Properties

Label 1089.3.b.d.485.1
Level $1089$
Weight $3$
Character 1089.485
Analytic conductor $29.673$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{15})\)
Defining polynomial: \( x^{4} + 16x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 485.1
Root \(-3.44572i\) of defining polynomial
Character \(\chi\) \(=\) 1089.485
Dual form 1089.3.b.d.485.4

$q$-expansion

\(f(q)\) \(=\) \(q-3.44572i q^{2} -7.87298 q^{4} -6.27415i q^{5} +8.87298 q^{7} +13.3452i q^{8} +O(q^{10})\) \(q-3.44572i q^{2} -7.87298 q^{4} -6.27415i q^{5} +8.87298 q^{7} +13.3452i q^{8} -21.6190 q^{10} +24.7460 q^{13} -30.5738i q^{14} +14.4919 q^{16} -4.68030i q^{17} -29.2379 q^{19} +49.3963i q^{20} -27.7454i q^{23} -14.3649 q^{25} -85.2677i q^{26} -69.8569 q^{28} -34.5584i q^{29} +24.7298 q^{31} +3.44572i q^{32} -16.1270 q^{34} -55.6704i q^{35} -1.36492 q^{37} +100.746i q^{38} +83.7298 q^{40} -50.9673i q^{41} -62.8407 q^{43} -95.6028 q^{46} -23.5027i q^{47} +29.7298 q^{49} +49.4975i q^{50} -194.825 q^{52} +4.32105i q^{53} +118.412i q^{56} -119.079 q^{58} +69.2965i q^{59} +36.9839 q^{61} -85.2121i q^{62} +69.8407 q^{64} -155.260i q^{65} -9.12702 q^{67} +36.8480i q^{68} -191.825 q^{70} +33.2454i q^{71} +8.76210 q^{73} +4.70312i q^{74} +230.190 q^{76} +3.01613 q^{79} -90.9245i q^{80} -175.619 q^{82} -43.9746i q^{83} -29.3649 q^{85} +216.531i q^{86} +105.449i q^{89} +219.571 q^{91} +218.439i q^{92} -80.9839 q^{94} +183.443i q^{95} +102.095 q^{97} -102.441i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{4} + 20 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 16 q^{4} + 20 q^{7} - 40 q^{10} + 68 q^{13} - 4 q^{16} - 24 q^{19} + 20 q^{25} - 140 q^{28} - 56 q^{31} - 80 q^{34} + 72 q^{37} + 180 q^{40} + 12 q^{43} - 212 q^{46} - 36 q^{49} - 392 q^{52} - 120 q^{58} + 24 q^{61} + 16 q^{64} - 52 q^{67} - 380 q^{70} + 128 q^{73} + 456 q^{76} + 136 q^{79} - 656 q^{82} - 40 q^{85} + 460 q^{91} - 200 q^{94} + 176 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 3.44572i − 1.72286i −0.507877 0.861430i \(-0.669569\pi\)
0.507877 0.861430i \(-0.330431\pi\)
\(3\) 0 0
\(4\) −7.87298 −1.96825
\(5\) − 6.27415i − 1.25483i −0.778685 0.627415i \(-0.784113\pi\)
0.778685 0.627415i \(-0.215887\pi\)
\(6\) 0 0
\(7\) 8.87298 1.26757 0.633785 0.773510i \(-0.281501\pi\)
0.633785 + 0.773510i \(0.281501\pi\)
\(8\) 13.3452i 1.66815i
\(9\) 0 0
\(10\) −21.6190 −2.16190
\(11\) 0 0
\(12\) 0 0
\(13\) 24.7460 1.90354 0.951768 0.306819i \(-0.0992646\pi\)
0.951768 + 0.306819i \(0.0992646\pi\)
\(14\) − 30.5738i − 2.18384i
\(15\) 0 0
\(16\) 14.4919 0.905746
\(17\) − 4.68030i − 0.275312i −0.990480 0.137656i \(-0.956043\pi\)
0.990480 0.137656i \(-0.0439568\pi\)
\(18\) 0 0
\(19\) −29.2379 −1.53884 −0.769418 0.638745i \(-0.779454\pi\)
−0.769418 + 0.638745i \(0.779454\pi\)
\(20\) 49.3963i 2.46981i
\(21\) 0 0
\(22\) 0 0
\(23\) − 27.7454i − 1.20632i −0.797620 0.603161i \(-0.793908\pi\)
0.797620 0.603161i \(-0.206092\pi\)
\(24\) 0 0
\(25\) −14.3649 −0.574597
\(26\) − 85.2677i − 3.27953i
\(27\) 0 0
\(28\) −69.8569 −2.49489
\(29\) − 34.5584i − 1.19167i −0.803107 0.595835i \(-0.796821\pi\)
0.803107 0.595835i \(-0.203179\pi\)
\(30\) 0 0
\(31\) 24.7298 0.797737 0.398868 0.917008i \(-0.369403\pi\)
0.398868 + 0.917008i \(0.369403\pi\)
\(32\) 3.44572i 0.107679i
\(33\) 0 0
\(34\) −16.1270 −0.474324
\(35\) − 55.6704i − 1.59058i
\(36\) 0 0
\(37\) −1.36492 −0.0368896 −0.0184448 0.999830i \(-0.505872\pi\)
−0.0184448 + 0.999830i \(0.505872\pi\)
\(38\) 100.746i 2.65120i
\(39\) 0 0
\(40\) 83.7298 2.09325
\(41\) − 50.9673i − 1.24310i −0.783373 0.621552i \(-0.786502\pi\)
0.783373 0.621552i \(-0.213498\pi\)
\(42\) 0 0
\(43\) −62.8407 −1.46141 −0.730706 0.682692i \(-0.760809\pi\)
−0.730706 + 0.682692i \(0.760809\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −95.6028 −2.07832
\(47\) − 23.5027i − 0.500058i −0.968238 0.250029i \(-0.919560\pi\)
0.968238 0.250029i \(-0.0804402\pi\)
\(48\) 0 0
\(49\) 29.7298 0.606731
\(50\) 49.4975i 0.989949i
\(51\) 0 0
\(52\) −194.825 −3.74663
\(53\) 4.32105i 0.0815292i 0.999169 + 0.0407646i \(0.0129794\pi\)
−0.999169 + 0.0407646i \(0.987021\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 118.412i 2.11450i
\(57\) 0 0
\(58\) −119.079 −2.05308
\(59\) 69.2965i 1.17452i 0.809400 + 0.587258i \(0.199793\pi\)
−0.809400 + 0.587258i \(0.800207\pi\)
\(60\) 0 0
\(61\) 36.9839 0.606293 0.303146 0.952944i \(-0.401963\pi\)
0.303146 + 0.952944i \(0.401963\pi\)
\(62\) − 85.2121i − 1.37439i
\(63\) 0 0
\(64\) 69.8407 1.09126
\(65\) − 155.260i − 2.38861i
\(66\) 0 0
\(67\) −9.12702 −0.136224 −0.0681121 0.997678i \(-0.521698\pi\)
−0.0681121 + 0.997678i \(0.521698\pi\)
\(68\) 36.8480i 0.541882i
\(69\) 0 0
\(70\) −191.825 −2.74035
\(71\) 33.2454i 0.468245i 0.972207 + 0.234123i \(0.0752218\pi\)
−0.972207 + 0.234123i \(0.924778\pi\)
\(72\) 0 0
\(73\) 8.76210 0.120029 0.0600144 0.998198i \(-0.480885\pi\)
0.0600144 + 0.998198i \(0.480885\pi\)
\(74\) 4.70312i 0.0635557i
\(75\) 0 0
\(76\) 230.190 3.02881
\(77\) 0 0
\(78\) 0 0
\(79\) 3.01613 0.0381789 0.0190895 0.999818i \(-0.493923\pi\)
0.0190895 + 0.999818i \(0.493923\pi\)
\(80\) − 90.9245i − 1.13656i
\(81\) 0 0
\(82\) −175.619 −2.14169
\(83\) − 43.9746i − 0.529815i −0.964274 0.264907i \(-0.914659\pi\)
0.964274 0.264907i \(-0.0853414\pi\)
\(84\) 0 0
\(85\) −29.3649 −0.345470
\(86\) 216.531i 2.51781i
\(87\) 0 0
\(88\) 0 0
\(89\) 105.449i 1.18482i 0.805638 + 0.592409i \(0.201823\pi\)
−0.805638 + 0.592409i \(0.798177\pi\)
\(90\) 0 0
\(91\) 219.571 2.41286
\(92\) 218.439i 2.37434i
\(93\) 0 0
\(94\) −80.9839 −0.861530
\(95\) 183.443i 1.93098i
\(96\) 0 0
\(97\) 102.095 1.05252 0.526262 0.850323i \(-0.323593\pi\)
0.526262 + 0.850323i \(0.323593\pi\)
\(98\) − 102.441i − 1.04531i
\(99\) 0 0
\(100\) 113.095 1.13095
\(101\) 90.8461i 0.899466i 0.893163 + 0.449733i \(0.148481\pi\)
−0.893163 + 0.449733i \(0.851519\pi\)
\(102\) 0 0
\(103\) −17.6351 −0.171214 −0.0856072 0.996329i \(-0.527283\pi\)
−0.0856072 + 0.996329i \(0.527283\pi\)
\(104\) 330.240i 3.17539i
\(105\) 0 0
\(106\) 14.8891 0.140463
\(107\) 4.08583i 0.0381853i 0.999818 + 0.0190927i \(0.00607775\pi\)
−0.999818 + 0.0190927i \(0.993922\pi\)
\(108\) 0 0
\(109\) −95.6976 −0.877959 −0.438980 0.898497i \(-0.644660\pi\)
−0.438980 + 0.898497i \(0.644660\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 128.587 1.14810
\(113\) 118.804i 1.05136i 0.850682 + 0.525681i \(0.176189\pi\)
−0.850682 + 0.525681i \(0.823811\pi\)
\(114\) 0 0
\(115\) −174.079 −1.51373
\(116\) 272.078i 2.34550i
\(117\) 0 0
\(118\) 238.776 2.02353
\(119\) − 41.5283i − 0.348977i
\(120\) 0 0
\(121\) 0 0
\(122\) − 127.436i − 1.04456i
\(123\) 0 0
\(124\) −194.698 −1.57014
\(125\) − 66.7261i − 0.533809i
\(126\) 0 0
\(127\) −112.508 −0.885890 −0.442945 0.896549i \(-0.646066\pi\)
−0.442945 + 0.896549i \(0.646066\pi\)
\(128\) − 226.869i − 1.77241i
\(129\) 0 0
\(130\) −534.982 −4.11524
\(131\) − 15.0403i − 0.114811i −0.998351 0.0574056i \(-0.981717\pi\)
0.998351 0.0574056i \(-0.0182828\pi\)
\(132\) 0 0
\(133\) −259.427 −1.95058
\(134\) 31.4491i 0.234695i
\(135\) 0 0
\(136\) 62.4597 0.459262
\(137\) − 199.470i − 1.45598i −0.685586 0.727991i \(-0.740454\pi\)
0.685586 0.727991i \(-0.259546\pi\)
\(138\) 0 0
\(139\) −152.808 −1.09934 −0.549671 0.835381i \(-0.685247\pi\)
−0.549671 + 0.835381i \(0.685247\pi\)
\(140\) 438.292i 3.13066i
\(141\) 0 0
\(142\) 114.554 0.806721
\(143\) 0 0
\(144\) 0 0
\(145\) −216.825 −1.49534
\(146\) − 30.1917i − 0.206793i
\(147\) 0 0
\(148\) 10.7460 0.0726079
\(149\) 270.664i 1.81653i 0.418391 + 0.908267i \(0.362594\pi\)
−0.418391 + 0.908267i \(0.637406\pi\)
\(150\) 0 0
\(151\) −94.2863 −0.624413 −0.312206 0.950014i \(-0.601068\pi\)
−0.312206 + 0.950014i \(0.601068\pi\)
\(152\) − 390.186i − 2.56701i
\(153\) 0 0
\(154\) 0 0
\(155\) − 155.159i − 1.00102i
\(156\) 0 0
\(157\) 186.157 1.18571 0.592857 0.805307i \(-0.298000\pi\)
0.592857 + 0.805307i \(0.298000\pi\)
\(158\) − 10.3927i − 0.0657769i
\(159\) 0 0
\(160\) 21.6190 0.135118
\(161\) − 246.184i − 1.52910i
\(162\) 0 0
\(163\) −44.3327 −0.271979 −0.135990 0.990710i \(-0.543421\pi\)
−0.135990 + 0.990710i \(0.543421\pi\)
\(164\) 401.265i 2.44673i
\(165\) 0 0
\(166\) −151.524 −0.912796
\(167\) − 152.039i − 0.910415i −0.890385 0.455208i \(-0.849565\pi\)
0.890385 0.455208i \(-0.150435\pi\)
\(168\) 0 0
\(169\) 443.363 2.62345
\(170\) 101.183i 0.595196i
\(171\) 0 0
\(172\) 494.744 2.87642
\(173\) 256.038i 1.47999i 0.672613 + 0.739995i \(0.265172\pi\)
−0.672613 + 0.739995i \(0.734828\pi\)
\(174\) 0 0
\(175\) −127.460 −0.728341
\(176\) 0 0
\(177\) 0 0
\(178\) 363.347 2.04127
\(179\) 113.405i 0.633548i 0.948501 + 0.316774i \(0.102600\pi\)
−0.948501 + 0.316774i \(0.897400\pi\)
\(180\) 0 0
\(181\) 160.270 0.885471 0.442735 0.896652i \(-0.354008\pi\)
0.442735 + 0.896652i \(0.354008\pi\)
\(182\) − 756.579i − 4.15702i
\(183\) 0 0
\(184\) 370.268 2.01233
\(185\) 8.56369i 0.0462902i
\(186\) 0 0
\(187\) 0 0
\(188\) 185.037i 0.984238i
\(189\) 0 0
\(190\) 632.093 3.32680
\(191\) − 75.0218i − 0.392784i −0.980525 0.196392i \(-0.937077\pi\)
0.980525 0.196392i \(-0.0629225\pi\)
\(192\) 0 0
\(193\) 118.903 0.616079 0.308039 0.951374i \(-0.400327\pi\)
0.308039 + 0.951374i \(0.400327\pi\)
\(194\) − 351.790i − 1.81335i
\(195\) 0 0
\(196\) −234.062 −1.19420
\(197\) − 187.271i − 0.950613i −0.879820 0.475306i \(-0.842337\pi\)
0.879820 0.475306i \(-0.157663\pi\)
\(198\) 0 0
\(199\) −29.5706 −0.148596 −0.0742979 0.997236i \(-0.523672\pi\)
−0.0742979 + 0.997236i \(0.523672\pi\)
\(200\) − 191.703i − 0.958514i
\(201\) 0 0
\(202\) 313.030 1.54965
\(203\) − 306.636i − 1.51052i
\(204\) 0 0
\(205\) −319.776 −1.55988
\(206\) 60.7656i 0.294978i
\(207\) 0 0
\(208\) 358.617 1.72412
\(209\) 0 0
\(210\) 0 0
\(211\) 393.825 1.86647 0.933234 0.359270i \(-0.116974\pi\)
0.933234 + 0.359270i \(0.116974\pi\)
\(212\) − 34.0195i − 0.160469i
\(213\) 0 0
\(214\) 14.0786 0.0657879
\(215\) 394.272i 1.83382i
\(216\) 0 0
\(217\) 219.427 1.01119
\(218\) 329.747i 1.51260i
\(219\) 0 0
\(220\) 0 0
\(221\) − 115.819i − 0.524066i
\(222\) 0 0
\(223\) −121.016 −0.542673 −0.271337 0.962485i \(-0.587466\pi\)
−0.271337 + 0.962485i \(0.587466\pi\)
\(224\) 30.5738i 0.136490i
\(225\) 0 0
\(226\) 409.365 1.81135
\(227\) − 160.388i − 0.706554i −0.935519 0.353277i \(-0.885067\pi\)
0.935519 0.353277i \(-0.114933\pi\)
\(228\) 0 0
\(229\) 201.000 0.877729 0.438865 0.898553i \(-0.355381\pi\)
0.438865 + 0.898553i \(0.355381\pi\)
\(230\) 599.826i 2.60794i
\(231\) 0 0
\(232\) 461.190 1.98789
\(233\) − 66.7261i − 0.286378i −0.989695 0.143189i \(-0.954264\pi\)
0.989695 0.143189i \(-0.0457357\pi\)
\(234\) 0 0
\(235\) −147.460 −0.627488
\(236\) − 545.570i − 2.31174i
\(237\) 0 0
\(238\) −143.095 −0.601238
\(239\) − 337.726i − 1.41308i −0.707673 0.706540i \(-0.750255\pi\)
0.707673 0.706540i \(-0.249745\pi\)
\(240\) 0 0
\(241\) −18.9516 −0.0786373 −0.0393187 0.999227i \(-0.512519\pi\)
−0.0393187 + 0.999227i \(0.512519\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −291.173 −1.19333
\(245\) − 186.529i − 0.761344i
\(246\) 0 0
\(247\) −723.520 −2.92923
\(248\) 330.025i 1.33075i
\(249\) 0 0
\(250\) −229.919 −0.919677
\(251\) − 247.487i − 0.986005i −0.870028 0.493003i \(-0.835899\pi\)
0.870028 0.493003i \(-0.164101\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 387.671i 1.52626i
\(255\) 0 0
\(256\) −502.363 −1.96235
\(257\) − 330.733i − 1.28690i −0.765488 0.643450i \(-0.777502\pi\)
0.765488 0.643450i \(-0.222498\pi\)
\(258\) 0 0
\(259\) −12.1109 −0.0467602
\(260\) 1222.36i 4.70138i
\(261\) 0 0
\(262\) −51.8246 −0.197804
\(263\) − 275.658i − 1.04813i −0.851679 0.524064i \(-0.824415\pi\)
0.851679 0.524064i \(-0.175585\pi\)
\(264\) 0 0
\(265\) 27.1109 0.102305
\(266\) 893.914i 3.36058i
\(267\) 0 0
\(268\) 71.8569 0.268123
\(269\) 141.004i 0.524177i 0.965044 + 0.262088i \(0.0844112\pi\)
−0.965044 + 0.262088i \(0.915589\pi\)
\(270\) 0 0
\(271\) 47.3165 0.174600 0.0872998 0.996182i \(-0.472176\pi\)
0.0872998 + 0.996182i \(0.472176\pi\)
\(272\) − 67.8267i − 0.249363i
\(273\) 0 0
\(274\) −687.317 −2.50845
\(275\) 0 0
\(276\) 0 0
\(277\) −257.919 −0.931117 −0.465558 0.885017i \(-0.654146\pi\)
−0.465558 + 0.885017i \(0.654146\pi\)
\(278\) 526.535i 1.89401i
\(279\) 0 0
\(280\) 742.933 2.65333
\(281\) 111.455i 0.396637i 0.980138 + 0.198318i \(0.0635480\pi\)
−0.980138 + 0.198318i \(0.936452\pi\)
\(282\) 0 0
\(283\) −116.490 −0.411625 −0.205813 0.978591i \(-0.565984\pi\)
−0.205813 + 0.978591i \(0.565984\pi\)
\(284\) − 261.741i − 0.921622i
\(285\) 0 0
\(286\) 0 0
\(287\) − 452.232i − 1.57572i
\(288\) 0 0
\(289\) 267.095 0.924203
\(290\) 747.117i 2.57626i
\(291\) 0 0
\(292\) −68.9839 −0.236246
\(293\) 274.233i 0.935950i 0.883742 + 0.467975i \(0.155016\pi\)
−0.883742 + 0.467975i \(0.844984\pi\)
\(294\) 0 0
\(295\) 434.776 1.47382
\(296\) − 18.2151i − 0.0615375i
\(297\) 0 0
\(298\) 932.631 3.12963
\(299\) − 686.586i − 2.29628i
\(300\) 0 0
\(301\) −557.585 −1.85244
\(302\) 324.884i 1.07578i
\(303\) 0 0
\(304\) −423.714 −1.39380
\(305\) − 232.042i − 0.760794i
\(306\) 0 0
\(307\) 506.587 1.65012 0.825060 0.565045i \(-0.191141\pi\)
0.825060 + 0.565045i \(0.191141\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −534.633 −1.72462
\(311\) 522.247i 1.67925i 0.543166 + 0.839625i \(0.317225\pi\)
−0.543166 + 0.839625i \(0.682775\pi\)
\(312\) 0 0
\(313\) 11.3186 0.0361616 0.0180808 0.999837i \(-0.494244\pi\)
0.0180808 + 0.999837i \(0.494244\pi\)
\(314\) − 641.446i − 2.04282i
\(315\) 0 0
\(316\) −23.7460 −0.0751455
\(317\) 5.85640i 0.0184745i 0.999957 + 0.00923723i \(0.00294034\pi\)
−0.999957 + 0.00923723i \(0.997060\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) − 438.191i − 1.36935i
\(321\) 0 0
\(322\) −848.282 −2.63442
\(323\) 136.842i 0.423660i
\(324\) 0 0
\(325\) −355.474 −1.09377
\(326\) 152.758i 0.468582i
\(327\) 0 0
\(328\) 680.169 2.07369
\(329\) − 208.539i − 0.633859i
\(330\) 0 0
\(331\) −599.617 −1.81153 −0.905766 0.423779i \(-0.860703\pi\)
−0.905766 + 0.423779i \(0.860703\pi\)
\(332\) 346.211i 1.04281i
\(333\) 0 0
\(334\) −523.885 −1.56852
\(335\) 57.2642i 0.170938i
\(336\) 0 0
\(337\) −544.571 −1.61594 −0.807968 0.589226i \(-0.799433\pi\)
−0.807968 + 0.589226i \(0.799433\pi\)
\(338\) − 1527.70i − 4.51983i
\(339\) 0 0
\(340\) 231.190 0.679969
\(341\) 0 0
\(342\) 0 0
\(343\) −170.984 −0.498495
\(344\) − 838.623i − 2.43786i
\(345\) 0 0
\(346\) 882.236 2.54981
\(347\) − 626.271i − 1.80482i −0.430882 0.902408i \(-0.641797\pi\)
0.430882 0.902408i \(-0.358203\pi\)
\(348\) 0 0
\(349\) −2.25198 −0.00645268 −0.00322634 0.999995i \(-0.501027\pi\)
−0.00322634 + 0.999995i \(0.501027\pi\)
\(350\) 439.190i 1.25483i
\(351\) 0 0
\(352\) 0 0
\(353\) 565.154i 1.60100i 0.599332 + 0.800501i \(0.295433\pi\)
−0.599332 + 0.800501i \(0.704567\pi\)
\(354\) 0 0
\(355\) 208.587 0.587568
\(356\) − 830.196i − 2.33201i
\(357\) 0 0
\(358\) 390.762 1.09151
\(359\) 151.210i 0.421197i 0.977573 + 0.210598i \(0.0675412\pi\)
−0.977573 + 0.210598i \(0.932459\pi\)
\(360\) 0 0
\(361\) 493.855 1.36802
\(362\) − 552.246i − 1.52554i
\(363\) 0 0
\(364\) −1728.68 −4.74911
\(365\) − 54.9747i − 0.150616i
\(366\) 0 0
\(367\) −8.45762 −0.0230453 −0.0115226 0.999934i \(-0.503668\pi\)
−0.0115226 + 0.999934i \(0.503668\pi\)
\(368\) − 402.084i − 1.09262i
\(369\) 0 0
\(370\) 29.5081 0.0797515
\(371\) 38.3406i 0.103344i
\(372\) 0 0
\(373\) −483.363 −1.29588 −0.647940 0.761692i \(-0.724369\pi\)
−0.647940 + 0.761692i \(0.724369\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 313.649 0.834173
\(377\) − 855.181i − 2.26839i
\(378\) 0 0
\(379\) 412.458 1.08828 0.544139 0.838995i \(-0.316856\pi\)
0.544139 + 0.838995i \(0.316856\pi\)
\(380\) − 1444.24i − 3.80064i
\(381\) 0 0
\(382\) −258.504 −0.676712
\(383\) − 529.141i − 1.38157i −0.723060 0.690785i \(-0.757265\pi\)
0.723060 0.690785i \(-0.242735\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) − 409.707i − 1.06142i
\(387\) 0 0
\(388\) −803.790 −2.07162
\(389\) 606.884i 1.56011i 0.625709 + 0.780057i \(0.284810\pi\)
−0.625709 + 0.780057i \(0.715190\pi\)
\(390\) 0 0
\(391\) −129.857 −0.332115
\(392\) 396.751i 1.01212i
\(393\) 0 0
\(394\) −645.282 −1.63777
\(395\) − 18.9237i − 0.0479080i
\(396\) 0 0
\(397\) 474.571 1.19539 0.597696 0.801723i \(-0.296083\pi\)
0.597696 + 0.801723i \(0.296083\pi\)
\(398\) 101.892i 0.256010i
\(399\) 0 0
\(400\) −208.175 −0.520439
\(401\) 115.682i 0.288483i 0.989543 + 0.144242i \(0.0460742\pi\)
−0.989543 + 0.144242i \(0.953926\pi\)
\(402\) 0 0
\(403\) 611.964 1.51852
\(404\) − 715.230i − 1.77037i
\(405\) 0 0
\(406\) −1056.58 −2.60242
\(407\) 0 0
\(408\) 0 0
\(409\) 56.9859 0.139330 0.0696649 0.997570i \(-0.477807\pi\)
0.0696649 + 0.997570i \(0.477807\pi\)
\(410\) 1101.86i 2.68746i
\(411\) 0 0
\(412\) 138.841 0.336992
\(413\) 614.866i 1.48878i
\(414\) 0 0
\(415\) −275.903 −0.664827
\(416\) 85.2677i 0.204970i
\(417\) 0 0
\(418\) 0 0
\(419\) 752.136i 1.79507i 0.440938 + 0.897537i \(0.354646\pi\)
−0.440938 + 0.897537i \(0.645354\pi\)
\(420\) 0 0
\(421\) 137.129 0.325722 0.162861 0.986649i \(-0.447928\pi\)
0.162861 + 0.986649i \(0.447928\pi\)
\(422\) − 1357.01i − 3.21566i
\(423\) 0 0
\(424\) −57.6653 −0.136003
\(425\) 67.2322i 0.158193i
\(426\) 0 0
\(427\) 328.157 0.768518
\(428\) − 32.1677i − 0.0751581i
\(429\) 0 0
\(430\) 1358.55 3.15942
\(431\) − 183.355i − 0.425416i −0.977116 0.212708i \(-0.931772\pi\)
0.977116 0.212708i \(-0.0682284\pi\)
\(432\) 0 0
\(433\) −36.4274 −0.0841279 −0.0420640 0.999115i \(-0.513393\pi\)
−0.0420640 + 0.999115i \(0.513393\pi\)
\(434\) − 756.085i − 1.74213i
\(435\) 0 0
\(436\) 753.425 1.72804
\(437\) 811.217i 1.85633i
\(438\) 0 0
\(439\) −120.730 −0.275011 −0.137506 0.990501i \(-0.543908\pi\)
−0.137506 + 0.990501i \(0.543908\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −399.079 −0.902893
\(443\) − 543.277i − 1.22636i −0.789943 0.613180i \(-0.789890\pi\)
0.789943 0.613180i \(-0.210110\pi\)
\(444\) 0 0
\(445\) 661.601 1.48674
\(446\) 416.988i 0.934950i
\(447\) 0 0
\(448\) 619.696 1.38325
\(449\) 85.2677i 0.189906i 0.995482 + 0.0949529i \(0.0302700\pi\)
−0.995482 + 0.0949529i \(0.969730\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) − 935.341i − 2.06934i
\(453\) 0 0
\(454\) −552.651 −1.21729
\(455\) − 1377.62i − 3.02773i
\(456\) 0 0
\(457\) 291.554 0.637975 0.318987 0.947759i \(-0.396657\pi\)
0.318987 + 0.947759i \(0.396657\pi\)
\(458\) − 692.590i − 1.51220i
\(459\) 0 0
\(460\) 1370.52 2.97939
\(461\) 61.4741i 0.133349i 0.997775 + 0.0666747i \(0.0212390\pi\)
−0.997775 + 0.0666747i \(0.978761\pi\)
\(462\) 0 0
\(463\) 71.6633 0.154780 0.0773901 0.997001i \(-0.475341\pi\)
0.0773901 + 0.997001i \(0.475341\pi\)
\(464\) − 500.818i − 1.07935i
\(465\) 0 0
\(466\) −229.919 −0.493389
\(467\) − 212.044i − 0.454055i −0.973888 0.227027i \(-0.927099\pi\)
0.973888 0.227027i \(-0.0729007\pi\)
\(468\) 0 0
\(469\) −80.9839 −0.172673
\(470\) 508.105i 1.08107i
\(471\) 0 0
\(472\) −924.776 −1.95927
\(473\) 0 0
\(474\) 0 0
\(475\) 420.000 0.884211
\(476\) 326.951i 0.686873i
\(477\) 0 0
\(478\) −1163.71 −2.43454
\(479\) − 8.84164i − 0.0184585i −0.999957 0.00922927i \(-0.997062\pi\)
0.999957 0.00922927i \(-0.00293781\pi\)
\(480\) 0 0
\(481\) −33.7762 −0.0702208
\(482\) 65.3019i 0.135481i
\(483\) 0 0
\(484\) 0 0
\(485\) − 640.557i − 1.32074i
\(486\) 0 0
\(487\) −483.252 −0.992304 −0.496152 0.868236i \(-0.665254\pi\)
−0.496152 + 0.868236i \(0.665254\pi\)
\(488\) 493.558i 1.01139i
\(489\) 0 0
\(490\) −642.728 −1.31169
\(491\) 317.006i 0.645634i 0.946461 + 0.322817i \(0.104630\pi\)
−0.946461 + 0.322817i \(0.895370\pi\)
\(492\) 0 0
\(493\) −161.744 −0.328081
\(494\) 2493.05i 5.04665i
\(495\) 0 0
\(496\) 358.383 0.722547
\(497\) 294.986i 0.593533i
\(498\) 0 0
\(499\) 503.631 1.00928 0.504640 0.863330i \(-0.331625\pi\)
0.504640 + 0.863330i \(0.331625\pi\)
\(500\) 525.333i 1.05067i
\(501\) 0 0
\(502\) −852.772 −1.69875
\(503\) 116.661i 0.231931i 0.993253 + 0.115965i \(0.0369962\pi\)
−0.993253 + 0.115965i \(0.963004\pi\)
\(504\) 0 0
\(505\) 569.982 1.12868
\(506\) 0 0
\(507\) 0 0
\(508\) 885.774 1.74365
\(509\) − 293.193i − 0.576017i −0.957628 0.288009i \(-0.907007\pi\)
0.957628 0.288009i \(-0.0929932\pi\)
\(510\) 0 0
\(511\) 77.7460 0.152145
\(512\) 823.527i 1.60845i
\(513\) 0 0
\(514\) −1139.61 −2.21715
\(515\) 110.645i 0.214845i
\(516\) 0 0
\(517\) 0 0
\(518\) 41.7307i 0.0805612i
\(519\) 0 0
\(520\) 2071.98 3.98457
\(521\) 96.5030i 0.185226i 0.995702 + 0.0926132i \(0.0295220\pi\)
−0.995702 + 0.0926132i \(0.970478\pi\)
\(522\) 0 0
\(523\) −201.427 −0.385138 −0.192569 0.981283i \(-0.561682\pi\)
−0.192569 + 0.981283i \(0.561682\pi\)
\(524\) 118.412i 0.225977i
\(525\) 0 0
\(526\) −949.839 −1.80578
\(527\) − 115.743i − 0.219626i
\(528\) 0 0
\(529\) −240.806 −0.455211
\(530\) − 93.4165i − 0.176258i
\(531\) 0 0
\(532\) 2042.47 3.83922
\(533\) − 1261.23i − 2.36629i
\(534\) 0 0
\(535\) 25.6351 0.0479160
\(536\) − 121.802i − 0.227243i
\(537\) 0 0
\(538\) 485.859 0.903083
\(539\) 0 0
\(540\) 0 0
\(541\) 866.375 1.60143 0.800716 0.599044i \(-0.204453\pi\)
0.800716 + 0.599044i \(0.204453\pi\)
\(542\) − 163.039i − 0.300811i
\(543\) 0 0
\(544\) 16.1270 0.0296453
\(545\) 600.421i 1.10169i
\(546\) 0 0
\(547\) −461.145 −0.843044 −0.421522 0.906818i \(-0.638504\pi\)
−0.421522 + 0.906818i \(0.638504\pi\)
\(548\) 1570.42i 2.86573i
\(549\) 0 0
\(550\) 0 0
\(551\) 1010.42i 1.83379i
\(552\) 0 0
\(553\) 26.7621 0.0483944
\(554\) 888.718i 1.60418i
\(555\) 0 0
\(556\) 1203.06 2.16377
\(557\) − 285.694i − 0.512916i −0.966555 0.256458i \(-0.917445\pi\)
0.966555 0.256458i \(-0.0825555\pi\)
\(558\) 0 0
\(559\) −1555.05 −2.78185
\(560\) − 806.772i − 1.44066i
\(561\) 0 0
\(562\) 384.042 0.683349
\(563\) 321.739i 0.571473i 0.958308 + 0.285736i \(0.0922382\pi\)
−0.958308 + 0.285736i \(0.907762\pi\)
\(564\) 0 0
\(565\) 745.393 1.31928
\(566\) 401.391i 0.709172i
\(567\) 0 0
\(568\) −443.667 −0.781104
\(569\) − 963.864i − 1.69396i −0.531624 0.846980i \(-0.678418\pi\)
0.531624 0.846980i \(-0.321582\pi\)
\(570\) 0 0
\(571\) 1014.45 1.77663 0.888313 0.459238i \(-0.151878\pi\)
0.888313 + 0.459238i \(0.151878\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −1558.26 −2.71475
\(575\) 398.560i 0.693148i
\(576\) 0 0
\(577\) 362.903 0.628948 0.314474 0.949266i \(-0.398172\pi\)
0.314474 + 0.949266i \(0.398172\pi\)
\(578\) − 920.334i − 1.59227i
\(579\) 0 0
\(580\) 1707.06 2.94320
\(581\) − 390.186i − 0.671577i
\(582\) 0 0
\(583\) 0 0
\(584\) 116.932i 0.200226i
\(585\) 0 0
\(586\) 944.931 1.61251
\(587\) 567.835i 0.967351i 0.875247 + 0.483676i \(0.160699\pi\)
−0.875247 + 0.483676i \(0.839301\pi\)
\(588\) 0 0
\(589\) −723.048 −1.22759
\(590\) − 1498.12i − 2.53918i
\(591\) 0 0
\(592\) −19.7803 −0.0334126
\(593\) − 203.728i − 0.343555i −0.985136 0.171777i \(-0.945049\pi\)
0.985136 0.171777i \(-0.0549510\pi\)
\(594\) 0 0
\(595\) −260.554 −0.437907
\(596\) − 2130.93i − 3.57539i
\(597\) 0 0
\(598\) −2365.78 −3.95616
\(599\) 591.566i 0.987589i 0.869579 + 0.493795i \(0.164391\pi\)
−0.869579 + 0.493795i \(0.835609\pi\)
\(600\) 0 0
\(601\) 719.157 1.19660 0.598301 0.801272i \(-0.295843\pi\)
0.598301 + 0.801272i \(0.295843\pi\)
\(602\) 1921.28i 3.19150i
\(603\) 0 0
\(604\) 742.314 1.22900
\(605\) 0 0
\(606\) 0 0
\(607\) 16.0000 0.0263591 0.0131796 0.999913i \(-0.495805\pi\)
0.0131796 + 0.999913i \(0.495805\pi\)
\(608\) − 100.746i − 0.165700i
\(609\) 0 0
\(610\) −799.552 −1.31074
\(611\) − 581.598i − 0.951879i
\(612\) 0 0
\(613\) 317.740 0.518336 0.259168 0.965832i \(-0.416552\pi\)
0.259168 + 0.965832i \(0.416552\pi\)
\(614\) − 1745.56i − 2.84292i
\(615\) 0 0
\(616\) 0 0
\(617\) − 629.469i − 1.02021i −0.860112 0.510105i \(-0.829607\pi\)
0.860112 0.510105i \(-0.170393\pi\)
\(618\) 0 0
\(619\) 243.982 0.394155 0.197077 0.980388i \(-0.436855\pi\)
0.197077 + 0.980388i \(0.436855\pi\)
\(620\) 1221.56i 1.97026i
\(621\) 0 0
\(622\) 1799.52 2.89311
\(623\) 935.645i 1.50184i
\(624\) 0 0
\(625\) −777.772 −1.24444
\(626\) − 39.0006i − 0.0623013i
\(627\) 0 0
\(628\) −1465.61 −2.33378
\(629\) 6.38823i 0.0101562i
\(630\) 0 0
\(631\) 658.236 1.04316 0.521581 0.853201i \(-0.325342\pi\)
0.521581 + 0.853201i \(0.325342\pi\)
\(632\) 40.2509i 0.0636882i
\(633\) 0 0
\(634\) 20.1795 0.0318289
\(635\) 705.892i 1.11164i
\(636\) 0 0
\(637\) 735.693 1.15493
\(638\) 0 0
\(639\) 0 0
\(640\) −1423.41 −2.22407
\(641\) − 464.074i − 0.723985i −0.932181 0.361993i \(-0.882097\pi\)
0.932181 0.361993i \(-0.117903\pi\)
\(642\) 0 0
\(643\) 218.000 0.339036 0.169518 0.985527i \(-0.445779\pi\)
0.169518 + 0.985527i \(0.445779\pi\)
\(644\) 1938.21i 3.00964i
\(645\) 0 0
\(646\) 471.520 0.729907
\(647\) − 1135.45i − 1.75494i −0.479629 0.877471i \(-0.659229\pi\)
0.479629 0.877471i \(-0.340771\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 1224.86i 1.88440i
\(651\) 0 0
\(652\) 349.030 0.535322
\(653\) 793.419i 1.21504i 0.794305 + 0.607519i \(0.207835\pi\)
−0.794305 + 0.607519i \(0.792165\pi\)
\(654\) 0 0
\(655\) −94.3649 −0.144069
\(656\) − 738.614i − 1.12594i
\(657\) 0 0
\(658\) −718.569 −1.09205
\(659\) − 87.6613i − 0.133022i −0.997786 0.0665109i \(-0.978813\pi\)
0.997786 0.0665109i \(-0.0211867\pi\)
\(660\) 0 0
\(661\) 1071.04 1.62033 0.810165 0.586202i \(-0.199377\pi\)
0.810165 + 0.586202i \(0.199377\pi\)
\(662\) 2066.11i 3.12101i
\(663\) 0 0
\(664\) 586.851 0.883811
\(665\) 1627.69i 2.44765i
\(666\) 0 0
\(667\) −958.837 −1.43754
\(668\) 1197.00i 1.79192i
\(669\) 0 0
\(670\) 197.317 0.294502
\(671\) 0 0
\(672\) 0 0
\(673\) 402.883 0.598637 0.299319 0.954153i \(-0.403241\pi\)
0.299319 + 0.954153i \(0.403241\pi\)
\(674\) 1876.44i 2.78403i
\(675\) 0 0
\(676\) −3490.59 −5.16359
\(677\) 123.859i 0.182953i 0.995807 + 0.0914765i \(0.0291586\pi\)
−0.995807 + 0.0914765i \(0.970841\pi\)
\(678\) 0 0
\(679\) 905.885 1.33415
\(680\) − 391.881i − 0.576296i
\(681\) 0 0
\(682\) 0 0
\(683\) 621.918i 0.910567i 0.890346 + 0.455284i \(0.150462\pi\)
−0.890346 + 0.455284i \(0.849538\pi\)
\(684\) 0 0
\(685\) −1251.50 −1.82701
\(686\) 589.162i 0.858837i
\(687\) 0 0
\(688\) −910.683 −1.32367
\(689\) 106.928i 0.155194i
\(690\) 0 0
\(691\) 1058.90 1.53242 0.766209 0.642591i \(-0.222141\pi\)
0.766209 + 0.642591i \(0.222141\pi\)
\(692\) − 2015.78i − 2.91298i
\(693\) 0 0
\(694\) −2157.96 −3.10945
\(695\) 958.743i 1.37949i
\(696\) 0 0
\(697\) −238.542 −0.342242
\(698\) 7.75971i 0.0111171i
\(699\) 0 0
\(700\) 1003.49 1.43355
\(701\) − 875.810i − 1.24937i −0.780876 0.624686i \(-0.785227\pi\)
0.780876 0.624686i \(-0.214773\pi\)
\(702\) 0 0
\(703\) 39.9073 0.0567671
\(704\) 0 0
\(705\) 0 0
\(706\) 1947.36 2.75830
\(707\) 806.076i 1.14014i
\(708\) 0 0
\(709\) 286.661 0.404318 0.202159 0.979353i \(-0.435204\pi\)
0.202159 + 0.979353i \(0.435204\pi\)
\(710\) − 718.731i − 1.01230i
\(711\) 0 0
\(712\) −1407.24 −1.97645
\(713\) − 686.139i − 0.962326i
\(714\) 0 0
\(715\) 0 0
\(716\) − 892.836i − 1.24698i
\(717\) 0 0
\(718\) 521.026 0.725663
\(719\) 273.950i 0.381015i 0.981686 + 0.190507i \(0.0610133\pi\)
−0.981686 + 0.190507i \(0.938987\pi\)
\(720\) 0 0
\(721\) −156.476 −0.217026
\(722\) − 1701.69i − 2.35690i
\(723\) 0 0
\(724\) −1261.80 −1.74282
\(725\) 496.429i 0.684729i
\(726\) 0 0
\(727\) 1276.84 1.75631 0.878154 0.478377i \(-0.158775\pi\)
0.878154 + 0.478377i \(0.158775\pi\)
\(728\) 2930.22i 4.02502i
\(729\) 0 0
\(730\) −189.427 −0.259490
\(731\) 294.114i 0.402344i
\(732\) 0 0
\(733\) 1064.27 1.45193 0.725966 0.687731i \(-0.241393\pi\)
0.725966 + 0.687731i \(0.241393\pi\)
\(734\) 29.1426i 0.0397038i
\(735\) 0 0
\(736\) 95.6028 0.129895
\(737\) 0 0
\(738\) 0 0
\(739\) 830.794 1.12421 0.562107 0.827064i \(-0.309991\pi\)
0.562107 + 0.827064i \(0.309991\pi\)
\(740\) − 67.4218i − 0.0911105i
\(741\) 0 0
\(742\) 132.111 0.178047
\(743\) 1335.35i 1.79724i 0.438729 + 0.898619i \(0.355429\pi\)
−0.438729 + 0.898619i \(0.644571\pi\)
\(744\) 0 0
\(745\) 1698.18 2.27944
\(746\) 1665.53i 2.23262i
\(747\) 0 0
\(748\) 0 0
\(749\) 36.2535i 0.0484025i
\(750\) 0 0
\(751\) −877.917 −1.16900 −0.584499 0.811395i \(-0.698709\pi\)
−0.584499 + 0.811395i \(0.698709\pi\)
\(752\) − 340.600i − 0.452926i
\(753\) 0 0
\(754\) −2946.72 −3.90811
\(755\) 591.566i 0.783531i
\(756\) 0 0
\(757\) 1075.54 1.42079 0.710394 0.703804i \(-0.248517\pi\)
0.710394 + 0.703804i \(0.248517\pi\)
\(758\) − 1421.21i − 1.87495i
\(759\) 0 0
\(760\) −2448.08 −3.22116
\(761\) − 351.679i − 0.462127i −0.972939 0.231064i \(-0.925779\pi\)
0.972939 0.231064i \(-0.0742205\pi\)
\(762\) 0 0
\(763\) −849.123 −1.11287
\(764\) 590.645i 0.773096i
\(765\) 0 0
\(766\) −1823.27 −2.38025
\(767\) 1714.81i 2.23573i
\(768\) 0 0
\(769\) 466.423 0.606532 0.303266 0.952906i \(-0.401923\pi\)
0.303266 + 0.952906i \(0.401923\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −936.123 −1.21259
\(773\) 258.978i 0.335030i 0.985870 + 0.167515i \(0.0535742\pi\)
−0.985870 + 0.167515i \(0.946426\pi\)
\(774\) 0 0
\(775\) −355.242 −0.458377
\(776\) 1362.48i 1.75577i
\(777\) 0 0
\(778\) 2091.15 2.68786
\(779\) 1490.18i 1.91293i
\(780\) 0 0
\(781\) 0 0
\(782\) 447.450i 0.572187i
\(783\) 0 0
\(784\) 430.843 0.549544
\(785\) − 1167.98i − 1.48787i
\(786\) 0 0
\(787\) −169.270 −0.215083 −0.107541 0.994201i \(-0.534298\pi\)
−0.107541 + 0.994201i \(0.534298\pi\)
\(788\) 1474.38i 1.87104i
\(789\) 0 0
\(790\) −65.2056 −0.0825388
\(791\) 1054.15i 1.33267i
\(792\) 0 0
\(793\) 915.202 1.15410
\(794\) − 1635.24i − 2.05949i
\(795\) 0 0
\(796\) 232.808 0.292473
\(797\) 888.999i 1.11543i 0.830032 + 0.557716i \(0.188322\pi\)
−0.830032 + 0.557716i \(0.811678\pi\)
\(798\) 0 0
\(799\) −110.000 −0.137672
\(800\) − 49.4975i − 0.0618718i
\(801\) 0 0
\(802\) 398.607 0.497016
\(803\) 0 0
\(804\) 0 0
\(805\) −1544.60 −1.91875
\(806\) − 2108.66i − 2.61620i
\(807\) 0 0
\(808\) −1212.36 −1.50045
\(809\) − 998.765i − 1.23457i −0.786740 0.617284i \(-0.788233\pi\)
0.786740 0.617284i \(-0.211767\pi\)
\(810\) 0 0
\(811\) −514.097 −0.633905 −0.316952 0.948441i \(-0.602660\pi\)
−0.316952 + 0.948441i \(0.602660\pi\)
\(812\) 2414.14i 2.97308i
\(813\) 0 0
\(814\) 0 0
\(815\) 278.150i 0.341288i
\(816\) 0 0
\(817\) 1837.33 2.24887
\(818\) − 196.357i − 0.240046i
\(819\) 0 0
\(820\) 2517.59 3.07023
\(821\) − 984.093i − 1.19865i −0.800505 0.599326i \(-0.795435\pi\)
0.800505 0.599326i \(-0.204565\pi\)
\(822\) 0 0
\(823\) −636.083 −0.772883 −0.386442 0.922314i \(-0.626296\pi\)
−0.386442 + 0.922314i \(0.626296\pi\)
\(824\) − 235.344i − 0.285612i
\(825\) 0 0
\(826\) 2118.66 2.56496
\(827\) − 994.996i − 1.20314i −0.798820 0.601570i \(-0.794542\pi\)
0.798820 0.601570i \(-0.205458\pi\)
\(828\) 0 0
\(829\) −768.996 −0.927619 −0.463809 0.885935i \(-0.653518\pi\)
−0.463809 + 0.885935i \(0.653518\pi\)
\(830\) 950.685i 1.14540i
\(831\) 0 0
\(832\) 1728.28 2.07725
\(833\) − 139.145i − 0.167040i
\(834\) 0 0
\(835\) −953.917 −1.14242
\(836\) 0 0
\(837\) 0 0
\(838\) 2591.65 3.09266
\(839\) − 927.065i − 1.10496i −0.833525 0.552482i \(-0.813681\pi\)
0.833525 0.552482i \(-0.186319\pi\)
\(840\) 0 0
\(841\) −353.284 −0.420076
\(842\) − 472.508i − 0.561174i
\(843\) 0 0
\(844\) −3100.57 −3.67367
\(845\) − 2781.72i − 3.29198i
\(846\) 0 0
\(847\) 0 0
\(848\) 62.6203i 0.0738447i
\(849\) 0 0
\(850\) 231.663 0.272545
\(851\) 37.8701i 0.0445008i
\(852\) 0 0
\(853\) −160.889 −0.188616 −0.0943078 0.995543i \(-0.530064\pi\)
−0.0943078 + 0.995543i \(0.530064\pi\)
\(854\) − 1130.74i − 1.32405i
\(855\) 0 0
\(856\) −54.5262 −0.0636989
\(857\) 808.391i 0.943280i 0.881791 + 0.471640i \(0.156338\pi\)
−0.881791 + 0.471640i \(0.843662\pi\)
\(858\) 0 0
\(859\) −1311.06 −1.52627 −0.763133 0.646241i \(-0.776340\pi\)
−0.763133 + 0.646241i \(0.776340\pi\)
\(860\) − 3104.10i − 3.60941i
\(861\) 0 0
\(862\) −631.788 −0.732933
\(863\) 183.805i 0.212984i 0.994314 + 0.106492i \(0.0339618\pi\)
−0.994314 + 0.106492i \(0.966038\pi\)
\(864\) 0 0
\(865\) 1606.42 1.85713
\(866\) 125.519i 0.144941i
\(867\) 0 0
\(868\) −1727.55 −1.99026
\(869\) 0 0
\(870\) 0 0
\(871\) −225.857 −0.259308
\(872\) − 1277.10i − 1.46457i
\(873\) 0 0
\(874\) 2795.23 3.19820
\(875\) − 592.059i − 0.676639i
\(876\) 0 0
\(877\) −1049.04 −1.19617 −0.598083 0.801434i \(-0.704071\pi\)
−0.598083 + 0.801434i \(0.704071\pi\)
\(878\) 416.001i 0.473805i
\(879\) 0 0
\(880\) 0 0
\(881\) − 624.821i − 0.709218i −0.935015 0.354609i \(-0.884614\pi\)
0.935015 0.354609i \(-0.115386\pi\)
\(882\) 0 0
\(883\) −1336.79 −1.51391 −0.756957 0.653465i \(-0.773315\pi\)
−0.756957 + 0.653465i \(0.773315\pi\)
\(884\) 911.838i 1.03149i
\(885\) 0 0
\(886\) −1871.98 −2.11285
\(887\) 1268.64i 1.43025i 0.698995 + 0.715127i \(0.253631\pi\)
−0.698995 + 0.715127i \(0.746369\pi\)
\(888\) 0 0
\(889\) −998.282 −1.12293
\(890\) − 2279.69i − 2.56145i
\(891\) 0 0
\(892\) 952.758 1.06811
\(893\) 687.171i 0.769508i
\(894\) 0 0
\(895\) 711.520 0.794995
\(896\) − 2013.00i − 2.24665i
\(897\) 0 0
\(898\) 293.808 0.327181
\(899\) − 854.624i − 0.950638i
\(900\) 0 0
\(901\) 20.2238 0.0224460
\(902\) 0 0
\(903\) 0 0
\(904\) −1585.46 −1.75383
\(905\) − 1005.56i − 1.11111i
\(906\) 0 0
\(907\) 1470.26 1.62102 0.810509 0.585726i \(-0.199190\pi\)
0.810509 + 0.585726i \(0.199190\pi\)
\(908\) 1262.73i 1.39067i
\(909\) 0 0
\(910\) −4746.88 −5.21636
\(911\) 1344.21i 1.47554i 0.675055 + 0.737768i \(0.264120\pi\)
−0.675055 + 0.737768i \(0.735880\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) − 1004.61i − 1.09914i
\(915\) 0 0
\(916\) −1582.47 −1.72759
\(917\) − 133.452i − 0.145531i
\(918\) 0 0
\(919\) 1013.26 1.10256 0.551282 0.834319i \(-0.314139\pi\)
0.551282 + 0.834319i \(0.314139\pi\)
\(920\) − 2323.12i − 2.52513i
\(921\) 0 0
\(922\) 211.823 0.229742
\(923\) 822.690i 0.891322i
\(924\) 0 0
\(925\) 19.6069 0.0211967
\(926\) − 246.931i − 0.266665i
\(927\) 0 0
\(928\) 119.079 0.128317
\(929\) − 578.763i − 0.622995i −0.950247 0.311498i \(-0.899169\pi\)
0.950247 0.311498i \(-0.100831\pi\)
\(930\) 0 0
\(931\) −869.238 −0.933660
\(932\) 525.333i 0.563662i
\(933\) 0 0
\(934\) −730.643 −0.782273
\(935\) 0 0
\(936\) 0 0
\(937\) 679.472 0.725157 0.362578 0.931953i \(-0.381897\pi\)
0.362578 + 0.931953i \(0.381897\pi\)
\(938\) 279.048i 0.297492i
\(939\) 0 0
\(940\) 1160.95 1.23505
\(941\) − 1046.59i − 1.11221i −0.831112 0.556106i \(-0.812295\pi\)
0.831112 0.556106i \(-0.187705\pi\)
\(942\) 0 0
\(943\) −1414.11 −1.49958
\(944\) 1004.24i 1.06381i
\(945\) 0 0
\(946\) 0 0
\(947\) − 377.752i − 0.398893i −0.979909 0.199447i \(-0.936086\pi\)
0.979909 0.199447i \(-0.0639144\pi\)
\(948\) 0 0
\(949\) 216.827 0.228479
\(950\) − 1447.20i − 1.52337i
\(951\) 0 0
\(952\) 554.204 0.582147
\(953\) − 939.364i − 0.985692i −0.870117 0.492846i \(-0.835957\pi\)
0.870117 0.492846i \(-0.164043\pi\)
\(954\) 0 0
\(955\) −470.698 −0.492877
\(956\) 2658.91i 2.78129i
\(957\) 0 0
\(958\) −30.4658 −0.0318015
\(959\) − 1769.89i − 1.84556i
\(960\) 0 0
\(961\) −349.435 −0.363616
\(962\) 116.383i 0.120981i
\(963\) 0 0
\(964\) 149.206 0.154778
\(965\) − 746.016i − 0.773074i
\(966\) 0 0
\(967\) 519.663 0.537397 0.268699 0.963224i \(-0.413406\pi\)
0.268699 + 0.963224i \(0.413406\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −2207.18 −2.27544
\(971\) − 461.769i − 0.475560i −0.971319 0.237780i \(-0.923580\pi\)
0.971319 0.237780i \(-0.0764198\pi\)
\(972\) 0 0
\(973\) −1355.87 −1.39349
\(974\) 1665.15i 1.70960i
\(975\) 0 0
\(976\) 535.968 0.549147
\(977\) 1613.28i 1.65126i 0.564211 + 0.825631i \(0.309181\pi\)
−0.564211 + 0.825631i \(0.690819\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 1468.54i 1.49851i
\(981\) 0 0
\(982\) 1092.31 1.11234
\(983\) − 1662.92i − 1.69168i −0.533437 0.845840i \(-0.679100\pi\)
0.533437 0.845840i \(-0.320900\pi\)
\(984\) 0 0
\(985\) −1174.96 −1.19286
\(986\) 557.324i 0.565238i
\(987\) 0 0
\(988\) 5696.26 5.76545
\(989\) 1743.54i 1.76293i
\(990\) 0 0
\(991\) −68.7903 −0.0694150 −0.0347075 0.999398i \(-0.511050\pi\)
−0.0347075 + 0.999398i \(0.511050\pi\)
\(992\) 85.2121i 0.0858993i
\(993\) 0 0
\(994\) 1016.44 1.02257
\(995\) 185.530i 0.186462i
\(996\) 0 0
\(997\) 758.510 0.760792 0.380396 0.924824i \(-0.375788\pi\)
0.380396 + 0.924824i \(0.375788\pi\)
\(998\) − 1735.37i − 1.73885i
\(999\) 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.d.485.1 yes 4
3.2 odd 2 inner 1089.3.b.d.485.4 yes 4
11.10 odd 2 1089.3.b.c.485.4 yes 4
33.32 even 2 1089.3.b.c.485.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1089.3.b.c.485.1 4 33.32 even 2
1089.3.b.c.485.4 yes 4 11.10 odd 2
1089.3.b.d.485.1 yes 4 1.1 even 1 trivial
1089.3.b.d.485.4 yes 4 3.2 odd 2 inner