Properties

Label 147.4.a.c.1.1
Level $147$
Weight $4$
Character 147.1
Self dual yes
Analytic conductor $8.673$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 147.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.67328077084\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 147.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +18.0000 q^{5} -9.00000 q^{6} +21.0000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +18.0000 q^{5} -9.00000 q^{6} +21.0000 q^{8} +9.00000 q^{9} -54.0000 q^{10} -36.0000 q^{11} +3.00000 q^{12} +34.0000 q^{13} +54.0000 q^{15} -71.0000 q^{16} -42.0000 q^{17} -27.0000 q^{18} +124.000 q^{19} +18.0000 q^{20} +108.000 q^{22} +63.0000 q^{24} +199.000 q^{25} -102.000 q^{26} +27.0000 q^{27} +102.000 q^{29} -162.000 q^{30} +160.000 q^{31} +45.0000 q^{32} -108.000 q^{33} +126.000 q^{34} +9.00000 q^{36} +398.000 q^{37} -372.000 q^{38} +102.000 q^{39} +378.000 q^{40} +318.000 q^{41} -268.000 q^{43} -36.0000 q^{44} +162.000 q^{45} -240.000 q^{47} -213.000 q^{48} -597.000 q^{50} -126.000 q^{51} +34.0000 q^{52} -498.000 q^{53} -81.0000 q^{54} -648.000 q^{55} +372.000 q^{57} -306.000 q^{58} +132.000 q^{59} +54.0000 q^{60} -398.000 q^{61} -480.000 q^{62} +433.000 q^{64} +612.000 q^{65} +324.000 q^{66} +92.0000 q^{67} -42.0000 q^{68} -720.000 q^{71} +189.000 q^{72} +502.000 q^{73} -1194.00 q^{74} +597.000 q^{75} +124.000 q^{76} -306.000 q^{78} -1024.00 q^{79} -1278.00 q^{80} +81.0000 q^{81} -954.000 q^{82} +204.000 q^{83} -756.000 q^{85} +804.000 q^{86} +306.000 q^{87} -756.000 q^{88} -354.000 q^{89} -486.000 q^{90} +480.000 q^{93} +720.000 q^{94} +2232.00 q^{95} +135.000 q^{96} +286.000 q^{97} -324.000 q^{99} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.00000 −1.06066 −0.530330 0.847791i \(-0.677932\pi\)
−0.530330 + 0.847791i \(0.677932\pi\)
\(3\) 3.00000 0.577350
\(4\) 1.00000 0.125000
\(5\) 18.0000 1.60997 0.804984 0.593296i \(-0.202174\pi\)
0.804984 + 0.593296i \(0.202174\pi\)
\(6\) −9.00000 −0.612372
\(7\) 0 0
\(8\) 21.0000 0.928078
\(9\) 9.00000 0.333333
\(10\) −54.0000 −1.70763
\(11\) −36.0000 −0.986764 −0.493382 0.869813i \(-0.664240\pi\)
−0.493382 + 0.869813i \(0.664240\pi\)
\(12\) 3.00000 0.0721688
\(13\) 34.0000 0.725377 0.362689 0.931910i \(-0.381859\pi\)
0.362689 + 0.931910i \(0.381859\pi\)
\(14\) 0 0
\(15\) 54.0000 0.929516
\(16\) −71.0000 −1.10938
\(17\) −42.0000 −0.599206 −0.299603 0.954064i \(-0.596854\pi\)
−0.299603 + 0.954064i \(0.596854\pi\)
\(18\) −27.0000 −0.353553
\(19\) 124.000 1.49724 0.748620 0.663000i \(-0.230717\pi\)
0.748620 + 0.663000i \(0.230717\pi\)
\(20\) 18.0000 0.201246
\(21\) 0 0
\(22\) 108.000 1.04662
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 63.0000 0.535826
\(25\) 199.000 1.59200
\(26\) −102.000 −0.769379
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 102.000 0.653135 0.326568 0.945174i \(-0.394108\pi\)
0.326568 + 0.945174i \(0.394108\pi\)
\(30\) −162.000 −0.985901
\(31\) 160.000 0.926995 0.463498 0.886098i \(-0.346594\pi\)
0.463498 + 0.886098i \(0.346594\pi\)
\(32\) 45.0000 0.248592
\(33\) −108.000 −0.569709
\(34\) 126.000 0.635554
\(35\) 0 0
\(36\) 9.00000 0.0416667
\(37\) 398.000 1.76840 0.884200 0.467109i \(-0.154704\pi\)
0.884200 + 0.467109i \(0.154704\pi\)
\(38\) −372.000 −1.58806
\(39\) 102.000 0.418797
\(40\) 378.000 1.49418
\(41\) 318.000 1.21130 0.605649 0.795732i \(-0.292913\pi\)
0.605649 + 0.795732i \(0.292913\pi\)
\(42\) 0 0
\(43\) −268.000 −0.950456 −0.475228 0.879863i \(-0.657634\pi\)
−0.475228 + 0.879863i \(0.657634\pi\)
\(44\) −36.0000 −0.123346
\(45\) 162.000 0.536656
\(46\) 0 0
\(47\) −240.000 −0.744843 −0.372421 0.928064i \(-0.621472\pi\)
−0.372421 + 0.928064i \(0.621472\pi\)
\(48\) −213.000 −0.640498
\(49\) 0 0
\(50\) −597.000 −1.68857
\(51\) −126.000 −0.345952
\(52\) 34.0000 0.0906721
\(53\) −498.000 −1.29067 −0.645335 0.763899i \(-0.723282\pi\)
−0.645335 + 0.763899i \(0.723282\pi\)
\(54\) −81.0000 −0.204124
\(55\) −648.000 −1.58866
\(56\) 0 0
\(57\) 372.000 0.864432
\(58\) −306.000 −0.692755
\(59\) 132.000 0.291270 0.145635 0.989338i \(-0.453477\pi\)
0.145635 + 0.989338i \(0.453477\pi\)
\(60\) 54.0000 0.116190
\(61\) −398.000 −0.835388 −0.417694 0.908588i \(-0.637162\pi\)
−0.417694 + 0.908588i \(0.637162\pi\)
\(62\) −480.000 −0.983227
\(63\) 0 0
\(64\) 433.000 0.845703
\(65\) 612.000 1.16783
\(66\) 324.000 0.604267
\(67\) 92.0000 0.167755 0.0838775 0.996476i \(-0.473270\pi\)
0.0838775 + 0.996476i \(0.473270\pi\)
\(68\) −42.0000 −0.0749007
\(69\) 0 0
\(70\) 0 0
\(71\) −720.000 −1.20350 −0.601748 0.798686i \(-0.705529\pi\)
−0.601748 + 0.798686i \(0.705529\pi\)
\(72\) 189.000 0.309359
\(73\) 502.000 0.804858 0.402429 0.915451i \(-0.368166\pi\)
0.402429 + 0.915451i \(0.368166\pi\)
\(74\) −1194.00 −1.87567
\(75\) 597.000 0.919142
\(76\) 124.000 0.187155
\(77\) 0 0
\(78\) −306.000 −0.444201
\(79\) −1024.00 −1.45834 −0.729171 0.684332i \(-0.760094\pi\)
−0.729171 + 0.684332i \(0.760094\pi\)
\(80\) −1278.00 −1.78606
\(81\) 81.0000 0.111111
\(82\) −954.000 −1.28478
\(83\) 204.000 0.269782 0.134891 0.990860i \(-0.456932\pi\)
0.134891 + 0.990860i \(0.456932\pi\)
\(84\) 0 0
\(85\) −756.000 −0.964703
\(86\) 804.000 1.00811
\(87\) 306.000 0.377088
\(88\) −756.000 −0.915794
\(89\) −354.000 −0.421617 −0.210809 0.977527i \(-0.567610\pi\)
−0.210809 + 0.977527i \(0.567610\pi\)
\(90\) −486.000 −0.569210
\(91\) 0 0
\(92\) 0 0
\(93\) 480.000 0.535201
\(94\) 720.000 0.790025
\(95\) 2232.00 2.41051
\(96\) 135.000 0.143525
\(97\) 286.000 0.299370 0.149685 0.988734i \(-0.452174\pi\)
0.149685 + 0.988734i \(0.452174\pi\)
\(98\) 0 0
\(99\) −324.000 −0.328921
\(100\) 199.000 0.199000
\(101\) −414.000 −0.407867 −0.203933 0.978985i \(-0.565373\pi\)
−0.203933 + 0.978985i \(0.565373\pi\)
\(102\) 378.000 0.366937
\(103\) −56.0000 −0.0535713 −0.0267857 0.999641i \(-0.508527\pi\)
−0.0267857 + 0.999641i \(0.508527\pi\)
\(104\) 714.000 0.673206
\(105\) 0 0
\(106\) 1494.00 1.36896
\(107\) 12.0000 0.0108419 0.00542095 0.999985i \(-0.498274\pi\)
0.00542095 + 0.999985i \(0.498274\pi\)
\(108\) 27.0000 0.0240563
\(109\) 1478.00 1.29878 0.649389 0.760457i \(-0.275025\pi\)
0.649389 + 0.760457i \(0.275025\pi\)
\(110\) 1944.00 1.68503
\(111\) 1194.00 1.02099
\(112\) 0 0
\(113\) 402.000 0.334664 0.167332 0.985901i \(-0.446485\pi\)
0.167332 + 0.985901i \(0.446485\pi\)
\(114\) −1116.00 −0.916868
\(115\) 0 0
\(116\) 102.000 0.0816419
\(117\) 306.000 0.241792
\(118\) −396.000 −0.308939
\(119\) 0 0
\(120\) 1134.00 0.862663
\(121\) −35.0000 −0.0262960
\(122\) 1194.00 0.886063
\(123\) 954.000 0.699344
\(124\) 160.000 0.115874
\(125\) 1332.00 0.953102
\(126\) 0 0
\(127\) 1280.00 0.894344 0.447172 0.894448i \(-0.352431\pi\)
0.447172 + 0.894448i \(0.352431\pi\)
\(128\) −1659.00 −1.14560
\(129\) −804.000 −0.548746
\(130\) −1836.00 −1.23868
\(131\) −1764.00 −1.17650 −0.588250 0.808679i \(-0.700183\pi\)
−0.588250 + 0.808679i \(0.700183\pi\)
\(132\) −108.000 −0.0712136
\(133\) 0 0
\(134\) −276.000 −0.177931
\(135\) 486.000 0.309839
\(136\) −882.000 −0.556109
\(137\) −2358.00 −1.47049 −0.735246 0.677800i \(-0.762934\pi\)
−0.735246 + 0.677800i \(0.762934\pi\)
\(138\) 0 0
\(139\) 52.0000 0.0317308 0.0158654 0.999874i \(-0.494950\pi\)
0.0158654 + 0.999874i \(0.494950\pi\)
\(140\) 0 0
\(141\) −720.000 −0.430035
\(142\) 2160.00 1.27650
\(143\) −1224.00 −0.715776
\(144\) −639.000 −0.369792
\(145\) 1836.00 1.05153
\(146\) −1506.00 −0.853681
\(147\) 0 0
\(148\) 398.000 0.221050
\(149\) −1746.00 −0.959986 −0.479993 0.877272i \(-0.659361\pi\)
−0.479993 + 0.877272i \(0.659361\pi\)
\(150\) −1791.00 −0.974897
\(151\) −232.000 −0.125032 −0.0625162 0.998044i \(-0.519913\pi\)
−0.0625162 + 0.998044i \(0.519913\pi\)
\(152\) 2604.00 1.38955
\(153\) −378.000 −0.199735
\(154\) 0 0
\(155\) 2880.00 1.49243
\(156\) 102.000 0.0523496
\(157\) −1694.00 −0.861120 −0.430560 0.902562i \(-0.641684\pi\)
−0.430560 + 0.902562i \(0.641684\pi\)
\(158\) 3072.00 1.54681
\(159\) −1494.00 −0.745169
\(160\) 810.000 0.400226
\(161\) 0 0
\(162\) −243.000 −0.117851
\(163\) −2932.00 −1.40891 −0.704454 0.709750i \(-0.748808\pi\)
−0.704454 + 0.709750i \(0.748808\pi\)
\(164\) 318.000 0.151412
\(165\) −1944.00 −0.917213
\(166\) −612.000 −0.286147
\(167\) −1176.00 −0.544920 −0.272460 0.962167i \(-0.587837\pi\)
−0.272460 + 0.962167i \(0.587837\pi\)
\(168\) 0 0
\(169\) −1041.00 −0.473828
\(170\) 2268.00 1.02322
\(171\) 1116.00 0.499080
\(172\) −268.000 −0.118807
\(173\) −870.000 −0.382340 −0.191170 0.981557i \(-0.561228\pi\)
−0.191170 + 0.981557i \(0.561228\pi\)
\(174\) −918.000 −0.399962
\(175\) 0 0
\(176\) 2556.00 1.09469
\(177\) 396.000 0.168165
\(178\) 1062.00 0.447193
\(179\) −2316.00 −0.967072 −0.483536 0.875324i \(-0.660648\pi\)
−0.483536 + 0.875324i \(0.660648\pi\)
\(180\) 162.000 0.0670820
\(181\) 106.000 0.0435299 0.0217650 0.999763i \(-0.493071\pi\)
0.0217650 + 0.999763i \(0.493071\pi\)
\(182\) 0 0
\(183\) −1194.00 −0.482312
\(184\) 0 0
\(185\) 7164.00 2.84707
\(186\) −1440.00 −0.567666
\(187\) 1512.00 0.591275
\(188\) −240.000 −0.0931053
\(189\) 0 0
\(190\) −6696.00 −2.55673
\(191\) −1128.00 −0.427326 −0.213663 0.976907i \(-0.568539\pi\)
−0.213663 + 0.976907i \(0.568539\pi\)
\(192\) 1299.00 0.488267
\(193\) 4034.00 1.50453 0.752263 0.658862i \(-0.228962\pi\)
0.752263 + 0.658862i \(0.228962\pi\)
\(194\) −858.000 −0.317530
\(195\) 1836.00 0.674250
\(196\) 0 0
\(197\) −1314.00 −0.475221 −0.237611 0.971360i \(-0.576364\pi\)
−0.237611 + 0.971360i \(0.576364\pi\)
\(198\) 972.000 0.348874
\(199\) −5096.00 −1.81531 −0.907653 0.419722i \(-0.862128\pi\)
−0.907653 + 0.419722i \(0.862128\pi\)
\(200\) 4179.00 1.47750
\(201\) 276.000 0.0968534
\(202\) 1242.00 0.432608
\(203\) 0 0
\(204\) −126.000 −0.0432439
\(205\) 5724.00 1.95015
\(206\) 168.000 0.0568209
\(207\) 0 0
\(208\) −2414.00 −0.804715
\(209\) −4464.00 −1.47742
\(210\) 0 0
\(211\) −3076.00 −1.00360 −0.501802 0.864982i \(-0.667330\pi\)
−0.501802 + 0.864982i \(0.667330\pi\)
\(212\) −498.000 −0.161334
\(213\) −2160.00 −0.694839
\(214\) −36.0000 −0.0114996
\(215\) −4824.00 −1.53020
\(216\) 567.000 0.178609
\(217\) 0 0
\(218\) −4434.00 −1.37756
\(219\) 1506.00 0.464685
\(220\) −648.000 −0.198583
\(221\) −1428.00 −0.434650
\(222\) −3582.00 −1.08292
\(223\) 1888.00 0.566950 0.283475 0.958980i \(-0.408513\pi\)
0.283475 + 0.958980i \(0.408513\pi\)
\(224\) 0 0
\(225\) 1791.00 0.530667
\(226\) −1206.00 −0.354964
\(227\) 4716.00 1.37891 0.689454 0.724330i \(-0.257851\pi\)
0.689454 + 0.724330i \(0.257851\pi\)
\(228\) 372.000 0.108054
\(229\) 1690.00 0.487678 0.243839 0.969816i \(-0.421593\pi\)
0.243839 + 0.969816i \(0.421593\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 2142.00 0.606160
\(233\) 138.000 0.0388012 0.0194006 0.999812i \(-0.493824\pi\)
0.0194006 + 0.999812i \(0.493824\pi\)
\(234\) −918.000 −0.256460
\(235\) −4320.00 −1.19917
\(236\) 132.000 0.0364088
\(237\) −3072.00 −0.841974
\(238\) 0 0
\(239\) 1896.00 0.513147 0.256573 0.966525i \(-0.417406\pi\)
0.256573 + 0.966525i \(0.417406\pi\)
\(240\) −3834.00 −1.03118
\(241\) 3598.00 0.961691 0.480846 0.876805i \(-0.340330\pi\)
0.480846 + 0.876805i \(0.340330\pi\)
\(242\) 105.000 0.0278911
\(243\) 243.000 0.0641500
\(244\) −398.000 −0.104424
\(245\) 0 0
\(246\) −2862.00 −0.741766
\(247\) 4216.00 1.08606
\(248\) 3360.00 0.860323
\(249\) 612.000 0.155759
\(250\) −3996.00 −1.01092
\(251\) 3060.00 0.769504 0.384752 0.923020i \(-0.374287\pi\)
0.384752 + 0.923020i \(0.374287\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −3840.00 −0.948595
\(255\) −2268.00 −0.556971
\(256\) 1513.00 0.369385
\(257\) 6822.00 1.65582 0.827908 0.560864i \(-0.189531\pi\)
0.827908 + 0.560864i \(0.189531\pi\)
\(258\) 2412.00 0.582033
\(259\) 0 0
\(260\) 612.000 0.145979
\(261\) 918.000 0.217712
\(262\) 5292.00 1.24787
\(263\) 2592.00 0.607717 0.303858 0.952717i \(-0.401725\pi\)
0.303858 + 0.952717i \(0.401725\pi\)
\(264\) −2268.00 −0.528734
\(265\) −8964.00 −2.07794
\(266\) 0 0
\(267\) −1062.00 −0.243421
\(268\) 92.0000 0.0209694
\(269\) −8214.00 −1.86177 −0.930886 0.365311i \(-0.880963\pi\)
−0.930886 + 0.365311i \(0.880963\pi\)
\(270\) −1458.00 −0.328634
\(271\) 5344.00 1.19788 0.598939 0.800795i \(-0.295589\pi\)
0.598939 + 0.800795i \(0.295589\pi\)
\(272\) 2982.00 0.664744
\(273\) 0 0
\(274\) 7074.00 1.55969
\(275\) −7164.00 −1.57093
\(276\) 0 0
\(277\) −6514.00 −1.41295 −0.706477 0.707736i \(-0.749717\pi\)
−0.706477 + 0.707736i \(0.749717\pi\)
\(278\) −156.000 −0.0336556
\(279\) 1440.00 0.308998
\(280\) 0 0
\(281\) 6618.00 1.40497 0.702485 0.711698i \(-0.252074\pi\)
0.702485 + 0.711698i \(0.252074\pi\)
\(282\) 2160.00 0.456121
\(283\) −3260.00 −0.684759 −0.342380 0.939562i \(-0.611233\pi\)
−0.342380 + 0.939562i \(0.611233\pi\)
\(284\) −720.000 −0.150437
\(285\) 6696.00 1.39171
\(286\) 3672.00 0.759195
\(287\) 0 0
\(288\) 405.000 0.0828641
\(289\) −3149.00 −0.640953
\(290\) −5508.00 −1.11531
\(291\) 858.000 0.172841
\(292\) 502.000 0.100607
\(293\) −5118.00 −1.02047 −0.510233 0.860036i \(-0.670441\pi\)
−0.510233 + 0.860036i \(0.670441\pi\)
\(294\) 0 0
\(295\) 2376.00 0.468936
\(296\) 8358.00 1.64121
\(297\) −972.000 −0.189903
\(298\) 5238.00 1.01822
\(299\) 0 0
\(300\) 597.000 0.114893
\(301\) 0 0
\(302\) 696.000 0.132617
\(303\) −1242.00 −0.235482
\(304\) −8804.00 −1.66100
\(305\) −7164.00 −1.34495
\(306\) 1134.00 0.211851
\(307\) −452.000 −0.0840293 −0.0420147 0.999117i \(-0.513378\pi\)
−0.0420147 + 0.999117i \(0.513378\pi\)
\(308\) 0 0
\(309\) −168.000 −0.0309294
\(310\) −8640.00 −1.58296
\(311\) −5016.00 −0.914570 −0.457285 0.889320i \(-0.651178\pi\)
−0.457285 + 0.889320i \(0.651178\pi\)
\(312\) 2142.00 0.388676
\(313\) −5402.00 −0.975524 −0.487762 0.872977i \(-0.662187\pi\)
−0.487762 + 0.872977i \(0.662187\pi\)
\(314\) 5082.00 0.913356
\(315\) 0 0
\(316\) −1024.00 −0.182293
\(317\) 10086.0 1.78702 0.893511 0.449041i \(-0.148234\pi\)
0.893511 + 0.449041i \(0.148234\pi\)
\(318\) 4482.00 0.790371
\(319\) −3672.00 −0.644491
\(320\) 7794.00 1.36156
\(321\) 36.0000 0.00625958
\(322\) 0 0
\(323\) −5208.00 −0.897154
\(324\) 81.0000 0.0138889
\(325\) 6766.00 1.15480
\(326\) 8796.00 1.49437
\(327\) 4434.00 0.749849
\(328\) 6678.00 1.12418
\(329\) 0 0
\(330\) 5832.00 0.972852
\(331\) −8044.00 −1.33577 −0.667883 0.744267i \(-0.732799\pi\)
−0.667883 + 0.744267i \(0.732799\pi\)
\(332\) 204.000 0.0337228
\(333\) 3582.00 0.589467
\(334\) 3528.00 0.577975
\(335\) 1656.00 0.270080
\(336\) 0 0
\(337\) 4178.00 0.675342 0.337671 0.941264i \(-0.390361\pi\)
0.337671 + 0.941264i \(0.390361\pi\)
\(338\) 3123.00 0.502570
\(339\) 1206.00 0.193218
\(340\) −756.000 −0.120588
\(341\) −5760.00 −0.914726
\(342\) −3348.00 −0.529354
\(343\) 0 0
\(344\) −5628.00 −0.882097
\(345\) 0 0
\(346\) 2610.00 0.405533
\(347\) 156.000 0.0241341 0.0120670 0.999927i \(-0.496159\pi\)
0.0120670 + 0.999927i \(0.496159\pi\)
\(348\) 306.000 0.0471360
\(349\) 12418.0 1.90464 0.952321 0.305097i \(-0.0986888\pi\)
0.952321 + 0.305097i \(0.0986888\pi\)
\(350\) 0 0
\(351\) 918.000 0.139599
\(352\) −1620.00 −0.245302
\(353\) 7830.00 1.18059 0.590296 0.807187i \(-0.299011\pi\)
0.590296 + 0.807187i \(0.299011\pi\)
\(354\) −1188.00 −0.178366
\(355\) −12960.0 −1.93759
\(356\) −354.000 −0.0527021
\(357\) 0 0
\(358\) 6948.00 1.02574
\(359\) −9312.00 −1.36899 −0.684497 0.729016i \(-0.739978\pi\)
−0.684497 + 0.729016i \(0.739978\pi\)
\(360\) 3402.00 0.498059
\(361\) 8517.00 1.24173
\(362\) −318.000 −0.0461705
\(363\) −105.000 −0.0151820
\(364\) 0 0
\(365\) 9036.00 1.29580
\(366\) 3582.00 0.511569
\(367\) 3760.00 0.534797 0.267398 0.963586i \(-0.413836\pi\)
0.267398 + 0.963586i \(0.413836\pi\)
\(368\) 0 0
\(369\) 2862.00 0.403766
\(370\) −21492.0 −3.01977
\(371\) 0 0
\(372\) 480.000 0.0669001
\(373\) 5870.00 0.814845 0.407422 0.913240i \(-0.366428\pi\)
0.407422 + 0.913240i \(0.366428\pi\)
\(374\) −4536.00 −0.627142
\(375\) 3996.00 0.550273
\(376\) −5040.00 −0.691272
\(377\) 3468.00 0.473769
\(378\) 0 0
\(379\) −1852.00 −0.251005 −0.125502 0.992093i \(-0.540054\pi\)
−0.125502 + 0.992093i \(0.540054\pi\)
\(380\) 2232.00 0.301314
\(381\) 3840.00 0.516350
\(382\) 3384.00 0.453247
\(383\) −2160.00 −0.288175 −0.144087 0.989565i \(-0.546025\pi\)
−0.144087 + 0.989565i \(0.546025\pi\)
\(384\) −4977.00 −0.661410
\(385\) 0 0
\(386\) −12102.0 −1.59579
\(387\) −2412.00 −0.316819
\(388\) 286.000 0.0374213
\(389\) −6786.00 −0.884483 −0.442241 0.896896i \(-0.645817\pi\)
−0.442241 + 0.896896i \(0.645817\pi\)
\(390\) −5508.00 −0.715150
\(391\) 0 0
\(392\) 0 0
\(393\) −5292.00 −0.679252
\(394\) 3942.00 0.504048
\(395\) −18432.0 −2.34788
\(396\) −324.000 −0.0411152
\(397\) 6514.00 0.823497 0.411748 0.911298i \(-0.364918\pi\)
0.411748 + 0.911298i \(0.364918\pi\)
\(398\) 15288.0 1.92542
\(399\) 0 0
\(400\) −14129.0 −1.76612
\(401\) 3330.00 0.414694 0.207347 0.978267i \(-0.433517\pi\)
0.207347 + 0.978267i \(0.433517\pi\)
\(402\) −828.000 −0.102729
\(403\) 5440.00 0.672421
\(404\) −414.000 −0.0509833
\(405\) 1458.00 0.178885
\(406\) 0 0
\(407\) −14328.0 −1.74499
\(408\) −2646.00 −0.321070
\(409\) 5398.00 0.652601 0.326301 0.945266i \(-0.394198\pi\)
0.326301 + 0.945266i \(0.394198\pi\)
\(410\) −17172.0 −2.06845
\(411\) −7074.00 −0.848990
\(412\) −56.0000 −0.00669641
\(413\) 0 0
\(414\) 0 0
\(415\) 3672.00 0.434341
\(416\) 1530.00 0.180323
\(417\) 156.000 0.0183198
\(418\) 13392.0 1.56704
\(419\) −13092.0 −1.52646 −0.763229 0.646128i \(-0.776387\pi\)
−0.763229 + 0.646128i \(0.776387\pi\)
\(420\) 0 0
\(421\) −322.000 −0.0372763 −0.0186381 0.999826i \(-0.505933\pi\)
−0.0186381 + 0.999826i \(0.505933\pi\)
\(422\) 9228.00 1.06448
\(423\) −2160.00 −0.248281
\(424\) −10458.0 −1.19784
\(425\) −8358.00 −0.953935
\(426\) 6480.00 0.736988
\(427\) 0 0
\(428\) 12.0000 0.00135524
\(429\) −3672.00 −0.413254
\(430\) 14472.0 1.62303
\(431\) 2616.00 0.292363 0.146181 0.989258i \(-0.453302\pi\)
0.146181 + 0.989258i \(0.453302\pi\)
\(432\) −1917.00 −0.213499
\(433\) −4322.00 −0.479681 −0.239841 0.970812i \(-0.577095\pi\)
−0.239841 + 0.970812i \(0.577095\pi\)
\(434\) 0 0
\(435\) 5508.00 0.607100
\(436\) 1478.00 0.162347
\(437\) 0 0
\(438\) −4518.00 −0.492873
\(439\) 9016.00 0.980205 0.490103 0.871665i \(-0.336959\pi\)
0.490103 + 0.871665i \(0.336959\pi\)
\(440\) −13608.0 −1.47440
\(441\) 0 0
\(442\) 4284.00 0.461016
\(443\) −5268.00 −0.564989 −0.282495 0.959269i \(-0.591162\pi\)
−0.282495 + 0.959269i \(0.591162\pi\)
\(444\) 1194.00 0.127623
\(445\) −6372.00 −0.678790
\(446\) −5664.00 −0.601341
\(447\) −5238.00 −0.554248
\(448\) 0 0
\(449\) −5310.00 −0.558117 −0.279058 0.960274i \(-0.590022\pi\)
−0.279058 + 0.960274i \(0.590022\pi\)
\(450\) −5373.00 −0.562857
\(451\) −11448.0 −1.19527
\(452\) 402.000 0.0418329
\(453\) −696.000 −0.0721875
\(454\) −14148.0 −1.46255
\(455\) 0 0
\(456\) 7812.00 0.802260
\(457\) 15770.0 1.61420 0.807100 0.590415i \(-0.201036\pi\)
0.807100 + 0.590415i \(0.201036\pi\)
\(458\) −5070.00 −0.517261
\(459\) −1134.00 −0.115317
\(460\) 0 0
\(461\) 5370.00 0.542529 0.271264 0.962505i \(-0.412558\pi\)
0.271264 + 0.962505i \(0.412558\pi\)
\(462\) 0 0
\(463\) −3328.00 −0.334050 −0.167025 0.985953i \(-0.553416\pi\)
−0.167025 + 0.985953i \(0.553416\pi\)
\(464\) −7242.00 −0.724572
\(465\) 8640.00 0.861657
\(466\) −414.000 −0.0411549
\(467\) −4548.00 −0.450656 −0.225328 0.974283i \(-0.572345\pi\)
−0.225328 + 0.974283i \(0.572345\pi\)
\(468\) 306.000 0.0302240
\(469\) 0 0
\(470\) 12960.0 1.27192
\(471\) −5082.00 −0.497168
\(472\) 2772.00 0.270321
\(473\) 9648.00 0.937876
\(474\) 9216.00 0.893048
\(475\) 24676.0 2.38361
\(476\) 0 0
\(477\) −4482.00 −0.430224
\(478\) −5688.00 −0.544274
\(479\) 8064.00 0.769214 0.384607 0.923080i \(-0.374337\pi\)
0.384607 + 0.923080i \(0.374337\pi\)
\(480\) 2430.00 0.231070
\(481\) 13532.0 1.28276
\(482\) −10794.0 −1.02003
\(483\) 0 0
\(484\) −35.0000 −0.00328700
\(485\) 5148.00 0.481977
\(486\) −729.000 −0.0680414
\(487\) 16616.0 1.54608 0.773042 0.634355i \(-0.218734\pi\)
0.773042 + 0.634355i \(0.218734\pi\)
\(488\) −8358.00 −0.775305
\(489\) −8796.00 −0.813433
\(490\) 0 0
\(491\) −7140.00 −0.656260 −0.328130 0.944633i \(-0.606418\pi\)
−0.328130 + 0.944633i \(0.606418\pi\)
\(492\) 954.000 0.0874180
\(493\) −4284.00 −0.391362
\(494\) −12648.0 −1.15194
\(495\) −5832.00 −0.529553
\(496\) −11360.0 −1.02839
\(497\) 0 0
\(498\) −1836.00 −0.165207
\(499\) −9124.00 −0.818530 −0.409265 0.912416i \(-0.634215\pi\)
−0.409265 + 0.912416i \(0.634215\pi\)
\(500\) 1332.00 0.119138
\(501\) −3528.00 −0.314610
\(502\) −9180.00 −0.816182
\(503\) 6552.00 0.580794 0.290397 0.956906i \(-0.406213\pi\)
0.290397 + 0.956906i \(0.406213\pi\)
\(504\) 0 0
\(505\) −7452.00 −0.656653
\(506\) 0 0
\(507\) −3123.00 −0.273565
\(508\) 1280.00 0.111793
\(509\) −2790.00 −0.242956 −0.121478 0.992594i \(-0.538763\pi\)
−0.121478 + 0.992594i \(0.538763\pi\)
\(510\) 6804.00 0.590757
\(511\) 0 0
\(512\) 8733.00 0.753804
\(513\) 3348.00 0.288144
\(514\) −20466.0 −1.75626
\(515\) −1008.00 −0.0862481
\(516\) −804.000 −0.0685933
\(517\) 8640.00 0.734984
\(518\) 0 0
\(519\) −2610.00 −0.220744
\(520\) 12852.0 1.08384
\(521\) 14862.0 1.24974 0.624871 0.780728i \(-0.285151\pi\)
0.624871 + 0.780728i \(0.285151\pi\)
\(522\) −2754.00 −0.230918
\(523\) −17660.0 −1.47652 −0.738258 0.674518i \(-0.764351\pi\)
−0.738258 + 0.674518i \(0.764351\pi\)
\(524\) −1764.00 −0.147062
\(525\) 0 0
\(526\) −7776.00 −0.644581
\(527\) −6720.00 −0.555461
\(528\) 7668.00 0.632021
\(529\) −12167.0 −1.00000
\(530\) 26892.0 2.20399
\(531\) 1188.00 0.0970900
\(532\) 0 0
\(533\) 10812.0 0.878649
\(534\) 3186.00 0.258187
\(535\) 216.000 0.0174551
\(536\) 1932.00 0.155690
\(537\) −6948.00 −0.558340
\(538\) 24642.0 1.97471
\(539\) 0 0
\(540\) 486.000 0.0387298
\(541\) −19834.0 −1.57621 −0.788106 0.615540i \(-0.788938\pi\)
−0.788106 + 0.615540i \(0.788938\pi\)
\(542\) −16032.0 −1.27054
\(543\) 318.000 0.0251320
\(544\) −1890.00 −0.148958
\(545\) 26604.0 2.09099
\(546\) 0 0
\(547\) 20972.0 1.63930 0.819651 0.572863i \(-0.194167\pi\)
0.819651 + 0.572863i \(0.194167\pi\)
\(548\) −2358.00 −0.183812
\(549\) −3582.00 −0.278463
\(550\) 21492.0 1.66622
\(551\) 12648.0 0.977900
\(552\) 0 0
\(553\) 0 0
\(554\) 19542.0 1.49866
\(555\) 21492.0 1.64376
\(556\) 52.0000 0.00396635
\(557\) 21174.0 1.61072 0.805360 0.592786i \(-0.201972\pi\)
0.805360 + 0.592786i \(0.201972\pi\)
\(558\) −4320.00 −0.327742
\(559\) −9112.00 −0.689439
\(560\) 0 0
\(561\) 4536.00 0.341373
\(562\) −19854.0 −1.49020
\(563\) 17772.0 1.33037 0.665187 0.746677i \(-0.268352\pi\)
0.665187 + 0.746677i \(0.268352\pi\)
\(564\) −720.000 −0.0537544
\(565\) 7236.00 0.538798
\(566\) 9780.00 0.726297
\(567\) 0 0
\(568\) −15120.0 −1.11694
\(569\) 8250.00 0.607835 0.303917 0.952698i \(-0.401705\pi\)
0.303917 + 0.952698i \(0.401705\pi\)
\(570\) −20088.0 −1.47613
\(571\) 20756.0 1.52121 0.760606 0.649214i \(-0.224902\pi\)
0.760606 + 0.649214i \(0.224902\pi\)
\(572\) −1224.00 −0.0894720
\(573\) −3384.00 −0.246717
\(574\) 0 0
\(575\) 0 0
\(576\) 3897.00 0.281901
\(577\) −2.00000 −0.000144300 0 −7.21500e−5 1.00000i \(-0.500023\pi\)
−7.21500e−5 1.00000i \(0.500023\pi\)
\(578\) 9447.00 0.679833
\(579\) 12102.0 0.868639
\(580\) 1836.00 0.131441
\(581\) 0 0
\(582\) −2574.00 −0.183326
\(583\) 17928.0 1.27359
\(584\) 10542.0 0.746971
\(585\) 5508.00 0.389278
\(586\) 15354.0 1.08237
\(587\) −26364.0 −1.85376 −0.926881 0.375354i \(-0.877521\pi\)
−0.926881 + 0.375354i \(0.877521\pi\)
\(588\) 0 0
\(589\) 19840.0 1.38793
\(590\) −7128.00 −0.497382
\(591\) −3942.00 −0.274369
\(592\) −28258.0 −1.96182
\(593\) −2298.00 −0.159136 −0.0795679 0.996829i \(-0.525354\pi\)
−0.0795679 + 0.996829i \(0.525354\pi\)
\(594\) 2916.00 0.201422
\(595\) 0 0
\(596\) −1746.00 −0.119998
\(597\) −15288.0 −1.04807
\(598\) 0 0
\(599\) 3072.00 0.209547 0.104773 0.994496i \(-0.466588\pi\)
0.104773 + 0.994496i \(0.466588\pi\)
\(600\) 12537.0 0.853035
\(601\) −24554.0 −1.66652 −0.833260 0.552881i \(-0.813528\pi\)
−0.833260 + 0.552881i \(0.813528\pi\)
\(602\) 0 0
\(603\) 828.000 0.0559184
\(604\) −232.000 −0.0156290
\(605\) −630.000 −0.0423358
\(606\) 3726.00 0.249766
\(607\) −16832.0 −1.12552 −0.562759 0.826621i \(-0.690260\pi\)
−0.562759 + 0.826621i \(0.690260\pi\)
\(608\) 5580.00 0.372202
\(609\) 0 0
\(610\) 21492.0 1.42653
\(611\) −8160.00 −0.540292
\(612\) −378.000 −0.0249669
\(613\) −2482.00 −0.163535 −0.0817676 0.996651i \(-0.526057\pi\)
−0.0817676 + 0.996651i \(0.526057\pi\)
\(614\) 1356.00 0.0891266
\(615\) 17172.0 1.12592
\(616\) 0 0
\(617\) −15798.0 −1.03080 −0.515400 0.856950i \(-0.672357\pi\)
−0.515400 + 0.856950i \(0.672357\pi\)
\(618\) 504.000 0.0328056
\(619\) 15460.0 1.00386 0.501930 0.864908i \(-0.332623\pi\)
0.501930 + 0.864908i \(0.332623\pi\)
\(620\) 2880.00 0.186554
\(621\) 0 0
\(622\) 15048.0 0.970048
\(623\) 0 0
\(624\) −7242.00 −0.464603
\(625\) −899.000 −0.0575360
\(626\) 16206.0 1.03470
\(627\) −13392.0 −0.852990
\(628\) −1694.00 −0.107640
\(629\) −16716.0 −1.05964
\(630\) 0 0
\(631\) −7720.00 −0.487050 −0.243525 0.969895i \(-0.578304\pi\)
−0.243525 + 0.969895i \(0.578304\pi\)
\(632\) −21504.0 −1.35345
\(633\) −9228.00 −0.579431
\(634\) −30258.0 −1.89542
\(635\) 23040.0 1.43987
\(636\) −1494.00 −0.0931462
\(637\) 0 0
\(638\) 11016.0 0.683586
\(639\) −6480.00 −0.401166
\(640\) −29862.0 −1.84437
\(641\) −17262.0 −1.06366 −0.531832 0.846850i \(-0.678496\pi\)
−0.531832 + 0.846850i \(0.678496\pi\)
\(642\) −108.000 −0.00663928
\(643\) 12220.0 0.749471 0.374735 0.927132i \(-0.377734\pi\)
0.374735 + 0.927132i \(0.377734\pi\)
\(644\) 0 0
\(645\) −14472.0 −0.883464
\(646\) 15624.0 0.951576
\(647\) −13560.0 −0.823955 −0.411977 0.911194i \(-0.635162\pi\)
−0.411977 + 0.911194i \(0.635162\pi\)
\(648\) 1701.00 0.103120
\(649\) −4752.00 −0.287415
\(650\) −20298.0 −1.22485
\(651\) 0 0
\(652\) −2932.00 −0.176113
\(653\) 23094.0 1.38398 0.691989 0.721908i \(-0.256735\pi\)
0.691989 + 0.721908i \(0.256735\pi\)
\(654\) −13302.0 −0.795335
\(655\) −31752.0 −1.89413
\(656\) −22578.0 −1.34378
\(657\) 4518.00 0.268286
\(658\) 0 0
\(659\) 22548.0 1.33285 0.666423 0.745574i \(-0.267825\pi\)
0.666423 + 0.745574i \(0.267825\pi\)
\(660\) −1944.00 −0.114652
\(661\) −17462.0 −1.02752 −0.513762 0.857933i \(-0.671748\pi\)
−0.513762 + 0.857933i \(0.671748\pi\)
\(662\) 24132.0 1.41679
\(663\) −4284.00 −0.250945
\(664\) 4284.00 0.250379
\(665\) 0 0
\(666\) −10746.0 −0.625224
\(667\) 0 0
\(668\) −1176.00 −0.0681150
\(669\) 5664.00 0.327329
\(670\) −4968.00 −0.286464
\(671\) 14328.0 0.824331
\(672\) 0 0
\(673\) −22462.0 −1.28655 −0.643274 0.765636i \(-0.722424\pi\)
−0.643274 + 0.765636i \(0.722424\pi\)
\(674\) −12534.0 −0.716308
\(675\) 5373.00 0.306381
\(676\) −1041.00 −0.0592285
\(677\) 25554.0 1.45069 0.725347 0.688383i \(-0.241679\pi\)
0.725347 + 0.688383i \(0.241679\pi\)
\(678\) −3618.00 −0.204939
\(679\) 0 0
\(680\) −15876.0 −0.895319
\(681\) 14148.0 0.796112
\(682\) 17280.0 0.970213
\(683\) 9276.00 0.519672 0.259836 0.965653i \(-0.416331\pi\)
0.259836 + 0.965653i \(0.416331\pi\)
\(684\) 1116.00 0.0623850
\(685\) −42444.0 −2.36745
\(686\) 0 0
\(687\) 5070.00 0.281561
\(688\) 19028.0 1.05441
\(689\) −16932.0 −0.936223
\(690\) 0 0
\(691\) −27380.0 −1.50736 −0.753679 0.657243i \(-0.771723\pi\)
−0.753679 + 0.657243i \(0.771723\pi\)
\(692\) −870.000 −0.0477925
\(693\) 0 0
\(694\) −468.000 −0.0255980
\(695\) 936.000 0.0510856
\(696\) 6426.00 0.349967
\(697\) −13356.0 −0.725817
\(698\) −37254.0 −2.02018
\(699\) 414.000 0.0224019
\(700\) 0 0
\(701\) 25830.0 1.39171 0.695853 0.718184i \(-0.255027\pi\)
0.695853 + 0.718184i \(0.255027\pi\)
\(702\) −2754.00 −0.148067
\(703\) 49352.0 2.64772
\(704\) −15588.0 −0.834510
\(705\) −12960.0 −0.692343
\(706\) −23490.0 −1.25221
\(707\) 0 0
\(708\) 396.000 0.0210206
\(709\) −6226.00 −0.329792 −0.164896 0.986311i \(-0.552729\pi\)
−0.164896 + 0.986311i \(0.552729\pi\)
\(710\) 38880.0 2.05513
\(711\) −9216.00 −0.486114
\(712\) −7434.00 −0.391293
\(713\) 0 0
\(714\) 0 0
\(715\) −22032.0 −1.15238
\(716\) −2316.00 −0.120884
\(717\) 5688.00 0.296265
\(718\) 27936.0 1.45204
\(719\) 15072.0 0.781767 0.390884 0.920440i \(-0.372169\pi\)
0.390884 + 0.920440i \(0.372169\pi\)
\(720\) −11502.0 −0.595353
\(721\) 0 0
\(722\) −25551.0 −1.31705
\(723\) 10794.0 0.555233
\(724\) 106.000 0.00544124
\(725\) 20298.0 1.03979
\(726\) 315.000 0.0161030
\(727\) 32920.0 1.67942 0.839708 0.543038i \(-0.182726\pi\)
0.839708 + 0.543038i \(0.182726\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) −27108.0 −1.37440
\(731\) 11256.0 0.569519
\(732\) −1194.00 −0.0602889
\(733\) 6946.00 0.350009 0.175004 0.984568i \(-0.444006\pi\)
0.175004 + 0.984568i \(0.444006\pi\)
\(734\) −11280.0 −0.567238
\(735\) 0 0
\(736\) 0 0
\(737\) −3312.00 −0.165535
\(738\) −8586.00 −0.428259
\(739\) −2356.00 −0.117276 −0.0586379 0.998279i \(-0.518676\pi\)
−0.0586379 + 0.998279i \(0.518676\pi\)
\(740\) 7164.00 0.355884
\(741\) 12648.0 0.627039
\(742\) 0 0
\(743\) −23520.0 −1.16133 −0.580663 0.814144i \(-0.697207\pi\)
−0.580663 + 0.814144i \(0.697207\pi\)
\(744\) 10080.0 0.496708
\(745\) −31428.0 −1.54555
\(746\) −17610.0 −0.864273
\(747\) 1836.00 0.0899273
\(748\) 1512.00 0.0739094
\(749\) 0 0
\(750\) −11988.0 −0.583653
\(751\) 3008.00 0.146156 0.0730782 0.997326i \(-0.476718\pi\)
0.0730782 + 0.997326i \(0.476718\pi\)
\(752\) 17040.0 0.826310
\(753\) 9180.00 0.444273
\(754\) −10404.0 −0.502508
\(755\) −4176.00 −0.201298
\(756\) 0 0
\(757\) −20770.0 −0.997224 −0.498612 0.866825i \(-0.666157\pi\)
−0.498612 + 0.866825i \(0.666157\pi\)
\(758\) 5556.00 0.266231
\(759\) 0 0
\(760\) 46872.0 2.23714
\(761\) −11538.0 −0.549609 −0.274804 0.961500i \(-0.588613\pi\)
−0.274804 + 0.961500i \(0.588613\pi\)
\(762\) −11520.0 −0.547671
\(763\) 0 0
\(764\) −1128.00 −0.0534157
\(765\) −6804.00 −0.321568
\(766\) 6480.00 0.305655
\(767\) 4488.00 0.211281
\(768\) 4539.00 0.213264
\(769\) −8498.00 −0.398499 −0.199249 0.979949i \(-0.563850\pi\)
−0.199249 + 0.979949i \(0.563850\pi\)
\(770\) 0 0
\(771\) 20466.0 0.955986
\(772\) 4034.00 0.188066
\(773\) 32322.0 1.50393 0.751967 0.659200i \(-0.229105\pi\)
0.751967 + 0.659200i \(0.229105\pi\)
\(774\) 7236.00 0.336037
\(775\) 31840.0 1.47578
\(776\) 6006.00 0.277839
\(777\) 0 0
\(778\) 20358.0 0.938136
\(779\) 39432.0 1.81360
\(780\) 1836.00 0.0842812
\(781\) 25920.0 1.18757
\(782\) 0 0
\(783\) 2754.00 0.125696
\(784\) 0 0
\(785\) −30492.0 −1.38638
\(786\) 15876.0 0.720456
\(787\) −26228.0 −1.18796 −0.593982 0.804479i \(-0.702445\pi\)
−0.593982 + 0.804479i \(0.702445\pi\)
\(788\) −1314.00 −0.0594027
\(789\) 7776.00 0.350866
\(790\) 55296.0 2.49031
\(791\) 0 0
\(792\) −6804.00 −0.305265
\(793\) −13532.0 −0.605972
\(794\) −19542.0 −0.873450
\(795\) −26892.0 −1.19970
\(796\) −5096.00 −0.226913
\(797\) 43338.0 1.92611 0.963056 0.269302i \(-0.0867931\pi\)
0.963056 + 0.269302i \(0.0867931\pi\)
\(798\) 0 0
\(799\) 10080.0 0.446314
\(800\) 8955.00 0.395759
\(801\) −3186.00 −0.140539
\(802\) −9990.00 −0.439849
\(803\) −18072.0 −0.794206
\(804\) 276.000 0.0121067
\(805\) 0 0
\(806\) −16320.0 −0.713210
\(807\) −24642.0 −1.07489
\(808\) −8694.00 −0.378532
\(809\) −28902.0 −1.25604 −0.628022 0.778195i \(-0.716135\pi\)
−0.628022 + 0.778195i \(0.716135\pi\)
\(810\) −4374.00 −0.189737
\(811\) −27164.0 −1.17615 −0.588075 0.808807i \(-0.700114\pi\)
−0.588075 + 0.808807i \(0.700114\pi\)
\(812\) 0 0
\(813\) 16032.0 0.691595
\(814\) 42984.0 1.85085
\(815\) −52776.0 −2.26830
\(816\) 8946.00 0.383790
\(817\) −33232.0 −1.42306
\(818\) −16194.0 −0.692188
\(819\) 0 0
\(820\) 5724.00 0.243769
\(821\) −17202.0 −0.731247 −0.365624 0.930763i \(-0.619144\pi\)
−0.365624 + 0.930763i \(0.619144\pi\)
\(822\) 21222.0 0.900489
\(823\) −5992.00 −0.253789 −0.126894 0.991916i \(-0.540501\pi\)
−0.126894 + 0.991916i \(0.540501\pi\)
\(824\) −1176.00 −0.0497183
\(825\) −21492.0 −0.906976
\(826\) 0 0
\(827\) 25884.0 1.08836 0.544181 0.838968i \(-0.316841\pi\)
0.544181 + 0.838968i \(0.316841\pi\)
\(828\) 0 0
\(829\) 1474.00 0.0617541 0.0308770 0.999523i \(-0.490170\pi\)
0.0308770 + 0.999523i \(0.490170\pi\)
\(830\) −11016.0 −0.460688
\(831\) −19542.0 −0.815770
\(832\) 14722.0 0.613454
\(833\) 0 0
\(834\) −468.000 −0.0194311
\(835\) −21168.0 −0.877304
\(836\) −4464.00 −0.184678
\(837\) 4320.00 0.178400
\(838\) 39276.0 1.61905
\(839\) −33528.0 −1.37964 −0.689818 0.723983i \(-0.742310\pi\)
−0.689818 + 0.723983i \(0.742310\pi\)
\(840\) 0 0
\(841\) −13985.0 −0.573414
\(842\) 966.000 0.0395375
\(843\) 19854.0 0.811160
\(844\) −3076.00 −0.125451
\(845\) −18738.0 −0.762848
\(846\) 6480.00 0.263342
\(847\) 0 0
\(848\) 35358.0 1.43184
\(849\) −9780.00 −0.395346
\(850\) 25074.0 1.01180
\(851\) 0 0
\(852\) −2160.00 −0.0868549
\(853\) −1190.00 −0.0477665 −0.0238832 0.999715i \(-0.507603\pi\)
−0.0238832 + 0.999715i \(0.507603\pi\)
\(854\) 0 0
\(855\) 20088.0 0.803503
\(856\) 252.000 0.0100621
\(857\) −34578.0 −1.37825 −0.689126 0.724642i \(-0.742005\pi\)
−0.689126 + 0.724642i \(0.742005\pi\)
\(858\) 11016.0 0.438322
\(859\) 44404.0 1.76373 0.881865 0.471501i \(-0.156288\pi\)
0.881865 + 0.471501i \(0.156288\pi\)
\(860\) −4824.00 −0.191276
\(861\) 0 0
\(862\) −7848.00 −0.310097
\(863\) −38328.0 −1.51182 −0.755910 0.654676i \(-0.772805\pi\)
−0.755910 + 0.654676i \(0.772805\pi\)
\(864\) 1215.00 0.0478416
\(865\) −15660.0 −0.615556
\(866\) 12966.0 0.508779
\(867\) −9447.00 −0.370054
\(868\) 0 0
\(869\) 36864.0 1.43904
\(870\) −16524.0 −0.643927
\(871\) 3128.00 0.121686
\(872\) 31038.0 1.20537
\(873\) 2574.00 0.0997900
\(874\) 0 0
\(875\) 0 0
\(876\) 1506.00 0.0580856
\(877\) −38842.0 −1.49555 −0.747777 0.663950i \(-0.768879\pi\)
−0.747777 + 0.663950i \(0.768879\pi\)
\(878\) −27048.0 −1.03966
\(879\) −15354.0 −0.589167
\(880\) 46008.0 1.76242
\(881\) 35046.0 1.34022 0.670108 0.742264i \(-0.266248\pi\)
0.670108 + 0.742264i \(0.266248\pi\)
\(882\) 0 0
\(883\) 14204.0 0.541339 0.270670 0.962672i \(-0.412755\pi\)
0.270670 + 0.962672i \(0.412755\pi\)
\(884\) −1428.00 −0.0543313
\(885\) 7128.00 0.270740
\(886\) 15804.0 0.599262
\(887\) 26136.0 0.989359 0.494679 0.869076i \(-0.335286\pi\)
0.494679 + 0.869076i \(0.335286\pi\)
\(888\) 25074.0 0.947554
\(889\) 0 0
\(890\) 19116.0 0.719966
\(891\) −2916.00 −0.109640
\(892\) 1888.00 0.0708687
\(893\) −29760.0 −1.11521
\(894\) 15714.0 0.587869
\(895\) −41688.0 −1.55696
\(896\) 0 0
\(897\) 0 0
\(898\) 15930.0 0.591972
\(899\) 16320.0 0.605453
\(900\) 1791.00 0.0663333
\(901\) 20916.0 0.773377
\(902\) 34344.0 1.26777
\(903\) 0 0
\(904\) 8442.00 0.310594
\(905\) 1908.00 0.0700818
\(906\) 2088.00 0.0765664
\(907\) −9052.00 −0.331386 −0.165693 0.986177i \(-0.552986\pi\)
−0.165693 + 0.986177i \(0.552986\pi\)
\(908\) 4716.00 0.172363
\(909\) −3726.00 −0.135956
\(910\) 0 0
\(911\) 5016.00 0.182423 0.0912116 0.995832i \(-0.470926\pi\)
0.0912116 + 0.995832i \(0.470926\pi\)
\(912\) −26412.0 −0.958979
\(913\) −7344.00 −0.266211
\(914\) −47310.0 −1.71212
\(915\) −21492.0 −0.776507
\(916\) 1690.00 0.0609598
\(917\) 0 0
\(918\) 3402.00 0.122312
\(919\) 44552.0 1.59917 0.799584 0.600555i \(-0.205054\pi\)
0.799584 + 0.600555i \(0.205054\pi\)
\(920\) 0 0
\(921\) −1356.00 −0.0485144
\(922\) −16110.0 −0.575439
\(923\) −24480.0 −0.872989
\(924\) 0 0
\(925\) 79202.0 2.81529
\(926\) 9984.00 0.354314
\(927\) −504.000 −0.0178571
\(928\) 4590.00 0.162364
\(929\) −24234.0 −0.855858 −0.427929 0.903812i \(-0.640757\pi\)
−0.427929 + 0.903812i \(0.640757\pi\)
\(930\) −25920.0 −0.913925
\(931\) 0 0
\(932\) 138.000 0.00485015
\(933\) −15048.0 −0.528027
\(934\) 13644.0 0.477993
\(935\) 27216.0 0.951934
\(936\) 6426.00 0.224402
\(937\) 13894.0 0.484415 0.242208 0.970224i \(-0.422128\pi\)
0.242208 + 0.970224i \(0.422128\pi\)
\(938\) 0 0
\(939\) −16206.0 −0.563219
\(940\) −4320.00 −0.149897
\(941\) −46758.0 −1.61984 −0.809919 0.586542i \(-0.800489\pi\)
−0.809919 + 0.586542i \(0.800489\pi\)
\(942\) 15246.0 0.527326
\(943\) 0 0
\(944\) −9372.00 −0.323128
\(945\) 0 0
\(946\) −28944.0 −0.994768
\(947\) 13812.0 0.473949 0.236974 0.971516i \(-0.423844\pi\)
0.236974 + 0.971516i \(0.423844\pi\)
\(948\) −3072.00 −0.105247
\(949\) 17068.0 0.583826
\(950\) −74028.0 −2.52820
\(951\) 30258.0 1.03174
\(952\) 0 0
\(953\) −58518.0 −1.98907 −0.994535 0.104402i \(-0.966707\pi\)
−0.994535 + 0.104402i \(0.966707\pi\)
\(954\) 13446.0 0.456321
\(955\) −20304.0 −0.687981
\(956\) 1896.00 0.0641433
\(957\) −11016.0 −0.372097
\(958\) −24192.0 −0.815875
\(959\) 0 0
\(960\) 23382.0 0.786095
\(961\) −4191.00 −0.140680
\(962\) −40596.0 −1.36057
\(963\) 108.000 0.00361397
\(964\) 3598.00 0.120211
\(965\) 72612.0 2.42224
\(966\) 0 0
\(967\) 19640.0 0.653133 0.326567 0.945174i \(-0.394108\pi\)
0.326567 + 0.945174i \(0.394108\pi\)
\(968\) −735.000 −0.0244047
\(969\) −15624.0 −0.517972
\(970\) −15444.0 −0.511213
\(971\) 58308.0 1.92708 0.963539 0.267568i \(-0.0862200\pi\)
0.963539 + 0.267568i \(0.0862200\pi\)
\(972\) 243.000 0.00801875
\(973\) 0 0
\(974\) −49848.0 −1.63987
\(975\) 20298.0 0.666724
\(976\) 28258.0 0.926759
\(977\) −23550.0 −0.771168 −0.385584 0.922673i \(-0.626000\pi\)
−0.385584 + 0.922673i \(0.626000\pi\)
\(978\) 26388.0 0.862776
\(979\) 12744.0 0.416037
\(980\) 0 0
\(981\) 13302.0 0.432926
\(982\) 21420.0 0.696069
\(983\) −15768.0 −0.511619 −0.255809 0.966727i \(-0.582342\pi\)
−0.255809 + 0.966727i \(0.582342\pi\)
\(984\) 20034.0 0.649045
\(985\) −23652.0 −0.765092
\(986\) 12852.0 0.415102
\(987\) 0 0
\(988\) 4216.00 0.135758
\(989\) 0 0
\(990\) 17496.0 0.561676
\(991\) 35264.0 1.13037 0.565186 0.824964i \(-0.308805\pi\)
0.565186 + 0.824964i \(0.308805\pi\)
\(992\) 7200.00 0.230444
\(993\) −24132.0 −0.771204
\(994\) 0 0
\(995\) −91728.0 −2.92259
\(996\) 612.000 0.0194698
\(997\) 29338.0 0.931940 0.465970 0.884801i \(-0.345706\pi\)
0.465970 + 0.884801i \(0.345706\pi\)
\(998\) 27372.0 0.868182
\(999\) 10746.0 0.340329
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 147.4.a.c.1.1 1
3.2 odd 2 441.4.a.j.1.1 1
4.3 odd 2 2352.4.a.r.1.1 1
7.2 even 3 147.4.e.g.67.1 2
7.3 odd 6 147.4.e.i.79.1 2
7.4 even 3 147.4.e.g.79.1 2
7.5 odd 6 147.4.e.i.67.1 2
7.6 odd 2 21.4.a.a.1.1 1
21.2 odd 6 441.4.e.d.361.1 2
21.5 even 6 441.4.e.b.361.1 2
21.11 odd 6 441.4.e.d.226.1 2
21.17 even 6 441.4.e.b.226.1 2
21.20 even 2 63.4.a.c.1.1 1
28.27 even 2 336.4.a.f.1.1 1
35.13 even 4 525.4.d.c.274.2 2
35.27 even 4 525.4.d.c.274.1 2
35.34 odd 2 525.4.a.g.1.1 1
56.13 odd 2 1344.4.a.ba.1.1 1
56.27 even 2 1344.4.a.n.1.1 1
84.83 odd 2 1008.4.a.v.1.1 1
105.104 even 2 1575.4.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.a.a.1.1 1 7.6 odd 2
63.4.a.c.1.1 1 21.20 even 2
147.4.a.c.1.1 1 1.1 even 1 trivial
147.4.e.g.67.1 2 7.2 even 3
147.4.e.g.79.1 2 7.4 even 3
147.4.e.i.67.1 2 7.5 odd 6
147.4.e.i.79.1 2 7.3 odd 6
336.4.a.f.1.1 1 28.27 even 2
441.4.a.j.1.1 1 3.2 odd 2
441.4.e.b.226.1 2 21.17 even 6
441.4.e.b.361.1 2 21.5 even 6
441.4.e.d.226.1 2 21.11 odd 6
441.4.e.d.361.1 2 21.2 odd 6
525.4.a.g.1.1 1 35.34 odd 2
525.4.d.c.274.1 2 35.27 even 4
525.4.d.c.274.2 2 35.13 even 4
1008.4.a.v.1.1 1 84.83 odd 2
1344.4.a.n.1.1 1 56.27 even 2
1344.4.a.ba.1.1 1 56.13 odd 2
1575.4.a.b.1.1 1 105.104 even 2
2352.4.a.r.1.1 1 4.3 odd 2