Properties

Label 1521.4.a.bc.1.4
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(89.7419051187\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 52x^{6} + 805x^{4} - 4210x^{2} + 4992 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 117)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.28670\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.28670 q^{2} -6.34441 q^{4} -12.4484 q^{5} -9.00273 q^{7} +18.4569 q^{8} +O(q^{10})\) \(q-1.28670 q^{2} -6.34441 q^{4} -12.4484 q^{5} -9.00273 q^{7} +18.4569 q^{8} +16.0173 q^{10} +51.0774 q^{11} +11.5838 q^{14} +27.0069 q^{16} +6.86199 q^{17} -83.1170 q^{19} +78.9779 q^{20} -65.7211 q^{22} -187.579 q^{23} +29.9631 q^{25} +57.1170 q^{28} +223.614 q^{29} +57.3882 q^{31} -182.405 q^{32} -8.82930 q^{34} +112.070 q^{35} -156.259 q^{37} +106.946 q^{38} -229.759 q^{40} -222.281 q^{41} +347.974 q^{43} -324.056 q^{44} +241.357 q^{46} +45.0185 q^{47} -261.951 q^{49} -38.5534 q^{50} -473.516 q^{53} -635.833 q^{55} -166.163 q^{56} -287.724 q^{58} -615.494 q^{59} +195.325 q^{61} -73.8412 q^{62} +18.6446 q^{64} -355.097 q^{67} -43.5353 q^{68} -144.200 q^{70} -763.420 q^{71} +331.595 q^{73} +201.057 q^{74} +527.329 q^{76} -459.836 q^{77} -207.777 q^{79} -336.193 q^{80} +286.008 q^{82} +251.185 q^{83} -85.4209 q^{85} -447.737 q^{86} +942.731 q^{88} -719.366 q^{89} +1190.08 q^{92} -57.9252 q^{94} +1034.68 q^{95} -1556.16 q^{97} +337.051 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 40 q^{4} - 22 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 40 q^{4} - 22 q^{7} + 36 q^{10} + 204 q^{16} + 244 q^{19} + 136 q^{22} + 354 q^{25} - 452 q^{28} + 242 q^{31} - 1292 q^{34} + 1018 q^{37} + 1700 q^{40} + 74 q^{43} - 896 q^{46} + 298 q^{49} + 1300 q^{55} + 812 q^{58} + 1148 q^{61} + 3636 q^{64} - 2198 q^{67} + 2200 q^{70} - 2176 q^{73} + 6936 q^{76} + 1862 q^{79} + 5436 q^{82} - 890 q^{85} + 3528 q^{88} - 3104 q^{94} - 4370 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.28670 −0.454916 −0.227458 0.973788i \(-0.573041\pi\)
−0.227458 + 0.973788i \(0.573041\pi\)
\(3\) 0 0
\(4\) −6.34441 −0.793052
\(5\) −12.4484 −1.11342 −0.556710 0.830707i \(-0.687937\pi\)
−0.556710 + 0.830707i \(0.687937\pi\)
\(6\) 0 0
\(7\) −9.00273 −0.486102 −0.243051 0.970014i \(-0.578148\pi\)
−0.243051 + 0.970014i \(0.578148\pi\)
\(8\) 18.4569 0.815687
\(9\) 0 0
\(10\) 16.0173 0.506512
\(11\) 51.0774 1.40004 0.700019 0.714124i \(-0.253175\pi\)
0.700019 + 0.714124i \(0.253175\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 11.5838 0.221135
\(15\) 0 0
\(16\) 27.0069 0.421983
\(17\) 6.86199 0.0978987 0.0489493 0.998801i \(-0.484413\pi\)
0.0489493 + 0.998801i \(0.484413\pi\)
\(18\) 0 0
\(19\) −83.1170 −1.00360 −0.501799 0.864984i \(-0.667328\pi\)
−0.501799 + 0.864984i \(0.667328\pi\)
\(20\) 78.9779 0.883000
\(21\) 0 0
\(22\) −65.7211 −0.636900
\(23\) −187.579 −1.70056 −0.850279 0.526332i \(-0.823567\pi\)
−0.850279 + 0.526332i \(0.823567\pi\)
\(24\) 0 0
\(25\) 29.9631 0.239705
\(26\) 0 0
\(27\) 0 0
\(28\) 57.1170 0.385504
\(29\) 223.614 1.43187 0.715933 0.698169i \(-0.246002\pi\)
0.715933 + 0.698169i \(0.246002\pi\)
\(30\) 0 0
\(31\) 57.3882 0.332491 0.166246 0.986084i \(-0.446836\pi\)
0.166246 + 0.986084i \(0.446836\pi\)
\(32\) −182.405 −1.00765
\(33\) 0 0
\(34\) −8.82930 −0.0445357
\(35\) 112.070 0.541236
\(36\) 0 0
\(37\) −156.259 −0.694291 −0.347146 0.937811i \(-0.612849\pi\)
−0.347146 + 0.937811i \(0.612849\pi\)
\(38\) 106.946 0.456552
\(39\) 0 0
\(40\) −229.759 −0.908203
\(41\) −222.281 −0.846695 −0.423348 0.905967i \(-0.639145\pi\)
−0.423348 + 0.905967i \(0.639145\pi\)
\(42\) 0 0
\(43\) 347.974 1.23408 0.617041 0.786931i \(-0.288331\pi\)
0.617041 + 0.786931i \(0.288331\pi\)
\(44\) −324.056 −1.11030
\(45\) 0 0
\(46\) 241.357 0.773611
\(47\) 45.0185 0.139715 0.0698577 0.997557i \(-0.477745\pi\)
0.0698577 + 0.997557i \(0.477745\pi\)
\(48\) 0 0
\(49\) −261.951 −0.763705
\(50\) −38.5534 −0.109045
\(51\) 0 0
\(52\) 0 0
\(53\) −473.516 −1.22722 −0.613608 0.789611i \(-0.710282\pi\)
−0.613608 + 0.789611i \(0.710282\pi\)
\(54\) 0 0
\(55\) −635.833 −1.55883
\(56\) −166.163 −0.396507
\(57\) 0 0
\(58\) −287.724 −0.651379
\(59\) −615.494 −1.35814 −0.679072 0.734071i \(-0.737618\pi\)
−0.679072 + 0.734071i \(0.737618\pi\)
\(60\) 0 0
\(61\) 195.325 0.409981 0.204990 0.978764i \(-0.434284\pi\)
0.204990 + 0.978764i \(0.434284\pi\)
\(62\) −73.8412 −0.151255
\(63\) 0 0
\(64\) 18.6446 0.0364151
\(65\) 0 0
\(66\) 0 0
\(67\) −355.097 −0.647492 −0.323746 0.946144i \(-0.604942\pi\)
−0.323746 + 0.946144i \(0.604942\pi\)
\(68\) −43.5353 −0.0776387
\(69\) 0 0
\(70\) −144.200 −0.246217
\(71\) −763.420 −1.27608 −0.638038 0.770005i \(-0.720254\pi\)
−0.638038 + 0.770005i \(0.720254\pi\)
\(72\) 0 0
\(73\) 331.595 0.531647 0.265824 0.964022i \(-0.414356\pi\)
0.265824 + 0.964022i \(0.414356\pi\)
\(74\) 201.057 0.315844
\(75\) 0 0
\(76\) 527.329 0.795905
\(77\) −459.836 −0.680561
\(78\) 0 0
\(79\) −207.777 −0.295908 −0.147954 0.988994i \(-0.547269\pi\)
−0.147954 + 0.988994i \(0.547269\pi\)
\(80\) −336.193 −0.469844
\(81\) 0 0
\(82\) 286.008 0.385175
\(83\) 251.185 0.332183 0.166091 0.986110i \(-0.446885\pi\)
0.166091 + 0.986110i \(0.446885\pi\)
\(84\) 0 0
\(85\) −85.4209 −0.109002
\(86\) −447.737 −0.561403
\(87\) 0 0
\(88\) 942.731 1.14199
\(89\) −719.366 −0.856771 −0.428385 0.903596i \(-0.640917\pi\)
−0.428385 + 0.903596i \(0.640917\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1190.08 1.34863
\(93\) 0 0
\(94\) −57.9252 −0.0635588
\(95\) 1034.68 1.11743
\(96\) 0 0
\(97\) −1556.16 −1.62891 −0.814454 0.580228i \(-0.802964\pi\)
−0.814454 + 0.580228i \(0.802964\pi\)
\(98\) 337.051 0.347421
\(99\) 0 0
\(100\) −190.098 −0.190098
\(101\) 783.385 0.771779 0.385890 0.922545i \(-0.373895\pi\)
0.385890 + 0.922545i \(0.373895\pi\)
\(102\) 0 0
\(103\) 1033.54 0.988720 0.494360 0.869257i \(-0.335402\pi\)
0.494360 + 0.869257i \(0.335402\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 609.271 0.558279
\(107\) 241.358 0.218065 0.109033 0.994038i \(-0.465225\pi\)
0.109033 + 0.994038i \(0.465225\pi\)
\(108\) 0 0
\(109\) 1763.93 1.55003 0.775016 0.631941i \(-0.217742\pi\)
0.775016 + 0.631941i \(0.217742\pi\)
\(110\) 818.124 0.709137
\(111\) 0 0
\(112\) −243.136 −0.205126
\(113\) −1867.83 −1.55496 −0.777482 0.628905i \(-0.783503\pi\)
−0.777482 + 0.628905i \(0.783503\pi\)
\(114\) 0 0
\(115\) 2335.06 1.89344
\(116\) −1418.70 −1.13554
\(117\) 0 0
\(118\) 791.954 0.617841
\(119\) −61.7767 −0.0475887
\(120\) 0 0
\(121\) 1277.90 0.960108
\(122\) −251.324 −0.186507
\(123\) 0 0
\(124\) −364.095 −0.263683
\(125\) 1183.06 0.846528
\(126\) 0 0
\(127\) 238.885 0.166910 0.0834551 0.996512i \(-0.473404\pi\)
0.0834551 + 0.996512i \(0.473404\pi\)
\(128\) 1435.25 0.991088
\(129\) 0 0
\(130\) 0 0
\(131\) 1158.72 0.772807 0.386403 0.922330i \(-0.373717\pi\)
0.386403 + 0.922330i \(0.373717\pi\)
\(132\) 0 0
\(133\) 748.280 0.487851
\(134\) 456.902 0.294554
\(135\) 0 0
\(136\) 126.651 0.0798547
\(137\) 912.520 0.569064 0.284532 0.958666i \(-0.408162\pi\)
0.284532 + 0.958666i \(0.408162\pi\)
\(138\) 0 0
\(139\) 2315.04 1.41265 0.706326 0.707886i \(-0.250351\pi\)
0.706326 + 0.707886i \(0.250351\pi\)
\(140\) −711.017 −0.429228
\(141\) 0 0
\(142\) 982.290 0.580507
\(143\) 0 0
\(144\) 0 0
\(145\) −2783.64 −1.59427
\(146\) −426.662 −0.241855
\(147\) 0 0
\(148\) 991.370 0.550609
\(149\) 925.408 0.508808 0.254404 0.967098i \(-0.418121\pi\)
0.254404 + 0.967098i \(0.418121\pi\)
\(150\) 0 0
\(151\) −3523.78 −1.89908 −0.949541 0.313644i \(-0.898450\pi\)
−0.949541 + 0.313644i \(0.898450\pi\)
\(152\) −1534.08 −0.818622
\(153\) 0 0
\(154\) 591.670 0.309598
\(155\) −714.392 −0.370202
\(156\) 0 0
\(157\) 1615.83 0.821384 0.410692 0.911774i \(-0.365287\pi\)
0.410692 + 0.911774i \(0.365287\pi\)
\(158\) 267.346 0.134613
\(159\) 0 0
\(160\) 2270.65 1.12194
\(161\) 1688.72 0.826644
\(162\) 0 0
\(163\) 1008.01 0.484375 0.242188 0.970229i \(-0.422135\pi\)
0.242188 + 0.970229i \(0.422135\pi\)
\(164\) 1410.24 0.671473
\(165\) 0 0
\(166\) −323.199 −0.151115
\(167\) −2354.04 −1.09078 −0.545392 0.838181i \(-0.683619\pi\)
−0.545392 + 0.838181i \(0.683619\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 109.911 0.0495869
\(171\) 0 0
\(172\) −2207.69 −0.978691
\(173\) −3132.32 −1.37657 −0.688284 0.725442i \(-0.741635\pi\)
−0.688284 + 0.725442i \(0.741635\pi\)
\(174\) 0 0
\(175\) −269.750 −0.116521
\(176\) 1379.44 0.590792
\(177\) 0 0
\(178\) 925.605 0.389759
\(179\) 3180.82 1.32819 0.664093 0.747650i \(-0.268818\pi\)
0.664093 + 0.747650i \(0.268818\pi\)
\(180\) 0 0
\(181\) 4230.52 1.73731 0.868653 0.495421i \(-0.164986\pi\)
0.868653 + 0.495421i \(0.164986\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −3462.12 −1.38712
\(185\) 1945.17 0.773038
\(186\) 0 0
\(187\) 350.493 0.137062
\(188\) −285.616 −0.110802
\(189\) 0 0
\(190\) −1331.31 −0.508335
\(191\) 3952.80 1.49746 0.748729 0.662876i \(-0.230665\pi\)
0.748729 + 0.662876i \(0.230665\pi\)
\(192\) 0 0
\(193\) 5042.45 1.88064 0.940319 0.340294i \(-0.110527\pi\)
0.940319 + 0.340294i \(0.110527\pi\)
\(194\) 2002.31 0.741016
\(195\) 0 0
\(196\) 1661.92 0.605658
\(197\) −3572.12 −1.29189 −0.645947 0.763382i \(-0.723537\pi\)
−0.645947 + 0.763382i \(0.723537\pi\)
\(198\) 0 0
\(199\) 5006.84 1.78355 0.891773 0.452482i \(-0.149461\pi\)
0.891773 + 0.452482i \(0.149461\pi\)
\(200\) 553.026 0.195524
\(201\) 0 0
\(202\) −1007.98 −0.351095
\(203\) −2013.14 −0.696033
\(204\) 0 0
\(205\) 2767.05 0.942728
\(206\) −1329.86 −0.449784
\(207\) 0 0
\(208\) 0 0
\(209\) −4245.40 −1.40508
\(210\) 0 0
\(211\) 1759.23 0.573983 0.286992 0.957933i \(-0.407345\pi\)
0.286992 + 0.957933i \(0.407345\pi\)
\(212\) 3004.18 0.973245
\(213\) 0 0
\(214\) −310.555 −0.0992014
\(215\) −4331.72 −1.37405
\(216\) 0 0
\(217\) −516.651 −0.161625
\(218\) −2269.64 −0.705134
\(219\) 0 0
\(220\) 4033.99 1.23623
\(221\) 0 0
\(222\) 0 0
\(223\) 1712.52 0.514255 0.257127 0.966377i \(-0.417224\pi\)
0.257127 + 0.966377i \(0.417224\pi\)
\(224\) 1642.14 0.489822
\(225\) 0 0
\(226\) 2403.33 0.707378
\(227\) −2576.65 −0.753385 −0.376693 0.926338i \(-0.622939\pi\)
−0.376693 + 0.926338i \(0.622939\pi\)
\(228\) 0 0
\(229\) −1963.10 −0.566486 −0.283243 0.959048i \(-0.591410\pi\)
−0.283243 + 0.959048i \(0.591410\pi\)
\(230\) −3004.51 −0.861354
\(231\) 0 0
\(232\) 4127.23 1.16796
\(233\) 1078.11 0.303131 0.151565 0.988447i \(-0.451569\pi\)
0.151565 + 0.988447i \(0.451569\pi\)
\(234\) 0 0
\(235\) −560.409 −0.155562
\(236\) 3904.95 1.07708
\(237\) 0 0
\(238\) 79.4878 0.0216489
\(239\) −3998.43 −1.08216 −0.541081 0.840970i \(-0.681985\pi\)
−0.541081 + 0.840970i \(0.681985\pi\)
\(240\) 0 0
\(241\) −3574.72 −0.955470 −0.477735 0.878504i \(-0.658542\pi\)
−0.477735 + 0.878504i \(0.658542\pi\)
\(242\) −1644.27 −0.436768
\(243\) 0 0
\(244\) −1239.22 −0.325136
\(245\) 3260.87 0.850325
\(246\) 0 0
\(247\) 0 0
\(248\) 1059.21 0.271209
\(249\) 0 0
\(250\) −1522.24 −0.385099
\(251\) −2150.69 −0.540839 −0.270419 0.962743i \(-0.587162\pi\)
−0.270419 + 0.962743i \(0.587162\pi\)
\(252\) 0 0
\(253\) −9581.03 −2.38085
\(254\) −307.372 −0.0759301
\(255\) 0 0
\(256\) −1995.89 −0.487277
\(257\) −4274.80 −1.03757 −0.518784 0.854906i \(-0.673615\pi\)
−0.518784 + 0.854906i \(0.673615\pi\)
\(258\) 0 0
\(259\) 1406.75 0.337496
\(260\) 0 0
\(261\) 0 0
\(262\) −1490.92 −0.351562
\(263\) −8371.65 −1.96281 −0.981403 0.191960i \(-0.938516\pi\)
−0.981403 + 0.191960i \(0.938516\pi\)
\(264\) 0 0
\(265\) 5894.52 1.36641
\(266\) −962.810 −0.221931
\(267\) 0 0
\(268\) 2252.88 0.513495
\(269\) 5079.64 1.15134 0.575672 0.817681i \(-0.304741\pi\)
0.575672 + 0.817681i \(0.304741\pi\)
\(270\) 0 0
\(271\) 4931.93 1.10551 0.552756 0.833343i \(-0.313576\pi\)
0.552756 + 0.833343i \(0.313576\pi\)
\(272\) 185.321 0.0413115
\(273\) 0 0
\(274\) −1174.14 −0.258876
\(275\) 1530.44 0.335596
\(276\) 0 0
\(277\) 5172.52 1.12197 0.560987 0.827825i \(-0.310422\pi\)
0.560987 + 0.827825i \(0.310422\pi\)
\(278\) −2978.75 −0.642638
\(279\) 0 0
\(280\) 2068.46 0.441479
\(281\) 217.515 0.0461773 0.0230887 0.999733i \(-0.492650\pi\)
0.0230887 + 0.999733i \(0.492650\pi\)
\(282\) 0 0
\(283\) 4922.98 1.03407 0.517033 0.855966i \(-0.327036\pi\)
0.517033 + 0.855966i \(0.327036\pi\)
\(284\) 4843.45 1.01199
\(285\) 0 0
\(286\) 0 0
\(287\) 2001.14 0.411580
\(288\) 0 0
\(289\) −4865.91 −0.990416
\(290\) 3581.70 0.725258
\(291\) 0 0
\(292\) −2103.78 −0.421624
\(293\) 2642.25 0.526832 0.263416 0.964682i \(-0.415151\pi\)
0.263416 + 0.964682i \(0.415151\pi\)
\(294\) 0 0
\(295\) 7661.93 1.51219
\(296\) −2884.05 −0.566325
\(297\) 0 0
\(298\) −1190.72 −0.231465
\(299\) 0 0
\(300\) 0 0
\(301\) −3132.72 −0.599889
\(302\) 4534.04 0.863922
\(303\) 0 0
\(304\) −2244.73 −0.423501
\(305\) −2431.49 −0.456481
\(306\) 0 0
\(307\) −3386.22 −0.629517 −0.314759 0.949172i \(-0.601924\pi\)
−0.314759 + 0.949172i \(0.601924\pi\)
\(308\) 2917.39 0.539720
\(309\) 0 0
\(310\) 919.206 0.168411
\(311\) 3.29813 0.000601349 0 0.000300675 1.00000i \(-0.499904\pi\)
0.000300675 1.00000i \(0.499904\pi\)
\(312\) 0 0
\(313\) 2882.62 0.520560 0.260280 0.965533i \(-0.416185\pi\)
0.260280 + 0.965533i \(0.416185\pi\)
\(314\) −2079.08 −0.373660
\(315\) 0 0
\(316\) 1318.22 0.234671
\(317\) 1442.70 0.255616 0.127808 0.991799i \(-0.459206\pi\)
0.127808 + 0.991799i \(0.459206\pi\)
\(318\) 0 0
\(319\) 11421.6 2.00467
\(320\) −232.095 −0.0405454
\(321\) 0 0
\(322\) −2172.87 −0.376054
\(323\) −570.349 −0.0982509
\(324\) 0 0
\(325\) 0 0
\(326\) −1297.00 −0.220350
\(327\) 0 0
\(328\) −4102.62 −0.690639
\(329\) −405.290 −0.0679159
\(330\) 0 0
\(331\) −3824.98 −0.635166 −0.317583 0.948230i \(-0.602871\pi\)
−0.317583 + 0.948230i \(0.602871\pi\)
\(332\) −1593.62 −0.263438
\(333\) 0 0
\(334\) 3028.93 0.496215
\(335\) 4420.39 0.720931
\(336\) 0 0
\(337\) 2290.22 0.370197 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 541.946 0.0864445
\(341\) 2931.24 0.465500
\(342\) 0 0
\(343\) 5446.21 0.857340
\(344\) 6422.52 1.00663
\(345\) 0 0
\(346\) 4030.35 0.626222
\(347\) 334.993 0.0518252 0.0259126 0.999664i \(-0.491751\pi\)
0.0259126 + 0.999664i \(0.491751\pi\)
\(348\) 0 0
\(349\) −8975.27 −1.37661 −0.688303 0.725424i \(-0.741644\pi\)
−0.688303 + 0.725424i \(0.741644\pi\)
\(350\) 347.086 0.0530072
\(351\) 0 0
\(352\) −9316.77 −1.41075
\(353\) 9925.87 1.49660 0.748302 0.663359i \(-0.230870\pi\)
0.748302 + 0.663359i \(0.230870\pi\)
\(354\) 0 0
\(355\) 9503.38 1.42081
\(356\) 4563.95 0.679463
\(357\) 0 0
\(358\) −4092.74 −0.604213
\(359\) −215.661 −0.0317052 −0.0158526 0.999874i \(-0.505046\pi\)
−0.0158526 + 0.999874i \(0.505046\pi\)
\(360\) 0 0
\(361\) 49.4437 0.00720858
\(362\) −5443.40 −0.790328
\(363\) 0 0
\(364\) 0 0
\(365\) −4127.83 −0.591947
\(366\) 0 0
\(367\) −2861.52 −0.407003 −0.203502 0.979075i \(-0.565232\pi\)
−0.203502 + 0.979075i \(0.565232\pi\)
\(368\) −5065.91 −0.717606
\(369\) 0 0
\(370\) −2502.85 −0.351667
\(371\) 4262.94 0.596551
\(372\) 0 0
\(373\) −17.0145 −0.00236186 −0.00118093 0.999999i \(-0.500376\pi\)
−0.00118093 + 0.999999i \(0.500376\pi\)
\(374\) −450.978 −0.0623516
\(375\) 0 0
\(376\) 830.902 0.113964
\(377\) 0 0
\(378\) 0 0
\(379\) −1901.97 −0.257777 −0.128888 0.991659i \(-0.541141\pi\)
−0.128888 + 0.991659i \(0.541141\pi\)
\(380\) −6564.41 −0.886177
\(381\) 0 0
\(382\) −5086.05 −0.681217
\(383\) 7729.78 1.03126 0.515631 0.856811i \(-0.327558\pi\)
0.515631 + 0.856811i \(0.327558\pi\)
\(384\) 0 0
\(385\) 5724.23 0.757751
\(386\) −6488.09 −0.855532
\(387\) 0 0
\(388\) 9872.92 1.29181
\(389\) 14445.5 1.88282 0.941408 0.337271i \(-0.109504\pi\)
0.941408 + 0.337271i \(0.109504\pi\)
\(390\) 0 0
\(391\) −1287.16 −0.166482
\(392\) −4834.80 −0.622945
\(393\) 0 0
\(394\) 4596.23 0.587703
\(395\) 2586.50 0.329470
\(396\) 0 0
\(397\) 1033.07 0.130600 0.0653000 0.997866i \(-0.479200\pi\)
0.0653000 + 0.997866i \(0.479200\pi\)
\(398\) −6442.29 −0.811364
\(399\) 0 0
\(400\) 809.210 0.101151
\(401\) 4611.73 0.574311 0.287155 0.957884i \(-0.407290\pi\)
0.287155 + 0.957884i \(0.407290\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −4970.12 −0.612061
\(405\) 0 0
\(406\) 2590.30 0.316636
\(407\) −7981.29 −0.972034
\(408\) 0 0
\(409\) 2668.56 0.322621 0.161311 0.986904i \(-0.448428\pi\)
0.161311 + 0.986904i \(0.448428\pi\)
\(410\) −3560.35 −0.428862
\(411\) 0 0
\(412\) −6557.23 −0.784106
\(413\) 5541.13 0.660197
\(414\) 0 0
\(415\) −3126.86 −0.369859
\(416\) 0 0
\(417\) 0 0
\(418\) 5462.55 0.639191
\(419\) −6189.83 −0.721701 −0.360851 0.932624i \(-0.617514\pi\)
−0.360851 + 0.932624i \(0.617514\pi\)
\(420\) 0 0
\(421\) 8991.13 1.04086 0.520429 0.853905i \(-0.325772\pi\)
0.520429 + 0.853905i \(0.325772\pi\)
\(422\) −2263.60 −0.261114
\(423\) 0 0
\(424\) −8739.63 −1.00102
\(425\) 205.606 0.0234668
\(426\) 0 0
\(427\) −1758.46 −0.199292
\(428\) −1531.28 −0.172937
\(429\) 0 0
\(430\) 5573.61 0.625078
\(431\) −2069.01 −0.231232 −0.115616 0.993294i \(-0.536884\pi\)
−0.115616 + 0.993294i \(0.536884\pi\)
\(432\) 0 0
\(433\) −14934.7 −1.65754 −0.828769 0.559592i \(-0.810958\pi\)
−0.828769 + 0.559592i \(0.810958\pi\)
\(434\) 664.772 0.0735256
\(435\) 0 0
\(436\) −11191.1 −1.22926
\(437\) 15591.0 1.70668
\(438\) 0 0
\(439\) −6000.24 −0.652336 −0.326168 0.945312i \(-0.605758\pi\)
−0.326168 + 0.945312i \(0.605758\pi\)
\(440\) −11735.5 −1.27152
\(441\) 0 0
\(442\) 0 0
\(443\) 5392.60 0.578352 0.289176 0.957276i \(-0.406619\pi\)
0.289176 + 0.957276i \(0.406619\pi\)
\(444\) 0 0
\(445\) 8954.96 0.953946
\(446\) −2203.49 −0.233943
\(447\) 0 0
\(448\) −167.852 −0.0177015
\(449\) 6046.60 0.635538 0.317769 0.948168i \(-0.397066\pi\)
0.317769 + 0.948168i \(0.397066\pi\)
\(450\) 0 0
\(451\) −11353.6 −1.18541
\(452\) 11850.3 1.23317
\(453\) 0 0
\(454\) 3315.37 0.342727
\(455\) 0 0
\(456\) 0 0
\(457\) 1014.57 0.103851 0.0519253 0.998651i \(-0.483464\pi\)
0.0519253 + 0.998651i \(0.483464\pi\)
\(458\) 2525.91 0.257704
\(459\) 0 0
\(460\) −14814.6 −1.50159
\(461\) 15978.9 1.61434 0.807171 0.590318i \(-0.200997\pi\)
0.807171 + 0.590318i \(0.200997\pi\)
\(462\) 0 0
\(463\) −128.805 −0.0129289 −0.00646445 0.999979i \(-0.502058\pi\)
−0.00646445 + 0.999979i \(0.502058\pi\)
\(464\) 6039.12 0.604223
\(465\) 0 0
\(466\) −1387.20 −0.137899
\(467\) −18832.4 −1.86608 −0.933039 0.359776i \(-0.882853\pi\)
−0.933039 + 0.359776i \(0.882853\pi\)
\(468\) 0 0
\(469\) 3196.84 0.314747
\(470\) 721.076 0.0707676
\(471\) 0 0
\(472\) −11360.1 −1.10782
\(473\) 17773.6 1.72776
\(474\) 0 0
\(475\) −2490.44 −0.240567
\(476\) 391.937 0.0377403
\(477\) 0 0
\(478\) 5144.76 0.492293
\(479\) 9231.59 0.880589 0.440294 0.897853i \(-0.354874\pi\)
0.440294 + 0.897853i \(0.354874\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 4599.58 0.434658
\(483\) 0 0
\(484\) −8107.55 −0.761415
\(485\) 19371.7 1.81366
\(486\) 0 0
\(487\) 7233.01 0.673016 0.336508 0.941681i \(-0.390754\pi\)
0.336508 + 0.941681i \(0.390754\pi\)
\(488\) 3605.10 0.334416
\(489\) 0 0
\(490\) −4195.75 −0.386826
\(491\) 655.131 0.0602152 0.0301076 0.999547i \(-0.490415\pi\)
0.0301076 + 0.999547i \(0.490415\pi\)
\(492\) 0 0
\(493\) 1534.44 0.140178
\(494\) 0 0
\(495\) 0 0
\(496\) 1549.88 0.140305
\(497\) 6872.87 0.620302
\(498\) 0 0
\(499\) −4560.07 −0.409092 −0.204546 0.978857i \(-0.565572\pi\)
−0.204546 + 0.978857i \(0.565572\pi\)
\(500\) −7505.82 −0.671341
\(501\) 0 0
\(502\) 2767.29 0.246036
\(503\) −9068.67 −0.803881 −0.401941 0.915666i \(-0.631664\pi\)
−0.401941 + 0.915666i \(0.631664\pi\)
\(504\) 0 0
\(505\) −9751.90 −0.859315
\(506\) 12327.9 1.08308
\(507\) 0 0
\(508\) −1515.58 −0.132368
\(509\) 15880.9 1.38292 0.691461 0.722413i \(-0.256967\pi\)
0.691461 + 0.722413i \(0.256967\pi\)
\(510\) 0 0
\(511\) −2985.26 −0.258435
\(512\) −8913.89 −0.769418
\(513\) 0 0
\(514\) 5500.37 0.472006
\(515\) −12866.0 −1.10086
\(516\) 0 0
\(517\) 2299.43 0.195607
\(518\) −1810.07 −0.153532
\(519\) 0 0
\(520\) 0 0
\(521\) 10998.0 0.924821 0.462410 0.886666i \(-0.346985\pi\)
0.462410 + 0.886666i \(0.346985\pi\)
\(522\) 0 0
\(523\) 18631.8 1.55776 0.778882 0.627171i \(-0.215787\pi\)
0.778882 + 0.627171i \(0.215787\pi\)
\(524\) −7351.39 −0.612875
\(525\) 0 0
\(526\) 10771.8 0.892911
\(527\) 393.797 0.0325504
\(528\) 0 0
\(529\) 23018.7 1.89190
\(530\) −7584.46 −0.621600
\(531\) 0 0
\(532\) −4747.40 −0.386891
\(533\) 0 0
\(534\) 0 0
\(535\) −3004.53 −0.242798
\(536\) −6553.99 −0.528151
\(537\) 0 0
\(538\) −6535.96 −0.523764
\(539\) −13379.8 −1.06922
\(540\) 0 0
\(541\) −14270.1 −1.13405 −0.567023 0.823702i \(-0.691905\pi\)
−0.567023 + 0.823702i \(0.691905\pi\)
\(542\) −6345.90 −0.502915
\(543\) 0 0
\(544\) −1251.66 −0.0986480
\(545\) −21958.1 −1.72584
\(546\) 0 0
\(547\) 14379.7 1.12400 0.562002 0.827136i \(-0.310031\pi\)
0.562002 + 0.827136i \(0.310031\pi\)
\(548\) −5789.40 −0.451298
\(549\) 0 0
\(550\) −1969.21 −0.152668
\(551\) −18586.2 −1.43702
\(552\) 0 0
\(553\) 1870.56 0.143842
\(554\) −6655.46 −0.510404
\(555\) 0 0
\(556\) −14687.5 −1.12031
\(557\) 9035.10 0.687306 0.343653 0.939097i \(-0.388336\pi\)
0.343653 + 0.939097i \(0.388336\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 3026.65 0.228392
\(561\) 0 0
\(562\) −279.875 −0.0210068
\(563\) 7293.47 0.545973 0.272987 0.962018i \(-0.411988\pi\)
0.272987 + 0.962018i \(0.411988\pi\)
\(564\) 0 0
\(565\) 23251.6 1.73133
\(566\) −6334.37 −0.470413
\(567\) 0 0
\(568\) −14090.4 −1.04088
\(569\) −26415.8 −1.94623 −0.973117 0.230310i \(-0.926026\pi\)
−0.973117 + 0.230310i \(0.926026\pi\)
\(570\) 0 0
\(571\) −23940.9 −1.75463 −0.877317 0.479912i \(-0.840669\pi\)
−0.877317 + 0.479912i \(0.840669\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −2574.86 −0.187234
\(575\) −5620.43 −0.407632
\(576\) 0 0
\(577\) 10182.8 0.734691 0.367346 0.930085i \(-0.380267\pi\)
0.367346 + 0.930085i \(0.380267\pi\)
\(578\) 6260.95 0.450556
\(579\) 0 0
\(580\) 17660.6 1.26434
\(581\) −2261.35 −0.161475
\(582\) 0 0
\(583\) −24186.0 −1.71815
\(584\) 6120.22 0.433658
\(585\) 0 0
\(586\) −3399.77 −0.239664
\(587\) −19107.5 −1.34353 −0.671763 0.740766i \(-0.734463\pi\)
−0.671763 + 0.740766i \(0.734463\pi\)
\(588\) 0 0
\(589\) −4769.94 −0.333687
\(590\) −9858.58 −0.687917
\(591\) 0 0
\(592\) −4220.06 −0.292979
\(593\) 13597.5 0.941621 0.470810 0.882234i \(-0.343962\pi\)
0.470810 + 0.882234i \(0.343962\pi\)
\(594\) 0 0
\(595\) 769.022 0.0529863
\(596\) −5871.17 −0.403511
\(597\) 0 0
\(598\) 0 0
\(599\) −4150.37 −0.283104 −0.141552 0.989931i \(-0.545209\pi\)
−0.141552 + 0.989931i \(0.545209\pi\)
\(600\) 0 0
\(601\) 19177.5 1.30161 0.650803 0.759247i \(-0.274433\pi\)
0.650803 + 0.759247i \(0.274433\pi\)
\(602\) 4030.85 0.272899
\(603\) 0 0
\(604\) 22356.3 1.50607
\(605\) −15907.9 −1.06900
\(606\) 0 0
\(607\) 7994.36 0.534565 0.267282 0.963618i \(-0.413874\pi\)
0.267282 + 0.963618i \(0.413874\pi\)
\(608\) 15161.0 1.01128
\(609\) 0 0
\(610\) 3128.59 0.207660
\(611\) 0 0
\(612\) 0 0
\(613\) 15613.4 1.02874 0.514371 0.857568i \(-0.328026\pi\)
0.514371 + 0.857568i \(0.328026\pi\)
\(614\) 4357.04 0.286377
\(615\) 0 0
\(616\) −8487.15 −0.555125
\(617\) −21636.2 −1.41174 −0.705868 0.708343i \(-0.749443\pi\)
−0.705868 + 0.708343i \(0.749443\pi\)
\(618\) 0 0
\(619\) 10394.0 0.674911 0.337456 0.941341i \(-0.390434\pi\)
0.337456 + 0.941341i \(0.390434\pi\)
\(620\) 4532.40 0.293590
\(621\) 0 0
\(622\) −4.24369 −0.000273563 0
\(623\) 6476.25 0.416478
\(624\) 0 0
\(625\) −18472.6 −1.18225
\(626\) −3709.06 −0.236811
\(627\) 0 0
\(628\) −10251.5 −0.651400
\(629\) −1072.25 −0.0679702
\(630\) 0 0
\(631\) 4644.14 0.292996 0.146498 0.989211i \(-0.453200\pi\)
0.146498 + 0.989211i \(0.453200\pi\)
\(632\) −3834.92 −0.241369
\(633\) 0 0
\(634\) −1856.32 −0.116284
\(635\) −2973.74 −0.185841
\(636\) 0 0
\(637\) 0 0
\(638\) −14696.2 −0.911955
\(639\) 0 0
\(640\) −17866.6 −1.10350
\(641\) −153.922 −0.00948451 −0.00474225 0.999989i \(-0.501510\pi\)
−0.00474225 + 0.999989i \(0.501510\pi\)
\(642\) 0 0
\(643\) −8438.05 −0.517518 −0.258759 0.965942i \(-0.583314\pi\)
−0.258759 + 0.965942i \(0.583314\pi\)
\(644\) −10713.9 −0.655572
\(645\) 0 0
\(646\) 733.865 0.0446959
\(647\) −665.236 −0.0404221 −0.0202111 0.999796i \(-0.506434\pi\)
−0.0202111 + 0.999796i \(0.506434\pi\)
\(648\) 0 0
\(649\) −31437.9 −1.90145
\(650\) 0 0
\(651\) 0 0
\(652\) −6395.21 −0.384134
\(653\) 22414.7 1.34327 0.671636 0.740882i \(-0.265592\pi\)
0.671636 + 0.740882i \(0.265592\pi\)
\(654\) 0 0
\(655\) −14424.2 −0.860458
\(656\) −6003.12 −0.357291
\(657\) 0 0
\(658\) 521.485 0.0308960
\(659\) 25585.6 1.51240 0.756202 0.654338i \(-0.227053\pi\)
0.756202 + 0.654338i \(0.227053\pi\)
\(660\) 0 0
\(661\) −13317.5 −0.783647 −0.391824 0.920040i \(-0.628156\pi\)
−0.391824 + 0.920040i \(0.628156\pi\)
\(662\) 4921.59 0.288947
\(663\) 0 0
\(664\) 4636.10 0.270957
\(665\) −9314.91 −0.543183
\(666\) 0 0
\(667\) −41945.2 −2.43497
\(668\) 14935.0 0.865048
\(669\) 0 0
\(670\) −5687.70 −0.327963
\(671\) 9976.71 0.573989
\(672\) 0 0
\(673\) 9778.28 0.560067 0.280034 0.959990i \(-0.409654\pi\)
0.280034 + 0.959990i \(0.409654\pi\)
\(674\) −2946.82 −0.168408
\(675\) 0 0
\(676\) 0 0
\(677\) −12392.8 −0.703537 −0.351769 0.936087i \(-0.614420\pi\)
−0.351769 + 0.936087i \(0.614420\pi\)
\(678\) 0 0
\(679\) 14009.7 0.791815
\(680\) −1576.61 −0.0889119
\(681\) 0 0
\(682\) −3771.62 −0.211763
\(683\) 18941.7 1.06118 0.530588 0.847630i \(-0.321971\pi\)
0.530588 + 0.847630i \(0.321971\pi\)
\(684\) 0 0
\(685\) −11359.4 −0.633608
\(686\) −7007.62 −0.390018
\(687\) 0 0
\(688\) 9397.69 0.520761
\(689\) 0 0
\(690\) 0 0
\(691\) 32377.6 1.78249 0.891247 0.453519i \(-0.149832\pi\)
0.891247 + 0.453519i \(0.149832\pi\)
\(692\) 19872.8 1.09169
\(693\) 0 0
\(694\) −431.034 −0.0235761
\(695\) −28818.5 −1.57288
\(696\) 0 0
\(697\) −1525.29 −0.0828903
\(698\) 11548.4 0.626240
\(699\) 0 0
\(700\) 1711.40 0.0924071
\(701\) −6072.34 −0.327174 −0.163587 0.986529i \(-0.552306\pi\)
−0.163587 + 0.986529i \(0.552306\pi\)
\(702\) 0 0
\(703\) 12987.8 0.696789
\(704\) 952.316 0.0509826
\(705\) 0 0
\(706\) −12771.6 −0.680829
\(707\) −7052.61 −0.375163
\(708\) 0 0
\(709\) −3398.52 −0.180020 −0.0900100 0.995941i \(-0.528690\pi\)
−0.0900100 + 0.995941i \(0.528690\pi\)
\(710\) −12228.0 −0.646348
\(711\) 0 0
\(712\) −13277.3 −0.698857
\(713\) −10764.8 −0.565421
\(714\) 0 0
\(715\) 0 0
\(716\) −20180.4 −1.05332
\(717\) 0 0
\(718\) 277.491 0.0144232
\(719\) −28833.6 −1.49556 −0.747782 0.663944i \(-0.768881\pi\)
−0.747782 + 0.663944i \(0.768881\pi\)
\(720\) 0 0
\(721\) −9304.72 −0.480619
\(722\) −63.6190 −0.00327930
\(723\) 0 0
\(724\) −26840.2 −1.37777
\(725\) 6700.17 0.343225
\(726\) 0 0
\(727\) −12894.7 −0.657822 −0.328911 0.944361i \(-0.606682\pi\)
−0.328911 + 0.944361i \(0.606682\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 5311.27 0.269286
\(731\) 2387.79 0.120815
\(732\) 0 0
\(733\) −23754.7 −1.19700 −0.598499 0.801124i \(-0.704236\pi\)
−0.598499 + 0.801124i \(0.704236\pi\)
\(734\) 3681.91 0.185152
\(735\) 0 0
\(736\) 34215.2 1.71357
\(737\) −18137.4 −0.906514
\(738\) 0 0
\(739\) −26047.2 −1.29656 −0.648282 0.761400i \(-0.724512\pi\)
−0.648282 + 0.761400i \(0.724512\pi\)
\(740\) −12341.0 −0.613059
\(741\) 0 0
\(742\) −5485.10 −0.271381
\(743\) −15919.3 −0.786033 −0.393016 0.919531i \(-0.628568\pi\)
−0.393016 + 0.919531i \(0.628568\pi\)
\(744\) 0 0
\(745\) −11519.9 −0.566517
\(746\) 21.8924 0.00107445
\(747\) 0 0
\(748\) −2223.67 −0.108697
\(749\) −2172.88 −0.106002
\(750\) 0 0
\(751\) 2237.59 0.108723 0.0543613 0.998521i \(-0.482688\pi\)
0.0543613 + 0.998521i \(0.482688\pi\)
\(752\) 1215.81 0.0589575
\(753\) 0 0
\(754\) 0 0
\(755\) 43865.5 2.11448
\(756\) 0 0
\(757\) 3263.91 0.156709 0.0783545 0.996926i \(-0.475033\pi\)
0.0783545 + 0.996926i \(0.475033\pi\)
\(758\) 2447.25 0.117267
\(759\) 0 0
\(760\) 19096.9 0.911471
\(761\) 1904.91 0.0907398 0.0453699 0.998970i \(-0.485553\pi\)
0.0453699 + 0.998970i \(0.485553\pi\)
\(762\) 0 0
\(763\) −15880.2 −0.753474
\(764\) −25078.2 −1.18756
\(765\) 0 0
\(766\) −9945.88 −0.469137
\(767\) 0 0
\(768\) 0 0
\(769\) 3429.87 0.160838 0.0804190 0.996761i \(-0.474374\pi\)
0.0804190 + 0.996761i \(0.474374\pi\)
\(770\) −7365.35 −0.344713
\(771\) 0 0
\(772\) −31991.4 −1.49144
\(773\) 28243.5 1.31416 0.657082 0.753819i \(-0.271790\pi\)
0.657082 + 0.753819i \(0.271790\pi\)
\(774\) 0 0
\(775\) 1719.53 0.0796997
\(776\) −28721.9 −1.32868
\(777\) 0 0
\(778\) −18587.0 −0.856522
\(779\) 18475.4 0.849741
\(780\) 0 0
\(781\) −38993.5 −1.78655
\(782\) 1656.19 0.0757355
\(783\) 0 0
\(784\) −7074.48 −0.322270
\(785\) −20114.5 −0.914545
\(786\) 0 0
\(787\) 6626.72 0.300149 0.150074 0.988675i \(-0.452049\pi\)
0.150074 + 0.988675i \(0.452049\pi\)
\(788\) 22663.0 1.02454
\(789\) 0 0
\(790\) −3328.03 −0.149881
\(791\) 16815.6 0.755871
\(792\) 0 0
\(793\) 0 0
\(794\) −1329.25 −0.0594120
\(795\) 0 0
\(796\) −31765.5 −1.41444
\(797\) 20750.8 0.922249 0.461125 0.887335i \(-0.347446\pi\)
0.461125 + 0.887335i \(0.347446\pi\)
\(798\) 0 0
\(799\) 308.917 0.0136780
\(800\) −5465.41 −0.241539
\(801\) 0 0
\(802\) −5933.89 −0.261263
\(803\) 16937.0 0.744327
\(804\) 0 0
\(805\) −21021.9 −0.920403
\(806\) 0 0
\(807\) 0 0
\(808\) 14458.9 0.629531
\(809\) −10005.3 −0.434818 −0.217409 0.976081i \(-0.569761\pi\)
−0.217409 + 0.976081i \(0.569761\pi\)
\(810\) 0 0
\(811\) 32773.5 1.41903 0.709515 0.704690i \(-0.248914\pi\)
0.709515 + 0.704690i \(0.248914\pi\)
\(812\) 12772.2 0.551990
\(813\) 0 0
\(814\) 10269.5 0.442194
\(815\) −12548.1 −0.539313
\(816\) 0 0
\(817\) −28922.6 −1.23852
\(818\) −3433.63 −0.146765
\(819\) 0 0
\(820\) −17555.3 −0.747632
\(821\) 3031.74 0.128878 0.0644388 0.997922i \(-0.479474\pi\)
0.0644388 + 0.997922i \(0.479474\pi\)
\(822\) 0 0
\(823\) 31057.0 1.31541 0.657704 0.753277i \(-0.271528\pi\)
0.657704 + 0.753277i \(0.271528\pi\)
\(824\) 19076.0 0.806486
\(825\) 0 0
\(826\) −7129.75 −0.300334
\(827\) 8770.13 0.368764 0.184382 0.982855i \(-0.440972\pi\)
0.184382 + 0.982855i \(0.440972\pi\)
\(828\) 0 0
\(829\) −38698.9 −1.62131 −0.810656 0.585523i \(-0.800889\pi\)
−0.810656 + 0.585523i \(0.800889\pi\)
\(830\) 4023.32 0.168255
\(831\) 0 0
\(832\) 0 0
\(833\) −1797.50 −0.0747657
\(834\) 0 0
\(835\) 29304.0 1.21450
\(836\) 26934.6 1.11430
\(837\) 0 0
\(838\) 7964.43 0.328313
\(839\) 22543.7 0.927647 0.463824 0.885928i \(-0.346477\pi\)
0.463824 + 0.885928i \(0.346477\pi\)
\(840\) 0 0
\(841\) 25614.3 1.05024
\(842\) −11568.9 −0.473502
\(843\) 0 0
\(844\) −11161.3 −0.455198
\(845\) 0 0
\(846\) 0 0
\(847\) −11504.6 −0.466710
\(848\) −12788.2 −0.517863
\(849\) 0 0
\(850\) −264.553 −0.0106754
\(851\) 29310.8 1.18068
\(852\) 0 0
\(853\) −7751.80 −0.311156 −0.155578 0.987824i \(-0.549724\pi\)
−0.155578 + 0.987824i \(0.549724\pi\)
\(854\) 2262.60 0.0906613
\(855\) 0 0
\(856\) 4454.73 0.177873
\(857\) −30235.6 −1.20517 −0.602583 0.798056i \(-0.705862\pi\)
−0.602583 + 0.798056i \(0.705862\pi\)
\(858\) 0 0
\(859\) 3568.09 0.141725 0.0708624 0.997486i \(-0.477425\pi\)
0.0708624 + 0.997486i \(0.477425\pi\)
\(860\) 27482.2 1.08969
\(861\) 0 0
\(862\) 2662.19 0.105191
\(863\) 406.509 0.0160345 0.00801723 0.999968i \(-0.497448\pi\)
0.00801723 + 0.999968i \(0.497448\pi\)
\(864\) 0 0
\(865\) 38992.5 1.53270
\(866\) 19216.4 0.754040
\(867\) 0 0
\(868\) 3277.85 0.128177
\(869\) −10612.7 −0.414283
\(870\) 0 0
\(871\) 0 0
\(872\) 32556.6 1.26434
\(873\) 0 0
\(874\) −20060.8 −0.776394
\(875\) −10650.8 −0.411499
\(876\) 0 0
\(877\) 18961.9 0.730102 0.365051 0.930988i \(-0.381052\pi\)
0.365051 + 0.930988i \(0.381052\pi\)
\(878\) 7720.48 0.296758
\(879\) 0 0
\(880\) −17171.9 −0.657800
\(881\) 31028.9 1.18660 0.593298 0.804983i \(-0.297826\pi\)
0.593298 + 0.804983i \(0.297826\pi\)
\(882\) 0 0
\(883\) 33896.2 1.29184 0.645922 0.763404i \(-0.276473\pi\)
0.645922 + 0.763404i \(0.276473\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −6938.64 −0.263102
\(887\) 1773.35 0.0671288 0.0335644 0.999437i \(-0.489314\pi\)
0.0335644 + 0.999437i \(0.489314\pi\)
\(888\) 0 0
\(889\) −2150.62 −0.0811354
\(890\) −11522.3 −0.433965
\(891\) 0 0
\(892\) −10864.9 −0.407831
\(893\) −3741.81 −0.140218
\(894\) 0 0
\(895\) −39596.1 −1.47883
\(896\) −12921.2 −0.481770
\(897\) 0 0
\(898\) −7780.14 −0.289116
\(899\) 12832.8 0.476083
\(900\) 0 0
\(901\) −3249.26 −0.120143
\(902\) 14608.6 0.539260
\(903\) 0 0
\(904\) −34474.4 −1.26836
\(905\) −52663.3 −1.93435
\(906\) 0 0
\(907\) 14401.6 0.527231 0.263615 0.964628i \(-0.415085\pi\)
0.263615 + 0.964628i \(0.415085\pi\)
\(908\) 16347.3 0.597473
\(909\) 0 0
\(910\) 0 0
\(911\) 19324.5 0.702799 0.351400 0.936226i \(-0.385706\pi\)
0.351400 + 0.936226i \(0.385706\pi\)
\(912\) 0 0
\(913\) 12829.9 0.465069
\(914\) −1305.45 −0.0472433
\(915\) 0 0
\(916\) 12454.7 0.449253
\(917\) −10431.6 −0.375663
\(918\) 0 0
\(919\) 18649.1 0.669399 0.334700 0.942325i \(-0.391365\pi\)
0.334700 + 0.942325i \(0.391365\pi\)
\(920\) 43097.9 1.54445
\(921\) 0 0
\(922\) −20560.0 −0.734390
\(923\) 0 0
\(924\) 0 0
\(925\) −4681.99 −0.166425
\(926\) 165.733 0.00588156
\(927\) 0 0
\(928\) −40788.3 −1.44283
\(929\) −27326.4 −0.965070 −0.482535 0.875877i \(-0.660284\pi\)
−0.482535 + 0.875877i \(0.660284\pi\)
\(930\) 0 0
\(931\) 21772.6 0.766453
\(932\) −6839.98 −0.240398
\(933\) 0 0
\(934\) 24231.5 0.848908
\(935\) −4363.08 −0.152608
\(936\) 0 0
\(937\) 28699.1 1.00060 0.500298 0.865853i \(-0.333224\pi\)
0.500298 + 0.865853i \(0.333224\pi\)
\(938\) −4113.36 −0.143183
\(939\) 0 0
\(940\) 3555.47 0.123369
\(941\) 55401.1 1.91926 0.959630 0.281264i \(-0.0907538\pi\)
0.959630 + 0.281264i \(0.0907538\pi\)
\(942\) 0 0
\(943\) 41695.2 1.43985
\(944\) −16622.6 −0.573113
\(945\) 0 0
\(946\) −22869.2 −0.785986
\(947\) −33258.4 −1.14124 −0.570619 0.821215i \(-0.693297\pi\)
−0.570619 + 0.821215i \(0.693297\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 3204.44 0.109438
\(951\) 0 0
\(952\) −1140.21 −0.0388175
\(953\) −4411.89 −0.149963 −0.0749817 0.997185i \(-0.523890\pi\)
−0.0749817 + 0.997185i \(0.523890\pi\)
\(954\) 0 0
\(955\) −49206.1 −1.66730
\(956\) 25367.7 0.858211
\(957\) 0 0
\(958\) −11878.3 −0.400594
\(959\) −8215.17 −0.276623
\(960\) 0 0
\(961\) −26497.6 −0.889450
\(962\) 0 0
\(963\) 0 0
\(964\) 22679.5 0.757737
\(965\) −62770.5 −2.09394
\(966\) 0 0
\(967\) 26014.3 0.865114 0.432557 0.901607i \(-0.357611\pi\)
0.432557 + 0.901607i \(0.357611\pi\)
\(968\) 23586.1 0.783148
\(969\) 0 0
\(970\) −24925.5 −0.825063
\(971\) 36204.9 1.19657 0.598286 0.801283i \(-0.295849\pi\)
0.598286 + 0.801283i \(0.295849\pi\)
\(972\) 0 0
\(973\) −20841.6 −0.686693
\(974\) −9306.69 −0.306166
\(975\) 0 0
\(976\) 5275.12 0.173005
\(977\) 13916.4 0.455706 0.227853 0.973696i \(-0.426830\pi\)
0.227853 + 0.973696i \(0.426830\pi\)
\(978\) 0 0
\(979\) −36743.3 −1.19951
\(980\) −20688.3 −0.674351
\(981\) 0 0
\(982\) −842.954 −0.0273928
\(983\) 13585.3 0.440796 0.220398 0.975410i \(-0.429264\pi\)
0.220398 + 0.975410i \(0.429264\pi\)
\(984\) 0 0
\(985\) 44467.2 1.43842
\(986\) −1974.36 −0.0637691
\(987\) 0 0
\(988\) 0 0
\(989\) −65272.4 −2.09863
\(990\) 0 0
\(991\) −1024.49 −0.0328396 −0.0164198 0.999865i \(-0.505227\pi\)
−0.0164198 + 0.999865i \(0.505227\pi\)
\(992\) −10467.9 −0.335036
\(993\) 0 0
\(994\) −8843.29 −0.282185
\(995\) −62327.3 −1.98584
\(996\) 0 0
\(997\) 18514.9 0.588136 0.294068 0.955784i \(-0.404991\pi\)
0.294068 + 0.955784i \(0.404991\pi\)
\(998\) 5867.43 0.186102
\(999\) 0 0
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 1521.4.a.bc.1.4 8
3.2 odd 2 inner 1521.4.a.bc.1.5 8
13.3 even 3 117.4.g.f.100.5 yes 16
13.9 even 3 117.4.g.f.55.5 yes 16
13.12 even 2 1521.4.a.bd.1.5 8
39.29 odd 6 117.4.g.f.100.4 yes 16
39.35 odd 6 117.4.g.f.55.4 16
39.38 odd 2 1521.4.a.bd.1.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.4.g.f.55.4 16 39.35 odd 6
117.4.g.f.55.5 yes 16 13.9 even 3
117.4.g.f.100.4 yes 16 39.29 odd 6
117.4.g.f.100.5 yes 16 13.3 even 3
1521.4.a.bc.1.4 8 1.1 even 1 trivial
1521.4.a.bc.1.5 8 3.2 odd 2 inner
1521.4.a.bd.1.4 8 39.38 odd 2
1521.4.a.bd.1.5 8 13.12 even 2