Properties

Label 1521.4.a.w.1.4
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(89.7419051187\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.1362828.1
Defining polynomial: \( x^{4} - 23x^{2} + 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(4.54739\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.54739 q^{2} +12.6788 q^{4} +12.9118 q^{5} -16.7289 q^{7} +21.2762 q^{8} +O(q^{10})\) \(q+4.54739 q^{2} +12.6788 q^{4} +12.9118 q^{5} -16.7289 q^{7} +21.2762 q^{8} +58.7151 q^{10} +24.9280 q^{11} -76.0727 q^{14} -4.67878 q^{16} +134.145 q^{17} +14.9376 q^{19} +163.706 q^{20} +113.358 q^{22} -72.0000 q^{23} +41.7151 q^{25} -212.101 q^{28} +206.145 q^{29} +249.142 q^{31} -191.486 q^{32} +610.012 q^{34} -216.000 q^{35} -293.955 q^{37} +67.9273 q^{38} +274.715 q^{40} +250.506 q^{41} +432.145 q^{43} +316.057 q^{44} -327.412 q^{46} -159.889 q^{47} -63.1454 q^{49} +189.695 q^{50} +194.581 q^{53} +321.866 q^{55} -355.927 q^{56} +937.424 q^{58} +232.647 q^{59} -185.006 q^{61} +1132.94 q^{62} -833.333 q^{64} +39.4393 q^{67} +1700.80 q^{68} -982.237 q^{70} +920.460 q^{71} +549.078 q^{73} -1336.73 q^{74} +189.391 q^{76} -417.018 q^{77} +933.140 q^{79} -60.4116 q^{80} +1139.15 q^{82} -1095.38 q^{83} +1732.06 q^{85} +1965.13 q^{86} +530.375 q^{88} -532.114 q^{89} -912.872 q^{92} -727.079 q^{94} +192.872 q^{95} +362.661 q^{97} -287.147 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 14 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 14 q^{4} + 88 q^{10} - 84 q^{14} + 18 q^{16} + 96 q^{17} + 380 q^{22} - 288 q^{23} + 20 q^{25} + 384 q^{29} - 864 q^{35} + 492 q^{38} + 952 q^{40} + 1288 q^{43} + 188 q^{49} - 984 q^{53} - 328 q^{55} - 1644 q^{56} + 288 q^{61} + 1668 q^{62} - 1314 q^{64} + 4380 q^{68} - 3144 q^{74} + 1416 q^{77} + 4320 q^{79} + 3088 q^{82} - 1036 q^{88} - 1008 q^{92} - 1660 q^{94} - 1872 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.54739 1.60775 0.803873 0.594801i \(-0.202769\pi\)
0.803873 + 0.594801i \(0.202769\pi\)
\(3\) 0 0
\(4\) 12.6788 1.58485
\(5\) 12.9118 1.15487 0.577434 0.816437i \(-0.304054\pi\)
0.577434 + 0.816437i \(0.304054\pi\)
\(6\) 0 0
\(7\) −16.7289 −0.903273 −0.451637 0.892202i \(-0.649160\pi\)
−0.451637 + 0.892202i \(0.649160\pi\)
\(8\) 21.2762 0.940286
\(9\) 0 0
\(10\) 58.7151 1.85674
\(11\) 24.9280 0.683280 0.341640 0.939831i \(-0.389018\pi\)
0.341640 + 0.939831i \(0.389018\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −76.0727 −1.45223
\(15\) 0 0
\(16\) −4.67878 −0.0731059
\(17\) 134.145 1.91383 0.956913 0.290376i \(-0.0937804\pi\)
0.956913 + 0.290376i \(0.0937804\pi\)
\(18\) 0 0
\(19\) 14.9376 0.180365 0.0901824 0.995925i \(-0.471255\pi\)
0.0901824 + 0.995925i \(0.471255\pi\)
\(20\) 163.706 1.83029
\(21\) 0 0
\(22\) 113.358 1.09854
\(23\) −72.0000 −0.652741 −0.326370 0.945242i \(-0.605826\pi\)
−0.326370 + 0.945242i \(0.605826\pi\)
\(24\) 0 0
\(25\) 41.7151 0.333721
\(26\) 0 0
\(27\) 0 0
\(28\) −212.101 −1.43155
\(29\) 206.145 1.32001 0.660004 0.751262i \(-0.270555\pi\)
0.660004 + 0.751262i \(0.270555\pi\)
\(30\) 0 0
\(31\) 249.142 1.44346 0.721728 0.692176i \(-0.243348\pi\)
0.721728 + 0.692176i \(0.243348\pi\)
\(32\) −191.486 −1.05782
\(33\) 0 0
\(34\) 610.012 3.07694
\(35\) −216.000 −1.04316
\(36\) 0 0
\(37\) −293.955 −1.30610 −0.653052 0.757313i \(-0.726512\pi\)
−0.653052 + 0.757313i \(0.726512\pi\)
\(38\) 67.9273 0.289981
\(39\) 0 0
\(40\) 274.715 1.08591
\(41\) 250.506 0.954208 0.477104 0.878847i \(-0.341686\pi\)
0.477104 + 0.878847i \(0.341686\pi\)
\(42\) 0 0
\(43\) 432.145 1.53259 0.766297 0.642486i \(-0.222097\pi\)
0.766297 + 0.642486i \(0.222097\pi\)
\(44\) 316.057 1.08290
\(45\) 0 0
\(46\) −327.412 −1.04944
\(47\) −159.889 −0.496217 −0.248109 0.968732i \(-0.579809\pi\)
−0.248109 + 0.968732i \(0.579809\pi\)
\(48\) 0 0
\(49\) −63.1454 −0.184097
\(50\) 189.695 0.536539
\(51\) 0 0
\(52\) 0 0
\(53\) 194.581 0.504298 0.252149 0.967688i \(-0.418863\pi\)
0.252149 + 0.967688i \(0.418863\pi\)
\(54\) 0 0
\(55\) 321.866 0.789099
\(56\) −355.927 −0.849336
\(57\) 0 0
\(58\) 937.424 2.12224
\(59\) 232.647 0.513358 0.256679 0.966497i \(-0.417372\pi\)
0.256679 + 0.966497i \(0.417372\pi\)
\(60\) 0 0
\(61\) −185.006 −0.388321 −0.194160 0.980970i \(-0.562198\pi\)
−0.194160 + 0.980970i \(0.562198\pi\)
\(62\) 1132.94 2.32071
\(63\) 0 0
\(64\) −833.333 −1.62760
\(65\) 0 0
\(66\) 0 0
\(67\) 39.4393 0.0719145 0.0359573 0.999353i \(-0.488552\pi\)
0.0359573 + 0.999353i \(0.488552\pi\)
\(68\) 1700.80 3.03312
\(69\) 0 0
\(70\) −982.237 −1.67714
\(71\) 920.460 1.53857 0.769286 0.638905i \(-0.220612\pi\)
0.769286 + 0.638905i \(0.220612\pi\)
\(72\) 0 0
\(73\) 549.078 0.880338 0.440169 0.897915i \(-0.354918\pi\)
0.440169 + 0.897915i \(0.354918\pi\)
\(74\) −1336.73 −2.09988
\(75\) 0 0
\(76\) 189.391 0.285851
\(77\) −417.018 −0.617189
\(78\) 0 0
\(79\) 933.140 1.32894 0.664471 0.747314i \(-0.268657\pi\)
0.664471 + 0.747314i \(0.268657\pi\)
\(80\) −60.4116 −0.0844277
\(81\) 0 0
\(82\) 1139.15 1.53412
\(83\) −1095.38 −1.44860 −0.724301 0.689484i \(-0.757837\pi\)
−0.724301 + 0.689484i \(0.757837\pi\)
\(84\) 0 0
\(85\) 1732.06 2.21022
\(86\) 1965.13 2.46402
\(87\) 0 0
\(88\) 530.375 0.642479
\(89\) −532.114 −0.633753 −0.316876 0.948467i \(-0.602634\pi\)
−0.316876 + 0.948467i \(0.602634\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −912.872 −1.03449
\(93\) 0 0
\(94\) −727.079 −0.797792
\(95\) 192.872 0.208298
\(96\) 0 0
\(97\) 362.661 0.379615 0.189808 0.981821i \(-0.439214\pi\)
0.189808 + 0.981821i \(0.439214\pi\)
\(98\) −287.147 −0.295982
\(99\) 0 0
\(100\) 528.897 0.528897
\(101\) −1490.58 −1.46850 −0.734249 0.678880i \(-0.762466\pi\)
−0.734249 + 0.678880i \(0.762466\pi\)
\(102\) 0 0
\(103\) 628.436 0.601181 0.300591 0.953753i \(-0.402816\pi\)
0.300591 + 0.953753i \(0.402816\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 884.838 0.810784
\(107\) −477.454 −0.431376 −0.215688 0.976462i \(-0.569199\pi\)
−0.215688 + 0.976462i \(0.569199\pi\)
\(108\) 0 0
\(109\) 378.207 0.332345 0.166173 0.986097i \(-0.446859\pi\)
0.166173 + 0.986097i \(0.446859\pi\)
\(110\) 1463.65 1.26867
\(111\) 0 0
\(112\) 78.2706 0.0660346
\(113\) −13.2732 −0.0110499 −0.00552495 0.999985i \(-0.501759\pi\)
−0.00552495 + 0.999985i \(0.501759\pi\)
\(114\) 0 0
\(115\) −929.651 −0.753830
\(116\) 2613.67 2.09201
\(117\) 0 0
\(118\) 1057.94 0.825349
\(119\) −2244.10 −1.72871
\(120\) 0 0
\(121\) −709.593 −0.533128
\(122\) −841.294 −0.624321
\(123\) 0 0
\(124\) 3158.81 2.28766
\(125\) −1075.36 −0.769465
\(126\) 0 0
\(127\) 145.988 0.102003 0.0510015 0.998699i \(-0.483759\pi\)
0.0510015 + 0.998699i \(0.483759\pi\)
\(128\) −2257.60 −1.55895
\(129\) 0 0
\(130\) 0 0
\(131\) −317.163 −0.211532 −0.105766 0.994391i \(-0.533729\pi\)
−0.105766 + 0.994391i \(0.533729\pi\)
\(132\) 0 0
\(133\) −249.890 −0.162919
\(134\) 179.346 0.115620
\(135\) 0 0
\(136\) 2854.11 1.79954
\(137\) 443.149 0.276356 0.138178 0.990407i \(-0.455875\pi\)
0.138178 + 0.990407i \(0.455875\pi\)
\(138\) 0 0
\(139\) 785.018 0.479024 0.239512 0.970893i \(-0.423013\pi\)
0.239512 + 0.970893i \(0.423013\pi\)
\(140\) −2738.62 −1.65325
\(141\) 0 0
\(142\) 4185.69 2.47363
\(143\) 0 0
\(144\) 0 0
\(145\) 2661.71 1.52444
\(146\) 2496.87 1.41536
\(147\) 0 0
\(148\) −3726.99 −2.06997
\(149\) 135.420 0.0744566 0.0372283 0.999307i \(-0.488147\pi\)
0.0372283 + 0.999307i \(0.488147\pi\)
\(150\) 0 0
\(151\) 2373.74 1.27929 0.639643 0.768672i \(-0.279082\pi\)
0.639643 + 0.768672i \(0.279082\pi\)
\(152\) 317.817 0.169594
\(153\) 0 0
\(154\) −1896.34 −0.992283
\(155\) 3216.87 1.66700
\(156\) 0 0
\(157\) −1166.73 −0.593089 −0.296544 0.955019i \(-0.595834\pi\)
−0.296544 + 0.955019i \(0.595834\pi\)
\(158\) 4243.35 2.13660
\(159\) 0 0
\(160\) −2472.44 −1.22165
\(161\) 1204.48 0.589603
\(162\) 0 0
\(163\) 2309.19 1.10963 0.554815 0.831974i \(-0.312789\pi\)
0.554815 + 0.831974i \(0.312789\pi\)
\(164\) 3176.12 1.51227
\(165\) 0 0
\(166\) −4981.14 −2.32898
\(167\) −600.788 −0.278386 −0.139193 0.990265i \(-0.544451\pi\)
−0.139193 + 0.990265i \(0.544451\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 7876.36 3.55347
\(171\) 0 0
\(172\) 5479.08 2.42893
\(173\) −3430.36 −1.50755 −0.753773 0.657135i \(-0.771768\pi\)
−0.753773 + 0.657135i \(0.771768\pi\)
\(174\) 0 0
\(175\) −697.846 −0.301441
\(176\) −116.633 −0.0499519
\(177\) 0 0
\(178\) −2419.73 −1.01891
\(179\) −978.837 −0.408725 −0.204362 0.978895i \(-0.565512\pi\)
−0.204362 + 0.978895i \(0.565512\pi\)
\(180\) 0 0
\(181\) −3839.09 −1.57656 −0.788279 0.615318i \(-0.789028\pi\)
−0.788279 + 0.615318i \(0.789028\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −1531.89 −0.613763
\(185\) −3795.49 −1.50838
\(186\) 0 0
\(187\) 3343.98 1.30768
\(188\) −2027.20 −0.786429
\(189\) 0 0
\(190\) 877.065 0.334890
\(191\) 487.709 0.184761 0.0923806 0.995724i \(-0.470552\pi\)
0.0923806 + 0.995724i \(0.470552\pi\)
\(192\) 0 0
\(193\) −4245.61 −1.58345 −0.791725 0.610878i \(-0.790817\pi\)
−0.791725 + 0.610878i \(0.790817\pi\)
\(194\) 1649.16 0.610325
\(195\) 0 0
\(196\) −800.606 −0.291766
\(197\) −2712.71 −0.981079 −0.490539 0.871419i \(-0.663200\pi\)
−0.490539 + 0.871419i \(0.663200\pi\)
\(198\) 0 0
\(199\) 3116.90 1.11031 0.555153 0.831748i \(-0.312660\pi\)
0.555153 + 0.831748i \(0.312660\pi\)
\(200\) 887.541 0.313793
\(201\) 0 0
\(202\) −6778.26 −2.36097
\(203\) −3448.58 −1.19233
\(204\) 0 0
\(205\) 3234.49 1.10198
\(206\) 2857.75 0.966546
\(207\) 0 0
\(208\) 0 0
\(209\) 372.366 0.123240
\(210\) 0 0
\(211\) −1051.22 −0.342981 −0.171491 0.985186i \(-0.554858\pi\)
−0.171491 + 0.985186i \(0.554858\pi\)
\(212\) 2467.06 0.799236
\(213\) 0 0
\(214\) −2171.17 −0.693542
\(215\) 5579.78 1.76994
\(216\) 0 0
\(217\) −4167.85 −1.30384
\(218\) 1719.85 0.534327
\(219\) 0 0
\(220\) 4080.87 1.25060
\(221\) 0 0
\(222\) 0 0
\(223\) −5496.12 −1.65044 −0.825218 0.564814i \(-0.808948\pi\)
−0.825218 + 0.564814i \(0.808948\pi\)
\(224\) 3203.35 0.955502
\(225\) 0 0
\(226\) −60.3585 −0.0177654
\(227\) −921.570 −0.269457 −0.134729 0.990883i \(-0.543016\pi\)
−0.134729 + 0.990883i \(0.543016\pi\)
\(228\) 0 0
\(229\) −192.941 −0.0556764 −0.0278382 0.999612i \(-0.508862\pi\)
−0.0278382 + 0.999612i \(0.508862\pi\)
\(230\) −4227.49 −1.21197
\(231\) 0 0
\(232\) 4386.00 1.24119
\(233\) −913.779 −0.256926 −0.128463 0.991714i \(-0.541004\pi\)
−0.128463 + 0.991714i \(0.541004\pi\)
\(234\) 0 0
\(235\) −2064.46 −0.573066
\(236\) 2949.68 0.813594
\(237\) 0 0
\(238\) −10204.8 −2.77932
\(239\) −1976.86 −0.535032 −0.267516 0.963553i \(-0.586203\pi\)
−0.267516 + 0.963553i \(0.586203\pi\)
\(240\) 0 0
\(241\) −3904.45 −1.04360 −0.521800 0.853068i \(-0.674739\pi\)
−0.521800 + 0.853068i \(0.674739\pi\)
\(242\) −3226.80 −0.857134
\(243\) 0 0
\(244\) −2345.65 −0.615429
\(245\) −815.322 −0.212608
\(246\) 0 0
\(247\) 0 0
\(248\) 5300.80 1.35726
\(249\) 0 0
\(250\) −4890.08 −1.23710
\(251\) 942.035 0.236895 0.118448 0.992960i \(-0.462208\pi\)
0.118448 + 0.992960i \(0.462208\pi\)
\(252\) 0 0
\(253\) −1794.82 −0.446005
\(254\) 663.866 0.163995
\(255\) 0 0
\(256\) −3599.54 −0.878794
\(257\) 812.616 0.197236 0.0986179 0.995125i \(-0.468558\pi\)
0.0986179 + 0.995125i \(0.468558\pi\)
\(258\) 0 0
\(259\) 4917.52 1.17977
\(260\) 0 0
\(261\) 0 0
\(262\) −1442.26 −0.340089
\(263\) 2608.29 0.611536 0.305768 0.952106i \(-0.401087\pi\)
0.305768 + 0.952106i \(0.401087\pi\)
\(264\) 0 0
\(265\) 2512.40 0.582398
\(266\) −1136.35 −0.261932
\(267\) 0 0
\(268\) 500.042 0.113974
\(269\) 4791.02 1.08592 0.542962 0.839757i \(-0.317303\pi\)
0.542962 + 0.839757i \(0.317303\pi\)
\(270\) 0 0
\(271\) 3663.62 0.821214 0.410607 0.911812i \(-0.365317\pi\)
0.410607 + 0.911812i \(0.365317\pi\)
\(272\) −627.637 −0.139912
\(273\) 0 0
\(274\) 2015.17 0.444311
\(275\) 1039.88 0.228025
\(276\) 0 0
\(277\) −624.326 −0.135423 −0.0677114 0.997705i \(-0.521570\pi\)
−0.0677114 + 0.997705i \(0.521570\pi\)
\(278\) 3569.78 0.770149
\(279\) 0 0
\(280\) −4595.67 −0.980871
\(281\) 5535.12 1.17508 0.587540 0.809195i \(-0.300096\pi\)
0.587540 + 0.809195i \(0.300096\pi\)
\(282\) 0 0
\(283\) 175.151 0.0367903 0.0183952 0.999831i \(-0.494144\pi\)
0.0183952 + 0.999831i \(0.494144\pi\)
\(284\) 11670.3 2.43840
\(285\) 0 0
\(286\) 0 0
\(287\) −4190.69 −0.861911
\(288\) 0 0
\(289\) 13082.0 2.66273
\(290\) 12103.8 2.45091
\(291\) 0 0
\(292\) 6961.64 1.39520
\(293\) −7774.33 −1.55011 −0.775054 0.631895i \(-0.782277\pi\)
−0.775054 + 0.631895i \(0.782277\pi\)
\(294\) 0 0
\(295\) 3003.90 0.592861
\(296\) −6254.25 −1.22811
\(297\) 0 0
\(298\) 615.808 0.119707
\(299\) 0 0
\(300\) 0 0
\(301\) −7229.30 −1.38435
\(302\) 10794.3 2.05677
\(303\) 0 0
\(304\) −69.8899 −0.0131857
\(305\) −2388.76 −0.448459
\(306\) 0 0
\(307\) −8022.85 −1.49149 −0.745746 0.666230i \(-0.767907\pi\)
−0.745746 + 0.666230i \(0.767907\pi\)
\(308\) −5287.27 −0.978150
\(309\) 0 0
\(310\) 14628.4 2.68012
\(311\) 9264.87 1.68927 0.844635 0.535343i \(-0.179818\pi\)
0.844635 + 0.535343i \(0.179818\pi\)
\(312\) 0 0
\(313\) 7423.57 1.34059 0.670296 0.742094i \(-0.266167\pi\)
0.670296 + 0.742094i \(0.266167\pi\)
\(314\) −5305.56 −0.953536
\(315\) 0 0
\(316\) 11831.1 2.10617
\(317\) 2641.04 0.467935 0.233968 0.972244i \(-0.424829\pi\)
0.233968 + 0.972244i \(0.424829\pi\)
\(318\) 0 0
\(319\) 5138.80 0.901936
\(320\) −10759.8 −1.87967
\(321\) 0 0
\(322\) 5477.23 0.947932
\(323\) 2003.82 0.345187
\(324\) 0 0
\(325\) 0 0
\(326\) 10500.8 1.78400
\(327\) 0 0
\(328\) 5329.84 0.897229
\(329\) 2674.76 0.448220
\(330\) 0 0
\(331\) −10779.5 −1.79001 −0.895007 0.446052i \(-0.852830\pi\)
−0.895007 + 0.446052i \(0.852830\pi\)
\(332\) −13888.1 −2.29581
\(333\) 0 0
\(334\) −2732.02 −0.447573
\(335\) 509.233 0.0830518
\(336\) 0 0
\(337\) −313.465 −0.0506693 −0.0253346 0.999679i \(-0.508065\pi\)
−0.0253346 + 0.999679i \(0.508065\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 21960.4 3.50286
\(341\) 6210.61 0.986286
\(342\) 0 0
\(343\) 6794.35 1.06956
\(344\) 9194.43 1.44108
\(345\) 0 0
\(346\) −15599.2 −2.42375
\(347\) 2849.23 0.440792 0.220396 0.975410i \(-0.429265\pi\)
0.220396 + 0.975410i \(0.429265\pi\)
\(348\) 0 0
\(349\) −6466.94 −0.991883 −0.495941 0.868356i \(-0.665177\pi\)
−0.495941 + 0.868356i \(0.665177\pi\)
\(350\) −3173.38 −0.484641
\(351\) 0 0
\(352\) −4773.38 −0.722789
\(353\) −2773.10 −0.418122 −0.209061 0.977903i \(-0.567041\pi\)
−0.209061 + 0.977903i \(0.567041\pi\)
\(354\) 0 0
\(355\) 11884.8 1.77685
\(356\) −6746.56 −1.00440
\(357\) 0 0
\(358\) −4451.16 −0.657126
\(359\) −1467.11 −0.215685 −0.107843 0.994168i \(-0.534394\pi\)
−0.107843 + 0.994168i \(0.534394\pi\)
\(360\) 0 0
\(361\) −6635.87 −0.967469
\(362\) −17457.8 −2.53471
\(363\) 0 0
\(364\) 0 0
\(365\) 7089.59 1.01667
\(366\) 0 0
\(367\) 4648.22 0.661130 0.330565 0.943783i \(-0.392761\pi\)
0.330565 + 0.943783i \(0.392761\pi\)
\(368\) 336.872 0.0477192
\(369\) 0 0
\(370\) −17259.6 −2.42509
\(371\) −3255.12 −0.455519
\(372\) 0 0
\(373\) 1763.72 0.244831 0.122416 0.992479i \(-0.460936\pi\)
0.122416 + 0.992479i \(0.460936\pi\)
\(374\) 15206.4 2.10242
\(375\) 0 0
\(376\) −3401.84 −0.466586
\(377\) 0 0
\(378\) 0 0
\(379\) −1930.47 −0.261640 −0.130820 0.991406i \(-0.541761\pi\)
−0.130820 + 0.991406i \(0.541761\pi\)
\(380\) 2445.38 0.330120
\(381\) 0 0
\(382\) 2217.81 0.297049
\(383\) −8845.93 −1.18017 −0.590086 0.807340i \(-0.700906\pi\)
−0.590086 + 0.807340i \(0.700906\pi\)
\(384\) 0 0
\(385\) −5384.46 −0.712772
\(386\) −19306.5 −2.54579
\(387\) 0 0
\(388\) 4598.10 0.601632
\(389\) 1598.08 0.208292 0.104146 0.994562i \(-0.466789\pi\)
0.104146 + 0.994562i \(0.466789\pi\)
\(390\) 0 0
\(391\) −9658.47 −1.24923
\(392\) −1343.50 −0.173104
\(393\) 0 0
\(394\) −12335.8 −1.57733
\(395\) 12048.5 1.53475
\(396\) 0 0
\(397\) −3578.82 −0.452433 −0.226217 0.974077i \(-0.572636\pi\)
−0.226217 + 0.974077i \(0.572636\pi\)
\(398\) 14173.7 1.78509
\(399\) 0 0
\(400\) −195.176 −0.0243970
\(401\) −3485.99 −0.434120 −0.217060 0.976158i \(-0.569647\pi\)
−0.217060 + 0.976158i \(0.569647\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −18898.8 −2.32735
\(405\) 0 0
\(406\) −15682.0 −1.91696
\(407\) −7327.71 −0.892435
\(408\) 0 0
\(409\) −14709.1 −1.77828 −0.889139 0.457637i \(-0.848696\pi\)
−0.889139 + 0.457637i \(0.848696\pi\)
\(410\) 14708.5 1.77171
\(411\) 0 0
\(412\) 7967.80 0.952780
\(413\) −3891.92 −0.463702
\(414\) 0 0
\(415\) −14143.4 −1.67294
\(416\) 0 0
\(417\) 0 0
\(418\) 1693.29 0.198138
\(419\) −3709.01 −0.432451 −0.216226 0.976343i \(-0.569375\pi\)
−0.216226 + 0.976343i \(0.569375\pi\)
\(420\) 0 0
\(421\) −794.029 −0.0919207 −0.0459603 0.998943i \(-0.514635\pi\)
−0.0459603 + 0.998943i \(0.514635\pi\)
\(422\) −4780.32 −0.551427
\(423\) 0 0
\(424\) 4139.96 0.474185
\(425\) 5595.89 0.638684
\(426\) 0 0
\(427\) 3094.94 0.350760
\(428\) −6053.53 −0.683664
\(429\) 0 0
\(430\) 25373.5 2.84562
\(431\) 2891.52 0.323155 0.161577 0.986860i \(-0.448342\pi\)
0.161577 + 0.986860i \(0.448342\pi\)
\(432\) 0 0
\(433\) 5560.94 0.617186 0.308593 0.951194i \(-0.400142\pi\)
0.308593 + 0.951194i \(0.400142\pi\)
\(434\) −18952.9 −2.09624
\(435\) 0 0
\(436\) 4795.20 0.526717
\(437\) −1075.51 −0.117731
\(438\) 0 0
\(439\) 15127.2 1.64460 0.822302 0.569051i \(-0.192689\pi\)
0.822302 + 0.569051i \(0.192689\pi\)
\(440\) 6848.11 0.741979
\(441\) 0 0
\(442\) 0 0
\(443\) −2357.89 −0.252883 −0.126441 0.991974i \(-0.540356\pi\)
−0.126441 + 0.991974i \(0.540356\pi\)
\(444\) 0 0
\(445\) −6870.56 −0.731901
\(446\) −24993.0 −2.65348
\(447\) 0 0
\(448\) 13940.7 1.47017
\(449\) −7165.06 −0.753096 −0.376548 0.926397i \(-0.622889\pi\)
−0.376548 + 0.926397i \(0.622889\pi\)
\(450\) 0 0
\(451\) 6244.63 0.651992
\(452\) −168.288 −0.0175124
\(453\) 0 0
\(454\) −4190.74 −0.433219
\(455\) 0 0
\(456\) 0 0
\(457\) −8020.96 −0.821017 −0.410508 0.911857i \(-0.634649\pi\)
−0.410508 + 0.911857i \(0.634649\pi\)
\(458\) −877.378 −0.0895135
\(459\) 0 0
\(460\) −11786.8 −1.19471
\(461\) −4146.59 −0.418928 −0.209464 0.977816i \(-0.567172\pi\)
−0.209464 + 0.977816i \(0.567172\pi\)
\(462\) 0 0
\(463\) −7118.21 −0.714495 −0.357248 0.934010i \(-0.616285\pi\)
−0.357248 + 0.934010i \(0.616285\pi\)
\(464\) −964.509 −0.0965004
\(465\) 0 0
\(466\) −4155.31 −0.413071
\(467\) 2128.22 0.210883 0.105441 0.994426i \(-0.466374\pi\)
0.105441 + 0.994426i \(0.466374\pi\)
\(468\) 0 0
\(469\) −659.774 −0.0649585
\(470\) −9387.91 −0.921344
\(471\) 0 0
\(472\) 4949.86 0.482703
\(473\) 10772.5 1.04719
\(474\) 0 0
\(475\) 623.125 0.0601915
\(476\) −28452.4 −2.73974
\(477\) 0 0
\(478\) −8989.57 −0.860196
\(479\) −3715.30 −0.354397 −0.177199 0.984175i \(-0.556704\pi\)
−0.177199 + 0.984175i \(0.556704\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −17755.0 −1.67784
\(483\) 0 0
\(484\) −8996.77 −0.844926
\(485\) 4682.62 0.438405
\(486\) 0 0
\(487\) 8139.28 0.757343 0.378671 0.925531i \(-0.376381\pi\)
0.378671 + 0.925531i \(0.376381\pi\)
\(488\) −3936.23 −0.365133
\(489\) 0 0
\(490\) −3707.59 −0.341820
\(491\) −18081.7 −1.66194 −0.830972 0.556315i \(-0.812215\pi\)
−0.830972 + 0.556315i \(0.812215\pi\)
\(492\) 0 0
\(493\) 27653.4 2.52626
\(494\) 0 0
\(495\) 0 0
\(496\) −1165.68 −0.105525
\(497\) −15398.3 −1.38975
\(498\) 0 0
\(499\) −11031.5 −0.989659 −0.494829 0.868990i \(-0.664769\pi\)
−0.494829 + 0.868990i \(0.664769\pi\)
\(500\) −13634.2 −1.21948
\(501\) 0 0
\(502\) 4283.80 0.380868
\(503\) −8016.14 −0.710581 −0.355290 0.934756i \(-0.615618\pi\)
−0.355290 + 0.934756i \(0.615618\pi\)
\(504\) 0 0
\(505\) −19246.1 −1.69592
\(506\) −8161.74 −0.717063
\(507\) 0 0
\(508\) 1850.95 0.161659
\(509\) 20173.9 1.75676 0.878382 0.477959i \(-0.158623\pi\)
0.878382 + 0.477959i \(0.158623\pi\)
\(510\) 0 0
\(511\) −9185.44 −0.795186
\(512\) 1692.30 0.146074
\(513\) 0 0
\(514\) 3695.29 0.317105
\(515\) 8114.25 0.694285
\(516\) 0 0
\(517\) −3985.72 −0.339056
\(518\) 22361.9 1.89677
\(519\) 0 0
\(520\) 0 0
\(521\) −9746.95 −0.819619 −0.409810 0.912171i \(-0.634405\pi\)
−0.409810 + 0.912171i \(0.634405\pi\)
\(522\) 0 0
\(523\) −18929.3 −1.58264 −0.791320 0.611402i \(-0.790606\pi\)
−0.791320 + 0.611402i \(0.790606\pi\)
\(524\) −4021.24 −0.335245
\(525\) 0 0
\(526\) 11860.9 0.983195
\(527\) 33421.2 2.76252
\(528\) 0 0
\(529\) −6983.00 −0.573929
\(530\) 11424.9 0.936349
\(531\) 0 0
\(532\) −3168.30 −0.258201
\(533\) 0 0
\(534\) 0 0
\(535\) −6164.80 −0.498182
\(536\) 839.120 0.0676203
\(537\) 0 0
\(538\) 21786.6 1.74589
\(539\) −1574.09 −0.125790
\(540\) 0 0
\(541\) 11366.8 0.903321 0.451661 0.892190i \(-0.350832\pi\)
0.451661 + 0.892190i \(0.350832\pi\)
\(542\) 16659.9 1.32030
\(543\) 0 0
\(544\) −25687.0 −2.02449
\(545\) 4883.34 0.383815
\(546\) 0 0
\(547\) 17495.4 1.36755 0.683775 0.729693i \(-0.260337\pi\)
0.683775 + 0.729693i \(0.260337\pi\)
\(548\) 5618.59 0.437983
\(549\) 0 0
\(550\) 4728.72 0.366606
\(551\) 3079.33 0.238083
\(552\) 0 0
\(553\) −15610.4 −1.20040
\(554\) −2839.05 −0.217725
\(555\) 0 0
\(556\) 9953.06 0.759180
\(557\) −11873.1 −0.903192 −0.451596 0.892223i \(-0.649145\pi\)
−0.451596 + 0.892223i \(0.649145\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 1010.62 0.0762613
\(561\) 0 0
\(562\) 25170.4 1.88923
\(563\) 2829.31 0.211796 0.105898 0.994377i \(-0.466228\pi\)
0.105898 + 0.994377i \(0.466228\pi\)
\(564\) 0 0
\(565\) −171.381 −0.0127612
\(566\) 796.481 0.0591495
\(567\) 0 0
\(568\) 19583.9 1.44670
\(569\) −16136.8 −1.18891 −0.594453 0.804130i \(-0.702632\pi\)
−0.594453 + 0.804130i \(0.702632\pi\)
\(570\) 0 0
\(571\) −17840.5 −1.30754 −0.653769 0.756695i \(-0.726813\pi\)
−0.653769 + 0.756695i \(0.726813\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −19056.7 −1.38573
\(575\) −3003.49 −0.217833
\(576\) 0 0
\(577\) −8516.17 −0.614442 −0.307221 0.951638i \(-0.599399\pi\)
−0.307221 + 0.951638i \(0.599399\pi\)
\(578\) 59488.9 4.28099
\(579\) 0 0
\(580\) 33747.3 2.41600
\(581\) 18324.5 1.30848
\(582\) 0 0
\(583\) 4850.53 0.344577
\(584\) 11682.3 0.827770
\(585\) 0 0
\(586\) −35352.9 −2.49218
\(587\) −20688.3 −1.45468 −0.727340 0.686277i \(-0.759244\pi\)
−0.727340 + 0.686277i \(0.759244\pi\)
\(588\) 0 0
\(589\) 3721.59 0.260349
\(590\) 13659.9 0.953169
\(591\) 0 0
\(592\) 1375.35 0.0954839
\(593\) 11435.9 0.791933 0.395966 0.918265i \(-0.370410\pi\)
0.395966 + 0.918265i \(0.370410\pi\)
\(594\) 0 0
\(595\) −28975.4 −1.99643
\(596\) 1716.96 0.118002
\(597\) 0 0
\(598\) 0 0
\(599\) −1260.80 −0.0860016 −0.0430008 0.999075i \(-0.513692\pi\)
−0.0430008 + 0.999075i \(0.513692\pi\)
\(600\) 0 0
\(601\) 6261.10 0.424951 0.212476 0.977166i \(-0.431847\pi\)
0.212476 + 0.977166i \(0.431847\pi\)
\(602\) −32874.5 −2.22569
\(603\) 0 0
\(604\) 30096.1 2.02747
\(605\) −9162.14 −0.615692
\(606\) 0 0
\(607\) 3230.33 0.216005 0.108003 0.994151i \(-0.465555\pi\)
0.108003 + 0.994151i \(0.465555\pi\)
\(608\) −2860.35 −0.190794
\(609\) 0 0
\(610\) −10862.6 −0.721009
\(611\) 0 0
\(612\) 0 0
\(613\) −14868.5 −0.979660 −0.489830 0.871818i \(-0.662941\pi\)
−0.489830 + 0.871818i \(0.662941\pi\)
\(614\) −36483.0 −2.39794
\(615\) 0 0
\(616\) −8872.57 −0.580334
\(617\) −19952.8 −1.30190 −0.650949 0.759121i \(-0.725629\pi\)
−0.650949 + 0.759121i \(0.725629\pi\)
\(618\) 0 0
\(619\) −8316.48 −0.540012 −0.270006 0.962859i \(-0.587026\pi\)
−0.270006 + 0.962859i \(0.587026\pi\)
\(620\) 40786.0 2.64194
\(621\) 0 0
\(622\) 42131.0 2.71592
\(623\) 8901.66 0.572452
\(624\) 0 0
\(625\) −19099.2 −1.22235
\(626\) 33757.9 2.15533
\(627\) 0 0
\(628\) −14792.7 −0.939955
\(629\) −39432.6 −2.49965
\(630\) 0 0
\(631\) 12605.9 0.795299 0.397649 0.917537i \(-0.369826\pi\)
0.397649 + 0.917537i \(0.369826\pi\)
\(632\) 19853.7 1.24959
\(633\) 0 0
\(634\) 12009.8 0.752321
\(635\) 1884.98 0.117800
\(636\) 0 0
\(637\) 0 0
\(638\) 23368.1 1.45008
\(639\) 0 0
\(640\) −29149.8 −1.80038
\(641\) 9224.04 0.568374 0.284187 0.958769i \(-0.408276\pi\)
0.284187 + 0.958769i \(0.408276\pi\)
\(642\) 0 0
\(643\) −4439.16 −0.272260 −0.136130 0.990691i \(-0.543467\pi\)
−0.136130 + 0.990691i \(0.543467\pi\)
\(644\) 15271.3 0.934431
\(645\) 0 0
\(646\) 9112.13 0.554972
\(647\) 9601.15 0.583401 0.291700 0.956510i \(-0.405779\pi\)
0.291700 + 0.956510i \(0.405779\pi\)
\(648\) 0 0
\(649\) 5799.44 0.350767
\(650\) 0 0
\(651\) 0 0
\(652\) 29277.7 1.75859
\(653\) −27112.8 −1.62482 −0.812410 0.583087i \(-0.801845\pi\)
−0.812410 + 0.583087i \(0.801845\pi\)
\(654\) 0 0
\(655\) −4095.15 −0.244291
\(656\) −1172.06 −0.0697583
\(657\) 0 0
\(658\) 12163.2 0.720624
\(659\) −5587.26 −0.330271 −0.165136 0.986271i \(-0.552806\pi\)
−0.165136 + 0.986271i \(0.552806\pi\)
\(660\) 0 0
\(661\) 3060.13 0.180069 0.0900343 0.995939i \(-0.471302\pi\)
0.0900343 + 0.995939i \(0.471302\pi\)
\(662\) −49018.6 −2.87789
\(663\) 0 0
\(664\) −23305.6 −1.36210
\(665\) −3226.53 −0.188150
\(666\) 0 0
\(667\) −14842.5 −0.861623
\(668\) −7617.26 −0.441199
\(669\) 0 0
\(670\) 2315.68 0.133526
\(671\) −4611.83 −0.265332
\(672\) 0 0
\(673\) 4121.55 0.236069 0.118034 0.993010i \(-0.462341\pi\)
0.118034 + 0.993010i \(0.462341\pi\)
\(674\) −1425.45 −0.0814633
\(675\) 0 0
\(676\) 0 0
\(677\) −22889.5 −1.29943 −0.649715 0.760178i \(-0.725112\pi\)
−0.649715 + 0.760178i \(0.725112\pi\)
\(678\) 0 0
\(679\) −6066.91 −0.342896
\(680\) 36851.8 2.07824
\(681\) 0 0
\(682\) 28242.1 1.58570
\(683\) −19297.9 −1.08113 −0.540566 0.841301i \(-0.681790\pi\)
−0.540566 + 0.841301i \(0.681790\pi\)
\(684\) 0 0
\(685\) 5721.87 0.319155
\(686\) 30896.6 1.71959
\(687\) 0 0
\(688\) −2021.91 −0.112042
\(689\) 0 0
\(690\) 0 0
\(691\) 30317.8 1.66910 0.834548 0.550935i \(-0.185729\pi\)
0.834548 + 0.550935i \(0.185729\pi\)
\(692\) −43492.8 −2.38923
\(693\) 0 0
\(694\) 12956.6 0.708682
\(695\) 10136.0 0.553210
\(696\) 0 0
\(697\) 33604.3 1.82619
\(698\) −29407.7 −1.59470
\(699\) 0 0
\(700\) −8847.84 −0.477738
\(701\) −9606.16 −0.517574 −0.258787 0.965934i \(-0.583323\pi\)
−0.258787 + 0.965934i \(0.583323\pi\)
\(702\) 0 0
\(703\) −4390.99 −0.235575
\(704\) −20773.4 −1.11211
\(705\) 0 0
\(706\) −12610.4 −0.672235
\(707\) 24935.7 1.32646
\(708\) 0 0
\(709\) 23398.8 1.23944 0.619718 0.784825i \(-0.287247\pi\)
0.619718 + 0.784825i \(0.287247\pi\)
\(710\) 54044.9 2.85672
\(711\) 0 0
\(712\) −11321.4 −0.595909
\(713\) −17938.2 −0.942203
\(714\) 0 0
\(715\) 0 0
\(716\) −12410.5 −0.647766
\(717\) 0 0
\(718\) −6671.51 −0.346767
\(719\) 23588.7 1.22352 0.611758 0.791045i \(-0.290463\pi\)
0.611758 + 0.791045i \(0.290463\pi\)
\(720\) 0 0
\(721\) −10513.0 −0.543031
\(722\) −30175.9 −1.55544
\(723\) 0 0
\(724\) −48674.9 −2.49861
\(725\) 8599.38 0.440514
\(726\) 0 0
\(727\) 15733.5 0.802644 0.401322 0.915937i \(-0.368551\pi\)
0.401322 + 0.915937i \(0.368551\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 32239.2 1.63455
\(731\) 57970.3 2.93312
\(732\) 0 0
\(733\) 17297.1 0.871598 0.435799 0.900044i \(-0.356466\pi\)
0.435799 + 0.900044i \(0.356466\pi\)
\(734\) 21137.3 1.06293
\(735\) 0 0
\(736\) 13787.0 0.690484
\(737\) 983.144 0.0491378
\(738\) 0 0
\(739\) −38749.2 −1.92884 −0.964419 0.264377i \(-0.914834\pi\)
−0.964419 + 0.264377i \(0.914834\pi\)
\(740\) −48122.2 −2.39055
\(741\) 0 0
\(742\) −14802.3 −0.732359
\(743\) −3139.76 −0.155029 −0.0775144 0.996991i \(-0.524698\pi\)
−0.0775144 + 0.996991i \(0.524698\pi\)
\(744\) 0 0
\(745\) 1748.52 0.0859876
\(746\) 8020.33 0.393626
\(747\) 0 0
\(748\) 42397.6 2.07247
\(749\) 7987.25 0.389650
\(750\) 0 0
\(751\) 40628.6 1.97411 0.987055 0.160380i \(-0.0512721\pi\)
0.987055 + 0.160380i \(0.0512721\pi\)
\(752\) 748.086 0.0362764
\(753\) 0 0
\(754\) 0 0
\(755\) 30649.3 1.47741
\(756\) 0 0
\(757\) 24004.9 1.15254 0.576271 0.817259i \(-0.304507\pi\)
0.576271 + 0.817259i \(0.304507\pi\)
\(758\) −8778.62 −0.420651
\(759\) 0 0
\(760\) 4103.60 0.195859
\(761\) 29540.1 1.40713 0.703566 0.710630i \(-0.251590\pi\)
0.703566 + 0.710630i \(0.251590\pi\)
\(762\) 0 0
\(763\) −6326.97 −0.300199
\(764\) 6183.56 0.292818
\(765\) 0 0
\(766\) −40225.9 −1.89742
\(767\) 0 0
\(768\) 0 0
\(769\) −7585.63 −0.355715 −0.177857 0.984056i \(-0.556917\pi\)
−0.177857 + 0.984056i \(0.556917\pi\)
\(770\) −24485.2 −1.14596
\(771\) 0 0
\(772\) −53829.2 −2.50953
\(773\) 3284.29 0.152817 0.0764086 0.997077i \(-0.475655\pi\)
0.0764086 + 0.997077i \(0.475655\pi\)
\(774\) 0 0
\(775\) 10393.0 0.481712
\(776\) 7716.07 0.356947
\(777\) 0 0
\(778\) 7267.08 0.334881
\(779\) 3741.98 0.172106
\(780\) 0 0
\(781\) 22945.3 1.05128
\(782\) −43920.8 −2.00845
\(783\) 0 0
\(784\) 295.443 0.0134586
\(785\) −15064.6 −0.684939
\(786\) 0 0
\(787\) 41624.1 1.88531 0.942654 0.333772i \(-0.108321\pi\)
0.942654 + 0.333772i \(0.108321\pi\)
\(788\) −34393.8 −1.55486
\(789\) 0 0
\(790\) 54789.4 2.46749
\(791\) 222.046 0.00998108
\(792\) 0 0
\(793\) 0 0
\(794\) −16274.3 −0.727398
\(795\) 0 0
\(796\) 39518.4 1.75967
\(797\) 30333.3 1.34813 0.674066 0.738671i \(-0.264546\pi\)
0.674066 + 0.738671i \(0.264546\pi\)
\(798\) 0 0
\(799\) −21448.4 −0.949674
\(800\) −7987.87 −0.353017
\(801\) 0 0
\(802\) −15852.2 −0.697954
\(803\) 13687.4 0.601518
\(804\) 0 0
\(805\) 15552.0 0.680914
\(806\) 0 0
\(807\) 0 0
\(808\) −31714.0 −1.38081
\(809\) 24853.9 1.08012 0.540060 0.841627i \(-0.318402\pi\)
0.540060 + 0.841627i \(0.318402\pi\)
\(810\) 0 0
\(811\) 17383.5 0.752674 0.376337 0.926483i \(-0.377184\pi\)
0.376337 + 0.926483i \(0.377184\pi\)
\(812\) −43723.7 −1.88966
\(813\) 0 0
\(814\) −33322.0 −1.43481
\(815\) 29815.8 1.28148
\(816\) 0 0
\(817\) 6455.23 0.276426
\(818\) −66887.8 −2.85902
\(819\) 0 0
\(820\) 41009.4 1.74648
\(821\) −31169.4 −1.32499 −0.662496 0.749065i \(-0.730503\pi\)
−0.662496 + 0.749065i \(0.730503\pi\)
\(822\) 0 0
\(823\) 5512.79 0.233492 0.116746 0.993162i \(-0.462754\pi\)
0.116746 + 0.993162i \(0.462754\pi\)
\(824\) 13370.8 0.565282
\(825\) 0 0
\(826\) −17698.1 −0.745516
\(827\) −13335.8 −0.560738 −0.280369 0.959892i \(-0.590457\pi\)
−0.280369 + 0.959892i \(0.590457\pi\)
\(828\) 0 0
\(829\) −28338.5 −1.18726 −0.593628 0.804739i \(-0.702305\pi\)
−0.593628 + 0.804739i \(0.702305\pi\)
\(830\) −64315.5 −2.68967
\(831\) 0 0
\(832\) 0 0
\(833\) −8470.66 −0.352330
\(834\) 0 0
\(835\) −7757.27 −0.321499
\(836\) 4721.15 0.195316
\(837\) 0 0
\(838\) −16866.3 −0.695272
\(839\) −27149.5 −1.11717 −0.558585 0.829447i \(-0.688656\pi\)
−0.558585 + 0.829447i \(0.688656\pi\)
\(840\) 0 0
\(841\) 18106.9 0.742421
\(842\) −3610.76 −0.147785
\(843\) 0 0
\(844\) −13328.2 −0.543573
\(845\) 0 0
\(846\) 0 0
\(847\) 11870.7 0.481560
\(848\) −910.404 −0.0368672
\(849\) 0 0
\(850\) 25446.7 1.02684
\(851\) 21164.7 0.852547
\(852\) 0 0
\(853\) −7978.22 −0.320245 −0.160123 0.987097i \(-0.551189\pi\)
−0.160123 + 0.987097i \(0.551189\pi\)
\(854\) 14073.9 0.563933
\(855\) 0 0
\(856\) −10158.4 −0.405616
\(857\) 13614.4 0.542657 0.271329 0.962487i \(-0.412537\pi\)
0.271329 + 0.962487i \(0.412537\pi\)
\(858\) 0 0
\(859\) −35007.7 −1.39051 −0.695255 0.718763i \(-0.744709\pi\)
−0.695255 + 0.718763i \(0.744709\pi\)
\(860\) 70744.8 2.80509
\(861\) 0 0
\(862\) 13148.9 0.519551
\(863\) −24461.5 −0.964867 −0.482434 0.875933i \(-0.660247\pi\)
−0.482434 + 0.875933i \(0.660247\pi\)
\(864\) 0 0
\(865\) −44292.2 −1.74102
\(866\) 25287.8 0.992278
\(867\) 0 0
\(868\) −52843.3 −2.06638
\(869\) 23261.3 0.908040
\(870\) 0 0
\(871\) 0 0
\(872\) 8046.82 0.312500
\(873\) 0 0
\(874\) −4890.77 −0.189282
\(875\) 17989.5 0.695037
\(876\) 0 0
\(877\) 43121.6 1.66034 0.830168 0.557514i \(-0.188245\pi\)
0.830168 + 0.557514i \(0.188245\pi\)
\(878\) 68789.3 2.64411
\(879\) 0 0
\(880\) −1505.94 −0.0576878
\(881\) 40824.5 1.56120 0.780598 0.625034i \(-0.214915\pi\)
0.780598 + 0.625034i \(0.214915\pi\)
\(882\) 0 0
\(883\) 8262.14 0.314885 0.157442 0.987528i \(-0.449675\pi\)
0.157442 + 0.987528i \(0.449675\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −10722.3 −0.406571
\(887\) 40858.6 1.54667 0.773336 0.633997i \(-0.218587\pi\)
0.773336 + 0.633997i \(0.218587\pi\)
\(888\) 0 0
\(889\) −2442.22 −0.0921365
\(890\) −31243.2 −1.17671
\(891\) 0 0
\(892\) −69684.1 −2.61569
\(893\) −2388.37 −0.0895001
\(894\) 0 0
\(895\) −12638.6 −0.472023
\(896\) 37767.1 1.40816
\(897\) 0 0
\(898\) −32582.3 −1.21079
\(899\) 51359.4 1.90537
\(900\) 0 0
\(901\) 26102.2 0.965139
\(902\) 28396.8 1.04824
\(903\) 0 0
\(904\) −282.404 −0.0103901
\(905\) −49569.6 −1.82072
\(906\) 0 0
\(907\) −32729.3 −1.19819 −0.599095 0.800678i \(-0.704473\pi\)
−0.599095 + 0.800678i \(0.704473\pi\)
\(908\) −11684.4 −0.427048
\(909\) 0 0
\(910\) 0 0
\(911\) −11065.9 −0.402447 −0.201223 0.979545i \(-0.564492\pi\)
−0.201223 + 0.979545i \(0.564492\pi\)
\(912\) 0 0
\(913\) −27305.7 −0.989801
\(914\) −36474.5 −1.31999
\(915\) 0 0
\(916\) −2446.26 −0.0882386
\(917\) 5305.77 0.191071
\(918\) 0 0
\(919\) 50682.2 1.81921 0.909604 0.415477i \(-0.136385\pi\)
0.909604 + 0.415477i \(0.136385\pi\)
\(920\) −19779.5 −0.708816
\(921\) 0 0
\(922\) −18856.2 −0.673531
\(923\) 0 0
\(924\) 0 0
\(925\) −12262.3 −0.435874
\(926\) −32369.3 −1.14873
\(927\) 0 0
\(928\) −39474.0 −1.39633
\(929\) 41045.5 1.44958 0.724790 0.688970i \(-0.241937\pi\)
0.724790 + 0.688970i \(0.241937\pi\)
\(930\) 0 0
\(931\) −943.243 −0.0332047
\(932\) −11585.6 −0.407188
\(933\) 0 0
\(934\) 9677.86 0.339046
\(935\) 43176.9 1.51020
\(936\) 0 0
\(937\) −788.985 −0.0275080 −0.0137540 0.999905i \(-0.504378\pi\)
−0.0137540 + 0.999905i \(0.504378\pi\)
\(938\) −3000.25 −0.104437
\(939\) 0 0
\(940\) −26174.8 −0.908222
\(941\) −25676.2 −0.889499 −0.444750 0.895655i \(-0.646707\pi\)
−0.444750 + 0.895655i \(0.646707\pi\)
\(942\) 0 0
\(943\) −18036.5 −0.622851
\(944\) −1088.51 −0.0375295
\(945\) 0 0
\(946\) 48986.9 1.68362
\(947\) −679.352 −0.0233115 −0.0116557 0.999932i \(-0.503710\pi\)
−0.0116557 + 0.999932i \(0.503710\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 2833.60 0.0967726
\(951\) 0 0
\(952\) −47746.0 −1.62548
\(953\) 20238.4 0.687917 0.343958 0.938985i \(-0.388232\pi\)
0.343958 + 0.938985i \(0.388232\pi\)
\(954\) 0 0
\(955\) 6297.21 0.213375
\(956\) −25064.2 −0.847944
\(957\) 0 0
\(958\) −16894.9 −0.569781
\(959\) −7413.38 −0.249625
\(960\) 0 0
\(961\) 32280.6 1.08357
\(962\) 0 0
\(963\) 0 0
\(964\) −49503.6 −1.65395
\(965\) −54818.6 −1.82868
\(966\) 0 0
\(967\) 6161.63 0.204906 0.102453 0.994738i \(-0.467331\pi\)
0.102453 + 0.994738i \(0.467331\pi\)
\(968\) −15097.5 −0.501293
\(969\) 0 0
\(970\) 21293.7 0.704845
\(971\) 48569.5 1.60522 0.802610 0.596504i \(-0.203444\pi\)
0.802610 + 0.596504i \(0.203444\pi\)
\(972\) 0 0
\(973\) −13132.4 −0.432689
\(974\) 37012.5 1.21761
\(975\) 0 0
\(976\) 865.602 0.0283886
\(977\) −44595.8 −1.46033 −0.730167 0.683269i \(-0.760558\pi\)
−0.730167 + 0.683269i \(0.760558\pi\)
\(978\) 0 0
\(979\) −13264.6 −0.433031
\(980\) −10337.3 −0.336951
\(981\) 0 0
\(982\) −82224.4 −2.67198
\(983\) −29599.0 −0.960389 −0.480195 0.877162i \(-0.659434\pi\)
−0.480195 + 0.877162i \(0.659434\pi\)
\(984\) 0 0
\(985\) −35026.0 −1.13302
\(986\) 125751. 4.06159
\(987\) 0 0
\(988\) 0 0
\(989\) −31114.5 −1.00039
\(990\) 0 0
\(991\) 5718.45 0.183302 0.0916511 0.995791i \(-0.470786\pi\)
0.0916511 + 0.995791i \(0.470786\pi\)
\(992\) −47707.2 −1.52692
\(993\) 0 0
\(994\) −70021.9 −2.23437
\(995\) 40244.8 1.28226
\(996\) 0 0
\(997\) −1491.44 −0.0473766 −0.0236883 0.999719i \(-0.507541\pi\)
−0.0236883 + 0.999719i \(0.507541\pi\)
\(998\) −50164.8 −1.59112
\(999\) 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 1521.4.a.w.1.4 4
3.2 odd 2 507.4.a.l.1.1 4
13.5 odd 4 117.4.b.e.64.1 4
13.8 odd 4 117.4.b.e.64.4 4
13.12 even 2 inner 1521.4.a.w.1.1 4
39.5 even 4 39.4.b.b.25.4 yes 4
39.8 even 4 39.4.b.b.25.1 4
39.38 odd 2 507.4.a.l.1.4 4
156.47 odd 4 624.4.c.c.337.1 4
156.83 odd 4 624.4.c.c.337.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.b.b.25.1 4 39.8 even 4
39.4.b.b.25.4 yes 4 39.5 even 4
117.4.b.e.64.1 4 13.5 odd 4
117.4.b.e.64.4 4 13.8 odd 4
507.4.a.l.1.1 4 3.2 odd 2
507.4.a.l.1.4 4 39.38 odd 2
624.4.c.c.337.1 4 156.47 odd 4
624.4.c.c.337.4 4 156.83 odd 4
1521.4.a.w.1.1 4 13.12 even 2 inner
1521.4.a.w.1.4 4 1.1 even 1 trivial