Properties

Label 1024.4.a.n.1.2
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 1.2
Root \(-0.357936\) of defining polynomial
Character \(\chi\) \(=\) 1024.1

$q$-expansion

\(f(q)\) \(=\) \(q-7.77277 q^{3} -6.59550 q^{5} +24.8965 q^{7} +33.4160 q^{9} +O(q^{10})\) \(q-7.77277 q^{3} -6.59550 q^{5} +24.8965 q^{7} +33.4160 q^{9} -31.5979 q^{11} -15.9402 q^{13} +51.2653 q^{15} +88.4846 q^{17} +53.4838 q^{19} -193.515 q^{21} +48.1224 q^{23} -81.4994 q^{25} -49.8702 q^{27} -14.7689 q^{29} -96.9578 q^{31} +245.604 q^{33} -164.205 q^{35} -230.911 q^{37} +123.899 q^{39} -360.519 q^{41} +141.774 q^{43} -220.395 q^{45} -220.669 q^{47} +276.837 q^{49} -687.771 q^{51} +248.551 q^{53} +208.404 q^{55} -415.717 q^{57} +572.767 q^{59} -939.852 q^{61} +831.942 q^{63} +105.133 q^{65} +151.854 q^{67} -374.045 q^{69} -215.050 q^{71} -668.587 q^{73} +633.476 q^{75} -786.679 q^{77} +822.956 q^{79} -514.603 q^{81} +462.269 q^{83} -583.600 q^{85} +114.795 q^{87} +262.733 q^{89} -396.855 q^{91} +753.631 q^{93} -352.752 q^{95} -150.801 q^{97} -1055.88 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 28q^{7} + 54q^{9} + O(q^{10}) \) \( 10q + 28q^{7} + 54q^{9} + 124q^{15} + 4q^{17} + 276q^{23} + 50q^{25} + 368q^{31} - 4q^{33} + 732q^{39} + 944q^{47} - 94q^{49} + 1380q^{55} + 108q^{57} + 2628q^{63} - 492q^{65} + 3468q^{71} - 296q^{73} + 4416q^{79} - 482q^{81} + 6036q^{87} + 88q^{89} + 6900q^{95} - 4q^{97} + O(q^{100}) \)

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
\(3\) −7.77277 −1.49587 −0.747935 0.663771i \(-0.768955\pi\)
−0.747935 + 0.663771i \(0.768955\pi\)
\(4\) 0 0
\(5\) −6.59550 −0.589919 −0.294960 0.955510i \(-0.595306\pi\)
−0.294960 + 0.955510i \(0.595306\pi\)
\(6\) 0 0
\(7\) 24.8965 1.34429 0.672143 0.740422i \(-0.265374\pi\)
0.672143 + 0.740422i \(0.265374\pi\)
\(8\) 0 0
\(9\) 33.4160 1.23763
\(10\) 0 0
\(11\) −31.5979 −0.866103 −0.433052 0.901369i \(-0.642563\pi\)
−0.433052 + 0.901369i \(0.642563\pi\)
\(12\) 0 0
\(13\) −15.9402 −0.340078 −0.170039 0.985437i \(-0.554389\pi\)
−0.170039 + 0.985437i \(0.554389\pi\)
\(14\) 0 0
\(15\) 51.2653 0.882443
\(16\) 0 0
\(17\) 88.4846 1.26239 0.631196 0.775623i \(-0.282564\pi\)
0.631196 + 0.775623i \(0.282564\pi\)
\(18\) 0 0
\(19\) 53.4838 0.645790 0.322895 0.946435i \(-0.395344\pi\)
0.322895 + 0.946435i \(0.395344\pi\)
\(20\) 0 0
\(21\) −193.515 −2.01088
\(22\) 0 0
\(23\) 48.1224 0.436270 0.218135 0.975919i \(-0.430003\pi\)
0.218135 + 0.975919i \(0.430003\pi\)
\(24\) 0 0
\(25\) −81.4994 −0.651995
\(26\) 0 0
\(27\) −49.8702 −0.355464
\(28\) 0 0
\(29\) −14.7689 −0.0945692 −0.0472846 0.998881i \(-0.515057\pi\)
−0.0472846 + 0.998881i \(0.515057\pi\)
\(30\) 0 0
\(31\) −96.9578 −0.561746 −0.280873 0.959745i \(-0.590624\pi\)
−0.280873 + 0.959745i \(0.590624\pi\)
\(32\) 0 0
\(33\) 245.604 1.29558
\(34\) 0 0
\(35\) −164.205 −0.793020
\(36\) 0 0
\(37\) −230.911 −1.02599 −0.512994 0.858392i \(-0.671464\pi\)
−0.512994 + 0.858392i \(0.671464\pi\)
\(38\) 0 0
\(39\) 123.899 0.508713
\(40\) 0 0
\(41\) −360.519 −1.37326 −0.686629 0.727008i \(-0.740910\pi\)
−0.686629 + 0.727008i \(0.740910\pi\)
\(42\) 0 0
\(43\) 141.774 0.502797 0.251398 0.967884i \(-0.419110\pi\)
0.251398 + 0.967884i \(0.419110\pi\)
\(44\) 0 0
\(45\) −220.395 −0.730102
\(46\) 0 0
\(47\) −220.669 −0.684849 −0.342425 0.939545i \(-0.611248\pi\)
−0.342425 + 0.939545i \(0.611248\pi\)
\(48\) 0 0
\(49\) 276.837 0.807104
\(50\) 0 0
\(51\) −687.771 −1.88838
\(52\) 0 0
\(53\) 248.551 0.644172 0.322086 0.946710i \(-0.395616\pi\)
0.322086 + 0.946710i \(0.395616\pi\)
\(54\) 0 0
\(55\) 208.404 0.510931
\(56\) 0 0
\(57\) −415.717 −0.966019
\(58\) 0 0
\(59\) 572.767 1.26386 0.631932 0.775024i \(-0.282262\pi\)
0.631932 + 0.775024i \(0.282262\pi\)
\(60\) 0 0
\(61\) −939.852 −1.97272 −0.986358 0.164614i \(-0.947362\pi\)
−0.986358 + 0.164614i \(0.947362\pi\)
\(62\) 0 0
\(63\) 831.942 1.66373
\(64\) 0 0
\(65\) 105.133 0.200619
\(66\) 0 0
\(67\) 151.854 0.276894 0.138447 0.990370i \(-0.455789\pi\)
0.138447 + 0.990370i \(0.455789\pi\)
\(68\) 0 0
\(69\) −374.045 −0.652604
\(70\) 0 0
\(71\) −215.050 −0.359461 −0.179731 0.983716i \(-0.557523\pi\)
−0.179731 + 0.983716i \(0.557523\pi\)
\(72\) 0 0
\(73\) −668.587 −1.07195 −0.535974 0.844235i \(-0.680055\pi\)
−0.535974 + 0.844235i \(0.680055\pi\)
\(74\) 0 0
\(75\) 633.476 0.975300
\(76\) 0 0
\(77\) −786.679 −1.16429
\(78\) 0 0
\(79\) 822.956 1.17202 0.586012 0.810303i \(-0.300697\pi\)
0.586012 + 0.810303i \(0.300697\pi\)
\(80\) 0 0
\(81\) −514.603 −0.705902
\(82\) 0 0
\(83\) 462.269 0.611332 0.305666 0.952139i \(-0.401121\pi\)
0.305666 + 0.952139i \(0.401121\pi\)
\(84\) 0 0
\(85\) −583.600 −0.744710
\(86\) 0 0
\(87\) 114.795 0.141463
\(88\) 0 0
\(89\) 262.733 0.312918 0.156459 0.987684i \(-0.449992\pi\)
0.156459 + 0.987684i \(0.449992\pi\)
\(90\) 0 0
\(91\) −396.855 −0.457162
\(92\) 0 0
\(93\) 753.631 0.840300
\(94\) 0 0
\(95\) −352.752 −0.380964
\(96\) 0 0
\(97\) −150.801 −0.157850 −0.0789251 0.996881i \(-0.525149\pi\)
−0.0789251 + 0.996881i \(0.525149\pi\)
\(98\) 0 0
\(99\) −1055.88 −1.07192
\(100\) 0 0
\(101\) 690.115 0.679891 0.339945 0.940445i \(-0.389591\pi\)
0.339945 + 0.940445i \(0.389591\pi\)
\(102\) 0 0
\(103\) 1840.58 1.76075 0.880377 0.474275i \(-0.157290\pi\)
0.880377 + 0.474275i \(0.157290\pi\)
\(104\) 0 0
\(105\) 1276.33 1.18626
\(106\) 0 0
\(107\) 112.302 0.101464 0.0507322 0.998712i \(-0.483845\pi\)
0.0507322 + 0.998712i \(0.483845\pi\)
\(108\) 0 0
\(109\) 1347.72 1.18429 0.592146 0.805831i \(-0.298281\pi\)
0.592146 + 0.805831i \(0.298281\pi\)
\(110\) 0 0
\(111\) 1794.82 1.53475
\(112\) 0 0
\(113\) 720.469 0.599788 0.299894 0.953973i \(-0.403049\pi\)
0.299894 + 0.953973i \(0.403049\pi\)
\(114\) 0 0
\(115\) −317.391 −0.257364
\(116\) 0 0
\(117\) −532.657 −0.420890
\(118\) 0 0
\(119\) 2202.96 1.69702
\(120\) 0 0
\(121\) −332.571 −0.249865
\(122\) 0 0
\(123\) 2802.23 2.05422
\(124\) 0 0
\(125\) 1361.97 0.974544
\(126\) 0 0
\(127\) 2622.35 1.83225 0.916124 0.400895i \(-0.131301\pi\)
0.916124 + 0.400895i \(0.131301\pi\)
\(128\) 0 0
\(129\) −1101.97 −0.752119
\(130\) 0 0
\(131\) −929.950 −0.620230 −0.310115 0.950699i \(-0.600368\pi\)
−0.310115 + 0.950699i \(0.600368\pi\)
\(132\) 0 0
\(133\) 1331.56 0.868127
\(134\) 0 0
\(135\) 328.919 0.209695
\(136\) 0 0
\(137\) 2511.52 1.56623 0.783117 0.621874i \(-0.213629\pi\)
0.783117 + 0.621874i \(0.213629\pi\)
\(138\) 0 0
\(139\) 1535.86 0.937195 0.468598 0.883412i \(-0.344759\pi\)
0.468598 + 0.883412i \(0.344759\pi\)
\(140\) 0 0
\(141\) 1715.21 1.02445
\(142\) 0 0
\(143\) 503.677 0.294543
\(144\) 0 0
\(145\) 97.4080 0.0557882
\(146\) 0 0
\(147\) −2151.79 −1.20732
\(148\) 0 0
\(149\) 3230.96 1.77645 0.888223 0.459413i \(-0.151940\pi\)
0.888223 + 0.459413i \(0.151940\pi\)
\(150\) 0 0
\(151\) −2814.39 −1.51677 −0.758383 0.651809i \(-0.774010\pi\)
−0.758383 + 0.651809i \(0.774010\pi\)
\(152\) 0 0
\(153\) 2956.80 1.56237
\(154\) 0 0
\(155\) 639.485 0.331385
\(156\) 0 0
\(157\) 1281.71 0.651541 0.325770 0.945449i \(-0.394376\pi\)
0.325770 + 0.945449i \(0.394376\pi\)
\(158\) 0 0
\(159\) −1931.93 −0.963598
\(160\) 0 0
\(161\) 1198.08 0.586472
\(162\) 0 0
\(163\) −1969.09 −0.946204 −0.473102 0.881008i \(-0.656866\pi\)
−0.473102 + 0.881008i \(0.656866\pi\)
\(164\) 0 0
\(165\) −1619.88 −0.764287
\(166\) 0 0
\(167\) 1221.66 0.566075 0.283038 0.959109i \(-0.408658\pi\)
0.283038 + 0.959109i \(0.408658\pi\)
\(168\) 0 0
\(169\) −1942.91 −0.884347
\(170\) 0 0
\(171\) 1787.21 0.799249
\(172\) 0 0
\(173\) −796.794 −0.350168 −0.175084 0.984553i \(-0.556020\pi\)
−0.175084 + 0.984553i \(0.556020\pi\)
\(174\) 0 0
\(175\) −2029.05 −0.876468
\(176\) 0 0
\(177\) −4451.99 −1.89058
\(178\) 0 0
\(179\) −3114.42 −1.30046 −0.650230 0.759737i \(-0.725328\pi\)
−0.650230 + 0.759737i \(0.725328\pi\)
\(180\) 0 0
\(181\) −171.535 −0.0704425 −0.0352213 0.999380i \(-0.511214\pi\)
−0.0352213 + 0.999380i \(0.511214\pi\)
\(182\) 0 0
\(183\) 7305.26 2.95093
\(184\) 0 0
\(185\) 1522.98 0.605251
\(186\) 0 0
\(187\) −2795.93 −1.09336
\(188\) 0 0
\(189\) −1241.59 −0.477845
\(190\) 0 0
\(191\) 3927.65 1.48793 0.743966 0.668218i \(-0.232942\pi\)
0.743966 + 0.668218i \(0.232942\pi\)
\(192\) 0 0
\(193\) −3249.02 −1.21176 −0.605880 0.795556i \(-0.707179\pi\)
−0.605880 + 0.795556i \(0.707179\pi\)
\(194\) 0 0
\(195\) −817.179 −0.300099
\(196\) 0 0
\(197\) 3423.68 1.23821 0.619103 0.785309i \(-0.287496\pi\)
0.619103 + 0.785309i \(0.287496\pi\)
\(198\) 0 0
\(199\) −1371.30 −0.488488 −0.244244 0.969714i \(-0.578540\pi\)
−0.244244 + 0.969714i \(0.578540\pi\)
\(200\) 0 0
\(201\) −1180.33 −0.414198
\(202\) 0 0
\(203\) −367.693 −0.127128
\(204\) 0 0
\(205\) 2377.80 0.810112
\(206\) 0 0
\(207\) 1608.06 0.539941
\(208\) 0 0
\(209\) −1689.98 −0.559321
\(210\) 0 0
\(211\) −2291.73 −0.747721 −0.373861 0.927485i \(-0.621966\pi\)
−0.373861 + 0.927485i \(0.621966\pi\)
\(212\) 0 0
\(213\) 1671.54 0.537708
\(214\) 0 0
\(215\) −935.068 −0.296610
\(216\) 0 0
\(217\) −2413.91 −0.755148
\(218\) 0 0
\(219\) 5196.77 1.60349
\(220\) 0 0
\(221\) −1410.46 −0.429312
\(222\) 0 0
\(223\) −419.617 −0.126007 −0.0630036 0.998013i \(-0.520068\pi\)
−0.0630036 + 0.998013i \(0.520068\pi\)
\(224\) 0 0
\(225\) −2723.38 −0.806929
\(226\) 0 0
\(227\) 3017.42 0.882262 0.441131 0.897443i \(-0.354577\pi\)
0.441131 + 0.897443i \(0.354577\pi\)
\(228\) 0 0
\(229\) −2226.56 −0.642513 −0.321256 0.946992i \(-0.604105\pi\)
−0.321256 + 0.946992i \(0.604105\pi\)
\(230\) 0 0
\(231\) 6114.67 1.74163
\(232\) 0 0
\(233\) 1194.86 0.335957 0.167978 0.985791i \(-0.446276\pi\)
0.167978 + 0.985791i \(0.446276\pi\)
\(234\) 0 0
\(235\) 1455.42 0.404006
\(236\) 0 0
\(237\) −6396.65 −1.75320
\(238\) 0 0
\(239\) 4241.03 1.14782 0.573911 0.818917i \(-0.305425\pi\)
0.573911 + 0.818917i \(0.305425\pi\)
\(240\) 0 0
\(241\) 5571.19 1.48910 0.744548 0.667569i \(-0.232665\pi\)
0.744548 + 0.667569i \(0.232665\pi\)
\(242\) 0 0
\(243\) 5346.38 1.41140
\(244\) 0 0
\(245\) −1825.88 −0.476126
\(246\) 0 0
\(247\) −852.541 −0.219619
\(248\) 0 0
\(249\) −3593.11 −0.914474
\(250\) 0 0
\(251\) −682.681 −0.171675 −0.0858375 0.996309i \(-0.527357\pi\)
−0.0858375 + 0.996309i \(0.527357\pi\)
\(252\) 0 0
\(253\) −1520.57 −0.377855
\(254\) 0 0
\(255\) 4536.19 1.11399
\(256\) 0 0
\(257\) 8093.12 1.96434 0.982169 0.188002i \(-0.0602013\pi\)
0.982169 + 0.188002i \(0.0602013\pi\)
\(258\) 0 0
\(259\) −5748.89 −1.37922
\(260\) 0 0
\(261\) −493.516 −0.117042
\(262\) 0 0
\(263\) 410.300 0.0961984 0.0480992 0.998843i \(-0.484684\pi\)
0.0480992 + 0.998843i \(0.484684\pi\)
\(264\) 0 0
\(265\) −1639.32 −0.380009
\(266\) 0 0
\(267\) −2042.17 −0.468085
\(268\) 0 0
\(269\) −6.75940 −0.00153207 −0.000766037 1.00000i \(-0.500244\pi\)
−0.000766037 1.00000i \(0.500244\pi\)
\(270\) 0 0
\(271\) 2833.98 0.635247 0.317623 0.948217i \(-0.397115\pi\)
0.317623 + 0.948217i \(0.397115\pi\)
\(272\) 0 0
\(273\) 3084.67 0.683855
\(274\) 0 0
\(275\) 2575.21 0.564695
\(276\) 0 0
\(277\) −2157.81 −0.468052 −0.234026 0.972230i \(-0.575190\pi\)
−0.234026 + 0.972230i \(0.575190\pi\)
\(278\) 0 0
\(279\) −3239.94 −0.695234
\(280\) 0 0
\(281\) −4750.23 −1.00845 −0.504226 0.863572i \(-0.668222\pi\)
−0.504226 + 0.863572i \(0.668222\pi\)
\(282\) 0 0
\(283\) 910.900 0.191334 0.0956668 0.995413i \(-0.469502\pi\)
0.0956668 + 0.995413i \(0.469502\pi\)
\(284\) 0 0
\(285\) 2741.86 0.569873
\(286\) 0 0
\(287\) −8975.67 −1.84605
\(288\) 0 0
\(289\) 2916.53 0.593635
\(290\) 0 0
\(291\) 1172.14 0.236124
\(292\) 0 0
\(293\) −2026.80 −0.404119 −0.202060 0.979373i \(-0.564763\pi\)
−0.202060 + 0.979373i \(0.564763\pi\)
\(294\) 0 0
\(295\) −3777.69 −0.745578
\(296\) 0 0
\(297\) 1575.79 0.307868
\(298\) 0 0
\(299\) −767.080 −0.148366
\(300\) 0 0
\(301\) 3529.67 0.675903
\(302\) 0 0
\(303\) −5364.10 −1.01703
\(304\) 0 0
\(305\) 6198.79 1.16374
\(306\) 0 0
\(307\) −326.981 −0.0607877 −0.0303938 0.999538i \(-0.509676\pi\)
−0.0303938 + 0.999538i \(0.509676\pi\)
\(308\) 0 0
\(309\) −14306.4 −2.63386
\(310\) 0 0
\(311\) 871.410 0.158885 0.0794423 0.996839i \(-0.474686\pi\)
0.0794423 + 0.996839i \(0.474686\pi\)
\(312\) 0 0
\(313\) 3515.02 0.634762 0.317381 0.948298i \(-0.397197\pi\)
0.317381 + 0.948298i \(0.397197\pi\)
\(314\) 0 0
\(315\) −5487.07 −0.981465
\(316\) 0 0
\(317\) −6680.42 −1.18363 −0.591814 0.806075i \(-0.701588\pi\)
−0.591814 + 0.806075i \(0.701588\pi\)
\(318\) 0 0
\(319\) 466.665 0.0819067
\(320\) 0 0
\(321\) −872.901 −0.151778
\(322\) 0 0
\(323\) 4732.49 0.815241
\(324\) 0 0
\(325\) 1299.12 0.221729
\(326\) 0 0
\(327\) −10475.5 −1.77155
\(328\) 0 0
\(329\) −5493.90 −0.920633
\(330\) 0 0
\(331\) 2574.24 0.427471 0.213736 0.976892i \(-0.431437\pi\)
0.213736 + 0.976892i \(0.431437\pi\)
\(332\) 0 0
\(333\) −7716.13 −1.26979
\(334\) 0 0
\(335\) −1001.55 −0.163345
\(336\) 0 0
\(337\) −74.0970 −0.0119772 −0.00598861 0.999982i \(-0.501906\pi\)
−0.00598861 + 0.999982i \(0.501906\pi\)
\(338\) 0 0
\(339\) −5600.04 −0.897205
\(340\) 0 0
\(341\) 3063.67 0.486530
\(342\) 0 0
\(343\) −1647.24 −0.259307
\(344\) 0 0
\(345\) 2467.01 0.384984
\(346\) 0 0
\(347\) 2973.72 0.460050 0.230025 0.973185i \(-0.426119\pi\)
0.230025 + 0.973185i \(0.426119\pi\)
\(348\) 0 0
\(349\) 9351.98 1.43438 0.717192 0.696875i \(-0.245427\pi\)
0.717192 + 0.696875i \(0.245427\pi\)
\(350\) 0 0
\(351\) 794.940 0.120885
\(352\) 0 0
\(353\) 2216.90 0.334259 0.167130 0.985935i \(-0.446550\pi\)
0.167130 + 0.985935i \(0.446550\pi\)
\(354\) 0 0
\(355\) 1418.36 0.212053
\(356\) 0 0
\(357\) −17123.1 −2.53852
\(358\) 0 0
\(359\) 2082.23 0.306117 0.153059 0.988217i \(-0.451088\pi\)
0.153059 + 0.988217i \(0.451088\pi\)
\(360\) 0 0
\(361\) −3998.49 −0.582955
\(362\) 0 0
\(363\) 2585.00 0.373766
\(364\) 0 0
\(365\) 4409.66 0.632362
\(366\) 0 0
\(367\) −4509.22 −0.641360 −0.320680 0.947188i \(-0.603911\pi\)
−0.320680 + 0.947188i \(0.603911\pi\)
\(368\) 0 0
\(369\) −12047.1 −1.69959
\(370\) 0 0
\(371\) 6188.05 0.865951
\(372\) 0 0
\(373\) 12249.3 1.70039 0.850194 0.526470i \(-0.176485\pi\)
0.850194 + 0.526470i \(0.176485\pi\)
\(374\) 0 0
\(375\) −10586.3 −1.45779
\(376\) 0 0
\(377\) 235.418 0.0321609
\(378\) 0 0
\(379\) 4981.51 0.675153 0.337576 0.941298i \(-0.390393\pi\)
0.337576 + 0.941298i \(0.390393\pi\)
\(380\) 0 0
\(381\) −20382.9 −2.74081
\(382\) 0 0
\(383\) 3044.88 0.406229 0.203115 0.979155i \(-0.434894\pi\)
0.203115 + 0.979155i \(0.434894\pi\)
\(384\) 0 0
\(385\) 5188.54 0.686837
\(386\) 0 0
\(387\) 4737.51 0.622277
\(388\) 0 0
\(389\) 2733.30 0.356256 0.178128 0.984007i \(-0.442996\pi\)
0.178128 + 0.984007i \(0.442996\pi\)
\(390\) 0 0
\(391\) 4258.09 0.550744
\(392\) 0 0
\(393\) 7228.29 0.927784
\(394\) 0 0
\(395\) −5427.81 −0.691399
\(396\) 0 0
\(397\) −950.997 −0.120225 −0.0601123 0.998192i \(-0.519146\pi\)
−0.0601123 + 0.998192i \(0.519146\pi\)
\(398\) 0 0
\(399\) −10349.9 −1.29861
\(400\) 0 0
\(401\) 7606.74 0.947288 0.473644 0.880716i \(-0.342938\pi\)
0.473644 + 0.880716i \(0.342938\pi\)
\(402\) 0 0
\(403\) 1545.53 0.191037
\(404\) 0 0
\(405\) 3394.06 0.416425
\(406\) 0 0
\(407\) 7296.32 0.888612
\(408\) 0 0
\(409\) −4981.58 −0.602257 −0.301129 0.953584i \(-0.597363\pi\)
−0.301129 + 0.953584i \(0.597363\pi\)
\(410\) 0 0
\(411\) −19521.5 −2.34288
\(412\) 0 0
\(413\) 14259.9 1.69899
\(414\) 0 0
\(415\) −3048.89 −0.360637
\(416\) 0 0
\(417\) −11937.9 −1.40192
\(418\) 0 0
\(419\) 4855.53 0.566129 0.283065 0.959101i \(-0.408649\pi\)
0.283065 + 0.959101i \(0.408649\pi\)
\(420\) 0 0
\(421\) 11276.5 1.30543 0.652714 0.757604i \(-0.273630\pi\)
0.652714 + 0.757604i \(0.273630\pi\)
\(422\) 0 0
\(423\) −7373.88 −0.847590
\(424\) 0 0
\(425\) −7211.44 −0.823074
\(426\) 0 0
\(427\) −23399.0 −2.65189
\(428\) 0 0
\(429\) −3914.97 −0.440598
\(430\) 0 0
\(431\) −4800.16 −0.536463 −0.268232 0.963354i \(-0.586439\pi\)
−0.268232 + 0.963354i \(0.586439\pi\)
\(432\) 0 0
\(433\) −6242.32 −0.692810 −0.346405 0.938085i \(-0.612598\pi\)
−0.346405 + 0.938085i \(0.612598\pi\)
\(434\) 0 0
\(435\) −757.130 −0.0834520
\(436\) 0 0
\(437\) 2573.77 0.281739
\(438\) 0 0
\(439\) 4929.27 0.535903 0.267951 0.963432i \(-0.413653\pi\)
0.267951 + 0.963432i \(0.413653\pi\)
\(440\) 0 0
\(441\) 9250.77 0.998896
\(442\) 0 0
\(443\) −10848.0 −1.16344 −0.581718 0.813390i \(-0.697619\pi\)
−0.581718 + 0.813390i \(0.697619\pi\)
\(444\) 0 0
\(445\) −1732.86 −0.184596
\(446\) 0 0
\(447\) −25113.5 −2.65733
\(448\) 0 0
\(449\) −11515.2 −1.21032 −0.605162 0.796102i \(-0.706892\pi\)
−0.605162 + 0.796102i \(0.706892\pi\)
\(450\) 0 0
\(451\) 11391.7 1.18938
\(452\) 0 0
\(453\) 21875.6 2.26889
\(454\) 0 0
\(455\) 2617.46 0.269689
\(456\) 0 0
\(457\) 4829.89 0.494383 0.247191 0.968967i \(-0.420492\pi\)
0.247191 + 0.968967i \(0.420492\pi\)
\(458\) 0 0
\(459\) −4412.74 −0.448735
\(460\) 0 0
\(461\) 11689.6 1.18100 0.590498 0.807039i \(-0.298931\pi\)
0.590498 + 0.807039i \(0.298931\pi\)
\(462\) 0 0
\(463\) 5043.86 0.506281 0.253141 0.967430i \(-0.418536\pi\)
0.253141 + 0.967430i \(0.418536\pi\)
\(464\) 0 0
\(465\) −4970.57 −0.495709
\(466\) 0 0
\(467\) 17591.1 1.74308 0.871542 0.490321i \(-0.163120\pi\)
0.871542 + 0.490321i \(0.163120\pi\)
\(468\) 0 0
\(469\) 3780.63 0.372225
\(470\) 0 0
\(471\) −9962.47 −0.974621
\(472\) 0 0
\(473\) −4479.75 −0.435474
\(474\) 0 0
\(475\) −4358.89 −0.421052
\(476\) 0 0
\(477\) 8305.58 0.797246
\(478\) 0 0
\(479\) 13059.7 1.24575 0.622875 0.782321i \(-0.285964\pi\)
0.622875 + 0.782321i \(0.285964\pi\)
\(480\) 0 0
\(481\) 3680.77 0.348916
\(482\) 0 0
\(483\) −9312.41 −0.877286
\(484\) 0 0
\(485\) 994.605 0.0931189
\(486\) 0 0
\(487\) −15549.3 −1.44683 −0.723414 0.690414i \(-0.757428\pi\)
−0.723414 + 0.690414i \(0.757428\pi\)
\(488\) 0 0
\(489\) 15305.3 1.41540
\(490\) 0 0
\(491\) 12202.3 1.12155 0.560777 0.827967i \(-0.310503\pi\)
0.560777 + 0.827967i \(0.310503\pi\)
\(492\) 0 0
\(493\) −1306.82 −0.119383
\(494\) 0 0
\(495\) 6964.03 0.632344
\(496\) 0 0
\(497\) −5354.00 −0.483219
\(498\) 0 0
\(499\) 2450.01 0.219794 0.109897 0.993943i \(-0.464948\pi\)
0.109897 + 0.993943i \(0.464948\pi\)
\(500\) 0 0
\(501\) −9495.65 −0.846775
\(502\) 0 0
\(503\) −16579.6 −1.46968 −0.734839 0.678241i \(-0.762742\pi\)
−0.734839 + 0.678241i \(0.762742\pi\)
\(504\) 0 0
\(505\) −4551.65 −0.401081
\(506\) 0 0
\(507\) 15101.8 1.32287
\(508\) 0 0
\(509\) 11074.6 0.964387 0.482193 0.876065i \(-0.339840\pi\)
0.482193 + 0.876065i \(0.339840\pi\)
\(510\) 0 0
\(511\) −16645.5 −1.44100
\(512\) 0 0
\(513\) −2667.24 −0.229555
\(514\) 0 0
\(515\) −12139.5 −1.03870
\(516\) 0 0
\(517\) 6972.69 0.593150
\(518\) 0 0
\(519\) 6193.30 0.523807
\(520\) 0 0
\(521\) −3400.02 −0.285907 −0.142953 0.989729i \(-0.545660\pi\)
−0.142953 + 0.989729i \(0.545660\pi\)
\(522\) 0 0
\(523\) 2856.01 0.238785 0.119393 0.992847i \(-0.461905\pi\)
0.119393 + 0.992847i \(0.461905\pi\)
\(524\) 0 0
\(525\) 15771.4 1.31108
\(526\) 0 0
\(527\) −8579.28 −0.709144
\(528\) 0 0
\(529\) −9851.23 −0.809668
\(530\) 0 0
\(531\) 19139.6 1.56420
\(532\) 0 0
\(533\) 5746.74 0.467015
\(534\) 0 0
\(535\) −740.691 −0.0598558
\(536\) 0 0
\(537\) 24207.7 1.94532
\(538\) 0 0
\(539\) −8747.47 −0.699035
\(540\) 0 0
\(541\) 18996.6 1.50966 0.754829 0.655921i \(-0.227720\pi\)
0.754829 + 0.655921i \(0.227720\pi\)
\(542\) 0 0
\(543\) 1333.30 0.105373
\(544\) 0 0
\(545\) −8888.86 −0.698637
\(546\) 0 0
\(547\) 7603.46 0.594333 0.297167 0.954826i \(-0.403958\pi\)
0.297167 + 0.954826i \(0.403958\pi\)
\(548\) 0 0
\(549\) −31406.1 −2.44149
\(550\) 0 0
\(551\) −789.894 −0.0610719
\(552\) 0 0
\(553\) 20488.7 1.57553
\(554\) 0 0
\(555\) −11837.7 −0.905377
\(556\) 0 0
\(557\) −2936.30 −0.223367 −0.111683 0.993744i \(-0.535624\pi\)
−0.111683 + 0.993744i \(0.535624\pi\)
\(558\) 0 0
\(559\) −2259.90 −0.170990
\(560\) 0 0
\(561\) 21732.1 1.63553
\(562\) 0 0
\(563\) −23536.9 −1.76192 −0.880961 0.473188i \(-0.843103\pi\)
−0.880961 + 0.473188i \(0.843103\pi\)
\(564\) 0 0
\(565\) −4751.85 −0.353826
\(566\) 0 0
\(567\) −12811.8 −0.948934
\(568\) 0 0
\(569\) −5659.60 −0.416982 −0.208491 0.978024i \(-0.566855\pi\)
−0.208491 + 0.978024i \(0.566855\pi\)
\(570\) 0 0
\(571\) 6891.00 0.505043 0.252521 0.967591i \(-0.418740\pi\)
0.252521 + 0.967591i \(0.418740\pi\)
\(572\) 0 0
\(573\) −30528.8 −2.22575
\(574\) 0 0
\(575\) −3921.95 −0.284446
\(576\) 0 0
\(577\) 10652.2 0.768556 0.384278 0.923217i \(-0.374450\pi\)
0.384278 + 0.923217i \(0.374450\pi\)
\(578\) 0 0
\(579\) 25253.9 1.81264
\(580\) 0 0
\(581\) 11508.9 0.821805
\(582\) 0 0
\(583\) −7853.69 −0.557919
\(584\) 0 0
\(585\) 3513.14 0.248291
\(586\) 0 0
\(587\) −14006.3 −0.984841 −0.492421 0.870357i \(-0.663888\pi\)
−0.492421 + 0.870357i \(0.663888\pi\)
\(588\) 0 0
\(589\) −5185.67 −0.362770
\(590\) 0 0
\(591\) −26611.5 −1.85220
\(592\) 0 0
\(593\) 3528.04 0.244316 0.122158 0.992511i \(-0.461019\pi\)
0.122158 + 0.992511i \(0.461019\pi\)
\(594\) 0 0
\(595\) −14529.6 −1.00110
\(596\) 0 0
\(597\) 10658.8 0.730715
\(598\) 0 0
\(599\) 19024.9 1.29772 0.648861 0.760907i \(-0.275246\pi\)
0.648861 + 0.760907i \(0.275246\pi\)
\(600\) 0 0
\(601\) −1065.92 −0.0723460 −0.0361730 0.999346i \(-0.511517\pi\)
−0.0361730 + 0.999346i \(0.511517\pi\)
\(602\) 0 0
\(603\) 5074.35 0.342692
\(604\) 0 0
\(605\) 2193.47 0.147400
\(606\) 0 0
\(607\) 12909.7 0.863242 0.431621 0.902055i \(-0.357942\pi\)
0.431621 + 0.902055i \(0.357942\pi\)
\(608\) 0 0
\(609\) 2858.00 0.190167
\(610\) 0 0
\(611\) 3517.51 0.232902
\(612\) 0 0
\(613\) −12393.3 −0.816578 −0.408289 0.912853i \(-0.633874\pi\)
−0.408289 + 0.912853i \(0.633874\pi\)
\(614\) 0 0
\(615\) −18482.1 −1.21182
\(616\) 0 0
\(617\) −18921.2 −1.23459 −0.617293 0.786733i \(-0.711771\pi\)
−0.617293 + 0.786733i \(0.711771\pi\)
\(618\) 0 0
\(619\) 21378.2 1.38814 0.694072 0.719906i \(-0.255815\pi\)
0.694072 + 0.719906i \(0.255815\pi\)
\(620\) 0 0
\(621\) −2399.87 −0.155078
\(622\) 0 0
\(623\) 6541.15 0.420651
\(624\) 0 0
\(625\) 1204.57 0.0770926
\(626\) 0 0
\(627\) 13135.8 0.836672
\(628\) 0 0
\(629\) −20432.1 −1.29520
\(630\) 0 0
\(631\) −9602.80 −0.605834 −0.302917 0.953017i \(-0.597961\pi\)
−0.302917 + 0.953017i \(0.597961\pi\)
\(632\) 0 0
\(633\) 17813.1 1.11849
\(634\) 0 0
\(635\) −17295.7 −1.08088
\(636\) 0 0
\(637\) −4412.83 −0.274478
\(638\) 0 0
\(639\) −7186.12 −0.444880
\(640\) 0 0
\(641\) 4450.84 0.274256 0.137128 0.990553i \(-0.456213\pi\)
0.137128 + 0.990553i \(0.456213\pi\)
\(642\) 0 0
\(643\) −9180.04 −0.563026 −0.281513 0.959557i \(-0.590836\pi\)
−0.281513 + 0.959557i \(0.590836\pi\)
\(644\) 0 0
\(645\) 7268.07 0.443690
\(646\) 0 0
\(647\) 5546.17 0.337005 0.168503 0.985701i \(-0.446107\pi\)
0.168503 + 0.985701i \(0.446107\pi\)
\(648\) 0 0
\(649\) −18098.3 −1.09464
\(650\) 0 0
\(651\) 18762.8 1.12960
\(652\) 0 0
\(653\) 8948.56 0.536270 0.268135 0.963381i \(-0.413593\pi\)
0.268135 + 0.963381i \(0.413593\pi\)
\(654\) 0 0
\(655\) 6133.49 0.365886
\(656\) 0 0
\(657\) −22341.5 −1.32667
\(658\) 0 0
\(659\) 21404.4 1.26525 0.632623 0.774460i \(-0.281978\pi\)
0.632623 + 0.774460i \(0.281978\pi\)
\(660\) 0 0
\(661\) 33177.6 1.95228 0.976140 0.217140i \(-0.0696728\pi\)
0.976140 + 0.217140i \(0.0696728\pi\)
\(662\) 0 0
\(663\) 10963.2 0.642195
\(664\) 0 0
\(665\) −8782.30 −0.512125
\(666\) 0 0
\(667\) −710.713 −0.0412577
\(668\) 0 0
\(669\) 3261.59 0.188491
\(670\) 0 0
\(671\) 29697.4 1.70858
\(672\) 0 0
\(673\) 30638.5 1.75487 0.877436 0.479694i \(-0.159252\pi\)
0.877436 + 0.479694i \(0.159252\pi\)
\(674\) 0 0
\(675\) 4064.39 0.231760
\(676\) 0 0
\(677\) −17633.7 −1.00106 −0.500529 0.865720i \(-0.666861\pi\)
−0.500529 + 0.865720i \(0.666861\pi\)
\(678\) 0 0
\(679\) −3754.41 −0.212196
\(680\) 0 0
\(681\) −23453.8 −1.31975
\(682\) 0 0
\(683\) −19570.6 −1.09641 −0.548206 0.836344i \(-0.684689\pi\)
−0.548206 + 0.836344i \(0.684689\pi\)
\(684\) 0 0
\(685\) −16564.8 −0.923952
\(686\) 0 0
\(687\) 17306.6 0.961116
\(688\) 0 0
\(689\) −3961.95 −0.219068
\(690\) 0 0
\(691\) −149.923 −0.00825375 −0.00412688 0.999991i \(-0.501314\pi\)
−0.00412688 + 0.999991i \(0.501314\pi\)
\(692\) 0 0
\(693\) −26287.7 −1.44096
\(694\) 0 0
\(695\) −10129.8 −0.552870
\(696\) 0 0
\(697\) −31900.4 −1.73359
\(698\) 0 0
\(699\) −9287.38 −0.502548
\(700\) 0 0
\(701\) −11086.5 −0.597335 −0.298667 0.954357i \(-0.596542\pi\)
−0.298667 + 0.954357i \(0.596542\pi\)
\(702\) 0 0
\(703\) −12350.0 −0.662574
\(704\) 0 0
\(705\) −11312.7 −0.604341
\(706\) 0 0
\(707\) 17181.5 0.913968
\(708\) 0 0
\(709\) 3858.40 0.204380 0.102190 0.994765i \(-0.467415\pi\)
0.102190 + 0.994765i \(0.467415\pi\)
\(710\) 0 0
\(711\) 27499.9 1.45053
\(712\) 0 0
\(713\) −4665.84 −0.245073
\(714\) 0 0
\(715\) −3322.00 −0.173756
\(716\) 0 0
\(717\) −32964.6 −1.71699
\(718\) 0 0
\(719\) 32717.8 1.69704 0.848519 0.529166i \(-0.177495\pi\)
0.848519 + 0.529166i \(0.177495\pi\)
\(720\) 0 0
\(721\) 45824.0 2.36696
\(722\) 0 0
\(723\) −43303.6 −2.22750
\(724\) 0 0
\(725\) 1203.65 0.0616587
\(726\) 0 0
\(727\) 25847.2 1.31859 0.659297 0.751883i \(-0.270854\pi\)
0.659297 + 0.751883i \(0.270854\pi\)
\(728\) 0 0
\(729\) −27662.0 −1.40537
\(730\) 0 0
\(731\) 12544.8 0.634727
\(732\) 0 0
\(733\) −14328.7 −0.722023 −0.361011 0.932561i \(-0.617568\pi\)
−0.361011 + 0.932561i \(0.617568\pi\)
\(734\) 0 0
\(735\) 14192.1 0.712223
\(736\) 0 0
\(737\) −4798.27 −0.239819
\(738\) 0 0
\(739\) −27101.5 −1.34905 −0.674524 0.738253i \(-0.735651\pi\)
−0.674524 + 0.738253i \(0.735651\pi\)
\(740\) 0 0
\(741\) 6626.61 0.328522
\(742\) 0 0
\(743\) 23322.1 1.15155 0.575777 0.817607i \(-0.304699\pi\)
0.575777 + 0.817607i \(0.304699\pi\)
\(744\) 0 0
\(745\) −21309.8 −1.04796
\(746\) 0 0
\(747\) 15447.2 0.756603
\(748\) 0 0
\(749\) 2795.94 0.136397
\(750\) 0 0
\(751\) −25994.0 −1.26303 −0.631515 0.775364i \(-0.717566\pi\)
−0.631515 + 0.775364i \(0.717566\pi\)
\(752\) 0 0
\(753\) 5306.32 0.256804
\(754\) 0 0
\(755\) 18562.3 0.894770
\(756\) 0 0
\(757\) −31317.8 −1.50365 −0.751825 0.659363i \(-0.770826\pi\)
−0.751825 + 0.659363i \(0.770826\pi\)
\(758\) 0 0
\(759\) 11819.0 0.565222
\(760\) 0 0
\(761\) −16497.5 −0.785853 −0.392926 0.919570i \(-0.628537\pi\)
−0.392926 + 0.919570i \(0.628537\pi\)
\(762\) 0 0
\(763\) 33553.4 1.59203
\(764\) 0 0
\(765\) −19501.6 −0.921675
\(766\) 0 0
\(767\) −9130.02 −0.429812
\(768\) 0 0
\(769\) −24867.3 −1.16611 −0.583055 0.812433i \(-0.698143\pi\)
−0.583055 + 0.812433i \(0.698143\pi\)
\(770\) 0 0
\(771\) −62905.9 −2.93839
\(772\) 0 0
\(773\) 2661.15 0.123823 0.0619114 0.998082i \(-0.480280\pi\)
0.0619114 + 0.998082i \(0.480280\pi\)
\(774\) 0 0
\(775\) 7902.00 0.366256
\(776\) 0 0
\(777\) 44684.8 2.06314
\(778\) 0 0
\(779\) −19281.9 −0.886837
\(780\) 0 0
\(781\) 6795.14 0.311331
\(782\) 0 0
\(783\) 736.525 0.0336159
\(784\) 0 0
\(785\) −8453.54 −0.384356
\(786\) 0 0
\(787\) −11018.1 −0.499049 −0.249524 0.968369i \(-0.580274\pi\)
−0.249524 + 0.968369i \(0.580274\pi\)
\(788\) 0 0
\(789\) −3189.17 −0.143900
\(790\) 0 0
\(791\) 17937.2 0.806286
\(792\) 0 0
\(793\) 14981.4 0.670877
\(794\) 0 0
\(795\) 12742.0 0.568445
\(796\) 0 0
\(797\) 5953.61 0.264602 0.132301 0.991210i \(-0.457763\pi\)
0.132301 + 0.991210i \(0.457763\pi\)
\(798\) 0 0
\(799\) −19525.8 −0.864549
\(800\) 0 0
\(801\) 8779.50 0.387276
\(802\) 0 0
\(803\) 21126.0 0.928417
\(804\) 0 0
\(805\) −7901.94 −0.345971
\(806\) 0 0
\(807\) 52.5393 0.00229178
\(808\) 0 0
\(809\) 27554.3 1.19747 0.598737 0.800946i \(-0.295670\pi\)
0.598737 + 0.800946i \(0.295670\pi\)
\(810\) 0 0
\(811\) −4817.57 −0.208592 −0.104296 0.994546i \(-0.533259\pi\)
−0.104296 + 0.994546i \(0.533259\pi\)
\(812\) 0 0
\(813\) −22027.9 −0.950247
\(814\) 0 0
\(815\) 12987.1 0.558184
\(816\) 0 0
\(817\) 7582.59 0.324701
\(818\) 0 0
\(819\) −13261.3 −0.565797
\(820\) 0 0
\(821\) 13914.6 0.591504 0.295752 0.955265i \(-0.404430\pi\)
0.295752 + 0.955265i \(0.404430\pi\)
\(822\) 0 0
\(823\) −36653.5 −1.55244 −0.776221 0.630461i \(-0.782866\pi\)
−0.776221 + 0.630461i \(0.782866\pi\)
\(824\) 0 0
\(825\) −20016.5 −0.844711
\(826\) 0 0
\(827\) −31428.3 −1.32149 −0.660743 0.750612i \(-0.729759\pi\)
−0.660743 + 0.750612i \(0.729759\pi\)
\(828\) 0 0
\(829\) −20810.9 −0.871885 −0.435942 0.899975i \(-0.643585\pi\)
−0.435942 + 0.899975i \(0.643585\pi\)
\(830\) 0 0
\(831\) 16772.2 0.700145
\(832\) 0 0
\(833\) 24495.8 1.01888
\(834\) 0 0
\(835\) −8057.43 −0.333939
\(836\) 0 0
\(837\) 4835.30 0.199680
\(838\) 0 0
\(839\) 11010.0 0.453050 0.226525 0.974005i \(-0.427263\pi\)
0.226525 + 0.974005i \(0.427263\pi\)
\(840\) 0 0
\(841\) −24170.9 −0.991057
\(842\) 0 0
\(843\) 36922.5 1.50852
\(844\) 0 0
\(845\) 12814.5 0.521694
\(846\) 0 0
\(847\) −8279.85 −0.335890
\(848\) 0 0
\(849\) −7080.22 −0.286210
\(850\) 0 0
\(851\) −11112.0 −0.447608
\(852\) 0 0
\(853\) −18116.0 −0.727175 −0.363587 0.931560i \(-0.618448\pi\)
−0.363587 + 0.931560i \(0.618448\pi\)
\(854\) 0 0
\(855\) −11787.6 −0.471493
\(856\) 0 0
\(857\) 38510.8 1.53501 0.767505 0.641043i \(-0.221498\pi\)
0.767505 + 0.641043i \(0.221498\pi\)
\(858\) 0 0
\(859\) 32858.7 1.30515 0.652576 0.757723i \(-0.273688\pi\)
0.652576 + 0.757723i \(0.273688\pi\)
\(860\) 0 0
\(861\) 69765.8 2.76146
\(862\) 0 0
\(863\) −22079.5 −0.870911 −0.435456 0.900210i \(-0.643413\pi\)
−0.435456 + 0.900210i \(0.643413\pi\)
\(864\) 0 0
\(865\) 5255.26 0.206571
\(866\) 0 0
\(867\) −22669.5 −0.888001
\(868\) 0 0
\(869\) −26003.7 −1.01509
\(870\) 0 0
\(871\) −2420.58 −0.0941656
\(872\) 0 0
\(873\) −5039.15 −0.195360
\(874\) 0 0
\(875\) 33908.2 1.31007
\(876\) 0 0
\(877\) 5773.00 0.222281 0.111140 0.993805i \(-0.464550\pi\)
0.111140 + 0.993805i \(0.464550\pi\)
\(878\) 0 0
\(879\) 15753.9 0.604510
\(880\) 0 0
\(881\) −7132.59 −0.272762 −0.136381 0.990656i \(-0.543547\pi\)
−0.136381 + 0.990656i \(0.543547\pi\)
\(882\) 0 0
\(883\) 27110.4 1.03323 0.516613 0.856219i \(-0.327193\pi\)
0.516613 + 0.856219i \(0.327193\pi\)
\(884\) 0 0
\(885\) 29363.1 1.11529
\(886\) 0 0
\(887\) 45045.7 1.70517 0.852585 0.522589i \(-0.175034\pi\)
0.852585 + 0.522589i \(0.175034\pi\)
\(888\) 0 0
\(889\) 65287.3 2.46306
\(890\) 0 0
\(891\) 16260.4 0.611384
\(892\) 0 0
\(893\) −11802.2 −0.442269
\(894\) 0 0
\(895\) 20541.1 0.767167
\(896\) 0 0
\(897\) 5962.34 0.221936
\(898\) 0 0
\(899\) 1431.96 0.0531239
\(900\) 0 0
\(901\) 21992.9 0.813197
\(902\) 0 0
\(903\) −27435.3 −1.01106
\(904\) 0 0
\(905\) 1131.36 0.0415554
\(906\) 0 0
\(907\) 47599.9 1.74259 0.871295 0.490760i \(-0.163281\pi\)
0.871295 + 0.490760i \(0.163281\pi\)
\(908\) 0 0
\(909\) 23060.9 0.841453
\(910\) 0 0
\(911\) 42503.8 1.54579 0.772895 0.634534i \(-0.218808\pi\)
0.772895 + 0.634534i \(0.218808\pi\)
\(912\) 0 0
\(913\) −14606.7 −0.529477
\(914\) 0 0
\(915\) −48181.8 −1.74081
\(916\) 0 0
\(917\) −23152.5 −0.833766
\(918\) 0 0
\(919\) 8819.41 0.316568 0.158284 0.987394i \(-0.449404\pi\)
0.158284 + 0.987394i \(0.449404\pi\)
\(920\) 0 0
\(921\) 2541.55 0.0909305
\(922\) 0 0
\(923\) 3427.94 0.122245
\(924\) 0 0
\(925\) 18819.1 0.668940
\(926\) 0 0
\(927\) 61504.8 2.17916
\(928\) 0 0
\(929\) −14155.6 −0.499925 −0.249963 0.968256i \(-0.580418\pi\)
−0.249963 + 0.968256i \(0.580418\pi\)
\(930\) 0 0
\(931\) 14806.3 0.521220
\(932\) 0 0
\(933\) −6773.27 −0.237671
\(934\) 0 0
\(935\) 18440.6 0.644996
\(936\) 0 0
\(937\) 38518.7 1.34296 0.671479 0.741023i \(-0.265659\pi\)
0.671479 + 0.741023i \(0.265659\pi\)
\(938\) 0 0
\(939\) −27321.4 −0.949523
\(940\) 0 0
\(941\) 39595.4 1.37170 0.685852 0.727741i \(-0.259430\pi\)
0.685852 + 0.727741i \(0.259430\pi\)
\(942\) 0 0
\(943\) −17349.0 −0.599112
\(944\) 0 0
\(945\) 8188.93 0.281890
\(946\) 0 0
\(947\) −46442.1 −1.59363 −0.796814 0.604225i \(-0.793483\pi\)
−0.796814 + 0.604225i \(0.793483\pi\)
\(948\) 0 0
\(949\) 10657.4 0.364545
\(950\) 0 0
\(951\) 51925.4 1.77055
\(952\) 0 0
\(953\) −20600.1 −0.700211 −0.350106 0.936710i \(-0.613854\pi\)
−0.350106 + 0.936710i \(0.613854\pi\)
\(954\) 0 0
\(955\) −25904.8 −0.877760
\(956\) 0 0
\(957\) −3627.28 −0.122522
\(958\) 0 0
\(959\) 62528.2 2.10547
\(960\) 0 0
\(961\) −20390.2 −0.684441
\(962\) 0 0
\(963\) 3752.70 0.125575
\(964\) 0 0
\(965\) 21428.9 0.714841
\(966\) 0 0
\(967\) 38210.9 1.27071 0.635356 0.772219i \(-0.280853\pi\)
0.635356 + 0.772219i \(0.280853\pi\)
\(968\) 0 0
\(969\) −36784.6 −1.21950
\(970\) 0 0
\(971\) −52643.9 −1.73988 −0.869941 0.493157i \(-0.835843\pi\)
−0.869941 + 0.493157i \(0.835843\pi\)
\(972\) 0 0
\(973\) 38237.6 1.25986
\(974\) 0 0
\(975\) −10097.7 −0.331678
\(976\) 0 0
\(977\) −7985.95 −0.261508 −0.130754 0.991415i \(-0.541740\pi\)
−0.130754 + 0.991415i \(0.541740\pi\)
\(978\) 0 0
\(979\) −8301.83 −0.271019
\(980\) 0 0
\(981\) 45035.3 1.46571
\(982\) 0 0
\(983\) 10703.1 0.347279 0.173639 0.984809i \(-0.444447\pi\)
0.173639 + 0.984809i \(0.444447\pi\)
\(984\) 0 0
\(985\) −22580.9 −0.730442
\(986\) 0 0
\(987\) 42702.8 1.37715
\(988\) 0 0
\(989\) 6822.49 0.219355
\(990\) 0 0
\(991\) −23945.4 −0.767558 −0.383779 0.923425i \(-0.625377\pi\)
−0.383779 + 0.923425i \(0.625377\pi\)
\(992\) 0 0
\(993\) −20009.0 −0.639442
\(994\) 0 0
\(995\) 9044.42 0.288168
\(996\) 0 0
\(997\) −20212.7 −0.642068 −0.321034 0.947068i \(-0.604030\pi\)
−0.321034 + 0.947068i \(0.604030\pi\)
\(998\) 0 0
\(999\) 11515.6 0.364702
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.1.2 10
4.3 odd 2 1024.4.a.m.1.9 10
8.3 odd 2 1024.4.a.m.1.2 10
8.5 even 2 inner 1024.4.a.n.1.9 10
16.3 odd 4 1024.4.b.k.513.9 10
16.5 even 4 1024.4.b.j.513.9 10
16.11 odd 4 1024.4.b.k.513.2 10
16.13 even 4 1024.4.b.j.513.2 10
32.3 odd 8 128.4.e.a.33.5 10
32.5 even 8 16.4.e.a.5.1 10
32.11 odd 8 128.4.e.a.97.5 10
32.13 even 8 16.4.e.a.13.1 yes 10
32.19 odd 8 64.4.e.a.17.1 10
32.21 even 8 128.4.e.b.97.1 10
32.27 odd 8 64.4.e.a.49.1 10
32.29 even 8 128.4.e.b.33.1 10
96.5 odd 8 144.4.k.a.37.5 10
96.59 even 8 576.4.k.a.433.4 10
96.77 odd 8 144.4.k.a.109.5 10
96.83 even 8 576.4.k.a.145.4 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.4.e.a.5.1 10 32.5 even 8
16.4.e.a.13.1 yes 10 32.13 even 8
64.4.e.a.17.1 10 32.19 odd 8
64.4.e.a.49.1 10 32.27 odd 8
128.4.e.a.33.5 10 32.3 odd 8
128.4.e.a.97.5 10 32.11 odd 8
128.4.e.b.33.1 10 32.29 even 8
128.4.e.b.97.1 10 32.21 even 8
144.4.k.a.37.5 10 96.5 odd 8
144.4.k.a.109.5 10 96.77 odd 8
576.4.k.a.145.4 10 96.83 even 8
576.4.k.a.433.4 10 96.59 even 8
1024.4.a.m.1.2 10 8.3 odd 2
1024.4.a.m.1.9 10 4.3 odd 2
1024.4.a.n.1.2 10 1.1 even 1 trivial
1024.4.a.n.1.9 10 8.5 even 2 inner
1024.4.b.j.513.2 10 16.13 even 4
1024.4.b.j.513.9 10 16.5 even 4
1024.4.b.k.513.2 10 16.11 odd 4
1024.4.b.k.513.9 10 16.3 odd 4