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Properties

Label 1344.4.a.bb.1.1
Level $1344$
Weight $4$
Character 1344.1
Self dual yes
Analytic conductor $79.299$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1344.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(79.2985670477\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 672)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1344.1

$q$-expansion

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

Display 100 coefficients 10 coefficients

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 \(-0.202174\pi\)
0.804984 + 0.593296i \(0.202174\pi\)
\(6\) 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 \(-0.712342\pi\)
−0.618704 + 0.785624i \(0.712342\pi\)
\(14\) 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 \(-0.624339\pi\)
−0.380765 + 0.924672i \(0.624339\pi\)
\(24\) 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 \(-0.204615\pi\)
0.800411 + 0.599452i \(0.204615\pi\)
\(30\) 0 0
\(31\) 72.0000 0.417148 0.208574 − 0.978007i \(-0.433118\pi\)
0.208574 + 0.978007i \(0.433118\pi\)
\(32\) 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 \(-0.211917\pi\)
0.786449 + 0.617655i \(0.211917\pi\)
\(38\) 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 \(-0.244135\pi\)
0.720014 + 0.693959i \(0.244135\pi\)
\(48\) 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 \(-0.301826\pi\)
0.583134 + 0.812376i \(0.301826\pi\)
\(54\) 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 \(-0.519388\pi\)
−0.0608700 + 0.998146i \(0.519388\pi\)
\(62\) 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 \(-0.212346\pi\)
0.785616 + 0.618714i \(0.212346\pi\)
\(72\) 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 \(-0.776447\pi\)
−0.763351 + 0.645984i \(0.776447\pi\)
\(80\) 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 \(-0.470795\pi\)
0.0916222 + 0.995794i \(0.470795\pi\)
\(102\) 0 0
\(103\) −472.000 −0.451530 −0.225765 − 0.974182i \(-0.572488\pi\)
−0.225765 + 0.974182i \(0.572488\pi\)
\(104\) 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 \(-0.291584\pi\)
0.608967 + 0.793196i \(0.291584\pi\)
\(110\) 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 \(-0.589237\pi\)
−0.276688 + 0.960960i \(0.589237\pi\)
\(128\) 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 \(-0.649468\pi\)
−0.452502 + 0.891763i \(0.649468\pi\)
\(150\) 0 0
\(151\) 1184.00 0.638096 0.319048 − 0.947738i \(-0.396637\pi\)
0.319048 + 0.947738i \(0.396637\pi\)
\(152\) 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 \(-0.405582\pi\)
0.292293 + 0.956329i \(0.405582\pi\)
\(158\) 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 \(-0.320764\pi\)
0.533799 + 0.845611i \(0.320764\pi\)
\(168\) 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 \(-0.705835\pi\)
−0.602516 + 0.798107i \(0.705835\pi\)
\(174\) 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 \(-0.632554\pi\)
−0.404500 + 0.914538i \(0.632554\pi\)
\(182\) 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 \(-0.471261\pi\)
0.0901627 + 0.995927i \(0.471261\pi\)
\(192\) 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 \(-0.378125\pi\)
0.373595 + 0.927592i \(0.378125\pi\)
\(198\) 0 0
\(199\) −768.000 −0.273578 −0.136789 − 0.990600i \(-0.543678\pi\)
−0.136789 + 0.990600i \(0.543678\pi\)
\(200\) 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 \(-0.324099\pi\)
0.524909 + 0.851158i \(0.324099\pi\)
\(224\) 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 \(-0.407684\pi\)
0.285970 + 0.958239i \(0.407684\pi\)
\(230\) 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 \(-0.374996\pi\)
0.382695 + 0.923875i \(0.374996\pi\)
\(240\) 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 \(-0.658575\pi\)
−0.477827 + 0.878454i \(0.658575\pi\)
\(264\) 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 \(-0.265028\pi\)
0.672948 + 0.739689i \(0.265028\pi\)
\(270\) 0 0
\(271\) −592.000 −0.132699 −0.0663495 − 0.997796i \(-0.521135\pi\)
−0.0663495 + 0.997796i \(0.521135\pi\)
\(272\) 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 \(-0.692988\pi\)
−0.569824 + 0.821767i \(0.692988\pi\)
\(278\) 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 \(-0.432149\pi\)
0.211550 + 0.977367i \(0.432149\pi\)
\(294\) 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 \(-0.580003\pi\)
−0.248699 + 0.968581i \(0.580003\pi\)
\(312\) 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 \(-0.377613\pi\)
0.375087 + 0.926990i \(0.377613\pi\)
\(318\) 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 \(-0.619626\pi\)
−0.367033 + 0.930208i \(0.619626\pi\)
\(350\) 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 \(-0.810344\pi\)
−0.827688 + 0.561188i \(0.810344\pi\)
\(360\) 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 \(-0.563626\pi\)
−0.198558 + 0.980089i \(0.563626\pi\)
\(368\) 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 \(-0.592266\pi\)
−0.285820 + 0.958283i \(0.592266\pi\)
\(374\) 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 \(-0.642473\pi\)
−0.432795 + 0.901492i \(0.642473\pi\)
\(384\) 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 \(-0.529207\pi\)
−0.0916286 + 0.995793i \(0.529207\pi\)
\(390\) 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 \(-0.701970\pi\)
−0.592781 + 0.805363i \(0.701970\pi\)
\(398\) 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 \(-0.511020\pi\)
−0.0346137 + 0.999401i \(0.511020\pi\)
\(422\) 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 \(-0.566436\pi\)
−0.207202 + 0.978298i \(0.566436\pi\)
\(432\) 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 \(-0.352320\pi\)
0.447485 + 0.894291i \(0.352320\pi\)
\(440\) 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 \(-0.702593\pi\)
−0.594357 + 0.804201i \(0.702593\pi\)
\(462\) 0 0
\(463\) 10240.0 1.02785 0.513923 − 0.857836i \(-0.328192\pi\)
0.513923 + 0.857836i \(0.328192\pi\)
\(464\) 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 \(-0.600014\pi\)
−0.309059 + 0.951043i \(0.600014\pi\)
\(480\) 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 \(-0.563205\pi\)
−0.197262 + 0.980351i \(0.563205\pi\)
\(488\) 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 \(-0.552380\pi\)
−0.163814 + 0.986491i \(0.552380\pi\)
\(504\) 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 \(-0.240087\pi\)
0.728781 + 0.684747i \(0.240087\pi\)
\(510\) 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 \(-0.263208\pi\)
0.677165 + 0.735831i \(0.263208\pi\)
\(542\) 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 \(-0.679289\pi\)
−0.533940 + 0.845522i \(0.679289\pi\)
\(558\) 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 \(-0.528220\pi\)
−0.0885390 + 0.996073i \(0.528220\pi\)
\(600\) 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 \(-0.570124\pi\)
−0.218524 + 0.975832i \(0.570124\pi\)
\(608\) 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 \(-0.549507\pi\)
−0.154904 + 0.987930i \(0.549507\pi\)
\(614\) 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 \(-0.819138\pi\)
−0.842874 + 0.538112i \(0.819138\pi\)
\(632\) 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 \(-0.338593\pi\)
0.485623 + 0.874168i \(0.338593\pi\)
\(648\) 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 \(-0.563504\pi\)
−0.198182 + 0.980165i \(0.563504\pi\)
\(654\) 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 \(-0.470457\pi\)
0.0926784 + 0.995696i \(0.470457\pi\)
\(662\) 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 \(-0.571564\pi\)
−0.222935 + 0.974833i \(0.571564\pi\)
\(678\) 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 \(-0.679230\pi\)
−0.533783 + 0.845621i \(0.679230\pi\)
\(702\) 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 \(-0.360840\pi\)
0.423390 + 0.905948i \(0.360840\pi\)
\(710\) 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 \(-0.697717\pi\)
−0.581969 + 0.813211i \(0.697717\pi\)
\(720\) 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 \(-0.584319\pi\)
−0.261809 + 0.965120i \(0.584319\pi\)
\(728\) 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 \(-0.575136\pi\)
−0.233860 + 0.972270i \(0.575136\pi\)
\(734\) 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 \(-0.718218\pi\)
−0.633100 + 0.774070i \(0.718218\pi\)
\(744\) 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 \(-0.464913\pi\)
0.110006 + 0.993931i \(0.464913\pi\)
\(752\) 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 \(-0.770736\pi\)
−0.751639 + 0.659575i \(0.770736\pi\)
\(758\) 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 \(-0.557226\pi\)
−0.178814 + 0.983883i \(0.557226\pi\)
\(774\) 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 \(-0.388445\pi\)
0.343329 + 0.939215i \(0.388445\pi\)
\(798\) 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 \(-0.669124\pi\)
−0.506670 + 0.862140i \(0.669124\pi\)
\(822\) 0 0
\(823\) −19136.0 −0.810497 −0.405248 − 0.914207i \(-0.632815\pi\)
−0.405248 + 0.914207i \(0.632815\pi\)
\(824\) 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 \(-0.256254\pi\)
0.693079 + 0.720862i \(0.256254\pi\)
\(830\) 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 \(-0.758256\pi\)
−0.725206 + 0.688532i \(0.758256\pi\)
\(840\) 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 \(-0.557307\pi\)
−0.179064 + 0.983837i \(0.557307\pi\)
\(854\) 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 \(-0.354992\pi\)
0.439961 + 0.898017i \(0.354992\pi\)
\(864\) 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 \(-0.264905\pi\)
0.673234 + 0.739429i \(0.264905\pi\)
\(878\) 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 \(-0.300708\pi\)
0.585984 + 0.810323i \(0.300708\pi\)
\(888\) 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 \(-0.651702\pi\)
−0.458749 + 0.888566i \(0.651702\pi\)
\(912\) 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 \(-0.387191\pi\)
0.347027 + 0.937855i \(0.387191\pi\)
\(920\) 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 \(-0.427782\pi\)
0.224937 + 0.974373i \(0.427782\pi\)
\(942\) 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 \(-0.0947517\pi\)
0.956022 + 0.293295i \(0.0947517\pi\)
\(968\) 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 \(-0.693720\pi\)
−0.571710 + 0.820456i \(0.693720\pi\)
\(984\) 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 \(-0.301272\pi\)
0.584547 + 0.811360i \(0.301272\pi\)
\(992\) 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 \(-0.630469\pi\)
−0.398500 + 0.917168i \(0.630469\pi\)
\(998\) 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 1344.4.a.bb.1.1 1
4.3 odd 2 1344.4.a.m.1.1 1
8.3 odd 2 672.4.a.c.1.1 yes 1
8.5 even 2 672.4.a.a.1.1 ✓ 1
24.5 odd 2 2016.4.a.f.1.1 1
24.11 even 2 2016.4.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
672.4.a.a.1.1 ✓ 1 8.5 even 2
672.4.a.c.1.1 yes 1 8.3 odd 2
1344.4.a.m.1.1 1 4.3 odd 2
1344.4.a.bb.1.1 1 1.1 even 1 trivial
2016.4.a.e.1.1 1 24.11 even 2
2016.4.a.f.1.1 1 24.5 odd 2