Properties

Label 507.4.a.l.1.4
Level $507$
Weight $4$
Character 507.1
Self dual yes
Analytic conductor $29.914$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 507.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.9139683729\)
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 \(\chi\) \(=\) 507.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.54739 q^{2} +3.00000 q^{3} +12.6788 q^{4} +12.9118 q^{5} +13.6422 q^{6} +16.7289 q^{7} +21.2762 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+4.54739 q^{2} +3.00000 q^{3} +12.6788 q^{4} +12.9118 q^{5} +13.6422 q^{6} +16.7289 q^{7} +21.2762 q^{8} +9.00000 q^{9} +58.7151 q^{10} +24.9280 q^{11} +38.0363 q^{12} +76.0727 q^{14} +38.7355 q^{15} -4.67878 q^{16} -134.145 q^{17} +40.9265 q^{18} -14.9376 q^{19} +163.706 q^{20} +50.1866 q^{21} +113.358 q^{22} +72.0000 q^{23} +63.8287 q^{24} +41.7151 q^{25} +27.0000 q^{27} +212.101 q^{28} -206.145 q^{29} +176.145 q^{30} -249.142 q^{31} -191.486 q^{32} +74.7841 q^{33} -610.012 q^{34} +216.000 q^{35} +114.109 q^{36} +293.955 q^{37} -67.9273 q^{38} +274.715 q^{40} +250.506 q^{41} +228.218 q^{42} +432.145 q^{43} +316.057 q^{44} +116.206 q^{45} +327.412 q^{46} -159.889 q^{47} -14.0363 q^{48} -63.1454 q^{49} +189.695 q^{50} -402.436 q^{51} -194.581 q^{53} +122.780 q^{54} +321.866 q^{55} +355.927 q^{56} -44.8129 q^{57} -937.424 q^{58} +232.647 q^{59} +491.118 q^{60} -185.006 q^{61} -1132.94 q^{62} +150.560 q^{63} -833.333 q^{64} +340.073 q^{66} -39.4393 q^{67} -1700.80 q^{68} +216.000 q^{69} +982.237 q^{70} +920.460 q^{71} +191.486 q^{72} -549.078 q^{73} +1336.73 q^{74} +125.145 q^{75} -189.391 q^{76} +417.018 q^{77} +933.140 q^{79} -60.4116 q^{80} +81.0000 q^{81} +1139.15 q^{82} -1095.38 q^{83} +636.304 q^{84} -1732.06 q^{85} +1965.13 q^{86} -618.436 q^{87} +530.375 q^{88} -532.114 q^{89} +528.436 q^{90} +912.872 q^{92} -747.425 q^{93} -727.079 q^{94} -192.872 q^{95} -574.459 q^{96} -362.661 q^{97} -287.147 q^{98} +224.352 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{3} + 14 q^{4} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{3} + 14 q^{4} + 36 q^{9} + 88 q^{10} + 42 q^{12} + 84 q^{14} + 18 q^{16} - 96 q^{17} + 380 q^{22} + 288 q^{23} + 20 q^{25} + 108 q^{27} - 384 q^{29} + 264 q^{30} + 864 q^{35} + 126 q^{36} - 492 q^{38} + 952 q^{40} + 252 q^{42} + 1288 q^{43} + 54 q^{48} + 188 q^{49} - 288 q^{51} + 984 q^{53} - 328 q^{55} + 1644 q^{56} + 288 q^{61} - 1668 q^{62} - 1314 q^{64} + 1140 q^{66} - 4380 q^{68} + 864 q^{69} + 3144 q^{74} + 60 q^{75} - 1416 q^{77} + 4320 q^{79} + 324 q^{81} + 3088 q^{82} - 1152 q^{87} - 1036 q^{88} + 792 q^{90} + 1008 q^{92} - 1660 q^{94} + 1872 q^{95}+O(q^{100}) \) 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 \(0.202769\pi\)
\(3\) 3.00000 0.577350
\(4\) 12.6788 1.58485
\(5\) 12.9118 1.15487 0.577434 0.816437i \(-0.304054\pi\)
0.577434 + 0.816437i \(0.304054\pi\)
\(6\) 13.6422 0.928233
\(7\) 16.7289 0.903273 0.451637 0.892202i \(-0.350840\pi\)
0.451637 + 0.892202i \(0.350840\pi\)
\(8\) 21.2762 0.940286
\(9\) 9.00000 0.333333
\(10\) 58.7151 1.85674
\(11\) 24.9280 0.683280 0.341640 0.939831i \(-0.389018\pi\)
0.341640 + 0.939831i \(0.389018\pi\)
\(12\) 38.0363 0.915012
\(13\) 0 0
\(14\) 76.0727 1.45223
\(15\) 38.7355 0.666764
\(16\) −4.67878 −0.0731059
\(17\) −134.145 −1.91383 −0.956913 0.290376i \(-0.906220\pi\)
−0.956913 + 0.290376i \(0.906220\pi\)
\(18\) 40.9265 0.535915
\(19\) −14.9376 −0.180365 −0.0901824 0.995925i \(-0.528745\pi\)
−0.0901824 + 0.995925i \(0.528745\pi\)
\(20\) 163.706 1.83029
\(21\) 50.1866 0.521505
\(22\) 113.358 1.09854
\(23\) 72.0000 0.652741 0.326370 0.945242i \(-0.394174\pi\)
0.326370 + 0.945242i \(0.394174\pi\)
\(24\) 63.8287 0.542875
\(25\) 41.7151 0.333721
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 212.101 1.43155
\(29\) −206.145 −1.32001 −0.660004 0.751262i \(-0.729445\pi\)
−0.660004 + 0.751262i \(0.729445\pi\)
\(30\) 176.145 1.07199
\(31\) −249.142 −1.44346 −0.721728 0.692176i \(-0.756652\pi\)
−0.721728 + 0.692176i \(0.756652\pi\)
\(32\) −191.486 −1.05782
\(33\) 74.7841 0.394492
\(34\) −610.012 −3.07694
\(35\) 216.000 1.04316
\(36\) 114.109 0.528282
\(37\) 293.955 1.30610 0.653052 0.757313i \(-0.273488\pi\)
0.653052 + 0.757313i \(0.273488\pi\)
\(38\) −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 \(0.341686\pi\)
\(42\) 228.218 0.838448
\(43\) 432.145 1.53259 0.766297 0.642486i \(-0.222097\pi\)
0.766297 + 0.642486i \(0.222097\pi\)
\(44\) 316.057 1.08290
\(45\) 116.206 0.384956
\(46\) 327.412 1.04944
\(47\) −159.889 −0.496217 −0.248109 0.968732i \(-0.579809\pi\)
−0.248109 + 0.968732i \(0.579809\pi\)
\(48\) −14.0363 −0.0422077
\(49\) −63.1454 −0.184097
\(50\) 189.695 0.536539
\(51\) −402.436 −1.10495
\(52\) 0 0
\(53\) −194.581 −0.504298 −0.252149 0.967688i \(-0.581137\pi\)
−0.252149 + 0.967688i \(0.581137\pi\)
\(54\) 122.780 0.309411
\(55\) 321.866 0.789099
\(56\) 355.927 0.849336
\(57\) −44.8129 −0.104134
\(58\) −937.424 −2.12224
\(59\) 232.647 0.513358 0.256679 0.966497i \(-0.417372\pi\)
0.256679 + 0.966497i \(0.417372\pi\)
\(60\) 491.118 1.05672
\(61\) −185.006 −0.388321 −0.194160 0.980970i \(-0.562198\pi\)
−0.194160 + 0.980970i \(0.562198\pi\)
\(62\) −1132.94 −2.32071
\(63\) 150.560 0.301091
\(64\) −833.333 −1.62760
\(65\) 0 0
\(66\) 340.073 0.634243
\(67\) −39.4393 −0.0719145 −0.0359573 0.999353i \(-0.511448\pi\)
−0.0359573 + 0.999353i \(0.511448\pi\)
\(68\) −1700.80 −3.03312
\(69\) 216.000 0.376860
\(70\) 982.237 1.67714
\(71\) 920.460 1.53857 0.769286 0.638905i \(-0.220612\pi\)
0.769286 + 0.638905i \(0.220612\pi\)
\(72\) 191.486 0.313429
\(73\) −549.078 −0.880338 −0.440169 0.897915i \(-0.645082\pi\)
−0.440169 + 0.897915i \(0.645082\pi\)
\(74\) 1336.73 2.09988
\(75\) 125.145 0.192674
\(76\) −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
\(81\) 81.0000 0.111111
\(82\) 1139.15 1.53412
\(83\) −1095.38 −1.44860 −0.724301 0.689484i \(-0.757837\pi\)
−0.724301 + 0.689484i \(0.757837\pi\)
\(84\) 636.304 0.826506
\(85\) −1732.06 −2.21022
\(86\) 1965.13 2.46402
\(87\) −618.436 −0.762107
\(88\) 530.375 0.642479
\(89\) −532.114 −0.633753 −0.316876 0.948467i \(-0.602634\pi\)
−0.316876 + 0.948467i \(0.602634\pi\)
\(90\) 528.436 0.618912
\(91\) 0 0
\(92\) 912.872 1.03449
\(93\) −747.425 −0.833380
\(94\) −727.079 −0.797792
\(95\) −192.872 −0.208298
\(96\) −574.459 −0.610734
\(97\) −362.661 −0.379615 −0.189808 0.981821i \(-0.560786\pi\)
−0.189808 + 0.981821i \(0.560786\pi\)
\(98\) −287.147 −0.295982
\(99\) 224.352 0.227760
\(100\) 528.897 0.528897
\(101\) 1490.58 1.46850 0.734249 0.678880i \(-0.237534\pi\)
0.734249 + 0.678880i \(0.237534\pi\)
\(102\) −1830.03 −1.77648
\(103\) 628.436 0.601181 0.300591 0.953753i \(-0.402816\pi\)
0.300591 + 0.953753i \(0.402816\pi\)
\(104\) 0 0
\(105\) 648.000 0.602270
\(106\) −884.838 −0.810784
\(107\) 477.454 0.431376 0.215688 0.976462i \(-0.430801\pi\)
0.215688 + 0.976462i \(0.430801\pi\)
\(108\) 342.327 0.305004
\(109\) −378.207 −0.332345 −0.166173 0.986097i \(-0.553141\pi\)
−0.166173 + 0.986097i \(0.553141\pi\)
\(110\) 1463.65 1.26867
\(111\) 881.864 0.754079
\(112\) −78.2706 −0.0660346
\(113\) 13.2732 0.0110499 0.00552495 0.999985i \(-0.498241\pi\)
0.00552495 + 0.999985i \(0.498241\pi\)
\(114\) −203.782 −0.167420
\(115\) 929.651 0.753830
\(116\) −2613.67 −2.09201
\(117\) 0 0
\(118\) 1057.94 0.825349
\(119\) −2244.10 −1.72871
\(120\) 824.145 0.626949
\(121\) −709.593 −0.533128
\(122\) −841.294 −0.624321
\(123\) 751.519 0.550912
\(124\) −3158.81 −2.28766
\(125\) −1075.36 −0.769465
\(126\) 684.654 0.484078
\(127\) 145.988 0.102003 0.0510015 0.998699i \(-0.483759\pi\)
0.0510015 + 0.998699i \(0.483759\pi\)
\(128\) −2257.60 −1.55895
\(129\) 1296.44 0.884844
\(130\) 0 0
\(131\) 317.163 0.211532 0.105766 0.994391i \(-0.466271\pi\)
0.105766 + 0.994391i \(0.466271\pi\)
\(132\) 948.171 0.625210
\(133\) −249.890 −0.162919
\(134\) −179.346 −0.115620
\(135\) 348.619 0.222255
\(136\) −2854.11 −1.79954
\(137\) 443.149 0.276356 0.138178 0.990407i \(-0.455875\pi\)
0.138178 + 0.990407i \(0.455875\pi\)
\(138\) 982.237 0.605895
\(139\) 785.018 0.479024 0.239512 0.970893i \(-0.423013\pi\)
0.239512 + 0.970893i \(0.423013\pi\)
\(140\) 2738.62 1.65325
\(141\) −479.667 −0.286491
\(142\) 4185.69 2.47363
\(143\) 0 0
\(144\) −42.1090 −0.0243686
\(145\) −2661.71 −1.52444
\(146\) −2496.87 −1.41536
\(147\) −189.436 −0.106289
\(148\) 3726.99 2.06997
\(149\) 135.420 0.0744566 0.0372283 0.999307i \(-0.488147\pi\)
0.0372283 + 0.999307i \(0.488147\pi\)
\(150\) 569.085 0.309771
\(151\) −2373.74 −1.27929 −0.639643 0.768672i \(-0.720918\pi\)
−0.639643 + 0.768672i \(0.720918\pi\)
\(152\) −317.817 −0.169594
\(153\) −1207.31 −0.637942
\(154\) 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
\(159\) −583.744 −0.291157
\(160\) −2472.44 −1.22165
\(161\) 1204.48 0.589603
\(162\) 368.339 0.178638
\(163\) −2309.19 −1.10963 −0.554815 0.831974i \(-0.687211\pi\)
−0.554815 + 0.831974i \(0.687211\pi\)
\(164\) 3176.12 1.51227
\(165\) 965.599 0.455587
\(166\) −4981.14 −2.32898
\(167\) −600.788 −0.278386 −0.139193 0.990265i \(-0.544451\pi\)
−0.139193 + 0.990265i \(0.544451\pi\)
\(168\) 1067.78 0.490364
\(169\) 0 0
\(170\) −7876.36 −3.55347
\(171\) −134.439 −0.0601216
\(172\) 5479.08 2.42893
\(173\) 3430.36 1.50755 0.753773 0.657135i \(-0.228232\pi\)
0.753773 + 0.657135i \(0.228232\pi\)
\(174\) −2812.27 −1.22527
\(175\) 697.846 0.301441
\(176\) −116.633 −0.0499519
\(177\) 697.942 0.296387
\(178\) −2419.73 −1.01891
\(179\) 978.837 0.408725 0.204362 0.978895i \(-0.434488\pi\)
0.204362 + 0.978895i \(0.434488\pi\)
\(180\) 1473.36 0.610097
\(181\) −3839.09 −1.57656 −0.788279 0.615318i \(-0.789028\pi\)
−0.788279 + 0.615318i \(0.789028\pi\)
\(182\) 0 0
\(183\) −555.018 −0.224197
\(184\) 1531.89 0.613763
\(185\) 3795.49 1.50838
\(186\) −3398.83 −1.33986
\(187\) −3343.98 −1.30768
\(188\) −2027.20 −0.786429
\(189\) 451.679 0.173835
\(190\) −877.065 −0.334890
\(191\) −487.709 −0.184761 −0.0923806 0.995724i \(-0.529448\pi\)
−0.0923806 + 0.995724i \(0.529448\pi\)
\(192\) −2500.00 −0.939697
\(193\) 4245.61 1.58345 0.791725 0.610878i \(-0.209183\pi\)
0.791725 + 0.610878i \(0.209183\pi\)
\(194\) −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 \(0.663200\pi\)
\(198\) 1020.22 0.366180
\(199\) 3116.90 1.11031 0.555153 0.831748i \(-0.312660\pi\)
0.555153 + 0.831748i \(0.312660\pi\)
\(200\) 887.541 0.313793
\(201\) −118.318 −0.0415199
\(202\) 6778.26 2.36097
\(203\) −3448.58 −1.19233
\(204\) −5102.40 −1.75117
\(205\) 3234.49 1.10198
\(206\) 2857.75 0.966546
\(207\) 648.000 0.217580
\(208\) 0 0
\(209\) −372.366 −0.123240
\(210\) 2946.71 0.968297
\(211\) −1051.22 −0.342981 −0.171491 0.985186i \(-0.554858\pi\)
−0.171491 + 0.985186i \(0.554858\pi\)
\(212\) −2467.06 −0.799236
\(213\) 2761.38 0.888294
\(214\) 2171.17 0.693542
\(215\) 5579.78 1.76994
\(216\) 574.459 0.180958
\(217\) −4167.85 −1.30384
\(218\) −1719.85 −0.534327
\(219\) −1647.23 −0.508264
\(220\) 4080.87 1.25060
\(221\) 0 0
\(222\) 4010.18 1.21237
\(223\) 5496.12 1.65044 0.825218 0.564814i \(-0.191052\pi\)
0.825218 + 0.564814i \(0.191052\pi\)
\(224\) −3203.35 −0.955502
\(225\) 375.436 0.111240
\(226\) 60.3585 0.0177654
\(227\) −921.570 −0.269457 −0.134729 0.990883i \(-0.543016\pi\)
−0.134729 + 0.990883i \(0.543016\pi\)
\(228\) −568.173 −0.165036
\(229\) 192.941 0.0556764 0.0278382 0.999612i \(-0.491138\pi\)
0.0278382 + 0.999612i \(0.491138\pi\)
\(230\) 4227.49 1.21197
\(231\) 1251.05 0.356334
\(232\) −4386.00 −1.24119
\(233\) 913.779 0.256926 0.128463 0.991714i \(-0.458996\pi\)
0.128463 + 0.991714i \(0.458996\pi\)
\(234\) 0 0
\(235\) −2064.46 −0.573066
\(236\) 2949.68 0.813594
\(237\) 2799.42 0.767265
\(238\) −10204.8 −2.77932
\(239\) −1976.86 −0.535032 −0.267516 0.963553i \(-0.586203\pi\)
−0.267516 + 0.963553i \(0.586203\pi\)
\(240\) −181.235 −0.0487444
\(241\) 3904.45 1.04360 0.521800 0.853068i \(-0.325261\pi\)
0.521800 + 0.853068i \(0.325261\pi\)
\(242\) −3226.80 −0.857134
\(243\) 243.000 0.0641500
\(244\) −2345.65 −0.615429
\(245\) −815.322 −0.212608
\(246\) 3417.45 0.885727
\(247\) 0 0
\(248\) −5300.80 −1.35726
\(249\) −3286.15 −0.836350
\(250\) −4890.08 −1.23710
\(251\) −942.035 −0.236895 −0.118448 0.992960i \(-0.537792\pi\)
−0.118448 + 0.992960i \(0.537792\pi\)
\(252\) 1908.91 0.477183
\(253\) 1794.82 0.446005
\(254\) 663.866 0.163995
\(255\) −5196.18 −1.27607
\(256\) −3599.54 −0.878794
\(257\) −812.616 −0.197236 −0.0986179 0.995125i \(-0.531442\pi\)
−0.0986179 + 0.995125i \(0.531442\pi\)
\(258\) 5895.40 1.42260
\(259\) 4917.52 1.17977
\(260\) 0 0
\(261\) −1855.31 −0.440003
\(262\) 1442.26 0.340089
\(263\) −2608.29 −0.611536 −0.305768 0.952106i \(-0.598913\pi\)
−0.305768 + 0.952106i \(0.598913\pi\)
\(264\) 1591.13 0.370936
\(265\) −2512.40 −0.582398
\(266\) −1136.35 −0.261932
\(267\) −1596.34 −0.365897
\(268\) −500.042 −0.113974
\(269\) −4791.02 −1.08592 −0.542962 0.839757i \(-0.682697\pi\)
−0.542962 + 0.839757i \(0.682697\pi\)
\(270\) 1585.31 0.357329
\(271\) −3663.62 −0.821214 −0.410607 0.911812i \(-0.634683\pi\)
−0.410607 + 0.911812i \(0.634683\pi\)
\(272\) 627.637 0.139912
\(273\) 0 0
\(274\) 2015.17 0.444311
\(275\) 1039.88 0.228025
\(276\) 2738.62 0.597266
\(277\) −624.326 −0.135423 −0.0677114 0.997705i \(-0.521570\pi\)
−0.0677114 + 0.997705i \(0.521570\pi\)
\(278\) 3569.78 0.770149
\(279\) −2242.27 −0.481152
\(280\) 4595.67 0.980871
\(281\) 5535.12 1.17508 0.587540 0.809195i \(-0.300096\pi\)
0.587540 + 0.809195i \(0.300096\pi\)
\(282\) −2181.24 −0.460605
\(283\) 175.151 0.0367903 0.0183952 0.999831i \(-0.494144\pi\)
0.0183952 + 0.999831i \(0.494144\pi\)
\(284\) 11670.3 2.43840
\(285\) −578.616 −0.120261
\(286\) 0 0
\(287\) 4190.69 0.861911
\(288\) −1723.38 −0.352607
\(289\) 13082.0 2.66273
\(290\) −12103.8 −2.45091
\(291\) −1087.98 −0.219171
\(292\) −6961.64 −1.39520
\(293\) −7774.33 −1.55011 −0.775054 0.631895i \(-0.782277\pi\)
−0.775054 + 0.631895i \(0.782277\pi\)
\(294\) −861.440 −0.170885
\(295\) 3003.90 0.592861
\(296\) 6254.25 1.22811
\(297\) 673.057 0.131497
\(298\) 615.808 0.119707
\(299\) 0 0
\(300\) 1586.69 0.305359
\(301\) 7229.30 1.38435
\(302\) −10794.3 −2.05677
\(303\) 4471.74 0.847838
\(304\) 69.8899 0.0131857
\(305\) −2388.76 −0.448459
\(306\) −5490.10 −1.02565
\(307\) 8022.85 1.49149 0.745746 0.666230i \(-0.232093\pi\)
0.745746 + 0.666230i \(0.232093\pi\)
\(308\) 5287.27 0.978150
\(309\) 1885.31 0.347092
\(310\) −14628.4 −2.68012
\(311\) −9264.87 −1.68927 −0.844635 0.535343i \(-0.820182\pi\)
−0.844635 + 0.535343i \(0.820182\pi\)
\(312\) 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
\(315\) 1944.00 0.347721
\(316\) 11831.1 2.10617
\(317\) 2641.04 0.467935 0.233968 0.972244i \(-0.424829\pi\)
0.233968 + 0.972244i \(0.424829\pi\)
\(318\) −2654.51 −0.468106
\(319\) −5138.80 −0.901936
\(320\) −10759.8 −1.87967
\(321\) 1432.36 0.249055
\(322\) 5477.23 0.947932
\(323\) 2003.82 0.345187
\(324\) 1026.98 0.176094
\(325\) 0 0
\(326\) −10500.8 −1.78400
\(327\) −1134.62 −0.191880
\(328\) 5329.84 0.897229
\(329\) −2674.76 −0.448220
\(330\) 4390.96 0.732467
\(331\) 10779.5 1.79001 0.895007 0.446052i \(-0.147170\pi\)
0.895007 + 0.446052i \(0.147170\pi\)
\(332\) −13888.1 −2.29581
\(333\) 2645.59 0.435368
\(334\) −2732.02 −0.447573
\(335\) −509.233 −0.0830518
\(336\) −234.812 −0.0381251
\(337\) −313.465 −0.0506693 −0.0253346 0.999679i \(-0.508065\pi\)
−0.0253346 + 0.999679i \(0.508065\pi\)
\(338\) 0 0
\(339\) 39.8196 0.00637966
\(340\) −21960.4 −3.50286
\(341\) −6210.61 −0.986286
\(342\) −611.346 −0.0966602
\(343\) −6794.35 −1.06956
\(344\) 9194.43 1.44108
\(345\) 2788.95 0.435224
\(346\) 15599.2 2.42375
\(347\) −2849.23 −0.440792 −0.220396 0.975410i \(-0.570735\pi\)
−0.220396 + 0.975410i \(0.570735\pi\)
\(348\) −7841.01 −1.20782
\(349\) 6466.94 0.991883 0.495941 0.868356i \(-0.334823\pi\)
0.495941 + 0.868356i \(0.334823\pi\)
\(350\) 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 \(0.567041\pi\)
\(354\) 3173.82 0.476515
\(355\) 11884.8 1.77685
\(356\) −6746.56 −1.00440
\(357\) −6732.30 −0.998070
\(358\) 4451.16 0.657126
\(359\) −1467.11 −0.215685 −0.107843 0.994168i \(-0.534394\pi\)
−0.107843 + 0.994168i \(0.534394\pi\)
\(360\) 2472.44 0.361969
\(361\) −6635.87 −0.967469
\(362\) −17457.8 −2.53471
\(363\) −2128.78 −0.307801
\(364\) 0 0
\(365\) −7089.59 −1.01667
\(366\) −2523.88 −0.360452
\(367\) 4648.22 0.661130 0.330565 0.943783i \(-0.392761\pi\)
0.330565 + 0.943783i \(0.392761\pi\)
\(368\) −336.872 −0.0477192
\(369\) 2254.56 0.318069
\(370\) 17259.6 2.42509
\(371\) −3255.12 −0.455519
\(372\) −9476.44 −1.32078
\(373\) 1763.72 0.244831 0.122416 0.992479i \(-0.460936\pi\)
0.122416 + 0.992479i \(0.460936\pi\)
\(374\) −15206.4 −2.10242
\(375\) −3226.08 −0.444251
\(376\) −3401.84 −0.466586
\(377\) 0 0
\(378\) 2053.96 0.279483
\(379\) 1930.47 0.261640 0.130820 0.991406i \(-0.458239\pi\)
0.130820 + 0.991406i \(0.458239\pi\)
\(380\) −2445.38 −0.330120
\(381\) 437.965 0.0588914
\(382\) −2217.81 −0.297049
\(383\) −8845.93 −1.18017 −0.590086 0.807340i \(-0.700906\pi\)
−0.590086 + 0.807340i \(0.700906\pi\)
\(384\) −6772.81 −0.900061
\(385\) 5384.46 0.712772
\(386\) 19306.5 2.54579
\(387\) 3889.31 0.510865
\(388\) −4598.10 −0.601632
\(389\) −1598.08 −0.208292 −0.104146 0.994562i \(-0.533211\pi\)
−0.104146 + 0.994562i \(0.533211\pi\)
\(390\) 0 0
\(391\) −9658.47 −1.24923
\(392\) −1343.50 −0.173104
\(393\) 951.489 0.122128
\(394\) −12335.8 −1.57733
\(395\) 12048.5 1.53475
\(396\) 2844.51 0.360965
\(397\) 3578.82 0.452433 0.226217 0.974077i \(-0.427364\pi\)
0.226217 + 0.974077i \(0.427364\pi\)
\(398\) 14173.7 1.78509
\(399\) −749.669 −0.0940611
\(400\) −195.176 −0.0243970
\(401\) −3485.99 −0.434120 −0.217060 0.976158i \(-0.569647\pi\)
−0.217060 + 0.976158i \(0.569647\pi\)
\(402\) −538.038 −0.0667534
\(403\) 0 0
\(404\) 18898.8 2.32735
\(405\) 1045.86 0.128319
\(406\) −15682.0 −1.91696
\(407\) 7327.71 0.892435
\(408\) −8562.33 −1.03897
\(409\) 14709.1 1.77828 0.889139 0.457637i \(-0.151304\pi\)
0.889139 + 0.457637i \(0.151304\pi\)
\(410\) 14708.5 1.77171
\(411\) 1329.45 0.159554
\(412\) 7967.80 0.952780
\(413\) 3891.92 0.463702
\(414\) 2946.71 0.349814
\(415\) −14143.4 −1.67294
\(416\) 0 0
\(417\) 2355.05 0.276565
\(418\) −1693.29 −0.198138
\(419\) 3709.01 0.432451 0.216226 0.976343i \(-0.430625\pi\)
0.216226 + 0.976343i \(0.430625\pi\)
\(420\) 8215.85 0.954506
\(421\) 794.029 0.0919207 0.0459603 0.998943i \(-0.485365\pi\)
0.0459603 + 0.998943i \(0.485365\pi\)
\(422\) −4780.32 −0.551427
\(423\) −1439.00 −0.165406
\(424\) −4139.96 −0.474185
\(425\) −5595.89 −0.638684
\(426\) 12557.1 1.42815
\(427\) −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 \(0.448342\pi\)
\(432\) −126.327 −0.0140692
\(433\) 5560.94 0.617186 0.308593 0.951194i \(-0.400142\pi\)
0.308593 + 0.951194i \(0.400142\pi\)
\(434\) −18952.9 −2.09624
\(435\) −7985.14 −0.880133
\(436\) −4795.20 −0.526717
\(437\) −1075.51 −0.117731
\(438\) −7490.62 −0.817159
\(439\) 15127.2 1.64460 0.822302 0.569051i \(-0.192689\pi\)
0.822302 + 0.569051i \(0.192689\pi\)
\(440\) 6848.11 0.741979
\(441\) −568.308 −0.0613658
\(442\) 0 0
\(443\) 2357.89 0.252883 0.126441 0.991974i \(-0.459644\pi\)
0.126441 + 0.991974i \(0.459644\pi\)
\(444\) 11181.0 1.19510
\(445\) −6870.56 −0.731901
\(446\) 24993.0 2.65348
\(447\) 406.260 0.0429876
\(448\) −13940.7 −1.47017
\(449\) −7165.06 −0.753096 −0.376548 0.926397i \(-0.622889\pi\)
−0.376548 + 0.926397i \(0.622889\pi\)
\(450\) 1707.26 0.178846
\(451\) 6244.63 0.651992
\(452\) 168.288 0.0175124
\(453\) −7121.22 −0.738596
\(454\) −4190.74 −0.433219
\(455\) 0 0
\(456\) −953.451 −0.0979154
\(457\) 8020.96 0.821017 0.410508 0.911857i \(-0.365351\pi\)
0.410508 + 0.911857i \(0.365351\pi\)
\(458\) 877.378 0.0895135
\(459\) −3621.92 −0.368316
\(460\) 11786.8 1.19471
\(461\) −4146.59 −0.418928 −0.209464 0.977816i \(-0.567172\pi\)
−0.209464 + 0.977816i \(0.567172\pi\)
\(462\) 5689.03 0.572895
\(463\) 7118.21 0.714495 0.357248 0.934010i \(-0.383715\pi\)
0.357248 + 0.934010i \(0.383715\pi\)
\(464\) 964.509 0.0965004
\(465\) −9650.62 −0.962444
\(466\) 4155.31 0.413071
\(467\) −2128.22 −0.210883 −0.105441 0.994426i \(-0.533626\pi\)
−0.105441 + 0.994426i \(0.533626\pi\)
\(468\) 0 0
\(469\) −659.774 −0.0649585
\(470\) −9387.91 −0.921344
\(471\) −3500.18 −0.342420
\(472\) 4949.86 0.482703
\(473\) 10772.5 1.04719
\(474\) 12730.1 1.23357
\(475\) −623.125 −0.0601915
\(476\) −28452.4 −2.73974
\(477\) −1751.23 −0.168099
\(478\) −8989.57 −0.860196
\(479\) −3715.30 −0.354397 −0.177199 0.984175i \(-0.556704\pi\)
−0.177199 + 0.984175i \(0.556704\pi\)
\(480\) −7417.31 −0.705317
\(481\) 0 0
\(482\) 17755.0 1.67784
\(483\) 3613.43 0.340408
\(484\) −8996.77 −0.844926
\(485\) −4682.62 −0.438405
\(486\) 1105.02 0.103137
\(487\) −8139.28 −0.757343 −0.378671 0.925531i \(-0.623619\pi\)
−0.378671 + 0.925531i \(0.623619\pi\)
\(488\) −3936.23 −0.365133
\(489\) −6927.57 −0.640645
\(490\) −3707.59 −0.341820
\(491\) 18081.7 1.66194 0.830972 0.556315i \(-0.187785\pi\)
0.830972 + 0.556315i \(0.187785\pi\)
\(492\) 9528.35 0.873112
\(493\) 27653.4 2.52626
\(494\) 0 0
\(495\) 2896.80 0.263033
\(496\) 1165.68 0.105525
\(497\) 15398.3 1.38975
\(498\) −14943.4 −1.34464
\(499\) 11031.5 0.989659 0.494829 0.868990i \(-0.335231\pi\)
0.494829 + 0.868990i \(0.335231\pi\)
\(500\) −13634.2 −1.21948
\(501\) −1802.37 −0.160726
\(502\) −4283.80 −0.380868
\(503\) 8016.14 0.710581 0.355290 0.934756i \(-0.384382\pi\)
0.355290 + 0.934756i \(0.384382\pi\)
\(504\) 3203.35 0.283112
\(505\) 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 \(0.158623\pi\)
\(510\) −23629.1 −2.05159
\(511\) −9185.44 −0.795186
\(512\) 1692.30 0.146074
\(513\) −403.316 −0.0347112
\(514\) −3695.29 −0.317105
\(515\) 8114.25 0.694285
\(516\) 16437.2 1.40234
\(517\) −3985.72 −0.339056
\(518\) 22361.9 1.89677
\(519\) 10291.1 0.870382
\(520\) 0 0
\(521\) 9746.95 0.819619 0.409810 0.912171i \(-0.365595\pi\)
0.409810 + 0.912171i \(0.365595\pi\)
\(522\) −8436.81 −0.707413
\(523\) −18929.3 −1.58264 −0.791320 0.611402i \(-0.790606\pi\)
−0.791320 + 0.611402i \(0.790606\pi\)
\(524\) 4021.24 0.335245
\(525\) 2093.54 0.174037
\(526\) −11860.9 −0.983195
\(527\) 33421.2 2.76252
\(528\) −349.898 −0.0288397
\(529\) −6983.00 −0.573929
\(530\) −11424.9 −0.936349
\(531\) 2093.83 0.171119
\(532\) −3168.30 −0.258201
\(533\) 0 0
\(534\) −7259.20 −0.588270
\(535\) 6164.80 0.498182
\(536\) −839.120 −0.0676203
\(537\) 2936.51 0.235977
\(538\) −21786.6 −1.74589
\(539\) −1574.09 −0.125790
\(540\) 4420.07 0.352240
\(541\) −11366.8 −0.903321 −0.451661 0.892190i \(-0.649168\pi\)
−0.451661 + 0.892190i \(0.649168\pi\)
\(542\) −16659.9 −1.32030
\(543\) −11517.3 −0.910227
\(544\) 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
\(549\) −1665.05 −0.129440
\(550\) 4728.72 0.366606
\(551\) 3079.33 0.238083
\(552\) 4595.67 0.354356
\(553\) 15610.4 1.20040
\(554\) −2839.05 −0.217725
\(555\) 11386.5 0.870862
\(556\) 9953.06 0.759180
\(557\) −11873.1 −0.903192 −0.451596 0.892223i \(-0.649145\pi\)
−0.451596 + 0.892223i \(0.649145\pi\)
\(558\) −10196.5 −0.773571
\(559\) 0 0
\(560\) −1010.62 −0.0762613
\(561\) −10031.9 −0.754989
\(562\) 25170.4 1.88923
\(563\) −2829.31 −0.211796 −0.105898 0.994377i \(-0.533772\pi\)
−0.105898 + 0.994377i \(0.533772\pi\)
\(564\) −6081.60 −0.454045
\(565\) 171.381 0.0127612
\(566\) 796.481 0.0591495
\(567\) 1355.04 0.100364
\(568\) 19583.9 1.44670
\(569\) 16136.8 1.18891 0.594453 0.804130i \(-0.297368\pi\)
0.594453 + 0.804130i \(0.297368\pi\)
\(570\) −2631.20 −0.193349
\(571\) −17840.5 −1.30754 −0.653769 0.756695i \(-0.726813\pi\)
−0.653769 + 0.756695i \(0.726813\pi\)
\(572\) 0 0
\(573\) −1463.13 −0.106672
\(574\) 19056.7 1.38573
\(575\) 3003.49 0.217833
\(576\) −7500.00 −0.542534
\(577\) 8516.17 0.614442 0.307221 0.951638i \(-0.400601\pi\)
0.307221 + 0.951638i \(0.400601\pi\)
\(578\) 59488.9 4.28099
\(579\) 12736.8 0.914205
\(580\) −33747.3 −2.41600
\(581\) −18324.5 −1.30848
\(582\) −4947.49 −0.352371
\(583\) −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 \(0.759244\pi\)
\(588\) −2401.82 −0.168451
\(589\) 3721.59 0.260349
\(590\) 13659.9 0.953169
\(591\) −8138.13 −0.566426
\(592\) −1375.35 −0.0954839
\(593\) 11435.9 0.791933 0.395966 0.918265i \(-0.370410\pi\)
0.395966 + 0.918265i \(0.370410\pi\)
\(594\) 3060.65 0.211414
\(595\) −28975.4 −1.99643
\(596\) 1716.96 0.118002
\(597\) 9350.69 0.641035
\(598\) 0 0
\(599\) 1260.80 0.0860016 0.0430008 0.999075i \(-0.486308\pi\)
0.0430008 + 0.999075i \(0.486308\pi\)
\(600\) 2662.62 0.181169
\(601\) 6261.10 0.424951 0.212476 0.977166i \(-0.431847\pi\)
0.212476 + 0.977166i \(0.431847\pi\)
\(602\) 32874.5 2.22569
\(603\) −354.954 −0.0239715
\(604\) −30096.1 −2.02747
\(605\) −9162.14 −0.615692
\(606\) 20334.8 1.36311
\(607\) 3230.33 0.216005 0.108003 0.994151i \(-0.465555\pi\)
0.108003 + 0.994151i \(0.465555\pi\)
\(608\) 2860.35 0.190794
\(609\) −10345.7 −0.688391
\(610\) −10862.6 −0.721009
\(611\) 0 0
\(612\) −15307.2 −1.01104
\(613\) 14868.5 0.979660 0.489830 0.871818i \(-0.337059\pi\)
0.489830 + 0.871818i \(0.337059\pi\)
\(614\) 36483.0 2.39794
\(615\) 9703.48 0.636231
\(616\) 8872.57 0.580334
\(617\) −19952.8 −1.30190 −0.650949 0.759121i \(-0.725629\pi\)
−0.650949 + 0.759121i \(0.725629\pi\)
\(618\) 8573.24 0.558036
\(619\) 8316.48 0.540012 0.270006 0.962859i \(-0.412974\pi\)
0.270006 + 0.962859i \(0.412974\pi\)
\(620\) −40786.0 −2.64194
\(621\) 1944.00 0.125620
\(622\) −42131.0 −2.71592
\(623\) −8901.66 −0.572452
\(624\) 0 0
\(625\) −19099.2 −1.22235
\(626\) 33757.9 2.15533
\(627\) −1117.10 −0.0711525
\(628\) −14792.7 −0.939955
\(629\) −39432.6 −2.49965
\(630\) 8840.13 0.559046
\(631\) −12605.9 −0.795299 −0.397649 0.917537i \(-0.630174\pi\)
−0.397649 + 0.917537i \(0.630174\pi\)
\(632\) 19853.7 1.24959
\(633\) −3153.66 −0.198020
\(634\) 12009.8 0.752321
\(635\) 1884.98 0.117800
\(636\) −7401.17 −0.461439
\(637\) 0 0
\(638\) −23368.1 −1.45008
\(639\) 8284.14 0.512857
\(640\) −29149.8 −1.80038
\(641\) −9224.04 −0.568374 −0.284187 0.958769i \(-0.591724\pi\)
−0.284187 + 0.958769i \(0.591724\pi\)
\(642\) 6513.51 0.400417
\(643\) 4439.16 0.272260 0.136130 0.990691i \(-0.456533\pi\)
0.136130 + 0.990691i \(0.456533\pi\)
\(644\) 15271.3 0.934431
\(645\) 16739.4 1.02188
\(646\) 9112.13 0.554972
\(647\) −9601.15 −0.583401 −0.291700 0.956510i \(-0.594221\pi\)
−0.291700 + 0.956510i \(0.594221\pi\)
\(648\) 1723.38 0.104476
\(649\) 5799.44 0.350767
\(650\) 0 0
\(651\) −12503.6 −0.752770
\(652\) −29277.7 −1.75859
\(653\) 27112.8 1.62482 0.812410 0.583087i \(-0.198155\pi\)
0.812410 + 0.583087i \(0.198155\pi\)
\(654\) −5159.56 −0.308494
\(655\) 4095.15 0.244291
\(656\) −1172.06 −0.0697583
\(657\) −4941.70 −0.293446
\(658\) −12163.2 −0.720624
\(659\) 5587.26 0.330271 0.165136 0.986271i \(-0.447194\pi\)
0.165136 + 0.986271i \(0.447194\pi\)
\(660\) 12242.6 0.722035
\(661\) −3060.13 −0.180069 −0.0900343 0.995939i \(-0.528698\pi\)
−0.0900343 + 0.995939i \(0.528698\pi\)
\(662\) 49018.6 2.87789
\(663\) 0 0
\(664\) −23305.6 −1.36210
\(665\) −3226.53 −0.188150
\(666\) 12030.5 0.699961
\(667\) −14842.5 −0.861623
\(668\) −7617.26 −0.441199
\(669\) 16488.4 0.952880
\(670\) −2315.68 −0.133526
\(671\) −4611.83 −0.265332
\(672\) −9610.04 −0.551660
\(673\) 4121.55 0.236069 0.118034 0.993010i \(-0.462341\pi\)
0.118034 + 0.993010i \(0.462341\pi\)
\(674\) −1425.45 −0.0814633
\(675\) 1126.31 0.0642246
\(676\) 0 0
\(677\) 22889.5 1.29943 0.649715 0.760178i \(-0.274888\pi\)
0.649715 + 0.760178i \(0.274888\pi\)
\(678\) 181.075 0.0102569
\(679\) −6066.91 −0.342896
\(680\) −36851.8 −2.07824
\(681\) −2764.71 −0.155571
\(682\) −28242.1 −1.58570
\(683\) −19297.9 −1.08113 −0.540566 0.841301i \(-0.681790\pi\)
−0.540566 + 0.841301i \(0.681790\pi\)
\(684\) −1704.52 −0.0952835
\(685\) 5721.87 0.319155
\(686\) −30896.6 −1.71959
\(687\) 578.823 0.0321448
\(688\) −2021.91 −0.112042
\(689\) 0 0
\(690\) 12682.5 0.699729
\(691\) −30317.8 −1.66910 −0.834548 0.550935i \(-0.814271\pi\)
−0.834548 + 0.550935i \(0.814271\pi\)
\(692\) 43492.8 2.38923
\(693\) 3753.16 0.205730
\(694\) −12956.6 −0.708682
\(695\) 10136.0 0.553210
\(696\) −13158.0 −0.716599
\(697\) −33604.3 −1.82619
\(698\) 29407.7 1.59470
\(699\) 2741.34 0.148336
\(700\) 8847.84 0.477738
\(701\) 9606.16 0.517574 0.258787 0.965934i \(-0.416677\pi\)
0.258787 + 0.965934i \(0.416677\pi\)
\(702\) 0 0
\(703\) −4390.99 −0.235575
\(704\) −20773.4 −1.11211
\(705\) −6193.38 −0.330860
\(706\) −12610.4 −0.672235
\(707\) 24935.7 1.32646
\(708\) 8849.05 0.469729
\(709\) −23398.8 −1.23944 −0.619718 0.784825i \(-0.712753\pi\)
−0.619718 + 0.784825i \(0.712753\pi\)
\(710\) 54044.9 2.85672
\(711\) 8398.26 0.442981
\(712\) −11321.4 −0.595909
\(713\) −17938.2 −0.942203
\(714\) −30614.4 −1.60464
\(715\) 0 0
\(716\) 12410.5 0.647766
\(717\) −5930.59 −0.308901
\(718\) −6671.51 −0.346767
\(719\) −23588.7 −1.22352 −0.611758 0.791045i \(-0.709537\pi\)
−0.611758 + 0.791045i \(0.709537\pi\)
\(720\) −543.704 −0.0281426
\(721\) 10513.0 0.543031
\(722\) −30175.9 −1.55544
\(723\) 11713.3 0.602522
\(724\) −48674.9 −2.49861
\(725\) −8599.38 −0.440514
\(726\) −9680.40 −0.494867
\(727\) 15733.5 0.802644 0.401322 0.915937i \(-0.368551\pi\)
0.401322 + 0.915937i \(0.368551\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) −32239.2 −1.63455
\(731\) −57970.3 −2.93312
\(732\) −7036.94 −0.355318
\(733\) −17297.1 −0.871598 −0.435799 0.900044i \(-0.643534\pi\)
−0.435799 + 0.900044i \(0.643534\pi\)
\(734\) 21137.3 1.06293
\(735\) −2445.96 −0.122749
\(736\) −13787.0 −0.690484
\(737\) −983.144 −0.0491378
\(738\) 10252.4 0.511375
\(739\) 38749.2 1.92884 0.964419 0.264377i \(-0.0851663\pi\)
0.964419 + 0.264377i \(0.0851663\pi\)
\(740\) 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 \(0.524698\pi\)
\(744\) −15902.4 −0.783616
\(745\) 1748.52 0.0859876
\(746\) 8020.33 0.393626
\(747\) −9858.45 −0.482867
\(748\) −42397.6 −2.07247
\(749\) 7987.25 0.389650
\(750\) −14670.2 −0.714242
\(751\) 40628.6 1.97411 0.987055 0.160380i \(-0.0512721\pi\)
0.987055 + 0.160380i \(0.0512721\pi\)
\(752\) 748.086 0.0362764
\(753\) −2826.11 −0.136772
\(754\) 0 0
\(755\) −30649.3 −1.47741
\(756\) 5726.74 0.275502
\(757\) 24004.9 1.15254 0.576271 0.817259i \(-0.304507\pi\)
0.576271 + 0.817259i \(0.304507\pi\)
\(758\) 8778.62 0.420651
\(759\) 5384.46 0.257501
\(760\) −4103.60 −0.195859
\(761\) 29540.1 1.40713 0.703566 0.710630i \(-0.251590\pi\)
0.703566 + 0.710630i \(0.251590\pi\)
\(762\) 1991.60 0.0946824
\(763\) −6326.97 −0.300199
\(764\) −6183.56 −0.292818
\(765\) −15588.5 −0.736739
\(766\) −40225.9 −1.89742
\(767\) 0 0
\(768\) −10798.6 −0.507372
\(769\) 7585.63 0.355715 0.177857 0.984056i \(-0.443083\pi\)
0.177857 + 0.984056i \(0.443083\pi\)
\(770\) 24485.2 1.14596
\(771\) −2437.85 −0.113874
\(772\) 53829.2 2.50953
\(773\) 3284.29 0.152817 0.0764086 0.997077i \(-0.475655\pi\)
0.0764086 + 0.997077i \(0.475655\pi\)
\(774\) 17686.2 0.821341
\(775\) −10393.0 −0.481712
\(776\) −7716.07 −0.356947
\(777\) 14752.6 0.681140
\(778\) −7267.08 −0.334881
\(779\) −3741.98 −0.172106
\(780\) 0 0
\(781\) 22945.3 1.05128
\(782\) −43920.8 −2.00845
\(783\) −5565.92 −0.254036
\(784\) 295.443 0.0134586
\(785\) −15064.6 −0.684939
\(786\) 4326.79 0.196351
\(787\) −41624.1 −1.88531 −0.942654 0.333772i \(-0.891679\pi\)
−0.942654 + 0.333772i \(0.891679\pi\)
\(788\) −34393.8 −1.55486
\(789\) −7824.87 −0.353071
\(790\) 54789.4 2.46749
\(791\) 222.046 0.00998108
\(792\) 4773.38 0.214160
\(793\) 0 0
\(794\) 16274.3 0.727398
\(795\) −7537.20 −0.336248
\(796\) 39518.4 1.75967
\(797\) −30333.3 −1.34813 −0.674066 0.738671i \(-0.735454\pi\)
−0.674066 + 0.738671i \(0.735454\pi\)
\(798\) −3409.04 −0.151226
\(799\) 21448.4 0.949674
\(800\) −7987.87 −0.353017
\(801\) −4789.03 −0.211251
\(802\) −15852.2 −0.697954
\(803\) −13687.4 −0.601518
\(804\) −1500.13 −0.0658027
\(805\) 15552.0 0.680914
\(806\) 0 0
\(807\) −14373.1 −0.626958
\(808\) 31714.0 1.38081
\(809\) −24853.9 −1.08012 −0.540060 0.841627i \(-0.681598\pi\)
−0.540060 + 0.841627i \(0.681598\pi\)
\(810\) 4755.92 0.206304
\(811\) −17383.5 −0.752674 −0.376337 0.926483i \(-0.622816\pi\)
−0.376337 + 0.926483i \(0.622816\pi\)
\(812\) −43723.7 −1.88966
\(813\) −10990.9 −0.474128
\(814\) 33322.0 1.43481
\(815\) −29815.8 −1.28148
\(816\) 1882.91 0.0807782
\(817\) −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 \(0.730503\pi\)
\(822\) 6045.52 0.256523
\(823\) 5512.79 0.233492 0.116746 0.993162i \(-0.462754\pi\)
0.116746 + 0.993162i \(0.462754\pi\)
\(824\) 13370.8 0.565282
\(825\) 3119.63 0.131650
\(826\) 17698.1 0.745516
\(827\) −13335.8 −0.560738 −0.280369 0.959892i \(-0.590457\pi\)
−0.280369 + 0.959892i \(0.590457\pi\)
\(828\) 8215.85 0.344832
\(829\) −28338.5 −1.18726 −0.593628 0.804739i \(-0.702305\pi\)
−0.593628 + 0.804739i \(0.702305\pi\)
\(830\) −64315.5 −2.68967
\(831\) −1872.98 −0.0781864
\(832\) 0 0
\(833\) 8470.66 0.352330
\(834\) 10709.3 0.444646
\(835\) −7757.27 −0.321499
\(836\) −4721.15 −0.195316
\(837\) −6726.82 −0.277793
\(838\) 16866.3 0.695272
\(839\) −27149.5 −1.11717 −0.558585 0.829447i \(-0.688656\pi\)
−0.558585 + 0.829447i \(0.688656\pi\)
\(840\) 13787.0 0.566306
\(841\) 18106.9 0.742421
\(842\) 3610.76 0.147785
\(843\) 16605.4 0.678433
\(844\) −13328.2 −0.543573
\(845\) 0 0
\(846\) −6543.71 −0.265931
\(847\) −11870.7 −0.481560
\(848\) 910.404 0.0368672
\(849\) 525.454 0.0212409
\(850\) −25446.7 −1.02684
\(851\) 21164.7 0.852547
\(852\) 35010.9 1.40781
\(853\) 7978.22 0.320245 0.160123 0.987097i \(-0.448811\pi\)
0.160123 + 0.987097i \(0.448811\pi\)
\(854\) −14073.9 −0.563933
\(855\) −1735.85 −0.0694325
\(856\) 10158.4 0.405616
\(857\) −13614.4 −0.542657 −0.271329 0.962487i \(-0.587463\pi\)
−0.271329 + 0.962487i \(0.587463\pi\)
\(858\) 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
\(861\) 12572.1 0.497624
\(862\) 13148.9 0.519551
\(863\) −24461.5 −0.964867 −0.482434 0.875933i \(-0.660247\pi\)
−0.482434 + 0.875933i \(0.660247\pi\)
\(864\) −5170.13 −0.203578
\(865\) 44292.2 1.74102
\(866\) 25287.8 0.992278
\(867\) 39245.9 1.53733
\(868\) −52843.3 −2.06638
\(869\) 23261.3 0.908040
\(870\) −36311.5 −1.41503
\(871\) 0 0
\(872\) −8046.82 −0.312500
\(873\) −3263.95 −0.126538
\(874\) −4890.77 −0.189282
\(875\) −17989.5 −0.695037
\(876\) −20884.9 −0.805520
\(877\) −43121.6 −1.66034 −0.830168 0.557514i \(-0.811755\pi\)
−0.830168 + 0.557514i \(0.811755\pi\)
\(878\) 68789.3 2.64411
\(879\) −23323.0 −0.894955
\(880\) −1505.94 −0.0576878
\(881\) −40824.5 −1.56120 −0.780598 0.625034i \(-0.785085\pi\)
−0.780598 + 0.625034i \(0.785085\pi\)
\(882\) −2584.32 −0.0986605
\(883\) 8262.14 0.314885 0.157442 0.987528i \(-0.449675\pi\)
0.157442 + 0.987528i \(0.449675\pi\)
\(884\) 0 0
\(885\) 9011.70 0.342288
\(886\) 10722.3 0.406571
\(887\) −40858.6 −1.54667 −0.773336 0.633997i \(-0.781413\pi\)
−0.773336 + 0.633997i \(0.781413\pi\)
\(888\) 18762.8 0.709050
\(889\) 2442.22 0.0921365
\(890\) −31243.2 −1.17671
\(891\) 2019.17 0.0759201
\(892\) 69684.1 2.61569
\(893\) 2388.37 0.0895001
\(894\) 1847.42 0.0691131
\(895\) 12638.6 0.472023
\(896\) −37767.1 −1.40816
\(897\) 0 0
\(898\) −32582.3 −1.21079
\(899\) 51359.4 1.90537
\(900\) 4760.07 0.176299
\(901\) 26102.2 0.965139
\(902\) 28396.8 1.04824
\(903\) 21687.9 0.799256
\(904\) 282.404 0.0103901
\(905\) −49569.6 −1.82072
\(906\) −32383.0 −1.18747
\(907\) −32729.3 −1.19819 −0.599095 0.800678i \(-0.704473\pi\)
−0.599095 + 0.800678i \(0.704473\pi\)
\(908\) −11684.4 −0.427048
\(909\) 13415.2 0.489500
\(910\) 0 0
\(911\) 11065.9 0.402447 0.201223 0.979545i \(-0.435508\pi\)
0.201223 + 0.979545i \(0.435508\pi\)
\(912\) 209.670 0.00761279
\(913\) −27305.7 −0.989801
\(914\) 36474.5 1.31999
\(915\) −7166.29 −0.258918
\(916\) 2446.26 0.0882386
\(917\) 5305.77 0.191071
\(918\) −16470.3 −0.592158
\(919\) 50682.2 1.81921 0.909604 0.415477i \(-0.136385\pi\)
0.909604 + 0.415477i \(0.136385\pi\)
\(920\) 19779.5 0.708816
\(921\) 24068.5 0.861114
\(922\) −18856.2 −0.673531
\(923\) 0 0
\(924\) 15861.8 0.564735
\(925\) 12262.3 0.435874
\(926\) 32369.3 1.14873
\(927\) 5655.92 0.200394
\(928\) 39474.0 1.39633
\(929\) 41045.5 1.44958 0.724790 0.688970i \(-0.241937\pi\)
0.724790 + 0.688970i \(0.241937\pi\)
\(930\) −43885.1 −1.54737
\(931\) 943.243 0.0332047
\(932\) 11585.6 0.407188
\(933\) −27794.6 −0.975300
\(934\) −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
\(939\) 22270.7 0.773991
\(940\) −26174.8 −0.908222
\(941\) −25676.2 −0.889499 −0.444750 0.895655i \(-0.646707\pi\)
−0.444750 + 0.895655i \(0.646707\pi\)
\(942\) −15916.7 −0.550524
\(943\) 18036.5 0.622851
\(944\) −1088.51 −0.0375295
\(945\) 5832.00 0.200757
\(946\) 48986.9 1.68362
\(947\) −679.352 −0.0233115 −0.0116557 0.999932i \(-0.503710\pi\)
−0.0116557 + 0.999932i \(0.503710\pi\)
\(948\) 35493.2 1.21600
\(949\) 0 0
\(950\) −2833.60 −0.0967726
\(951\) 7923.12 0.270163
\(952\) −47746.0 −1.62548
\(953\) −20238.4 −0.687917 −0.343958 0.938985i \(-0.611768\pi\)
−0.343958 + 0.938985i \(0.611768\pi\)
\(954\) −7963.54 −0.270261
\(955\) −6297.21 −0.213375
\(956\) −25064.2 −0.847944
\(957\) −15416.4 −0.520733
\(958\) −16894.9 −0.569781
\(959\) 7413.38 0.249625
\(960\) −32279.5 −1.08523
\(961\) 32280.6 1.08357
\(962\) 0 0
\(963\) 4297.08 0.143792
\(964\) 49503.6 1.65395
\(965\) 54818.6 1.82868
\(966\) 16431.7 0.547289
\(967\) −6161.63 −0.204906 −0.102453 0.994738i \(-0.532669\pi\)
−0.102453 + 0.994738i \(0.532669\pi\)
\(968\) −15097.5 −0.501293
\(969\) 6011.45 0.199294
\(970\) −21293.7 −0.704845
\(971\) −48569.5 −1.60522 −0.802610 0.596504i \(-0.796556\pi\)
−0.802610 + 0.596504i \(0.796556\pi\)
\(972\) 3080.94 0.101668
\(973\) 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 \(0.760558\pi\)
\(978\) −31502.4 −1.03000
\(979\) −13264.6 −0.433031
\(980\) −10337.3 −0.336951
\(981\) −3403.86 −0.110782
\(982\) 82224.4 2.67198
\(983\) −29599.0 −0.960389 −0.480195 0.877162i \(-0.659434\pi\)
−0.480195 + 0.877162i \(0.659434\pi\)
\(984\) 15989.5 0.518015
\(985\) −35026.0 −1.13302
\(986\) 125751. 4.06159
\(987\) −8024.29 −0.258780
\(988\) 0 0
\(989\) 31114.5 1.00039
\(990\) 13172.9 0.422890
\(991\) 5718.45 0.183302 0.0916511 0.995791i \(-0.470786\pi\)
0.0916511 + 0.995791i \(0.470786\pi\)
\(992\) 47707.2 1.52692
\(993\) 32338.5 1.03347
\(994\) 70021.9 2.23437
\(995\) 40244.8 1.28226
\(996\) −41664.4 −1.32549
\(997\) −1491.44 −0.0473766 −0.0236883 0.999719i \(-0.507541\pi\)
−0.0236883 + 0.999719i \(0.507541\pi\)
\(998\) 50164.8 1.59112
\(999\) 7936.77 0.251360
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 507.4.a.l.1.4 4
3.2 odd 2 1521.4.a.w.1.1 4
13.5 odd 4 39.4.b.b.25.1 4
13.8 odd 4 39.4.b.b.25.4 yes 4
13.12 even 2 inner 507.4.a.l.1.1 4
39.5 even 4 117.4.b.e.64.4 4
39.8 even 4 117.4.b.e.64.1 4
39.38 odd 2 1521.4.a.w.1.4 4
52.31 even 4 624.4.c.c.337.1 4
52.47 even 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 13.5 odd 4
39.4.b.b.25.4 yes 4 13.8 odd 4
117.4.b.e.64.1 4 39.8 even 4
117.4.b.e.64.4 4 39.5 even 4
507.4.a.l.1.1 4 13.12 even 2 inner
507.4.a.l.1.4 4 1.1 even 1 trivial
624.4.c.c.337.1 4 52.31 even 4
624.4.c.c.337.4 4 52.47 even 4
1521.4.a.w.1.1 4 3.2 odd 2
1521.4.a.w.1.4 4 39.38 odd 2