Properties

Label 210.4.a.a.1.1
Level 210
Weight 4
Character 210.1
Self dual yes
Analytic conductor 12.390
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 210.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.3904011012\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0\) of \(x\)
Character \(\chi\) \(=\) 210.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} +6.00000 q^{6} +7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} +6.00000 q^{6} +7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +10.0000 q^{10} +12.0000 q^{11} -12.0000 q^{12} +2.00000 q^{13} -14.0000 q^{14} +15.0000 q^{15} +16.0000 q^{16} -18.0000 q^{17} -18.0000 q^{18} +56.0000 q^{19} -20.0000 q^{20} -21.0000 q^{21} -24.0000 q^{22} -156.000 q^{23} +24.0000 q^{24} +25.0000 q^{25} -4.00000 q^{26} -27.0000 q^{27} +28.0000 q^{28} -186.000 q^{29} -30.0000 q^{30} -52.0000 q^{31} -32.0000 q^{32} -36.0000 q^{33} +36.0000 q^{34} -35.0000 q^{35} +36.0000 q^{36} -178.000 q^{37} -112.000 q^{38} -6.00000 q^{39} +40.0000 q^{40} -138.000 q^{41} +42.0000 q^{42} -412.000 q^{43} +48.0000 q^{44} -45.0000 q^{45} +312.000 q^{46} -456.000 q^{47} -48.0000 q^{48} +49.0000 q^{49} -50.0000 q^{50} +54.0000 q^{51} +8.00000 q^{52} -198.000 q^{53} +54.0000 q^{54} -60.0000 q^{55} -56.0000 q^{56} -168.000 q^{57} +372.000 q^{58} +348.000 q^{59} +60.0000 q^{60} +110.000 q^{61} +104.000 q^{62} +63.0000 q^{63} +64.0000 q^{64} -10.0000 q^{65} +72.0000 q^{66} -196.000 q^{67} -72.0000 q^{68} +468.000 q^{69} +70.0000 q^{70} -936.000 q^{71} -72.0000 q^{72} +542.000 q^{73} +356.000 q^{74} -75.0000 q^{75} +224.000 q^{76} +84.0000 q^{77} +12.0000 q^{78} +992.000 q^{79} -80.0000 q^{80} +81.0000 q^{81} +276.000 q^{82} -276.000 q^{83} -84.0000 q^{84} +90.0000 q^{85} +824.000 q^{86} +558.000 q^{87} -96.0000 q^{88} +630.000 q^{89} +90.0000 q^{90} +14.0000 q^{91} -624.000 q^{92} +156.000 q^{93} +912.000 q^{94} -280.000 q^{95} +96.0000 q^{96} +110.000 q^{97} -98.0000 q^{98} +108.000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) −3.00000 −0.577350
\(4\) 4.00000 0.500000
\(5\) −5.00000 −0.447214
\(6\) 6.00000 0.408248
\(7\) 7.00000 0.377964
\(8\) −8.00000 −0.353553
\(9\) 9.00000 0.333333
\(10\) 10.0000 0.316228
\(11\) 12.0000 0.328921 0.164461 0.986384i \(-0.447412\pi\)
0.164461 + 0.986384i \(0.447412\pi\)
\(12\) −12.0000 −0.288675
\(13\) 2.00000 0.0426692 0.0213346 0.999772i \(-0.493208\pi\)
0.0213346 + 0.999772i \(0.493208\pi\)
\(14\) −14.0000 −0.267261
\(15\) 15.0000 0.258199
\(16\) 16.0000 0.250000
\(17\) −18.0000 −0.256802 −0.128401 0.991722i \(-0.540985\pi\)
−0.128401 + 0.991722i \(0.540985\pi\)
\(18\) −18.0000 −0.235702
\(19\) 56.0000 0.676173 0.338086 0.941115i \(-0.390220\pi\)
0.338086 + 0.941115i \(0.390220\pi\)
\(20\) −20.0000 −0.223607
\(21\) −21.0000 −0.218218
\(22\) −24.0000 −0.232583
\(23\) −156.000 −1.41427 −0.707136 0.707078i \(-0.750013\pi\)
−0.707136 + 0.707078i \(0.750013\pi\)
\(24\) 24.0000 0.204124
\(25\) 25.0000 0.200000
\(26\) −4.00000 −0.0301717
\(27\) −27.0000 −0.192450
\(28\) 28.0000 0.188982
\(29\) −186.000 −1.19101 −0.595506 0.803351i \(-0.703048\pi\)
−0.595506 + 0.803351i \(0.703048\pi\)
\(30\) −30.0000 −0.182574
\(31\) −52.0000 −0.301273 −0.150637 0.988589i \(-0.548132\pi\)
−0.150637 + 0.988589i \(0.548132\pi\)
\(32\) −32.0000 −0.176777
\(33\) −36.0000 −0.189903
\(34\) 36.0000 0.181587
\(35\) −35.0000 −0.169031
\(36\) 36.0000 0.166667
\(37\) −178.000 −0.790892 −0.395446 0.918489i \(-0.629410\pi\)
−0.395446 + 0.918489i \(0.629410\pi\)
\(38\) −112.000 −0.478126
\(39\) −6.00000 −0.0246351
\(40\) 40.0000 0.158114
\(41\) −138.000 −0.525658 −0.262829 0.964842i \(-0.584656\pi\)
−0.262829 + 0.964842i \(0.584656\pi\)
\(42\) 42.0000 0.154303
\(43\) −412.000 −1.46115 −0.730575 0.682833i \(-0.760748\pi\)
−0.730575 + 0.682833i \(0.760748\pi\)
\(44\) 48.0000 0.164461
\(45\) −45.0000 −0.149071
\(46\) 312.000 1.00004
\(47\) −456.000 −1.41520 −0.707600 0.706613i \(-0.750222\pi\)
−0.707600 + 0.706613i \(0.750222\pi\)
\(48\) −48.0000 −0.144338
\(49\) 49.0000 0.142857
\(50\) −50.0000 −0.141421
\(51\) 54.0000 0.148265
\(52\) 8.00000 0.0213346
\(53\) −198.000 −0.513158 −0.256579 0.966523i \(-0.582595\pi\)
−0.256579 + 0.966523i \(0.582595\pi\)
\(54\) 54.0000 0.136083
\(55\) −60.0000 −0.147098
\(56\) −56.0000 −0.133631
\(57\) −168.000 −0.390388
\(58\) 372.000 0.842172
\(59\) 348.000 0.767894 0.383947 0.923355i \(-0.374565\pi\)
0.383947 + 0.923355i \(0.374565\pi\)
\(60\) 60.0000 0.129099
\(61\) 110.000 0.230886 0.115443 0.993314i \(-0.463171\pi\)
0.115443 + 0.993314i \(0.463171\pi\)
\(62\) 104.000 0.213032
\(63\) 63.0000 0.125988
\(64\) 64.0000 0.125000
\(65\) −10.0000 −0.0190823
\(66\) 72.0000 0.134282
\(67\) −196.000 −0.357391 −0.178696 0.983904i \(-0.557188\pi\)
−0.178696 + 0.983904i \(0.557188\pi\)
\(68\) −72.0000 −0.128401
\(69\) 468.000 0.816530
\(70\) 70.0000 0.119523
\(71\) −936.000 −1.56455 −0.782273 0.622936i \(-0.785940\pi\)
−0.782273 + 0.622936i \(0.785940\pi\)
\(72\) −72.0000 −0.117851
\(73\) 542.000 0.868990 0.434495 0.900674i \(-0.356927\pi\)
0.434495 + 0.900674i \(0.356927\pi\)
\(74\) 356.000 0.559245
\(75\) −75.0000 −0.115470
\(76\) 224.000 0.338086
\(77\) 84.0000 0.124321
\(78\) 12.0000 0.0174196
\(79\) 992.000 1.41277 0.706384 0.707829i \(-0.250325\pi\)
0.706384 + 0.707829i \(0.250325\pi\)
\(80\) −80.0000 −0.111803
\(81\) 81.0000 0.111111
\(82\) 276.000 0.371696
\(83\) −276.000 −0.364999 −0.182500 0.983206i \(-0.558419\pi\)
−0.182500 + 0.983206i \(0.558419\pi\)
\(84\) −84.0000 −0.109109
\(85\) 90.0000 0.114846
\(86\) 824.000 1.03319
\(87\) 558.000 0.687631
\(88\) −96.0000 −0.116291
\(89\) 630.000 0.750336 0.375168 0.926957i \(-0.377585\pi\)
0.375168 + 0.926957i \(0.377585\pi\)
\(90\) 90.0000 0.105409
\(91\) 14.0000 0.0161275
\(92\) −624.000 −0.707136
\(93\) 156.000 0.173940
\(94\) 912.000 1.00070
\(95\) −280.000 −0.302394
\(96\) 96.0000 0.102062
\(97\) 110.000 0.115142 0.0575712 0.998341i \(-0.481664\pi\)
0.0575712 + 0.998341i \(0.481664\pi\)
\(98\) −98.0000 −0.101015
\(99\) 108.000 0.109640
\(100\) 100.000 0.100000
\(101\) 570.000 0.561556 0.280778 0.959773i \(-0.409408\pi\)
0.280778 + 0.959773i \(0.409408\pi\)
\(102\) −108.000 −0.104839
\(103\) −304.000 −0.290816 −0.145408 0.989372i \(-0.546449\pi\)
−0.145408 + 0.989372i \(0.546449\pi\)
\(104\) −16.0000 −0.0150859
\(105\) 105.000 0.0975900
\(106\) 396.000 0.362858
\(107\) 216.000 0.195154 0.0975771 0.995228i \(-0.468891\pi\)
0.0975771 + 0.995228i \(0.468891\pi\)
\(108\) −108.000 −0.0962250
\(109\) 614.000 0.539546 0.269773 0.962924i \(-0.413051\pi\)
0.269773 + 0.962924i \(0.413051\pi\)
\(110\) 120.000 0.104014
\(111\) 534.000 0.456622
\(112\) 112.000 0.0944911
\(113\) −498.000 −0.414583 −0.207292 0.978279i \(-0.566465\pi\)
−0.207292 + 0.978279i \(0.566465\pi\)
\(114\) 336.000 0.276046
\(115\) 780.000 0.632482
\(116\) −744.000 −0.595506
\(117\) 18.0000 0.0142231
\(118\) −696.000 −0.542983
\(119\) −126.000 −0.0970622
\(120\) −120.000 −0.0912871
\(121\) −1187.00 −0.891811
\(122\) −220.000 −0.163261
\(123\) 414.000 0.303489
\(124\) −208.000 −0.150637
\(125\) −125.000 −0.0894427
\(126\) −126.000 −0.0890871
\(127\) −1888.00 −1.31916 −0.659578 0.751636i \(-0.729265\pi\)
−0.659578 + 0.751636i \(0.729265\pi\)
\(128\) −128.000 −0.0883883
\(129\) 1236.00 0.843595
\(130\) 20.0000 0.0134932
\(131\) −2892.00 −1.92882 −0.964409 0.264414i \(-0.914821\pi\)
−0.964409 + 0.264414i \(0.914821\pi\)
\(132\) −144.000 −0.0949514
\(133\) 392.000 0.255569
\(134\) 392.000 0.252714
\(135\) 135.000 0.0860663
\(136\) 144.000 0.0907934
\(137\) 822.000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) −936.000 −0.577374
\(139\) −376.000 −0.229438 −0.114719 0.993398i \(-0.536597\pi\)
−0.114719 + 0.993398i \(0.536597\pi\)
\(140\) −140.000 −0.0845154
\(141\) 1368.00 0.817067
\(142\) 1872.00 1.10630
\(143\) 24.0000 0.0140348
\(144\) 144.000 0.0833333
\(145\) 930.000 0.532637
\(146\) −1084.00 −0.614469
\(147\) −147.000 −0.0824786
\(148\) −712.000 −0.395446
\(149\) 3390.00 1.86389 0.931945 0.362600i \(-0.118111\pi\)
0.931945 + 0.362600i \(0.118111\pi\)
\(150\) 150.000 0.0816497
\(151\) −2968.00 −1.59955 −0.799776 0.600298i \(-0.795049\pi\)
−0.799776 + 0.600298i \(0.795049\pi\)
\(152\) −448.000 −0.239063
\(153\) −162.000 −0.0856008
\(154\) −168.000 −0.0879080
\(155\) 260.000 0.134734
\(156\) −24.0000 −0.0123176
\(157\) 1874.00 0.952621 0.476310 0.879277i \(-0.341974\pi\)
0.476310 + 0.879277i \(0.341974\pi\)
\(158\) −1984.00 −0.998978
\(159\) 594.000 0.296272
\(160\) 160.000 0.0790569
\(161\) −1092.00 −0.534544
\(162\) −162.000 −0.0785674
\(163\) 452.000 0.217199 0.108599 0.994086i \(-0.465363\pi\)
0.108599 + 0.994086i \(0.465363\pi\)
\(164\) −552.000 −0.262829
\(165\) 180.000 0.0849272
\(166\) 552.000 0.258093
\(167\) −1416.00 −0.656128 −0.328064 0.944656i \(-0.606396\pi\)
−0.328064 + 0.944656i \(0.606396\pi\)
\(168\) 168.000 0.0771517
\(169\) −2193.00 −0.998179
\(170\) −180.000 −0.0812081
\(171\) 504.000 0.225391
\(172\) −1648.00 −0.730575
\(173\) 426.000 0.187215 0.0936075 0.995609i \(-0.470160\pi\)
0.0936075 + 0.995609i \(0.470160\pi\)
\(174\) −1116.00 −0.486228
\(175\) 175.000 0.0755929
\(176\) 192.000 0.0822304
\(177\) −1044.00 −0.443344
\(178\) −1260.00 −0.530567
\(179\) 2700.00 1.12742 0.563708 0.825974i \(-0.309374\pi\)
0.563708 + 0.825974i \(0.309374\pi\)
\(180\) −180.000 −0.0745356
\(181\) −1978.00 −0.812285 −0.406142 0.913810i \(-0.633126\pi\)
−0.406142 + 0.913810i \(0.633126\pi\)
\(182\) −28.0000 −0.0114038
\(183\) −330.000 −0.133302
\(184\) 1248.00 0.500021
\(185\) 890.000 0.353698
\(186\) −312.000 −0.122994
\(187\) −216.000 −0.0844678
\(188\) −1824.00 −0.707600
\(189\) −189.000 −0.0727393
\(190\) 560.000 0.213825
\(191\) −2328.00 −0.881928 −0.440964 0.897525i \(-0.645363\pi\)
−0.440964 + 0.897525i \(0.645363\pi\)
\(192\) −192.000 −0.0721688
\(193\) −3166.00 −1.18080 −0.590398 0.807112i \(-0.701029\pi\)
−0.590398 + 0.807112i \(0.701029\pi\)
\(194\) −220.000 −0.0814179
\(195\) 30.0000 0.0110172
\(196\) 196.000 0.0714286
\(197\) −414.000 −0.149727 −0.0748637 0.997194i \(-0.523852\pi\)
−0.0748637 + 0.997194i \(0.523852\pi\)
\(198\) −216.000 −0.0775275
\(199\) −1636.00 −0.582779 −0.291389 0.956605i \(-0.594118\pi\)
−0.291389 + 0.956605i \(0.594118\pi\)
\(200\) −200.000 −0.0707107
\(201\) 588.000 0.206340
\(202\) −1140.00 −0.397080
\(203\) −1302.00 −0.450160
\(204\) 216.000 0.0741325
\(205\) 690.000 0.235081
\(206\) 608.000 0.205638
\(207\) −1404.00 −0.471424
\(208\) 32.0000 0.0106673
\(209\) 672.000 0.222408
\(210\) −210.000 −0.0690066
\(211\) −2860.00 −0.933130 −0.466565 0.884487i \(-0.654509\pi\)
−0.466565 + 0.884487i \(0.654509\pi\)
\(212\) −792.000 −0.256579
\(213\) 2808.00 0.903291
\(214\) −432.000 −0.137995
\(215\) 2060.00 0.653446
\(216\) 216.000 0.0680414
\(217\) −364.000 −0.113871
\(218\) −1228.00 −0.381517
\(219\) −1626.00 −0.501712
\(220\) −240.000 −0.0735491
\(221\) −36.0000 −0.0109576
\(222\) −1068.00 −0.322880
\(223\) −1096.00 −0.329119 −0.164560 0.986367i \(-0.552620\pi\)
−0.164560 + 0.986367i \(0.552620\pi\)
\(224\) −224.000 −0.0668153
\(225\) 225.000 0.0666667
\(226\) 996.000 0.293155
\(227\) 6276.00 1.83503 0.917517 0.397696i \(-0.130190\pi\)
0.917517 + 0.397696i \(0.130190\pi\)
\(228\) −672.000 −0.195194
\(229\) −754.000 −0.217580 −0.108790 0.994065i \(-0.534698\pi\)
−0.108790 + 0.994065i \(0.534698\pi\)
\(230\) −1560.00 −0.447232
\(231\) −252.000 −0.0717765
\(232\) 1488.00 0.421086
\(233\) 3870.00 1.08812 0.544060 0.839046i \(-0.316886\pi\)
0.544060 + 0.839046i \(0.316886\pi\)
\(234\) −36.0000 −0.0100572
\(235\) 2280.00 0.632897
\(236\) 1392.00 0.383947
\(237\) −2976.00 −0.815662
\(238\) 252.000 0.0686333
\(239\) 744.000 0.201361 0.100681 0.994919i \(-0.467898\pi\)
0.100681 + 0.994919i \(0.467898\pi\)
\(240\) 240.000 0.0645497
\(241\) 5474.00 1.46312 0.731559 0.681778i \(-0.238793\pi\)
0.731559 + 0.681778i \(0.238793\pi\)
\(242\) 2374.00 0.630605
\(243\) −243.000 −0.0641500
\(244\) 440.000 0.115443
\(245\) −245.000 −0.0638877
\(246\) −828.000 −0.214599
\(247\) 112.000 0.0288518
\(248\) 416.000 0.106516
\(249\) 828.000 0.210732
\(250\) 250.000 0.0632456
\(251\) 1980.00 0.497914 0.248957 0.968514i \(-0.419912\pi\)
0.248957 + 0.968514i \(0.419912\pi\)
\(252\) 252.000 0.0629941
\(253\) −1872.00 −0.465184
\(254\) 3776.00 0.932785
\(255\) −270.000 −0.0663061
\(256\) 256.000 0.0625000
\(257\) −210.000 −0.0509706 −0.0254853 0.999675i \(-0.508113\pi\)
−0.0254853 + 0.999675i \(0.508113\pi\)
\(258\) −2472.00 −0.596512
\(259\) −1246.00 −0.298929
\(260\) −40.0000 −0.00954113
\(261\) −1674.00 −0.397004
\(262\) 5784.00 1.36388
\(263\) −1428.00 −0.334807 −0.167404 0.985888i \(-0.553538\pi\)
−0.167404 + 0.985888i \(0.553538\pi\)
\(264\) 288.000 0.0671408
\(265\) 990.000 0.229491
\(266\) −784.000 −0.180715
\(267\) −1890.00 −0.433206
\(268\) −784.000 −0.178696
\(269\) 4122.00 0.934285 0.467143 0.884182i \(-0.345283\pi\)
0.467143 + 0.884182i \(0.345283\pi\)
\(270\) −270.000 −0.0608581
\(271\) 5780.00 1.29561 0.647804 0.761807i \(-0.275687\pi\)
0.647804 + 0.761807i \(0.275687\pi\)
\(272\) −288.000 −0.0642006
\(273\) −42.0000 −0.00931119
\(274\) −1644.00 −0.362473
\(275\) 300.000 0.0657843
\(276\) 1872.00 0.408265
\(277\) 4574.00 0.992148 0.496074 0.868280i \(-0.334774\pi\)
0.496074 + 0.868280i \(0.334774\pi\)
\(278\) 752.000 0.162237
\(279\) −468.000 −0.100424
\(280\) 280.000 0.0597614
\(281\) 3450.00 0.732419 0.366210 0.930532i \(-0.380655\pi\)
0.366210 + 0.930532i \(0.380655\pi\)
\(282\) −2736.00 −0.577753
\(283\) −700.000 −0.147034 −0.0735171 0.997294i \(-0.523422\pi\)
−0.0735171 + 0.997294i \(0.523422\pi\)
\(284\) −3744.00 −0.782273
\(285\) 840.000 0.174587
\(286\) −48.0000 −0.00992412
\(287\) −966.000 −0.198680
\(288\) −288.000 −0.0589256
\(289\) −4589.00 −0.934053
\(290\) −1860.00 −0.376631
\(291\) −330.000 −0.0664775
\(292\) 2168.00 0.434495
\(293\) 7170.00 1.42961 0.714805 0.699324i \(-0.246515\pi\)
0.714805 + 0.699324i \(0.246515\pi\)
\(294\) 294.000 0.0583212
\(295\) −1740.00 −0.343413
\(296\) 1424.00 0.279623
\(297\) −324.000 −0.0633010
\(298\) −6780.00 −1.31797
\(299\) −312.000 −0.0603459
\(300\) −300.000 −0.0577350
\(301\) −2884.00 −0.552262
\(302\) 5936.00 1.13105
\(303\) −1710.00 −0.324214
\(304\) 896.000 0.169043
\(305\) −550.000 −0.103255
\(306\) 324.000 0.0605289
\(307\) 6644.00 1.23516 0.617578 0.786509i \(-0.288114\pi\)
0.617578 + 0.786509i \(0.288114\pi\)
\(308\) 336.000 0.0621603
\(309\) 912.000 0.167902
\(310\) −520.000 −0.0952710
\(311\) −5376.00 −0.980209 −0.490104 0.871664i \(-0.663041\pi\)
−0.490104 + 0.871664i \(0.663041\pi\)
\(312\) 48.0000 0.00870982
\(313\) 2126.00 0.383925 0.191963 0.981402i \(-0.438515\pi\)
0.191963 + 0.981402i \(0.438515\pi\)
\(314\) −3748.00 −0.673605
\(315\) −315.000 −0.0563436
\(316\) 3968.00 0.706384
\(317\) 1074.00 0.190290 0.0951449 0.995463i \(-0.469669\pi\)
0.0951449 + 0.995463i \(0.469669\pi\)
\(318\) −1188.00 −0.209496
\(319\) −2232.00 −0.391749
\(320\) −320.000 −0.0559017
\(321\) −648.000 −0.112672
\(322\) 2184.00 0.377980
\(323\) −1008.00 −0.173643
\(324\) 324.000 0.0555556
\(325\) 50.0000 0.00853385
\(326\) −904.000 −0.153583
\(327\) −1842.00 −0.311507
\(328\) 1104.00 0.185848
\(329\) −3192.00 −0.534896
\(330\) −360.000 −0.0600526
\(331\) −2788.00 −0.462968 −0.231484 0.972839i \(-0.574358\pi\)
−0.231484 + 0.972839i \(0.574358\pi\)
\(332\) −1104.00 −0.182500
\(333\) −1602.00 −0.263631
\(334\) 2832.00 0.463953
\(335\) 980.000 0.159830
\(336\) −336.000 −0.0545545
\(337\) −6334.00 −1.02384 −0.511921 0.859032i \(-0.671066\pi\)
−0.511921 + 0.859032i \(0.671066\pi\)
\(338\) 4386.00 0.705819
\(339\) 1494.00 0.239360
\(340\) 360.000 0.0574228
\(341\) −624.000 −0.0990953
\(342\) −1008.00 −0.159375
\(343\) 343.000 0.0539949
\(344\) 3296.00 0.516594
\(345\) −2340.00 −0.365163
\(346\) −852.000 −0.132381
\(347\) −7032.00 −1.08789 −0.543945 0.839121i \(-0.683070\pi\)
−0.543945 + 0.839121i \(0.683070\pi\)
\(348\) 2232.00 0.343815
\(349\) −1474.00 −0.226079 −0.113039 0.993591i \(-0.536059\pi\)
−0.113039 + 0.993591i \(0.536059\pi\)
\(350\) −350.000 −0.0534522
\(351\) −54.0000 −0.00821170
\(352\) −384.000 −0.0581456
\(353\) 7950.00 1.19868 0.599342 0.800493i \(-0.295429\pi\)
0.599342 + 0.800493i \(0.295429\pi\)
\(354\) 2088.00 0.313491
\(355\) 4680.00 0.699686
\(356\) 2520.00 0.375168
\(357\) 378.000 0.0560389
\(358\) −5400.00 −0.797204
\(359\) 6624.00 0.973820 0.486910 0.873452i \(-0.338124\pi\)
0.486910 + 0.873452i \(0.338124\pi\)
\(360\) 360.000 0.0527046
\(361\) −3723.00 −0.542790
\(362\) 3956.00 0.574372
\(363\) 3561.00 0.514887
\(364\) 56.0000 0.00806373
\(365\) −2710.00 −0.388624
\(366\) 660.000 0.0942589
\(367\) 1784.00 0.253744 0.126872 0.991919i \(-0.459506\pi\)
0.126872 + 0.991919i \(0.459506\pi\)
\(368\) −2496.00 −0.353568
\(369\) −1242.00 −0.175219
\(370\) −1780.00 −0.250102
\(371\) −1386.00 −0.193956
\(372\) 624.000 0.0869701
\(373\) −1978.00 −0.274576 −0.137288 0.990531i \(-0.543839\pi\)
−0.137288 + 0.990531i \(0.543839\pi\)
\(374\) 432.000 0.0597278
\(375\) 375.000 0.0516398
\(376\) 3648.00 0.500349
\(377\) −372.000 −0.0508196
\(378\) 378.000 0.0514344
\(379\) −10780.0 −1.46103 −0.730516 0.682895i \(-0.760721\pi\)
−0.730516 + 0.682895i \(0.760721\pi\)
\(380\) −1120.00 −0.151197
\(381\) 5664.00 0.761616
\(382\) 4656.00 0.623617
\(383\) −5880.00 −0.784475 −0.392238 0.919864i \(-0.628299\pi\)
−0.392238 + 0.919864i \(0.628299\pi\)
\(384\) 384.000 0.0510310
\(385\) −420.000 −0.0555979
\(386\) 6332.00 0.834949
\(387\) −3708.00 −0.487050
\(388\) 440.000 0.0575712
\(389\) 6438.00 0.839125 0.419562 0.907726i \(-0.362184\pi\)
0.419562 + 0.907726i \(0.362184\pi\)
\(390\) −60.0000 −0.00779030
\(391\) 2808.00 0.363188
\(392\) −392.000 −0.0505076
\(393\) 8676.00 1.11360
\(394\) 828.000 0.105873
\(395\) −4960.00 −0.631809
\(396\) 432.000 0.0548202
\(397\) 2954.00 0.373443 0.186722 0.982413i \(-0.440214\pi\)
0.186722 + 0.982413i \(0.440214\pi\)
\(398\) 3272.00 0.412087
\(399\) −1176.00 −0.147553
\(400\) 400.000 0.0500000
\(401\) −5574.00 −0.694145 −0.347073 0.937838i \(-0.612824\pi\)
−0.347073 + 0.937838i \(0.612824\pi\)
\(402\) −1176.00 −0.145904
\(403\) −104.000 −0.0128551
\(404\) 2280.00 0.280778
\(405\) −405.000 −0.0496904
\(406\) 2604.00 0.318311
\(407\) −2136.00 −0.260141
\(408\) −432.000 −0.0524196
\(409\) −15406.0 −1.86254 −0.931269 0.364333i \(-0.881297\pi\)
−0.931269 + 0.364333i \(0.881297\pi\)
\(410\) −1380.00 −0.166228
\(411\) −2466.00 −0.295958
\(412\) −1216.00 −0.145408
\(413\) 2436.00 0.290237
\(414\) 2808.00 0.333347
\(415\) 1380.00 0.163233
\(416\) −64.0000 −0.00754293
\(417\) 1128.00 0.132466
\(418\) −1344.00 −0.157266
\(419\) 2940.00 0.342789 0.171394 0.985203i \(-0.445173\pi\)
0.171394 + 0.985203i \(0.445173\pi\)
\(420\) 420.000 0.0487950
\(421\) 254.000 0.0294043 0.0147021 0.999892i \(-0.495320\pi\)
0.0147021 + 0.999892i \(0.495320\pi\)
\(422\) 5720.00 0.659823
\(423\) −4104.00 −0.471734
\(424\) 1584.00 0.181429
\(425\) −450.000 −0.0513605
\(426\) −5616.00 −0.638723
\(427\) 770.000 0.0872668
\(428\) 864.000 0.0975771
\(429\) −72.0000 −0.00810301
\(430\) −4120.00 −0.462056
\(431\) −13248.0 −1.48059 −0.740294 0.672283i \(-0.765314\pi\)
−0.740294 + 0.672283i \(0.765314\pi\)
\(432\) −432.000 −0.0481125
\(433\) 16598.0 1.84215 0.921073 0.389391i \(-0.127314\pi\)
0.921073 + 0.389391i \(0.127314\pi\)
\(434\) 728.000 0.0805187
\(435\) −2790.00 −0.307518
\(436\) 2456.00 0.269773
\(437\) −8736.00 −0.956292
\(438\) 3252.00 0.354764
\(439\) −6532.00 −0.710149 −0.355074 0.934838i \(-0.615544\pi\)
−0.355074 + 0.934838i \(0.615544\pi\)
\(440\) 480.000 0.0520071
\(441\) 441.000 0.0476190
\(442\) 72.0000 0.00774817
\(443\) 12216.0 1.31016 0.655079 0.755561i \(-0.272635\pi\)
0.655079 + 0.755561i \(0.272635\pi\)
\(444\) 2136.00 0.228311
\(445\) −3150.00 −0.335560
\(446\) 2192.00 0.232722
\(447\) −10170.0 −1.07612
\(448\) 448.000 0.0472456
\(449\) 306.000 0.0321627 0.0160813 0.999871i \(-0.494881\pi\)
0.0160813 + 0.999871i \(0.494881\pi\)
\(450\) −450.000 −0.0471405
\(451\) −1656.00 −0.172900
\(452\) −1992.00 −0.207292
\(453\) 8904.00 0.923502
\(454\) −12552.0 −1.29757
\(455\) −70.0000 −0.00721242
\(456\) 1344.00 0.138023
\(457\) −6046.00 −0.618862 −0.309431 0.950922i \(-0.600139\pi\)
−0.309431 + 0.950922i \(0.600139\pi\)
\(458\) 1508.00 0.153852
\(459\) 486.000 0.0494217
\(460\) 3120.00 0.316241
\(461\) 7122.00 0.719533 0.359766 0.933042i \(-0.382856\pi\)
0.359766 + 0.933042i \(0.382856\pi\)
\(462\) 504.000 0.0507537
\(463\) −11248.0 −1.12903 −0.564513 0.825424i \(-0.690936\pi\)
−0.564513 + 0.825424i \(0.690936\pi\)
\(464\) −2976.00 −0.297753
\(465\) −780.000 −0.0777885
\(466\) −7740.00 −0.769418
\(467\) −18252.0 −1.80857 −0.904285 0.426930i \(-0.859595\pi\)
−0.904285 + 0.426930i \(0.859595\pi\)
\(468\) 72.0000 0.00711154
\(469\) −1372.00 −0.135081
\(470\) −4560.00 −0.447526
\(471\) −5622.00 −0.549996
\(472\) −2784.00 −0.271491
\(473\) −4944.00 −0.480603
\(474\) 5952.00 0.576760
\(475\) 1400.00 0.135235
\(476\) −504.000 −0.0485311
\(477\) −1782.00 −0.171053
\(478\) −1488.00 −0.142384
\(479\) −18168.0 −1.73302 −0.866511 0.499159i \(-0.833642\pi\)
−0.866511 + 0.499159i \(0.833642\pi\)
\(480\) −480.000 −0.0456435
\(481\) −356.000 −0.0337468
\(482\) −10948.0 −1.03458
\(483\) 3276.00 0.308619
\(484\) −4748.00 −0.445905
\(485\) −550.000 −0.0514932
\(486\) 486.000 0.0453609
\(487\) 19856.0 1.84756 0.923780 0.382925i \(-0.125083\pi\)
0.923780 + 0.382925i \(0.125083\pi\)
\(488\) −880.000 −0.0816306
\(489\) −1356.00 −0.125400
\(490\) 490.000 0.0451754
\(491\) 11220.0 1.03127 0.515633 0.856810i \(-0.327557\pi\)
0.515633 + 0.856810i \(0.327557\pi\)
\(492\) 1656.00 0.151744
\(493\) 3348.00 0.305855
\(494\) −224.000 −0.0204013
\(495\) −540.000 −0.0490327
\(496\) −832.000 −0.0753184
\(497\) −6552.00 −0.591343
\(498\) −1656.00 −0.149010
\(499\) −9268.00 −0.831448 −0.415724 0.909491i \(-0.636472\pi\)
−0.415724 + 0.909491i \(0.636472\pi\)
\(500\) −500.000 −0.0447214
\(501\) 4248.00 0.378816
\(502\) −3960.00 −0.352079
\(503\) 18576.0 1.64665 0.823323 0.567573i \(-0.192118\pi\)
0.823323 + 0.567573i \(0.192118\pi\)
\(504\) −504.000 −0.0445435
\(505\) −2850.00 −0.251135
\(506\) 3744.00 0.328935
\(507\) 6579.00 0.576299
\(508\) −7552.00 −0.659578
\(509\) −11190.0 −0.974436 −0.487218 0.873280i \(-0.661988\pi\)
−0.487218 + 0.873280i \(0.661988\pi\)
\(510\) 540.000 0.0468855
\(511\) 3794.00 0.328448
\(512\) −512.000 −0.0441942
\(513\) −1512.00 −0.130129
\(514\) 420.000 0.0360416
\(515\) 1520.00 0.130057
\(516\) 4944.00 0.421797
\(517\) −5472.00 −0.465490
\(518\) 2492.00 0.211375
\(519\) −1278.00 −0.108089
\(520\) 80.0000 0.00674660
\(521\) −306.000 −0.0257315 −0.0128657 0.999917i \(-0.504095\pi\)
−0.0128657 + 0.999917i \(0.504095\pi\)
\(522\) 3348.00 0.280724
\(523\) 17444.0 1.45846 0.729228 0.684270i \(-0.239879\pi\)
0.729228 + 0.684270i \(0.239879\pi\)
\(524\) −11568.0 −0.964409
\(525\) −525.000 −0.0436436
\(526\) 2856.00 0.236744
\(527\) 936.000 0.0773677
\(528\) −576.000 −0.0474757
\(529\) 12169.0 1.00016
\(530\) −1980.00 −0.162275
\(531\) 3132.00 0.255965
\(532\) 1568.00 0.127785
\(533\) −276.000 −0.0224294
\(534\) 3780.00 0.306323
\(535\) −1080.00 −0.0872756
\(536\) 1568.00 0.126357
\(537\) −8100.00 −0.650914
\(538\) −8244.00 −0.660640
\(539\) 588.000 0.0469888
\(540\) 540.000 0.0430331
\(541\) −538.000 −0.0427549 −0.0213775 0.999771i \(-0.506805\pi\)
−0.0213775 + 0.999771i \(0.506805\pi\)
\(542\) −11560.0 −0.916134
\(543\) 5934.00 0.468973
\(544\) 576.000 0.0453967
\(545\) −3070.00 −0.241292
\(546\) 84.0000 0.00658401
\(547\) 19820.0 1.54925 0.774627 0.632418i \(-0.217938\pi\)
0.774627 + 0.632418i \(0.217938\pi\)
\(548\) 3288.00 0.256307
\(549\) 990.000 0.0769621
\(550\) −600.000 −0.0465165
\(551\) −10416.0 −0.805329
\(552\) −3744.00 −0.288687
\(553\) 6944.00 0.533976
\(554\) −9148.00 −0.701555
\(555\) −2670.00 −0.204208
\(556\) −1504.00 −0.114719
\(557\) −16686.0 −1.26932 −0.634658 0.772794i \(-0.718859\pi\)
−0.634658 + 0.772794i \(0.718859\pi\)
\(558\) 936.000 0.0710108
\(559\) −824.000 −0.0623461
\(560\) −560.000 −0.0422577
\(561\) 648.000 0.0487675
\(562\) −6900.00 −0.517898
\(563\) −16788.0 −1.25671 −0.628357 0.777925i \(-0.716272\pi\)
−0.628357 + 0.777925i \(0.716272\pi\)
\(564\) 5472.00 0.408533
\(565\) 2490.00 0.185407
\(566\) 1400.00 0.103969
\(567\) 567.000 0.0419961
\(568\) 7488.00 0.553151
\(569\) 15906.0 1.17191 0.585953 0.810345i \(-0.300720\pi\)
0.585953 + 0.810345i \(0.300720\pi\)
\(570\) −1680.00 −0.123452
\(571\) 17084.0 1.25209 0.626045 0.779787i \(-0.284673\pi\)
0.626045 + 0.779787i \(0.284673\pi\)
\(572\) 96.0000 0.00701742
\(573\) 6984.00 0.509181
\(574\) 1932.00 0.140488
\(575\) −3900.00 −0.282854
\(576\) 576.000 0.0416667
\(577\) 25382.0 1.83131 0.915656 0.401964i \(-0.131672\pi\)
0.915656 + 0.401964i \(0.131672\pi\)
\(578\) 9178.00 0.660475
\(579\) 9498.00 0.681733
\(580\) 3720.00 0.266318
\(581\) −1932.00 −0.137957
\(582\) 660.000 0.0470067
\(583\) −2376.00 −0.168789
\(584\) −4336.00 −0.307235
\(585\) −90.0000 −0.00636076
\(586\) −14340.0 −1.01089
\(587\) 13764.0 0.967804 0.483902 0.875122i \(-0.339219\pi\)
0.483902 + 0.875122i \(0.339219\pi\)
\(588\) −588.000 −0.0412393
\(589\) −2912.00 −0.203713
\(590\) 3480.00 0.242829
\(591\) 1242.00 0.0864451
\(592\) −2848.00 −0.197723
\(593\) −13266.0 −0.918667 −0.459333 0.888264i \(-0.651912\pi\)
−0.459333 + 0.888264i \(0.651912\pi\)
\(594\) 648.000 0.0447605
\(595\) 630.000 0.0434075
\(596\) 13560.0 0.931945
\(597\) 4908.00 0.336467
\(598\) 624.000 0.0426710
\(599\) 9600.00 0.654834 0.327417 0.944880i \(-0.393822\pi\)
0.327417 + 0.944880i \(0.393822\pi\)
\(600\) 600.000 0.0408248
\(601\) −19582.0 −1.32906 −0.664531 0.747261i \(-0.731369\pi\)
−0.664531 + 0.747261i \(0.731369\pi\)
\(602\) 5768.00 0.390509
\(603\) −1764.00 −0.119130
\(604\) −11872.0 −0.799776
\(605\) 5935.00 0.398830
\(606\) 3420.00 0.229254
\(607\) 3944.00 0.263727 0.131863 0.991268i \(-0.457904\pi\)
0.131863 + 0.991268i \(0.457904\pi\)
\(608\) −1792.00 −0.119532
\(609\) 3906.00 0.259900
\(610\) 1100.00 0.0730126
\(611\) −912.000 −0.0603855
\(612\) −648.000 −0.0428004
\(613\) 20846.0 1.37351 0.686755 0.726889i \(-0.259034\pi\)
0.686755 + 0.726889i \(0.259034\pi\)
\(614\) −13288.0 −0.873388
\(615\) −2070.00 −0.135724
\(616\) −672.000 −0.0439540
\(617\) 15342.0 1.00105 0.500523 0.865723i \(-0.333141\pi\)
0.500523 + 0.865723i \(0.333141\pi\)
\(618\) −1824.00 −0.118725
\(619\) −5128.00 −0.332975 −0.166488 0.986044i \(-0.553243\pi\)
−0.166488 + 0.986044i \(0.553243\pi\)
\(620\) 1040.00 0.0673668
\(621\) 4212.00 0.272177
\(622\) 10752.0 0.693112
\(623\) 4410.00 0.283600
\(624\) −96.0000 −0.00615878
\(625\) 625.000 0.0400000
\(626\) −4252.00 −0.271476
\(627\) −2016.00 −0.128407
\(628\) 7496.00 0.476310
\(629\) 3204.00 0.203103
\(630\) 630.000 0.0398410
\(631\) 31016.0 1.95678 0.978389 0.206771i \(-0.0662954\pi\)
0.978389 + 0.206771i \(0.0662954\pi\)
\(632\) −7936.00 −0.499489
\(633\) 8580.00 0.538743
\(634\) −2148.00 −0.134555
\(635\) 9440.00 0.589945
\(636\) 2376.00 0.148136
\(637\) 98.0000 0.00609561
\(638\) 4464.00 0.277009
\(639\) −8424.00 −0.521515
\(640\) 640.000 0.0395285
\(641\) 15474.0 0.953489 0.476744 0.879042i \(-0.341817\pi\)
0.476744 + 0.879042i \(0.341817\pi\)
\(642\) 1296.00 0.0796714
\(643\) 24644.0 1.51145 0.755727 0.654887i \(-0.227284\pi\)
0.755727 + 0.654887i \(0.227284\pi\)
\(644\) −4368.00 −0.267272
\(645\) −6180.00 −0.377267
\(646\) 2016.00 0.122784
\(647\) −16632.0 −1.01062 −0.505310 0.862938i \(-0.668622\pi\)
−0.505310 + 0.862938i \(0.668622\pi\)
\(648\) −648.000 −0.0392837
\(649\) 4176.00 0.252577
\(650\) −100.000 −0.00603434
\(651\) 1092.00 0.0657432
\(652\) 1808.00 0.108599
\(653\) −10542.0 −0.631762 −0.315881 0.948799i \(-0.602300\pi\)
−0.315881 + 0.948799i \(0.602300\pi\)
\(654\) 3684.00 0.220269
\(655\) 14460.0 0.862594
\(656\) −2208.00 −0.131415
\(657\) 4878.00 0.289663
\(658\) 6384.00 0.378228
\(659\) 15276.0 0.902987 0.451494 0.892274i \(-0.350891\pi\)
0.451494 + 0.892274i \(0.350891\pi\)
\(660\) 720.000 0.0424636
\(661\) 1478.00 0.0869706 0.0434853 0.999054i \(-0.486154\pi\)
0.0434853 + 0.999054i \(0.486154\pi\)
\(662\) 5576.00 0.327368
\(663\) 108.000 0.00632635
\(664\) 2208.00 0.129047
\(665\) −1960.00 −0.114294
\(666\) 3204.00 0.186415
\(667\) 29016.0 1.68441
\(668\) −5664.00 −0.328064
\(669\) 3288.00 0.190017
\(670\) −1960.00 −0.113017
\(671\) 1320.00 0.0759434
\(672\) 672.000 0.0385758
\(673\) −19366.0 −1.10922 −0.554610 0.832111i \(-0.687132\pi\)
−0.554610 + 0.832111i \(0.687132\pi\)
\(674\) 12668.0 0.723966
\(675\) −675.000 −0.0384900
\(676\) −8772.00 −0.499090
\(677\) −21390.0 −1.21430 −0.607152 0.794585i \(-0.707688\pi\)
−0.607152 + 0.794585i \(0.707688\pi\)
\(678\) −2988.00 −0.169253
\(679\) 770.000 0.0435197
\(680\) −720.000 −0.0406040
\(681\) −18828.0 −1.05946
\(682\) 1248.00 0.0700710
\(683\) −21672.0 −1.21414 −0.607069 0.794649i \(-0.707655\pi\)
−0.607069 + 0.794649i \(0.707655\pi\)
\(684\) 2016.00 0.112695
\(685\) −4110.00 −0.229248
\(686\) −686.000 −0.0381802
\(687\) 2262.00 0.125620
\(688\) −6592.00 −0.365287
\(689\) −396.000 −0.0218961
\(690\) 4680.00 0.258210
\(691\) −5992.00 −0.329879 −0.164940 0.986304i \(-0.552743\pi\)
−0.164940 + 0.986304i \(0.552743\pi\)
\(692\) 1704.00 0.0936075
\(693\) 756.000 0.0414402
\(694\) 14064.0 0.769254
\(695\) 1880.00 0.102608
\(696\) −4464.00 −0.243114
\(697\) 2484.00 0.134990
\(698\) 2948.00 0.159862
\(699\) −11610.0 −0.628227
\(700\) 700.000 0.0377964
\(701\) 17766.0 0.957222 0.478611 0.878027i \(-0.341140\pi\)
0.478611 + 0.878027i \(0.341140\pi\)
\(702\) 108.000 0.00580655
\(703\) −9968.00 −0.534780
\(704\) 768.000 0.0411152
\(705\) −6840.00 −0.365403
\(706\) −15900.0 −0.847598
\(707\) 3990.00 0.212248
\(708\) −4176.00 −0.221672
\(709\) −24514.0 −1.29851 −0.649254 0.760571i \(-0.724919\pi\)
−0.649254 + 0.760571i \(0.724919\pi\)
\(710\) −9360.00 −0.494753
\(711\) 8928.00 0.470923
\(712\) −5040.00 −0.265284
\(713\) 8112.00 0.426082
\(714\) −756.000 −0.0396255
\(715\) −120.000 −0.00627657
\(716\) 10800.0 0.563708
\(717\) −2232.00 −0.116256
\(718\) −13248.0 −0.688595
\(719\) −13176.0 −0.683424 −0.341712 0.939805i \(-0.611007\pi\)
−0.341712 + 0.939805i \(0.611007\pi\)
\(720\) −720.000 −0.0372678
\(721\) −2128.00 −0.109918
\(722\) 7446.00 0.383811
\(723\) −16422.0 −0.844731
\(724\) −7912.00 −0.406142
\(725\) −4650.00 −0.238202
\(726\) −7122.00 −0.364080
\(727\) 20792.0 1.06071 0.530353 0.847777i \(-0.322060\pi\)
0.530353 + 0.847777i \(0.322060\pi\)
\(728\) −112.000 −0.00570192
\(729\) 729.000 0.0370370
\(730\) 5420.00 0.274799
\(731\) 7416.00 0.375227
\(732\) −1320.00 −0.0666511
\(733\) −21742.0 −1.09558 −0.547789 0.836616i \(-0.684530\pi\)
−0.547789 + 0.836616i \(0.684530\pi\)
\(734\) −3568.00 −0.179424
\(735\) 735.000 0.0368856
\(736\) 4992.00 0.250010
\(737\) −2352.00 −0.117554
\(738\) 2484.00 0.123899
\(739\) 39044.0 1.94351 0.971757 0.235984i \(-0.0758313\pi\)
0.971757 + 0.235984i \(0.0758313\pi\)
\(740\) 3560.00 0.176849
\(741\) −336.000 −0.0166576
\(742\) 2772.00 0.137147
\(743\) −31116.0 −1.53639 −0.768193 0.640218i \(-0.778844\pi\)
−0.768193 + 0.640218i \(0.778844\pi\)
\(744\) −1248.00 −0.0614972
\(745\) −16950.0 −0.833557
\(746\) 3956.00 0.194155
\(747\) −2484.00 −0.121666
\(748\) −864.000 −0.0422339
\(749\) 1512.00 0.0737614
\(750\) −750.000 −0.0365148
\(751\) −29320.0 −1.42464 −0.712318 0.701857i \(-0.752355\pi\)
−0.712318 + 0.701857i \(0.752355\pi\)
\(752\) −7296.00 −0.353800
\(753\) −5940.00 −0.287471
\(754\) 744.000 0.0359349
\(755\) 14840.0 0.715342
\(756\) −756.000 −0.0363696
\(757\) −2266.00 −0.108797 −0.0543984 0.998519i \(-0.517324\pi\)
−0.0543984 + 0.998519i \(0.517324\pi\)
\(758\) 21560.0 1.03311
\(759\) 5616.00 0.268574
\(760\) 2240.00 0.106912
\(761\) −29946.0 −1.42647 −0.713234 0.700926i \(-0.752770\pi\)
−0.713234 + 0.700926i \(0.752770\pi\)
\(762\) −11328.0 −0.538543
\(763\) 4298.00 0.203929
\(764\) −9312.00 −0.440964
\(765\) 810.000 0.0382818
\(766\) 11760.0 0.554708
\(767\) 696.000 0.0327655
\(768\) −768.000 −0.0360844
\(769\) −23110.0 −1.08370 −0.541852 0.840474i \(-0.682277\pi\)
−0.541852 + 0.840474i \(0.682277\pi\)
\(770\) 840.000 0.0393136
\(771\) 630.000 0.0294279
\(772\) −12664.0 −0.590398
\(773\) −31950.0 −1.48663 −0.743313 0.668944i \(-0.766747\pi\)
−0.743313 + 0.668944i \(0.766747\pi\)
\(774\) 7416.00 0.344396
\(775\) −1300.00 −0.0602547
\(776\) −880.000 −0.0407090
\(777\) 3738.00 0.172587
\(778\) −12876.0 −0.593351
\(779\) −7728.00 −0.355436
\(780\) 120.000 0.00550858
\(781\) −11232.0 −0.514613
\(782\) −5616.00 −0.256813
\(783\) 5022.00 0.229210
\(784\) 784.000 0.0357143
\(785\) −9370.00 −0.426025
\(786\) −17352.0 −0.787437
\(787\) 6284.00 0.284626 0.142313 0.989822i \(-0.454546\pi\)
0.142313 + 0.989822i \(0.454546\pi\)
\(788\) −1656.00 −0.0748637
\(789\) 4284.00 0.193301
\(790\) 9920.00 0.446757
\(791\) −3486.00 −0.156698
\(792\) −864.000 −0.0387638
\(793\) 220.000 0.00985174
\(794\) −5908.00 −0.264064
\(795\) −2970.00 −0.132497
\(796\) −6544.00 −0.291389
\(797\) 5946.00 0.264264 0.132132 0.991232i \(-0.457818\pi\)
0.132132 + 0.991232i \(0.457818\pi\)
\(798\) 2352.00 0.104336
\(799\) 8208.00 0.363427
\(800\) −800.000 −0.0353553
\(801\) 5670.00 0.250112
\(802\) 11148.0 0.490835
\(803\) 6504.00 0.285830
\(804\) 2352.00 0.103170
\(805\) 5460.00 0.239056
\(806\) 208.000 0.00908993
\(807\) −12366.0 −0.539410
\(808\) −4560.00 −0.198540
\(809\) 27090.0 1.17730 0.588649 0.808389i \(-0.299660\pi\)
0.588649 + 0.808389i \(0.299660\pi\)
\(810\) 810.000 0.0351364
\(811\) −20104.0 −0.870465 −0.435232 0.900318i \(-0.643334\pi\)
−0.435232 + 0.900318i \(0.643334\pi\)
\(812\) −5208.00 −0.225080
\(813\) −17340.0 −0.748020
\(814\) 4272.00 0.183948
\(815\) −2260.00 −0.0971342
\(816\) 864.000 0.0370662
\(817\) −23072.0 −0.987989
\(818\) 30812.0 1.31701
\(819\) 126.000 0.00537582
\(820\) 2760.00 0.117541
\(821\) 7302.00 0.310404 0.155202 0.987883i \(-0.450397\pi\)
0.155202 + 0.987883i \(0.450397\pi\)
\(822\) 4932.00 0.209274
\(823\) −24136.0 −1.02227 −0.511135 0.859500i \(-0.670775\pi\)
−0.511135 + 0.859500i \(0.670775\pi\)
\(824\) 2432.00 0.102819
\(825\) −900.000 −0.0379806
\(826\) −4872.00 −0.205228
\(827\) 22680.0 0.953641 0.476820 0.879001i \(-0.341789\pi\)
0.476820 + 0.879001i \(0.341789\pi\)
\(828\) −5616.00 −0.235712
\(829\) −20338.0 −0.852072 −0.426036 0.904706i \(-0.640090\pi\)
−0.426036 + 0.904706i \(0.640090\pi\)
\(830\) −2760.00 −0.115423
\(831\) −13722.0 −0.572817
\(832\) 128.000 0.00533366
\(833\) −882.000 −0.0366861
\(834\) −2256.00 −0.0936677
\(835\) 7080.00 0.293429
\(836\) 2688.00 0.111204
\(837\) 1404.00 0.0579801
\(838\) −5880.00 −0.242388
\(839\) 6600.00 0.271582 0.135791 0.990738i \(-0.456642\pi\)
0.135791 + 0.990738i \(0.456642\pi\)
\(840\) −840.000 −0.0345033
\(841\) 10207.0 0.418508
\(842\) −508.000 −0.0207920
\(843\) −10350.0 −0.422862
\(844\) −11440.0 −0.466565
\(845\) 10965.0 0.446399
\(846\) 8208.00 0.333566
\(847\) −8309.00 −0.337073
\(848\) −3168.00 −0.128290
\(849\) 2100.00 0.0848902
\(850\) 900.000 0.0363173
\(851\) 27768.0 1.11854
\(852\) 11232.0 0.451646
\(853\) −40174.0 −1.61258 −0.806290 0.591520i \(-0.798528\pi\)
−0.806290 + 0.591520i \(0.798528\pi\)
\(854\) −1540.00 −0.0617069
\(855\) −2520.00 −0.100798
\(856\) −1728.00 −0.0689975
\(857\) −20778.0 −0.828195 −0.414097 0.910233i \(-0.635903\pi\)
−0.414097 + 0.910233i \(0.635903\pi\)
\(858\) 144.000 0.00572970
\(859\) 7400.00 0.293929 0.146964 0.989142i \(-0.453050\pi\)
0.146964 + 0.989142i \(0.453050\pi\)
\(860\) 8240.00 0.326723
\(861\) 2898.00 0.114708
\(862\) 26496.0 1.04693
\(863\) 684.000 0.0269799 0.0134899 0.999909i \(-0.495706\pi\)
0.0134899 + 0.999909i \(0.495706\pi\)
\(864\) 864.000 0.0340207
\(865\) −2130.00 −0.0837251
\(866\) −33196.0 −1.30259
\(867\) 13767.0 0.539275
\(868\) −1456.00 −0.0569353
\(869\) 11904.0 0.464690
\(870\) 5580.00 0.217448
\(871\) −392.000 −0.0152496
\(872\) −4912.00 −0.190758
\(873\) 990.000 0.0383808
\(874\) 17472.0 0.676200
\(875\) −875.000 −0.0338062
\(876\) −6504.00 −0.250856
\(877\) −9754.00 −0.375563 −0.187782 0.982211i \(-0.560130\pi\)
−0.187782 + 0.982211i \(0.560130\pi\)
\(878\) 13064.0 0.502151
\(879\) −21510.0 −0.825386
\(880\) −960.000 −0.0367745
\(881\) 14310.0 0.547237 0.273619 0.961838i \(-0.411779\pi\)
0.273619 + 0.961838i \(0.411779\pi\)
\(882\) −882.000 −0.0336718
\(883\) −14092.0 −0.537071 −0.268535 0.963270i \(-0.586540\pi\)
−0.268535 + 0.963270i \(0.586540\pi\)
\(884\) −144.000 −0.00547878
\(885\) 5220.00 0.198269
\(886\) −24432.0 −0.926421
\(887\) 45600.0 1.72615 0.863077 0.505073i \(-0.168534\pi\)
0.863077 + 0.505073i \(0.168534\pi\)
\(888\) −4272.00 −0.161440
\(889\) −13216.0 −0.498594
\(890\) 6300.00 0.237277
\(891\) 972.000 0.0365468
\(892\) −4384.00 −0.164560
\(893\) −25536.0 −0.956920
\(894\) 20340.0 0.760930
\(895\) −13500.0 −0.504196
\(896\) −896.000 −0.0334077
\(897\) 936.000 0.0348407
\(898\) −612.000 −0.0227424
\(899\) 9672.00 0.358820
\(900\) 900.000 0.0333333
\(901\) 3564.00 0.131780
\(902\) 3312.00 0.122259
\(903\) 8652.00 0.318849
\(904\) 3984.00 0.146577
\(905\) 9890.00 0.363265
\(906\) −17808.0 −0.653015
\(907\) 8012.00 0.293312 0.146656 0.989188i \(-0.453149\pi\)
0.146656 + 0.989188i \(0.453149\pi\)
\(908\) 25104.0 0.917517
\(909\) 5130.00 0.187185
\(910\) 140.000 0.00509995
\(911\) 2136.00 0.0776826 0.0388413 0.999245i \(-0.487633\pi\)
0.0388413 + 0.999245i \(0.487633\pi\)
\(912\) −2688.00 −0.0975971
\(913\) −3312.00 −0.120056
\(914\) 12092.0 0.437602
\(915\) 1650.00 0.0596146
\(916\) −3016.00 −0.108790
\(917\) −20244.0 −0.729025
\(918\) −972.000 −0.0349464
\(919\) −15280.0 −0.548466 −0.274233 0.961663i \(-0.588424\pi\)
−0.274233 + 0.961663i \(0.588424\pi\)
\(920\) −6240.00 −0.223616
\(921\) −19932.0 −0.713118
\(922\) −14244.0 −0.508787
\(923\) −1872.00 −0.0667580
\(924\) −1008.00 −0.0358883
\(925\) −4450.00 −0.158178
\(926\) 22496.0 0.798342
\(927\) −2736.00 −0.0969385
\(928\) 5952.00 0.210543
\(929\) 20910.0 0.738466 0.369233 0.929337i \(-0.379620\pi\)
0.369233 + 0.929337i \(0.379620\pi\)
\(930\) 1560.00 0.0550047
\(931\) 2744.00 0.0965961
\(932\) 15480.0 0.544060
\(933\) 16128.0 0.565924
\(934\) 36504.0 1.27885
\(935\) 1080.00 0.0377752
\(936\) −144.000 −0.00502862
\(937\) −38122.0 −1.32913 −0.664563 0.747232i \(-0.731382\pi\)
−0.664563 + 0.747232i \(0.731382\pi\)
\(938\) 2744.00 0.0955168
\(939\) −6378.00 −0.221659
\(940\) 9120.00 0.316449
\(941\) 42810.0 1.48307 0.741534 0.670916i \(-0.234099\pi\)
0.741534 + 0.670916i \(0.234099\pi\)
\(942\) 11244.0 0.388906
\(943\) 21528.0 0.743423
\(944\) 5568.00 0.191973
\(945\) 945.000 0.0325300
\(946\) 9888.00 0.339838
\(947\) 39864.0 1.36790 0.683952 0.729527i \(-0.260260\pi\)
0.683952 + 0.729527i \(0.260260\pi\)
\(948\) −11904.0 −0.407831
\(949\) 1084.00 0.0370792
\(950\) −2800.00 −0.0956253
\(951\) −3222.00 −0.109864
\(952\) 1008.00 0.0343167
\(953\) −23850.0 −0.810679 −0.405340 0.914166i \(-0.632847\pi\)
−0.405340 + 0.914166i \(0.632847\pi\)
\(954\) 3564.00 0.120953
\(955\) 11640.0 0.394410
\(956\) 2976.00 0.100681
\(957\) 6696.00 0.226177
\(958\) 36336.0 1.22543
\(959\) 5754.00 0.193750
\(960\) 960.000 0.0322749
\(961\) −27087.0 −0.909234
\(962\) 712.000 0.0238626
\(963\) 1944.00 0.0650514
\(964\) 21896.0 0.731559
\(965\) 15830.0 0.528068
\(966\) −6552.00 −0.218227
\(967\) −12832.0 −0.426731 −0.213366 0.976972i \(-0.568443\pi\)
−0.213366 + 0.976972i \(0.568443\pi\)
\(968\) 9496.00 0.315303
\(969\) 3024.00 0.100253
\(970\) 1100.00 0.0364112
\(971\) −15804.0 −0.522322 −0.261161 0.965295i \(-0.584105\pi\)
−0.261161 + 0.965295i \(0.584105\pi\)
\(972\) −972.000 −0.0320750
\(973\) −2632.00 −0.0867195
\(974\) −39712.0 −1.30642
\(975\) −150.000 −0.00492702
\(976\) 1760.00 0.0577215
\(977\) −33114.0 −1.08435 −0.542175 0.840265i \(-0.682399\pi\)
−0.542175 + 0.840265i \(0.682399\pi\)
\(978\) 2712.00 0.0886710
\(979\) 7560.00 0.246801
\(980\) −980.000 −0.0319438
\(981\) 5526.00 0.179849
\(982\) −22440.0 −0.729215
\(983\) −58632.0 −1.90241 −0.951206 0.308558i \(-0.900154\pi\)
−0.951206 + 0.308558i \(0.900154\pi\)
\(984\) −3312.00 −0.107299
\(985\) 2070.00 0.0669601
\(986\) −6696.00 −0.216272
\(987\) 9576.00 0.308822
\(988\) 448.000 0.0144259
\(989\) 64272.0 2.06646
\(990\) 1080.00 0.0346714
\(991\) 55784.0 1.78813 0.894065 0.447937i \(-0.147841\pi\)
0.894065 + 0.447937i \(0.147841\pi\)
\(992\) 1664.00 0.0532581
\(993\) 8364.00 0.267295
\(994\) 13104.0 0.418142
\(995\) 8180.00 0.260627
\(996\) 3312.00 0.105366
\(997\) −23326.0 −0.740965 −0.370482 0.928840i \(-0.620808\pi\)
−0.370482 + 0.928840i \(0.620808\pi\)
\(998\) 18536.0 0.587923
\(999\) 4806.00 0.152207
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 210.4.a.a.1.1 1
3.2 odd 2 630.4.a.v.1.1 1
4.3 odd 2 1680.4.a.n.1.1 1
5.2 odd 4 1050.4.g.o.799.1 2
5.3 odd 4 1050.4.g.o.799.2 2
5.4 even 2 1050.4.a.t.1.1 1
7.6 odd 2 1470.4.a.n.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
210.4.a.a.1.1 1 1.1 even 1 trivial
630.4.a.v.1.1 1 3.2 odd 2
1050.4.a.t.1.1 1 5.4 even 2
1050.4.g.o.799.1 2 5.2 odd 4
1050.4.g.o.799.2 2 5.3 odd 4
1470.4.a.n.1.1 1 7.6 odd 2
1680.4.a.n.1.1 1 4.3 odd 2