Properties

Label 1024.4.a.n.1.1
Level 1024
Weight 4
Character 1024.1
Self dual yes
Analytic conductor 60.418
Analytic rank 0
Dimension 10
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 1024 = 2^{10} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1024.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.4179558459\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 36 x^{8} + 405 x^{6} - 1380 x^{4} + 420 x^{2} - 32\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{22} \)
Twist minimal: no (minimal twist has level 16)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.34476\) of defining polynomial
Character \(\chi\) \(=\) 1024.1

$q$-expansion

\(f(q)\) \(=\) \(q-8.43597 q^{3} -12.2748 q^{5} +1.63924 q^{7} +44.1656 q^{9} +O(q^{10})\) \(q-8.43597 q^{3} -12.2748 q^{5} +1.63924 q^{7} +44.1656 q^{9} +25.7416 q^{11} +13.2187 q^{13} +103.550 q^{15} -53.6113 q^{17} -100.391 q^{19} -13.8286 q^{21} +25.1189 q^{23} +25.6706 q^{25} -144.809 q^{27} -256.105 q^{29} +132.684 q^{31} -217.155 q^{33} -20.1214 q^{35} -247.306 q^{37} -111.512 q^{39} +198.660 q^{41} -404.064 q^{43} -542.124 q^{45} -78.3629 q^{47} -340.313 q^{49} +452.264 q^{51} -743.559 q^{53} -315.973 q^{55} +846.894 q^{57} -65.8036 q^{59} +273.392 q^{61} +72.3982 q^{63} -162.256 q^{65} +399.066 q^{67} -211.903 q^{69} +727.536 q^{71} +106.065 q^{73} -216.556 q^{75} +42.1968 q^{77} +58.9970 q^{79} +29.1298 q^{81} +580.049 q^{83} +658.068 q^{85} +2160.50 q^{87} -768.959 q^{89} +21.6686 q^{91} -1119.32 q^{93} +1232.28 q^{95} -809.953 q^{97} +1136.89 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 28q^{7} + 54q^{9} + O(q^{10}) \) \( 10q + 28q^{7} + 54q^{9} + 124q^{15} + 4q^{17} + 276q^{23} + 50q^{25} + 368q^{31} - 4q^{33} + 732q^{39} + 944q^{47} - 94q^{49} + 1380q^{55} + 108q^{57} + 2628q^{63} - 492q^{65} + 3468q^{71} - 296q^{73} + 4416q^{79} - 482q^{81} + 6036q^{87} + 88q^{89} + 6900q^{95} - 4q^{97} + O(q^{100}) \)

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
\(3\) −8.43597 −1.62350 −0.811752 0.584003i \(-0.801486\pi\)
−0.811752 + 0.584003i \(0.801486\pi\)
\(4\) 0 0
\(5\) −12.2748 −1.09789 −0.548945 0.835858i \(-0.684971\pi\)
−0.548945 + 0.835858i \(0.684971\pi\)
\(6\) 0 0
\(7\) 1.63924 0.0885109 0.0442554 0.999020i \(-0.485908\pi\)
0.0442554 + 0.999020i \(0.485908\pi\)
\(8\) 0 0
\(9\) 44.1656 1.63576
\(10\) 0 0
\(11\) 25.7416 0.705580 0.352790 0.935702i \(-0.385233\pi\)
0.352790 + 0.935702i \(0.385233\pi\)
\(12\) 0 0
\(13\) 13.2187 0.282015 0.141008 0.990009i \(-0.454966\pi\)
0.141008 + 0.990009i \(0.454966\pi\)
\(14\) 0 0
\(15\) 103.550 1.78243
\(16\) 0 0
\(17\) −53.6113 −0.764862 −0.382431 0.923984i \(-0.624913\pi\)
−0.382431 + 0.923984i \(0.624913\pi\)
\(18\) 0 0
\(19\) −100.391 −1.21217 −0.606085 0.795400i \(-0.707261\pi\)
−0.606085 + 0.795400i \(0.707261\pi\)
\(20\) 0 0
\(21\) −13.8286 −0.143698
\(22\) 0 0
\(23\) 25.1189 0.227724 0.113862 0.993497i \(-0.463678\pi\)
0.113862 + 0.993497i \(0.463678\pi\)
\(24\) 0 0
\(25\) 25.6706 0.205365
\(26\) 0 0
\(27\) −144.809 −1.03216
\(28\) 0 0
\(29\) −256.105 −1.63992 −0.819958 0.572424i \(-0.806003\pi\)
−0.819958 + 0.572424i \(0.806003\pi\)
\(30\) 0 0
\(31\) 132.684 0.768733 0.384367 0.923180i \(-0.374420\pi\)
0.384367 + 0.923180i \(0.374420\pi\)
\(32\) 0 0
\(33\) −217.155 −1.14551
\(34\) 0 0
\(35\) −20.1214 −0.0971753
\(36\) 0 0
\(37\) −247.306 −1.09884 −0.549418 0.835548i \(-0.685151\pi\)
−0.549418 + 0.835548i \(0.685151\pi\)
\(38\) 0 0
\(39\) −111.512 −0.457853
\(40\) 0 0
\(41\) 198.660 0.756720 0.378360 0.925658i \(-0.376488\pi\)
0.378360 + 0.925658i \(0.376488\pi\)
\(42\) 0 0
\(43\) −404.064 −1.43300 −0.716502 0.697585i \(-0.754258\pi\)
−0.716502 + 0.697585i \(0.754258\pi\)
\(44\) 0 0
\(45\) −542.124 −1.79589
\(46\) 0 0
\(47\) −78.3629 −0.243200 −0.121600 0.992579i \(-0.538803\pi\)
−0.121600 + 0.992579i \(0.538803\pi\)
\(48\) 0 0
\(49\) −340.313 −0.992166
\(50\) 0 0
\(51\) 452.264 1.24176
\(52\) 0 0
\(53\) −743.559 −1.92709 −0.963544 0.267549i \(-0.913786\pi\)
−0.963544 + 0.267549i \(0.913786\pi\)
\(54\) 0 0
\(55\) −315.973 −0.774650
\(56\) 0 0
\(57\) 846.894 1.96796
\(58\) 0 0
\(59\) −65.8036 −0.145202 −0.0726008 0.997361i \(-0.523130\pi\)
−0.0726008 + 0.997361i \(0.523130\pi\)
\(60\) 0 0
\(61\) 273.392 0.573841 0.286921 0.957954i \(-0.407368\pi\)
0.286921 + 0.957954i \(0.407368\pi\)
\(62\) 0 0
\(63\) 72.3982 0.144783
\(64\) 0 0
\(65\) −162.256 −0.309622
\(66\) 0 0
\(67\) 399.066 0.727667 0.363834 0.931464i \(-0.381468\pi\)
0.363834 + 0.931464i \(0.381468\pi\)
\(68\) 0 0
\(69\) −211.903 −0.369711
\(70\) 0 0
\(71\) 727.536 1.21609 0.608046 0.793901i \(-0.291953\pi\)
0.608046 + 0.793901i \(0.291953\pi\)
\(72\) 0 0
\(73\) 106.065 0.170054 0.0850270 0.996379i \(-0.472902\pi\)
0.0850270 + 0.996379i \(0.472902\pi\)
\(74\) 0 0
\(75\) −216.556 −0.333410
\(76\) 0 0
\(77\) 42.1968 0.0624515
\(78\) 0 0
\(79\) 58.9970 0.0840213 0.0420107 0.999117i \(-0.486624\pi\)
0.0420107 + 0.999117i \(0.486624\pi\)
\(80\) 0 0
\(81\) 29.1298 0.0399586
\(82\) 0 0
\(83\) 580.049 0.767092 0.383546 0.923522i \(-0.374703\pi\)
0.383546 + 0.923522i \(0.374703\pi\)
\(84\) 0 0
\(85\) 658.068 0.839735
\(86\) 0 0
\(87\) 2160.50 2.66241
\(88\) 0 0
\(89\) −768.959 −0.915837 −0.457918 0.888994i \(-0.651405\pi\)
−0.457918 + 0.888994i \(0.651405\pi\)
\(90\) 0 0
\(91\) 21.6686 0.0249614
\(92\) 0 0
\(93\) −1119.32 −1.24804
\(94\) 0 0
\(95\) 1232.28 1.33083
\(96\) 0 0
\(97\) −809.953 −0.847817 −0.423908 0.905705i \(-0.639342\pi\)
−0.423908 + 0.905705i \(0.639342\pi\)
\(98\) 0 0
\(99\) 1136.89 1.15416
\(100\) 0 0
\(101\) 428.775 0.422422 0.211211 0.977440i \(-0.432259\pi\)
0.211211 + 0.977440i \(0.432259\pi\)
\(102\) 0 0
\(103\) 962.201 0.920471 0.460235 0.887797i \(-0.347765\pi\)
0.460235 + 0.887797i \(0.347765\pi\)
\(104\) 0 0
\(105\) 169.743 0.157764
\(106\) 0 0
\(107\) 1030.02 0.930618 0.465309 0.885148i \(-0.345943\pi\)
0.465309 + 0.885148i \(0.345943\pi\)
\(108\) 0 0
\(109\) 838.993 0.737256 0.368628 0.929577i \(-0.379828\pi\)
0.368628 + 0.929577i \(0.379828\pi\)
\(110\) 0 0
\(111\) 2086.27 1.78396
\(112\) 0 0
\(113\) −351.938 −0.292987 −0.146493 0.989212i \(-0.546799\pi\)
−0.146493 + 0.989212i \(0.546799\pi\)
\(114\) 0 0
\(115\) −308.330 −0.250017
\(116\) 0 0
\(117\) 583.810 0.461310
\(118\) 0 0
\(119\) −87.8821 −0.0676986
\(120\) 0 0
\(121\) −668.370 −0.502157
\(122\) 0 0
\(123\) −1675.89 −1.22854
\(124\) 0 0
\(125\) 1219.25 0.872423
\(126\) 0 0
\(127\) −2365.81 −1.65301 −0.826504 0.562931i \(-0.809674\pi\)
−0.826504 + 0.562931i \(0.809674\pi\)
\(128\) 0 0
\(129\) 3408.67 2.32649
\(130\) 0 0
\(131\) 570.570 0.380541 0.190271 0.981732i \(-0.439063\pi\)
0.190271 + 0.981732i \(0.439063\pi\)
\(132\) 0 0
\(133\) −164.565 −0.107290
\(134\) 0 0
\(135\) 1777.50 1.13320
\(136\) 0 0
\(137\) −856.850 −0.534348 −0.267174 0.963648i \(-0.586090\pi\)
−0.267174 + 0.963648i \(0.586090\pi\)
\(138\) 0 0
\(139\) −2389.08 −1.45783 −0.728916 0.684603i \(-0.759976\pi\)
−0.728916 + 0.684603i \(0.759976\pi\)
\(140\) 0 0
\(141\) 661.067 0.394836
\(142\) 0 0
\(143\) 340.269 0.198984
\(144\) 0 0
\(145\) 3143.64 1.80045
\(146\) 0 0
\(147\) 2870.87 1.61078
\(148\) 0 0
\(149\) −45.5671 −0.0250537 −0.0125269 0.999922i \(-0.503988\pi\)
−0.0125269 + 0.999922i \(0.503988\pi\)
\(150\) 0 0
\(151\) 1077.06 0.580460 0.290230 0.956957i \(-0.406268\pi\)
0.290230 + 0.956957i \(0.406268\pi\)
\(152\) 0 0
\(153\) −2367.78 −1.25113
\(154\) 0 0
\(155\) −1628.67 −0.843986
\(156\) 0 0
\(157\) 2694.94 1.36994 0.684968 0.728573i \(-0.259816\pi\)
0.684968 + 0.728573i \(0.259816\pi\)
\(158\) 0 0
\(159\) 6272.64 3.12863
\(160\) 0 0
\(161\) 41.1761 0.0201561
\(162\) 0 0
\(163\) 1003.85 0.482377 0.241189 0.970478i \(-0.422463\pi\)
0.241189 + 0.970478i \(0.422463\pi\)
\(164\) 0 0
\(165\) 2665.54 1.25765
\(166\) 0 0
\(167\) −3460.66 −1.60356 −0.801778 0.597623i \(-0.796112\pi\)
−0.801778 + 0.597623i \(0.796112\pi\)
\(168\) 0 0
\(169\) −2022.27 −0.920467
\(170\) 0 0
\(171\) −4433.82 −1.98282
\(172\) 0 0
\(173\) 1843.49 0.810162 0.405081 0.914281i \(-0.367243\pi\)
0.405081 + 0.914281i \(0.367243\pi\)
\(174\) 0 0
\(175\) 42.0803 0.0181770
\(176\) 0 0
\(177\) 555.117 0.235735
\(178\) 0 0
\(179\) 1530.67 0.639149 0.319575 0.947561i \(-0.396460\pi\)
0.319575 + 0.947561i \(0.396460\pi\)
\(180\) 0 0
\(181\) 4163.35 1.70972 0.854859 0.518860i \(-0.173643\pi\)
0.854859 + 0.518860i \(0.173643\pi\)
\(182\) 0 0
\(183\) −2306.33 −0.931633
\(184\) 0 0
\(185\) 3035.64 1.20640
\(186\) 0 0
\(187\) −1380.04 −0.539672
\(188\) 0 0
\(189\) −237.377 −0.0913577
\(190\) 0 0
\(191\) −430.650 −0.163145 −0.0815726 0.996667i \(-0.525994\pi\)
−0.0815726 + 0.996667i \(0.525994\pi\)
\(192\) 0 0
\(193\) −2266.98 −0.845497 −0.422749 0.906247i \(-0.638935\pi\)
−0.422749 + 0.906247i \(0.638935\pi\)
\(194\) 0 0
\(195\) 1368.79 0.502672
\(196\) 0 0
\(197\) 1469.95 0.531621 0.265810 0.964025i \(-0.414360\pi\)
0.265810 + 0.964025i \(0.414360\pi\)
\(198\) 0 0
\(199\) 4989.44 1.77735 0.888674 0.458540i \(-0.151627\pi\)
0.888674 + 0.458540i \(0.151627\pi\)
\(200\) 0 0
\(201\) −3366.51 −1.18137
\(202\) 0 0
\(203\) −419.819 −0.145150
\(204\) 0 0
\(205\) −2438.51 −0.830796
\(206\) 0 0
\(207\) 1109.39 0.372503
\(208\) 0 0
\(209\) −2584.22 −0.855283
\(210\) 0 0
\(211\) −3749.79 −1.22344 −0.611720 0.791074i \(-0.709522\pi\)
−0.611720 + 0.791074i \(0.709522\pi\)
\(212\) 0 0
\(213\) −6137.47 −1.97433
\(214\) 0 0
\(215\) 4959.80 1.57328
\(216\) 0 0
\(217\) 217.501 0.0680413
\(218\) 0 0
\(219\) −894.760 −0.276083
\(220\) 0 0
\(221\) −708.670 −0.215703
\(222\) 0 0
\(223\) 3690.85 1.10833 0.554165 0.832407i \(-0.313038\pi\)
0.554165 + 0.832407i \(0.313038\pi\)
\(224\) 0 0
\(225\) 1133.76 0.335928
\(226\) 0 0
\(227\) −2418.90 −0.707259 −0.353630 0.935385i \(-0.615053\pi\)
−0.353630 + 0.935385i \(0.615053\pi\)
\(228\) 0 0
\(229\) 129.827 0.0374637 0.0187318 0.999825i \(-0.494037\pi\)
0.0187318 + 0.999825i \(0.494037\pi\)
\(230\) 0 0
\(231\) −355.971 −0.101390
\(232\) 0 0
\(233\) 4259.71 1.19769 0.598847 0.800863i \(-0.295626\pi\)
0.598847 + 0.800863i \(0.295626\pi\)
\(234\) 0 0
\(235\) 961.889 0.267007
\(236\) 0 0
\(237\) −497.697 −0.136409
\(238\) 0 0
\(239\) 5053.12 1.36761 0.683806 0.729664i \(-0.260324\pi\)
0.683806 + 0.729664i \(0.260324\pi\)
\(240\) 0 0
\(241\) −48.8379 −0.0130536 −0.00652681 0.999979i \(-0.502078\pi\)
−0.00652681 + 0.999979i \(0.502078\pi\)
\(242\) 0 0
\(243\) 3664.09 0.967291
\(244\) 0 0
\(245\) 4177.27 1.08929
\(246\) 0 0
\(247\) −1327.03 −0.341850
\(248\) 0 0
\(249\) −4893.28 −1.24538
\(250\) 0 0
\(251\) 3683.69 0.926345 0.463172 0.886268i \(-0.346711\pi\)
0.463172 + 0.886268i \(0.346711\pi\)
\(252\) 0 0
\(253\) 646.602 0.160678
\(254\) 0 0
\(255\) −5551.44 −1.36331
\(256\) 0 0
\(257\) 739.054 0.179381 0.0896905 0.995970i \(-0.471412\pi\)
0.0896905 + 0.995970i \(0.471412\pi\)
\(258\) 0 0
\(259\) −405.396 −0.0972589
\(260\) 0 0
\(261\) −11311.0 −2.68251
\(262\) 0 0
\(263\) 2448.30 0.574025 0.287012 0.957927i \(-0.407338\pi\)
0.287012 + 0.957927i \(0.407338\pi\)
\(264\) 0 0
\(265\) 9127.03 2.11573
\(266\) 0 0
\(267\) 6486.91 1.48686
\(268\) 0 0
\(269\) 1173.73 0.266035 0.133018 0.991114i \(-0.457533\pi\)
0.133018 + 0.991114i \(0.457533\pi\)
\(270\) 0 0
\(271\) −1404.85 −0.314902 −0.157451 0.987527i \(-0.550328\pi\)
−0.157451 + 0.987527i \(0.550328\pi\)
\(272\) 0 0
\(273\) −182.796 −0.0405249
\(274\) 0 0
\(275\) 660.801 0.144901
\(276\) 0 0
\(277\) −3175.88 −0.688881 −0.344440 0.938808i \(-0.611931\pi\)
−0.344440 + 0.938808i \(0.611931\pi\)
\(278\) 0 0
\(279\) 5860.07 1.25747
\(280\) 0 0
\(281\) 6045.97 1.28353 0.641766 0.766900i \(-0.278202\pi\)
0.641766 + 0.766900i \(0.278202\pi\)
\(282\) 0 0
\(283\) −3478.45 −0.730644 −0.365322 0.930881i \(-0.619041\pi\)
−0.365322 + 0.930881i \(0.619041\pi\)
\(284\) 0 0
\(285\) −10395.4 −2.16061
\(286\) 0 0
\(287\) 325.653 0.0669780
\(288\) 0 0
\(289\) −2038.82 −0.414986
\(290\) 0 0
\(291\) 6832.74 1.37643
\(292\) 0 0
\(293\) 2619.82 0.522360 0.261180 0.965290i \(-0.415888\pi\)
0.261180 + 0.965290i \(0.415888\pi\)
\(294\) 0 0
\(295\) 807.725 0.159415
\(296\) 0 0
\(297\) −3727.60 −0.728275
\(298\) 0 0
\(299\) 332.039 0.0642217
\(300\) 0 0
\(301\) −662.360 −0.126837
\(302\) 0 0
\(303\) −3617.13 −0.685804
\(304\) 0 0
\(305\) −3355.83 −0.630015
\(306\) 0 0
\(307\) −2980.24 −0.554044 −0.277022 0.960864i \(-0.589347\pi\)
−0.277022 + 0.960864i \(0.589347\pi\)
\(308\) 0 0
\(309\) −8117.10 −1.49439
\(310\) 0 0
\(311\) −5294.90 −0.965422 −0.482711 0.875780i \(-0.660348\pi\)
−0.482711 + 0.875780i \(0.660348\pi\)
\(312\) 0 0
\(313\) 4005.87 0.723403 0.361702 0.932294i \(-0.382196\pi\)
0.361702 + 0.932294i \(0.382196\pi\)
\(314\) 0 0
\(315\) −888.673 −0.158956
\(316\) 0 0
\(317\) 1144.44 0.202770 0.101385 0.994847i \(-0.467673\pi\)
0.101385 + 0.994847i \(0.467673\pi\)
\(318\) 0 0
\(319\) −6592.56 −1.15709
\(320\) 0 0
\(321\) −8689.25 −1.51086
\(322\) 0 0
\(323\) 5382.08 0.927143
\(324\) 0 0
\(325\) 339.331 0.0579159
\(326\) 0 0
\(327\) −7077.72 −1.19694
\(328\) 0 0
\(329\) −128.456 −0.0215259
\(330\) 0 0
\(331\) −5981.64 −0.993295 −0.496648 0.867952i \(-0.665436\pi\)
−0.496648 + 0.867952i \(0.665436\pi\)
\(332\) 0 0
\(333\) −10922.4 −1.79744
\(334\) 0 0
\(335\) −4898.46 −0.798899
\(336\) 0 0
\(337\) −10002.6 −1.61684 −0.808419 0.588607i \(-0.799677\pi\)
−0.808419 + 0.588607i \(0.799677\pi\)
\(338\) 0 0
\(339\) 2968.94 0.475665
\(340\) 0 0
\(341\) 3415.50 0.542403
\(342\) 0 0
\(343\) −1120.12 −0.176328
\(344\) 0 0
\(345\) 2601.06 0.405903
\(346\) 0 0
\(347\) 9064.38 1.40231 0.701155 0.713009i \(-0.252668\pi\)
0.701155 + 0.713009i \(0.252668\pi\)
\(348\) 0 0
\(349\) −7782.74 −1.19370 −0.596849 0.802354i \(-0.703581\pi\)
−0.596849 + 0.802354i \(0.703581\pi\)
\(350\) 0 0
\(351\) −1914.18 −0.291086
\(352\) 0 0
\(353\) −1411.35 −0.212800 −0.106400 0.994323i \(-0.533932\pi\)
−0.106400 + 0.994323i \(0.533932\pi\)
\(354\) 0 0
\(355\) −8930.35 −1.33514
\(356\) 0 0
\(357\) 741.371 0.109909
\(358\) 0 0
\(359\) −2160.73 −0.317658 −0.158829 0.987306i \(-0.550772\pi\)
−0.158829 + 0.987306i \(0.550772\pi\)
\(360\) 0 0
\(361\) 3219.31 0.469355
\(362\) 0 0
\(363\) 5638.35 0.815253
\(364\) 0 0
\(365\) −1301.92 −0.186701
\(366\) 0 0
\(367\) −10757.7 −1.53010 −0.765052 0.643969i \(-0.777287\pi\)
−0.765052 + 0.643969i \(0.777287\pi\)
\(368\) 0 0
\(369\) 8773.95 1.23782
\(370\) 0 0
\(371\) −1218.87 −0.170568
\(372\) 0 0
\(373\) 1989.79 0.276213 0.138107 0.990417i \(-0.455898\pi\)
0.138107 + 0.990417i \(0.455898\pi\)
\(374\) 0 0
\(375\) −10285.5 −1.41638
\(376\) 0 0
\(377\) −3385.37 −0.462481
\(378\) 0 0
\(379\) 1622.04 0.219838 0.109919 0.993941i \(-0.464941\pi\)
0.109919 + 0.993941i \(0.464941\pi\)
\(380\) 0 0
\(381\) 19957.9 2.68366
\(382\) 0 0
\(383\) 9042.17 1.20635 0.603176 0.797608i \(-0.293902\pi\)
0.603176 + 0.797608i \(0.293902\pi\)
\(384\) 0 0
\(385\) −517.957 −0.0685650
\(386\) 0 0
\(387\) −17845.7 −2.34406
\(388\) 0 0
\(389\) −3642.08 −0.474706 −0.237353 0.971423i \(-0.576280\pi\)
−0.237353 + 0.971423i \(0.576280\pi\)
\(390\) 0 0
\(391\) −1346.66 −0.174178
\(392\) 0 0
\(393\) −4813.31 −0.617810
\(394\) 0 0
\(395\) −724.176 −0.0922462
\(396\) 0 0
\(397\) 10071.3 1.27320 0.636602 0.771192i \(-0.280339\pi\)
0.636602 + 0.771192i \(0.280339\pi\)
\(398\) 0 0
\(399\) 1388.27 0.174186
\(400\) 0 0
\(401\) 3025.14 0.376729 0.188365 0.982099i \(-0.439681\pi\)
0.188365 + 0.982099i \(0.439681\pi\)
\(402\) 0 0
\(403\) 1753.90 0.216794
\(404\) 0 0
\(405\) −357.562 −0.0438702
\(406\) 0 0
\(407\) −6366.06 −0.775317
\(408\) 0 0
\(409\) 9440.21 1.14129 0.570646 0.821196i \(-0.306693\pi\)
0.570646 + 0.821196i \(0.306693\pi\)
\(410\) 0 0
\(411\) 7228.36 0.867515
\(412\) 0 0
\(413\) −107.868 −0.0128519
\(414\) 0 0
\(415\) −7119.98 −0.842183
\(416\) 0 0
\(417\) 20154.2 2.36680
\(418\) 0 0
\(419\) −4604.25 −0.536831 −0.268416 0.963303i \(-0.586500\pi\)
−0.268416 + 0.963303i \(0.586500\pi\)
\(420\) 0 0
\(421\) 13347.4 1.54516 0.772580 0.634917i \(-0.218966\pi\)
0.772580 + 0.634917i \(0.218966\pi\)
\(422\) 0 0
\(423\) −3460.95 −0.397818
\(424\) 0 0
\(425\) −1376.23 −0.157076
\(426\) 0 0
\(427\) 448.157 0.0507912
\(428\) 0 0
\(429\) −2870.50 −0.323052
\(430\) 0 0
\(431\) 10617.7 1.18663 0.593314 0.804971i \(-0.297819\pi\)
0.593314 + 0.804971i \(0.297819\pi\)
\(432\) 0 0
\(433\) −706.479 −0.0784093 −0.0392046 0.999231i \(-0.512482\pi\)
−0.0392046 + 0.999231i \(0.512482\pi\)
\(434\) 0 0
\(435\) −26519.7 −2.92303
\(436\) 0 0
\(437\) −2521.71 −0.276041
\(438\) 0 0
\(439\) 13611.8 1.47985 0.739926 0.672688i \(-0.234860\pi\)
0.739926 + 0.672688i \(0.234860\pi\)
\(440\) 0 0
\(441\) −15030.1 −1.62295
\(442\) 0 0
\(443\) −4422.21 −0.474279 −0.237139 0.971476i \(-0.576210\pi\)
−0.237139 + 0.971476i \(0.576210\pi\)
\(444\) 0 0
\(445\) 9438.81 1.00549
\(446\) 0 0
\(447\) 384.403 0.0406748
\(448\) 0 0
\(449\) −5231.76 −0.549893 −0.274947 0.961460i \(-0.588660\pi\)
−0.274947 + 0.961460i \(0.588660\pi\)
\(450\) 0 0
\(451\) 5113.83 0.533927
\(452\) 0 0
\(453\) −9086.01 −0.942379
\(454\) 0 0
\(455\) −265.978 −0.0274049
\(456\) 0 0
\(457\) −6833.10 −0.699429 −0.349715 0.936856i \(-0.613721\pi\)
−0.349715 + 0.936856i \(0.613721\pi\)
\(458\) 0 0
\(459\) 7763.38 0.789463
\(460\) 0 0
\(461\) 8451.19 0.853821 0.426910 0.904294i \(-0.359602\pi\)
0.426910 + 0.904294i \(0.359602\pi\)
\(462\) 0 0
\(463\) 4273.38 0.428943 0.214472 0.976730i \(-0.431197\pi\)
0.214472 + 0.976730i \(0.431197\pi\)
\(464\) 0 0
\(465\) 13739.4 1.37021
\(466\) 0 0
\(467\) −17317.9 −1.71601 −0.858004 0.513643i \(-0.828296\pi\)
−0.858004 + 0.513643i \(0.828296\pi\)
\(468\) 0 0
\(469\) 654.167 0.0644065
\(470\) 0 0
\(471\) −22734.5 −2.22410
\(472\) 0 0
\(473\) −10401.3 −1.01110
\(474\) 0 0
\(475\) −2577.09 −0.248937
\(476\) 0 0
\(477\) −32839.7 −3.15226
\(478\) 0 0
\(479\) −4067.97 −0.388038 −0.194019 0.980998i \(-0.562152\pi\)
−0.194019 + 0.980998i \(0.562152\pi\)
\(480\) 0 0
\(481\) −3269.06 −0.309888
\(482\) 0 0
\(483\) −347.360 −0.0327235
\(484\) 0 0
\(485\) 9942.00 0.930811
\(486\) 0 0
\(487\) −16174.3 −1.50499 −0.752493 0.658600i \(-0.771149\pi\)
−0.752493 + 0.658600i \(0.771149\pi\)
\(488\) 0 0
\(489\) −8468.44 −0.783141
\(490\) 0 0
\(491\) 19228.6 1.76736 0.883680 0.468091i \(-0.155058\pi\)
0.883680 + 0.468091i \(0.155058\pi\)
\(492\) 0 0
\(493\) 13730.1 1.25431
\(494\) 0 0
\(495\) −13955.1 −1.26714
\(496\) 0 0
\(497\) 1192.61 0.107637
\(498\) 0 0
\(499\) 20713.6 1.85825 0.929127 0.369762i \(-0.120561\pi\)
0.929127 + 0.369762i \(0.120561\pi\)
\(500\) 0 0
\(501\) 29194.0 2.60338
\(502\) 0 0
\(503\) 9828.84 0.871265 0.435632 0.900125i \(-0.356525\pi\)
0.435632 + 0.900125i \(0.356525\pi\)
\(504\) 0 0
\(505\) −5263.12 −0.463774
\(506\) 0 0
\(507\) 17059.8 1.49438
\(508\) 0 0
\(509\) 19029.8 1.65714 0.828568 0.559889i \(-0.189156\pi\)
0.828568 + 0.559889i \(0.189156\pi\)
\(510\) 0 0
\(511\) 173.866 0.0150516
\(512\) 0 0
\(513\) 14537.5 1.25116
\(514\) 0 0
\(515\) −11810.8 −1.01058
\(516\) 0 0
\(517\) −2017.19 −0.171597
\(518\) 0 0
\(519\) −15551.6 −1.31530
\(520\) 0 0
\(521\) 10607.1 0.891950 0.445975 0.895045i \(-0.352857\pi\)
0.445975 + 0.895045i \(0.352857\pi\)
\(522\) 0 0
\(523\) 5519.88 0.461506 0.230753 0.973012i \(-0.425881\pi\)
0.230753 + 0.973012i \(0.425881\pi\)
\(524\) 0 0
\(525\) −354.989 −0.0295104
\(526\) 0 0
\(527\) −7113.36 −0.587975
\(528\) 0 0
\(529\) −11536.0 −0.948142
\(530\) 0 0
\(531\) −2906.25 −0.237515
\(532\) 0 0
\(533\) 2626.02 0.213407
\(534\) 0 0
\(535\) −12643.3 −1.02172
\(536\) 0 0
\(537\) −12912.7 −1.03766
\(538\) 0 0
\(539\) −8760.20 −0.700053
\(540\) 0 0
\(541\) 13481.4 1.07137 0.535683 0.844419i \(-0.320054\pi\)
0.535683 + 0.844419i \(0.320054\pi\)
\(542\) 0 0
\(543\) −35121.9 −2.77573
\(544\) 0 0
\(545\) −10298.5 −0.809427
\(546\) 0 0
\(547\) −1743.55 −0.136287 −0.0681434 0.997676i \(-0.521708\pi\)
−0.0681434 + 0.997676i \(0.521708\pi\)
\(548\) 0 0
\(549\) 12074.5 0.938668
\(550\) 0 0
\(551\) 25710.6 1.98786
\(552\) 0 0
\(553\) 96.7105 0.00743680
\(554\) 0 0
\(555\) −25608.5 −1.95860
\(556\) 0 0
\(557\) 4086.47 0.310861 0.155430 0.987847i \(-0.450324\pi\)
0.155430 + 0.987847i \(0.450324\pi\)
\(558\) 0 0
\(559\) −5341.19 −0.404129
\(560\) 0 0
\(561\) 11642.0 0.876159
\(562\) 0 0
\(563\) −99.1014 −0.00741852 −0.00370926 0.999993i \(-0.501181\pi\)
−0.00370926 + 0.999993i \(0.501181\pi\)
\(564\) 0 0
\(565\) 4319.96 0.321668
\(566\) 0 0
\(567\) 47.7509 0.00353677
\(568\) 0 0
\(569\) 8915.23 0.656847 0.328423 0.944531i \(-0.393483\pi\)
0.328423 + 0.944531i \(0.393483\pi\)
\(570\) 0 0
\(571\) −6995.13 −0.512674 −0.256337 0.966587i \(-0.582516\pi\)
−0.256337 + 0.966587i \(0.582516\pi\)
\(572\) 0 0
\(573\) 3632.95 0.264867
\(574\) 0 0
\(575\) 644.818 0.0467665
\(576\) 0 0
\(577\) 17911.5 1.29232 0.646159 0.763203i \(-0.276374\pi\)
0.646159 + 0.763203i \(0.276374\pi\)
\(578\) 0 0
\(579\) 19124.2 1.37267
\(580\) 0 0
\(581\) 950.842 0.0678960
\(582\) 0 0
\(583\) −19140.4 −1.35972
\(584\) 0 0
\(585\) −7166.15 −0.506468
\(586\) 0 0
\(587\) 11229.2 0.789573 0.394787 0.918773i \(-0.370819\pi\)
0.394787 + 0.918773i \(0.370819\pi\)
\(588\) 0 0
\(589\) −13320.2 −0.931835
\(590\) 0 0
\(591\) −12400.4 −0.863088
\(592\) 0 0
\(593\) 7006.26 0.485181 0.242591 0.970129i \(-0.422003\pi\)
0.242591 + 0.970129i \(0.422003\pi\)
\(594\) 0 0
\(595\) 1078.73 0.0743257
\(596\) 0 0
\(597\) −42090.8 −2.88553
\(598\) 0 0
\(599\) −8502.74 −0.579987 −0.289994 0.957029i \(-0.593653\pi\)
−0.289994 + 0.957029i \(0.593653\pi\)
\(600\) 0 0
\(601\) 11936.2 0.810127 0.405063 0.914289i \(-0.367249\pi\)
0.405063 + 0.914289i \(0.367249\pi\)
\(602\) 0 0
\(603\) 17625.0 1.19029
\(604\) 0 0
\(605\) 8204.11 0.551313
\(606\) 0 0
\(607\) −3850.00 −0.257441 −0.128721 0.991681i \(-0.541087\pi\)
−0.128721 + 0.991681i \(0.541087\pi\)
\(608\) 0 0
\(609\) 3541.58 0.235652
\(610\) 0 0
\(611\) −1035.85 −0.0685861
\(612\) 0 0
\(613\) 8938.34 0.588933 0.294467 0.955662i \(-0.404858\pi\)
0.294467 + 0.955662i \(0.404858\pi\)
\(614\) 0 0
\(615\) 20571.2 1.34880
\(616\) 0 0
\(617\) −2585.09 −0.168674 −0.0843370 0.996437i \(-0.526877\pi\)
−0.0843370 + 0.996437i \(0.526877\pi\)
\(618\) 0 0
\(619\) 10359.1 0.672648 0.336324 0.941746i \(-0.390816\pi\)
0.336324 + 0.941746i \(0.390816\pi\)
\(620\) 0 0
\(621\) −3637.44 −0.235049
\(622\) 0 0
\(623\) −1260.51 −0.0810615
\(624\) 0 0
\(625\) −18174.8 −1.16319
\(626\) 0 0
\(627\) 21800.4 1.38855
\(628\) 0 0
\(629\) 13258.4 0.840458
\(630\) 0 0
\(631\) 14411.5 0.909210 0.454605 0.890693i \(-0.349780\pi\)
0.454605 + 0.890693i \(0.349780\pi\)
\(632\) 0 0
\(633\) 31633.1 1.98626
\(634\) 0 0
\(635\) 29039.9 1.81482
\(636\) 0 0
\(637\) −4498.48 −0.279806
\(638\) 0 0
\(639\) 32132.1 1.98924
\(640\) 0 0
\(641\) −25724.0 −1.58508 −0.792542 0.609818i \(-0.791243\pi\)
−0.792542 + 0.609818i \(0.791243\pi\)
\(642\) 0 0
\(643\) −11081.4 −0.679640 −0.339820 0.940491i \(-0.610366\pi\)
−0.339820 + 0.940491i \(0.610366\pi\)
\(644\) 0 0
\(645\) −41840.8 −2.55423
\(646\) 0 0
\(647\) 1247.43 0.0757981 0.0378991 0.999282i \(-0.487933\pi\)
0.0378991 + 0.999282i \(0.487933\pi\)
\(648\) 0 0
\(649\) −1693.89 −0.102451
\(650\) 0 0
\(651\) −1834.84 −0.110465
\(652\) 0 0
\(653\) −11741.1 −0.703621 −0.351811 0.936071i \(-0.614434\pi\)
−0.351811 + 0.936071i \(0.614434\pi\)
\(654\) 0 0
\(655\) −7003.63 −0.417793
\(656\) 0 0
\(657\) 4684.42 0.278168
\(658\) 0 0
\(659\) −2398.73 −0.141792 −0.0708961 0.997484i \(-0.522586\pi\)
−0.0708961 + 0.997484i \(0.522586\pi\)
\(660\) 0 0
\(661\) −12428.5 −0.731337 −0.365669 0.930745i \(-0.619160\pi\)
−0.365669 + 0.930745i \(0.619160\pi\)
\(662\) 0 0
\(663\) 5978.32 0.350194
\(664\) 0 0
\(665\) 2020.00 0.117793
\(666\) 0 0
\(667\) −6433.09 −0.373449
\(668\) 0 0
\(669\) −31135.9 −1.79938
\(670\) 0 0
\(671\) 7037.56 0.404891
\(672\) 0 0
\(673\) −23869.3 −1.36716 −0.683578 0.729878i \(-0.739577\pi\)
−0.683578 + 0.729878i \(0.739577\pi\)
\(674\) 0 0
\(675\) −3717.32 −0.211970
\(676\) 0 0
\(677\) 8009.12 0.454676 0.227338 0.973816i \(-0.426998\pi\)
0.227338 + 0.973816i \(0.426998\pi\)
\(678\) 0 0
\(679\) −1327.71 −0.0750410
\(680\) 0 0
\(681\) 20405.8 1.14824
\(682\) 0 0
\(683\) −12944.0 −0.725167 −0.362583 0.931951i \(-0.618105\pi\)
−0.362583 + 0.931951i \(0.618105\pi\)
\(684\) 0 0
\(685\) 10517.7 0.586655
\(686\) 0 0
\(687\) −1095.21 −0.0608224
\(688\) 0 0
\(689\) −9828.85 −0.543468
\(690\) 0 0
\(691\) −24123.5 −1.32807 −0.664037 0.747699i \(-0.731158\pi\)
−0.664037 + 0.747699i \(0.731158\pi\)
\(692\) 0 0
\(693\) 1863.65 0.102156
\(694\) 0 0
\(695\) 29325.4 1.60054
\(696\) 0 0
\(697\) −10650.4 −0.578787
\(698\) 0 0
\(699\) −35934.8 −1.94446
\(700\) 0 0
\(701\) −10918.3 −0.588274 −0.294137 0.955763i \(-0.595032\pi\)
−0.294137 + 0.955763i \(0.595032\pi\)
\(702\) 0 0
\(703\) 24827.3 1.33198
\(704\) 0 0
\(705\) −8114.47 −0.433487
\(706\) 0 0
\(707\) 702.866 0.0373890
\(708\) 0 0
\(709\) −6473.79 −0.342917 −0.171459 0.985191i \(-0.554848\pi\)
−0.171459 + 0.985191i \(0.554848\pi\)
\(710\) 0 0
\(711\) 2605.64 0.137439
\(712\) 0 0
\(713\) 3332.88 0.175059
\(714\) 0 0
\(715\) −4176.74 −0.218463
\(716\) 0 0
\(717\) −42628.0 −2.22032
\(718\) 0 0
\(719\) 30210.0 1.56696 0.783479 0.621418i \(-0.213443\pi\)
0.783479 + 0.621418i \(0.213443\pi\)
\(720\) 0 0
\(721\) 1577.28 0.0814717
\(722\) 0 0
\(723\) 411.995 0.0211926
\(724\) 0 0
\(725\) −6574.37 −0.336781
\(726\) 0 0
\(727\) −20721.3 −1.05710 −0.528549 0.848903i \(-0.677264\pi\)
−0.528549 + 0.848903i \(0.677264\pi\)
\(728\) 0 0
\(729\) −31696.7 −1.61036
\(730\) 0 0
\(731\) 21662.4 1.09605
\(732\) 0 0
\(733\) −19629.0 −0.989107 −0.494553 0.869147i \(-0.664668\pi\)
−0.494553 + 0.869147i \(0.664668\pi\)
\(734\) 0 0
\(735\) −35239.3 −1.76847
\(736\) 0 0
\(737\) 10272.6 0.513428
\(738\) 0 0
\(739\) 12436.5 0.619058 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(740\) 0 0
\(741\) 11194.8 0.554995
\(742\) 0 0
\(743\) 7669.27 0.378678 0.189339 0.981912i \(-0.439365\pi\)
0.189339 + 0.981912i \(0.439365\pi\)
\(744\) 0 0
\(745\) 559.327 0.0275062
\(746\) 0 0
\(747\) 25618.2 1.25478
\(748\) 0 0
\(749\) 1688.46 0.0823698
\(750\) 0 0
\(751\) 26531.8 1.28916 0.644580 0.764537i \(-0.277032\pi\)
0.644580 + 0.764537i \(0.277032\pi\)
\(752\) 0 0
\(753\) −31075.5 −1.50392
\(754\) 0 0
\(755\) −13220.6 −0.637282
\(756\) 0 0
\(757\) −112.316 −0.00539258 −0.00269629 0.999996i \(-0.500858\pi\)
−0.00269629 + 0.999996i \(0.500858\pi\)
\(758\) 0 0
\(759\) −5454.71 −0.260861
\(760\) 0 0
\(761\) 36991.3 1.76207 0.881033 0.473055i \(-0.156849\pi\)
0.881033 + 0.473055i \(0.156849\pi\)
\(762\) 0 0
\(763\) 1375.31 0.0652552
\(764\) 0 0
\(765\) 29064.0 1.37361
\(766\) 0 0
\(767\) −869.835 −0.0409490
\(768\) 0 0
\(769\) 26637.0 1.24910 0.624548 0.780987i \(-0.285283\pi\)
0.624548 + 0.780987i \(0.285283\pi\)
\(770\) 0 0
\(771\) −6234.64 −0.291226
\(772\) 0 0
\(773\) −27921.7 −1.29919 −0.649596 0.760280i \(-0.725062\pi\)
−0.649596 + 0.760280i \(0.725062\pi\)
\(774\) 0 0
\(775\) 3406.07 0.157871
\(776\) 0 0
\(777\) 3419.91 0.157900
\(778\) 0 0
\(779\) −19943.7 −0.917273
\(780\) 0 0
\(781\) 18727.9 0.858051
\(782\) 0 0
\(783\) 37086.2 1.69266
\(784\) 0 0
\(785\) −33079.9 −1.50404
\(786\) 0 0
\(787\) 40839.8 1.84979 0.924893 0.380229i \(-0.124155\pi\)
0.924893 + 0.380229i \(0.124155\pi\)
\(788\) 0 0
\(789\) −20653.8 −0.931931
\(790\) 0 0
\(791\) −576.912 −0.0259325
\(792\) 0 0
\(793\) 3613.88 0.161832
\(794\) 0 0
\(795\) −76995.4 −3.43490
\(796\) 0 0
\(797\) −23555.1 −1.04688 −0.523440 0.852062i \(-0.675352\pi\)
−0.523440 + 0.852062i \(0.675352\pi\)
\(798\) 0 0
\(799\) 4201.14 0.186015
\(800\) 0 0
\(801\) −33961.5 −1.49809
\(802\) 0 0
\(803\) 2730.28 0.119987
\(804\) 0 0
\(805\) −505.428 −0.0221292
\(806\) 0 0
\(807\) −9901.55 −0.431910
\(808\) 0 0
\(809\) 34940.4 1.51847 0.759233 0.650819i \(-0.225574\pi\)
0.759233 + 0.650819i \(0.225574\pi\)
\(810\) 0 0
\(811\) −21451.0 −0.928789 −0.464395 0.885628i \(-0.653728\pi\)
−0.464395 + 0.885628i \(0.653728\pi\)
\(812\) 0 0
\(813\) 11851.3 0.511244
\(814\) 0 0
\(815\) −12322.0 −0.529597
\(816\) 0 0
\(817\) 40564.3 1.73705
\(818\) 0 0
\(819\) 957.008 0.0408310
\(820\) 0 0
\(821\) 12318.6 0.523656 0.261828 0.965115i \(-0.415675\pi\)
0.261828 + 0.965115i \(0.415675\pi\)
\(822\) 0 0
\(823\) −24493.5 −1.03741 −0.518707 0.854952i \(-0.673586\pi\)
−0.518707 + 0.854952i \(0.673586\pi\)
\(824\) 0 0
\(825\) −5574.50 −0.235248
\(826\) 0 0
\(827\) −37233.4 −1.56558 −0.782789 0.622287i \(-0.786204\pi\)
−0.782789 + 0.622287i \(0.786204\pi\)
\(828\) 0 0
\(829\) 12881.0 0.539658 0.269829 0.962908i \(-0.413033\pi\)
0.269829 + 0.962908i \(0.413033\pi\)
\(830\) 0 0
\(831\) 26791.6 1.11840
\(832\) 0 0
\(833\) 18244.6 0.758870
\(834\) 0 0
\(835\) 42478.9 1.76053
\(836\) 0 0
\(837\) −19213.8 −0.793459
\(838\) 0 0
\(839\) 1394.89 0.0573982 0.0286991 0.999588i \(-0.490864\pi\)
0.0286991 + 0.999588i \(0.490864\pi\)
\(840\) 0 0
\(841\) 41200.9 1.68932
\(842\) 0 0
\(843\) −51003.7 −2.08382
\(844\) 0 0
\(845\) 24822.9 1.01057
\(846\) 0 0
\(847\) −1095.62 −0.0444463
\(848\) 0 0
\(849\) 29344.1 1.18620
\(850\) 0 0
\(851\) −6212.08 −0.250232
\(852\) 0 0
\(853\) 21615.8 0.867658 0.433829 0.900995i \(-0.357162\pi\)
0.433829 + 0.900995i \(0.357162\pi\)
\(854\) 0 0
\(855\) 54424.2 2.17692
\(856\) 0 0
\(857\) −2273.70 −0.0906277 −0.0453139 0.998973i \(-0.514429\pi\)
−0.0453139 + 0.998973i \(0.514429\pi\)
\(858\) 0 0
\(859\) 30652.1 1.21750 0.608752 0.793361i \(-0.291671\pi\)
0.608752 + 0.793361i \(0.291671\pi\)
\(860\) 0 0
\(861\) −2747.20 −0.108739
\(862\) 0 0
\(863\) 23721.7 0.935686 0.467843 0.883812i \(-0.345031\pi\)
0.467843 + 0.883812i \(0.345031\pi\)
\(864\) 0 0
\(865\) −22628.5 −0.889469
\(866\) 0 0
\(867\) 17199.5 0.673731
\(868\) 0 0
\(869\) 1518.68 0.0592838
\(870\) 0 0
\(871\) 5275.12 0.205213
\(872\) 0 0
\(873\) −35772.1 −1.38683
\(874\) 0 0
\(875\) 1998.65 0.0772189
\(876\) 0 0
\(877\) −31720.2 −1.22134 −0.610670 0.791885i \(-0.709100\pi\)
−0.610670 + 0.791885i \(0.709100\pi\)
\(878\) 0 0
\(879\) −22100.7 −0.848054
\(880\) 0 0
\(881\) −24603.0 −0.940859 −0.470429 0.882438i \(-0.655901\pi\)
−0.470429 + 0.882438i \(0.655901\pi\)
\(882\) 0 0
\(883\) 33215.2 1.26589 0.632945 0.774197i \(-0.281846\pi\)
0.632945 + 0.774197i \(0.281846\pi\)
\(884\) 0 0
\(885\) −6813.95 −0.258812
\(886\) 0 0
\(887\) 39722.9 1.50368 0.751841 0.659345i \(-0.229166\pi\)
0.751841 + 0.659345i \(0.229166\pi\)
\(888\) 0 0
\(889\) −3878.15 −0.146309
\(890\) 0 0
\(891\) 749.848 0.0281940
\(892\) 0 0
\(893\) 7866.92 0.294800
\(894\) 0 0
\(895\) −18788.7 −0.701716
\(896\) 0 0
\(897\) −2801.07 −0.104264
\(898\) 0 0
\(899\) −33981.1 −1.26066
\(900\) 0 0
\(901\) 39863.2 1.47396
\(902\) 0 0
\(903\) 5587.65 0.205920
\(904\) 0 0
\(905\) −51104.2 −1.87708
\(906\) 0 0
\(907\) 6456.50 0.236367 0.118183 0.992992i \(-0.462293\pi\)
0.118183 + 0.992992i \(0.462293\pi\)
\(908\) 0 0
\(909\) 18937.1 0.690983
\(910\) 0 0
\(911\) 2013.95 0.0732438 0.0366219 0.999329i \(-0.488340\pi\)
0.0366219 + 0.999329i \(0.488340\pi\)
\(912\) 0 0
\(913\) 14931.4 0.541245
\(914\) 0 0
\(915\) 28309.7 1.02283
\(916\) 0 0
\(917\) 935.303 0.0336820
\(918\) 0 0
\(919\) −37746.5 −1.35489 −0.677443 0.735575i \(-0.736912\pi\)
−0.677443 + 0.735575i \(0.736912\pi\)
\(920\) 0 0
\(921\) 25141.2 0.899492
\(922\) 0 0
\(923\) 9617.05 0.342957
\(924\) 0 0
\(925\) −6348.50 −0.225662
\(926\) 0 0
\(927\) 42496.2 1.50567
\(928\) 0 0
\(929\) 45643.5 1.61197 0.805983 0.591939i \(-0.201637\pi\)
0.805983 + 0.591939i \(0.201637\pi\)
\(930\) 0 0
\(931\) 34164.3 1.20267
\(932\) 0 0
\(933\) 44667.6 1.56737
\(934\) 0 0
\(935\) 16939.7 0.592501
\(936\) 0 0
\(937\) −47317.5 −1.64973 −0.824864 0.565331i \(-0.808748\pi\)
−0.824864 + 0.565331i \(0.808748\pi\)
\(938\) 0 0
\(939\) −33793.4 −1.17445
\(940\) 0 0
\(941\) −21318.9 −0.738550 −0.369275 0.929320i \(-0.620394\pi\)
−0.369275 + 0.929320i \(0.620394\pi\)
\(942\) 0 0
\(943\) 4990.14 0.172324
\(944\) 0 0
\(945\) 2913.75 0.100301
\(946\) 0 0
\(947\) −20601.8 −0.706937 −0.353468 0.935446i \(-0.614998\pi\)
−0.353468 + 0.935446i \(0.614998\pi\)
\(948\) 0 0
\(949\) 1402.03 0.0479578
\(950\) 0 0
\(951\) −9654.45 −0.329198
\(952\) 0 0
\(953\) −42987.2 −1.46117 −0.730583 0.682824i \(-0.760752\pi\)
−0.730583 + 0.682824i \(0.760752\pi\)
\(954\) 0 0
\(955\) 5286.14 0.179116
\(956\) 0 0
\(957\) 55614.6 1.87854
\(958\) 0 0
\(959\) −1404.59 −0.0472956
\(960\) 0 0
\(961\) −12186.0 −0.409049
\(962\) 0 0
\(963\) 45491.6 1.52227
\(964\) 0 0
\(965\) 27826.7 0.928264
\(966\) 0 0
\(967\) 44030.7 1.46425 0.732126 0.681170i \(-0.238528\pi\)
0.732126 + 0.681170i \(0.238528\pi\)
\(968\) 0 0
\(969\) −45403.1 −1.50522
\(970\) 0 0
\(971\) 50487.1 1.66860 0.834298 0.551313i \(-0.185873\pi\)
0.834298 + 0.551313i \(0.185873\pi\)
\(972\) 0 0
\(973\) −3916.28 −0.129034
\(974\) 0 0
\(975\) −2862.58 −0.0940267
\(976\) 0 0
\(977\) 49515.3 1.62143 0.810714 0.585442i \(-0.199079\pi\)
0.810714 + 0.585442i \(0.199079\pi\)
\(978\) 0 0
\(979\) −19794.2 −0.646196
\(980\) 0 0
\(981\) 37054.6 1.20598
\(982\) 0 0
\(983\) 40046.2 1.29936 0.649682 0.760206i \(-0.274902\pi\)
0.649682 + 0.760206i \(0.274902\pi\)
\(984\) 0 0
\(985\) −18043.3 −0.583662
\(986\) 0 0
\(987\) 1083.65 0.0349473
\(988\) 0 0
\(989\) −10149.7 −0.326330
\(990\) 0 0
\(991\) 18673.2 0.598560 0.299280 0.954165i \(-0.403254\pi\)
0.299280 + 0.954165i \(0.403254\pi\)
\(992\) 0 0
\(993\) 50460.9 1.61262
\(994\) 0 0
\(995\) −61244.4 −1.95133
\(996\) 0 0
\(997\) −31087.3 −0.987508 −0.493754 0.869602i \(-0.664376\pi\)
−0.493754 + 0.869602i \(0.664376\pi\)
\(998\) 0 0
\(999\) 35812.1 1.13418
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 1024.4.a.n.1.1 10
4.3 odd 2 1024.4.a.m.1.10 10
8.3 odd 2 1024.4.a.m.1.1 10
8.5 even 2 inner 1024.4.a.n.1.10 10
16.3 odd 4 1024.4.b.k.513.10 10
16.5 even 4 1024.4.b.j.513.10 10
16.11 odd 4 1024.4.b.k.513.1 10
16.13 even 4 1024.4.b.j.513.1 10
32.3 odd 8 64.4.e.a.17.5 10
32.5 even 8 128.4.e.b.97.5 10
32.11 odd 8 64.4.e.a.49.5 10
32.13 even 8 128.4.e.b.33.5 10
32.19 odd 8 128.4.e.a.33.1 10
32.21 even 8 16.4.e.a.5.2 10
32.27 odd 8 128.4.e.a.97.1 10
32.29 even 8 16.4.e.a.13.2 yes 10
96.11 even 8 576.4.k.a.433.1 10
96.29 odd 8 144.4.k.a.109.4 10
96.35 even 8 576.4.k.a.145.1 10
96.53 odd 8 144.4.k.a.37.4 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.4.e.a.5.2 10 32.21 even 8
16.4.e.a.13.2 yes 10 32.29 even 8
64.4.e.a.17.5 10 32.3 odd 8
64.4.e.a.49.5 10 32.11 odd 8
128.4.e.a.33.1 10 32.19 odd 8
128.4.e.a.97.1 10 32.27 odd 8
128.4.e.b.33.5 10 32.13 even 8
128.4.e.b.97.5 10 32.5 even 8
144.4.k.a.37.4 10 96.53 odd 8
144.4.k.a.109.4 10 96.29 odd 8
576.4.k.a.145.1 10 96.35 even 8
576.4.k.a.433.1 10 96.11 even 8
1024.4.a.m.1.1 10 8.3 odd 2
1024.4.a.m.1.10 10 4.3 odd 2
1024.4.a.n.1.1 10 1.1 even 1 trivial
1024.4.a.n.1.10 10 8.5 even 2 inner
1024.4.b.j.513.1 10 16.13 even 4
1024.4.b.j.513.10 10 16.5 even 4
1024.4.b.k.513.1 10 16.11 odd 4
1024.4.b.k.513.10 10 16.3 odd 4