Properties

Label 1849.4.a.i.1.48
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 1.48
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 \(-0.177049\pi\)
0.849258 + 0.527977i \(0.177049\pi\)
\(3\) −2.56623 −0.493871 −0.246936 0.969032i \(-0.579424\pi\)
−0.246936 + 0.969032i \(0.579424\pi\)
\(4\) 15.0797 1.88496
\(5\) −6.34124 −0.567178 −0.283589 0.958946i \(-0.591525\pi\)
−0.283589 + 0.958946i \(0.591525\pi\)
\(6\) −12.3285 −0.838849
\(7\) 6.17767 0.333563 0.166782 0.985994i \(-0.446663\pi\)
0.166782 + 0.985994i \(0.446663\pi\)
\(8\) 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 \(-0.472451\pi\)
0.0864400 + 0.996257i \(0.472451\pi\)
\(20\) −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 \(-0.543162\pi\)
−0.135181 + 0.990821i \(0.543162\pi\)
\(30\) 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 \(-0.806825\pi\)
−0.821433 + 0.570305i \(0.806825\pi\)
\(38\) 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 \(-0.796430\pi\)
−0.802374 + 0.596822i \(0.796430\pi\)
\(62\) 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 \(-0.185714\pi\)
0.834573 + 0.550897i \(0.185714\pi\)
\(72\) −694.330 −1.13649
\(73\) 160.169 0.256799 0.128400 0.991723i \(-0.459016\pi\)
0.128400 + 0.991723i \(0.459016\pi\)
\(74\) −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 \(-0.638185\pi\)
−0.420614 + 0.907240i \(0.638185\pi\)
\(90\) 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 \(-0.798665\pi\)
−0.806545 + 0.591172i \(0.798665\pi\)
\(114\) −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 \(-0.651072\pi\)
−0.456988 + 0.889473i \(0.651072\pi\)
\(132\) −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 \(-0.535400\pi\)
−0.110982 + 0.993822i \(0.535400\pi\)
\(138\) 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 \(-0.390046\pi\)
0.338601 + 0.940930i \(0.390046\pi\)
\(150\) 1045.32 0.568999
\(151\) 1504.88 0.811032 0.405516 0.914088i \(-0.367092\pi\)
0.405516 + 0.914088i \(0.367092\pi\)
\(152\) 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 \(-0.645900\pi\)
−0.442477 + 0.896780i \(0.645900\pi\)
\(158\) −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 \(-0.409170\pi\)
0.281493 + 0.959563i \(0.409170\pi\)
\(164\) 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 \(-0.130595\pi\)
0.917011 + 0.398863i \(0.130595\pi\)
\(180\) 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 \(-0.430776\pi\)
0.215764 + 0.976446i \(0.430776\pi\)
\(192\) 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 \(-0.598960\pi\)
−0.305908 + 0.952061i \(0.598960\pi\)
\(200\) −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 \(-0.722464\pi\)
−0.643370 + 0.765555i \(0.722464\pi\)
\(212\) 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 \(-0.412981\pi\)
0.269985 + 0.962864i \(0.412981\pi\)
\(224\) −411.881 −0.122857
\(225\) 1730.91 0.512863
\(226\) −9308.75 −2.73986
\(227\) 6340.20 1.85381 0.926903 0.375301i \(-0.122461\pi\)
0.926903 + 0.375301i \(0.122461\pi\)
\(228\) −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 \(-0.389995\pi\)
0.338752 + 0.940876i \(0.389995\pi\)
\(234\) 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 \(-0.671411\pi\)
−0.512852 + 0.858477i \(0.671411\pi\)
\(242\) −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 \(-0.402019\pi\)
0.302980 + 0.952997i \(0.402019\pi\)
\(258\) 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 \(-0.431748\pi\)
0.212781 + 0.977100i \(0.431748\pi\)
\(264\) −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 \(-0.571099\pi\)
−0.221512 + 0.975158i \(0.571099\pi\)
\(278\) −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 \(-0.555068\pi\)
−0.172140 + 0.985072i \(0.555068\pi\)
\(314\) −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 \(-0.477717\pi\)
0.0699465 + 0.997551i \(0.477717\pi\)
\(332\) −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 \(-0.155530\pi\)
0.882985 + 0.469402i \(0.155530\pi\)
\(348\) 1633.92 0.251688
\(349\) −2729.17 −0.418594 −0.209297 0.977852i \(-0.567118\pi\)
−0.209297 + 0.977852i \(0.567118\pi\)
\(350\) −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 \(-0.266300\pi\)
0.669985 + 0.742374i \(0.266300\pi\)
\(374\) 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 \(-0.274803\pi\)
0.649919 + 0.760003i \(0.274803\pi\)
\(384\) 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 \(-0.482285\pi\)
0.0556238 + 0.998452i \(0.482285\pi\)
\(390\) −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 \(-0.243058\pi\)
0.722358 + 0.691520i \(0.243058\pi\)
\(410\) −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 \(-0.553081\pi\)
−0.165987 + 0.986128i \(0.553081\pi\)
\(420\) 1515.96 0.176122
\(421\) 10205.8 1.18147 0.590737 0.806864i \(-0.298837\pi\)
0.590737 + 0.806864i \(0.298837\pi\)
\(422\) −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 \(-0.660728\pi\)
−0.483756 + 0.875203i \(0.660728\pi\)
\(434\) 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 \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(450\) 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 \(-0.475195\pi\)
0.0778469 + 0.996965i \(0.475195\pi\)
\(458\) −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 \(-0.582329\pi\)
−0.255771 + 0.966737i \(0.582329\pi\)
\(464\) −1805.40 −0.180633
\(465\) 1.64018 0.000163573 0
\(466\) 11576.1 1.15075
\(467\) −5931.03 −0.587699 −0.293850 0.955852i \(-0.594936\pi\)
−0.293850 + 0.955852i \(0.594936\pi\)
\(468\) 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 \(-0.398037\pi\)
0.314875 + 0.949133i \(0.398037\pi\)
\(492\) −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 \(-0.543453\pi\)
−0.136088 + 0.990697i \(0.543453\pi\)
\(500\) 20060.8 1.79429
\(501\) −3991.62 −0.355953
\(502\) −35525.6 −3.15854
\(503\) −6789.54 −0.601850 −0.300925 0.953648i \(-0.597295\pi\)
−0.300925 + 0.953648i \(0.597295\pi\)
\(504\) −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 \(-0.240347\pi\)
0.728222 + 0.685341i \(0.240347\pi\)
\(522\) 4140.91 0.347208
\(523\) 11154.5 0.932601 0.466300 0.884626i \(-0.345587\pi\)
0.466300 + 0.884626i \(0.345587\pi\)
\(524\) −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 \(-0.232700\pi\)
0.744475 + 0.667650i \(0.232700\pi\)
\(572\) −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 \(-0.347763\pi\)
0.460242 + 0.887793i \(0.347763\pi\)
\(578\) 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 \(-0.377410\pi\)
0.375677 + 0.926751i \(0.377410\pi\)
\(588\) 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 \(-0.470894\pi\)
0.0913130 + 0.995822i \(0.470894\pi\)
\(594\) 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 \(-0.509169\pi\)
−0.0287998 + 0.999585i \(0.509169\pi\)
\(602\) 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 \(-0.732619\pi\)
−0.667461 + 0.744645i \(0.732619\pi\)
\(608\) −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 \(-0.160576\pi\)
0.875433 + 0.483339i \(0.160576\pi\)
\(632\) −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 \(-0.521784\pi\)
−0.0683816 + 0.997659i \(0.521784\pi\)
\(642\) −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 \(-0.327648\pi\)
0.515389 + 0.856957i \(0.327648\pi\)
\(648\) 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 \(-0.450347\pi\)
0.155358 + 0.987858i \(0.450347\pi\)
\(654\) −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 \(-0.616345\pi\)
−0.357425 + 0.933942i \(0.616345\pi\)
\(674\) −14693.3 −0.839712
\(675\) −10316.8 −0.588286
\(676\) −15285.6 −0.869685
\(677\) 30422.7 1.72709 0.863546 0.504270i \(-0.168238\pi\)
0.863546 + 0.504270i \(0.168238\pi\)
\(678\) 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 \(-0.626321\pi\)
−0.386516 + 0.922283i \(0.626321\pi\)
\(692\) 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 \(-0.0731767\pi\)
0.973691 + 0.227872i \(0.0731767\pi\)
\(728\) −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 \(-0.797106\pi\)
−0.803639 + 0.595117i \(0.797106\pi\)
\(734\) −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 \(-0.469643\pi\)
0.0952245 + 0.995456i \(0.469643\pi\)
\(740\) 35356.7 1.75640
\(741\) 1263.94 0.0626613
\(742\) 13048.0 0.645560
\(743\) 17425.8 0.860418 0.430209 0.902729i \(-0.358440\pi\)
0.430209 + 0.902729i \(0.358440\pi\)
\(744\) −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 \(-0.377948\pi\)
0.374112 + 0.927384i \(0.377948\pi\)
\(752\) −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 \(-0.621168\pi\)
−0.371532 + 0.928420i \(0.621168\pi\)
\(758\) −6902.16 −0.330736
\(759\) 13853.0 0.662494
\(760\) −3088.01 −0.147386
\(761\) −19666.4 −0.936802 −0.468401 0.883516i \(-0.655170\pi\)
−0.468401 + 0.883516i \(0.655170\pi\)
\(762\) 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 \(-0.660534\pi\)
−0.483222 + 0.875498i \(0.660534\pi\)
\(774\) 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 \(-0.556552\pi\)
−0.176729 + 0.984259i \(0.556552\pi\)
\(812\) −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 \(-0.616138\pi\)
−0.356818 + 0.934174i \(0.616138\pi\)
\(830\) 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 \(-0.452030\pi\)
0.150131 + 0.988666i \(0.452030\pi\)
\(840\) 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 \(-0.735842\pi\)
−0.674966 + 0.737849i \(0.735842\pi\)
\(860\) 0 0
\(861\) −7135.32 −0.282429
\(862\) −66437.6 −2.62514
\(863\) 10162.6 0.400855 0.200427 0.979709i \(-0.435767\pi\)
0.200427 + 0.979709i \(0.435767\pi\)
\(864\) −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 \(-0.740757\pi\)
−0.686279 + 0.727339i \(0.740757\pi\)
\(888\) 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 \(-0.656983\pi\)
−0.473426 + 0.880834i \(0.656983\pi\)
\(912\) −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 \(-0.658474\pi\)
−0.477546 + 0.878607i \(0.658474\pi\)
\(930\) 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 \(-0.713255\pi\)
−0.620956 + 0.783845i \(0.713255\pi\)
\(938\) −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 \(-0.726550\pi\)
−0.653143 + 0.757235i \(0.726550\pi\)
\(954\) −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 \(-0.532238\pi\)
−0.101105 + 0.994876i \(0.532238\pi\)
\(984\) −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 \(-0.271757\pi\)
0.657161 + 0.753751i \(0.271757\pi\)
\(992\) −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 \(-0.479049\pi\)
0.0657712 + 0.997835i \(0.479049\pi\)
\(998\) −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.i.1.48 50
43.42 odd 2 1849.4.a.j.1.3 yes 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.48 50 1.1 even 1 trivial
1849.4.a.j.1.3 yes 50 43.42 odd 2