Properties

Label 4864.2.a.bf.1.2
Level $4864$
Weight $2$
Character 4864.1
Self dual yes
Analytic conductor $38.839$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4864 = 2^{8} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4864.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.8392355432\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.892.1
Defining polynomial: \(x^{3} - x^{2} - 8 x + 10\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2432)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.31955\) of defining polynomial
Character \(\chi\) \(=\) 4864.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.31955 q^{3} -0.319551 q^{5} +1.93923 q^{7} -1.25879 q^{9} +O(q^{10})\) \(q+1.31955 q^{3} -0.319551 q^{5} +1.93923 q^{7} -1.25879 q^{9} -3.25879 q^{11} +4.25879 q^{13} -0.421664 q^{15} +3.00000 q^{17} -1.00000 q^{19} +2.55892 q^{21} -5.61968 q^{23} -4.89789 q^{25} -5.61968 q^{27} -6.55892 q^{29} -6.93923 q^{31} -4.30013 q^{33} -0.619684 q^{35} -7.57834 q^{37} +5.61968 q^{39} -4.30013 q^{41} +7.13726 q^{43} +0.402246 q^{45} -6.31955 q^{47} -3.23937 q^{49} +3.95865 q^{51} -5.31955 q^{53} +1.04135 q^{55} -1.31955 q^{57} -1.74121 q^{59} +2.19802 q^{61} -2.44108 q^{63} -1.36090 q^{65} +8.68045 q^{67} -7.41546 q^{69} +9.15667 q^{71} +15.9136 q^{73} -6.46301 q^{75} -6.31955 q^{77} -6.93923 q^{79} -3.63910 q^{81} +15.1178 q^{83} -0.958652 q^{85} -8.65483 q^{87} -5.69987 q^{89} +8.25879 q^{91} -9.15667 q^{93} +0.319551 q^{95} +17.1178 q^{97} +4.10211 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + q^{3} + 2q^{5} - 3q^{7} + 8q^{9} + O(q^{10}) \) \( 3q + q^{3} + 2q^{5} - 3q^{7} + 8q^{9} + 2q^{11} + q^{13} - 16q^{15} + 9q^{17} - 3q^{19} - 7q^{21} - 11q^{23} + 3q^{25} - 11q^{27} - 5q^{29} - 12q^{31} - 10q^{33} + 4q^{35} - 8q^{37} + 11q^{39} - 10q^{41} - 8q^{43} + 16q^{45} - 16q^{47} + 2q^{49} + 3q^{51} - 13q^{53} + 12q^{55} - q^{57} - 17q^{59} - 14q^{61} - 22q^{63} - 10q^{65} + 29q^{67} + 19q^{69} - 2q^{71} - 17q^{73} - 43q^{75} - 16q^{77} - 12q^{79} - 5q^{81} + 16q^{83} + 6q^{85} + 27q^{87} - 20q^{89} + 13q^{91} + 2q^{93} - 2q^{95} + 22q^{97} + 30q^{99} + O(q^{100}) \)

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
\(3\) 1.31955 0.761843 0.380922 0.924607i \(-0.375607\pi\)
0.380922 + 0.924607i \(0.375607\pi\)
\(4\) 0 0
\(5\) −0.319551 −0.142907 −0.0714537 0.997444i \(-0.522764\pi\)
−0.0714537 + 0.997444i \(0.522764\pi\)
\(6\) 0 0
\(7\) 1.93923 0.732962 0.366481 0.930426i \(-0.380562\pi\)
0.366481 + 0.930426i \(0.380562\pi\)
\(8\) 0 0
\(9\) −1.25879 −0.419595
\(10\) 0 0
\(11\) −3.25879 −0.982561 −0.491280 0.871001i \(-0.663471\pi\)
−0.491280 + 0.871001i \(0.663471\pi\)
\(12\) 0 0
\(13\) 4.25879 1.18117 0.590587 0.806974i \(-0.298896\pi\)
0.590587 + 0.806974i \(0.298896\pi\)
\(14\) 0 0
\(15\) −0.421664 −0.108873
\(16\) 0 0
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 2.55892 0.558402
\(22\) 0 0
\(23\) −5.61968 −1.17179 −0.585893 0.810389i \(-0.699256\pi\)
−0.585893 + 0.810389i \(0.699256\pi\)
\(24\) 0 0
\(25\) −4.89789 −0.979577
\(26\) 0 0
\(27\) −5.61968 −1.08151
\(28\) 0 0
\(29\) −6.55892 −1.21796 −0.608980 0.793185i \(-0.708421\pi\)
−0.608980 + 0.793185i \(0.708421\pi\)
\(30\) 0 0
\(31\) −6.93923 −1.24632 −0.623162 0.782093i \(-0.714152\pi\)
−0.623162 + 0.782093i \(0.714152\pi\)
\(32\) 0 0
\(33\) −4.30013 −0.748557
\(34\) 0 0
\(35\) −0.619684 −0.104746
\(36\) 0 0
\(37\) −7.57834 −1.24587 −0.622935 0.782273i \(-0.714060\pi\)
−0.622935 + 0.782273i \(0.714060\pi\)
\(38\) 0 0
\(39\) 5.61968 0.899870
\(40\) 0 0
\(41\) −4.30013 −0.671568 −0.335784 0.941939i \(-0.609001\pi\)
−0.335784 + 0.941939i \(0.609001\pi\)
\(42\) 0 0
\(43\) 7.13726 1.08842 0.544211 0.838949i \(-0.316829\pi\)
0.544211 + 0.838949i \(0.316829\pi\)
\(44\) 0 0
\(45\) 0.402246 0.0599633
\(46\) 0 0
\(47\) −6.31955 −0.921801 −0.460901 0.887452i \(-0.652474\pi\)
−0.460901 + 0.887452i \(0.652474\pi\)
\(48\) 0 0
\(49\) −3.23937 −0.462767
\(50\) 0 0
\(51\) 3.95865 0.554322
\(52\) 0 0
\(53\) −5.31955 −0.730696 −0.365348 0.930871i \(-0.619050\pi\)
−0.365348 + 0.930871i \(0.619050\pi\)
\(54\) 0 0
\(55\) 1.04135 0.140415
\(56\) 0 0
\(57\) −1.31955 −0.174779
\(58\) 0 0
\(59\) −1.74121 −0.226687 −0.113343 0.993556i \(-0.536156\pi\)
−0.113343 + 0.993556i \(0.536156\pi\)
\(60\) 0 0
\(61\) 2.19802 0.281428 0.140714 0.990050i \(-0.455060\pi\)
0.140714 + 0.990050i \(0.455060\pi\)
\(62\) 0 0
\(63\) −2.44108 −0.307547
\(64\) 0 0
\(65\) −1.36090 −0.168799
\(66\) 0 0
\(67\) 8.68045 1.06049 0.530243 0.847846i \(-0.322101\pi\)
0.530243 + 0.847846i \(0.322101\pi\)
\(68\) 0 0
\(69\) −7.41546 −0.892716
\(70\) 0 0
\(71\) 9.15667 1.08670 0.543349 0.839507i \(-0.317156\pi\)
0.543349 + 0.839507i \(0.317156\pi\)
\(72\) 0 0
\(73\) 15.9136 1.86255 0.931274 0.364320i \(-0.118699\pi\)
0.931274 + 0.364320i \(0.118699\pi\)
\(74\) 0 0
\(75\) −6.46301 −0.746284
\(76\) 0 0
\(77\) −6.31955 −0.720180
\(78\) 0 0
\(79\) −6.93923 −0.780725 −0.390362 0.920661i \(-0.627650\pi\)
−0.390362 + 0.920661i \(0.627650\pi\)
\(80\) 0 0
\(81\) −3.63910 −0.404345
\(82\) 0 0
\(83\) 15.1178 1.65940 0.829699 0.558211i \(-0.188512\pi\)
0.829699 + 0.558211i \(0.188512\pi\)
\(84\) 0 0
\(85\) −0.958652 −0.103980
\(86\) 0 0
\(87\) −8.65483 −0.927895
\(88\) 0 0
\(89\) −5.69987 −0.604185 −0.302092 0.953279i \(-0.597685\pi\)
−0.302092 + 0.953279i \(0.597685\pi\)
\(90\) 0 0
\(91\) 8.25879 0.865756
\(92\) 0 0
\(93\) −9.15667 −0.949503
\(94\) 0 0
\(95\) 0.319551 0.0327852
\(96\) 0 0
\(97\) 17.1178 1.73805 0.869027 0.494765i \(-0.164746\pi\)
0.869027 + 0.494765i \(0.164746\pi\)
\(98\) 0 0
\(99\) 4.10211 0.412278
\(100\) 0 0
\(101\) 7.36090 0.732437 0.366218 0.930529i \(-0.380652\pi\)
0.366218 + 0.930529i \(0.380652\pi\)
\(102\) 0 0
\(103\) −15.1178 −1.48960 −0.744802 0.667285i \(-0.767456\pi\)
−0.744802 + 0.667285i \(0.767456\pi\)
\(104\) 0 0
\(105\) −0.817705 −0.0797998
\(106\) 0 0
\(107\) −12.1373 −1.17335 −0.586676 0.809821i \(-0.699564\pi\)
−0.586676 + 0.809821i \(0.699564\pi\)
\(108\) 0 0
\(109\) 9.07649 0.869370 0.434685 0.900583i \(-0.356860\pi\)
0.434685 + 0.900583i \(0.356860\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) 0 0
\(113\) −4.81770 −0.453212 −0.226606 0.973987i \(-0.572763\pi\)
−0.226606 + 0.973987i \(0.572763\pi\)
\(114\) 0 0
\(115\) 1.79577 0.167457
\(116\) 0 0
\(117\) −5.36090 −0.495615
\(118\) 0 0
\(119\) 5.81770 0.533308
\(120\) 0 0
\(121\) −0.380316 −0.0345742
\(122\) 0 0
\(123\) −5.67424 −0.511629
\(124\) 0 0
\(125\) 3.16288 0.282896
\(126\) 0 0
\(127\) −18.7350 −1.66246 −0.831232 0.555926i \(-0.812364\pi\)
−0.831232 + 0.555926i \(0.812364\pi\)
\(128\) 0 0
\(129\) 9.41797 0.829206
\(130\) 0 0
\(131\) −12.4155 −1.08474 −0.542372 0.840139i \(-0.682474\pi\)
−0.542372 + 0.840139i \(0.682474\pi\)
\(132\) 0 0
\(133\) −1.93923 −0.168153
\(134\) 0 0
\(135\) 1.79577 0.154556
\(136\) 0 0
\(137\) 12.2782 1.04900 0.524499 0.851411i \(-0.324253\pi\)
0.524499 + 0.851411i \(0.324253\pi\)
\(138\) 0 0
\(139\) −13.2588 −1.12460 −0.562298 0.826935i \(-0.690083\pi\)
−0.562298 + 0.826935i \(0.690083\pi\)
\(140\) 0 0
\(141\) −8.33897 −0.702268
\(142\) 0 0
\(143\) −13.8785 −1.16058
\(144\) 0 0
\(145\) 2.09591 0.174056
\(146\) 0 0
\(147\) −4.27451 −0.352556
\(148\) 0 0
\(149\) −7.55892 −0.619251 −0.309625 0.950859i \(-0.600204\pi\)
−0.309625 + 0.950859i \(0.600204\pi\)
\(150\) 0 0
\(151\) 0.396041 0.0322294 0.0161147 0.999870i \(-0.494870\pi\)
0.0161147 + 0.999870i \(0.494870\pi\)
\(152\) 0 0
\(153\) −3.77636 −0.305300
\(154\) 0 0
\(155\) 2.21744 0.178109
\(156\) 0 0
\(157\) −11.0351 −0.880700 −0.440350 0.897826i \(-0.645146\pi\)
−0.440350 + 0.897826i \(0.645146\pi\)
\(158\) 0 0
\(159\) −7.01942 −0.556676
\(160\) 0 0
\(161\) −10.8979 −0.858874
\(162\) 0 0
\(163\) 3.03514 0.237731 0.118865 0.992910i \(-0.462074\pi\)
0.118865 + 0.992910i \(0.462074\pi\)
\(164\) 0 0
\(165\) 1.37411 0.106974
\(166\) 0 0
\(167\) −9.97438 −0.771841 −0.385920 0.922532i \(-0.626116\pi\)
−0.385920 + 0.922532i \(0.626116\pi\)
\(168\) 0 0
\(169\) 5.13726 0.395173
\(170\) 0 0
\(171\) 1.25879 0.0962617
\(172\) 0 0
\(173\) −1.75694 −0.133578 −0.0667888 0.997767i \(-0.521275\pi\)
−0.0667888 + 0.997767i \(0.521275\pi\)
\(174\) 0 0
\(175\) −9.49815 −0.717993
\(176\) 0 0
\(177\) −2.29762 −0.172700
\(178\) 0 0
\(179\) 4.30013 0.321407 0.160704 0.987003i \(-0.448624\pi\)
0.160704 + 0.987003i \(0.448624\pi\)
\(180\) 0 0
\(181\) 14.4349 1.07294 0.536468 0.843921i \(-0.319758\pi\)
0.536468 + 0.843921i \(0.319758\pi\)
\(182\) 0 0
\(183\) 2.90040 0.214404
\(184\) 0 0
\(185\) 2.42166 0.178044
\(186\) 0 0
\(187\) −9.77636 −0.714918
\(188\) 0 0
\(189\) −10.8979 −0.792705
\(190\) 0 0
\(191\) −23.7350 −1.71741 −0.858703 0.512474i \(-0.828729\pi\)
−0.858703 + 0.512474i \(0.828729\pi\)
\(192\) 0 0
\(193\) −20.6962 −1.48974 −0.744872 0.667208i \(-0.767489\pi\)
−0.744872 + 0.667208i \(0.767489\pi\)
\(194\) 0 0
\(195\) −1.79577 −0.128598
\(196\) 0 0
\(197\) 15.7569 1.12264 0.561318 0.827600i \(-0.310295\pi\)
0.561318 + 0.827600i \(0.310295\pi\)
\(198\) 0 0
\(199\) −9.85654 −0.698712 −0.349356 0.936990i \(-0.613600\pi\)
−0.349356 + 0.936990i \(0.613600\pi\)
\(200\) 0 0
\(201\) 11.4543 0.807924
\(202\) 0 0
\(203\) −12.7193 −0.892719
\(204\) 0 0
\(205\) 1.37411 0.0959721
\(206\) 0 0
\(207\) 7.07398 0.491675
\(208\) 0 0
\(209\) 3.25879 0.225415
\(210\) 0 0
\(211\) 12.1373 0.835563 0.417782 0.908548i \(-0.362808\pi\)
0.417782 + 0.908548i \(0.362808\pi\)
\(212\) 0 0
\(213\) 12.0827 0.827893
\(214\) 0 0
\(215\) −2.28072 −0.155544
\(216\) 0 0
\(217\) −13.4568 −0.913508
\(218\) 0 0
\(219\) 20.9988 1.41897
\(220\) 0 0
\(221\) 12.7764 0.859431
\(222\) 0 0
\(223\) 18.5746 1.24385 0.621925 0.783077i \(-0.286351\pi\)
0.621925 + 0.783077i \(0.286351\pi\)
\(224\) 0 0
\(225\) 6.16539 0.411026
\(226\) 0 0
\(227\) −21.2939 −1.41333 −0.706664 0.707549i \(-0.749801\pi\)
−0.706664 + 0.707549i \(0.749801\pi\)
\(228\) 0 0
\(229\) 11.4762 0.758370 0.379185 0.925321i \(-0.376204\pi\)
0.379185 + 0.925321i \(0.376204\pi\)
\(230\) 0 0
\(231\) −8.33897 −0.548664
\(232\) 0 0
\(233\) −17.6548 −1.15661 −0.578303 0.815822i \(-0.696285\pi\)
−0.578303 + 0.815822i \(0.696285\pi\)
\(234\) 0 0
\(235\) 2.01942 0.131732
\(236\) 0 0
\(237\) −9.15667 −0.594790
\(238\) 0 0
\(239\) −11.5432 −0.746667 −0.373334 0.927697i \(-0.621785\pi\)
−0.373334 + 0.927697i \(0.621785\pi\)
\(240\) 0 0
\(241\) 9.57834 0.616995 0.308497 0.951225i \(-0.400174\pi\)
0.308497 + 0.951225i \(0.400174\pi\)
\(242\) 0 0
\(243\) 12.0571 0.773462
\(244\) 0 0
\(245\) 1.03514 0.0661328
\(246\) 0 0
\(247\) −4.25879 −0.270980
\(248\) 0 0
\(249\) 19.9488 1.26420
\(250\) 0 0
\(251\) 0.223643 0.0141162 0.00705811 0.999975i \(-0.497753\pi\)
0.00705811 + 0.999975i \(0.497753\pi\)
\(252\) 0 0
\(253\) 18.3133 1.15135
\(254\) 0 0
\(255\) −1.26499 −0.0792168
\(256\) 0 0
\(257\) −0.0826952 −0.00515838 −0.00257919 0.999997i \(-0.500821\pi\)
−0.00257919 + 0.999997i \(0.500821\pi\)
\(258\) 0 0
\(259\) −14.6962 −0.913176
\(260\) 0 0
\(261\) 8.25627 0.511050
\(262\) 0 0
\(263\) −1.04135 −0.0642122 −0.0321061 0.999484i \(-0.510221\pi\)
−0.0321061 + 0.999484i \(0.510221\pi\)
\(264\) 0 0
\(265\) 1.69987 0.104422
\(266\) 0 0
\(267\) −7.52126 −0.460294
\(268\) 0 0
\(269\) 13.8528 0.844623 0.422312 0.906451i \(-0.361219\pi\)
0.422312 + 0.906451i \(0.361219\pi\)
\(270\) 0 0
\(271\) 12.2976 0.747027 0.373514 0.927625i \(-0.378153\pi\)
0.373514 + 0.927625i \(0.378153\pi\)
\(272\) 0 0
\(273\) 10.8979 0.659570
\(274\) 0 0
\(275\) 15.9612 0.962494
\(276\) 0 0
\(277\) 11.6804 0.701810 0.350905 0.936411i \(-0.385874\pi\)
0.350905 + 0.936411i \(0.385874\pi\)
\(278\) 0 0
\(279\) 8.73501 0.522951
\(280\) 0 0
\(281\) 1.23937 0.0739345 0.0369673 0.999316i \(-0.488230\pi\)
0.0369673 + 0.999316i \(0.488230\pi\)
\(282\) 0 0
\(283\) 14.6585 0.871359 0.435679 0.900102i \(-0.356508\pi\)
0.435679 + 0.900102i \(0.356508\pi\)
\(284\) 0 0
\(285\) 0.421664 0.0249772
\(286\) 0 0
\(287\) −8.33897 −0.492234
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 22.5879 1.32412
\(292\) 0 0
\(293\) −20.0157 −1.16933 −0.584666 0.811274i \(-0.698774\pi\)
−0.584666 + 0.811274i \(0.698774\pi\)
\(294\) 0 0
\(295\) 0.556406 0.0323952
\(296\) 0 0
\(297\) 18.3133 1.06265
\(298\) 0 0
\(299\) −23.9330 −1.38408
\(300\) 0 0
\(301\) 13.8408 0.797771
\(302\) 0 0
\(303\) 9.71308 0.558002
\(304\) 0 0
\(305\) −0.702379 −0.0402181
\(306\) 0 0
\(307\) −19.4180 −1.10824 −0.554121 0.832436i \(-0.686946\pi\)
−0.554121 + 0.832436i \(0.686946\pi\)
\(308\) 0 0
\(309\) −19.9488 −1.13485
\(310\) 0 0
\(311\) −29.8565 −1.69301 −0.846505 0.532382i \(-0.821297\pi\)
−0.846505 + 0.532382i \(0.821297\pi\)
\(312\) 0 0
\(313\) −17.4543 −0.986575 −0.493288 0.869866i \(-0.664205\pi\)
−0.493288 + 0.869866i \(0.664205\pi\)
\(314\) 0 0
\(315\) 0.780049 0.0439508
\(316\) 0 0
\(317\) 11.5370 0.647982 0.323991 0.946060i \(-0.394975\pi\)
0.323991 + 0.946060i \(0.394975\pi\)
\(318\) 0 0
\(319\) 21.3741 1.19672
\(320\) 0 0
\(321\) −16.0157 −0.893911
\(322\) 0 0
\(323\) −3.00000 −0.166924
\(324\) 0 0
\(325\) −20.8591 −1.15705
\(326\) 0 0
\(327\) 11.9769 0.662324
\(328\) 0 0
\(329\) −12.2551 −0.675645
\(330\) 0 0
\(331\) 11.3196 0.622179 0.311089 0.950381i \(-0.399306\pi\)
0.311089 + 0.950381i \(0.399306\pi\)
\(332\) 0 0
\(333\) 9.53950 0.522761
\(334\) 0 0
\(335\) −2.77384 −0.151551
\(336\) 0 0
\(337\) 4.51757 0.246088 0.123044 0.992401i \(-0.460734\pi\)
0.123044 + 0.992401i \(0.460734\pi\)
\(338\) 0 0
\(339\) −6.35721 −0.345276
\(340\) 0 0
\(341\) 22.6135 1.22459
\(342\) 0 0
\(343\) −19.8565 −1.07215
\(344\) 0 0
\(345\) 2.36962 0.127576
\(346\) 0 0
\(347\) 17.6937 0.949846 0.474923 0.880027i \(-0.342476\pi\)
0.474923 + 0.880027i \(0.342476\pi\)
\(348\) 0 0
\(349\) 11.7193 0.627319 0.313659 0.949536i \(-0.398445\pi\)
0.313659 + 0.949536i \(0.398445\pi\)
\(350\) 0 0
\(351\) −23.9330 −1.27745
\(352\) 0 0
\(353\) −9.37662 −0.499067 −0.249534 0.968366i \(-0.580277\pi\)
−0.249534 + 0.968366i \(0.580277\pi\)
\(354\) 0 0
\(355\) −2.92602 −0.155297
\(356\) 0 0
\(357\) 7.67676 0.406297
\(358\) 0 0
\(359\) −28.4568 −1.50189 −0.750946 0.660363i \(-0.770402\pi\)
−0.750946 + 0.660363i \(0.770402\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −0.501846 −0.0263401
\(364\) 0 0
\(365\) −5.08521 −0.266172
\(366\) 0 0
\(367\) 8.60027 0.448930 0.224465 0.974482i \(-0.427937\pi\)
0.224465 + 0.974482i \(0.427937\pi\)
\(368\) 0 0
\(369\) 5.41295 0.281787
\(370\) 0 0
\(371\) −10.3159 −0.535573
\(372\) 0 0
\(373\) −12.0157 −0.622151 −0.311075 0.950385i \(-0.600689\pi\)
−0.311075 + 0.950385i \(0.600689\pi\)
\(374\) 0 0
\(375\) 4.17358 0.215523
\(376\) 0 0
\(377\) −27.9330 −1.43862
\(378\) 0 0
\(379\) 23.5370 1.20901 0.604507 0.796600i \(-0.293370\pi\)
0.604507 + 0.796600i \(0.293370\pi\)
\(380\) 0 0
\(381\) −24.7218 −1.26654
\(382\) 0 0
\(383\) −12.0571 −0.616088 −0.308044 0.951372i \(-0.599674\pi\)
−0.308044 + 0.951372i \(0.599674\pi\)
\(384\) 0 0
\(385\) 2.01942 0.102919
\(386\) 0 0
\(387\) −8.98427 −0.456696
\(388\) 0 0
\(389\) −17.3986 −0.882142 −0.441071 0.897472i \(-0.645401\pi\)
−0.441071 + 0.897472i \(0.645401\pi\)
\(390\) 0 0
\(391\) −16.8591 −0.852599
\(392\) 0 0
\(393\) −16.3828 −0.826404
\(394\) 0 0
\(395\) 2.21744 0.111571
\(396\) 0 0
\(397\) 14.8371 0.744654 0.372327 0.928102i \(-0.378560\pi\)
0.372327 + 0.928102i \(0.378560\pi\)
\(398\) 0 0
\(399\) −2.55892 −0.128106
\(400\) 0 0
\(401\) −2.81770 −0.140709 −0.0703547 0.997522i \(-0.522413\pi\)
−0.0703547 + 0.997522i \(0.522413\pi\)
\(402\) 0 0
\(403\) −29.5527 −1.47213
\(404\) 0 0
\(405\) 1.16288 0.0577839
\(406\) 0 0
\(407\) 24.6962 1.22414
\(408\) 0 0
\(409\) 3.33528 0.164919 0.0824594 0.996594i \(-0.473723\pi\)
0.0824594 + 0.996594i \(0.473723\pi\)
\(410\) 0 0
\(411\) 16.2017 0.799172
\(412\) 0 0
\(413\) −3.37662 −0.166153
\(414\) 0 0
\(415\) −4.83092 −0.237140
\(416\) 0 0
\(417\) −17.4956 −0.856765
\(418\) 0 0
\(419\) 20.2357 0.988577 0.494289 0.869298i \(-0.335429\pi\)
0.494289 + 0.869298i \(0.335429\pi\)
\(420\) 0 0
\(421\) 5.83712 0.284484 0.142242 0.989832i \(-0.454569\pi\)
0.142242 + 0.989832i \(0.454569\pi\)
\(422\) 0 0
\(423\) 7.95496 0.386783
\(424\) 0 0
\(425\) −14.6937 −0.712747
\(426\) 0 0
\(427\) 4.26248 0.206276
\(428\) 0 0
\(429\) −18.3133 −0.884177
\(430\) 0 0
\(431\) −13.1955 −0.635605 −0.317803 0.948157i \(-0.602945\pi\)
−0.317803 + 0.948157i \(0.602945\pi\)
\(432\) 0 0
\(433\) 16.9781 0.815914 0.407957 0.913001i \(-0.366241\pi\)
0.407957 + 0.913001i \(0.366241\pi\)
\(434\) 0 0
\(435\) 2.76566 0.132603
\(436\) 0 0
\(437\) 5.61968 0.268826
\(438\) 0 0
\(439\) 6.38283 0.304636 0.152318 0.988332i \(-0.451326\pi\)
0.152318 + 0.988332i \(0.451326\pi\)
\(440\) 0 0
\(441\) 4.07767 0.194175
\(442\) 0 0
\(443\) 18.3766 0.873100 0.436550 0.899680i \(-0.356200\pi\)
0.436550 + 0.899680i \(0.356200\pi\)
\(444\) 0 0
\(445\) 1.82140 0.0863425
\(446\) 0 0
\(447\) −9.97438 −0.471772
\(448\) 0 0
\(449\) 31.7569 1.49870 0.749351 0.662173i \(-0.230365\pi\)
0.749351 + 0.662173i \(0.230365\pi\)
\(450\) 0 0
\(451\) 14.0132 0.659856
\(452\) 0 0
\(453\) 0.522596 0.0245537
\(454\) 0 0
\(455\) −2.63910 −0.123723
\(456\) 0 0
\(457\) 30.9173 1.44625 0.723125 0.690717i \(-0.242705\pi\)
0.723125 + 0.690717i \(0.242705\pi\)
\(458\) 0 0
\(459\) −16.8591 −0.786913
\(460\) 0 0
\(461\) 24.2369 1.12882 0.564411 0.825494i \(-0.309103\pi\)
0.564411 + 0.825494i \(0.309103\pi\)
\(462\) 0 0
\(463\) 20.5941 0.957087 0.478544 0.878064i \(-0.341165\pi\)
0.478544 + 0.878064i \(0.341165\pi\)
\(464\) 0 0
\(465\) 2.92602 0.135691
\(466\) 0 0
\(467\) −1.85905 −0.0860267 −0.0430133 0.999074i \(-0.513696\pi\)
−0.0430133 + 0.999074i \(0.513696\pi\)
\(468\) 0 0
\(469\) 16.8334 0.777296
\(470\) 0 0
\(471\) −14.5614 −0.670955
\(472\) 0 0
\(473\) −23.2588 −1.06944
\(474\) 0 0
\(475\) 4.89789 0.224730
\(476\) 0 0
\(477\) 6.69617 0.306597
\(478\) 0 0
\(479\) 6.91361 0.315891 0.157946 0.987448i \(-0.449513\pi\)
0.157946 + 0.987448i \(0.449513\pi\)
\(480\) 0 0
\(481\) −32.2745 −1.47159
\(482\) 0 0
\(483\) −14.3803 −0.654327
\(484\) 0 0
\(485\) −5.47002 −0.248381
\(486\) 0 0
\(487\) −39.5527 −1.79230 −0.896152 0.443747i \(-0.853649\pi\)
−0.896152 + 0.443747i \(0.853649\pi\)
\(488\) 0 0
\(489\) 4.00502 0.181113
\(490\) 0 0
\(491\) −11.7958 −0.532336 −0.266168 0.963927i \(-0.585758\pi\)
−0.266168 + 0.963927i \(0.585758\pi\)
\(492\) 0 0
\(493\) −19.6768 −0.886197
\(494\) 0 0
\(495\) −1.31083 −0.0589176
\(496\) 0 0
\(497\) 17.7569 0.796508
\(498\) 0 0
\(499\) 1.01573 0.0454701 0.0227351 0.999742i \(-0.492763\pi\)
0.0227351 + 0.999742i \(0.492763\pi\)
\(500\) 0 0
\(501\) −13.1617 −0.588021
\(502\) 0 0
\(503\) −43.0897 −1.92127 −0.960637 0.277805i \(-0.910393\pi\)
−0.960637 + 0.277805i \(0.910393\pi\)
\(504\) 0 0
\(505\) −2.35218 −0.104671
\(506\) 0 0
\(507\) 6.77887 0.301060
\(508\) 0 0
\(509\) 35.1881 1.55969 0.779843 0.625975i \(-0.215299\pi\)
0.779843 + 0.625975i \(0.215299\pi\)
\(510\) 0 0
\(511\) 30.8602 1.36518
\(512\) 0 0
\(513\) 5.61968 0.248115
\(514\) 0 0
\(515\) 4.83092 0.212876
\(516\) 0 0
\(517\) 20.5941 0.905726
\(518\) 0 0
\(519\) −2.31837 −0.101765
\(520\) 0 0
\(521\) 14.2174 0.622877 0.311439 0.950266i \(-0.399189\pi\)
0.311439 + 0.950266i \(0.399189\pi\)
\(522\) 0 0
\(523\) −4.38032 −0.191538 −0.0957689 0.995404i \(-0.530531\pi\)
−0.0957689 + 0.995404i \(0.530531\pi\)
\(524\) 0 0
\(525\) −12.5333 −0.546998
\(526\) 0 0
\(527\) −20.8177 −0.906833
\(528\) 0 0
\(529\) 8.58085 0.373080
\(530\) 0 0
\(531\) 2.19182 0.0951167
\(532\) 0 0
\(533\) −18.3133 −0.793239
\(534\) 0 0
\(535\) 3.87847 0.167681
\(536\) 0 0
\(537\) 5.67424 0.244862
\(538\) 0 0
\(539\) 10.5564 0.454697
\(540\) 0 0
\(541\) 14.9198 0.641453 0.320727 0.947172i \(-0.396073\pi\)
0.320727 + 0.947172i \(0.396073\pi\)
\(542\) 0 0
\(543\) 19.0476 0.817409
\(544\) 0 0
\(545\) −2.90040 −0.124239
\(546\) 0 0
\(547\) −46.0703 −1.96982 −0.984912 0.173058i \(-0.944635\pi\)
−0.984912 + 0.173058i \(0.944635\pi\)
\(548\) 0 0
\(549\) −2.76684 −0.118086
\(550\) 0 0
\(551\) 6.55892 0.279419
\(552\) 0 0
\(553\) −13.4568 −0.572242
\(554\) 0 0
\(555\) 3.19551 0.135642
\(556\) 0 0
\(557\) −44.2295 −1.87406 −0.937031 0.349245i \(-0.886438\pi\)
−0.937031 + 0.349245i \(0.886438\pi\)
\(558\) 0 0
\(559\) 30.3960 1.28562
\(560\) 0 0
\(561\) −12.9004 −0.544655
\(562\) 0 0
\(563\) −2.61348 −0.110145 −0.0550725 0.998482i \(-0.517539\pi\)
−0.0550725 + 0.998482i \(0.517539\pi\)
\(564\) 0 0
\(565\) 1.53950 0.0647673
\(566\) 0 0
\(567\) −7.05707 −0.296369
\(568\) 0 0
\(569\) 24.0000 1.00613 0.503066 0.864248i \(-0.332205\pi\)
0.503066 + 0.864248i \(0.332205\pi\)
\(570\) 0 0
\(571\) 33.4312 1.39905 0.699526 0.714607i \(-0.253395\pi\)
0.699526 + 0.714607i \(0.253395\pi\)
\(572\) 0 0
\(573\) −31.3196 −1.30839
\(574\) 0 0
\(575\) 27.5246 1.14785
\(576\) 0 0
\(577\) 12.6742 0.527636 0.263818 0.964573i \(-0.415018\pi\)
0.263818 + 0.964573i \(0.415018\pi\)
\(578\) 0 0
\(579\) −27.3097 −1.13495
\(580\) 0 0
\(581\) 29.3170 1.21628
\(582\) 0 0
\(583\) 17.3353 0.717954
\(584\) 0 0
\(585\) 1.71308 0.0708271
\(586\) 0 0
\(587\) 21.5721 0.890377 0.445189 0.895437i \(-0.353137\pi\)
0.445189 + 0.895437i \(0.353137\pi\)
\(588\) 0 0
\(589\) 6.93923 0.285926
\(590\) 0 0
\(591\) 20.7921 0.855272
\(592\) 0 0
\(593\) −4.76063 −0.195496 −0.0977479 0.995211i \(-0.531164\pi\)
−0.0977479 + 0.995211i \(0.531164\pi\)
\(594\) 0 0
\(595\) −1.85905 −0.0762137
\(596\) 0 0
\(597\) −13.0062 −0.532309
\(598\) 0 0
\(599\) 40.9268 1.67222 0.836112 0.548558i \(-0.184823\pi\)
0.836112 + 0.548558i \(0.184823\pi\)
\(600\) 0 0
\(601\) −25.5139 −1.04073 −0.520366 0.853943i \(-0.674205\pi\)
−0.520366 + 0.853943i \(0.674205\pi\)
\(602\) 0 0
\(603\) −10.9268 −0.444975
\(604\) 0 0
\(605\) 0.121530 0.00494091
\(606\) 0 0
\(607\) 33.1955 1.34736 0.673682 0.739021i \(-0.264712\pi\)
0.673682 + 0.739021i \(0.264712\pi\)
\(608\) 0 0
\(609\) −16.7837 −0.680112
\(610\) 0 0
\(611\) −26.9136 −1.08881
\(612\) 0 0
\(613\) 39.3159 1.58795 0.793976 0.607949i \(-0.208007\pi\)
0.793976 + 0.607949i \(0.208007\pi\)
\(614\) 0 0
\(615\) 1.81321 0.0731157
\(616\) 0 0
\(617\) 2.45932 0.0990084 0.0495042 0.998774i \(-0.484236\pi\)
0.0495042 + 0.998774i \(0.484236\pi\)
\(618\) 0 0
\(619\) −45.5139 −1.82936 −0.914679 0.404182i \(-0.867556\pi\)
−0.914679 + 0.404182i \(0.867556\pi\)
\(620\) 0 0
\(621\) 31.5808 1.26730
\(622\) 0 0
\(623\) −11.0534 −0.442844
\(624\) 0 0
\(625\) 23.4787 0.939149
\(626\) 0 0
\(627\) 4.30013 0.171731
\(628\) 0 0
\(629\) −22.7350 −0.906504
\(630\) 0 0
\(631\) −24.6329 −0.980620 −0.490310 0.871548i \(-0.663116\pi\)
−0.490310 + 0.871548i \(0.663116\pi\)
\(632\) 0 0
\(633\) 16.0157 0.636568
\(634\) 0 0
\(635\) 5.98679 0.237578
\(636\) 0 0
\(637\) −13.7958 −0.546608
\(638\) 0 0
\(639\) −11.5263 −0.455973
\(640\) 0 0
\(641\) 30.2357 1.19424 0.597119 0.802153i \(-0.296312\pi\)
0.597119 + 0.802153i \(0.296312\pi\)
\(642\) 0 0
\(643\) −23.4945 −0.926531 −0.463266 0.886220i \(-0.653322\pi\)
−0.463266 + 0.886220i \(0.653322\pi\)
\(644\) 0 0
\(645\) −3.00952 −0.118500
\(646\) 0 0
\(647\) −31.0959 −1.22251 −0.611253 0.791435i \(-0.709334\pi\)
−0.611253 + 0.791435i \(0.709334\pi\)
\(648\) 0 0
\(649\) 5.67424 0.222734
\(650\) 0 0
\(651\) −17.7569 −0.695949
\(652\) 0 0
\(653\) 21.9550 0.859164 0.429582 0.903028i \(-0.358661\pi\)
0.429582 + 0.903028i \(0.358661\pi\)
\(654\) 0 0
\(655\) 3.96737 0.155018
\(656\) 0 0
\(657\) −20.0318 −0.781516
\(658\) 0 0
\(659\) 32.2976 1.25814 0.629068 0.777350i \(-0.283437\pi\)
0.629068 + 0.777350i \(0.283437\pi\)
\(660\) 0 0
\(661\) −19.5808 −0.761607 −0.380803 0.924656i \(-0.624353\pi\)
−0.380803 + 0.924656i \(0.624353\pi\)
\(662\) 0 0
\(663\) 16.8591 0.654751
\(664\) 0 0
\(665\) 0.619684 0.0240303
\(666\) 0 0
\(667\) 36.8591 1.42719
\(668\) 0 0
\(669\) 24.5102 0.947619
\(670\) 0 0
\(671\) −7.16288 −0.276520
\(672\) 0 0
\(673\) 40.8309 1.57392 0.786958 0.617006i \(-0.211655\pi\)
0.786958 + 0.617006i \(0.211655\pi\)
\(674\) 0 0
\(675\) 27.5246 1.05942
\(676\) 0 0
\(677\) 4.43739 0.170543 0.0852714 0.996358i \(-0.472824\pi\)
0.0852714 + 0.996358i \(0.472824\pi\)
\(678\) 0 0
\(679\) 33.1955 1.27393
\(680\) 0 0
\(681\) −28.0984 −1.07673
\(682\) 0 0
\(683\) 13.8785 0.531045 0.265522 0.964105i \(-0.414456\pi\)
0.265522 + 0.964105i \(0.414456\pi\)
\(684\) 0 0
\(685\) −3.92351 −0.149910
\(686\) 0 0
\(687\) 15.1435 0.577759
\(688\) 0 0
\(689\) −22.6548 −0.863080
\(690\) 0 0
\(691\) −36.4543 −1.38679 −0.693393 0.720559i \(-0.743885\pi\)
−0.693393 + 0.720559i \(0.743885\pi\)
\(692\) 0 0
\(693\) 7.95496 0.302184
\(694\) 0 0
\(695\) 4.23686 0.160713
\(696\) 0 0
\(697\) −12.9004 −0.488637
\(698\) 0 0
\(699\) −23.2964 −0.881152
\(700\) 0 0
\(701\) 3.11784 0.117759 0.0588796 0.998265i \(-0.481247\pi\)
0.0588796 + 0.998265i \(0.481247\pi\)
\(702\) 0 0
\(703\) 7.57834 0.285822
\(704\) 0 0
\(705\) 2.66472 0.100359
\(706\) 0 0
\(707\) 14.2745 0.536848
\(708\) 0 0
\(709\) −24.3133 −0.913107 −0.456553 0.889696i \(-0.650916\pi\)
−0.456553 + 0.889696i \(0.650916\pi\)
\(710\) 0 0
\(711\) 8.73501 0.327588
\(712\) 0 0
\(713\) 38.9963 1.46042
\(714\) 0 0
\(715\) 4.43488 0.165855
\(716\) 0 0
\(717\) −15.2318 −0.568843
\(718\) 0 0
\(719\) −18.0608 −0.673553 −0.336776 0.941585i \(-0.609337\pi\)
−0.336776 + 0.941585i \(0.609337\pi\)
\(720\) 0 0
\(721\) −29.3170 −1.09182
\(722\) 0 0
\(723\) 12.6391 0.470053
\(724\) 0 0
\(725\) 32.1248 1.19309
\(726\) 0 0
\(727\) −27.4531 −1.01818 −0.509090 0.860713i \(-0.670018\pi\)
−0.509090 + 0.860713i \(0.670018\pi\)
\(728\) 0 0
\(729\) 26.8272 0.993601
\(730\) 0 0
\(731\) 21.4118 0.791943
\(732\) 0 0
\(733\) −22.6003 −0.834760 −0.417380 0.908732i \(-0.637052\pi\)
−0.417380 + 0.908732i \(0.637052\pi\)
\(734\) 0 0
\(735\) 1.36592 0.0503828
\(736\) 0 0
\(737\) −28.2877 −1.04199
\(738\) 0 0
\(739\) 6.13356 0.225627 0.112813 0.993616i \(-0.464014\pi\)
0.112813 + 0.993616i \(0.464014\pi\)
\(740\) 0 0
\(741\) −5.61968 −0.206444
\(742\) 0 0
\(743\) 0.817705 0.0299987 0.0149993 0.999888i \(-0.495225\pi\)
0.0149993 + 0.999888i \(0.495225\pi\)
\(744\) 0 0
\(745\) 2.41546 0.0884956
\(746\) 0 0
\(747\) −19.0301 −0.696276
\(748\) 0 0
\(749\) −23.5370 −0.860023
\(750\) 0 0
\(751\) 28.6003 1.04364 0.521819 0.853056i \(-0.325254\pi\)
0.521819 + 0.853056i \(0.325254\pi\)
\(752\) 0 0
\(753\) 0.295108 0.0107543
\(754\) 0 0
\(755\) −0.126555 −0.00460582
\(756\) 0 0
\(757\) −22.9901 −0.835589 −0.417795 0.908541i \(-0.637197\pi\)
−0.417795 + 0.908541i \(0.637197\pi\)
\(758\) 0 0
\(759\) 24.1654 0.877148
\(760\) 0 0
\(761\) 46.6669 1.69167 0.845836 0.533443i \(-0.179102\pi\)
0.845836 + 0.533443i \(0.179102\pi\)
\(762\) 0 0
\(763\) 17.6014 0.637215
\(764\) 0 0
\(765\) 1.20674 0.0436297
\(766\) 0 0
\(767\) −7.41546 −0.267757
\(768\) 0 0
\(769\) −0.674244 −0.0243139 −0.0121569 0.999926i \(-0.503870\pi\)
−0.0121569 + 0.999926i \(0.503870\pi\)
\(770\) 0 0
\(771\) −0.109120 −0.00392988
\(772\) 0 0
\(773\) −51.5684 −1.85479 −0.927394 0.374086i \(-0.877956\pi\)
−0.927394 + 0.374086i \(0.877956\pi\)
\(774\) 0 0
\(775\) 33.9876 1.22087
\(776\) 0 0
\(777\) −19.3923 −0.695697
\(778\) 0 0
\(779\) 4.30013 0.154068
\(780\) 0 0
\(781\) −29.8396 −1.06775
\(782\) 0 0
\(783\) 36.8591 1.31724
\(784\) 0 0
\(785\) 3.52629 0.125859
\(786\) 0 0
\(787\) 40.2902 1.43619 0.718096 0.695944i \(-0.245014\pi\)
0.718096 + 0.695944i \(0.245014\pi\)
\(788\) 0 0
\(789\) −1.37411 −0.0489196
\(790\) 0 0
\(791\) −9.34266 −0.332187
\(792\) 0 0
\(793\) 9.36090 0.332415
\(794\) 0 0
\(795\) 2.24306 0.0795532
\(796\) 0 0
\(797\) 16.0728 0.569328 0.284664 0.958627i \(-0.408118\pi\)
0.284664 + 0.958627i \(0.408118\pi\)
\(798\) 0 0
\(799\) −18.9587 −0.670709
\(800\) 0 0
\(801\) 7.17491 0.253513
\(802\) 0 0
\(803\) −51.8591 −1.83007
\(804\) 0 0
\(805\) 3.48243 0.122739
\(806\) 0 0
\(807\) 18.2795 0.643470
\(808\) 0 0
\(809\) 43.7094 1.53674 0.768370 0.640006i \(-0.221068\pi\)
0.768370 + 0.640006i \(0.221068\pi\)
\(810\) 0 0
\(811\) 6.30265 0.221316 0.110658 0.993859i \(-0.464704\pi\)
0.110658 + 0.993859i \(0.464704\pi\)
\(812\) 0 0
\(813\) 16.2273 0.569117
\(814\) 0 0
\(815\) −0.969882 −0.0339735
\(816\) 0 0
\(817\) −7.13726 −0.249701
\(818\) 0 0
\(819\) −10.3960 −0.363267
\(820\) 0 0
\(821\) 49.1505 1.71536 0.857682 0.514181i \(-0.171904\pi\)
0.857682 + 0.514181i \(0.171904\pi\)
\(822\) 0 0
\(823\) 44.9231 1.56592 0.782961 0.622071i \(-0.213708\pi\)
0.782961 + 0.622071i \(0.213708\pi\)
\(824\) 0 0
\(825\) 21.0616 0.733270
\(826\) 0 0
\(827\) −44.3547 −1.54236 −0.771182 0.636615i \(-0.780334\pi\)
−0.771182 + 0.636615i \(0.780334\pi\)
\(828\) 0 0
\(829\) −33.4857 −1.16301 −0.581504 0.813544i \(-0.697536\pi\)
−0.581504 + 0.813544i \(0.697536\pi\)
\(830\) 0 0
\(831\) 15.4129 0.534669
\(832\) 0 0
\(833\) −9.71810 −0.336712
\(834\) 0 0
\(835\) 3.18732 0.110302
\(836\) 0 0
\(837\) 38.9963 1.34791
\(838\) 0 0
\(839\) −16.9136 −0.583923 −0.291961 0.956430i \(-0.594308\pi\)
−0.291961 + 0.956430i \(0.594308\pi\)
\(840\) 0 0
\(841\) 14.0194 0.483428
\(842\) 0 0
\(843\) 1.63541 0.0563265
\(844\) 0 0
\(845\) −1.64161 −0.0564732
\(846\) 0 0
\(847\) −0.737522 −0.0253416
\(848\) 0 0
\(849\) 19.3427 0.663838
\(850\) 0 0
\(851\) 42.5879 1.45989
\(852\) 0 0
\(853\) −44.1530 −1.51177 −0.755885 0.654705i \(-0.772793\pi\)
−0.755885 + 0.654705i \(0.772793\pi\)
\(854\) 0 0
\(855\) −0.402246 −0.0137565
\(856\) 0 0
\(857\) −46.3887 −1.58461 −0.792303 0.610128i \(-0.791118\pi\)
−0.792303 + 0.610128i \(0.791118\pi\)
\(858\) 0 0
\(859\) 34.3378 1.17159 0.585795 0.810459i \(-0.300782\pi\)
0.585795 + 0.810459i \(0.300782\pi\)
\(860\) 0 0
\(861\) −11.0037 −0.375005
\(862\) 0 0
\(863\) 24.0132 0.817419 0.408710 0.912664i \(-0.365979\pi\)
0.408710 + 0.912664i \(0.365979\pi\)
\(864\) 0 0
\(865\) 0.561431 0.0190892
\(866\) 0 0
\(867\) −10.5564 −0.358514
\(868\) 0 0
\(869\) 22.6135 0.767110
\(870\) 0 0
\(871\) 36.9682 1.25262
\(872\) 0 0
\(873\) −21.5477 −0.729279
\(874\) 0 0
\(875\) 6.13356 0.207352
\(876\) 0 0
\(877\) −44.7325 −1.51051 −0.755255 0.655432i \(-0.772487\pi\)
−0.755255 + 0.655432i \(0.772487\pi\)
\(878\) 0 0
\(879\) −26.4118 −0.890847
\(880\) 0 0
\(881\) −24.6073 −0.829040 −0.414520 0.910040i \(-0.636051\pi\)
−0.414520 + 0.910040i \(0.636051\pi\)
\(882\) 0 0
\(883\) −27.9806 −0.941622 −0.470811 0.882234i \(-0.656039\pi\)
−0.470811 + 0.882234i \(0.656039\pi\)
\(884\) 0 0
\(885\) 0.734207 0.0246801
\(886\) 0 0
\(887\) −46.3060 −1.55480 −0.777401 0.629005i \(-0.783462\pi\)
−0.777401 + 0.629005i \(0.783462\pi\)
\(888\) 0 0
\(889\) −36.3316 −1.21852
\(890\) 0 0
\(891\) 11.8591 0.397293
\(892\) 0 0
\(893\) 6.31955 0.211476
\(894\) 0 0
\(895\) −1.37411 −0.0459315
\(896\) 0 0
\(897\) −31.5808 −1.05445
\(898\) 0 0
\(899\) 45.5139 1.51797
\(900\) 0 0
\(901\) −15.9587 −0.531660
\(902\) 0 0
\(903\) 18.2637 0.607776
\(904\) 0 0
\(905\) −4.61268 −0.153331
\(906\) 0 0
\(907\) −6.99380 −0.232225 −0.116113 0.993236i \(-0.537043\pi\)
−0.116113 + 0.993236i \(0.537043\pi\)
\(908\) 0 0
\(909\) −9.26579 −0.307327
\(910\) 0 0
\(911\) 2.19182 0.0726181 0.0363090 0.999341i \(-0.488440\pi\)
0.0363090 + 0.999341i \(0.488440\pi\)
\(912\) 0 0
\(913\) −49.2658 −1.63046
\(914\) 0 0
\(915\) −0.926825 −0.0306399
\(916\) 0 0
\(917\) −24.0765 −0.795076
\(918\) 0 0
\(919\) −29.3716 −0.968880 −0.484440 0.874825i \(-0.660977\pi\)
−0.484440 + 0.874825i \(0.660977\pi\)
\(920\) 0 0
\(921\) −25.6230 −0.844307
\(922\) 0 0
\(923\) 38.9963 1.28358
\(924\) 0 0
\(925\) 37.1178 1.22043
\(926\) 0 0
\(927\) 19.0301 0.625031
\(928\) 0 0
\(929\) 21.7288 0.712899 0.356449 0.934315i \(-0.383987\pi\)
0.356449 + 0.934315i \(0.383987\pi\)
\(930\) 0 0
\(931\) 3.23937 0.106166
\(932\) 0 0
\(933\) −39.3972 −1.28981
\(934\) 0 0
\(935\) 3.12404 0.102167
\(936\) 0 0
\(937\) −0.917305 −0.0299670 −0.0149835 0.999888i \(-0.504770\pi\)
−0.0149835 + 0.999888i \(0.504770\pi\)
\(938\) 0 0
\(939\) −23.0318 −0.751615
\(940\) 0 0
\(941\) −19.3716 −0.631496 −0.315748 0.948843i \(-0.602255\pi\)
−0.315748 + 0.948843i \(0.602255\pi\)
\(942\) 0 0
\(943\) 24.1654 0.786933
\(944\) 0 0
\(945\) 3.48243 0.113283
\(946\) 0 0
\(947\) 46.7532 1.51928 0.759638 0.650346i \(-0.225376\pi\)
0.759638 + 0.650346i \(0.225376\pi\)
\(948\) 0 0
\(949\) 67.7727 2.19999
\(950\) 0 0
\(951\) 15.2236 0.493660
\(952\) 0 0
\(953\) −13.7313 −0.444801 −0.222400 0.974955i \(-0.571389\pi\)
−0.222400 + 0.974955i \(0.571389\pi\)
\(954\) 0 0
\(955\) 7.58454 0.245430
\(956\) 0 0
\(957\) 28.2042 0.911713
\(958\) 0 0
\(959\) 23.8103 0.768875
\(960\) 0 0
\(961\) 17.1530 0.553322
\(962\) 0 0
\(963\) 15.2782 0.492333
\(964\) 0 0
\(965\) 6.61348 0.212895
\(966\) 0 0
\(967\) 8.47371 0.272496 0.136248 0.990675i \(-0.456496\pi\)
0.136248 + 0.990675i \(0.456496\pi\)
\(968\) 0 0
\(969\) −3.95865 −0.127170
\(970\) 0 0
\(971\) −59.1054 −1.89678 −0.948392 0.317101i \(-0.897291\pi\)
−0.948392 + 0.317101i \(0.897291\pi\)
\(972\) 0 0
\(973\) −25.7119 −0.824286
\(974\) 0 0
\(975\) −27.5246 −0.881492
\(976\) 0 0
\(977\) −38.9268 −1.24538 −0.622690 0.782469i \(-0.713960\pi\)
−0.622690 + 0.782469i \(0.713960\pi\)
\(978\) 0 0
\(979\) 18.5746 0.593648
\(980\) 0 0
\(981\) −11.4254 −0.364784
\(982\) 0 0
\(983\) 51.5089 1.64288 0.821439 0.570297i \(-0.193172\pi\)
0.821439 + 0.570297i \(0.193172\pi\)
\(984\) 0 0
\(985\) −5.03514 −0.160433
\(986\) 0 0
\(987\) −16.1712 −0.514736
\(988\) 0 0
\(989\) −40.1091 −1.27540
\(990\) 0 0
\(991\) 28.9707 0.920284 0.460142 0.887845i \(-0.347798\pi\)
0.460142 + 0.887845i \(0.347798\pi\)
\(992\) 0 0
\(993\) 14.9367 0.474003
\(994\) 0 0
\(995\) 3.14967 0.0998511
\(996\) 0 0
\(997\) −27.9111 −0.883953 −0.441977 0.897027i \(-0.645723\pi\)
−0.441977 + 0.897027i \(0.645723\pi\)
\(998\) 0 0
\(999\) 42.5879 1.34742
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 4864.2.a.bf.1.2 3
4.3 odd 2 4864.2.a.bd.1.2 3
8.3 odd 2 4864.2.a.be.1.2 3
8.5 even 2 4864.2.a.bc.1.2 3
16.3 odd 4 2432.2.c.f.1217.3 6
16.5 even 4 2432.2.c.g.1217.3 yes 6
16.11 odd 4 2432.2.c.f.1217.4 yes 6
16.13 even 4 2432.2.c.g.1217.4 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2432.2.c.f.1217.3 6 16.3 odd 4
2432.2.c.f.1217.4 yes 6 16.11 odd 4
2432.2.c.g.1217.3 yes 6 16.5 even 4
2432.2.c.g.1217.4 yes 6 16.13 even 4
4864.2.a.bc.1.2 3 8.5 even 2
4864.2.a.bd.1.2 3 4.3 odd 2
4864.2.a.be.1.2 3 8.3 odd 2
4864.2.a.bf.1.2 3 1.1 even 1 trivial