Properties

Label 1143.3.b.a.890.4
Level $1143$
Weight $3$
Character 1143.890
Analytic conductor $31.144$
Analytic rank $0$
Dimension $84$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1143 = 3^{2} \cdot 127 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1143.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(31.1444942164\)
Analytic rank: \(0\)
Dimension: \(84\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 890.4
Character \(\chi\) \(=\) 1143.890
Dual form 1143.3.b.a.890.81

$q$-expansion

\(f(q)\) \(=\) \(q-3.66638i q^{2} -9.44232 q^{4} -0.851756i q^{5} +2.76278 q^{7} +19.9536i q^{8} +O(q^{10})\) \(q-3.66638i q^{2} -9.44232 q^{4} -0.851756i q^{5} +2.76278 q^{7} +19.9536i q^{8} -3.12286 q^{10} -14.2731i q^{11} +7.64232 q^{13} -10.1294i q^{14} +35.3881 q^{16} -9.98646i q^{17} +24.6909 q^{19} +8.04255i q^{20} -52.3304 q^{22} -26.3758i q^{23} +24.2745 q^{25} -28.0196i q^{26} -26.0870 q^{28} -32.4798i q^{29} -4.59199 q^{31} -49.9317i q^{32} -36.6141 q^{34} -2.35321i q^{35} +8.59256 q^{37} -90.5262i q^{38} +16.9956 q^{40} +60.9097i q^{41} -34.3172 q^{43} +134.771i q^{44} -96.7035 q^{46} -1.72698i q^{47} -41.3670 q^{49} -88.9995i q^{50} -72.1612 q^{52} -15.5905i q^{53} -12.1572 q^{55} +55.1274i q^{56} -119.083 q^{58} -59.5018i q^{59} +31.0892 q^{61} +16.8360i q^{62} -41.5161 q^{64} -6.50940i q^{65} -124.484 q^{67} +94.2954i q^{68} -8.62777 q^{70} +9.71062i q^{71} -98.4398 q^{73} -31.5035i q^{74} -233.140 q^{76} -39.4333i q^{77} +35.2163 q^{79} -30.1420i q^{80} +223.318 q^{82} -50.8065i q^{83} -8.50603 q^{85} +125.820i q^{86} +284.799 q^{88} +12.0062i q^{89} +21.1141 q^{91} +249.048i q^{92} -6.33175 q^{94} -21.0307i q^{95} +35.5414 q^{97} +151.667i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 84q - 160q^{4} + O(q^{10}) \) \( 84q - 160q^{4} - 48q^{10} + 16q^{13} + 360q^{16} + 64q^{19} - 8q^{22} - 388q^{25} - 120q^{28} - 160q^{31} + 192q^{34} - 152q^{37} + 208q^{40} - 24q^{43} + 56q^{46} + 564q^{49} - 80q^{52} + 136q^{55} - 136q^{58} + 168q^{61} - 736q^{64} + 168q^{67} - 608q^{70} + 80q^{73} - 32q^{76} - 168q^{79} + 528q^{82} + 288q^{85} - 392q^{88} + 176q^{91} + 176q^{94} - 120q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1143\mathbb{Z}\right)^\times\).

\(n\) \(128\) \(892\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 3.66638i − 1.83319i −0.399819 0.916594i \(-0.630927\pi\)
0.399819 0.916594i \(-0.369073\pi\)
\(3\) 0 0
\(4\) −9.44232 −2.36058
\(5\) − 0.851756i − 0.170351i −0.996366 0.0851756i \(-0.972855\pi\)
0.996366 0.0851756i \(-0.0271451\pi\)
\(6\) 0 0
\(7\) 2.76278 0.394683 0.197341 0.980335i \(-0.436769\pi\)
0.197341 + 0.980335i \(0.436769\pi\)
\(8\) 19.9536i 2.49420i
\(9\) 0 0
\(10\) −3.12286 −0.312286
\(11\) − 14.2731i − 1.29755i −0.760980 0.648775i \(-0.775282\pi\)
0.760980 0.648775i \(-0.224718\pi\)
\(12\) 0 0
\(13\) 7.64232 0.587871 0.293936 0.955825i \(-0.405035\pi\)
0.293936 + 0.955825i \(0.405035\pi\)
\(14\) − 10.1294i − 0.723528i
\(15\) 0 0
\(16\) 35.3881 2.21176
\(17\) − 9.98646i − 0.587439i −0.955892 0.293719i \(-0.905107\pi\)
0.955892 0.293719i \(-0.0948932\pi\)
\(18\) 0 0
\(19\) 24.6909 1.29952 0.649761 0.760138i \(-0.274869\pi\)
0.649761 + 0.760138i \(0.274869\pi\)
\(20\) 8.04255i 0.402128i
\(21\) 0 0
\(22\) −52.3304 −2.37865
\(23\) − 26.3758i − 1.14677i −0.819285 0.573386i \(-0.805629\pi\)
0.819285 0.573386i \(-0.194371\pi\)
\(24\) 0 0
\(25\) 24.2745 0.970980
\(26\) − 28.0196i − 1.07768i
\(27\) 0 0
\(28\) −26.0870 −0.931680
\(29\) − 32.4798i − 1.11999i −0.828495 0.559997i \(-0.810802\pi\)
0.828495 0.559997i \(-0.189198\pi\)
\(30\) 0 0
\(31\) −4.59199 −0.148129 −0.0740644 0.997253i \(-0.523597\pi\)
−0.0740644 + 0.997253i \(0.523597\pi\)
\(32\) − 49.9317i − 1.56037i
\(33\) 0 0
\(34\) −36.6141 −1.07689
\(35\) − 2.35321i − 0.0672347i
\(36\) 0 0
\(37\) 8.59256 0.232231 0.116116 0.993236i \(-0.462956\pi\)
0.116116 + 0.993236i \(0.462956\pi\)
\(38\) − 90.5262i − 2.38227i
\(39\) 0 0
\(40\) 16.9956 0.424890
\(41\) 60.9097i 1.48560i 0.669512 + 0.742802i \(0.266503\pi\)
−0.669512 + 0.742802i \(0.733497\pi\)
\(42\) 0 0
\(43\) −34.3172 −0.798074 −0.399037 0.916935i \(-0.630656\pi\)
−0.399037 + 0.916935i \(0.630656\pi\)
\(44\) 134.771i 3.06297i
\(45\) 0 0
\(46\) −96.7035 −2.10225
\(47\) − 1.72698i − 0.0367442i −0.999831 0.0183721i \(-0.994152\pi\)
0.999831 0.0183721i \(-0.00584835\pi\)
\(48\) 0 0
\(49\) −41.3670 −0.844225
\(50\) − 88.9995i − 1.77999i
\(51\) 0 0
\(52\) −72.1612 −1.38772
\(53\) − 15.5905i − 0.294160i −0.989125 0.147080i \(-0.953013\pi\)
0.989125 0.147080i \(-0.0469874\pi\)
\(54\) 0 0
\(55\) −12.1572 −0.221039
\(56\) 55.1274i 0.984417i
\(57\) 0 0
\(58\) −119.083 −2.05316
\(59\) − 59.5018i − 1.00851i −0.863556 0.504253i \(-0.831768\pi\)
0.863556 0.504253i \(-0.168232\pi\)
\(60\) 0 0
\(61\) 31.0892 0.509659 0.254830 0.966986i \(-0.417981\pi\)
0.254830 + 0.966986i \(0.417981\pi\)
\(62\) 16.8360i 0.271548i
\(63\) 0 0
\(64\) −41.5161 −0.648690
\(65\) − 6.50940i − 0.100145i
\(66\) 0 0
\(67\) −124.484 −1.85797 −0.928984 0.370119i \(-0.879317\pi\)
−0.928984 + 0.370119i \(0.879317\pi\)
\(68\) 94.2954i 1.38670i
\(69\) 0 0
\(70\) −8.62777 −0.123254
\(71\) 9.71062i 0.136769i 0.997659 + 0.0683847i \(0.0217845\pi\)
−0.997659 + 0.0683847i \(0.978215\pi\)
\(72\) 0 0
\(73\) −98.4398 −1.34849 −0.674245 0.738508i \(-0.735531\pi\)
−0.674245 + 0.738508i \(0.735531\pi\)
\(74\) − 31.5035i − 0.425724i
\(75\) 0 0
\(76\) −233.140 −3.06763
\(77\) − 39.4333i − 0.512121i
\(78\) 0 0
\(79\) 35.2163 0.445776 0.222888 0.974844i \(-0.428452\pi\)
0.222888 + 0.974844i \(0.428452\pi\)
\(80\) − 30.1420i − 0.376775i
\(81\) 0 0
\(82\) 223.318 2.72339
\(83\) − 50.8065i − 0.612126i −0.952011 0.306063i \(-0.900988\pi\)
0.952011 0.306063i \(-0.0990119\pi\)
\(84\) 0 0
\(85\) −8.50603 −0.100071
\(86\) 125.820i 1.46302i
\(87\) 0 0
\(88\) 284.799 3.23635
\(89\) 12.0062i 0.134901i 0.997723 + 0.0674507i \(0.0214865\pi\)
−0.997723 + 0.0674507i \(0.978513\pi\)
\(90\) 0 0
\(91\) 21.1141 0.232023
\(92\) 249.048i 2.70705i
\(93\) 0 0
\(94\) −6.33175 −0.0673591
\(95\) − 21.0307i − 0.221375i
\(96\) 0 0
\(97\) 35.5414 0.366406 0.183203 0.983075i \(-0.441353\pi\)
0.183203 + 0.983075i \(0.441353\pi\)
\(98\) 151.667i 1.54762i
\(99\) 0 0
\(100\) −229.208 −2.29208
\(101\) 85.3748i 0.845295i 0.906294 + 0.422647i \(0.138899\pi\)
−0.906294 + 0.422647i \(0.861101\pi\)
\(102\) 0 0
\(103\) −5.67316 −0.0550792 −0.0275396 0.999621i \(-0.508767\pi\)
−0.0275396 + 0.999621i \(0.508767\pi\)
\(104\) 152.492i 1.46627i
\(105\) 0 0
\(106\) −57.1605 −0.539250
\(107\) 163.390i 1.52701i 0.645801 + 0.763506i \(0.276524\pi\)
−0.645801 + 0.763506i \(0.723476\pi\)
\(108\) 0 0
\(109\) 1.24644 0.0114352 0.00571762 0.999984i \(-0.498180\pi\)
0.00571762 + 0.999984i \(0.498180\pi\)
\(110\) 44.5727i 0.405207i
\(111\) 0 0
\(112\) 97.7695 0.872942
\(113\) − 38.9817i − 0.344970i −0.985012 0.172485i \(-0.944820\pi\)
0.985012 0.172485i \(-0.0551797\pi\)
\(114\) 0 0
\(115\) −22.4657 −0.195354
\(116\) 306.685i 2.64383i
\(117\) 0 0
\(118\) −218.156 −1.84878
\(119\) − 27.5904i − 0.231852i
\(120\) 0 0
\(121\) −82.7200 −0.683636
\(122\) − 113.985i − 0.934302i
\(123\) 0 0
\(124\) 43.3590 0.349670
\(125\) − 41.9699i − 0.335759i
\(126\) 0 0
\(127\) −11.2694 −0.0887357
\(128\) − 47.5131i − 0.371196i
\(129\) 0 0
\(130\) −23.8659 −0.183584
\(131\) 61.7794i 0.471599i 0.971802 + 0.235799i \(0.0757708\pi\)
−0.971802 + 0.235799i \(0.924229\pi\)
\(132\) 0 0
\(133\) 68.2156 0.512899
\(134\) 456.405i 3.40601i
\(135\) 0 0
\(136\) 199.266 1.46519
\(137\) − 232.233i − 1.69513i −0.530692 0.847565i \(-0.678068\pi\)
0.530692 0.847565i \(-0.321932\pi\)
\(138\) 0 0
\(139\) 84.4304 0.607413 0.303706 0.952766i \(-0.401776\pi\)
0.303706 + 0.952766i \(0.401776\pi\)
\(140\) 22.2198i 0.158713i
\(141\) 0 0
\(142\) 35.6028 0.250724
\(143\) − 109.079i − 0.762792i
\(144\) 0 0
\(145\) −27.6649 −0.190792
\(146\) 360.917i 2.47204i
\(147\) 0 0
\(148\) −81.1337 −0.548200
\(149\) − 144.774i − 0.971636i −0.874060 0.485818i \(-0.838522\pi\)
0.874060 0.485818i \(-0.161478\pi\)
\(150\) 0 0
\(151\) −260.071 −1.72232 −0.861162 0.508331i \(-0.830263\pi\)
−0.861162 + 0.508331i \(0.830263\pi\)
\(152\) 492.672i 3.24127i
\(153\) 0 0
\(154\) −144.577 −0.938814
\(155\) 3.91126i 0.0252339i
\(156\) 0 0
\(157\) 259.318 1.65171 0.825853 0.563885i \(-0.190694\pi\)
0.825853 + 0.563885i \(0.190694\pi\)
\(158\) − 129.116i − 0.817191i
\(159\) 0 0
\(160\) −42.5297 −0.265810
\(161\) − 72.8704i − 0.452611i
\(162\) 0 0
\(163\) 103.310 0.633802 0.316901 0.948459i \(-0.397358\pi\)
0.316901 + 0.948459i \(0.397358\pi\)
\(164\) − 575.129i − 3.50688i
\(165\) 0 0
\(166\) −186.276 −1.12214
\(167\) − 289.782i − 1.73522i −0.497245 0.867610i \(-0.665655\pi\)
0.497245 0.867610i \(-0.334345\pi\)
\(168\) 0 0
\(169\) −110.595 −0.654408
\(170\) 31.1863i 0.183449i
\(171\) 0 0
\(172\) 324.034 1.88392
\(173\) 128.657i 0.743685i 0.928296 + 0.371842i \(0.121274\pi\)
−0.928296 + 0.371842i \(0.878726\pi\)
\(174\) 0 0
\(175\) 67.0651 0.383229
\(176\) − 505.096i − 2.86986i
\(177\) 0 0
\(178\) 44.0194 0.247300
\(179\) − 258.062i − 1.44169i −0.693097 0.720844i \(-0.743754\pi\)
0.693097 0.720844i \(-0.256246\pi\)
\(180\) 0 0
\(181\) 51.5156 0.284616 0.142308 0.989822i \(-0.454548\pi\)
0.142308 + 0.989822i \(0.454548\pi\)
\(182\) − 77.4121i − 0.425341i
\(183\) 0 0
\(184\) 526.291 2.86028
\(185\) − 7.31876i − 0.0395609i
\(186\) 0 0
\(187\) −142.537 −0.762231
\(188\) 16.3067i 0.0867376i
\(189\) 0 0
\(190\) −77.1063 −0.405823
\(191\) − 128.808i − 0.674390i −0.941435 0.337195i \(-0.890522\pi\)
0.941435 0.337195i \(-0.109478\pi\)
\(192\) 0 0
\(193\) −60.3990 −0.312948 −0.156474 0.987682i \(-0.550013\pi\)
−0.156474 + 0.987682i \(0.550013\pi\)
\(194\) − 130.308i − 0.671692i
\(195\) 0 0
\(196\) 390.601 1.99286
\(197\) 251.511i 1.27671i 0.769744 + 0.638353i \(0.220384\pi\)
−0.769744 + 0.638353i \(0.779616\pi\)
\(198\) 0 0
\(199\) −272.046 −1.36707 −0.683533 0.729920i \(-0.739557\pi\)
−0.683533 + 0.729920i \(0.739557\pi\)
\(200\) 484.364i 2.42182i
\(201\) 0 0
\(202\) 313.016 1.54958
\(203\) − 89.7346i − 0.442042i
\(204\) 0 0
\(205\) 51.8802 0.253074
\(206\) 20.7999i 0.100971i
\(207\) 0 0
\(208\) 270.447 1.30023
\(209\) − 352.415i − 1.68620i
\(210\) 0 0
\(211\) 5.09269 0.0241360 0.0120680 0.999927i \(-0.496159\pi\)
0.0120680 + 0.999927i \(0.496159\pi\)
\(212\) 147.210i 0.694387i
\(213\) 0 0
\(214\) 599.050 2.79930
\(215\) 29.2299i 0.135953i
\(216\) 0 0
\(217\) −12.6867 −0.0584639
\(218\) − 4.56992i − 0.0209629i
\(219\) 0 0
\(220\) 114.792 0.521781
\(221\) − 76.3198i − 0.345338i
\(222\) 0 0
\(223\) 75.0319 0.336466 0.168233 0.985747i \(-0.446194\pi\)
0.168233 + 0.985747i \(0.446194\pi\)
\(224\) − 137.950i − 0.615850i
\(225\) 0 0
\(226\) −142.921 −0.632396
\(227\) 221.592i 0.976176i 0.872794 + 0.488088i \(0.162306\pi\)
−0.872794 + 0.488088i \(0.837694\pi\)
\(228\) 0 0
\(229\) −327.428 −1.42981 −0.714907 0.699219i \(-0.753531\pi\)
−0.714907 + 0.699219i \(0.753531\pi\)
\(230\) 82.3678i 0.358121i
\(231\) 0 0
\(232\) 648.089 2.79349
\(233\) 52.8126i 0.226663i 0.993557 + 0.113332i \(0.0361523\pi\)
−0.993557 + 0.113332i \(0.963848\pi\)
\(234\) 0 0
\(235\) −1.47096 −0.00625942
\(236\) 561.835i 2.38066i
\(237\) 0 0
\(238\) −101.157 −0.425028
\(239\) − 74.9951i − 0.313787i −0.987616 0.156894i \(-0.949852\pi\)
0.987616 0.156894i \(-0.0501480\pi\)
\(240\) 0 0
\(241\) 205.835 0.854085 0.427043 0.904232i \(-0.359555\pi\)
0.427043 + 0.904232i \(0.359555\pi\)
\(242\) 303.283i 1.25323i
\(243\) 0 0
\(244\) −293.554 −1.20309
\(245\) 35.2346i 0.143815i
\(246\) 0 0
\(247\) 188.696 0.763952
\(248\) − 91.6267i − 0.369462i
\(249\) 0 0
\(250\) −153.877 −0.615509
\(251\) − 93.4507i − 0.372314i −0.982520 0.186157i \(-0.940397\pi\)
0.982520 0.186157i \(-0.0596032\pi\)
\(252\) 0 0
\(253\) −376.463 −1.48799
\(254\) 41.3180i 0.162669i
\(255\) 0 0
\(256\) −340.265 −1.32916
\(257\) 230.227i 0.895823i 0.894078 + 0.447912i \(0.147832\pi\)
−0.894078 + 0.447912i \(0.852168\pi\)
\(258\) 0 0
\(259\) 23.7393 0.0916577
\(260\) 61.4638i 0.236399i
\(261\) 0 0
\(262\) 226.507 0.864529
\(263\) 428.852i 1.63061i 0.579028 + 0.815307i \(0.303432\pi\)
−0.579028 + 0.815307i \(0.696568\pi\)
\(264\) 0 0
\(265\) −13.2793 −0.0501105
\(266\) − 250.104i − 0.940241i
\(267\) 0 0
\(268\) 1175.42 4.38588
\(269\) − 279.561i − 1.03926i −0.854391 0.519630i \(-0.826070\pi\)
0.854391 0.519630i \(-0.173930\pi\)
\(270\) 0 0
\(271\) 145.582 0.537201 0.268601 0.963252i \(-0.413439\pi\)
0.268601 + 0.963252i \(0.413439\pi\)
\(272\) − 353.402i − 1.29927i
\(273\) 0 0
\(274\) −851.453 −3.10749
\(275\) − 346.471i − 1.25990i
\(276\) 0 0
\(277\) 392.296 1.41623 0.708116 0.706096i \(-0.249545\pi\)
0.708116 + 0.706096i \(0.249545\pi\)
\(278\) − 309.554i − 1.11350i
\(279\) 0 0
\(280\) 46.9551 0.167697
\(281\) 43.1317i 0.153494i 0.997051 + 0.0767468i \(0.0244533\pi\)
−0.997051 + 0.0767468i \(0.975547\pi\)
\(282\) 0 0
\(283\) 87.6603 0.309754 0.154877 0.987934i \(-0.450502\pi\)
0.154877 + 0.987934i \(0.450502\pi\)
\(284\) − 91.6908i − 0.322855i
\(285\) 0 0
\(286\) −399.926 −1.39834
\(287\) 168.280i 0.586342i
\(288\) 0 0
\(289\) 189.271 0.654915
\(290\) 101.430i 0.349758i
\(291\) 0 0
\(292\) 929.499 3.18322
\(293\) 158.686i 0.541590i 0.962637 + 0.270795i \(0.0872865\pi\)
−0.962637 + 0.270795i \(0.912713\pi\)
\(294\) 0 0
\(295\) −50.6811 −0.171800
\(296\) 171.452i 0.579231i
\(297\) 0 0
\(298\) −530.795 −1.78119
\(299\) − 201.572i − 0.674154i
\(300\) 0 0
\(301\) −94.8108 −0.314986
\(302\) 953.518i 3.15734i
\(303\) 0 0
\(304\) 873.765 2.87423
\(305\) − 26.4804i − 0.0868211i
\(306\) 0 0
\(307\) 120.792 0.393460 0.196730 0.980458i \(-0.436968\pi\)
0.196730 + 0.980458i \(0.436968\pi\)
\(308\) 372.342i 1.20890i
\(309\) 0 0
\(310\) 14.3401 0.0462585
\(311\) − 62.7402i − 0.201737i −0.994900 0.100868i \(-0.967838\pi\)
0.994900 0.100868i \(-0.0321621\pi\)
\(312\) 0 0
\(313\) −177.785 −0.568003 −0.284001 0.958824i \(-0.591662\pi\)
−0.284001 + 0.958824i \(0.591662\pi\)
\(314\) − 950.757i − 3.02789i
\(315\) 0 0
\(316\) −332.523 −1.05229
\(317\) − 202.295i − 0.638155i −0.947729 0.319078i \(-0.896627\pi\)
0.947729 0.319078i \(-0.103373\pi\)
\(318\) 0 0
\(319\) −463.586 −1.45325
\(320\) 35.3616i 0.110505i
\(321\) 0 0
\(322\) −267.170 −0.829722
\(323\) − 246.575i − 0.763390i
\(324\) 0 0
\(325\) 185.514 0.570811
\(326\) − 378.772i − 1.16188i
\(327\) 0 0
\(328\) −1215.37 −3.70539
\(329\) − 4.77126i − 0.0145023i
\(330\) 0 0
\(331\) −147.920 −0.446888 −0.223444 0.974717i \(-0.571730\pi\)
−0.223444 + 0.974717i \(0.571730\pi\)
\(332\) 479.731i 1.44497i
\(333\) 0 0
\(334\) −1062.45 −3.18099
\(335\) 106.030i 0.316507i
\(336\) 0 0
\(337\) −459.988 −1.36495 −0.682474 0.730909i \(-0.739096\pi\)
−0.682474 + 0.730909i \(0.739096\pi\)
\(338\) 405.483i 1.19965i
\(339\) 0 0
\(340\) 80.3167 0.236225
\(341\) 65.5417i 0.192204i
\(342\) 0 0
\(343\) −249.664 −0.727884
\(344\) − 684.751i − 1.99056i
\(345\) 0 0
\(346\) 471.707 1.36331
\(347\) − 547.855i − 1.57883i −0.613858 0.789416i \(-0.710383\pi\)
0.613858 0.789416i \(-0.289617\pi\)
\(348\) 0 0
\(349\) −30.6558 −0.0878389 −0.0439195 0.999035i \(-0.513985\pi\)
−0.0439195 + 0.999035i \(0.513985\pi\)
\(350\) − 245.886i − 0.702531i
\(351\) 0 0
\(352\) −712.678 −2.02465
\(353\) 162.085i 0.459166i 0.973289 + 0.229583i \(0.0737362\pi\)
−0.973289 + 0.229583i \(0.926264\pi\)
\(354\) 0 0
\(355\) 8.27108 0.0232988
\(356\) − 113.367i − 0.318446i
\(357\) 0 0
\(358\) −946.153 −2.64289
\(359\) 119.205i 0.332046i 0.986122 + 0.166023i \(0.0530926\pi\)
−0.986122 + 0.166023i \(0.946907\pi\)
\(360\) 0 0
\(361\) 248.642 0.688758
\(362\) − 188.875i − 0.521755i
\(363\) 0 0
\(364\) −199.366 −0.547708
\(365\) 83.8467i 0.229717i
\(366\) 0 0
\(367\) 507.592 1.38309 0.691543 0.722336i \(-0.256931\pi\)
0.691543 + 0.722336i \(0.256931\pi\)
\(368\) − 933.388i − 2.53638i
\(369\) 0 0
\(370\) −26.8333 −0.0725226
\(371\) − 43.0730i − 0.116100i
\(372\) 0 0
\(373\) −499.234 −1.33843 −0.669215 0.743069i \(-0.733369\pi\)
−0.669215 + 0.743069i \(0.733369\pi\)
\(374\) 522.595i 1.39731i
\(375\) 0 0
\(376\) 34.4594 0.0916473
\(377\) − 248.221i − 0.658412i
\(378\) 0 0
\(379\) 26.0902 0.0688395 0.0344198 0.999407i \(-0.489042\pi\)
0.0344198 + 0.999407i \(0.489042\pi\)
\(380\) 198.578i 0.522574i
\(381\) 0 0
\(382\) −472.260 −1.23628
\(383\) − 327.847i − 0.855998i −0.903779 0.427999i \(-0.859219\pi\)
0.903779 0.427999i \(-0.140781\pi\)
\(384\) 0 0
\(385\) −33.5876 −0.0872404
\(386\) 221.446i 0.573693i
\(387\) 0 0
\(388\) −335.593 −0.864931
\(389\) − 80.5907i − 0.207174i −0.994620 0.103587i \(-0.966968\pi\)
0.994620 0.103587i \(-0.0330320\pi\)
\(390\) 0 0
\(391\) −263.401 −0.673659
\(392\) − 825.421i − 2.10567i
\(393\) 0 0
\(394\) 922.134 2.34044
\(395\) − 29.9957i − 0.0759384i
\(396\) 0 0
\(397\) −71.5458 −0.180216 −0.0901080 0.995932i \(-0.528721\pi\)
−0.0901080 + 0.995932i \(0.528721\pi\)
\(398\) 997.423i 2.50609i
\(399\) 0 0
\(400\) 859.029 2.14757
\(401\) 139.812i 0.348658i 0.984687 + 0.174329i \(0.0557756\pi\)
−0.984687 + 0.174329i \(0.944224\pi\)
\(402\) 0 0
\(403\) −35.0935 −0.0870806
\(404\) − 806.136i − 1.99539i
\(405\) 0 0
\(406\) −329.001 −0.810347
\(407\) − 122.642i − 0.301332i
\(408\) 0 0
\(409\) 658.477 1.60997 0.804984 0.593297i \(-0.202174\pi\)
0.804984 + 0.593297i \(0.202174\pi\)
\(410\) − 190.213i − 0.463933i
\(411\) 0 0
\(412\) 53.5678 0.130019
\(413\) − 164.390i − 0.398040i
\(414\) 0 0
\(415\) −43.2747 −0.104276
\(416\) − 381.594i − 0.917294i
\(417\) 0 0
\(418\) −1292.09 −3.09111
\(419\) 83.3121i 0.198835i 0.995046 + 0.0994177i \(0.0316980\pi\)
−0.995046 + 0.0994177i \(0.968302\pi\)
\(420\) 0 0
\(421\) −44.9743 −0.106827 −0.0534137 0.998572i \(-0.517010\pi\)
−0.0534137 + 0.998572i \(0.517010\pi\)
\(422\) − 18.6717i − 0.0442458i
\(423\) 0 0
\(424\) 311.086 0.733693
\(425\) − 242.417i − 0.570392i
\(426\) 0 0
\(427\) 85.8927 0.201154
\(428\) − 1542.78i − 3.60463i
\(429\) 0 0
\(430\) 107.168 0.249227
\(431\) 488.409i 1.13320i 0.823993 + 0.566599i \(0.191741\pi\)
−0.823993 + 0.566599i \(0.808259\pi\)
\(432\) 0 0
\(433\) −659.033 −1.52202 −0.761009 0.648742i \(-0.775296\pi\)
−0.761009 + 0.648742i \(0.775296\pi\)
\(434\) 46.5141i 0.107175i
\(435\) 0 0
\(436\) −11.7693 −0.0269938
\(437\) − 651.242i − 1.49026i
\(438\) 0 0
\(439\) 86.1838 0.196319 0.0981593 0.995171i \(-0.468705\pi\)
0.0981593 + 0.995171i \(0.468705\pi\)
\(440\) − 242.579i − 0.551316i
\(441\) 0 0
\(442\) −279.817 −0.633070
\(443\) − 521.457i − 1.17710i −0.808459 0.588552i \(-0.799698\pi\)
0.808459 0.588552i \(-0.200302\pi\)
\(444\) 0 0
\(445\) 10.2264 0.0229806
\(446\) − 275.095i − 0.616805i
\(447\) 0 0
\(448\) −114.700 −0.256027
\(449\) − 543.281i − 1.20998i −0.796233 0.604990i \(-0.793177\pi\)
0.796233 0.604990i \(-0.206823\pi\)
\(450\) 0 0
\(451\) 869.368 1.92764
\(452\) 368.077i 0.814330i
\(453\) 0 0
\(454\) 812.440 1.78951
\(455\) − 17.9840i − 0.0395253i
\(456\) 0 0
\(457\) 179.592 0.392981 0.196491 0.980506i \(-0.437045\pi\)
0.196491 + 0.980506i \(0.437045\pi\)
\(458\) 1200.47i 2.62112i
\(459\) 0 0
\(460\) 212.128 0.461149
\(461\) 156.830i 0.340196i 0.985427 + 0.170098i \(0.0544084\pi\)
−0.985427 + 0.170098i \(0.945592\pi\)
\(462\) 0 0
\(463\) 488.330 1.05471 0.527354 0.849646i \(-0.323184\pi\)
0.527354 + 0.849646i \(0.323184\pi\)
\(464\) − 1149.40i − 2.47715i
\(465\) 0 0
\(466\) 193.631 0.415517
\(467\) 314.991i 0.674498i 0.941415 + 0.337249i \(0.109496\pi\)
−0.941415 + 0.337249i \(0.890504\pi\)
\(468\) 0 0
\(469\) −343.922 −0.733308
\(470\) 5.39311i 0.0114747i
\(471\) 0 0
\(472\) 1187.28 2.51541
\(473\) 489.811i 1.03554i
\(474\) 0 0
\(475\) 599.360 1.26181
\(476\) 260.517i 0.547305i
\(477\) 0 0
\(478\) −274.960 −0.575231
\(479\) 683.816i 1.42759i 0.700354 + 0.713796i \(0.253026\pi\)
−0.700354 + 0.713796i \(0.746974\pi\)
\(480\) 0 0
\(481\) 65.6671 0.136522
\(482\) − 754.667i − 1.56570i
\(483\) 0 0
\(484\) 781.068 1.61378
\(485\) − 30.2726i − 0.0624178i
\(486\) 0 0
\(487\) 690.839 1.41856 0.709280 0.704927i \(-0.249020\pi\)
0.709280 + 0.704927i \(0.249020\pi\)
\(488\) 620.341i 1.27119i
\(489\) 0 0
\(490\) 129.183 0.263640
\(491\) − 525.752i − 1.07078i −0.844606 0.535389i \(-0.820165\pi\)
0.844606 0.535389i \(-0.179835\pi\)
\(492\) 0 0
\(493\) −324.358 −0.657928
\(494\) − 691.831i − 1.40047i
\(495\) 0 0
\(496\) −162.502 −0.327624
\(497\) 26.8283i 0.0539805i
\(498\) 0 0
\(499\) −99.2375 −0.198873 −0.0994364 0.995044i \(-0.531704\pi\)
−0.0994364 + 0.995044i \(0.531704\pi\)
\(500\) 396.293i 0.792586i
\(501\) 0 0
\(502\) −342.626 −0.682521
\(503\) − 776.756i − 1.54425i −0.635472 0.772124i \(-0.719195\pi\)
0.635472 0.772124i \(-0.280805\pi\)
\(504\) 0 0
\(505\) 72.7185 0.143997
\(506\) 1380.25i 2.72777i
\(507\) 0 0
\(508\) 106.410 0.209468
\(509\) 644.281i 1.26578i 0.774242 + 0.632889i \(0.218131\pi\)
−0.774242 + 0.632889i \(0.781869\pi\)
\(510\) 0 0
\(511\) −271.967 −0.532226
\(512\) 1057.49i 2.06541i
\(513\) 0 0
\(514\) 844.097 1.64221
\(515\) 4.83215i 0.00938282i
\(516\) 0 0
\(517\) −24.6492 −0.0476774
\(518\) − 87.0374i − 0.168026i
\(519\) 0 0
\(520\) 129.886 0.249780
\(521\) − 498.797i − 0.957384i −0.877983 0.478692i \(-0.841111\pi\)
0.877983 0.478692i \(-0.158889\pi\)
\(522\) 0 0
\(523\) 552.330 1.05608 0.528040 0.849219i \(-0.322927\pi\)
0.528040 + 0.849219i \(0.322927\pi\)
\(524\) − 583.341i − 1.11325i
\(525\) 0 0
\(526\) 1572.33 2.98922
\(527\) 45.8577i 0.0870166i
\(528\) 0 0
\(529\) −166.681 −0.315087
\(530\) 48.6868i 0.0918620i
\(531\) 0 0
\(532\) −644.113 −1.21074
\(533\) 465.492i 0.873343i
\(534\) 0 0
\(535\) 139.169 0.260128
\(536\) − 2483.90i − 4.63414i
\(537\) 0 0
\(538\) −1024.98 −1.90516
\(539\) 590.434i 1.09542i
\(540\) 0 0
\(541\) 307.368 0.568149 0.284074 0.958802i \(-0.408314\pi\)
0.284074 + 0.958802i \(0.408314\pi\)
\(542\) − 533.757i − 0.984791i
\(543\) 0 0
\(544\) −498.641 −0.916620
\(545\) − 1.06166i − 0.00194801i
\(546\) 0 0
\(547\) −1037.81 −1.89727 −0.948637 0.316367i \(-0.897537\pi\)
−0.948637 + 0.316367i \(0.897537\pi\)
\(548\) 2192.82i 4.00149i
\(549\) 0 0
\(550\) −1270.29 −2.30963
\(551\) − 801.957i − 1.45546i
\(552\) 0 0
\(553\) 97.2948 0.175940
\(554\) − 1438.31i − 2.59622i
\(555\) 0 0
\(556\) −797.218 −1.43385
\(557\) − 94.2544i − 0.169218i −0.996414 0.0846090i \(-0.973036\pi\)
0.996414 0.0846090i \(-0.0269641\pi\)
\(558\) 0 0
\(559\) −262.263 −0.469165
\(560\) − 83.2758i − 0.148707i
\(561\) 0 0
\(562\) 158.137 0.281383
\(563\) 917.628i 1.62989i 0.579538 + 0.814945i \(0.303233\pi\)
−0.579538 + 0.814945i \(0.696767\pi\)
\(564\) 0 0
\(565\) −33.2029 −0.0587662
\(566\) − 321.396i − 0.567837i
\(567\) 0 0
\(568\) −193.762 −0.341130
\(569\) 97.4460i 0.171258i 0.996327 + 0.0856292i \(0.0272900\pi\)
−0.996327 + 0.0856292i \(0.972710\pi\)
\(570\) 0 0
\(571\) 299.153 0.523911 0.261955 0.965080i \(-0.415633\pi\)
0.261955 + 0.965080i \(0.415633\pi\)
\(572\) 1029.96i 1.80063i
\(573\) 0 0
\(574\) 616.978 1.07488
\(575\) − 640.259i − 1.11349i
\(576\) 0 0
\(577\) 532.221 0.922393 0.461196 0.887298i \(-0.347420\pi\)
0.461196 + 0.887298i \(0.347420\pi\)
\(578\) − 693.937i − 1.20058i
\(579\) 0 0
\(580\) 261.221 0.450380
\(581\) − 140.367i − 0.241596i
\(582\) 0 0
\(583\) −222.524 −0.381687
\(584\) − 1964.23i − 3.36340i
\(585\) 0 0
\(586\) 581.802 0.992836
\(587\) 475.281i 0.809677i 0.914388 + 0.404839i \(0.132672\pi\)
−0.914388 + 0.404839i \(0.867328\pi\)
\(588\) 0 0
\(589\) −113.380 −0.192497
\(590\) 185.816i 0.314942i
\(591\) 0 0
\(592\) 304.074 0.513639
\(593\) − 321.262i − 0.541758i −0.962613 0.270879i \(-0.912686\pi\)
0.962613 0.270879i \(-0.0873143\pi\)
\(594\) 0 0
\(595\) −23.5003 −0.0394963
\(596\) 1367.00i 2.29362i
\(597\) 0 0
\(598\) −739.039 −1.23585
\(599\) 231.313i 0.386166i 0.981182 + 0.193083i \(0.0618486\pi\)
−0.981182 + 0.193083i \(0.938151\pi\)
\(600\) 0 0
\(601\) 980.113 1.63080 0.815402 0.578895i \(-0.196516\pi\)
0.815402 + 0.578895i \(0.196516\pi\)
\(602\) 347.612i 0.577429i
\(603\) 0 0
\(604\) 2455.67 4.06568
\(605\) 70.4573i 0.116458i
\(606\) 0 0
\(607\) 533.312 0.878602 0.439301 0.898340i \(-0.355226\pi\)
0.439301 + 0.898340i \(0.355226\pi\)
\(608\) − 1232.86i − 2.02773i
\(609\) 0 0
\(610\) −97.0873 −0.159159
\(611\) − 13.1981i − 0.0216009i
\(612\) 0 0
\(613\) −826.976 −1.34906 −0.674532 0.738246i \(-0.735655\pi\)
−0.674532 + 0.738246i \(0.735655\pi\)
\(614\) − 442.870i − 0.721287i
\(615\) 0 0
\(616\) 786.836 1.27733
\(617\) 1225.99i 1.98701i 0.113785 + 0.993505i \(0.463703\pi\)
−0.113785 + 0.993505i \(0.536297\pi\)
\(618\) 0 0
\(619\) 804.256 1.29928 0.649641 0.760241i \(-0.274919\pi\)
0.649641 + 0.760241i \(0.274919\pi\)
\(620\) − 36.9313i − 0.0595667i
\(621\) 0 0
\(622\) −230.029 −0.369822
\(623\) 33.1706i 0.0532433i
\(624\) 0 0
\(625\) 571.115 0.913783
\(626\) 651.826i 1.04126i
\(627\) 0 0
\(628\) −2448.56 −3.89898
\(629\) − 85.8092i − 0.136422i
\(630\) 0 0
\(631\) −541.991 −0.858939 −0.429470 0.903081i \(-0.641299\pi\)
−0.429470 + 0.903081i \(0.641299\pi\)
\(632\) 702.691i 1.11185i
\(633\) 0 0
\(634\) −741.690 −1.16986
\(635\) 9.59881i 0.0151162i
\(636\) 0 0
\(637\) −316.140 −0.496296
\(638\) 1699.68i 2.66408i
\(639\) 0 0
\(640\) −40.4696 −0.0632337
\(641\) − 427.172i − 0.666416i −0.942853 0.333208i \(-0.891869\pi\)
0.942853 0.333208i \(-0.108131\pi\)
\(642\) 0 0
\(643\) 1273.41 1.98042 0.990210 0.139589i \(-0.0445781\pi\)
0.990210 + 0.139589i \(0.0445781\pi\)
\(644\) 688.066i 1.06842i
\(645\) 0 0
\(646\) −904.037 −1.39944
\(647\) − 344.452i − 0.532383i −0.963920 0.266191i \(-0.914235\pi\)
0.963920 0.266191i \(-0.0857653\pi\)
\(648\) 0 0
\(649\) −849.273 −1.30859
\(650\) − 680.163i − 1.04640i
\(651\) 0 0
\(652\) −975.483 −1.49614
\(653\) − 132.502i − 0.202913i −0.994840 0.101456i \(-0.967650\pi\)
0.994840 0.101456i \(-0.0323502\pi\)
\(654\) 0 0
\(655\) 52.6210 0.0803374
\(656\) 2155.48i 3.28579i
\(657\) 0 0
\(658\) −17.4932 −0.0265855
\(659\) − 731.028i − 1.10930i −0.832084 0.554650i \(-0.812852\pi\)
0.832084 0.554650i \(-0.187148\pi\)
\(660\) 0 0
\(661\) 513.775 0.777270 0.388635 0.921392i \(-0.372947\pi\)
0.388635 + 0.921392i \(0.372947\pi\)
\(662\) 542.330i 0.819230i
\(663\) 0 0
\(664\) 1013.77 1.52676
\(665\) − 58.1031i − 0.0873730i
\(666\) 0 0
\(667\) −856.680 −1.28438
\(668\) 2736.21i 4.09613i
\(669\) 0 0
\(670\) 388.746 0.580218
\(671\) − 443.738i − 0.661309i
\(672\) 0 0
\(673\) −216.699 −0.321990 −0.160995 0.986955i \(-0.551470\pi\)
−0.160995 + 0.986955i \(0.551470\pi\)
\(674\) 1686.49i 2.50221i
\(675\) 0 0
\(676\) 1044.27 1.54478
\(677\) − 676.400i − 0.999114i −0.866281 0.499557i \(-0.833496\pi\)
0.866281 0.499557i \(-0.166504\pi\)
\(678\) 0 0
\(679\) 98.1931 0.144614
\(680\) − 169.726i − 0.249597i
\(681\) 0 0
\(682\) 240.301 0.352347
\(683\) 96.7435i 0.141645i 0.997489 + 0.0708225i \(0.0225624\pi\)
−0.997489 + 0.0708225i \(0.977438\pi\)
\(684\) 0 0
\(685\) −197.806 −0.288767
\(686\) 915.363i 1.33435i
\(687\) 0 0
\(688\) −1214.42 −1.76515
\(689\) − 119.147i − 0.172928i
\(690\) 0 0
\(691\) −152.026 −0.220008 −0.110004 0.993931i \(-0.535086\pi\)
−0.110004 + 0.993931i \(0.535086\pi\)
\(692\) − 1214.82i − 1.75553i
\(693\) 0 0
\(694\) −2008.64 −2.89430
\(695\) − 71.9141i − 0.103474i
\(696\) 0 0
\(697\) 608.273 0.872701
\(698\) 112.396i 0.161025i
\(699\) 0 0
\(700\) −633.250 −0.904643
\(701\) 516.572i 0.736907i 0.929646 + 0.368454i \(0.120113\pi\)
−0.929646 + 0.368454i \(0.879887\pi\)
\(702\) 0 0
\(703\) 212.158 0.301790
\(704\) 592.562i 0.841707i
\(705\) 0 0
\(706\) 594.266 0.841737
\(707\) 235.872i 0.333623i
\(708\) 0 0
\(709\) 652.207 0.919897 0.459949 0.887946i \(-0.347868\pi\)
0.459949 + 0.887946i \(0.347868\pi\)
\(710\) − 30.3249i − 0.0427111i
\(711\) 0 0
\(712\) −239.567 −0.336471
\(713\) 121.117i 0.169870i
\(714\) 0 0
\(715\) −92.9090 −0.129943
\(716\) 2436.71i 3.40322i
\(717\) 0 0
\(718\) 437.049 0.608703
\(719\) 1380.67i 1.92026i 0.279552 + 0.960131i \(0.409814\pi\)
−0.279552 + 0.960131i \(0.590186\pi\)
\(720\) 0 0
\(721\) −15.6737 −0.0217388
\(722\) − 911.614i − 1.26262i
\(723\) 0 0
\(724\) −486.426 −0.671859
\(725\) − 788.432i − 1.08749i
\(726\) 0 0
\(727\) 733.171 1.00849 0.504244 0.863561i \(-0.331771\pi\)
0.504244 + 0.863561i \(0.331771\pi\)
\(728\) 421.301i 0.578710i
\(729\) 0 0
\(730\) 307.414 0.421114
\(731\) 342.707i 0.468820i
\(732\) 0 0
\(733\) 8.24724 0.0112513 0.00562567 0.999984i \(-0.498209\pi\)
0.00562567 + 0.999984i \(0.498209\pi\)
\(734\) − 1861.02i − 2.53546i
\(735\) 0 0
\(736\) −1316.99 −1.78938
\(737\) 1776.77i 2.41081i
\(738\) 0 0
\(739\) 944.003 1.27741 0.638703 0.769453i \(-0.279471\pi\)
0.638703 + 0.769453i \(0.279471\pi\)
\(740\) 69.1061i 0.0933866i
\(741\) 0 0
\(742\) −157.922 −0.212833
\(743\) − 467.178i − 0.628773i −0.949295 0.314386i \(-0.898201\pi\)
0.949295 0.314386i \(-0.101799\pi\)
\(744\) 0 0
\(745\) −123.312 −0.165519
\(746\) 1830.38i 2.45359i
\(747\) 0 0
\(748\) 1345.88 1.79931
\(749\) 451.411i 0.602686i
\(750\) 0 0
\(751\) 721.896 0.961247 0.480623 0.876927i \(-0.340410\pi\)
0.480623 + 0.876927i \(0.340410\pi\)
\(752\) − 61.1144i − 0.0812692i
\(753\) 0 0
\(754\) −910.073 −1.20699
\(755\) 221.517i 0.293400i
\(756\) 0 0
\(757\) 1140.81 1.50701 0.753507 0.657440i \(-0.228361\pi\)
0.753507 + 0.657440i \(0.228361\pi\)
\(758\) − 95.6564i − 0.126196i
\(759\) 0 0
\(760\) 419.637 0.552154
\(761\) 646.929i 0.850103i 0.905169 + 0.425052i \(0.139744\pi\)
−0.905169 + 0.425052i \(0.860256\pi\)
\(762\) 0 0
\(763\) 3.44364 0.00451329
\(764\) 1216.25i 1.59195i
\(765\) 0 0
\(766\) −1202.01 −1.56921
\(767\) − 454.732i − 0.592871i
\(768\) 0 0
\(769\) 868.292 1.12912 0.564559 0.825393i \(-0.309046\pi\)
0.564559 + 0.825393i \(0.309046\pi\)
\(770\) 123.145i 0.159928i
\(771\) 0 0
\(772\) 570.307 0.738739
\(773\) − 273.329i − 0.353596i −0.984247 0.176798i \(-0.943426\pi\)
0.984247 0.176798i \(-0.0565739\pi\)
\(774\) 0 0
\(775\) −111.468 −0.143830
\(776\) 709.178i 0.913890i
\(777\) 0 0
\(778\) −295.476 −0.379789
\(779\) 1503.92i 1.93057i
\(780\) 0 0
\(781\) 138.600 0.177465
\(782\) 965.726i 1.23494i
\(783\) 0 0
\(784\) −1463.90 −1.86722
\(785\) − 220.876i − 0.281370i
\(786\) 0 0
\(787\) 1042.66 1.32485 0.662425 0.749128i \(-0.269527\pi\)
0.662425 + 0.749128i \(0.269527\pi\)
\(788\) − 2374.85i − 3.01376i
\(789\) 0 0
\(790\) −109.975 −0.139209
\(791\) − 107.698i − 0.136154i
\(792\) 0 0
\(793\) 237.594 0.299614
\(794\) 262.314i 0.330370i
\(795\) 0 0
\(796\) 2568.75 3.22707
\(797\) 67.4321i 0.0846074i 0.999105 + 0.0423037i \(0.0134697\pi\)
−0.999105 + 0.0423037i \(0.986530\pi\)
\(798\) 0 0
\(799\) −17.2464 −0.0215850
\(800\) − 1212.07i − 1.51509i
\(801\) 0 0
\(802\) 512.602 0.639155
\(803\) 1405.04i 1.74973i
\(804\) 0 0
\(805\) −62.0678 −0.0771029
\(806\) 128.666i 0.159635i
\(807\) 0 0
\(808\) −1703.53 −2.10833
\(809\) 1496.12i 1.84935i 0.380758 + 0.924675i \(0.375663\pi\)
−0.380758 + 0.924675i \(0.624337\pi\)
\(810\) 0 0
\(811\) 217.664 0.268389 0.134195 0.990955i \(-0.457155\pi\)
0.134195 + 0.990955i \(0.457155\pi\)
\(812\) 847.302i 1.04348i
\(813\) 0 0
\(814\) −449.652 −0.552398
\(815\) − 87.9947i − 0.107969i
\(816\) 0 0
\(817\) −847.323 −1.03712
\(818\) − 2414.22i − 2.95137i
\(819\) 0 0
\(820\) −489.870 −0.597402
\(821\) − 352.519i − 0.429378i −0.976682 0.214689i \(-0.931126\pi\)
0.976682 0.214689i \(-0.0688738\pi\)
\(822\) 0 0
\(823\) −1222.88 −1.48588 −0.742941 0.669357i \(-0.766569\pi\)
−0.742941 + 0.669357i \(0.766569\pi\)
\(824\) − 113.200i − 0.137379i
\(825\) 0 0
\(826\) −602.717 −0.729682
\(827\) − 454.985i − 0.550163i −0.961421 0.275082i \(-0.911295\pi\)
0.961421 0.275082i \(-0.0887049\pi\)
\(828\) 0 0
\(829\) 1307.17 1.57681 0.788404 0.615157i \(-0.210908\pi\)
0.788404 + 0.615157i \(0.210908\pi\)
\(830\) 158.662i 0.191158i
\(831\) 0 0
\(832\) −317.280 −0.381346
\(833\) 413.110i 0.495931i
\(834\) 0 0
\(835\) −246.824 −0.295597
\(836\) 3327.61i 3.98040i
\(837\) 0 0
\(838\) 305.453 0.364503
\(839\) − 1460.75i − 1.74106i −0.492116 0.870529i \(-0.663777\pi\)
0.492116 0.870529i \(-0.336223\pi\)
\(840\) 0 0
\(841\) −213.938 −0.254386
\(842\) 164.893i 0.195835i
\(843\) 0 0
\(844\) −48.0868 −0.0569749
\(845\) 94.1999i 0.111479i
\(846\) 0 0
\(847\) −228.537 −0.269819
\(848\) − 551.717i − 0.650609i
\(849\) 0 0
\(850\) −888.790 −1.04564
\(851\) − 226.635i − 0.266316i
\(852\) 0 0
\(853\) 815.030 0.955487 0.477744 0.878499i \(-0.341455\pi\)
0.477744 + 0.878499i \(0.341455\pi\)
\(854\) − 314.915i − 0.368753i
\(855\) 0 0
\(856\) −3260.22 −3.80867
\(857\) 409.071i 0.477329i 0.971102 + 0.238665i \(0.0767096\pi\)
−0.971102 + 0.238665i \(0.923290\pi\)
\(858\) 0 0
\(859\) 1179.16 1.37271 0.686357 0.727265i \(-0.259209\pi\)
0.686357 + 0.727265i \(0.259209\pi\)
\(860\) − 275.998i − 0.320928i
\(861\) 0 0
\(862\) 1790.69 2.07737
\(863\) 1131.09i 1.31065i 0.755347 + 0.655325i \(0.227468\pi\)
−0.755347 + 0.655325i \(0.772532\pi\)
\(864\) 0 0
\(865\) 109.585 0.126688
\(866\) 2416.26i 2.79014i
\(867\) 0 0
\(868\) 119.791 0.138009
\(869\) − 502.644i − 0.578416i
\(870\) 0 0
\(871\) −951.346 −1.09225
\(872\) 24.8710i 0.0285217i
\(873\) 0 0
\(874\) −2387.70 −2.73192
\(875\) − 115.954i − 0.132518i
\(876\) 0 0
\(877\) −647.836 −0.738695 −0.369348 0.929291i \(-0.620419\pi\)
−0.369348 + 0.929291i \(0.620419\pi\)
\(878\) − 315.982i − 0.359889i
\(879\) 0 0
\(880\) −430.219 −0.488885
\(881\) 1224.05i 1.38939i 0.719304 + 0.694696i \(0.244461\pi\)
−0.719304 + 0.694696i \(0.755539\pi\)
\(882\) 0 0
\(883\) 1114.32 1.26197 0.630986 0.775794i \(-0.282650\pi\)
0.630986 + 0.775794i \(0.282650\pi\)
\(884\) 720.636i 0.815199i
\(885\) 0 0
\(886\) −1911.86 −2.15785
\(887\) − 1180.73i − 1.33115i −0.746333 0.665573i \(-0.768187\pi\)
0.746333 0.665573i \(-0.231813\pi\)
\(888\) 0 0
\(889\) −31.1349 −0.0350224
\(890\) − 37.4938i − 0.0421278i
\(891\) 0 0
\(892\) −708.475 −0.794255
\(893\) − 42.6407i − 0.0477499i
\(894\) 0 0
\(895\) −219.806 −0.245593
\(896\) − 131.268i − 0.146505i
\(897\) 0 0
\(898\) −1991.87 −2.21812
\(899\) 149.147i 0.165903i
\(900\) 0 0
\(901\) −155.694 −0.172801
\(902\) − 3187.43i − 3.53374i
\(903\) 0 0
\(904\) 777.824 0.860425
\(905\) − 43.8787i − 0.0484847i
\(906\) 0 0
\(907\) 688.839 0.759470 0.379735 0.925095i \(-0.376015\pi\)
0.379735 + 0.925095i \(0.376015\pi\)
\(908\) − 2092.34i − 2.30434i
\(909\) 0 0
\(910\) −65.9362 −0.0724574
\(911\) − 904.227i − 0.992565i −0.868161 0.496283i \(-0.834698\pi\)
0.868161 0.496283i \(-0.165302\pi\)
\(912\) 0 0
\(913\) −725.164 −0.794265
\(914\) − 658.453i − 0.720408i
\(915\) 0 0
\(916\) 3091.67 3.37519
\(917\) 170.683i 0.186132i
\(918\) 0 0
\(919\) 591.385 0.643509 0.321754 0.946823i \(-0.395727\pi\)
0.321754 + 0.946823i \(0.395727\pi\)
\(920\) − 448.272i − 0.487252i
\(921\) 0 0
\(922\) 574.999 0.623643
\(923\) 74.2117i 0.0804027i
\(924\) 0 0
\(925\) 208.580 0.225492
\(926\) − 1790.40i − 1.93348i
\(927\) 0 0
\(928\) −1621.77 −1.74760
\(929\) − 1652.28i − 1.77856i −0.457364 0.889280i \(-0.651206\pi\)
0.457364 0.889280i \(-0.348794\pi\)
\(930\) 0 0
\(931\) −1021.39 −1.09709
\(932\) − 498.673i − 0.535057i
\(933\) 0 0
\(934\) 1154.87 1.23648
\(935\) 121.407i 0.129847i
\(936\) 0 0
\(937\) −1538.88 −1.64235 −0.821174 0.570678i \(-0.806680\pi\)
−0.821174 + 0.570678i \(0.806680\pi\)
\(938\) 1260.95i 1.34429i
\(939\) 0 0
\(940\) 13.8893 0.0147759
\(941\) 1733.41i 1.84210i 0.389450 + 0.921048i \(0.372665\pi\)
−0.389450 + 0.921048i \(0.627335\pi\)
\(942\) 0 0
\(943\) 1606.54 1.70365
\(944\) − 2105.66i − 2.23057i
\(945\) 0 0
\(946\) 1795.83 1.89834
\(947\) 160.332i 0.169305i 0.996411 + 0.0846527i \(0.0269781\pi\)
−0.996411 + 0.0846527i \(0.973022\pi\)
\(948\) 0 0
\(949\) −752.309 −0.792738
\(950\) − 2197.48i − 2.31314i
\(951\) 0 0
\(952\) 550.527 0.578285
\(953\) − 102.479i − 0.107533i −0.998554 0.0537667i \(-0.982877\pi\)
0.998554 0.0537667i \(-0.0171227\pi\)
\(954\) 0 0
\(955\) −109.713 −0.114883
\(956\) 708.128i 0.740719i
\(957\) 0 0
\(958\) 2507.13 2.61704
\(959\) − 641.608i − 0.669039i
\(960\) 0 0
\(961\) −939.914 −0.978058
\(962\) − 240.760i − 0.250271i
\(963\) 0 0
\(964\) −1943.55 −2.01614
\(965\) 51.4453i 0.0533111i
\(966\) 0 0
\(967\) −483.412 −0.499908 −0.249954 0.968258i \(-0.580416\pi\)
−0.249954 + 0.968258i \(0.580416\pi\)
\(968\) − 1650.56i − 1.70512i
\(969\) 0 0
\(970\) −110.991 −0.114424
\(971\) − 1891.66i − 1.94816i −0.226206 0.974079i \(-0.572632\pi\)
0.226206 0.974079i \(-0.427368\pi\)
\(972\) 0 0
\(973\) 233.263 0.239735
\(974\) − 2532.88i − 2.60049i
\(975\) 0 0
\(976\) 1100.19 1.12724
\(977\) − 1540.72i − 1.57699i −0.615043 0.788493i \(-0.710861\pi\)
0.615043 0.788493i \(-0.289139\pi\)
\(978\) 0 0
\(979\) 171.366 0.175041
\(980\) − 332.697i − 0.339486i
\(981\) 0 0
\(982\) −1927.60 −1.96294
\(983\) − 1251.38i − 1.27302i −0.771266 0.636512i \(-0.780376\pi\)
0.771266 0.636512i \(-0.219624\pi\)
\(984\) 0 0
\(985\) 214.226 0.217488
\(986\) 1189.22i 1.20611i
\(987\) 0 0
\(988\) −1781.73 −1.80337
\(989\) 905.142i 0.915210i
\(990\) 0 0
\(991\) −857.100 −0.864884 −0.432442 0.901662i \(-0.642348\pi\)
−0.432442 + 0.901662i \(0.642348\pi\)
\(992\) 229.286i 0.231135i
\(993\) 0 0
\(994\) 98.3627 0.0989564
\(995\) 231.717i 0.232881i
\(996\) 0 0
\(997\) −1671.70 −1.67673 −0.838364 0.545111i \(-0.816487\pi\)
−0.838364 + 0.545111i \(0.816487\pi\)
\(998\) 363.842i 0.364571i
\(999\) 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 1143.3.b.a.890.4 84
3.2 odd 2 inner 1143.3.b.a.890.81 yes 84
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1143.3.b.a.890.4 84 1.1 even 1 trivial
1143.3.b.a.890.81 yes 84 3.2 odd 2 inner