Properties

Label 80.4.a.e.1.1
Level $80$
Weight $4$
Character 80.1
Self dual yes
Analytic conductor $4.720$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 80.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.72015280046\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 40)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 80.1

$q$-expansion

\(f(q)\) \(=\) \(q+6.00000 q^{3} -5.00000 q^{5} +34.0000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+6.00000 q^{3} -5.00000 q^{5} +34.0000 q^{7} +9.00000 q^{9} -16.0000 q^{11} +58.0000 q^{13} -30.0000 q^{15} -70.0000 q^{17} -4.00000 q^{19} +204.000 q^{21} +134.000 q^{23} +25.0000 q^{25} -108.000 q^{27} -242.000 q^{29} -100.000 q^{31} -96.0000 q^{33} -170.000 q^{35} -438.000 q^{37} +348.000 q^{39} -138.000 q^{41} -178.000 q^{43} -45.0000 q^{45} -22.0000 q^{47} +813.000 q^{49} -420.000 q^{51} +162.000 q^{53} +80.0000 q^{55} -24.0000 q^{57} +268.000 q^{59} +250.000 q^{61} +306.000 q^{63} -290.000 q^{65} -422.000 q^{67} +804.000 q^{69} +852.000 q^{71} +306.000 q^{73} +150.000 q^{75} -544.000 q^{77} +456.000 q^{79} -891.000 q^{81} -434.000 q^{83} +350.000 q^{85} -1452.00 q^{87} -726.000 q^{89} +1972.00 q^{91} -600.000 q^{93} +20.0000 q^{95} +1378.00 q^{97} -144.000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 6.00000 1.15470 0.577350 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 0 0
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 34.0000 1.83583 0.917914 0.396780i \(-0.129872\pi\)
0.917914 + 0.396780i \(0.129872\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −16.0000 −0.438562 −0.219281 0.975662i \(-0.570371\pi\)
−0.219281 + 0.975662i \(0.570371\pi\)
\(12\) 0 0
\(13\) 58.0000 1.23741 0.618704 0.785624i \(-0.287658\pi\)
0.618704 + 0.785624i \(0.287658\pi\)
\(14\) 0 0
\(15\) −30.0000 −0.516398
\(16\) 0 0
\(17\) −70.0000 −0.998676 −0.499338 0.866407i \(-0.666423\pi\)
−0.499338 + 0.866407i \(0.666423\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.0482980 −0.0241490 0.999708i \(-0.507688\pi\)
−0.0241490 + 0.999708i \(0.507688\pi\)
\(20\) 0 0
\(21\) 204.000 2.11983
\(22\) 0 0
\(23\) 134.000 1.21482 0.607412 0.794387i \(-0.292208\pi\)
0.607412 + 0.794387i \(0.292208\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) −108.000 −0.769800
\(28\) 0 0
\(29\) −242.000 −1.54960 −0.774798 0.632209i \(-0.782148\pi\)
−0.774798 + 0.632209i \(0.782148\pi\)
\(30\) 0 0
\(31\) −100.000 −0.579372 −0.289686 0.957122i \(-0.593551\pi\)
−0.289686 + 0.957122i \(0.593551\pi\)
\(32\) 0 0
\(33\) −96.0000 −0.506408
\(34\) 0 0
\(35\) −170.000 −0.821007
\(36\) 0 0
\(37\) −438.000 −1.94613 −0.973064 0.230534i \(-0.925953\pi\)
−0.973064 + 0.230534i \(0.925953\pi\)
\(38\) 0 0
\(39\) 348.000 1.42884
\(40\) 0 0
\(41\) −138.000 −0.525658 −0.262829 0.964842i \(-0.584656\pi\)
−0.262829 + 0.964842i \(0.584656\pi\)
\(42\) 0 0
\(43\) −178.000 −0.631273 −0.315637 0.948880i \(-0.602218\pi\)
−0.315637 + 0.948880i \(0.602218\pi\)
\(44\) 0 0
\(45\) −45.0000 −0.149071
\(46\) 0 0
\(47\) −22.0000 −0.0682772 −0.0341386 0.999417i \(-0.510869\pi\)
−0.0341386 + 0.999417i \(0.510869\pi\)
\(48\) 0 0
\(49\) 813.000 2.37026
\(50\) 0 0
\(51\) −420.000 −1.15317
\(52\) 0 0
\(53\) 162.000 0.419857 0.209928 0.977717i \(-0.432677\pi\)
0.209928 + 0.977717i \(0.432677\pi\)
\(54\) 0 0
\(55\) 80.0000 0.196131
\(56\) 0 0
\(57\) −24.0000 −0.0557698
\(58\) 0 0
\(59\) 268.000 0.591367 0.295683 0.955286i \(-0.404453\pi\)
0.295683 + 0.955286i \(0.404453\pi\)
\(60\) 0 0
\(61\) 250.000 0.524741 0.262371 0.964967i \(-0.415496\pi\)
0.262371 + 0.964967i \(0.415496\pi\)
\(62\) 0 0
\(63\) 306.000 0.611942
\(64\) 0 0
\(65\) −290.000 −0.553386
\(66\) 0 0
\(67\) −422.000 −0.769485 −0.384743 0.923024i \(-0.625710\pi\)
−0.384743 + 0.923024i \(0.625710\pi\)
\(68\) 0 0
\(69\) 804.000 1.40276
\(70\) 0 0
\(71\) 852.000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) 306.000 0.490611 0.245305 0.969446i \(-0.421112\pi\)
0.245305 + 0.969446i \(0.421112\pi\)
\(74\) 0 0
\(75\) 150.000 0.230940
\(76\) 0 0
\(77\) −544.000 −0.805124
\(78\) 0 0
\(79\) 456.000 0.649418 0.324709 0.945814i \(-0.394734\pi\)
0.324709 + 0.945814i \(0.394734\pi\)
\(80\) 0 0
\(81\) −891.000 −1.22222
\(82\) 0 0
\(83\) −434.000 −0.573948 −0.286974 0.957938i \(-0.592649\pi\)
−0.286974 + 0.957938i \(0.592649\pi\)
\(84\) 0 0
\(85\) 350.000 0.446622
\(86\) 0 0
\(87\) −1452.00 −1.78932
\(88\) 0 0
\(89\) −726.000 −0.864672 −0.432336 0.901712i \(-0.642311\pi\)
−0.432336 + 0.901712i \(0.642311\pi\)
\(90\) 0 0
\(91\) 1972.00 2.27167
\(92\) 0 0
\(93\) −600.000 −0.669001
\(94\) 0 0
\(95\) 20.0000 0.0215995
\(96\) 0 0
\(97\) 1378.00 1.44242 0.721210 0.692717i \(-0.243586\pi\)
0.721210 + 0.692717i \(0.243586\pi\)
\(98\) 0 0
\(99\) −144.000 −0.146187
\(100\) 0 0
\(101\) 126.000 0.124133 0.0620667 0.998072i \(-0.480231\pi\)
0.0620667 + 0.998072i \(0.480231\pi\)
\(102\) 0 0
\(103\) 1262.00 1.20727 0.603634 0.797262i \(-0.293719\pi\)
0.603634 + 0.797262i \(0.293719\pi\)
\(104\) 0 0
\(105\) −1020.00 −0.948017
\(106\) 0 0
\(107\) −510.000 −0.460781 −0.230390 0.973098i \(-0.574000\pi\)
−0.230390 + 0.973098i \(0.574000\pi\)
\(108\) 0 0
\(109\) 26.0000 0.0228472 0.0114236 0.999935i \(-0.496364\pi\)
0.0114236 + 0.999935i \(0.496364\pi\)
\(110\) 0 0
\(111\) −2628.00 −2.24720
\(112\) 0 0
\(113\) 1242.00 1.03396 0.516980 0.855997i \(-0.327056\pi\)
0.516980 + 0.855997i \(0.327056\pi\)
\(114\) 0 0
\(115\) −670.000 −0.543285
\(116\) 0 0
\(117\) 522.000 0.412469
\(118\) 0 0
\(119\) −2380.00 −1.83340
\(120\) 0 0
\(121\) −1075.00 −0.807663
\(122\) 0 0
\(123\) −828.000 −0.606978
\(124\) 0 0
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) 978.000 0.683334 0.341667 0.939821i \(-0.389008\pi\)
0.341667 + 0.939821i \(0.389008\pi\)
\(128\) 0 0
\(129\) −1068.00 −0.728931
\(130\) 0 0
\(131\) 912.000 0.608258 0.304129 0.952631i \(-0.401635\pi\)
0.304129 + 0.952631i \(0.401635\pi\)
\(132\) 0 0
\(133\) −136.000 −0.0886669
\(134\) 0 0
\(135\) 540.000 0.344265
\(136\) 0 0
\(137\) −926.000 −0.577471 −0.288735 0.957409i \(-0.593235\pi\)
−0.288735 + 0.957409i \(0.593235\pi\)
\(138\) 0 0
\(139\) −516.000 −0.314867 −0.157434 0.987530i \(-0.550322\pi\)
−0.157434 + 0.987530i \(0.550322\pi\)
\(140\) 0 0
\(141\) −132.000 −0.0788398
\(142\) 0 0
\(143\) −928.000 −0.542680
\(144\) 0 0
\(145\) 1210.00 0.693000
\(146\) 0 0
\(147\) 4878.00 2.73694
\(148\) 0 0
\(149\) −958.000 −0.526728 −0.263364 0.964697i \(-0.584832\pi\)
−0.263364 + 0.964697i \(0.584832\pi\)
\(150\) 0 0
\(151\) −332.000 −0.178926 −0.0894628 0.995990i \(-0.528515\pi\)
−0.0894628 + 0.995990i \(0.528515\pi\)
\(152\) 0 0
\(153\) −630.000 −0.332892
\(154\) 0 0
\(155\) 500.000 0.259103
\(156\) 0 0
\(157\) −1022.00 −0.519519 −0.259759 0.965673i \(-0.583643\pi\)
−0.259759 + 0.965673i \(0.583643\pi\)
\(158\) 0 0
\(159\) 972.000 0.484809
\(160\) 0 0
\(161\) 4556.00 2.23021
\(162\) 0 0
\(163\) 926.000 0.444969 0.222484 0.974936i \(-0.428583\pi\)
0.222484 + 0.974936i \(0.428583\pi\)
\(164\) 0 0
\(165\) 480.000 0.226472
\(166\) 0 0
\(167\) −654.000 −0.303042 −0.151521 0.988454i \(-0.548417\pi\)
−0.151521 + 0.988454i \(0.548417\pi\)
\(168\) 0 0
\(169\) 1167.00 0.531179
\(170\) 0 0
\(171\) −36.0000 −0.0160993
\(172\) 0 0
\(173\) −1294.00 −0.568676 −0.284338 0.958724i \(-0.591774\pi\)
−0.284338 + 0.958724i \(0.591774\pi\)
\(174\) 0 0
\(175\) 850.000 0.367165
\(176\) 0 0
\(177\) 1608.00 0.682851
\(178\) 0 0
\(179\) 2836.00 1.18420 0.592102 0.805863i \(-0.298298\pi\)
0.592102 + 0.805863i \(0.298298\pi\)
\(180\) 0 0
\(181\) 1742.00 0.715369 0.357685 0.933842i \(-0.383566\pi\)
0.357685 + 0.933842i \(0.383566\pi\)
\(182\) 0 0
\(183\) 1500.00 0.605919
\(184\) 0 0
\(185\) 2190.00 0.870335
\(186\) 0 0
\(187\) 1120.00 0.437981
\(188\) 0 0
\(189\) −3672.00 −1.41322
\(190\) 0 0
\(191\) −4460.00 −1.68960 −0.844802 0.535079i \(-0.820282\pi\)
−0.844802 + 0.535079i \(0.820282\pi\)
\(192\) 0 0
\(193\) −3782.00 −1.41054 −0.705270 0.708939i \(-0.749174\pi\)
−0.705270 + 0.708939i \(0.749174\pi\)
\(194\) 0 0
\(195\) −1740.00 −0.638995
\(196\) 0 0
\(197\) 4474.00 1.61807 0.809034 0.587762i \(-0.199991\pi\)
0.809034 + 0.587762i \(0.199991\pi\)
\(198\) 0 0
\(199\) −3608.00 −1.28525 −0.642624 0.766182i \(-0.722154\pi\)
−0.642624 + 0.766182i \(0.722154\pi\)
\(200\) 0 0
\(201\) −2532.00 −0.888525
\(202\) 0 0
\(203\) −8228.00 −2.84479
\(204\) 0 0
\(205\) 690.000 0.235081
\(206\) 0 0
\(207\) 1206.00 0.404941
\(208\) 0 0
\(209\) 64.0000 0.0211817
\(210\) 0 0
\(211\) 256.000 0.0835250 0.0417625 0.999128i \(-0.486703\pi\)
0.0417625 + 0.999128i \(0.486703\pi\)
\(212\) 0 0
\(213\) 5112.00 1.64445
\(214\) 0 0
\(215\) 890.000 0.282314
\(216\) 0 0
\(217\) −3400.00 −1.06363
\(218\) 0 0
\(219\) 1836.00 0.566509
\(220\) 0 0
\(221\) −4060.00 −1.23577
\(222\) 0 0
\(223\) 5158.00 1.54890 0.774451 0.632634i \(-0.218026\pi\)
0.774451 + 0.632634i \(0.218026\pi\)
\(224\) 0 0
\(225\) 225.000 0.0666667
\(226\) 0 0
\(227\) 2226.00 0.650858 0.325429 0.945566i \(-0.394491\pi\)
0.325429 + 0.945566i \(0.394491\pi\)
\(228\) 0 0
\(229\) 2086.00 0.601951 0.300975 0.953632i \(-0.402688\pi\)
0.300975 + 0.953632i \(0.402688\pi\)
\(230\) 0 0
\(231\) −3264.00 −0.929677
\(232\) 0 0
\(233\) −5718.00 −1.60772 −0.803860 0.594819i \(-0.797224\pi\)
−0.803860 + 0.594819i \(0.797224\pi\)
\(234\) 0 0
\(235\) 110.000 0.0305345
\(236\) 0 0
\(237\) 2736.00 0.749883
\(238\) 0 0
\(239\) 3624.00 0.980825 0.490412 0.871491i \(-0.336846\pi\)
0.490412 + 0.871491i \(0.336846\pi\)
\(240\) 0 0
\(241\) −82.0000 −0.0219174 −0.0109587 0.999940i \(-0.503488\pi\)
−0.0109587 + 0.999940i \(0.503488\pi\)
\(242\) 0 0
\(243\) −2430.00 −0.641500
\(244\) 0 0
\(245\) −4065.00 −1.06001
\(246\) 0 0
\(247\) −232.000 −0.0597644
\(248\) 0 0
\(249\) −2604.00 −0.662738
\(250\) 0 0
\(251\) 5040.00 1.26742 0.633709 0.773571i \(-0.281532\pi\)
0.633709 + 0.773571i \(0.281532\pi\)
\(252\) 0 0
\(253\) −2144.00 −0.532775
\(254\) 0 0
\(255\) 2100.00 0.515714
\(256\) 0 0
\(257\) −2310.00 −0.560676 −0.280338 0.959901i \(-0.590447\pi\)
−0.280338 + 0.959901i \(0.590447\pi\)
\(258\) 0 0
\(259\) −14892.0 −3.57276
\(260\) 0 0
\(261\) −2178.00 −0.516532
\(262\) 0 0
\(263\) 4110.00 0.963625 0.481813 0.876274i \(-0.339979\pi\)
0.481813 + 0.876274i \(0.339979\pi\)
\(264\) 0 0
\(265\) −810.000 −0.187766
\(266\) 0 0
\(267\) −4356.00 −0.998438
\(268\) 0 0
\(269\) 746.000 0.169087 0.0845435 0.996420i \(-0.473057\pi\)
0.0845435 + 0.996420i \(0.473057\pi\)
\(270\) 0 0
\(271\) 4596.00 1.03021 0.515105 0.857127i \(-0.327753\pi\)
0.515105 + 0.857127i \(0.327753\pi\)
\(272\) 0 0
\(273\) 11832.0 2.62310
\(274\) 0 0
\(275\) −400.000 −0.0877124
\(276\) 0 0
\(277\) −2206.00 −0.478504 −0.239252 0.970957i \(-0.576902\pi\)
−0.239252 + 0.970957i \(0.576902\pi\)
\(278\) 0 0
\(279\) −900.000 −0.193124
\(280\) 0 0
\(281\) 8278.00 1.75738 0.878691 0.477392i \(-0.158418\pi\)
0.878691 + 0.477392i \(0.158418\pi\)
\(282\) 0 0
\(283\) −1178.00 −0.247438 −0.123719 0.992317i \(-0.539482\pi\)
−0.123719 + 0.992317i \(0.539482\pi\)
\(284\) 0 0
\(285\) 120.000 0.0249410
\(286\) 0 0
\(287\) −4692.00 −0.965017
\(288\) 0 0
\(289\) −13.0000 −0.00264604
\(290\) 0 0
\(291\) 8268.00 1.66556
\(292\) 0 0
\(293\) 106.000 0.0211351 0.0105676 0.999944i \(-0.496636\pi\)
0.0105676 + 0.999944i \(0.496636\pi\)
\(294\) 0 0
\(295\) −1340.00 −0.264467
\(296\) 0 0
\(297\) 1728.00 0.337605
\(298\) 0 0
\(299\) 7772.00 1.50323
\(300\) 0 0
\(301\) −6052.00 −1.15891
\(302\) 0 0
\(303\) 756.000 0.143337
\(304\) 0 0
\(305\) −1250.00 −0.234671
\(306\) 0 0
\(307\) −8134.00 −1.51216 −0.756078 0.654482i \(-0.772887\pi\)
−0.756078 + 0.654482i \(0.772887\pi\)
\(308\) 0 0
\(309\) 7572.00 1.39403
\(310\) 0 0
\(311\) 4396.00 0.801525 0.400763 0.916182i \(-0.368745\pi\)
0.400763 + 0.916182i \(0.368745\pi\)
\(312\) 0 0
\(313\) 4826.00 0.871507 0.435753 0.900066i \(-0.356482\pi\)
0.435753 + 0.900066i \(0.356482\pi\)
\(314\) 0 0
\(315\) −1530.00 −0.273669
\(316\) 0 0
\(317\) 7026.00 1.24486 0.622428 0.782677i \(-0.286146\pi\)
0.622428 + 0.782677i \(0.286146\pi\)
\(318\) 0 0
\(319\) 3872.00 0.679594
\(320\) 0 0
\(321\) −3060.00 −0.532064
\(322\) 0 0
\(323\) 280.000 0.0482341
\(324\) 0 0
\(325\) 1450.00 0.247482
\(326\) 0 0
\(327\) 156.000 0.0263817
\(328\) 0 0
\(329\) −748.000 −0.125345
\(330\) 0 0
\(331\) −8808.00 −1.46263 −0.731316 0.682038i \(-0.761094\pi\)
−0.731316 + 0.682038i \(0.761094\pi\)
\(332\) 0 0
\(333\) −3942.00 −0.648710
\(334\) 0 0
\(335\) 2110.00 0.344124
\(336\) 0 0
\(337\) 5602.00 0.905520 0.452760 0.891632i \(-0.350439\pi\)
0.452760 + 0.891632i \(0.350439\pi\)
\(338\) 0 0
\(339\) 7452.00 1.19391
\(340\) 0 0
\(341\) 1600.00 0.254090
\(342\) 0 0
\(343\) 15980.0 2.51557
\(344\) 0 0
\(345\) −4020.00 −0.627332
\(346\) 0 0
\(347\) 6634.00 1.02632 0.513158 0.858294i \(-0.328475\pi\)
0.513158 + 0.858294i \(0.328475\pi\)
\(348\) 0 0
\(349\) 3198.00 0.490501 0.245251 0.969460i \(-0.421130\pi\)
0.245251 + 0.969460i \(0.421130\pi\)
\(350\) 0 0
\(351\) −6264.00 −0.952557
\(352\) 0 0
\(353\) −5230.00 −0.788569 −0.394284 0.918988i \(-0.629008\pi\)
−0.394284 + 0.918988i \(0.629008\pi\)
\(354\) 0 0
\(355\) −4260.00 −0.636894
\(356\) 0 0
\(357\) −14280.0 −2.11702
\(358\) 0 0
\(359\) 312.000 0.0458683 0.0229342 0.999737i \(-0.492699\pi\)
0.0229342 + 0.999737i \(0.492699\pi\)
\(360\) 0 0
\(361\) −6843.00 −0.997667
\(362\) 0 0
\(363\) −6450.00 −0.932609
\(364\) 0 0
\(365\) −1530.00 −0.219408
\(366\) 0 0
\(367\) −10790.0 −1.53470 −0.767348 0.641231i \(-0.778424\pi\)
−0.767348 + 0.641231i \(0.778424\pi\)
\(368\) 0 0
\(369\) −1242.00 −0.175219
\(370\) 0 0
\(371\) 5508.00 0.770785
\(372\) 0 0
\(373\) −4190.00 −0.581635 −0.290818 0.956778i \(-0.593927\pi\)
−0.290818 + 0.956778i \(0.593927\pi\)
\(374\) 0 0
\(375\) −750.000 −0.103280
\(376\) 0 0
\(377\) −14036.0 −1.91748
\(378\) 0 0
\(379\) 6980.00 0.946012 0.473006 0.881059i \(-0.343169\pi\)
0.473006 + 0.881059i \(0.343169\pi\)
\(380\) 0 0
\(381\) 5868.00 0.789047
\(382\) 0 0
\(383\) −13962.0 −1.86273 −0.931364 0.364089i \(-0.881380\pi\)
−0.931364 + 0.364089i \(0.881380\pi\)
\(384\) 0 0
\(385\) 2720.00 0.360062
\(386\) 0 0
\(387\) −1602.00 −0.210424
\(388\) 0 0
\(389\) 3810.00 0.496593 0.248296 0.968684i \(-0.420129\pi\)
0.248296 + 0.968684i \(0.420129\pi\)
\(390\) 0 0
\(391\) −9380.00 −1.21321
\(392\) 0 0
\(393\) 5472.00 0.702356
\(394\) 0 0
\(395\) −2280.00 −0.290428
\(396\) 0 0
\(397\) −9158.00 −1.15775 −0.578875 0.815416i \(-0.696508\pi\)
−0.578875 + 0.815416i \(0.696508\pi\)
\(398\) 0 0
\(399\) −816.000 −0.102384
\(400\) 0 0
\(401\) 4866.00 0.605976 0.302988 0.952994i \(-0.402016\pi\)
0.302988 + 0.952994i \(0.402016\pi\)
\(402\) 0 0
\(403\) −5800.00 −0.716920
\(404\) 0 0
\(405\) 4455.00 0.546594
\(406\) 0 0
\(407\) 7008.00 0.853498
\(408\) 0 0
\(409\) 13486.0 1.63042 0.815208 0.579169i \(-0.196623\pi\)
0.815208 + 0.579169i \(0.196623\pi\)
\(410\) 0 0
\(411\) −5556.00 −0.666806
\(412\) 0 0
\(413\) 9112.00 1.08565
\(414\) 0 0
\(415\) 2170.00 0.256677
\(416\) 0 0
\(417\) −3096.00 −0.363577
\(418\) 0 0
\(419\) −5628.00 −0.656195 −0.328098 0.944644i \(-0.606407\pi\)
−0.328098 + 0.944644i \(0.606407\pi\)
\(420\) 0 0
\(421\) 7938.00 0.918942 0.459471 0.888193i \(-0.348039\pi\)
0.459471 + 0.888193i \(0.348039\pi\)
\(422\) 0 0
\(423\) −198.000 −0.0227591
\(424\) 0 0
\(425\) −1750.00 −0.199735
\(426\) 0 0
\(427\) 8500.00 0.963334
\(428\) 0 0
\(429\) −5568.00 −0.626633
\(430\) 0 0
\(431\) −1916.00 −0.214131 −0.107066 0.994252i \(-0.534145\pi\)
−0.107066 + 0.994252i \(0.534145\pi\)
\(432\) 0 0
\(433\) −16510.0 −1.83238 −0.916189 0.400746i \(-0.868751\pi\)
−0.916189 + 0.400746i \(0.868751\pi\)
\(434\) 0 0
\(435\) 7260.00 0.800208
\(436\) 0 0
\(437\) −536.000 −0.0586736
\(438\) 0 0
\(439\) 1256.00 0.136550 0.0682752 0.997667i \(-0.478250\pi\)
0.0682752 + 0.997667i \(0.478250\pi\)
\(440\) 0 0
\(441\) 7317.00 0.790087
\(442\) 0 0
\(443\) 12222.0 1.31080 0.655400 0.755282i \(-0.272500\pi\)
0.655400 + 0.755282i \(0.272500\pi\)
\(444\) 0 0
\(445\) 3630.00 0.386693
\(446\) 0 0
\(447\) −5748.00 −0.608213
\(448\) 0 0
\(449\) −5946.00 −0.624965 −0.312482 0.949924i \(-0.601160\pi\)
−0.312482 + 0.949924i \(0.601160\pi\)
\(450\) 0 0
\(451\) 2208.00 0.230534
\(452\) 0 0
\(453\) −1992.00 −0.206606
\(454\) 0 0
\(455\) −9860.00 −1.01592
\(456\) 0 0
\(457\) 1258.00 0.128768 0.0643838 0.997925i \(-0.479492\pi\)
0.0643838 + 0.997925i \(0.479492\pi\)
\(458\) 0 0
\(459\) 7560.00 0.768781
\(460\) 0 0
\(461\) 16422.0 1.65911 0.829554 0.558426i \(-0.188595\pi\)
0.829554 + 0.558426i \(0.188595\pi\)
\(462\) 0 0
\(463\) −2658.00 −0.266799 −0.133399 0.991062i \(-0.542589\pi\)
−0.133399 + 0.991062i \(0.542589\pi\)
\(464\) 0 0
\(465\) 3000.00 0.299186
\(466\) 0 0
\(467\) −3686.00 −0.365241 −0.182621 0.983183i \(-0.558458\pi\)
−0.182621 + 0.983183i \(0.558458\pi\)
\(468\) 0 0
\(469\) −14348.0 −1.41264
\(470\) 0 0
\(471\) −6132.00 −0.599889
\(472\) 0 0
\(473\) 2848.00 0.276852
\(474\) 0 0
\(475\) −100.000 −0.00965961
\(476\) 0 0
\(477\) 1458.00 0.139952
\(478\) 0 0
\(479\) −88.0000 −0.00839420 −0.00419710 0.999991i \(-0.501336\pi\)
−0.00419710 + 0.999991i \(0.501336\pi\)
\(480\) 0 0
\(481\) −25404.0 −2.40816
\(482\) 0 0
\(483\) 27336.0 2.57522
\(484\) 0 0
\(485\) −6890.00 −0.645070
\(486\) 0 0
\(487\) 14714.0 1.36911 0.684553 0.728963i \(-0.259997\pi\)
0.684553 + 0.728963i \(0.259997\pi\)
\(488\) 0 0
\(489\) 5556.00 0.513806
\(490\) 0 0
\(491\) 7344.00 0.675010 0.337505 0.941324i \(-0.390417\pi\)
0.337505 + 0.941324i \(0.390417\pi\)
\(492\) 0 0
\(493\) 16940.0 1.54754
\(494\) 0 0
\(495\) 720.000 0.0653770
\(496\) 0 0
\(497\) 28968.0 2.61447
\(498\) 0 0
\(499\) −1604.00 −0.143898 −0.0719488 0.997408i \(-0.522922\pi\)
−0.0719488 + 0.997408i \(0.522922\pi\)
\(500\) 0 0
\(501\) −3924.00 −0.349923
\(502\) 0 0
\(503\) −14802.0 −1.31210 −0.656052 0.754715i \(-0.727775\pi\)
−0.656052 + 0.754715i \(0.727775\pi\)
\(504\) 0 0
\(505\) −630.000 −0.0555141
\(506\) 0 0
\(507\) 7002.00 0.613353
\(508\) 0 0
\(509\) −22514.0 −1.96054 −0.980271 0.197660i \(-0.936666\pi\)
−0.980271 + 0.197660i \(0.936666\pi\)
\(510\) 0 0
\(511\) 10404.0 0.900677
\(512\) 0 0
\(513\) 432.000 0.0371799
\(514\) 0 0
\(515\) −6310.00 −0.539906
\(516\) 0 0
\(517\) 352.000 0.0299438
\(518\) 0 0
\(519\) −7764.00 −0.656651
\(520\) 0 0
\(521\) −6710.00 −0.564243 −0.282121 0.959379i \(-0.591038\pi\)
−0.282121 + 0.959379i \(0.591038\pi\)
\(522\) 0 0
\(523\) −7930.00 −0.663011 −0.331505 0.943453i \(-0.607557\pi\)
−0.331505 + 0.943453i \(0.607557\pi\)
\(524\) 0 0
\(525\) 5100.00 0.423966
\(526\) 0 0
\(527\) 7000.00 0.578605
\(528\) 0 0
\(529\) 5789.00 0.475795
\(530\) 0 0
\(531\) 2412.00 0.197122
\(532\) 0 0
\(533\) −8004.00 −0.650454
\(534\) 0 0
\(535\) 2550.00 0.206068
\(536\) 0 0
\(537\) 17016.0 1.36740
\(538\) 0 0
\(539\) −13008.0 −1.03951
\(540\) 0 0
\(541\) 4918.00 0.390834 0.195417 0.980720i \(-0.437394\pi\)
0.195417 + 0.980720i \(0.437394\pi\)
\(542\) 0 0
\(543\) 10452.0 0.826037
\(544\) 0 0
\(545\) −130.000 −0.0102176
\(546\) 0 0
\(547\) 3922.00 0.306568 0.153284 0.988182i \(-0.451015\pi\)
0.153284 + 0.988182i \(0.451015\pi\)
\(548\) 0 0
\(549\) 2250.00 0.174914
\(550\) 0 0
\(551\) 968.000 0.0748424
\(552\) 0 0
\(553\) 15504.0 1.19222
\(554\) 0 0
\(555\) 13140.0 1.00498
\(556\) 0 0
\(557\) 17786.0 1.35299 0.676496 0.736446i \(-0.263497\pi\)
0.676496 + 0.736446i \(0.263497\pi\)
\(558\) 0 0
\(559\) −10324.0 −0.781143
\(560\) 0 0
\(561\) 6720.00 0.505737
\(562\) 0 0
\(563\) −20266.0 −1.51707 −0.758535 0.651633i \(-0.774084\pi\)
−0.758535 + 0.651633i \(0.774084\pi\)
\(564\) 0 0
\(565\) −6210.00 −0.462401
\(566\) 0 0
\(567\) −30294.0 −2.24379
\(568\) 0 0
\(569\) 13358.0 0.984177 0.492088 0.870545i \(-0.336234\pi\)
0.492088 + 0.870545i \(0.336234\pi\)
\(570\) 0 0
\(571\) −16360.0 −1.19903 −0.599514 0.800364i \(-0.704639\pi\)
−0.599514 + 0.800364i \(0.704639\pi\)
\(572\) 0 0
\(573\) −26760.0 −1.95099
\(574\) 0 0
\(575\) 3350.00 0.242965
\(576\) 0 0
\(577\) −15574.0 −1.12366 −0.561832 0.827251i \(-0.689903\pi\)
−0.561832 + 0.827251i \(0.689903\pi\)
\(578\) 0 0
\(579\) −22692.0 −1.62875
\(580\) 0 0
\(581\) −14756.0 −1.05367
\(582\) 0 0
\(583\) −2592.00 −0.184133
\(584\) 0 0
\(585\) −2610.00 −0.184462
\(586\) 0 0
\(587\) −6654.00 −0.467870 −0.233935 0.972252i \(-0.575160\pi\)
−0.233935 + 0.972252i \(0.575160\pi\)
\(588\) 0 0
\(589\) 400.000 0.0279825
\(590\) 0 0
\(591\) 26844.0 1.86838
\(592\) 0 0
\(593\) −17742.0 −1.22863 −0.614314 0.789062i \(-0.710567\pi\)
−0.614314 + 0.789062i \(0.710567\pi\)
\(594\) 0 0
\(595\) 11900.0 0.819920
\(596\) 0 0
\(597\) −21648.0 −1.48408
\(598\) 0 0
\(599\) −15840.0 −1.08048 −0.540238 0.841512i \(-0.681666\pi\)
−0.540238 + 0.841512i \(0.681666\pi\)
\(600\) 0 0
\(601\) −3002.00 −0.203751 −0.101875 0.994797i \(-0.532484\pi\)
−0.101875 + 0.994797i \(0.532484\pi\)
\(602\) 0 0
\(603\) −3798.00 −0.256495
\(604\) 0 0
\(605\) 5375.00 0.361198
\(606\) 0 0
\(607\) 23610.0 1.57875 0.789374 0.613912i \(-0.210405\pi\)
0.789374 + 0.613912i \(0.210405\pi\)
\(608\) 0 0
\(609\) −49368.0 −3.28488
\(610\) 0 0
\(611\) −1276.00 −0.0844868
\(612\) 0 0
\(613\) 23850.0 1.57144 0.785720 0.618583i \(-0.212293\pi\)
0.785720 + 0.618583i \(0.212293\pi\)
\(614\) 0 0
\(615\) 4140.00 0.271449
\(616\) 0 0
\(617\) −5334.00 −0.348037 −0.174018 0.984742i \(-0.555675\pi\)
−0.174018 + 0.984742i \(0.555675\pi\)
\(618\) 0 0
\(619\) 2164.00 0.140515 0.0702573 0.997529i \(-0.477618\pi\)
0.0702573 + 0.997529i \(0.477618\pi\)
\(620\) 0 0
\(621\) −14472.0 −0.935171
\(622\) 0 0
\(623\) −24684.0 −1.58739
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) 384.000 0.0244585
\(628\) 0 0
\(629\) 30660.0 1.94355
\(630\) 0 0
\(631\) 25220.0 1.59111 0.795557 0.605879i \(-0.207179\pi\)
0.795557 + 0.605879i \(0.207179\pi\)
\(632\) 0 0
\(633\) 1536.00 0.0964463
\(634\) 0 0
\(635\) −4890.00 −0.305596
\(636\) 0 0
\(637\) 47154.0 2.93298
\(638\) 0 0
\(639\) 7668.00 0.474713
\(640\) 0 0
\(641\) −12306.0 −0.758280 −0.379140 0.925339i \(-0.623780\pi\)
−0.379140 + 0.925339i \(0.623780\pi\)
\(642\) 0 0
\(643\) 27414.0 1.68134 0.840671 0.541547i \(-0.182161\pi\)
0.840671 + 0.541547i \(0.182161\pi\)
\(644\) 0 0
\(645\) 5340.00 0.325988
\(646\) 0 0
\(647\) 21834.0 1.32671 0.663356 0.748304i \(-0.269131\pi\)
0.663356 + 0.748304i \(0.269131\pi\)
\(648\) 0 0
\(649\) −4288.00 −0.259351
\(650\) 0 0
\(651\) −20400.0 −1.22817
\(652\) 0 0
\(653\) −23998.0 −1.43815 −0.719077 0.694931i \(-0.755435\pi\)
−0.719077 + 0.694931i \(0.755435\pi\)
\(654\) 0 0
\(655\) −4560.00 −0.272021
\(656\) 0 0
\(657\) 2754.00 0.163537
\(658\) 0 0
\(659\) 32004.0 1.89180 0.945902 0.324452i \(-0.105180\pi\)
0.945902 + 0.324452i \(0.105180\pi\)
\(660\) 0 0
\(661\) −8526.00 −0.501699 −0.250849 0.968026i \(-0.580710\pi\)
−0.250849 + 0.968026i \(0.580710\pi\)
\(662\) 0 0
\(663\) −24360.0 −1.42694
\(664\) 0 0
\(665\) 680.000 0.0396530
\(666\) 0 0
\(667\) −32428.0 −1.88248
\(668\) 0 0
\(669\) 30948.0 1.78852
\(670\) 0 0
\(671\) −4000.00 −0.230132
\(672\) 0 0
\(673\) 8178.00 0.468408 0.234204 0.972187i \(-0.424752\pi\)
0.234204 + 0.972187i \(0.424752\pi\)
\(674\) 0 0
\(675\) −2700.00 −0.153960
\(676\) 0 0
\(677\) −16646.0 −0.944989 −0.472495 0.881334i \(-0.656646\pi\)
−0.472495 + 0.881334i \(0.656646\pi\)
\(678\) 0 0
\(679\) 46852.0 2.64803
\(680\) 0 0
\(681\) 13356.0 0.751546
\(682\) 0 0
\(683\) 22446.0 1.25750 0.628750 0.777608i \(-0.283567\pi\)
0.628750 + 0.777608i \(0.283567\pi\)
\(684\) 0 0
\(685\) 4630.00 0.258253
\(686\) 0 0
\(687\) 12516.0 0.695073
\(688\) 0 0
\(689\) 9396.00 0.519534
\(690\) 0 0
\(691\) −35336.0 −1.94536 −0.972681 0.232147i \(-0.925425\pi\)
−0.972681 + 0.232147i \(0.925425\pi\)
\(692\) 0 0
\(693\) −4896.00 −0.268375
\(694\) 0 0
\(695\) 2580.00 0.140813
\(696\) 0 0
\(697\) 9660.00 0.524962
\(698\) 0 0
\(699\) −34308.0 −1.85643
\(700\) 0 0
\(701\) 3482.00 0.187608 0.0938041 0.995591i \(-0.470097\pi\)
0.0938041 + 0.995591i \(0.470097\pi\)
\(702\) 0 0
\(703\) 1752.00 0.0939942
\(704\) 0 0
\(705\) 660.000 0.0352582
\(706\) 0 0
\(707\) 4284.00 0.227887
\(708\) 0 0
\(709\) −19402.0 −1.02773 −0.513863 0.857872i \(-0.671786\pi\)
−0.513863 + 0.857872i \(0.671786\pi\)
\(710\) 0 0
\(711\) 4104.00 0.216473
\(712\) 0 0
\(713\) −13400.0 −0.703834
\(714\) 0 0
\(715\) 4640.00 0.242694
\(716\) 0 0
\(717\) 21744.0 1.13256
\(718\) 0 0
\(719\) 9896.00 0.513294 0.256647 0.966505i \(-0.417382\pi\)
0.256647 + 0.966505i \(0.417382\pi\)
\(720\) 0 0
\(721\) 42908.0 2.21633
\(722\) 0 0
\(723\) −492.000 −0.0253080
\(724\) 0 0
\(725\) −6050.00 −0.309919
\(726\) 0 0
\(727\) −494.000 −0.0252014 −0.0126007 0.999921i \(-0.504011\pi\)
−0.0126007 + 0.999921i \(0.504011\pi\)
\(728\) 0 0
\(729\) 9477.00 0.481481
\(730\) 0 0
\(731\) 12460.0 0.630437
\(732\) 0 0
\(733\) 9282.00 0.467720 0.233860 0.972270i \(-0.424864\pi\)
0.233860 + 0.972270i \(0.424864\pi\)
\(734\) 0 0
\(735\) −24390.0 −1.22400
\(736\) 0 0
\(737\) 6752.00 0.337467
\(738\) 0 0
\(739\) 3252.00 0.161877 0.0809383 0.996719i \(-0.474208\pi\)
0.0809383 + 0.996719i \(0.474208\pi\)
\(740\) 0 0
\(741\) −1392.00 −0.0690100
\(742\) 0 0
\(743\) 4710.00 0.232561 0.116281 0.993216i \(-0.462903\pi\)
0.116281 + 0.993216i \(0.462903\pi\)
\(744\) 0 0
\(745\) 4790.00 0.235560
\(746\) 0 0
\(747\) −3906.00 −0.191316
\(748\) 0 0
\(749\) −17340.0 −0.845914
\(750\) 0 0
\(751\) −25764.0 −1.25185 −0.625927 0.779882i \(-0.715279\pi\)
−0.625927 + 0.779882i \(0.715279\pi\)
\(752\) 0 0
\(753\) 30240.0 1.46349
\(754\) 0 0
\(755\) 1660.00 0.0800180
\(756\) 0 0
\(757\) −30094.0 −1.44489 −0.722447 0.691426i \(-0.756983\pi\)
−0.722447 + 0.691426i \(0.756983\pi\)
\(758\) 0 0
\(759\) −12864.0 −0.615196
\(760\) 0 0
\(761\) 22362.0 1.06521 0.532603 0.846365i \(-0.321214\pi\)
0.532603 + 0.846365i \(0.321214\pi\)
\(762\) 0 0
\(763\) 884.000 0.0419436
\(764\) 0 0
\(765\) 3150.00 0.148874
\(766\) 0 0
\(767\) 15544.0 0.731762
\(768\) 0 0
\(769\) −30398.0 −1.42546 −0.712731 0.701438i \(-0.752542\pi\)
−0.712731 + 0.701438i \(0.752542\pi\)
\(770\) 0 0
\(771\) −13860.0 −0.647413
\(772\) 0 0
\(773\) 1290.00 0.0600234 0.0300117 0.999550i \(-0.490446\pi\)
0.0300117 + 0.999550i \(0.490446\pi\)
\(774\) 0 0
\(775\) −2500.00 −0.115874
\(776\) 0 0
\(777\) −89352.0 −4.12546
\(778\) 0 0
\(779\) 552.000 0.0253883
\(780\) 0 0
\(781\) −13632.0 −0.624573
\(782\) 0 0
\(783\) 26136.0 1.19288
\(784\) 0 0
\(785\) 5110.00 0.232336
\(786\) 0 0
\(787\) −14.0000 −0.000634112 0 −0.000317056 1.00000i \(-0.500101\pi\)
−0.000317056 1.00000i \(0.500101\pi\)
\(788\) 0 0
\(789\) 24660.0 1.11270
\(790\) 0 0
\(791\) 42228.0 1.89817
\(792\) 0 0
\(793\) 14500.0 0.649319
\(794\) 0 0
\(795\) −4860.00 −0.216813
\(796\) 0 0
\(797\) −38814.0 −1.72505 −0.862523 0.506017i \(-0.831117\pi\)
−0.862523 + 0.506017i \(0.831117\pi\)
\(798\) 0 0
\(799\) 1540.00 0.0681868
\(800\) 0 0
\(801\) −6534.00 −0.288224
\(802\) 0 0
\(803\) −4896.00 −0.215163
\(804\) 0 0
\(805\) −22780.0 −0.997378
\(806\) 0 0
\(807\) 4476.00 0.195245
\(808\) 0 0
\(809\) 27402.0 1.19086 0.595428 0.803408i \(-0.296982\pi\)
0.595428 + 0.803408i \(0.296982\pi\)
\(810\) 0 0
\(811\) 28576.0 1.23729 0.618643 0.785672i \(-0.287683\pi\)
0.618643 + 0.785672i \(0.287683\pi\)
\(812\) 0 0
\(813\) 27576.0 1.18958
\(814\) 0 0
\(815\) −4630.00 −0.198996
\(816\) 0 0
\(817\) 712.000 0.0304893
\(818\) 0 0
\(819\) 17748.0 0.757223
\(820\) 0 0
\(821\) 31762.0 1.35018 0.675092 0.737733i \(-0.264104\pi\)
0.675092 + 0.737733i \(0.264104\pi\)
\(822\) 0 0
\(823\) −20506.0 −0.868523 −0.434261 0.900787i \(-0.642991\pi\)
−0.434261 + 0.900787i \(0.642991\pi\)
\(824\) 0 0
\(825\) −2400.00 −0.101282
\(826\) 0 0
\(827\) −13014.0 −0.547208 −0.273604 0.961842i \(-0.588216\pi\)
−0.273604 + 0.961842i \(0.588216\pi\)
\(828\) 0 0
\(829\) −22790.0 −0.954800 −0.477400 0.878686i \(-0.658421\pi\)
−0.477400 + 0.878686i \(0.658421\pi\)
\(830\) 0 0
\(831\) −13236.0 −0.552529
\(832\) 0 0
\(833\) −56910.0 −2.36712
\(834\) 0 0
\(835\) 3270.00 0.135525
\(836\) 0 0
\(837\) 10800.0 0.446001
\(838\) 0 0
\(839\) −23696.0 −0.975062 −0.487531 0.873106i \(-0.662102\pi\)
−0.487531 + 0.873106i \(0.662102\pi\)
\(840\) 0 0
\(841\) 34175.0 1.40125
\(842\) 0 0
\(843\) 49668.0 2.02925
\(844\) 0 0
\(845\) −5835.00 −0.237550
\(846\) 0 0
\(847\) −36550.0 −1.48273
\(848\) 0 0
\(849\) −7068.00 −0.285716
\(850\) 0 0
\(851\) −58692.0 −2.36420
\(852\) 0 0
\(853\) 5306.00 0.212982 0.106491 0.994314i \(-0.466038\pi\)
0.106491 + 0.994314i \(0.466038\pi\)
\(854\) 0 0
\(855\) 180.000 0.00719985
\(856\) 0 0
\(857\) −21054.0 −0.839196 −0.419598 0.907710i \(-0.637829\pi\)
−0.419598 + 0.907710i \(0.637829\pi\)
\(858\) 0 0
\(859\) −7364.00 −0.292499 −0.146249 0.989248i \(-0.546720\pi\)
−0.146249 + 0.989248i \(0.546720\pi\)
\(860\) 0 0
\(861\) −28152.0 −1.11431
\(862\) 0 0
\(863\) −17226.0 −0.679467 −0.339733 0.940522i \(-0.610337\pi\)
−0.339733 + 0.940522i \(0.610337\pi\)
\(864\) 0 0
\(865\) 6470.00 0.254320
\(866\) 0 0
\(867\) −78.0000 −0.00305539
\(868\) 0 0
\(869\) −7296.00 −0.284810
\(870\) 0 0
\(871\) −24476.0 −0.952167
\(872\) 0 0
\(873\) 12402.0 0.480807
\(874\) 0 0
\(875\) −4250.00 −0.164201
\(876\) 0 0
\(877\) 21202.0 0.816352 0.408176 0.912903i \(-0.366165\pi\)
0.408176 + 0.912903i \(0.366165\pi\)
\(878\) 0 0
\(879\) 636.000 0.0244047
\(880\) 0 0
\(881\) −29490.0 −1.12774 −0.563872 0.825862i \(-0.690689\pi\)
−0.563872 + 0.825862i \(0.690689\pi\)
\(882\) 0 0
\(883\) −2570.00 −0.0979472 −0.0489736 0.998800i \(-0.515595\pi\)
−0.0489736 + 0.998800i \(0.515595\pi\)
\(884\) 0 0
\(885\) −8040.00 −0.305380
\(886\) 0 0
\(887\) −36334.0 −1.37540 −0.687698 0.725997i \(-0.741379\pi\)
−0.687698 + 0.725997i \(0.741379\pi\)
\(888\) 0 0
\(889\) 33252.0 1.25448
\(890\) 0 0
\(891\) 14256.0 0.536020
\(892\) 0 0
\(893\) 88.0000 0.00329766
\(894\) 0 0
\(895\) −14180.0 −0.529592
\(896\) 0 0
\(897\) 46632.0 1.73578
\(898\) 0 0
\(899\) 24200.0 0.897792
\(900\) 0 0
\(901\) −11340.0 −0.419301
\(902\) 0 0
\(903\) −36312.0 −1.33819
\(904\) 0 0
\(905\) −8710.00 −0.319923
\(906\) 0 0
\(907\) 12474.0 0.456662 0.228331 0.973584i \(-0.426673\pi\)
0.228331 + 0.973584i \(0.426673\pi\)
\(908\) 0 0
\(909\) 1134.00 0.0413778
\(910\) 0 0
\(911\) 41132.0 1.49590 0.747949 0.663756i \(-0.231039\pi\)
0.747949 + 0.663756i \(0.231039\pi\)
\(912\) 0 0
\(913\) 6944.00 0.251712
\(914\) 0 0
\(915\) −7500.00 −0.270975
\(916\) 0 0
\(917\) 31008.0 1.11666
\(918\) 0 0
\(919\) 38416.0 1.37892 0.689460 0.724324i \(-0.257848\pi\)
0.689460 + 0.724324i \(0.257848\pi\)
\(920\) 0 0
\(921\) −48804.0 −1.74609
\(922\) 0 0
\(923\) 49416.0 1.76224
\(924\) 0 0
\(925\) −10950.0 −0.389226
\(926\) 0 0
\(927\) 11358.0 0.402423
\(928\) 0 0
\(929\) 41302.0 1.45864 0.729319 0.684174i \(-0.239837\pi\)
0.729319 + 0.684174i \(0.239837\pi\)
\(930\) 0 0
\(931\) −3252.00 −0.114479
\(932\) 0 0
\(933\) 26376.0 0.925521
\(934\) 0 0
\(935\) −5600.00 −0.195871
\(936\) 0 0
\(937\) −26150.0 −0.911722 −0.455861 0.890051i \(-0.650669\pi\)
−0.455861 + 0.890051i \(0.650669\pi\)
\(938\) 0 0
\(939\) 28956.0 1.00633
\(940\) 0 0
\(941\) 35254.0 1.22130 0.610652 0.791899i \(-0.290907\pi\)
0.610652 + 0.791899i \(0.290907\pi\)
\(942\) 0 0
\(943\) −18492.0 −0.638582
\(944\) 0 0
\(945\) 18360.0 0.632011
\(946\) 0 0
\(947\) −18550.0 −0.636530 −0.318265 0.948002i \(-0.603100\pi\)
−0.318265 + 0.948002i \(0.603100\pi\)
\(948\) 0 0
\(949\) 17748.0 0.607086
\(950\) 0 0
\(951\) 42156.0 1.43744
\(952\) 0 0
\(953\) 17322.0 0.588788 0.294394 0.955684i \(-0.404882\pi\)
0.294394 + 0.955684i \(0.404882\pi\)
\(954\) 0 0
\(955\) 22300.0 0.755614
\(956\) 0 0
\(957\) 23232.0 0.784727
\(958\) 0 0
\(959\) −31484.0 −1.06014
\(960\) 0 0
\(961\) −19791.0 −0.664328
\(962\) 0 0
\(963\) −4590.00 −0.153594
\(964\) 0 0
\(965\) 18910.0 0.630813
\(966\) 0 0
\(967\) −35190.0 −1.17025 −0.585126 0.810942i \(-0.698955\pi\)
−0.585126 + 0.810942i \(0.698955\pi\)
\(968\) 0 0
\(969\) 1680.00 0.0556960
\(970\) 0 0
\(971\) 40696.0 1.34500 0.672501 0.740096i \(-0.265220\pi\)
0.672501 + 0.740096i \(0.265220\pi\)
\(972\) 0 0
\(973\) −17544.0 −0.578042
\(974\) 0 0
\(975\) 8700.00 0.285767
\(976\) 0 0
\(977\) 44306.0 1.45084 0.725422 0.688304i \(-0.241645\pi\)
0.725422 + 0.688304i \(0.241645\pi\)
\(978\) 0 0
\(979\) 11616.0 0.379212
\(980\) 0 0
\(981\) 234.000 0.00761574
\(982\) 0 0
\(983\) 18798.0 0.609932 0.304966 0.952363i \(-0.401355\pi\)
0.304966 + 0.952363i \(0.401355\pi\)
\(984\) 0 0
\(985\) −22370.0 −0.723622
\(986\) 0 0
\(987\) −4488.00 −0.144736
\(988\) 0 0
\(989\) −23852.0 −0.766885
\(990\) 0 0
\(991\) −2468.00 −0.0791106 −0.0395553 0.999217i \(-0.512594\pi\)
−0.0395553 + 0.999217i \(0.512594\pi\)
\(992\) 0 0
\(993\) −52848.0 −1.68890
\(994\) 0 0
\(995\) 18040.0 0.574780
\(996\) 0 0
\(997\) −61086.0 −1.94043 −0.970217 0.242237i \(-0.922119\pi\)
−0.970217 + 0.242237i \(0.922119\pi\)
\(998\) 0 0
\(999\) 47304.0 1.49813
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 80.4.a.e.1.1 1
3.2 odd 2 720.4.a.bd.1.1 1
4.3 odd 2 40.4.a.a.1.1 1
5.2 odd 4 400.4.c.f.49.1 2
5.3 odd 4 400.4.c.f.49.2 2
5.4 even 2 400.4.a.e.1.1 1
8.3 odd 2 320.4.a.l.1.1 1
8.5 even 2 320.4.a.c.1.1 1
12.11 even 2 360.4.a.h.1.1 1
16.3 odd 4 1280.4.d.p.641.1 2
16.5 even 4 1280.4.d.a.641.1 2
16.11 odd 4 1280.4.d.p.641.2 2
16.13 even 4 1280.4.d.a.641.2 2
20.3 even 4 200.4.c.c.49.1 2
20.7 even 4 200.4.c.c.49.2 2
20.19 odd 2 200.4.a.i.1.1 1
28.27 even 2 1960.4.a.h.1.1 1
40.19 odd 2 1600.4.a.j.1.1 1
40.29 even 2 1600.4.a.br.1.1 1
60.23 odd 4 1800.4.f.j.649.2 2
60.47 odd 4 1800.4.f.j.649.1 2
60.59 even 2 1800.4.a.bi.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.4.a.a.1.1 1 4.3 odd 2
80.4.a.e.1.1 1 1.1 even 1 trivial
200.4.a.i.1.1 1 20.19 odd 2
200.4.c.c.49.1 2 20.3 even 4
200.4.c.c.49.2 2 20.7 even 4
320.4.a.c.1.1 1 8.5 even 2
320.4.a.l.1.1 1 8.3 odd 2
360.4.a.h.1.1 1 12.11 even 2
400.4.a.e.1.1 1 5.4 even 2
400.4.c.f.49.1 2 5.2 odd 4
400.4.c.f.49.2 2 5.3 odd 4
720.4.a.bd.1.1 1 3.2 odd 2
1280.4.d.a.641.1 2 16.5 even 4
1280.4.d.a.641.2 2 16.13 even 4
1280.4.d.p.641.1 2 16.3 odd 4
1280.4.d.p.641.2 2 16.11 odd 4
1600.4.a.j.1.1 1 40.19 odd 2
1600.4.a.br.1.1 1 40.29 even 2
1800.4.a.bi.1.1 1 60.59 even 2
1800.4.f.j.649.1 2 60.47 odd 4
1800.4.f.j.649.2 2 60.23 odd 4
1960.4.a.h.1.1 1 28.27 even 2