Properties

Label 7728.2.a.ce.1.4
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.7083906820\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.15317.1
Defining polynomial: \(x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 483)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.69353\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +3.51256 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +3.51256 q^{5} +1.00000 q^{7} +1.00000 q^{9} +1.74252 q^{11} +6.33805 q^{13} +3.51256 q^{15} -5.94860 q^{17} -1.74252 q^{19} +1.00000 q^{21} -1.00000 q^{23} +7.33805 q^{25} +1.00000 q^{27} -5.68466 q^{29} -7.94860 q^{31} +1.74252 q^{33} +3.51256 q^{35} +1.53644 q^{37} +6.33805 q^{39} +12.1547 q^{41} -6.43605 q^{43} +3.51256 q^{45} -3.59313 q^{47} +1.00000 q^{49} -5.94860 q^{51} +12.9397 q^{53} +6.12071 q^{55} -1.74252 q^{57} +8.69353 q^{59} +8.47618 q^{61} +1.00000 q^{63} +22.2628 q^{65} +4.46971 q^{67} -1.00000 q^{69} +13.4612 q^{71} +4.29761 q^{73} +7.33805 q^{75} +1.74252 q^{77} +7.42071 q^{79} +1.00000 q^{81} +7.75262 q^{83} -20.8948 q^{85} -5.68466 q^{87} +4.18503 q^{89} +6.33805 q^{91} -7.94860 q^{93} -6.12071 q^{95} +5.53644 q^{97} +1.74252 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + 5q^{5} + 4q^{7} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} + 5q^{5} + 4q^{7} + 4q^{9} - q^{11} + 7q^{13} + 5q^{15} + 2q^{17} + q^{19} + 4q^{21} - 4q^{23} + 11q^{25} + 4q^{27} + 2q^{29} - 6q^{31} - q^{33} + 5q^{35} + 16q^{37} + 7q^{39} + 5q^{41} - 9q^{43} + 5q^{45} + 21q^{47} + 4q^{49} + 2q^{51} + 10q^{53} - 17q^{55} + q^{57} + 26q^{59} + 2q^{61} + 4q^{63} + 26q^{65} - 5q^{67} - 4q^{69} + 19q^{71} + 10q^{73} + 11q^{75} - q^{77} + 6q^{79} + 4q^{81} + 2q^{83} - 7q^{85} + 2q^{87} - 17q^{89} + 7q^{91} - 6q^{93} + 17q^{95} + 32q^{97} - q^{99} + O(q^{100}) \)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 3.51256 1.57086 0.785432 0.618949i \(-0.212441\pi\)
0.785432 + 0.618949i \(0.212441\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.74252 0.525390 0.262695 0.964879i \(-0.415389\pi\)
0.262695 + 0.964879i \(0.415389\pi\)
\(12\) 0 0
\(13\) 6.33805 1.75786 0.878930 0.476951i \(-0.158258\pi\)
0.878930 + 0.476951i \(0.158258\pi\)
\(14\) 0 0
\(15\) 3.51256 0.906938
\(16\) 0 0
\(17\) −5.94860 −1.44275 −0.721374 0.692546i \(-0.756489\pi\)
−0.721374 + 0.692546i \(0.756489\pi\)
\(18\) 0 0
\(19\) −1.74252 −0.399762 −0.199881 0.979820i \(-0.564056\pi\)
−0.199881 + 0.979820i \(0.564056\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 7.33805 1.46761
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −5.68466 −1.05561 −0.527807 0.849364i \(-0.676986\pi\)
−0.527807 + 0.849364i \(0.676986\pi\)
\(30\) 0 0
\(31\) −7.94860 −1.42761 −0.713806 0.700344i \(-0.753030\pi\)
−0.713806 + 0.700344i \(0.753030\pi\)
\(32\) 0 0
\(33\) 1.74252 0.303334
\(34\) 0 0
\(35\) 3.51256 0.593730
\(36\) 0 0
\(37\) 1.53644 0.252589 0.126295 0.991993i \(-0.459692\pi\)
0.126295 + 0.991993i \(0.459692\pi\)
\(38\) 0 0
\(39\) 6.33805 1.01490
\(40\) 0 0
\(41\) 12.1547 1.89824 0.949121 0.314910i \(-0.101974\pi\)
0.949121 + 0.314910i \(0.101974\pi\)
\(42\) 0 0
\(43\) −6.43605 −0.981488 −0.490744 0.871304i \(-0.663275\pi\)
−0.490744 + 0.871304i \(0.663275\pi\)
\(44\) 0 0
\(45\) 3.51256 0.523621
\(46\) 0 0
\(47\) −3.59313 −0.524112 −0.262056 0.965053i \(-0.584401\pi\)
−0.262056 + 0.965053i \(0.584401\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −5.94860 −0.832971
\(52\) 0 0
\(53\) 12.9397 1.77741 0.888705 0.458480i \(-0.151606\pi\)
0.888705 + 0.458480i \(0.151606\pi\)
\(54\) 0 0
\(55\) 6.12071 0.825316
\(56\) 0 0
\(57\) −1.74252 −0.230803
\(58\) 0 0
\(59\) 8.69353 1.13180 0.565900 0.824474i \(-0.308529\pi\)
0.565900 + 0.824474i \(0.308529\pi\)
\(60\) 0 0
\(61\) 8.47618 1.08526 0.542632 0.839971i \(-0.317428\pi\)
0.542632 + 0.839971i \(0.317428\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 22.2628 2.76136
\(66\) 0 0
\(67\) 4.46971 0.546062 0.273031 0.962005i \(-0.411974\pi\)
0.273031 + 0.962005i \(0.411974\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 13.4612 1.59755 0.798773 0.601633i \(-0.205483\pi\)
0.798773 + 0.601633i \(0.205483\pi\)
\(72\) 0 0
\(73\) 4.29761 0.502997 0.251498 0.967858i \(-0.419077\pi\)
0.251498 + 0.967858i \(0.419077\pi\)
\(74\) 0 0
\(75\) 7.33805 0.847326
\(76\) 0 0
\(77\) 1.74252 0.198579
\(78\) 0 0
\(79\) 7.42071 0.834896 0.417448 0.908701i \(-0.362925\pi\)
0.417448 + 0.908701i \(0.362925\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 7.75262 0.850960 0.425480 0.904968i \(-0.360105\pi\)
0.425480 + 0.904968i \(0.360105\pi\)
\(84\) 0 0
\(85\) −20.8948 −2.26636
\(86\) 0 0
\(87\) −5.68466 −0.609459
\(88\) 0 0
\(89\) 4.18503 0.443613 0.221806 0.975091i \(-0.428805\pi\)
0.221806 + 0.975091i \(0.428805\pi\)
\(90\) 0 0
\(91\) 6.33805 0.664409
\(92\) 0 0
\(93\) −7.94860 −0.824232
\(94\) 0 0
\(95\) −6.12071 −0.627971
\(96\) 0 0
\(97\) 5.53644 0.562140 0.281070 0.959687i \(-0.409311\pi\)
0.281070 + 0.959687i \(0.409311\pi\)
\(98\) 0 0
\(99\) 1.74252 0.175130
\(100\) 0 0
\(101\) −15.9778 −1.58985 −0.794924 0.606709i \(-0.792490\pi\)
−0.794924 + 0.606709i \(0.792490\pi\)
\(102\) 0 0
\(103\) −8.96725 −0.883569 −0.441785 0.897121i \(-0.645654\pi\)
−0.441785 + 0.897121i \(0.645654\pi\)
\(104\) 0 0
\(105\) 3.51256 0.342790
\(106\) 0 0
\(107\) −20.2353 −1.95622 −0.978108 0.208097i \(-0.933273\pi\)
−0.978108 + 0.208097i \(0.933273\pi\)
\(108\) 0 0
\(109\) −3.31657 −0.317670 −0.158835 0.987305i \(-0.550774\pi\)
−0.158835 + 0.987305i \(0.550774\pi\)
\(110\) 0 0
\(111\) 1.53644 0.145832
\(112\) 0 0
\(113\) 16.1207 1.51651 0.758254 0.651959i \(-0.226053\pi\)
0.758254 + 0.651959i \(0.226053\pi\)
\(114\) 0 0
\(115\) −3.51256 −0.327548
\(116\) 0 0
\(117\) 6.33805 0.585953
\(118\) 0 0
\(119\) −5.94860 −0.545308
\(120\) 0 0
\(121\) −7.96362 −0.723965
\(122\) 0 0
\(123\) 12.1547 1.09595
\(124\) 0 0
\(125\) 8.21255 0.734553
\(126\) 0 0
\(127\) −0.480977 −0.0426798 −0.0213399 0.999772i \(-0.506793\pi\)
−0.0213399 + 0.999772i \(0.506793\pi\)
\(128\) 0 0
\(129\) −6.43605 −0.566662
\(130\) 0 0
\(131\) 6.71741 0.586903 0.293451 0.955974i \(-0.405196\pi\)
0.293451 + 0.955974i \(0.405196\pi\)
\(132\) 0 0
\(133\) −1.74252 −0.151096
\(134\) 0 0
\(135\) 3.51256 0.302313
\(136\) 0 0
\(137\) −2.96362 −0.253199 −0.126600 0.991954i \(-0.540406\pi\)
−0.126600 + 0.991954i \(0.540406\pi\)
\(138\) 0 0
\(139\) 8.80161 0.746543 0.373272 0.927722i \(-0.378236\pi\)
0.373272 + 0.927722i \(0.378236\pi\)
\(140\) 0 0
\(141\) −3.59313 −0.302596
\(142\) 0 0
\(143\) 11.0442 0.923562
\(144\) 0 0
\(145\) −19.9677 −1.65823
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) −11.2441 −0.921155 −0.460577 0.887620i \(-0.652358\pi\)
−0.460577 + 0.887620i \(0.652358\pi\)
\(150\) 0 0
\(151\) −5.01502 −0.408116 −0.204058 0.978959i \(-0.565413\pi\)
−0.204058 + 0.978959i \(0.565413\pi\)
\(152\) 0 0
\(153\) −5.94860 −0.480916
\(154\) 0 0
\(155\) −27.9199 −2.24258
\(156\) 0 0
\(157\) 14.8190 1.18269 0.591344 0.806420i \(-0.298598\pi\)
0.591344 + 0.806420i \(0.298598\pi\)
\(158\) 0 0
\(159\) 12.9397 1.02619
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) −2.11541 −0.165692 −0.0828458 0.996562i \(-0.526401\pi\)
−0.0828458 + 0.996562i \(0.526401\pi\)
\(164\) 0 0
\(165\) 6.12071 0.476496
\(166\) 0 0
\(167\) 14.4186 1.11575 0.557874 0.829926i \(-0.311617\pi\)
0.557874 + 0.829926i \(0.311617\pi\)
\(168\) 0 0
\(169\) 27.1709 2.09007
\(170\) 0 0
\(171\) −1.74252 −0.133254
\(172\) 0 0
\(173\) −9.73969 −0.740495 −0.370247 0.928933i \(-0.620727\pi\)
−0.370247 + 0.928933i \(0.620727\pi\)
\(174\) 0 0
\(175\) 7.33805 0.554705
\(176\) 0 0
\(177\) 8.69353 0.653445
\(178\) 0 0
\(179\) 0.376519 0.0281423 0.0140712 0.999901i \(-0.495521\pi\)
0.0140712 + 0.999901i \(0.495521\pi\)
\(180\) 0 0
\(181\) −22.2580 −1.65442 −0.827211 0.561891i \(-0.810074\pi\)
−0.827211 + 0.561891i \(0.810074\pi\)
\(182\) 0 0
\(183\) 8.47618 0.626577
\(184\) 0 0
\(185\) 5.39683 0.396783
\(186\) 0 0
\(187\) −10.3656 −0.758005
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −2.98498 −0.215986 −0.107993 0.994152i \(-0.534442\pi\)
−0.107993 + 0.994152i \(0.534442\pi\)
\(192\) 0 0
\(193\) 4.98498 0.358827 0.179413 0.983774i \(-0.442580\pi\)
0.179413 + 0.983774i \(0.442580\pi\)
\(194\) 0 0
\(195\) 22.2628 1.59427
\(196\) 0 0
\(197\) 3.41573 0.243361 0.121681 0.992569i \(-0.461172\pi\)
0.121681 + 0.992569i \(0.461172\pi\)
\(198\) 0 0
\(199\) 3.91462 0.277500 0.138750 0.990327i \(-0.455692\pi\)
0.138750 + 0.990327i \(0.455692\pi\)
\(200\) 0 0
\(201\) 4.46971 0.315269
\(202\) 0 0
\(203\) −5.68466 −0.398985
\(204\) 0 0
\(205\) 42.6940 2.98188
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) −3.03638 −0.210031
\(210\) 0 0
\(211\) 1.59430 0.109756 0.0548782 0.998493i \(-0.482523\pi\)
0.0548782 + 0.998493i \(0.482523\pi\)
\(212\) 0 0
\(213\) 13.4612 0.922343
\(214\) 0 0
\(215\) −22.6070 −1.54178
\(216\) 0 0
\(217\) −7.94860 −0.539586
\(218\) 0 0
\(219\) 4.29761 0.290405
\(220\) 0 0
\(221\) −37.7026 −2.53615
\(222\) 0 0
\(223\) −2.88667 −0.193306 −0.0966530 0.995318i \(-0.530814\pi\)
−0.0966530 + 0.995318i \(0.530814\pi\)
\(224\) 0 0
\(225\) 7.33805 0.489204
\(226\) 0 0
\(227\) −2.14324 −0.142252 −0.0711259 0.997467i \(-0.522659\pi\)
−0.0711259 + 0.997467i \(0.522659\pi\)
\(228\) 0 0
\(229\) −14.1308 −0.933790 −0.466895 0.884313i \(-0.654627\pi\)
−0.466895 + 0.884313i \(0.654627\pi\)
\(230\) 0 0
\(231\) 1.74252 0.114649
\(232\) 0 0
\(233\) −2.88187 −0.188798 −0.0943989 0.995534i \(-0.530093\pi\)
−0.0943989 + 0.995534i \(0.530093\pi\)
\(234\) 0 0
\(235\) −12.6211 −0.823309
\(236\) 0 0
\(237\) 7.42071 0.482027
\(238\) 0 0
\(239\) −16.4956 −1.06701 −0.533505 0.845797i \(-0.679125\pi\)
−0.533505 + 0.845797i \(0.679125\pi\)
\(240\) 0 0
\(241\) −18.0017 −1.15959 −0.579795 0.814763i \(-0.696867\pi\)
−0.579795 + 0.814763i \(0.696867\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 3.51256 0.224409
\(246\) 0 0
\(247\) −11.0442 −0.702725
\(248\) 0 0
\(249\) 7.75262 0.491302
\(250\) 0 0
\(251\) −11.4373 −0.721915 −0.360957 0.932582i \(-0.617550\pi\)
−0.360957 + 0.932582i \(0.617550\pi\)
\(252\) 0 0
\(253\) −1.74252 −0.109551
\(254\) 0 0
\(255\) −20.8948 −1.30848
\(256\) 0 0
\(257\) −30.4045 −1.89658 −0.948291 0.317402i \(-0.897190\pi\)
−0.948291 + 0.317402i \(0.897190\pi\)
\(258\) 0 0
\(259\) 1.53644 0.0954697
\(260\) 0 0
\(261\) −5.68466 −0.351872
\(262\) 0 0
\(263\) 18.2226 1.12366 0.561828 0.827254i \(-0.310098\pi\)
0.561828 + 0.827254i \(0.310098\pi\)
\(264\) 0 0
\(265\) 45.4516 2.79207
\(266\) 0 0
\(267\) 4.18503 0.256120
\(268\) 0 0
\(269\) 7.65714 0.466864 0.233432 0.972373i \(-0.425004\pi\)
0.233432 + 0.972373i \(0.425004\pi\)
\(270\) 0 0
\(271\) −10.2628 −0.623419 −0.311710 0.950177i \(-0.600902\pi\)
−0.311710 + 0.950177i \(0.600902\pi\)
\(272\) 0 0
\(273\) 6.33805 0.383596
\(274\) 0 0
\(275\) 12.7867 0.771068
\(276\) 0 0
\(277\) 7.83319 0.470651 0.235326 0.971917i \(-0.424384\pi\)
0.235326 + 0.971917i \(0.424384\pi\)
\(278\) 0 0
\(279\) −7.94860 −0.475870
\(280\) 0 0
\(281\) −2.34420 −0.139843 −0.0699217 0.997552i \(-0.522275\pi\)
−0.0699217 + 0.997552i \(0.522275\pi\)
\(282\) 0 0
\(283\) −0.517476 −0.0307607 −0.0153804 0.999882i \(-0.504896\pi\)
−0.0153804 + 0.999882i \(0.504896\pi\)
\(284\) 0 0
\(285\) −6.12071 −0.362559
\(286\) 0 0
\(287\) 12.1547 0.717468
\(288\) 0 0
\(289\) 18.3859 1.08152
\(290\) 0 0
\(291\) 5.53644 0.324552
\(292\) 0 0
\(293\) −9.15314 −0.534732 −0.267366 0.963595i \(-0.586153\pi\)
−0.267366 + 0.963595i \(0.586153\pi\)
\(294\) 0 0
\(295\) 30.5365 1.77790
\(296\) 0 0
\(297\) 1.74252 0.101111
\(298\) 0 0
\(299\) −6.33805 −0.366539
\(300\) 0 0
\(301\) −6.43605 −0.370968
\(302\) 0 0
\(303\) −15.9778 −0.917900
\(304\) 0 0
\(305\) 29.7730 1.70480
\(306\) 0 0
\(307\) 33.0492 1.88622 0.943108 0.332487i \(-0.107888\pi\)
0.943108 + 0.332487i \(0.107888\pi\)
\(308\) 0 0
\(309\) −8.96725 −0.510129
\(310\) 0 0
\(311\) 28.4591 1.61377 0.806883 0.590711i \(-0.201153\pi\)
0.806883 + 0.590711i \(0.201153\pi\)
\(312\) 0 0
\(313\) 0.0903560 0.00510722 0.00255361 0.999997i \(-0.499187\pi\)
0.00255361 + 0.999997i \(0.499187\pi\)
\(314\) 0 0
\(315\) 3.51256 0.197910
\(316\) 0 0
\(317\) −13.1029 −0.735930 −0.367965 0.929840i \(-0.619945\pi\)
−0.367965 + 0.929840i \(0.619945\pi\)
\(318\) 0 0
\(319\) −9.90564 −0.554609
\(320\) 0 0
\(321\) −20.2353 −1.12942
\(322\) 0 0
\(323\) 10.3656 0.576756
\(324\) 0 0
\(325\) 46.5090 2.57985
\(326\) 0 0
\(327\) −3.31657 −0.183407
\(328\) 0 0
\(329\) −3.59313 −0.198096
\(330\) 0 0
\(331\) −4.04013 −0.222066 −0.111033 0.993817i \(-0.535416\pi\)
−0.111033 + 0.993817i \(0.535416\pi\)
\(332\) 0 0
\(333\) 1.53644 0.0841964
\(334\) 0 0
\(335\) 15.7001 0.857789
\(336\) 0 0
\(337\) −2.24486 −0.122285 −0.0611427 0.998129i \(-0.519474\pi\)
−0.0611427 + 0.998129i \(0.519474\pi\)
\(338\) 0 0
\(339\) 16.1207 0.875557
\(340\) 0 0
\(341\) −13.8506 −0.750053
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −3.51256 −0.189110
\(346\) 0 0
\(347\) −14.4458 −0.775493 −0.387746 0.921766i \(-0.626746\pi\)
−0.387746 + 0.921766i \(0.626746\pi\)
\(348\) 0 0
\(349\) −29.2438 −1.56539 −0.782693 0.622408i \(-0.786154\pi\)
−0.782693 + 0.622408i \(0.786154\pi\)
\(350\) 0 0
\(351\) 6.33805 0.338300
\(352\) 0 0
\(353\) 10.6595 0.567350 0.283675 0.958920i \(-0.408446\pi\)
0.283675 + 0.958920i \(0.408446\pi\)
\(354\) 0 0
\(355\) 47.2831 2.50953
\(356\) 0 0
\(357\) −5.94860 −0.314833
\(358\) 0 0
\(359\) 5.44460 0.287355 0.143677 0.989625i \(-0.454107\pi\)
0.143677 + 0.989625i \(0.454107\pi\)
\(360\) 0 0
\(361\) −15.9636 −0.840190
\(362\) 0 0
\(363\) −7.96362 −0.417982
\(364\) 0 0
\(365\) 15.0956 0.790139
\(366\) 0 0
\(367\) −11.4640 −0.598416 −0.299208 0.954188i \(-0.596722\pi\)
−0.299208 + 0.954188i \(0.596722\pi\)
\(368\) 0 0
\(369\) 12.1547 0.632748
\(370\) 0 0
\(371\) 12.9397 0.671798
\(372\) 0 0
\(373\) −27.9859 −1.44905 −0.724527 0.689246i \(-0.757942\pi\)
−0.724527 + 0.689246i \(0.757942\pi\)
\(374\) 0 0
\(375\) 8.21255 0.424094
\(376\) 0 0
\(377\) −36.0297 −1.85562
\(378\) 0 0
\(379\) −11.1604 −0.573271 −0.286636 0.958040i \(-0.592537\pi\)
−0.286636 + 0.958040i \(0.592537\pi\)
\(380\) 0 0
\(381\) −0.480977 −0.0246412
\(382\) 0 0
\(383\) 17.2677 0.882338 0.441169 0.897424i \(-0.354564\pi\)
0.441169 + 0.897424i \(0.354564\pi\)
\(384\) 0 0
\(385\) 6.12071 0.311940
\(386\) 0 0
\(387\) −6.43605 −0.327163
\(388\) 0 0
\(389\) 27.8158 1.41032 0.705158 0.709050i \(-0.250876\pi\)
0.705158 + 0.709050i \(0.250876\pi\)
\(390\) 0 0
\(391\) 5.94860 0.300834
\(392\) 0 0
\(393\) 6.71741 0.338848
\(394\) 0 0
\(395\) 26.0657 1.31151
\(396\) 0 0
\(397\) −36.4276 −1.82825 −0.914123 0.405436i \(-0.867120\pi\)
−0.914123 + 0.405436i \(0.867120\pi\)
\(398\) 0 0
\(399\) −1.74252 −0.0872352
\(400\) 0 0
\(401\) −1.41605 −0.0707142 −0.0353571 0.999375i \(-0.511257\pi\)
−0.0353571 + 0.999375i \(0.511257\pi\)
\(402\) 0 0
\(403\) −50.3787 −2.50954
\(404\) 0 0
\(405\) 3.51256 0.174540
\(406\) 0 0
\(407\) 2.67728 0.132708
\(408\) 0 0
\(409\) −0.339704 −0.0167973 −0.00839863 0.999965i \(-0.502673\pi\)
−0.00839863 + 0.999965i \(0.502673\pi\)
\(410\) 0 0
\(411\) −2.96362 −0.146185
\(412\) 0 0
\(413\) 8.69353 0.427780
\(414\) 0 0
\(415\) 27.2315 1.33674
\(416\) 0 0
\(417\) 8.80161 0.431017
\(418\) 0 0
\(419\) −36.7313 −1.79444 −0.897221 0.441582i \(-0.854418\pi\)
−0.897221 + 0.441582i \(0.854418\pi\)
\(420\) 0 0
\(421\) 19.6701 0.958661 0.479330 0.877635i \(-0.340880\pi\)
0.479330 + 0.877635i \(0.340880\pi\)
\(422\) 0 0
\(423\) −3.59313 −0.174704
\(424\) 0 0
\(425\) −43.6512 −2.11739
\(426\) 0 0
\(427\) 8.47618 0.410191
\(428\) 0 0
\(429\) 11.0442 0.533219
\(430\) 0 0
\(431\) 2.51827 0.121301 0.0606504 0.998159i \(-0.480683\pi\)
0.0606504 + 0.998159i \(0.480683\pi\)
\(432\) 0 0
\(433\) −27.8393 −1.33787 −0.668937 0.743319i \(-0.733250\pi\)
−0.668937 + 0.743319i \(0.733250\pi\)
\(434\) 0 0
\(435\) −19.9677 −0.957377
\(436\) 0 0
\(437\) 1.74252 0.0833561
\(438\) 0 0
\(439\) 28.7859 1.37387 0.686937 0.726717i \(-0.258955\pi\)
0.686937 + 0.726717i \(0.258955\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 12.3920 0.588760 0.294380 0.955688i \(-0.404887\pi\)
0.294380 + 0.955688i \(0.404887\pi\)
\(444\) 0 0
\(445\) 14.7002 0.696855
\(446\) 0 0
\(447\) −11.2441 −0.531829
\(448\) 0 0
\(449\) 0.359419 0.0169620 0.00848101 0.999964i \(-0.497300\pi\)
0.00848101 + 0.999964i \(0.497300\pi\)
\(450\) 0 0
\(451\) 21.1798 0.997318
\(452\) 0 0
\(453\) −5.01502 −0.235626
\(454\) 0 0
\(455\) 22.2628 1.04369
\(456\) 0 0
\(457\) −27.3677 −1.28021 −0.640103 0.768289i \(-0.721108\pi\)
−0.640103 + 0.768289i \(0.721108\pi\)
\(458\) 0 0
\(459\) −5.94860 −0.277657
\(460\) 0 0
\(461\) −7.43230 −0.346157 −0.173078 0.984908i \(-0.555371\pi\)
−0.173078 + 0.984908i \(0.555371\pi\)
\(462\) 0 0
\(463\) −6.45346 −0.299918 −0.149959 0.988692i \(-0.547914\pi\)
−0.149959 + 0.988692i \(0.547914\pi\)
\(464\) 0 0
\(465\) −27.9199 −1.29476
\(466\) 0 0
\(467\) −42.2953 −1.95719 −0.978596 0.205792i \(-0.934023\pi\)
−0.978596 + 0.205792i \(0.934023\pi\)
\(468\) 0 0
\(469\) 4.46971 0.206392
\(470\) 0 0
\(471\) 14.8190 0.682825
\(472\) 0 0
\(473\) −11.2149 −0.515664
\(474\) 0 0
\(475\) −12.7867 −0.586695
\(476\) 0 0
\(477\) 12.9397 0.592470
\(478\) 0 0
\(479\) 21.6417 0.988834 0.494417 0.869225i \(-0.335382\pi\)
0.494417 + 0.869225i \(0.335382\pi\)
\(480\) 0 0
\(481\) 9.73804 0.444016
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) 19.4471 0.883045
\(486\) 0 0
\(487\) −7.21180 −0.326798 −0.163399 0.986560i \(-0.552246\pi\)
−0.163399 + 0.986560i \(0.552246\pi\)
\(488\) 0 0
\(489\) −2.11541 −0.0956621
\(490\) 0 0
\(491\) 7.42387 0.335034 0.167517 0.985869i \(-0.446425\pi\)
0.167517 + 0.985869i \(0.446425\pi\)
\(492\) 0 0
\(493\) 33.8158 1.52299
\(494\) 0 0
\(495\) 6.12071 0.275105
\(496\) 0 0
\(497\) 13.4612 0.603816
\(498\) 0 0
\(499\) 29.8147 1.33469 0.667344 0.744750i \(-0.267431\pi\)
0.667344 + 0.744750i \(0.267431\pi\)
\(500\) 0 0
\(501\) 14.4186 0.644177
\(502\) 0 0
\(503\) −32.1036 −1.43143 −0.715715 0.698393i \(-0.753899\pi\)
−0.715715 + 0.698393i \(0.753899\pi\)
\(504\) 0 0
\(505\) −56.1229 −2.49743
\(506\) 0 0
\(507\) 27.1709 1.20670
\(508\) 0 0
\(509\) 19.4401 0.861668 0.430834 0.902431i \(-0.358219\pi\)
0.430834 + 0.902431i \(0.358219\pi\)
\(510\) 0 0
\(511\) 4.29761 0.190115
\(512\) 0 0
\(513\) −1.74252 −0.0769342
\(514\) 0 0
\(515\) −31.4980 −1.38797
\(516\) 0 0
\(517\) −6.26111 −0.275363
\(518\) 0 0
\(519\) −9.73969 −0.427525
\(520\) 0 0
\(521\) −9.59843 −0.420515 −0.210257 0.977646i \(-0.567430\pi\)
−0.210257 + 0.977646i \(0.567430\pi\)
\(522\) 0 0
\(523\) −14.6384 −0.640094 −0.320047 0.947402i \(-0.603699\pi\)
−0.320047 + 0.947402i \(0.603699\pi\)
\(524\) 0 0
\(525\) 7.33805 0.320259
\(526\) 0 0
\(527\) 47.2831 2.05968
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 8.69353 0.377267
\(532\) 0 0
\(533\) 77.0371 3.33685
\(534\) 0 0
\(535\) −71.0775 −3.07295
\(536\) 0 0
\(537\) 0.376519 0.0162480
\(538\) 0 0
\(539\) 1.74252 0.0750557
\(540\) 0 0
\(541\) 5.49994 0.236461 0.118230 0.992986i \(-0.462278\pi\)
0.118230 + 0.992986i \(0.462278\pi\)
\(542\) 0 0
\(543\) −22.2580 −0.955181
\(544\) 0 0
\(545\) −11.6496 −0.499016
\(546\) 0 0
\(547\) −6.59357 −0.281921 −0.140960 0.990015i \(-0.545019\pi\)
−0.140960 + 0.990015i \(0.545019\pi\)
\(548\) 0 0
\(549\) 8.47618 0.361754
\(550\) 0 0
\(551\) 9.90564 0.421994
\(552\) 0 0
\(553\) 7.42071 0.315561
\(554\) 0 0
\(555\) 5.39683 0.229083
\(556\) 0 0
\(557\) −5.45021 −0.230933 −0.115466 0.993311i \(-0.536836\pi\)
−0.115466 + 0.993311i \(0.536836\pi\)
\(558\) 0 0
\(559\) −40.7920 −1.72532
\(560\) 0 0
\(561\) −10.3656 −0.437635
\(562\) 0 0
\(563\) 19.3154 0.814047 0.407024 0.913418i \(-0.366567\pi\)
0.407024 + 0.913418i \(0.366567\pi\)
\(564\) 0 0
\(565\) 56.6249 2.38223
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −20.2263 −0.847930 −0.423965 0.905679i \(-0.639362\pi\)
−0.423965 + 0.905679i \(0.639362\pi\)
\(570\) 0 0
\(571\) −34.6328 −1.44934 −0.724670 0.689096i \(-0.758008\pi\)
−0.724670 + 0.689096i \(0.758008\pi\)
\(572\) 0 0
\(573\) −2.98498 −0.124699
\(574\) 0 0
\(575\) −7.33805 −0.306018
\(576\) 0 0
\(577\) −36.7645 −1.53053 −0.765263 0.643718i \(-0.777391\pi\)
−0.765263 + 0.643718i \(0.777391\pi\)
\(578\) 0 0
\(579\) 4.98498 0.207169
\(580\) 0 0
\(581\) 7.75262 0.321633
\(582\) 0 0
\(583\) 22.5478 0.933833
\(584\) 0 0
\(585\) 22.2628 0.920452
\(586\) 0 0
\(587\) −4.37818 −0.180707 −0.0903535 0.995910i \(-0.528800\pi\)
−0.0903535 + 0.995910i \(0.528800\pi\)
\(588\) 0 0
\(589\) 13.8506 0.570704
\(590\) 0 0
\(591\) 3.41573 0.140505
\(592\) 0 0
\(593\) −23.5701 −0.967908 −0.483954 0.875093i \(-0.660800\pi\)
−0.483954 + 0.875093i \(0.660800\pi\)
\(594\) 0 0
\(595\) −20.8948 −0.856604
\(596\) 0 0
\(597\) 3.91462 0.160215
\(598\) 0 0
\(599\) 22.1721 0.905928 0.452964 0.891529i \(-0.350367\pi\)
0.452964 + 0.891529i \(0.350367\pi\)
\(600\) 0 0
\(601\) 34.7190 1.41622 0.708109 0.706103i \(-0.249548\pi\)
0.708109 + 0.706103i \(0.249548\pi\)
\(602\) 0 0
\(603\) 4.46971 0.182021
\(604\) 0 0
\(605\) −27.9727 −1.13725
\(606\) 0 0
\(607\) 5.60938 0.227678 0.113839 0.993499i \(-0.463685\pi\)
0.113839 + 0.993499i \(0.463685\pi\)
\(608\) 0 0
\(609\) −5.68466 −0.230354
\(610\) 0 0
\(611\) −22.7735 −0.921316
\(612\) 0 0
\(613\) 14.1194 0.570275 0.285138 0.958487i \(-0.407961\pi\)
0.285138 + 0.958487i \(0.407961\pi\)
\(614\) 0 0
\(615\) 42.6940 1.72159
\(616\) 0 0
\(617\) 37.0094 1.48994 0.744970 0.667098i \(-0.232464\pi\)
0.744970 + 0.667098i \(0.232464\pi\)
\(618\) 0 0
\(619\) −42.2855 −1.69960 −0.849799 0.527107i \(-0.823277\pi\)
−0.849799 + 0.527107i \(0.823277\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 4.18503 0.167670
\(624\) 0 0
\(625\) −7.84323 −0.313729
\(626\) 0 0
\(627\) −3.03638 −0.121261
\(628\) 0 0
\(629\) −9.13967 −0.364422
\(630\) 0 0
\(631\) −31.1770 −1.24114 −0.620568 0.784153i \(-0.713098\pi\)
−0.620568 + 0.784153i \(0.713098\pi\)
\(632\) 0 0
\(633\) 1.59430 0.0633678
\(634\) 0 0
\(635\) −1.68946 −0.0670442
\(636\) 0 0
\(637\) 6.33805 0.251123
\(638\) 0 0
\(639\) 13.4612 0.532515
\(640\) 0 0
\(641\) 35.9135 1.41850 0.709248 0.704959i \(-0.249035\pi\)
0.709248 + 0.704959i \(0.249035\pi\)
\(642\) 0 0
\(643\) 23.3808 0.922047 0.461024 0.887388i \(-0.347482\pi\)
0.461024 + 0.887388i \(0.347482\pi\)
\(644\) 0 0
\(645\) −22.6070 −0.890149
\(646\) 0 0
\(647\) 3.09559 0.121700 0.0608501 0.998147i \(-0.480619\pi\)
0.0608501 + 0.998147i \(0.480619\pi\)
\(648\) 0 0
\(649\) 15.1487 0.594637
\(650\) 0 0
\(651\) −7.94860 −0.311530
\(652\) 0 0
\(653\) 31.9927 1.25197 0.625985 0.779835i \(-0.284697\pi\)
0.625985 + 0.779835i \(0.284697\pi\)
\(654\) 0 0
\(655\) 23.5953 0.921944
\(656\) 0 0
\(657\) 4.29761 0.167666
\(658\) 0 0
\(659\) 4.45396 0.173502 0.0867508 0.996230i \(-0.472352\pi\)
0.0867508 + 0.996230i \(0.472352\pi\)
\(660\) 0 0
\(661\) −18.8131 −0.731743 −0.365872 0.930665i \(-0.619229\pi\)
−0.365872 + 0.930665i \(0.619229\pi\)
\(662\) 0 0
\(663\) −37.7026 −1.46425
\(664\) 0 0
\(665\) −6.12071 −0.237351
\(666\) 0 0
\(667\) 5.68466 0.220111
\(668\) 0 0
\(669\) −2.88667 −0.111605
\(670\) 0 0
\(671\) 14.7699 0.570186
\(672\) 0 0
\(673\) −34.6850 −1.33701 −0.668505 0.743708i \(-0.733065\pi\)
−0.668505 + 0.743708i \(0.733065\pi\)
\(674\) 0 0
\(675\) 7.33805 0.282442
\(676\) 0 0
\(677\) 12.9841 0.499021 0.249510 0.968372i \(-0.419730\pi\)
0.249510 + 0.968372i \(0.419730\pi\)
\(678\) 0 0
\(679\) 5.53644 0.212469
\(680\) 0 0
\(681\) −2.14324 −0.0821291
\(682\) 0 0
\(683\) −8.41814 −0.322111 −0.161055 0.986945i \(-0.551490\pi\)
−0.161055 + 0.986945i \(0.551490\pi\)
\(684\) 0 0
\(685\) −10.4099 −0.397741
\(686\) 0 0
\(687\) −14.1308 −0.539124
\(688\) 0 0
\(689\) 82.0128 3.12444
\(690\) 0 0
\(691\) −37.8312 −1.43917 −0.719584 0.694406i \(-0.755667\pi\)
−0.719584 + 0.694406i \(0.755667\pi\)
\(692\) 0 0
\(693\) 1.74252 0.0661929
\(694\) 0 0
\(695\) 30.9162 1.17272
\(696\) 0 0
\(697\) −72.3034 −2.73869
\(698\) 0 0
\(699\) −2.88187 −0.109002
\(700\) 0 0
\(701\) 7.93899 0.299851 0.149926 0.988697i \(-0.452097\pi\)
0.149926 + 0.988697i \(0.452097\pi\)
\(702\) 0 0
\(703\) −2.67728 −0.100975
\(704\) 0 0
\(705\) −12.6211 −0.475337
\(706\) 0 0
\(707\) −15.9778 −0.600906
\(708\) 0 0
\(709\) 1.33730 0.0502235 0.0251117 0.999685i \(-0.492006\pi\)
0.0251117 + 0.999685i \(0.492006\pi\)
\(710\) 0 0
\(711\) 7.42071 0.278299
\(712\) 0 0
\(713\) 7.94860 0.297678
\(714\) 0 0
\(715\) 38.7934 1.45079
\(716\) 0 0
\(717\) −16.4956 −0.616039
\(718\) 0 0
\(719\) −43.6532 −1.62799 −0.813994 0.580873i \(-0.802711\pi\)
−0.813994 + 0.580873i \(0.802711\pi\)
\(720\) 0 0
\(721\) −8.96725 −0.333958
\(722\) 0 0
\(723\) −18.0017 −0.669489
\(724\) 0 0
\(725\) −41.7143 −1.54923
\(726\) 0 0
\(727\) −13.1675 −0.488356 −0.244178 0.969730i \(-0.578518\pi\)
−0.244178 + 0.969730i \(0.578518\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 38.2855 1.41604
\(732\) 0 0
\(733\) −34.4447 −1.27224 −0.636121 0.771589i \(-0.719462\pi\)
−0.636121 + 0.771589i \(0.719462\pi\)
\(734\) 0 0
\(735\) 3.51256 0.129563
\(736\) 0 0
\(737\) 7.78856 0.286895
\(738\) 0 0
\(739\) 7.05398 0.259485 0.129742 0.991548i \(-0.458585\pi\)
0.129742 + 0.991548i \(0.458585\pi\)
\(740\) 0 0
\(741\) −11.0442 −0.405719
\(742\) 0 0
\(743\) 6.49211 0.238172 0.119086 0.992884i \(-0.462004\pi\)
0.119086 + 0.992884i \(0.462004\pi\)
\(744\) 0 0
\(745\) −39.4956 −1.44701
\(746\) 0 0
\(747\) 7.75262 0.283653
\(748\) 0 0
\(749\) −20.2353 −0.739380
\(750\) 0 0
\(751\) −2.31669 −0.0845372 −0.0422686 0.999106i \(-0.513459\pi\)
−0.0422686 + 0.999106i \(0.513459\pi\)
\(752\) 0 0
\(753\) −11.4373 −0.416798
\(754\) 0 0
\(755\) −17.6155 −0.641095
\(756\) 0 0
\(757\) −10.8685 −0.395021 −0.197510 0.980301i \(-0.563286\pi\)
−0.197510 + 0.980301i \(0.563286\pi\)
\(758\) 0 0
\(759\) −1.74252 −0.0632495
\(760\) 0 0
\(761\) −14.1760 −0.513881 −0.256941 0.966427i \(-0.582715\pi\)
−0.256941 + 0.966427i \(0.582715\pi\)
\(762\) 0 0
\(763\) −3.31657 −0.120068
\(764\) 0 0
\(765\) −20.8948 −0.755453
\(766\) 0 0
\(767\) 55.1000 1.98955
\(768\) 0 0
\(769\) 11.0939 0.400058 0.200029 0.979790i \(-0.435896\pi\)
0.200029 + 0.979790i \(0.435896\pi\)
\(770\) 0 0
\(771\) −30.4045 −1.09499
\(772\) 0 0
\(773\) 26.3956 0.949384 0.474692 0.880152i \(-0.342559\pi\)
0.474692 + 0.880152i \(0.342559\pi\)
\(774\) 0 0
\(775\) −58.3273 −2.09518
\(776\) 0 0
\(777\) 1.53644 0.0551195
\(778\) 0 0
\(779\) −21.1798 −0.758845
\(780\) 0 0
\(781\) 23.4564 0.839335
\(782\) 0 0
\(783\) −5.68466 −0.203153
\(784\) 0 0
\(785\) 52.0527 1.85784
\(786\) 0 0
\(787\) −21.0254 −0.749476 −0.374738 0.927131i \(-0.622267\pi\)
−0.374738 + 0.927131i \(0.622267\pi\)
\(788\) 0 0
\(789\) 18.2226 0.648743
\(790\) 0 0
\(791\) 16.1207 0.573186
\(792\) 0 0
\(793\) 53.7225 1.90774
\(794\) 0 0
\(795\) 45.4516 1.61200
\(796\) 0 0
\(797\) −3.60323 −0.127633 −0.0638165 0.997962i \(-0.520327\pi\)
−0.0638165 + 0.997962i \(0.520327\pi\)
\(798\) 0 0
\(799\) 21.3741 0.756162
\(800\) 0 0
\(801\) 4.18503 0.147871
\(802\) 0 0
\(803\) 7.48867 0.264270
\(804\) 0 0
\(805\) −3.51256 −0.123801
\(806\) 0 0
\(807\) 7.65714 0.269544
\(808\) 0 0
\(809\) −46.2734 −1.62689 −0.813443 0.581644i \(-0.802410\pi\)
−0.813443 + 0.581644i \(0.802410\pi\)
\(810\) 0 0
\(811\) −23.3657 −0.820480 −0.410240 0.911978i \(-0.634555\pi\)
−0.410240 + 0.911978i \(0.634555\pi\)
\(812\) 0 0
\(813\) −10.2628 −0.359931
\(814\) 0 0
\(815\) −7.43049 −0.260279
\(816\) 0 0
\(817\) 11.2149 0.392361
\(818\) 0 0
\(819\) 6.33805 0.221470
\(820\) 0 0
\(821\) −29.6333 −1.03421 −0.517104 0.855923i \(-0.672990\pi\)
−0.517104 + 0.855923i \(0.672990\pi\)
\(822\) 0 0
\(823\) −3.93716 −0.137241 −0.0686203 0.997643i \(-0.521860\pi\)
−0.0686203 + 0.997643i \(0.521860\pi\)
\(824\) 0 0
\(825\) 12.7867 0.445176
\(826\) 0 0
\(827\) 21.6408 0.752526 0.376263 0.926513i \(-0.377209\pi\)
0.376263 + 0.926513i \(0.377209\pi\)
\(828\) 0 0
\(829\) 20.5616 0.714132 0.357066 0.934079i \(-0.383777\pi\)
0.357066 + 0.934079i \(0.383777\pi\)
\(830\) 0 0
\(831\) 7.83319 0.271730
\(832\) 0 0
\(833\) −5.94860 −0.206107
\(834\) 0 0
\(835\) 50.6463 1.75269
\(836\) 0 0
\(837\) −7.94860 −0.274744
\(838\) 0 0
\(839\) −14.3203 −0.494392 −0.247196 0.968965i \(-0.579509\pi\)
−0.247196 + 0.968965i \(0.579509\pi\)
\(840\) 0 0
\(841\) 3.31534 0.114322
\(842\) 0 0
\(843\) −2.34420 −0.0807387
\(844\) 0 0
\(845\) 95.4394 3.28322
\(846\) 0 0
\(847\) −7.96362 −0.273633
\(848\) 0 0
\(849\) −0.517476 −0.0177597
\(850\) 0 0
\(851\) −1.53644 −0.0526685
\(852\) 0 0
\(853\) 6.04210 0.206877 0.103439 0.994636i \(-0.467015\pi\)
0.103439 + 0.994636i \(0.467015\pi\)
\(854\) 0 0
\(855\) −6.12071 −0.209324
\(856\) 0 0
\(857\) 7.86521 0.268670 0.134335 0.990936i \(-0.457110\pi\)
0.134335 + 0.990936i \(0.457110\pi\)
\(858\) 0 0
\(859\) 54.5758 1.86210 0.931051 0.364890i \(-0.118893\pi\)
0.931051 + 0.364890i \(0.118893\pi\)
\(860\) 0 0
\(861\) 12.1547 0.414231
\(862\) 0 0
\(863\) 41.3174 1.40646 0.703230 0.710962i \(-0.251740\pi\)
0.703230 + 0.710962i \(0.251740\pi\)
\(864\) 0 0
\(865\) −34.2112 −1.16322
\(866\) 0 0
\(867\) 18.3859 0.624417
\(868\) 0 0
\(869\) 12.9308 0.438646
\(870\) 0 0
\(871\) 28.3293 0.959900
\(872\) 0 0
\(873\) 5.53644 0.187380
\(874\) 0 0
\(875\) 8.21255 0.277635
\(876\) 0 0
\(877\) 22.5939 0.762941 0.381471 0.924381i \(-0.375418\pi\)
0.381471 + 0.924381i \(0.375418\pi\)
\(878\) 0 0
\(879\) −9.15314 −0.308728
\(880\) 0 0
\(881\) −32.4203 −1.09227 −0.546134 0.837698i \(-0.683901\pi\)
−0.546134 + 0.837698i \(0.683901\pi\)
\(882\) 0 0
\(883\) −46.9261 −1.57919 −0.789595 0.613628i \(-0.789709\pi\)
−0.789595 + 0.613628i \(0.789709\pi\)
\(884\) 0 0
\(885\) 30.5365 1.02647
\(886\) 0 0
\(887\) −37.9121 −1.27296 −0.636482 0.771291i \(-0.719611\pi\)
−0.636482 + 0.771291i \(0.719611\pi\)
\(888\) 0 0
\(889\) −0.480977 −0.0161315
\(890\) 0 0
\(891\) 1.74252 0.0583767
\(892\) 0 0
\(893\) 6.26111 0.209520
\(894\) 0 0
\(895\) 1.32254 0.0442078
\(896\) 0 0
\(897\) −6.33805 −0.211621
\(898\) 0 0
\(899\) 45.1851 1.50701
\(900\) 0 0
\(901\) −76.9734 −2.56435
\(902\) 0 0
\(903\) −6.43605 −0.214178
\(904\) 0 0
\(905\) −78.1824 −2.59887
\(906\) 0 0
\(907\) 44.4382 1.47555 0.737773 0.675049i \(-0.235877\pi\)
0.737773 + 0.675049i \(0.235877\pi\)
\(908\) 0 0
\(909\) −15.9778 −0.529950
\(910\) 0 0
\(911\) 32.9725 1.09243 0.546215 0.837645i \(-0.316068\pi\)
0.546215 + 0.837645i \(0.316068\pi\)
\(912\) 0 0
\(913\) 13.5091 0.447086
\(914\) 0 0
\(915\) 29.7730 0.984267
\(916\) 0 0
\(917\) 6.71741 0.221828
\(918\) 0 0
\(919\) 41.8741 1.38130 0.690650 0.723189i \(-0.257324\pi\)
0.690650 + 0.723189i \(0.257324\pi\)
\(920\) 0 0
\(921\) 33.0492 1.08901
\(922\) 0 0
\(923\) 85.3176 2.80826
\(924\) 0 0
\(925\) 11.2745 0.370703
\(926\) 0 0
\(927\) −8.96725 −0.294523
\(928\) 0 0
\(929\) 21.7717 0.714306 0.357153 0.934046i \(-0.383747\pi\)
0.357153 + 0.934046i \(0.383747\pi\)
\(930\) 0 0
\(931\) −1.74252 −0.0571088
\(932\) 0 0
\(933\) 28.4591 0.931708
\(934\) 0 0
\(935\) −36.4096 −1.19072
\(936\) 0 0
\(937\) 51.2836 1.67536 0.837681 0.546160i \(-0.183911\pi\)
0.837681 + 0.546160i \(0.183911\pi\)
\(938\) 0 0
\(939\) 0.0903560 0.00294866
\(940\) 0 0
\(941\) −50.8018 −1.65609 −0.828046 0.560661i \(-0.810547\pi\)
−0.828046 + 0.560661i \(0.810547\pi\)
\(942\) 0 0
\(943\) −12.1547 −0.395811
\(944\) 0 0
\(945\) 3.51256 0.114263
\(946\) 0 0
\(947\) −40.5255 −1.31690 −0.658452 0.752623i \(-0.728788\pi\)
−0.658452 + 0.752623i \(0.728788\pi\)
\(948\) 0 0
\(949\) 27.2385 0.884198
\(950\) 0 0
\(951\) −13.1029 −0.424889
\(952\) 0 0
\(953\) −6.05417 −0.196114 −0.0980570 0.995181i \(-0.531263\pi\)
−0.0980570 + 0.995181i \(0.531263\pi\)
\(954\) 0 0
\(955\) −10.4849 −0.339284
\(956\) 0 0
\(957\) −9.90564 −0.320204
\(958\) 0 0
\(959\) −2.96362 −0.0957003
\(960\) 0 0
\(961\) 32.1803 1.03807
\(962\) 0 0
\(963\) −20.2353 −0.652072
\(964\) 0 0
\(965\) 17.5100 0.563668
\(966\) 0 0
\(967\) −40.4901 −1.30207 −0.651037 0.759046i \(-0.725666\pi\)
−0.651037 + 0.759046i \(0.725666\pi\)
\(968\) 0 0
\(969\) 10.3656 0.332990
\(970\) 0 0
\(971\) 17.9470 0.575945 0.287973 0.957639i \(-0.407019\pi\)
0.287973 + 0.957639i \(0.407019\pi\)
\(972\) 0 0
\(973\) 8.80161 0.282167
\(974\) 0 0
\(975\) 46.5090 1.48948
\(976\) 0 0
\(977\) 15.5013 0.495930 0.247965 0.968769i \(-0.420238\pi\)
0.247965 + 0.968769i \(0.420238\pi\)
\(978\) 0 0
\(979\) 7.29251 0.233070
\(980\) 0 0
\(981\) −3.31657 −0.105890
\(982\) 0 0
\(983\) 24.1422 0.770018 0.385009 0.922913i \(-0.374198\pi\)
0.385009 + 0.922913i \(0.374198\pi\)
\(984\) 0 0
\(985\) 11.9980 0.382287
\(986\) 0 0
\(987\) −3.59313 −0.114371
\(988\) 0 0
\(989\) 6.43605 0.204654
\(990\) 0 0
\(991\) 3.26567 0.103737 0.0518687 0.998654i \(-0.483482\pi\)
0.0518687 + 0.998654i \(0.483482\pi\)
\(992\) 0 0
\(993\) −4.04013 −0.128210
\(994\) 0 0
\(995\) 13.7503 0.435915
\(996\) 0 0
\(997\) −16.6749 −0.528101 −0.264050 0.964509i \(-0.585058\pi\)
−0.264050 + 0.964509i \(0.585058\pi\)
\(998\) 0 0
\(999\) 1.53644 0.0486108
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 7728.2.a.ce.1.4 4
4.3 odd 2 483.2.a.j.1.1 4
12.11 even 2 1449.2.a.o.1.4 4
28.27 even 2 3381.2.a.x.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.a.j.1.1 4 4.3 odd 2
1449.2.a.o.1.4 4 12.11 even 2
3381.2.a.x.1.1 4 28.27 even 2
7728.2.a.ce.1.4 4 1.1 even 1 trivial