Properties

Label 507.4.a.l.1.1
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 1.1
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 \(-0.797231\pi\)
−0.803873 + 0.594801i \(0.797231\pi\)
\(3\) 3.00000 0.577350
\(4\) 12.6788 1.58485
\(5\) −12.9118 −1.15487 −0.577434 0.816437i \(-0.695946\pi\)
−0.577434 + 0.816437i \(0.695946\pi\)
\(6\) −13.6422 −0.928233
\(7\) −16.7289 −0.903273 −0.451637 0.892202i \(-0.649160\pi\)
−0.451637 + 0.892202i \(0.649160\pi\)
\(8\) −21.2762 −0.940286
\(9\) 9.00000 0.333333
\(10\) 58.7151 1.85674
\(11\) −24.9280 −0.683280 −0.341640 0.939831i \(-0.610982\pi\)
−0.341640 + 0.939831i \(0.610982\pi\)
\(12\) 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 \(-0.471255\pi\)
0.0901824 + 0.995925i \(0.471255\pi\)
\(20\) −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 \(-0.243348\pi\)
0.721728 + 0.692176i \(0.243348\pi\)
\(32\) 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 \(-0.726512\pi\)
−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 \(-0.658314\pi\)
−0.477104 + 0.878847i \(0.658314\pi\)
\(42\) 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 \(-0.420191\pi\)
0.248109 + 0.968732i \(0.420191\pi\)
\(48\) −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 \(-0.582628\pi\)
−0.256679 + 0.966497i \(0.582628\pi\)
\(60\) −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 \(-0.488552\pi\)
0.0359573 + 0.999353i \(0.488552\pi\)
\(68\) −1700.80 −3.03312
\(69\) 216.000 0.376860
\(70\) −982.237 −1.67714
\(71\) −920.460 −1.53857 −0.769286 0.638905i \(-0.779388\pi\)
−0.769286 + 0.638905i \(0.779388\pi\)
\(72\) −191.486 −0.313429
\(73\) 549.078 0.880338 0.440169 0.897915i \(-0.354918\pi\)
0.440169 + 0.897915i \(0.354918\pi\)
\(74\) 1336.73 2.09988
\(75\) 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 \(-0.242163\pi\)
0.724301 + 0.689484i \(0.242163\pi\)
\(84\) −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 \(-0.397366\pi\)
0.316876 + 0.948467i \(0.397366\pi\)
\(90\) 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 \(-0.439214\pi\)
0.189808 + 0.981821i \(0.439214\pi\)
\(98\) 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 \(-0.446859\pi\)
0.166173 + 0.986097i \(0.446859\pi\)
\(110\) −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 \(-0.544125\pi\)
−0.138178 + 0.990407i \(0.544125\pi\)
\(138\) −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 \(-0.511853\pi\)
−0.0372283 + 0.999307i \(0.511853\pi\)
\(150\) −569.085 −0.309771
\(151\) 2373.74 1.27929 0.639643 0.768672i \(-0.279082\pi\)
0.639643 + 0.768672i \(0.279082\pi\)
\(152\) −317.817 −0.169594
\(153\) −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 \(-0.312789\pi\)
0.554815 + 0.831974i \(0.312789\pi\)
\(164\) −3176.12 −1.51227
\(165\) 965.599 0.455587
\(166\) −4981.14 −2.32898
\(167\) 600.788 0.278386 0.139193 0.990265i \(-0.455549\pi\)
0.139193 + 0.990265i \(0.455549\pi\)
\(168\) 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 \(-0.790817\pi\)
−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 \(-0.336800\pi\)
0.490539 + 0.871419i \(0.336800\pi\)
\(198\) 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 \(-0.808948\pi\)
−0.825218 + 0.564814i \(0.808948\pi\)
\(224\) −3203.35 −0.955502
\(225\) 375.436 0.111240
\(226\) −60.3585 −0.0177654
\(227\) 921.570 0.269457 0.134729 0.990883i \(-0.456984\pi\)
0.134729 + 0.990883i \(0.456984\pi\)
\(228\) 568.173 0.165036
\(229\) −192.941 −0.0556764 −0.0278382 0.999612i \(-0.508862\pi\)
−0.0278382 + 0.999612i \(0.508862\pi\)
\(230\) 4227.49 1.21197
\(231\) 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 \(-0.413797\pi\)
0.267516 + 0.963553i \(0.413797\pi\)
\(240\) 181.235 0.0487444
\(241\) −3904.45 −1.04360 −0.521800 0.853068i \(-0.674739\pi\)
−0.521800 + 0.853068i \(0.674739\pi\)
\(242\) 3226.80 0.857134
\(243\) 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 \(-0.365317\pi\)
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
\(276\) 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 \(-0.699904\pi\)
−0.587540 + 0.809195i \(0.699904\pi\)
\(282\) −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 \(-0.217723\pi\)
0.775054 + 0.631895i \(0.217723\pi\)
\(294\) 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 \(-0.767907\pi\)
−0.745746 + 0.666230i \(0.767907\pi\)
\(308\) 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 \(-0.575171\pi\)
−0.233968 + 0.972244i \(0.575171\pi\)
\(318\) 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 \(-0.852830\pi\)
−0.895007 + 0.446052i \(0.852830\pi\)
\(332\) 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 \(-0.665177\pi\)
−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 \(-0.432959\pi\)
0.209061 + 0.977903i \(0.432959\pi\)
\(354\) 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 \(-0.465606\pi\)
0.107843 + 0.994168i \(0.465606\pi\)
\(360\) 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 \(-0.541761\pi\)
−0.130820 + 0.991406i \(0.541761\pi\)
\(380\) −2445.38 −0.330120
\(381\) 437.965 0.0588914
\(382\) 2217.81 0.297049
\(383\) 8845.93 1.18017 0.590086 0.807340i \(-0.299094\pi\)
0.590086 + 0.807340i \(0.299094\pi\)
\(384\) 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 \(-0.572636\pi\)
−0.226217 + 0.974077i \(0.572636\pi\)
\(398\) −14173.7 −1.78509
\(399\) −749.669 −0.0940611
\(400\) −195.176 −0.0243970
\(401\) 3485.99 0.434120 0.217060 0.976158i \(-0.430353\pi\)
0.217060 + 0.976158i \(0.430353\pi\)
\(402\) −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 \(-0.848696\pi\)
−0.889139 + 0.457637i \(0.848696\pi\)
\(410\) −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 \(-0.514635\pi\)
−0.0459603 + 0.998943i \(0.514635\pi\)
\(422\) 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 \(-0.551658\pi\)
−0.161577 + 0.986860i \(0.551658\pi\)
\(432\) −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 \(-0.377111\pi\)
0.376548 + 0.926397i \(0.377111\pi\)
\(450\) −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 \(-0.634649\pi\)
−0.410508 + 0.911857i \(0.634649\pi\)
\(458\) 877.378 0.0895135
\(459\) −3621.92 −0.368316
\(460\) −11786.8 −1.19471
\(461\) 4146.59 0.418928 0.209464 0.977816i \(-0.432828\pi\)
0.209464 + 0.977816i \(0.432828\pi\)
\(462\) −5689.03 −0.572895
\(463\) −7118.21 −0.714495 −0.357248 0.934010i \(-0.616285\pi\)
−0.357248 + 0.934010i \(0.616285\pi\)
\(464\) 964.509 0.0965004
\(465\) −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 \(-0.443296\pi\)
0.177199 + 0.984175i \(0.443296\pi\)
\(480\) −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 \(-0.376381\pi\)
0.378671 + 0.925531i \(0.376381\pi\)
\(488\) 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 \(-0.664769\pi\)
−0.494829 + 0.868990i \(0.664769\pi\)
\(500\) 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 \(-0.841377\pi\)
−0.878382 + 0.477959i \(0.841377\pi\)
\(510\) −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 \(-0.350832\pi\)
0.451661 + 0.892190i \(0.350832\pi\)
\(542\) −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 \(-0.350855\pi\)
0.451596 + 0.892223i \(0.350855\pi\)
\(558\) −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 \(-0.599399\pi\)
−0.307221 + 0.951638i \(0.599399\pi\)
\(578\) −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 \(-0.240756\pi\)
0.727340 + 0.686277i \(0.240756\pi\)
\(588\) −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 \(-0.629590\pi\)
−0.395966 + 0.918265i \(0.629590\pi\)
\(594\) 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 \(-0.662941\pi\)
−0.489830 + 0.871818i \(0.662941\pi\)
\(614\) 36483.0 2.39794
\(615\) 9703.48 0.636231
\(616\) −8872.57 −0.580334
\(617\) 19952.8 1.30190 0.650949 0.759121i \(-0.274371\pi\)
0.650949 + 0.759121i \(0.274371\pi\)
\(618\) −8573.24 −0.558036
\(619\) −8316.48 −0.540012 −0.270006 0.962859i \(-0.587026\pi\)
−0.270006 + 0.962859i \(0.587026\pi\)
\(620\) −40786.0 −2.64194
\(621\) 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 \(-0.369826\pi\)
0.397649 + 0.917537i \(0.369826\pi\)
\(632\) −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 \(-0.543467\pi\)
−0.136130 + 0.990691i \(0.543467\pi\)
\(644\) −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 \(-0.471302\pi\)
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
\(666\) 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 \(-0.318210\pi\)
0.540566 + 0.841301i \(0.318210\pi\)
\(684\) 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 \(-0.185729\pi\)
0.834548 + 0.550935i \(0.185729\pi\)
\(692\) 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 \(-0.287247\pi\)
0.619718 + 0.784825i \(0.287247\pi\)
\(710\) −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 \(-0.356466\pi\)
0.435799 + 0.900044i \(0.356466\pi\)
\(734\) −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 \(-0.914834\pi\)
−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 \(-0.475302\pi\)
0.0775144 + 0.996991i \(0.475302\pi\)
\(744\) −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 \(-0.748410\pi\)
−0.703566 + 0.710630i \(0.748410\pi\)
\(762\) −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 \(-0.556917\pi\)
−0.177857 + 0.984056i \(0.556917\pi\)
\(770\) 24485.2 1.14596
\(771\) −2437.85 −0.113874
\(772\) −53829.2 −2.50953
\(773\) −3284.29 −0.152817 −0.0764086 0.997077i \(-0.524345\pi\)
−0.0764086 + 0.997077i \(0.524345\pi\)
\(774\) −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 \(-0.108321\pi\)
0.942654 + 0.333772i \(0.108321\pi\)
\(788\) 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 \(-0.377184\pi\)
0.376337 + 0.926483i \(0.377184\pi\)
\(812\) 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 \(-0.269497\pi\)
0.662496 + 0.749065i \(0.269497\pi\)
\(822\) 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 \(-0.409543\pi\)
0.280369 + 0.959892i \(0.409543\pi\)
\(828\) 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 \(-0.311344\pi\)
0.558585 + 0.829447i \(0.311344\pi\)
\(840\) −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 \(-0.551189\pi\)
−0.160123 + 0.987097i \(0.551189\pi\)
\(854\) −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 \(-0.339753\pi\)
0.482434 + 0.875933i \(0.339753\pi\)
\(864\) 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 \(-0.188245\pi\)
0.830168 + 0.557514i \(0.188245\pi\)
\(878\) −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 \(-0.758063\pi\)
−0.724790 + 0.688970i \(0.758063\pi\)
\(930\) 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 \(-0.353293\pi\)
0.444750 + 0.895655i \(0.353293\pi\)
\(942\) 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 \(-0.496290\pi\)
0.0116557 + 0.999932i \(0.496290\pi\)
\(948\) 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 \(-0.467331\pi\)
0.102453 + 0.994738i \(0.467331\pi\)
\(968\) 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 \(-0.239442\pi\)
0.730167 + 0.683269i \(0.239442\pi\)
\(978\) −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 \(-0.340566\pi\)
0.480195 + 0.877162i \(0.340566\pi\)
\(984\) 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.1.1 4
3.2 odd 2 1521.4.a.w.1.4 4
13.5 odd 4 39.4.b.b.25.4 yes 4
13.8 odd 4 39.4.b.b.25.1 4
13.12 even 2 inner 507.4.a.l.1.4 4
39.5 even 4 117.4.b.e.64.1 4
39.8 even 4 117.4.b.e.64.4 4
39.38 odd 2 1521.4.a.w.1.1 4
52.31 even 4 624.4.c.c.337.4 4
52.47 even 4 624.4.c.c.337.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.b.b.25.1 4 13.8 odd 4
39.4.b.b.25.4 yes 4 13.5 odd 4
117.4.b.e.64.1 4 39.5 even 4
117.4.b.e.64.4 4 39.8 even 4
507.4.a.l.1.1 4 1.1 even 1 trivial
507.4.a.l.1.4 4 13.12 even 2 inner
624.4.c.c.337.1 4 52.47 even 4
624.4.c.c.337.4 4 52.31 even 4
1521.4.a.w.1.1 4 39.38 odd 2
1521.4.a.w.1.4 4 3.2 odd 2