Properties

Label 1089.3.b.i.485.15
Level $1089$
Weight $3$
Character 1089.485
Analytic conductor $29.673$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(29.6731007888\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 48 x^{14} + 921 x^{12} + 8986 x^{10} + 46812 x^{8} + 125072 x^{6} + 152129 x^{4} + 65614 x^{2} + 5041\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 99)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 485.15
Root \(3.25431i\) of defining polynomial
Character \(\chi\) \(=\) 1089.485
Dual form 1089.3.b.i.485.2

$q$-expansion

\(f(q)\) \(=\) \(q+3.25431i q^{2} -6.59053 q^{4} -1.59288i q^{5} -10.8798 q^{7} -8.43039i q^{8} +O(q^{10})\) \(q+3.25431i q^{2} -6.59053 q^{4} -1.59288i q^{5} -10.8798 q^{7} -8.43039i q^{8} +5.18373 q^{10} -9.23818 q^{13} -35.4062i q^{14} +1.07299 q^{16} -4.81648i q^{17} -25.0886 q^{19} +10.4979i q^{20} +36.6892i q^{23} +22.4627 q^{25} -30.0639i q^{26} +71.7035 q^{28} -24.4515i q^{29} +58.1821 q^{31} -30.2297i q^{32} +15.6743 q^{34} +17.3302i q^{35} +12.7978 q^{37} -81.6462i q^{38} -13.4286 q^{40} +12.4167i q^{41} +19.6984 q^{43} -119.398 q^{46} -64.9463i q^{47} +69.3696 q^{49} +73.1007i q^{50} +60.8845 q^{52} -49.5059i q^{53} +91.7208i q^{56} +79.5728 q^{58} +24.0917i q^{59} +44.2401 q^{61} +189.343i q^{62} +102.669 q^{64} +14.7153i q^{65} +33.0898 q^{67} +31.7432i q^{68} -56.3978 q^{70} +74.9390i q^{71} +41.3762 q^{73} +41.6480i q^{74} +165.347 q^{76} +63.9703 q^{79} -1.70914i q^{80} -40.4076 q^{82} -42.1858i q^{83} -7.67208 q^{85} +64.1046i q^{86} -111.111i q^{89} +100.509 q^{91} -241.801i q^{92} +211.355 q^{94} +39.9632i q^{95} -108.103 q^{97} +225.750i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 32q^{4} - 8q^{7} + O(q^{10}) \) \( 16q - 32q^{4} - 8q^{7} + 24q^{10} + 4q^{13} + 28q^{16} - 20q^{19} - 44q^{25} + 16q^{28} + 28q^{31} + 148q^{34} - 148q^{37} + 224q^{40} + 272q^{43} + 208q^{46} + 348q^{49} + 520q^{52} - 44q^{58} + 224q^{61} + 436q^{64} + 24q^{67} - 664q^{70} - 4q^{73} + 1052q^{76} + 216q^{79} + 348q^{82} + 416q^{85} - 168q^{91} + 1140q^{94} - 44q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.25431i 1.62715i 0.581457 + 0.813577i \(0.302483\pi\)
−0.581457 + 0.813577i \(0.697517\pi\)
\(3\) 0 0
\(4\) −6.59053 −1.64763
\(5\) − 1.59288i − 0.318576i −0.987232 0.159288i \(-0.949080\pi\)
0.987232 0.159288i \(-0.0509198\pi\)
\(6\) 0 0
\(7\) −10.8798 −1.55425 −0.777127 0.629344i \(-0.783324\pi\)
−0.777127 + 0.629344i \(0.783324\pi\)
\(8\) − 8.43039i − 1.05380i
\(9\) 0 0
\(10\) 5.18373 0.518373
\(11\) 0 0
\(12\) 0 0
\(13\) −9.23818 −0.710629 −0.355315 0.934747i \(-0.615626\pi\)
−0.355315 + 0.934747i \(0.615626\pi\)
\(14\) − 35.4062i − 2.52901i
\(15\) 0 0
\(16\) 1.07299 0.0670616
\(17\) − 4.81648i − 0.283322i −0.989915 0.141661i \(-0.954756\pi\)
0.989915 0.141661i \(-0.0452444\pi\)
\(18\) 0 0
\(19\) −25.0886 −1.32045 −0.660227 0.751066i \(-0.729540\pi\)
−0.660227 + 0.751066i \(0.729540\pi\)
\(20\) 10.4979i 0.524896i
\(21\) 0 0
\(22\) 0 0
\(23\) 36.6892i 1.59518i 0.603199 + 0.797590i \(0.293892\pi\)
−0.603199 + 0.797590i \(0.706108\pi\)
\(24\) 0 0
\(25\) 22.4627 0.898509
\(26\) − 30.0639i − 1.15630i
\(27\) 0 0
\(28\) 71.7035 2.56084
\(29\) − 24.4515i − 0.843156i −0.906792 0.421578i \(-0.861476\pi\)
0.906792 0.421578i \(-0.138524\pi\)
\(30\) 0 0
\(31\) 58.1821 1.87684 0.938421 0.345493i \(-0.112288\pi\)
0.938421 + 0.345493i \(0.112288\pi\)
\(32\) − 30.2297i − 0.944680i
\(33\) 0 0
\(34\) 15.6743 0.461009
\(35\) 17.3302i 0.495148i
\(36\) 0 0
\(37\) 12.7978 0.345887 0.172943 0.984932i \(-0.444672\pi\)
0.172943 + 0.984932i \(0.444672\pi\)
\(38\) − 81.6462i − 2.14858i
\(39\) 0 0
\(40\) −13.4286 −0.335715
\(41\) 12.4167i 0.302845i 0.988469 + 0.151423i \(0.0483854\pi\)
−0.988469 + 0.151423i \(0.951615\pi\)
\(42\) 0 0
\(43\) 19.6984 0.458102 0.229051 0.973414i \(-0.426438\pi\)
0.229051 + 0.973414i \(0.426438\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −119.398 −2.59561
\(47\) − 64.9463i − 1.38184i −0.722933 0.690918i \(-0.757207\pi\)
0.722933 0.690918i \(-0.242793\pi\)
\(48\) 0 0
\(49\) 69.3696 1.41571
\(50\) 73.1007i 1.46201i
\(51\) 0 0
\(52\) 60.8845 1.17086
\(53\) − 49.5059i − 0.934073i −0.884238 0.467037i \(-0.845322\pi\)
0.884238 0.467037i \(-0.154678\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 91.7208i 1.63787i
\(57\) 0 0
\(58\) 79.5728 1.37195
\(59\) 24.0917i 0.408333i 0.978936 + 0.204167i \(0.0654484\pi\)
−0.978936 + 0.204167i \(0.934552\pi\)
\(60\) 0 0
\(61\) 44.2401 0.725248 0.362624 0.931936i \(-0.381881\pi\)
0.362624 + 0.931936i \(0.381881\pi\)
\(62\) 189.343i 3.05391i
\(63\) 0 0
\(64\) 102.669 1.60420
\(65\) 14.7153i 0.226389i
\(66\) 0 0
\(67\) 33.0898 0.493878 0.246939 0.969031i \(-0.420575\pi\)
0.246939 + 0.969031i \(0.420575\pi\)
\(68\) 31.7432i 0.466811i
\(69\) 0 0
\(70\) −56.3978 −0.805683
\(71\) 74.9390i 1.05548i 0.849406 + 0.527739i \(0.176960\pi\)
−0.849406 + 0.527739i \(0.823040\pi\)
\(72\) 0 0
\(73\) 41.3762 0.566797 0.283399 0.959002i \(-0.408538\pi\)
0.283399 + 0.959002i \(0.408538\pi\)
\(74\) 41.6480i 0.562811i
\(75\) 0 0
\(76\) 165.347 2.17562
\(77\) 0 0
\(78\) 0 0
\(79\) 63.9703 0.809751 0.404875 0.914372i \(-0.367315\pi\)
0.404875 + 0.914372i \(0.367315\pi\)
\(80\) − 1.70914i − 0.0213642i
\(81\) 0 0
\(82\) −40.4076 −0.492776
\(83\) − 42.1858i − 0.508262i −0.967170 0.254131i \(-0.918211\pi\)
0.967170 0.254131i \(-0.0817895\pi\)
\(84\) 0 0
\(85\) −7.67208 −0.0902597
\(86\) 64.1046i 0.745402i
\(87\) 0 0
\(88\) 0 0
\(89\) − 111.111i − 1.24844i −0.781249 0.624220i \(-0.785417\pi\)
0.781249 0.624220i \(-0.214583\pi\)
\(90\) 0 0
\(91\) 100.509 1.10450
\(92\) − 241.801i − 2.62827i
\(93\) 0 0
\(94\) 211.355 2.24846
\(95\) 39.9632i 0.420665i
\(96\) 0 0
\(97\) −108.103 −1.11447 −0.557233 0.830356i \(-0.688137\pi\)
−0.557233 + 0.830356i \(0.688137\pi\)
\(98\) 225.750i 2.30357i
\(99\) 0 0
\(100\) −148.041 −1.48041
\(101\) − 116.418i − 1.15265i −0.817221 0.576324i \(-0.804486\pi\)
0.817221 0.576324i \(-0.195514\pi\)
\(102\) 0 0
\(103\) −109.280 −1.06097 −0.530484 0.847695i \(-0.677990\pi\)
−0.530484 + 0.847695i \(0.677990\pi\)
\(104\) 77.8815i 0.748861i
\(105\) 0 0
\(106\) 161.107 1.51988
\(107\) 43.1743i 0.403498i 0.979437 + 0.201749i \(0.0646626\pi\)
−0.979437 + 0.201749i \(0.935337\pi\)
\(108\) 0 0
\(109\) 75.3063 0.690884 0.345442 0.938440i \(-0.387729\pi\)
0.345442 + 0.938440i \(0.387729\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −11.6738 −0.104231
\(113\) 3.52202i 0.0311683i 0.999879 + 0.0155842i \(0.00496080\pi\)
−0.999879 + 0.0155842i \(0.995039\pi\)
\(114\) 0 0
\(115\) 58.4414 0.508186
\(116\) 161.149i 1.38921i
\(117\) 0 0
\(118\) −78.4017 −0.664421
\(119\) 52.4023i 0.440355i
\(120\) 0 0
\(121\) 0 0
\(122\) 143.971i 1.18009i
\(123\) 0 0
\(124\) −383.451 −3.09235
\(125\) − 75.6024i − 0.604820i
\(126\) 0 0
\(127\) −237.588 −1.87077 −0.935385 0.353630i \(-0.884947\pi\)
−0.935385 + 0.353630i \(0.884947\pi\)
\(128\) 213.197i 1.66561i
\(129\) 0 0
\(130\) −47.8882 −0.368371
\(131\) 64.9832i 0.496055i 0.968753 + 0.248027i \(0.0797823\pi\)
−0.968753 + 0.248027i \(0.920218\pi\)
\(132\) 0 0
\(133\) 272.959 2.05232
\(134\) 107.685i 0.803616i
\(135\) 0 0
\(136\) −40.6048 −0.298565
\(137\) − 125.513i − 0.916152i −0.888913 0.458076i \(-0.848539\pi\)
0.888913 0.458076i \(-0.151461\pi\)
\(138\) 0 0
\(139\) 83.9520 0.603971 0.301986 0.953312i \(-0.402351\pi\)
0.301986 + 0.953312i \(0.402351\pi\)
\(140\) − 114.215i − 0.815822i
\(141\) 0 0
\(142\) −243.875 −1.71743
\(143\) 0 0
\(144\) 0 0
\(145\) −38.9483 −0.268609
\(146\) 134.651i 0.922267i
\(147\) 0 0
\(148\) −84.3444 −0.569894
\(149\) 208.842i 1.40162i 0.713346 + 0.700812i \(0.247179\pi\)
−0.713346 + 0.700812i \(0.752821\pi\)
\(150\) 0 0
\(151\) 62.1096 0.411322 0.205661 0.978623i \(-0.434066\pi\)
0.205661 + 0.978623i \(0.434066\pi\)
\(152\) 211.507i 1.39149i
\(153\) 0 0
\(154\) 0 0
\(155\) − 92.6771i − 0.597917i
\(156\) 0 0
\(157\) −107.364 −0.683847 −0.341923 0.939728i \(-0.611078\pi\)
−0.341923 + 0.939728i \(0.611078\pi\)
\(158\) 208.179i 1.31759i
\(159\) 0 0
\(160\) −48.1524 −0.300952
\(161\) − 399.170i − 2.47932i
\(162\) 0 0
\(163\) 203.397 1.24783 0.623916 0.781491i \(-0.285541\pi\)
0.623916 + 0.781491i \(0.285541\pi\)
\(164\) − 81.8324i − 0.498978i
\(165\) 0 0
\(166\) 137.286 0.827021
\(167\) − 316.071i − 1.89264i −0.323230 0.946320i \(-0.604769\pi\)
0.323230 0.946320i \(-0.395231\pi\)
\(168\) 0 0
\(169\) −83.6560 −0.495006
\(170\) − 24.9673i − 0.146867i
\(171\) 0 0
\(172\) −129.823 −0.754783
\(173\) 114.001i 0.658963i 0.944162 + 0.329482i \(0.106874\pi\)
−0.944162 + 0.329482i \(0.893126\pi\)
\(174\) 0 0
\(175\) −244.390 −1.39651
\(176\) 0 0
\(177\) 0 0
\(178\) 361.590 2.03140
\(179\) − 145.784i − 0.814436i −0.913331 0.407218i \(-0.866499\pi\)
0.913331 0.407218i \(-0.133501\pi\)
\(180\) 0 0
\(181\) −347.607 −1.92048 −0.960241 0.279171i \(-0.909940\pi\)
−0.960241 + 0.279171i \(0.909940\pi\)
\(182\) 327.089i 1.79719i
\(183\) 0 0
\(184\) 309.304 1.68100
\(185\) − 20.3854i − 0.110191i
\(186\) 0 0
\(187\) 0 0
\(188\) 428.030i 2.27676i
\(189\) 0 0
\(190\) −130.053 −0.684487
\(191\) − 74.8885i − 0.392086i −0.980595 0.196043i \(-0.937191\pi\)
0.980595 0.196043i \(-0.0628093\pi\)
\(192\) 0 0
\(193\) −31.8200 −0.164871 −0.0824353 0.996596i \(-0.526270\pi\)
−0.0824353 + 0.996596i \(0.526270\pi\)
\(194\) − 351.801i − 1.81341i
\(195\) 0 0
\(196\) −457.183 −2.33256
\(197\) − 165.104i − 0.838089i −0.907966 0.419045i \(-0.862365\pi\)
0.907966 0.419045i \(-0.137635\pi\)
\(198\) 0 0
\(199\) 195.093 0.980365 0.490182 0.871620i \(-0.336930\pi\)
0.490182 + 0.871620i \(0.336930\pi\)
\(200\) − 189.370i − 0.946848i
\(201\) 0 0
\(202\) 378.859 1.87554
\(203\) 266.027i 1.31048i
\(204\) 0 0
\(205\) 19.7782 0.0964792
\(206\) − 355.630i − 1.72636i
\(207\) 0 0
\(208\) −9.91243 −0.0476559
\(209\) 0 0
\(210\) 0 0
\(211\) 182.295 0.863959 0.431980 0.901883i \(-0.357815\pi\)
0.431980 + 0.901883i \(0.357815\pi\)
\(212\) 326.270i 1.53901i
\(213\) 0 0
\(214\) −140.503 −0.656554
\(215\) − 31.3771i − 0.145940i
\(216\) 0 0
\(217\) −633.009 −2.91709
\(218\) 245.070i 1.12417i
\(219\) 0 0
\(220\) 0 0
\(221\) 44.4955i 0.201337i
\(222\) 0 0
\(223\) −114.587 −0.513843 −0.256921 0.966432i \(-0.582708\pi\)
−0.256921 + 0.966432i \(0.582708\pi\)
\(224\) 328.893i 1.46827i
\(225\) 0 0
\(226\) −11.4618 −0.0507157
\(227\) 233.643i 1.02926i 0.857411 + 0.514632i \(0.172071\pi\)
−0.857411 + 0.514632i \(0.827929\pi\)
\(228\) 0 0
\(229\) 139.279 0.608206 0.304103 0.952639i \(-0.401643\pi\)
0.304103 + 0.952639i \(0.401643\pi\)
\(230\) 190.187i 0.826898i
\(231\) 0 0
\(232\) −206.136 −0.888517
\(233\) − 118.544i − 0.508772i −0.967103 0.254386i \(-0.918127\pi\)
0.967103 0.254386i \(-0.0818734\pi\)
\(234\) 0 0
\(235\) −103.452 −0.440220
\(236\) − 158.777i − 0.672783i
\(237\) 0 0
\(238\) −170.533 −0.716526
\(239\) 288.314i 1.20634i 0.797614 + 0.603168i \(0.206095\pi\)
−0.797614 + 0.603168i \(0.793905\pi\)
\(240\) 0 0
\(241\) −115.517 −0.479324 −0.239662 0.970856i \(-0.577037\pi\)
−0.239662 + 0.970856i \(0.577037\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −291.566 −1.19494
\(245\) − 110.497i − 0.451010i
\(246\) 0 0
\(247\) 231.773 0.938354
\(248\) − 490.498i − 1.97782i
\(249\) 0 0
\(250\) 246.034 0.984135
\(251\) − 34.4392i − 0.137208i −0.997644 0.0686040i \(-0.978145\pi\)
0.997644 0.0686040i \(-0.0218545\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) − 773.185i − 3.04403i
\(255\) 0 0
\(256\) −283.135 −1.10600
\(257\) − 208.411i − 0.810937i −0.914109 0.405469i \(-0.867108\pi\)
0.914109 0.405469i \(-0.132892\pi\)
\(258\) 0 0
\(259\) −139.237 −0.537596
\(260\) − 96.9817i − 0.373007i
\(261\) 0 0
\(262\) −211.475 −0.807158
\(263\) − 230.901i − 0.877952i −0.898499 0.438976i \(-0.855341\pi\)
0.898499 0.438976i \(-0.144659\pi\)
\(264\) 0 0
\(265\) −78.8569 −0.297573
\(266\) 888.293i 3.33945i
\(267\) 0 0
\(268\) −218.080 −0.813730
\(269\) − 11.0843i − 0.0412055i −0.999788 0.0206027i \(-0.993441\pi\)
0.999788 0.0206027i \(-0.00655852\pi\)
\(270\) 0 0
\(271\) 100.794 0.371935 0.185968 0.982556i \(-0.440458\pi\)
0.185968 + 0.982556i \(0.440458\pi\)
\(272\) − 5.16801i − 0.0190000i
\(273\) 0 0
\(274\) 408.458 1.49072
\(275\) 0 0
\(276\) 0 0
\(277\) −53.4162 −0.192838 −0.0964191 0.995341i \(-0.530739\pi\)
−0.0964191 + 0.995341i \(0.530739\pi\)
\(278\) 273.206i 0.982755i
\(279\) 0 0
\(280\) 146.100 0.521787
\(281\) − 369.254i − 1.31407i −0.753859 0.657036i \(-0.771810\pi\)
0.753859 0.657036i \(-0.228190\pi\)
\(282\) 0 0
\(283\) 241.063 0.851814 0.425907 0.904767i \(-0.359955\pi\)
0.425907 + 0.904767i \(0.359955\pi\)
\(284\) − 493.888i − 1.73904i
\(285\) 0 0
\(286\) 0 0
\(287\) − 135.090i − 0.470699i
\(288\) 0 0
\(289\) 265.802 0.919728
\(290\) − 126.750i − 0.437069i
\(291\) 0 0
\(292\) −272.691 −0.933874
\(293\) 55.1299i 0.188157i 0.995565 + 0.0940783i \(0.0299904\pi\)
−0.995565 + 0.0940783i \(0.970010\pi\)
\(294\) 0 0
\(295\) 38.3751 0.130085
\(296\) − 107.891i − 0.364495i
\(297\) 0 0
\(298\) −679.637 −2.28066
\(299\) − 338.941i − 1.13358i
\(300\) 0 0
\(301\) −214.314 −0.712006
\(302\) 202.124i 0.669285i
\(303\) 0 0
\(304\) −26.9197 −0.0885518
\(305\) − 70.4692i − 0.231047i
\(306\) 0 0
\(307\) −12.5104 −0.0407504 −0.0203752 0.999792i \(-0.506486\pi\)
−0.0203752 + 0.999792i \(0.506486\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 301.600 0.972904
\(311\) − 215.665i − 0.693458i −0.937965 0.346729i \(-0.887292\pi\)
0.937965 0.346729i \(-0.112708\pi\)
\(312\) 0 0
\(313\) 435.646 1.39184 0.695920 0.718120i \(-0.254997\pi\)
0.695920 + 0.718120i \(0.254997\pi\)
\(314\) − 349.396i − 1.11272i
\(315\) 0 0
\(316\) −421.598 −1.33417
\(317\) − 313.816i − 0.989956i −0.868905 0.494978i \(-0.835176\pi\)
0.868905 0.494978i \(-0.164824\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) − 163.539i − 0.511060i
\(321\) 0 0
\(322\) 1299.02 4.03423
\(323\) 120.839i 0.374114i
\(324\) 0 0
\(325\) −207.515 −0.638507
\(326\) 661.916i 2.03042i
\(327\) 0 0
\(328\) 104.677 0.319138
\(329\) 706.601i 2.14772i
\(330\) 0 0
\(331\) 69.5045 0.209983 0.104992 0.994473i \(-0.466518\pi\)
0.104992 + 0.994473i \(0.466518\pi\)
\(332\) 278.027i 0.837430i
\(333\) 0 0
\(334\) 1028.59 3.07962
\(335\) − 52.7081i − 0.157338i
\(336\) 0 0
\(337\) −108.794 −0.322832 −0.161416 0.986886i \(-0.551606\pi\)
−0.161416 + 0.986886i \(0.551606\pi\)
\(338\) − 272.243i − 0.805452i
\(339\) 0 0
\(340\) 50.5631 0.148715
\(341\) 0 0
\(342\) 0 0
\(343\) −221.617 −0.646113
\(344\) − 166.065i − 0.482747i
\(345\) 0 0
\(346\) −370.993 −1.07224
\(347\) − 382.854i − 1.10332i −0.834068 0.551662i \(-0.813994\pi\)
0.834068 0.551662i \(-0.186006\pi\)
\(348\) 0 0
\(349\) 441.623 1.26540 0.632698 0.774399i \(-0.281948\pi\)
0.632698 + 0.774399i \(0.281948\pi\)
\(350\) − 795.319i − 2.27234i
\(351\) 0 0
\(352\) 0 0
\(353\) − 237.587i − 0.673051i −0.941674 0.336525i \(-0.890748\pi\)
0.941674 0.336525i \(-0.109252\pi\)
\(354\) 0 0
\(355\) 119.369 0.336250
\(356\) 732.281i 2.05697i
\(357\) 0 0
\(358\) 474.427 1.32521
\(359\) − 385.744i − 1.07450i −0.843424 0.537248i \(-0.819464\pi\)
0.843424 0.537248i \(-0.180536\pi\)
\(360\) 0 0
\(361\) 268.440 0.743600
\(362\) − 1131.22i − 3.12492i
\(363\) 0 0
\(364\) −662.410 −1.81981
\(365\) − 65.9073i − 0.180568i
\(366\) 0 0
\(367\) 154.272 0.420361 0.210180 0.977663i \(-0.432595\pi\)
0.210180 + 0.977663i \(0.432595\pi\)
\(368\) 39.3669i 0.106975i
\(369\) 0 0
\(370\) 66.3403 0.179298
\(371\) 538.613i 1.45179i
\(372\) 0 0
\(373\) 440.608 1.18125 0.590627 0.806944i \(-0.298880\pi\)
0.590627 + 0.806944i \(0.298880\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −547.523 −1.45618
\(377\) 225.888i 0.599171i
\(378\) 0 0
\(379\) 556.553 1.46848 0.734239 0.678891i \(-0.237539\pi\)
0.734239 + 0.678891i \(0.237539\pi\)
\(380\) − 263.379i − 0.693102i
\(381\) 0 0
\(382\) 243.710 0.637985
\(383\) 125.668i 0.328115i 0.986451 + 0.164057i \(0.0524582\pi\)
−0.986451 + 0.164057i \(0.947542\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) − 103.552i − 0.268270i
\(387\) 0 0
\(388\) 712.457 1.83623
\(389\) 67.1506i 0.172624i 0.996268 + 0.0863119i \(0.0275081\pi\)
−0.996268 + 0.0863119i \(0.972492\pi\)
\(390\) 0 0
\(391\) 176.713 0.451951
\(392\) − 584.813i − 1.49187i
\(393\) 0 0
\(394\) 537.298 1.36370
\(395\) − 101.897i − 0.257967i
\(396\) 0 0
\(397\) 342.588 0.862942 0.431471 0.902127i \(-0.357995\pi\)
0.431471 + 0.902127i \(0.357995\pi\)
\(398\) 634.892i 1.59521i
\(399\) 0 0
\(400\) 24.1022 0.0602555
\(401\) 370.533i 0.924022i 0.886874 + 0.462011i \(0.152872\pi\)
−0.886874 + 0.462011i \(0.847128\pi\)
\(402\) 0 0
\(403\) −537.497 −1.33374
\(404\) 767.254i 1.89914i
\(405\) 0 0
\(406\) −865.735 −2.13235
\(407\) 0 0
\(408\) 0 0
\(409\) −320.164 −0.782797 −0.391398 0.920221i \(-0.628009\pi\)
−0.391398 + 0.920221i \(0.628009\pi\)
\(410\) 64.3645i 0.156987i
\(411\) 0 0
\(412\) 720.211 1.74808
\(413\) − 262.112i − 0.634653i
\(414\) 0 0
\(415\) −67.1969 −0.161920
\(416\) 279.268i 0.671317i
\(417\) 0 0
\(418\) 0 0
\(419\) 574.973i 1.37225i 0.727483 + 0.686126i \(0.240690\pi\)
−0.727483 + 0.686126i \(0.759310\pi\)
\(420\) 0 0
\(421\) 141.737 0.336668 0.168334 0.985730i \(-0.446161\pi\)
0.168334 + 0.985730i \(0.446161\pi\)
\(422\) 593.246i 1.40580i
\(423\) 0 0
\(424\) −417.354 −0.984326
\(425\) − 108.191i − 0.254568i
\(426\) 0 0
\(427\) −481.323 −1.12722
\(428\) − 284.542i − 0.664817i
\(429\) 0 0
\(430\) 102.111 0.237467
\(431\) 391.659i 0.908722i 0.890818 + 0.454361i \(0.150132\pi\)
−0.890818 + 0.454361i \(0.849868\pi\)
\(432\) 0 0
\(433\) −343.995 −0.794447 −0.397223 0.917722i \(-0.630026\pi\)
−0.397223 + 0.917722i \(0.630026\pi\)
\(434\) − 2060.01i − 4.74656i
\(435\) 0 0
\(436\) −496.309 −1.13832
\(437\) − 920.481i − 2.10636i
\(438\) 0 0
\(439\) 36.1143 0.0822650 0.0411325 0.999154i \(-0.486903\pi\)
0.0411325 + 0.999154i \(0.486903\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −144.802 −0.327607
\(443\) − 382.066i − 0.862452i −0.902244 0.431226i \(-0.858081\pi\)
0.902244 0.431226i \(-0.141919\pi\)
\(444\) 0 0
\(445\) −176.987 −0.397723
\(446\) − 372.901i − 0.836101i
\(447\) 0 0
\(448\) −1117.02 −2.49334
\(449\) 30.7104i 0.0683974i 0.999415 + 0.0341987i \(0.0108879\pi\)
−0.999415 + 0.0341987i \(0.989112\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) − 23.2120i − 0.0513540i
\(453\) 0 0
\(454\) −760.346 −1.67477
\(455\) − 160.099i − 0.351867i
\(456\) 0 0
\(457\) 412.699 0.903061 0.451531 0.892256i \(-0.350878\pi\)
0.451531 + 0.892256i \(0.350878\pi\)
\(458\) 453.258i 0.989645i
\(459\) 0 0
\(460\) −385.160 −0.837305
\(461\) − 765.072i − 1.65959i −0.558066 0.829797i \(-0.688456\pi\)
0.558066 0.829797i \(-0.311544\pi\)
\(462\) 0 0
\(463\) 113.587 0.245328 0.122664 0.992448i \(-0.460856\pi\)
0.122664 + 0.992448i \(0.460856\pi\)
\(464\) − 26.2361i − 0.0565434i
\(465\) 0 0
\(466\) 385.779 0.827852
\(467\) 756.817i 1.62059i 0.586019 + 0.810297i \(0.300694\pi\)
−0.586019 + 0.810297i \(0.699306\pi\)
\(468\) 0 0
\(469\) −360.010 −0.767612
\(470\) − 336.664i − 0.716305i
\(471\) 0 0
\(472\) 203.102 0.430301
\(473\) 0 0
\(474\) 0 0
\(475\) −563.559 −1.18644
\(476\) − 345.359i − 0.725544i
\(477\) 0 0
\(478\) −938.264 −1.96290
\(479\) 364.563i 0.761091i 0.924762 + 0.380546i \(0.124264\pi\)
−0.924762 + 0.380546i \(0.875736\pi\)
\(480\) 0 0
\(481\) −118.228 −0.245797
\(482\) − 375.929i − 0.779935i
\(483\) 0 0
\(484\) 0 0
\(485\) 172.195i 0.355042i
\(486\) 0 0
\(487\) −240.050 −0.492915 −0.246458 0.969154i \(-0.579267\pi\)
−0.246458 + 0.969154i \(0.579267\pi\)
\(488\) − 372.962i − 0.764266i
\(489\) 0 0
\(490\) 359.593 0.733863
\(491\) − 837.796i − 1.70630i −0.521662 0.853152i \(-0.674688\pi\)
0.521662 0.853152i \(-0.325312\pi\)
\(492\) 0 0
\(493\) −117.770 −0.238885
\(494\) 754.262i 1.52685i
\(495\) 0 0
\(496\) 62.4286 0.125864
\(497\) − 815.320i − 1.64048i
\(498\) 0 0
\(499\) 644.663 1.29191 0.645955 0.763376i \(-0.276460\pi\)
0.645955 + 0.763376i \(0.276460\pi\)
\(500\) 498.260i 0.996521i
\(501\) 0 0
\(502\) 112.076 0.223259
\(503\) 10.5076i 0.0208898i 0.999945 + 0.0104449i \(0.00332478\pi\)
−0.999945 + 0.0104449i \(0.996675\pi\)
\(504\) 0 0
\(505\) −185.439 −0.367206
\(506\) 0 0
\(507\) 0 0
\(508\) 1565.83 3.08234
\(509\) − 915.705i − 1.79903i −0.436893 0.899514i \(-0.643921\pi\)
0.436893 0.899514i \(-0.356079\pi\)
\(510\) 0 0
\(511\) −450.164 −0.880947
\(512\) − 68.6188i − 0.134021i
\(513\) 0 0
\(514\) 678.233 1.31952
\(515\) 174.069i 0.337999i
\(516\) 0 0
\(517\) 0 0
\(518\) − 453.121i − 0.874752i
\(519\) 0 0
\(520\) 124.056 0.238569
\(521\) − 235.989i − 0.452953i −0.974017 0.226477i \(-0.927279\pi\)
0.974017 0.226477i \(-0.0727207\pi\)
\(522\) 0 0
\(523\) −151.331 −0.289351 −0.144676 0.989479i \(-0.546214\pi\)
−0.144676 + 0.989479i \(0.546214\pi\)
\(524\) − 428.274i − 0.817316i
\(525\) 0 0
\(526\) 751.425 1.42856
\(527\) − 280.233i − 0.531752i
\(528\) 0 0
\(529\) −817.095 −1.54460
\(530\) − 256.625i − 0.484198i
\(531\) 0 0
\(532\) −1798.94 −3.38147
\(533\) − 114.707i − 0.215211i
\(534\) 0 0
\(535\) 68.7715 0.128545
\(536\) − 278.960i − 0.520448i
\(537\) 0 0
\(538\) 36.0716 0.0670477
\(539\) 0 0
\(540\) 0 0
\(541\) 300.434 0.555331 0.277665 0.960678i \(-0.410439\pi\)
0.277665 + 0.960678i \(0.410439\pi\)
\(542\) 328.016i 0.605196i
\(543\) 0 0
\(544\) −145.601 −0.267649
\(545\) − 119.954i − 0.220099i
\(546\) 0 0
\(547\) −324.960 −0.594077 −0.297038 0.954866i \(-0.595999\pi\)
−0.297038 + 0.954866i \(0.595999\pi\)
\(548\) 827.196i 1.50948i
\(549\) 0 0
\(550\) 0 0
\(551\) 613.455i 1.11335i
\(552\) 0 0
\(553\) −695.983 −1.25856
\(554\) − 173.833i − 0.313778i
\(555\) 0 0
\(556\) −553.289 −0.995123
\(557\) − 45.2931i − 0.0813162i −0.999173 0.0406581i \(-0.987055\pi\)
0.999173 0.0406581i \(-0.0129454\pi\)
\(558\) 0 0
\(559\) −181.977 −0.325540
\(560\) 18.5950i 0.0332054i
\(561\) 0 0
\(562\) 1201.67 2.13820
\(563\) 728.842i 1.29457i 0.762249 + 0.647284i \(0.224095\pi\)
−0.762249 + 0.647284i \(0.775905\pi\)
\(564\) 0 0
\(565\) 5.61016 0.00992948
\(566\) 784.495i 1.38603i
\(567\) 0 0
\(568\) 631.765 1.11226
\(569\) − 146.146i − 0.256847i −0.991719 0.128423i \(-0.959008\pi\)
0.991719 0.128423i \(-0.0409916\pi\)
\(570\) 0 0
\(571\) 398.941 0.698672 0.349336 0.936998i \(-0.386407\pi\)
0.349336 + 0.936998i \(0.386407\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 439.626 0.765899
\(575\) 824.139i 1.43328i
\(576\) 0 0
\(577\) 252.489 0.437589 0.218795 0.975771i \(-0.429788\pi\)
0.218795 + 0.975771i \(0.429788\pi\)
\(578\) 865.000i 1.49654i
\(579\) 0 0
\(580\) 256.690 0.442569
\(581\) 458.972i 0.789969i
\(582\) 0 0
\(583\) 0 0
\(584\) − 348.818i − 0.597290i
\(585\) 0 0
\(586\) −179.410 −0.306160
\(587\) − 985.570i − 1.67900i −0.543363 0.839498i \(-0.682849\pi\)
0.543363 0.839498i \(-0.317151\pi\)
\(588\) 0 0
\(589\) −1459.71 −2.47829
\(590\) 124.884i 0.211669i
\(591\) 0 0
\(592\) 13.7319 0.0231957
\(593\) 795.418i 1.34135i 0.741753 + 0.670673i \(0.233995\pi\)
−0.741753 + 0.670673i \(0.766005\pi\)
\(594\) 0 0
\(595\) 83.4705 0.140287
\(596\) − 1376.38i − 2.30936i
\(597\) 0 0
\(598\) 1103.02 1.84451
\(599\) 114.903i 0.191825i 0.995390 + 0.0959126i \(0.0305769\pi\)
−0.995390 + 0.0959126i \(0.969423\pi\)
\(600\) 0 0
\(601\) −294.608 −0.490196 −0.245098 0.969498i \(-0.578820\pi\)
−0.245098 + 0.969498i \(0.578820\pi\)
\(602\) − 697.444i − 1.15854i
\(603\) 0 0
\(604\) −409.336 −0.677708
\(605\) 0 0
\(606\) 0 0
\(607\) −435.121 −0.716838 −0.358419 0.933561i \(-0.616684\pi\)
−0.358419 + 0.933561i \(0.616684\pi\)
\(608\) 758.423i 1.24741i
\(609\) 0 0
\(610\) 229.329 0.375949
\(611\) 599.985i 0.981973i
\(612\) 0 0
\(613\) 628.161 1.02473 0.512366 0.858767i \(-0.328769\pi\)
0.512366 + 0.858767i \(0.328769\pi\)
\(614\) − 40.7126i − 0.0663072i
\(615\) 0 0
\(616\) 0 0
\(617\) 215.080i 0.348590i 0.984694 + 0.174295i \(0.0557646\pi\)
−0.984694 + 0.174295i \(0.944235\pi\)
\(618\) 0 0
\(619\) −956.887 −1.54586 −0.772929 0.634492i \(-0.781209\pi\)
−0.772929 + 0.634492i \(0.781209\pi\)
\(620\) 610.792i 0.985148i
\(621\) 0 0
\(622\) 701.842 1.12836
\(623\) 1208.86i 1.94039i
\(624\) 0 0
\(625\) 441.143 0.705828
\(626\) 1417.73i 2.26474i
\(627\) 0 0
\(628\) 707.585 1.12673
\(629\) − 61.6404i − 0.0979975i
\(630\) 0 0
\(631\) −637.996 −1.01109 −0.505544 0.862801i \(-0.668708\pi\)
−0.505544 + 0.862801i \(0.668708\pi\)
\(632\) − 539.295i − 0.853315i
\(633\) 0 0
\(634\) 1021.25 1.61081
\(635\) 378.449i 0.595983i
\(636\) 0 0
\(637\) −640.849 −1.00604
\(638\) 0 0
\(639\) 0 0
\(640\) 339.598 0.530622
\(641\) 958.209i 1.49487i 0.664337 + 0.747433i \(0.268714\pi\)
−0.664337 + 0.747433i \(0.731286\pi\)
\(642\) 0 0
\(643\) −294.864 −0.458575 −0.229288 0.973359i \(-0.573640\pi\)
−0.229288 + 0.973359i \(0.573640\pi\)
\(644\) 2630.74i 4.08500i
\(645\) 0 0
\(646\) −393.247 −0.608742
\(647\) − 1005.11i − 1.55350i −0.629809 0.776750i \(-0.716867\pi\)
0.629809 0.776750i \(-0.283133\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) − 675.317i − 1.03895i
\(651\) 0 0
\(652\) −1340.49 −2.05597
\(653\) 1035.63i 1.58595i 0.609253 + 0.792976i \(0.291469\pi\)
−0.609253 + 0.792976i \(0.708531\pi\)
\(654\) 0 0
\(655\) 103.510 0.158031
\(656\) 13.3229i 0.0203093i
\(657\) 0 0
\(658\) −2299.50 −3.49468
\(659\) 572.524i 0.868777i 0.900726 + 0.434388i \(0.143035\pi\)
−0.900726 + 0.434388i \(0.856965\pi\)
\(660\) 0 0
\(661\) −717.443 −1.08539 −0.542695 0.839930i \(-0.682596\pi\)
−0.542695 + 0.839930i \(0.682596\pi\)
\(662\) 226.189i 0.341675i
\(663\) 0 0
\(664\) −355.643 −0.535606
\(665\) − 434.791i − 0.653821i
\(666\) 0 0
\(667\) 897.106 1.34499
\(668\) 2083.08i 3.11838i
\(669\) 0 0
\(670\) 171.529 0.256013
\(671\) 0 0
\(672\) 0 0
\(673\) −872.035 −1.29574 −0.647872 0.761749i \(-0.724341\pi\)
−0.647872 + 0.761749i \(0.724341\pi\)
\(674\) − 354.050i − 0.525297i
\(675\) 0 0
\(676\) 551.338 0.815588
\(677\) − 1276.05i − 1.88487i −0.334394 0.942433i \(-0.608532\pi\)
0.334394 0.942433i \(-0.391468\pi\)
\(678\) 0 0
\(679\) 1176.14 1.73216
\(680\) 64.6786i 0.0951156i
\(681\) 0 0
\(682\) 0 0
\(683\) 411.727i 0.602822i 0.953494 + 0.301411i \(0.0974575\pi\)
−0.953494 + 0.301411i \(0.902542\pi\)
\(684\) 0 0
\(685\) −199.927 −0.291864
\(686\) − 721.210i − 1.05133i
\(687\) 0 0
\(688\) 21.1361 0.0307210
\(689\) 457.344i 0.663780i
\(690\) 0 0
\(691\) −832.518 −1.20480 −0.602401 0.798194i \(-0.705789\pi\)
−0.602401 + 0.798194i \(0.705789\pi\)
\(692\) − 751.325i − 1.08573i
\(693\) 0 0
\(694\) 1245.92 1.79528
\(695\) − 133.726i − 0.192411i
\(696\) 0 0
\(697\) 59.8046 0.0858029
\(698\) 1437.18i 2.05899i
\(699\) 0 0
\(700\) 1610.66 2.30094
\(701\) 90.8137i 0.129549i 0.997900 + 0.0647744i \(0.0206328\pi\)
−0.997900 + 0.0647744i \(0.979367\pi\)
\(702\) 0 0
\(703\) −321.080 −0.456728
\(704\) 0 0
\(705\) 0 0
\(706\) 773.182 1.09516
\(707\) 1266.60i 1.79151i
\(708\) 0 0
\(709\) −684.473 −0.965406 −0.482703 0.875784i \(-0.660345\pi\)
−0.482703 + 0.875784i \(0.660345\pi\)
\(710\) 388.463i 0.547131i
\(711\) 0 0
\(712\) −936.711 −1.31560
\(713\) 2134.65i 2.99390i
\(714\) 0 0
\(715\) 0 0
\(716\) 960.795i 1.34189i
\(717\) 0 0
\(718\) 1255.33 1.74837
\(719\) − 1246.85i − 1.73415i −0.498182 0.867073i \(-0.665999\pi\)
0.498182 0.867073i \(-0.334001\pi\)
\(720\) 0 0
\(721\) 1188.94 1.64901
\(722\) 873.586i 1.20995i
\(723\) 0 0
\(724\) 2290.92 3.16425
\(725\) − 549.248i − 0.757583i
\(726\) 0 0
\(727\) −837.811 −1.15242 −0.576211 0.817301i \(-0.695469\pi\)
−0.576211 + 0.817301i \(0.695469\pi\)
\(728\) − 847.334i − 1.16392i
\(729\) 0 0
\(730\) 214.483 0.293812
\(731\) − 94.8768i − 0.129790i
\(732\) 0 0
\(733\) −62.9218 −0.0858415 −0.0429208 0.999078i \(-0.513666\pi\)
−0.0429208 + 0.999078i \(0.513666\pi\)
\(734\) 502.050i 0.683992i
\(735\) 0 0
\(736\) 1109.10 1.50694
\(737\) 0 0
\(738\) 0 0
\(739\) 1180.19 1.59701 0.798504 0.601989i \(-0.205625\pi\)
0.798504 + 0.601989i \(0.205625\pi\)
\(740\) 134.350i 0.181555i
\(741\) 0 0
\(742\) −1752.81 −2.36228
\(743\) − 496.072i − 0.667661i −0.942633 0.333830i \(-0.891659\pi\)
0.942633 0.333830i \(-0.108341\pi\)
\(744\) 0 0
\(745\) 332.660 0.446524
\(746\) 1433.88i 1.92208i
\(747\) 0 0
\(748\) 0 0
\(749\) − 469.727i − 0.627139i
\(750\) 0 0
\(751\) 408.480 0.543914 0.271957 0.962309i \(-0.412329\pi\)
0.271957 + 0.962309i \(0.412329\pi\)
\(752\) − 69.6864i − 0.0926681i
\(753\) 0 0
\(754\) −735.108 −0.974944
\(755\) − 98.9332i − 0.131037i
\(756\) 0 0
\(757\) −757.992 −1.00131 −0.500655 0.865647i \(-0.666907\pi\)
−0.500655 + 0.865647i \(0.666907\pi\)
\(758\) 1811.20i 2.38944i
\(759\) 0 0
\(760\) 336.905 0.443297
\(761\) − 1353.55i − 1.77865i −0.457278 0.889324i \(-0.651175\pi\)
0.457278 0.889324i \(-0.348825\pi\)
\(762\) 0 0
\(763\) −819.316 −1.07381
\(764\) 493.555i 0.646014i
\(765\) 0 0
\(766\) −408.962 −0.533893
\(767\) − 222.563i − 0.290173i
\(768\) 0 0
\(769\) 550.877 0.716355 0.358178 0.933654i \(-0.383398\pi\)
0.358178 + 0.933654i \(0.383398\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 209.711 0.271646
\(773\) − 885.139i − 1.14507i −0.819880 0.572535i \(-0.805960\pi\)
0.819880 0.572535i \(-0.194040\pi\)
\(774\) 0 0
\(775\) 1306.93 1.68636
\(776\) 911.352i 1.17442i
\(777\) 0 0
\(778\) −218.529 −0.280886
\(779\) − 311.517i − 0.399893i
\(780\) 0 0
\(781\) 0 0
\(782\) 575.078i 0.735394i
\(783\) 0 0
\(784\) 74.4326 0.0949395
\(785\) 171.018i 0.217857i
\(786\) 0 0
\(787\) 1481.78 1.88283 0.941413 0.337255i \(-0.109498\pi\)
0.941413 + 0.337255i \(0.109498\pi\)
\(788\) 1088.12i 1.38086i
\(789\) 0 0
\(790\) 331.605 0.419753
\(791\) − 38.3188i − 0.0484435i
\(792\) 0 0
\(793\) −408.698 −0.515382
\(794\) 1114.89i 1.40414i
\(795\) 0 0
\(796\) −1285.76 −1.61528
\(797\) 69.4562i 0.0871471i 0.999050 + 0.0435735i \(0.0138743\pi\)
−0.999050 + 0.0435735i \(0.986126\pi\)
\(798\) 0 0
\(799\) −312.812 −0.391505
\(800\) − 679.043i − 0.848803i
\(801\) 0 0
\(802\) −1205.83 −1.50353
\(803\) 0 0
\(804\) 0 0
\(805\) −635.830 −0.789851
\(806\) − 1749.18i − 2.17020i
\(807\) 0 0
\(808\) −981.446 −1.21466
\(809\) 881.324i 1.08940i 0.838631 + 0.544700i \(0.183356\pi\)
−0.838631 + 0.544700i \(0.816644\pi\)
\(810\) 0 0
\(811\) 1048.72 1.29312 0.646560 0.762863i \(-0.276207\pi\)
0.646560 + 0.762863i \(0.276207\pi\)
\(812\) − 1753.26i − 2.15919i
\(813\) 0 0
\(814\) 0 0
\(815\) − 323.986i − 0.397529i
\(816\) 0 0
\(817\) −494.205 −0.604902
\(818\) − 1041.91i − 1.27373i
\(819\) 0 0
\(820\) −130.349 −0.158962
\(821\) 983.682i 1.19815i 0.800692 + 0.599076i \(0.204465\pi\)
−0.800692 + 0.599076i \(0.795535\pi\)
\(822\) 0 0
\(823\) −977.362 −1.18756 −0.593780 0.804627i \(-0.702365\pi\)
−0.593780 + 0.804627i \(0.702365\pi\)
\(824\) 921.270i 1.11805i
\(825\) 0 0
\(826\) 852.993 1.03268
\(827\) 23.7343i 0.0286993i 0.999897 + 0.0143496i \(0.00456779\pi\)
−0.999897 + 0.0143496i \(0.995432\pi\)
\(828\) 0 0
\(829\) 908.389 1.09576 0.547882 0.836555i \(-0.315434\pi\)
0.547882 + 0.836555i \(0.315434\pi\)
\(830\) − 218.679i − 0.263469i
\(831\) 0 0
\(832\) −948.474 −1.13999
\(833\) − 334.117i − 0.401101i
\(834\) 0 0
\(835\) −503.463 −0.602950
\(836\) 0 0
\(837\) 0 0
\(838\) −1871.14 −2.23287
\(839\) − 780.010i − 0.929690i −0.885392 0.464845i \(-0.846110\pi\)
0.885392 0.464845i \(-0.153890\pi\)
\(840\) 0 0
\(841\) 243.123 0.289088
\(842\) 461.256i 0.547810i
\(843\) 0 0
\(844\) −1201.42 −1.42349
\(845\) 133.254i 0.157697i
\(846\) 0 0
\(847\) 0 0
\(848\) − 53.1191i − 0.0626404i
\(849\) 0 0
\(850\) 352.088 0.414221
\(851\) 469.541i 0.551752i
\(852\) 0 0
\(853\) −642.554 −0.753287 −0.376644 0.926358i \(-0.622922\pi\)
−0.376644 + 0.926358i \(0.622922\pi\)
\(854\) − 1566.37i − 1.83416i
\(855\) 0 0
\(856\) 363.977 0.425206
\(857\) − 442.968i − 0.516883i −0.966027 0.258441i \(-0.916791\pi\)
0.966027 0.258441i \(-0.0832088\pi\)
\(858\) 0 0
\(859\) −95.7522 −0.111469 −0.0557347 0.998446i \(-0.517750\pi\)
−0.0557347 + 0.998446i \(0.517750\pi\)
\(860\) 206.792i 0.240456i
\(861\) 0 0
\(862\) −1274.58 −1.47863
\(863\) − 576.575i − 0.668105i −0.942554 0.334053i \(-0.891584\pi\)
0.942554 0.334053i \(-0.108416\pi\)
\(864\) 0 0
\(865\) 181.589 0.209930
\(866\) − 1119.47i − 1.29269i
\(867\) 0 0
\(868\) 4171.86 4.80630
\(869\) 0 0
\(870\) 0 0
\(871\) −305.690 −0.350964
\(872\) − 634.862i − 0.728053i
\(873\) 0 0
\(874\) 2995.53 3.42738
\(875\) 822.538i 0.940043i
\(876\) 0 0
\(877\) −909.291 −1.03682 −0.518410 0.855132i \(-0.673476\pi\)
−0.518410 + 0.855132i \(0.673476\pi\)
\(878\) 117.527i 0.133858i
\(879\) 0 0
\(880\) 0 0
\(881\) 864.536i 0.981312i 0.871353 + 0.490656i \(0.163243\pi\)
−0.871353 + 0.490656i \(0.836757\pi\)
\(882\) 0 0
\(883\) −1.93310 −0.00218924 −0.00109462 0.999999i \(-0.500348\pi\)
−0.00109462 + 0.999999i \(0.500348\pi\)
\(884\) − 293.249i − 0.331730i
\(885\) 0 0
\(886\) 1243.36 1.40334
\(887\) − 743.312i − 0.838007i −0.907985 0.419004i \(-0.862379\pi\)
0.907985 0.419004i \(-0.137621\pi\)
\(888\) 0 0
\(889\) 2584.90 2.90765
\(890\) − 575.970i − 0.647157i
\(891\) 0 0
\(892\) 755.189 0.846624
\(893\) 1629.41i 1.82465i
\(894\) 0 0
\(895\) −232.217 −0.259460
\(896\) − 2319.54i − 2.58877i
\(897\) 0 0
\(898\) −99.9413 −0.111293
\(899\) − 1422.64i − 1.58247i
\(900\) 0 0
\(901\) −238.444 −0.264644
\(902\) 0 0
\(903\) 0 0
\(904\) 29.6920 0.0328452
\(905\) 553.697i 0.611820i
\(906\) 0 0
\(907\) −715.123 −0.788449 −0.394225 0.919014i \(-0.628987\pi\)
−0.394225 + 0.919014i \(0.628987\pi\)
\(908\) − 1539.83i − 1.69585i
\(909\) 0 0
\(910\) 521.013 0.572542
\(911\) − 382.652i − 0.420035i −0.977698 0.210018i \(-0.932648\pi\)
0.977698 0.210018i \(-0.0673521\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 1343.05i 1.46942i
\(915\) 0 0
\(916\) −917.924 −1.00210
\(917\) − 707.003i − 0.770995i
\(918\) 0 0
\(919\) 383.352 0.417140 0.208570 0.978007i \(-0.433119\pi\)
0.208570 + 0.978007i \(0.433119\pi\)
\(920\) − 492.684i − 0.535526i
\(921\) 0 0
\(922\) 2489.78 2.70042
\(923\) − 692.300i − 0.750054i
\(924\) 0 0
\(925\) 287.474 0.310782
\(926\) 369.646i 0.399186i
\(927\) 0 0
\(928\) −739.163 −0.796512
\(929\) 216.559i 0.233110i 0.993184 + 0.116555i \(0.0371851\pi\)
−0.993184 + 0.116555i \(0.962815\pi\)
\(930\) 0 0
\(931\) −1740.39 −1.86938
\(932\) 781.268i 0.838270i
\(933\) 0 0
\(934\) −2462.92 −2.63696
\(935\) 0 0
\(936\) 0 0
\(937\) −931.108 −0.993712 −0.496856 0.867833i \(-0.665512\pi\)
−0.496856 + 0.867833i \(0.665512\pi\)
\(938\) − 1171.58i − 1.24902i
\(939\) 0 0
\(940\) 681.801 0.725320
\(941\) − 1364.86i − 1.45043i −0.688522 0.725215i \(-0.741740\pi\)
0.688522 0.725215i \(-0.258260\pi\)
\(942\) 0 0
\(943\) −455.557 −0.483093
\(944\) 25.8500i 0.0273835i
\(945\) 0 0
\(946\) 0 0
\(947\) 564.949i 0.596567i 0.954477 + 0.298283i \(0.0964141\pi\)
−0.954477 + 0.298283i \(0.903586\pi\)
\(948\) 0 0
\(949\) −382.241 −0.402783
\(950\) − 1834.00i − 1.93052i
\(951\) 0 0
\(952\) 441.772 0.464046
\(953\) − 199.814i − 0.209668i −0.994490 0.104834i \(-0.966569\pi\)
0.994490 0.104834i \(-0.0334311\pi\)
\(954\) 0 0
\(955\) −119.288 −0.124909
\(956\) − 1900.15i − 1.98760i
\(957\) 0 0
\(958\) −1186.40 −1.23841
\(959\) 1365.55i 1.42393i
\(960\) 0 0
\(961\) 2424.16 2.52254
\(962\) − 384.752i − 0.399950i
\(963\) 0 0
\(964\) 761.319 0.789751
\(965\) 50.6855i 0.0525238i
\(966\) 0 0
\(967\) 267.827 0.276967 0.138483 0.990365i \(-0.455777\pi\)
0.138483 + 0.990365i \(0.455777\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −560.377 −0.577708
\(971\) − 1748.88i − 1.80111i −0.434745 0.900554i \(-0.643161\pi\)
0.434745 0.900554i \(-0.356839\pi\)
\(972\) 0 0
\(973\) −913.380 −0.938725
\(974\) − 781.196i − 0.802050i
\(975\) 0 0
\(976\) 47.4690 0.0486363
\(977\) 917.940i 0.939549i 0.882786 + 0.469775i \(0.155665\pi\)
−0.882786 + 0.469775i \(0.844335\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 728.237i 0.743099i
\(981\) 0 0
\(982\) 2726.45 2.77642
\(983\) 652.287i 0.663568i 0.943355 + 0.331784i \(0.107650\pi\)
−0.943355 + 0.331784i \(0.892350\pi\)
\(984\) 0 0
\(985\) −262.990 −0.266995
\(986\) − 383.261i − 0.388703i
\(987\) 0 0
\(988\) −1527.51 −1.54606
\(989\) 722.717i 0.730755i
\(990\) 0 0
\(991\) 573.705 0.578915 0.289458 0.957191i \(-0.406525\pi\)
0.289458 + 0.957191i \(0.406525\pi\)
\(992\) − 1758.83i − 1.77302i
\(993\) 0 0
\(994\) 2653.30 2.66932
\(995\) − 310.759i − 0.312321i
\(996\) 0 0
\(997\) 333.026 0.334028 0.167014 0.985955i \(-0.446587\pi\)
0.167014 + 0.985955i \(0.446587\pi\)
\(998\) 2097.93i 2.10214i
\(999\) 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 1089.3.b.i.485.15 16
3.2 odd 2 inner 1089.3.b.i.485.2 16
11.5 even 5 99.3.l.a.80.1 yes 32
11.9 even 5 99.3.l.a.26.8 yes 32
11.10 odd 2 1089.3.b.j.485.2 16
33.5 odd 10 99.3.l.a.80.8 yes 32
33.20 odd 10 99.3.l.a.26.1 32
33.32 even 2 1089.3.b.j.485.15 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
99.3.l.a.26.1 32 33.20 odd 10
99.3.l.a.26.8 yes 32 11.9 even 5
99.3.l.a.80.1 yes 32 11.5 even 5
99.3.l.a.80.8 yes 32 33.5 odd 10
1089.3.b.i.485.2 16 3.2 odd 2 inner
1089.3.b.i.485.15 16 1.1 even 1 trivial
1089.3.b.j.485.2 16 11.10 odd 2
1089.3.b.j.485.15 16 33.32 even 2