Properties

Label 1089.3.b.d.485.4
Level $1089$
Weight $3$
Character 1089.485
Analytic conductor $29.673$
Analytic rank $0$
Dimension $4$
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: \(4\)
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 485.4
Root \(3.44572i\) of defining polynomial
Character \(\chi\) \(=\) 1089.485
Dual form 1089.3.b.d.485.1

$q$-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 \(0.330431\pi\)
−0.507877 + 0.861430i \(0.669569\pi\)
\(3\) 0 0
\(4\) −7.87298 −1.96825
\(5\) 6.27415i 1.25483i 0.778685 + 0.627415i \(0.215887\pi\)
−0.778685 + 0.627415i \(0.784113\pi\)
\(6\) 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 \(0.0439568\pi\)
−0.990480 + 0.137656i \(0.956043\pi\)
\(18\) 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 \(0.206092\pi\)
−0.797620 + 0.603161i \(0.793908\pi\)
\(24\) 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 \(0.203179\pi\)
−0.803107 + 0.595835i \(0.796821\pi\)
\(30\) 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 \(0.213498\pi\)
−0.783373 + 0.621552i \(0.786502\pi\)
\(42\) 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 \(0.0804402\pi\)
−0.968238 + 0.250029i \(0.919560\pi\)
\(48\) 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 \(-0.987021\pi\)
0.999169 0.0407646i \(-0.0129794\pi\)
\(54\) 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 \(-0.800207\pi\)
0.809400 0.587258i \(-0.199793\pi\)
\(60\) 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 \(-0.924778\pi\)
0.972207 0.234123i \(-0.0752218\pi\)
\(72\) 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 \(0.0853414\pi\)
−0.964274 + 0.264907i \(0.914659\pi\)
\(84\) 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 \(-0.798177\pi\)
0.805638 0.592409i \(-0.201823\pi\)
\(90\) 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 \(-0.851519\pi\)
0.893163 0.449733i \(-0.148481\pi\)
\(102\) 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 \(-0.993922\pi\)
0.999818 0.0190927i \(-0.00607775\pi\)
\(108\) 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 \(-0.823811\pi\)
0.850682 0.525681i \(-0.176189\pi\)
\(114\) 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 \(0.0182828\pi\)
−0.998351 + 0.0574056i \(0.981717\pi\)
\(132\) 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 \(0.259546\pi\)
−0.685586 + 0.727991i \(0.740454\pi\)
\(138\) 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 \(-0.637406\pi\)
0.418391 0.908267i \(-0.362594\pi\)
\(150\) 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 \(0.150435\pi\)
−0.890385 + 0.455208i \(0.849565\pi\)
\(168\) 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 \(-0.734828\pi\)
0.672613 0.739995i \(-0.265172\pi\)
\(174\) 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 \(-0.897400\pi\)
0.948501 0.316774i \(-0.102600\pi\)
\(180\) 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 \(0.0629225\pi\)
−0.980525 + 0.196392i \(0.937077\pi\)
\(192\) 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 \(0.157663\pi\)
−0.879820 + 0.475306i \(0.842337\pi\)
\(198\) 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 \(0.114933\pi\)
−0.935519 + 0.353277i \(0.885067\pi\)
\(228\) 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 \(0.0457357\pi\)
−0.989695 + 0.143189i \(0.954264\pi\)
\(234\) 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 \(0.249745\pi\)
−0.707673 + 0.706540i \(0.750255\pi\)
\(240\) 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 \(0.164101\pi\)
−0.870028 + 0.493003i \(0.835899\pi\)
\(252\) 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 \(0.222498\pi\)
−0.765488 + 0.643450i \(0.777502\pi\)
\(258\) 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 \(0.175585\pi\)
−0.851679 + 0.524064i \(0.824415\pi\)
\(264\) 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 \(-0.915589\pi\)
0.965044 0.262088i \(-0.0844112\pi\)
\(270\) 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 \(-0.936452\pi\)
0.980138 0.198318i \(-0.0635480\pi\)
\(282\) 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 \(-0.844984\pi\)
0.883742 0.467975i \(-0.155016\pi\)
\(294\) 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 \(-0.682775\pi\)
0.543166 0.839625i \(-0.317225\pi\)
\(312\) 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 \(-0.997060\pi\)
0.999957 0.00923723i \(-0.00294034\pi\)
\(318\) 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 \(0.358203\pi\)
−0.430882 + 0.902408i \(0.641797\pi\)
\(348\) 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 \(-0.704567\pi\)
0.599332 0.800501i \(-0.295433\pi\)
\(354\) 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 \(-0.932459\pi\)
0.977573 0.210598i \(-0.0675412\pi\)
\(360\) 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 \(0.242735\pi\)
−0.723060 + 0.690785i \(0.757265\pi\)
\(384\) 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 \(-0.715190\pi\)
0.625709 0.780057i \(-0.284810\pi\)
\(390\) 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 \(-0.953926\pi\)
0.989543 0.144242i \(-0.0460742\pi\)
\(402\) 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 \(-0.645354\pi\)
0.440938 0.897537i \(-0.354646\pi\)
\(420\) 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 \(0.0682284\pi\)
−0.977116 + 0.212708i \(0.931772\pi\)
\(432\) 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 \(0.210110\pi\)
−0.789943 + 0.613180i \(0.789890\pi\)
\(444\) 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 \(-0.969730\pi\)
0.995482 0.0949529i \(-0.0302700\pi\)
\(450\) 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 \(-0.978761\pi\)
0.997775 0.0666747i \(-0.0212390\pi\)
\(462\) 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 \(0.0729007\pi\)
−0.973888 + 0.227027i \(0.927099\pi\)
\(468\) 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 \(0.00293781\pi\)
−0.999957 + 0.00922927i \(0.997062\pi\)
\(480\) 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 \(-0.895370\pi\)
0.946461 0.322817i \(-0.104630\pi\)
\(492\) 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 \(-0.963004\pi\)
0.993253 0.115965i \(-0.0369962\pi\)
\(504\) 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 \(0.0929932\pi\)
−0.957628 + 0.288009i \(0.907007\pi\)
\(510\) 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 \(-0.970478\pi\)
0.995702 0.0926132i \(-0.0295220\pi\)
\(522\) 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 \(0.0825555\pi\)
−0.966555 + 0.256458i \(0.917445\pi\)
\(558\) 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 \(-0.907762\pi\)
0.958308 0.285736i \(-0.0922382\pi\)
\(564\) 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 \(0.321582\pi\)
−0.531624 + 0.846980i \(0.678418\pi\)
\(570\) 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 \(-0.839301\pi\)
0.875247 0.483676i \(-0.160699\pi\)
\(588\) 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 \(0.0549510\pi\)
−0.985136 + 0.171777i \(0.945049\pi\)
\(594\) 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 \(-0.835609\pi\)
0.869579 0.493795i \(-0.164391\pi\)
\(600\) 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 \(0.170393\pi\)
−0.860112 + 0.510105i \(0.829607\pi\)
\(618\) 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 \(0.117903\pi\)
−0.932181 + 0.361993i \(0.882097\pi\)
\(642\) 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 \(0.340771\pi\)
−0.479629 + 0.877471i \(0.659229\pi\)
\(648\) 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 \(-0.792165\pi\)
0.794305 0.607519i \(-0.207835\pi\)
\(654\) 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 \(0.0211867\pi\)
−0.997786 + 0.0665109i \(0.978813\pi\)
\(660\) 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 \(-0.970841\pi\)
0.995807 0.0914765i \(-0.0291586\pi\)
\(678\) 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 \(-0.849538\pi\)
0.890346 0.455284i \(-0.150462\pi\)
\(684\) 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 \(0.214773\pi\)
−0.780876 + 0.624686i \(0.785227\pi\)
\(702\) 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 \(-0.938987\pi\)
0.981686 0.190507i \(-0.0610133\pi\)
\(720\) 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 \(-0.644571\pi\)
0.438729 0.898619i \(-0.355429\pi\)
\(744\) 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 \(0.0742205\pi\)
−0.972939 + 0.231064i \(0.925779\pi\)
\(762\) 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 \(-0.946426\pi\)
0.985870 0.167515i \(-0.0535742\pi\)
\(774\) 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 \(-0.811678\pi\)
0.830032 0.557716i \(-0.188322\pi\)
\(798\) 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 \(0.211767\pi\)
−0.786740 + 0.617284i \(0.788233\pi\)
\(810\) 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 \(0.204565\pi\)
−0.800505 + 0.599326i \(0.795435\pi\)
\(822\) 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 \(0.205458\pi\)
−0.798820 + 0.601570i \(0.794542\pi\)
\(828\) 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 \(0.186319\pi\)
−0.833525 + 0.552482i \(0.813681\pi\)
\(840\) 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 \(-0.843662\pi\)
0.881791 0.471640i \(-0.156338\pi\)
\(858\) 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 \(-0.966038\pi\)
0.994314 0.106492i \(-0.0339618\pi\)
\(864\) 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 \(0.115386\pi\)
−0.935015 + 0.354609i \(0.884614\pi\)
\(882\) 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 \(-0.746369\pi\)
0.698995 0.715127i \(-0.253631\pi\)
\(888\) 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 \(-0.735880\pi\)
0.675055 0.737768i \(-0.264120\pi\)
\(912\) 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 \(0.100831\pi\)
−0.950247 + 0.311498i \(0.899169\pi\)
\(930\) 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 \(0.187705\pi\)
−0.831112 + 0.556106i \(0.812295\pi\)
\(942\) 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 \(0.0639144\pi\)
−0.979909 + 0.199447i \(0.936086\pi\)
\(948\) 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 \(0.164043\pi\)
−0.870117 + 0.492846i \(0.835957\pi\)
\(954\) 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 \(0.0764198\pi\)
−0.971319 + 0.237780i \(0.923580\pi\)
\(972\) 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 \(-0.690819\pi\)
0.564211 0.825631i \(-0.309181\pi\)
\(978\) 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 \(0.320900\pi\)
−0.533437 + 0.845840i \(0.679100\pi\)
\(984\) 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.485.4 yes 4
3.2 odd 2 inner 1089.3.b.d.485.1 yes 4
11.10 odd 2 1089.3.b.c.485.1 4
33.32 even 2 1089.3.b.c.485.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1089.3.b.c.485.1 4 11.10 odd 2
1089.3.b.c.485.4 yes 4 33.32 even 2
1089.3.b.d.485.1 yes 4 3.2 odd 2 inner
1089.3.b.d.485.4 yes 4 1.1 even 1 trivial