Properties

Label 1849.4.a.i.1.18
Level $1849$
Weight $4$
Character 1849.1
Self dual yes
Analytic conductor $109.095$
Analytic rank $1$
Dimension $50$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1849 = 43^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1849.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(109.094531601\)
Analytic rank: \(1\)
Dimension: \(50\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.18
Character \(\chi\) \(=\) 1849.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.25115 q^{2} +4.02813 q^{3} -2.93234 q^{4} +4.98765 q^{5} -9.06790 q^{6} +5.25109 q^{7} +24.6103 q^{8} -10.7742 q^{9} +O(q^{10})\) \(q-2.25115 q^{2} +4.02813 q^{3} -2.93234 q^{4} +4.98765 q^{5} -9.06790 q^{6} +5.25109 q^{7} +24.6103 q^{8} -10.7742 q^{9} -11.2279 q^{10} +25.9553 q^{11} -11.8118 q^{12} +14.2068 q^{13} -11.8210 q^{14} +20.0909 q^{15} -31.9427 q^{16} +20.3859 q^{17} +24.2543 q^{18} -2.21919 q^{19} -14.6255 q^{20} +21.1520 q^{21} -58.4292 q^{22} +39.1220 q^{23} +99.1334 q^{24} -100.123 q^{25} -31.9817 q^{26} -152.159 q^{27} -15.3980 q^{28} -277.025 q^{29} -45.2275 q^{30} -47.2599 q^{31} -124.975 q^{32} +104.551 q^{33} -45.8917 q^{34} +26.1906 q^{35} +31.5936 q^{36} -43.9224 q^{37} +4.99572 q^{38} +57.2270 q^{39} +122.748 q^{40} +98.7920 q^{41} -47.6164 q^{42} -76.1098 q^{44} -53.7380 q^{45} -88.0694 q^{46} +488.535 q^{47} -128.669 q^{48} -315.426 q^{49} +225.392 q^{50} +82.1171 q^{51} -41.6593 q^{52} -230.167 q^{53} +342.533 q^{54} +129.456 q^{55} +129.231 q^{56} -8.93917 q^{57} +623.623 q^{58} -400.770 q^{59} -58.9133 q^{60} -13.3239 q^{61} +106.389 q^{62} -56.5763 q^{63} +536.878 q^{64} +70.8588 q^{65} -235.360 q^{66} -640.592 q^{67} -59.7785 q^{68} +157.588 q^{69} -58.9589 q^{70} -251.714 q^{71} -265.156 q^{72} +526.799 q^{73} +98.8758 q^{74} -403.309 q^{75} +6.50741 q^{76} +136.294 q^{77} -128.826 q^{78} +978.288 q^{79} -159.319 q^{80} -322.013 q^{81} -222.395 q^{82} -37.2953 q^{83} -62.0250 q^{84} +101.678 q^{85} -1115.89 q^{87} +638.768 q^{88} -437.612 q^{89} +120.972 q^{90} +74.6014 q^{91} -114.719 q^{92} -190.369 q^{93} -1099.76 q^{94} -11.0685 q^{95} -503.414 q^{96} -1121.71 q^{97} +710.070 q^{98} -279.648 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50q - 10q^{2} - 2q^{3} + 186q^{4} + 8q^{5} - 51q^{6} - 6q^{7} - 138q^{8} + 360q^{9} + O(q^{10}) \) \( 50q - 10q^{2} - 2q^{3} + 186q^{4} + 8q^{5} - 51q^{6} - 6q^{7} - 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} - 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} - 386q^{18} + 12q^{19} + 108q^{20} - 408q^{21} + 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} + 1493q^{26} + 10q^{27} - 242q^{28} + 208q^{29} - 48q^{30} - 932q^{31} - 1124q^{32} + 254q^{33} + 765q^{34} - 1452q^{35} + 747q^{36} - 90q^{37} - 1213q^{38} - 1610q^{39} - 1693q^{40} - 1354q^{41} - 16q^{42} - 2704q^{44} + 4508q^{45} + 233q^{46} - 3484q^{47} - 376q^{48} + 1324q^{49} - 408q^{50} + 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} - 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} + 1172q^{61} - 1546q^{62} - 3686q^{63} + 606q^{64} + 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} + 136q^{69} - 1310q^{70} + 162q^{71} - 5814q^{72} + 746q^{73} - 4332q^{74} + 236q^{75} + 1338q^{76} + 2024q^{77} - 2782q^{78} - 2656q^{79} - 5713q^{80} - 86q^{81} - 4168q^{82} - 3514q^{83} - 4269q^{84} - 7558q^{85} - 10278q^{87} + 11692q^{88} + 2640q^{89} - 8286q^{90} - 5946q^{91} - 4271q^{92} - 2q^{93} + 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} - 2826q^{98} - 8174q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.25115 −0.795900 −0.397950 0.917407i \(-0.630278\pi\)
−0.397950 + 0.917407i \(0.630278\pi\)
\(3\) 4.02813 0.775213 0.387607 0.921825i \(-0.373302\pi\)
0.387607 + 0.921825i \(0.373302\pi\)
\(4\) −2.93234 −0.366543
\(5\) 4.98765 0.446109 0.223054 0.974806i \(-0.428397\pi\)
0.223054 + 0.974806i \(0.428397\pi\)
\(6\) −9.06790 −0.616992
\(7\) 5.25109 0.283532 0.141766 0.989900i \(-0.454722\pi\)
0.141766 + 0.989900i \(0.454722\pi\)
\(8\) 24.6103 1.08763
\(9\) −10.7742 −0.399045
\(10\) −11.2279 −0.355058
\(11\) 25.9553 0.711438 0.355719 0.934593i \(-0.384236\pi\)
0.355719 + 0.934593i \(0.384236\pi\)
\(12\) −11.8118 −0.284149
\(13\) 14.2068 0.303098 0.151549 0.988450i \(-0.451574\pi\)
0.151549 + 0.988450i \(0.451574\pi\)
\(14\) −11.8210 −0.225663
\(15\) 20.0909 0.345830
\(16\) −31.9427 −0.499104
\(17\) 20.3859 0.290842 0.145421 0.989370i \(-0.453546\pi\)
0.145421 + 0.989370i \(0.453546\pi\)
\(18\) 24.2543 0.317600
\(19\) −2.21919 −0.0267956 −0.0133978 0.999910i \(-0.504265\pi\)
−0.0133978 + 0.999910i \(0.504265\pi\)
\(20\) −14.6255 −0.163518
\(21\) 21.1520 0.219798
\(22\) −58.4292 −0.566234
\(23\) 39.1220 0.354674 0.177337 0.984150i \(-0.443252\pi\)
0.177337 + 0.984150i \(0.443252\pi\)
\(24\) 99.1334 0.843146
\(25\) −100.123 −0.800987
\(26\) −31.9817 −0.241236
\(27\) −152.159 −1.08456
\(28\) −15.3980 −0.103927
\(29\) −277.025 −1.77387 −0.886934 0.461896i \(-0.847169\pi\)
−0.886934 + 0.461896i \(0.847169\pi\)
\(30\) −45.2275 −0.275246
\(31\) −47.2599 −0.273810 −0.136905 0.990584i \(-0.543716\pi\)
−0.136905 + 0.990584i \(0.543716\pi\)
\(32\) −124.975 −0.690395
\(33\) 104.551 0.551516
\(34\) −45.8917 −0.231481
\(35\) 26.1906 0.126486
\(36\) 31.5936 0.146267
\(37\) −43.9224 −0.195157 −0.0975784 0.995228i \(-0.531110\pi\)
−0.0975784 + 0.995228i \(0.531110\pi\)
\(38\) 4.99572 0.0213266
\(39\) 57.2270 0.234965
\(40\) 122.748 0.485202
\(41\) 98.7920 0.376310 0.188155 0.982139i \(-0.439749\pi\)
0.188155 + 0.982139i \(0.439749\pi\)
\(42\) −47.6164 −0.174937
\(43\) 0 0
\(44\) −76.1098 −0.260772
\(45\) −53.7380 −0.178017
\(46\) −88.0694 −0.282285
\(47\) 488.535 1.51617 0.758086 0.652154i \(-0.226134\pi\)
0.758086 + 0.652154i \(0.226134\pi\)
\(48\) −128.669 −0.386912
\(49\) −315.426 −0.919609
\(50\) 225.392 0.637506
\(51\) 82.1171 0.225465
\(52\) −41.6593 −0.111098
\(53\) −230.167 −0.596525 −0.298263 0.954484i \(-0.596407\pi\)
−0.298263 + 0.954484i \(0.596407\pi\)
\(54\) 342.533 0.863200
\(55\) 129.456 0.317379
\(56\) 129.231 0.308379
\(57\) −8.93917 −0.0207723
\(58\) 623.623 1.41182
\(59\) −400.770 −0.884335 −0.442167 0.896933i \(-0.645790\pi\)
−0.442167 + 0.896933i \(0.645790\pi\)
\(60\) −58.9133 −0.126761
\(61\) −13.3239 −0.0279665 −0.0139832 0.999902i \(-0.504451\pi\)
−0.0139832 + 0.999902i \(0.504451\pi\)
\(62\) 106.389 0.217926
\(63\) −56.5763 −0.113142
\(64\) 536.878 1.04859
\(65\) 70.8588 0.135215
\(66\) −235.360 −0.438952
\(67\) −640.592 −1.16807 −0.584036 0.811728i \(-0.698527\pi\)
−0.584036 + 0.811728i \(0.698527\pi\)
\(68\) −59.7785 −0.106606
\(69\) 157.588 0.274948
\(70\) −58.9589 −0.100670
\(71\) −251.714 −0.420746 −0.210373 0.977621i \(-0.567468\pi\)
−0.210373 + 0.977621i \(0.567468\pi\)
\(72\) −265.156 −0.434014
\(73\) 526.799 0.844618 0.422309 0.906452i \(-0.361220\pi\)
0.422309 + 0.906452i \(0.361220\pi\)
\(74\) 98.8758 0.155325
\(75\) −403.309 −0.620935
\(76\) 6.50741 0.00982173
\(77\) 136.294 0.201716
\(78\) −128.826 −0.187009
\(79\) 978.288 1.39324 0.696620 0.717440i \(-0.254686\pi\)
0.696620 + 0.717440i \(0.254686\pi\)
\(80\) −159.319 −0.222655
\(81\) −322.013 −0.441719
\(82\) −222.395 −0.299505
\(83\) −37.2953 −0.0493215 −0.0246608 0.999696i \(-0.507851\pi\)
−0.0246608 + 0.999696i \(0.507851\pi\)
\(84\) −62.0250 −0.0805653
\(85\) 101.678 0.129747
\(86\) 0 0
\(87\) −1115.89 −1.37513
\(88\) 638.768 0.773783
\(89\) −437.612 −0.521200 −0.260600 0.965447i \(-0.583920\pi\)
−0.260600 + 0.965447i \(0.583920\pi\)
\(90\) 120.972 0.141684
\(91\) 74.6014 0.0859380
\(92\) −114.719 −0.130003
\(93\) −190.369 −0.212261
\(94\) −1099.76 −1.20672
\(95\) −11.0685 −0.0119538
\(96\) −503.414 −0.535203
\(97\) −1121.71 −1.17415 −0.587075 0.809533i \(-0.699721\pi\)
−0.587075 + 0.809533i \(0.699721\pi\)
\(98\) 710.070 0.731918
\(99\) −279.648 −0.283896
\(100\) 293.596 0.293596
\(101\) 1581.80 1.55837 0.779183 0.626796i \(-0.215634\pi\)
0.779183 + 0.626796i \(0.215634\pi\)
\(102\) −184.858 −0.179447
\(103\) −972.456 −0.930281 −0.465141 0.885237i \(-0.653996\pi\)
−0.465141 + 0.885237i \(0.653996\pi\)
\(104\) 349.635 0.329659
\(105\) 105.499 0.0980538
\(106\) 518.139 0.474775
\(107\) 1961.57 1.77226 0.886132 0.463433i \(-0.153383\pi\)
0.886132 + 0.463433i \(0.153383\pi\)
\(108\) 446.183 0.397537
\(109\) −2028.69 −1.78269 −0.891347 0.453322i \(-0.850239\pi\)
−0.891347 + 0.453322i \(0.850239\pi\)
\(110\) −291.424 −0.252602
\(111\) −176.925 −0.151288
\(112\) −167.734 −0.141512
\(113\) −1359.59 −1.13185 −0.565925 0.824457i \(-0.691481\pi\)
−0.565925 + 0.824457i \(0.691481\pi\)
\(114\) 20.1234 0.0165327
\(115\) 195.127 0.158223
\(116\) 812.330 0.650198
\(117\) −153.067 −0.120950
\(118\) 902.191 0.703842
\(119\) 107.048 0.0824631
\(120\) 494.443 0.376135
\(121\) −657.322 −0.493856
\(122\) 29.9941 0.0222585
\(123\) 397.946 0.291720
\(124\) 138.582 0.100363
\(125\) −1122.84 −0.803436
\(126\) 127.362 0.0900498
\(127\) 641.350 0.448115 0.224057 0.974576i \(-0.428070\pi\)
0.224057 + 0.974576i \(0.428070\pi\)
\(128\) −208.792 −0.144178
\(129\) 0 0
\(130\) −159.513 −0.107617
\(131\) 1243.19 0.829146 0.414573 0.910016i \(-0.363931\pi\)
0.414573 + 0.910016i \(0.363931\pi\)
\(132\) −306.580 −0.202154
\(133\) −11.6532 −0.00759742
\(134\) 1442.07 0.929669
\(135\) −758.917 −0.483831
\(136\) 501.704 0.316329
\(137\) 2114.51 1.31865 0.659323 0.751860i \(-0.270843\pi\)
0.659323 + 0.751860i \(0.270843\pi\)
\(138\) −354.755 −0.218831
\(139\) −3137.34 −1.91443 −0.957216 0.289374i \(-0.906553\pi\)
−0.957216 + 0.289374i \(0.906553\pi\)
\(140\) −76.7997 −0.0463626
\(141\) 1967.88 1.17536
\(142\) 566.645 0.334872
\(143\) 368.743 0.215635
\(144\) 344.157 0.199165
\(145\) −1381.70 −0.791338
\(146\) −1185.90 −0.672232
\(147\) −1270.58 −0.712893
\(148\) 128.795 0.0715332
\(149\) 1460.78 0.803167 0.401584 0.915822i \(-0.368460\pi\)
0.401584 + 0.915822i \(0.368460\pi\)
\(150\) 907.908 0.494203
\(151\) 3084.79 1.66249 0.831246 0.555905i \(-0.187628\pi\)
0.831246 + 0.555905i \(0.187628\pi\)
\(152\) −54.6149 −0.0291438
\(153\) −219.642 −0.116059
\(154\) −306.817 −0.160546
\(155\) −235.716 −0.122149
\(156\) −167.809 −0.0861248
\(157\) −2713.54 −1.37939 −0.689695 0.724100i \(-0.742255\pi\)
−0.689695 + 0.724100i \(0.742255\pi\)
\(158\) −2202.27 −1.10888
\(159\) −927.141 −0.462434
\(160\) −623.331 −0.307991
\(161\) 205.433 0.100562
\(162\) 724.898 0.351564
\(163\) 1564.36 0.751721 0.375860 0.926676i \(-0.377347\pi\)
0.375860 + 0.926676i \(0.377347\pi\)
\(164\) −289.692 −0.137934
\(165\) 521.465 0.246036
\(166\) 83.9571 0.0392550
\(167\) 508.284 0.235522 0.117761 0.993042i \(-0.462428\pi\)
0.117761 + 0.993042i \(0.462428\pi\)
\(168\) 520.558 0.239059
\(169\) −1995.17 −0.908132
\(170\) −228.892 −0.103266
\(171\) 23.9100 0.0106926
\(172\) 0 0
\(173\) −2615.65 −1.14950 −0.574751 0.818328i \(-0.694901\pi\)
−0.574751 + 0.818328i \(0.694901\pi\)
\(174\) 2512.03 1.09446
\(175\) −525.757 −0.227106
\(176\) −829.082 −0.355082
\(177\) −1614.35 −0.685548
\(178\) 985.129 0.414823
\(179\) −1299.11 −0.542458 −0.271229 0.962515i \(-0.587430\pi\)
−0.271229 + 0.962515i \(0.587430\pi\)
\(180\) 157.578 0.0652509
\(181\) −1263.14 −0.518721 −0.259360 0.965781i \(-0.583512\pi\)
−0.259360 + 0.965781i \(0.583512\pi\)
\(182\) −167.939 −0.0683981
\(183\) −53.6705 −0.0216800
\(184\) 962.805 0.385755
\(185\) −219.070 −0.0870612
\(186\) 428.548 0.168939
\(187\) 529.123 0.206916
\(188\) −1432.55 −0.555742
\(189\) −799.002 −0.307507
\(190\) 24.9169 0.00951400
\(191\) −1507.21 −0.570985 −0.285493 0.958381i \(-0.592157\pi\)
−0.285493 + 0.958381i \(0.592157\pi\)
\(192\) 2162.61 0.812880
\(193\) 643.966 0.240175 0.120087 0.992763i \(-0.461683\pi\)
0.120087 + 0.992763i \(0.461683\pi\)
\(194\) 2525.14 0.934506
\(195\) 285.428 0.104820
\(196\) 924.937 0.337076
\(197\) −3128.98 −1.13163 −0.565813 0.824534i \(-0.691438\pi\)
−0.565813 + 0.824534i \(0.691438\pi\)
\(198\) 629.528 0.225953
\(199\) −4879.50 −1.73818 −0.869092 0.494650i \(-0.835296\pi\)
−0.869092 + 0.494650i \(0.835296\pi\)
\(200\) −2464.07 −0.871179
\(201\) −2580.39 −0.905505
\(202\) −3560.86 −1.24030
\(203\) −1454.68 −0.502949
\(204\) −240.795 −0.0826424
\(205\) 492.740 0.167875
\(206\) 2189.14 0.740411
\(207\) −421.509 −0.141531
\(208\) −453.804 −0.151277
\(209\) −57.5997 −0.0190634
\(210\) −237.494 −0.0780411
\(211\) −79.4254 −0.0259141 −0.0129570 0.999916i \(-0.504124\pi\)
−0.0129570 + 0.999916i \(0.504124\pi\)
\(212\) 674.927 0.218652
\(213\) −1013.94 −0.326168
\(214\) −4415.78 −1.41055
\(215\) 0 0
\(216\) −3744.68 −1.17960
\(217\) −248.166 −0.0776341
\(218\) 4566.89 1.41885
\(219\) 2122.01 0.654759
\(220\) −379.609 −0.116333
\(221\) 289.620 0.0881536
\(222\) 398.284 0.120410
\(223\) 2831.34 0.850226 0.425113 0.905140i \(-0.360234\pi\)
0.425113 + 0.905140i \(0.360234\pi\)
\(224\) −656.254 −0.195749
\(225\) 1078.75 0.319629
\(226\) 3060.63 0.900840
\(227\) 3030.54 0.886097 0.443049 0.896498i \(-0.353897\pi\)
0.443049 + 0.896498i \(0.353897\pi\)
\(228\) 26.2127 0.00761393
\(229\) 6084.93 1.75591 0.877956 0.478742i \(-0.158907\pi\)
0.877956 + 0.478742i \(0.158907\pi\)
\(230\) −439.260 −0.125930
\(231\) 549.008 0.156373
\(232\) −6817.66 −1.92932
\(233\) −5872.34 −1.65111 −0.825557 0.564319i \(-0.809139\pi\)
−0.825557 + 0.564319i \(0.809139\pi\)
\(234\) 344.577 0.0962638
\(235\) 2436.64 0.676378
\(236\) 1175.19 0.324146
\(237\) 3940.66 1.08006
\(238\) −240.982 −0.0656324
\(239\) −1168.26 −0.316187 −0.158093 0.987424i \(-0.550535\pi\)
−0.158093 + 0.987424i \(0.550535\pi\)
\(240\) −641.756 −0.172605
\(241\) 2150.85 0.574890 0.287445 0.957797i \(-0.407194\pi\)
0.287445 + 0.957797i \(0.407194\pi\)
\(242\) 1479.73 0.393060
\(243\) 2811.19 0.742132
\(244\) 39.0703 0.0102509
\(245\) −1573.23 −0.410246
\(246\) −895.836 −0.232180
\(247\) −31.5277 −0.00812169
\(248\) −1163.08 −0.297805
\(249\) −150.230 −0.0382347
\(250\) 2527.67 0.639455
\(251\) −982.679 −0.247116 −0.123558 0.992337i \(-0.539431\pi\)
−0.123558 + 0.992337i \(0.539431\pi\)
\(252\) 165.901 0.0414714
\(253\) 1015.42 0.252329
\(254\) −1443.77 −0.356655
\(255\) 409.572 0.100582
\(256\) −3825.00 −0.933838
\(257\) 585.060 0.142004 0.0710020 0.997476i \(-0.477380\pi\)
0.0710020 + 0.997476i \(0.477380\pi\)
\(258\) 0 0
\(259\) −230.640 −0.0553332
\(260\) −207.782 −0.0495619
\(261\) 2984.72 0.707852
\(262\) −2798.60 −0.659917
\(263\) −104.375 −0.0244715 −0.0122358 0.999925i \(-0.503895\pi\)
−0.0122358 + 0.999925i \(0.503895\pi\)
\(264\) 2573.04 0.599847
\(265\) −1147.99 −0.266115
\(266\) 26.2329 0.00604679
\(267\) −1762.76 −0.404041
\(268\) 1878.44 0.428148
\(269\) 5940.70 1.34651 0.673254 0.739411i \(-0.264896\pi\)
0.673254 + 0.739411i \(0.264896\pi\)
\(270\) 1708.43 0.385081
\(271\) 3256.54 0.729965 0.364982 0.931014i \(-0.381075\pi\)
0.364982 + 0.931014i \(0.381075\pi\)
\(272\) −651.181 −0.145160
\(273\) 300.504 0.0666202
\(274\) −4760.06 −1.04951
\(275\) −2598.73 −0.569853
\(276\) −462.103 −0.100780
\(277\) −5348.68 −1.16018 −0.580092 0.814551i \(-0.696983\pi\)
−0.580092 + 0.814551i \(0.696983\pi\)
\(278\) 7062.62 1.52370
\(279\) 509.187 0.109263
\(280\) 644.558 0.137570
\(281\) −3582.54 −0.760557 −0.380279 0.924872i \(-0.624172\pi\)
−0.380279 + 0.924872i \(0.624172\pi\)
\(282\) −4429.98 −0.935467
\(283\) −7997.08 −1.67978 −0.839889 0.542759i \(-0.817380\pi\)
−0.839889 + 0.542759i \(0.817380\pi\)
\(284\) 738.111 0.154221
\(285\) −44.5854 −0.00926671
\(286\) −830.095 −0.171624
\(287\) 518.765 0.106696
\(288\) 1346.50 0.275498
\(289\) −4497.41 −0.915411
\(290\) 3110.41 0.629827
\(291\) −4518.39 −0.910216
\(292\) −1544.75 −0.309589
\(293\) −8020.77 −1.59924 −0.799621 0.600504i \(-0.794966\pi\)
−0.799621 + 0.600504i \(0.794966\pi\)
\(294\) 2860.25 0.567392
\(295\) −1998.90 −0.394510
\(296\) −1080.94 −0.212259
\(297\) −3949.34 −0.771596
\(298\) −3288.43 −0.639241
\(299\) 555.801 0.107501
\(300\) 1182.64 0.227599
\(301\) 0 0
\(302\) −6944.30 −1.32318
\(303\) 6371.69 1.20807
\(304\) 70.8867 0.0133738
\(305\) −66.4552 −0.0124761
\(306\) 494.447 0.0923714
\(307\) 4940.98 0.918556 0.459278 0.888292i \(-0.348108\pi\)
0.459278 + 0.888292i \(0.348108\pi\)
\(308\) −399.659 −0.0739374
\(309\) −3917.18 −0.721166
\(310\) 530.630 0.0972186
\(311\) 3934.00 0.717288 0.358644 0.933474i \(-0.383239\pi\)
0.358644 + 0.933474i \(0.383239\pi\)
\(312\) 1408.37 0.255556
\(313\) 2101.15 0.379437 0.189719 0.981839i \(-0.439242\pi\)
0.189719 + 0.981839i \(0.439242\pi\)
\(314\) 6108.58 1.09786
\(315\) −282.183 −0.0504737
\(316\) −2868.67 −0.510682
\(317\) 5062.65 0.896993 0.448497 0.893785i \(-0.351960\pi\)
0.448497 + 0.893785i \(0.351960\pi\)
\(318\) 2087.13 0.368051
\(319\) −7190.26 −1.26200
\(320\) 2677.76 0.467785
\(321\) 7901.45 1.37388
\(322\) −462.460 −0.0800370
\(323\) −45.2402 −0.00779329
\(324\) 944.252 0.161909
\(325\) −1422.44 −0.242777
\(326\) −3521.61 −0.598295
\(327\) −8171.83 −1.38197
\(328\) 2431.30 0.409287
\(329\) 2565.34 0.429884
\(330\) −1173.89 −0.195820
\(331\) −7137.29 −1.18520 −0.592599 0.805497i \(-0.701898\pi\)
−0.592599 + 0.805497i \(0.701898\pi\)
\(332\) 109.362 0.0180784
\(333\) 473.229 0.0778762
\(334\) −1144.22 −0.187452
\(335\) −3195.05 −0.521087
\(336\) −675.653 −0.109702
\(337\) −6668.67 −1.07794 −0.538970 0.842325i \(-0.681186\pi\)
−0.538970 + 0.842325i \(0.681186\pi\)
\(338\) 4491.41 0.722782
\(339\) −5476.58 −0.877425
\(340\) −298.154 −0.0475579
\(341\) −1226.64 −0.194799
\(342\) −53.8249 −0.00851028
\(343\) −3457.45 −0.544271
\(344\) 0 0
\(345\) 785.996 0.122657
\(346\) 5888.20 0.914889
\(347\) 7743.07 1.19790 0.598948 0.800788i \(-0.295586\pi\)
0.598948 + 0.800788i \(0.295586\pi\)
\(348\) 3272.17 0.504042
\(349\) 928.356 0.142389 0.0711945 0.997462i \(-0.477319\pi\)
0.0711945 + 0.997462i \(0.477319\pi\)
\(350\) 1183.56 0.180753
\(351\) −2161.70 −0.328727
\(352\) −3243.76 −0.491173
\(353\) −4940.92 −0.744982 −0.372491 0.928036i \(-0.621496\pi\)
−0.372491 + 0.928036i \(0.621496\pi\)
\(354\) 3634.14 0.545628
\(355\) −1255.46 −0.187699
\(356\) 1283.23 0.191042
\(357\) 431.204 0.0639265
\(358\) 2924.49 0.431743
\(359\) −308.245 −0.0453163 −0.0226581 0.999743i \(-0.507213\pi\)
−0.0226581 + 0.999743i \(0.507213\pi\)
\(360\) −1322.51 −0.193617
\(361\) −6854.08 −0.999282
\(362\) 2843.51 0.412850
\(363\) −2647.78 −0.382843
\(364\) −218.757 −0.0314999
\(365\) 2627.49 0.376792
\(366\) 120.820 0.0172551
\(367\) −8207.98 −1.16745 −0.583724 0.811952i \(-0.698405\pi\)
−0.583724 + 0.811952i \(0.698405\pi\)
\(368\) −1249.66 −0.177019
\(369\) −1064.40 −0.150164
\(370\) 493.158 0.0692920
\(371\) −1208.63 −0.169134
\(372\) 558.226 0.0778028
\(373\) 6480.94 0.899653 0.449826 0.893116i \(-0.351486\pi\)
0.449826 + 0.893116i \(0.351486\pi\)
\(374\) −1191.13 −0.164685
\(375\) −4522.93 −0.622834
\(376\) 12023.0 1.64904
\(377\) −3935.65 −0.537655
\(378\) 1798.67 0.244745
\(379\) −7927.62 −1.07444 −0.537222 0.843441i \(-0.680526\pi\)
−0.537222 + 0.843441i \(0.680526\pi\)
\(380\) 32.4567 0.00438156
\(381\) 2583.44 0.347384
\(382\) 3392.96 0.454447
\(383\) −13330.3 −1.77845 −0.889227 0.457467i \(-0.848757\pi\)
−0.889227 + 0.457467i \(0.848757\pi\)
\(384\) −841.041 −0.111769
\(385\) 679.785 0.0899872
\(386\) −1449.66 −0.191155
\(387\) 0 0
\(388\) 3289.24 0.430376
\(389\) −83.9976 −0.0109482 −0.00547410 0.999985i \(-0.501742\pi\)
−0.00547410 + 0.999985i \(0.501742\pi\)
\(390\) −642.540 −0.0834264
\(391\) 797.540 0.103154
\(392\) −7762.73 −1.00020
\(393\) 5007.73 0.642765
\(394\) 7043.78 0.900661
\(395\) 4879.36 0.621537
\(396\) 820.023 0.104060
\(397\) −3720.21 −0.470307 −0.235154 0.971958i \(-0.575559\pi\)
−0.235154 + 0.971958i \(0.575559\pi\)
\(398\) 10984.5 1.38342
\(399\) −46.9404 −0.00588962
\(400\) 3198.21 0.399776
\(401\) −10601.0 −1.32018 −0.660088 0.751188i \(-0.729481\pi\)
−0.660088 + 0.751188i \(0.729481\pi\)
\(402\) 5808.83 0.720692
\(403\) −671.414 −0.0829913
\(404\) −4638.38 −0.571208
\(405\) −1606.09 −0.197055
\(406\) 3274.70 0.400297
\(407\) −1140.02 −0.138842
\(408\) 2020.93 0.245223
\(409\) −3919.36 −0.473839 −0.236919 0.971529i \(-0.576138\pi\)
−0.236919 + 0.971529i \(0.576138\pi\)
\(410\) −1109.23 −0.133612
\(411\) 8517.50 1.02223
\(412\) 2851.57 0.340988
\(413\) −2104.48 −0.250737
\(414\) 948.878 0.112644
\(415\) −186.016 −0.0220028
\(416\) −1775.50 −0.209257
\(417\) −12637.6 −1.48409
\(418\) 129.665 0.0151726
\(419\) 4415.24 0.514793 0.257397 0.966306i \(-0.417135\pi\)
0.257397 + 0.966306i \(0.417135\pi\)
\(420\) −309.359 −0.0359409
\(421\) −7715.34 −0.893165 −0.446583 0.894742i \(-0.647359\pi\)
−0.446583 + 0.894742i \(0.647359\pi\)
\(422\) 178.798 0.0206250
\(423\) −5263.57 −0.605021
\(424\) −5664.47 −0.648800
\(425\) −2041.11 −0.232961
\(426\) 2282.52 0.259597
\(427\) −69.9652 −0.00792940
\(428\) −5751.99 −0.649610
\(429\) 1485.34 0.167163
\(430\) 0 0
\(431\) 10963.5 1.22527 0.612636 0.790365i \(-0.290109\pi\)
0.612636 + 0.790365i \(0.290109\pi\)
\(432\) 4860.37 0.541307
\(433\) −3896.75 −0.432485 −0.216242 0.976340i \(-0.569380\pi\)
−0.216242 + 0.976340i \(0.569380\pi\)
\(434\) 558.657 0.0617890
\(435\) −5565.67 −0.613456
\(436\) 5948.82 0.653433
\(437\) −86.8191 −0.00950371
\(438\) −4776.96 −0.521123
\(439\) −7414.58 −0.806102 −0.403051 0.915178i \(-0.632050\pi\)
−0.403051 + 0.915178i \(0.632050\pi\)
\(440\) 3185.95 0.345191
\(441\) 3398.46 0.366965
\(442\) −651.977 −0.0701615
\(443\) −8192.89 −0.878681 −0.439341 0.898321i \(-0.644788\pi\)
−0.439341 + 0.898321i \(0.644788\pi\)
\(444\) 518.804 0.0554535
\(445\) −2182.66 −0.232512
\(446\) −6373.76 −0.676695
\(447\) 5884.21 0.622626
\(448\) 2819.19 0.297309
\(449\) −8163.71 −0.858061 −0.429031 0.903290i \(-0.641145\pi\)
−0.429031 + 0.903290i \(0.641145\pi\)
\(450\) −2428.42 −0.254393
\(451\) 2564.18 0.267721
\(452\) 3986.77 0.414871
\(453\) 12425.9 1.28879
\(454\) −6822.19 −0.705245
\(455\) 372.086 0.0383377
\(456\) −219.996 −0.0225926
\(457\) −7695.45 −0.787698 −0.393849 0.919175i \(-0.628857\pi\)
−0.393849 + 0.919175i \(0.628857\pi\)
\(458\) −13698.1 −1.39753
\(459\) −3101.91 −0.315435
\(460\) −572.179 −0.0579956
\(461\) −2082.47 −0.210392 −0.105196 0.994452i \(-0.533547\pi\)
−0.105196 + 0.994452i \(0.533547\pi\)
\(462\) −1235.90 −0.124457
\(463\) 2804.81 0.281535 0.140767 0.990043i \(-0.455043\pi\)
0.140767 + 0.990043i \(0.455043\pi\)
\(464\) 8848.90 0.885345
\(465\) −949.492 −0.0946917
\(466\) 13219.5 1.31412
\(467\) −17860.1 −1.76974 −0.884869 0.465839i \(-0.845752\pi\)
−0.884869 + 0.465839i \(0.845752\pi\)
\(468\) 448.846 0.0443331
\(469\) −3363.81 −0.331186
\(470\) −5485.23 −0.538330
\(471\) −10930.5 −1.06932
\(472\) −9863.06 −0.961831
\(473\) 0 0
\(474\) −8871.01 −0.859618
\(475\) 222.192 0.0214629
\(476\) −313.902 −0.0302262
\(477\) 2479.86 0.238040
\(478\) 2629.93 0.251653
\(479\) 920.735 0.0878277 0.0439138 0.999035i \(-0.486017\pi\)
0.0439138 + 0.999035i \(0.486017\pi\)
\(480\) −2510.85 −0.238759
\(481\) −623.999 −0.0591516
\(482\) −4841.88 −0.457555
\(483\) 827.511 0.0779567
\(484\) 1927.49 0.181019
\(485\) −5594.70 −0.523799
\(486\) −6328.40 −0.590663
\(487\) 1931.99 0.179768 0.0898839 0.995952i \(-0.471350\pi\)
0.0898839 + 0.995952i \(0.471350\pi\)
\(488\) −327.906 −0.0304172
\(489\) 6301.46 0.582744
\(490\) 3541.58 0.326515
\(491\) 14234.0 1.30830 0.654148 0.756367i \(-0.273027\pi\)
0.654148 + 0.756367i \(0.273027\pi\)
\(492\) −1166.91 −0.106928
\(493\) −5647.41 −0.515916
\(494\) 70.9734 0.00646405
\(495\) −1394.79 −0.126648
\(496\) 1509.61 0.136660
\(497\) −1321.77 −0.119295
\(498\) 338.190 0.0304310
\(499\) −477.809 −0.0428651 −0.0214325 0.999770i \(-0.506823\pi\)
−0.0214325 + 0.999770i \(0.506823\pi\)
\(500\) 3292.54 0.294494
\(501\) 2047.43 0.182580
\(502\) 2212.15 0.196680
\(503\) −2881.02 −0.255384 −0.127692 0.991814i \(-0.540757\pi\)
−0.127692 + 0.991814i \(0.540757\pi\)
\(504\) −1392.36 −0.123057
\(505\) 7889.47 0.695201
\(506\) −2285.87 −0.200829
\(507\) −8036.78 −0.703996
\(508\) −1880.66 −0.164253
\(509\) −15177.9 −1.32171 −0.660854 0.750514i \(-0.729806\pi\)
−0.660854 + 0.750514i \(0.729806\pi\)
\(510\) −922.005 −0.0800531
\(511\) 2766.27 0.239476
\(512\) 10281.0 0.887420
\(513\) 337.670 0.0290614
\(514\) −1317.05 −0.113021
\(515\) −4850.27 −0.415007
\(516\) 0 0
\(517\) 12680.1 1.07866
\(518\) 519.205 0.0440397
\(519\) −10536.1 −0.891109
\(520\) 1743.86 0.147064
\(521\) 5817.50 0.489192 0.244596 0.969625i \(-0.421345\pi\)
0.244596 + 0.969625i \(0.421345\pi\)
\(522\) −6719.04 −0.563380
\(523\) 7595.81 0.635070 0.317535 0.948247i \(-0.397145\pi\)
0.317535 + 0.948247i \(0.397145\pi\)
\(524\) −3645.46 −0.303917
\(525\) −2117.81 −0.176055
\(526\) 234.962 0.0194769
\(527\) −963.437 −0.0796356
\(528\) −3339.64 −0.275264
\(529\) −10636.5 −0.874206
\(530\) 2584.30 0.211801
\(531\) 4317.97 0.352889
\(532\) 34.1710 0.00278478
\(533\) 1403.52 0.114059
\(534\) 3968.22 0.321576
\(535\) 9783.63 0.790623
\(536\) −15765.2 −1.27043
\(537\) −5232.98 −0.420521
\(538\) −13373.4 −1.07169
\(539\) −8186.98 −0.654245
\(540\) 2225.40 0.177345
\(541\) 5132.13 0.407851 0.203925 0.978986i \(-0.434630\pi\)
0.203925 + 0.978986i \(0.434630\pi\)
\(542\) −7330.94 −0.580979
\(543\) −5088.09 −0.402119
\(544\) −2547.73 −0.200796
\(545\) −10118.4 −0.795276
\(546\) −676.478 −0.0530231
\(547\) 6569.33 0.513500 0.256750 0.966478i \(-0.417348\pi\)
0.256750 + 0.966478i \(0.417348\pi\)
\(548\) −6200.45 −0.483340
\(549\) 143.555 0.0111599
\(550\) 5850.13 0.453546
\(551\) 614.769 0.0475319
\(552\) 3878.30 0.299042
\(553\) 5137.08 0.395028
\(554\) 12040.7 0.923390
\(555\) −882.440 −0.0674910
\(556\) 9199.76 0.701721
\(557\) 7594.11 0.577689 0.288845 0.957376i \(-0.406729\pi\)
0.288845 + 0.957376i \(0.406729\pi\)
\(558\) −1146.26 −0.0869621
\(559\) 0 0
\(560\) −836.597 −0.0631298
\(561\) 2131.38 0.160404
\(562\) 8064.83 0.605328
\(563\) 12214.0 0.914318 0.457159 0.889385i \(-0.348867\pi\)
0.457159 + 0.889385i \(0.348867\pi\)
\(564\) −5770.49 −0.430818
\(565\) −6781.14 −0.504929
\(566\) 18002.6 1.33694
\(567\) −1690.92 −0.125241
\(568\) −6194.76 −0.457617
\(569\) −11696.6 −0.861772 −0.430886 0.902406i \(-0.641799\pi\)
−0.430886 + 0.902406i \(0.641799\pi\)
\(570\) 100.368 0.00737538
\(571\) 17353.9 1.27187 0.635936 0.771742i \(-0.280614\pi\)
0.635936 + 0.771742i \(0.280614\pi\)
\(572\) −1081.28 −0.0790395
\(573\) −6071.25 −0.442635
\(574\) −1167.82 −0.0849194
\(575\) −3917.03 −0.284089
\(576\) −5784.43 −0.418434
\(577\) −14209.3 −1.02520 −0.512602 0.858626i \(-0.671318\pi\)
−0.512602 + 0.858626i \(0.671318\pi\)
\(578\) 10124.3 0.728576
\(579\) 2593.98 0.186186
\(580\) 4051.62 0.290059
\(581\) −195.841 −0.0139842
\(582\) 10171.6 0.724441
\(583\) −5974.05 −0.424391
\(584\) 12964.7 0.918634
\(585\) −763.447 −0.0539567
\(586\) 18055.9 1.27284
\(587\) 12326.7 0.866740 0.433370 0.901216i \(-0.357324\pi\)
0.433370 + 0.901216i \(0.357324\pi\)
\(588\) 3725.76 0.261306
\(589\) 104.878 0.00733691
\(590\) 4499.81 0.313990
\(591\) −12603.9 −0.877251
\(592\) 1403.00 0.0974035
\(593\) −18961.8 −1.31310 −0.656548 0.754284i \(-0.727984\pi\)
−0.656548 + 0.754284i \(0.727984\pi\)
\(594\) 8890.54 0.614113
\(595\) 533.920 0.0367875
\(596\) −4283.51 −0.294395
\(597\) −19655.3 −1.34746
\(598\) −1251.19 −0.0855601
\(599\) 14749.8 1.00611 0.503054 0.864255i \(-0.332210\pi\)
0.503054 + 0.864255i \(0.332210\pi\)
\(600\) −9925.56 −0.675349
\(601\) 14740.4 1.00045 0.500227 0.865894i \(-0.333250\pi\)
0.500227 + 0.865894i \(0.333250\pi\)
\(602\) 0 0
\(603\) 6901.87 0.466113
\(604\) −9045.64 −0.609374
\(605\) −3278.49 −0.220313
\(606\) −14343.6 −0.961501
\(607\) 28252.6 1.88919 0.944595 0.328239i \(-0.106455\pi\)
0.944595 + 0.328239i \(0.106455\pi\)
\(608\) 277.342 0.0184995
\(609\) −5859.64 −0.389892
\(610\) 149.600 0.00992973
\(611\) 6940.54 0.459549
\(612\) 644.066 0.0425406
\(613\) −24635.3 −1.62318 −0.811591 0.584226i \(-0.801398\pi\)
−0.811591 + 0.584226i \(0.801398\pi\)
\(614\) −11122.9 −0.731079
\(615\) 1984.82 0.130139
\(616\) 3354.23 0.219392
\(617\) −2636.49 −0.172028 −0.0860140 0.996294i \(-0.527413\pi\)
−0.0860140 + 0.996294i \(0.527413\pi\)
\(618\) 8818.14 0.573977
\(619\) 9682.05 0.628683 0.314341 0.949310i \(-0.398216\pi\)
0.314341 + 0.949310i \(0.398216\pi\)
\(620\) 691.198 0.0447729
\(621\) −5952.78 −0.384665
\(622\) −8856.01 −0.570890
\(623\) −2297.94 −0.147777
\(624\) −1827.98 −0.117272
\(625\) 6915.10 0.442567
\(626\) −4729.99 −0.301994
\(627\) −232.019 −0.0147782
\(628\) 7957.03 0.505605
\(629\) −895.400 −0.0567598
\(630\) 635.235 0.0401720
\(631\) −6750.28 −0.425871 −0.212935 0.977066i \(-0.568302\pi\)
−0.212935 + 0.977066i \(0.568302\pi\)
\(632\) 24075.9 1.51533
\(633\) −319.936 −0.0200889
\(634\) −11396.8 −0.713917
\(635\) 3198.83 0.199908
\(636\) 2718.69 0.169502
\(637\) −4481.21 −0.278732
\(638\) 16186.3 1.00442
\(639\) 2712.02 0.167896
\(640\) −1041.38 −0.0643191
\(641\) −4005.28 −0.246801 −0.123400 0.992357i \(-0.539380\pi\)
−0.123400 + 0.992357i \(0.539380\pi\)
\(642\) −17787.3 −1.09347
\(643\) 21183.8 1.29923 0.649617 0.760262i \(-0.274929\pi\)
0.649617 + 0.760262i \(0.274929\pi\)
\(644\) −602.401 −0.0368601
\(645\) 0 0
\(646\) 101.842 0.00620268
\(647\) −8425.30 −0.511952 −0.255976 0.966683i \(-0.582397\pi\)
−0.255976 + 0.966683i \(0.582397\pi\)
\(648\) −7924.84 −0.480427
\(649\) −10402.1 −0.629150
\(650\) 3202.11 0.193227
\(651\) −999.643 −0.0601829
\(652\) −4587.25 −0.275538
\(653\) 15075.9 0.903467 0.451733 0.892153i \(-0.350806\pi\)
0.451733 + 0.892153i \(0.350806\pi\)
\(654\) 18396.0 1.09991
\(655\) 6200.60 0.369889
\(656\) −3155.68 −0.187818
\(657\) −5675.84 −0.337040
\(658\) −5774.96 −0.342145
\(659\) −15177.4 −0.897159 −0.448580 0.893743i \(-0.648070\pi\)
−0.448580 + 0.893743i \(0.648070\pi\)
\(660\) −1529.11 −0.0901828
\(661\) 8032.86 0.472681 0.236340 0.971670i \(-0.424052\pi\)
0.236340 + 0.971670i \(0.424052\pi\)
\(662\) 16067.1 0.943300
\(663\) 1166.63 0.0683378
\(664\) −917.848 −0.0536437
\(665\) −58.1218 −0.00338928
\(666\) −1065.31 −0.0619817
\(667\) −10837.8 −0.629145
\(668\) −1490.46 −0.0863288
\(669\) 11405.0 0.659107
\(670\) 7192.53 0.414734
\(671\) −345.827 −0.0198964
\(672\) −2643.47 −0.151747
\(673\) 2275.45 0.130330 0.0651651 0.997874i \(-0.479243\pi\)
0.0651651 + 0.997874i \(0.479243\pi\)
\(674\) 15012.1 0.857932
\(675\) 15234.7 0.868716
\(676\) 5850.50 0.332869
\(677\) −10618.2 −0.602793 −0.301397 0.953499i \(-0.597453\pi\)
−0.301397 + 0.953499i \(0.597453\pi\)
\(678\) 12328.6 0.698343
\(679\) −5890.21 −0.332909
\(680\) 2502.32 0.141117
\(681\) 12207.4 0.686914
\(682\) 2761.36 0.155041
\(683\) −19742.9 −1.10606 −0.553031 0.833161i \(-0.686529\pi\)
−0.553031 + 0.833161i \(0.686529\pi\)
\(684\) −70.1122 −0.00391931
\(685\) 10546.4 0.588260
\(686\) 7783.24 0.433186
\(687\) 24510.9 1.36121
\(688\) 0 0
\(689\) −3269.94 −0.180805
\(690\) −1769.39 −0.0976226
\(691\) −8138.94 −0.448075 −0.224037 0.974581i \(-0.571924\pi\)
−0.224037 + 0.974581i \(0.571924\pi\)
\(692\) 7669.96 0.421341
\(693\) −1468.46 −0.0804935
\(694\) −17430.8 −0.953406
\(695\) −15648.0 −0.854045
\(696\) −27462.4 −1.49563
\(697\) 2013.97 0.109447
\(698\) −2089.86 −0.113327
\(699\) −23654.5 −1.27997
\(700\) 1541.70 0.0832438
\(701\) 30481.7 1.64234 0.821169 0.570685i \(-0.193322\pi\)
0.821169 + 0.570685i \(0.193322\pi\)
\(702\) 4866.31 0.261634
\(703\) 97.4720 0.00522934
\(704\) 13934.8 0.746007
\(705\) 9815.09 0.524337
\(706\) 11122.7 0.592931
\(707\) 8306.18 0.441847
\(708\) 4733.82 0.251283
\(709\) −84.5679 −0.00447957 −0.00223978 0.999997i \(-0.500713\pi\)
−0.00223978 + 0.999997i \(0.500713\pi\)
\(710\) 2826.23 0.149389
\(711\) −10540.3 −0.555965
\(712\) −10769.8 −0.566873
\(713\) −1848.90 −0.0971135
\(714\) −970.704 −0.0508791
\(715\) 1839.16 0.0961968
\(716\) 3809.43 0.198834
\(717\) −4705.91 −0.245112
\(718\) 693.905 0.0360673
\(719\) 19339.8 1.00314 0.501568 0.865118i \(-0.332757\pi\)
0.501568 + 0.865118i \(0.332757\pi\)
\(720\) 1716.53 0.0888492
\(721\) −5106.46 −0.263765
\(722\) 15429.5 0.795329
\(723\) 8663.90 0.445662
\(724\) 3703.96 0.190133
\(725\) 27736.6 1.42084
\(726\) 5960.53 0.304705
\(727\) 25267.6 1.28903 0.644514 0.764592i \(-0.277060\pi\)
0.644514 + 0.764592i \(0.277060\pi\)
\(728\) 1835.96 0.0934688
\(729\) 20018.2 1.01703
\(730\) −5914.86 −0.299889
\(731\) 0 0
\(732\) 157.380 0.00794664
\(733\) −22425.0 −1.13000 −0.564998 0.825092i \(-0.691123\pi\)
−0.564998 + 0.825092i \(0.691123\pi\)
\(734\) 18477.4 0.929172
\(735\) −6337.19 −0.318028
\(736\) −4889.27 −0.244865
\(737\) −16626.8 −0.831011
\(738\) 2396.13 0.119516
\(739\) 31297.4 1.55791 0.778953 0.627083i \(-0.215751\pi\)
0.778953 + 0.627083i \(0.215751\pi\)
\(740\) 642.387 0.0319116
\(741\) −126.997 −0.00629604
\(742\) 2720.79 0.134614
\(743\) −26708.5 −1.31876 −0.659380 0.751810i \(-0.729181\pi\)
−0.659380 + 0.751810i \(0.729181\pi\)
\(744\) −4685.03 −0.230862
\(745\) 7285.87 0.358300
\(746\) −14589.6 −0.716034
\(747\) 401.827 0.0196815
\(748\) −1551.57 −0.0758436
\(749\) 10300.4 0.502494
\(750\) 10181.8 0.495714
\(751\) −36421.1 −1.76967 −0.884837 0.465901i \(-0.845730\pi\)
−0.884837 + 0.465901i \(0.845730\pi\)
\(752\) −15605.1 −0.756728
\(753\) −3958.35 −0.191568
\(754\) 8859.71 0.427920
\(755\) 15385.8 0.741653
\(756\) 2342.95 0.112714
\(757\) 16912.5 0.812016 0.406008 0.913870i \(-0.366921\pi\)
0.406008 + 0.913870i \(0.366921\pi\)
\(758\) 17846.2 0.855151
\(759\) 4090.26 0.195609
\(760\) −272.400 −0.0130013
\(761\) 39596.2 1.88615 0.943075 0.332581i \(-0.107919\pi\)
0.943075 + 0.332581i \(0.107919\pi\)
\(762\) −5815.69 −0.276483
\(763\) −10652.9 −0.505451
\(764\) 4419.66 0.209290
\(765\) −1095.50 −0.0517750
\(766\) 30008.5 1.41547
\(767\) −5693.67 −0.268040
\(768\) −15407.6 −0.723924
\(769\) 35365.1 1.65839 0.829193 0.558962i \(-0.188800\pi\)
0.829193 + 0.558962i \(0.188800\pi\)
\(770\) −1530.30 −0.0716208
\(771\) 2356.69 0.110083
\(772\) −1888.33 −0.0880342
\(773\) 3602.72 0.167634 0.0838169 0.996481i \(-0.473289\pi\)
0.0838169 + 0.996481i \(0.473289\pi\)
\(774\) 0 0
\(775\) 4731.82 0.219318
\(776\) −27605.6 −1.27704
\(777\) −929.049 −0.0428950
\(778\) 189.091 0.00871367
\(779\) −219.238 −0.0100835
\(780\) −836.972 −0.0384210
\(781\) −6533.32 −0.299335
\(782\) −1795.38 −0.0821005
\(783\) 42151.8 1.92386
\(784\) 10075.5 0.458981
\(785\) −13534.2 −0.615358
\(786\) −11273.1 −0.511577
\(787\) −18197.3 −0.824224 −0.412112 0.911133i \(-0.635209\pi\)
−0.412112 + 0.911133i \(0.635209\pi\)
\(788\) 9175.22 0.414789
\(789\) −420.434 −0.0189706
\(790\) −10984.1 −0.494681
\(791\) −7139.31 −0.320916
\(792\) −6882.22 −0.308774
\(793\) −189.291 −0.00847658
\(794\) 8374.74 0.374318
\(795\) −4624.25 −0.206296
\(796\) 14308.4 0.637119
\(797\) −5337.43 −0.237216 −0.118608 0.992941i \(-0.537843\pi\)
−0.118608 + 0.992941i \(0.537843\pi\)
\(798\) 105.670 0.00468755
\(799\) 9959.24 0.440967
\(800\) 12512.9 0.552997
\(801\) 4714.92 0.207982
\(802\) 23864.5 1.05073
\(803\) 13673.2 0.600894
\(804\) 7566.57 0.331906
\(805\) 1024.63 0.0448614
\(806\) 1511.45 0.0660528
\(807\) 23929.9 1.04383
\(808\) 38928.6 1.69493
\(809\) 29582.7 1.28563 0.642813 0.766023i \(-0.277767\pi\)
0.642813 + 0.766023i \(0.277767\pi\)
\(810\) 3615.54 0.156836
\(811\) 28394.0 1.22941 0.614703 0.788759i \(-0.289276\pi\)
0.614703 + 0.788759i \(0.289276\pi\)
\(812\) 4265.62 0.184352
\(813\) 13117.7 0.565878
\(814\) 2566.35 0.110504
\(815\) 7802.50 0.335349
\(816\) −2623.04 −0.112530
\(817\) 0 0
\(818\) 8823.06 0.377129
\(819\) −803.771 −0.0342931
\(820\) −1444.88 −0.0615334
\(821\) −11641.0 −0.494852 −0.247426 0.968907i \(-0.579585\pi\)
−0.247426 + 0.968907i \(0.579585\pi\)
\(822\) −19174.1 −0.813594
\(823\) −12556.6 −0.531827 −0.265914 0.963997i \(-0.585674\pi\)
−0.265914 + 0.963997i \(0.585674\pi\)
\(824\) −23932.4 −1.01180
\(825\) −10468.0 −0.441757
\(826\) 4737.49 0.199562
\(827\) −38940.8 −1.63737 −0.818684 0.574245i \(-0.805296\pi\)
−0.818684 + 0.574245i \(0.805296\pi\)
\(828\) 1236.01 0.0518771
\(829\) 23760.5 0.995461 0.497731 0.867332i \(-0.334167\pi\)
0.497731 + 0.867332i \(0.334167\pi\)
\(830\) 418.749 0.0175120
\(831\) −21545.1 −0.899389
\(832\) 7627.34 0.317825
\(833\) −6430.26 −0.267461
\(834\) 28449.1 1.18119
\(835\) 2535.14 0.105068
\(836\) 168.902 0.00698755
\(837\) 7191.02 0.296963
\(838\) −9939.34 −0.409724
\(839\) 1611.44 0.0663087 0.0331544 0.999450i \(-0.489445\pi\)
0.0331544 + 0.999450i \(0.489445\pi\)
\(840\) 2596.36 0.106646
\(841\) 52353.6 2.14661
\(842\) 17368.4 0.710871
\(843\) −14430.9 −0.589594
\(844\) 232.902 0.00949862
\(845\) −9951.19 −0.405126
\(846\) 11849.1 0.481536
\(847\) −3451.66 −0.140024
\(848\) 7352.14 0.297728
\(849\) −32213.2 −1.30219
\(850\) 4594.83 0.185414
\(851\) −1718.33 −0.0692171
\(852\) 2973.21 0.119554
\(853\) −38499.2 −1.54536 −0.772678 0.634799i \(-0.781083\pi\)
−0.772678 + 0.634799i \(0.781083\pi\)
\(854\) 157.502 0.00631101
\(855\) 119.255 0.00477008
\(856\) 48274.8 1.92757
\(857\) −1365.17 −0.0544146 −0.0272073 0.999630i \(-0.508661\pi\)
−0.0272073 + 0.999630i \(0.508661\pi\)
\(858\) −3343.73 −0.133045
\(859\) 35044.7 1.39198 0.695989 0.718052i \(-0.254966\pi\)
0.695989 + 0.718052i \(0.254966\pi\)
\(860\) 0 0
\(861\) 2089.65 0.0827121
\(862\) −24680.4 −0.975195
\(863\) −41315.3 −1.62965 −0.814826 0.579706i \(-0.803167\pi\)
−0.814826 + 0.579706i \(0.803167\pi\)
\(864\) 19016.1 0.748773
\(865\) −13045.9 −0.512803
\(866\) 8772.16 0.344215
\(867\) −18116.1 −0.709638
\(868\) 727.707 0.0284562
\(869\) 25391.8 0.991204
\(870\) 12529.1 0.488250
\(871\) −9100.80 −0.354040
\(872\) −49926.8 −1.93891
\(873\) 12085.5 0.468538
\(874\) 195.443 0.00756401
\(875\) −5896.11 −0.227800
\(876\) −6222.46 −0.239997
\(877\) 27801.4 1.07045 0.535227 0.844708i \(-0.320226\pi\)
0.535227 + 0.844708i \(0.320226\pi\)
\(878\) 16691.3 0.641577
\(879\) −32308.7 −1.23975
\(880\) −4135.17 −0.158405
\(881\) −34464.0 −1.31796 −0.658980 0.752161i \(-0.729012\pi\)
−0.658980 + 0.752161i \(0.729012\pi\)
\(882\) −7650.44 −0.292068
\(883\) 37024.3 1.41106 0.705531 0.708679i \(-0.250709\pi\)
0.705531 + 0.708679i \(0.250709\pi\)
\(884\) −849.264 −0.0323120
\(885\) −8051.81 −0.305829
\(886\) 18443.4 0.699343
\(887\) −13945.5 −0.527898 −0.263949 0.964537i \(-0.585025\pi\)
−0.263949 + 0.964537i \(0.585025\pi\)
\(888\) −4354.18 −0.164546
\(889\) 3367.78 0.127055
\(890\) 4913.48 0.185056
\(891\) −8357.95 −0.314256
\(892\) −8302.45 −0.311644
\(893\) −1084.15 −0.0406268
\(894\) −13246.2 −0.495548
\(895\) −6479.50 −0.241995
\(896\) −1096.39 −0.0408791
\(897\) 2238.84 0.0833362
\(898\) 18377.7 0.682931
\(899\) 13092.1 0.485703
\(900\) −3163.26 −0.117158
\(901\) −4692.17 −0.173495
\(902\) −5772.34 −0.213080
\(903\) 0 0
\(904\) −33459.8 −1.23104
\(905\) −6300.10 −0.231406
\(906\) −27972.5 −1.02574
\(907\) −24719.4 −0.904955 −0.452478 0.891776i \(-0.649460\pi\)
−0.452478 + 0.891776i \(0.649460\pi\)
\(908\) −8886.58 −0.324792
\(909\) −17042.6 −0.621858
\(910\) −837.619 −0.0305130
\(911\) 29271.0 1.06454 0.532268 0.846576i \(-0.321340\pi\)
0.532268 + 0.846576i \(0.321340\pi\)
\(912\) 285.541 0.0103675
\(913\) −968.011 −0.0350892
\(914\) 17323.6 0.626929
\(915\) −267.690 −0.00967164
\(916\) −17843.1 −0.643616
\(917\) 6528.11 0.235090
\(918\) 6982.85 0.251055
\(919\) 8933.98 0.320680 0.160340 0.987062i \(-0.448741\pi\)
0.160340 + 0.987062i \(0.448741\pi\)
\(920\) 4802.13 0.172089
\(921\) 19902.9 0.712077
\(922\) 4687.95 0.167451
\(923\) −3576.06 −0.127527
\(924\) −1609.88 −0.0573172
\(925\) 4397.66 0.156318
\(926\) −6314.04 −0.224074
\(927\) 10477.4 0.371224
\(928\) 34621.1 1.22467
\(929\) 34274.1 1.21044 0.605218 0.796060i \(-0.293086\pi\)
0.605218 + 0.796060i \(0.293086\pi\)
\(930\) 2137.45 0.0753652
\(931\) 699.989 0.0246415
\(932\) 17219.7 0.605204
\(933\) 15846.6 0.556051
\(934\) 40205.7 1.40854
\(935\) 2639.08 0.0923072
\(936\) −3767.04 −0.131549
\(937\) 15930.4 0.555414 0.277707 0.960666i \(-0.410425\pi\)
0.277707 + 0.960666i \(0.410425\pi\)
\(938\) 7572.42 0.263591
\(939\) 8463.68 0.294145
\(940\) −7145.06 −0.247921
\(941\) −10078.6 −0.349154 −0.174577 0.984644i \(-0.555856\pi\)
−0.174577 + 0.984644i \(0.555856\pi\)
\(942\) 24606.1 0.851073
\(943\) 3864.94 0.133468
\(944\) 12801.6 0.441375
\(945\) −3985.14 −0.137182
\(946\) 0 0
\(947\) 17097.1 0.586675 0.293338 0.956009i \(-0.405234\pi\)
0.293338 + 0.956009i \(0.405234\pi\)
\(948\) −11555.4 −0.395887
\(949\) 7484.15 0.256002
\(950\) −500.188 −0.0170824
\(951\) 20393.0 0.695361
\(952\) 2634.49 0.0896895
\(953\) 3511.84 0.119370 0.0596850 0.998217i \(-0.480990\pi\)
0.0596850 + 0.998217i \(0.480990\pi\)
\(954\) −5582.54 −0.189456
\(955\) −7517.46 −0.254722
\(956\) 3425.74 0.115896
\(957\) −28963.3 −0.978317
\(958\) −2072.71 −0.0699021
\(959\) 11103.5 0.373878
\(960\) 10786.3 0.362633
\(961\) −27557.5 −0.925028
\(962\) 1404.71 0.0470787
\(963\) −21134.4 −0.707212
\(964\) −6307.03 −0.210722
\(965\) 3211.88 0.107144
\(966\) −1862.85 −0.0620457
\(967\) −24101.7 −0.801509 −0.400755 0.916185i \(-0.631252\pi\)
−0.400755 + 0.916185i \(0.631252\pi\)
\(968\) −16176.9 −0.537133
\(969\) −182.233 −0.00604146
\(970\) 12594.5 0.416892
\(971\) −43736.1 −1.44548 −0.722739 0.691121i \(-0.757117\pi\)
−0.722739 + 0.691121i \(0.757117\pi\)
\(972\) −8243.37 −0.272023
\(973\) −16474.5 −0.542803
\(974\) −4349.20 −0.143077
\(975\) −5729.75 −0.188204
\(976\) 425.602 0.0139582
\(977\) 10718.7 0.350995 0.175497 0.984480i \(-0.443847\pi\)
0.175497 + 0.984480i \(0.443847\pi\)
\(978\) −14185.5 −0.463806
\(979\) −11358.4 −0.370801
\(980\) 4613.26 0.150373
\(981\) 21857.6 0.711374
\(982\) −32042.9 −1.04127
\(983\) 47327.5 1.53562 0.767809 0.640679i \(-0.221347\pi\)
0.767809 + 0.640679i \(0.221347\pi\)
\(984\) 9793.58 0.317284
\(985\) −15606.2 −0.504828
\(986\) 12713.1 0.410617
\(987\) 10333.5 0.333252
\(988\) 92.4498 0.00297694
\(989\) 0 0
\(990\) 3139.87 0.100799
\(991\) 20633.6 0.661402 0.330701 0.943736i \(-0.392715\pi\)
0.330701 + 0.943736i \(0.392715\pi\)
\(992\) 5906.29 0.189037
\(993\) −28749.9 −0.918781
\(994\) 2975.50 0.0949470
\(995\) −24337.3 −0.775420
\(996\) 440.526 0.0140146
\(997\) 18221.2 0.578806 0.289403 0.957207i \(-0.406543\pi\)
0.289403 + 0.957207i \(0.406543\pi\)
\(998\) 1075.62 0.0341163
\(999\) 6683.20 0.211659
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 1849.4.a.i.1.18 50
43.42 odd 2 1849.4.a.j.1.33 yes 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.18 50 1.1 even 1 trivial
1849.4.a.j.1.33 yes 50 43.42 odd 2