Properties

Label 147.4.a.d.1.1
Level $147$
Weight $4$
Character 147.1
Self dual yes
Analytic conductor $8.673$
Analytic rank $1$
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: \(1\)
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
Character \(\chi\) \(=\) 147.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -3.00000 q^{3} -7.00000 q^{4} +12.0000 q^{5} +3.00000 q^{6} +15.0000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -3.00000 q^{3} -7.00000 q^{4} +12.0000 q^{5} +3.00000 q^{6} +15.0000 q^{8} +9.00000 q^{9} -12.0000 q^{10} +20.0000 q^{11} +21.0000 q^{12} -84.0000 q^{13} -36.0000 q^{15} +41.0000 q^{16} -96.0000 q^{17} -9.00000 q^{18} +12.0000 q^{19} -84.0000 q^{20} -20.0000 q^{22} -176.000 q^{23} -45.0000 q^{24} +19.0000 q^{25} +84.0000 q^{26} -27.0000 q^{27} +58.0000 q^{29} +36.0000 q^{30} -264.000 q^{31} -161.000 q^{32} -60.0000 q^{33} +96.0000 q^{34} -63.0000 q^{36} +258.000 q^{37} -12.0000 q^{38} +252.000 q^{39} +180.000 q^{40} +156.000 q^{43} -140.000 q^{44} +108.000 q^{45} +176.000 q^{46} -408.000 q^{47} -123.000 q^{48} -19.0000 q^{50} +288.000 q^{51} +588.000 q^{52} -722.000 q^{53} +27.0000 q^{54} +240.000 q^{55} -36.0000 q^{57} -58.0000 q^{58} +492.000 q^{59} +252.000 q^{60} -492.000 q^{61} +264.000 q^{62} -167.000 q^{64} -1008.00 q^{65} +60.0000 q^{66} +412.000 q^{67} +672.000 q^{68} +528.000 q^{69} +296.000 q^{71} +135.000 q^{72} +240.000 q^{73} -258.000 q^{74} -57.0000 q^{75} -84.0000 q^{76} -252.000 q^{78} +776.000 q^{79} +492.000 q^{80} +81.0000 q^{81} +924.000 q^{83} -1152.00 q^{85} -156.000 q^{86} -174.000 q^{87} +300.000 q^{88} -744.000 q^{89} -108.000 q^{90} +1232.00 q^{92} +792.000 q^{93} +408.000 q^{94} +144.000 q^{95} +483.000 q^{96} -168.000 q^{97} +180.000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.353553 −0.176777 0.984251i \(-0.556567\pi\)
−0.176777 + 0.984251i \(0.556567\pi\)
\(3\) −3.00000 −0.577350
\(4\) −7.00000 −0.875000
\(5\) 12.0000 1.07331 0.536656 0.843801i \(-0.319687\pi\)
0.536656 + 0.843801i \(0.319687\pi\)
\(6\) 3.00000 0.204124
\(7\) 0 0
\(8\) 15.0000 0.662913
\(9\) 9.00000 0.333333
\(10\) −12.0000 −0.379473
\(11\) 20.0000 0.548202 0.274101 0.961701i \(-0.411620\pi\)
0.274101 + 0.961701i \(0.411620\pi\)
\(12\) 21.0000 0.505181
\(13\) −84.0000 −1.79211 −0.896054 0.443945i \(-0.853579\pi\)
−0.896054 + 0.443945i \(0.853579\pi\)
\(14\) 0 0
\(15\) −36.0000 −0.619677
\(16\) 41.0000 0.640625
\(17\) −96.0000 −1.36961 −0.684806 0.728725i \(-0.740113\pi\)
−0.684806 + 0.728725i \(0.740113\pi\)
\(18\) −9.00000 −0.117851
\(19\) 12.0000 0.144894 0.0724471 0.997372i \(-0.476919\pi\)
0.0724471 + 0.997372i \(0.476919\pi\)
\(20\) −84.0000 −0.939149
\(21\) 0 0
\(22\) −20.0000 −0.193819
\(23\) −176.000 −1.59559 −0.797794 0.602930i \(-0.794000\pi\)
−0.797794 + 0.602930i \(0.794000\pi\)
\(24\) −45.0000 −0.382733
\(25\) 19.0000 0.152000
\(26\) 84.0000 0.633606
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 58.0000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 36.0000 0.219089
\(31\) −264.000 −1.52954 −0.764771 0.644302i \(-0.777148\pi\)
−0.764771 + 0.644302i \(0.777148\pi\)
\(32\) −161.000 −0.889408
\(33\) −60.0000 −0.316505
\(34\) 96.0000 0.484231
\(35\) 0 0
\(36\) −63.0000 −0.291667
\(37\) 258.000 1.14635 0.573175 0.819433i \(-0.305712\pi\)
0.573175 + 0.819433i \(0.305712\pi\)
\(38\) −12.0000 −0.0512278
\(39\) 252.000 1.03467
\(40\) 180.000 0.711512
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 156.000 0.553251 0.276625 0.960978i \(-0.410784\pi\)
0.276625 + 0.960978i \(0.410784\pi\)
\(44\) −140.000 −0.479677
\(45\) 108.000 0.357771
\(46\) 176.000 0.564126
\(47\) −408.000 −1.26623 −0.633116 0.774057i \(-0.718224\pi\)
−0.633116 + 0.774057i \(0.718224\pi\)
\(48\) −123.000 −0.369865
\(49\) 0 0
\(50\) −19.0000 −0.0537401
\(51\) 288.000 0.790746
\(52\) 588.000 1.56809
\(53\) −722.000 −1.87121 −0.935607 0.353044i \(-0.885147\pi\)
−0.935607 + 0.353044i \(0.885147\pi\)
\(54\) 27.0000 0.0680414
\(55\) 240.000 0.588393
\(56\) 0 0
\(57\) −36.0000 −0.0836547
\(58\) −58.0000 −0.131306
\(59\) 492.000 1.08564 0.542822 0.839848i \(-0.317356\pi\)
0.542822 + 0.839848i \(0.317356\pi\)
\(60\) 252.000 0.542218
\(61\) −492.000 −1.03269 −0.516345 0.856380i \(-0.672708\pi\)
−0.516345 + 0.856380i \(0.672708\pi\)
\(62\) 264.000 0.540775
\(63\) 0 0
\(64\) −167.000 −0.326172
\(65\) −1008.00 −1.92349
\(66\) 60.0000 0.111901
\(67\) 412.000 0.751251 0.375625 0.926772i \(-0.377428\pi\)
0.375625 + 0.926772i \(0.377428\pi\)
\(68\) 672.000 1.19841
\(69\) 528.000 0.921213
\(70\) 0 0
\(71\) 296.000 0.494771 0.247385 0.968917i \(-0.420429\pi\)
0.247385 + 0.968917i \(0.420429\pi\)
\(72\) 135.000 0.220971
\(73\) 240.000 0.384793 0.192396 0.981317i \(-0.438374\pi\)
0.192396 + 0.981317i \(0.438374\pi\)
\(74\) −258.000 −0.405296
\(75\) −57.0000 −0.0877572
\(76\) −84.0000 −0.126782
\(77\) 0 0
\(78\) −252.000 −0.365813
\(79\) 776.000 1.10515 0.552575 0.833463i \(-0.313645\pi\)
0.552575 + 0.833463i \(0.313645\pi\)
\(80\) 492.000 0.687591
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 924.000 1.22195 0.610977 0.791648i \(-0.290777\pi\)
0.610977 + 0.791648i \(0.290777\pi\)
\(84\) 0 0
\(85\) −1152.00 −1.47002
\(86\) −156.000 −0.195604
\(87\) −174.000 −0.214423
\(88\) 300.000 0.363410
\(89\) −744.000 −0.886111 −0.443055 0.896494i \(-0.646105\pi\)
−0.443055 + 0.896494i \(0.646105\pi\)
\(90\) −108.000 −0.126491
\(91\) 0 0
\(92\) 1232.00 1.39614
\(93\) 792.000 0.883081
\(94\) 408.000 0.447681
\(95\) 144.000 0.155517
\(96\) 483.000 0.513500
\(97\) −168.000 −0.175854 −0.0879269 0.996127i \(-0.528024\pi\)
−0.0879269 + 0.996127i \(0.528024\pi\)
\(98\) 0 0
\(99\) 180.000 0.182734
\(100\) −133.000 −0.133000
\(101\) −1524.00 −1.50142 −0.750711 0.660630i \(-0.770289\pi\)
−0.750711 + 0.660630i \(0.770289\pi\)
\(102\) −288.000 −0.279571
\(103\) −408.000 −0.390305 −0.195153 0.980773i \(-0.562520\pi\)
−0.195153 + 0.980773i \(0.562520\pi\)
\(104\) −1260.00 −1.18801
\(105\) 0 0
\(106\) 722.000 0.661574
\(107\) −820.000 −0.740863 −0.370432 0.928860i \(-0.620790\pi\)
−0.370432 + 0.928860i \(0.620790\pi\)
\(108\) 189.000 0.168394
\(109\) −918.000 −0.806683 −0.403342 0.915050i \(-0.632151\pi\)
−0.403342 + 0.915050i \(0.632151\pi\)
\(110\) −240.000 −0.208028
\(111\) −774.000 −0.661845
\(112\) 0 0
\(113\) −110.000 −0.0915746 −0.0457873 0.998951i \(-0.514580\pi\)
−0.0457873 + 0.998951i \(0.514580\pi\)
\(114\) 36.0000 0.0295764
\(115\) −2112.00 −1.71257
\(116\) −406.000 −0.324967
\(117\) −756.000 −0.597369
\(118\) −492.000 −0.383833
\(119\) 0 0
\(120\) −540.000 −0.410792
\(121\) −931.000 −0.699474
\(122\) 492.000 0.365111
\(123\) 0 0
\(124\) 1848.00 1.33835
\(125\) −1272.00 −0.910169
\(126\) 0 0
\(127\) 16.0000 0.0111793 0.00558965 0.999984i \(-0.498221\pi\)
0.00558965 + 0.999984i \(0.498221\pi\)
\(128\) 1455.00 1.00473
\(129\) −468.000 −0.319419
\(130\) 1008.00 0.680057
\(131\) 1692.00 1.12848 0.564239 0.825611i \(-0.309169\pi\)
0.564239 + 0.825611i \(0.309169\pi\)
\(132\) 420.000 0.276942
\(133\) 0 0
\(134\) −412.000 −0.265607
\(135\) −324.000 −0.206559
\(136\) −1440.00 −0.907934
\(137\) 1126.00 0.702195 0.351097 0.936339i \(-0.385809\pi\)
0.351097 + 0.936339i \(0.385809\pi\)
\(138\) −528.000 −0.325698
\(139\) 1092.00 0.666347 0.333173 0.942866i \(-0.391881\pi\)
0.333173 + 0.942866i \(0.391881\pi\)
\(140\) 0 0
\(141\) 1224.00 0.731060
\(142\) −296.000 −0.174928
\(143\) −1680.00 −0.982438
\(144\) 369.000 0.213542
\(145\) 696.000 0.398618
\(146\) −240.000 −0.136045
\(147\) 0 0
\(148\) −1806.00 −1.00306
\(149\) 1070.00 0.588307 0.294154 0.955758i \(-0.404962\pi\)
0.294154 + 0.955758i \(0.404962\pi\)
\(150\) 57.0000 0.0310269
\(151\) −120.000 −0.0646719 −0.0323360 0.999477i \(-0.510295\pi\)
−0.0323360 + 0.999477i \(0.510295\pi\)
\(152\) 180.000 0.0960522
\(153\) −864.000 −0.456538
\(154\) 0 0
\(155\) −3168.00 −1.64168
\(156\) −1764.00 −0.905340
\(157\) 1836.00 0.933304 0.466652 0.884441i \(-0.345460\pi\)
0.466652 + 0.884441i \(0.345460\pi\)
\(158\) −776.000 −0.390729
\(159\) 2166.00 1.08035
\(160\) −1932.00 −0.954613
\(161\) 0 0
\(162\) −81.0000 −0.0392837
\(163\) 916.000 0.440164 0.220082 0.975481i \(-0.429368\pi\)
0.220082 + 0.975481i \(0.429368\pi\)
\(164\) 0 0
\(165\) −720.000 −0.339709
\(166\) −924.000 −0.432026
\(167\) 504.000 0.233537 0.116769 0.993159i \(-0.462746\pi\)
0.116769 + 0.993159i \(0.462746\pi\)
\(168\) 0 0
\(169\) 4859.00 2.21165
\(170\) 1152.00 0.519732
\(171\) 108.000 0.0482980
\(172\) −1092.00 −0.484094
\(173\) −1836.00 −0.806870 −0.403435 0.915008i \(-0.632184\pi\)
−0.403435 + 0.915008i \(0.632184\pi\)
\(174\) 174.000 0.0758098
\(175\) 0 0
\(176\) 820.000 0.351192
\(177\) −1476.00 −0.626796
\(178\) 744.000 0.313287
\(179\) 2372.00 0.990456 0.495228 0.868763i \(-0.335085\pi\)
0.495228 + 0.868763i \(0.335085\pi\)
\(180\) −756.000 −0.313050
\(181\) −1092.00 −0.448440 −0.224220 0.974539i \(-0.571983\pi\)
−0.224220 + 0.974539i \(0.571983\pi\)
\(182\) 0 0
\(183\) 1476.00 0.596224
\(184\) −2640.00 −1.05774
\(185\) 3096.00 1.23039
\(186\) −792.000 −0.312216
\(187\) −1920.00 −0.750825
\(188\) 2856.00 1.10795
\(189\) 0 0
\(190\) −144.000 −0.0549835
\(191\) 2512.00 0.951633 0.475817 0.879545i \(-0.342153\pi\)
0.475817 + 0.879545i \(0.342153\pi\)
\(192\) 501.000 0.188315
\(193\) −2430.00 −0.906297 −0.453148 0.891435i \(-0.649699\pi\)
−0.453148 + 0.891435i \(0.649699\pi\)
\(194\) 168.000 0.0621737
\(195\) 3024.00 1.11053
\(196\) 0 0
\(197\) −1762.00 −0.637245 −0.318623 0.947882i \(-0.603220\pi\)
−0.318623 + 0.947882i \(0.603220\pi\)
\(198\) −180.000 −0.0646063
\(199\) 3096.00 1.10286 0.551431 0.834220i \(-0.314082\pi\)
0.551431 + 0.834220i \(0.314082\pi\)
\(200\) 285.000 0.100763
\(201\) −1236.00 −0.433735
\(202\) 1524.00 0.530833
\(203\) 0 0
\(204\) −2016.00 −0.691903
\(205\) 0 0
\(206\) 408.000 0.137994
\(207\) −1584.00 −0.531863
\(208\) −3444.00 −1.14807
\(209\) 240.000 0.0794313
\(210\) 0 0
\(211\) 156.000 0.0508980 0.0254490 0.999676i \(-0.491898\pi\)
0.0254490 + 0.999676i \(0.491898\pi\)
\(212\) 5054.00 1.63731
\(213\) −888.000 −0.285656
\(214\) 820.000 0.261935
\(215\) 1872.00 0.593811
\(216\) −405.000 −0.127578
\(217\) 0 0
\(218\) 918.000 0.285206
\(219\) −720.000 −0.222160
\(220\) −1680.00 −0.514844
\(221\) 8064.00 2.45449
\(222\) 774.000 0.233998
\(223\) 5040.00 1.51347 0.756734 0.653723i \(-0.226794\pi\)
0.756734 + 0.653723i \(0.226794\pi\)
\(224\) 0 0
\(225\) 171.000 0.0506667
\(226\) 110.000 0.0323765
\(227\) 2172.00 0.635069 0.317535 0.948247i \(-0.397145\pi\)
0.317535 + 0.948247i \(0.397145\pi\)
\(228\) 252.000 0.0731978
\(229\) 2700.00 0.779131 0.389566 0.920999i \(-0.372625\pi\)
0.389566 + 0.920999i \(0.372625\pi\)
\(230\) 2112.00 0.605483
\(231\) 0 0
\(232\) 870.000 0.246200
\(233\) −3802.00 −1.06900 −0.534501 0.845168i \(-0.679500\pi\)
−0.534501 + 0.845168i \(0.679500\pi\)
\(234\) 756.000 0.211202
\(235\) −4896.00 −1.35906
\(236\) −3444.00 −0.949938
\(237\) −2328.00 −0.638058
\(238\) 0 0
\(239\) −4408.00 −1.19301 −0.596506 0.802609i \(-0.703445\pi\)
−0.596506 + 0.802609i \(0.703445\pi\)
\(240\) −1476.00 −0.396981
\(241\) 3096.00 0.827514 0.413757 0.910387i \(-0.364216\pi\)
0.413757 + 0.910387i \(0.364216\pi\)
\(242\) 931.000 0.247301
\(243\) −243.000 −0.0641500
\(244\) 3444.00 0.903605
\(245\) 0 0
\(246\) 0 0
\(247\) −1008.00 −0.259666
\(248\) −3960.00 −1.01395
\(249\) −2772.00 −0.705495
\(250\) 1272.00 0.321793
\(251\) −924.000 −0.232360 −0.116180 0.993228i \(-0.537065\pi\)
−0.116180 + 0.993228i \(0.537065\pi\)
\(252\) 0 0
\(253\) −3520.00 −0.874706
\(254\) −16.0000 −0.00395248
\(255\) 3456.00 0.848718
\(256\) −119.000 −0.0290527
\(257\) −2760.00 −0.669899 −0.334950 0.942236i \(-0.608719\pi\)
−0.334950 + 0.942236i \(0.608719\pi\)
\(258\) 468.000 0.112932
\(259\) 0 0
\(260\) 7056.00 1.68306
\(261\) 522.000 0.123797
\(262\) −1692.00 −0.398978
\(263\) −2360.00 −0.553323 −0.276661 0.960967i \(-0.589228\pi\)
−0.276661 + 0.960967i \(0.589228\pi\)
\(264\) −900.000 −0.209815
\(265\) −8664.00 −2.00840
\(266\) 0 0
\(267\) 2232.00 0.511596
\(268\) −2884.00 −0.657345
\(269\) 4020.00 0.911166 0.455583 0.890193i \(-0.349431\pi\)
0.455583 + 0.890193i \(0.349431\pi\)
\(270\) 324.000 0.0730297
\(271\) 4800.00 1.07594 0.537969 0.842965i \(-0.319192\pi\)
0.537969 + 0.842965i \(0.319192\pi\)
\(272\) −3936.00 −0.877408
\(273\) 0 0
\(274\) −1126.00 −0.248263
\(275\) 380.000 0.0833268
\(276\) −3696.00 −0.806062
\(277\) 6446.00 1.39820 0.699102 0.715022i \(-0.253583\pi\)
0.699102 + 0.715022i \(0.253583\pi\)
\(278\) −1092.00 −0.235589
\(279\) −2376.00 −0.509847
\(280\) 0 0
\(281\) −2602.00 −0.552393 −0.276196 0.961101i \(-0.589074\pi\)
−0.276196 + 0.961101i \(0.589074\pi\)
\(282\) −1224.00 −0.258469
\(283\) −6900.00 −1.44934 −0.724669 0.689098i \(-0.758007\pi\)
−0.724669 + 0.689098i \(0.758007\pi\)
\(284\) −2072.00 −0.432925
\(285\) −432.000 −0.0897876
\(286\) 1680.00 0.347344
\(287\) 0 0
\(288\) −1449.00 −0.296469
\(289\) 4303.00 0.875840
\(290\) −696.000 −0.140933
\(291\) 504.000 0.101529
\(292\) −1680.00 −0.336694
\(293\) −4452.00 −0.887674 −0.443837 0.896107i \(-0.646383\pi\)
−0.443837 + 0.896107i \(0.646383\pi\)
\(294\) 0 0
\(295\) 5904.00 1.16523
\(296\) 3870.00 0.759930
\(297\) −540.000 −0.105502
\(298\) −1070.00 −0.207998
\(299\) 14784.0 2.85947
\(300\) 399.000 0.0767876
\(301\) 0 0
\(302\) 120.000 0.0228650
\(303\) 4572.00 0.866847
\(304\) 492.000 0.0928228
\(305\) −5904.00 −1.10840
\(306\) 864.000 0.161410
\(307\) −2436.00 −0.452866 −0.226433 0.974027i \(-0.572706\pi\)
−0.226433 + 0.974027i \(0.572706\pi\)
\(308\) 0 0
\(309\) 1224.00 0.225343
\(310\) 3168.00 0.580420
\(311\) −7488.00 −1.36529 −0.682646 0.730750i \(-0.739171\pi\)
−0.682646 + 0.730750i \(0.739171\pi\)
\(312\) 3780.00 0.685899
\(313\) −1752.00 −0.316386 −0.158193 0.987408i \(-0.550567\pi\)
−0.158193 + 0.987408i \(0.550567\pi\)
\(314\) −1836.00 −0.329973
\(315\) 0 0
\(316\) −5432.00 −0.967006
\(317\) −1562.00 −0.276753 −0.138376 0.990380i \(-0.544188\pi\)
−0.138376 + 0.990380i \(0.544188\pi\)
\(318\) −2166.00 −0.381960
\(319\) 1160.00 0.203597
\(320\) −2004.00 −0.350084
\(321\) 2460.00 0.427738
\(322\) 0 0
\(323\) −1152.00 −0.198449
\(324\) −567.000 −0.0972222
\(325\) −1596.00 −0.272400
\(326\) −916.000 −0.155621
\(327\) 2754.00 0.465739
\(328\) 0 0
\(329\) 0 0
\(330\) 720.000 0.120105
\(331\) −7092.00 −1.17768 −0.588839 0.808250i \(-0.700415\pi\)
−0.588839 + 0.808250i \(0.700415\pi\)
\(332\) −6468.00 −1.06921
\(333\) 2322.00 0.382117
\(334\) −504.000 −0.0825678
\(335\) 4944.00 0.806327
\(336\) 0 0
\(337\) 366.000 0.0591611 0.0295805 0.999562i \(-0.490583\pi\)
0.0295805 + 0.999562i \(0.490583\pi\)
\(338\) −4859.00 −0.781937
\(339\) 330.000 0.0528706
\(340\) 8064.00 1.28627
\(341\) −5280.00 −0.838499
\(342\) −108.000 −0.0170759
\(343\) 0 0
\(344\) 2340.00 0.366757
\(345\) 6336.00 0.988750
\(346\) 1836.00 0.285272
\(347\) −6364.00 −0.984546 −0.492273 0.870441i \(-0.663834\pi\)
−0.492273 + 0.870441i \(0.663834\pi\)
\(348\) 1218.00 0.187620
\(349\) −10500.0 −1.61046 −0.805232 0.592960i \(-0.797959\pi\)
−0.805232 + 0.592960i \(0.797959\pi\)
\(350\) 0 0
\(351\) 2268.00 0.344891
\(352\) −3220.00 −0.487576
\(353\) 408.000 0.0615174 0.0307587 0.999527i \(-0.490208\pi\)
0.0307587 + 0.999527i \(0.490208\pi\)
\(354\) 1476.00 0.221606
\(355\) 3552.00 0.531044
\(356\) 5208.00 0.775347
\(357\) 0 0
\(358\) −2372.00 −0.350179
\(359\) −11936.0 −1.75476 −0.877379 0.479798i \(-0.840710\pi\)
−0.877379 + 0.479798i \(0.840710\pi\)
\(360\) 1620.00 0.237171
\(361\) −6715.00 −0.979006
\(362\) 1092.00 0.158548
\(363\) 2793.00 0.403842
\(364\) 0 0
\(365\) 2880.00 0.413003
\(366\) −1476.00 −0.210797
\(367\) −2448.00 −0.348187 −0.174093 0.984729i \(-0.555699\pi\)
−0.174093 + 0.984729i \(0.555699\pi\)
\(368\) −7216.00 −1.02217
\(369\) 0 0
\(370\) −3096.00 −0.435009
\(371\) 0 0
\(372\) −5544.00 −0.772696
\(373\) 11374.0 1.57888 0.789442 0.613826i \(-0.210370\pi\)
0.789442 + 0.613826i \(0.210370\pi\)
\(374\) 1920.00 0.265457
\(375\) 3816.00 0.525486
\(376\) −6120.00 −0.839401
\(377\) −4872.00 −0.665572
\(378\) 0 0
\(379\) −5892.00 −0.798553 −0.399277 0.916830i \(-0.630739\pi\)
−0.399277 + 0.916830i \(0.630739\pi\)
\(380\) −1008.00 −0.136077
\(381\) −48.0000 −0.00645437
\(382\) −2512.00 −0.336453
\(383\) −10488.0 −1.39925 −0.699624 0.714511i \(-0.746649\pi\)
−0.699624 + 0.714511i \(0.746649\pi\)
\(384\) −4365.00 −0.580079
\(385\) 0 0
\(386\) 2430.00 0.320424
\(387\) 1404.00 0.184417
\(388\) 1176.00 0.153872
\(389\) 4514.00 0.588352 0.294176 0.955751i \(-0.404955\pi\)
0.294176 + 0.955751i \(0.404955\pi\)
\(390\) −3024.00 −0.392631
\(391\) 16896.0 2.18534
\(392\) 0 0
\(393\) −5076.00 −0.651528
\(394\) 1762.00 0.225300
\(395\) 9312.00 1.18617
\(396\) −1260.00 −0.159892
\(397\) −6036.00 −0.763068 −0.381534 0.924355i \(-0.624604\pi\)
−0.381534 + 0.924355i \(0.624604\pi\)
\(398\) −3096.00 −0.389921
\(399\) 0 0
\(400\) 779.000 0.0973750
\(401\) −6770.00 −0.843086 −0.421543 0.906808i \(-0.638511\pi\)
−0.421543 + 0.906808i \(0.638511\pi\)
\(402\) 1236.00 0.153348
\(403\) 22176.0 2.74110
\(404\) 10668.0 1.31374
\(405\) 972.000 0.119257
\(406\) 0 0
\(407\) 5160.00 0.628432
\(408\) 4320.00 0.524196
\(409\) 12504.0 1.51169 0.755847 0.654748i \(-0.227225\pi\)
0.755847 + 0.654748i \(0.227225\pi\)
\(410\) 0 0
\(411\) −3378.00 −0.405412
\(412\) 2856.00 0.341517
\(413\) 0 0
\(414\) 1584.00 0.188042
\(415\) 11088.0 1.31154
\(416\) 13524.0 1.59392
\(417\) −3276.00 −0.384716
\(418\) −240.000 −0.0280832
\(419\) 9492.00 1.10672 0.553359 0.832943i \(-0.313346\pi\)
0.553359 + 0.832943i \(0.313346\pi\)
\(420\) 0 0
\(421\) 5182.00 0.599894 0.299947 0.953956i \(-0.403031\pi\)
0.299947 + 0.953956i \(0.403031\pi\)
\(422\) −156.000 −0.0179952
\(423\) −3672.00 −0.422077
\(424\) −10830.0 −1.24045
\(425\) −1824.00 −0.208181
\(426\) 888.000 0.100995
\(427\) 0 0
\(428\) 5740.00 0.648256
\(429\) 5040.00 0.567211
\(430\) −1872.00 −0.209944
\(431\) −5720.00 −0.639264 −0.319632 0.947542i \(-0.603559\pi\)
−0.319632 + 0.947542i \(0.603559\pi\)
\(432\) −1107.00 −0.123288
\(433\) 13608.0 1.51030 0.755149 0.655554i \(-0.227565\pi\)
0.755149 + 0.655554i \(0.227565\pi\)
\(434\) 0 0
\(435\) −2088.00 −0.230142
\(436\) 6426.00 0.705848
\(437\) −2112.00 −0.231191
\(438\) 720.000 0.0785455
\(439\) 12864.0 1.39855 0.699277 0.714851i \(-0.253505\pi\)
0.699277 + 0.714851i \(0.253505\pi\)
\(440\) 3600.00 0.390053
\(441\) 0 0
\(442\) −8064.00 −0.867795
\(443\) −13252.0 −1.42127 −0.710634 0.703562i \(-0.751592\pi\)
−0.710634 + 0.703562i \(0.751592\pi\)
\(444\) 5418.00 0.579115
\(445\) −8928.00 −0.951074
\(446\) −5040.00 −0.535092
\(447\) −3210.00 −0.339659
\(448\) 0 0
\(449\) 226.000 0.0237541 0.0118771 0.999929i \(-0.496219\pi\)
0.0118771 + 0.999929i \(0.496219\pi\)
\(450\) −171.000 −0.0179134
\(451\) 0 0
\(452\) 770.000 0.0801278
\(453\) 360.000 0.0373384
\(454\) −2172.00 −0.224531
\(455\) 0 0
\(456\) −540.000 −0.0554557
\(457\) −11334.0 −1.16014 −0.580068 0.814568i \(-0.696974\pi\)
−0.580068 + 0.814568i \(0.696974\pi\)
\(458\) −2700.00 −0.275464
\(459\) 2592.00 0.263582
\(460\) 14784.0 1.49849
\(461\) −1596.00 −0.161243 −0.0806216 0.996745i \(-0.525691\pi\)
−0.0806216 + 0.996745i \(0.525691\pi\)
\(462\) 0 0
\(463\) 12728.0 1.27758 0.638791 0.769380i \(-0.279435\pi\)
0.638791 + 0.769380i \(0.279435\pi\)
\(464\) 2378.00 0.237922
\(465\) 9504.00 0.947822
\(466\) 3802.00 0.377949
\(467\) −3012.00 −0.298456 −0.149228 0.988803i \(-0.547679\pi\)
−0.149228 + 0.988803i \(0.547679\pi\)
\(468\) 5292.00 0.522698
\(469\) 0 0
\(470\) 4896.00 0.480501
\(471\) −5508.00 −0.538843
\(472\) 7380.00 0.719687
\(473\) 3120.00 0.303293
\(474\) 2328.00 0.225588
\(475\) 228.000 0.0220239
\(476\) 0 0
\(477\) −6498.00 −0.623738
\(478\) 4408.00 0.421793
\(479\) −4296.00 −0.409790 −0.204895 0.978784i \(-0.565685\pi\)
−0.204895 + 0.978784i \(0.565685\pi\)
\(480\) 5796.00 0.551146
\(481\) −21672.0 −2.05438
\(482\) −3096.00 −0.292570
\(483\) 0 0
\(484\) 6517.00 0.612040
\(485\) −2016.00 −0.188746
\(486\) 243.000 0.0226805
\(487\) −8184.00 −0.761504 −0.380752 0.924677i \(-0.624335\pi\)
−0.380752 + 0.924677i \(0.624335\pi\)
\(488\) −7380.00 −0.684584
\(489\) −2748.00 −0.254129
\(490\) 0 0
\(491\) −12164.0 −1.11803 −0.559016 0.829157i \(-0.688821\pi\)
−0.559016 + 0.829157i \(0.688821\pi\)
\(492\) 0 0
\(493\) −5568.00 −0.508661
\(494\) 1008.00 0.0918058
\(495\) 2160.00 0.196131
\(496\) −10824.0 −0.979863
\(497\) 0 0
\(498\) 2772.00 0.249430
\(499\) 972.000 0.0871998 0.0435999 0.999049i \(-0.486117\pi\)
0.0435999 + 0.999049i \(0.486117\pi\)
\(500\) 8904.00 0.796398
\(501\) −1512.00 −0.134833
\(502\) 924.000 0.0821517
\(503\) 7728.00 0.685039 0.342519 0.939511i \(-0.388720\pi\)
0.342519 + 0.939511i \(0.388720\pi\)
\(504\) 0 0
\(505\) −18288.0 −1.61150
\(506\) 3520.00 0.309255
\(507\) −14577.0 −1.27690
\(508\) −112.000 −0.00978188
\(509\) 11604.0 1.01049 0.505244 0.862977i \(-0.331403\pi\)
0.505244 + 0.862977i \(0.331403\pi\)
\(510\) −3456.00 −0.300067
\(511\) 0 0
\(512\) −11521.0 −0.994455
\(513\) −324.000 −0.0278849
\(514\) 2760.00 0.236845
\(515\) −4896.00 −0.418919
\(516\) 3276.00 0.279492
\(517\) −8160.00 −0.694152
\(518\) 0 0
\(519\) 5508.00 0.465847
\(520\) −15120.0 −1.27511
\(521\) −10848.0 −0.912206 −0.456103 0.889927i \(-0.650755\pi\)
−0.456103 + 0.889927i \(0.650755\pi\)
\(522\) −522.000 −0.0437688
\(523\) −18132.0 −1.51598 −0.757989 0.652267i \(-0.773818\pi\)
−0.757989 + 0.652267i \(0.773818\pi\)
\(524\) −11844.0 −0.987419
\(525\) 0 0
\(526\) 2360.00 0.195629
\(527\) 25344.0 2.09488
\(528\) −2460.00 −0.202761
\(529\) 18809.0 1.54590
\(530\) 8664.00 0.710076
\(531\) 4428.00 0.361881
\(532\) 0 0
\(533\) 0 0
\(534\) −2232.00 −0.180877
\(535\) −9840.00 −0.795178
\(536\) 6180.00 0.498014
\(537\) −7116.00 −0.571840
\(538\) −4020.00 −0.322146
\(539\) 0 0
\(540\) 2268.00 0.180739
\(541\) 6950.00 0.552318 0.276159 0.961112i \(-0.410938\pi\)
0.276159 + 0.961112i \(0.410938\pi\)
\(542\) −4800.00 −0.380402
\(543\) 3276.00 0.258907
\(544\) 15456.0 1.21814
\(545\) −11016.0 −0.865823
\(546\) 0 0
\(547\) 17012.0 1.32976 0.664882 0.746949i \(-0.268482\pi\)
0.664882 + 0.746949i \(0.268482\pi\)
\(548\) −7882.00 −0.614420
\(549\) −4428.00 −0.344230
\(550\) −380.000 −0.0294605
\(551\) 696.000 0.0538123
\(552\) 7920.00 0.610684
\(553\) 0 0
\(554\) −6446.00 −0.494340
\(555\) −9288.00 −0.710367
\(556\) −7644.00 −0.583054
\(557\) 3926.00 0.298653 0.149327 0.988788i \(-0.452289\pi\)
0.149327 + 0.988788i \(0.452289\pi\)
\(558\) 2376.00 0.180258
\(559\) −13104.0 −0.991485
\(560\) 0 0
\(561\) 5760.00 0.433489
\(562\) 2602.00 0.195300
\(563\) −18828.0 −1.40942 −0.704712 0.709494i \(-0.748924\pi\)
−0.704712 + 0.709494i \(0.748924\pi\)
\(564\) −8568.00 −0.639677
\(565\) −1320.00 −0.0982882
\(566\) 6900.00 0.512418
\(567\) 0 0
\(568\) 4440.00 0.327990
\(569\) 11990.0 0.883387 0.441693 0.897166i \(-0.354378\pi\)
0.441693 + 0.897166i \(0.354378\pi\)
\(570\) 432.000 0.0317447
\(571\) −15716.0 −1.15183 −0.575914 0.817510i \(-0.695354\pi\)
−0.575914 + 0.817510i \(0.695354\pi\)
\(572\) 11760.0 0.859633
\(573\) −7536.00 −0.549426
\(574\) 0 0
\(575\) −3344.00 −0.242529
\(576\) −1503.00 −0.108724
\(577\) −13872.0 −1.00086 −0.500432 0.865776i \(-0.666826\pi\)
−0.500432 + 0.865776i \(0.666826\pi\)
\(578\) −4303.00 −0.309656
\(579\) 7290.00 0.523251
\(580\) −4872.00 −0.348791
\(581\) 0 0
\(582\) −504.000 −0.0358960
\(583\) −14440.0 −1.02580
\(584\) 3600.00 0.255084
\(585\) −9072.00 −0.641164
\(586\) 4452.00 0.313840
\(587\) 8820.00 0.620171 0.310085 0.950709i \(-0.399642\pi\)
0.310085 + 0.950709i \(0.399642\pi\)
\(588\) 0 0
\(589\) −3168.00 −0.221622
\(590\) −5904.00 −0.411973
\(591\) 5286.00 0.367914
\(592\) 10578.0 0.734380
\(593\) −16872.0 −1.16838 −0.584191 0.811617i \(-0.698588\pi\)
−0.584191 + 0.811617i \(0.698588\pi\)
\(594\) 540.000 0.0373005
\(595\) 0 0
\(596\) −7490.00 −0.514769
\(597\) −9288.00 −0.636738
\(598\) −14784.0 −1.01097
\(599\) −6056.00 −0.413091 −0.206545 0.978437i \(-0.566222\pi\)
−0.206545 + 0.978437i \(0.566222\pi\)
\(600\) −855.000 −0.0581754
\(601\) 10752.0 0.729756 0.364878 0.931055i \(-0.381111\pi\)
0.364878 + 0.931055i \(0.381111\pi\)
\(602\) 0 0
\(603\) 3708.00 0.250417
\(604\) 840.000 0.0565879
\(605\) −11172.0 −0.750754
\(606\) −4572.00 −0.306477
\(607\) 20256.0 1.35447 0.677237 0.735765i \(-0.263177\pi\)
0.677237 + 0.735765i \(0.263177\pi\)
\(608\) −1932.00 −0.128870
\(609\) 0 0
\(610\) 5904.00 0.391879
\(611\) 34272.0 2.26923
\(612\) 6048.00 0.399470
\(613\) −28190.0 −1.85740 −0.928698 0.370838i \(-0.879071\pi\)
−0.928698 + 0.370838i \(0.879071\pi\)
\(614\) 2436.00 0.160112
\(615\) 0 0
\(616\) 0 0
\(617\) 29318.0 1.91296 0.956482 0.291793i \(-0.0942518\pi\)
0.956482 + 0.291793i \(0.0942518\pi\)
\(618\) −1224.00 −0.0796707
\(619\) 24348.0 1.58098 0.790492 0.612473i \(-0.209825\pi\)
0.790492 + 0.612473i \(0.209825\pi\)
\(620\) 22176.0 1.43647
\(621\) 4752.00 0.307071
\(622\) 7488.00 0.482703
\(623\) 0 0
\(624\) 10332.0 0.662838
\(625\) −17639.0 −1.12890
\(626\) 1752.00 0.111859
\(627\) −720.000 −0.0458597
\(628\) −12852.0 −0.816641
\(629\) −24768.0 −1.57006
\(630\) 0 0
\(631\) −25184.0 −1.58884 −0.794421 0.607368i \(-0.792226\pi\)
−0.794421 + 0.607368i \(0.792226\pi\)
\(632\) 11640.0 0.732618
\(633\) −468.000 −0.0293860
\(634\) 1562.00 0.0978469
\(635\) 192.000 0.0119989
\(636\) −15162.0 −0.945303
\(637\) 0 0
\(638\) −1160.00 −0.0719825
\(639\) 2664.00 0.164924
\(640\) 17460.0 1.07839
\(641\) 32318.0 1.99140 0.995698 0.0926628i \(-0.0295379\pi\)
0.995698 + 0.0926628i \(0.0295379\pi\)
\(642\) −2460.00 −0.151228
\(643\) −3948.00 −0.242137 −0.121068 0.992644i \(-0.538632\pi\)
−0.121068 + 0.992644i \(0.538632\pi\)
\(644\) 0 0
\(645\) −5616.00 −0.342837
\(646\) 1152.00 0.0701623
\(647\) 13848.0 0.841454 0.420727 0.907187i \(-0.361775\pi\)
0.420727 + 0.907187i \(0.361775\pi\)
\(648\) 1215.00 0.0736570
\(649\) 9840.00 0.595152
\(650\) 1596.00 0.0963081
\(651\) 0 0
\(652\) −6412.00 −0.385143
\(653\) −3158.00 −0.189253 −0.0946264 0.995513i \(-0.530166\pi\)
−0.0946264 + 0.995513i \(0.530166\pi\)
\(654\) −2754.00 −0.164663
\(655\) 20304.0 1.21121
\(656\) 0 0
\(657\) 2160.00 0.128264
\(658\) 0 0
\(659\) −24596.0 −1.45391 −0.726953 0.686687i \(-0.759064\pi\)
−0.726953 + 0.686687i \(0.759064\pi\)
\(660\) 5040.00 0.297245
\(661\) −15468.0 −0.910190 −0.455095 0.890443i \(-0.650395\pi\)
−0.455095 + 0.890443i \(0.650395\pi\)
\(662\) 7092.00 0.416372
\(663\) −24192.0 −1.41710
\(664\) 13860.0 0.810049
\(665\) 0 0
\(666\) −2322.00 −0.135099
\(667\) −10208.0 −0.592587
\(668\) −3528.00 −0.204345
\(669\) −15120.0 −0.873801
\(670\) −4944.00 −0.285080
\(671\) −9840.00 −0.566124
\(672\) 0 0
\(673\) 13470.0 0.771516 0.385758 0.922600i \(-0.373940\pi\)
0.385758 + 0.922600i \(0.373940\pi\)
\(674\) −366.000 −0.0209166
\(675\) −513.000 −0.0292524
\(676\) −34013.0 −1.93520
\(677\) −9564.00 −0.542946 −0.271473 0.962446i \(-0.587511\pi\)
−0.271473 + 0.962446i \(0.587511\pi\)
\(678\) −330.000 −0.0186926
\(679\) 0 0
\(680\) −17280.0 −0.974497
\(681\) −6516.00 −0.366657
\(682\) 5280.00 0.296454
\(683\) 13852.0 0.776035 0.388018 0.921652i \(-0.373160\pi\)
0.388018 + 0.921652i \(0.373160\pi\)
\(684\) −756.000 −0.0422608
\(685\) 13512.0 0.753674
\(686\) 0 0
\(687\) −8100.00 −0.449832
\(688\) 6396.00 0.354426
\(689\) 60648.0 3.35342
\(690\) −6336.00 −0.349576
\(691\) −324.000 −0.0178373 −0.00891863 0.999960i \(-0.502839\pi\)
−0.00891863 + 0.999960i \(0.502839\pi\)
\(692\) 12852.0 0.706011
\(693\) 0 0
\(694\) 6364.00 0.348090
\(695\) 13104.0 0.715199
\(696\) −2610.00 −0.142143
\(697\) 0 0
\(698\) 10500.0 0.569385
\(699\) 11406.0 0.617188
\(700\) 0 0
\(701\) 24922.0 1.34278 0.671392 0.741103i \(-0.265697\pi\)
0.671392 + 0.741103i \(0.265697\pi\)
\(702\) −2268.00 −0.121938
\(703\) 3096.00 0.166099
\(704\) −3340.00 −0.178808
\(705\) 14688.0 0.784655
\(706\) −408.000 −0.0217497
\(707\) 0 0
\(708\) 10332.0 0.548447
\(709\) −17886.0 −0.947423 −0.473711 0.880680i \(-0.657086\pi\)
−0.473711 + 0.880680i \(0.657086\pi\)
\(710\) −3552.00 −0.187752
\(711\) 6984.00 0.368383
\(712\) −11160.0 −0.587414
\(713\) 46464.0 2.44052
\(714\) 0 0
\(715\) −20160.0 −1.05446
\(716\) −16604.0 −0.866649
\(717\) 13224.0 0.688786
\(718\) 11936.0 0.620401
\(719\) −6792.00 −0.352293 −0.176147 0.984364i \(-0.556363\pi\)
−0.176147 + 0.984364i \(0.556363\pi\)
\(720\) 4428.00 0.229197
\(721\) 0 0
\(722\) 6715.00 0.346131
\(723\) −9288.00 −0.477765
\(724\) 7644.00 0.392385
\(725\) 1102.00 0.0564514
\(726\) −2793.00 −0.142780
\(727\) 1512.00 0.0771348 0.0385674 0.999256i \(-0.487721\pi\)
0.0385674 + 0.999256i \(0.487721\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) −2880.00 −0.146019
\(731\) −14976.0 −0.757739
\(732\) −10332.0 −0.521696
\(733\) −11244.0 −0.566585 −0.283292 0.959034i \(-0.591427\pi\)
−0.283292 + 0.959034i \(0.591427\pi\)
\(734\) 2448.00 0.123103
\(735\) 0 0
\(736\) 28336.0 1.41913
\(737\) 8240.00 0.411838
\(738\) 0 0
\(739\) −1996.00 −0.0993559 −0.0496780 0.998765i \(-0.515820\pi\)
−0.0496780 + 0.998765i \(0.515820\pi\)
\(740\) −21672.0 −1.07659
\(741\) 3024.00 0.149918
\(742\) 0 0
\(743\) −656.000 −0.0323907 −0.0161954 0.999869i \(-0.505155\pi\)
−0.0161954 + 0.999869i \(0.505155\pi\)
\(744\) 11880.0 0.585406
\(745\) 12840.0 0.631438
\(746\) −11374.0 −0.558219
\(747\) 8316.00 0.407318
\(748\) 13440.0 0.656972
\(749\) 0 0
\(750\) −3816.00 −0.185787
\(751\) 1056.00 0.0513102 0.0256551 0.999671i \(-0.491833\pi\)
0.0256551 + 0.999671i \(0.491833\pi\)
\(752\) −16728.0 −0.811180
\(753\) 2772.00 0.134153
\(754\) 4872.00 0.235315
\(755\) −1440.00 −0.0694132
\(756\) 0 0
\(757\) −18702.0 −0.897934 −0.448967 0.893548i \(-0.648208\pi\)
−0.448967 + 0.893548i \(0.648208\pi\)
\(758\) 5892.00 0.282331
\(759\) 10560.0 0.505011
\(760\) 2160.00 0.103094
\(761\) 17904.0 0.852851 0.426425 0.904523i \(-0.359773\pi\)
0.426425 + 0.904523i \(0.359773\pi\)
\(762\) 48.0000 0.00228196
\(763\) 0 0
\(764\) −17584.0 −0.832679
\(765\) −10368.0 −0.490008
\(766\) 10488.0 0.494709
\(767\) −41328.0 −1.94559
\(768\) 357.000 0.0167736
\(769\) −7560.00 −0.354513 −0.177257 0.984165i \(-0.556722\pi\)
−0.177257 + 0.984165i \(0.556722\pi\)
\(770\) 0 0
\(771\) 8280.00 0.386766
\(772\) 17010.0 0.793009
\(773\) −14292.0 −0.665003 −0.332502 0.943103i \(-0.607893\pi\)
−0.332502 + 0.943103i \(0.607893\pi\)
\(774\) −1404.00 −0.0652012
\(775\) −5016.00 −0.232490
\(776\) −2520.00 −0.116576
\(777\) 0 0
\(778\) −4514.00 −0.208014
\(779\) 0 0
\(780\) −21168.0 −0.971713
\(781\) 5920.00 0.271235
\(782\) −16896.0 −0.772634
\(783\) −1566.00 −0.0714742
\(784\) 0 0
\(785\) 22032.0 1.00173
\(786\) 5076.00 0.230350
\(787\) 26364.0 1.19412 0.597062 0.802195i \(-0.296335\pi\)
0.597062 + 0.802195i \(0.296335\pi\)
\(788\) 12334.0 0.557590
\(789\) 7080.00 0.319461
\(790\) −9312.00 −0.419375
\(791\) 0 0
\(792\) 2700.00 0.121137
\(793\) 41328.0 1.85069
\(794\) 6036.00 0.269785
\(795\) 25992.0 1.15955
\(796\) −21672.0 −0.965005
\(797\) 17220.0 0.765325 0.382662 0.923888i \(-0.375007\pi\)
0.382662 + 0.923888i \(0.375007\pi\)
\(798\) 0 0
\(799\) 39168.0 1.73425
\(800\) −3059.00 −0.135190
\(801\) −6696.00 −0.295370
\(802\) 6770.00 0.298076
\(803\) 4800.00 0.210944
\(804\) 8652.00 0.379518
\(805\) 0 0
\(806\) −22176.0 −0.969127
\(807\) −12060.0 −0.526062
\(808\) −22860.0 −0.995312
\(809\) 16442.0 0.714549 0.357274 0.933999i \(-0.383706\pi\)
0.357274 + 0.933999i \(0.383706\pi\)
\(810\) −972.000 −0.0421637
\(811\) −31332.0 −1.35662 −0.678308 0.734778i \(-0.737286\pi\)
−0.678308 + 0.734778i \(0.737286\pi\)
\(812\) 0 0
\(813\) −14400.0 −0.621193
\(814\) −5160.00 −0.222184
\(815\) 10992.0 0.472433
\(816\) 11808.0 0.506572
\(817\) 1872.00 0.0801628
\(818\) −12504.0 −0.534465
\(819\) 0 0
\(820\) 0 0
\(821\) −25810.0 −1.09717 −0.548584 0.836095i \(-0.684833\pi\)
−0.548584 + 0.836095i \(0.684833\pi\)
\(822\) 3378.00 0.143335
\(823\) 12368.0 0.523841 0.261921 0.965089i \(-0.415644\pi\)
0.261921 + 0.965089i \(0.415644\pi\)
\(824\) −6120.00 −0.258738
\(825\) −1140.00 −0.0481087
\(826\) 0 0
\(827\) 6316.00 0.265573 0.132786 0.991145i \(-0.457608\pi\)
0.132786 + 0.991145i \(0.457608\pi\)
\(828\) 11088.0 0.465380
\(829\) −23868.0 −0.999964 −0.499982 0.866036i \(-0.666660\pi\)
−0.499982 + 0.866036i \(0.666660\pi\)
\(830\) −11088.0 −0.463699
\(831\) −19338.0 −0.807254
\(832\) 14028.0 0.584535
\(833\) 0 0
\(834\) 3276.00 0.136018
\(835\) 6048.00 0.250658
\(836\) −1680.00 −0.0695024
\(837\) 7128.00 0.294360
\(838\) −9492.00 −0.391284
\(839\) −48216.0 −1.98403 −0.992015 0.126120i \(-0.959748\pi\)
−0.992015 + 0.126120i \(0.959748\pi\)
\(840\) 0 0
\(841\) −21025.0 −0.862069
\(842\) −5182.00 −0.212094
\(843\) 7806.00 0.318924
\(844\) −1092.00 −0.0445358
\(845\) 58308.0 2.37379
\(846\) 3672.00 0.149227
\(847\) 0 0
\(848\) −29602.0 −1.19875
\(849\) 20700.0 0.836775
\(850\) 1824.00 0.0736032
\(851\) −45408.0 −1.82910
\(852\) 6216.00 0.249949
\(853\) −27300.0 −1.09582 −0.547910 0.836537i \(-0.684576\pi\)
−0.547910 + 0.836537i \(0.684576\pi\)
\(854\) 0 0
\(855\) 1296.00 0.0518389
\(856\) −12300.0 −0.491128
\(857\) 8640.00 0.344384 0.172192 0.985063i \(-0.444915\pi\)
0.172192 + 0.985063i \(0.444915\pi\)
\(858\) −5040.00 −0.200539
\(859\) 24372.0 0.968058 0.484029 0.875052i \(-0.339173\pi\)
0.484029 + 0.875052i \(0.339173\pi\)
\(860\) −13104.0 −0.519585
\(861\) 0 0
\(862\) 5720.00 0.226014
\(863\) 2176.00 0.0858307 0.0429154 0.999079i \(-0.486335\pi\)
0.0429154 + 0.999079i \(0.486335\pi\)
\(864\) 4347.00 0.171167
\(865\) −22032.0 −0.866024
\(866\) −13608.0 −0.533971
\(867\) −12909.0 −0.505666
\(868\) 0 0
\(869\) 15520.0 0.605846
\(870\) 2088.00 0.0813676
\(871\) −34608.0 −1.34632
\(872\) −13770.0 −0.534760
\(873\) −1512.00 −0.0586179
\(874\) 2112.00 0.0817385
\(875\) 0 0
\(876\) 5040.00 0.194390
\(877\) −27574.0 −1.06170 −0.530848 0.847467i \(-0.678127\pi\)
−0.530848 + 0.847467i \(0.678127\pi\)
\(878\) −12864.0 −0.494464
\(879\) 13356.0 0.512499
\(880\) 9840.00 0.376939
\(881\) 16968.0 0.648884 0.324442 0.945906i \(-0.394824\pi\)
0.324442 + 0.945906i \(0.394824\pi\)
\(882\) 0 0
\(883\) −1860.00 −0.0708879 −0.0354439 0.999372i \(-0.511285\pi\)
−0.0354439 + 0.999372i \(0.511285\pi\)
\(884\) −56448.0 −2.14768
\(885\) −17712.0 −0.672748
\(886\) 13252.0 0.502494
\(887\) 2280.00 0.0863077 0.0431538 0.999068i \(-0.486259\pi\)
0.0431538 + 0.999068i \(0.486259\pi\)
\(888\) −11610.0 −0.438746
\(889\) 0 0
\(890\) 8928.00 0.336255
\(891\) 1620.00 0.0609114
\(892\) −35280.0 −1.32428
\(893\) −4896.00 −0.183470
\(894\) 3210.00 0.120088
\(895\) 28464.0 1.06307
\(896\) 0 0
\(897\) −44352.0 −1.65091
\(898\) −226.000 −0.00839835
\(899\) −15312.0 −0.568058
\(900\) −1197.00 −0.0443333
\(901\) 69312.0 2.56284
\(902\) 0 0
\(903\) 0 0
\(904\) −1650.00 −0.0607060
\(905\) −13104.0 −0.481317
\(906\) −360.000 −0.0132011
\(907\) 36084.0 1.32100 0.660501 0.750825i \(-0.270344\pi\)
0.660501 + 0.750825i \(0.270344\pi\)
\(908\) −15204.0 −0.555686
\(909\) −13716.0 −0.500474
\(910\) 0 0
\(911\) 24152.0 0.878366 0.439183 0.898398i \(-0.355268\pi\)
0.439183 + 0.898398i \(0.355268\pi\)
\(912\) −1476.00 −0.0535913
\(913\) 18480.0 0.669878
\(914\) 11334.0 0.410170
\(915\) 17712.0 0.639935
\(916\) −18900.0 −0.681740
\(917\) 0 0
\(918\) −2592.00 −0.0931904
\(919\) 36336.0 1.30426 0.652130 0.758108i \(-0.273876\pi\)
0.652130 + 0.758108i \(0.273876\pi\)
\(920\) −31680.0 −1.13528
\(921\) 7308.00 0.261462
\(922\) 1596.00 0.0570081
\(923\) −24864.0 −0.886683
\(924\) 0 0
\(925\) 4902.00 0.174245
\(926\) −12728.0 −0.451693
\(927\) −3672.00 −0.130102
\(928\) −9338.00 −0.330318
\(929\) 432.000 0.0152567 0.00762834 0.999971i \(-0.497572\pi\)
0.00762834 + 0.999971i \(0.497572\pi\)
\(930\) −9504.00 −0.335106
\(931\) 0 0
\(932\) 26614.0 0.935376
\(933\) 22464.0 0.788251
\(934\) 3012.00 0.105520
\(935\) −23040.0 −0.805870
\(936\) −11340.0 −0.396004
\(937\) 22176.0 0.773168 0.386584 0.922254i \(-0.373655\pi\)
0.386584 + 0.922254i \(0.373655\pi\)
\(938\) 0 0
\(939\) 5256.00 0.182666
\(940\) 34272.0 1.18918
\(941\) −43524.0 −1.50780 −0.753901 0.656988i \(-0.771830\pi\)
−0.753901 + 0.656988i \(0.771830\pi\)
\(942\) 5508.00 0.190510
\(943\) 0 0
\(944\) 20172.0 0.695490
\(945\) 0 0
\(946\) −3120.00 −0.107230
\(947\) 1868.00 0.0640991 0.0320495 0.999486i \(-0.489797\pi\)
0.0320495 + 0.999486i \(0.489797\pi\)
\(948\) 16296.0 0.558301
\(949\) −20160.0 −0.689590
\(950\) −228.000 −0.00778663
\(951\) 4686.00 0.159783
\(952\) 0 0
\(953\) −9238.00 −0.314006 −0.157003 0.987598i \(-0.550183\pi\)
−0.157003 + 0.987598i \(0.550183\pi\)
\(954\) 6498.00 0.220525
\(955\) 30144.0 1.02140
\(956\) 30856.0 1.04389
\(957\) −3480.00 −0.117547
\(958\) 4296.00 0.144883
\(959\) 0 0
\(960\) 6012.00 0.202121
\(961\) 39905.0 1.33950
\(962\) 21672.0 0.726334
\(963\) −7380.00 −0.246954
\(964\) −21672.0 −0.724075
\(965\) −29160.0 −0.972739
\(966\) 0 0
\(967\) −30616.0 −1.01814 −0.509071 0.860724i \(-0.670011\pi\)
−0.509071 + 0.860724i \(0.670011\pi\)
\(968\) −13965.0 −0.463690
\(969\) 3456.00 0.114575
\(970\) 2016.00 0.0667318
\(971\) −27540.0 −0.910196 −0.455098 0.890441i \(-0.650396\pi\)
−0.455098 + 0.890441i \(0.650396\pi\)
\(972\) 1701.00 0.0561313
\(973\) 0 0
\(974\) 8184.00 0.269232
\(975\) 4788.00 0.157270
\(976\) −20172.0 −0.661568
\(977\) −16402.0 −0.537100 −0.268550 0.963266i \(-0.586544\pi\)
−0.268550 + 0.963266i \(0.586544\pi\)
\(978\) 2748.00 0.0898480
\(979\) −14880.0 −0.485768
\(980\) 0 0
\(981\) −8262.00 −0.268894
\(982\) 12164.0 0.395284
\(983\) 55176.0 1.79028 0.895138 0.445789i \(-0.147077\pi\)
0.895138 + 0.445789i \(0.147077\pi\)
\(984\) 0 0
\(985\) −21144.0 −0.683963
\(986\) 5568.00 0.179839
\(987\) 0 0
\(988\) 7056.00 0.227208
\(989\) −27456.0 −0.882760
\(990\) −2160.00 −0.0693427
\(991\) 27096.0 0.868550 0.434275 0.900780i \(-0.357005\pi\)
0.434275 + 0.900780i \(0.357005\pi\)
\(992\) 42504.0 1.36039
\(993\) 21276.0 0.679933
\(994\) 0 0
\(995\) 37152.0 1.18372
\(996\) 19404.0 0.617309
\(997\) −16812.0 −0.534044 −0.267022 0.963691i \(-0.586040\pi\)
−0.267022 + 0.963691i \(0.586040\pi\)
\(998\) −972.000 −0.0308298
\(999\) −6966.00 −0.220615
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.d.1.1 1
3.2 odd 2 441.4.a.g.1.1 1
4.3 odd 2 2352.4.a.bi.1.1 1
7.2 even 3 147.4.e.f.67.1 2
7.3 odd 6 147.4.e.e.79.1 2
7.4 even 3 147.4.e.f.79.1 2
7.5 odd 6 147.4.e.e.67.1 2
7.6 odd 2 147.4.a.e.1.1 yes 1
21.2 odd 6 441.4.e.g.361.1 2
21.5 even 6 441.4.e.f.361.1 2
21.11 odd 6 441.4.e.g.226.1 2
21.17 even 6 441.4.e.f.226.1 2
21.20 even 2 441.4.a.h.1.1 1
28.27 even 2 2352.4.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
147.4.a.d.1.1 1 1.1 even 1 trivial
147.4.a.e.1.1 yes 1 7.6 odd 2
147.4.e.e.67.1 2 7.5 odd 6
147.4.e.e.79.1 2 7.3 odd 6
147.4.e.f.67.1 2 7.2 even 3
147.4.e.f.79.1 2 7.4 even 3
441.4.a.g.1.1 1 3.2 odd 2
441.4.a.h.1.1 1 21.20 even 2
441.4.e.f.226.1 2 21.17 even 6
441.4.e.f.361.1 2 21.5 even 6
441.4.e.g.226.1 2 21.11 odd 6
441.4.e.g.361.1 2 21.2 odd 6
2352.4.a.b.1.1 1 28.27 even 2
2352.4.a.bi.1.1 1 4.3 odd 2