Properties

Label 1024.4.a.m.1.6
Level $1024$
Weight $4$
Character 1024.1
Self dual yes
Analytic conductor $60.418$
Analytic rank $1$
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: \(1\)
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 1.6
Root \(3.94652\) of defining polynomial
Character \(\chi\) \(=\) 1024.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.07024 q^{3} +11.6331 q^{5} -2.67171 q^{7} -25.8546 q^{9} +O(q^{10})\) \(q+1.07024 q^{3} +11.6331 q^{5} -2.67171 q^{7} -25.8546 q^{9} -63.9525 q^{11} +50.0586 q^{13} +12.4503 q^{15} +72.4991 q^{17} -27.4961 q^{19} -2.85937 q^{21} +139.462 q^{23} +10.3299 q^{25} -56.5672 q^{27} -93.3995 q^{29} -188.682 q^{31} -68.4447 q^{33} -31.0803 q^{35} -118.886 q^{37} +53.5748 q^{39} +104.629 q^{41} -44.5275 q^{43} -300.770 q^{45} -488.151 q^{47} -335.862 q^{49} +77.5916 q^{51} -211.510 q^{53} -743.968 q^{55} -29.4275 q^{57} -402.624 q^{59} +322.538 q^{61} +69.0758 q^{63} +582.338 q^{65} +196.789 q^{67} +149.258 q^{69} -453.655 q^{71} -259.747 q^{73} +11.0555 q^{75} +170.862 q^{77} +323.190 q^{79} +637.533 q^{81} +797.471 q^{83} +843.392 q^{85} -99.9602 q^{87} -866.853 q^{89} -133.742 q^{91} -201.935 q^{93} -319.866 q^{95} -936.077 q^{97} +1653.47 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 28q^{7} + 54q^{9} + O(q^{10}) \) \( 10q - 28q^{7} + 54q^{9} - 124q^{15} + 4q^{17} - 276q^{23} + 50q^{25} - 368q^{31} - 4q^{33} - 732q^{39} - 944q^{47} - 94q^{49} - 1380q^{55} + 108q^{57} - 2628q^{63} - 492q^{65} - 3468q^{71} - 296q^{73} - 4416q^{79} - 482q^{81} - 6036q^{87} + 88q^{89} - 6900q^{95} - 4q^{97} + O(q^{100}) \)

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
\(3\) 1.07024 0.205968 0.102984 0.994683i \(-0.467161\pi\)
0.102984 + 0.994683i \(0.467161\pi\)
\(4\) 0 0
\(5\) 11.6331 1.04050 0.520250 0.854014i \(-0.325839\pi\)
0.520250 + 0.854014i \(0.325839\pi\)
\(6\) 0 0
\(7\) −2.67171 −0.144259 −0.0721293 0.997395i \(-0.522979\pi\)
−0.0721293 + 0.997395i \(0.522979\pi\)
\(8\) 0 0
\(9\) −25.8546 −0.957577
\(10\) 0 0
\(11\) −63.9525 −1.75295 −0.876473 0.481451i \(-0.840110\pi\)
−0.876473 + 0.481451i \(0.840110\pi\)
\(12\) 0 0
\(13\) 50.0586 1.06798 0.533990 0.845491i \(-0.320692\pi\)
0.533990 + 0.845491i \(0.320692\pi\)
\(14\) 0 0
\(15\) 12.4503 0.214310
\(16\) 0 0
\(17\) 72.4991 1.03433 0.517165 0.855886i \(-0.326987\pi\)
0.517165 + 0.855886i \(0.326987\pi\)
\(18\) 0 0
\(19\) −27.4961 −0.332002 −0.166001 0.986126i \(-0.553085\pi\)
−0.166001 + 0.986126i \(0.553085\pi\)
\(20\) 0 0
\(21\) −2.85937 −0.0297127
\(22\) 0 0
\(23\) 139.462 1.26434 0.632170 0.774830i \(-0.282165\pi\)
0.632170 + 0.774830i \(0.282165\pi\)
\(24\) 0 0
\(25\) 10.3299 0.0826390
\(26\) 0 0
\(27\) −56.5672 −0.403199
\(28\) 0 0
\(29\) −93.3995 −0.598064 −0.299032 0.954243i \(-0.596664\pi\)
−0.299032 + 0.954243i \(0.596664\pi\)
\(30\) 0 0
\(31\) −188.682 −1.09317 −0.546584 0.837404i \(-0.684072\pi\)
−0.546584 + 0.837404i \(0.684072\pi\)
\(32\) 0 0
\(33\) −68.4447 −0.361051
\(34\) 0 0
\(35\) −31.0803 −0.150101
\(36\) 0 0
\(37\) −118.886 −0.528238 −0.264119 0.964490i \(-0.585081\pi\)
−0.264119 + 0.964490i \(0.585081\pi\)
\(38\) 0 0
\(39\) 53.5748 0.219970
\(40\) 0 0
\(41\) 104.629 0.398545 0.199272 0.979944i \(-0.436142\pi\)
0.199272 + 0.979944i \(0.436142\pi\)
\(42\) 0 0
\(43\) −44.5275 −0.157916 −0.0789579 0.996878i \(-0.525159\pi\)
−0.0789579 + 0.996878i \(0.525159\pi\)
\(44\) 0 0
\(45\) −300.770 −0.996358
\(46\) 0 0
\(47\) −488.151 −1.51498 −0.757491 0.652846i \(-0.773575\pi\)
−0.757491 + 0.652846i \(0.773575\pi\)
\(48\) 0 0
\(49\) −335.862 −0.979189
\(50\) 0 0
\(51\) 77.5916 0.213039
\(52\) 0 0
\(53\) −211.510 −0.548173 −0.274087 0.961705i \(-0.588376\pi\)
−0.274087 + 0.961705i \(0.588376\pi\)
\(54\) 0 0
\(55\) −743.968 −1.82394
\(56\) 0 0
\(57\) −29.4275 −0.0683819
\(58\) 0 0
\(59\) −402.624 −0.888426 −0.444213 0.895921i \(-0.646517\pi\)
−0.444213 + 0.895921i \(0.646517\pi\)
\(60\) 0 0
\(61\) 322.538 0.676997 0.338498 0.940967i \(-0.390081\pi\)
0.338498 + 0.940967i \(0.390081\pi\)
\(62\) 0 0
\(63\) 69.0758 0.138139
\(64\) 0 0
\(65\) 582.338 1.11123
\(66\) 0 0
\(67\) 196.789 0.358829 0.179415 0.983774i \(-0.442580\pi\)
0.179415 + 0.983774i \(0.442580\pi\)
\(68\) 0 0
\(69\) 149.258 0.260414
\(70\) 0 0
\(71\) −453.655 −0.758294 −0.379147 0.925336i \(-0.623783\pi\)
−0.379147 + 0.925336i \(0.623783\pi\)
\(72\) 0 0
\(73\) −259.747 −0.416454 −0.208227 0.978081i \(-0.566769\pi\)
−0.208227 + 0.978081i \(0.566769\pi\)
\(74\) 0 0
\(75\) 11.0555 0.0170210
\(76\) 0 0
\(77\) 170.862 0.252877
\(78\) 0 0
\(79\) 323.190 0.460275 0.230138 0.973158i \(-0.426082\pi\)
0.230138 + 0.973158i \(0.426082\pi\)
\(80\) 0 0
\(81\) 637.533 0.874531
\(82\) 0 0
\(83\) 797.471 1.05462 0.527312 0.849672i \(-0.323200\pi\)
0.527312 + 0.849672i \(0.323200\pi\)
\(84\) 0 0
\(85\) 843.392 1.07622
\(86\) 0 0
\(87\) −99.9602 −0.123182
\(88\) 0 0
\(89\) −866.853 −1.03243 −0.516215 0.856459i \(-0.672659\pi\)
−0.516215 + 0.856459i \(0.672659\pi\)
\(90\) 0 0
\(91\) −133.742 −0.154065
\(92\) 0 0
\(93\) −201.935 −0.225158
\(94\) 0 0
\(95\) −319.866 −0.345448
\(96\) 0 0
\(97\) −936.077 −0.979837 −0.489919 0.871768i \(-0.662974\pi\)
−0.489919 + 0.871768i \(0.662974\pi\)
\(98\) 0 0
\(99\) 1653.47 1.67858
\(100\) 0 0
\(101\) −2.24639 −0.00221311 −0.00110656 0.999999i \(-0.500352\pi\)
−0.00110656 + 0.999999i \(0.500352\pi\)
\(102\) 0 0
\(103\) −1388.28 −1.32807 −0.664036 0.747700i \(-0.731158\pi\)
−0.664036 + 0.747700i \(0.731158\pi\)
\(104\) 0 0
\(105\) −33.2635 −0.0309160
\(106\) 0 0
\(107\) −1161.81 −1.04969 −0.524845 0.851198i \(-0.675877\pi\)
−0.524845 + 0.851198i \(0.675877\pi\)
\(108\) 0 0
\(109\) 753.489 0.662121 0.331060 0.943610i \(-0.392594\pi\)
0.331060 + 0.943610i \(0.392594\pi\)
\(110\) 0 0
\(111\) −127.237 −0.108800
\(112\) 0 0
\(113\) 67.2680 0.0560003 0.0280002 0.999608i \(-0.491086\pi\)
0.0280002 + 0.999608i \(0.491086\pi\)
\(114\) 0 0
\(115\) 1622.38 1.31554
\(116\) 0 0
\(117\) −1294.24 −1.02267
\(118\) 0 0
\(119\) −193.696 −0.149211
\(120\) 0 0
\(121\) 2758.92 2.07282
\(122\) 0 0
\(123\) 111.979 0.0820876
\(124\) 0 0
\(125\) −1333.97 −0.954514
\(126\) 0 0
\(127\) −1903.59 −1.33005 −0.665026 0.746820i \(-0.731579\pi\)
−0.665026 + 0.746820i \(0.731579\pi\)
\(128\) 0 0
\(129\) −47.6552 −0.0325257
\(130\) 0 0
\(131\) 1298.86 0.866271 0.433136 0.901329i \(-0.357407\pi\)
0.433136 + 0.901329i \(0.357407\pi\)
\(132\) 0 0
\(133\) 73.4615 0.0478941
\(134\) 0 0
\(135\) −658.054 −0.419528
\(136\) 0 0
\(137\) 477.234 0.297612 0.148806 0.988866i \(-0.452457\pi\)
0.148806 + 0.988866i \(0.452457\pi\)
\(138\) 0 0
\(139\) −2140.97 −1.30643 −0.653217 0.757171i \(-0.726581\pi\)
−0.653217 + 0.757171i \(0.726581\pi\)
\(140\) 0 0
\(141\) −522.440 −0.312038
\(142\) 0 0
\(143\) −3201.37 −1.87211
\(144\) 0 0
\(145\) −1086.53 −0.622285
\(146\) 0 0
\(147\) −359.454 −0.201682
\(148\) 0 0
\(149\) −530.829 −0.291861 −0.145930 0.989295i \(-0.546618\pi\)
−0.145930 + 0.989295i \(0.546618\pi\)
\(150\) 0 0
\(151\) −2997.52 −1.61546 −0.807730 0.589553i \(-0.799304\pi\)
−0.807730 + 0.589553i \(0.799304\pi\)
\(152\) 0 0
\(153\) −1874.43 −0.990451
\(154\) 0 0
\(155\) −2194.96 −1.13744
\(156\) 0 0
\(157\) 2134.06 1.08482 0.542409 0.840115i \(-0.317512\pi\)
0.542409 + 0.840115i \(0.317512\pi\)
\(158\) 0 0
\(159\) −226.368 −0.112906
\(160\) 0 0
\(161\) −372.601 −0.182392
\(162\) 0 0
\(163\) −2015.52 −0.968515 −0.484258 0.874925i \(-0.660910\pi\)
−0.484258 + 0.874925i \(0.660910\pi\)
\(164\) 0 0
\(165\) −796.227 −0.375674
\(166\) 0 0
\(167\) −792.415 −0.367179 −0.183590 0.983003i \(-0.558772\pi\)
−0.183590 + 0.983003i \(0.558772\pi\)
\(168\) 0 0
\(169\) 308.861 0.140583
\(170\) 0 0
\(171\) 710.900 0.317917
\(172\) 0 0
\(173\) −1094.03 −0.480794 −0.240397 0.970675i \(-0.577278\pi\)
−0.240397 + 0.970675i \(0.577278\pi\)
\(174\) 0 0
\(175\) −27.5984 −0.0119214
\(176\) 0 0
\(177\) −430.905 −0.182988
\(178\) 0 0
\(179\) −602.526 −0.251592 −0.125796 0.992056i \(-0.540148\pi\)
−0.125796 + 0.992056i \(0.540148\pi\)
\(180\) 0 0
\(181\) 3702.50 1.52047 0.760234 0.649649i \(-0.225084\pi\)
0.760234 + 0.649649i \(0.225084\pi\)
\(182\) 0 0
\(183\) 345.194 0.139440
\(184\) 0 0
\(185\) −1383.02 −0.549631
\(186\) 0 0
\(187\) −4636.50 −1.81312
\(188\) 0 0
\(189\) 151.131 0.0581649
\(190\) 0 0
\(191\) −3216.39 −1.21848 −0.609240 0.792986i \(-0.708525\pi\)
−0.609240 + 0.792986i \(0.708525\pi\)
\(192\) 0 0
\(193\) 2852.57 1.06390 0.531950 0.846776i \(-0.321459\pi\)
0.531950 + 0.846776i \(0.321459\pi\)
\(194\) 0 0
\(195\) 623.243 0.228879
\(196\) 0 0
\(197\) 2275.50 0.822956 0.411478 0.911420i \(-0.365013\pi\)
0.411478 + 0.911420i \(0.365013\pi\)
\(198\) 0 0
\(199\) 747.136 0.266146 0.133073 0.991106i \(-0.457516\pi\)
0.133073 + 0.991106i \(0.457516\pi\)
\(200\) 0 0
\(201\) 210.612 0.0739074
\(202\) 0 0
\(203\) 249.536 0.0862758
\(204\) 0 0
\(205\) 1217.17 0.414685
\(206\) 0 0
\(207\) −3605.73 −1.21070
\(208\) 0 0
\(209\) 1758.44 0.581981
\(210\) 0 0
\(211\) 3149.64 1.02763 0.513816 0.857901i \(-0.328231\pi\)
0.513816 + 0.857901i \(0.328231\pi\)
\(212\) 0 0
\(213\) −485.520 −0.156185
\(214\) 0 0
\(215\) −517.995 −0.164311
\(216\) 0 0
\(217\) 504.102 0.157699
\(218\) 0 0
\(219\) −277.993 −0.0857763
\(220\) 0 0
\(221\) 3629.20 1.10464
\(222\) 0 0
\(223\) 358.053 0.107520 0.0537601 0.998554i \(-0.482879\pi\)
0.0537601 + 0.998554i \(0.482879\pi\)
\(224\) 0 0
\(225\) −267.075 −0.0791332
\(226\) 0 0
\(227\) 4886.67 1.42881 0.714404 0.699733i \(-0.246698\pi\)
0.714404 + 0.699733i \(0.246698\pi\)
\(228\) 0 0
\(229\) −2022.37 −0.583589 −0.291794 0.956481i \(-0.594252\pi\)
−0.291794 + 0.956481i \(0.594252\pi\)
\(230\) 0 0
\(231\) 182.864 0.0520847
\(232\) 0 0
\(233\) −926.479 −0.260496 −0.130248 0.991481i \(-0.541577\pi\)
−0.130248 + 0.991481i \(0.541577\pi\)
\(234\) 0 0
\(235\) −5678.73 −1.57634
\(236\) 0 0
\(237\) 345.892 0.0948021
\(238\) 0 0
\(239\) −792.472 −0.214480 −0.107240 0.994233i \(-0.534201\pi\)
−0.107240 + 0.994233i \(0.534201\pi\)
\(240\) 0 0
\(241\) −1449.01 −0.387299 −0.193650 0.981071i \(-0.562033\pi\)
−0.193650 + 0.981071i \(0.562033\pi\)
\(242\) 0 0
\(243\) 2209.63 0.583325
\(244\) 0 0
\(245\) −3907.13 −1.01885
\(246\) 0 0
\(247\) −1376.42 −0.354572
\(248\) 0 0
\(249\) 853.487 0.217219
\(250\) 0 0
\(251\) 5062.94 1.27319 0.636594 0.771199i \(-0.280343\pi\)
0.636594 + 0.771199i \(0.280343\pi\)
\(252\) 0 0
\(253\) −8918.93 −2.21632
\(254\) 0 0
\(255\) 902.634 0.221667
\(256\) 0 0
\(257\) −4708.87 −1.14292 −0.571461 0.820629i \(-0.693623\pi\)
−0.571461 + 0.820629i \(0.693623\pi\)
\(258\) 0 0
\(259\) 317.629 0.0762028
\(260\) 0 0
\(261\) 2414.81 0.572692
\(262\) 0 0
\(263\) −2967.82 −0.695830 −0.347915 0.937526i \(-0.613110\pi\)
−0.347915 + 0.937526i \(0.613110\pi\)
\(264\) 0 0
\(265\) −2460.53 −0.570374
\(266\) 0 0
\(267\) −927.743 −0.212648
\(268\) 0 0
\(269\) −938.519 −0.212723 −0.106362 0.994328i \(-0.533920\pi\)
−0.106362 + 0.994328i \(0.533920\pi\)
\(270\) 0 0
\(271\) 8058.74 1.80640 0.903199 0.429223i \(-0.141212\pi\)
0.903199 + 0.429223i \(0.141212\pi\)
\(272\) 0 0
\(273\) −143.136 −0.0317326
\(274\) 0 0
\(275\) −660.622 −0.144862
\(276\) 0 0
\(277\) −682.325 −0.148003 −0.0740017 0.997258i \(-0.523577\pi\)
−0.0740017 + 0.997258i \(0.523577\pi\)
\(278\) 0 0
\(279\) 4878.29 1.04679
\(280\) 0 0
\(281\) 5899.10 1.25235 0.626175 0.779682i \(-0.284619\pi\)
0.626175 + 0.779682i \(0.284619\pi\)
\(282\) 0 0
\(283\) 961.519 0.201966 0.100983 0.994888i \(-0.467801\pi\)
0.100983 + 0.994888i \(0.467801\pi\)
\(284\) 0 0
\(285\) −342.334 −0.0711513
\(286\) 0 0
\(287\) −279.538 −0.0574935
\(288\) 0 0
\(289\) 343.118 0.0698388
\(290\) 0 0
\(291\) −1001.83 −0.201815
\(292\) 0 0
\(293\) −5024.51 −1.00183 −0.500913 0.865498i \(-0.667002\pi\)
−0.500913 + 0.865498i \(0.667002\pi\)
\(294\) 0 0
\(295\) −4683.78 −0.924407
\(296\) 0 0
\(297\) 3617.62 0.706786
\(298\) 0 0
\(299\) 6981.26 1.35029
\(300\) 0 0
\(301\) 118.964 0.0227807
\(302\) 0 0
\(303\) −2.40419 −0.000455831 0
\(304\) 0 0
\(305\) 3752.13 0.704415
\(306\) 0 0
\(307\) −3868.67 −0.719207 −0.359603 0.933105i \(-0.617088\pi\)
−0.359603 + 0.933105i \(0.617088\pi\)
\(308\) 0 0
\(309\) −1485.80 −0.273541
\(310\) 0 0
\(311\) −5796.70 −1.05692 −0.528458 0.848960i \(-0.677229\pi\)
−0.528458 + 0.848960i \(0.677229\pi\)
\(312\) 0 0
\(313\) −8362.62 −1.51017 −0.755085 0.655627i \(-0.772404\pi\)
−0.755085 + 0.655627i \(0.772404\pi\)
\(314\) 0 0
\(315\) 803.569 0.143733
\(316\) 0 0
\(317\) 487.064 0.0862973 0.0431487 0.999069i \(-0.486261\pi\)
0.0431487 + 0.999069i \(0.486261\pi\)
\(318\) 0 0
\(319\) 5973.13 1.04837
\(320\) 0 0
\(321\) −1243.42 −0.216203
\(322\) 0 0
\(323\) −1993.44 −0.343400
\(324\) 0 0
\(325\) 517.099 0.0882569
\(326\) 0 0
\(327\) 806.416 0.136376
\(328\) 0 0
\(329\) 1304.20 0.218549
\(330\) 0 0
\(331\) 3801.21 0.631218 0.315609 0.948889i \(-0.397791\pi\)
0.315609 + 0.948889i \(0.397791\pi\)
\(332\) 0 0
\(333\) 3073.76 0.505828
\(334\) 0 0
\(335\) 2289.27 0.373361
\(336\) 0 0
\(337\) 1795.31 0.290199 0.145099 0.989417i \(-0.453650\pi\)
0.145099 + 0.989417i \(0.453650\pi\)
\(338\) 0 0
\(339\) 71.9931 0.0115343
\(340\) 0 0
\(341\) 12066.7 1.91627
\(342\) 0 0
\(343\) 1813.72 0.285515
\(344\) 0 0
\(345\) 1736.34 0.270960
\(346\) 0 0
\(347\) 2782.23 0.430426 0.215213 0.976567i \(-0.430955\pi\)
0.215213 + 0.976567i \(0.430955\pi\)
\(348\) 0 0
\(349\) 10413.4 1.59718 0.798589 0.601876i \(-0.205580\pi\)
0.798589 + 0.601876i \(0.205580\pi\)
\(350\) 0 0
\(351\) −2831.68 −0.430609
\(352\) 0 0
\(353\) 10644.3 1.60493 0.802466 0.596698i \(-0.203521\pi\)
0.802466 + 0.596698i \(0.203521\pi\)
\(354\) 0 0
\(355\) −5277.43 −0.789005
\(356\) 0 0
\(357\) −207.302 −0.0307327
\(358\) 0 0
\(359\) 7459.42 1.09664 0.548319 0.836269i \(-0.315268\pi\)
0.548319 + 0.836269i \(0.315268\pi\)
\(360\) 0 0
\(361\) −6102.96 −0.889775
\(362\) 0 0
\(363\) 2952.72 0.426935
\(364\) 0 0
\(365\) −3021.68 −0.433320
\(366\) 0 0
\(367\) −6251.35 −0.889149 −0.444574 0.895742i \(-0.646645\pi\)
−0.444574 + 0.895742i \(0.646645\pi\)
\(368\) 0 0
\(369\) −2705.14 −0.381637
\(370\) 0 0
\(371\) 565.094 0.0790787
\(372\) 0 0
\(373\) −12603.3 −1.74953 −0.874763 0.484552i \(-0.838983\pi\)
−0.874763 + 0.484552i \(0.838983\pi\)
\(374\) 0 0
\(375\) −1427.68 −0.196600
\(376\) 0 0
\(377\) −4675.45 −0.638721
\(378\) 0 0
\(379\) −1674.47 −0.226943 −0.113472 0.993541i \(-0.536197\pi\)
−0.113472 + 0.993541i \(0.536197\pi\)
\(380\) 0 0
\(381\) −2037.31 −0.273949
\(382\) 0 0
\(383\) 2880.38 0.384283 0.192142 0.981367i \(-0.438457\pi\)
0.192142 + 0.981367i \(0.438457\pi\)
\(384\) 0 0
\(385\) 1987.66 0.263119
\(386\) 0 0
\(387\) 1151.24 0.151217
\(388\) 0 0
\(389\) 13073.3 1.70397 0.851984 0.523567i \(-0.175399\pi\)
0.851984 + 0.523567i \(0.175399\pi\)
\(390\) 0 0
\(391\) 10110.9 1.30774
\(392\) 0 0
\(393\) 1390.09 0.178424
\(394\) 0 0
\(395\) 3759.72 0.478916
\(396\) 0 0
\(397\) −6021.44 −0.761228 −0.380614 0.924734i \(-0.624287\pi\)
−0.380614 + 0.924734i \(0.624287\pi\)
\(398\) 0 0
\(399\) 78.6216 0.00986467
\(400\) 0 0
\(401\) −12722.6 −1.58437 −0.792187 0.610278i \(-0.791058\pi\)
−0.792187 + 0.610278i \(0.791058\pi\)
\(402\) 0 0
\(403\) −9445.14 −1.16748
\(404\) 0 0
\(405\) 7416.51 0.909949
\(406\) 0 0
\(407\) 7603.08 0.925972
\(408\) 0 0
\(409\) −232.991 −0.0281678 −0.0140839 0.999901i \(-0.504483\pi\)
−0.0140839 + 0.999901i \(0.504483\pi\)
\(410\) 0 0
\(411\) 510.756 0.0612987
\(412\) 0 0
\(413\) 1075.69 0.128163
\(414\) 0 0
\(415\) 9277.08 1.09734
\(416\) 0 0
\(417\) −2291.35 −0.269084
\(418\) 0 0
\(419\) −8663.03 −1.01006 −0.505032 0.863101i \(-0.668519\pi\)
−0.505032 + 0.863101i \(0.668519\pi\)
\(420\) 0 0
\(421\) −11749.9 −1.36023 −0.680113 0.733107i \(-0.738069\pi\)
−0.680113 + 0.733107i \(0.738069\pi\)
\(422\) 0 0
\(423\) 12620.9 1.45071
\(424\) 0 0
\(425\) 748.907 0.0854760
\(426\) 0 0
\(427\) −861.727 −0.0976626
\(428\) 0 0
\(429\) −3426.24 −0.385596
\(430\) 0 0
\(431\) 8737.57 0.976506 0.488253 0.872702i \(-0.337634\pi\)
0.488253 + 0.872702i \(0.337634\pi\)
\(432\) 0 0
\(433\) 11627.5 1.29049 0.645247 0.763974i \(-0.276755\pi\)
0.645247 + 0.763974i \(0.276755\pi\)
\(434\) 0 0
\(435\) −1162.85 −0.128171
\(436\) 0 0
\(437\) −3834.66 −0.419763
\(438\) 0 0
\(439\) 17631.8 1.91690 0.958450 0.285261i \(-0.0920804\pi\)
0.958450 + 0.285261i \(0.0920804\pi\)
\(440\) 0 0
\(441\) 8683.57 0.937649
\(442\) 0 0
\(443\) 6434.41 0.690085 0.345043 0.938587i \(-0.387864\pi\)
0.345043 + 0.938587i \(0.387864\pi\)
\(444\) 0 0
\(445\) −10084.2 −1.07424
\(446\) 0 0
\(447\) −568.116 −0.0601140
\(448\) 0 0
\(449\) −12926.5 −1.35867 −0.679334 0.733830i \(-0.737731\pi\)
−0.679334 + 0.733830i \(0.737731\pi\)
\(450\) 0 0
\(451\) −6691.30 −0.698627
\(452\) 0 0
\(453\) −3208.07 −0.332734
\(454\) 0 0
\(455\) −1555.84 −0.160305
\(456\) 0 0
\(457\) 9320.32 0.954018 0.477009 0.878898i \(-0.341721\pi\)
0.477009 + 0.878898i \(0.341721\pi\)
\(458\) 0 0
\(459\) −4101.07 −0.417041
\(460\) 0 0
\(461\) 18222.1 1.84098 0.920488 0.390771i \(-0.127792\pi\)
0.920488 + 0.390771i \(0.127792\pi\)
\(462\) 0 0
\(463\) 7038.37 0.706482 0.353241 0.935532i \(-0.385080\pi\)
0.353241 + 0.935532i \(0.385080\pi\)
\(464\) 0 0
\(465\) −2349.14 −0.234277
\(466\) 0 0
\(467\) −8487.77 −0.841043 −0.420522 0.907283i \(-0.638153\pi\)
−0.420522 + 0.907283i \(0.638153\pi\)
\(468\) 0 0
\(469\) −525.761 −0.0517642
\(470\) 0 0
\(471\) 2283.96 0.223438
\(472\) 0 0
\(473\) 2847.65 0.276818
\(474\) 0 0
\(475\) −284.031 −0.0274363
\(476\) 0 0
\(477\) 5468.51 0.524918
\(478\) 0 0
\(479\) 587.317 0.0560234 0.0280117 0.999608i \(-0.491082\pi\)
0.0280117 + 0.999608i \(0.491082\pi\)
\(480\) 0 0
\(481\) −5951.28 −0.564148
\(482\) 0 0
\(483\) −398.773 −0.0375669
\(484\) 0 0
\(485\) −10889.5 −1.01952
\(486\) 0 0
\(487\) −8366.45 −0.778481 −0.389240 0.921136i \(-0.627262\pi\)
−0.389240 + 0.921136i \(0.627262\pi\)
\(488\) 0 0
\(489\) −2157.10 −0.199483
\(490\) 0 0
\(491\) 2162.76 0.198786 0.0993929 0.995048i \(-0.468310\pi\)
0.0993929 + 0.995048i \(0.468310\pi\)
\(492\) 0 0
\(493\) −6771.38 −0.618596
\(494\) 0 0
\(495\) 19235.0 1.74656
\(496\) 0 0
\(497\) 1212.03 0.109390
\(498\) 0 0
\(499\) −16071.8 −1.44183 −0.720913 0.693026i \(-0.756277\pi\)
−0.720913 + 0.693026i \(0.756277\pi\)
\(500\) 0 0
\(501\) −848.077 −0.0756273
\(502\) 0 0
\(503\) 12570.2 1.11427 0.557137 0.830421i \(-0.311900\pi\)
0.557137 + 0.830421i \(0.311900\pi\)
\(504\) 0 0
\(505\) −26.1326 −0.00230274
\(506\) 0 0
\(507\) 330.556 0.0289556
\(508\) 0 0
\(509\) −16801.4 −1.46308 −0.731542 0.681797i \(-0.761199\pi\)
−0.731542 + 0.681797i \(0.761199\pi\)
\(510\) 0 0
\(511\) 693.968 0.0600770
\(512\) 0 0
\(513\) 1555.38 0.133863
\(514\) 0 0
\(515\) −16150.1 −1.38186
\(516\) 0 0
\(517\) 31218.5 2.65568
\(518\) 0 0
\(519\) −1170.87 −0.0990283
\(520\) 0 0
\(521\) −6612.98 −0.556085 −0.278042 0.960569i \(-0.589686\pi\)
−0.278042 + 0.960569i \(0.589686\pi\)
\(522\) 0 0
\(523\) 7253.92 0.606485 0.303243 0.952913i \(-0.401931\pi\)
0.303243 + 0.952913i \(0.401931\pi\)
\(524\) 0 0
\(525\) −29.5370 −0.00245543
\(526\) 0 0
\(527\) −13679.3 −1.13070
\(528\) 0 0
\(529\) 7282.60 0.598553
\(530\) 0 0
\(531\) 10409.7 0.850737
\(532\) 0 0
\(533\) 5237.59 0.425638
\(534\) 0 0
\(535\) −13515.5 −1.09220
\(536\) 0 0
\(537\) −644.849 −0.0518199
\(538\) 0 0
\(539\) 21479.2 1.71647
\(540\) 0 0
\(541\) 15511.9 1.23273 0.616365 0.787461i \(-0.288605\pi\)
0.616365 + 0.787461i \(0.288605\pi\)
\(542\) 0 0
\(543\) 3962.57 0.313168
\(544\) 0 0
\(545\) 8765.44 0.688936
\(546\) 0 0
\(547\) 18510.4 1.44689 0.723445 0.690382i \(-0.242558\pi\)
0.723445 + 0.690382i \(0.242558\pi\)
\(548\) 0 0
\(549\) −8339.09 −0.648277
\(550\) 0 0
\(551\) 2568.12 0.198558
\(552\) 0 0
\(553\) −863.469 −0.0663986
\(554\) 0 0
\(555\) −1480.17 −0.113207
\(556\) 0 0
\(557\) 7141.59 0.543266 0.271633 0.962401i \(-0.412436\pi\)
0.271633 + 0.962401i \(0.412436\pi\)
\(558\) 0 0
\(559\) −2228.98 −0.168651
\(560\) 0 0
\(561\) −4962.18 −0.373446
\(562\) 0 0
\(563\) 4594.86 0.343961 0.171981 0.985100i \(-0.444983\pi\)
0.171981 + 0.985100i \(0.444983\pi\)
\(564\) 0 0
\(565\) 782.538 0.0582683
\(566\) 0 0
\(567\) −1703.30 −0.126159
\(568\) 0 0
\(569\) −2806.05 −0.206741 −0.103371 0.994643i \(-0.532963\pi\)
−0.103371 + 0.994643i \(0.532963\pi\)
\(570\) 0 0
\(571\) 17025.4 1.24779 0.623897 0.781506i \(-0.285548\pi\)
0.623897 + 0.781506i \(0.285548\pi\)
\(572\) 0 0
\(573\) −3442.31 −0.250968
\(574\) 0 0
\(575\) 1440.62 0.104484
\(576\) 0 0
\(577\) 7206.84 0.519973 0.259987 0.965612i \(-0.416282\pi\)
0.259987 + 0.965612i \(0.416282\pi\)
\(578\) 0 0
\(579\) 3052.95 0.219130
\(580\) 0 0
\(581\) −2130.61 −0.152138
\(582\) 0 0
\(583\) 13526.6 0.960918
\(584\) 0 0
\(585\) −15056.1 −1.06409
\(586\) 0 0
\(587\) 14675.2 1.03188 0.515938 0.856626i \(-0.327443\pi\)
0.515938 + 0.856626i \(0.327443\pi\)
\(588\) 0 0
\(589\) 5188.01 0.362934
\(590\) 0 0
\(591\) 2435.33 0.169503
\(592\) 0 0
\(593\) 4758.60 0.329531 0.164766 0.986333i \(-0.447313\pi\)
0.164766 + 0.986333i \(0.447313\pi\)
\(594\) 0 0
\(595\) −2253.29 −0.155254
\(596\) 0 0
\(597\) 799.617 0.0548176
\(598\) 0 0
\(599\) −14256.4 −0.972455 −0.486227 0.873832i \(-0.661627\pi\)
−0.486227 + 0.873832i \(0.661627\pi\)
\(600\) 0 0
\(601\) −10385.2 −0.704862 −0.352431 0.935838i \(-0.614645\pi\)
−0.352431 + 0.935838i \(0.614645\pi\)
\(602\) 0 0
\(603\) −5087.89 −0.343606
\(604\) 0 0
\(605\) 32094.9 2.15677
\(606\) 0 0
\(607\) 16243.6 1.08618 0.543088 0.839676i \(-0.317255\pi\)
0.543088 + 0.839676i \(0.317255\pi\)
\(608\) 0 0
\(609\) 267.064 0.0177701
\(610\) 0 0
\(611\) −24436.1 −1.61797
\(612\) 0 0
\(613\) −707.817 −0.0466369 −0.0233185 0.999728i \(-0.507423\pi\)
−0.0233185 + 0.999728i \(0.507423\pi\)
\(614\) 0 0
\(615\) 1302.66 0.0854121
\(616\) 0 0
\(617\) 11575.9 0.755316 0.377658 0.925945i \(-0.376729\pi\)
0.377658 + 0.925945i \(0.376729\pi\)
\(618\) 0 0
\(619\) 25959.4 1.68562 0.842808 0.538214i \(-0.180901\pi\)
0.842808 + 0.538214i \(0.180901\pi\)
\(620\) 0 0
\(621\) −7888.97 −0.509780
\(622\) 0 0
\(623\) 2315.98 0.148937
\(624\) 0 0
\(625\) −16809.5 −1.07581
\(626\) 0 0
\(627\) 1881.96 0.119870
\(628\) 0 0
\(629\) −8619.15 −0.546372
\(630\) 0 0
\(631\) 10224.8 0.645079 0.322539 0.946556i \(-0.395463\pi\)
0.322539 + 0.946556i \(0.395463\pi\)
\(632\) 0 0
\(633\) 3370.88 0.211660
\(634\) 0 0
\(635\) −22144.8 −1.38392
\(636\) 0 0
\(637\) −16812.8 −1.04576
\(638\) 0 0
\(639\) 11729.0 0.726125
\(640\) 0 0
\(641\) 19804.4 1.22032 0.610162 0.792277i \(-0.291104\pi\)
0.610162 + 0.792277i \(0.291104\pi\)
\(642\) 0 0
\(643\) 22175.9 1.36008 0.680041 0.733174i \(-0.261962\pi\)
0.680041 + 0.733174i \(0.261962\pi\)
\(644\) 0 0
\(645\) −554.380 −0.0338429
\(646\) 0 0
\(647\) −9232.26 −0.560985 −0.280493 0.959856i \(-0.590498\pi\)
−0.280493 + 0.959856i \(0.590498\pi\)
\(648\) 0 0
\(649\) 25748.8 1.55736
\(650\) 0 0
\(651\) 539.511 0.0324810
\(652\) 0 0
\(653\) −27857.1 −1.66942 −0.834710 0.550690i \(-0.814365\pi\)
−0.834710 + 0.550690i \(0.814365\pi\)
\(654\) 0 0
\(655\) 15109.8 0.901355
\(656\) 0 0
\(657\) 6715.66 0.398786
\(658\) 0 0
\(659\) −5498.55 −0.325028 −0.162514 0.986706i \(-0.551960\pi\)
−0.162514 + 0.986706i \(0.551960\pi\)
\(660\) 0 0
\(661\) 11469.6 0.674908 0.337454 0.941342i \(-0.390434\pi\)
0.337454 + 0.941342i \(0.390434\pi\)
\(662\) 0 0
\(663\) 3884.13 0.227522
\(664\) 0 0
\(665\) 854.587 0.0498338
\(666\) 0 0
\(667\) −13025.7 −0.756156
\(668\) 0 0
\(669\) 383.204 0.0221458
\(670\) 0 0
\(671\) −20627.1 −1.18674
\(672\) 0 0
\(673\) −28428.2 −1.62827 −0.814135 0.580676i \(-0.802788\pi\)
−0.814135 + 0.580676i \(0.802788\pi\)
\(674\) 0 0
\(675\) −584.333 −0.0333200
\(676\) 0 0
\(677\) −23995.5 −1.36222 −0.681110 0.732181i \(-0.738502\pi\)
−0.681110 + 0.732181i \(0.738502\pi\)
\(678\) 0 0
\(679\) 2500.92 0.141350
\(680\) 0 0
\(681\) 5229.92 0.294289
\(682\) 0 0
\(683\) −13506.0 −0.756650 −0.378325 0.925673i \(-0.623500\pi\)
−0.378325 + 0.925673i \(0.623500\pi\)
\(684\) 0 0
\(685\) 5551.73 0.309665
\(686\) 0 0
\(687\) −2164.42 −0.120201
\(688\) 0 0
\(689\) −10587.9 −0.585439
\(690\) 0 0
\(691\) 29500.1 1.62408 0.812038 0.583605i \(-0.198358\pi\)
0.812038 + 0.583605i \(0.198358\pi\)
\(692\) 0 0
\(693\) −4417.57 −0.242150
\(694\) 0 0
\(695\) −24906.1 −1.35934
\(696\) 0 0
\(697\) 7585.52 0.412227
\(698\) 0 0
\(699\) −991.557 −0.0536540
\(700\) 0 0
\(701\) −33227.5 −1.79028 −0.895139 0.445787i \(-0.852924\pi\)
−0.895139 + 0.445787i \(0.852924\pi\)
\(702\) 0 0
\(703\) 3268.91 0.175376
\(704\) 0 0
\(705\) −6077.62 −0.324676
\(706\) 0 0
\(707\) 6.00170 0.000319261 0
\(708\) 0 0
\(709\) 6448.04 0.341553 0.170777 0.985310i \(-0.445372\pi\)
0.170777 + 0.985310i \(0.445372\pi\)
\(710\) 0 0
\(711\) −8355.95 −0.440749
\(712\) 0 0
\(713\) −26313.9 −1.38214
\(714\) 0 0
\(715\) −37242.0 −1.94793
\(716\) 0 0
\(717\) −848.138 −0.0441761
\(718\) 0 0
\(719\) −6494.67 −0.336871 −0.168436 0.985713i \(-0.553872\pi\)
−0.168436 + 0.985713i \(0.553872\pi\)
\(720\) 0 0
\(721\) 3709.08 0.191586
\(722\) 0 0
\(723\) −1550.80 −0.0797714
\(724\) 0 0
\(725\) −964.806 −0.0494234
\(726\) 0 0
\(727\) 24866.4 1.26856 0.634280 0.773103i \(-0.281296\pi\)
0.634280 + 0.773103i \(0.281296\pi\)
\(728\) 0 0
\(729\) −14848.5 −0.754384
\(730\) 0 0
\(731\) −3228.20 −0.163337
\(732\) 0 0
\(733\) 21092.1 1.06283 0.531414 0.847112i \(-0.321661\pi\)
0.531414 + 0.847112i \(0.321661\pi\)
\(734\) 0 0
\(735\) −4181.58 −0.209850
\(736\) 0 0
\(737\) −12585.1 −0.629008
\(738\) 0 0
\(739\) 11952.7 0.594977 0.297489 0.954725i \(-0.403851\pi\)
0.297489 + 0.954725i \(0.403851\pi\)
\(740\) 0 0
\(741\) −1473.10 −0.0730305
\(742\) 0 0
\(743\) −5622.43 −0.277614 −0.138807 0.990319i \(-0.544327\pi\)
−0.138807 + 0.990319i \(0.544327\pi\)
\(744\) 0 0
\(745\) −6175.21 −0.303681
\(746\) 0 0
\(747\) −20618.3 −1.00988
\(748\) 0 0
\(749\) 3104.02 0.151427
\(750\) 0 0
\(751\) −32314.9 −1.57016 −0.785079 0.619396i \(-0.787378\pi\)
−0.785079 + 0.619396i \(0.787378\pi\)
\(752\) 0 0
\(753\) 5418.58 0.262236
\(754\) 0 0
\(755\) −34870.5 −1.68089
\(756\) 0 0
\(757\) −17950.4 −0.861847 −0.430923 0.902389i \(-0.641812\pi\)
−0.430923 + 0.902389i \(0.641812\pi\)
\(758\) 0 0
\(759\) −9545.42 −0.456491
\(760\) 0 0
\(761\) 13108.2 0.624404 0.312202 0.950016i \(-0.398933\pi\)
0.312202 + 0.950016i \(0.398933\pi\)
\(762\) 0 0
\(763\) −2013.10 −0.0955166
\(764\) 0 0
\(765\) −21805.5 −1.03056
\(766\) 0 0
\(767\) −20154.8 −0.948822
\(768\) 0 0
\(769\) 23661.2 1.10955 0.554776 0.832000i \(-0.312804\pi\)
0.554776 + 0.832000i \(0.312804\pi\)
\(770\) 0 0
\(771\) −5039.63 −0.235406
\(772\) 0 0
\(773\) 30222.4 1.40624 0.703120 0.711071i \(-0.251790\pi\)
0.703120 + 0.711071i \(0.251790\pi\)
\(774\) 0 0
\(775\) −1949.06 −0.0903384
\(776\) 0 0
\(777\) 339.940 0.0156954
\(778\) 0 0
\(779\) −2876.89 −0.132318
\(780\) 0 0
\(781\) 29012.3 1.32925
\(782\) 0 0
\(783\) 5283.35 0.241139
\(784\) 0 0
\(785\) 24825.8 1.12875
\(786\) 0 0
\(787\) 29544.2 1.33817 0.669083 0.743188i \(-0.266687\pi\)
0.669083 + 0.743188i \(0.266687\pi\)
\(788\) 0 0
\(789\) −3176.28 −0.143319
\(790\) 0 0
\(791\) −179.720 −0.00807853
\(792\) 0 0
\(793\) 16145.8 0.723020
\(794\) 0 0
\(795\) −2633.36 −0.117479
\(796\) 0 0
\(797\) 9665.91 0.429591 0.214795 0.976659i \(-0.431092\pi\)
0.214795 + 0.976659i \(0.431092\pi\)
\(798\) 0 0
\(799\) −35390.5 −1.56699
\(800\) 0 0
\(801\) 22412.1 0.988631
\(802\) 0 0
\(803\) 16611.5 0.730021
\(804\) 0 0
\(805\) −4334.52 −0.189778
\(806\) 0 0
\(807\) −1004.44 −0.0438142
\(808\) 0 0
\(809\) −15807.0 −0.686952 −0.343476 0.939162i \(-0.611604\pi\)
−0.343476 + 0.939162i \(0.611604\pi\)
\(810\) 0 0
\(811\) −16295.5 −0.705565 −0.352783 0.935705i \(-0.614764\pi\)
−0.352783 + 0.935705i \(0.614764\pi\)
\(812\) 0 0
\(813\) 8624.81 0.372061
\(814\) 0 0
\(815\) −23446.9 −1.00774
\(816\) 0 0
\(817\) 1224.33 0.0524284
\(818\) 0 0
\(819\) 3457.84 0.147529
\(820\) 0 0
\(821\) −436.743 −0.0185657 −0.00928286 0.999957i \(-0.502955\pi\)
−0.00928286 + 0.999957i \(0.502955\pi\)
\(822\) 0 0
\(823\) 5633.49 0.238604 0.119302 0.992858i \(-0.461934\pi\)
0.119302 + 0.992858i \(0.461934\pi\)
\(824\) 0 0
\(825\) −707.026 −0.0298369
\(826\) 0 0
\(827\) 22394.7 0.941645 0.470822 0.882228i \(-0.343957\pi\)
0.470822 + 0.882228i \(0.343957\pi\)
\(828\) 0 0
\(829\) 8768.39 0.367357 0.183678 0.982986i \(-0.441199\pi\)
0.183678 + 0.982986i \(0.441199\pi\)
\(830\) 0 0
\(831\) −730.253 −0.0304840
\(832\) 0 0
\(833\) −24349.7 −1.01281
\(834\) 0 0
\(835\) −9218.27 −0.382050
\(836\) 0 0
\(837\) 10673.2 0.440764
\(838\) 0 0
\(839\) −30644.3 −1.26098 −0.630488 0.776199i \(-0.717145\pi\)
−0.630488 + 0.776199i \(0.717145\pi\)
\(840\) 0 0
\(841\) −15665.5 −0.642319
\(842\) 0 0
\(843\) 6313.47 0.257945
\(844\) 0 0
\(845\) 3593.02 0.146277
\(846\) 0 0
\(847\) −7371.03 −0.299022
\(848\) 0 0
\(849\) 1029.06 0.0415986
\(850\) 0 0
\(851\) −16580.1 −0.667871
\(852\) 0 0
\(853\) 43559.2 1.74846 0.874232 0.485508i \(-0.161365\pi\)
0.874232 + 0.485508i \(0.161365\pi\)
\(854\) 0 0
\(855\) 8270.00 0.330793
\(856\) 0 0
\(857\) −41788.5 −1.66566 −0.832828 0.553533i \(-0.813279\pi\)
−0.832828 + 0.553533i \(0.813279\pi\)
\(858\) 0 0
\(859\) −16849.6 −0.669267 −0.334633 0.942348i \(-0.608612\pi\)
−0.334633 + 0.942348i \(0.608612\pi\)
\(860\) 0 0
\(861\) −299.174 −0.0118418
\(862\) 0 0
\(863\) −27636.7 −1.09011 −0.545054 0.838401i \(-0.683491\pi\)
−0.545054 + 0.838401i \(0.683491\pi\)
\(864\) 0 0
\(865\) −12727.0 −0.500266
\(866\) 0 0
\(867\) 367.220 0.0143846
\(868\) 0 0
\(869\) −20668.8 −0.806837
\(870\) 0 0
\(871\) 9850.95 0.383223
\(872\) 0 0
\(873\) 24201.9 0.938270
\(874\) 0 0
\(875\) 3563.98 0.137697
\(876\) 0 0
\(877\) −16855.0 −0.648977 −0.324488 0.945890i \(-0.605192\pi\)
−0.324488 + 0.945890i \(0.605192\pi\)
\(878\) 0 0
\(879\) −5377.45 −0.206344
\(880\) 0 0
\(881\) 13330.0 0.509759 0.254880 0.966973i \(-0.417964\pi\)
0.254880 + 0.966973i \(0.417964\pi\)
\(882\) 0 0
\(883\) 35598.7 1.35673 0.678365 0.734725i \(-0.262689\pi\)
0.678365 + 0.734725i \(0.262689\pi\)
\(884\) 0 0
\(885\) −5012.78 −0.190399
\(886\) 0 0
\(887\) −48821.3 −1.84810 −0.924048 0.382278i \(-0.875140\pi\)
−0.924048 + 0.382278i \(0.875140\pi\)
\(888\) 0 0
\(889\) 5085.84 0.191871
\(890\) 0 0
\(891\) −40771.8 −1.53301
\(892\) 0 0
\(893\) 13422.2 0.502977
\(894\) 0 0
\(895\) −7009.27 −0.261781
\(896\) 0 0
\(897\) 7471.64 0.278117
\(898\) 0 0
\(899\) 17622.8 0.653785
\(900\) 0 0
\(901\) −15334.3 −0.566992
\(902\) 0 0
\(903\) 127.321 0.00469210
\(904\) 0 0
\(905\) 43071.7 1.58205
\(906\) 0 0
\(907\) 7523.90 0.275443 0.137722 0.990471i \(-0.456022\pi\)
0.137722 + 0.990471i \(0.456022\pi\)
\(908\) 0 0
\(909\) 58.0796 0.00211923
\(910\) 0 0
\(911\) −26016.0 −0.946155 −0.473077 0.881021i \(-0.656857\pi\)
−0.473077 + 0.881021i \(0.656857\pi\)
\(912\) 0 0
\(913\) −51000.2 −1.84870
\(914\) 0 0
\(915\) 4015.69 0.145087
\(916\) 0 0
\(917\) −3470.16 −0.124967
\(918\) 0 0
\(919\) −24082.1 −0.864413 −0.432206 0.901775i \(-0.642265\pi\)
−0.432206 + 0.901775i \(0.642265\pi\)
\(920\) 0 0
\(921\) −4140.41 −0.148134
\(922\) 0 0
\(923\) −22709.3 −0.809844
\(924\) 0 0
\(925\) −1228.08 −0.0436530
\(926\) 0 0
\(927\) 35893.5 1.27173
\(928\) 0 0
\(929\) 9324.93 0.329323 0.164661 0.986350i \(-0.447347\pi\)
0.164661 + 0.986350i \(0.447347\pi\)
\(930\) 0 0
\(931\) 9234.89 0.325093
\(932\) 0 0
\(933\) −6203.87 −0.217691
\(934\) 0 0
\(935\) −53937.0 −1.88656
\(936\) 0 0
\(937\) −15535.2 −0.541636 −0.270818 0.962631i \(-0.587294\pi\)
−0.270818 + 0.962631i \(0.587294\pi\)
\(938\) 0 0
\(939\) −8950.03 −0.311047
\(940\) 0 0
\(941\) −48118.3 −1.66696 −0.833482 0.552547i \(-0.813656\pi\)
−0.833482 + 0.552547i \(0.813656\pi\)
\(942\) 0 0
\(943\) 14591.8 0.503896
\(944\) 0 0
\(945\) 1758.13 0.0605205
\(946\) 0 0
\(947\) 5396.67 0.185183 0.0925915 0.995704i \(-0.470485\pi\)
0.0925915 + 0.995704i \(0.470485\pi\)
\(948\) 0 0
\(949\) −13002.6 −0.444765
\(950\) 0 0
\(951\) 521.277 0.0177745
\(952\) 0 0
\(953\) 21759.7 0.739629 0.369815 0.929106i \(-0.379421\pi\)
0.369815 + 0.929106i \(0.379421\pi\)
\(954\) 0 0
\(955\) −37416.7 −1.26783
\(956\) 0 0
\(957\) 6392.70 0.215932
\(958\) 0 0
\(959\) −1275.03 −0.0429331
\(960\) 0 0
\(961\) 5809.78 0.195018
\(962\) 0 0
\(963\) 30038.2 1.00516
\(964\) 0 0
\(965\) 33184.4 1.10699
\(966\) 0 0
\(967\) 19338.5 0.643108 0.321554 0.946891i \(-0.395795\pi\)
0.321554 + 0.946891i \(0.395795\pi\)
\(968\) 0 0
\(969\) −2133.47 −0.0707294
\(970\) 0 0
\(971\) 5918.37 0.195602 0.0978009 0.995206i \(-0.468819\pi\)
0.0978009 + 0.995206i \(0.468819\pi\)
\(972\) 0 0
\(973\) 5720.03 0.188464
\(974\) 0 0
\(975\) 553.421 0.0181781
\(976\) 0 0
\(977\) −17841.9 −0.584249 −0.292125 0.956380i \(-0.594362\pi\)
−0.292125 + 0.956380i \(0.594362\pi\)
\(978\) 0 0
\(979\) 55437.4 1.80979
\(980\) 0 0
\(981\) −19481.1 −0.634032
\(982\) 0 0
\(983\) −61512.0 −1.99586 −0.997929 0.0643304i \(-0.979509\pi\)
−0.997929 + 0.0643304i \(0.979509\pi\)
\(984\) 0 0
\(985\) 26471.2 0.856286
\(986\) 0 0
\(987\) 1395.81 0.0450142
\(988\) 0 0
\(989\) −6209.89 −0.199659
\(990\) 0 0
\(991\) −1827.73 −0.0585870 −0.0292935 0.999571i \(-0.509326\pi\)
−0.0292935 + 0.999571i \(0.509326\pi\)
\(992\) 0 0
\(993\) 4068.21 0.130011
\(994\) 0 0
\(995\) 8691.53 0.276925
\(996\) 0 0
\(997\) −35379.4 −1.12385 −0.561925 0.827188i \(-0.689939\pi\)
−0.561925 + 0.827188i \(0.689939\pi\)
\(998\) 0 0
\(999\) 6725.07 0.212985
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.m.1.6 10
4.3 odd 2 1024.4.a.n.1.5 10
8.3 odd 2 1024.4.a.n.1.6 10
8.5 even 2 inner 1024.4.a.m.1.5 10
16.3 odd 4 1024.4.b.j.513.5 10
16.5 even 4 1024.4.b.k.513.5 10
16.11 odd 4 1024.4.b.j.513.6 10
16.13 even 4 1024.4.b.k.513.6 10
32.3 odd 8 128.4.e.b.33.3 10
32.5 even 8 64.4.e.a.49.3 10
32.11 odd 8 128.4.e.b.97.3 10
32.13 even 8 64.4.e.a.17.3 10
32.19 odd 8 16.4.e.a.13.3 yes 10
32.21 even 8 128.4.e.a.97.3 10
32.27 odd 8 16.4.e.a.5.3 10
32.29 even 8 128.4.e.a.33.3 10
96.5 odd 8 576.4.k.a.433.2 10
96.59 even 8 144.4.k.a.37.3 10
96.77 odd 8 576.4.k.a.145.2 10
96.83 even 8 144.4.k.a.109.3 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.4.e.a.5.3 10 32.27 odd 8
16.4.e.a.13.3 yes 10 32.19 odd 8
64.4.e.a.17.3 10 32.13 even 8
64.4.e.a.49.3 10 32.5 even 8
128.4.e.a.33.3 10 32.29 even 8
128.4.e.a.97.3 10 32.21 even 8
128.4.e.b.33.3 10 32.3 odd 8
128.4.e.b.97.3 10 32.11 odd 8
144.4.k.a.37.3 10 96.59 even 8
144.4.k.a.109.3 10 96.83 even 8
576.4.k.a.145.2 10 96.77 odd 8
576.4.k.a.433.2 10 96.5 odd 8
1024.4.a.m.1.5 10 8.5 even 2 inner
1024.4.a.m.1.6 10 1.1 even 1 trivial
1024.4.a.n.1.5 10 4.3 odd 2
1024.4.a.n.1.6 10 8.3 odd 2
1024.4.b.j.513.5 10 16.3 odd 4
1024.4.b.j.513.6 10 16.11 odd 4
1024.4.b.k.513.5 10 16.5 even 4
1024.4.b.k.513.6 10 16.13 even 4