Properties

Label 1521.4.a.x.1.3
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.5054412.1
Defining polynomial: \( x^{4} - 29x^{2} + 48 \) 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 39)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.32750\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.32750 q^{2} -6.23774 q^{4} +15.4241 q^{5} +7.96501 q^{7} -18.9006 q^{8} +O(q^{10})\) \(q+1.32750 q^{2} -6.23774 q^{4} +15.4241 q^{5} +7.96501 q^{7} -18.9006 q^{8} +20.4755 q^{10} +12.7691 q^{11} +10.5736 q^{14} +24.8113 q^{16} +54.0000 q^{17} +84.5794 q^{19} -96.2113 q^{20} +16.9510 q^{22} -122.853 q^{23} +112.902 q^{25} -49.6837 q^{28} -140.853 q^{29} +116.439 q^{31} +184.142 q^{32} +71.6851 q^{34} +122.853 q^{35} +433.898 q^{37} +112.279 q^{38} -291.525 q^{40} -205.823 q^{41} -418.853 q^{43} -79.6501 q^{44} -163.087 q^{46} +485.861 q^{47} -279.559 q^{49} +149.877 q^{50} +674.559 q^{53} +196.951 q^{55} -150.544 q^{56} -186.982 q^{58} +186.226 q^{59} -671.902 q^{61} +154.574 q^{62} +45.9584 q^{64} -14.0364 q^{67} -336.838 q^{68} +163.087 q^{70} -346.789 q^{71} +832.900 q^{73} +576.000 q^{74} -527.584 q^{76} +101.706 q^{77} -335.608 q^{79} +382.691 q^{80} -273.230 q^{82} +568.797 q^{83} +832.900 q^{85} -556.028 q^{86} -241.343 q^{88} -236.671 q^{89} +766.324 q^{92} +644.981 q^{94} +1304.56 q^{95} +1278.94 q^{97} -371.115 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 26 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 26 q^{4} - 20 q^{10} + 348 q^{14} + 354 q^{16} + 216 q^{17} - 136 q^{22} + 120 q^{23} + 44 q^{25} + 48 q^{29} - 120 q^{35} - 468 q^{38} - 1268 q^{40} - 1064 q^{43} + 716 q^{49} + 864 q^{53} + 584 q^{55} + 3372 q^{56} - 2280 q^{61} + 924 q^{62} + 1050 q^{64} + 1404 q^{68} + 2304 q^{74} - 816 q^{77} + 288 q^{79} + 28 q^{82} - 2392 q^{88} + 8568 q^{92} + 6656 q^{94} + 3384 q^{95}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.32750 0.469343 0.234671 0.972075i \(-0.424599\pi\)
0.234671 + 0.972075i \(0.424599\pi\)
\(3\) 0 0
\(4\) −6.23774 −0.779717
\(5\) 15.4241 1.37957 0.689785 0.724014i \(-0.257705\pi\)
0.689785 + 0.724014i \(0.257705\pi\)
\(6\) 0 0
\(7\) 7.96501 0.430070 0.215035 0.976606i \(-0.431013\pi\)
0.215035 + 0.976606i \(0.431013\pi\)
\(8\) −18.9006 −0.835297
\(9\) 0 0
\(10\) 20.4755 0.647491
\(11\) 12.7691 0.350002 0.175001 0.984568i \(-0.444007\pi\)
0.175001 + 0.984568i \(0.444007\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 10.5736 0.201850
\(15\) 0 0
\(16\) 24.8113 0.387677
\(17\) 54.0000 0.770407 0.385204 0.922832i \(-0.374131\pi\)
0.385204 + 0.922832i \(0.374131\pi\)
\(18\) 0 0
\(19\) 84.5794 1.02126 0.510628 0.859802i \(-0.329413\pi\)
0.510628 + 0.859802i \(0.329413\pi\)
\(20\) −96.2113 −1.07568
\(21\) 0 0
\(22\) 16.9510 0.164271
\(23\) −122.853 −1.11376 −0.556882 0.830591i \(-0.688003\pi\)
−0.556882 + 0.830591i \(0.688003\pi\)
\(24\) 0 0
\(25\) 112.902 0.903215
\(26\) 0 0
\(27\) 0 0
\(28\) −49.6837 −0.335333
\(29\) −140.853 −0.901921 −0.450961 0.892544i \(-0.648919\pi\)
−0.450961 + 0.892544i \(0.648919\pi\)
\(30\) 0 0
\(31\) 116.439 0.674617 0.337309 0.941394i \(-0.390483\pi\)
0.337309 + 0.941394i \(0.390483\pi\)
\(32\) 184.142 1.01725
\(33\) 0 0
\(34\) 71.6851 0.361585
\(35\) 122.853 0.593312
\(36\) 0 0
\(37\) 433.898 1.92790 0.963951 0.266081i \(-0.0857289\pi\)
0.963951 + 0.266081i \(0.0857289\pi\)
\(38\) 112.279 0.479319
\(39\) 0 0
\(40\) −291.525 −1.15235
\(41\) −205.823 −0.784003 −0.392002 0.919965i \(-0.628217\pi\)
−0.392002 + 0.919965i \(0.628217\pi\)
\(42\) 0 0
\(43\) −418.853 −1.48545 −0.742726 0.669595i \(-0.766468\pi\)
−0.742726 + 0.669595i \(0.766468\pi\)
\(44\) −79.6501 −0.272902
\(45\) 0 0
\(46\) −163.087 −0.522737
\(47\) 485.861 1.50787 0.753937 0.656947i \(-0.228152\pi\)
0.753937 + 0.656947i \(0.228152\pi\)
\(48\) 0 0
\(49\) −279.559 −0.815040
\(50\) 149.877 0.423918
\(51\) 0 0
\(52\) 0 0
\(53\) 674.559 1.74826 0.874130 0.485693i \(-0.161433\pi\)
0.874130 + 0.485693i \(0.161433\pi\)
\(54\) 0 0
\(55\) 196.951 0.482852
\(56\) −150.544 −0.359236
\(57\) 0 0
\(58\) −186.982 −0.423310
\(59\) 186.226 0.410925 0.205462 0.978665i \(-0.434130\pi\)
0.205462 + 0.978665i \(0.434130\pi\)
\(60\) 0 0
\(61\) −671.902 −1.41030 −0.705149 0.709059i \(-0.749120\pi\)
−0.705149 + 0.709059i \(0.749120\pi\)
\(62\) 154.574 0.316627
\(63\) 0 0
\(64\) 45.9584 0.0897626
\(65\) 0 0
\(66\) 0 0
\(67\) −14.0364 −0.0255944 −0.0127972 0.999918i \(-0.504074\pi\)
−0.0127972 + 0.999918i \(0.504074\pi\)
\(68\) −336.838 −0.600700
\(69\) 0 0
\(70\) 163.087 0.278467
\(71\) −346.789 −0.579665 −0.289833 0.957077i \(-0.593600\pi\)
−0.289833 + 0.957077i \(0.593600\pi\)
\(72\) 0 0
\(73\) 832.900 1.33539 0.667695 0.744435i \(-0.267281\pi\)
0.667695 + 0.744435i \(0.267281\pi\)
\(74\) 576.000 0.904846
\(75\) 0 0
\(76\) −527.584 −0.796290
\(77\) 101.706 0.150525
\(78\) 0 0
\(79\) −335.608 −0.477960 −0.238980 0.971025i \(-0.576813\pi\)
−0.238980 + 0.971025i \(0.576813\pi\)
\(80\) 382.691 0.534827
\(81\) 0 0
\(82\) −273.230 −0.367966
\(83\) 568.797 0.752212 0.376106 0.926577i \(-0.377263\pi\)
0.376106 + 0.926577i \(0.377263\pi\)
\(84\) 0 0
\(85\) 832.900 1.06283
\(86\) −556.028 −0.697186
\(87\) 0 0
\(88\) −241.343 −0.292355
\(89\) −236.671 −0.281877 −0.140939 0.990018i \(-0.545012\pi\)
−0.140939 + 0.990018i \(0.545012\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 766.324 0.868422
\(93\) 0 0
\(94\) 644.981 0.707710
\(95\) 1304.56 1.40889
\(96\) 0 0
\(97\) 1278.94 1.33873 0.669365 0.742934i \(-0.266566\pi\)
0.669365 + 0.742934i \(0.266566\pi\)
\(98\) −371.115 −0.382533
\(99\) 0 0
\(100\) −704.253 −0.704253
\(101\) −632.264 −0.622898 −0.311449 0.950263i \(-0.600814\pi\)
−0.311449 + 0.950263i \(0.600814\pi\)
\(102\) 0 0
\(103\) 1506.26 1.44094 0.720469 0.693487i \(-0.243927\pi\)
0.720469 + 0.693487i \(0.243927\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 895.478 0.820533
\(107\) 1268.56 1.14613 0.573066 0.819509i \(-0.305754\pi\)
0.573066 + 0.819509i \(0.305754\pi\)
\(108\) 0 0
\(109\) −347.425 −0.305296 −0.152648 0.988281i \(-0.548780\pi\)
−0.152648 + 0.988281i \(0.548780\pi\)
\(110\) 261.453 0.226623
\(111\) 0 0
\(112\) 197.622 0.166728
\(113\) −659.706 −0.549203 −0.274601 0.961558i \(-0.588546\pi\)
−0.274601 + 0.961558i \(0.588546\pi\)
\(114\) 0 0
\(115\) −1894.89 −1.53652
\(116\) 878.603 0.703244
\(117\) 0 0
\(118\) 247.215 0.192865
\(119\) 430.111 0.331329
\(120\) 0 0
\(121\) −1167.95 −0.877499
\(122\) −891.951 −0.661913
\(123\) 0 0
\(124\) −726.319 −0.526011
\(125\) −186.602 −0.133521
\(126\) 0 0
\(127\) 275.019 0.192157 0.0960787 0.995374i \(-0.469370\pi\)
0.0960787 + 0.995374i \(0.469370\pi\)
\(128\) −1412.13 −0.975121
\(129\) 0 0
\(130\) 0 0
\(131\) 1183.97 0.789648 0.394824 0.918757i \(-0.370805\pi\)
0.394824 + 0.918757i \(0.370805\pi\)
\(132\) 0 0
\(133\) 673.676 0.439211
\(134\) −18.6334 −0.0120125
\(135\) 0 0
\(136\) −1020.63 −0.643519
\(137\) 2557.36 1.59482 0.797410 0.603438i \(-0.206203\pi\)
0.797410 + 0.603438i \(0.206203\pi\)
\(138\) 0 0
\(139\) 545.736 0.333012 0.166506 0.986040i \(-0.446751\pi\)
0.166506 + 0.986040i \(0.446751\pi\)
\(140\) −766.324 −0.462616
\(141\) 0 0
\(142\) −460.362 −0.272062
\(143\) 0 0
\(144\) 0 0
\(145\) −2172.52 −1.24426
\(146\) 1105.68 0.626756
\(147\) 0 0
\(148\) −2706.54 −1.50322
\(149\) 1376.78 0.756981 0.378491 0.925605i \(-0.376443\pi\)
0.378491 + 0.925605i \(0.376443\pi\)
\(150\) 0 0
\(151\) −2733.47 −1.47316 −0.736579 0.676352i \(-0.763560\pi\)
−0.736579 + 0.676352i \(0.763560\pi\)
\(152\) −1598.60 −0.853052
\(153\) 0 0
\(154\) 135.015 0.0706479
\(155\) 1795.97 0.930683
\(156\) 0 0
\(157\) 1029.97 0.523570 0.261785 0.965126i \(-0.415689\pi\)
0.261785 + 0.965126i \(0.415689\pi\)
\(158\) −445.520 −0.224327
\(159\) 0 0
\(160\) 2840.22 1.40337
\(161\) −978.524 −0.478997
\(162\) 0 0
\(163\) 2882.91 1.38532 0.692660 0.721264i \(-0.256439\pi\)
0.692660 + 0.721264i \(0.256439\pi\)
\(164\) 1283.87 0.611301
\(165\) 0 0
\(166\) 755.079 0.353045
\(167\) 1153.90 0.534679 0.267340 0.963602i \(-0.413855\pi\)
0.267340 + 0.963602i \(0.413855\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 1105.68 0.498832
\(171\) 0 0
\(172\) 2612.69 1.15823
\(173\) 1688.85 0.742203 0.371101 0.928592i \(-0.378980\pi\)
0.371101 + 0.928592i \(0.378980\pi\)
\(174\) 0 0
\(175\) 899.265 0.388446
\(176\) 316.817 0.135687
\(177\) 0 0
\(178\) −314.181 −0.132297
\(179\) 942.793 0.393674 0.196837 0.980436i \(-0.436933\pi\)
0.196837 + 0.980436i \(0.436933\pi\)
\(180\) 0 0
\(181\) 482.030 0.197950 0.0989751 0.995090i \(-0.468444\pi\)
0.0989751 + 0.995090i \(0.468444\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 2322.00 0.930325
\(185\) 6692.47 2.65968
\(186\) 0 0
\(187\) 689.530 0.269644
\(188\) −3030.67 −1.17572
\(189\) 0 0
\(190\) 1731.80 0.661254
\(191\) −4223.32 −1.59994 −0.799971 0.600039i \(-0.795152\pi\)
−0.799971 + 0.600039i \(0.795152\pi\)
\(192\) 0 0
\(193\) 229.092 0.0854424 0.0427212 0.999087i \(-0.486397\pi\)
0.0427212 + 0.999087i \(0.486397\pi\)
\(194\) 1697.80 0.628323
\(195\) 0 0
\(196\) 1743.81 0.635501
\(197\) −228.335 −0.0825798 −0.0412899 0.999147i \(-0.513147\pi\)
−0.0412899 + 0.999147i \(0.513147\pi\)
\(198\) 0 0
\(199\) 2939.02 1.04694 0.523471 0.852043i \(-0.324637\pi\)
0.523471 + 0.852043i \(0.324637\pi\)
\(200\) −2133.92 −0.754453
\(201\) 0 0
\(202\) −839.332 −0.292352
\(203\) −1121.89 −0.387889
\(204\) 0 0
\(205\) −3174.63 −1.08159
\(206\) 1999.57 0.676294
\(207\) 0 0
\(208\) 0 0
\(209\) 1080.00 0.357441
\(210\) 0 0
\(211\) −1607.02 −0.524321 −0.262161 0.965024i \(-0.584435\pi\)
−0.262161 + 0.965024i \(0.584435\pi\)
\(212\) −4207.72 −1.36315
\(213\) 0 0
\(214\) 1684.01 0.537929
\(215\) −6460.42 −2.04929
\(216\) 0 0
\(217\) 927.441 0.290133
\(218\) −461.207 −0.143288
\(219\) 0 0
\(220\) −1228.53 −0.376488
\(221\) 0 0
\(222\) 0 0
\(223\) −130.867 −0.0392981 −0.0196490 0.999807i \(-0.506255\pi\)
−0.0196490 + 0.999807i \(0.506255\pi\)
\(224\) 1466.69 0.437489
\(225\) 0 0
\(226\) −875.761 −0.257764
\(227\) 4325.19 1.26464 0.632319 0.774708i \(-0.282103\pi\)
0.632319 + 0.774708i \(0.282103\pi\)
\(228\) 0 0
\(229\) 2621.57 0.756499 0.378250 0.925704i \(-0.376526\pi\)
0.378250 + 0.925704i \(0.376526\pi\)
\(230\) −2515.47 −0.721153
\(231\) 0 0
\(232\) 2662.21 0.753373
\(233\) 4643.12 1.30550 0.652748 0.757575i \(-0.273616\pi\)
0.652748 + 0.757575i \(0.273616\pi\)
\(234\) 0 0
\(235\) 7493.95 2.08022
\(236\) −1161.63 −0.320405
\(237\) 0 0
\(238\) 570.972 0.155507
\(239\) 6696.69 1.81244 0.906219 0.422809i \(-0.138956\pi\)
0.906219 + 0.422809i \(0.138956\pi\)
\(240\) 0 0
\(241\) −2301.47 −0.615148 −0.307574 0.951524i \(-0.599517\pi\)
−0.307574 + 0.951524i \(0.599517\pi\)
\(242\) −1550.46 −0.411848
\(243\) 0 0
\(244\) 4191.15 1.09963
\(245\) −4311.93 −1.12440
\(246\) 0 0
\(247\) 0 0
\(248\) −2200.78 −0.563506
\(249\) 0 0
\(250\) −247.714 −0.0626673
\(251\) −828.000 −0.208219 −0.104109 0.994566i \(-0.533199\pi\)
−0.104109 + 0.994566i \(0.533199\pi\)
\(252\) 0 0
\(253\) −1568.72 −0.389820
\(254\) 365.088 0.0901877
\(255\) 0 0
\(256\) −2242.27 −0.547429
\(257\) −884.763 −0.214747 −0.107374 0.994219i \(-0.534244\pi\)
−0.107374 + 0.994219i \(0.534244\pi\)
\(258\) 0 0
\(259\) 3456.00 0.829133
\(260\) 0 0
\(261\) 0 0
\(262\) 1571.72 0.370616
\(263\) 8343.94 1.95631 0.978155 0.207878i \(-0.0666557\pi\)
0.978155 + 0.207878i \(0.0666557\pi\)
\(264\) 0 0
\(265\) 10404.4 2.41185
\(266\) 894.306 0.206141
\(267\) 0 0
\(268\) 87.5556 0.0199564
\(269\) −2762.56 −0.626157 −0.313078 0.949727i \(-0.601360\pi\)
−0.313078 + 0.949727i \(0.601360\pi\)
\(270\) 0 0
\(271\) 3116.54 0.698585 0.349293 0.937014i \(-0.386422\pi\)
0.349293 + 0.937014i \(0.386422\pi\)
\(272\) 1339.81 0.298669
\(273\) 0 0
\(274\) 3394.91 0.748517
\(275\) 1441.65 0.316127
\(276\) 0 0
\(277\) −502.060 −0.108902 −0.0544510 0.998516i \(-0.517341\pi\)
−0.0544510 + 0.998516i \(0.517341\pi\)
\(278\) 724.465 0.156297
\(279\) 0 0
\(280\) −2322.00 −0.495592
\(281\) 6607.56 1.40275 0.701377 0.712791i \(-0.252569\pi\)
0.701377 + 0.712791i \(0.252569\pi\)
\(282\) 0 0
\(283\) −4368.98 −0.917699 −0.458850 0.888514i \(-0.651738\pi\)
−0.458850 + 0.888514i \(0.651738\pi\)
\(284\) 2163.18 0.451975
\(285\) 0 0
\(286\) 0 0
\(287\) −1639.38 −0.337176
\(288\) 0 0
\(289\) −1997.00 −0.406473
\(290\) −2884.03 −0.583986
\(291\) 0 0
\(292\) −5195.41 −1.04123
\(293\) 5348.12 1.06635 0.533175 0.846005i \(-0.320999\pi\)
0.533175 + 0.846005i \(0.320999\pi\)
\(294\) 0 0
\(295\) 2872.36 0.566900
\(296\) −8200.94 −1.61037
\(297\) 0 0
\(298\) 1827.68 0.355284
\(299\) 0 0
\(300\) 0 0
\(301\) −3336.17 −0.638849
\(302\) −3628.69 −0.691416
\(303\) 0 0
\(304\) 2098.53 0.395917
\(305\) −10363.5 −1.94561
\(306\) 0 0
\(307\) −4502.46 −0.837032 −0.418516 0.908210i \(-0.637450\pi\)
−0.418516 + 0.908210i \(0.637450\pi\)
\(308\) −634.414 −0.117367
\(309\) 0 0
\(310\) 2384.15 0.436809
\(311\) 7447.20 1.35785 0.678926 0.734207i \(-0.262446\pi\)
0.678926 + 0.734207i \(0.262446\pi\)
\(312\) 0 0
\(313\) −6508.93 −1.17542 −0.587710 0.809072i \(-0.699970\pi\)
−0.587710 + 0.809072i \(0.699970\pi\)
\(314\) 1367.29 0.245734
\(315\) 0 0
\(316\) 2093.43 0.372673
\(317\) −2465.57 −0.436846 −0.218423 0.975854i \(-0.570091\pi\)
−0.218423 + 0.975854i \(0.570091\pi\)
\(318\) 0 0
\(319\) −1798.56 −0.315674
\(320\) 708.866 0.123834
\(321\) 0 0
\(322\) −1298.99 −0.224814
\(323\) 4567.29 0.786782
\(324\) 0 0
\(325\) 0 0
\(326\) 3827.07 0.650190
\(327\) 0 0
\(328\) 3890.18 0.654876
\(329\) 3869.89 0.648491
\(330\) 0 0
\(331\) −4114.84 −0.683300 −0.341650 0.939827i \(-0.610986\pi\)
−0.341650 + 0.939827i \(0.610986\pi\)
\(332\) −3548.01 −0.586513
\(333\) 0 0
\(334\) 1531.80 0.250948
\(335\) −216.499 −0.0353092
\(336\) 0 0
\(337\) 4798.05 0.775568 0.387784 0.921750i \(-0.373241\pi\)
0.387784 + 0.921750i \(0.373241\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −5195.41 −0.828708
\(341\) 1486.82 0.236117
\(342\) 0 0
\(343\) −4958.69 −0.780594
\(344\) 7916.58 1.24079
\(345\) 0 0
\(346\) 2241.96 0.348348
\(347\) 3314.76 0.512812 0.256406 0.966569i \(-0.417462\pi\)
0.256406 + 0.966569i \(0.417462\pi\)
\(348\) 0 0
\(349\) −371.740 −0.0570166 −0.0285083 0.999594i \(-0.509076\pi\)
−0.0285083 + 0.999594i \(0.509076\pi\)
\(350\) 1193.78 0.182314
\(351\) 0 0
\(352\) 2351.32 0.356039
\(353\) −7539.10 −1.13673 −0.568365 0.822776i \(-0.692424\pi\)
−0.568365 + 0.822776i \(0.692424\pi\)
\(354\) 0 0
\(355\) −5348.89 −0.799689
\(356\) 1476.29 0.219785
\(357\) 0 0
\(358\) 1251.56 0.184768
\(359\) −12741.5 −1.87317 −0.936586 0.350437i \(-0.886033\pi\)
−0.936586 + 0.350437i \(0.886033\pi\)
\(360\) 0 0
\(361\) 294.676 0.0429619
\(362\) 639.896 0.0929065
\(363\) 0 0
\(364\) 0 0
\(365\) 12846.7 1.84227
\(366\) 0 0
\(367\) 7187.26 1.02227 0.511134 0.859501i \(-0.329226\pi\)
0.511134 + 0.859501i \(0.329226\pi\)
\(368\) −3048.14 −0.431781
\(369\) 0 0
\(370\) 8884.26 1.24830
\(371\) 5372.87 0.751874
\(372\) 0 0
\(373\) −2087.99 −0.289845 −0.144922 0.989443i \(-0.546293\pi\)
−0.144922 + 0.989443i \(0.546293\pi\)
\(374\) 915.352 0.126555
\(375\) 0 0
\(376\) −9183.07 −1.25952
\(377\) 0 0
\(378\) 0 0
\(379\) 3982.08 0.539699 0.269850 0.962902i \(-0.413026\pi\)
0.269850 + 0.962902i \(0.413026\pi\)
\(380\) −8137.50 −1.09854
\(381\) 0 0
\(382\) −5606.47 −0.750921
\(383\) −8638.43 −1.15249 −0.576244 0.817278i \(-0.695482\pi\)
−0.576244 + 0.817278i \(0.695482\pi\)
\(384\) 0 0
\(385\) 1568.72 0.207660
\(386\) 304.120 0.0401018
\(387\) 0 0
\(388\) −7977.70 −1.04383
\(389\) 1275.74 0.166279 0.0831393 0.996538i \(-0.473505\pi\)
0.0831393 + 0.996538i \(0.473505\pi\)
\(390\) 0 0
\(391\) −6634.06 −0.858053
\(392\) 5283.83 0.680801
\(393\) 0 0
\(394\) −303.115 −0.0387582
\(395\) −5176.44 −0.659379
\(396\) 0 0
\(397\) −4622.65 −0.584394 −0.292197 0.956358i \(-0.594386\pi\)
−0.292197 + 0.956358i \(0.594386\pi\)
\(398\) 3901.55 0.491375
\(399\) 0 0
\(400\) 2801.24 0.350155
\(401\) 138.075 0.0171949 0.00859743 0.999963i \(-0.497263\pi\)
0.00859743 + 0.999963i \(0.497263\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 3943.90 0.485684
\(405\) 0 0
\(406\) −1489.32 −0.182053
\(407\) 5540.47 0.674769
\(408\) 0 0
\(409\) −1204.64 −0.145637 −0.0728186 0.997345i \(-0.523199\pi\)
−0.0728186 + 0.997345i \(0.523199\pi\)
\(410\) −4214.32 −0.507635
\(411\) 0 0
\(412\) −9395.68 −1.12352
\(413\) 1483.29 0.176726
\(414\) 0 0
\(415\) 8773.16 1.03773
\(416\) 0 0
\(417\) 0 0
\(418\) 1433.70 0.167762
\(419\) −5199.85 −0.606275 −0.303138 0.952947i \(-0.598034\pi\)
−0.303138 + 0.952947i \(0.598034\pi\)
\(420\) 0 0
\(421\) −14136.5 −1.63651 −0.818256 0.574854i \(-0.805059\pi\)
−0.818256 + 0.574854i \(0.805059\pi\)
\(422\) −2133.32 −0.246086
\(423\) 0 0
\(424\) −12749.6 −1.46032
\(425\) 6096.70 0.695844
\(426\) 0 0
\(427\) −5351.71 −0.606527
\(428\) −7912.94 −0.893660
\(429\) 0 0
\(430\) −8576.21 −0.961818
\(431\) 2279.83 0.254793 0.127396 0.991852i \(-0.459338\pi\)
0.127396 + 0.991852i \(0.459338\pi\)
\(432\) 0 0
\(433\) −13298.7 −1.47597 −0.737984 0.674819i \(-0.764222\pi\)
−0.737984 + 0.674819i \(0.764222\pi\)
\(434\) 1231.18 0.136172
\(435\) 0 0
\(436\) 2167.14 0.238045
\(437\) −10390.8 −1.13744
\(438\) 0 0
\(439\) −10452.3 −1.13635 −0.568177 0.822907i \(-0.692351\pi\)
−0.568177 + 0.822907i \(0.692351\pi\)
\(440\) −3722.50 −0.403325
\(441\) 0 0
\(442\) 0 0
\(443\) −5363.50 −0.575232 −0.287616 0.957746i \(-0.592863\pi\)
−0.287616 + 0.957746i \(0.592863\pi\)
\(444\) 0 0
\(445\) −3650.43 −0.388870
\(446\) −173.726 −0.0184443
\(447\) 0 0
\(448\) 366.059 0.0386042
\(449\) −9681.73 −1.01762 −0.508808 0.860880i \(-0.669914\pi\)
−0.508808 + 0.860880i \(0.669914\pi\)
\(450\) 0 0
\(451\) −2628.17 −0.274402
\(452\) 4115.07 0.428223
\(453\) 0 0
\(454\) 5741.69 0.593548
\(455\) 0 0
\(456\) 0 0
\(457\) −3537.94 −0.362139 −0.181070 0.983470i \(-0.557956\pi\)
−0.181070 + 0.983470i \(0.557956\pi\)
\(458\) 3480.14 0.355057
\(459\) 0 0
\(460\) 11819.8 1.19805
\(461\) 15074.9 1.52302 0.761508 0.648156i \(-0.224459\pi\)
0.761508 + 0.648156i \(0.224459\pi\)
\(462\) 0 0
\(463\) −11070.0 −1.11116 −0.555580 0.831463i \(-0.687504\pi\)
−0.555580 + 0.831463i \(0.687504\pi\)
\(464\) −3494.74 −0.349654
\(465\) 0 0
\(466\) 6163.75 0.612725
\(467\) −13252.8 −1.31320 −0.656600 0.754239i \(-0.728006\pi\)
−0.656600 + 0.754239i \(0.728006\pi\)
\(468\) 0 0
\(469\) −111.800 −0.0110074
\(470\) 9948.23 0.976335
\(471\) 0 0
\(472\) −3519.79 −0.343244
\(473\) −5348.36 −0.519911
\(474\) 0 0
\(475\) 9549.18 0.922413
\(476\) −2682.92 −0.258343
\(477\) 0 0
\(478\) 8889.86 0.850654
\(479\) −12241.4 −1.16769 −0.583843 0.811866i \(-0.698452\pi\)
−0.583843 + 0.811866i \(0.698452\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −3055.20 −0.288715
\(483\) 0 0
\(484\) 7285.37 0.684201
\(485\) 19726.5 1.84687
\(486\) 0 0
\(487\) −13413.3 −1.24808 −0.624041 0.781392i \(-0.714510\pi\)
−0.624041 + 0.781392i \(0.714510\pi\)
\(488\) 12699.4 1.17802
\(489\) 0 0
\(490\) −5724.10 −0.527731
\(491\) 737.885 0.0678214 0.0339107 0.999425i \(-0.489204\pi\)
0.0339107 + 0.999425i \(0.489204\pi\)
\(492\) 0 0
\(493\) −7606.06 −0.694847
\(494\) 0 0
\(495\) 0 0
\(496\) 2889.01 0.261533
\(497\) −2762.17 −0.249297
\(498\) 0 0
\(499\) 1865.65 0.167370 0.0836852 0.996492i \(-0.473331\pi\)
0.0836852 + 0.996492i \(0.473331\pi\)
\(500\) 1163.97 0.104109
\(501\) 0 0
\(502\) −1099.17 −0.0977259
\(503\) 16632.0 1.47432 0.737161 0.675717i \(-0.236166\pi\)
0.737161 + 0.675717i \(0.236166\pi\)
\(504\) 0 0
\(505\) −9752.09 −0.859331
\(506\) −2082.47 −0.182959
\(507\) 0 0
\(508\) −1715.50 −0.149829
\(509\) −4128.91 −0.359549 −0.179775 0.983708i \(-0.557537\pi\)
−0.179775 + 0.983708i \(0.557537\pi\)
\(510\) 0 0
\(511\) 6634.06 0.574312
\(512\) 8320.40 0.718190
\(513\) 0 0
\(514\) −1174.52 −0.100790
\(515\) 23232.7 1.98788
\(516\) 0 0
\(517\) 6203.99 0.527758
\(518\) 4587.85 0.389147
\(519\) 0 0
\(520\) 0 0
\(521\) 988.234 0.0831005 0.0415502 0.999136i \(-0.486770\pi\)
0.0415502 + 0.999136i \(0.486770\pi\)
\(522\) 0 0
\(523\) −9441.62 −0.789394 −0.394697 0.918811i \(-0.629150\pi\)
−0.394697 + 0.918811i \(0.629150\pi\)
\(524\) −7385.30 −0.615703
\(525\) 0 0
\(526\) 11076.6 0.918180
\(527\) 6287.73 0.519730
\(528\) 0 0
\(529\) 2925.83 0.240472
\(530\) 13811.9 1.13198
\(531\) 0 0
\(532\) −4202.21 −0.342461
\(533\) 0 0
\(534\) 0 0
\(535\) 19566.3 1.58117
\(536\) 265.297 0.0213789
\(537\) 0 0
\(538\) −3667.30 −0.293882
\(539\) −3569.70 −0.285265
\(540\) 0 0
\(541\) 14001.5 1.11270 0.556351 0.830948i \(-0.312201\pi\)
0.556351 + 0.830948i \(0.312201\pi\)
\(542\) 4137.22 0.327876
\(543\) 0 0
\(544\) 9943.67 0.783697
\(545\) −5358.70 −0.421177
\(546\) 0 0
\(547\) −4244.85 −0.331804 −0.165902 0.986142i \(-0.553053\pi\)
−0.165902 + 0.986142i \(0.553053\pi\)
\(548\) −15952.2 −1.24351
\(549\) 0 0
\(550\) 1913.80 0.148372
\(551\) −11913.3 −0.921092
\(552\) 0 0
\(553\) −2673.12 −0.205556
\(554\) −666.485 −0.0511124
\(555\) 0 0
\(556\) −3404.16 −0.259655
\(557\) 11732.2 0.892473 0.446236 0.894915i \(-0.352764\pi\)
0.446236 + 0.894915i \(0.352764\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 3048.14 0.230013
\(561\) 0 0
\(562\) 8771.54 0.658372
\(563\) 9941.80 0.744222 0.372111 0.928188i \(-0.378634\pi\)
0.372111 + 0.928188i \(0.378634\pi\)
\(564\) 0 0
\(565\) −10175.3 −0.757664
\(566\) −5799.83 −0.430716
\(567\) 0 0
\(568\) 6554.52 0.484193
\(569\) −3690.77 −0.271925 −0.135962 0.990714i \(-0.543413\pi\)
−0.135962 + 0.990714i \(0.543413\pi\)
\(570\) 0 0
\(571\) −5685.09 −0.416661 −0.208331 0.978058i \(-0.566803\pi\)
−0.208331 + 0.978058i \(0.566803\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −2176.28 −0.158251
\(575\) −13870.3 −1.00597
\(576\) 0 0
\(577\) 7746.50 0.558910 0.279455 0.960159i \(-0.409846\pi\)
0.279455 + 0.960159i \(0.409846\pi\)
\(578\) −2651.02 −0.190775
\(579\) 0 0
\(580\) 13551.6 0.970175
\(581\) 4530.47 0.323504
\(582\) 0 0
\(583\) 8613.48 0.611894
\(584\) −15742.3 −1.11545
\(585\) 0 0
\(586\) 7099.64 0.500484
\(587\) 2766.54 0.194527 0.0972635 0.995259i \(-0.468991\pi\)
0.0972635 + 0.995259i \(0.468991\pi\)
\(588\) 0 0
\(589\) 9848.38 0.688957
\(590\) 3813.07 0.266070
\(591\) 0 0
\(592\) 10765.6 0.747402
\(593\) 1440.79 0.0997743 0.0498871 0.998755i \(-0.484114\pi\)
0.0498871 + 0.998755i \(0.484114\pi\)
\(594\) 0 0
\(595\) 6634.06 0.457092
\(596\) −8587.99 −0.590231
\(597\) 0 0
\(598\) 0 0
\(599\) −23837.5 −1.62600 −0.813001 0.582263i \(-0.802168\pi\)
−0.813001 + 0.582263i \(0.802168\pi\)
\(600\) 0 0
\(601\) −6694.23 −0.454348 −0.227174 0.973854i \(-0.572949\pi\)
−0.227174 + 0.973854i \(0.572949\pi\)
\(602\) −4428.77 −0.299839
\(603\) 0 0
\(604\) 17050.7 1.14865
\(605\) −18014.6 −1.21057
\(606\) 0 0
\(607\) 3330.50 0.222703 0.111352 0.993781i \(-0.464482\pi\)
0.111352 + 0.993781i \(0.464482\pi\)
\(608\) 15574.6 1.03887
\(609\) 0 0
\(610\) −13757.5 −0.913156
\(611\) 0 0
\(612\) 0 0
\(613\) −13490.3 −0.888857 −0.444428 0.895814i \(-0.646593\pi\)
−0.444428 + 0.895814i \(0.646593\pi\)
\(614\) −5977.02 −0.392855
\(615\) 0 0
\(616\) −1922.30 −0.125733
\(617\) −7470.76 −0.487458 −0.243729 0.969843i \(-0.578371\pi\)
−0.243729 + 0.969843i \(0.578371\pi\)
\(618\) 0 0
\(619\) 24806.9 1.61078 0.805389 0.592746i \(-0.201956\pi\)
0.805389 + 0.592746i \(0.201956\pi\)
\(620\) −11202.8 −0.725669
\(621\) 0 0
\(622\) 9886.17 0.637298
\(623\) −1885.09 −0.121227
\(624\) 0 0
\(625\) −16990.9 −1.08742
\(626\) −8640.61 −0.551675
\(627\) 0 0
\(628\) −6424.68 −0.408237
\(629\) 23430.5 1.48527
\(630\) 0 0
\(631\) 314.333 0.0198311 0.00991554 0.999951i \(-0.496844\pi\)
0.00991554 + 0.999951i \(0.496844\pi\)
\(632\) 6343.19 0.399238
\(633\) 0 0
\(634\) −3273.05 −0.205031
\(635\) 4241.91 0.265095
\(636\) 0 0
\(637\) 0 0
\(638\) −2387.59 −0.148159
\(639\) 0 0
\(640\) −21780.7 −1.34525
\(641\) −5550.41 −0.342009 −0.171005 0.985270i \(-0.554701\pi\)
−0.171005 + 0.985270i \(0.554701\pi\)
\(642\) 0 0
\(643\) 5479.48 0.336065 0.168032 0.985781i \(-0.446259\pi\)
0.168032 + 0.985781i \(0.446259\pi\)
\(644\) 6103.78 0.373482
\(645\) 0 0
\(646\) 6063.08 0.369271
\(647\) −4724.83 −0.287098 −0.143549 0.989643i \(-0.545851\pi\)
−0.143549 + 0.989643i \(0.545851\pi\)
\(648\) 0 0
\(649\) 2377.93 0.143824
\(650\) 0 0
\(651\) 0 0
\(652\) −17982.9 −1.08016
\(653\) −3463.91 −0.207585 −0.103793 0.994599i \(-0.533098\pi\)
−0.103793 + 0.994599i \(0.533098\pi\)
\(654\) 0 0
\(655\) 18261.6 1.08938
\(656\) −5106.73 −0.303940
\(657\) 0 0
\(658\) 5137.28 0.304365
\(659\) 2606.35 0.154065 0.0770327 0.997029i \(-0.475455\pi\)
0.0770327 + 0.997029i \(0.475455\pi\)
\(660\) 0 0
\(661\) −22436.7 −1.32025 −0.660127 0.751154i \(-0.729498\pi\)
−0.660127 + 0.751154i \(0.729498\pi\)
\(662\) −5462.46 −0.320702
\(663\) 0 0
\(664\) −10750.6 −0.628321
\(665\) 10390.8 0.605923
\(666\) 0 0
\(667\) 17304.2 1.00453
\(668\) −7197.73 −0.416899
\(669\) 0 0
\(670\) −287.403 −0.0165721
\(671\) −8579.56 −0.493607
\(672\) 0 0
\(673\) −633.970 −0.0363117 −0.0181558 0.999835i \(-0.505779\pi\)
−0.0181558 + 0.999835i \(0.505779\pi\)
\(674\) 6369.42 0.364007
\(675\) 0 0
\(676\) 0 0
\(677\) −24457.4 −1.38844 −0.694221 0.719762i \(-0.744251\pi\)
−0.694221 + 0.719762i \(0.744251\pi\)
\(678\) 0 0
\(679\) 10186.8 0.575747
\(680\) −15742.3 −0.887780
\(681\) 0 0
\(682\) 1973.76 0.110820
\(683\) 12367.6 0.692875 0.346437 0.938073i \(-0.387391\pi\)
0.346437 + 0.938073i \(0.387391\pi\)
\(684\) 0 0
\(685\) 39445.0 2.20017
\(686\) −6582.66 −0.366366
\(687\) 0 0
\(688\) −10392.3 −0.575875
\(689\) 0 0
\(690\) 0 0
\(691\) 1050.99 0.0578605 0.0289302 0.999581i \(-0.490790\pi\)
0.0289302 + 0.999581i \(0.490790\pi\)
\(692\) −10534.6 −0.578709
\(693\) 0 0
\(694\) 4400.35 0.240685
\(695\) 8417.46 0.459414
\(696\) 0 0
\(697\) −11114.4 −0.604002
\(698\) −493.486 −0.0267603
\(699\) 0 0
\(700\) −5609.38 −0.302878
\(701\) 24294.1 1.30895 0.654476 0.756083i \(-0.272889\pi\)
0.654476 + 0.756083i \(0.272889\pi\)
\(702\) 0 0
\(703\) 36698.8 1.96888
\(704\) 586.846 0.0314170
\(705\) 0 0
\(706\) −10008.2 −0.533516
\(707\) −5035.99 −0.267890
\(708\) 0 0
\(709\) −27465.9 −1.45487 −0.727436 0.686176i \(-0.759288\pi\)
−0.727436 + 0.686176i \(0.759288\pi\)
\(710\) −7100.66 −0.375328
\(711\) 0 0
\(712\) 4473.23 0.235451
\(713\) −14304.9 −0.751365
\(714\) 0 0
\(715\) 0 0
\(716\) −5880.90 −0.306955
\(717\) 0 0
\(718\) −16914.3 −0.879160
\(719\) −36433.5 −1.88976 −0.944882 0.327411i \(-0.893824\pi\)
−0.944882 + 0.327411i \(0.893824\pi\)
\(720\) 0 0
\(721\) 11997.4 0.619704
\(722\) 391.183 0.0201639
\(723\) 0 0
\(724\) −3006.78 −0.154345
\(725\) −15902.6 −0.814629
\(726\) 0 0
\(727\) 551.608 0.0281403 0.0140701 0.999901i \(-0.495521\pi\)
0.0140701 + 0.999901i \(0.495521\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 17054.0 0.864654
\(731\) −22618.1 −1.14440
\(732\) 0 0
\(733\) −20317.2 −1.02379 −0.511893 0.859049i \(-0.671055\pi\)
−0.511893 + 0.859049i \(0.671055\pi\)
\(734\) 9541.10 0.479794
\(735\) 0 0
\(736\) −22622.4 −1.13298
\(737\) −179.232 −0.00895807
\(738\) 0 0
\(739\) 29931.6 1.48992 0.744962 0.667107i \(-0.232468\pi\)
0.744962 + 0.667107i \(0.232468\pi\)
\(740\) −41745.9 −2.07380
\(741\) 0 0
\(742\) 7132.49 0.352887
\(743\) 24376.4 1.20361 0.601805 0.798643i \(-0.294448\pi\)
0.601805 + 0.798643i \(0.294448\pi\)
\(744\) 0 0
\(745\) 21235.5 1.04431
\(746\) −2771.81 −0.136037
\(747\) 0 0
\(748\) −4301.11 −0.210246
\(749\) 10104.1 0.492917
\(750\) 0 0
\(751\) 22692.2 1.10260 0.551298 0.834308i \(-0.314133\pi\)
0.551298 + 0.834308i \(0.314133\pi\)
\(752\) 12054.8 0.584567
\(753\) 0 0
\(754\) 0 0
\(755\) −42161.3 −2.03232
\(756\) 0 0
\(757\) −33063.5 −1.58747 −0.793734 0.608265i \(-0.791866\pi\)
−0.793734 + 0.608265i \(0.791866\pi\)
\(758\) 5286.22 0.253304
\(759\) 0 0
\(760\) −24657.0 −1.17685
\(761\) −216.324 −0.0103045 −0.00515226 0.999987i \(-0.501640\pi\)
−0.00515226 + 0.999987i \(0.501640\pi\)
\(762\) 0 0
\(763\) −2767.24 −0.131299
\(764\) 26344.0 1.24750
\(765\) 0 0
\(766\) −11467.5 −0.540912
\(767\) 0 0
\(768\) 0 0
\(769\) −19214.4 −0.901025 −0.450512 0.892770i \(-0.648759\pi\)
−0.450512 + 0.892770i \(0.648759\pi\)
\(770\) 2082.47 0.0974638
\(771\) 0 0
\(772\) −1429.01 −0.0666209
\(773\) 33175.6 1.54365 0.771826 0.635834i \(-0.219344\pi\)
0.771826 + 0.635834i \(0.219344\pi\)
\(774\) 0 0
\(775\) 13146.2 0.609325
\(776\) −24172.8 −1.11824
\(777\) 0 0
\(778\) 1693.54 0.0780416
\(779\) −17408.4 −0.800667
\(780\) 0 0
\(781\) −4428.17 −0.202884
\(782\) −8806.72 −0.402721
\(783\) 0 0
\(784\) −6936.21 −0.315972
\(785\) 15886.3 0.722302
\(786\) 0 0
\(787\) 14887.9 0.674330 0.337165 0.941446i \(-0.390532\pi\)
0.337165 + 0.941446i \(0.390532\pi\)
\(788\) 1424.30 0.0643889
\(789\) 0 0
\(790\) −6871.73 −0.309475
\(791\) −5254.56 −0.236196
\(792\) 0 0
\(793\) 0 0
\(794\) −6136.58 −0.274281
\(795\) 0 0
\(796\) −18332.8 −0.816319
\(797\) −12954.5 −0.575751 −0.287876 0.957668i \(-0.592949\pi\)
−0.287876 + 0.957668i \(0.592949\pi\)
\(798\) 0 0
\(799\) 26236.5 1.16168
\(800\) 20790.0 0.918796
\(801\) 0 0
\(802\) 183.295 0.00807028
\(803\) 10635.4 0.467389
\(804\) 0 0
\(805\) −15092.8 −0.660810
\(806\) 0 0
\(807\) 0 0
\(808\) 11950.2 0.520305
\(809\) −8275.59 −0.359647 −0.179823 0.983699i \(-0.557553\pi\)
−0.179823 + 0.983699i \(0.557553\pi\)
\(810\) 0 0
\(811\) −26327.1 −1.13991 −0.569956 0.821675i \(-0.693040\pi\)
−0.569956 + 0.821675i \(0.693040\pi\)
\(812\) 6998.09 0.302444
\(813\) 0 0
\(814\) 7354.98 0.316698
\(815\) 44466.3 1.91115
\(816\) 0 0
\(817\) −35426.3 −1.51703
\(818\) −1599.16 −0.0683538
\(819\) 0 0
\(820\) 19802.5 0.843333
\(821\) −34439.0 −1.46398 −0.731992 0.681314i \(-0.761409\pi\)
−0.731992 + 0.681314i \(0.761409\pi\)
\(822\) 0 0
\(823\) −13870.5 −0.587479 −0.293739 0.955886i \(-0.594900\pi\)
−0.293739 + 0.955886i \(0.594900\pi\)
\(824\) −28469.3 −1.20361
\(825\) 0 0
\(826\) 1969.07 0.0829453
\(827\) 2132.30 0.0896583 0.0448292 0.998995i \(-0.485726\pi\)
0.0448292 + 0.998995i \(0.485726\pi\)
\(828\) 0 0
\(829\) −6212.39 −0.260272 −0.130136 0.991496i \(-0.541541\pi\)
−0.130136 + 0.991496i \(0.541541\pi\)
\(830\) 11646.4 0.487051
\(831\) 0 0
\(832\) 0 0
\(833\) −15096.2 −0.627913
\(834\) 0 0
\(835\) 17797.8 0.737628
\(836\) −6736.76 −0.278703
\(837\) 0 0
\(838\) −6902.81 −0.284551
\(839\) −4550.52 −0.187248 −0.0936242 0.995608i \(-0.529845\pi\)
−0.0936242 + 0.995608i \(0.529845\pi\)
\(840\) 0 0
\(841\) −4549.47 −0.186538
\(842\) −18766.2 −0.768085
\(843\) 0 0
\(844\) 10024.2 0.408822
\(845\) 0 0
\(846\) 0 0
\(847\) −9302.74 −0.377386
\(848\) 16736.7 0.677759
\(849\) 0 0
\(850\) 8093.38 0.326589
\(851\) −53305.6 −2.14723
\(852\) 0 0
\(853\) −12262.8 −0.492228 −0.246114 0.969241i \(-0.579154\pi\)
−0.246114 + 0.969241i \(0.579154\pi\)
\(854\) −7104.40 −0.284669
\(855\) 0 0
\(856\) −23976.5 −0.957362
\(857\) 34949.1 1.39304 0.696521 0.717536i \(-0.254730\pi\)
0.696521 + 0.717536i \(0.254730\pi\)
\(858\) 0 0
\(859\) −21762.1 −0.864394 −0.432197 0.901779i \(-0.642261\pi\)
−0.432197 + 0.901779i \(0.642261\pi\)
\(860\) 40298.4 1.59786
\(861\) 0 0
\(862\) 3026.48 0.119585
\(863\) −19811.5 −0.781450 −0.390725 0.920507i \(-0.627776\pi\)
−0.390725 + 0.920507i \(0.627776\pi\)
\(864\) 0 0
\(865\) 26049.0 1.02392
\(866\) −17654.0 −0.692735
\(867\) 0 0
\(868\) −5785.14 −0.226222
\(869\) −4285.40 −0.167287
\(870\) 0 0
\(871\) 0 0
\(872\) 6566.54 0.255013
\(873\) 0 0
\(874\) −13793.8 −0.533848
\(875\) −1486.28 −0.0574235
\(876\) 0 0
\(877\) 25716.8 0.990186 0.495093 0.868840i \(-0.335134\pi\)
0.495093 + 0.868840i \(0.335134\pi\)
\(878\) −13875.4 −0.533339
\(879\) 0 0
\(880\) 4886.61 0.187190
\(881\) −34709.6 −1.32735 −0.663676 0.748020i \(-0.731005\pi\)
−0.663676 + 0.748020i \(0.731005\pi\)
\(882\) 0 0
\(883\) −3848.68 −0.146680 −0.0733400 0.997307i \(-0.523366\pi\)
−0.0733400 + 0.997307i \(0.523366\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −7120.06 −0.269981
\(887\) −32804.8 −1.24180 −0.620900 0.783890i \(-0.713233\pi\)
−0.620900 + 0.783890i \(0.713233\pi\)
\(888\) 0 0
\(889\) 2190.53 0.0826412
\(890\) −4845.95 −0.182513
\(891\) 0 0
\(892\) 816.311 0.0306414
\(893\) 41093.8 1.53992
\(894\) 0 0
\(895\) 14541.7 0.543101
\(896\) −11247.6 −0.419371
\(897\) 0 0
\(898\) −12852.5 −0.477610
\(899\) −16400.8 −0.608452
\(900\) 0 0
\(901\) 36426.2 1.34687
\(902\) −3488.90 −0.128789
\(903\) 0 0
\(904\) 12468.8 0.458748
\(905\) 7434.86 0.273086
\(906\) 0 0
\(907\) −22262.2 −0.814999 −0.407500 0.913205i \(-0.633599\pi\)
−0.407500 + 0.913205i \(0.633599\pi\)
\(908\) −26979.4 −0.986060
\(909\) 0 0
\(910\) 0 0
\(911\) 13515.3 0.491528 0.245764 0.969330i \(-0.420961\pi\)
0.245764 + 0.969330i \(0.420961\pi\)
\(912\) 0 0
\(913\) 7263.01 0.263275
\(914\) −4696.62 −0.169968
\(915\) 0 0
\(916\) −16352.7 −0.589855
\(917\) 9430.33 0.339604
\(918\) 0 0
\(919\) 26600.6 0.954811 0.477405 0.878683i \(-0.341577\pi\)
0.477405 + 0.878683i \(0.341577\pi\)
\(920\) 35814.6 1.28345
\(921\) 0 0
\(922\) 20012.0 0.714816
\(923\) 0 0
\(924\) 0 0
\(925\) 48987.9 1.74131
\(926\) −14695.5 −0.521515
\(927\) 0 0
\(928\) −25936.9 −0.917480
\(929\) −32887.7 −1.16148 −0.580738 0.814090i \(-0.697236\pi\)
−0.580738 + 0.814090i \(0.697236\pi\)
\(930\) 0 0
\(931\) −23644.9 −0.832363
\(932\) −28962.6 −1.01792
\(933\) 0 0
\(934\) −17593.1 −0.616341
\(935\) 10635.4 0.371993
\(936\) 0 0
\(937\) −9261.78 −0.322913 −0.161456 0.986880i \(-0.551619\pi\)
−0.161456 + 0.986880i \(0.551619\pi\)
\(938\) −148.415 −0.00516623
\(939\) 0 0
\(940\) −46745.3 −1.62198
\(941\) 12054.9 0.417619 0.208810 0.977956i \(-0.433041\pi\)
0.208810 + 0.977956i \(0.433041\pi\)
\(942\) 0 0
\(943\) 25285.9 0.873195
\(944\) 4620.51 0.159306
\(945\) 0 0
\(946\) −7099.96 −0.244016
\(947\) 20221.4 0.693885 0.346942 0.937886i \(-0.387220\pi\)
0.346942 + 0.937886i \(0.387220\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 12676.5 0.432928
\(951\) 0 0
\(952\) −8129.36 −0.276758
\(953\) 20331.5 0.691083 0.345542 0.938403i \(-0.387695\pi\)
0.345542 + 0.938403i \(0.387695\pi\)
\(954\) 0 0
\(955\) −65140.8 −2.20723
\(956\) −41772.2 −1.41319
\(957\) 0 0
\(958\) −16250.4 −0.548045
\(959\) 20369.4 0.685885
\(960\) 0 0
\(961\) −16232.9 −0.544891
\(962\) 0 0
\(963\) 0 0
\(964\) 14356.0 0.479641
\(965\) 3533.53 0.117874
\(966\) 0 0
\(967\) 11082.2 0.368541 0.184270 0.982876i \(-0.441008\pi\)
0.184270 + 0.982876i \(0.441008\pi\)
\(968\) 22075.0 0.732973
\(969\) 0 0
\(970\) 26186.9 0.866816
\(971\) 36694.1 1.21274 0.606369 0.795184i \(-0.292626\pi\)
0.606369 + 0.795184i \(0.292626\pi\)
\(972\) 0 0
\(973\) 4346.79 0.143219
\(974\) −17806.2 −0.585778
\(975\) 0 0
\(976\) −16670.8 −0.546740
\(977\) 17155.3 0.561767 0.280883 0.959742i \(-0.409373\pi\)
0.280883 + 0.959742i \(0.409373\pi\)
\(978\) 0 0
\(979\) −3022.07 −0.0986575
\(980\) 26896.7 0.876718
\(981\) 0 0
\(982\) 979.544 0.0318315
\(983\) −38419.7 −1.24659 −0.623296 0.781986i \(-0.714207\pi\)
−0.623296 + 0.781986i \(0.714207\pi\)
\(984\) 0 0
\(985\) −3521.86 −0.113925
\(986\) −10097.1 −0.326121
\(987\) 0 0
\(988\) 0 0
\(989\) 51457.3 1.65445
\(990\) 0 0
\(991\) 51728.9 1.65815 0.829073 0.559140i \(-0.188869\pi\)
0.829073 + 0.559140i \(0.188869\pi\)
\(992\) 21441.4 0.686255
\(993\) 0 0
\(994\) −3666.79 −0.117006
\(995\) 45331.6 1.44433
\(996\) 0 0
\(997\) −26846.8 −0.852805 −0.426403 0.904533i \(-0.640219\pi\)
−0.426403 + 0.904533i \(0.640219\pi\)
\(998\) 2476.65 0.0785541
\(999\) 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.x.1.3 4
3.2 odd 2 507.4.a.j.1.2 4
13.5 odd 4 117.4.b.d.64.2 4
13.8 odd 4 117.4.b.d.64.3 4
13.12 even 2 inner 1521.4.a.x.1.2 4
39.5 even 4 39.4.b.a.25.3 yes 4
39.8 even 4 39.4.b.a.25.2 4
39.38 odd 2 507.4.a.j.1.3 4
156.47 odd 4 624.4.c.e.337.1 4
156.83 odd 4 624.4.c.e.337.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.b.a.25.2 4 39.8 even 4
39.4.b.a.25.3 yes 4 39.5 even 4
117.4.b.d.64.2 4 13.5 odd 4
117.4.b.d.64.3 4 13.8 odd 4
507.4.a.j.1.2 4 3.2 odd 2
507.4.a.j.1.3 4 39.38 odd 2
624.4.c.e.337.1 4 156.47 odd 4
624.4.c.e.337.4 4 156.83 odd 4
1521.4.a.x.1.2 4 13.12 even 2 inner
1521.4.a.x.1.3 4 1.1 even 1 trivial