Properties

Label 1849.4.a.j.1.11
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 1.11
Character \(\chi\) \(=\) 1849.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.61756 q^{2} -2.47684 q^{3} +5.08677 q^{4} -17.9947 q^{5} +8.96012 q^{6} +10.9253 q^{7} +10.5388 q^{8} -20.8653 q^{9} +O(q^{10})\) \(q-3.61756 q^{2} -2.47684 q^{3} +5.08677 q^{4} -17.9947 q^{5} +8.96012 q^{6} +10.9253 q^{7} +10.5388 q^{8} -20.8653 q^{9} +65.0971 q^{10} -60.2496 q^{11} -12.5991 q^{12} +83.3256 q^{13} -39.5231 q^{14} +44.5701 q^{15} -78.8189 q^{16} -55.2843 q^{17} +75.4815 q^{18} -24.4261 q^{19} -91.5351 q^{20} -27.0603 q^{21} +217.957 q^{22} -128.082 q^{23} -26.1029 q^{24} +198.811 q^{25} -301.436 q^{26} +118.555 q^{27} +55.5746 q^{28} +119.404 q^{29} -161.235 q^{30} -124.927 q^{31} +200.822 q^{32} +149.229 q^{33} +199.994 q^{34} -196.598 q^{35} -106.137 q^{36} -279.623 q^{37} +88.3631 q^{38} -206.384 q^{39} -189.643 q^{40} +88.4105 q^{41} +97.8923 q^{42} -306.476 q^{44} +375.465 q^{45} +463.344 q^{46} +102.372 q^{47} +195.222 q^{48} -223.637 q^{49} -719.210 q^{50} +136.930 q^{51} +423.858 q^{52} +184.241 q^{53} -428.879 q^{54} +1084.18 q^{55} +115.140 q^{56} +60.4996 q^{57} -431.951 q^{58} -539.091 q^{59} +226.718 q^{60} +709.639 q^{61} +451.931 q^{62} -227.960 q^{63} -95.9353 q^{64} -1499.42 q^{65} -539.844 q^{66} +74.0370 q^{67} -281.218 q^{68} +317.238 q^{69} +711.207 q^{70} +555.772 q^{71} -219.895 q^{72} +659.489 q^{73} +1011.56 q^{74} -492.422 q^{75} -124.250 q^{76} -658.247 q^{77} +746.607 q^{78} -531.541 q^{79} +1418.33 q^{80} +269.722 q^{81} -319.831 q^{82} -731.654 q^{83} -137.649 q^{84} +994.826 q^{85} -295.744 q^{87} -634.959 q^{88} +927.680 q^{89} -1358.27 q^{90} +910.360 q^{91} -651.522 q^{92} +309.424 q^{93} -370.337 q^{94} +439.542 q^{95} -497.404 q^{96} +1297.66 q^{97} +809.022 q^{98} +1257.12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} + O(q^{10}) \) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} + 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} + 386q^{18} - 12q^{19} - 108q^{20} - 408q^{21} - 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} - 1493q^{26} - 10q^{27} + 242q^{28} - 208q^{29} + 48q^{30} - 932q^{31} + 1124q^{32} - 254q^{33} - 765q^{34} - 1452q^{35} + 747q^{36} + 90q^{37} - 1213q^{38} + 1610q^{39} - 1693q^{40} - 1354q^{41} + 16q^{42} - 2704q^{44} - 4508q^{45} - 233q^{46} - 3484q^{47} + 376q^{48} + 1324q^{49} + 408q^{50} - 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} + 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} - 1172q^{61} + 1546q^{62} + 3686q^{63} + 606q^{64} - 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} - 136q^{69} + 1310q^{70} - 162q^{71} + 5814q^{72} - 746q^{73} - 4332q^{74} - 236q^{75} - 1338q^{76} - 2024q^{77} - 2782q^{78} - 2656q^{79} + 5713q^{80} - 86q^{81} + 4168q^{82} - 3514q^{83} - 4269q^{84} + 7558q^{85} - 10278q^{87} - 11692q^{88} - 2640q^{89} - 8286q^{90} + 5946q^{91} - 4271q^{92} + 2q^{93} - 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} + 2826q^{98} - 8174q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.61756 −1.27900 −0.639501 0.768790i \(-0.720859\pi\)
−0.639501 + 0.768790i \(0.720859\pi\)
\(3\) −2.47684 −0.476668 −0.238334 0.971183i \(-0.576601\pi\)
−0.238334 + 0.971183i \(0.576601\pi\)
\(4\) 5.08677 0.635846
\(5\) −17.9947 −1.60950 −0.804749 0.593615i \(-0.797700\pi\)
−0.804749 + 0.593615i \(0.797700\pi\)
\(6\) 8.96012 0.609659
\(7\) 10.9253 0.589912 0.294956 0.955511i \(-0.404695\pi\)
0.294956 + 0.955511i \(0.404695\pi\)
\(8\) 10.5388 0.465754
\(9\) −20.8653 −0.772788
\(10\) 65.0971 2.05855
\(11\) −60.2496 −1.65145 −0.825725 0.564073i \(-0.809234\pi\)
−0.825725 + 0.564073i \(0.809234\pi\)
\(12\) −12.5991 −0.303087
\(13\) 83.3256 1.77772 0.888860 0.458179i \(-0.151498\pi\)
0.888860 + 0.458179i \(0.151498\pi\)
\(14\) −39.5231 −0.754499
\(15\) 44.5701 0.767196
\(16\) −78.8189 −1.23155
\(17\) −55.2843 −0.788729 −0.394365 0.918954i \(-0.629035\pi\)
−0.394365 + 0.918954i \(0.629035\pi\)
\(18\) 75.4815 0.988397
\(19\) −24.4261 −0.294934 −0.147467 0.989067i \(-0.547112\pi\)
−0.147467 + 0.989067i \(0.547112\pi\)
\(20\) −91.5351 −1.02339
\(21\) −27.0603 −0.281192
\(22\) 217.957 2.11221
\(23\) −128.082 −1.16117 −0.580584 0.814200i \(-0.697176\pi\)
−0.580584 + 0.814200i \(0.697176\pi\)
\(24\) −26.1029 −0.222010
\(25\) 198.811 1.59048
\(26\) −301.436 −2.27371
\(27\) 118.555 0.845031
\(28\) 55.5746 0.375093
\(29\) 119.404 0.764577 0.382288 0.924043i \(-0.375136\pi\)
0.382288 + 0.924043i \(0.375136\pi\)
\(30\) −161.235 −0.981245
\(31\) −124.927 −0.723791 −0.361895 0.932219i \(-0.617870\pi\)
−0.361895 + 0.932219i \(0.617870\pi\)
\(32\) 200.822 1.10940
\(33\) 149.229 0.787193
\(34\) 199.994 1.00879
\(35\) −196.598 −0.949463
\(36\) −106.137 −0.491374
\(37\) −279.623 −1.24243 −0.621214 0.783641i \(-0.713360\pi\)
−0.621214 + 0.783641i \(0.713360\pi\)
\(38\) 88.3631 0.377221
\(39\) −206.384 −0.847382
\(40\) −189.643 −0.749630
\(41\) 88.4105 0.336766 0.168383 0.985722i \(-0.446145\pi\)
0.168383 + 0.985722i \(0.446145\pi\)
\(42\) 97.8923 0.359645
\(43\) 0 0
\(44\) −306.476 −1.05007
\(45\) 375.465 1.24380
\(46\) 463.344 1.48514
\(47\) 102.372 0.317713 0.158856 0.987302i \(-0.449219\pi\)
0.158856 + 0.987302i \(0.449219\pi\)
\(48\) 195.222 0.587038
\(49\) −223.637 −0.652003
\(50\) −719.210 −2.03423
\(51\) 136.930 0.375962
\(52\) 423.858 1.13036
\(53\) 184.241 0.477498 0.238749 0.971081i \(-0.423263\pi\)
0.238749 + 0.971081i \(0.423263\pi\)
\(54\) −428.879 −1.08080
\(55\) 1084.18 2.65800
\(56\) 115.140 0.274754
\(57\) 60.4996 0.140585
\(58\) −431.951 −0.977895
\(59\) −539.091 −1.18955 −0.594776 0.803891i \(-0.702759\pi\)
−0.594776 + 0.803891i \(0.702759\pi\)
\(60\) 226.718 0.487818
\(61\) 709.639 1.48951 0.744754 0.667339i \(-0.232567\pi\)
0.744754 + 0.667339i \(0.232567\pi\)
\(62\) 451.931 0.925730
\(63\) −227.960 −0.455877
\(64\) −95.9353 −0.187374
\(65\) −1499.42 −2.86124
\(66\) −539.844 −1.00682
\(67\) 74.0370 0.135001 0.0675005 0.997719i \(-0.478498\pi\)
0.0675005 + 0.997719i \(0.478498\pi\)
\(68\) −281.218 −0.501511
\(69\) 317.238 0.553492
\(70\) 711.207 1.21436
\(71\) 555.772 0.928986 0.464493 0.885577i \(-0.346237\pi\)
0.464493 + 0.885577i \(0.346237\pi\)
\(72\) −219.895 −0.359929
\(73\) 659.489 1.05736 0.528681 0.848821i \(-0.322687\pi\)
0.528681 + 0.848821i \(0.322687\pi\)
\(74\) 1011.56 1.58907
\(75\) −492.422 −0.758133
\(76\) −124.250 −0.187532
\(77\) −658.247 −0.974211
\(78\) 746.607 1.08380
\(79\) −531.541 −0.757001 −0.378501 0.925601i \(-0.623560\pi\)
−0.378501 + 0.925601i \(0.623560\pi\)
\(80\) 1418.33 1.98217
\(81\) 269.722 0.369989
\(82\) −319.831 −0.430724
\(83\) −731.654 −0.967584 −0.483792 0.875183i \(-0.660741\pi\)
−0.483792 + 0.875183i \(0.660741\pi\)
\(84\) −137.649 −0.178795
\(85\) 994.826 1.26946
\(86\) 0 0
\(87\) −295.744 −0.364449
\(88\) −634.959 −0.769169
\(89\) 927.680 1.10488 0.552438 0.833554i \(-0.313698\pi\)
0.552438 + 0.833554i \(0.313698\pi\)
\(90\) −1358.27 −1.59082
\(91\) 910.360 1.04870
\(92\) −651.522 −0.738324
\(93\) 309.424 0.345008
\(94\) −370.337 −0.406355
\(95\) 439.542 0.474695
\(96\) −497.404 −0.528813
\(97\) 1297.66 1.35833 0.679164 0.733987i \(-0.262343\pi\)
0.679164 + 0.733987i \(0.262343\pi\)
\(98\) 809.022 0.833914
\(99\) 1257.12 1.27622
\(100\) 1011.30 1.01130
\(101\) 315.009 0.310342 0.155171 0.987888i \(-0.450407\pi\)
0.155171 + 0.987888i \(0.450407\pi\)
\(102\) −495.354 −0.480856
\(103\) 1434.80 1.37258 0.686288 0.727330i \(-0.259239\pi\)
0.686288 + 0.727330i \(0.259239\pi\)
\(104\) 878.152 0.827980
\(105\) 486.943 0.452578
\(106\) −666.502 −0.610721
\(107\) −719.465 −0.650031 −0.325015 0.945709i \(-0.605369\pi\)
−0.325015 + 0.945709i \(0.605369\pi\)
\(108\) 603.059 0.537309
\(109\) 1089.84 0.957688 0.478844 0.877900i \(-0.341056\pi\)
0.478844 + 0.877900i \(0.341056\pi\)
\(110\) −3922.08 −3.39959
\(111\) 692.582 0.592225
\(112\) −861.123 −0.726504
\(113\) 281.293 0.234176 0.117088 0.993122i \(-0.462644\pi\)
0.117088 + 0.993122i \(0.462644\pi\)
\(114\) −218.861 −0.179809
\(115\) 2304.80 1.86890
\(116\) 607.379 0.486153
\(117\) −1738.61 −1.37380
\(118\) 1950.19 1.52144
\(119\) −603.999 −0.465281
\(120\) 469.715 0.357324
\(121\) 2299.02 1.72729
\(122\) −2567.17 −1.90508
\(123\) −218.979 −0.160525
\(124\) −635.474 −0.460220
\(125\) −1328.20 −0.950384
\(126\) 824.660 0.583068
\(127\) −687.188 −0.480142 −0.240071 0.970755i \(-0.577171\pi\)
−0.240071 + 0.970755i \(0.577171\pi\)
\(128\) −1259.52 −0.869745
\(129\) 0 0
\(130\) 5424.26 3.65953
\(131\) −551.794 −0.368019 −0.184009 0.982924i \(-0.558908\pi\)
−0.184009 + 0.982924i \(0.558908\pi\)
\(132\) 759.091 0.500533
\(133\) −266.864 −0.173985
\(134\) −267.834 −0.172666
\(135\) −2133.36 −1.36008
\(136\) −582.630 −0.367354
\(137\) 1879.78 1.17227 0.586134 0.810214i \(-0.300649\pi\)
0.586134 + 0.810214i \(0.300649\pi\)
\(138\) −1147.63 −0.707917
\(139\) 1766.14 1.07771 0.538855 0.842399i \(-0.318857\pi\)
0.538855 + 0.842399i \(0.318857\pi\)
\(140\) −1000.05 −0.603712
\(141\) −253.559 −0.151443
\(142\) −2010.54 −1.18817
\(143\) −5020.34 −2.93582
\(144\) 1644.58 0.951724
\(145\) −2148.64 −1.23059
\(146\) −2385.74 −1.35237
\(147\) 553.913 0.310789
\(148\) −1422.38 −0.789992
\(149\) 1233.08 0.677974 0.338987 0.940791i \(-0.389916\pi\)
0.338987 + 0.940791i \(0.389916\pi\)
\(150\) 1781.37 0.969653
\(151\) 1127.73 0.607771 0.303885 0.952709i \(-0.401716\pi\)
0.303885 + 0.952709i \(0.401716\pi\)
\(152\) −257.422 −0.137366
\(153\) 1153.52 0.609521
\(154\) 2381.25 1.24602
\(155\) 2248.03 1.16494
\(156\) −1049.83 −0.538804
\(157\) −605.410 −0.307751 −0.153876 0.988090i \(-0.549176\pi\)
−0.153876 + 0.988090i \(0.549176\pi\)
\(158\) 1922.89 0.968206
\(159\) −456.334 −0.227608
\(160\) −3613.74 −1.78557
\(161\) −1399.33 −0.684988
\(162\) −975.736 −0.473217
\(163\) −2491.95 −1.19745 −0.598725 0.800955i \(-0.704326\pi\)
−0.598725 + 0.800955i \(0.704326\pi\)
\(164\) 449.724 0.214131
\(165\) −2685.33 −1.26699
\(166\) 2646.81 1.23754
\(167\) −2713.05 −1.25714 −0.628569 0.777754i \(-0.716359\pi\)
−0.628569 + 0.777754i \(0.716359\pi\)
\(168\) −285.183 −0.130966
\(169\) 4746.15 2.16029
\(170\) −3598.85 −1.62364
\(171\) 509.658 0.227921
\(172\) 0 0
\(173\) 1135.63 0.499077 0.249538 0.968365i \(-0.419721\pi\)
0.249538 + 0.968365i \(0.419721\pi\)
\(174\) 1069.87 0.466131
\(175\) 2172.07 0.938246
\(176\) 4748.81 2.03384
\(177\) 1335.24 0.567022
\(178\) −3355.94 −1.41314
\(179\) −3005.54 −1.25500 −0.627500 0.778617i \(-0.715922\pi\)
−0.627500 + 0.778617i \(0.715922\pi\)
\(180\) 1909.90 0.790866
\(181\) 1380.70 0.566997 0.283499 0.958973i \(-0.408505\pi\)
0.283499 + 0.958973i \(0.408505\pi\)
\(182\) −3293.28 −1.34129
\(183\) −1757.66 −0.710001
\(184\) −1349.83 −0.540818
\(185\) 5031.75 1.99968
\(186\) −1119.36 −0.441266
\(187\) 3330.86 1.30255
\(188\) 520.743 0.202016
\(189\) 1295.25 0.498494
\(190\) −1590.07 −0.607136
\(191\) −197.941 −0.0749870 −0.0374935 0.999297i \(-0.511937\pi\)
−0.0374935 + 0.999297i \(0.511937\pi\)
\(192\) 237.616 0.0893150
\(193\) 731.734 0.272909 0.136454 0.990646i \(-0.456429\pi\)
0.136454 + 0.990646i \(0.456429\pi\)
\(194\) −4694.38 −1.73730
\(195\) 3713.83 1.36386
\(196\) −1137.59 −0.414574
\(197\) 2219.55 0.802723 0.401362 0.915920i \(-0.368537\pi\)
0.401362 + 0.915920i \(0.368537\pi\)
\(198\) −4547.73 −1.63229
\(199\) 2457.75 0.875505 0.437753 0.899095i \(-0.355775\pi\)
0.437753 + 0.899095i \(0.355775\pi\)
\(200\) 2095.23 0.740774
\(201\) −183.378 −0.0643506
\(202\) −1139.56 −0.396928
\(203\) 1304.53 0.451033
\(204\) 696.532 0.239054
\(205\) −1590.92 −0.542024
\(206\) −5190.49 −1.75553
\(207\) 2672.46 0.897337
\(208\) −6567.63 −2.18934
\(209\) 1471.67 0.487068
\(210\) −1761.55 −0.578849
\(211\) 152.246 0.0496731 0.0248366 0.999692i \(-0.492093\pi\)
0.0248366 + 0.999692i \(0.492093\pi\)
\(212\) 937.189 0.303615
\(213\) −1376.56 −0.442817
\(214\) 2602.71 0.831390
\(215\) 0 0
\(216\) 1249.42 0.393576
\(217\) −1364.87 −0.426973
\(218\) −3942.58 −1.22488
\(219\) −1633.45 −0.504010
\(220\) 5514.95 1.69008
\(221\) −4606.59 −1.40214
\(222\) −2505.46 −0.757457
\(223\) −4272.05 −1.28286 −0.641429 0.767182i \(-0.721658\pi\)
−0.641429 + 0.767182i \(0.721658\pi\)
\(224\) 2194.05 0.654446
\(225\) −4148.24 −1.22911
\(226\) −1017.60 −0.299511
\(227\) 2146.82 0.627706 0.313853 0.949472i \(-0.398380\pi\)
0.313853 + 0.949472i \(0.398380\pi\)
\(228\) 307.747 0.0893906
\(229\) 5101.80 1.47221 0.736106 0.676867i \(-0.236663\pi\)
0.736106 + 0.676867i \(0.236663\pi\)
\(230\) −8337.75 −2.39033
\(231\) 1630.37 0.464375
\(232\) 1258.37 0.356104
\(233\) −5124.12 −1.44074 −0.720369 0.693591i \(-0.756028\pi\)
−0.720369 + 0.693591i \(0.756028\pi\)
\(234\) 6289.54 1.75709
\(235\) −1842.16 −0.511358
\(236\) −2742.23 −0.756373
\(237\) 1316.54 0.360838
\(238\) 2185.00 0.595096
\(239\) −5810.98 −1.57273 −0.786363 0.617765i \(-0.788038\pi\)
−0.786363 + 0.617765i \(0.788038\pi\)
\(240\) −3512.96 −0.944837
\(241\) 2897.00 0.774325 0.387163 0.922011i \(-0.373455\pi\)
0.387163 + 0.922011i \(0.373455\pi\)
\(242\) −8316.84 −2.20920
\(243\) −3869.03 −1.02139
\(244\) 3609.77 0.947098
\(245\) 4024.29 1.04940
\(246\) 792.169 0.205312
\(247\) −2035.32 −0.524309
\(248\) −1316.58 −0.337108
\(249\) 1812.19 0.461216
\(250\) 4804.85 1.21554
\(251\) 5952.10 1.49679 0.748393 0.663255i \(-0.230826\pi\)
0.748393 + 0.663255i \(0.230826\pi\)
\(252\) −1159.58 −0.289868
\(253\) 7716.87 1.91761
\(254\) 2485.95 0.614103
\(255\) −2464.02 −0.605110
\(256\) 5323.89 1.29978
\(257\) 1636.32 0.397162 0.198581 0.980084i \(-0.436367\pi\)
0.198581 + 0.980084i \(0.436367\pi\)
\(258\) 0 0
\(259\) −3054.98 −0.732923
\(260\) −7627.21 −1.81931
\(261\) −2491.39 −0.590856
\(262\) 1996.15 0.470697
\(263\) 5471.20 1.28277 0.641385 0.767219i \(-0.278360\pi\)
0.641385 + 0.767219i \(0.278360\pi\)
\(264\) 1572.69 0.366638
\(265\) −3315.36 −0.768532
\(266\) 965.396 0.222527
\(267\) −2297.71 −0.526659
\(268\) 376.609 0.0858398
\(269\) −5056.33 −1.14606 −0.573030 0.819535i \(-0.694232\pi\)
−0.573030 + 0.819535i \(0.694232\pi\)
\(270\) 7717.56 1.73954
\(271\) 2645.05 0.592899 0.296449 0.955049i \(-0.404197\pi\)
0.296449 + 0.955049i \(0.404197\pi\)
\(272\) 4357.45 0.971357
\(273\) −2254.81 −0.499881
\(274\) −6800.23 −1.49933
\(275\) −11978.3 −2.62660
\(276\) 1613.71 0.351935
\(277\) 1722.07 0.373535 0.186768 0.982404i \(-0.440199\pi\)
0.186768 + 0.982404i \(0.440199\pi\)
\(278\) −6389.11 −1.37839
\(279\) 2606.63 0.559337
\(280\) −2071.91 −0.442216
\(281\) −8409.61 −1.78532 −0.892661 0.450729i \(-0.851164\pi\)
−0.892661 + 0.450729i \(0.851164\pi\)
\(282\) 917.265 0.193696
\(283\) −7474.29 −1.56997 −0.784983 0.619518i \(-0.787328\pi\)
−0.784983 + 0.619518i \(0.787328\pi\)
\(284\) 2827.08 0.590692
\(285\) −1088.67 −0.226272
\(286\) 18161.4 3.75491
\(287\) 965.914 0.198662
\(288\) −4190.21 −0.857328
\(289\) −1856.65 −0.377906
\(290\) 7772.84 1.57392
\(291\) −3214.10 −0.647471
\(292\) 3354.67 0.672319
\(293\) 2830.74 0.564415 0.282207 0.959353i \(-0.408933\pi\)
0.282207 + 0.959353i \(0.408933\pi\)
\(294\) −2003.82 −0.397500
\(295\) 9700.79 1.91458
\(296\) −2946.90 −0.578665
\(297\) −7142.87 −1.39553
\(298\) −4460.76 −0.867130
\(299\) −10672.5 −2.06423
\(300\) −2504.83 −0.482056
\(301\) 0 0
\(302\) −4079.64 −0.777340
\(303\) −780.225 −0.147930
\(304\) 1925.24 0.363224
\(305\) −12769.8 −2.39736
\(306\) −4172.94 −0.779578
\(307\) 1054.02 0.195948 0.0979739 0.995189i \(-0.468764\pi\)
0.0979739 + 0.995189i \(0.468764\pi\)
\(308\) −3348.35 −0.619448
\(309\) −3553.77 −0.654263
\(310\) −8132.38 −1.48996
\(311\) 9646.08 1.75877 0.879387 0.476107i \(-0.157953\pi\)
0.879387 + 0.476107i \(0.157953\pi\)
\(312\) −2175.04 −0.394671
\(313\) −1750.95 −0.316197 −0.158098 0.987423i \(-0.550536\pi\)
−0.158098 + 0.987423i \(0.550536\pi\)
\(314\) 2190.11 0.393615
\(315\) 4102.08 0.733733
\(316\) −2703.83 −0.481336
\(317\) 704.472 0.124817 0.0624086 0.998051i \(-0.480122\pi\)
0.0624086 + 0.998051i \(0.480122\pi\)
\(318\) 1650.82 0.291111
\(319\) −7194.03 −1.26266
\(320\) 1726.33 0.301578
\(321\) 1782.00 0.309849
\(322\) 5062.18 0.876101
\(323\) 1350.38 0.232623
\(324\) 1372.01 0.235256
\(325\) 16566.0 2.82744
\(326\) 9014.77 1.53154
\(327\) −2699.36 −0.456499
\(328\) 931.741 0.156850
\(329\) 1118.45 0.187423
\(330\) 9714.35 1.62048
\(331\) −5792.53 −0.961892 −0.480946 0.876750i \(-0.659707\pi\)
−0.480946 + 0.876750i \(0.659707\pi\)
\(332\) −3721.76 −0.615235
\(333\) 5834.42 0.960133
\(334\) 9814.63 1.60788
\(335\) −1332.28 −0.217284
\(336\) 2132.86 0.346301
\(337\) −5869.09 −0.948694 −0.474347 0.880338i \(-0.657316\pi\)
−0.474347 + 0.880338i \(0.657316\pi\)
\(338\) −17169.5 −2.76301
\(339\) −696.718 −0.111624
\(340\) 5060.45 0.807180
\(341\) 7526.79 1.19530
\(342\) −1843.72 −0.291512
\(343\) −6190.70 −0.974537
\(344\) 0 0
\(345\) −5708.61 −0.890844
\(346\) −4108.21 −0.638320
\(347\) 7730.08 1.19589 0.597943 0.801538i \(-0.295985\pi\)
0.597943 + 0.801538i \(0.295985\pi\)
\(348\) −1504.38 −0.231734
\(349\) −406.164 −0.0622965 −0.0311482 0.999515i \(-0.509916\pi\)
−0.0311482 + 0.999515i \(0.509916\pi\)
\(350\) −7857.61 −1.20002
\(351\) 9878.63 1.50223
\(352\) −12099.5 −1.83211
\(353\) −5126.15 −0.772911 −0.386456 0.922308i \(-0.626301\pi\)
−0.386456 + 0.922308i \(0.626301\pi\)
\(354\) −4830.32 −0.725222
\(355\) −10001.0 −1.49520
\(356\) 4718.89 0.702531
\(357\) 1496.01 0.221785
\(358\) 10872.7 1.60515
\(359\) −1617.42 −0.237783 −0.118892 0.992907i \(-0.537934\pi\)
−0.118892 + 0.992907i \(0.537934\pi\)
\(360\) 3956.95 0.579305
\(361\) −6262.36 −0.913014
\(362\) −4994.76 −0.725190
\(363\) −5694.29 −0.823341
\(364\) 4630.79 0.666811
\(365\) −11867.3 −1.70182
\(366\) 6358.45 0.908092
\(367\) −10719.8 −1.52471 −0.762354 0.647161i \(-0.775956\pi\)
−0.762354 + 0.647161i \(0.775956\pi\)
\(368\) 10095.3 1.43003
\(369\) −1844.71 −0.260249
\(370\) −18202.7 −2.55760
\(371\) 2012.89 0.281682
\(372\) 1573.97 0.219372
\(373\) −8000.84 −1.11064 −0.555319 0.831637i \(-0.687404\pi\)
−0.555319 + 0.831637i \(0.687404\pi\)
\(374\) −12049.6 −1.66596
\(375\) 3289.74 0.453017
\(376\) 1078.88 0.147976
\(377\) 9949.39 1.35920
\(378\) −4685.64 −0.637575
\(379\) −10844.9 −1.46982 −0.734911 0.678163i \(-0.762776\pi\)
−0.734911 + 0.678163i \(0.762776\pi\)
\(380\) 2235.85 0.301833
\(381\) 1702.05 0.228868
\(382\) 716.065 0.0959085
\(383\) 12651.9 1.68794 0.843972 0.536388i \(-0.180212\pi\)
0.843972 + 0.536388i \(0.180212\pi\)
\(384\) 3119.64 0.414579
\(385\) 11845.0 1.56799
\(386\) −2647.09 −0.349051
\(387\) 0 0
\(388\) 6600.91 0.863687
\(389\) 2845.89 0.370931 0.185466 0.982651i \(-0.440621\pi\)
0.185466 + 0.982651i \(0.440621\pi\)
\(390\) −13435.0 −1.74438
\(391\) 7080.90 0.915848
\(392\) −2356.87 −0.303673
\(393\) 1366.71 0.175423
\(394\) −8029.37 −1.02668
\(395\) 9564.95 1.21839
\(396\) 6394.70 0.811479
\(397\) 5007.64 0.633063 0.316532 0.948582i \(-0.397482\pi\)
0.316532 + 0.948582i \(0.397482\pi\)
\(398\) −8891.08 −1.11977
\(399\) 660.978 0.0829330
\(400\) −15670.0 −1.95875
\(401\) −1390.02 −0.173103 −0.0865513 0.996247i \(-0.527585\pi\)
−0.0865513 + 0.996247i \(0.527585\pi\)
\(402\) 663.381 0.0823045
\(403\) −10409.6 −1.28670
\(404\) 1602.38 0.197330
\(405\) −4853.57 −0.595496
\(406\) −4719.21 −0.576873
\(407\) 16847.2 2.05181
\(408\) 1443.08 0.175106
\(409\) −402.096 −0.0486121 −0.0243060 0.999705i \(-0.507738\pi\)
−0.0243060 + 0.999705i \(0.507738\pi\)
\(410\) 5755.27 0.693250
\(411\) −4655.92 −0.558782
\(412\) 7298.51 0.872747
\(413\) −5889.74 −0.701732
\(414\) −9667.79 −1.14770
\(415\) 13165.9 1.55732
\(416\) 16733.6 1.97220
\(417\) −4374.43 −0.513710
\(418\) −5323.84 −0.622961
\(419\) −4606.62 −0.537107 −0.268554 0.963265i \(-0.586546\pi\)
−0.268554 + 0.963265i \(0.586546\pi\)
\(420\) 2476.96 0.287770
\(421\) −5395.50 −0.624609 −0.312305 0.949982i \(-0.601101\pi\)
−0.312305 + 0.949982i \(0.601101\pi\)
\(422\) −550.759 −0.0635320
\(423\) −2136.02 −0.245524
\(424\) 1941.67 0.222396
\(425\) −10991.1 −1.25446
\(426\) 4979.78 0.566364
\(427\) 7753.04 0.878679
\(428\) −3659.75 −0.413319
\(429\) 12434.6 1.39941
\(430\) 0 0
\(431\) −934.590 −0.104449 −0.0522246 0.998635i \(-0.516631\pi\)
−0.0522246 + 0.998635i \(0.516631\pi\)
\(432\) −9344.34 −1.04069
\(433\) −6512.81 −0.722831 −0.361415 0.932405i \(-0.617706\pi\)
−0.361415 + 0.932405i \(0.617706\pi\)
\(434\) 4937.49 0.546100
\(435\) 5321.83 0.586580
\(436\) 5543.78 0.608942
\(437\) 3128.54 0.342468
\(438\) 5909.10 0.644630
\(439\) −4162.25 −0.452513 −0.226257 0.974068i \(-0.572649\pi\)
−0.226257 + 0.974068i \(0.572649\pi\)
\(440\) 11425.9 1.23798
\(441\) 4666.25 0.503860
\(442\) 16664.6 1.79334
\(443\) −5508.69 −0.590803 −0.295402 0.955373i \(-0.595453\pi\)
−0.295402 + 0.955373i \(0.595453\pi\)
\(444\) 3523.00 0.376564
\(445\) −16693.4 −1.77830
\(446\) 15454.4 1.64078
\(447\) −3054.15 −0.323168
\(448\) −1048.13 −0.110534
\(449\) −11101.7 −1.16686 −0.583430 0.812163i \(-0.698290\pi\)
−0.583430 + 0.812163i \(0.698290\pi\)
\(450\) 15006.5 1.57203
\(451\) −5326.70 −0.556152
\(452\) 1430.87 0.148900
\(453\) −2793.21 −0.289705
\(454\) −7766.25 −0.802837
\(455\) −16381.7 −1.68788
\(456\) 637.593 0.0654781
\(457\) −11637.9 −1.19124 −0.595621 0.803265i \(-0.703094\pi\)
−0.595621 + 0.803265i \(0.703094\pi\)
\(458\) −18456.1 −1.88296
\(459\) −6554.20 −0.666501
\(460\) 11724.0 1.18833
\(461\) 3771.32 0.381015 0.190507 0.981686i \(-0.438987\pi\)
0.190507 + 0.981686i \(0.438987\pi\)
\(462\) −5897.97 −0.593936
\(463\) −11682.2 −1.17261 −0.586306 0.810090i \(-0.699418\pi\)
−0.586306 + 0.810090i \(0.699418\pi\)
\(464\) −9411.28 −0.941611
\(465\) −5568.00 −0.555289
\(466\) 18536.8 1.84271
\(467\) 2086.46 0.206745 0.103372 0.994643i \(-0.467037\pi\)
0.103372 + 0.994643i \(0.467037\pi\)
\(468\) −8843.91 −0.873526
\(469\) 808.879 0.0796387
\(470\) 6664.12 0.654028
\(471\) 1499.50 0.146695
\(472\) −5681.37 −0.554039
\(473\) 0 0
\(474\) −4762.68 −0.461513
\(475\) −4856.17 −0.469087
\(476\) −3072.40 −0.295847
\(477\) −3844.23 −0.369005
\(478\) 21021.6 2.01152
\(479\) 15342.3 1.46348 0.731742 0.681582i \(-0.238708\pi\)
0.731742 + 0.681582i \(0.238708\pi\)
\(480\) 8950.65 0.851124
\(481\) −23299.8 −2.20869
\(482\) −10480.1 −0.990364
\(483\) 3465.93 0.326512
\(484\) 11694.6 1.09829
\(485\) −23351.1 −2.18623
\(486\) 13996.5 1.30636
\(487\) −1046.75 −0.0973977 −0.0486989 0.998814i \(-0.515507\pi\)
−0.0486989 + 0.998814i \(0.515507\pi\)
\(488\) 7478.75 0.693744
\(489\) 6172.15 0.570785
\(490\) −14558.1 −1.34218
\(491\) −7721.24 −0.709684 −0.354842 0.934926i \(-0.615465\pi\)
−0.354842 + 0.934926i \(0.615465\pi\)
\(492\) −1113.89 −0.102069
\(493\) −6601.15 −0.603044
\(494\) 7362.91 0.670593
\(495\) −22621.6 −2.05407
\(496\) 9846.60 0.891382
\(497\) 6071.99 0.548020
\(498\) −6555.71 −0.589896
\(499\) 17819.9 1.59865 0.799326 0.600897i \(-0.205190\pi\)
0.799326 + 0.600897i \(0.205190\pi\)
\(500\) −6756.25 −0.604298
\(501\) 6719.79 0.599238
\(502\) −21532.1 −1.91439
\(503\) −10784.0 −0.955938 −0.477969 0.878377i \(-0.658627\pi\)
−0.477969 + 0.878377i \(0.658627\pi\)
\(504\) −2402.43 −0.212326
\(505\) −5668.50 −0.499495
\(506\) −27916.3 −2.45263
\(507\) −11755.5 −1.02974
\(508\) −3495.57 −0.305297
\(509\) −15169.1 −1.32094 −0.660472 0.750851i \(-0.729644\pi\)
−0.660472 + 0.750851i \(0.729644\pi\)
\(510\) 8913.76 0.773937
\(511\) 7205.14 0.623750
\(512\) −9183.33 −0.792675
\(513\) −2895.83 −0.249228
\(514\) −5919.48 −0.507971
\(515\) −25818.9 −2.20916
\(516\) 0 0
\(517\) −6167.87 −0.524686
\(518\) 11051.6 0.937410
\(519\) −2812.77 −0.237894
\(520\) −15802.1 −1.33263
\(521\) −2765.08 −0.232515 −0.116257 0.993219i \(-0.537090\pi\)
−0.116257 + 0.993219i \(0.537090\pi\)
\(522\) 9012.77 0.755706
\(523\) 2271.48 0.189914 0.0949568 0.995481i \(-0.469729\pi\)
0.0949568 + 0.995481i \(0.469729\pi\)
\(524\) −2806.85 −0.234003
\(525\) −5379.87 −0.447232
\(526\) −19792.4 −1.64067
\(527\) 6906.49 0.570875
\(528\) −11762.0 −0.969464
\(529\) 4237.92 0.348312
\(530\) 11993.5 0.982954
\(531\) 11248.3 0.919272
\(532\) −1357.47 −0.110628
\(533\) 7366.86 0.598675
\(534\) 8312.13 0.673597
\(535\) 12946.6 1.04622
\(536\) 780.261 0.0628772
\(537\) 7444.25 0.598218
\(538\) 18291.6 1.46581
\(539\) 13474.1 1.07675
\(540\) −10851.9 −0.864799
\(541\) −4266.25 −0.339040 −0.169520 0.985527i \(-0.554222\pi\)
−0.169520 + 0.985527i \(0.554222\pi\)
\(542\) −9568.65 −0.758319
\(543\) −3419.77 −0.270269
\(544\) −11102.3 −0.875013
\(545\) −19611.4 −1.54140
\(546\) 8156.93 0.639349
\(547\) 23063.3 1.80277 0.901386 0.433017i \(-0.142551\pi\)
0.901386 + 0.433017i \(0.142551\pi\)
\(548\) 9562.02 0.745381
\(549\) −14806.8 −1.15107
\(550\) 43332.1 3.35943
\(551\) −2916.57 −0.225499
\(552\) 3343.30 0.257791
\(553\) −5807.27 −0.446564
\(554\) −6229.71 −0.477752
\(555\) −12462.8 −0.953185
\(556\) 8983.92 0.685258
\(557\) −3638.87 −0.276811 −0.138406 0.990376i \(-0.544198\pi\)
−0.138406 + 0.990376i \(0.544198\pi\)
\(558\) −9429.66 −0.715393
\(559\) 0 0
\(560\) 15495.7 1.16931
\(561\) −8249.99 −0.620882
\(562\) 30422.3 2.28343
\(563\) −6527.20 −0.488612 −0.244306 0.969698i \(-0.578560\pi\)
−0.244306 + 0.969698i \(0.578560\pi\)
\(564\) −1289.80 −0.0962946
\(565\) −5061.80 −0.376905
\(566\) 27038.7 2.00799
\(567\) 2946.80 0.218261
\(568\) 5857.17 0.432678
\(569\) −20981.2 −1.54583 −0.772917 0.634507i \(-0.781203\pi\)
−0.772917 + 0.634507i \(0.781203\pi\)
\(570\) 3938.35 0.289402
\(571\) 15555.5 1.14006 0.570032 0.821622i \(-0.306931\pi\)
0.570032 + 0.821622i \(0.306931\pi\)
\(572\) −25537.3 −1.86673
\(573\) 490.268 0.0357439
\(574\) −3494.26 −0.254090
\(575\) −25464.0 −1.84682
\(576\) 2001.72 0.144800
\(577\) −5345.98 −0.385713 −0.192856 0.981227i \(-0.561775\pi\)
−0.192856 + 0.981227i \(0.561775\pi\)
\(578\) 6716.55 0.483342
\(579\) −1812.39 −0.130087
\(580\) −10929.6 −0.782463
\(581\) −7993.56 −0.570790
\(582\) 11627.2 0.828117
\(583\) −11100.4 −0.788564
\(584\) 6950.22 0.492470
\(585\) 31285.9 2.21113
\(586\) −10240.4 −0.721887
\(587\) 1744.52 0.122665 0.0613324 0.998117i \(-0.480465\pi\)
0.0613324 + 0.998117i \(0.480465\pi\)
\(588\) 2817.63 0.197614
\(589\) 3051.48 0.213470
\(590\) −35093.2 −2.44876
\(591\) −5497.47 −0.382632
\(592\) 22039.6 1.53011
\(593\) −9301.56 −0.644130 −0.322065 0.946718i \(-0.604377\pi\)
−0.322065 + 0.946718i \(0.604377\pi\)
\(594\) 25839.8 1.78488
\(595\) 10868.8 0.748869
\(596\) 6272.41 0.431087
\(597\) −6087.46 −0.417325
\(598\) 38608.4 2.64016
\(599\) 1942.24 0.132484 0.0662420 0.997804i \(-0.478899\pi\)
0.0662420 + 0.997804i \(0.478899\pi\)
\(600\) −5189.53 −0.353103
\(601\) 21909.3 1.48702 0.743509 0.668725i \(-0.233160\pi\)
0.743509 + 0.668725i \(0.233160\pi\)
\(602\) 0 0
\(603\) −1544.80 −0.104327
\(604\) 5736.50 0.386449
\(605\) −41370.2 −2.78006
\(606\) 2822.52 0.189203
\(607\) 975.955 0.0652599 0.0326300 0.999468i \(-0.489612\pi\)
0.0326300 + 0.999468i \(0.489612\pi\)
\(608\) −4905.31 −0.327198
\(609\) −3231.10 −0.214993
\(610\) 46195.5 3.06623
\(611\) 8530.21 0.564804
\(612\) 5867.69 0.387561
\(613\) 5980.96 0.394076 0.197038 0.980396i \(-0.436868\pi\)
0.197038 + 0.980396i \(0.436868\pi\)
\(614\) −3812.98 −0.250618
\(615\) 3940.46 0.258365
\(616\) −6937.14 −0.453742
\(617\) 27743.2 1.81021 0.905105 0.425188i \(-0.139792\pi\)
0.905105 + 0.425188i \(0.139792\pi\)
\(618\) 12856.0 0.836803
\(619\) 5498.82 0.357053 0.178527 0.983935i \(-0.442867\pi\)
0.178527 + 0.983935i \(0.442867\pi\)
\(620\) 11435.2 0.740723
\(621\) −15184.7 −0.981223
\(622\) −34895.3 −2.24948
\(623\) 10135.2 0.651780
\(624\) 16267.0 1.04359
\(625\) −950.684 −0.0608438
\(626\) 6334.18 0.404416
\(627\) −3645.08 −0.232170
\(628\) −3079.58 −0.195682
\(629\) 15458.8 0.979939
\(630\) −14839.5 −0.938446
\(631\) −2676.32 −0.168847 −0.0844235 0.996430i \(-0.526905\pi\)
−0.0844235 + 0.996430i \(0.526905\pi\)
\(632\) −5601.81 −0.352576
\(633\) −377.088 −0.0236776
\(634\) −2548.47 −0.159642
\(635\) 12365.8 0.772788
\(636\) −2321.27 −0.144724
\(637\) −18634.7 −1.15908
\(638\) 26024.9 1.61494
\(639\) −11596.3 −0.717909
\(640\) 22664.8 1.39985
\(641\) −21148.3 −1.30313 −0.651566 0.758592i \(-0.725888\pi\)
−0.651566 + 0.758592i \(0.725888\pi\)
\(642\) −6446.49 −0.396297
\(643\) −246.770 −0.0151348 −0.00756739 0.999971i \(-0.502409\pi\)
−0.00756739 + 0.999971i \(0.502409\pi\)
\(644\) −7118.09 −0.435547
\(645\) 0 0
\(646\) −4885.09 −0.297525
\(647\) 32803.7 1.99327 0.996634 0.0819754i \(-0.0261229\pi\)
0.996634 + 0.0819754i \(0.0261229\pi\)
\(648\) 2842.55 0.172324
\(649\) 32480.0 1.96449
\(650\) −59928.6 −3.61630
\(651\) 3380.55 0.203524
\(652\) −12675.9 −0.761393
\(653\) 8579.88 0.514176 0.257088 0.966388i \(-0.417237\pi\)
0.257088 + 0.966388i \(0.417237\pi\)
\(654\) 9765.12 0.583863
\(655\) 9929.39 0.592326
\(656\) −6968.42 −0.414743
\(657\) −13760.4 −0.817116
\(658\) −4046.06 −0.239714
\(659\) −22596.8 −1.33573 −0.667866 0.744281i \(-0.732792\pi\)
−0.667866 + 0.744281i \(0.732792\pi\)
\(660\) −13659.6 −0.805608
\(661\) 8017.05 0.471750 0.235875 0.971783i \(-0.424204\pi\)
0.235875 + 0.971783i \(0.424204\pi\)
\(662\) 20954.8 1.23026
\(663\) 11409.8 0.668355
\(664\) −7710.76 −0.450656
\(665\) 4802.14 0.280029
\(666\) −21106.4 −1.22801
\(667\) −15293.4 −0.887803
\(668\) −13800.7 −0.799347
\(669\) 10581.2 0.611497
\(670\) 4819.60 0.277906
\(671\) −42755.5 −2.45985
\(672\) −5434.30 −0.311954
\(673\) 25565.1 1.46428 0.732140 0.681154i \(-0.238522\pi\)
0.732140 + 0.681154i \(0.238522\pi\)
\(674\) 21231.8 1.21338
\(675\) 23569.9 1.34401
\(676\) 24142.6 1.37361
\(677\) −103.167 −0.00585674 −0.00292837 0.999996i \(-0.500932\pi\)
−0.00292837 + 0.999996i \(0.500932\pi\)
\(678\) 2520.42 0.142767
\(679\) 14177.4 0.801294
\(680\) 10484.3 0.591255
\(681\) −5317.32 −0.299207
\(682\) −27228.7 −1.52880
\(683\) −18832.6 −1.05507 −0.527533 0.849534i \(-0.676883\pi\)
−0.527533 + 0.849534i \(0.676883\pi\)
\(684\) 2592.51 0.144923
\(685\) −33826.2 −1.88676
\(686\) 22395.2 1.24644
\(687\) −12636.3 −0.701756
\(688\) 0 0
\(689\) 15352.0 0.848858
\(690\) 20651.3 1.13939
\(691\) 26.0919 0.00143644 0.000718222 1.00000i \(-0.499771\pi\)
0.000718222 1.00000i \(0.499771\pi\)
\(692\) 5776.68 0.317336
\(693\) 13734.5 0.752858
\(694\) −27964.1 −1.52954
\(695\) −31781.1 −1.73457
\(696\) −3116.79 −0.169744
\(697\) −4887.71 −0.265617
\(698\) 1469.33 0.0796773
\(699\) 12691.6 0.686754
\(700\) 11048.8 0.596580
\(701\) 30741.8 1.65635 0.828176 0.560468i \(-0.189379\pi\)
0.828176 + 0.560468i \(0.189379\pi\)
\(702\) −35736.6 −1.92135
\(703\) 6830.12 0.366434
\(704\) 5780.07 0.309438
\(705\) 4562.73 0.243748
\(706\) 18544.2 0.988555
\(707\) 3441.57 0.183075
\(708\) 6792.06 0.360538
\(709\) 32760.2 1.73531 0.867656 0.497165i \(-0.165626\pi\)
0.867656 + 0.497165i \(0.165626\pi\)
\(710\) 36179.1 1.91236
\(711\) 11090.8 0.585001
\(712\) 9776.64 0.514600
\(713\) 16000.8 0.840443
\(714\) −5411.90 −0.283663
\(715\) 90339.6 4.72519
\(716\) −15288.5 −0.797986
\(717\) 14392.9 0.749667
\(718\) 5851.13 0.304126
\(719\) 27352.4 1.41874 0.709370 0.704837i \(-0.248980\pi\)
0.709370 + 0.704837i \(0.248980\pi\)
\(720\) −29593.8 −1.53180
\(721\) 15675.7 0.809700
\(722\) 22654.5 1.16775
\(723\) −7175.41 −0.369096
\(724\) 7023.29 0.360523
\(725\) 23738.7 1.21605
\(726\) 20599.5 1.05306
\(727\) 6560.47 0.334683 0.167342 0.985899i \(-0.446482\pi\)
0.167342 + 0.985899i \(0.446482\pi\)
\(728\) 9594.10 0.488435
\(729\) 2300.47 0.116876
\(730\) 42930.8 2.17663
\(731\) 0 0
\(732\) −8940.82 −0.451451
\(733\) −4640.51 −0.233835 −0.116918 0.993142i \(-0.537301\pi\)
−0.116918 + 0.993142i \(0.537301\pi\)
\(734\) 38779.5 1.95010
\(735\) −9967.52 −0.500214
\(736\) −25721.6 −1.28820
\(737\) −4460.70 −0.222947
\(738\) 6673.35 0.332858
\(739\) −25462.4 −1.26745 −0.633727 0.773557i \(-0.718476\pi\)
−0.633727 + 0.773557i \(0.718476\pi\)
\(740\) 25595.3 1.27149
\(741\) 5041.16 0.249921
\(742\) −7281.75 −0.360272
\(743\) −31591.6 −1.55987 −0.779935 0.625861i \(-0.784748\pi\)
−0.779935 + 0.625861i \(0.784748\pi\)
\(744\) 3260.95 0.160689
\(745\) −22189.0 −1.09120
\(746\) 28943.6 1.42051
\(747\) 15266.2 0.747737
\(748\) 16943.3 0.828219
\(749\) −7860.39 −0.383461
\(750\) −11900.8 −0.579410
\(751\) 11251.7 0.546712 0.273356 0.961913i \(-0.411866\pi\)
0.273356 + 0.961913i \(0.411866\pi\)
\(752\) −8068.85 −0.391278
\(753\) −14742.4 −0.713470
\(754\) −35992.6 −1.73842
\(755\) −20293.2 −0.978206
\(756\) 6588.62 0.316966
\(757\) 30940.9 1.48556 0.742779 0.669536i \(-0.233507\pi\)
0.742779 + 0.669536i \(0.233507\pi\)
\(758\) 39232.0 1.87991
\(759\) −19113.4 −0.914063
\(760\) 4632.24 0.221091
\(761\) 9097.60 0.433361 0.216681 0.976243i \(-0.430477\pi\)
0.216681 + 0.976243i \(0.430477\pi\)
\(762\) −6157.29 −0.292723
\(763\) 11906.9 0.564952
\(764\) −1006.88 −0.0476802
\(765\) −20757.3 −0.981022
\(766\) −45769.1 −2.15888
\(767\) −44920.1 −2.11469
\(768\) −13186.4 −0.619563
\(769\) 27674.9 1.29777 0.648883 0.760888i \(-0.275236\pi\)
0.648883 + 0.760888i \(0.275236\pi\)
\(770\) −42850.0 −2.00546
\(771\) −4052.89 −0.189314
\(772\) 3722.16 0.173528
\(773\) 40171.3 1.86916 0.934581 0.355750i \(-0.115775\pi\)
0.934581 + 0.355750i \(0.115775\pi\)
\(774\) 0 0
\(775\) −24836.8 −1.15118
\(776\) 13675.8 0.632646
\(777\) 7566.69 0.349361
\(778\) −10295.2 −0.474422
\(779\) −2159.53 −0.0993236
\(780\) 18891.4 0.867205
\(781\) −33485.0 −1.53417
\(782\) −25615.6 −1.17137
\(783\) 14155.9 0.646091
\(784\) 17626.8 0.802972
\(785\) 10894.2 0.495325
\(786\) −4944.14 −0.224366
\(787\) 22928.4 1.03851 0.519256 0.854619i \(-0.326209\pi\)
0.519256 + 0.854619i \(0.326209\pi\)
\(788\) 11290.3 0.510408
\(789\) −13551.3 −0.611455
\(790\) −34601.8 −1.55833
\(791\) 3073.22 0.138143
\(792\) 13248.6 0.594404
\(793\) 59131.1 2.64793
\(794\) −18115.4 −0.809689
\(795\) 8211.61 0.366334
\(796\) 12502.0 0.556687
\(797\) −23583.5 −1.04814 −0.524072 0.851674i \(-0.675588\pi\)
−0.524072 + 0.851674i \(0.675588\pi\)
\(798\) −2391.13 −0.106072
\(799\) −5659.56 −0.250589
\(800\) 39925.6 1.76448
\(801\) −19356.3 −0.853834
\(802\) 5028.48 0.221399
\(803\) −39734.0 −1.74618
\(804\) −932.800 −0.0409171
\(805\) 25180.7 1.10249
\(806\) 37657.4 1.64569
\(807\) 12523.7 0.546290
\(808\) 3319.81 0.144543
\(809\) 12806.6 0.556560 0.278280 0.960500i \(-0.410236\pi\)
0.278280 + 0.960500i \(0.410236\pi\)
\(810\) 17558.1 0.761641
\(811\) 7841.91 0.339540 0.169770 0.985484i \(-0.445698\pi\)
0.169770 + 0.985484i \(0.445698\pi\)
\(812\) 6635.82 0.286788
\(813\) −6551.37 −0.282616
\(814\) −60945.8 −2.62426
\(815\) 44841.9 1.92729
\(816\) −10792.7 −0.463014
\(817\) 0 0
\(818\) 1454.61 0.0621750
\(819\) −18994.9 −0.810422
\(820\) −8092.66 −0.344644
\(821\) 29489.6 1.25358 0.626792 0.779186i \(-0.284367\pi\)
0.626792 + 0.779186i \(0.284367\pi\)
\(822\) 16843.1 0.714683
\(823\) 46394.4 1.96501 0.982507 0.186225i \(-0.0596253\pi\)
0.982507 + 0.186225i \(0.0596253\pi\)
\(824\) 15121.1 0.639282
\(825\) 29668.2 1.25202
\(826\) 21306.5 0.897517
\(827\) −10323.6 −0.434082 −0.217041 0.976163i \(-0.569641\pi\)
−0.217041 + 0.976163i \(0.569641\pi\)
\(828\) 13594.2 0.570568
\(829\) 2230.44 0.0934454 0.0467227 0.998908i \(-0.485122\pi\)
0.0467227 + 0.998908i \(0.485122\pi\)
\(830\) −47628.6 −1.99182
\(831\) −4265.29 −0.178052
\(832\) −7993.87 −0.333098
\(833\) 12363.6 0.514254
\(834\) 15824.8 0.657036
\(835\) 48820.6 2.02336
\(836\) 7486.02 0.309700
\(837\) −14810.6 −0.611626
\(838\) 16664.7 0.686961
\(839\) −883.921 −0.0363723 −0.0181861 0.999835i \(-0.505789\pi\)
−0.0181861 + 0.999835i \(0.505789\pi\)
\(840\) 5131.79 0.210790
\(841\) −10131.7 −0.415422
\(842\) 19518.6 0.798876
\(843\) 20829.2 0.851005
\(844\) 774.439 0.0315845
\(845\) −85405.8 −3.47698
\(846\) 7727.19 0.314026
\(847\) 25117.5 1.01895
\(848\) −14521.6 −0.588061
\(849\) 18512.6 0.748352
\(850\) 39761.0 1.60446
\(851\) 35814.6 1.44267
\(852\) −7002.23 −0.281564
\(853\) 15341.3 0.615797 0.307899 0.951419i \(-0.400374\pi\)
0.307899 + 0.951419i \(0.400374\pi\)
\(854\) −28047.1 −1.12383
\(855\) −9171.16 −0.366839
\(856\) −7582.30 −0.302754
\(857\) −27250.3 −1.08618 −0.543088 0.839676i \(-0.682745\pi\)
−0.543088 + 0.839676i \(0.682745\pi\)
\(858\) −44982.8 −1.78985
\(859\) 6254.63 0.248435 0.124217 0.992255i \(-0.460358\pi\)
0.124217 + 0.992255i \(0.460358\pi\)
\(860\) 0 0
\(861\) −2392.41 −0.0946959
\(862\) 3380.94 0.133591
\(863\) 34856.0 1.37487 0.687434 0.726247i \(-0.258737\pi\)
0.687434 + 0.726247i \(0.258737\pi\)
\(864\) 23808.4 0.937474
\(865\) −20435.3 −0.803263
\(866\) 23560.5 0.924502
\(867\) 4598.62 0.180136
\(868\) −6942.76 −0.271489
\(869\) 32025.2 1.25015
\(870\) −19252.1 −0.750237
\(871\) 6169.18 0.239994
\(872\) 11485.6 0.446047
\(873\) −27076.1 −1.04970
\(874\) −11317.7 −0.438017
\(875\) −14511.0 −0.560643
\(876\) −8308.97 −0.320473
\(877\) 6804.93 0.262014 0.131007 0.991381i \(-0.458179\pi\)
0.131007 + 0.991381i \(0.458179\pi\)
\(878\) 15057.2 0.578766
\(879\) −7011.28 −0.269038
\(880\) −85453.6 −3.27346
\(881\) −47274.9 −1.80787 −0.903935 0.427670i \(-0.859334\pi\)
−0.903935 + 0.427670i \(0.859334\pi\)
\(882\) −16880.5 −0.644438
\(883\) −31363.8 −1.19533 −0.597665 0.801746i \(-0.703905\pi\)
−0.597665 + 0.801746i \(0.703905\pi\)
\(884\) −23432.7 −0.891545
\(885\) −24027.3 −0.912620
\(886\) 19928.0 0.755638
\(887\) −27624.7 −1.04571 −0.522856 0.852421i \(-0.675134\pi\)
−0.522856 + 0.852421i \(0.675134\pi\)
\(888\) 7298.99 0.275831
\(889\) −7507.76 −0.283242
\(890\) 60389.3 2.27444
\(891\) −16250.6 −0.611018
\(892\) −21730.9 −0.815700
\(893\) −2500.55 −0.0937041
\(894\) 11048.6 0.413333
\(895\) 54084.0 2.01992
\(896\) −13760.7 −0.513073
\(897\) 26434.0 0.983953
\(898\) 40161.0 1.49242
\(899\) −14916.7 −0.553394
\(900\) −21101.1 −0.781523
\(901\) −10185.6 −0.376617
\(902\) 19269.7 0.711319
\(903\) 0 0
\(904\) 2964.49 0.109068
\(905\) −24845.3 −0.912581
\(906\) 10104.6 0.370533
\(907\) −30861.9 −1.12983 −0.564914 0.825150i \(-0.691091\pi\)
−0.564914 + 0.825150i \(0.691091\pi\)
\(908\) 10920.4 0.399124
\(909\) −6572.74 −0.239828
\(910\) 59261.8 2.15880
\(911\) −4012.75 −0.145937 −0.0729684 0.997334i \(-0.523247\pi\)
−0.0729684 + 0.997334i \(0.523247\pi\)
\(912\) −4768.51 −0.173137
\(913\) 44081.9 1.59792
\(914\) 42100.8 1.52360
\(915\) 31628.7 1.14274
\(916\) 25951.7 0.936100
\(917\) −6028.53 −0.217099
\(918\) 23710.2 0.852456
\(919\) −20261.8 −0.727285 −0.363643 0.931539i \(-0.618467\pi\)
−0.363643 + 0.931539i \(0.618467\pi\)
\(920\) 24289.8 0.870446
\(921\) −2610.63 −0.0934020
\(922\) −13643.0 −0.487318
\(923\) 46310.0 1.65148
\(924\) 8293.32 0.295271
\(925\) −55592.1 −1.97606
\(926\) 42261.2 1.49977
\(927\) −29937.5 −1.06071
\(928\) 23978.9 0.848218
\(929\) 10715.5 0.378432 0.189216 0.981935i \(-0.439405\pi\)
0.189216 + 0.981935i \(0.439405\pi\)
\(930\) 20142.6 0.710216
\(931\) 5462.59 0.192298
\(932\) −26065.2 −0.916088
\(933\) −23891.8 −0.838351
\(934\) −7547.90 −0.264427
\(935\) −59937.9 −2.09645
\(936\) −18322.9 −0.639853
\(937\) −26739.1 −0.932262 −0.466131 0.884716i \(-0.654352\pi\)
−0.466131 + 0.884716i \(0.654352\pi\)
\(938\) −2926.17 −0.101858
\(939\) 4336.82 0.150721
\(940\) −9370.63 −0.325145
\(941\) 40264.8 1.39489 0.697446 0.716637i \(-0.254320\pi\)
0.697446 + 0.716637i \(0.254320\pi\)
\(942\) −5424.54 −0.187623
\(943\) −11323.8 −0.391042
\(944\) 42490.6 1.46499
\(945\) −23307.6 −0.802325
\(946\) 0 0
\(947\) 12014.0 0.412253 0.206126 0.978525i \(-0.433914\pi\)
0.206126 + 0.978525i \(0.433914\pi\)
\(948\) 6696.94 0.229437
\(949\) 54952.3 1.87969
\(950\) 17567.5 0.599964
\(951\) −1744.86 −0.0594964
\(952\) −6365.42 −0.216706
\(953\) 14553.4 0.494681 0.247341 0.968929i \(-0.420443\pi\)
0.247341 + 0.968929i \(0.420443\pi\)
\(954\) 13906.7 0.471958
\(955\) 3561.90 0.120691
\(956\) −29559.1 −1.00001
\(957\) 17818.5 0.601869
\(958\) −55501.8 −1.87180
\(959\) 20537.2 0.691535
\(960\) −4275.84 −0.143752
\(961\) −14184.3 −0.476127
\(962\) 84288.5 2.82492
\(963\) 15011.8 0.502336
\(964\) 14736.4 0.492352
\(965\) −13167.4 −0.439246
\(966\) −12538.2 −0.417609
\(967\) −39923.2 −1.32766 −0.663828 0.747885i \(-0.731069\pi\)
−0.663828 + 0.747885i \(0.731069\pi\)
\(968\) 24228.9 0.804490
\(969\) −3344.67 −0.110884
\(970\) 84474.2 2.79619
\(971\) −12312.2 −0.406919 −0.203460 0.979083i \(-0.565219\pi\)
−0.203460 + 0.979083i \(0.565219\pi\)
\(972\) −19680.9 −0.649448
\(973\) 19295.6 0.635754
\(974\) 3786.68 0.124572
\(975\) −41031.3 −1.34775
\(976\) −55933.0 −1.83440
\(977\) −11000.6 −0.360225 −0.180112 0.983646i \(-0.557646\pi\)
−0.180112 + 0.983646i \(0.557646\pi\)
\(978\) −22328.1 −0.730036
\(979\) −55892.4 −1.82465
\(980\) 20470.6 0.667256
\(981\) −22739.9 −0.740090
\(982\) 27932.1 0.907687
\(983\) 44584.6 1.44662 0.723310 0.690524i \(-0.242620\pi\)
0.723310 + 0.690524i \(0.242620\pi\)
\(984\) −2307.77 −0.0747653
\(985\) −39940.2 −1.29198
\(986\) 23880.1 0.771295
\(987\) −2770.21 −0.0893383
\(988\) −10353.2 −0.333380
\(989\) 0 0
\(990\) 81835.2 2.62716
\(991\) 52956.1 1.69748 0.848742 0.528808i \(-0.177361\pi\)
0.848742 + 0.528808i \(0.177361\pi\)
\(992\) −25088.1 −0.802971
\(993\) 14347.2 0.458503
\(994\) −21965.8 −0.700919
\(995\) −44226.6 −1.40912
\(996\) 9218.19 0.293262
\(997\) −26517.6 −0.842347 −0.421174 0.906980i \(-0.638382\pi\)
−0.421174 + 0.906980i \(0.638382\pi\)
\(998\) −64464.6 −2.04468
\(999\) −33150.6 −1.04989
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.j.1.11 yes 50
43.42 odd 2 1849.4.a.i.1.40 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.40 50 43.42 odd 2
1849.4.a.j.1.11 yes 50 1.1 even 1 trivial