Properties

Label 441.4.a.h.1.1
Level $441$
Weight $4$
Character 441.1
Self dual yes
Analytic conductor $26.020$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.0198423125\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 147)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 441.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -7.00000 q^{4} +12.0000 q^{5} -15.0000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} -7.00000 q^{4} +12.0000 q^{5} -15.0000 q^{8} +12.0000 q^{10} -20.0000 q^{11} +84.0000 q^{13} +41.0000 q^{16} -96.0000 q^{17} -12.0000 q^{19} -84.0000 q^{20} -20.0000 q^{22} +176.000 q^{23} +19.0000 q^{25} +84.0000 q^{26} -58.0000 q^{29} +264.000 q^{31} +161.000 q^{32} -96.0000 q^{34} +258.000 q^{37} -12.0000 q^{38} -180.000 q^{40} +156.000 q^{43} +140.000 q^{44} +176.000 q^{46} -408.000 q^{47} +19.0000 q^{50} -588.000 q^{52} +722.000 q^{53} -240.000 q^{55} -58.0000 q^{58} +492.000 q^{59} +492.000 q^{61} +264.000 q^{62} -167.000 q^{64} +1008.00 q^{65} +412.000 q^{67} +672.000 q^{68} -296.000 q^{71} -240.000 q^{73} +258.000 q^{74} +84.0000 q^{76} +776.000 q^{79} +492.000 q^{80} +924.000 q^{83} -1152.00 q^{85} +156.000 q^{86} +300.000 q^{88} -744.000 q^{89} -1232.00 q^{92} -408.000 q^{94} -144.000 q^{95} +168.000 q^{97} +O(q^{100})\)

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 \(-0.443433\pi\)
0.176777 + 0.984251i \(0.443433\pi\)
\(3\) 0 0
\(4\) −7.00000 −0.875000
\(5\) 12.0000 1.07331 0.536656 0.843801i \(-0.319687\pi\)
0.536656 + 0.843801i \(0.319687\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −15.0000 −0.662913
\(9\) 0 0
\(10\) 12.0000 0.379473
\(11\) −20.0000 −0.548202 −0.274101 0.961701i \(-0.588380\pi\)
−0.274101 + 0.961701i \(0.588380\pi\)
\(12\) 0 0
\(13\) 84.0000 1.79211 0.896054 0.443945i \(-0.146421\pi\)
0.896054 + 0.443945i \(0.146421\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 41.0000 0.640625
\(17\) −96.0000 −1.36961 −0.684806 0.728725i \(-0.740113\pi\)
−0.684806 + 0.728725i \(0.740113\pi\)
\(18\) 0 0
\(19\) −12.0000 −0.144894 −0.0724471 0.997372i \(-0.523081\pi\)
−0.0724471 + 0.997372i \(0.523081\pi\)
\(20\) −84.0000 −0.939149
\(21\) 0 0
\(22\) −20.0000 −0.193819
\(23\) 176.000 1.59559 0.797794 0.602930i \(-0.206000\pi\)
0.797794 + 0.602930i \(0.206000\pi\)
\(24\) 0 0
\(25\) 19.0000 0.152000
\(26\) 84.0000 0.633606
\(27\) 0 0
\(28\) 0 0
\(29\) −58.0000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 264.000 1.52954 0.764771 0.644302i \(-0.222852\pi\)
0.764771 + 0.644302i \(0.222852\pi\)
\(32\) 161.000 0.889408
\(33\) 0 0
\(34\) −96.0000 −0.484231
\(35\) 0 0
\(36\) 0 0
\(37\) 258.000 1.14635 0.573175 0.819433i \(-0.305712\pi\)
0.573175 + 0.819433i \(0.305712\pi\)
\(38\) −12.0000 −0.0512278
\(39\) 0 0
\(40\) −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
\(45\) 0 0
\(46\) 176.000 0.564126
\(47\) −408.000 −1.26623 −0.633116 0.774057i \(-0.718224\pi\)
−0.633116 + 0.774057i \(0.718224\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 19.0000 0.0537401
\(51\) 0 0
\(52\) −588.000 −1.56809
\(53\) 722.000 1.87121 0.935607 0.353044i \(-0.114853\pi\)
0.935607 + 0.353044i \(0.114853\pi\)
\(54\) 0 0
\(55\) −240.000 −0.588393
\(56\) 0 0
\(57\) 0 0
\(58\) −58.0000 −0.131306
\(59\) 492.000 1.08564 0.542822 0.839848i \(-0.317356\pi\)
0.542822 + 0.839848i \(0.317356\pi\)
\(60\) 0 0
\(61\) 492.000 1.03269 0.516345 0.856380i \(-0.327292\pi\)
0.516345 + 0.856380i \(0.327292\pi\)
\(62\) 264.000 0.540775
\(63\) 0 0
\(64\) −167.000 −0.326172
\(65\) 1008.00 1.92349
\(66\) 0 0
\(67\) 412.000 0.751251 0.375625 0.926772i \(-0.377428\pi\)
0.375625 + 0.926772i \(0.377428\pi\)
\(68\) 672.000 1.19841
\(69\) 0 0
\(70\) 0 0
\(71\) −296.000 −0.494771 −0.247385 0.968917i \(-0.579571\pi\)
−0.247385 + 0.968917i \(0.579571\pi\)
\(72\) 0 0
\(73\) −240.000 −0.384793 −0.192396 0.981317i \(-0.561626\pi\)
−0.192396 + 0.981317i \(0.561626\pi\)
\(74\) 258.000 0.405296
\(75\) 0 0
\(76\) 84.0000 0.126782
\(77\) 0 0
\(78\) 0 0
\(79\) 776.000 1.10515 0.552575 0.833463i \(-0.313645\pi\)
0.552575 + 0.833463i \(0.313645\pi\)
\(80\) 492.000 0.687591
\(81\) 0 0
\(82\) 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
\(87\) 0 0
\(88\) 300.000 0.363410
\(89\) −744.000 −0.886111 −0.443055 0.896494i \(-0.646105\pi\)
−0.443055 + 0.896494i \(0.646105\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1232.00 −1.39614
\(93\) 0 0
\(94\) −408.000 −0.447681
\(95\) −144.000 −0.155517
\(96\) 0 0
\(97\) 168.000 0.175854 0.0879269 0.996127i \(-0.471976\pi\)
0.0879269 + 0.996127i \(0.471976\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −133.000 −0.133000
\(101\) −1524.00 −1.50142 −0.750711 0.660630i \(-0.770289\pi\)
−0.750711 + 0.660630i \(0.770289\pi\)
\(102\) 0 0
\(103\) 408.000 0.390305 0.195153 0.980773i \(-0.437480\pi\)
0.195153 + 0.980773i \(0.437480\pi\)
\(104\) −1260.00 −1.18801
\(105\) 0 0
\(106\) 722.000 0.661574
\(107\) 820.000 0.740863 0.370432 0.928860i \(-0.379210\pi\)
0.370432 + 0.928860i \(0.379210\pi\)
\(108\) 0 0
\(109\) −918.000 −0.806683 −0.403342 0.915050i \(-0.632151\pi\)
−0.403342 + 0.915050i \(0.632151\pi\)
\(110\) −240.000 −0.208028
\(111\) 0 0
\(112\) 0 0
\(113\) 110.000 0.0915746 0.0457873 0.998951i \(-0.485420\pi\)
0.0457873 + 0.998951i \(0.485420\pi\)
\(114\) 0 0
\(115\) 2112.00 1.71257
\(116\) 406.000 0.324967
\(117\) 0 0
\(118\) 492.000 0.383833
\(119\) 0 0
\(120\) 0 0
\(121\) −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
\(129\) 0 0
\(130\) 1008.00 0.680057
\(131\) 1692.00 1.12848 0.564239 0.825611i \(-0.309169\pi\)
0.564239 + 0.825611i \(0.309169\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 412.000 0.265607
\(135\) 0 0
\(136\) 1440.00 0.907934
\(137\) −1126.00 −0.702195 −0.351097 0.936339i \(-0.614191\pi\)
−0.351097 + 0.936339i \(0.614191\pi\)
\(138\) 0 0
\(139\) −1092.00 −0.666347 −0.333173 0.942866i \(-0.608119\pi\)
−0.333173 + 0.942866i \(0.608119\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −296.000 −0.174928
\(143\) −1680.00 −0.982438
\(144\) 0 0
\(145\) −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 \(-0.595038\pi\)
−0.294154 + 0.955758i \(0.595038\pi\)
\(150\) 0 0
\(151\) −120.000 −0.0646719 −0.0323360 0.999477i \(-0.510295\pi\)
−0.0323360 + 0.999477i \(0.510295\pi\)
\(152\) 180.000 0.0960522
\(153\) 0 0
\(154\) 0 0
\(155\) 3168.00 1.64168
\(156\) 0 0
\(157\) −1836.00 −0.933304 −0.466652 0.884441i \(-0.654540\pi\)
−0.466652 + 0.884441i \(0.654540\pi\)
\(158\) 776.000 0.390729
\(159\) 0 0
\(160\) 1932.00 0.954613
\(161\) 0 0
\(162\) 0 0
\(163\) 916.000 0.440164 0.220082 0.975481i \(-0.429368\pi\)
0.220082 + 0.975481i \(0.429368\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 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
\(171\) 0 0
\(172\) −1092.00 −0.484094
\(173\) −1836.00 −0.806870 −0.403435 0.915008i \(-0.632184\pi\)
−0.403435 + 0.915008i \(0.632184\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −820.000 −0.351192
\(177\) 0 0
\(178\) −744.000 −0.313287
\(179\) −2372.00 −0.990456 −0.495228 0.868763i \(-0.664915\pi\)
−0.495228 + 0.868763i \(0.664915\pi\)
\(180\) 0 0
\(181\) 1092.00 0.448440 0.224220 0.974539i \(-0.428017\pi\)
0.224220 + 0.974539i \(0.428017\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −2640.00 −1.05774
\(185\) 3096.00 1.23039
\(186\) 0 0
\(187\) 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 \(-0.657847\pi\)
−0.475817 + 0.879545i \(0.657847\pi\)
\(192\) 0 0
\(193\) −2430.00 −0.906297 −0.453148 0.891435i \(-0.649699\pi\)
−0.453148 + 0.891435i \(0.649699\pi\)
\(194\) 168.000 0.0621737
\(195\) 0 0
\(196\) 0 0
\(197\) 1762.00 0.637245 0.318623 0.947882i \(-0.396780\pi\)
0.318623 + 0.947882i \(0.396780\pi\)
\(198\) 0 0
\(199\) −3096.00 −1.10286 −0.551431 0.834220i \(-0.685918\pi\)
−0.551431 + 0.834220i \(0.685918\pi\)
\(200\) −285.000 −0.100763
\(201\) 0 0
\(202\) −1524.00 −0.530833
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 408.000 0.137994
\(207\) 0 0
\(208\) 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
\(213\) 0 0
\(214\) 820.000 0.261935
\(215\) 1872.00 0.593811
\(216\) 0 0
\(217\) 0 0
\(218\) −918.000 −0.285206
\(219\) 0 0
\(220\) 1680.00 0.514844
\(221\) −8064.00 −2.45449
\(222\) 0 0
\(223\) −5040.00 −1.51347 −0.756734 0.653723i \(-0.773206\pi\)
−0.756734 + 0.653723i \(0.773206\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 110.000 0.0323765
\(227\) 2172.00 0.635069 0.317535 0.948247i \(-0.397145\pi\)
0.317535 + 0.948247i \(0.397145\pi\)
\(228\) 0 0
\(229\) −2700.00 −0.779131 −0.389566 0.920999i \(-0.627375\pi\)
−0.389566 + 0.920999i \(0.627375\pi\)
\(230\) 2112.00 0.605483
\(231\) 0 0
\(232\) 870.000 0.246200
\(233\) 3802.00 1.06900 0.534501 0.845168i \(-0.320500\pi\)
0.534501 + 0.845168i \(0.320500\pi\)
\(234\) 0 0
\(235\) −4896.00 −1.35906
\(236\) −3444.00 −0.949938
\(237\) 0 0
\(238\) 0 0
\(239\) 4408.00 1.19301 0.596506 0.802609i \(-0.296555\pi\)
0.596506 + 0.802609i \(0.296555\pi\)
\(240\) 0 0
\(241\) −3096.00 −0.827514 −0.413757 0.910387i \(-0.635784\pi\)
−0.413757 + 0.910387i \(0.635784\pi\)
\(242\) −931.000 −0.247301
\(243\) 0 0
\(244\) −3444.00 −0.903605
\(245\) 0 0
\(246\) 0 0
\(247\) −1008.00 −0.259666
\(248\) −3960.00 −1.01395
\(249\) 0 0
\(250\) −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
\(255\) 0 0
\(256\) −119.000 −0.0290527
\(257\) −2760.00 −0.669899 −0.334950 0.942236i \(-0.608719\pi\)
−0.334950 + 0.942236i \(0.608719\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −7056.00 −1.68306
\(261\) 0 0
\(262\) 1692.00 0.398978
\(263\) 2360.00 0.553323 0.276661 0.960967i \(-0.410772\pi\)
0.276661 + 0.960967i \(0.410772\pi\)
\(264\) 0 0
\(265\) 8664.00 2.00840
\(266\) 0 0
\(267\) 0 0
\(268\) −2884.00 −0.657345
\(269\) 4020.00 0.911166 0.455583 0.890193i \(-0.349431\pi\)
0.455583 + 0.890193i \(0.349431\pi\)
\(270\) 0 0
\(271\) −4800.00 −1.07594 −0.537969 0.842965i \(-0.680808\pi\)
−0.537969 + 0.842965i \(0.680808\pi\)
\(272\) −3936.00 −0.877408
\(273\) 0 0
\(274\) −1126.00 −0.248263
\(275\) −380.000 −0.0833268
\(276\) 0 0
\(277\) 6446.00 1.39820 0.699102 0.715022i \(-0.253583\pi\)
0.699102 + 0.715022i \(0.253583\pi\)
\(278\) −1092.00 −0.235589
\(279\) 0 0
\(280\) 0 0
\(281\) 2602.00 0.552393 0.276196 0.961101i \(-0.410926\pi\)
0.276196 + 0.961101i \(0.410926\pi\)
\(282\) 0 0
\(283\) 6900.00 1.44934 0.724669 0.689098i \(-0.241993\pi\)
0.724669 + 0.689098i \(0.241993\pi\)
\(284\) 2072.00 0.432925
\(285\) 0 0
\(286\) −1680.00 −0.347344
\(287\) 0 0
\(288\) 0 0
\(289\) 4303.00 0.875840
\(290\) −696.000 −0.140933
\(291\) 0 0
\(292\) 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
\(297\) 0 0
\(298\) −1070.00 −0.207998
\(299\) 14784.0 2.85947
\(300\) 0 0
\(301\) 0 0
\(302\) −120.000 −0.0228650
\(303\) 0 0
\(304\) −492.000 −0.0928228
\(305\) 5904.00 1.10840
\(306\) 0 0
\(307\) 2436.00 0.452866 0.226433 0.974027i \(-0.427294\pi\)
0.226433 + 0.974027i \(0.427294\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 3168.00 0.580420
\(311\) −7488.00 −1.36529 −0.682646 0.730750i \(-0.739171\pi\)
−0.682646 + 0.730750i \(0.739171\pi\)
\(312\) 0 0
\(313\) 1752.00 0.316386 0.158193 0.987408i \(-0.449433\pi\)
0.158193 + 0.987408i \(0.449433\pi\)
\(314\) −1836.00 −0.329973
\(315\) 0 0
\(316\) −5432.00 −0.967006
\(317\) 1562.00 0.276753 0.138376 0.990380i \(-0.455812\pi\)
0.138376 + 0.990380i \(0.455812\pi\)
\(318\) 0 0
\(319\) 1160.00 0.203597
\(320\) −2004.00 −0.350084
\(321\) 0 0
\(322\) 0 0
\(323\) 1152.00 0.198449
\(324\) 0 0
\(325\) 1596.00 0.272400
\(326\) 916.000 0.155621
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −7092.00 −1.17768 −0.588839 0.808250i \(-0.700415\pi\)
−0.588839 + 0.808250i \(0.700415\pi\)
\(332\) −6468.00 −1.06921
\(333\) 0 0
\(334\) 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
\(339\) 0 0
\(340\) 8064.00 1.28627
\(341\) −5280.00 −0.838499
\(342\) 0 0
\(343\) 0 0
\(344\) −2340.00 −0.366757
\(345\) 0 0
\(346\) −1836.00 −0.285272
\(347\) 6364.00 0.984546 0.492273 0.870441i \(-0.336166\pi\)
0.492273 + 0.870441i \(0.336166\pi\)
\(348\) 0 0
\(349\) 10500.0 1.61046 0.805232 0.592960i \(-0.202041\pi\)
0.805232 + 0.592960i \(0.202041\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −3220.00 −0.487576
\(353\) 408.000 0.0615174 0.0307587 0.999527i \(-0.490208\pi\)
0.0307587 + 0.999527i \(0.490208\pi\)
\(354\) 0 0
\(355\) −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 \(-0.159290\pi\)
0.877379 + 0.479798i \(0.159290\pi\)
\(360\) 0 0
\(361\) −6715.00 −0.979006
\(362\) 1092.00 0.158548
\(363\) 0 0
\(364\) 0 0
\(365\) −2880.00 −0.413003
\(366\) 0 0
\(367\) 2448.00 0.348187 0.174093 0.984729i \(-0.444301\pi\)
0.174093 + 0.984729i \(0.444301\pi\)
\(368\) 7216.00 1.02217
\(369\) 0 0
\(370\) 3096.00 0.435009
\(371\) 0 0
\(372\) 0 0
\(373\) 11374.0 1.57888 0.789442 0.613826i \(-0.210370\pi\)
0.789442 + 0.613826i \(0.210370\pi\)
\(374\) 1920.00 0.265457
\(375\) 0 0
\(376\) 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
\(381\) 0 0
\(382\) −2512.00 −0.336453
\(383\) −10488.0 −1.39925 −0.699624 0.714511i \(-0.746649\pi\)
−0.699624 + 0.714511i \(0.746649\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −2430.00 −0.320424
\(387\) 0 0
\(388\) −1176.00 −0.153872
\(389\) −4514.00 −0.588352 −0.294176 0.955751i \(-0.595045\pi\)
−0.294176 + 0.955751i \(0.595045\pi\)
\(390\) 0 0
\(391\) −16896.0 −2.18534
\(392\) 0 0
\(393\) 0 0
\(394\) 1762.00 0.225300
\(395\) 9312.00 1.18617
\(396\) 0 0
\(397\) 6036.00 0.763068 0.381534 0.924355i \(-0.375396\pi\)
0.381534 + 0.924355i \(0.375396\pi\)
\(398\) −3096.00 −0.389921
\(399\) 0 0
\(400\) 779.000 0.0973750
\(401\) 6770.00 0.843086 0.421543 0.906808i \(-0.361489\pi\)
0.421543 + 0.906808i \(0.361489\pi\)
\(402\) 0 0
\(403\) 22176.0 2.74110
\(404\) 10668.0 1.31374
\(405\) 0 0
\(406\) 0 0
\(407\) −5160.00 −0.628432
\(408\) 0 0
\(409\) −12504.0 −1.51169 −0.755847 0.654748i \(-0.772775\pi\)
−0.755847 + 0.654748i \(0.772775\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −2856.00 −0.341517
\(413\) 0 0
\(414\) 0 0
\(415\) 11088.0 1.31154
\(416\) 13524.0 1.59392
\(417\) 0 0
\(418\) 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
\(423\) 0 0
\(424\) −10830.0 −1.24045
\(425\) −1824.00 −0.208181
\(426\) 0 0
\(427\) 0 0
\(428\) −5740.00 −0.648256
\(429\) 0 0
\(430\) 1872.00 0.209944
\(431\) 5720.00 0.639264 0.319632 0.947542i \(-0.396441\pi\)
0.319632 + 0.947542i \(0.396441\pi\)
\(432\) 0 0
\(433\) −13608.0 −1.51030 −0.755149 0.655554i \(-0.772435\pi\)
−0.755149 + 0.655554i \(0.772435\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 6426.00 0.705848
\(437\) −2112.00 −0.231191
\(438\) 0 0
\(439\) −12864.0 −1.39855 −0.699277 0.714851i \(-0.746495\pi\)
−0.699277 + 0.714851i \(0.746495\pi\)
\(440\) 3600.00 0.390053
\(441\) 0 0
\(442\) −8064.00 −0.867795
\(443\) 13252.0 1.42127 0.710634 0.703562i \(-0.248408\pi\)
0.710634 + 0.703562i \(0.248408\pi\)
\(444\) 0 0
\(445\) −8928.00 −0.951074
\(446\) −5040.00 −0.535092
\(447\) 0 0
\(448\) 0 0
\(449\) −226.000 −0.0237541 −0.0118771 0.999929i \(-0.503781\pi\)
−0.0118771 + 0.999929i \(0.503781\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −770.000 −0.0801278
\(453\) 0 0
\(454\) 2172.00 0.224531
\(455\) 0 0
\(456\) 0 0
\(457\) −11334.0 −1.16014 −0.580068 0.814568i \(-0.696974\pi\)
−0.580068 + 0.814568i \(0.696974\pi\)
\(458\) −2700.00 −0.275464
\(459\) 0 0
\(460\) −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
\(465\) 0 0
\(466\) 3802.00 0.377949
\(467\) −3012.00 −0.298456 −0.149228 0.988803i \(-0.547679\pi\)
−0.149228 + 0.988803i \(0.547679\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −4896.00 −0.480501
\(471\) 0 0
\(472\) −7380.00 −0.719687
\(473\) −3120.00 −0.303293
\(474\) 0 0
\(475\) −228.000 −0.0220239
\(476\) 0 0
\(477\) 0 0
\(478\) 4408.00 0.421793
\(479\) −4296.00 −0.409790 −0.204895 0.978784i \(-0.565685\pi\)
−0.204895 + 0.978784i \(0.565685\pi\)
\(480\) 0 0
\(481\) 21672.0 2.05438
\(482\) −3096.00 −0.292570
\(483\) 0 0
\(484\) 6517.00 0.612040
\(485\) 2016.00 0.188746
\(486\) 0 0
\(487\) −8184.00 −0.761504 −0.380752 0.924677i \(-0.624335\pi\)
−0.380752 + 0.924677i \(0.624335\pi\)
\(488\) −7380.00 −0.684584
\(489\) 0 0
\(490\) 0 0
\(491\) 12164.0 1.11803 0.559016 0.829157i \(-0.311179\pi\)
0.559016 + 0.829157i \(0.311179\pi\)
\(492\) 0 0
\(493\) 5568.00 0.508661
\(494\) −1008.00 −0.0918058
\(495\) 0 0
\(496\) 10824.0 0.979863
\(497\) 0 0
\(498\) 0 0
\(499\) 972.000 0.0871998 0.0435999 0.999049i \(-0.486117\pi\)
0.0435999 + 0.999049i \(0.486117\pi\)
\(500\) 8904.00 0.796398
\(501\) 0 0
\(502\) −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
\(507\) 0 0
\(508\) −112.000 −0.00978188
\(509\) 11604.0 1.01049 0.505244 0.862977i \(-0.331403\pi\)
0.505244 + 0.862977i \(0.331403\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 11521.0 0.994455
\(513\) 0 0
\(514\) −2760.00 −0.236845
\(515\) 4896.00 0.418919
\(516\) 0 0
\(517\) 8160.00 0.694152
\(518\) 0 0
\(519\) 0 0
\(520\) −15120.0 −1.27511
\(521\) −10848.0 −0.912206 −0.456103 0.889927i \(-0.650755\pi\)
−0.456103 + 0.889927i \(0.650755\pi\)
\(522\) 0 0
\(523\) 18132.0 1.51598 0.757989 0.652267i \(-0.226182\pi\)
0.757989 + 0.652267i \(0.226182\pi\)
\(524\) −11844.0 −0.987419
\(525\) 0 0
\(526\) 2360.00 0.195629
\(527\) −25344.0 −2.09488
\(528\) 0 0
\(529\) 18809.0 1.54590
\(530\) 8664.00 0.710076
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 9840.00 0.795178
\(536\) −6180.00 −0.498014
\(537\) 0 0
\(538\) 4020.00 0.322146
\(539\) 0 0
\(540\) 0 0
\(541\) 6950.00 0.552318 0.276159 0.961112i \(-0.410938\pi\)
0.276159 + 0.961112i \(0.410938\pi\)
\(542\) −4800.00 −0.380402
\(543\) 0 0
\(544\) −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
\(549\) 0 0
\(550\) −380.000 −0.0294605
\(551\) 696.000 0.0538123
\(552\) 0 0
\(553\) 0 0
\(554\) 6446.00 0.494340
\(555\) 0 0
\(556\) 7644.00 0.583054
\(557\) −3926.00 −0.298653 −0.149327 0.988788i \(-0.547711\pi\)
−0.149327 + 0.988788i \(0.547711\pi\)
\(558\) 0 0
\(559\) 13104.0 0.991485
\(560\) 0 0
\(561\) 0 0
\(562\) 2602.00 0.195300
\(563\) −18828.0 −1.40942 −0.704712 0.709494i \(-0.748924\pi\)
−0.704712 + 0.709494i \(0.748924\pi\)
\(564\) 0 0
\(565\) 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 \(-0.645622\pi\)
−0.441693 + 0.897166i \(0.645622\pi\)
\(570\) 0 0
\(571\) −15716.0 −1.15183 −0.575914 0.817510i \(-0.695354\pi\)
−0.575914 + 0.817510i \(0.695354\pi\)
\(572\) 11760.0 0.859633
\(573\) 0 0
\(574\) 0 0
\(575\) 3344.00 0.242529
\(576\) 0 0
\(577\) 13872.0 1.00086 0.500432 0.865776i \(-0.333174\pi\)
0.500432 + 0.865776i \(0.333174\pi\)
\(578\) 4303.00 0.309656
\(579\) 0 0
\(580\) 4872.00 0.348791
\(581\) 0 0
\(582\) 0 0
\(583\) −14440.0 −1.02580
\(584\) 3600.00 0.255084
\(585\) 0 0
\(586\) −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
\(591\) 0 0
\(592\) 10578.0 0.734380
\(593\) −16872.0 −1.16838 −0.584191 0.811617i \(-0.698588\pi\)
−0.584191 + 0.811617i \(0.698588\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 7490.00 0.514769
\(597\) 0 0
\(598\) 14784.0 1.01097
\(599\) 6056.00 0.413091 0.206545 0.978437i \(-0.433778\pi\)
0.206545 + 0.978437i \(0.433778\pi\)
\(600\) 0 0
\(601\) −10752.0 −0.729756 −0.364878 0.931055i \(-0.618889\pi\)
−0.364878 + 0.931055i \(0.618889\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 840.000 0.0565879
\(605\) −11172.0 −0.750754
\(606\) 0 0
\(607\) −20256.0 −1.35447 −0.677237 0.735765i \(-0.736823\pi\)
−0.677237 + 0.735765i \(0.736823\pi\)
\(608\) −1932.00 −0.128870
\(609\) 0 0
\(610\) 5904.00 0.391879
\(611\) −34272.0 −2.26923
\(612\) 0 0
\(613\) −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 \(-0.905748\pi\)
−0.956482 + 0.291793i \(0.905748\pi\)
\(618\) 0 0
\(619\) −24348.0 −1.58098 −0.790492 0.612473i \(-0.790175\pi\)
−0.790492 + 0.612473i \(0.790175\pi\)
\(620\) −22176.0 −1.43647
\(621\) 0 0
\(622\) −7488.00 −0.482703
\(623\) 0 0
\(624\) 0 0
\(625\) −17639.0 −1.12890
\(626\) 1752.00 0.111859
\(627\) 0 0
\(628\) 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
\(633\) 0 0
\(634\) 1562.00 0.0978469
\(635\) 192.000 0.0119989
\(636\) 0 0
\(637\) 0 0
\(638\) 1160.00 0.0719825
\(639\) 0 0
\(640\) −17460.0 −1.07839
\(641\) −32318.0 −1.99140 −0.995698 0.0926628i \(-0.970462\pi\)
−0.995698 + 0.0926628i \(0.970462\pi\)
\(642\) 0 0
\(643\) 3948.00 0.242137 0.121068 0.992644i \(-0.461368\pi\)
0.121068 + 0.992644i \(0.461368\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 1152.00 0.0701623
\(647\) 13848.0 0.841454 0.420727 0.907187i \(-0.361775\pi\)
0.420727 + 0.907187i \(0.361775\pi\)
\(648\) 0 0
\(649\) −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 \(-0.469834\pi\)
0.0946264 + 0.995513i \(0.469834\pi\)
\(654\) 0 0
\(655\) 20304.0 1.21121
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 24596.0 1.45391 0.726953 0.686687i \(-0.240936\pi\)
0.726953 + 0.686687i \(0.240936\pi\)
\(660\) 0 0
\(661\) 15468.0 0.910190 0.455095 0.890443i \(-0.349605\pi\)
0.455095 + 0.890443i \(0.349605\pi\)
\(662\) −7092.00 −0.416372
\(663\) 0 0
\(664\) −13860.0 −0.810049
\(665\) 0 0
\(666\) 0 0
\(667\) −10208.0 −0.592587
\(668\) −3528.00 −0.204345
\(669\) 0 0
\(670\) 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
\(675\) 0 0
\(676\) −34013.0 −1.93520
\(677\) −9564.00 −0.542946 −0.271473 0.962446i \(-0.587511\pi\)
−0.271473 + 0.962446i \(0.587511\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 17280.0 0.974497
\(681\) 0 0
\(682\) −5280.00 −0.296454
\(683\) −13852.0 −0.776035 −0.388018 0.921652i \(-0.626840\pi\)
−0.388018 + 0.921652i \(0.626840\pi\)
\(684\) 0 0
\(685\) −13512.0 −0.753674
\(686\) 0 0
\(687\) 0 0
\(688\) 6396.00 0.354426
\(689\) 60648.0 3.35342
\(690\) 0 0
\(691\) 324.000 0.0178373 0.00891863 0.999960i \(-0.497161\pi\)
0.00891863 + 0.999960i \(0.497161\pi\)
\(692\) 12852.0 0.706011
\(693\) 0 0
\(694\) 6364.00 0.348090
\(695\) −13104.0 −0.715199
\(696\) 0 0
\(697\) 0 0
\(698\) 10500.0 0.569385
\(699\) 0 0
\(700\) 0 0
\(701\) −24922.0 −1.34278 −0.671392 0.741103i \(-0.734303\pi\)
−0.671392 + 0.741103i \(0.734303\pi\)
\(702\) 0 0
\(703\) −3096.00 −0.166099
\(704\) 3340.00 0.178808
\(705\) 0 0
\(706\) 408.000 0.0217497
\(707\) 0 0
\(708\) 0 0
\(709\) −17886.0 −0.947423 −0.473711 0.880680i \(-0.657086\pi\)
−0.473711 + 0.880680i \(0.657086\pi\)
\(710\) −3552.00 −0.187752
\(711\) 0 0
\(712\) 11160.0 0.587414
\(713\) 46464.0 2.44052
\(714\) 0 0
\(715\) −20160.0 −1.05446
\(716\) 16604.0 0.866649
\(717\) 0 0
\(718\) 11936.0 0.620401
\(719\) −6792.00 −0.352293 −0.176147 0.984364i \(-0.556363\pi\)
−0.176147 + 0.984364i \(0.556363\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −6715.00 −0.346131
\(723\) 0 0
\(724\) −7644.00 −0.392385
\(725\) −1102.00 −0.0564514
\(726\) 0 0
\(727\) −1512.00 −0.0771348 −0.0385674 0.999256i \(-0.512279\pi\)
−0.0385674 + 0.999256i \(0.512279\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −2880.00 −0.146019
\(731\) −14976.0 −0.757739
\(732\) 0 0
\(733\) 11244.0 0.566585 0.283292 0.959034i \(-0.408573\pi\)
0.283292 + 0.959034i \(0.408573\pi\)
\(734\) 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
\(741\) 0 0
\(742\) 0 0
\(743\) 656.000 0.0323907 0.0161954 0.999869i \(-0.494845\pi\)
0.0161954 + 0.999869i \(0.494845\pi\)
\(744\) 0 0
\(745\) −12840.0 −0.631438
\(746\) 11374.0 0.558219
\(747\) 0 0
\(748\) −13440.0 −0.656972
\(749\) 0 0
\(750\) 0 0
\(751\) 1056.00 0.0513102 0.0256551 0.999671i \(-0.491833\pi\)
0.0256551 + 0.999671i \(0.491833\pi\)
\(752\) −16728.0 −0.811180
\(753\) 0 0
\(754\) −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
\(759\) 0 0
\(760\) 2160.00 0.103094
\(761\) 17904.0 0.852851 0.426425 0.904523i \(-0.359773\pi\)
0.426425 + 0.904523i \(0.359773\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 17584.0 0.832679
\(765\) 0 0
\(766\) −10488.0 −0.494709
\(767\) 41328.0 1.94559
\(768\) 0 0
\(769\) 7560.00 0.354513 0.177257 0.984165i \(-0.443278\pi\)
0.177257 + 0.984165i \(0.443278\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 17010.0 0.793009
\(773\) −14292.0 −0.665003 −0.332502 0.943103i \(-0.607893\pi\)
−0.332502 + 0.943103i \(0.607893\pi\)
\(774\) 0 0
\(775\) 5016.00 0.232490
\(776\) −2520.00 −0.116576
\(777\) 0 0
\(778\) −4514.00 −0.208014
\(779\) 0 0
\(780\) 0 0
\(781\) 5920.00 0.271235
\(782\) −16896.0 −0.772634
\(783\) 0 0
\(784\) 0 0
\(785\) −22032.0 −1.00173
\(786\) 0 0
\(787\) −26364.0 −1.19412 −0.597062 0.802195i \(-0.703665\pi\)
−0.597062 + 0.802195i \(0.703665\pi\)
\(788\) −12334.0 −0.557590
\(789\) 0 0
\(790\) 9312.00 0.419375
\(791\) 0 0
\(792\) 0 0
\(793\) 41328.0 1.85069
\(794\) 6036.00 0.269785
\(795\) 0 0
\(796\) 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
\(801\) 0 0
\(802\) 6770.00 0.298076
\(803\) 4800.00 0.210944
\(804\) 0 0
\(805\) 0 0
\(806\) 22176.0 0.969127
\(807\) 0 0
\(808\) 22860.0 0.995312
\(809\) −16442.0 −0.714549 −0.357274 0.933999i \(-0.616294\pi\)
−0.357274 + 0.933999i \(0.616294\pi\)
\(810\) 0 0
\(811\) 31332.0 1.35662 0.678308 0.734778i \(-0.262714\pi\)
0.678308 + 0.734778i \(0.262714\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −5160.00 −0.222184
\(815\) 10992.0 0.472433
\(816\) 0 0
\(817\) −1872.00 −0.0801628
\(818\) −12504.0 −0.534465
\(819\) 0 0
\(820\) 0 0
\(821\) 25810.0 1.09717 0.548584 0.836095i \(-0.315167\pi\)
0.548584 + 0.836095i \(0.315167\pi\)
\(822\) 0 0
\(823\) 12368.0 0.523841 0.261921 0.965089i \(-0.415644\pi\)
0.261921 + 0.965089i \(0.415644\pi\)
\(824\) −6120.00 −0.258738
\(825\) 0 0
\(826\) 0 0
\(827\) −6316.00 −0.265573 −0.132786 0.991145i \(-0.542392\pi\)
−0.132786 + 0.991145i \(0.542392\pi\)
\(828\) 0 0
\(829\) 23868.0 0.999964 0.499982 0.866036i \(-0.333340\pi\)
0.499982 + 0.866036i \(0.333340\pi\)
\(830\) 11088.0 0.463699
\(831\) 0 0
\(832\) −14028.0 −0.584535
\(833\) 0 0
\(834\) 0 0
\(835\) 6048.00 0.250658
\(836\) −1680.00 −0.0695024
\(837\) 0 0
\(838\) 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
\(843\) 0 0
\(844\) −1092.00 −0.0445358
\(845\) 58308.0 2.37379
\(846\) 0 0
\(847\) 0 0
\(848\) 29602.0 1.19875
\(849\) 0 0
\(850\) −1824.00 −0.0736032
\(851\) 45408.0 1.82910
\(852\) 0 0
\(853\) 27300.0 1.09582 0.547910 0.836537i \(-0.315424\pi\)
0.547910 + 0.836537i \(0.315424\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −12300.0 −0.491128
\(857\) 8640.00 0.344384 0.172192 0.985063i \(-0.444915\pi\)
0.172192 + 0.985063i \(0.444915\pi\)
\(858\) 0 0
\(859\) −24372.0 −0.968058 −0.484029 0.875052i \(-0.660827\pi\)
−0.484029 + 0.875052i \(0.660827\pi\)
\(860\) −13104.0 −0.519585
\(861\) 0 0
\(862\) 5720.00 0.226014
\(863\) −2176.00 −0.0858307 −0.0429154 0.999079i \(-0.513665\pi\)
−0.0429154 + 0.999079i \(0.513665\pi\)
\(864\) 0 0
\(865\) −22032.0 −0.866024
\(866\) −13608.0 −0.533971
\(867\) 0 0
\(868\) 0 0
\(869\) −15520.0 −0.605846
\(870\) 0 0
\(871\) 34608.0 1.34632
\(872\) 13770.0 0.534760
\(873\) 0 0
\(874\) −2112.00 −0.0817385
\(875\) 0 0
\(876\) 0 0
\(877\) −27574.0 −1.06170 −0.530848 0.847467i \(-0.678127\pi\)
−0.530848 + 0.847467i \(0.678127\pi\)
\(878\) −12864.0 −0.494464
\(879\) 0 0
\(880\) −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
\(885\) 0 0
\(886\) 13252.0 0.502494
\(887\) 2280.00 0.0863077 0.0431538 0.999068i \(-0.486259\pi\)
0.0431538 + 0.999068i \(0.486259\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −8928.00 −0.336255
\(891\) 0 0
\(892\) 35280.0 1.32428
\(893\) 4896.00 0.183470
\(894\) 0 0
\(895\) −28464.0 −1.06307
\(896\) 0 0
\(897\) 0 0
\(898\) −226.000 −0.00839835
\(899\) −15312.0 −0.568058
\(900\) 0 0
\(901\) −69312.0 −2.56284
\(902\) 0 0
\(903\) 0 0
\(904\) −1650.00 −0.0607060
\(905\) 13104.0 0.481317
\(906\) 0 0
\(907\) 36084.0 1.32100 0.660501 0.750825i \(-0.270344\pi\)
0.660501 + 0.750825i \(0.270344\pi\)
\(908\) −15204.0 −0.555686
\(909\) 0 0
\(910\) 0 0
\(911\) −24152.0 −0.878366 −0.439183 0.898398i \(-0.644732\pi\)
−0.439183 + 0.898398i \(0.644732\pi\)
\(912\) 0 0
\(913\) −18480.0 −0.669878
\(914\) −11334.0 −0.410170
\(915\) 0 0
\(916\) 18900.0 0.681740
\(917\) 0 0
\(918\) 0 0
\(919\) 36336.0 1.30426 0.652130 0.758108i \(-0.273876\pi\)
0.652130 + 0.758108i \(0.273876\pi\)
\(920\) −31680.0 −1.13528
\(921\) 0 0
\(922\) −1596.00 −0.0570081
\(923\) −24864.0 −0.886683
\(924\) 0 0
\(925\) 4902.00 0.174245
\(926\) 12728.0 0.451693
\(927\) 0 0
\(928\) −9338.00 −0.330318
\(929\) 432.000 0.0152567 0.00762834 0.999971i \(-0.497572\pi\)
0.00762834 + 0.999971i \(0.497572\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −26614.0 −0.935376
\(933\) 0 0
\(934\) −3012.00 −0.105520
\(935\) 23040.0 0.805870
\(936\) 0 0
\(937\) −22176.0 −0.773168 −0.386584 0.922254i \(-0.626345\pi\)
−0.386584 + 0.922254i \(0.626345\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 34272.0 1.18918
\(941\) −43524.0 −1.50780 −0.753901 0.656988i \(-0.771830\pi\)
−0.753901 + 0.656988i \(0.771830\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 20172.0 0.695490
\(945\) 0 0
\(946\) −3120.00 −0.107230
\(947\) −1868.00 −0.0640991 −0.0320495 0.999486i \(-0.510203\pi\)
−0.0320495 + 0.999486i \(0.510203\pi\)
\(948\) 0 0
\(949\) −20160.0 −0.689590
\(950\) −228.000 −0.00778663
\(951\) 0 0
\(952\) 0 0
\(953\) 9238.00 0.314006 0.157003 0.987598i \(-0.449817\pi\)
0.157003 + 0.987598i \(0.449817\pi\)
\(954\) 0 0
\(955\) −30144.0 −1.02140
\(956\) −30856.0 −1.04389
\(957\) 0 0
\(958\) −4296.00 −0.144883
\(959\) 0 0
\(960\) 0 0
\(961\) 39905.0 1.33950
\(962\) 21672.0 0.726334
\(963\) 0 0
\(964\) 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
\(969\) 0 0
\(970\) 2016.00 0.0667318
\(971\) −27540.0 −0.910196 −0.455098 0.890441i \(-0.650396\pi\)
−0.455098 + 0.890441i \(0.650396\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −8184.00 −0.269232
\(975\) 0 0
\(976\) 20172.0 0.661568
\(977\) 16402.0 0.537100 0.268550 0.963266i \(-0.413456\pi\)
0.268550 + 0.963266i \(0.413456\pi\)
\(978\) 0 0
\(979\) 14880.0 0.485768
\(980\) 0 0
\(981\) 0 0
\(982\) 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
\(990\) 0 0
\(991\) 27096.0 0.868550 0.434275 0.900780i \(-0.357005\pi\)
0.434275 + 0.900780i \(0.357005\pi\)
\(992\) 42504.0 1.36039
\(993\) 0 0
\(994\) 0 0
\(995\) −37152.0 −1.18372
\(996\) 0 0
\(997\) 16812.0 0.534044 0.267022 0.963691i \(-0.413960\pi\)
0.267022 + 0.963691i \(0.413960\pi\)
\(998\) 972.000 0.0308298
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 441.4.a.h.1.1 1
3.2 odd 2 147.4.a.e.1.1 yes 1
7.2 even 3 441.4.e.f.361.1 2
7.3 odd 6 441.4.e.g.226.1 2
7.4 even 3 441.4.e.f.226.1 2
7.5 odd 6 441.4.e.g.361.1 2
7.6 odd 2 441.4.a.g.1.1 1
12.11 even 2 2352.4.a.b.1.1 1
21.2 odd 6 147.4.e.e.67.1 2
21.5 even 6 147.4.e.f.67.1 2
21.11 odd 6 147.4.e.e.79.1 2
21.17 even 6 147.4.e.f.79.1 2
21.20 even 2 147.4.a.d.1.1 1
84.83 odd 2 2352.4.a.bi.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
147.4.a.d.1.1 1 21.20 even 2
147.4.a.e.1.1 yes 1 3.2 odd 2
147.4.e.e.67.1 2 21.2 odd 6
147.4.e.e.79.1 2 21.11 odd 6
147.4.e.f.67.1 2 21.5 even 6
147.4.e.f.79.1 2 21.17 even 6
441.4.a.g.1.1 1 7.6 odd 2
441.4.a.h.1.1 1 1.1 even 1 trivial
441.4.e.f.226.1 2 7.4 even 3
441.4.e.f.361.1 2 7.2 even 3
441.4.e.g.226.1 2 7.3 odd 6
441.4.e.g.361.1 2 7.5 odd 6
2352.4.a.b.1.1 1 12.11 even 2
2352.4.a.bi.1.1 1 84.83 odd 2