Properties

Label 672.4.a.c.1.1
Level $672$
Weight $4$
Character 672.1
Self dual yes
Analytic conductor $39.649$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 672.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(39.6492835239\)
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 \(\chi\) \(=\) 672.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{3} -18.0000 q^{5} -7.00000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -18.0000 q^{5} -7.00000 q^{7} +9.00000 q^{9} +44.0000 q^{11} +58.0000 q^{13} -54.0000 q^{15} -130.000 q^{17} +92.0000 q^{19} -21.0000 q^{21} +84.0000 q^{23} +199.000 q^{25} +27.0000 q^{27} -250.000 q^{29} -72.0000 q^{31} +132.000 q^{33} +126.000 q^{35} -354.000 q^{37} +174.000 q^{39} +334.000 q^{41} -416.000 q^{43} -162.000 q^{45} -464.000 q^{47} +49.0000 q^{49} -390.000 q^{51} -450.000 q^{53} -792.000 q^{55} +276.000 q^{57} -516.000 q^{59} +58.0000 q^{61} -63.0000 q^{63} -1044.00 q^{65} -656.000 q^{67} +252.000 q^{69} -940.000 q^{71} +178.000 q^{73} +597.000 q^{75} -308.000 q^{77} +1072.00 q^{79} +81.0000 q^{81} +660.000 q^{83} +2340.00 q^{85} -750.000 q^{87} +1254.00 q^{89} -406.000 q^{91} -216.000 q^{93} -1656.00 q^{95} +210.000 q^{97} +396.000 q^{99} +O(q^{100})\)

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
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) −18.0000 −1.60997 −0.804984 0.593296i \(-0.797826\pi\)
−0.804984 + 0.593296i \(0.797826\pi\)
\(6\) 0 0
\(7\) −7.00000 −0.377964
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 44.0000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) 58.0000 1.23741 0.618704 0.785624i \(-0.287658\pi\)
0.618704 + 0.785624i \(0.287658\pi\)
\(14\) 0 0
\(15\) −54.0000 −0.929516
\(16\) 0 0
\(17\) −130.000 −1.85468 −0.927342 0.374215i \(-0.877912\pi\)
−0.927342 + 0.374215i \(0.877912\pi\)
\(18\) 0 0
\(19\) 92.0000 1.11086 0.555428 0.831565i \(-0.312555\pi\)
0.555428 + 0.831565i \(0.312555\pi\)
\(20\) 0 0
\(21\) −21.0000 −0.218218
\(22\) 0 0
\(23\) 84.0000 0.761531 0.380765 0.924672i \(-0.375661\pi\)
0.380765 + 0.924672i \(0.375661\pi\)
\(24\) 0 0
\(25\) 199.000 1.59200
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −250.000 −1.60082 −0.800411 0.599452i \(-0.795385\pi\)
−0.800411 + 0.599452i \(0.795385\pi\)
\(30\) 0 0
\(31\) −72.0000 −0.417148 −0.208574 0.978007i \(-0.566882\pi\)
−0.208574 + 0.978007i \(0.566882\pi\)
\(32\) 0 0
\(33\) 132.000 0.696311
\(34\) 0 0
\(35\) 126.000 0.608511
\(36\) 0 0
\(37\) −354.000 −1.57290 −0.786449 0.617655i \(-0.788083\pi\)
−0.786449 + 0.617655i \(0.788083\pi\)
\(38\) 0 0
\(39\) 174.000 0.714418
\(40\) 0 0
\(41\) 334.000 1.27224 0.636122 0.771588i \(-0.280537\pi\)
0.636122 + 0.771588i \(0.280537\pi\)
\(42\) 0 0
\(43\) −416.000 −1.47534 −0.737668 0.675164i \(-0.764073\pi\)
−0.737668 + 0.675164i \(0.764073\pi\)
\(44\) 0 0
\(45\) −162.000 −0.536656
\(46\) 0 0
\(47\) −464.000 −1.44003 −0.720014 0.693959i \(-0.755865\pi\)
−0.720014 + 0.693959i \(0.755865\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) −390.000 −1.07080
\(52\) 0 0
\(53\) −450.000 −1.16627 −0.583134 0.812376i \(-0.698174\pi\)
−0.583134 + 0.812376i \(0.698174\pi\)
\(54\) 0 0
\(55\) −792.000 −1.94170
\(56\) 0 0
\(57\) 276.000 0.641353
\(58\) 0 0
\(59\) −516.000 −1.13860 −0.569301 0.822129i \(-0.692786\pi\)
−0.569301 + 0.822129i \(0.692786\pi\)
\(60\) 0 0
\(61\) 58.0000 0.121740 0.0608700 0.998146i \(-0.480612\pi\)
0.0608700 + 0.998146i \(0.480612\pi\)
\(62\) 0 0
\(63\) −63.0000 −0.125988
\(64\) 0 0
\(65\) −1044.00 −1.99219
\(66\) 0 0
\(67\) −656.000 −1.19617 −0.598083 0.801434i \(-0.704071\pi\)
−0.598083 + 0.801434i \(0.704071\pi\)
\(68\) 0 0
\(69\) 252.000 0.439670
\(70\) 0 0
\(71\) −940.000 −1.57123 −0.785616 0.618714i \(-0.787654\pi\)
−0.785616 + 0.618714i \(0.787654\pi\)
\(72\) 0 0
\(73\) 178.000 0.285388 0.142694 0.989767i \(-0.454424\pi\)
0.142694 + 0.989767i \(0.454424\pi\)
\(74\) 0 0
\(75\) 597.000 0.919142
\(76\) 0 0
\(77\) −308.000 −0.455842
\(78\) 0 0
\(79\) 1072.00 1.52670 0.763351 0.645984i \(-0.223553\pi\)
0.763351 + 0.645984i \(0.223553\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 660.000 0.872824 0.436412 0.899747i \(-0.356249\pi\)
0.436412 + 0.899747i \(0.356249\pi\)
\(84\) 0 0
\(85\) 2340.00 2.98598
\(86\) 0 0
\(87\) −750.000 −0.924235
\(88\) 0 0
\(89\) 1254.00 1.49353 0.746763 0.665091i \(-0.231607\pi\)
0.746763 + 0.665091i \(0.231607\pi\)
\(90\) 0 0
\(91\) −406.000 −0.467696
\(92\) 0 0
\(93\) −216.000 −0.240840
\(94\) 0 0
\(95\) −1656.00 −1.78844
\(96\) 0 0
\(97\) 210.000 0.219817 0.109909 0.993942i \(-0.464944\pi\)
0.109909 + 0.993942i \(0.464944\pi\)
\(98\) 0 0
\(99\) 396.000 0.402015
\(100\) 0 0
\(101\) −186.000 −0.183244 −0.0916222 0.995794i \(-0.529205\pi\)
−0.0916222 + 0.995794i \(0.529205\pi\)
\(102\) 0 0
\(103\) 472.000 0.451530 0.225765 0.974182i \(-0.427512\pi\)
0.225765 + 0.974182i \(0.427512\pi\)
\(104\) 0 0
\(105\) 378.000 0.351324
\(106\) 0 0
\(107\) −1212.00 −1.09503 −0.547516 0.836795i \(-0.684427\pi\)
−0.547516 + 0.836795i \(0.684427\pi\)
\(108\) 0 0
\(109\) −1386.00 −1.21793 −0.608967 0.793196i \(-0.708416\pi\)
−0.608967 + 0.793196i \(0.708416\pi\)
\(110\) 0 0
\(111\) −1062.00 −0.908113
\(112\) 0 0
\(113\) 114.000 0.0949046 0.0474523 0.998874i \(-0.484890\pi\)
0.0474523 + 0.998874i \(0.484890\pi\)
\(114\) 0 0
\(115\) −1512.00 −1.22604
\(116\) 0 0
\(117\) 522.000 0.412469
\(118\) 0 0
\(119\) 910.000 0.701005
\(120\) 0 0
\(121\) 605.000 0.454545
\(122\) 0 0
\(123\) 1002.00 0.734531
\(124\) 0 0
\(125\) −1332.00 −0.953102
\(126\) 0 0
\(127\) 792.000 0.553375 0.276688 0.960960i \(-0.410763\pi\)
0.276688 + 0.960960i \(0.410763\pi\)
\(128\) 0 0
\(129\) −1248.00 −0.851785
\(130\) 0 0
\(131\) −428.000 −0.285454 −0.142727 0.989762i \(-0.545587\pi\)
−0.142727 + 0.989762i \(0.545587\pi\)
\(132\) 0 0
\(133\) −644.000 −0.419864
\(134\) 0 0
\(135\) −486.000 −0.309839
\(136\) 0 0
\(137\) −2238.00 −1.39566 −0.697829 0.716264i \(-0.745851\pi\)
−0.697829 + 0.716264i \(0.745851\pi\)
\(138\) 0 0
\(139\) 300.000 0.183062 0.0915312 0.995802i \(-0.470824\pi\)
0.0915312 + 0.995802i \(0.470824\pi\)
\(140\) 0 0
\(141\) −1392.00 −0.831401
\(142\) 0 0
\(143\) 2552.00 1.49237
\(144\) 0 0
\(145\) 4500.00 2.57727
\(146\) 0 0
\(147\) 147.000 0.0824786
\(148\) 0 0
\(149\) 1646.00 0.905004 0.452502 0.891763i \(-0.350532\pi\)
0.452502 + 0.891763i \(0.350532\pi\)
\(150\) 0 0
\(151\) −1184.00 −0.638096 −0.319048 0.947738i \(-0.603363\pi\)
−0.319048 + 0.947738i \(0.603363\pi\)
\(152\) 0 0
\(153\) −1170.00 −0.618228
\(154\) 0 0
\(155\) 1296.00 0.671595
\(156\) 0 0
\(157\) −1150.00 −0.584586 −0.292293 0.956329i \(-0.594418\pi\)
−0.292293 + 0.956329i \(0.594418\pi\)
\(158\) 0 0
\(159\) −1350.00 −0.673346
\(160\) 0 0
\(161\) −588.000 −0.287832
\(162\) 0 0
\(163\) −344.000 −0.165302 −0.0826508 0.996579i \(-0.526339\pi\)
−0.0826508 + 0.996579i \(0.526339\pi\)
\(164\) 0 0
\(165\) −2376.00 −1.12104
\(166\) 0 0
\(167\) −2304.00 −1.06760 −0.533799 0.845611i \(-0.679236\pi\)
−0.533799 + 0.845611i \(0.679236\pi\)
\(168\) 0 0
\(169\) 1167.00 0.531179
\(170\) 0 0
\(171\) 828.000 0.370285
\(172\) 0 0
\(173\) 2742.00 1.20503 0.602516 0.798107i \(-0.294165\pi\)
0.602516 + 0.798107i \(0.294165\pi\)
\(174\) 0 0
\(175\) −1393.00 −0.601719
\(176\) 0 0
\(177\) −1548.00 −0.657372
\(178\) 0 0
\(179\) −3940.00 −1.64519 −0.822596 0.568626i \(-0.807475\pi\)
−0.822596 + 0.568626i \(0.807475\pi\)
\(180\) 0 0
\(181\) 1970.00 0.809000 0.404500 0.914538i \(-0.367446\pi\)
0.404500 + 0.914538i \(0.367446\pi\)
\(182\) 0 0
\(183\) 174.000 0.0702866
\(184\) 0 0
\(185\) 6372.00 2.53232
\(186\) 0 0
\(187\) −5720.00 −2.23683
\(188\) 0 0
\(189\) −189.000 −0.0727393
\(190\) 0 0
\(191\) −476.000 −0.180325 −0.0901627 0.995927i \(-0.528739\pi\)
−0.0901627 + 0.995927i \(0.528739\pi\)
\(192\) 0 0
\(193\) −782.000 −0.291656 −0.145828 0.989310i \(-0.546585\pi\)
−0.145828 + 0.989310i \(0.546585\pi\)
\(194\) 0 0
\(195\) −3132.00 −1.15019
\(196\) 0 0
\(197\) −2066.00 −0.747190 −0.373595 0.927592i \(-0.621875\pi\)
−0.373595 + 0.927592i \(0.621875\pi\)
\(198\) 0 0
\(199\) 768.000 0.273578 0.136789 0.990600i \(-0.456322\pi\)
0.136789 + 0.990600i \(0.456322\pi\)
\(200\) 0 0
\(201\) −1968.00 −0.690607
\(202\) 0 0
\(203\) 1750.00 0.605054
\(204\) 0 0
\(205\) −6012.00 −2.04827
\(206\) 0 0
\(207\) 756.000 0.253844
\(208\) 0 0
\(209\) 4048.00 1.33974
\(210\) 0 0
\(211\) 4248.00 1.38599 0.692996 0.720941i \(-0.256290\pi\)
0.692996 + 0.720941i \(0.256290\pi\)
\(212\) 0 0
\(213\) −2820.00 −0.907151
\(214\) 0 0
\(215\) 7488.00 2.37524
\(216\) 0 0
\(217\) 504.000 0.157667
\(218\) 0 0
\(219\) 534.000 0.164769
\(220\) 0 0
\(221\) −7540.00 −2.29500
\(222\) 0 0
\(223\) −3496.00 −1.04982 −0.524909 0.851158i \(-0.675901\pi\)
−0.524909 + 0.851158i \(0.675901\pi\)
\(224\) 0 0
\(225\) 1791.00 0.530667
\(226\) 0 0
\(227\) −5620.00 −1.64323 −0.821613 0.570045i \(-0.806926\pi\)
−0.821613 + 0.570045i \(0.806926\pi\)
\(228\) 0 0
\(229\) −1982.00 −0.571940 −0.285970 0.958239i \(-0.592316\pi\)
−0.285970 + 0.958239i \(0.592316\pi\)
\(230\) 0 0
\(231\) −924.000 −0.263181
\(232\) 0 0
\(233\) −1342.00 −0.377328 −0.188664 0.982042i \(-0.560416\pi\)
−0.188664 + 0.982042i \(0.560416\pi\)
\(234\) 0 0
\(235\) 8352.00 2.31840
\(236\) 0 0
\(237\) 3216.00 0.881442
\(238\) 0 0
\(239\) −2828.00 −0.765390 −0.382695 0.923875i \(-0.625004\pi\)
−0.382695 + 0.923875i \(0.625004\pi\)
\(240\) 0 0
\(241\) 2002.00 0.535104 0.267552 0.963543i \(-0.413785\pi\)
0.267552 + 0.963543i \(0.413785\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) −882.000 −0.229996
\(246\) 0 0
\(247\) 5336.00 1.37458
\(248\) 0 0
\(249\) 1980.00 0.503925
\(250\) 0 0
\(251\) 1188.00 0.298749 0.149374 0.988781i \(-0.452274\pi\)
0.149374 + 0.988781i \(0.452274\pi\)
\(252\) 0 0
\(253\) 3696.00 0.918441
\(254\) 0 0
\(255\) 7020.00 1.72396
\(256\) 0 0
\(257\) −5506.00 −1.33640 −0.668200 0.743982i \(-0.732935\pi\)
−0.668200 + 0.743982i \(0.732935\pi\)
\(258\) 0 0
\(259\) 2478.00 0.594500
\(260\) 0 0
\(261\) −2250.00 −0.533607
\(262\) 0 0
\(263\) 4076.00 0.955654 0.477827 0.878454i \(-0.341425\pi\)
0.477827 + 0.878454i \(0.341425\pi\)
\(264\) 0 0
\(265\) 8100.00 1.87766
\(266\) 0 0
\(267\) 3762.00 0.862287
\(268\) 0 0
\(269\) −5938.00 −1.34590 −0.672948 0.739689i \(-0.734972\pi\)
−0.672948 + 0.739689i \(0.734972\pi\)
\(270\) 0 0
\(271\) 592.000 0.132699 0.0663495 0.997796i \(-0.478865\pi\)
0.0663495 + 0.997796i \(0.478865\pi\)
\(272\) 0 0
\(273\) −1218.00 −0.270025
\(274\) 0 0
\(275\) 8756.00 1.92002
\(276\) 0 0
\(277\) 5254.00 1.13965 0.569824 0.821767i \(-0.307012\pi\)
0.569824 + 0.821767i \(0.307012\pi\)
\(278\) 0 0
\(279\) −648.000 −0.139049
\(280\) 0 0
\(281\) 3410.00 0.723927 0.361964 0.932192i \(-0.382106\pi\)
0.361964 + 0.932192i \(0.382106\pi\)
\(282\) 0 0
\(283\) −2212.00 −0.464628 −0.232314 0.972641i \(-0.574630\pi\)
−0.232314 + 0.972641i \(0.574630\pi\)
\(284\) 0 0
\(285\) −4968.00 −1.03256
\(286\) 0 0
\(287\) −2338.00 −0.480863
\(288\) 0 0
\(289\) 11987.0 2.43985
\(290\) 0 0
\(291\) 630.000 0.126912
\(292\) 0 0
\(293\) −2122.00 −0.423101 −0.211550 0.977367i \(-0.567851\pi\)
−0.211550 + 0.977367i \(0.567851\pi\)
\(294\) 0 0
\(295\) 9288.00 1.83311
\(296\) 0 0
\(297\) 1188.00 0.232104
\(298\) 0 0
\(299\) 4872.00 0.942325
\(300\) 0 0
\(301\) 2912.00 0.557624
\(302\) 0 0
\(303\) −558.000 −0.105796
\(304\) 0 0
\(305\) −1044.00 −0.195998
\(306\) 0 0
\(307\) 2588.00 0.481124 0.240562 0.970634i \(-0.422668\pi\)
0.240562 + 0.970634i \(0.422668\pi\)
\(308\) 0 0
\(309\) 1416.00 0.260691
\(310\) 0 0
\(311\) 2728.00 0.497398 0.248699 0.968581i \(-0.419997\pi\)
0.248699 + 0.968581i \(0.419997\pi\)
\(312\) 0 0
\(313\) −6446.00 −1.16406 −0.582028 0.813169i \(-0.697741\pi\)
−0.582028 + 0.813169i \(0.697741\pi\)
\(314\) 0 0
\(315\) 1134.00 0.202837
\(316\) 0 0
\(317\) −4234.00 −0.750174 −0.375087 0.926990i \(-0.622387\pi\)
−0.375087 + 0.926990i \(0.622387\pi\)
\(318\) 0 0
\(319\) −11000.0 −1.93066
\(320\) 0 0
\(321\) −3636.00 −0.632217
\(322\) 0 0
\(323\) −11960.0 −2.06029
\(324\) 0 0
\(325\) 11542.0 1.96995
\(326\) 0 0
\(327\) −4158.00 −0.703174
\(328\) 0 0
\(329\) 3248.00 0.544280
\(330\) 0 0
\(331\) −4592.00 −0.762535 −0.381268 0.924465i \(-0.624512\pi\)
−0.381268 + 0.924465i \(0.624512\pi\)
\(332\) 0 0
\(333\) −3186.00 −0.524299
\(334\) 0 0
\(335\) 11808.0 1.92579
\(336\) 0 0
\(337\) −1006.00 −0.162612 −0.0813061 0.996689i \(-0.525909\pi\)
−0.0813061 + 0.996689i \(0.525909\pi\)
\(338\) 0 0
\(339\) 342.000 0.0547932
\(340\) 0 0
\(341\) −3168.00 −0.503099
\(342\) 0 0
\(343\) −343.000 −0.0539949
\(344\) 0 0
\(345\) −4536.00 −0.707855
\(346\) 0 0
\(347\) 4644.00 0.718452 0.359226 0.933251i \(-0.383041\pi\)
0.359226 + 0.933251i \(0.383041\pi\)
\(348\) 0 0
\(349\) 4786.00 0.734065 0.367033 0.930208i \(-0.380374\pi\)
0.367033 + 0.930208i \(0.380374\pi\)
\(350\) 0 0
\(351\) 1566.00 0.238139
\(352\) 0 0
\(353\) 1302.00 0.196313 0.0981565 0.995171i \(-0.468705\pi\)
0.0981565 + 0.995171i \(0.468705\pi\)
\(354\) 0 0
\(355\) 16920.0 2.52963
\(356\) 0 0
\(357\) 2730.00 0.404725
\(358\) 0 0
\(359\) 11260.0 1.65538 0.827688 0.561188i \(-0.189656\pi\)
0.827688 + 0.561188i \(0.189656\pi\)
\(360\) 0 0
\(361\) 1605.00 0.233999
\(362\) 0 0
\(363\) 1815.00 0.262432
\(364\) 0 0
\(365\) −3204.00 −0.459466
\(366\) 0 0
\(367\) 2792.00 0.397115 0.198558 0.980089i \(-0.436374\pi\)
0.198558 + 0.980089i \(0.436374\pi\)
\(368\) 0 0
\(369\) 3006.00 0.424082
\(370\) 0 0
\(371\) 3150.00 0.440808
\(372\) 0 0
\(373\) 4118.00 0.571641 0.285820 0.958283i \(-0.407734\pi\)
0.285820 + 0.958283i \(0.407734\pi\)
\(374\) 0 0
\(375\) −3996.00 −0.550273
\(376\) 0 0
\(377\) −14500.0 −1.98087
\(378\) 0 0
\(379\) 8624.00 1.16883 0.584413 0.811456i \(-0.301325\pi\)
0.584413 + 0.811456i \(0.301325\pi\)
\(380\) 0 0
\(381\) 2376.00 0.319491
\(382\) 0 0
\(383\) 6488.00 0.865591 0.432795 0.901492i \(-0.357527\pi\)
0.432795 + 0.901492i \(0.357527\pi\)
\(384\) 0 0
\(385\) 5544.00 0.733892
\(386\) 0 0
\(387\) −3744.00 −0.491778
\(388\) 0 0
\(389\) 1406.00 0.183257 0.0916286 0.995793i \(-0.470793\pi\)
0.0916286 + 0.995793i \(0.470793\pi\)
\(390\) 0 0
\(391\) −10920.0 −1.41240
\(392\) 0 0
\(393\) −1284.00 −0.164807
\(394\) 0 0
\(395\) −19296.0 −2.45794
\(396\) 0 0
\(397\) 9378.00 1.18556 0.592781 0.805363i \(-0.298030\pi\)
0.592781 + 0.805363i \(0.298030\pi\)
\(398\) 0 0
\(399\) −1932.00 −0.242408
\(400\) 0 0
\(401\) 2890.00 0.359900 0.179950 0.983676i \(-0.442406\pi\)
0.179950 + 0.983676i \(0.442406\pi\)
\(402\) 0 0
\(403\) −4176.00 −0.516182
\(404\) 0 0
\(405\) −1458.00 −0.178885
\(406\) 0 0
\(407\) −15576.0 −1.89699
\(408\) 0 0
\(409\) −10582.0 −1.27933 −0.639665 0.768654i \(-0.720927\pi\)
−0.639665 + 0.768654i \(0.720927\pi\)
\(410\) 0 0
\(411\) −6714.00 −0.805784
\(412\) 0 0
\(413\) 3612.00 0.430351
\(414\) 0 0
\(415\) −11880.0 −1.40522
\(416\) 0 0
\(417\) 900.000 0.105691
\(418\) 0 0
\(419\) −9500.00 −1.10765 −0.553825 0.832633i \(-0.686832\pi\)
−0.553825 + 0.832633i \(0.686832\pi\)
\(420\) 0 0
\(421\) 598.000 0.0692274 0.0346137 0.999401i \(-0.488980\pi\)
0.0346137 + 0.999401i \(0.488980\pi\)
\(422\) 0 0
\(423\) −4176.00 −0.480010
\(424\) 0 0
\(425\) −25870.0 −2.95266
\(426\) 0 0
\(427\) −406.000 −0.0460134
\(428\) 0 0
\(429\) 7656.00 0.861620
\(430\) 0 0
\(431\) 3708.00 0.414404 0.207202 0.978298i \(-0.433564\pi\)
0.207202 + 0.978298i \(0.433564\pi\)
\(432\) 0 0
\(433\) 13706.0 1.52117 0.760587 0.649236i \(-0.224911\pi\)
0.760587 + 0.649236i \(0.224911\pi\)
\(434\) 0 0
\(435\) 13500.0 1.48799
\(436\) 0 0
\(437\) 7728.00 0.845951
\(438\) 0 0
\(439\) −8232.00 −0.894970 −0.447485 0.894291i \(-0.647680\pi\)
−0.447485 + 0.894291i \(0.647680\pi\)
\(440\) 0 0
\(441\) 441.000 0.0476190
\(442\) 0 0
\(443\) 2524.00 0.270697 0.135349 0.990798i \(-0.456785\pi\)
0.135349 + 0.990798i \(0.456785\pi\)
\(444\) 0 0
\(445\) −22572.0 −2.40453
\(446\) 0 0
\(447\) 4938.00 0.522504
\(448\) 0 0
\(449\) −3630.00 −0.381537 −0.190769 0.981635i \(-0.561098\pi\)
−0.190769 + 0.981635i \(0.561098\pi\)
\(450\) 0 0
\(451\) 14696.0 1.53438
\(452\) 0 0
\(453\) −3552.00 −0.368405
\(454\) 0 0
\(455\) 7308.00 0.752977
\(456\) 0 0
\(457\) 5386.00 0.551305 0.275653 0.961257i \(-0.411106\pi\)
0.275653 + 0.961257i \(0.411106\pi\)
\(458\) 0 0
\(459\) −3510.00 −0.356934
\(460\) 0 0
\(461\) 11766.0 1.18871 0.594357 0.804201i \(-0.297407\pi\)
0.594357 + 0.804201i \(0.297407\pi\)
\(462\) 0 0
\(463\) −10240.0 −1.02785 −0.513923 0.857836i \(-0.671808\pi\)
−0.513923 + 0.857836i \(0.671808\pi\)
\(464\) 0 0
\(465\) 3888.00 0.387746
\(466\) 0 0
\(467\) 6076.00 0.602064 0.301032 0.953614i \(-0.402669\pi\)
0.301032 + 0.953614i \(0.402669\pi\)
\(468\) 0 0
\(469\) 4592.00 0.452108
\(470\) 0 0
\(471\) −3450.00 −0.337511
\(472\) 0 0
\(473\) −18304.0 −1.77932
\(474\) 0 0
\(475\) 18308.0 1.76848
\(476\) 0 0
\(477\) −4050.00 −0.388756
\(478\) 0 0
\(479\) 6480.00 0.618118 0.309059 0.951043i \(-0.399986\pi\)
0.309059 + 0.951043i \(0.399986\pi\)
\(480\) 0 0
\(481\) −20532.0 −1.94632
\(482\) 0 0
\(483\) −1764.00 −0.166180
\(484\) 0 0
\(485\) −3780.00 −0.353899
\(486\) 0 0
\(487\) 4240.00 0.394523 0.197262 0.980351i \(-0.436795\pi\)
0.197262 + 0.980351i \(0.436795\pi\)
\(488\) 0 0
\(489\) −1032.00 −0.0954369
\(490\) 0 0
\(491\) −17892.0 −1.64451 −0.822255 0.569119i \(-0.807284\pi\)
−0.822255 + 0.569119i \(0.807284\pi\)
\(492\) 0 0
\(493\) 32500.0 2.96902
\(494\) 0 0
\(495\) −7128.00 −0.647232
\(496\) 0 0
\(497\) 6580.00 0.593870
\(498\) 0 0
\(499\) −4616.00 −0.414109 −0.207055 0.978329i \(-0.566388\pi\)
−0.207055 + 0.978329i \(0.566388\pi\)
\(500\) 0 0
\(501\) −6912.00 −0.616378
\(502\) 0 0
\(503\) 3696.00 0.327627 0.163814 0.986491i \(-0.447620\pi\)
0.163814 + 0.986491i \(0.447620\pi\)
\(504\) 0 0
\(505\) 3348.00 0.295018
\(506\) 0 0
\(507\) 3501.00 0.306676
\(508\) 0 0
\(509\) −16738.0 −1.45756 −0.728781 0.684747i \(-0.759913\pi\)
−0.728781 + 0.684747i \(0.759913\pi\)
\(510\) 0 0
\(511\) −1246.00 −0.107867
\(512\) 0 0
\(513\) 2484.00 0.213784
\(514\) 0 0
\(515\) −8496.00 −0.726949
\(516\) 0 0
\(517\) −20416.0 −1.73674
\(518\) 0 0
\(519\) 8226.00 0.695725
\(520\) 0 0
\(521\) 19062.0 1.60292 0.801460 0.598048i \(-0.204057\pi\)
0.801460 + 0.598048i \(0.204057\pi\)
\(522\) 0 0
\(523\) −12268.0 −1.02570 −0.512851 0.858478i \(-0.671411\pi\)
−0.512851 + 0.858478i \(0.671411\pi\)
\(524\) 0 0
\(525\) −4179.00 −0.347403
\(526\) 0 0
\(527\) 9360.00 0.773677
\(528\) 0 0
\(529\) −5111.00 −0.420071
\(530\) 0 0
\(531\) −4644.00 −0.379534
\(532\) 0 0
\(533\) 19372.0 1.57429
\(534\) 0 0
\(535\) 21816.0 1.76297
\(536\) 0 0
\(537\) −11820.0 −0.949852
\(538\) 0 0
\(539\) 2156.00 0.172292
\(540\) 0 0
\(541\) −17042.0 −1.35433 −0.677165 0.735831i \(-0.736792\pi\)
−0.677165 + 0.735831i \(0.736792\pi\)
\(542\) 0 0
\(543\) 5910.00 0.467076
\(544\) 0 0
\(545\) 24948.0 1.96083
\(546\) 0 0
\(547\) −3656.00 −0.285776 −0.142888 0.989739i \(-0.545639\pi\)
−0.142888 + 0.989739i \(0.545639\pi\)
\(548\) 0 0
\(549\) 522.000 0.0405800
\(550\) 0 0
\(551\) −23000.0 −1.77828
\(552\) 0 0
\(553\) −7504.00 −0.577039
\(554\) 0 0
\(555\) 19116.0 1.46203
\(556\) 0 0
\(557\) 14038.0 1.06788 0.533940 0.845522i \(-0.320711\pi\)
0.533940 + 0.845522i \(0.320711\pi\)
\(558\) 0 0
\(559\) −24128.0 −1.82559
\(560\) 0 0
\(561\) −17160.0 −1.29144
\(562\) 0 0
\(563\) −18332.0 −1.37229 −0.686147 0.727463i \(-0.740699\pi\)
−0.686147 + 0.727463i \(0.740699\pi\)
\(564\) 0 0
\(565\) −2052.00 −0.152793
\(566\) 0 0
\(567\) −567.000 −0.0419961
\(568\) 0 0
\(569\) −10046.0 −0.740159 −0.370079 0.929000i \(-0.620669\pi\)
−0.370079 + 0.929000i \(0.620669\pi\)
\(570\) 0 0
\(571\) −5704.00 −0.418047 −0.209024 0.977911i \(-0.567029\pi\)
−0.209024 + 0.977911i \(0.567029\pi\)
\(572\) 0 0
\(573\) −1428.00 −0.104111
\(574\) 0 0
\(575\) 16716.0 1.21236
\(576\) 0 0
\(577\) 24610.0 1.77561 0.887806 0.460219i \(-0.152229\pi\)
0.887806 + 0.460219i \(0.152229\pi\)
\(578\) 0 0
\(579\) −2346.00 −0.168388
\(580\) 0 0
\(581\) −4620.00 −0.329897
\(582\) 0 0
\(583\) −19800.0 −1.40657
\(584\) 0 0
\(585\) −9396.00 −0.664063
\(586\) 0 0
\(587\) 18516.0 1.30194 0.650969 0.759105i \(-0.274363\pi\)
0.650969 + 0.759105i \(0.274363\pi\)
\(588\) 0 0
\(589\) −6624.00 −0.463391
\(590\) 0 0
\(591\) −6198.00 −0.431390
\(592\) 0 0
\(593\) 20038.0 1.38763 0.693813 0.720155i \(-0.255929\pi\)
0.693813 + 0.720155i \(0.255929\pi\)
\(594\) 0 0
\(595\) −16380.0 −1.12860
\(596\) 0 0
\(597\) 2304.00 0.157950
\(598\) 0 0
\(599\) 2596.00 0.177078 0.0885390 0.996073i \(-0.471780\pi\)
0.0885390 + 0.996073i \(0.471780\pi\)
\(600\) 0 0
\(601\) −5190.00 −0.352254 −0.176127 0.984367i \(-0.556357\pi\)
−0.176127 + 0.984367i \(0.556357\pi\)
\(602\) 0 0
\(603\) −5904.00 −0.398722
\(604\) 0 0
\(605\) −10890.0 −0.731804
\(606\) 0 0
\(607\) 6536.00 0.437048 0.218524 0.975832i \(-0.429876\pi\)
0.218524 + 0.975832i \(0.429876\pi\)
\(608\) 0 0
\(609\) 5250.00 0.349328
\(610\) 0 0
\(611\) −26912.0 −1.78190
\(612\) 0 0
\(613\) 4702.00 0.309807 0.154904 0.987930i \(-0.450493\pi\)
0.154904 + 0.987930i \(0.450493\pi\)
\(614\) 0 0
\(615\) −18036.0 −1.18257
\(616\) 0 0
\(617\) −8638.00 −0.563619 −0.281809 0.959470i \(-0.590935\pi\)
−0.281809 + 0.959470i \(0.590935\pi\)
\(618\) 0 0
\(619\) −19676.0 −1.27762 −0.638809 0.769366i \(-0.720572\pi\)
−0.638809 + 0.769366i \(0.720572\pi\)
\(620\) 0 0
\(621\) 2268.00 0.146557
\(622\) 0 0
\(623\) −8778.00 −0.564499
\(624\) 0 0
\(625\) −899.000 −0.0575360
\(626\) 0 0
\(627\) 12144.0 0.773500
\(628\) 0 0
\(629\) 46020.0 2.91723
\(630\) 0 0
\(631\) 26720.0 1.68575 0.842874 0.538112i \(-0.180862\pi\)
0.842874 + 0.538112i \(0.180862\pi\)
\(632\) 0 0
\(633\) 12744.0 0.800203
\(634\) 0 0
\(635\) −14256.0 −0.890917
\(636\) 0 0
\(637\) 2842.00 0.176773
\(638\) 0 0
\(639\) −8460.00 −0.523744
\(640\) 0 0
\(641\) −1990.00 −0.122621 −0.0613107 0.998119i \(-0.519528\pi\)
−0.0613107 + 0.998119i \(0.519528\pi\)
\(642\) 0 0
\(643\) −6956.00 −0.426622 −0.213311 0.976984i \(-0.568425\pi\)
−0.213311 + 0.976984i \(0.568425\pi\)
\(644\) 0 0
\(645\) 22464.0 1.37135
\(646\) 0 0
\(647\) −15984.0 −0.971246 −0.485623 0.874168i \(-0.661407\pi\)
−0.485623 + 0.874168i \(0.661407\pi\)
\(648\) 0 0
\(649\) −22704.0 −1.37320
\(650\) 0 0
\(651\) 1512.00 0.0910291
\(652\) 0 0
\(653\) 6614.00 0.396364 0.198182 0.980165i \(-0.436496\pi\)
0.198182 + 0.980165i \(0.436496\pi\)
\(654\) 0 0
\(655\) 7704.00 0.459573
\(656\) 0 0
\(657\) 1602.00 0.0951293
\(658\) 0 0
\(659\) −29364.0 −1.73575 −0.867875 0.496783i \(-0.834515\pi\)
−0.867875 + 0.496783i \(0.834515\pi\)
\(660\) 0 0
\(661\) −3150.00 −0.185357 −0.0926784 0.995696i \(-0.529543\pi\)
−0.0926784 + 0.995696i \(0.529543\pi\)
\(662\) 0 0
\(663\) −22620.0 −1.32502
\(664\) 0 0
\(665\) 11592.0 0.675968
\(666\) 0 0
\(667\) −21000.0 −1.21908
\(668\) 0 0
\(669\) −10488.0 −0.606113
\(670\) 0 0
\(671\) 2552.00 0.146824
\(672\) 0 0
\(673\) 8402.00 0.481238 0.240619 0.970620i \(-0.422650\pi\)
0.240619 + 0.970620i \(0.422650\pi\)
\(674\) 0 0
\(675\) 5373.00 0.306381
\(676\) 0 0
\(677\) 7854.00 0.445870 0.222935 0.974833i \(-0.428436\pi\)
0.222935 + 0.974833i \(0.428436\pi\)
\(678\) 0 0
\(679\) −1470.00 −0.0830831
\(680\) 0 0
\(681\) −16860.0 −0.948717
\(682\) 0 0
\(683\) −14244.0 −0.797996 −0.398998 0.916952i \(-0.630642\pi\)
−0.398998 + 0.916952i \(0.630642\pi\)
\(684\) 0 0
\(685\) 40284.0 2.24697
\(686\) 0 0
\(687\) −5946.00 −0.330210
\(688\) 0 0
\(689\) −26100.0 −1.44315
\(690\) 0 0
\(691\) 22420.0 1.23429 0.617147 0.786848i \(-0.288288\pi\)
0.617147 + 0.786848i \(0.288288\pi\)
\(692\) 0 0
\(693\) −2772.00 −0.151947
\(694\) 0 0
\(695\) −5400.00 −0.294725
\(696\) 0 0
\(697\) −43420.0 −2.35961
\(698\) 0 0
\(699\) −4026.00 −0.217850
\(700\) 0 0
\(701\) 19814.0 1.06757 0.533783 0.845621i \(-0.320770\pi\)
0.533783 + 0.845621i \(0.320770\pi\)
\(702\) 0 0
\(703\) −32568.0 −1.74726
\(704\) 0 0
\(705\) 25056.0 1.33853
\(706\) 0 0
\(707\) 1302.00 0.0692599
\(708\) 0 0
\(709\) −15986.0 −0.846780 −0.423390 0.905948i \(-0.639160\pi\)
−0.423390 + 0.905948i \(0.639160\pi\)
\(710\) 0 0
\(711\) 9648.00 0.508901
\(712\) 0 0
\(713\) −6048.00 −0.317671
\(714\) 0 0
\(715\) −45936.0 −2.40267
\(716\) 0 0
\(717\) −8484.00 −0.441898
\(718\) 0 0
\(719\) 22440.0 1.16394 0.581969 0.813211i \(-0.302283\pi\)
0.581969 + 0.813211i \(0.302283\pi\)
\(720\) 0 0
\(721\) −3304.00 −0.170662
\(722\) 0 0
\(723\) 6006.00 0.308943
\(724\) 0 0
\(725\) −49750.0 −2.54851
\(726\) 0 0
\(727\) 10264.0 0.523619 0.261809 0.965120i \(-0.415681\pi\)
0.261809 + 0.965120i \(0.415681\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 54080.0 2.73628
\(732\) 0 0
\(733\) 9282.00 0.467720 0.233860 0.972270i \(-0.424864\pi\)
0.233860 + 0.972270i \(0.424864\pi\)
\(734\) 0 0
\(735\) −2646.00 −0.132788
\(736\) 0 0
\(737\) −28864.0 −1.44263
\(738\) 0 0
\(739\) 12792.0 0.636754 0.318377 0.947964i \(-0.396862\pi\)
0.318377 + 0.947964i \(0.396862\pi\)
\(740\) 0 0
\(741\) 16008.0 0.793615
\(742\) 0 0
\(743\) 25644.0 1.26620 0.633100 0.774070i \(-0.281782\pi\)
0.633100 + 0.774070i \(0.281782\pi\)
\(744\) 0 0
\(745\) −29628.0 −1.45703
\(746\) 0 0
\(747\) 5940.00 0.290941
\(748\) 0 0
\(749\) 8484.00 0.413883
\(750\) 0 0
\(751\) −4528.00 −0.220012 −0.110006 0.993931i \(-0.535087\pi\)
−0.110006 + 0.993931i \(0.535087\pi\)
\(752\) 0 0
\(753\) 3564.00 0.172483
\(754\) 0 0
\(755\) 21312.0 1.02732
\(756\) 0 0
\(757\) 31310.0 1.50328 0.751639 0.659575i \(-0.229264\pi\)
0.751639 + 0.659575i \(0.229264\pi\)
\(758\) 0 0
\(759\) 11088.0 0.530262
\(760\) 0 0
\(761\) 16622.0 0.791783 0.395892 0.918297i \(-0.370436\pi\)
0.395892 + 0.918297i \(0.370436\pi\)
\(762\) 0 0
\(763\) 9702.00 0.460335
\(764\) 0 0
\(765\) 21060.0 0.995328
\(766\) 0 0
\(767\) −29928.0 −1.40891
\(768\) 0 0
\(769\) −9814.00 −0.460211 −0.230105 0.973166i \(-0.573907\pi\)
−0.230105 + 0.973166i \(0.573907\pi\)
\(770\) 0 0
\(771\) −16518.0 −0.771571
\(772\) 0 0
\(773\) 7686.00 0.357628 0.178814 0.983883i \(-0.442774\pi\)
0.178814 + 0.983883i \(0.442774\pi\)
\(774\) 0 0
\(775\) −14328.0 −0.664099
\(776\) 0 0
\(777\) 7434.00 0.343235
\(778\) 0 0
\(779\) 30728.0 1.41328
\(780\) 0 0
\(781\) −41360.0 −1.89498
\(782\) 0 0
\(783\) −6750.00 −0.308078
\(784\) 0 0
\(785\) 20700.0 0.941165
\(786\) 0 0
\(787\) 5860.00 0.265421 0.132711 0.991155i \(-0.457632\pi\)
0.132711 + 0.991155i \(0.457632\pi\)
\(788\) 0 0
\(789\) 12228.0 0.551747
\(790\) 0 0
\(791\) −798.000 −0.0358706
\(792\) 0 0
\(793\) 3364.00 0.150642
\(794\) 0 0
\(795\) 24300.0 1.08407
\(796\) 0 0
\(797\) −15450.0 −0.686659 −0.343329 0.939215i \(-0.611555\pi\)
−0.343329 + 0.939215i \(0.611555\pi\)
\(798\) 0 0
\(799\) 60320.0 2.67080
\(800\) 0 0
\(801\) 11286.0 0.497842
\(802\) 0 0
\(803\) 7832.00 0.344191
\(804\) 0 0
\(805\) 10584.0 0.463400
\(806\) 0 0
\(807\) −17814.0 −0.777054
\(808\) 0 0
\(809\) −26726.0 −1.16148 −0.580739 0.814090i \(-0.697236\pi\)
−0.580739 + 0.814090i \(0.697236\pi\)
\(810\) 0 0
\(811\) −3052.00 −0.132146 −0.0660729 0.997815i \(-0.521047\pi\)
−0.0660729 + 0.997815i \(0.521047\pi\)
\(812\) 0 0
\(813\) 1776.00 0.0766138
\(814\) 0 0
\(815\) 6192.00 0.266130
\(816\) 0 0
\(817\) −38272.0 −1.63888
\(818\) 0 0
\(819\) −3654.00 −0.155899
\(820\) 0 0
\(821\) 23838.0 1.01334 0.506670 0.862140i \(-0.330876\pi\)
0.506670 + 0.862140i \(0.330876\pi\)
\(822\) 0 0
\(823\) 19136.0 0.810497 0.405248 0.914207i \(-0.367185\pi\)
0.405248 + 0.914207i \(0.367185\pi\)
\(824\) 0 0
\(825\) 26268.0 1.10853
\(826\) 0 0
\(827\) −32556.0 −1.36890 −0.684452 0.729058i \(-0.739958\pi\)
−0.684452 + 0.729058i \(0.739958\pi\)
\(828\) 0 0
\(829\) −33086.0 −1.38616 −0.693079 0.720862i \(-0.743746\pi\)
−0.693079 + 0.720862i \(0.743746\pi\)
\(830\) 0 0
\(831\) 15762.0 0.657976
\(832\) 0 0
\(833\) −6370.00 −0.264955
\(834\) 0 0
\(835\) 41472.0 1.71880
\(836\) 0 0
\(837\) −1944.00 −0.0802801
\(838\) 0 0
\(839\) 35248.0 1.45041 0.725206 0.688532i \(-0.241744\pi\)
0.725206 + 0.688532i \(0.241744\pi\)
\(840\) 0 0
\(841\) 38111.0 1.56263
\(842\) 0 0
\(843\) 10230.0 0.417960
\(844\) 0 0
\(845\) −21006.0 −0.855182
\(846\) 0 0
\(847\) −4235.00 −0.171802
\(848\) 0 0
\(849\) −6636.00 −0.268253
\(850\) 0 0
\(851\) −29736.0 −1.19781
\(852\) 0 0
\(853\) 8922.00 0.358128 0.179064 0.983837i \(-0.442693\pi\)
0.179064 + 0.983837i \(0.442693\pi\)
\(854\) 0 0
\(855\) −14904.0 −0.596147
\(856\) 0 0
\(857\) 28126.0 1.12108 0.560540 0.828127i \(-0.310594\pi\)
0.560540 + 0.828127i \(0.310594\pi\)
\(858\) 0 0
\(859\) 28916.0 1.14855 0.574273 0.818664i \(-0.305285\pi\)
0.574273 + 0.818664i \(0.305285\pi\)
\(860\) 0 0
\(861\) −7014.00 −0.277627
\(862\) 0 0
\(863\) −22308.0 −0.879923 −0.439961 0.898017i \(-0.645008\pi\)
−0.439961 + 0.898017i \(0.645008\pi\)
\(864\) 0 0
\(865\) −49356.0 −1.94006
\(866\) 0 0
\(867\) 35961.0 1.40865
\(868\) 0 0
\(869\) 47168.0 1.84127
\(870\) 0 0
\(871\) −38048.0 −1.48015
\(872\) 0 0
\(873\) 1890.00 0.0732724
\(874\) 0 0
\(875\) 9324.00 0.360239
\(876\) 0 0
\(877\) −34970.0 −1.34647 −0.673234 0.739429i \(-0.735095\pi\)
−0.673234 + 0.739429i \(0.735095\pi\)
\(878\) 0 0
\(879\) −6366.00 −0.244277
\(880\) 0 0
\(881\) −858.000 −0.0328113 −0.0164056 0.999865i \(-0.505222\pi\)
−0.0164056 + 0.999865i \(0.505222\pi\)
\(882\) 0 0
\(883\) 24088.0 0.918036 0.459018 0.888427i \(-0.348201\pi\)
0.459018 + 0.888427i \(0.348201\pi\)
\(884\) 0 0
\(885\) 27864.0 1.05835
\(886\) 0 0
\(887\) −30960.0 −1.17197 −0.585984 0.810323i \(-0.699292\pi\)
−0.585984 + 0.810323i \(0.699292\pi\)
\(888\) 0 0
\(889\) −5544.00 −0.209156
\(890\) 0 0
\(891\) 3564.00 0.134005
\(892\) 0 0
\(893\) −42688.0 −1.59966
\(894\) 0 0
\(895\) 70920.0 2.64871
\(896\) 0 0
\(897\) 14616.0 0.544051
\(898\) 0 0
\(899\) 18000.0 0.667779
\(900\) 0 0
\(901\) 58500.0 2.16306
\(902\) 0 0
\(903\) 8736.00 0.321944
\(904\) 0 0
\(905\) −35460.0 −1.30246
\(906\) 0 0
\(907\) −37048.0 −1.35629 −0.678147 0.734926i \(-0.737217\pi\)
−0.678147 + 0.734926i \(0.737217\pi\)
\(908\) 0 0
\(909\) −1674.00 −0.0610815
\(910\) 0 0
\(911\) 25228.0 0.917498 0.458749 0.888566i \(-0.348298\pi\)
0.458749 + 0.888566i \(0.348298\pi\)
\(912\) 0 0
\(913\) 29040.0 1.05267
\(914\) 0 0
\(915\) −3132.00 −0.113159
\(916\) 0 0
\(917\) 2996.00 0.107892
\(918\) 0 0
\(919\) −19336.0 −0.694054 −0.347027 0.937855i \(-0.612809\pi\)
−0.347027 + 0.937855i \(0.612809\pi\)
\(920\) 0 0
\(921\) 7764.00 0.277777
\(922\) 0 0
\(923\) −54520.0 −1.94426
\(924\) 0 0
\(925\) −70446.0 −2.50405
\(926\) 0 0
\(927\) 4248.00 0.150510
\(928\) 0 0
\(929\) 11926.0 0.421183 0.210592 0.977574i \(-0.432461\pi\)
0.210592 + 0.977574i \(0.432461\pi\)
\(930\) 0 0
\(931\) 4508.00 0.158694
\(932\) 0 0
\(933\) 8184.00 0.287173
\(934\) 0 0
\(935\) 102960. 3.60123
\(936\) 0 0
\(937\) 4698.00 0.163796 0.0818981 0.996641i \(-0.473902\pi\)
0.0818981 + 0.996641i \(0.473902\pi\)
\(938\) 0 0
\(939\) −19338.0 −0.672068
\(940\) 0 0
\(941\) −12986.0 −0.449874 −0.224937 0.974373i \(-0.572218\pi\)
−0.224937 + 0.974373i \(0.572218\pi\)
\(942\) 0 0
\(943\) 28056.0 0.968854
\(944\) 0 0
\(945\) 3402.00 0.117108
\(946\) 0 0
\(947\) −17972.0 −0.616696 −0.308348 0.951274i \(-0.599776\pi\)
−0.308348 + 0.951274i \(0.599776\pi\)
\(948\) 0 0
\(949\) 10324.0 0.353141
\(950\) 0 0
\(951\) −12702.0 −0.433113
\(952\) 0 0
\(953\) −5414.00 −0.184026 −0.0920129 0.995758i \(-0.529330\pi\)
−0.0920129 + 0.995758i \(0.529330\pi\)
\(954\) 0 0
\(955\) 8568.00 0.290318
\(956\) 0 0
\(957\) −33000.0 −1.11467
\(958\) 0 0
\(959\) 15666.0 0.527509
\(960\) 0 0
\(961\) −24607.0 −0.825988
\(962\) 0 0
\(963\) −10908.0 −0.365011
\(964\) 0 0
\(965\) 14076.0 0.469557
\(966\) 0 0
\(967\) −57496.0 −1.91204 −0.956022 0.293295i \(-0.905248\pi\)
−0.956022 + 0.293295i \(0.905248\pi\)
\(968\) 0 0
\(969\) −35880.0 −1.18951
\(970\) 0 0
\(971\) −36812.0 −1.21664 −0.608318 0.793693i \(-0.708155\pi\)
−0.608318 + 0.793693i \(0.708155\pi\)
\(972\) 0 0
\(973\) −2100.00 −0.0691911
\(974\) 0 0
\(975\) 34626.0 1.13735
\(976\) 0 0
\(977\) 26442.0 0.865870 0.432935 0.901425i \(-0.357478\pi\)
0.432935 + 0.901425i \(0.357478\pi\)
\(978\) 0 0
\(979\) 55176.0 1.80126
\(980\) 0 0
\(981\) −12474.0 −0.405978
\(982\) 0 0
\(983\) 35240.0 1.14342 0.571710 0.820456i \(-0.306280\pi\)
0.571710 + 0.820456i \(0.306280\pi\)
\(984\) 0 0
\(985\) 37188.0 1.20295
\(986\) 0 0
\(987\) 9744.00 0.314240
\(988\) 0 0
\(989\) −34944.0 −1.12351
\(990\) 0 0
\(991\) −36472.0 −1.16909 −0.584547 0.811360i \(-0.698728\pi\)
−0.584547 + 0.811360i \(0.698728\pi\)
\(992\) 0 0
\(993\) −13776.0 −0.440250
\(994\) 0 0
\(995\) −13824.0 −0.440453
\(996\) 0 0
\(997\) 25090.0 0.796999 0.398500 0.917168i \(-0.369531\pi\)
0.398500 + 0.917168i \(0.369531\pi\)
\(998\) 0 0
\(999\) −9558.00 −0.302704
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 672.4.a.c.1.1 yes 1
3.2 odd 2 2016.4.a.e.1.1 1
4.3 odd 2 672.4.a.a.1.1 1
8.3 odd 2 1344.4.a.bb.1.1 1
8.5 even 2 1344.4.a.m.1.1 1
12.11 even 2 2016.4.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
672.4.a.a.1.1 1 4.3 odd 2
672.4.a.c.1.1 yes 1 1.1 even 1 trivial
1344.4.a.m.1.1 1 8.5 even 2
1344.4.a.bb.1.1 1 8.3 odd 2
2016.4.a.e.1.1 1 3.2 odd 2
2016.4.a.f.1.1 1 12.11 even 2