Properties

Label 2070.4.a.bj.1.1
Level $2070$
Weight $4$
Character 2070.1
Self dual yes
Analytic conductor $122.134$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(122.133953712\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \( x^{4} - 68x^{2} - 111x + 342 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-6.57209\) of defining polynomial
Character \(\chi\) \(=\) 2070.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} -35.4229 q^{7} +8.00000 q^{8} +O(q^{10})\) \(q+2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} -35.4229 q^{7} +8.00000 q^{8} +10.0000 q^{10} +16.6298 q^{11} -79.9132 q^{13} -70.8458 q^{14} +16.0000 q^{16} +46.8219 q^{17} -110.653 q^{19} +20.0000 q^{20} +33.2597 q^{22} +23.0000 q^{23} +25.0000 q^{25} -159.826 q^{26} -141.692 q^{28} +0.836422 q^{29} -119.836 q^{31} +32.0000 q^{32} +93.6438 q^{34} -177.114 q^{35} +368.201 q^{37} -221.307 q^{38} +40.0000 q^{40} +95.7927 q^{41} +331.961 q^{43} +66.5193 q^{44} +46.0000 q^{46} +535.037 q^{47} +911.782 q^{49} +50.0000 q^{50} -319.653 q^{52} -409.345 q^{53} +83.1492 q^{55} -283.383 q^{56} +1.67284 q^{58} +352.950 q^{59} -507.223 q^{61} -239.672 q^{62} +64.0000 q^{64} -399.566 q^{65} -820.056 q^{67} +187.288 q^{68} -354.229 q^{70} +733.770 q^{71} -91.4599 q^{73} +736.402 q^{74} -442.613 q^{76} -589.077 q^{77} +329.381 q^{79} +80.0000 q^{80} +191.585 q^{82} +753.834 q^{83} +234.110 q^{85} +663.922 q^{86} +133.039 q^{88} +1050.14 q^{89} +2830.76 q^{91} +92.0000 q^{92} +1070.07 q^{94} -553.267 q^{95} -271.928 q^{97} +1823.56 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{2} + 16 q^{4} + 20 q^{5} - q^{7} + 32 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{2} + 16 q^{4} + 20 q^{5} - q^{7} + 32 q^{8} + 40 q^{10} + 39 q^{11} - 20 q^{13} - 2 q^{14} + 64 q^{16} + 23 q^{17} + 53 q^{19} + 80 q^{20} + 78 q^{22} + 92 q^{23} + 100 q^{25} - 40 q^{26} - 4 q^{28} - 161 q^{29} + 388 q^{31} + 128 q^{32} + 46 q^{34} - 5 q^{35} + 466 q^{37} + 106 q^{38} + 160 q^{40} - 484 q^{41} + 894 q^{43} + 156 q^{44} + 184 q^{46} + 265 q^{47} + 1643 q^{49} + 200 q^{50} - 80 q^{52} - 576 q^{53} + 195 q^{55} - 8 q^{56} - 322 q^{58} + 94 q^{59} + 1153 q^{61} + 776 q^{62} + 256 q^{64} - 100 q^{65} - 1472 q^{67} + 92 q^{68} - 10 q^{70} - 200 q^{71} + 1147 q^{73} + 932 q^{74} + 212 q^{76} + 2176 q^{77} - 908 q^{79} + 320 q^{80} - 968 q^{82} + 1048 q^{83} + 115 q^{85} + 1788 q^{86} + 312 q^{88} + 1784 q^{89} + 2329 q^{91} + 368 q^{92} + 530 q^{94} + 265 q^{95} - 2047 q^{97} + 3286 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 0.707107
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) −35.4229 −1.91266 −0.956328 0.292294i \(-0.905581\pi\)
−0.956328 + 0.292294i \(0.905581\pi\)
\(8\) 8.00000 0.353553
\(9\) 0 0
\(10\) 10.0000 0.316228
\(11\) 16.6298 0.455826 0.227913 0.973682i \(-0.426810\pi\)
0.227913 + 0.973682i \(0.426810\pi\)
\(12\) 0 0
\(13\) −79.9132 −1.70492 −0.852459 0.522795i \(-0.824889\pi\)
−0.852459 + 0.522795i \(0.824889\pi\)
\(14\) −70.8458 −1.35245
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 46.8219 0.667999 0.333999 0.942573i \(-0.391602\pi\)
0.333999 + 0.942573i \(0.391602\pi\)
\(18\) 0 0
\(19\) −110.653 −1.33608 −0.668042 0.744123i \(-0.732868\pi\)
−0.668042 + 0.744123i \(0.732868\pi\)
\(20\) 20.0000 0.223607
\(21\) 0 0
\(22\) 33.2597 0.322318
\(23\) 23.0000 0.208514
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) −159.826 −1.20556
\(27\) 0 0
\(28\) −141.692 −0.956328
\(29\) 0.836422 0.00535585 0.00267793 0.999996i \(-0.499148\pi\)
0.00267793 + 0.999996i \(0.499148\pi\)
\(30\) 0 0
\(31\) −119.836 −0.694295 −0.347147 0.937811i \(-0.612850\pi\)
−0.347147 + 0.937811i \(0.612850\pi\)
\(32\) 32.0000 0.176777
\(33\) 0 0
\(34\) 93.6438 0.472346
\(35\) −177.114 −0.855366
\(36\) 0 0
\(37\) 368.201 1.63600 0.817998 0.575221i \(-0.195084\pi\)
0.817998 + 0.575221i \(0.195084\pi\)
\(38\) −221.307 −0.944755
\(39\) 0 0
\(40\) 40.0000 0.158114
\(41\) 95.7927 0.364886 0.182443 0.983216i \(-0.441600\pi\)
0.182443 + 0.983216i \(0.441600\pi\)
\(42\) 0 0
\(43\) 331.961 1.17729 0.588647 0.808390i \(-0.299661\pi\)
0.588647 + 0.808390i \(0.299661\pi\)
\(44\) 66.5193 0.227913
\(45\) 0 0
\(46\) 46.0000 0.147442
\(47\) 535.037 1.66049 0.830246 0.557397i \(-0.188200\pi\)
0.830246 + 0.557397i \(0.188200\pi\)
\(48\) 0 0
\(49\) 911.782 2.65826
\(50\) 50.0000 0.141421
\(51\) 0 0
\(52\) −319.653 −0.852459
\(53\) −409.345 −1.06090 −0.530452 0.847715i \(-0.677978\pi\)
−0.530452 + 0.847715i \(0.677978\pi\)
\(54\) 0 0
\(55\) 83.1492 0.203852
\(56\) −283.383 −0.676226
\(57\) 0 0
\(58\) 1.67284 0.00378716
\(59\) 352.950 0.778817 0.389408 0.921065i \(-0.372679\pi\)
0.389408 + 0.921065i \(0.372679\pi\)
\(60\) 0 0
\(61\) −507.223 −1.06464 −0.532322 0.846542i \(-0.678680\pi\)
−0.532322 + 0.846542i \(0.678680\pi\)
\(62\) −239.672 −0.490941
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) −399.566 −0.762462
\(66\) 0 0
\(67\) −820.056 −1.49531 −0.747655 0.664087i \(-0.768820\pi\)
−0.747655 + 0.664087i \(0.768820\pi\)
\(68\) 187.288 0.333999
\(69\) 0 0
\(70\) −354.229 −0.604835
\(71\) 733.770 1.22651 0.613257 0.789884i \(-0.289859\pi\)
0.613257 + 0.789884i \(0.289859\pi\)
\(72\) 0 0
\(73\) −91.4599 −0.146638 −0.0733190 0.997309i \(-0.523359\pi\)
−0.0733190 + 0.997309i \(0.523359\pi\)
\(74\) 736.402 1.15682
\(75\) 0 0
\(76\) −442.613 −0.668042
\(77\) −589.077 −0.871838
\(78\) 0 0
\(79\) 329.381 0.469092 0.234546 0.972105i \(-0.424640\pi\)
0.234546 + 0.972105i \(0.424640\pi\)
\(80\) 80.0000 0.111803
\(81\) 0 0
\(82\) 191.585 0.258013
\(83\) 753.834 0.996916 0.498458 0.866914i \(-0.333900\pi\)
0.498458 + 0.866914i \(0.333900\pi\)
\(84\) 0 0
\(85\) 234.110 0.298738
\(86\) 663.922 0.832472
\(87\) 0 0
\(88\) 133.039 0.161159
\(89\) 1050.14 1.25073 0.625365 0.780332i \(-0.284950\pi\)
0.625365 + 0.780332i \(0.284950\pi\)
\(90\) 0 0
\(91\) 2830.76 3.26092
\(92\) 92.0000 0.104257
\(93\) 0 0
\(94\) 1070.07 1.17415
\(95\) −553.267 −0.597515
\(96\) 0 0
\(97\) −271.928 −0.284641 −0.142320 0.989821i \(-0.545456\pi\)
−0.142320 + 0.989821i \(0.545456\pi\)
\(98\) 1823.56 1.87967
\(99\) 0 0
\(100\) 100.000 0.100000
\(101\) −1658.68 −1.63410 −0.817052 0.576563i \(-0.804393\pi\)
−0.817052 + 0.576563i \(0.804393\pi\)
\(102\) 0 0
\(103\) 1735.52 1.66025 0.830123 0.557580i \(-0.188270\pi\)
0.830123 + 0.557580i \(0.188270\pi\)
\(104\) −639.305 −0.602779
\(105\) 0 0
\(106\) −818.691 −0.750172
\(107\) 1629.88 1.47258 0.736292 0.676664i \(-0.236575\pi\)
0.736292 + 0.676664i \(0.236575\pi\)
\(108\) 0 0
\(109\) −432.151 −0.379748 −0.189874 0.981808i \(-0.560808\pi\)
−0.189874 + 0.981808i \(0.560808\pi\)
\(110\) 166.298 0.144145
\(111\) 0 0
\(112\) −566.766 −0.478164
\(113\) 240.115 0.199895 0.0999476 0.994993i \(-0.468132\pi\)
0.0999476 + 0.994993i \(0.468132\pi\)
\(114\) 0 0
\(115\) 115.000 0.0932505
\(116\) 3.34569 0.00267793
\(117\) 0 0
\(118\) 705.900 0.550707
\(119\) −1658.57 −1.27765
\(120\) 0 0
\(121\) −1054.45 −0.792223
\(122\) −1014.45 −0.752817
\(123\) 0 0
\(124\) −479.343 −0.347147
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) −210.993 −0.147422 −0.0737110 0.997280i \(-0.523484\pi\)
−0.0737110 + 0.997280i \(0.523484\pi\)
\(128\) 128.000 0.0883883
\(129\) 0 0
\(130\) −799.132 −0.539142
\(131\) 746.178 0.497663 0.248832 0.968547i \(-0.419953\pi\)
0.248832 + 0.968547i \(0.419953\pi\)
\(132\) 0 0
\(133\) 3919.66 2.55547
\(134\) −1640.11 −1.05734
\(135\) 0 0
\(136\) 374.575 0.236173
\(137\) 2420.07 1.50920 0.754601 0.656184i \(-0.227830\pi\)
0.754601 + 0.656184i \(0.227830\pi\)
\(138\) 0 0
\(139\) 924.030 0.563850 0.281925 0.959436i \(-0.409027\pi\)
0.281925 + 0.959436i \(0.409027\pi\)
\(140\) −708.458 −0.427683
\(141\) 0 0
\(142\) 1467.54 0.867276
\(143\) −1328.94 −0.777145
\(144\) 0 0
\(145\) 4.18211 0.00239521
\(146\) −182.920 −0.103689
\(147\) 0 0
\(148\) 1472.80 0.817998
\(149\) −430.614 −0.236760 −0.118380 0.992968i \(-0.537770\pi\)
−0.118380 + 0.992968i \(0.537770\pi\)
\(150\) 0 0
\(151\) −25.4118 −0.0136953 −0.00684763 0.999977i \(-0.502180\pi\)
−0.00684763 + 0.999977i \(0.502180\pi\)
\(152\) −885.226 −0.472377
\(153\) 0 0
\(154\) −1178.15 −0.616483
\(155\) −599.179 −0.310498
\(156\) 0 0
\(157\) −1580.29 −0.803317 −0.401658 0.915790i \(-0.631566\pi\)
−0.401658 + 0.915790i \(0.631566\pi\)
\(158\) 658.762 0.331698
\(159\) 0 0
\(160\) 160.000 0.0790569
\(161\) −814.727 −0.398817
\(162\) 0 0
\(163\) 458.739 0.220437 0.110218 0.993907i \(-0.464845\pi\)
0.110218 + 0.993907i \(0.464845\pi\)
\(164\) 383.171 0.182443
\(165\) 0 0
\(166\) 1507.67 0.704926
\(167\) 561.275 0.260076 0.130038 0.991509i \(-0.458490\pi\)
0.130038 + 0.991509i \(0.458490\pi\)
\(168\) 0 0
\(169\) 4189.11 1.90674
\(170\) 468.219 0.211240
\(171\) 0 0
\(172\) 1327.84 0.588647
\(173\) −1573.95 −0.691705 −0.345852 0.938289i \(-0.612410\pi\)
−0.345852 + 0.938289i \(0.612410\pi\)
\(174\) 0 0
\(175\) −885.572 −0.382531
\(176\) 266.077 0.113956
\(177\) 0 0
\(178\) 2100.29 0.884400
\(179\) −2215.59 −0.925145 −0.462572 0.886581i \(-0.653073\pi\)
−0.462572 + 0.886581i \(0.653073\pi\)
\(180\) 0 0
\(181\) 873.497 0.358710 0.179355 0.983784i \(-0.442599\pi\)
0.179355 + 0.983784i \(0.442599\pi\)
\(182\) 5661.51 2.30582
\(183\) 0 0
\(184\) 184.000 0.0737210
\(185\) 1841.00 0.731640
\(186\) 0 0
\(187\) 778.641 0.304491
\(188\) 2140.15 0.830246
\(189\) 0 0
\(190\) −1106.53 −0.422507
\(191\) 2497.74 0.946230 0.473115 0.881001i \(-0.343130\pi\)
0.473115 + 0.881001i \(0.343130\pi\)
\(192\) 0 0
\(193\) 909.155 0.339080 0.169540 0.985523i \(-0.445772\pi\)
0.169540 + 0.985523i \(0.445772\pi\)
\(194\) −543.857 −0.201271
\(195\) 0 0
\(196\) 3647.13 1.32913
\(197\) −608.627 −0.220116 −0.110058 0.993925i \(-0.535104\pi\)
−0.110058 + 0.993925i \(0.535104\pi\)
\(198\) 0 0
\(199\) −2304.98 −0.821083 −0.410542 0.911842i \(-0.634660\pi\)
−0.410542 + 0.911842i \(0.634660\pi\)
\(200\) 200.000 0.0707107
\(201\) 0 0
\(202\) −3317.36 −1.15549
\(203\) −29.6285 −0.0102439
\(204\) 0 0
\(205\) 478.963 0.163182
\(206\) 3471.03 1.17397
\(207\) 0 0
\(208\) −1278.61 −0.426229
\(209\) −1840.15 −0.609022
\(210\) 0 0
\(211\) −4373.37 −1.42690 −0.713448 0.700708i \(-0.752868\pi\)
−0.713448 + 0.700708i \(0.752868\pi\)
\(212\) −1637.38 −0.530452
\(213\) 0 0
\(214\) 3259.76 1.04127
\(215\) 1659.81 0.526502
\(216\) 0 0
\(217\) 4244.93 1.32795
\(218\) −864.303 −0.268523
\(219\) 0 0
\(220\) 332.597 0.101926
\(221\) −3741.69 −1.13888
\(222\) 0 0
\(223\) 5439.50 1.63343 0.816717 0.577038i \(-0.195792\pi\)
0.816717 + 0.577038i \(0.195792\pi\)
\(224\) −1133.53 −0.338113
\(225\) 0 0
\(226\) 480.231 0.141347
\(227\) −216.498 −0.0633017 −0.0316508 0.999499i \(-0.510076\pi\)
−0.0316508 + 0.999499i \(0.510076\pi\)
\(228\) 0 0
\(229\) −924.664 −0.266828 −0.133414 0.991060i \(-0.542594\pi\)
−0.133414 + 0.991060i \(0.542594\pi\)
\(230\) 230.000 0.0659380
\(231\) 0 0
\(232\) 6.69138 0.00189358
\(233\) 2369.69 0.666280 0.333140 0.942877i \(-0.391892\pi\)
0.333140 + 0.942877i \(0.391892\pi\)
\(234\) 0 0
\(235\) 2675.18 0.742595
\(236\) 1411.80 0.389408
\(237\) 0 0
\(238\) −3317.14 −0.903437
\(239\) 2769.60 0.749583 0.374791 0.927109i \(-0.377714\pi\)
0.374791 + 0.927109i \(0.377714\pi\)
\(240\) 0 0
\(241\) −5334.10 −1.42572 −0.712862 0.701305i \(-0.752601\pi\)
−0.712862 + 0.701305i \(0.752601\pi\)
\(242\) −2108.90 −0.560186
\(243\) 0 0
\(244\) −2028.89 −0.532322
\(245\) 4558.91 1.18881
\(246\) 0 0
\(247\) 8842.66 2.27791
\(248\) −958.686 −0.245470
\(249\) 0 0
\(250\) 250.000 0.0632456
\(251\) −4677.13 −1.17617 −0.588084 0.808800i \(-0.700118\pi\)
−0.588084 + 0.808800i \(0.700118\pi\)
\(252\) 0 0
\(253\) 382.486 0.0950463
\(254\) −421.986 −0.104243
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) 2602.99 0.631789 0.315895 0.948794i \(-0.397695\pi\)
0.315895 + 0.948794i \(0.397695\pi\)
\(258\) 0 0
\(259\) −13042.7 −3.12910
\(260\) −1598.26 −0.381231
\(261\) 0 0
\(262\) 1492.36 0.351901
\(263\) 3411.30 0.799809 0.399904 0.916557i \(-0.369043\pi\)
0.399904 + 0.916557i \(0.369043\pi\)
\(264\) 0 0
\(265\) −2046.73 −0.474451
\(266\) 7839.32 1.80699
\(267\) 0 0
\(268\) −3280.22 −0.747655
\(269\) −4366.56 −0.989718 −0.494859 0.868973i \(-0.664780\pi\)
−0.494859 + 0.868973i \(0.664780\pi\)
\(270\) 0 0
\(271\) 5682.45 1.27374 0.636872 0.770970i \(-0.280228\pi\)
0.636872 + 0.770970i \(0.280228\pi\)
\(272\) 749.150 0.167000
\(273\) 0 0
\(274\) 4840.14 1.06717
\(275\) 415.746 0.0911652
\(276\) 0 0
\(277\) −1884.62 −0.408794 −0.204397 0.978888i \(-0.565523\pi\)
−0.204397 + 0.978888i \(0.565523\pi\)
\(278\) 1848.06 0.398702
\(279\) 0 0
\(280\) −1416.92 −0.302418
\(281\) 2706.42 0.574561 0.287281 0.957846i \(-0.407249\pi\)
0.287281 + 0.957846i \(0.407249\pi\)
\(282\) 0 0
\(283\) −5963.64 −1.25266 −0.626328 0.779560i \(-0.715443\pi\)
−0.626328 + 0.779560i \(0.715443\pi\)
\(284\) 2935.08 0.613257
\(285\) 0 0
\(286\) −2657.89 −0.549525
\(287\) −3393.26 −0.697901
\(288\) 0 0
\(289\) −2720.71 −0.553778
\(290\) 8.36422 0.00169367
\(291\) 0 0
\(292\) −365.840 −0.0733190
\(293\) 4735.85 0.944272 0.472136 0.881526i \(-0.343483\pi\)
0.472136 + 0.881526i \(0.343483\pi\)
\(294\) 0 0
\(295\) 1764.75 0.348297
\(296\) 2945.61 0.578412
\(297\) 0 0
\(298\) −861.227 −0.167415
\(299\) −1838.00 −0.355500
\(300\) 0 0
\(301\) −11759.0 −2.25176
\(302\) −50.8236 −0.00968401
\(303\) 0 0
\(304\) −1770.45 −0.334021
\(305\) −2536.12 −0.476123
\(306\) 0 0
\(307\) −769.241 −0.143006 −0.0715031 0.997440i \(-0.522780\pi\)
−0.0715031 + 0.997440i \(0.522780\pi\)
\(308\) −2356.31 −0.435919
\(309\) 0 0
\(310\) −1198.36 −0.219555
\(311\) 5592.71 1.01972 0.509861 0.860257i \(-0.329697\pi\)
0.509861 + 0.860257i \(0.329697\pi\)
\(312\) 0 0
\(313\) 9777.19 1.76562 0.882811 0.469729i \(-0.155648\pi\)
0.882811 + 0.469729i \(0.155648\pi\)
\(314\) −3160.58 −0.568031
\(315\) 0 0
\(316\) 1317.52 0.234546
\(317\) −1868.51 −0.331060 −0.165530 0.986205i \(-0.552934\pi\)
−0.165530 + 0.986205i \(0.552934\pi\)
\(318\) 0 0
\(319\) 13.9096 0.00244134
\(320\) 320.000 0.0559017
\(321\) 0 0
\(322\) −1629.45 −0.282006
\(323\) −5181.00 −0.892503
\(324\) 0 0
\(325\) −1997.83 −0.340983
\(326\) 917.477 0.155872
\(327\) 0 0
\(328\) 766.342 0.129007
\(329\) −18952.6 −3.17595
\(330\) 0 0
\(331\) 6966.91 1.15691 0.578454 0.815715i \(-0.303656\pi\)
0.578454 + 0.815715i \(0.303656\pi\)
\(332\) 3015.34 0.498458
\(333\) 0 0
\(334\) 1122.55 0.183902
\(335\) −4100.28 −0.668723
\(336\) 0 0
\(337\) −17.5487 −0.00283661 −0.00141830 0.999999i \(-0.500451\pi\)
−0.00141830 + 0.999999i \(0.500451\pi\)
\(338\) 8378.23 1.34827
\(339\) 0 0
\(340\) 936.438 0.149369
\(341\) −1992.85 −0.316477
\(342\) 0 0
\(343\) −20147.9 −3.17167
\(344\) 2655.69 0.416236
\(345\) 0 0
\(346\) −3147.89 −0.489109
\(347\) 9859.30 1.52529 0.762644 0.646818i \(-0.223901\pi\)
0.762644 + 0.646818i \(0.223901\pi\)
\(348\) 0 0
\(349\) 7200.25 1.10436 0.552178 0.833726i \(-0.313797\pi\)
0.552178 + 0.833726i \(0.313797\pi\)
\(350\) −1771.14 −0.270491
\(351\) 0 0
\(352\) 532.155 0.0805794
\(353\) 6054.58 0.912897 0.456449 0.889750i \(-0.349121\pi\)
0.456449 + 0.889750i \(0.349121\pi\)
\(354\) 0 0
\(355\) 3668.85 0.548513
\(356\) 4200.57 0.625365
\(357\) 0 0
\(358\) −4431.18 −0.654176
\(359\) −2734.26 −0.401974 −0.200987 0.979594i \(-0.564415\pi\)
−0.200987 + 0.979594i \(0.564415\pi\)
\(360\) 0 0
\(361\) 5385.16 0.785123
\(362\) 1746.99 0.253646
\(363\) 0 0
\(364\) 11323.0 1.63046
\(365\) −457.300 −0.0655785
\(366\) 0 0
\(367\) −2465.50 −0.350676 −0.175338 0.984508i \(-0.556102\pi\)
−0.175338 + 0.984508i \(0.556102\pi\)
\(368\) 368.000 0.0521286
\(369\) 0 0
\(370\) 3682.01 0.517347
\(371\) 14500.2 2.02915
\(372\) 0 0
\(373\) 5704.45 0.791863 0.395932 0.918280i \(-0.370422\pi\)
0.395932 + 0.918280i \(0.370422\pi\)
\(374\) 1557.28 0.215308
\(375\) 0 0
\(376\) 4280.29 0.587073
\(377\) −66.8411 −0.00913128
\(378\) 0 0
\(379\) 10231.9 1.38675 0.693376 0.720576i \(-0.256122\pi\)
0.693376 + 0.720576i \(0.256122\pi\)
\(380\) −2213.07 −0.298758
\(381\) 0 0
\(382\) 4995.47 0.669086
\(383\) −6321.34 −0.843356 −0.421678 0.906746i \(-0.638559\pi\)
−0.421678 + 0.906746i \(0.638559\pi\)
\(384\) 0 0
\(385\) −2945.38 −0.389898
\(386\) 1818.31 0.239766
\(387\) 0 0
\(388\) −1087.71 −0.142320
\(389\) 925.594 0.120641 0.0603207 0.998179i \(-0.480788\pi\)
0.0603207 + 0.998179i \(0.480788\pi\)
\(390\) 0 0
\(391\) 1076.90 0.139287
\(392\) 7294.25 0.939835
\(393\) 0 0
\(394\) −1217.25 −0.155646
\(395\) 1646.91 0.209784
\(396\) 0 0
\(397\) −2706.75 −0.342187 −0.171093 0.985255i \(-0.554730\pi\)
−0.171093 + 0.985255i \(0.554730\pi\)
\(398\) −4609.96 −0.580594
\(399\) 0 0
\(400\) 400.000 0.0500000
\(401\) 14095.3 1.75532 0.877661 0.479281i \(-0.159103\pi\)
0.877661 + 0.479281i \(0.159103\pi\)
\(402\) 0 0
\(403\) 9576.45 1.18372
\(404\) −6634.71 −0.817052
\(405\) 0 0
\(406\) −59.2570 −0.00724353
\(407\) 6123.12 0.745729
\(408\) 0 0
\(409\) −10846.0 −1.31125 −0.655625 0.755087i \(-0.727595\pi\)
−0.655625 + 0.755087i \(0.727595\pi\)
\(410\) 957.927 0.115387
\(411\) 0 0
\(412\) 6942.06 0.830123
\(413\) −12502.5 −1.48961
\(414\) 0 0
\(415\) 3769.17 0.445834
\(416\) −2557.22 −0.301390
\(417\) 0 0
\(418\) −3680.29 −0.430644
\(419\) 5626.55 0.656026 0.328013 0.944673i \(-0.393621\pi\)
0.328013 + 0.944673i \(0.393621\pi\)
\(420\) 0 0
\(421\) −7109.09 −0.822983 −0.411491 0.911414i \(-0.634992\pi\)
−0.411491 + 0.911414i \(0.634992\pi\)
\(422\) −8746.74 −1.00897
\(423\) 0 0
\(424\) −3274.76 −0.375086
\(425\) 1170.55 0.133600
\(426\) 0 0
\(427\) 17967.3 2.03630
\(428\) 6519.52 0.736292
\(429\) 0 0
\(430\) 3319.61 0.372293
\(431\) −4464.14 −0.498909 −0.249455 0.968387i \(-0.580251\pi\)
−0.249455 + 0.968387i \(0.580251\pi\)
\(432\) 0 0
\(433\) 11009.4 1.22189 0.610947 0.791672i \(-0.290789\pi\)
0.610947 + 0.791672i \(0.290789\pi\)
\(434\) 8489.86 0.939001
\(435\) 0 0
\(436\) −1728.61 −0.189874
\(437\) −2545.03 −0.278593
\(438\) 0 0
\(439\) −3052.70 −0.331885 −0.165943 0.986135i \(-0.553067\pi\)
−0.165943 + 0.986135i \(0.553067\pi\)
\(440\) 665.193 0.0720724
\(441\) 0 0
\(442\) −7483.37 −0.805312
\(443\) 4465.82 0.478956 0.239478 0.970902i \(-0.423024\pi\)
0.239478 + 0.970902i \(0.423024\pi\)
\(444\) 0 0
\(445\) 5250.72 0.559343
\(446\) 10879.0 1.15501
\(447\) 0 0
\(448\) −2267.07 −0.239082
\(449\) −9040.14 −0.950179 −0.475090 0.879937i \(-0.657584\pi\)
−0.475090 + 0.879937i \(0.657584\pi\)
\(450\) 0 0
\(451\) 1593.02 0.166324
\(452\) 960.462 0.0999476
\(453\) 0 0
\(454\) −432.996 −0.0447610
\(455\) 14153.8 1.45833
\(456\) 0 0
\(457\) 7756.88 0.793986 0.396993 0.917822i \(-0.370054\pi\)
0.396993 + 0.917822i \(0.370054\pi\)
\(458\) −1849.33 −0.188676
\(459\) 0 0
\(460\) 460.000 0.0466252
\(461\) −7064.19 −0.713692 −0.356846 0.934163i \(-0.616148\pi\)
−0.356846 + 0.934163i \(0.616148\pi\)
\(462\) 0 0
\(463\) 12599.7 1.26470 0.632350 0.774682i \(-0.282090\pi\)
0.632350 + 0.774682i \(0.282090\pi\)
\(464\) 13.3828 0.00133896
\(465\) 0 0
\(466\) 4739.37 0.471131
\(467\) −14746.0 −1.46117 −0.730583 0.682824i \(-0.760751\pi\)
−0.730583 + 0.682824i \(0.760751\pi\)
\(468\) 0 0
\(469\) 29048.8 2.86002
\(470\) 5350.37 0.525094
\(471\) 0 0
\(472\) 2823.60 0.275353
\(473\) 5520.46 0.536641
\(474\) 0 0
\(475\) −2766.33 −0.267217
\(476\) −6634.27 −0.638826
\(477\) 0 0
\(478\) 5539.19 0.530035
\(479\) 17411.2 1.66084 0.830418 0.557141i \(-0.188102\pi\)
0.830418 + 0.557141i \(0.188102\pi\)
\(480\) 0 0
\(481\) −29424.1 −2.78924
\(482\) −10668.2 −1.00814
\(483\) 0 0
\(484\) −4217.79 −0.396111
\(485\) −1359.64 −0.127295
\(486\) 0 0
\(487\) 1320.34 0.122855 0.0614276 0.998112i \(-0.480435\pi\)
0.0614276 + 0.998112i \(0.480435\pi\)
\(488\) −4057.78 −0.376408
\(489\) 0 0
\(490\) 9117.82 0.840614
\(491\) −17115.8 −1.57317 −0.786583 0.617485i \(-0.788152\pi\)
−0.786583 + 0.617485i \(0.788152\pi\)
\(492\) 0 0
\(493\) 39.1629 0.00357770
\(494\) 17685.3 1.61073
\(495\) 0 0
\(496\) −1917.37 −0.173574
\(497\) −25992.3 −2.34590
\(498\) 0 0
\(499\) 17540.6 1.57360 0.786798 0.617210i \(-0.211737\pi\)
0.786798 + 0.617210i \(0.211737\pi\)
\(500\) 500.000 0.0447214
\(501\) 0 0
\(502\) −9354.27 −0.831676
\(503\) 4934.98 0.437455 0.218727 0.975786i \(-0.429809\pi\)
0.218727 + 0.975786i \(0.429809\pi\)
\(504\) 0 0
\(505\) −8293.39 −0.730794
\(506\) 764.972 0.0672079
\(507\) 0 0
\(508\) −843.972 −0.0737110
\(509\) −11927.3 −1.03865 −0.519323 0.854578i \(-0.673816\pi\)
−0.519323 + 0.854578i \(0.673816\pi\)
\(510\) 0 0
\(511\) 3239.78 0.280468
\(512\) 512.000 0.0441942
\(513\) 0 0
\(514\) 5205.97 0.446743
\(515\) 8677.58 0.742485
\(516\) 0 0
\(517\) 8897.57 0.756895
\(518\) −26085.5 −2.21261
\(519\) 0 0
\(520\) −3196.53 −0.269571
\(521\) 9592.76 0.806653 0.403327 0.915056i \(-0.367854\pi\)
0.403327 + 0.915056i \(0.367854\pi\)
\(522\) 0 0
\(523\) 6525.20 0.545558 0.272779 0.962077i \(-0.412057\pi\)
0.272779 + 0.962077i \(0.412057\pi\)
\(524\) 2984.71 0.248832
\(525\) 0 0
\(526\) 6822.60 0.565550
\(527\) −5610.94 −0.463788
\(528\) 0 0
\(529\) 529.000 0.0434783
\(530\) −4093.45 −0.335487
\(531\) 0 0
\(532\) 15678.6 1.27774
\(533\) −7655.10 −0.622100
\(534\) 0 0
\(535\) 8149.40 0.658560
\(536\) −6560.45 −0.528672
\(537\) 0 0
\(538\) −8733.13 −0.699836
\(539\) 15162.8 1.21170
\(540\) 0 0
\(541\) 5634.05 0.447739 0.223870 0.974619i \(-0.428131\pi\)
0.223870 + 0.974619i \(0.428131\pi\)
\(542\) 11364.9 0.900673
\(543\) 0 0
\(544\) 1498.30 0.118087
\(545\) −2160.76 −0.169829
\(546\) 0 0
\(547\) 6093.08 0.476273 0.238136 0.971232i \(-0.423463\pi\)
0.238136 + 0.971232i \(0.423463\pi\)
\(548\) 9680.29 0.754601
\(549\) 0 0
\(550\) 831.492 0.0644635
\(551\) −92.5529 −0.00715587
\(552\) 0 0
\(553\) −11667.6 −0.897212
\(554\) −3769.24 −0.289061
\(555\) 0 0
\(556\) 3696.12 0.281925
\(557\) −21461.9 −1.63262 −0.816311 0.577612i \(-0.803985\pi\)
−0.816311 + 0.577612i \(0.803985\pi\)
\(558\) 0 0
\(559\) −26528.1 −2.00719
\(560\) −2833.83 −0.213842
\(561\) 0 0
\(562\) 5412.85 0.406276
\(563\) 11038.5 0.826316 0.413158 0.910659i \(-0.364426\pi\)
0.413158 + 0.910659i \(0.364426\pi\)
\(564\) 0 0
\(565\) 1200.58 0.0893959
\(566\) −11927.3 −0.885761
\(567\) 0 0
\(568\) 5870.16 0.433638
\(569\) 19210.3 1.41535 0.707677 0.706536i \(-0.249743\pi\)
0.707677 + 0.706536i \(0.249743\pi\)
\(570\) 0 0
\(571\) −11967.0 −0.877066 −0.438533 0.898715i \(-0.644502\pi\)
−0.438533 + 0.898715i \(0.644502\pi\)
\(572\) −5315.77 −0.388573
\(573\) 0 0
\(574\) −6786.51 −0.493490
\(575\) 575.000 0.0417029
\(576\) 0 0
\(577\) −14982.3 −1.08097 −0.540487 0.841352i \(-0.681760\pi\)
−0.540487 + 0.841352i \(0.681760\pi\)
\(578\) −5441.42 −0.391580
\(579\) 0 0
\(580\) 16.7284 0.00119760
\(581\) −26703.0 −1.90676
\(582\) 0 0
\(583\) −6807.35 −0.483587
\(584\) −731.679 −0.0518444
\(585\) 0 0
\(586\) 9471.71 0.667701
\(587\) −24002.9 −1.68774 −0.843871 0.536547i \(-0.819729\pi\)
−0.843871 + 0.536547i \(0.819729\pi\)
\(588\) 0 0
\(589\) 13260.2 0.927637
\(590\) 3529.50 0.246283
\(591\) 0 0
\(592\) 5891.21 0.408999
\(593\) −5124.67 −0.354882 −0.177441 0.984131i \(-0.556782\pi\)
−0.177441 + 0.984131i \(0.556782\pi\)
\(594\) 0 0
\(595\) −8292.84 −0.571384
\(596\) −1722.45 −0.118380
\(597\) 0 0
\(598\) −3676.01 −0.251376
\(599\) 23776.8 1.62186 0.810928 0.585146i \(-0.198963\pi\)
0.810928 + 0.585146i \(0.198963\pi\)
\(600\) 0 0
\(601\) −25435.3 −1.72633 −0.863166 0.504920i \(-0.831522\pi\)
−0.863166 + 0.504920i \(0.831522\pi\)
\(602\) −23518.1 −1.59223
\(603\) 0 0
\(604\) −101.647 −0.00684763
\(605\) −5272.24 −0.354293
\(606\) 0 0
\(607\) −7445.94 −0.497894 −0.248947 0.968517i \(-0.580084\pi\)
−0.248947 + 0.968517i \(0.580084\pi\)
\(608\) −3540.91 −0.236189
\(609\) 0 0
\(610\) −5072.23 −0.336670
\(611\) −42756.5 −2.83100
\(612\) 0 0
\(613\) −12874.4 −0.848275 −0.424138 0.905598i \(-0.639423\pi\)
−0.424138 + 0.905598i \(0.639423\pi\)
\(614\) −1538.48 −0.101121
\(615\) 0 0
\(616\) −4712.62 −0.308241
\(617\) 19247.2 1.25585 0.627927 0.778272i \(-0.283904\pi\)
0.627927 + 0.778272i \(0.283904\pi\)
\(618\) 0 0
\(619\) −14496.4 −0.941293 −0.470647 0.882322i \(-0.655979\pi\)
−0.470647 + 0.882322i \(0.655979\pi\)
\(620\) −2396.72 −0.155249
\(621\) 0 0
\(622\) 11185.4 0.721052
\(623\) −37199.1 −2.39222
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 19554.4 1.24848
\(627\) 0 0
\(628\) −6321.15 −0.401658
\(629\) 17239.9 1.09284
\(630\) 0 0
\(631\) −18030.9 −1.13756 −0.568779 0.822490i \(-0.692584\pi\)
−0.568779 + 0.822490i \(0.692584\pi\)
\(632\) 2635.05 0.165849
\(633\) 0 0
\(634\) −3737.03 −0.234095
\(635\) −1054.96 −0.0659291
\(636\) 0 0
\(637\) −72863.4 −4.53211
\(638\) 27.8191 0.00172628
\(639\) 0 0
\(640\) 640.000 0.0395285
\(641\) −11776.4 −0.725646 −0.362823 0.931858i \(-0.618187\pi\)
−0.362823 + 0.931858i \(0.618187\pi\)
\(642\) 0 0
\(643\) 20207.4 1.23935 0.619676 0.784858i \(-0.287264\pi\)
0.619676 + 0.784858i \(0.287264\pi\)
\(644\) −3258.91 −0.199408
\(645\) 0 0
\(646\) −10362.0 −0.631095
\(647\) −21452.8 −1.30355 −0.651776 0.758412i \(-0.725976\pi\)
−0.651776 + 0.758412i \(0.725976\pi\)
\(648\) 0 0
\(649\) 5869.50 0.355005
\(650\) −3995.66 −0.241112
\(651\) 0 0
\(652\) 1834.95 0.110218
\(653\) 15095.2 0.904624 0.452312 0.891860i \(-0.350599\pi\)
0.452312 + 0.891860i \(0.350599\pi\)
\(654\) 0 0
\(655\) 3730.89 0.222562
\(656\) 1532.68 0.0912214
\(657\) 0 0
\(658\) −37905.1 −2.24574
\(659\) −6964.68 −0.411693 −0.205846 0.978584i \(-0.565995\pi\)
−0.205846 + 0.978584i \(0.565995\pi\)
\(660\) 0 0
\(661\) −17380.4 −1.02272 −0.511361 0.859366i \(-0.670858\pi\)
−0.511361 + 0.859366i \(0.670858\pi\)
\(662\) 13933.8 0.818057
\(663\) 0 0
\(664\) 6030.67 0.352463
\(665\) 19598.3 1.14284
\(666\) 0 0
\(667\) 19.2377 0.00111677
\(668\) 2245.10 0.130038
\(669\) 0 0
\(670\) −8200.56 −0.472859
\(671\) −8435.04 −0.485292
\(672\) 0 0
\(673\) 12383.2 0.709271 0.354635 0.935005i \(-0.384605\pi\)
0.354635 + 0.935005i \(0.384605\pi\)
\(674\) −35.0973 −0.00200578
\(675\) 0 0
\(676\) 16756.5 0.953371
\(677\) 8375.74 0.475489 0.237744 0.971328i \(-0.423592\pi\)
0.237744 + 0.971328i \(0.423592\pi\)
\(678\) 0 0
\(679\) 9632.49 0.544420
\(680\) 1872.88 0.105620
\(681\) 0 0
\(682\) −3985.70 −0.223783
\(683\) 4434.66 0.248444 0.124222 0.992254i \(-0.460356\pi\)
0.124222 + 0.992254i \(0.460356\pi\)
\(684\) 0 0
\(685\) 12100.4 0.674936
\(686\) −40295.8 −2.24271
\(687\) 0 0
\(688\) 5311.38 0.294323
\(689\) 32712.1 1.80875
\(690\) 0 0
\(691\) 11032.8 0.607389 0.303694 0.952770i \(-0.401780\pi\)
0.303694 + 0.952770i \(0.401780\pi\)
\(692\) −6295.79 −0.345852
\(693\) 0 0
\(694\) 19718.6 1.07854
\(695\) 4620.15 0.252162
\(696\) 0 0
\(697\) 4485.20 0.243743
\(698\) 14400.5 0.780898
\(699\) 0 0
\(700\) −3542.29 −0.191266
\(701\) −3708.40 −0.199806 −0.0999032 0.994997i \(-0.531853\pi\)
−0.0999032 + 0.994997i \(0.531853\pi\)
\(702\) 0 0
\(703\) −40742.6 −2.18583
\(704\) 1064.31 0.0569782
\(705\) 0 0
\(706\) 12109.2 0.645516
\(707\) 58755.2 3.12548
\(708\) 0 0
\(709\) −6315.04 −0.334508 −0.167254 0.985914i \(-0.553490\pi\)
−0.167254 + 0.985914i \(0.553490\pi\)
\(710\) 7337.70 0.387858
\(711\) 0 0
\(712\) 8401.15 0.442200
\(713\) −2756.22 −0.144770
\(714\) 0 0
\(715\) −6644.71 −0.347550
\(716\) −8862.36 −0.462572
\(717\) 0 0
\(718\) −5468.52 −0.284239
\(719\) −24736.9 −1.28307 −0.641536 0.767093i \(-0.721703\pi\)
−0.641536 + 0.767093i \(0.721703\pi\)
\(720\) 0 0
\(721\) −61477.0 −3.17548
\(722\) 10770.3 0.555165
\(723\) 0 0
\(724\) 3493.99 0.179355
\(725\) 20.9106 0.00107117
\(726\) 0 0
\(727\) 33283.6 1.69797 0.848983 0.528420i \(-0.177215\pi\)
0.848983 + 0.528420i \(0.177215\pi\)
\(728\) 22646.0 1.15291
\(729\) 0 0
\(730\) −914.599 −0.0463710
\(731\) 15543.1 0.786430
\(732\) 0 0
\(733\) −33636.5 −1.69494 −0.847471 0.530841i \(-0.821876\pi\)
−0.847471 + 0.530841i \(0.821876\pi\)
\(734\) −4931.00 −0.247965
\(735\) 0 0
\(736\) 736.000 0.0368605
\(737\) −13637.4 −0.681601
\(738\) 0 0
\(739\) 36575.5 1.82064 0.910319 0.413907i \(-0.135836\pi\)
0.910319 + 0.413907i \(0.135836\pi\)
\(740\) 7364.02 0.365820
\(741\) 0 0
\(742\) 29000.4 1.43482
\(743\) 17704.7 0.874190 0.437095 0.899415i \(-0.356007\pi\)
0.437095 + 0.899415i \(0.356007\pi\)
\(744\) 0 0
\(745\) −2153.07 −0.105882
\(746\) 11408.9 0.559932
\(747\) 0 0
\(748\) 3114.56 0.152246
\(749\) −57735.1 −2.81655
\(750\) 0 0
\(751\) 8858.37 0.430421 0.215211 0.976568i \(-0.430956\pi\)
0.215211 + 0.976568i \(0.430956\pi\)
\(752\) 8560.59 0.415123
\(753\) 0 0
\(754\) −133.682 −0.00645679
\(755\) −127.059 −0.00612470
\(756\) 0 0
\(757\) 7899.26 0.379265 0.189633 0.981855i \(-0.439270\pi\)
0.189633 + 0.981855i \(0.439270\pi\)
\(758\) 20463.9 0.980582
\(759\) 0 0
\(760\) −4426.13 −0.211254
\(761\) −23437.4 −1.11643 −0.558216 0.829696i \(-0.688514\pi\)
−0.558216 + 0.829696i \(0.688514\pi\)
\(762\) 0 0
\(763\) 15308.1 0.726329
\(764\) 9990.95 0.473115
\(765\) 0 0
\(766\) −12642.7 −0.596343
\(767\) −28205.4 −1.32782
\(768\) 0 0
\(769\) 31447.7 1.47468 0.737342 0.675519i \(-0.236081\pi\)
0.737342 + 0.675519i \(0.236081\pi\)
\(770\) −5890.77 −0.275699
\(771\) 0 0
\(772\) 3636.62 0.169540
\(773\) −2397.42 −0.111551 −0.0557756 0.998443i \(-0.517763\pi\)
−0.0557756 + 0.998443i \(0.517763\pi\)
\(774\) 0 0
\(775\) −2995.89 −0.138859
\(776\) −2175.43 −0.100636
\(777\) 0 0
\(778\) 1851.19 0.0853063
\(779\) −10599.8 −0.487518
\(780\) 0 0
\(781\) 12202.5 0.559077
\(782\) 2153.81 0.0984911
\(783\) 0 0
\(784\) 14588.5 0.664564
\(785\) −7901.44 −0.359254
\(786\) 0 0
\(787\) −19407.0 −0.879014 −0.439507 0.898239i \(-0.644847\pi\)
−0.439507 + 0.898239i \(0.644847\pi\)
\(788\) −2434.51 −0.110058
\(789\) 0 0
\(790\) 3293.81 0.148340
\(791\) −8505.58 −0.382331
\(792\) 0 0
\(793\) 40533.8 1.81513
\(794\) −5413.51 −0.241962
\(795\) 0 0
\(796\) −9219.91 −0.410542
\(797\) 28631.1 1.27248 0.636240 0.771491i \(-0.280489\pi\)
0.636240 + 0.771491i \(0.280489\pi\)
\(798\) 0 0
\(799\) 25051.4 1.10921
\(800\) 800.000 0.0353553
\(801\) 0 0
\(802\) 28190.6 1.24120
\(803\) −1520.96 −0.0668414
\(804\) 0 0
\(805\) −4073.63 −0.178356
\(806\) 19152.9 0.837013
\(807\) 0 0
\(808\) −13269.4 −0.577743
\(809\) 9191.25 0.399440 0.199720 0.979853i \(-0.435997\pi\)
0.199720 + 0.979853i \(0.435997\pi\)
\(810\) 0 0
\(811\) −567.477 −0.0245706 −0.0122853 0.999925i \(-0.503911\pi\)
−0.0122853 + 0.999925i \(0.503911\pi\)
\(812\) −118.514 −0.00512195
\(813\) 0 0
\(814\) 12246.2 0.527310
\(815\) 2293.69 0.0985823
\(816\) 0 0
\(817\) −36732.6 −1.57296
\(818\) −21692.0 −0.927193
\(819\) 0 0
\(820\) 1915.85 0.0815909
\(821\) 6057.35 0.257494 0.128747 0.991677i \(-0.458904\pi\)
0.128747 + 0.991677i \(0.458904\pi\)
\(822\) 0 0
\(823\) −21327.0 −0.903294 −0.451647 0.892197i \(-0.649163\pi\)
−0.451647 + 0.892197i \(0.649163\pi\)
\(824\) 13884.1 0.586986
\(825\) 0 0
\(826\) −25005.0 −1.05331
\(827\) 27553.8 1.15857 0.579286 0.815124i \(-0.303332\pi\)
0.579286 + 0.815124i \(0.303332\pi\)
\(828\) 0 0
\(829\) 11181.6 0.468461 0.234231 0.972181i \(-0.424743\pi\)
0.234231 + 0.972181i \(0.424743\pi\)
\(830\) 7538.34 0.315253
\(831\) 0 0
\(832\) −5114.44 −0.213115
\(833\) 42691.4 1.77571
\(834\) 0 0
\(835\) 2806.37 0.116310
\(836\) −7360.59 −0.304511
\(837\) 0 0
\(838\) 11253.1 0.463881
\(839\) −9811.83 −0.403745 −0.201872 0.979412i \(-0.564703\pi\)
−0.201872 + 0.979412i \(0.564703\pi\)
\(840\) 0 0
\(841\) −24388.3 −0.999971
\(842\) −14218.2 −0.581937
\(843\) 0 0
\(844\) −17493.5 −0.713448
\(845\) 20945.6 0.852721
\(846\) 0 0
\(847\) 37351.6 1.51525
\(848\) −6549.53 −0.265226
\(849\) 0 0
\(850\) 2341.10 0.0944693
\(851\) 8468.62 0.341129
\(852\) 0 0
\(853\) −24030.8 −0.964594 −0.482297 0.876008i \(-0.660197\pi\)
−0.482297 + 0.876008i \(0.660197\pi\)
\(854\) 35934.6 1.43988
\(855\) 0 0
\(856\) 13039.0 0.520637
\(857\) 16934.8 0.675008 0.337504 0.941324i \(-0.390417\pi\)
0.337504 + 0.941324i \(0.390417\pi\)
\(858\) 0 0
\(859\) −31057.0 −1.23359 −0.616793 0.787125i \(-0.711568\pi\)
−0.616793 + 0.787125i \(0.711568\pi\)
\(860\) 6639.22 0.263251
\(861\) 0 0
\(862\) −8928.27 −0.352782
\(863\) 16157.8 0.637332 0.318666 0.947867i \(-0.396765\pi\)
0.318666 + 0.947867i \(0.396765\pi\)
\(864\) 0 0
\(865\) −7869.73 −0.309340
\(866\) 22018.9 0.864009
\(867\) 0 0
\(868\) 16979.7 0.663974
\(869\) 5477.55 0.213824
\(870\) 0 0
\(871\) 65533.3 2.54938
\(872\) −3457.21 −0.134261
\(873\) 0 0
\(874\) −5090.05 −0.196995
\(875\) −4427.86 −0.171073
\(876\) 0 0
\(877\) −32473.9 −1.25036 −0.625180 0.780480i \(-0.714975\pi\)
−0.625180 + 0.780480i \(0.714975\pi\)
\(878\) −6105.41 −0.234678
\(879\) 0 0
\(880\) 1330.39 0.0509629
\(881\) 15703.2 0.600517 0.300258 0.953858i \(-0.402927\pi\)
0.300258 + 0.953858i \(0.402927\pi\)
\(882\) 0 0
\(883\) −14330.4 −0.546155 −0.273078 0.961992i \(-0.588042\pi\)
−0.273078 + 0.961992i \(0.588042\pi\)
\(884\) −14966.7 −0.569441
\(885\) 0 0
\(886\) 8931.65 0.338673
\(887\) −23238.2 −0.879664 −0.439832 0.898080i \(-0.644962\pi\)
−0.439832 + 0.898080i \(0.644962\pi\)
\(888\) 0 0
\(889\) 7473.98 0.281968
\(890\) 10501.4 0.395515
\(891\) 0 0
\(892\) 21758.0 0.816717
\(893\) −59203.6 −2.21856
\(894\) 0 0
\(895\) −11077.9 −0.413737
\(896\) −4534.13 −0.169057
\(897\) 0 0
\(898\) −18080.3 −0.671878
\(899\) −100.233 −0.00371854
\(900\) 0 0
\(901\) −19166.3 −0.708683
\(902\) 3186.03 0.117609
\(903\) 0 0
\(904\) 1920.92 0.0706736
\(905\) 4367.48 0.160420
\(906\) 0 0
\(907\) −3667.46 −0.134263 −0.0671313 0.997744i \(-0.521385\pi\)
−0.0671313 + 0.997744i \(0.521385\pi\)
\(908\) −865.992 −0.0316508
\(909\) 0 0
\(910\) 28307.6 1.03119
\(911\) −21457.9 −0.780385 −0.390192 0.920733i \(-0.627591\pi\)
−0.390192 + 0.920733i \(0.627591\pi\)
\(912\) 0 0
\(913\) 12536.1 0.454420
\(914\) 15513.8 0.561433
\(915\) 0 0
\(916\) −3698.66 −0.133414
\(917\) −26431.8 −0.951859
\(918\) 0 0
\(919\) −11109.3 −0.398762 −0.199381 0.979922i \(-0.563893\pi\)
−0.199381 + 0.979922i \(0.563893\pi\)
\(920\) 920.000 0.0329690
\(921\) 0 0
\(922\) −14128.4 −0.504657
\(923\) −58637.9 −2.09110
\(924\) 0 0
\(925\) 9205.02 0.327199
\(926\) 25199.3 0.894279
\(927\) 0 0
\(928\) 26.7655 0.000946790 0
\(929\) 20804.3 0.734732 0.367366 0.930076i \(-0.380260\pi\)
0.367366 + 0.930076i \(0.380260\pi\)
\(930\) 0 0
\(931\) −100892. −3.55166
\(932\) 9478.75 0.333140
\(933\) 0 0
\(934\) −29492.0 −1.03320
\(935\) 3893.20 0.136173
\(936\) 0 0
\(937\) −35550.5 −1.23947 −0.619735 0.784811i \(-0.712760\pi\)
−0.619735 + 0.784811i \(0.712760\pi\)
\(938\) 58097.5 2.02234
\(939\) 0 0
\(940\) 10700.7 0.371297
\(941\) −49674.4 −1.72087 −0.860436 0.509558i \(-0.829809\pi\)
−0.860436 + 0.509558i \(0.829809\pi\)
\(942\) 0 0
\(943\) 2203.23 0.0760839
\(944\) 5647.20 0.194704
\(945\) 0 0
\(946\) 11040.9 0.379462
\(947\) −33466.2 −1.14837 −0.574184 0.818726i \(-0.694681\pi\)
−0.574184 + 0.818726i \(0.694681\pi\)
\(948\) 0 0
\(949\) 7308.85 0.250006
\(950\) −5532.67 −0.188951
\(951\) 0 0
\(952\) −13268.5 −0.451718
\(953\) 17298.4 0.587984 0.293992 0.955808i \(-0.405016\pi\)
0.293992 + 0.955808i \(0.405016\pi\)
\(954\) 0 0
\(955\) 12488.7 0.423167
\(956\) 11078.4 0.374791
\(957\) 0 0
\(958\) 34822.5 1.17439
\(959\) −85726.0 −2.88659
\(960\) 0 0
\(961\) −15430.4 −0.517955
\(962\) −58848.2 −1.97229
\(963\) 0 0
\(964\) −21336.4 −0.712862
\(965\) 4545.78 0.151641
\(966\) 0 0
\(967\) −14869.7 −0.494495 −0.247248 0.968952i \(-0.579526\pi\)
−0.247248 + 0.968952i \(0.579526\pi\)
\(968\) −8435.59 −0.280093
\(969\) 0 0
\(970\) −2719.28 −0.0900113
\(971\) −26634.9 −0.880283 −0.440142 0.897928i \(-0.645072\pi\)
−0.440142 + 0.897928i \(0.645072\pi\)
\(972\) 0 0
\(973\) −32731.8 −1.07845
\(974\) 2640.69 0.0868717
\(975\) 0 0
\(976\) −8115.57 −0.266161
\(977\) −40803.6 −1.33615 −0.668077 0.744092i \(-0.732882\pi\)
−0.668077 + 0.744092i \(0.732882\pi\)
\(978\) 0 0
\(979\) 17463.7 0.570115
\(980\) 18235.6 0.594404
\(981\) 0 0
\(982\) −34231.6 −1.11240
\(983\) 18615.8 0.604021 0.302011 0.953305i \(-0.402342\pi\)
0.302011 + 0.953305i \(0.402342\pi\)
\(984\) 0 0
\(985\) −3043.14 −0.0984389
\(986\) 78.3257 0.00252982
\(987\) 0 0
\(988\) 35370.6 1.13896
\(989\) 7635.11 0.245483
\(990\) 0 0
\(991\) 46185.1 1.48044 0.740220 0.672364i \(-0.234721\pi\)
0.740220 + 0.672364i \(0.234721\pi\)
\(992\) −3834.74 −0.122735
\(993\) 0 0
\(994\) −51984.5 −1.65880
\(995\) −11524.9 −0.367200
\(996\) 0 0
\(997\) 16544.3 0.525541 0.262771 0.964858i \(-0.415364\pi\)
0.262771 + 0.964858i \(0.415364\pi\)
\(998\) 35081.2 1.11270
\(999\) 0 0
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 2070.4.a.bj.1.1 4
3.2 odd 2 230.4.a.h.1.1 4
12.11 even 2 1840.4.a.m.1.4 4
15.2 even 4 1150.4.b.n.599.4 8
15.8 even 4 1150.4.b.n.599.5 8
15.14 odd 2 1150.4.a.p.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.a.h.1.1 4 3.2 odd 2
1150.4.a.p.1.4 4 15.14 odd 2
1150.4.b.n.599.4 8 15.2 even 4
1150.4.b.n.599.5 8 15.8 even 4
1840.4.a.m.1.4 4 12.11 even 2
2070.4.a.bj.1.1 4 1.1 even 1 trivial