Properties

Label 1849.4.a.j.1.3
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 1.3
Character \(\chi\) \(=\) 1849.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.80413 q^{2} +2.56623 q^{3} +15.0797 q^{4} +6.34124 q^{5} -12.3285 q^{6} -6.17767 q^{7} -34.0117 q^{8} -20.4145 q^{9} +O(q^{10})\) \(q-4.80413 q^{2} +2.56623 q^{3} +15.0797 q^{4} +6.34124 q^{5} -12.3285 q^{6} -6.17767 q^{7} -34.0117 q^{8} -20.4145 q^{9} -30.4642 q^{10} +33.6300 q^{11} +38.6979 q^{12} -34.3998 q^{13} +29.6784 q^{14} +16.2731 q^{15} +42.7593 q^{16} +105.843 q^{17} +98.0737 q^{18} -14.3178 q^{19} +95.6239 q^{20} -15.8533 q^{21} -161.563 q^{22} -160.517 q^{23} -87.2819 q^{24} -84.7886 q^{25} +165.261 q^{26} -121.676 q^{27} -93.1573 q^{28} +42.2224 q^{29} -78.1781 q^{30} +0.100791 q^{31} +66.6725 q^{32} +86.3024 q^{33} -508.483 q^{34} -39.1741 q^{35} -307.843 q^{36} +369.747 q^{37} +68.7844 q^{38} -88.2778 q^{39} -215.676 q^{40} +450.083 q^{41} +76.1615 q^{42} +507.130 q^{44} -129.453 q^{45} +771.146 q^{46} -288.931 q^{47} +109.730 q^{48} -304.836 q^{49} +407.336 q^{50} +271.617 q^{51} -518.738 q^{52} +439.646 q^{53} +584.550 q^{54} +213.256 q^{55} +210.113 q^{56} -36.7427 q^{57} -202.842 q^{58} -16.1085 q^{59} +245.393 q^{60} +764.542 q^{61} -0.484211 q^{62} +126.114 q^{63} -662.378 q^{64} -218.137 q^{65} -414.608 q^{66} -415.572 q^{67} +1596.08 q^{68} -411.925 q^{69} +188.198 q^{70} -998.578 q^{71} +694.330 q^{72} -160.169 q^{73} -1776.31 q^{74} -217.587 q^{75} -215.907 q^{76} -207.755 q^{77} +424.098 q^{78} -1319.53 q^{79} +271.147 q^{80} +238.940 q^{81} -2162.26 q^{82} -374.027 q^{83} -239.063 q^{84} +671.175 q^{85} +108.352 q^{87} -1143.81 q^{88} +706.315 q^{89} +621.909 q^{90} +212.511 q^{91} -2420.55 q^{92} +0.258652 q^{93} +1388.06 q^{94} -90.7924 q^{95} +171.097 q^{96} -1260.72 q^{97} +1464.47 q^{98} -686.539 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} + O(q^{10}) \) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} + 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} + 386q^{18} - 12q^{19} - 108q^{20} - 408q^{21} - 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} - 1493q^{26} - 10q^{27} + 242q^{28} - 208q^{29} + 48q^{30} - 932q^{31} + 1124q^{32} - 254q^{33} - 765q^{34} - 1452q^{35} + 747q^{36} + 90q^{37} - 1213q^{38} + 1610q^{39} - 1693q^{40} - 1354q^{41} + 16q^{42} - 2704q^{44} - 4508q^{45} - 233q^{46} - 3484q^{47} + 376q^{48} + 1324q^{49} + 408q^{50} - 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} + 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} - 1172q^{61} + 1546q^{62} + 3686q^{63} + 606q^{64} - 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} - 136q^{69} + 1310q^{70} - 162q^{71} + 5814q^{72} - 746q^{73} - 4332q^{74} - 236q^{75} - 1338q^{76} - 2024q^{77} - 2782q^{78} - 2656q^{79} + 5713q^{80} - 86q^{81} + 4168q^{82} - 3514q^{83} - 4269q^{84} + 7558q^{85} - 10278q^{87} - 11692q^{88} - 2640q^{89} - 8286q^{90} + 5946q^{91} - 4271q^{92} + 2q^{93} - 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} + 2826q^{98} - 8174q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.80413 −1.69852 −0.849258 0.527977i \(-0.822951\pi\)
−0.849258 + 0.527977i \(0.822951\pi\)
\(3\) 2.56623 0.493871 0.246936 0.969032i \(-0.420576\pi\)
0.246936 + 0.969032i \(0.420576\pi\)
\(4\) 15.0797 1.88496
\(5\) 6.34124 0.567178 0.283589 0.958946i \(-0.408475\pi\)
0.283589 + 0.958946i \(0.408475\pi\)
\(6\) −12.3285 −0.838849
\(7\) −6.17767 −0.333563 −0.166782 0.985994i \(-0.553337\pi\)
−0.166782 + 0.985994i \(0.553337\pi\)
\(8\) −34.0117 −1.50312
\(9\) −20.4145 −0.756091
\(10\) −30.4642 −0.963361
\(11\) 33.6300 0.921803 0.460902 0.887451i \(-0.347526\pi\)
0.460902 + 0.887451i \(0.347526\pi\)
\(12\) 38.6979 0.930928
\(13\) −34.3998 −0.733906 −0.366953 0.930239i \(-0.619599\pi\)
−0.366953 + 0.930239i \(0.619599\pi\)
\(14\) 29.6784 0.566562
\(15\) 16.2731 0.280113
\(16\) 42.7593 0.668114
\(17\) 105.843 1.51004 0.755020 0.655702i \(-0.227627\pi\)
0.755020 + 0.655702i \(0.227627\pi\)
\(18\) 98.0737 1.28423
\(19\) −14.3178 −0.172880 −0.0864400 0.996257i \(-0.527549\pi\)
−0.0864400 + 0.996257i \(0.527549\pi\)
\(20\) 95.6239 1.06911
\(21\) −15.8533 −0.164737
\(22\) −161.563 −1.56570
\(23\) −160.517 −1.45522 −0.727612 0.685988i \(-0.759370\pi\)
−0.727612 + 0.685988i \(0.759370\pi\)
\(24\) −87.2819 −0.742348
\(25\) −84.7886 −0.678309
\(26\) 165.261 1.24655
\(27\) −121.676 −0.867283
\(28\) −93.1573 −0.628753
\(29\) 42.2224 0.270362 0.135181 0.990821i \(-0.456838\pi\)
0.135181 + 0.990821i \(0.456838\pi\)
\(30\) −78.1781 −0.475777
\(31\) 0.100791 0.000583952 0 0.000291976 1.00000i \(-0.499907\pi\)
0.000291976 1.00000i \(0.499907\pi\)
\(32\) 66.6725 0.368317
\(33\) 86.3024 0.455252
\(34\) −508.483 −2.56483
\(35\) −39.1741 −0.189190
\(36\) −307.843 −1.42520
\(37\) 369.747 1.64287 0.821433 0.570305i \(-0.193175\pi\)
0.821433 + 0.570305i \(0.193175\pi\)
\(38\) 68.7844 0.293640
\(39\) −88.2778 −0.362455
\(40\) −215.676 −0.852536
\(41\) 450.083 1.71442 0.857210 0.514967i \(-0.172196\pi\)
0.857210 + 0.514967i \(0.172196\pi\)
\(42\) 76.1615 0.279809
\(43\) 0 0
\(44\) 507.130 1.73756
\(45\) −129.453 −0.428838
\(46\) 771.146 2.47172
\(47\) −288.931 −0.896700 −0.448350 0.893858i \(-0.647988\pi\)
−0.448350 + 0.893858i \(0.647988\pi\)
\(48\) 109.730 0.329962
\(49\) −304.836 −0.888736
\(50\) 407.336 1.15212
\(51\) 271.617 0.745765
\(52\) −518.738 −1.38338
\(53\) 439.646 1.13943 0.569717 0.821841i \(-0.307053\pi\)
0.569717 + 0.821841i \(0.307053\pi\)
\(54\) 584.550 1.47310
\(55\) 213.256 0.522826
\(56\) 210.113 0.501385
\(57\) −36.7427 −0.0853805
\(58\) −202.842 −0.459215
\(59\) −16.1085 −0.0355450 −0.0177725 0.999842i \(-0.505657\pi\)
−0.0177725 + 0.999842i \(0.505657\pi\)
\(60\) 245.393 0.528002
\(61\) 764.542 1.60475 0.802374 0.596822i \(-0.203570\pi\)
0.802374 + 0.596822i \(0.203570\pi\)
\(62\) −0.484211 −0.000991853 0
\(63\) 126.114 0.252204
\(64\) −662.378 −1.29371
\(65\) −218.137 −0.416256
\(66\) −414.608 −0.773254
\(67\) −415.572 −0.757764 −0.378882 0.925445i \(-0.623691\pi\)
−0.378882 + 0.925445i \(0.623691\pi\)
\(68\) 1596.08 2.84636
\(69\) −411.925 −0.718694
\(70\) 188.198 0.321342
\(71\) −998.578 −1.66915 −0.834573 0.550897i \(-0.814286\pi\)
−0.834573 + 0.550897i \(0.814286\pi\)
\(72\) 694.330 1.13649
\(73\) −160.169 −0.256799 −0.128400 0.991723i \(-0.540984\pi\)
−0.128400 + 0.991723i \(0.540984\pi\)
\(74\) −1776.31 −2.79044
\(75\) −217.587 −0.334998
\(76\) −215.907 −0.325872
\(77\) −207.755 −0.307479
\(78\) 424.098 0.615637
\(79\) −1319.53 −1.87922 −0.939609 0.342251i \(-0.888811\pi\)
−0.939609 + 0.342251i \(0.888811\pi\)
\(80\) 271.147 0.378939
\(81\) 238.940 0.327765
\(82\) −2162.26 −2.91197
\(83\) −374.027 −0.494636 −0.247318 0.968934i \(-0.579549\pi\)
−0.247318 + 0.968934i \(0.579549\pi\)
\(84\) −239.063 −0.310523
\(85\) 671.175 0.856461
\(86\) 0 0
\(87\) 108.352 0.133524
\(88\) −1143.81 −1.38558
\(89\) 706.315 0.841227 0.420614 0.907240i \(-0.361815\pi\)
0.420614 + 0.907240i \(0.361815\pi\)
\(90\) 621.909 0.728389
\(91\) 212.511 0.244804
\(92\) −2420.55 −2.74304
\(93\) 0.258652 0.000288397 0
\(94\) 1388.06 1.52306
\(95\) −90.7924 −0.0980538
\(96\) 171.097 0.181901
\(97\) −1260.72 −1.31966 −0.659829 0.751416i \(-0.729371\pi\)
−0.659829 + 0.751416i \(0.729371\pi\)
\(98\) 1464.47 1.50953
\(99\) −686.539 −0.696967
\(100\) −1278.59 −1.27859
\(101\) −617.433 −0.608286 −0.304143 0.952626i \(-0.598370\pi\)
−0.304143 + 0.952626i \(0.598370\pi\)
\(102\) −1304.89 −1.26669
\(103\) −217.291 −0.207867 −0.103934 0.994584i \(-0.533143\pi\)
−0.103934 + 0.994584i \(0.533143\pi\)
\(104\) 1170.00 1.10315
\(105\) −100.530 −0.0934353
\(106\) −2112.12 −1.93535
\(107\) 322.466 0.291346 0.145673 0.989333i \(-0.453465\pi\)
0.145673 + 0.989333i \(0.453465\pi\)
\(108\) −1834.84 −1.63479
\(109\) 74.8798 0.0657999 0.0328999 0.999459i \(-0.489526\pi\)
0.0328999 + 0.999459i \(0.489526\pi\)
\(110\) −1024.51 −0.888030
\(111\) 948.856 0.811365
\(112\) −264.153 −0.222858
\(113\) 1937.66 1.61309 0.806545 0.591172i \(-0.201335\pi\)
0.806545 + 0.591172i \(0.201335\pi\)
\(114\) 176.517 0.145020
\(115\) −1017.88 −0.825371
\(116\) 636.700 0.509622
\(117\) 702.253 0.554900
\(118\) 77.3876 0.0603737
\(119\) −653.863 −0.503693
\(120\) −553.476 −0.421043
\(121\) −200.021 −0.150279
\(122\) −3672.96 −2.72569
\(123\) 1155.02 0.846703
\(124\) 1.51989 0.00110073
\(125\) −1330.32 −0.951900
\(126\) −605.867 −0.428373
\(127\) −85.5015 −0.0597404 −0.0298702 0.999554i \(-0.509509\pi\)
−0.0298702 + 0.999554i \(0.509509\pi\)
\(128\) 2648.77 1.82906
\(129\) 0 0
\(130\) 1047.96 0.707017
\(131\) 1370.38 0.913975 0.456988 0.889473i \(-0.348928\pi\)
0.456988 + 0.889473i \(0.348928\pi\)
\(132\) 1301.41 0.858132
\(133\) 88.4505 0.0576664
\(134\) 1996.46 1.28708
\(135\) −771.580 −0.491904
\(136\) −3599.90 −2.26977
\(137\) 355.929 0.221964 0.110982 0.993822i \(-0.464600\pi\)
0.110982 + 0.993822i \(0.464600\pi\)
\(138\) 1978.94 1.22071
\(139\) −2301.37 −1.40431 −0.702156 0.712024i \(-0.747779\pi\)
−0.702156 + 0.712024i \(0.747779\pi\)
\(140\) −590.733 −0.356615
\(141\) −741.463 −0.442855
\(142\) 4797.30 2.83507
\(143\) −1156.87 −0.676517
\(144\) −872.907 −0.505155
\(145\) 267.743 0.153343
\(146\) 769.472 0.436178
\(147\) −782.281 −0.438921
\(148\) 5575.67 3.09674
\(149\) −1231.68 −0.677202 −0.338601 0.940930i \(-0.609954\pi\)
−0.338601 + 0.940930i \(0.609954\pi\)
\(150\) 1045.32 0.568999
\(151\) −1504.88 −0.811032 −0.405516 0.914088i \(-0.632908\pi\)
−0.405516 + 0.914088i \(0.632908\pi\)
\(152\) 486.972 0.259859
\(153\) −2160.72 −1.14173
\(154\) 998.084 0.522259
\(155\) 0.639138 0.000331205 0
\(156\) −1331.20 −0.683214
\(157\) 1740.89 0.884955 0.442477 0.896780i \(-0.354100\pi\)
0.442477 + 0.896780i \(0.354100\pi\)
\(158\) 6339.17 3.19188
\(159\) 1128.23 0.562734
\(160\) 422.786 0.208901
\(161\) 991.623 0.485409
\(162\) −1147.90 −0.556714
\(163\) −1171.60 −0.562986 −0.281493 0.959563i \(-0.590830\pi\)
−0.281493 + 0.959563i \(0.590830\pi\)
\(164\) 6787.11 3.23161
\(165\) 547.265 0.258209
\(166\) 1796.88 0.840148
\(167\) 1555.44 0.720741 0.360370 0.932809i \(-0.382650\pi\)
0.360370 + 0.932809i \(0.382650\pi\)
\(168\) 539.199 0.247620
\(169\) −1013.65 −0.461381
\(170\) −3224.41 −1.45471
\(171\) 292.289 0.130713
\(172\) 0 0
\(173\) 2048.52 0.900268 0.450134 0.892961i \(-0.351376\pi\)
0.450134 + 0.892961i \(0.351376\pi\)
\(174\) −520.539 −0.226793
\(175\) 523.797 0.226259
\(176\) 1438.00 0.615869
\(177\) −41.3382 −0.0175547
\(178\) −3393.23 −1.42884
\(179\) −4392.22 −1.83402 −0.917011 0.398863i \(-0.869405\pi\)
−0.917011 + 0.398863i \(0.869405\pi\)
\(180\) −1952.11 −0.808343
\(181\) 2563.35 1.05266 0.526332 0.850279i \(-0.323567\pi\)
0.526332 + 0.850279i \(0.323567\pi\)
\(182\) −1020.93 −0.415804
\(183\) 1961.99 0.792539
\(184\) 5459.47 2.18738
\(185\) 2344.66 0.931797
\(186\) −1.24260 −0.000489848 0
\(187\) 3559.50 1.39196
\(188\) −4356.98 −1.69024
\(189\) 751.677 0.289294
\(190\) 436.179 0.166546
\(191\) −1139.09 −0.431528 −0.215764 0.976446i \(-0.569224\pi\)
−0.215764 + 0.976446i \(0.569224\pi\)
\(192\) −1699.81 −0.638924
\(193\) 2118.90 0.790267 0.395133 0.918624i \(-0.370698\pi\)
0.395133 + 0.918624i \(0.370698\pi\)
\(194\) 6056.67 2.24146
\(195\) −559.791 −0.205577
\(196\) −4596.83 −1.67523
\(197\) −4044.27 −1.46265 −0.731326 0.682028i \(-0.761098\pi\)
−0.731326 + 0.682028i \(0.761098\pi\)
\(198\) 3298.22 1.18381
\(199\) 1717.52 0.611817 0.305908 0.952061i \(-0.401040\pi\)
0.305908 + 0.952061i \(0.401040\pi\)
\(200\) 2883.81 1.01958
\(201\) −1066.45 −0.374238
\(202\) 2966.23 1.03318
\(203\) −260.836 −0.0901828
\(204\) 4095.90 1.40574
\(205\) 2854.09 0.972381
\(206\) 1043.90 0.353066
\(207\) 3276.87 1.10028
\(208\) −1470.91 −0.490333
\(209\) −481.507 −0.159361
\(210\) 482.959 0.158702
\(211\) 3943.80 1.28674 0.643370 0.765555i \(-0.277536\pi\)
0.643370 + 0.765555i \(0.277536\pi\)
\(212\) 6629.71 2.14779
\(213\) −2562.58 −0.824344
\(214\) −1549.17 −0.494856
\(215\) 0 0
\(216\) 4138.42 1.30363
\(217\) −0.622651 −0.000194785 0
\(218\) −359.732 −0.111762
\(219\) −411.030 −0.126826
\(220\) 3215.83 0.985507
\(221\) −3640.97 −1.10823
\(222\) −4558.43 −1.37812
\(223\) −1798.16 −0.539971 −0.269985 0.962864i \(-0.587019\pi\)
−0.269985 + 0.962864i \(0.587019\pi\)
\(224\) −411.881 −0.122857
\(225\) 1730.91 0.512863
\(226\) −9308.75 −2.73986
\(227\) −6340.20 −1.85381 −0.926903 0.375301i \(-0.877539\pi\)
−0.926903 + 0.375301i \(0.877539\pi\)
\(228\) −554.068 −0.160939
\(229\) −1568.74 −0.452686 −0.226343 0.974048i \(-0.572677\pi\)
−0.226343 + 0.974048i \(0.572677\pi\)
\(230\) 4890.03 1.40191
\(231\) −533.148 −0.151855
\(232\) −1436.06 −0.406387
\(233\) −2409.61 −0.677505 −0.338752 0.940876i \(-0.610005\pi\)
−0.338752 + 0.940876i \(0.610005\pi\)
\(234\) −3373.72 −0.942507
\(235\) −1832.18 −0.508588
\(236\) −242.912 −0.0670009
\(237\) −3386.21 −0.928092
\(238\) 3141.24 0.855531
\(239\) 2111.13 0.571371 0.285685 0.958324i \(-0.407779\pi\)
0.285685 + 0.958324i \(0.407779\pi\)
\(240\) 695.826 0.187147
\(241\) 3837.49 1.02570 0.512852 0.858477i \(-0.328589\pi\)
0.512852 + 0.858477i \(0.328589\pi\)
\(242\) 960.929 0.255252
\(243\) 3898.44 1.02916
\(244\) 11529.0 3.02488
\(245\) −1933.04 −0.504071
\(246\) −5548.86 −1.43814
\(247\) 492.528 0.126878
\(248\) −3.42806 −0.000877750 0
\(249\) −959.840 −0.244287
\(250\) 6391.04 1.61682
\(251\) −7394.81 −1.85959 −0.929793 0.368083i \(-0.880014\pi\)
−0.929793 + 0.368083i \(0.880014\pi\)
\(252\) 1901.76 0.475394
\(253\) −5398.20 −1.34143
\(254\) 410.760 0.101470
\(255\) 1722.39 0.422982
\(256\) −7426.01 −1.81299
\(257\) −2496.57 −0.605959 −0.302980 0.952997i \(-0.597981\pi\)
−0.302980 + 0.952997i \(0.597981\pi\)
\(258\) 0 0
\(259\) −2284.18 −0.547999
\(260\) −3289.44 −0.784625
\(261\) −861.947 −0.204418
\(262\) −6583.49 −1.55240
\(263\) −1815.08 −0.425562 −0.212781 0.977100i \(-0.568252\pi\)
−0.212781 + 0.977100i \(0.568252\pi\)
\(264\) −2935.29 −0.684298
\(265\) 2787.90 0.646261
\(266\) −424.928 −0.0979473
\(267\) 1812.57 0.415458
\(268\) −6266.69 −1.42836
\(269\) −673.864 −0.152737 −0.0763684 0.997080i \(-0.524333\pi\)
−0.0763684 + 0.997080i \(0.524333\pi\)
\(270\) 3706.77 0.835507
\(271\) 2582.80 0.578944 0.289472 0.957186i \(-0.406520\pi\)
0.289472 + 0.957186i \(0.406520\pi\)
\(272\) 4525.76 1.00888
\(273\) 545.351 0.120902
\(274\) −1709.93 −0.377009
\(275\) −2851.44 −0.625267
\(276\) −6211.69 −1.35471
\(277\) 2042.43 0.443024 0.221512 0.975158i \(-0.428901\pi\)
0.221512 + 0.975158i \(0.428901\pi\)
\(278\) 11056.1 2.38525
\(279\) −2.05759 −0.000441521 0
\(280\) 1332.38 0.284375
\(281\) 5529.75 1.17394 0.586970 0.809608i \(-0.300321\pi\)
0.586970 + 0.809608i \(0.300321\pi\)
\(282\) 3562.09 0.752196
\(283\) 213.594 0.0448653 0.0224326 0.999748i \(-0.492859\pi\)
0.0224326 + 0.999748i \(0.492859\pi\)
\(284\) −15058.2 −3.14627
\(285\) −232.994 −0.0484260
\(286\) 5557.73 1.14908
\(287\) −2780.47 −0.571867
\(288\) −1361.08 −0.278481
\(289\) 6289.71 1.28022
\(290\) −1286.27 −0.260456
\(291\) −3235.30 −0.651741
\(292\) −2415.29 −0.484056
\(293\) −6029.67 −1.20224 −0.601121 0.799158i \(-0.705279\pi\)
−0.601121 + 0.799158i \(0.705279\pi\)
\(294\) 3758.18 0.745515
\(295\) −102.148 −0.0201603
\(296\) −12575.7 −2.46942
\(297\) −4091.98 −0.799464
\(298\) 5917.15 1.15024
\(299\) 5521.76 1.06800
\(300\) −3281.15 −0.631457
\(301\) 0 0
\(302\) 7229.66 1.37755
\(303\) −1584.47 −0.300415
\(304\) −612.217 −0.115504
\(305\) 4848.15 0.910177
\(306\) 10380.4 1.93924
\(307\) 9957.66 1.85118 0.925592 0.378523i \(-0.123568\pi\)
0.925592 + 0.378523i \(0.123568\pi\)
\(308\) −3132.88 −0.579586
\(309\) −557.619 −0.102660
\(310\) −3.07050 −0.000562557 0
\(311\) 9950.53 1.81429 0.907143 0.420824i \(-0.138259\pi\)
0.907143 + 0.420824i \(0.138259\pi\)
\(312\) 3002.48 0.544814
\(313\) 1906.47 0.344281 0.172140 0.985072i \(-0.444932\pi\)
0.172140 + 0.985072i \(0.444932\pi\)
\(314\) −8363.45 −1.50311
\(315\) 799.718 0.143045
\(316\) −19898.0 −3.54225
\(317\) −2985.80 −0.529019 −0.264510 0.964383i \(-0.585210\pi\)
−0.264510 + 0.964383i \(0.585210\pi\)
\(318\) −5420.18 −0.955812
\(319\) 1419.94 0.249221
\(320\) −4200.30 −0.733762
\(321\) 827.523 0.143887
\(322\) −4763.89 −0.824476
\(323\) −1515.43 −0.261056
\(324\) 3603.14 0.617823
\(325\) 2916.71 0.497815
\(326\) 5628.52 0.956242
\(327\) 192.159 0.0324967
\(328\) −15308.1 −2.57698
\(329\) 1784.92 0.299106
\(330\) −2629.13 −0.438572
\(331\) −842.438 −0.139893 −0.0699465 0.997551i \(-0.522283\pi\)
−0.0699465 + 0.997551i \(0.522283\pi\)
\(332\) −5640.21 −0.932370
\(333\) −7548.18 −1.24216
\(334\) −7472.55 −1.22419
\(335\) −2635.24 −0.429787
\(336\) −677.877 −0.110063
\(337\) −3058.48 −0.494380 −0.247190 0.968967i \(-0.579507\pi\)
−0.247190 + 0.968967i \(0.579507\pi\)
\(338\) 4869.73 0.783664
\(339\) 4972.47 0.796659
\(340\) 10121.1 1.61439
\(341\) 3.38959 0.000538289 0
\(342\) −1404.20 −0.222018
\(343\) 4002.12 0.630012
\(344\) 0 0
\(345\) −2612.11 −0.407627
\(346\) −9841.37 −1.52912
\(347\) −11415.0 −1.76597 −0.882985 0.469402i \(-0.844470\pi\)
−0.882985 + 0.469402i \(0.844470\pi\)
\(348\) 1633.92 0.251688
\(349\) 2729.17 0.418594 0.209297 0.977852i \(-0.432882\pi\)
0.209297 + 0.977852i \(0.432882\pi\)
\(350\) −2516.39 −0.384304
\(351\) 4185.64 0.636505
\(352\) 2242.20 0.339516
\(353\) −11443.4 −1.72541 −0.862703 0.505710i \(-0.831230\pi\)
−0.862703 + 0.505710i \(0.831230\pi\)
\(354\) 198.594 0.0298169
\(355\) −6332.23 −0.946703
\(356\) 10651.0 1.58568
\(357\) −1677.96 −0.248760
\(358\) 21100.8 3.11512
\(359\) −3938.99 −0.579086 −0.289543 0.957165i \(-0.593503\pi\)
−0.289543 + 0.957165i \(0.593503\pi\)
\(360\) 4402.92 0.644595
\(361\) −6654.00 −0.970112
\(362\) −12314.7 −1.78797
\(363\) −513.301 −0.0742185
\(364\) 3204.59 0.461446
\(365\) −1015.67 −0.145651
\(366\) −9425.67 −1.34614
\(367\) −7300.33 −1.03835 −0.519175 0.854668i \(-0.673761\pi\)
−0.519175 + 0.854668i \(0.673761\pi\)
\(368\) −6863.60 −0.972256
\(369\) −9188.21 −1.29626
\(370\) −11264.0 −1.58267
\(371\) −2715.99 −0.380073
\(372\) 3.90039 0.000543618 0
\(373\) −9652.92 −1.33997 −0.669985 0.742374i \(-0.733700\pi\)
−0.669985 + 0.742374i \(0.733700\pi\)
\(374\) −17100.3 −2.36427
\(375\) −3413.91 −0.470116
\(376\) 9827.03 1.34785
\(377\) −1452.44 −0.198421
\(378\) −3611.16 −0.491370
\(379\) −1436.71 −0.194720 −0.0973601 0.995249i \(-0.531040\pi\)
−0.0973601 + 0.995249i \(0.531040\pi\)
\(380\) −1369.12 −0.184827
\(381\) −219.417 −0.0295041
\(382\) 5472.35 0.732958
\(383\) −9742.88 −1.29984 −0.649919 0.760003i \(-0.725197\pi\)
−0.649919 + 0.760003i \(0.725197\pi\)
\(384\) 6797.35 0.903323
\(385\) −1317.43 −0.174396
\(386\) −10179.5 −1.34228
\(387\) 0 0
\(388\) −19011.3 −2.48750
\(389\) −853.522 −0.111248 −0.0556238 0.998452i \(-0.517715\pi\)
−0.0556238 + 0.998452i \(0.517715\pi\)
\(390\) 2689.31 0.349176
\(391\) −16989.6 −2.19745
\(392\) 10368.0 1.33588
\(393\) 3516.71 0.451386
\(394\) 19429.2 2.48434
\(395\) −8367.43 −1.06585
\(396\) −10352.8 −1.31375
\(397\) 7093.70 0.896782 0.448391 0.893837i \(-0.351997\pi\)
0.448391 + 0.893837i \(0.351997\pi\)
\(398\) −8251.18 −1.03918
\(399\) 226.984 0.0284798
\(400\) −3625.50 −0.453188
\(401\) −2816.61 −0.350761 −0.175380 0.984501i \(-0.556115\pi\)
−0.175380 + 0.984501i \(0.556115\pi\)
\(402\) 5123.38 0.635650
\(403\) −3.46718 −0.000428567 0
\(404\) −9310.69 −1.14659
\(405\) 1515.18 0.185901
\(406\) 1253.09 0.153177
\(407\) 12434.6 1.51440
\(408\) −9238.17 −1.12097
\(409\) −11950.0 −1.44472 −0.722358 0.691520i \(-0.756942\pi\)
−0.722358 + 0.691520i \(0.756942\pi\)
\(410\) −13711.4 −1.65161
\(411\) 913.395 0.109622
\(412\) −3276.68 −0.391822
\(413\) 99.5133 0.0118565
\(414\) −15742.5 −1.86885
\(415\) −2371.80 −0.280547
\(416\) −2293.52 −0.270310
\(417\) −5905.84 −0.693549
\(418\) 2313.22 0.270678
\(419\) 2847.25 0.331974 0.165987 0.986128i \(-0.446919\pi\)
0.165987 + 0.986128i \(0.446919\pi\)
\(420\) −1515.96 −0.176122
\(421\) −10205.8 −1.18147 −0.590737 0.806864i \(-0.701163\pi\)
−0.590737 + 0.806864i \(0.701163\pi\)
\(422\) −18946.5 −2.18555
\(423\) 5898.37 0.677987
\(424\) −14953.1 −1.71270
\(425\) −8974.27 −1.02427
\(426\) 12311.0 1.40016
\(427\) −4723.09 −0.535284
\(428\) 4862.69 0.549175
\(429\) −2968.78 −0.334113
\(430\) 0 0
\(431\) −13829.3 −1.54555 −0.772775 0.634680i \(-0.781132\pi\)
−0.772775 + 0.634680i \(0.781132\pi\)
\(432\) −5202.80 −0.579444
\(433\) 8717.43 0.967512 0.483756 0.875203i \(-0.339272\pi\)
0.483756 + 0.875203i \(0.339272\pi\)
\(434\) 2.99130 0.000330846 0
\(435\) 687.089 0.0757320
\(436\) 1129.16 0.124030
\(437\) 2298.25 0.251579
\(438\) 1974.64 0.215416
\(439\) −14904.6 −1.62041 −0.810203 0.586149i \(-0.800643\pi\)
−0.810203 + 0.586149i \(0.800643\pi\)
\(440\) −7253.21 −0.785870
\(441\) 6223.07 0.671965
\(442\) 17491.7 1.88234
\(443\) 16289.2 1.74700 0.873502 0.486820i \(-0.161843\pi\)
0.873502 + 0.486820i \(0.161843\pi\)
\(444\) 14308.4 1.52939
\(445\) 4478.91 0.477126
\(446\) 8638.58 0.917149
\(447\) −3160.77 −0.334451
\(448\) 4091.95 0.431532
\(449\) −12947.9 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(450\) −8315.54 −0.871107
\(451\) 15136.3 1.58036
\(452\) 29219.2 3.04061
\(453\) −3861.88 −0.400545
\(454\) 30459.2 3.14872
\(455\) 1347.58 0.138847
\(456\) 1249.68 0.128337
\(457\) −1521.06 −0.155694 −0.0778469 0.996965i \(-0.524805\pi\)
−0.0778469 + 0.996965i \(0.524805\pi\)
\(458\) 7536.42 0.768895
\(459\) −12878.6 −1.30963
\(460\) −15349.3 −1.55579
\(461\) −4500.11 −0.454644 −0.227322 0.973820i \(-0.572997\pi\)
−0.227322 + 0.973820i \(0.572997\pi\)
\(462\) 2561.31 0.257929
\(463\) 5096.27 0.511542 0.255771 0.966737i \(-0.417671\pi\)
0.255771 + 0.966737i \(0.417671\pi\)
\(464\) 1805.40 0.180633
\(465\) 1.64018 0.000163573 0
\(466\) 11576.1 1.15075
\(467\) 5931.03 0.587699 0.293850 0.955852i \(-0.405064\pi\)
0.293850 + 0.955852i \(0.405064\pi\)
\(468\) 10589.7 1.04596
\(469\) 2567.27 0.252762
\(470\) 8802.04 0.863846
\(471\) 4467.52 0.437054
\(472\) 547.879 0.0534283
\(473\) 0 0
\(474\) 16267.8 1.57638
\(475\) 1213.98 0.117266
\(476\) −9860.04 −0.949441
\(477\) −8975.13 −0.861515
\(478\) −10142.1 −0.970482
\(479\) −13403.9 −1.27858 −0.639292 0.768964i \(-0.720772\pi\)
−0.639292 + 0.768964i \(0.720772\pi\)
\(480\) 1084.97 0.103170
\(481\) −12719.2 −1.20571
\(482\) −18435.8 −1.74218
\(483\) 2544.74 0.239730
\(484\) −3016.26 −0.283270
\(485\) −7994.54 −0.748481
\(486\) −18728.6 −1.74804
\(487\) 9526.78 0.886447 0.443223 0.896411i \(-0.353835\pi\)
0.443223 + 0.896411i \(0.353835\pi\)
\(488\) −26003.4 −2.41213
\(489\) −3006.60 −0.278043
\(490\) 9286.58 0.856174
\(491\) −6851.57 −0.629750 −0.314875 0.949133i \(-0.601963\pi\)
−0.314875 + 0.949133i \(0.601963\pi\)
\(492\) 17417.3 1.59600
\(493\) 4468.94 0.408257
\(494\) −2366.17 −0.215504
\(495\) −4353.51 −0.395304
\(496\) 4.30973 0.000390147 0
\(497\) 6168.89 0.556766
\(498\) 4611.20 0.414925
\(499\) 3033.89 0.272176 0.136088 0.990697i \(-0.456547\pi\)
0.136088 + 0.990697i \(0.456547\pi\)
\(500\) −20060.8 −1.79429
\(501\) 3991.62 0.355953
\(502\) 35525.6 3.15854
\(503\) 6789.54 0.601850 0.300925 0.953648i \(-0.402705\pi\)
0.300925 + 0.953648i \(0.402705\pi\)
\(504\) −4289.35 −0.379093
\(505\) −3915.29 −0.345006
\(506\) 25933.7 2.27844
\(507\) −2601.27 −0.227863
\(508\) −1289.33 −0.112608
\(509\) −2197.89 −0.191395 −0.0956974 0.995410i \(-0.530508\pi\)
−0.0956974 + 0.995410i \(0.530508\pi\)
\(510\) −8274.59 −0.718441
\(511\) 989.470 0.0856587
\(512\) 14485.4 1.25033
\(513\) 1742.14 0.149936
\(514\) 11993.8 1.02923
\(515\) −1377.90 −0.117898
\(516\) 0 0
\(517\) −9716.75 −0.826581
\(518\) 10973.5 0.930786
\(519\) 5256.98 0.444617
\(520\) 7419.22 0.625682
\(521\) −17320.1 −1.45644 −0.728222 0.685341i \(-0.759653\pi\)
−0.728222 + 0.685341i \(0.759653\pi\)
\(522\) 4140.91 0.347208
\(523\) −11154.5 −0.932601 −0.466300 0.884626i \(-0.654413\pi\)
−0.466300 + 0.884626i \(0.654413\pi\)
\(524\) 20664.9 1.72281
\(525\) 1344.18 0.111743
\(526\) 8719.90 0.722824
\(527\) 10.6680 0.000881791 0
\(528\) 3690.23 0.304160
\(529\) 13598.8 1.11768
\(530\) −13393.4 −1.09769
\(531\) 328.847 0.0268752
\(532\) 1333.80 0.108699
\(533\) −15482.8 −1.25822
\(534\) −8707.81 −0.705663
\(535\) 2044.84 0.165245
\(536\) 14134.3 1.13901
\(537\) −11271.4 −0.905771
\(538\) 3237.33 0.259426
\(539\) −10251.7 −0.819239
\(540\) −11635.2 −0.927219
\(541\) −18613.0 −1.47918 −0.739591 0.673057i \(-0.764981\pi\)
−0.739591 + 0.673057i \(0.764981\pi\)
\(542\) −12408.1 −0.983347
\(543\) 6578.14 0.519881
\(544\) 7056.81 0.556173
\(545\) 474.831 0.0373202
\(546\) −2619.94 −0.205354
\(547\) 19378.5 1.51474 0.757371 0.652985i \(-0.226484\pi\)
0.757371 + 0.652985i \(0.226484\pi\)
\(548\) 5367.29 0.418393
\(549\) −15607.7 −1.21333
\(550\) 13698.7 1.06203
\(551\) −604.531 −0.0467402
\(552\) 14010.3 1.08028
\(553\) 8151.59 0.626837
\(554\) −9812.11 −0.752485
\(555\) 6016.93 0.460188
\(556\) −34703.8 −2.64707
\(557\) −8962.10 −0.681753 −0.340877 0.940108i \(-0.610724\pi\)
−0.340877 + 0.940108i \(0.610724\pi\)
\(558\) 9.88491 0.000749931 0
\(559\) 0 0
\(560\) −1675.06 −0.126400
\(561\) 9134.50 0.687449
\(562\) −26565.7 −1.99396
\(563\) 12298.5 0.920640 0.460320 0.887753i \(-0.347735\pi\)
0.460320 + 0.887753i \(0.347735\pi\)
\(564\) −11181.0 −0.834763
\(565\) 12287.1 0.914910
\(566\) −1026.14 −0.0762044
\(567\) −1476.10 −0.109330
\(568\) 33963.3 2.50893
\(569\) 5282.29 0.389183 0.194592 0.980884i \(-0.437662\pi\)
0.194592 + 0.980884i \(0.437662\pi\)
\(570\) 1119.34 0.0822523
\(571\) −20315.8 −1.48895 −0.744475 0.667650i \(-0.767300\pi\)
−0.744475 + 0.667650i \(0.767300\pi\)
\(572\) −17445.2 −1.27521
\(573\) −2923.18 −0.213119
\(574\) 13357.7 0.971326
\(575\) 13610.0 0.987092
\(576\) 13522.1 0.978160
\(577\) −12757.9 −0.920484 −0.460242 0.887793i \(-0.652237\pi\)
−0.460242 + 0.887793i \(0.652237\pi\)
\(578\) −30216.6 −2.17447
\(579\) 5437.58 0.390290
\(580\) 4037.47 0.289046
\(581\) 2310.62 0.164992
\(582\) 15542.8 1.10699
\(583\) 14785.3 1.05033
\(584\) 5447.61 0.386000
\(585\) 4453.16 0.314727
\(586\) 28967.3 2.04203
\(587\) −10685.7 −0.751354 −0.375677 0.926751i \(-0.622590\pi\)
−0.375677 + 0.926751i \(0.622590\pi\)
\(588\) −11796.5 −0.827349
\(589\) −1.44310 −0.000100954 0
\(590\) 490.733 0.0342427
\(591\) −10378.5 −0.722362
\(592\) 15810.1 1.09762
\(593\) −2637.21 −0.182626 −0.0913130 0.995822i \(-0.529106\pi\)
−0.0913130 + 0.995822i \(0.529106\pi\)
\(594\) 19658.4 1.35790
\(595\) −4146.30 −0.285684
\(596\) −18573.3 −1.27650
\(597\) 4407.55 0.302159
\(598\) −26527.3 −1.81401
\(599\) 20472.5 1.39647 0.698234 0.715869i \(-0.253969\pi\)
0.698234 + 0.715869i \(0.253969\pi\)
\(600\) 7400.51 0.503541
\(601\) 848.655 0.0575996 0.0287998 0.999585i \(-0.490831\pi\)
0.0287998 + 0.999585i \(0.490831\pi\)
\(602\) 0 0
\(603\) 8483.68 0.572939
\(604\) −22693.2 −1.52876
\(605\) −1268.38 −0.0852350
\(606\) 7612.03 0.510260
\(607\) 19963.6 1.33492 0.667461 0.744645i \(-0.267381\pi\)
0.667461 + 0.744645i \(0.267381\pi\)
\(608\) −954.601 −0.0636746
\(609\) −669.366 −0.0445387
\(610\) −23291.1 −1.54595
\(611\) 9939.16 0.658094
\(612\) −32583.0 −2.15211
\(613\) −7530.10 −0.496147 −0.248073 0.968741i \(-0.579797\pi\)
−0.248073 + 0.968741i \(0.579797\pi\)
\(614\) −47837.9 −3.14427
\(615\) 7324.25 0.480231
\(616\) 7066.11 0.462178
\(617\) 10468.8 0.683075 0.341538 0.939868i \(-0.389052\pi\)
0.341538 + 0.939868i \(0.389052\pi\)
\(618\) 2678.88 0.174369
\(619\) 5756.07 0.373757 0.186879 0.982383i \(-0.440163\pi\)
0.186879 + 0.982383i \(0.440163\pi\)
\(620\) 9.63799 0.000624308 0
\(621\) 19531.2 1.26209
\(622\) −47803.6 −3.08159
\(623\) −4363.38 −0.280602
\(624\) −3774.69 −0.242161
\(625\) 2162.69 0.138412
\(626\) −9158.91 −0.584766
\(627\) −1235.66 −0.0787040
\(628\) 26252.0 1.66810
\(629\) 39135.1 2.48079
\(630\) −3841.95 −0.242964
\(631\) −27752.2 −1.75087 −0.875433 0.483339i \(-0.839424\pi\)
−0.875433 + 0.483339i \(0.839424\pi\)
\(632\) 44879.3 2.82469
\(633\) 10120.7 0.635484
\(634\) 14344.2 0.898548
\(635\) −542.186 −0.0338834
\(636\) 17013.4 1.06073
\(637\) 10486.3 0.652249
\(638\) −6821.58 −0.423306
\(639\) 20385.4 1.26203
\(640\) 16796.5 1.03741
\(641\) 2219.51 0.136763 0.0683816 0.997659i \(-0.478216\pi\)
0.0683816 + 0.997659i \(0.478216\pi\)
\(642\) −3975.53 −0.244395
\(643\) −20580.1 −1.26221 −0.631104 0.775698i \(-0.717398\pi\)
−0.631104 + 0.775698i \(0.717398\pi\)
\(644\) 14953.4 0.914977
\(645\) 0 0
\(646\) 7280.34 0.443407
\(647\) −16963.7 −1.03078 −0.515389 0.856957i \(-0.672352\pi\)
−0.515389 + 0.856957i \(0.672352\pi\)
\(648\) −8126.77 −0.492669
\(649\) −541.731 −0.0327655
\(650\) −14012.3 −0.845548
\(651\) −1.59787 −9.61987e−5 0
\(652\) −17667.3 −1.06121
\(653\) −5184.83 −0.310717 −0.155358 0.987858i \(-0.549653\pi\)
−0.155358 + 0.987858i \(0.549653\pi\)
\(654\) −923.157 −0.0551961
\(655\) 8689.92 0.518387
\(656\) 19245.2 1.14543
\(657\) 3269.76 0.194163
\(658\) −8574.99 −0.508037
\(659\) −30887.9 −1.82583 −0.912915 0.408149i \(-0.866174\pi\)
−0.912915 + 0.408149i \(0.866174\pi\)
\(660\) 8252.57 0.486714
\(661\) −5445.52 −0.320433 −0.160216 0.987082i \(-0.551219\pi\)
−0.160216 + 0.987082i \(0.551219\pi\)
\(662\) 4047.18 0.237611
\(663\) −9343.58 −0.547322
\(664\) 12721.3 0.743497
\(665\) 560.886 0.0327071
\(666\) 36262.5 2.10982
\(667\) −6777.43 −0.393438
\(668\) 23455.6 1.35857
\(669\) −4614.48 −0.266676
\(670\) 12660.1 0.730001
\(671\) 25711.6 1.47926
\(672\) −1056.98 −0.0606755
\(673\) 12480.7 0.714850 0.357425 0.933942i \(-0.383655\pi\)
0.357425 + 0.933942i \(0.383655\pi\)
\(674\) 14693.3 0.839712
\(675\) 10316.8 0.588286
\(676\) −15285.6 −0.869685
\(677\) −30422.7 −1.72709 −0.863546 0.504270i \(-0.831762\pi\)
−0.863546 + 0.504270i \(0.831762\pi\)
\(678\) −23888.4 −1.35314
\(679\) 7788.32 0.440189
\(680\) −22827.8 −1.28736
\(681\) −16270.4 −0.915542
\(682\) −16.2840 −0.000914293 0
\(683\) −9567.17 −0.535985 −0.267992 0.963421i \(-0.586360\pi\)
−0.267992 + 0.963421i \(0.586360\pi\)
\(684\) 4407.63 0.246389
\(685\) 2257.03 0.125893
\(686\) −19226.7 −1.07009
\(687\) −4025.74 −0.223569
\(688\) 0 0
\(689\) −15123.7 −0.836237
\(690\) 12548.9 0.692362
\(691\) 14041.5 0.773031 0.386516 0.922283i \(-0.373679\pi\)
0.386516 + 0.922283i \(0.373679\pi\)
\(692\) 30891.1 1.69697
\(693\) 4241.21 0.232482
\(694\) 54839.3 2.99953
\(695\) −14593.5 −0.796494
\(696\) −3685.25 −0.200703
\(697\) 47638.1 2.58884
\(698\) −13111.3 −0.710989
\(699\) −6183.61 −0.334600
\(700\) 7898.68 0.426489
\(701\) 4386.14 0.236323 0.118161 0.992994i \(-0.462300\pi\)
0.118161 + 0.992994i \(0.462300\pi\)
\(702\) −20108.4 −1.08111
\(703\) −5293.95 −0.284019
\(704\) −22275.8 −1.19254
\(705\) −4701.80 −0.251177
\(706\) 54975.4 2.93063
\(707\) 3814.30 0.202902
\(708\) −623.368 −0.0330898
\(709\) −2852.53 −0.151099 −0.0755494 0.997142i \(-0.524071\pi\)
−0.0755494 + 0.997142i \(0.524071\pi\)
\(710\) 30420.8 1.60799
\(711\) 26937.4 1.42086
\(712\) −24023.0 −1.26447
\(713\) −16.1786 −0.000849782 0
\(714\) 8061.15 0.422523
\(715\) −7335.97 −0.383706
\(716\) −66233.2 −3.45706
\(717\) 5417.64 0.282184
\(718\) 18923.4 0.983588
\(719\) 608.604 0.0315676 0.0157838 0.999875i \(-0.494976\pi\)
0.0157838 + 0.999875i \(0.494976\pi\)
\(720\) −5535.32 −0.286513
\(721\) 1342.35 0.0693369
\(722\) 31966.7 1.64775
\(723\) 9847.90 0.506566
\(724\) 38654.5 1.98423
\(725\) −3579.98 −0.183389
\(726\) 2465.97 0.126061
\(727\) −38172.7 −1.94738 −0.973691 0.227872i \(-0.926823\pi\)
−0.973691 + 0.227872i \(0.926823\pi\)
\(728\) −7227.85 −0.367970
\(729\) 3552.91 0.180507
\(730\) 4879.41 0.247390
\(731\) 0 0
\(732\) 29586.2 1.49390
\(733\) 31896.8 1.60728 0.803639 0.595117i \(-0.202894\pi\)
0.803639 + 0.595117i \(0.202894\pi\)
\(734\) 35071.8 1.76365
\(735\) −4960.63 −0.248946
\(736\) −10702.1 −0.535984
\(737\) −13975.7 −0.698509
\(738\) 44141.4 2.20172
\(739\) −3826.00 −0.190449 −0.0952245 0.995456i \(-0.530357\pi\)
−0.0952245 + 0.995456i \(0.530357\pi\)
\(740\) 35356.7 1.75640
\(741\) 1263.94 0.0626613
\(742\) 13048.0 0.645560
\(743\) −17425.8 −0.860418 −0.430209 0.902729i \(-0.641560\pi\)
−0.430209 + 0.902729i \(0.641560\pi\)
\(744\) −8.79720 −0.000433496 0
\(745\) −7810.38 −0.384094
\(746\) 46373.9 2.27596
\(747\) 7635.56 0.373990
\(748\) 53676.1 2.62379
\(749\) −1992.09 −0.0971822
\(750\) 16400.9 0.798500
\(751\) −15398.9 −0.748223 −0.374112 0.927384i \(-0.622052\pi\)
−0.374112 + 0.927384i \(0.622052\pi\)
\(752\) −12354.5 −0.599097
\(753\) −18976.8 −0.918396
\(754\) 6977.72 0.337021
\(755\) −9542.84 −0.459999
\(756\) 11335.1 0.545307
\(757\) 15476.4 0.743065 0.371532 0.928420i \(-0.378832\pi\)
0.371532 + 0.928420i \(0.378832\pi\)
\(758\) 6902.16 0.330736
\(759\) −13853.0 −0.662494
\(760\) 3088.01 0.147386
\(761\) 19666.4 0.936802 0.468401 0.883516i \(-0.344830\pi\)
0.468401 + 0.883516i \(0.344830\pi\)
\(762\) 1054.11 0.0501132
\(763\) −462.583 −0.0219484
\(764\) −17177.2 −0.813413
\(765\) −13701.7 −0.647562
\(766\) 46806.1 2.20780
\(767\) 554.130 0.0260867
\(768\) −19056.9 −0.895385
\(769\) 18122.5 0.849821 0.424911 0.905235i \(-0.360306\pi\)
0.424911 + 0.905235i \(0.360306\pi\)
\(770\) 6329.09 0.296214
\(771\) −6406.77 −0.299266
\(772\) 31952.3 1.48962
\(773\) 20770.5 0.966444 0.483222 0.875498i \(-0.339466\pi\)
0.483222 + 0.875498i \(0.339466\pi\)
\(774\) 0 0
\(775\) −8.54590 −0.000396100 0
\(776\) 42879.3 1.98360
\(777\) −5861.72 −0.270641
\(778\) 4100.43 0.188956
\(779\) −6444.19 −0.296389
\(780\) −8441.47 −0.387504
\(781\) −33582.2 −1.53862
\(782\) 81620.3 3.73240
\(783\) −5137.47 −0.234481
\(784\) −13034.6 −0.593776
\(785\) 11039.4 0.501927
\(786\) −16894.8 −0.766687
\(787\) 10239.3 0.463776 0.231888 0.972743i \(-0.425510\pi\)
0.231888 + 0.972743i \(0.425510\pi\)
\(788\) −60986.3 −2.75704
\(789\) −4657.92 −0.210173
\(790\) 40198.2 1.81037
\(791\) −11970.2 −0.538067
\(792\) 23350.4 1.04762
\(793\) −26300.1 −1.17773
\(794\) −34079.1 −1.52320
\(795\) 7154.39 0.319170
\(796\) 25899.6 1.15325
\(797\) −25982.1 −1.15475 −0.577373 0.816481i \(-0.695922\pi\)
−0.577373 + 0.816481i \(0.695922\pi\)
\(798\) −1090.46 −0.0483734
\(799\) −30581.3 −1.35405
\(800\) −5653.07 −0.249833
\(801\) −14419.0 −0.636044
\(802\) 13531.4 0.595773
\(803\) −5386.48 −0.236718
\(804\) −16081.8 −0.705424
\(805\) 6288.12 0.275313
\(806\) 16.6568 0.000727927 0
\(807\) −1729.29 −0.0754324
\(808\) 20999.9 0.914326
\(809\) −22187.4 −0.964236 −0.482118 0.876106i \(-0.660132\pi\)
−0.482118 + 0.876106i \(0.660132\pi\)
\(810\) −7279.12 −0.315756
\(811\) 8163.38 0.353459 0.176729 0.984259i \(-0.443448\pi\)
0.176729 + 0.984259i \(0.443448\pi\)
\(812\) −3933.33 −0.169991
\(813\) 6628.06 0.285924
\(814\) −59737.5 −2.57223
\(815\) −7429.40 −0.319313
\(816\) 11614.2 0.498256
\(817\) 0 0
\(818\) 57409.3 2.45387
\(819\) −4338.29 −0.185094
\(820\) 43038.7 1.83290
\(821\) −1562.98 −0.0664415 −0.0332208 0.999448i \(-0.510576\pi\)
−0.0332208 + 0.999448i \(0.510576\pi\)
\(822\) −4388.07 −0.186194
\(823\) −35023.3 −1.48340 −0.741699 0.670733i \(-0.765980\pi\)
−0.741699 + 0.670733i \(0.765980\pi\)
\(824\) 7390.44 0.312449
\(825\) −7317.47 −0.308802
\(826\) −478.075 −0.0201384
\(827\) 19516.9 0.820638 0.410319 0.911942i \(-0.365417\pi\)
0.410319 + 0.911942i \(0.365417\pi\)
\(828\) 49414.2 2.07399
\(829\) 17033.7 0.713635 0.356818 0.934174i \(-0.383862\pi\)
0.356818 + 0.934174i \(0.383862\pi\)
\(830\) 11394.4 0.476514
\(831\) 5241.35 0.218797
\(832\) 22785.6 0.949459
\(833\) −32264.8 −1.34203
\(834\) 28372.4 1.17800
\(835\) 9863.43 0.408788
\(836\) −7260.97 −0.300390
\(837\) −12.2638 −0.000506452 0
\(838\) −13678.6 −0.563864
\(839\) −7296.99 −0.300262 −0.150131 0.988666i \(-0.547970\pi\)
−0.150131 + 0.988666i \(0.547970\pi\)
\(840\) 3419.19 0.140444
\(841\) −22606.3 −0.926904
\(842\) 49030.1 2.00675
\(843\) 14190.6 0.579776
\(844\) 59471.2 2.42545
\(845\) −6427.83 −0.261685
\(846\) −28336.5 −1.15157
\(847\) 1235.67 0.0501275
\(848\) 18798.9 0.761271
\(849\) 548.133 0.0221577
\(850\) 43113.6 1.73975
\(851\) −59350.8 −2.39074
\(852\) −38642.9 −1.55386
\(853\) −9689.81 −0.388948 −0.194474 0.980908i \(-0.562300\pi\)
−0.194474 + 0.980908i \(0.562300\pi\)
\(854\) 22690.4 0.909189
\(855\) 1853.48 0.0741376
\(856\) −10967.6 −0.437927
\(857\) 25619.7 1.02118 0.510590 0.859824i \(-0.329427\pi\)
0.510590 + 0.859824i \(0.329427\pi\)
\(858\) 14262.4 0.567496
\(859\) 33986.1 1.34993 0.674966 0.737849i \(-0.264158\pi\)
0.674966 + 0.737849i \(0.264158\pi\)
\(860\) 0 0
\(861\) −7135.32 −0.282429
\(862\) 66437.6 2.62514
\(863\) −10162.6 −0.400855 −0.200427 0.979709i \(-0.564233\pi\)
−0.200427 + 0.979709i \(0.564233\pi\)
\(864\) −8112.47 −0.319435
\(865\) 12990.2 0.510612
\(866\) −41879.7 −1.64334
\(867\) 16140.9 0.632263
\(868\) −9.38938 −0.000367162 0
\(869\) −44375.7 −1.73227
\(870\) −3300.87 −0.128632
\(871\) 14295.6 0.556128
\(872\) −2546.79 −0.0989050
\(873\) 25736.9 0.997781
\(874\) −11041.1 −0.427312
\(875\) 8218.29 0.317519
\(876\) −6198.20 −0.239061
\(877\) 33751.4 1.29955 0.649774 0.760127i \(-0.274863\pi\)
0.649774 + 0.760127i \(0.274863\pi\)
\(878\) 71603.7 2.75229
\(879\) −15473.5 −0.593753
\(880\) 9118.68 0.349307
\(881\) 18540.8 0.709030 0.354515 0.935050i \(-0.384646\pi\)
0.354515 + 0.935050i \(0.384646\pi\)
\(882\) −29896.4 −1.14134
\(883\) −33606.6 −1.28081 −0.640403 0.768039i \(-0.721233\pi\)
−0.640403 + 0.768039i \(0.721233\pi\)
\(884\) −54904.7 −2.08896
\(885\) −262.136 −0.00995661
\(886\) −78255.4 −2.96732
\(887\) 36259.0 1.37256 0.686279 0.727339i \(-0.259243\pi\)
0.686279 + 0.727339i \(0.259243\pi\)
\(888\) −32272.2 −1.21958
\(889\) 528.200 0.0199272
\(890\) −21517.3 −0.810406
\(891\) 8035.57 0.302134
\(892\) −27115.6 −1.01782
\(893\) 4136.84 0.155022
\(894\) 15184.8 0.568070
\(895\) −27852.1 −1.04022
\(896\) −16363.2 −0.610108
\(897\) 14170.1 0.527454
\(898\) 62203.6 2.31154
\(899\) 4.25562 0.000157879 0
\(900\) 26101.6 0.966727
\(901\) 46533.3 1.72059
\(902\) −72716.8 −2.68426
\(903\) 0 0
\(904\) −65903.0 −2.42467
\(905\) 16254.8 0.597048
\(906\) 18553.0 0.680333
\(907\) 2435.81 0.0891730 0.0445865 0.999006i \(-0.485803\pi\)
0.0445865 + 0.999006i \(0.485803\pi\)
\(908\) −95608.2 −3.49435
\(909\) 12604.6 0.459919
\(910\) −6473.96 −0.235835
\(911\) 26035.1 0.946851 0.473426 0.880834i \(-0.343017\pi\)
0.473426 + 0.880834i \(0.343017\pi\)
\(912\) −1571.09 −0.0570439
\(913\) −12578.5 −0.455957
\(914\) 7307.36 0.264449
\(915\) 12441.5 0.449511
\(916\) −23656.1 −0.853295
\(917\) −8465.76 −0.304868
\(918\) 61870.4 2.22443
\(919\) −37812.0 −1.35724 −0.678620 0.734490i \(-0.737422\pi\)
−0.678620 + 0.734490i \(0.737422\pi\)
\(920\) 34619.8 1.24063
\(921\) 25553.7 0.914247
\(922\) 21619.1 0.772221
\(923\) 34350.9 1.22500
\(924\) −8039.70 −0.286241
\(925\) −31350.3 −1.11437
\(926\) −24483.2 −0.868862
\(927\) 4435.88 0.157167
\(928\) 2815.07 0.0995790
\(929\) 27043.9 0.955093 0.477546 0.878607i \(-0.341526\pi\)
0.477546 + 0.878607i \(0.341526\pi\)
\(930\) −7.87962 −0.000277831 0
\(931\) 4364.58 0.153645
\(932\) −36336.1 −1.27707
\(933\) 25535.4 0.896024
\(934\) −28493.5 −0.998217
\(935\) 22571.6 0.789488
\(936\) −23884.8 −0.834081
\(937\) 35620.5 1.24191 0.620956 0.783845i \(-0.286745\pi\)
0.620956 + 0.783845i \(0.286745\pi\)
\(938\) −12333.5 −0.429321
\(939\) 4892.43 0.170030
\(940\) −27628.7 −0.958669
\(941\) −54098.1 −1.87412 −0.937060 0.349168i \(-0.886464\pi\)
−0.937060 + 0.349168i \(0.886464\pi\)
\(942\) −21462.5 −0.742343
\(943\) −72246.2 −2.49487
\(944\) −688.790 −0.0237481
\(945\) 4766.57 0.164081
\(946\) 0 0
\(947\) −1896.92 −0.0650914 −0.0325457 0.999470i \(-0.510361\pi\)
−0.0325457 + 0.999470i \(0.510361\pi\)
\(948\) −51062.9 −1.74942
\(949\) 5509.77 0.188467
\(950\) −5832.14 −0.199178
\(951\) −7662.25 −0.261268
\(952\) 22239.0 0.757111
\(953\) 38430.6 1.30629 0.653143 0.757235i \(-0.273450\pi\)
0.653143 + 0.757235i \(0.273450\pi\)
\(954\) 43117.7 1.46330
\(955\) −7223.27 −0.244753
\(956\) 31835.1 1.07701
\(957\) 3643.90 0.123083
\(958\) 64394.3 2.17170
\(959\) −2198.81 −0.0740389
\(960\) −10778.9 −0.362384
\(961\) −29791.0 −1.00000
\(962\) 61104.8 2.04792
\(963\) −6582.98 −0.220284
\(964\) 57868.2 1.93341
\(965\) 13436.4 0.448222
\(966\) −12225.2 −0.407185
\(967\) −43807.6 −1.45683 −0.728417 0.685135i \(-0.759743\pi\)
−0.728417 + 0.685135i \(0.759743\pi\)
\(968\) 6803.07 0.225887
\(969\) −3888.95 −0.128928
\(970\) 38406.8 1.27131
\(971\) 13880.3 0.458742 0.229371 0.973339i \(-0.426333\pi\)
0.229371 + 0.973339i \(0.426333\pi\)
\(972\) 58787.2 1.93992
\(973\) 14217.1 0.468426
\(974\) −45767.9 −1.50564
\(975\) 7484.96 0.245857
\(976\) 32691.3 1.07215
\(977\) 16948.3 0.554990 0.277495 0.960727i \(-0.410496\pi\)
0.277495 + 0.960727i \(0.410496\pi\)
\(978\) 14444.1 0.472260
\(979\) 23753.4 0.775446
\(980\) −29149.6 −0.950154
\(981\) −1528.63 −0.0497507
\(982\) 32915.8 1.06964
\(983\) 6232.10 0.202211 0.101105 0.994876i \(-0.467762\pi\)
0.101105 + 0.994876i \(0.467762\pi\)
\(984\) −39284.1 −1.27270
\(985\) −25645.7 −0.829584
\(986\) −21469.4 −0.693432
\(987\) 4580.52 0.147720
\(988\) 7427.17 0.239160
\(989\) 0 0
\(990\) 20914.8 0.671431
\(991\) −41002.7 −1.31432 −0.657161 0.753751i \(-0.728243\pi\)
−0.657161 + 0.753751i \(0.728243\pi\)
\(992\) 6.71996 0.000215080 0
\(993\) −2161.89 −0.0690892
\(994\) −29636.2 −0.945676
\(995\) 10891.2 0.347009
\(996\) −14474.1 −0.460471
\(997\) −4141.03 −0.131542 −0.0657712 0.997835i \(-0.520951\pi\)
−0.0657712 + 0.997835i \(0.520951\pi\)
\(998\) −14575.2 −0.462295
\(999\) −44989.5 −1.42483
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.1.3 yes 50
43.42 odd 2 1849.4.a.i.1.48 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.48 50 43.42 odd 2
1849.4.a.j.1.3 yes 50 1.1 even 1 trivial