Properties

Label 7728.2.a.bn.1.2
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.61803 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.61803 q^{5} +1.00000 q^{7} +1.00000 q^{9} +2.23607 q^{11} -4.61803 q^{13} +1.61803 q^{15} -6.70820 q^{17} -5.47214 q^{19} +1.00000 q^{21} +1.00000 q^{23} -2.38197 q^{25} +1.00000 q^{27} -3.76393 q^{29} +6.70820 q^{31} +2.23607 q^{33} +1.61803 q^{35} -11.0000 q^{37} -4.61803 q^{39} +7.47214 q^{41} -0.618034 q^{43} +1.61803 q^{45} +2.76393 q^{47} +1.00000 q^{49} -6.70820 q^{51} -1.90983 q^{53} +3.61803 q^{55} -5.47214 q^{57} -11.6180 q^{59} +1.85410 q^{61} +1.00000 q^{63} -7.47214 q^{65} -6.09017 q^{67} +1.00000 q^{69} -6.61803 q^{71} +0.708204 q^{73} -2.38197 q^{75} +2.23607 q^{77} +0.527864 q^{79} +1.00000 q^{81} -13.1803 q^{83} -10.8541 q^{85} -3.76393 q^{87} +9.38197 q^{89} -4.61803 q^{91} +6.70820 q^{93} -8.85410 q^{95} -16.4164 q^{97} +2.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9} - 7 q^{13} + q^{15} - 2 q^{19} + 2 q^{21} + 2 q^{23} - 7 q^{25} + 2 q^{27} - 12 q^{29} + q^{35} - 22 q^{37} - 7 q^{39} + 6 q^{41} + q^{43} + q^{45} + 10 q^{47} + 2 q^{49} - 15 q^{53} + 5 q^{55} - 2 q^{57} - 21 q^{59} - 3 q^{61} + 2 q^{63} - 6 q^{65} - q^{67} + 2 q^{69} - 11 q^{71} - 12 q^{73} - 7 q^{75} + 10 q^{79} + 2 q^{81} - 4 q^{83} - 15 q^{85} - 12 q^{87} + 21 q^{89} - 7 q^{91} - 11 q^{95} - 6 q^{97} + O(q^{100}) \)

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
\(5\) 1.61803 0.723607 0.361803 0.932254i \(-0.382161\pi\)
0.361803 + 0.932254i \(0.382161\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.23607 0.674200 0.337100 0.941469i \(-0.390554\pi\)
0.337100 + 0.941469i \(0.390554\pi\)
\(12\) 0 0
\(13\) −4.61803 −1.28081 −0.640406 0.768036i \(-0.721234\pi\)
−0.640406 + 0.768036i \(0.721234\pi\)
\(14\) 0 0
\(15\) 1.61803 0.417775
\(16\) 0 0
\(17\) −6.70820 −1.62698 −0.813489 0.581580i \(-0.802435\pi\)
−0.813489 + 0.581580i \(0.802435\pi\)
\(18\) 0 0
\(19\) −5.47214 −1.25539 −0.627697 0.778458i \(-0.716002\pi\)
−0.627697 + 0.778458i \(0.716002\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −2.38197 −0.476393
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −3.76393 −0.698945 −0.349472 0.936947i \(-0.613639\pi\)
−0.349472 + 0.936947i \(0.613639\pi\)
\(30\) 0 0
\(31\) 6.70820 1.20483 0.602414 0.798183i \(-0.294205\pi\)
0.602414 + 0.798183i \(0.294205\pi\)
\(32\) 0 0
\(33\) 2.23607 0.389249
\(34\) 0 0
\(35\) 1.61803 0.273498
\(36\) 0 0
\(37\) −11.0000 −1.80839 −0.904194 0.427121i \(-0.859528\pi\)
−0.904194 + 0.427121i \(0.859528\pi\)
\(38\) 0 0
\(39\) −4.61803 −0.739477
\(40\) 0 0
\(41\) 7.47214 1.16695 0.583476 0.812131i \(-0.301692\pi\)
0.583476 + 0.812131i \(0.301692\pi\)
\(42\) 0 0
\(43\) −0.618034 −0.0942493 −0.0471246 0.998889i \(-0.515006\pi\)
−0.0471246 + 0.998889i \(0.515006\pi\)
\(44\) 0 0
\(45\) 1.61803 0.241202
\(46\) 0 0
\(47\) 2.76393 0.403161 0.201580 0.979472i \(-0.435392\pi\)
0.201580 + 0.979472i \(0.435392\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −6.70820 −0.939336
\(52\) 0 0
\(53\) −1.90983 −0.262335 −0.131168 0.991360i \(-0.541873\pi\)
−0.131168 + 0.991360i \(0.541873\pi\)
\(54\) 0 0
\(55\) 3.61803 0.487856
\(56\) 0 0
\(57\) −5.47214 −0.724802
\(58\) 0 0
\(59\) −11.6180 −1.51254 −0.756270 0.654260i \(-0.772980\pi\)
−0.756270 + 0.654260i \(0.772980\pi\)
\(60\) 0 0
\(61\) 1.85410 0.237393 0.118697 0.992931i \(-0.462128\pi\)
0.118697 + 0.992931i \(0.462128\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) −7.47214 −0.926804
\(66\) 0 0
\(67\) −6.09017 −0.744033 −0.372016 0.928226i \(-0.621333\pi\)
−0.372016 + 0.928226i \(0.621333\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −6.61803 −0.785416 −0.392708 0.919663i \(-0.628462\pi\)
−0.392708 + 0.919663i \(0.628462\pi\)
\(72\) 0 0
\(73\) 0.708204 0.0828890 0.0414445 0.999141i \(-0.486804\pi\)
0.0414445 + 0.999141i \(0.486804\pi\)
\(74\) 0 0
\(75\) −2.38197 −0.275046
\(76\) 0 0
\(77\) 2.23607 0.254824
\(78\) 0 0
\(79\) 0.527864 0.0593893 0.0296947 0.999559i \(-0.490547\pi\)
0.0296947 + 0.999559i \(0.490547\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −13.1803 −1.44673 −0.723365 0.690466i \(-0.757406\pi\)
−0.723365 + 0.690466i \(0.757406\pi\)
\(84\) 0 0
\(85\) −10.8541 −1.17729
\(86\) 0 0
\(87\) −3.76393 −0.403536
\(88\) 0 0
\(89\) 9.38197 0.994486 0.497243 0.867611i \(-0.334346\pi\)
0.497243 + 0.867611i \(0.334346\pi\)
\(90\) 0 0
\(91\) −4.61803 −0.484102
\(92\) 0 0
\(93\) 6.70820 0.695608
\(94\) 0 0
\(95\) −8.85410 −0.908412
\(96\) 0 0
\(97\) −16.4164 −1.66683 −0.833417 0.552645i \(-0.813619\pi\)
−0.833417 + 0.552645i \(0.813619\pi\)
\(98\) 0 0
\(99\) 2.23607 0.224733
\(100\) 0 0
\(101\) 10.8541 1.08002 0.540012 0.841657i \(-0.318420\pi\)
0.540012 + 0.841657i \(0.318420\pi\)
\(102\) 0 0
\(103\) −19.4164 −1.91316 −0.956578 0.291477i \(-0.905853\pi\)
−0.956578 + 0.291477i \(0.905853\pi\)
\(104\) 0 0
\(105\) 1.61803 0.157904
\(106\) 0 0
\(107\) 12.3262 1.19162 0.595811 0.803125i \(-0.296831\pi\)
0.595811 + 0.803125i \(0.296831\pi\)
\(108\) 0 0
\(109\) −14.2705 −1.36687 −0.683433 0.730013i \(-0.739514\pi\)
−0.683433 + 0.730013i \(0.739514\pi\)
\(110\) 0 0
\(111\) −11.0000 −1.04407
\(112\) 0 0
\(113\) −12.0902 −1.13735 −0.568674 0.822563i \(-0.692543\pi\)
−0.568674 + 0.822563i \(0.692543\pi\)
\(114\) 0 0
\(115\) 1.61803 0.150882
\(116\) 0 0
\(117\) −4.61803 −0.426937
\(118\) 0 0
\(119\) −6.70820 −0.614940
\(120\) 0 0
\(121\) −6.00000 −0.545455
\(122\) 0 0
\(123\) 7.47214 0.673740
\(124\) 0 0
\(125\) −11.9443 −1.06833
\(126\) 0 0
\(127\) 11.2705 1.00010 0.500048 0.865998i \(-0.333316\pi\)
0.500048 + 0.865998i \(0.333316\pi\)
\(128\) 0 0
\(129\) −0.618034 −0.0544149
\(130\) 0 0
\(131\) 7.18034 0.627349 0.313675 0.949531i \(-0.398440\pi\)
0.313675 + 0.949531i \(0.398440\pi\)
\(132\) 0 0
\(133\) −5.47214 −0.474494
\(134\) 0 0
\(135\) 1.61803 0.139258
\(136\) 0 0
\(137\) 10.4164 0.889934 0.444967 0.895547i \(-0.353215\pi\)
0.444967 + 0.895547i \(0.353215\pi\)
\(138\) 0 0
\(139\) 13.6180 1.15507 0.577533 0.816367i \(-0.304015\pi\)
0.577533 + 0.816367i \(0.304015\pi\)
\(140\) 0 0
\(141\) 2.76393 0.232765
\(142\) 0 0
\(143\) −10.3262 −0.863523
\(144\) 0 0
\(145\) −6.09017 −0.505761
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) −14.7639 −1.20951 −0.604754 0.796412i \(-0.706729\pi\)
−0.604754 + 0.796412i \(0.706729\pi\)
\(150\) 0 0
\(151\) 10.7639 0.875956 0.437978 0.898986i \(-0.355695\pi\)
0.437978 + 0.898986i \(0.355695\pi\)
\(152\) 0 0
\(153\) −6.70820 −0.542326
\(154\) 0 0
\(155\) 10.8541 0.871822
\(156\) 0 0
\(157\) −13.7082 −1.09403 −0.547017 0.837122i \(-0.684237\pi\)
−0.547017 + 0.837122i \(0.684237\pi\)
\(158\) 0 0
\(159\) −1.90983 −0.151459
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) −1.61803 −0.126734 −0.0633671 0.997990i \(-0.520184\pi\)
−0.0633671 + 0.997990i \(0.520184\pi\)
\(164\) 0 0
\(165\) 3.61803 0.281664
\(166\) 0 0
\(167\) −15.1803 −1.17469 −0.587345 0.809337i \(-0.699827\pi\)
−0.587345 + 0.809337i \(0.699827\pi\)
\(168\) 0 0
\(169\) 8.32624 0.640480
\(170\) 0 0
\(171\) −5.47214 −0.418465
\(172\) 0 0
\(173\) 5.47214 0.416039 0.208019 0.978125i \(-0.433298\pi\)
0.208019 + 0.978125i \(0.433298\pi\)
\(174\) 0 0
\(175\) −2.38197 −0.180060
\(176\) 0 0
\(177\) −11.6180 −0.873265
\(178\) 0 0
\(179\) 4.85410 0.362813 0.181406 0.983408i \(-0.441935\pi\)
0.181406 + 0.983408i \(0.441935\pi\)
\(180\) 0 0
\(181\) 15.9443 1.18513 0.592564 0.805523i \(-0.298116\pi\)
0.592564 + 0.805523i \(0.298116\pi\)
\(182\) 0 0
\(183\) 1.85410 0.137059
\(184\) 0 0
\(185\) −17.7984 −1.30856
\(186\) 0 0
\(187\) −15.0000 −1.09691
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 6.18034 0.447194 0.223597 0.974682i \(-0.428220\pi\)
0.223597 + 0.974682i \(0.428220\pi\)
\(192\) 0 0
\(193\) 21.7082 1.56259 0.781295 0.624161i \(-0.214559\pi\)
0.781295 + 0.624161i \(0.214559\pi\)
\(194\) 0 0
\(195\) −7.47214 −0.535091
\(196\) 0 0
\(197\) 12.5066 0.891057 0.445528 0.895268i \(-0.353016\pi\)
0.445528 + 0.895268i \(0.353016\pi\)
\(198\) 0 0
\(199\) 18.0902 1.28238 0.641189 0.767383i \(-0.278441\pi\)
0.641189 + 0.767383i \(0.278441\pi\)
\(200\) 0 0
\(201\) −6.09017 −0.429567
\(202\) 0 0
\(203\) −3.76393 −0.264176
\(204\) 0 0
\(205\) 12.0902 0.844414
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −12.2361 −0.846387
\(210\) 0 0
\(211\) 16.4164 1.13015 0.565076 0.825039i \(-0.308847\pi\)
0.565076 + 0.825039i \(0.308847\pi\)
\(212\) 0 0
\(213\) −6.61803 −0.453460
\(214\) 0 0
\(215\) −1.00000 −0.0681994
\(216\) 0 0
\(217\) 6.70820 0.455383
\(218\) 0 0
\(219\) 0.708204 0.0478560
\(220\) 0 0
\(221\) 30.9787 2.08385
\(222\) 0 0
\(223\) 11.1459 0.746385 0.373192 0.927754i \(-0.378263\pi\)
0.373192 + 0.927754i \(0.378263\pi\)
\(224\) 0 0
\(225\) −2.38197 −0.158798
\(226\) 0 0
\(227\) −8.67376 −0.575698 −0.287849 0.957676i \(-0.592940\pi\)
−0.287849 + 0.957676i \(0.592940\pi\)
\(228\) 0 0
\(229\) −0.673762 −0.0445235 −0.0222617 0.999752i \(-0.507087\pi\)
−0.0222617 + 0.999752i \(0.507087\pi\)
\(230\) 0 0
\(231\) 2.23607 0.147122
\(232\) 0 0
\(233\) 0.0901699 0.00590723 0.00295361 0.999996i \(-0.499060\pi\)
0.00295361 + 0.999996i \(0.499060\pi\)
\(234\) 0 0
\(235\) 4.47214 0.291730
\(236\) 0 0
\(237\) 0.527864 0.0342885
\(238\) 0 0
\(239\) −19.7984 −1.28065 −0.640325 0.768104i \(-0.721200\pi\)
−0.640325 + 0.768104i \(0.721200\pi\)
\(240\) 0 0
\(241\) −11.0000 −0.708572 −0.354286 0.935137i \(-0.615276\pi\)
−0.354286 + 0.935137i \(0.615276\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 1.61803 0.103372
\(246\) 0 0
\(247\) 25.2705 1.60792
\(248\) 0 0
\(249\) −13.1803 −0.835270
\(250\) 0 0
\(251\) 17.1246 1.08090 0.540448 0.841377i \(-0.318255\pi\)
0.540448 + 0.841377i \(0.318255\pi\)
\(252\) 0 0
\(253\) 2.23607 0.140580
\(254\) 0 0
\(255\) −10.8541 −0.679710
\(256\) 0 0
\(257\) 1.23607 0.0771038 0.0385519 0.999257i \(-0.487726\pi\)
0.0385519 + 0.999257i \(0.487726\pi\)
\(258\) 0 0
\(259\) −11.0000 −0.683507
\(260\) 0 0
\(261\) −3.76393 −0.232982
\(262\) 0 0
\(263\) 17.9443 1.10649 0.553246 0.833018i \(-0.313389\pi\)
0.553246 + 0.833018i \(0.313389\pi\)
\(264\) 0 0
\(265\) −3.09017 −0.189828
\(266\) 0 0
\(267\) 9.38197 0.574167
\(268\) 0 0
\(269\) −20.9098 −1.27489 −0.637447 0.770494i \(-0.720010\pi\)
−0.637447 + 0.770494i \(0.720010\pi\)
\(270\) 0 0
\(271\) −4.05573 −0.246368 −0.123184 0.992384i \(-0.539311\pi\)
−0.123184 + 0.992384i \(0.539311\pi\)
\(272\) 0 0
\(273\) −4.61803 −0.279496
\(274\) 0 0
\(275\) −5.32624 −0.321184
\(276\) 0 0
\(277\) −29.2705 −1.75869 −0.879347 0.476181i \(-0.842021\pi\)
−0.879347 + 0.476181i \(0.842021\pi\)
\(278\) 0 0
\(279\) 6.70820 0.401610
\(280\) 0 0
\(281\) 9.70820 0.579143 0.289571 0.957156i \(-0.406487\pi\)
0.289571 + 0.957156i \(0.406487\pi\)
\(282\) 0 0
\(283\) −25.2148 −1.49886 −0.749432 0.662082i \(-0.769673\pi\)
−0.749432 + 0.662082i \(0.769673\pi\)
\(284\) 0 0
\(285\) −8.85410 −0.524472
\(286\) 0 0
\(287\) 7.47214 0.441066
\(288\) 0 0
\(289\) 28.0000 1.64706
\(290\) 0 0
\(291\) −16.4164 −0.962347
\(292\) 0 0
\(293\) 12.0000 0.701047 0.350524 0.936554i \(-0.386004\pi\)
0.350524 + 0.936554i \(0.386004\pi\)
\(294\) 0 0
\(295\) −18.7984 −1.09448
\(296\) 0 0
\(297\) 2.23607 0.129750
\(298\) 0 0
\(299\) −4.61803 −0.267068
\(300\) 0 0
\(301\) −0.618034 −0.0356229
\(302\) 0 0
\(303\) 10.8541 0.623552
\(304\) 0 0
\(305\) 3.00000 0.171780
\(306\) 0 0
\(307\) 24.1246 1.37686 0.688432 0.725301i \(-0.258299\pi\)
0.688432 + 0.725301i \(0.258299\pi\)
\(308\) 0 0
\(309\) −19.4164 −1.10456
\(310\) 0 0
\(311\) 0.326238 0.0184993 0.00924963 0.999957i \(-0.497056\pi\)
0.00924963 + 0.999957i \(0.497056\pi\)
\(312\) 0 0
\(313\) 6.47214 0.365827 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(314\) 0 0
\(315\) 1.61803 0.0911659
\(316\) 0 0
\(317\) −23.5066 −1.32026 −0.660130 0.751151i \(-0.729499\pi\)
−0.660130 + 0.751151i \(0.729499\pi\)
\(318\) 0 0
\(319\) −8.41641 −0.471228
\(320\) 0 0
\(321\) 12.3262 0.687984
\(322\) 0 0
\(323\) 36.7082 2.04250
\(324\) 0 0
\(325\) 11.0000 0.610170
\(326\) 0 0
\(327\) −14.2705 −0.789161
\(328\) 0 0
\(329\) 2.76393 0.152381
\(330\) 0 0
\(331\) −13.4164 −0.737432 −0.368716 0.929542i \(-0.620203\pi\)
−0.368716 + 0.929542i \(0.620203\pi\)
\(332\) 0 0
\(333\) −11.0000 −0.602796
\(334\) 0 0
\(335\) −9.85410 −0.538387
\(336\) 0 0
\(337\) 30.5066 1.66180 0.830900 0.556422i \(-0.187826\pi\)
0.830900 + 0.556422i \(0.187826\pi\)
\(338\) 0 0
\(339\) −12.0902 −0.656648
\(340\) 0 0
\(341\) 15.0000 0.812296
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 1.61803 0.0871120
\(346\) 0 0
\(347\) 17.1803 0.922289 0.461144 0.887325i \(-0.347439\pi\)
0.461144 + 0.887325i \(0.347439\pi\)
\(348\) 0 0
\(349\) −5.14590 −0.275454 −0.137727 0.990470i \(-0.543980\pi\)
−0.137727 + 0.990470i \(0.543980\pi\)
\(350\) 0 0
\(351\) −4.61803 −0.246492
\(352\) 0 0
\(353\) −32.3050 −1.71942 −0.859710 0.510783i \(-0.829355\pi\)
−0.859710 + 0.510783i \(0.829355\pi\)
\(354\) 0 0
\(355\) −10.7082 −0.568332
\(356\) 0 0
\(357\) −6.70820 −0.355036
\(358\) 0 0
\(359\) −11.9098 −0.628577 −0.314288 0.949328i \(-0.601766\pi\)
−0.314288 + 0.949328i \(0.601766\pi\)
\(360\) 0 0
\(361\) 10.9443 0.576014
\(362\) 0 0
\(363\) −6.00000 −0.314918
\(364\) 0 0
\(365\) 1.14590 0.0599790
\(366\) 0 0
\(367\) −4.14590 −0.216414 −0.108207 0.994128i \(-0.534511\pi\)
−0.108207 + 0.994128i \(0.534511\pi\)
\(368\) 0 0
\(369\) 7.47214 0.388984
\(370\) 0 0
\(371\) −1.90983 −0.0991534
\(372\) 0 0
\(373\) −2.41641 −0.125117 −0.0625584 0.998041i \(-0.519926\pi\)
−0.0625584 + 0.998041i \(0.519926\pi\)
\(374\) 0 0
\(375\) −11.9443 −0.616800
\(376\) 0 0
\(377\) 17.3820 0.895217
\(378\) 0 0
\(379\) 19.4164 0.997354 0.498677 0.866788i \(-0.333819\pi\)
0.498677 + 0.866788i \(0.333819\pi\)
\(380\) 0 0
\(381\) 11.2705 0.577406
\(382\) 0 0
\(383\) −12.7082 −0.649359 −0.324679 0.945824i \(-0.605256\pi\)
−0.324679 + 0.945824i \(0.605256\pi\)
\(384\) 0 0
\(385\) 3.61803 0.184392
\(386\) 0 0
\(387\) −0.618034 −0.0314164
\(388\) 0 0
\(389\) 18.7082 0.948544 0.474272 0.880378i \(-0.342711\pi\)
0.474272 + 0.880378i \(0.342711\pi\)
\(390\) 0 0
\(391\) −6.70820 −0.339248
\(392\) 0 0
\(393\) 7.18034 0.362200
\(394\) 0 0
\(395\) 0.854102 0.0429745
\(396\) 0 0
\(397\) −36.0689 −1.81025 −0.905123 0.425150i \(-0.860221\pi\)
−0.905123 + 0.425150i \(0.860221\pi\)
\(398\) 0 0
\(399\) −5.47214 −0.273949
\(400\) 0 0
\(401\) 10.8197 0.540308 0.270154 0.962817i \(-0.412925\pi\)
0.270154 + 0.962817i \(0.412925\pi\)
\(402\) 0 0
\(403\) −30.9787 −1.54316
\(404\) 0 0
\(405\) 1.61803 0.0804008
\(406\) 0 0
\(407\) −24.5967 −1.21922
\(408\) 0 0
\(409\) −20.4164 −1.00953 −0.504763 0.863258i \(-0.668420\pi\)
−0.504763 + 0.863258i \(0.668420\pi\)
\(410\) 0 0
\(411\) 10.4164 0.513804
\(412\) 0 0
\(413\) −11.6180 −0.571686
\(414\) 0 0
\(415\) −21.3262 −1.04686
\(416\) 0 0
\(417\) 13.6180 0.666878
\(418\) 0 0
\(419\) −31.3262 −1.53039 −0.765193 0.643800i \(-0.777357\pi\)
−0.765193 + 0.643800i \(0.777357\pi\)
\(420\) 0 0
\(421\) 23.2705 1.13414 0.567068 0.823671i \(-0.308078\pi\)
0.567068 + 0.823671i \(0.308078\pi\)
\(422\) 0 0
\(423\) 2.76393 0.134387
\(424\) 0 0
\(425\) 15.9787 0.775081
\(426\) 0 0
\(427\) 1.85410 0.0897263
\(428\) 0 0
\(429\) −10.3262 −0.498555
\(430\) 0 0
\(431\) −16.3262 −0.786407 −0.393204 0.919451i \(-0.628633\pi\)
−0.393204 + 0.919451i \(0.628633\pi\)
\(432\) 0 0
\(433\) −21.1246 −1.01518 −0.507592 0.861598i \(-0.669464\pi\)
−0.507592 + 0.861598i \(0.669464\pi\)
\(434\) 0 0
\(435\) −6.09017 −0.292001
\(436\) 0 0
\(437\) −5.47214 −0.261768
\(438\) 0 0
\(439\) 7.65248 0.365233 0.182616 0.983184i \(-0.441543\pi\)
0.182616 + 0.983184i \(0.441543\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 18.4721 0.877638 0.438819 0.898576i \(-0.355397\pi\)
0.438819 + 0.898576i \(0.355397\pi\)
\(444\) 0 0
\(445\) 15.1803 0.719617
\(446\) 0 0
\(447\) −14.7639 −0.698310
\(448\) 0 0
\(449\) −38.5623 −1.81987 −0.909934 0.414753i \(-0.863868\pi\)
−0.909934 + 0.414753i \(0.863868\pi\)
\(450\) 0 0
\(451\) 16.7082 0.786759
\(452\) 0 0
\(453\) 10.7639 0.505734
\(454\) 0 0
\(455\) −7.47214 −0.350299
\(456\) 0 0
\(457\) 1.32624 0.0620388 0.0310194 0.999519i \(-0.490125\pi\)
0.0310194 + 0.999519i \(0.490125\pi\)
\(458\) 0 0
\(459\) −6.70820 −0.313112
\(460\) 0 0
\(461\) −30.4508 −1.41824 −0.709119 0.705089i \(-0.750907\pi\)
−0.709119 + 0.705089i \(0.750907\pi\)
\(462\) 0 0
\(463\) −30.7082 −1.42713 −0.713566 0.700588i \(-0.752921\pi\)
−0.713566 + 0.700588i \(0.752921\pi\)
\(464\) 0 0
\(465\) 10.8541 0.503347
\(466\) 0 0
\(467\) −7.18034 −0.332267 −0.166133 0.986103i \(-0.553128\pi\)
−0.166133 + 0.986103i \(0.553128\pi\)
\(468\) 0 0
\(469\) −6.09017 −0.281218
\(470\) 0 0
\(471\) −13.7082 −0.631641
\(472\) 0 0
\(473\) −1.38197 −0.0635429
\(474\) 0 0
\(475\) 13.0344 0.598061
\(476\) 0 0
\(477\) −1.90983 −0.0874451
\(478\) 0 0
\(479\) −21.0000 −0.959514 −0.479757 0.877401i \(-0.659275\pi\)
−0.479757 + 0.877401i \(0.659275\pi\)
\(480\) 0 0
\(481\) 50.7984 2.31621
\(482\) 0 0
\(483\) 1.00000 0.0455016
\(484\) 0 0
\(485\) −26.5623 −1.20613
\(486\) 0 0
\(487\) 42.7082 1.93529 0.967647 0.252309i \(-0.0811899\pi\)
0.967647 + 0.252309i \(0.0811899\pi\)
\(488\) 0 0
\(489\) −1.61803 −0.0731700
\(490\) 0 0
\(491\) 14.7984 0.667841 0.333921 0.942601i \(-0.391628\pi\)
0.333921 + 0.942601i \(0.391628\pi\)
\(492\) 0 0
\(493\) 25.2492 1.13717
\(494\) 0 0
\(495\) 3.61803 0.162619
\(496\) 0 0
\(497\) −6.61803 −0.296859
\(498\) 0 0
\(499\) −35.3951 −1.58450 −0.792252 0.610195i \(-0.791091\pi\)
−0.792252 + 0.610195i \(0.791091\pi\)
\(500\) 0 0
\(501\) −15.1803 −0.678208
\(502\) 0 0
\(503\) −20.6738 −0.921797 −0.460899 0.887453i \(-0.652473\pi\)
−0.460899 + 0.887453i \(0.652473\pi\)
\(504\) 0 0
\(505\) 17.5623 0.781512
\(506\) 0 0
\(507\) 8.32624 0.369781
\(508\) 0 0
\(509\) 33.5967 1.48915 0.744575 0.667539i \(-0.232652\pi\)
0.744575 + 0.667539i \(0.232652\pi\)
\(510\) 0 0
\(511\) 0.708204 0.0313291
\(512\) 0 0
\(513\) −5.47214 −0.241601
\(514\) 0 0
\(515\) −31.4164 −1.38437
\(516\) 0 0
\(517\) 6.18034 0.271811
\(518\) 0 0
\(519\) 5.47214 0.240200
\(520\) 0 0
\(521\) 12.4721 0.546414 0.273207 0.961955i \(-0.411916\pi\)
0.273207 + 0.961955i \(0.411916\pi\)
\(522\) 0 0
\(523\) 5.41641 0.236843 0.118421 0.992963i \(-0.462217\pi\)
0.118421 + 0.992963i \(0.462217\pi\)
\(524\) 0 0
\(525\) −2.38197 −0.103958
\(526\) 0 0
\(527\) −45.0000 −1.96023
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −11.6180 −0.504180
\(532\) 0 0
\(533\) −34.5066 −1.49465
\(534\) 0 0
\(535\) 19.9443 0.862266
\(536\) 0 0
\(537\) 4.85410 0.209470
\(538\) 0 0
\(539\) 2.23607 0.0963143
\(540\) 0 0
\(541\) 35.1246 1.51013 0.755063 0.655653i \(-0.227606\pi\)
0.755063 + 0.655653i \(0.227606\pi\)
\(542\) 0 0
\(543\) 15.9443 0.684234
\(544\) 0 0
\(545\) −23.0902 −0.989074
\(546\) 0 0
\(547\) 11.7984 0.504462 0.252231 0.967667i \(-0.418836\pi\)
0.252231 + 0.967667i \(0.418836\pi\)
\(548\) 0 0
\(549\) 1.85410 0.0791311
\(550\) 0 0
\(551\) 20.5967 0.877451
\(552\) 0 0
\(553\) 0.527864 0.0224471
\(554\) 0 0
\(555\) −17.7984 −0.755499
\(556\) 0 0
\(557\) −28.1803 −1.19404 −0.597020 0.802227i \(-0.703649\pi\)
−0.597020 + 0.802227i \(0.703649\pi\)
\(558\) 0 0
\(559\) 2.85410 0.120716
\(560\) 0 0
\(561\) −15.0000 −0.633300
\(562\) 0 0
\(563\) −14.5066 −0.611379 −0.305690 0.952131i \(-0.598887\pi\)
−0.305690 + 0.952131i \(0.598887\pi\)
\(564\) 0 0
\(565\) −19.5623 −0.822992
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 28.3607 1.18894 0.594471 0.804117i \(-0.297362\pi\)
0.594471 + 0.804117i \(0.297362\pi\)
\(570\) 0 0
\(571\) 17.5836 0.735850 0.367925 0.929855i \(-0.380068\pi\)
0.367925 + 0.929855i \(0.380068\pi\)
\(572\) 0 0
\(573\) 6.18034 0.258187
\(574\) 0 0
\(575\) −2.38197 −0.0993348
\(576\) 0 0
\(577\) 16.8328 0.700759 0.350380 0.936608i \(-0.386053\pi\)
0.350380 + 0.936608i \(0.386053\pi\)
\(578\) 0 0
\(579\) 21.7082 0.902162
\(580\) 0 0
\(581\) −13.1803 −0.546813
\(582\) 0 0
\(583\) −4.27051 −0.176866
\(584\) 0 0
\(585\) −7.47214 −0.308935
\(586\) 0 0
\(587\) 1.96556 0.0811273 0.0405636 0.999177i \(-0.487085\pi\)
0.0405636 + 0.999177i \(0.487085\pi\)
\(588\) 0 0
\(589\) −36.7082 −1.51254
\(590\) 0 0
\(591\) 12.5066 0.514452
\(592\) 0 0
\(593\) 26.0689 1.07052 0.535260 0.844687i \(-0.320213\pi\)
0.535260 + 0.844687i \(0.320213\pi\)
\(594\) 0 0
\(595\) −10.8541 −0.444975
\(596\) 0 0
\(597\) 18.0902 0.740381
\(598\) 0 0
\(599\) −22.5066 −0.919594 −0.459797 0.888024i \(-0.652078\pi\)
−0.459797 + 0.888024i \(0.652078\pi\)
\(600\) 0 0
\(601\) 12.0902 0.493168 0.246584 0.969121i \(-0.420692\pi\)
0.246584 + 0.969121i \(0.420692\pi\)
\(602\) 0 0
\(603\) −6.09017 −0.248011
\(604\) 0 0
\(605\) −9.70820 −0.394695
\(606\) 0 0
\(607\) 9.20163 0.373482 0.186741 0.982409i \(-0.440207\pi\)
0.186741 + 0.982409i \(0.440207\pi\)
\(608\) 0 0
\(609\) −3.76393 −0.152522
\(610\) 0 0
\(611\) −12.7639 −0.516373
\(612\) 0 0
\(613\) 19.6525 0.793756 0.396878 0.917871i \(-0.370094\pi\)
0.396878 + 0.917871i \(0.370094\pi\)
\(614\) 0 0
\(615\) 12.0902 0.487523
\(616\) 0 0
\(617\) −18.3262 −0.737787 −0.368893 0.929472i \(-0.620263\pi\)
−0.368893 + 0.929472i \(0.620263\pi\)
\(618\) 0 0
\(619\) 17.3820 0.698640 0.349320 0.937003i \(-0.386413\pi\)
0.349320 + 0.937003i \(0.386413\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 9.38197 0.375881
\(624\) 0 0
\(625\) −7.41641 −0.296656
\(626\) 0 0
\(627\) −12.2361 −0.488661
\(628\) 0 0
\(629\) 73.7902 2.94221
\(630\) 0 0
\(631\) 24.1246 0.960386 0.480193 0.877163i \(-0.340567\pi\)
0.480193 + 0.877163i \(0.340567\pi\)
\(632\) 0 0
\(633\) 16.4164 0.652494
\(634\) 0 0
\(635\) 18.2361 0.723676
\(636\) 0 0
\(637\) −4.61803 −0.182973
\(638\) 0 0
\(639\) −6.61803 −0.261805
\(640\) 0 0
\(641\) −6.79837 −0.268520 −0.134260 0.990946i \(-0.542866\pi\)
−0.134260 + 0.990946i \(0.542866\pi\)
\(642\) 0 0
\(643\) −34.9787 −1.37943 −0.689713 0.724083i \(-0.742263\pi\)
−0.689713 + 0.724083i \(0.742263\pi\)
\(644\) 0 0
\(645\) −1.00000 −0.0393750
\(646\) 0 0
\(647\) −5.03444 −0.197924 −0.0989622 0.995091i \(-0.531552\pi\)
−0.0989622 + 0.995091i \(0.531552\pi\)
\(648\) 0 0
\(649\) −25.9787 −1.01975
\(650\) 0 0
\(651\) 6.70820 0.262915
\(652\) 0 0
\(653\) −14.7426 −0.576924 −0.288462 0.957491i \(-0.593144\pi\)
−0.288462 + 0.957491i \(0.593144\pi\)
\(654\) 0 0
\(655\) 11.6180 0.453954
\(656\) 0 0
\(657\) 0.708204 0.0276297
\(658\) 0 0
\(659\) 21.6525 0.843461 0.421730 0.906721i \(-0.361423\pi\)
0.421730 + 0.906721i \(0.361423\pi\)
\(660\) 0 0
\(661\) −12.4164 −0.482942 −0.241471 0.970408i \(-0.577630\pi\)
−0.241471 + 0.970408i \(0.577630\pi\)
\(662\) 0 0
\(663\) 30.9787 1.20311
\(664\) 0 0
\(665\) −8.85410 −0.343347
\(666\) 0 0
\(667\) −3.76393 −0.145740
\(668\) 0 0
\(669\) 11.1459 0.430925
\(670\) 0 0
\(671\) 4.14590 0.160051
\(672\) 0 0
\(673\) −14.4164 −0.555712 −0.277856 0.960623i \(-0.589624\pi\)
−0.277856 + 0.960623i \(0.589624\pi\)
\(674\) 0 0
\(675\) −2.38197 −0.0916819
\(676\) 0 0
\(677\) −43.7426 −1.68117 −0.840583 0.541682i \(-0.817788\pi\)
−0.840583 + 0.541682i \(0.817788\pi\)
\(678\) 0 0
\(679\) −16.4164 −0.630004
\(680\) 0 0
\(681\) −8.67376 −0.332379
\(682\) 0 0
\(683\) 50.0132 1.91370 0.956850 0.290582i \(-0.0938489\pi\)
0.956850 + 0.290582i \(0.0938489\pi\)
\(684\) 0 0
\(685\) 16.8541 0.643962
\(686\) 0 0
\(687\) −0.673762 −0.0257056
\(688\) 0 0
\(689\) 8.81966 0.336002
\(690\) 0 0
\(691\) 41.2705 1.57000 0.785002 0.619493i \(-0.212662\pi\)
0.785002 + 0.619493i \(0.212662\pi\)
\(692\) 0 0
\(693\) 2.23607 0.0849412
\(694\) 0 0
\(695\) 22.0344 0.835814
\(696\) 0 0
\(697\) −50.1246 −1.89861
\(698\) 0 0
\(699\) 0.0901699 0.00341054
\(700\) 0 0
\(701\) −18.4508 −0.696879 −0.348439 0.937331i \(-0.613288\pi\)
−0.348439 + 0.937331i \(0.613288\pi\)
\(702\) 0 0
\(703\) 60.1935 2.27024
\(704\) 0 0
\(705\) 4.47214 0.168430
\(706\) 0 0
\(707\) 10.8541 0.408211
\(708\) 0 0
\(709\) 17.5623 0.659566 0.329783 0.944057i \(-0.393024\pi\)
0.329783 + 0.944057i \(0.393024\pi\)
\(710\) 0 0
\(711\) 0.527864 0.0197964
\(712\) 0 0
\(713\) 6.70820 0.251224
\(714\) 0 0
\(715\) −16.7082 −0.624851
\(716\) 0 0
\(717\) −19.7984 −0.739384
\(718\) 0 0
\(719\) 3.00000 0.111881 0.0559406 0.998434i \(-0.482184\pi\)
0.0559406 + 0.998434i \(0.482184\pi\)
\(720\) 0 0
\(721\) −19.4164 −0.723105
\(722\) 0 0
\(723\) −11.0000 −0.409094
\(724\) 0 0
\(725\) 8.96556 0.332972
\(726\) 0 0
\(727\) 16.8885 0.626361 0.313181 0.949694i \(-0.398605\pi\)
0.313181 + 0.949694i \(0.398605\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 4.14590 0.153342
\(732\) 0 0
\(733\) −23.5967 −0.871566 −0.435783 0.900052i \(-0.643528\pi\)
−0.435783 + 0.900052i \(0.643528\pi\)
\(734\) 0 0
\(735\) 1.61803 0.0596821
\(736\) 0 0
\(737\) −13.6180 −0.501627
\(738\) 0 0
\(739\) 35.2492 1.29666 0.648332 0.761358i \(-0.275467\pi\)
0.648332 + 0.761358i \(0.275467\pi\)
\(740\) 0 0
\(741\) 25.2705 0.928335
\(742\) 0 0
\(743\) −27.3262 −1.00250 −0.501251 0.865302i \(-0.667127\pi\)
−0.501251 + 0.865302i \(0.667127\pi\)
\(744\) 0 0
\(745\) −23.8885 −0.875209
\(746\) 0 0
\(747\) −13.1803 −0.482243
\(748\) 0 0
\(749\) 12.3262 0.450391
\(750\) 0 0
\(751\) 13.9787 0.510091 0.255045 0.966929i \(-0.417910\pi\)
0.255045 + 0.966929i \(0.417910\pi\)
\(752\) 0 0
\(753\) 17.1246 0.624056
\(754\) 0 0
\(755\) 17.4164 0.633848
\(756\) 0 0
\(757\) −17.0000 −0.617876 −0.308938 0.951082i \(-0.599973\pi\)
−0.308938 + 0.951082i \(0.599973\pi\)
\(758\) 0 0
\(759\) 2.23607 0.0811641
\(760\) 0 0
\(761\) −5.52786 −0.200385 −0.100192 0.994968i \(-0.531946\pi\)
−0.100192 + 0.994968i \(0.531946\pi\)
\(762\) 0 0
\(763\) −14.2705 −0.516627
\(764\) 0 0
\(765\) −10.8541 −0.392431
\(766\) 0 0
\(767\) 53.6525 1.93728
\(768\) 0 0
\(769\) 28.7639 1.03725 0.518627 0.855001i \(-0.326443\pi\)
0.518627 + 0.855001i \(0.326443\pi\)
\(770\) 0 0
\(771\) 1.23607 0.0445159
\(772\) 0 0
\(773\) −2.05573 −0.0739394 −0.0369697 0.999316i \(-0.511771\pi\)
−0.0369697 + 0.999316i \(0.511771\pi\)
\(774\) 0 0
\(775\) −15.9787 −0.573972
\(776\) 0 0
\(777\) −11.0000 −0.394623
\(778\) 0 0
\(779\) −40.8885 −1.46498
\(780\) 0 0
\(781\) −14.7984 −0.529527
\(782\) 0 0
\(783\) −3.76393 −0.134512
\(784\) 0 0
\(785\) −22.1803 −0.791650
\(786\) 0 0
\(787\) −47.5755 −1.69588 −0.847941 0.530091i \(-0.822158\pi\)
−0.847941 + 0.530091i \(0.822158\pi\)
\(788\) 0 0
\(789\) 17.9443 0.638833
\(790\) 0 0
\(791\) −12.0902 −0.429877
\(792\) 0 0
\(793\) −8.56231 −0.304056
\(794\) 0 0
\(795\) −3.09017 −0.109597
\(796\) 0 0
\(797\) 21.8197 0.772892 0.386446 0.922312i \(-0.373703\pi\)
0.386446 + 0.922312i \(0.373703\pi\)
\(798\) 0 0
\(799\) −18.5410 −0.655934
\(800\) 0 0
\(801\) 9.38197 0.331495
\(802\) 0 0
\(803\) 1.58359 0.0558838
\(804\) 0 0
\(805\) 1.61803 0.0570282
\(806\) 0 0
\(807\) −20.9098 −0.736061
\(808\) 0 0
\(809\) −29.5623 −1.03936 −0.519678 0.854362i \(-0.673948\pi\)
−0.519678 + 0.854362i \(0.673948\pi\)
\(810\) 0 0
\(811\) −17.4721 −0.613530 −0.306765 0.951785i \(-0.599247\pi\)
−0.306765 + 0.951785i \(0.599247\pi\)
\(812\) 0 0
\(813\) −4.05573 −0.142241
\(814\) 0 0
\(815\) −2.61803 −0.0917057
\(816\) 0 0
\(817\) 3.38197 0.118320
\(818\) 0 0
\(819\) −4.61803 −0.161367
\(820\) 0 0
\(821\) 11.1246 0.388252 0.194126 0.980977i \(-0.437813\pi\)
0.194126 + 0.980977i \(0.437813\pi\)
\(822\) 0 0
\(823\) 33.1459 1.15539 0.577697 0.816252i \(-0.303952\pi\)
0.577697 + 0.816252i \(0.303952\pi\)
\(824\) 0 0
\(825\) −5.32624 −0.185436
\(826\) 0 0
\(827\) −23.2148 −0.807257 −0.403629 0.914923i \(-0.632251\pi\)
−0.403629 + 0.914923i \(0.632251\pi\)
\(828\) 0 0
\(829\) −3.58359 −0.124463 −0.0622316 0.998062i \(-0.519822\pi\)
−0.0622316 + 0.998062i \(0.519822\pi\)
\(830\) 0 0
\(831\) −29.2705 −1.01538
\(832\) 0 0
\(833\) −6.70820 −0.232425
\(834\) 0 0
\(835\) −24.5623 −0.850014
\(836\) 0 0
\(837\) 6.70820 0.231869
\(838\) 0 0
\(839\) 25.7426 0.888735 0.444367 0.895845i \(-0.353428\pi\)
0.444367 + 0.895845i \(0.353428\pi\)
\(840\) 0 0
\(841\) −14.8328 −0.511476
\(842\) 0 0
\(843\) 9.70820 0.334368
\(844\) 0 0
\(845\) 13.4721 0.463456
\(846\) 0 0
\(847\) −6.00000 −0.206162
\(848\) 0 0
\(849\) −25.2148 −0.865369
\(850\) 0 0
\(851\) −11.0000 −0.377075
\(852\) 0 0
\(853\) −6.87539 −0.235409 −0.117704 0.993049i \(-0.537554\pi\)
−0.117704 + 0.993049i \(0.537554\pi\)
\(854\) 0 0
\(855\) −8.85410 −0.302804
\(856\) 0 0
\(857\) 26.2918 0.898111 0.449055 0.893504i \(-0.351761\pi\)
0.449055 + 0.893504i \(0.351761\pi\)
\(858\) 0 0
\(859\) 14.0000 0.477674 0.238837 0.971060i \(-0.423234\pi\)
0.238837 + 0.971060i \(0.423234\pi\)
\(860\) 0 0
\(861\) 7.47214 0.254650
\(862\) 0 0
\(863\) −50.0689 −1.70436 −0.852182 0.523245i \(-0.824721\pi\)
−0.852182 + 0.523245i \(0.824721\pi\)
\(864\) 0 0
\(865\) 8.85410 0.301048
\(866\) 0 0
\(867\) 28.0000 0.950930
\(868\) 0 0
\(869\) 1.18034 0.0400403
\(870\) 0 0
\(871\) 28.1246 0.952966
\(872\) 0 0
\(873\) −16.4164 −0.555611
\(874\) 0 0
\(875\) −11.9443 −0.403790
\(876\) 0 0
\(877\) 43.4164 1.46607 0.733034 0.680192i \(-0.238104\pi\)
0.733034 + 0.680192i \(0.238104\pi\)
\(878\) 0 0
\(879\) 12.0000 0.404750
\(880\) 0 0
\(881\) 24.1803 0.814656 0.407328 0.913282i \(-0.366461\pi\)
0.407328 + 0.913282i \(0.366461\pi\)
\(882\) 0 0
\(883\) 15.0902 0.507825 0.253912 0.967227i \(-0.418283\pi\)
0.253912 + 0.967227i \(0.418283\pi\)
\(884\) 0 0
\(885\) −18.7984 −0.631900
\(886\) 0 0
\(887\) 14.7984 0.496881 0.248440 0.968647i \(-0.420082\pi\)
0.248440 + 0.968647i \(0.420082\pi\)
\(888\) 0 0
\(889\) 11.2705 0.378001
\(890\) 0 0
\(891\) 2.23607 0.0749111
\(892\) 0 0
\(893\) −15.1246 −0.506126
\(894\) 0 0
\(895\) 7.85410 0.262534
\(896\) 0 0
\(897\) −4.61803 −0.154192
\(898\) 0 0
\(899\) −25.2492 −0.842109
\(900\) 0 0
\(901\) 12.8115 0.426814
\(902\) 0 0
\(903\) −0.618034 −0.0205669
\(904\) 0 0
\(905\) 25.7984 0.857567
\(906\) 0 0
\(907\) −2.02129 −0.0671157 −0.0335579 0.999437i \(-0.510684\pi\)
−0.0335579 + 0.999437i \(0.510684\pi\)
\(908\) 0 0
\(909\) 10.8541 0.360008
\(910\) 0 0
\(911\) −41.5967 −1.37816 −0.689081 0.724684i \(-0.741986\pi\)
−0.689081 + 0.724684i \(0.741986\pi\)
\(912\) 0 0
\(913\) −29.4721 −0.975385
\(914\) 0 0
\(915\) 3.00000 0.0991769
\(916\) 0 0
\(917\) 7.18034 0.237116
\(918\) 0 0
\(919\) 9.87539 0.325759 0.162879 0.986646i \(-0.447922\pi\)
0.162879 + 0.986646i \(0.447922\pi\)
\(920\) 0 0
\(921\) 24.1246 0.794933
\(922\) 0 0
\(923\) 30.5623 1.00597
\(924\) 0 0
\(925\) 26.2016 0.861504
\(926\) 0 0
\(927\) −19.4164 −0.637719
\(928\) 0 0
\(929\) 40.0902 1.31532 0.657658 0.753317i \(-0.271547\pi\)
0.657658 + 0.753317i \(0.271547\pi\)
\(930\) 0 0
\(931\) −5.47214 −0.179342
\(932\) 0 0
\(933\) 0.326238 0.0106806
\(934\) 0 0
\(935\) −24.2705 −0.793731
\(936\) 0 0
\(937\) −10.7082 −0.349822 −0.174911 0.984584i \(-0.555964\pi\)
−0.174911 + 0.984584i \(0.555964\pi\)
\(938\) 0 0
\(939\) 6.47214 0.211210
\(940\) 0 0
\(941\) −32.9443 −1.07395 −0.536976 0.843597i \(-0.680434\pi\)
−0.536976 + 0.843597i \(0.680434\pi\)
\(942\) 0 0
\(943\) 7.47214 0.243326
\(944\) 0 0
\(945\) 1.61803 0.0526346
\(946\) 0 0
\(947\) 1.05573 0.0343066 0.0171533 0.999853i \(-0.494540\pi\)
0.0171533 + 0.999853i \(0.494540\pi\)
\(948\) 0 0
\(949\) −3.27051 −0.106165
\(950\) 0 0
\(951\) −23.5066 −0.762253
\(952\) 0 0
\(953\) −57.1591 −1.85156 −0.925782 0.378059i \(-0.876592\pi\)
−0.925782 + 0.378059i \(0.876592\pi\)
\(954\) 0 0
\(955\) 10.0000 0.323592
\(956\) 0 0
\(957\) −8.41641 −0.272064
\(958\) 0 0
\(959\) 10.4164 0.336363
\(960\) 0 0
\(961\) 14.0000 0.451613
\(962\) 0 0
\(963\) 12.3262 0.397207
\(964\) 0 0
\(965\) 35.1246 1.13070
\(966\) 0 0
\(967\) −5.63932 −0.181348 −0.0906742 0.995881i \(-0.528902\pi\)
−0.0906742 + 0.995881i \(0.528902\pi\)
\(968\) 0 0
\(969\) 36.7082 1.17924
\(970\) 0 0
\(971\) −40.6869 −1.30571 −0.652853 0.757485i \(-0.726428\pi\)
−0.652853 + 0.757485i \(0.726428\pi\)
\(972\) 0 0
\(973\) 13.6180 0.436574
\(974\) 0 0
\(975\) 11.0000 0.352282
\(976\) 0 0
\(977\) 25.2016 0.806271 0.403136 0.915140i \(-0.367920\pi\)
0.403136 + 0.915140i \(0.367920\pi\)
\(978\) 0 0
\(979\) 20.9787 0.670483
\(980\) 0 0
\(981\) −14.2705 −0.455622
\(982\) 0 0
\(983\) 56.5967 1.80516 0.902578 0.430526i \(-0.141672\pi\)
0.902578 + 0.430526i \(0.141672\pi\)
\(984\) 0 0
\(985\) 20.2361 0.644775
\(986\) 0 0
\(987\) 2.76393 0.0879769
\(988\) 0 0
\(989\) −0.618034 −0.0196523
\(990\) 0 0
\(991\) −60.5755 −1.92424 −0.962121 0.272621i \(-0.912109\pi\)
−0.962121 + 0.272621i \(0.912109\pi\)
\(992\) 0 0
\(993\) −13.4164 −0.425757
\(994\) 0 0
\(995\) 29.2705 0.927938
\(996\) 0 0
\(997\) −31.7214 −1.00463 −0.502313 0.864686i \(-0.667517\pi\)
−0.502313 + 0.864686i \(0.667517\pi\)
\(998\) 0 0
\(999\) −11.0000 −0.348025
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.bn.1.2 2
4.3 odd 2 483.2.a.d.1.2 2
12.11 even 2 1449.2.a.h.1.1 2
28.27 even 2 3381.2.a.r.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.a.d.1.2 2 4.3 odd 2
1449.2.a.h.1.1 2 12.11 even 2
3381.2.a.r.1.2 2 28.27 even 2
7728.2.a.bn.1.2 2 1.1 even 1 trivial