Properties

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

$q$-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 \(-0.697517\pi\)
0.581457 0.813577i \(-0.302483\pi\)
\(3\) 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 \(-0.216676\pi\)
0.777127 + 0.629344i \(0.216676\pi\)
\(8\) 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 \(-0.384374\pi\)
0.355315 + 0.934747i \(0.384374\pi\)
\(14\) − 35.4062i − 2.52901i
\(15\) 0 0
\(16\) 1.07299 0.0670616
\(17\) 4.81648i 0.283322i 0.989915 + 0.141661i \(0.0452444\pi\)
−0.989915 + 0.141661i \(0.954756\pi\)
\(18\) 0 0
\(19\) 25.0886 1.32045 0.660227 0.751066i \(-0.270460\pi\)
0.660227 + 0.751066i \(0.270460\pi\)
\(20\) 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 \(0.138524\pi\)
−0.906792 + 0.421578i \(0.861476\pi\)
\(30\) 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 \(-0.951615\pi\)
0.988469 0.151423i \(-0.0483854\pi\)
\(42\) 0 0
\(43\) −19.6984 −0.458102 −0.229051 0.973414i \(-0.573562\pi\)
−0.229051 + 0.973414i \(0.573562\pi\)
\(44\) 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 \(-0.618119\pi\)
−0.362624 + 0.931936i \(0.618119\pi\)
\(62\) − 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 \(-0.591462\pi\)
−0.283399 + 0.959002i \(0.591462\pi\)
\(74\) − 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 \(-0.632685\pi\)
−0.404875 + 0.914372i \(0.632685\pi\)
\(80\) − 1.70914i − 0.0213642i
\(81\) 0 0
\(82\) −40.4076 −0.492776
\(83\) 42.1858i 0.508262i 0.967170 + 0.254131i \(0.0817895\pi\)
−0.967170 + 0.254131i \(0.918211\pi\)
\(84\) 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 \(0.195514\pi\)
−0.817221 + 0.576324i \(0.804486\pi\)
\(102\) 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 \(-0.935337\pi\)
0.979437 0.201749i \(-0.0646626\pi\)
\(108\) 0 0
\(109\) −75.3063 −0.690884 −0.345442 0.938440i \(-0.612271\pi\)
−0.345442 + 0.938440i \(0.612271\pi\)
\(110\) 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 \(-0.115053\pi\)
0.935385 + 0.353630i \(0.115053\pi\)
\(128\) − 213.197i − 1.66561i
\(129\) 0 0
\(130\) −47.8882 −0.368371
\(131\) − 64.9832i − 0.496055i −0.968753 0.248027i \(-0.920218\pi\)
0.968753 0.248027i \(-0.0797823\pi\)
\(132\) 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 \(-0.597649\pi\)
−0.301986 + 0.953312i \(0.597649\pi\)
\(140\) 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 \(-0.752821\pi\)
0.713346 0.700812i \(-0.247179\pi\)
\(150\) 0 0
\(151\) −62.1096 −0.411322 −0.205661 0.978623i \(-0.565934\pi\)
−0.205661 + 0.978623i \(0.565934\pi\)
\(152\) 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 \(0.395231\pi\)
−0.323230 + 0.946320i \(0.604769\pi\)
\(168\) 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 \(-0.893126\pi\)
0.944162 0.329482i \(-0.106874\pi\)
\(174\) 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 \(-0.473730\pi\)
0.0824353 + 0.996596i \(0.473730\pi\)
\(194\) 351.801i 1.81341i
\(195\) 0 0
\(196\) −457.183 −2.33256
\(197\) 165.104i 0.838089i 0.907966 + 0.419045i \(0.137635\pi\)
−0.907966 + 0.419045i \(0.862365\pi\)
\(198\) 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 \(-0.642185\pi\)
−0.431980 + 0.901883i \(0.642185\pi\)
\(212\) 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 \(-0.827929\pi\)
0.857411 0.514632i \(-0.172071\pi\)
\(228\) 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 \(0.0818734\pi\)
−0.967103 + 0.254386i \(0.918127\pi\)
\(234\) 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 \(-0.793905\pi\)
0.797614 0.603168i \(-0.206095\pi\)
\(240\) 0 0
\(241\) 115.517 0.479324 0.239662 0.970856i \(-0.422963\pi\)
0.239662 + 0.970856i \(0.422963\pi\)
\(242\) 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 \(0.144659\pi\)
−0.898499 + 0.438976i \(0.855341\pi\)
\(264\) 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 \(-0.559542\pi\)
−0.185968 + 0.982556i \(0.559542\pi\)
\(272\) 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 \(-0.469261\pi\)
0.0964191 + 0.995341i \(0.469261\pi\)
\(278\) 273.206i 0.982755i
\(279\) 0 0
\(280\) 146.100 0.521787
\(281\) 369.254i 1.31407i 0.753859 + 0.657036i \(0.228190\pi\)
−0.753859 + 0.657036i \(0.771810\pi\)
\(282\) 0 0
\(283\) −241.063 −0.851814 −0.425907 0.904767i \(-0.640045\pi\)
−0.425907 + 0.904767i \(0.640045\pi\)
\(284\) − 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 \(-0.970010\pi\)
0.995565 0.0940783i \(-0.0299904\pi\)
\(294\) 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 \(-0.493514\pi\)
0.0203752 + 0.999792i \(0.493514\pi\)
\(308\) 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 \(-0.448394\pi\)
0.161416 + 0.986886i \(0.448394\pi\)
\(338\) 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 \(0.186006\pi\)
−0.834068 + 0.551662i \(0.813994\pi\)
\(348\) 0 0
\(349\) −441.623 −1.26540 −0.632698 0.774399i \(-0.718052\pi\)
−0.632698 + 0.774399i \(0.718052\pi\)
\(350\) − 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 \(0.180536\pi\)
−0.843424 + 0.537248i \(0.819464\pi\)
\(360\) 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 \(-0.701120\pi\)
−0.590627 + 0.806944i \(0.701120\pi\)
\(374\) 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 \(-0.371991\pi\)
0.391398 + 0.920221i \(0.371991\pi\)
\(410\) 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 \(-0.849868\pi\)
0.890818 0.454361i \(-0.150132\pi\)
\(432\) 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 \(-0.513097\pi\)
−0.0411325 + 0.999154i \(0.513097\pi\)
\(440\) 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 \(-0.649122\pi\)
−0.451531 + 0.892256i \(0.649122\pi\)
\(458\) − 453.258i − 0.989645i
\(459\) 0 0
\(460\) −385.160 −0.837305
\(461\) 765.072i 1.65959i 0.558066 + 0.829797i \(0.311544\pi\)
−0.558066 + 0.829797i \(0.688456\pi\)
\(462\) 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 \(-0.875736\pi\)
0.924762 0.380546i \(-0.124264\pi\)
\(480\) 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 \(0.325312\pi\)
−0.521662 + 0.853152i \(0.674688\pi\)
\(492\) 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 \(-0.996675\pi\)
0.999945 0.0104449i \(-0.00332478\pi\)
\(504\) 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 \(-0.453786\pi\)
0.144676 + 0.989479i \(0.453786\pi\)
\(524\) 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 \(-0.589561\pi\)
−0.277665 + 0.960678i \(0.589561\pi\)
\(542\) 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 \(-0.404001\pi\)
0.297038 + 0.954866i \(0.404001\pi\)
\(548\) 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 \(0.0129454\pi\)
−0.999173 + 0.0406581i \(0.987055\pi\)
\(558\) 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 \(-0.775905\pi\)
0.762249 0.647284i \(-0.224095\pi\)
\(564\) 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 \(0.0409916\pi\)
−0.991719 + 0.128423i \(0.959008\pi\)
\(570\) 0 0
\(571\) −398.941 −0.698672 −0.349336 0.936998i \(-0.613593\pi\)
−0.349336 + 0.936998i \(0.613593\pi\)
\(572\) 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 \(-0.766005\pi\)
0.741753 0.670673i \(-0.233995\pi\)
\(594\) 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 \(-0.421180\pi\)
0.245098 + 0.969498i \(0.421180\pi\)
\(602\) 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 \(-0.383316\pi\)
0.358419 + 0.933561i \(0.383316\pi\)
\(608\) 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 \(-0.671231\pi\)
−0.512366 + 0.858767i \(0.671231\pi\)
\(614\) − 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 \(-0.856965\pi\)
0.900726 0.434388i \(-0.143035\pi\)
\(660\) 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 \(-0.275659\pi\)
0.647872 + 0.761749i \(0.275659\pi\)
\(674\) − 354.050i − 0.525297i
\(675\) 0 0
\(676\) 551.338 0.815588
\(677\) 1276.05i 1.88487i 0.334394 + 0.942433i \(0.391468\pi\)
−0.334394 + 0.942433i \(0.608532\pi\)
\(678\) 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 \(-0.979367\pi\)
0.997900 0.0647744i \(-0.0206328\pi\)
\(702\) 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 \(-0.486334\pi\)
0.0429208 + 0.999078i \(0.486334\pi\)
\(734\) − 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 \(-0.794375\pi\)
−0.798504 + 0.601989i \(0.794375\pi\)
\(740\) 134.350i 0.181555i
\(741\) 0 0
\(742\) −1752.81 −2.36228
\(743\) 496.072i 0.667661i 0.942633 + 0.333830i \(0.108341\pi\)
−0.942633 + 0.333830i \(0.891659\pi\)
\(744\) 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 \(0.348825\pi\)
−0.457278 + 0.889324i \(0.651175\pi\)
\(762\) 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 \(-0.616602\pi\)
−0.358178 + 0.933654i \(0.616602\pi\)
\(770\) 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 \(-0.890502\pi\)
−0.941413 + 0.337255i \(0.890502\pi\)
\(788\) − 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 \(-0.816644\pi\)
0.838631 0.544700i \(-0.183356\pi\)
\(810\) 0 0
\(811\) −1048.72 −1.29312 −0.646560 0.762863i \(-0.723793\pi\)
−0.646560 + 0.762863i \(0.723793\pi\)
\(812\) − 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 \(-0.795535\pi\)
0.800692 0.599076i \(-0.204465\pi\)
\(822\) 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 \(-0.995432\pi\)
0.999897 0.0143496i \(-0.00456779\pi\)
\(828\) 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 \(-0.377078\pi\)
0.376644 + 0.926358i \(0.377078\pi\)
\(854\) 1566.37i 1.83416i
\(855\) 0 0
\(856\) 363.977 0.425206
\(857\) 442.968i 0.516883i 0.966027 + 0.258441i \(0.0832088\pi\)
−0.966027 + 0.258441i \(0.916791\pi\)
\(858\) 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 \(-0.326524\pi\)
0.518410 + 0.855132i \(0.326524\pi\)
\(878\) 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 \(0.137621\pi\)
−0.907985 + 0.419004i \(0.862379\pi\)
\(888\) 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 \(-0.566881\pi\)
−0.208570 + 0.978007i \(0.566881\pi\)
\(920\) 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 \(-0.334488\pi\)
0.496856 + 0.867833i \(0.334488\pi\)
\(938\) − 1171.58i − 1.24902i
\(939\) 0 0
\(940\) 681.801 0.725320
\(941\) 1364.86i 1.45043i 0.688522 + 0.725215i \(0.258260\pi\)
−0.688522 + 0.725215i \(0.741740\pi\)
\(942\) 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 \(0.0334311\pi\)
−0.994490 + 0.104834i \(0.966569\pi\)
\(954\) 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 \(-0.544223\pi\)
−0.138483 + 0.990365i \(0.544223\pi\)
\(968\) 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 \(-0.553413\pi\)
−0.167014 + 0.985955i \(0.553413\pi\)
\(998\) − 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.j.485.2 16
3.2 odd 2 inner 1089.3.b.j.485.15 16
11.2 odd 10 99.3.l.a.26.8 yes 32
11.6 odd 10 99.3.l.a.80.1 yes 32
11.10 odd 2 1089.3.b.i.485.15 16
33.2 even 10 99.3.l.a.26.1 32
33.17 even 10 99.3.l.a.80.8 yes 32
33.32 even 2 1089.3.b.i.485.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
99.3.l.a.26.1 32 33.2 even 10
99.3.l.a.26.8 yes 32 11.2 odd 10
99.3.l.a.80.1 yes 32 11.6 odd 10
99.3.l.a.80.8 yes 32 33.17 even 10
1089.3.b.i.485.2 16 33.32 even 2
1089.3.b.i.485.15 16 11.10 odd 2
1089.3.b.j.485.2 16 1.1 even 1 trivial
1089.3.b.j.485.15 16 3.2 odd 2 inner