i − 1.23850i
\(189\) 0 0
\(190\) −0.171161 −0.0124173
\(191\) 13.6683i 0.989000i 0.869178 + 0.494500i \(0.164649\pi\)
−0.869178 + 0.494500i \(0.835351\pi\)
\(192\) 0 0
\(193\) 25.0349i 1.80205i 0.433767 + 0.901025i \(0.357184\pi\)
−0.433767 + 0.901025i \(0.642816\pi\)
\(194\) −2.08431 −0.149645
\(195\) 0 0
\(196\) −10.5908 −0.756485
\(197\) −21.0442 −1.49934 −0.749668 0.661814i \(-0.769787\pi\)
−0.749668 + 0.661814i \(0.769787\pi\)
\(198\) 0 0
\(199\) −10.3709 −0.735176 −0.367588 0.929989i \(-0.619816\pi\)
−0.367588 + 0.929989i \(0.619816\pi\)
\(200\) −12.1315 −0.857830
\(201\) 0 0
\(202\) 4.78498 0.336670
\(203\) 1.38130i 0.0969481i
\(204\) 0 0
\(205\) − 0.385161i − 0.0269008i
\(206\) 9.02997 0.629148
\(207\) 0 0
\(208\) − 3.18062i − 0.220537i
\(209\) 0 0
\(210\) 0 0
\(211\) − 18.7436i − 1.29036i −0.764029 0.645182i \(-0.776782\pi\)
0.764029 0.645182i \(-0.223218\pi\)
\(212\) − 13.6871i − 0.940033i
\(213\) 0 0
\(214\) −0.941908 −0.0643875
\(215\) 0.0413787 0.00282201
\(216\) 0 0
\(217\) 1.18422i 0.0803904i
\(218\) 5.05518i 0.342380i
\(219\) 0 0
\(220\) 0 0
\(221\) − 9.84179i − 0.662031i
\(222\) 0 0
\(223\) 9.66894 0.647480 0.323740 0.946146i \(-0.395060\pi\)
0.323740 + 0.946146i \(0.395060\pi\)
\(224\) 1.43305i 0.0957494i
\(225\) 0 0
\(226\) 0.363791i 0.0241990i
\(227\) −11.7162 −0.777630 −0.388815 0.921316i \(-0.627116\pi\)
−0.388815 + 0.921316i \(0.627116\pi\)
\(228\) 0 0
\(229\) 20.4802 1.35337 0.676684 0.736274i \(-0.263416\pi\)
0.676684 + 0.736274i \(0.263416\pi\)
\(230\) −0.187720 −0.0123779
\(231\) 0 0
\(232\) −13.5814 −0.891661
\(233\) −16.1429 −1.05755 −0.528777 0.848761i \(-0.677349\pi\)
−0.528777 + 0.848761i \(0.677349\pi\)
\(234\) 0 0
\(235\) −0.446258 −0.0291107
\(236\) − 4.25569i − 0.277022i
\(237\) 0 0
\(238\) 0.726543i 0.0470948i
\(239\) 28.3875 1.83624 0.918118 0.396306i \(-0.129708\pi\)
0.918118 + 0.396306i \(0.129708\pi\)
\(240\) 0 0
\(241\) 14.3654i 0.925357i 0.886526 + 0.462679i \(0.153112\pi\)
−0.886526 + 0.462679i \(0.846888\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 12.9495i 0.829007i
\(245\) 0.278317i 0.0177810i
\(246\) 0 0
\(247\) 14.2695 0.907948
\(248\) −11.6437 −0.739374
\(249\) 0 0
\(250\) 0.276021i 0.0174571i
\(251\) − 6.94278i − 0.438225i −0.975700 0.219112i \(-0.929684\pi\)
0.975700 0.219112i \(-0.0703161\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) − 8.79077i − 0.551582i
\(255\) 0 0
\(256\) −9.87170 −0.616981
\(257\) 10.4183i 0.649875i 0.945735 + 0.324938i \(0.105343\pi\)
−0.945735 + 0.324938i \(0.894657\pi\)
\(258\) 0 0
\(259\) − 0.995006i − 0.0618267i
\(260\) −0.140890 −0.00873765
\(261\) 0 0
\(262\) −12.4948 −0.771931
\(263\) 4.26110 0.262751 0.131375 0.991333i \(-0.458061\pi\)
0.131375 + 0.991333i \(0.458061\pi\)
\(264\) 0 0
\(265\) −0.359686 −0.0220953
\(266\) −1.05341 −0.0645885
\(267\) 0 0
\(268\) −12.1274 −0.740801
\(269\) − 11.0638i − 0.674569i −0.941403 0.337285i \(-0.890492\pi\)
0.941403 0.337285i \(-0.109508\pi\)
\(270\) 0 0
\(271\) 15.8957i 0.965596i 0.875732 + 0.482798i \(0.160379\pi\)
−0.875732 + 0.482798i \(0.839621\pi\)
\(272\) 5.90958 0.358321
\(273\) 0 0
\(274\) − 11.0629i − 0.668334i
\(275\) 0 0
\(276\) 0 0
\(277\) 11.3410i 0.681417i 0.940169 + 0.340709i \(0.110667\pi\)
−0.940169 + 0.340709i \(0.889333\pi\)
\(278\) 2.65495i 0.159233i
\(279\) 0 0
\(280\) 0.0240301 0.00143607
\(281\) −1.56030 −0.0930794 −0.0465397 0.998916i \(-0.514819\pi\)
−0.0465397 + 0.998916i \(0.514819\pi\)
\(282\) 0 0
\(283\) − 26.5654i − 1.57915i −0.613653 0.789576i \(-0.710301\pi\)
0.613653 0.789576i \(-0.289699\pi\)
\(284\) − 5.08014i − 0.301451i
\(285\) 0 0
\(286\) 0 0
\(287\) − 2.37046i − 0.139924i
\(288\) 0 0
\(289\) 1.28598 0.0756456
\(290\) 0.154479i 0.00907131i
\(291\) 0 0
\(292\) 18.0918i 1.05874i
\(293\) 26.3345 1.53848 0.769240 0.638960i \(-0.220635\pi\)
0.769240 + 0.638960i \(0.220635\pi\)
\(294\) 0 0
\(295\) −0.111836 −0.00651134
\(296\) 9.78322 0.568638
\(297\) 0 0
\(298\) −7.06874 −0.409481
\(299\) 15.6500 0.905062
\(300\) 0 0
\(301\) 0.254664 0.0146786
\(302\) − 6.03660i − 0.347367i
\(303\) 0 0
\(304\) 8.56824i 0.491422i
\(305\) 0.340302 0.0194857
\(306\) 0 0
\(307\) 26.0083i 1.48437i 0.670195 + 0.742185i \(0.266211\pi\)
−0.670195 + 0.742185i \(0.733789\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0.132439i 0.00752202i
\(311\) − 13.3248i − 0.755578i −0.925892 0.377789i \(-0.876684\pi\)
0.925892 0.377789i \(-0.123316\pi\)
\(312\) 0 0
\(313\) −11.2483 −0.635792 −0.317896 0.948126i \(-0.602976\pi\)
−0.317896 + 0.948126i \(0.602976\pi\)
\(314\) −8.96901 −0.506151
\(315\) 0 0
\(316\) 4.59556i 0.258520i
\(317\) − 10.9526i − 0.615161i −0.951522 0.307580i \(-0.900481\pi\)
0.951522 0.307580i \(-0.0995193\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0.0494082i 0.00276200i
\(321\) 0 0
\(322\) −1.15532 −0.0643832
\(323\) 26.5127i 1.47520i
\(324\) 0 0
\(325\) − 11.5039i − 0.638122i
\(326\) −6.31695 −0.349864
\(327\) 0 0
\(328\) 23.3072 1.28692
\(329\) −2.74648 −0.151419
\(330\) 0 0
\(331\) 4.23285 0.232659 0.116329 0.993211i \(-0.462887\pi\)
0.116329 + 0.993211i \(0.462887\pi\)
\(332\) −8.07968 −0.443430
\(333\) 0 0
\(334\) 5.23744 0.286580
\(335\) 0.318699i 0.0174124i
\(336\) 0 0
\(337\) − 2.66238i − 0.145029i −0.997367 0.0725146i \(-0.976898\pi\)
0.997367 0.0725146i \(-0.0231024\pi\)
\(338\) 5.30191 0.288386
\(339\) 0 0
\(340\) − 0.261773i − 0.0141966i
\(341\) 0 0
\(342\) 0 0
\(343\) 3.44083i 0.185787i
\(344\) 2.50394i 0.135003i
\(345\) 0 0
\(346\) 11.4397 0.615001
\(347\) 16.6032 0.891304 0.445652 0.895206i \(-0.352972\pi\)
0.445652 + 0.895206i \(0.352972\pi\)
\(348\) 0 0
\(349\) 13.7748i 0.737346i 0.929559 + 0.368673i \(0.120188\pi\)
−0.929559 + 0.368673i \(0.879812\pi\)
\(350\) 0.849243i 0.0453940i
\(351\) 0 0
\(352\) 0 0
\(353\) − 11.0249i − 0.586795i −0.955990 0.293398i \(-0.905214\pi\)
0.955990 0.293398i \(-0.0947860\pi\)
\(354\) 0 0
\(355\) −0.133502 −0.00708555
\(356\) − 13.0407i − 0.691154i
\(357\) 0 0
\(358\) − 4.85231i − 0.256452i
\(359\) −19.0142 −1.00353 −0.501766 0.865003i \(-0.667316\pi\)
−0.501766 + 0.865003i \(0.667316\pi\)
\(360\) 0 0
\(361\) −19.4404 −1.02318
\(362\) −14.5392 −0.764163
\(363\) 0 0
\(364\) −0.867106 −0.0454487
\(365\) 0.475437 0.0248855
\(366\) 0 0
\(367\) −5.69718 −0.297390 −0.148695 0.988883i \(-0.547507\pi\)
−0.148695 + 0.988883i \(0.547507\pi\)
\(368\) 9.39715i 0.489860i
\(369\) 0 0
\(370\) − 0.111277i − 0.00578504i
\(371\) −2.21367 −0.114928
\(372\) 0 0
\(373\) − 22.1594i − 1.14737i −0.819076 0.573684i \(-0.805514\pi\)
0.819076 0.573684i \(-0.194486\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) − 27.0043i − 1.39264i
\(377\) − 12.8787i − 0.663288i
\(378\) 0 0
\(379\) −0.381352 −0.0195887 −0.00979436 0.999952i \(-0.503118\pi\)
−0.00979436 + 0.999952i \(0.503118\pi\)
\(380\) 0.379542 0.0194701
\(381\) 0 0
\(382\) 9.40774i 0.481342i
\(383\) 12.6644i 0.647122i 0.946207 + 0.323561i \(0.104880\pi\)
−0.946207 + 0.323561i \(0.895120\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 17.2313i 0.877049i
\(387\) 0 0
\(388\) 4.62187 0.234640
\(389\) − 23.7986i − 1.20664i −0.797500 0.603318i \(-0.793845\pi\)
0.797500 0.603318i \(-0.206155\pi\)
\(390\) 0 0
\(391\) 29.0775i 1.47051i
\(392\) −16.8417 −0.850636
\(393\) 0 0
\(394\) −14.4845 −0.729720
\(395\) 0.120768 0.00607647
\(396\) 0 0
\(397\) 5.00497 0.251192 0.125596 0.992081i \(-0.459916\pi\)
0.125596 + 0.992081i \(0.459916\pi\)
\(398\) −7.13823 −0.357807
\(399\) 0 0
\(400\) 6.90761 0.345380
\(401\) − 4.24415i − 0.211943i −0.994369 0.105971i \(-0.966205\pi\)
0.994369 0.105971i \(-0.0337952\pi\)
\(402\) 0 0
\(403\) − 11.0413i − 0.550005i
\(404\) −10.6105 −0.527892
\(405\) 0 0
\(406\) 0.950735i 0.0471842i
\(407\) 0 0
\(408\) 0 0
\(409\) − 11.8056i − 0.583748i −0.956457 0.291874i \(-0.905721\pi\)
0.956457 0.291874i \(-0.0942788\pi\)
\(410\) − 0.265103i − 0.0130925i
\(411\) 0 0
\(412\) −20.0236 −0.986490
\(413\) −0.688291 −0.0338686
\(414\) 0 0
\(415\) 0.212328i 0.0104227i
\(416\) − 13.3612i − 0.655087i
\(417\) 0 0
\(418\) 0 0
\(419\) 20.9795i 1.02492i 0.858712 + 0.512459i \(0.171265\pi\)
−0.858712 + 0.512459i \(0.828735\pi\)
\(420\) 0 0
\(421\) −2.77978 −0.135478 −0.0677390 0.997703i \(-0.521579\pi\)
−0.0677390 + 0.997703i \(0.521579\pi\)
\(422\) − 12.9011i − 0.628014i
\(423\) 0 0
\(424\) − 21.7656i − 1.05703i
\(425\) 21.3742 1.03680
\(426\) 0 0
\(427\) 2.09438 0.101354
\(428\) 2.08864 0.100958
\(429\) 0 0
\(430\) 0.0284806 0.00137346
\(431\) 27.9292 1.34530 0.672652 0.739959i \(-0.265155\pi\)
0.672652 + 0.739959i \(0.265155\pi\)
\(432\) 0 0
\(433\) −9.16809 −0.440590 −0.220295 0.975433i \(-0.570702\pi\)
−0.220295 + 0.975433i \(0.570702\pi\)
\(434\) 0.815091i 0.0391256i
\(435\) 0 0
\(436\) − 11.2096i − 0.536845i
\(437\) −42.1592 −2.01675
\(438\) 0 0
\(439\) 2.93111i 0.139894i 0.997551 + 0.0699472i \(0.0222831\pi\)
−0.997551 + 0.0699472i \(0.977717\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 6.77402i − 0.322207i
\(443\) 35.8414i 1.70288i 0.524455 + 0.851438i \(0.324269\pi\)
−0.524455 + 0.851438i \(0.675731\pi\)
\(444\) 0 0
\(445\) −0.342698 −0.0162454
\(446\) 6.65504 0.315125
\(447\) 0 0
\(448\) 0.304081i 0.0143665i
\(449\) 14.2141i 0.670806i 0.942075 + 0.335403i \(0.108872\pi\)
−0.942075 + 0.335403i \(0.891128\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) − 0.806691i − 0.0379435i
\(453\) 0 0
\(454\) −8.06414 −0.378469
\(455\) 0.0227868i 0.00106826i
\(456\) 0 0
\(457\) − 20.3993i − 0.954238i −0.878839 0.477119i \(-0.841681\pi\)
0.878839 0.477119i \(-0.158319\pi\)
\(458\) 14.0963 0.658678
\(459\) 0 0
\(460\) 0.416260 0.0194082
\(461\) 0.770354 0.0358790 0.0179395 0.999839i \(-0.494289\pi\)
0.0179395 + 0.999839i \(0.494289\pi\)
\(462\) 0 0
\(463\) 37.7948 1.75647 0.878236 0.478228i \(-0.158721\pi\)
0.878236 + 0.478228i \(0.158721\pi\)
\(464\) 7.73312 0.359001
\(465\) 0 0
\(466\) −11.1110 −0.514707
\(467\) 17.6026i 0.814551i 0.913305 + 0.407276i \(0.133521\pi\)
−0.913305 + 0.407276i \(0.866479\pi\)
\(468\) 0 0
\(469\) 1.96143i 0.0905702i
\(470\) −0.307156 −0.0141680
\(471\) 0 0
\(472\) − 6.76750i − 0.311499i
\(473\) 0 0
\(474\) 0 0
\(475\) 30.9902i 1.42193i
\(476\) − 1.61108i − 0.0738436i
\(477\) 0 0
\(478\) 19.5389 0.893688
\(479\) −2.40586 −0.109927 −0.0549633 0.998488i \(-0.517504\pi\)
−0.0549633 + 0.998488i \(0.517504\pi\)
\(480\) 0 0
\(481\) 9.27708i 0.422998i
\(482\) 9.88758i 0.450367i
\(483\) 0 0
\(484\) 0 0
\(485\) − 0.121459i − 0.00551517i
\(486\) 0 0
\(487\) 14.5978 0.661491 0.330745 0.943720i \(-0.392700\pi\)
0.330745 + 0.943720i \(0.392700\pi\)
\(488\) 20.5926i 0.932185i
\(489\) 0 0
\(490\) 0.191563i 0.00865395i
\(491\) −25.0424 −1.13015 −0.565073 0.825041i \(-0.691152\pi\)
−0.565073 + 0.825041i \(0.691152\pi\)
\(492\) 0 0
\(493\) 23.9286 1.07769
\(494\) 9.82158 0.441894
\(495\) 0 0
\(496\) 6.62982 0.297688
\(497\) −0.821634 −0.0368553
\(498\) 0 0
\(499\) 22.7116 1.01671 0.508355 0.861148i \(-0.330254\pi\)
0.508355 + 0.861148i \(0.330254\pi\)
\(500\) − 0.612063i − 0.0273723i
\(501\) 0 0
\(502\) − 4.77866i − 0.213282i
\(503\) −10.4071 −0.464030 −0.232015 0.972712i \(-0.574532\pi\)
−0.232015 + 0.972712i \(0.574532\pi\)
\(504\) 0 0
\(505\) 0.278835i 0.0124080i
\(506\) 0 0
\(507\) 0 0
\(508\) 19.4931i 0.864868i
\(509\) 8.08550i 0.358383i 0.983814 + 0.179192i \(0.0573482\pi\)
−0.983814 + 0.179192i \(0.942652\pi\)
\(510\) 0 0
\(511\) 2.92606 0.129441
\(512\) 14.7311 0.651031
\(513\) 0 0
\(514\) 7.17082i 0.316291i
\(515\) 0.526203i 0.0231873i
\(516\) 0 0
\(517\) 0 0
\(518\) − 0.684854i − 0.0300908i
\(519\) 0 0
\(520\) −0.224047 −0.00982513
\(521\) − 35.0206i − 1.53428i −0.641479 0.767141i \(-0.721679\pi\)
0.641479 0.767141i \(-0.278321\pi\)
\(522\) 0 0
\(523\) − 43.2956i − 1.89318i −0.322434 0.946592i \(-0.604501\pi\)
0.322434 0.946592i \(-0.395499\pi\)
\(524\) 27.7067 1.21037
\(525\) 0 0
\(526\) 2.93288 0.127880
\(527\) 20.5146 0.893630
\(528\) 0 0
\(529\) −23.2378 −1.01034
\(530\) −0.247568 −0.0107537
\(531\) 0 0
\(532\) 2.33588 0.101273
\(533\) 22.1013i 0.957316i
\(534\) 0 0
\(535\) − 0.0548877i − 0.00237300i
\(536\) −19.2854 −0.833001
\(537\) 0 0
\(538\) − 7.61509i − 0.328310i
\(539\) 0 0
\(540\) 0 0
\(541\) − 0.226329i − 0.00973066i −0.999988 0.00486533i \(-0.998451\pi\)
0.999988 0.00486533i \(-0.00154869\pi\)
\(542\) 10.9409i 0.469951i
\(543\) 0 0
\(544\) 24.8250 1.06436
\(545\) −0.294580 −0.0126184
\(546\) 0 0
\(547\) 16.4633i 0.703918i 0.936015 + 0.351959i \(0.114484\pi\)
−0.936015 + 0.351959i \(0.885516\pi\)
\(548\) 24.5315i 1.04793i
\(549\) 0 0
\(550\) 0 0
\(551\) 34.6938i 1.47801i
\(552\) 0 0
\(553\) 0.743260 0.0316066
\(554\) 7.80594i 0.331643i
\(555\) 0 0
\(556\) − 5.88723i − 0.249674i
\(557\) 26.2818 1.11360 0.556798 0.830648i \(-0.312030\pi\)
0.556798 + 0.830648i \(0.312030\pi\)
\(558\) 0 0
\(559\) −2.37440 −0.100426
\(560\) −0.0136825 −0.000578192 0
\(561\) 0 0
\(562\) −1.07394 −0.0453013
\(563\) −36.3658 −1.53263 −0.766317 0.642463i \(-0.777913\pi\)
−0.766317 + 0.642463i \(0.777913\pi\)
\(564\) 0 0
\(565\) −0.0211992 −0.000891856 0
\(566\) − 18.2848i − 0.768565i
\(567\) 0 0
\(568\) − 8.07857i − 0.338969i
\(569\) −20.2307 −0.848114 −0.424057 0.905636i \(-0.639394\pi\)
−0.424057 + 0.905636i \(0.639394\pi\)
\(570\) 0 0
\(571\) − 24.4002i − 1.02112i −0.859843 0.510558i \(-0.829439\pi\)
0.859843 0.510558i \(-0.170561\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) − 1.63157i − 0.0681004i
\(575\) 33.9883i 1.41741i
\(576\) 0 0
\(577\) −19.3030 −0.803593 −0.401796 0.915729i \(-0.631614\pi\)
−0.401796 + 0.915729i \(0.631614\pi\)
\(578\) 0.885126 0.0368164
\(579\) 0 0
\(580\) − 0.342550i − 0.0142236i
\(581\) 1.30676i 0.0542137i
\(582\) 0 0
\(583\) 0 0
\(584\) 28.7700i 1.19051i
\(585\) 0 0
\(586\) 18.1258 0.748771
\(587\) − 19.1363i − 0.789840i −0.918715 0.394920i \(-0.870772\pi\)
0.918715 0.394920i \(-0.129228\pi\)
\(588\) 0 0
\(589\) 29.7439i 1.22558i
\(590\) −0.0769757 −0.00316904
\(591\) 0 0
\(592\) −5.57049 −0.228946
\(593\) 14.9885 0.615503 0.307751 0.951467i \(-0.400423\pi\)
0.307751 + 0.951467i \(0.400423\pi\)
\(594\) 0 0
\(595\) −0.0423378 −0.00173568
\(596\) 15.6746 0.642058
\(597\) 0 0
\(598\) 10.7717 0.440489
\(599\) 33.3409i 1.36227i 0.732158 + 0.681135i \(0.238513\pi\)
−0.732158 + 0.681135i \(0.761487\pi\)
\(600\) 0 0
\(601\) − 42.2730i − 1.72435i −0.506610 0.862175i \(-0.669102\pi\)
0.506610 0.862175i \(-0.330898\pi\)
\(602\) 0.175283 0.00714401
\(603\) 0 0
\(604\) 13.3859i 0.544664i
\(605\) 0 0
\(606\) 0 0
\(607\) − 30.8859i − 1.25362i −0.779172 0.626810i \(-0.784360\pi\)
0.779172 0.626810i \(-0.215640\pi\)
\(608\) 35.9935i 1.45973i
\(609\) 0 0
\(610\) 0.234227 0.00948358
\(611\) 25.6072 1.03596
\(612\) 0 0
\(613\) 15.9745i 0.645204i 0.946535 + 0.322602i \(0.104558\pi\)
−0.946535 + 0.322602i \(0.895442\pi\)
\(614\) 17.9013i 0.722436i
\(615\) 0 0
\(616\) 0 0
\(617\) 2.63374i 0.106030i 0.998594 + 0.0530151i \(0.0168831\pi\)
−0.998594 + 0.0530151i \(0.983117\pi\)
\(618\) 0 0
\(619\) 20.2495 0.813894 0.406947 0.913452i \(-0.366593\pi\)
0.406947 + 0.913452i \(0.366593\pi\)
\(620\) − 0.293677i − 0.0117944i
\(621\) 0 0
\(622\) − 9.17132i − 0.367736i
\(623\) −2.10913 −0.0845003
\(624\) 0 0
\(625\) 24.9759 0.999035
\(626\) −7.74212 −0.309437
\(627\) 0 0
\(628\) 19.8884 0.793633
\(629\) −17.2367 −0.687274
\(630\) 0 0
\(631\) 8.44044 0.336009 0.168004 0.985786i \(-0.446268\pi\)
0.168004 + 0.985786i \(0.446268\pi\)
\(632\) 7.30797i 0.290696i
\(633\) 0 0
\(634\) − 7.53859i − 0.299396i
\(635\) 0.512264 0.0203286
\(636\) 0 0
\(637\) − 15.9704i − 0.632771i
\(638\) 0 0
\(639\) 0 0
\(640\) − 0.431685i − 0.0170639i
\(641\) − 32.3145i − 1.27634i −0.769894 0.638172i \(-0.779691\pi\)
0.769894 0.638172i \(-0.220309\pi\)
\(642\) 0 0
\(643\) 13.7767 0.543300 0.271650 0.962396i \(-0.412431\pi\)
0.271650 + 0.962396i \(0.412431\pi\)
\(644\) 2.56186 0.100951
\(645\) 0 0
\(646\) 18.2484i 0.717975i
\(647\) − 8.52739i − 0.335246i −0.985851 0.167623i \(-0.946391\pi\)
0.985851 0.167623i \(-0.0536092\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) − 7.91804i − 0.310571i
\(651\) 0 0
\(652\) 14.0076 0.548579
\(653\) − 1.20799i − 0.0472722i −0.999721 0.0236361i \(-0.992476\pi\)
0.999721 0.0236361i \(-0.00752430\pi\)
\(654\) 0 0
\(655\) − 0.728108i − 0.0284496i
\(656\) −13.2709 −0.518142
\(657\) 0 0
\(658\) −1.89038 −0.0736947
\(659\) 9.41054 0.366583 0.183291 0.983059i \(-0.441325\pi\)
0.183291 + 0.983059i \(0.441325\pi\)
\(660\) 0 0
\(661\) 15.1027 0.587425 0.293713 0.955894i \(-0.405109\pi\)
0.293713 + 0.955894i \(0.405109\pi\)
\(662\) 2.91343 0.113234
\(663\) 0 0
\(664\) −12.8485 −0.498619
\(665\) − 0.0613851i − 0.00238041i
\(666\) 0 0
\(667\) 38.0501i 1.47331i
\(668\) −11.6138 −0.449351
\(669\) 0 0
\(670\) 0.219358i 0.00847453i
\(671\) 0 0
\(672\) 0 0
\(673\) 41.5989i 1.60352i 0.597647 + 0.801759i \(0.296102\pi\)
−0.597647 + 0.801759i \(0.703898\pi\)
\(674\) − 1.83249i − 0.0705850i
\(675\) 0 0
\(676\) −11.7567 −0.452182
\(677\) 4.52056 0.173739 0.0868696 0.996220i \(-0.472314\pi\)
0.0868696 + 0.996220i \(0.472314\pi\)
\(678\) 0 0
\(679\) − 0.747516i − 0.0286870i
\(680\) − 0.416279i − 0.0159636i
\(681\) 0 0
\(682\) 0 0
\(683\) − 34.7783i − 1.33075i −0.746507 0.665377i \(-0.768271\pi\)
0.746507 0.665377i \(-0.231729\pi\)
\(684\) 0 0
\(685\) 0.644667 0.0246315
\(686\) 2.36829i 0.0904219i
\(687\) 0 0
\(688\) − 1.42572i − 0.0543552i
\(689\) 20.6395 0.786302
\(690\) 0 0
\(691\) −17.8835 −0.680320 −0.340160 0.940368i \(-0.610481\pi\)
−0.340160 + 0.940368i \(0.610481\pi\)
\(692\) −25.3670 −0.964308
\(693\) 0 0
\(694\) 11.4278 0.433794
\(695\) −0.154712 −0.00586854
\(696\) 0 0
\(697\) −41.0641 −1.55542
\(698\) 9.48105i 0.358863i
\(699\) 0 0
\(700\) − 1.88316i − 0.0711768i
\(701\) −17.4712 −0.659878 −0.329939 0.944002i \(-0.607028\pi\)
−0.329939 + 0.944002i \(0.607028\pi\)
\(702\) 0 0
\(703\) − 24.9914i − 0.942568i
\(704\) 0 0
\(705\) 0 0
\(706\) − 7.58833i − 0.285591i
\(707\) 1.71608i 0.0645399i
\(708\) 0 0
\(709\) 8.33849 0.313158 0.156579 0.987665i \(-0.449953\pi\)
0.156579 + 0.987665i \(0.449953\pi\)
\(710\) −0.0918882 −0.00344850
\(711\) 0 0
\(712\) − 20.7376i − 0.777174i
\(713\) 32.6214i 1.22168i
\(714\) 0 0
\(715\) 0 0
\(716\) 10.7598i 0.402112i
\(717\) 0 0
\(718\) −13.0873 −0.488415
\(719\) 23.6031i 0.880249i 0.897937 + 0.440124i \(0.145066\pi\)
−0.897937 + 0.440124i \(0.854934\pi\)
\(720\) 0 0
\(721\) 3.23850i 0.120608i
\(722\) −13.3807 −0.497977
\(723\) 0 0
\(724\) 32.2400 1.19819
\(725\) 27.9697 1.03877
\(726\) 0 0
\(727\) 1.90462 0.0706386 0.0353193 0.999376i \(-0.488755\pi\)
0.0353193 + 0.999376i \(0.488755\pi\)
\(728\) −1.37889 −0.0511052
\(729\) 0 0
\(730\) 0.327239 0.0121117
\(731\) − 4.41161i − 0.163169i
\(732\) 0 0
\(733\) − 17.6118i − 0.650506i −0.945627 0.325253i \(-0.894550\pi\)
0.945627 0.325253i \(-0.105450\pi\)
\(734\) −3.92132 −0.144738
\(735\) 0 0
\(736\) 39.4756i 1.45509i
\(737\) 0 0
\(738\) 0 0
\(739\) − 4.77737i − 0.175738i −0.996132 0.0878692i \(-0.971994\pi\)
0.996132 0.0878692i \(-0.0280058\pi\)
\(740\) 0.246753i 0.00907082i
\(741\) 0 0
\(742\) −1.52365 −0.0559350
\(743\) −1.83777 −0.0674211 −0.0337105 0.999432i \(-0.510732\pi\)
−0.0337105 + 0.999432i \(0.510732\pi\)
\(744\) 0 0
\(745\) − 0.411916i − 0.0150914i
\(746\) − 15.2521i − 0.558419i
\(747\) 0 0
\(748\) 0 0
\(749\) − 0.337805i − 0.0123431i
\(750\) 0 0
\(751\) −37.9446 −1.38462 −0.692308 0.721602i \(-0.743406\pi\)
−0.692308 + 0.721602i \(0.743406\pi\)
\(752\) 15.3760i 0.560707i
\(753\) 0 0
\(754\) − 8.86431i − 0.322819i
\(755\) 0.351770 0.0128022
\(756\) 0 0
\(757\) −30.6265 −1.11314 −0.556569 0.830802i \(-0.687882\pi\)
−0.556569 + 0.830802i \(0.687882\pi\)
\(758\) −0.262481 −0.00953373
\(759\) 0 0
\(760\) 0.603558 0.0218934
\(761\) 30.8652 1.11886 0.559432 0.828876i \(-0.311019\pi\)
0.559432 + 0.828876i \(0.311019\pi\)
\(762\) 0 0
\(763\) −1.81299 −0.0656345
\(764\) − 20.8613i − 0.754734i
\(765\) 0 0
\(766\) 8.71682i 0.314951i
\(767\) 6.41738 0.231718
\(768\) 0 0
\(769\) 19.3154i 0.696532i 0.937396 + 0.348266i \(0.113229\pi\)
−0.937396 + 0.348266i \(0.886771\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 38.2096i − 1.37519i
\(773\) − 2.94203i − 0.105818i −0.998599 0.0529088i \(-0.983151\pi\)
0.998599 0.0529088i \(-0.0168493\pi\)
\(774\) 0 0
\(775\) 23.9792 0.861358
\(776\) 7.34982 0.263843
\(777\) 0 0
\(778\) − 16.3804i − 0.587265i
\(779\) − 59.5385i − 2.13319i
\(780\) 0 0
\(781\) 0 0
\(782\) 20.0138i 0.715693i
\(783\) 0 0
\(784\) 9.58955 0.342484
\(785\) − 0.522651i − 0.0186542i
\(786\) 0 0
\(787\) 8.04569i 0.286798i 0.989665 + 0.143399i \(0.0458032\pi\)
−0.989665 + 0.143399i \(0.954197\pi\)
\(788\) 32.1188 1.14419
\(789\) 0 0
\(790\) 0.0831232 0.00295739
\(791\) −0.130470 −0.00463897
\(792\) 0 0
\(793\) −19.5272 −0.693433
\(794\) 3.44487 0.122254
\(795\) 0 0
\(796\) 15.8287 0.561033
\(797\) − 54.9219i − 1.94543i −0.231996 0.972717i \(-0.574526\pi\)
0.231996 0.972717i \(-0.425474\pi\)
\(798\) 0 0
\(799\) 47.5780i 1.68319i
\(800\) 29.0175 1.02592
\(801\) 0 0
\(802\) − 2.92121i − 0.103152i
\(803\) 0 0
\(804\) 0 0
\(805\) − 0.0673236i − 0.00237285i
\(806\) − 7.59961i − 0.267685i
\(807\) 0 0
\(808\) −16.8731 −0.593593
\(809\) −24.3526 −0.856193 −0.428096 0.903733i \(-0.640816\pi\)
−0.428096 + 0.903733i \(0.640816\pi\)
\(810\) 0 0
\(811\) 20.7604i 0.728996i 0.931204 + 0.364498i \(0.118759\pi\)
−0.931204 + 0.364498i \(0.881241\pi\)
\(812\) − 2.10821i − 0.0739838i
\(813\) 0 0
\(814\) 0 0
\(815\) − 0.368107i − 0.0128942i
\(816\) 0 0
\(817\) 6.39635 0.223780
\(818\) − 8.12567i − 0.284107i
\(819\) 0 0
\(820\) 0.587855i 0.0205288i
\(821\) −28.6373 −0.999449 −0.499725 0.866184i \(-0.666565\pi\)
−0.499725 + 0.866184i \(0.666565\pi\)
\(822\) 0 0
\(823\) 2.48784 0.0867207 0.0433603 0.999059i \(-0.486194\pi\)
0.0433603 + 0.999059i \(0.486194\pi\)
\(824\) −31.8420 −1.10927
\(825\) 0 0
\(826\) −0.473745 −0.0164837
\(827\) 23.8333 0.828765 0.414383 0.910103i \(-0.363998\pi\)
0.414383 + 0.910103i \(0.363998\pi\)
\(828\) 0 0
\(829\) −40.6340 −1.41128 −0.705639 0.708572i \(-0.749340\pi\)
−0.705639 + 0.708572i \(0.749340\pi\)
\(830\) 0.146143i 0.00507270i
\(831\) 0 0
\(832\) − 2.83514i − 0.0982909i
\(833\) 29.6729 1.02811
\(834\) 0 0
\(835\) 0.305201i 0.0105619i
\(836\) 0 0
\(837\) 0 0
\(838\) 14.4400i 0.498822i
\(839\) − 57.2232i − 1.97556i −0.155842 0.987782i \(-0.549809\pi\)
0.155842 0.987782i \(-0.450191\pi\)
\(840\) 0 0
\(841\) 2.31232 0.0797352
\(842\) −1.91330 −0.0659365
\(843\) 0 0
\(844\) 28.6075i 0.984712i
\(845\) 0.308958i 0.0106285i
\(846\) 0 0
\(847\) 0 0
\(848\) 12.3931i 0.425582i
\(849\) 0 0
\(850\) 14.7117 0.504606
\(851\) − 27.4091i − 0.939572i
\(852\) 0 0
\(853\) 30.5412i 1.04571i 0.852422 + 0.522855i \(0.175133\pi\)
−0.852422 + 0.522855i \(0.824867\pi\)
\(854\) 1.44154 0.0493286
\(855\) 0 0
\(856\) 3.32141 0.113523
\(857\) 6.14781 0.210005 0.105003 0.994472i \(-0.466515\pi\)
0.105003 + 0.994472i \(0.466515\pi\)
\(858\) 0 0
\(859\) −27.5988 −0.941660 −0.470830 0.882224i \(-0.656045\pi\)
−0.470830 + 0.882224i \(0.656045\pi\)
\(860\) −0.0631545 −0.00215355
\(861\) 0 0
\(862\) 19.2234 0.654753
\(863\) 42.8009i 1.45696i 0.685067 + 0.728481i \(0.259773\pi\)
−0.685067 + 0.728481i \(0.740227\pi\)
\(864\) 0 0
\(865\) 0.666624i 0.0226659i
\(866\) −6.31032 −0.214433
\(867\) 0 0
\(868\) − 1.80743i − 0.0613481i
\(869\) 0 0
\(870\) 0 0
\(871\) − 18.2876i − 0.619652i
\(872\) − 17.8259i − 0.603660i
\(873\) 0 0
\(874\) −29.0178 −0.981543
\(875\) −0.0989917 −0.00334653
\(876\) 0 0
\(877\) − 11.2305i − 0.379227i −0.981859 0.189613i \(-0.939277\pi\)
0.981859 0.189613i \(-0.0607235\pi\)
\(878\) 2.01746i 0.0680860i
\(879\) 0 0
\(880\) 0 0
\(881\) − 47.3136i − 1.59403i −0.603957 0.797017i \(-0.706410\pi\)
0.603957 0.797017i \(-0.293590\pi\)
\(882\) 0 0
\(883\) 18.2617 0.614555 0.307278 0.951620i \(-0.400582\pi\)
0.307278 + 0.951620i \(0.400582\pi\)
\(884\) 15.0211i 0.505214i
\(885\) 0 0
\(886\) 24.6693i 0.828782i
\(887\) −40.7104 −1.36692 −0.683460 0.729988i \(-0.739526\pi\)
−0.683460 + 0.729988i \(0.739526\pi\)
\(888\) 0 0
\(889\) 3.15271 0.105739
\(890\) −0.235876 −0.00790658
\(891\) 0 0
\(892\) −14.7573 −0.494110
\(893\) −68.9829 −2.30842
\(894\) 0 0
\(895\) 0.282758 0.00945157
\(896\) − 2.65680i − 0.0887573i
\(897\) 0 0
\(898\) 9.78345i 0.326478i
\(899\) 26.8449 0.895327
\(900\) 0 0
\(901\) 38.3480i 1.27756i
\(902\) 0 0
\(903\) 0 0
\(904\) − 1.28282i − 0.0426660i
\(905\) − 0.847242i − 0.0281633i
\(906\) 0 0
\(907\) 1.18171 0.0392380 0.0196190 0.999808i \(-0.493755\pi\)
0.0196190 + 0.999808i \(0.493755\pi\)
\(908\) 17.8819 0.593431
\(909\) 0 0
\(910\) 0.0156840i 0 0.000519919i
\(911\) − 21.1641i − 0.701198i −0.936526 0.350599i \(-0.885978\pi\)
0.936526 0.350599i \(-0.114022\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) − 14.0406i − 0.464423i
\(915\) 0 0
\(916\) −31.2580 −1.03279
\(917\) − 4.48112i − 0.147980i
\(918\) 0 0
\(919\) 14.0906i 0.464805i 0.972620 + 0.232402i \(0.0746586\pi\)
−0.972620 + 0.232402i \(0.925341\pi\)
\(920\) 0.661948 0.0218238
\(921\) 0 0
\(922\) 0.530228 0.0174621
\(923\) 7.66061 0.252152
\(924\) 0 0
\(925\) −20.1477 −0.662454
\(926\) 26.0138 0.854867
\(927\) 0 0
\(928\) 32.4854 1.06638
\(929\) − 34.0857i − 1.11832i −0.829061 0.559158i \(-0.811125\pi\)
0.829061 0.559158i \(-0.188875\pi\)
\(930\) 0 0
\(931\) 43.0225i 1.41000i
\(932\) 24.6381 0.807049
\(933\) 0 0
\(934\) 12.1157i 0.396438i
\(935\) 0 0
\(936\) 0 0
\(937\) 6.75011i 0.220517i 0.993903 + 0.110258i \(0.0351678\pi\)
−0.993903 + 0.110258i \(0.964832\pi\)
\(938\) 1.35003i 0.0440801i
\(939\) 0 0
\(940\) 0.681104 0.0222152
\(941\) −33.2979 −1.08548 −0.542740 0.839901i \(-0.682613\pi\)
−0.542740 + 0.839901i \(0.682613\pi\)
\(942\) 0 0
\(943\) − 65.2984i − 2.12641i
\(944\) 3.85336i 0.125416i
\(945\) 0 0
\(946\) 0 0
\(947\) − 0.751232i − 0.0244117i −0.999926 0.0122059i \(-0.996115\pi\)
0.999926 0.0122059i \(-0.00388535\pi\)
\(948\) 0 0
\(949\) −27.2815 −0.885596
\(950\) 21.3303i 0.692046i
\(951\) 0 0
\(952\) − 2.56197i − 0.0830341i
\(953\) −4.65369 −0.150748 −0.0753739 0.997155i \(-0.524015\pi\)
−0.0753739 + 0.997155i \(0.524015\pi\)
\(954\) 0 0
\(955\) −0.548217 −0.0177399
\(956\) −43.3266 −1.40128
\(957\) 0 0
\(958\) −1.65593 −0.0535007
\(959\) 3.96758 0.128120
\(960\) 0 0
\(961\) −7.98515 −0.257585
\(962\) 6.38533i 0.205871i
\(963\) 0 0
\(964\) − 21.9253i − 0.706166i
\(965\) −1.00412 −0.0323237
\(966\) 0 0
\(967\) − 21.7529i − 0.699525i −0.936838 0.349763i \(-0.886262\pi\)
0.936838 0.349763i \(-0.113738\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) − 0.0835991i − 0.00268421i
\(971\) 5.87314i 0.188478i 0.995550 + 0.0942391i \(0.0300418\pi\)
−0.995550 + 0.0942391i \(0.969958\pi\)
\(972\) 0 0
\(973\) −0.952168 −0.0305251
\(974\) 10.0476 0.321944
\(975\) 0 0
\(976\) − 11.7253i − 0.375317i
\(977\) 0.919993i 0.0294332i 0.999892 + 0.0147166i \(0.00468460\pi\)
−0.999892 + 0.0147166i \(0.995315\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) − 0.424783i − 0.0135692i
\(981\) 0 0
\(982\) −17.2364 −0.550037
\(983\) 46.6359i 1.48745i 0.668483 + 0.743727i \(0.266944\pi\)
−0.668483 + 0.743727i \(0.733056\pi\)
\(984\) 0 0
\(985\) − 0.844056i − 0.0268939i
\(986\) 16.4698 0.524506
\(987\) 0 0
\(988\) −21.7789 −0.692880
\(989\) 7.01515 0.223069
\(990\) 0 0
\(991\) 6.21090 0.197296 0.0986478 0.995122i \(-0.468548\pi\)
0.0986478 + 0.995122i \(0.468548\pi\)
\(992\) 27.8506 0.884257
\(993\) 0 0
\(994\) −0.565523 −0.0179373
\(995\) − 0.415965i − 0.0131870i
\(996\) 0 0
\(997\) − 28.5668i − 0.904719i −0.891836 0.452360i \(-0.850582\pi\)
0.891836 0.452360i \(-0.149418\pi\)
\(998\) 15.6322 0.494828
\(999\) 0 0
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1089.2.d.g.1088.12 16
3.2 odd 2 inner 1089.2.d.g.1088.5 16
11.3 even 5 99.2.j.a.35.3 yes 16
11.7 odd 10 99.2.j.a.17.2 16
11.10 odd 2 inner 1089.2.d.g.1088.6 16
33.14 odd 10 99.2.j.a.35.2 yes 16
33.29 even 10 99.2.j.a.17.3 yes 16
33.32 even 2 inner 1089.2.d.g.1088.11 16
44.3 odd 10 1584.2.cd.c.1025.2 16
44.7 even 10 1584.2.cd.c.17.3 16
99.7 odd 30 891.2.u.c.215.2 32
99.14 odd 30 891.2.u.c.431.2 32
99.25 even 15 891.2.u.c.134.2 32
99.29 even 30 891.2.u.c.215.3 32
99.40 odd 30 891.2.u.c.512.3 32
99.47 odd 30 891.2.u.c.134.3 32
99.58 even 15 891.2.u.c.431.3 32
99.95 even 30 891.2.u.c.512.2 32
132.47 even 10 1584.2.cd.c.1025.3 16
132.95 odd 10 1584.2.cd.c.17.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
99.2.j.a.17.2 16 11.7 odd 10
99.2.j.a.17.3 yes 16 33.29 even 10
99.2.j.a.35.2 yes 16 33.14 odd 10
99.2.j.a.35.3 yes 16 11.3 even 5
891.2.u.c.134.2 32 99.25 even 15
891.2.u.c.134.3 32 99.47 odd 30
891.2.u.c.215.2 32 99.7 odd 30
891.2.u.c.215.3 32 99.29 even 30
891.2.u.c.431.2 32 99.14 odd 30
891.2.u.c.431.3 32 99.58 even 15
891.2.u.c.512.2 32 99.95 even 30
891.2.u.c.512.3 32 99.40 odd 30
1089.2.d.g.1088.5 16 3.2 odd 2 inner
1089.2.d.g.1088.6 16 11.10 odd 2 inner
1089.2.d.g.1088.11 16 33.32 even 2 inner
1089.2.d.g.1088.12 16 1.1 even 1 trivial
1584.2.cd.c.17.2 16 132.95 odd 10
1584.2.cd.c.17.3 16 44.7 even 10
1584.2.cd.c.1025.2 16 44.3 odd 10
1584.2.cd.c.1025.3 16 132.47 even 10