Properties

Label 1521.4.a.w.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: 4.4.1362828.1
Defining polynomial: \( x^{4} - 23x^{2} + 48 \) 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 \(1.52356\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.52356 q^{2} -5.67878 q^{4} -9.65841 q^{5} +22.3639 q^{7} -20.8404 q^{8} +O(q^{10})\) \(q+1.52356 q^{2} -5.67878 q^{4} -9.65841 q^{5} +22.3639 q^{7} -20.8404 q^{8} -14.7151 q^{10} +50.3050 q^{11} +34.0727 q^{14} +13.6788 q^{16} -86.1454 q^{17} +116.880 q^{19} +54.8480 q^{20} +76.6424 q^{22} -72.0000 q^{23} -31.7151 q^{25} -127.000 q^{28} -14.1454 q^{29} -196.215 q^{31} +187.563 q^{32} -131.247 q^{34} -216.000 q^{35} -154.424 q^{37} +178.073 q^{38} +201.285 q^{40} +265.726 q^{41} +211.855 q^{43} -285.671 q^{44} -109.696 q^{46} -67.5535 q^{47} +157.145 q^{49} -48.3197 q^{50} -686.581 q^{53} -485.866 q^{55} -466.073 q^{56} -21.5512 q^{58} +91.9304 q^{59} +329.006 q^{61} -298.945 q^{62} +176.333 q^{64} +768.370 q^{67} +489.201 q^{68} -329.088 q^{70} -264.969 q^{71} +771.306 q^{73} -235.273 q^{74} -663.734 q^{76} +1125.02 q^{77} +1226.86 q^{79} -132.115 q^{80} +404.849 q^{82} +514.019 q^{83} +832.027 q^{85} +322.772 q^{86} -1048.38 q^{88} -527.889 q^{89} +408.872 q^{92} -102.921 q^{94} -1128.87 q^{95} -74.2755 q^{97} +239.420 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 14 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 14 q^{4} + 88 q^{10} - 84 q^{14} + 18 q^{16} + 96 q^{17} + 380 q^{22} - 288 q^{23} + 20 q^{25} + 384 q^{29} - 864 q^{35} + 492 q^{38} + 952 q^{40} + 1288 q^{43} + 188 q^{49} - 984 q^{53} - 328 q^{55} - 1644 q^{56} + 288 q^{61} + 1668 q^{62} - 1314 q^{64} + 4380 q^{68} - 3144 q^{74} + 1416 q^{77} + 4320 q^{79} + 3088 q^{82} - 1036 q^{88} - 1008 q^{92} - 1660 q^{94} - 1872 q^{95}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.52356 0.538658 0.269329 0.963048i \(-0.413198\pi\)
0.269329 + 0.963048i \(0.413198\pi\)
\(3\) 0 0
\(4\) −5.67878 −0.709847
\(5\) −9.65841 −0.863874 −0.431937 0.901904i \(-0.642170\pi\)
−0.431937 + 0.901904i \(0.642170\pi\)
\(6\) 0 0
\(7\) 22.3639 1.20754 0.603769 0.797159i \(-0.293665\pi\)
0.603769 + 0.797159i \(0.293665\pi\)
\(8\) −20.8404 −0.921023
\(9\) 0 0
\(10\) −14.7151 −0.465333
\(11\) 50.3050 1.37887 0.689433 0.724349i \(-0.257860\pi\)
0.689433 + 0.724349i \(0.257860\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 34.0727 0.650450
\(15\) 0 0
\(16\) 13.6788 0.213731
\(17\) −86.1454 −1.22902 −0.614509 0.788910i \(-0.710646\pi\)
−0.614509 + 0.788910i \(0.710646\pi\)
\(18\) 0 0
\(19\) 116.880 1.41127 0.705633 0.708578i \(-0.250663\pi\)
0.705633 + 0.708578i \(0.250663\pi\)
\(20\) 54.8480 0.613219
\(21\) 0 0
\(22\) 76.6424 0.742737
\(23\) −72.0000 −0.652741 −0.326370 0.945242i \(-0.605826\pi\)
−0.326370 + 0.945242i \(0.605826\pi\)
\(24\) 0 0
\(25\) −31.7151 −0.253721
\(26\) 0 0
\(27\) 0 0
\(28\) −127.000 −0.857168
\(29\) −14.1454 −0.0905768 −0.0452884 0.998974i \(-0.514421\pi\)
−0.0452884 + 0.998974i \(0.514421\pi\)
\(30\) 0 0
\(31\) −196.215 −1.13682 −0.568408 0.822747i \(-0.692441\pi\)
−0.568408 + 0.822747i \(0.692441\pi\)
\(32\) 187.563 1.03615
\(33\) 0 0
\(34\) −131.247 −0.662021
\(35\) −216.000 −1.04316
\(36\) 0 0
\(37\) −154.424 −0.686138 −0.343069 0.939310i \(-0.611467\pi\)
−0.343069 + 0.939310i \(0.611467\pi\)
\(38\) 178.073 0.760190
\(39\) 0 0
\(40\) 201.285 0.795648
\(41\) 265.726 1.01218 0.506091 0.862480i \(-0.331090\pi\)
0.506091 + 0.862480i \(0.331090\pi\)
\(42\) 0 0
\(43\) 211.855 0.751338 0.375669 0.926754i \(-0.377413\pi\)
0.375669 + 0.926754i \(0.377413\pi\)
\(44\) −285.671 −0.978785
\(45\) 0 0
\(46\) −109.696 −0.351604
\(47\) −67.5535 −0.209653 −0.104827 0.994491i \(-0.533429\pi\)
−0.104827 + 0.994491i \(0.533429\pi\)
\(48\) 0 0
\(49\) 157.145 0.458150
\(50\) −48.3197 −0.136669
\(51\) 0 0
\(52\) 0 0
\(53\) −686.581 −1.77942 −0.889710 0.456527i \(-0.849093\pi\)
−0.889710 + 0.456527i \(0.849093\pi\)
\(54\) 0 0
\(55\) −485.866 −1.19117
\(56\) −466.073 −1.11217
\(57\) 0 0
\(58\) −21.5512 −0.0487899
\(59\) 91.9304 0.202853 0.101426 0.994843i \(-0.467659\pi\)
0.101426 + 0.994843i \(0.467659\pi\)
\(60\) 0 0
\(61\) 329.006 0.690572 0.345286 0.938498i \(-0.387782\pi\)
0.345286 + 0.938498i \(0.387782\pi\)
\(62\) −298.945 −0.612355
\(63\) 0 0
\(64\) 176.333 0.344400
\(65\) 0 0
\(66\) 0 0
\(67\) 768.370 1.40106 0.700532 0.713621i \(-0.252946\pi\)
0.700532 + 0.713621i \(0.252946\pi\)
\(68\) 489.201 0.872416
\(69\) 0 0
\(70\) −329.088 −0.561908
\(71\) −264.969 −0.442902 −0.221451 0.975171i \(-0.571079\pi\)
−0.221451 + 0.975171i \(0.571079\pi\)
\(72\) 0 0
\(73\) 771.306 1.23664 0.618319 0.785927i \(-0.287814\pi\)
0.618319 + 0.785927i \(0.287814\pi\)
\(74\) −235.273 −0.369594
\(75\) 0 0
\(76\) −663.734 −1.00178
\(77\) 1125.02 1.66503
\(78\) 0 0
\(79\) 1226.86 1.74725 0.873624 0.486602i \(-0.161764\pi\)
0.873624 + 0.486602i \(0.161764\pi\)
\(80\) −132.115 −0.184637
\(81\) 0 0
\(82\) 404.849 0.545220
\(83\) 514.019 0.679771 0.339885 0.940467i \(-0.389612\pi\)
0.339885 + 0.940467i \(0.389612\pi\)
\(84\) 0 0
\(85\) 832.027 1.06172
\(86\) 322.772 0.404714
\(87\) 0 0
\(88\) −1048.38 −1.26997
\(89\) −527.889 −0.628720 −0.314360 0.949304i \(-0.601790\pi\)
−0.314360 + 0.949304i \(0.601790\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 408.872 0.463346
\(93\) 0 0
\(94\) −102.921 −0.112931
\(95\) −1128.87 −1.21916
\(96\) 0 0
\(97\) −74.2755 −0.0777478 −0.0388739 0.999244i \(-0.512377\pi\)
−0.0388739 + 0.999244i \(0.512377\pi\)
\(98\) 239.420 0.246786
\(99\) 0 0
\(100\) 180.103 0.180103
\(101\) −609.419 −0.600390 −0.300195 0.953878i \(-0.597052\pi\)
−0.300195 + 0.953878i \(0.597052\pi\)
\(102\) 0 0
\(103\) −32.4361 −0.0310293 −0.0155147 0.999880i \(-0.504939\pi\)
−0.0155147 + 0.999880i \(0.504939\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −1046.04 −0.958498
\(107\) 1725.45 1.55893 0.779467 0.626444i \(-0.215490\pi\)
0.779467 + 0.626444i \(0.215490\pi\)
\(108\) 0 0
\(109\) 1273.43 1.11902 0.559508 0.828825i \(-0.310990\pi\)
0.559508 + 0.828825i \(0.310990\pi\)
\(110\) −740.244 −0.641632
\(111\) 0 0
\(112\) 305.911 0.258088
\(113\) −1114.73 −0.928006 −0.464003 0.885834i \(-0.653587\pi\)
−0.464003 + 0.885834i \(0.653587\pi\)
\(114\) 0 0
\(115\) 695.406 0.563886
\(116\) 80.3284 0.0642957
\(117\) 0 0
\(118\) 140.061 0.109268
\(119\) −1926.55 −1.48409
\(120\) 0 0
\(121\) 1199.59 0.901272
\(122\) 501.259 0.371982
\(123\) 0 0
\(124\) 1114.26 0.806966
\(125\) 1513.62 1.08306
\(126\) 0 0
\(127\) 1174.01 0.820289 0.410144 0.912021i \(-0.365478\pi\)
0.410144 + 0.912021i \(0.365478\pi\)
\(128\) −1231.85 −0.850637
\(129\) 0 0
\(130\) 0 0
\(131\) 1445.16 0.963851 0.481925 0.876212i \(-0.339938\pi\)
0.481925 + 0.876212i \(0.339938\pi\)
\(132\) 0 0
\(133\) 2613.89 1.70416
\(134\) 1170.65 0.754695
\(135\) 0 0
\(136\) 1795.30 1.13195
\(137\) −508.793 −0.317293 −0.158646 0.987335i \(-0.550713\pi\)
−0.158646 + 0.987335i \(0.550713\pi\)
\(138\) 0 0
\(139\) −757.018 −0.461938 −0.230969 0.972961i \(-0.574190\pi\)
−0.230969 + 0.972961i \(0.574190\pi\)
\(140\) 1226.62 0.740486
\(141\) 0 0
\(142\) −403.695 −0.238573
\(143\) 0 0
\(144\) 0 0
\(145\) 136.622 0.0782470
\(146\) 1175.13 0.666125
\(147\) 0 0
\(148\) 876.939 0.487054
\(149\) 3247.79 1.78570 0.892851 0.450352i \(-0.148702\pi\)
0.892851 + 0.450352i \(0.148702\pi\)
\(150\) 0 0
\(151\) 795.296 0.428611 0.214305 0.976767i \(-0.431251\pi\)
0.214305 + 0.976767i \(0.431251\pi\)
\(152\) −2435.82 −1.29981
\(153\) 0 0
\(154\) 1714.03 0.896884
\(155\) 1895.13 0.982067
\(156\) 0 0
\(157\) −65.2732 −0.0331807 −0.0165903 0.999862i \(-0.505281\pi\)
−0.0165903 + 0.999862i \(0.505281\pi\)
\(158\) 1869.19 0.941169
\(159\) 0 0
\(160\) −1811.56 −0.895104
\(161\) −1610.20 −0.788210
\(162\) 0 0
\(163\) −1855.39 −0.891568 −0.445784 0.895140i \(-0.647075\pi\)
−0.445784 + 0.895140i \(0.647075\pi\)
\(164\) −1509.00 −0.718495
\(165\) 0 0
\(166\) 783.137 0.366164
\(167\) 3532.54 1.63686 0.818432 0.574604i \(-0.194844\pi\)
0.818432 + 0.574604i \(0.194844\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 1267.64 0.571903
\(171\) 0 0
\(172\) −1203.08 −0.533335
\(173\) 3178.36 1.39680 0.698400 0.715708i \(-0.253896\pi\)
0.698400 + 0.715708i \(0.253896\pi\)
\(174\) 0 0
\(175\) −709.275 −0.306378
\(176\) 688.111 0.294706
\(177\) 0 0
\(178\) −804.267 −0.338665
\(179\) −2741.16 −1.14460 −0.572302 0.820043i \(-0.693950\pi\)
−0.572302 + 0.820043i \(0.693950\pi\)
\(180\) 0 0
\(181\) 3871.09 1.58970 0.794850 0.606806i \(-0.207550\pi\)
0.794850 + 0.606806i \(0.207550\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 1500.51 0.601189
\(185\) 1491.49 0.592737
\(186\) 0 0
\(187\) −4333.54 −1.69465
\(188\) 383.621 0.148822
\(189\) 0 0
\(190\) −1719.90 −0.656708
\(191\) 928.291 0.351669 0.175834 0.984420i \(-0.443738\pi\)
0.175834 + 0.984420i \(0.443738\pi\)
\(192\) 0 0
\(193\) −2261.51 −0.843456 −0.421728 0.906722i \(-0.638576\pi\)
−0.421728 + 0.906722i \(0.638576\pi\)
\(194\) −113.163 −0.0418795
\(195\) 0 0
\(196\) −892.394 −0.325216
\(197\) 2265.59 0.819374 0.409687 0.912226i \(-0.365638\pi\)
0.409687 + 0.912226i \(0.365638\pi\)
\(198\) 0 0
\(199\) −260.895 −0.0929366 −0.0464683 0.998920i \(-0.514797\pi\)
−0.0464683 + 0.998920i \(0.514797\pi\)
\(200\) 660.955 0.233683
\(201\) 0 0
\(202\) −928.483 −0.323405
\(203\) −316.346 −0.109375
\(204\) 0 0
\(205\) −2566.49 −0.874399
\(206\) −49.4181 −0.0167142
\(207\) 0 0
\(208\) 0 0
\(209\) 5879.63 1.94595
\(210\) 0 0
\(211\) 5851.22 1.90907 0.954537 0.298092i \(-0.0963502\pi\)
0.954537 + 0.298092i \(0.0963502\pi\)
\(212\) 3898.94 1.26312
\(213\) 0 0
\(214\) 2628.82 0.839732
\(215\) −2046.18 −0.649062
\(216\) 0 0
\(217\) −4388.15 −1.37275
\(218\) 1940.15 0.602767
\(219\) 0 0
\(220\) 2759.13 0.845547
\(221\) 0 0
\(222\) 0 0
\(223\) −3463.60 −1.04009 −0.520045 0.854139i \(-0.674085\pi\)
−0.520045 + 0.854139i \(0.674085\pi\)
\(224\) 4194.65 1.25119
\(225\) 0 0
\(226\) −1698.35 −0.499878
\(227\) −5329.15 −1.55819 −0.779093 0.626908i \(-0.784320\pi\)
−0.779093 + 0.626908i \(0.784320\pi\)
\(228\) 0 0
\(229\) 4773.95 1.37761 0.688803 0.724949i \(-0.258137\pi\)
0.688803 + 0.724949i \(0.258137\pi\)
\(230\) 1059.49 0.303742
\(231\) 0 0
\(232\) 294.795 0.0834233
\(233\) 4813.78 1.35348 0.676741 0.736221i \(-0.263392\pi\)
0.676741 + 0.736221i \(0.263392\pi\)
\(234\) 0 0
\(235\) 652.459 0.181114
\(236\) −522.052 −0.143995
\(237\) 0 0
\(238\) −2935.20 −0.799416
\(239\) −1683.19 −0.455549 −0.227775 0.973714i \(-0.573145\pi\)
−0.227775 + 0.973714i \(0.573145\pi\)
\(240\) 0 0
\(241\) −664.861 −0.177707 −0.0888537 0.996045i \(-0.528320\pi\)
−0.0888537 + 0.996045i \(0.528320\pi\)
\(242\) 1827.65 0.485477
\(243\) 0 0
\(244\) −1868.35 −0.490201
\(245\) −1517.77 −0.395784
\(246\) 0 0
\(247\) 0 0
\(248\) 4089.20 1.04703
\(249\) 0 0
\(250\) 2306.08 0.583398
\(251\) −2142.04 −0.538662 −0.269331 0.963048i \(-0.586802\pi\)
−0.269331 + 0.963048i \(0.586802\pi\)
\(252\) 0 0
\(253\) −3621.96 −0.900042
\(254\) 1788.67 0.441855
\(255\) 0 0
\(256\) −3287.46 −0.802603
\(257\) −3152.62 −0.765194 −0.382597 0.923915i \(-0.624970\pi\)
−0.382597 + 0.923915i \(0.624970\pi\)
\(258\) 0 0
\(259\) −3453.52 −0.828539
\(260\) 0 0
\(261\) 0 0
\(262\) 2201.79 0.519186
\(263\) 2167.71 0.508238 0.254119 0.967173i \(-0.418214\pi\)
0.254119 + 0.967173i \(0.418214\pi\)
\(264\) 0 0
\(265\) 6631.28 1.53719
\(266\) 3982.40 0.917958
\(267\) 0 0
\(268\) −4363.40 −0.994542
\(269\) 3248.98 0.736409 0.368204 0.929745i \(-0.379973\pi\)
0.368204 + 0.929745i \(0.379973\pi\)
\(270\) 0 0
\(271\) −4897.70 −1.09784 −0.548919 0.835876i \(-0.684960\pi\)
−0.548919 + 0.835876i \(0.684960\pi\)
\(272\) −1178.36 −0.262679
\(273\) 0 0
\(274\) −775.175 −0.170912
\(275\) −1595.43 −0.349847
\(276\) 0 0
\(277\) 2900.33 0.629111 0.314555 0.949239i \(-0.398145\pi\)
0.314555 + 0.949239i \(0.398145\pi\)
\(278\) −1153.36 −0.248827
\(279\) 0 0
\(280\) 4501.52 0.960776
\(281\) −4396.53 −0.933364 −0.466682 0.884425i \(-0.654551\pi\)
−0.466682 + 0.884425i \(0.654551\pi\)
\(282\) 0 0
\(283\) −559.151 −0.117449 −0.0587245 0.998274i \(-0.518703\pi\)
−0.0587245 + 0.998274i \(0.518703\pi\)
\(284\) 1504.70 0.314393
\(285\) 0 0
\(286\) 0 0
\(287\) 5942.69 1.22225
\(288\) 0 0
\(289\) 2508.02 0.510487
\(290\) 208.151 0.0421484
\(291\) 0 0
\(292\) −4380.08 −0.877825
\(293\) 6918.65 1.37950 0.689748 0.724050i \(-0.257722\pi\)
0.689748 + 0.724050i \(0.257722\pi\)
\(294\) 0 0
\(295\) −887.901 −0.175239
\(296\) 3218.25 0.631949
\(297\) 0 0
\(298\) 4948.19 0.961883
\(299\) 0 0
\(300\) 0 0
\(301\) 4737.90 0.907270
\(302\) 1211.68 0.230875
\(303\) 0 0
\(304\) 1598.77 0.301631
\(305\) −3177.67 −0.596567
\(306\) 0 0
\(307\) −8980.94 −1.66961 −0.834803 0.550548i \(-0.814419\pi\)
−0.834803 + 0.550548i \(0.814419\pi\)
\(308\) −6388.73 −1.18192
\(309\) 0 0
\(310\) 2887.33 0.528998
\(311\) 7943.13 1.44827 0.724137 0.689656i \(-0.242238\pi\)
0.724137 + 0.689656i \(0.242238\pi\)
\(312\) 0 0
\(313\) −5059.57 −0.913686 −0.456843 0.889547i \(-0.651020\pi\)
−0.456843 + 0.889547i \(0.651020\pi\)
\(314\) −99.4473 −0.0178730
\(315\) 0 0
\(316\) −6967.07 −1.24028
\(317\) 8702.12 1.54183 0.770914 0.636939i \(-0.219800\pi\)
0.770914 + 0.636939i \(0.219800\pi\)
\(318\) 0 0
\(319\) −711.582 −0.124893
\(320\) −1703.10 −0.297519
\(321\) 0 0
\(322\) −2453.23 −0.424576
\(323\) −10068.6 −1.73447
\(324\) 0 0
\(325\) 0 0
\(326\) −2826.79 −0.480250
\(327\) 0 0
\(328\) −5537.84 −0.932244
\(329\) −1510.76 −0.253164
\(330\) 0 0
\(331\) −1737.65 −0.288549 −0.144275 0.989538i \(-0.546085\pi\)
−0.144275 + 0.989538i \(0.546085\pi\)
\(332\) −2919.00 −0.482533
\(333\) 0 0
\(334\) 5382.02 0.881710
\(335\) −7421.23 −1.21034
\(336\) 0 0
\(337\) 2917.47 0.471586 0.235793 0.971803i \(-0.424231\pi\)
0.235793 + 0.971803i \(0.424231\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −4724.90 −0.753658
\(341\) −9870.61 −1.56752
\(342\) 0 0
\(343\) −4156.44 −0.654305
\(344\) −4415.13 −0.692000
\(345\) 0 0
\(346\) 4842.41 0.752397
\(347\) −5081.23 −0.786095 −0.393047 0.919518i \(-0.628579\pi\)
−0.393047 + 0.919518i \(0.628579\pi\)
\(348\) 0 0
\(349\) 4266.14 0.654330 0.327165 0.944967i \(-0.393907\pi\)
0.327165 + 0.944967i \(0.393907\pi\)
\(350\) −1080.62 −0.165033
\(351\) 0 0
\(352\) 9435.38 1.42871
\(353\) −3264.49 −0.492213 −0.246106 0.969243i \(-0.579151\pi\)
−0.246106 + 0.969243i \(0.579151\pi\)
\(354\) 0 0
\(355\) 2559.18 0.382612
\(356\) 2997.76 0.446295
\(357\) 0 0
\(358\) −4176.31 −0.616550
\(359\) 4416.98 0.649357 0.324679 0.945824i \(-0.394744\pi\)
0.324679 + 0.945824i \(0.394744\pi\)
\(360\) 0 0
\(361\) 6801.87 0.991670
\(362\) 5897.82 0.856305
\(363\) 0 0
\(364\) 0 0
\(365\) −7449.59 −1.06830
\(366\) 0 0
\(367\) −1740.22 −0.247516 −0.123758 0.992312i \(-0.539495\pi\)
−0.123758 + 0.992312i \(0.539495\pi\)
\(368\) −984.872 −0.139511
\(369\) 0 0
\(370\) 2272.37 0.319283
\(371\) −15354.7 −2.14872
\(372\) 0 0
\(373\) 1176.28 0.163285 0.0816427 0.996662i \(-0.473983\pi\)
0.0816427 + 0.996662i \(0.473983\pi\)
\(374\) −6602.39 −0.912838
\(375\) 0 0
\(376\) 1407.84 0.193095
\(377\) 0 0
\(378\) 0 0
\(379\) −7135.54 −0.967092 −0.483546 0.875319i \(-0.660651\pi\)
−0.483546 + 0.875319i \(0.660651\pi\)
\(380\) 6410.62 0.865415
\(381\) 0 0
\(382\) 1414.30 0.189429
\(383\) −1942.87 −0.259207 −0.129603 0.991566i \(-0.541370\pi\)
−0.129603 + 0.991566i \(0.541370\pi\)
\(384\) 0 0
\(385\) −10865.9 −1.43838
\(386\) −3445.53 −0.454334
\(387\) 0 0
\(388\) 421.794 0.0551891
\(389\) 7545.92 0.983531 0.491766 0.870728i \(-0.336352\pi\)
0.491766 + 0.870728i \(0.336352\pi\)
\(390\) 0 0
\(391\) 6202.47 0.802231
\(392\) −3274.97 −0.421967
\(393\) 0 0
\(394\) 3451.75 0.441362
\(395\) −11849.5 −1.50940
\(396\) 0 0
\(397\) −415.922 −0.0525806 −0.0262903 0.999654i \(-0.508369\pi\)
−0.0262903 + 0.999654i \(0.508369\pi\)
\(398\) −397.489 −0.0500611
\(399\) 0 0
\(400\) −433.824 −0.0542280
\(401\) −958.178 −0.119324 −0.0596622 0.998219i \(-0.519002\pi\)
−0.0596622 + 0.998219i \(0.519002\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 3460.75 0.426185
\(405\) 0 0
\(406\) −481.970 −0.0589157
\(407\) −7768.29 −0.946093
\(408\) 0 0
\(409\) −4284.83 −0.518022 −0.259011 0.965874i \(-0.583397\pi\)
−0.259011 + 0.965874i \(0.583397\pi\)
\(410\) −3910.20 −0.471002
\(411\) 0 0
\(412\) 184.197 0.0220261
\(413\) 2055.92 0.244953
\(414\) 0 0
\(415\) −4964.61 −0.587236
\(416\) 0 0
\(417\) 0 0
\(418\) 8957.95 1.04820
\(419\) 9949.01 1.16000 0.580001 0.814616i \(-0.303052\pi\)
0.580001 + 0.814616i \(0.303052\pi\)
\(420\) 0 0
\(421\) 377.250 0.0436724 0.0218362 0.999762i \(-0.493049\pi\)
0.0218362 + 0.999762i \(0.493049\pi\)
\(422\) 8914.66 1.02834
\(423\) 0 0
\(424\) 14308.6 1.63889
\(425\) 2732.11 0.311828
\(426\) 0 0
\(427\) 7357.86 0.833892
\(428\) −9798.47 −1.10661
\(429\) 0 0
\(430\) −3117.47 −0.349622
\(431\) 2437.13 0.272373 0.136186 0.990683i \(-0.456515\pi\)
0.136186 + 0.990683i \(0.456515\pi\)
\(432\) 0 0
\(433\) 11215.1 1.24471 0.622357 0.782733i \(-0.286175\pi\)
0.622357 + 0.782733i \(0.286175\pi\)
\(434\) −6685.58 −0.739443
\(435\) 0 0
\(436\) −7231.55 −0.794331
\(437\) −8415.34 −0.921191
\(438\) 0 0
\(439\) −1835.19 −0.199519 −0.0997596 0.995012i \(-0.531807\pi\)
−0.0997596 + 0.995012i \(0.531807\pi\)
\(440\) 10125.6 1.09709
\(441\) 0 0
\(442\) 0 0
\(443\) −11610.1 −1.24518 −0.622588 0.782550i \(-0.713919\pi\)
−0.622588 + 0.782550i \(0.713919\pi\)
\(444\) 0 0
\(445\) 5098.56 0.543135
\(446\) −5276.99 −0.560253
\(447\) 0 0
\(448\) 3943.50 0.415877
\(449\) 14087.0 1.48064 0.740319 0.672255i \(-0.234674\pi\)
0.740319 + 0.672255i \(0.234674\pi\)
\(450\) 0 0
\(451\) 13367.4 1.39566
\(452\) 6330.29 0.658743
\(453\) 0 0
\(454\) −8119.26 −0.839330
\(455\) 0 0
\(456\) 0 0
\(457\) 2375.01 0.243103 0.121552 0.992585i \(-0.461213\pi\)
0.121552 + 0.992585i \(0.461213\pi\)
\(458\) 7273.38 0.742058
\(459\) 0 0
\(460\) −3949.05 −0.400273
\(461\) 6372.06 0.643766 0.321883 0.946779i \(-0.395684\pi\)
0.321883 + 0.946779i \(0.395684\pi\)
\(462\) 0 0
\(463\) −63.4732 −0.00637117 −0.00318558 0.999995i \(-0.501014\pi\)
−0.00318558 + 0.999995i \(0.501014\pi\)
\(464\) −193.491 −0.0193591
\(465\) 0 0
\(466\) 7334.06 0.729064
\(467\) 7855.78 0.778420 0.389210 0.921149i \(-0.372748\pi\)
0.389210 + 0.921149i \(0.372748\pi\)
\(468\) 0 0
\(469\) 17183.8 1.69184
\(470\) 994.058 0.0975584
\(471\) 0 0
\(472\) −1915.86 −0.186832
\(473\) 10657.3 1.03599
\(474\) 0 0
\(475\) −3706.85 −0.358068
\(476\) 10940.4 1.05348
\(477\) 0 0
\(478\) −2564.43 −0.245385
\(479\) 13033.3 1.24323 0.621613 0.783324i \(-0.286477\pi\)
0.621613 + 0.783324i \(0.286477\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −1012.95 −0.0957235
\(483\) 0 0
\(484\) −6812.23 −0.639766
\(485\) 717.384 0.0671643
\(486\) 0 0
\(487\) −69.8976 −0.00650383 −0.00325191 0.999995i \(-0.501035\pi\)
−0.00325191 + 0.999995i \(0.501035\pi\)
\(488\) −6856.60 −0.636033
\(489\) 0 0
\(490\) −2312.41 −0.213192
\(491\) 2625.66 0.241333 0.120667 0.992693i \(-0.461497\pi\)
0.120667 + 0.992693i \(0.461497\pi\)
\(492\) 0 0
\(493\) 1218.56 0.111321
\(494\) 0 0
\(495\) 0 0
\(496\) −2683.99 −0.242973
\(497\) −5925.75 −0.534821
\(498\) 0 0
\(499\) 7631.34 0.684621 0.342310 0.939587i \(-0.388791\pi\)
0.342310 + 0.939587i \(0.388791\pi\)
\(500\) −8595.51 −0.768806
\(501\) 0 0
\(502\) −3263.51 −0.290154
\(503\) 4320.14 0.382953 0.191477 0.981497i \(-0.438672\pi\)
0.191477 + 0.981497i \(0.438672\pi\)
\(504\) 0 0
\(505\) 5886.01 0.518662
\(506\) −5518.26 −0.484815
\(507\) 0 0
\(508\) −6666.95 −0.582280
\(509\) 12450.7 1.08422 0.542109 0.840308i \(-0.317626\pi\)
0.542109 + 0.840308i \(0.317626\pi\)
\(510\) 0 0
\(511\) 17249.4 1.49329
\(512\) 4846.21 0.418309
\(513\) 0 0
\(514\) −4803.18 −0.412178
\(515\) 313.281 0.0268054
\(516\) 0 0
\(517\) −3398.28 −0.289083
\(518\) −5261.63 −0.446299
\(519\) 0 0
\(520\) 0 0
\(521\) −14373.1 −1.20863 −0.604314 0.796746i \(-0.706553\pi\)
−0.604314 + 0.796746i \(0.706553\pi\)
\(522\) 0 0
\(523\) −16946.7 −1.41688 −0.708439 0.705772i \(-0.750600\pi\)
−0.708439 + 0.705772i \(0.750600\pi\)
\(524\) −8206.76 −0.684187
\(525\) 0 0
\(526\) 3302.62 0.273767
\(527\) 16903.0 1.39717
\(528\) 0 0
\(529\) −6983.00 −0.573929
\(530\) 10103.1 0.828022
\(531\) 0 0
\(532\) −14843.7 −1.20969
\(533\) 0 0
\(534\) 0 0
\(535\) −16665.1 −1.34672
\(536\) −16013.1 −1.29041
\(537\) 0 0
\(538\) 4950.00 0.396673
\(539\) 7905.20 0.631727
\(540\) 0 0
\(541\) 815.667 0.0648212 0.0324106 0.999475i \(-0.489682\pi\)
0.0324106 + 0.999475i \(0.489682\pi\)
\(542\) −7461.92 −0.591359
\(543\) 0 0
\(544\) −16157.7 −1.27345
\(545\) −12299.3 −0.966689
\(546\) 0 0
\(547\) −17971.4 −1.40476 −0.702378 0.711804i \(-0.747878\pi\)
−0.702378 + 0.711804i \(0.747878\pi\)
\(548\) 2889.32 0.225230
\(549\) 0 0
\(550\) −2430.72 −0.188448
\(551\) −1653.31 −0.127828
\(552\) 0 0
\(553\) 27437.4 2.10987
\(554\) 4418.81 0.338876
\(555\) 0 0
\(556\) 4298.94 0.327906
\(557\) −6760.83 −0.514301 −0.257151 0.966371i \(-0.582784\pi\)
−0.257151 + 0.966371i \(0.582784\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −2954.62 −0.222956
\(561\) 0 0
\(562\) −6698.36 −0.502764
\(563\) 12962.7 0.970359 0.485179 0.874415i \(-0.338754\pi\)
0.485179 + 0.874415i \(0.338754\pi\)
\(564\) 0 0
\(565\) 10766.5 0.801681
\(566\) −851.898 −0.0632649
\(567\) 0 0
\(568\) 5522.05 0.407923
\(569\) 164.757 0.0121388 0.00606938 0.999982i \(-0.498068\pi\)
0.00606938 + 0.999982i \(0.498068\pi\)
\(570\) 0 0
\(571\) 5216.55 0.382322 0.191161 0.981559i \(-0.438775\pi\)
0.191161 + 0.981559i \(0.438775\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 9054.01 0.658375
\(575\) 2283.49 0.165614
\(576\) 0 0
\(577\) 12753.3 0.920151 0.460076 0.887880i \(-0.347822\pi\)
0.460076 + 0.887880i \(0.347822\pi\)
\(578\) 3821.11 0.274978
\(579\) 0 0
\(580\) −775.844 −0.0555434
\(581\) 11495.5 0.820849
\(582\) 0 0
\(583\) −34538.5 −2.45358
\(584\) −16074.3 −1.13897
\(585\) 0 0
\(586\) 10540.9 0.743076
\(587\) −1575.38 −0.110771 −0.0553857 0.998465i \(-0.517639\pi\)
−0.0553857 + 0.998465i \(0.517639\pi\)
\(588\) 0 0
\(589\) −22933.6 −1.60435
\(590\) −1352.77 −0.0943941
\(591\) 0 0
\(592\) −2112.33 −0.146649
\(593\) −3845.95 −0.266331 −0.133165 0.991094i \(-0.542514\pi\)
−0.133165 + 0.991094i \(0.542514\pi\)
\(594\) 0 0
\(595\) 18607.4 1.28207
\(596\) −18443.5 −1.26758
\(597\) 0 0
\(598\) 0 0
\(599\) −6107.20 −0.416583 −0.208292 0.978067i \(-0.566790\pi\)
−0.208292 + 0.978067i \(0.566790\pi\)
\(600\) 0 0
\(601\) 9638.90 0.654208 0.327104 0.944988i \(-0.393927\pi\)
0.327104 + 0.944988i \(0.393927\pi\)
\(602\) 7218.46 0.488708
\(603\) 0 0
\(604\) −4516.31 −0.304248
\(605\) −11586.2 −0.778586
\(606\) 0 0
\(607\) 11821.7 0.790489 0.395244 0.918576i \(-0.370660\pi\)
0.395244 + 0.918576i \(0.370660\pi\)
\(608\) 21922.4 1.46228
\(609\) 0 0
\(610\) −4841.36 −0.321346
\(611\) 0 0
\(612\) 0 0
\(613\) 8107.86 0.534214 0.267107 0.963667i \(-0.413932\pi\)
0.267107 + 0.963667i \(0.413932\pi\)
\(614\) −13683.0 −0.899347
\(615\) 0 0
\(616\) −23445.8 −1.53354
\(617\) −27647.6 −1.80397 −0.901985 0.431768i \(-0.857890\pi\)
−0.901985 + 0.431768i \(0.857890\pi\)
\(618\) 0 0
\(619\) 29181.9 1.89486 0.947432 0.319956i \(-0.103668\pi\)
0.947432 + 0.319956i \(0.103668\pi\)
\(620\) −10762.0 −0.697118
\(621\) 0 0
\(622\) 12101.8 0.780125
\(623\) −11805.7 −0.759204
\(624\) 0 0
\(625\) −10654.8 −0.681905
\(626\) −7708.53 −0.492165
\(627\) 0 0
\(628\) 370.672 0.0235532
\(629\) 13302.9 0.843277
\(630\) 0 0
\(631\) −2209.34 −0.139386 −0.0696928 0.997569i \(-0.522202\pi\)
−0.0696928 + 0.997569i \(0.522202\pi\)
\(632\) −25568.2 −1.60926
\(633\) 0 0
\(634\) 13258.2 0.830518
\(635\) −11339.1 −0.708627
\(636\) 0 0
\(637\) 0 0
\(638\) −1084.13 −0.0672748
\(639\) 0 0
\(640\) 11897.8 0.734844
\(641\) 18256.0 1.12491 0.562455 0.826828i \(-0.309857\pi\)
0.562455 + 0.826828i \(0.309857\pi\)
\(642\) 0 0
\(643\) 1281.61 0.0786033 0.0393016 0.999227i \(-0.487487\pi\)
0.0393016 + 0.999227i \(0.487487\pi\)
\(644\) 9143.99 0.559509
\(645\) 0 0
\(646\) −15340.1 −0.934287
\(647\) −16393.2 −0.996107 −0.498054 0.867146i \(-0.665952\pi\)
−0.498054 + 0.867146i \(0.665952\pi\)
\(648\) 0 0
\(649\) 4624.56 0.279707
\(650\) 0 0
\(651\) 0 0
\(652\) 10536.4 0.632878
\(653\) −16759.2 −1.00434 −0.502172 0.864768i \(-0.667466\pi\)
−0.502172 + 0.864768i \(0.667466\pi\)
\(654\) 0 0
\(655\) −13958.0 −0.832646
\(656\) 3634.81 0.216335
\(657\) 0 0
\(658\) −2301.73 −0.136369
\(659\) 29659.3 1.75320 0.876601 0.481217i \(-0.159805\pi\)
0.876601 + 0.481217i \(0.159805\pi\)
\(660\) 0 0
\(661\) −10386.0 −0.611147 −0.305573 0.952169i \(-0.598848\pi\)
−0.305573 + 0.952169i \(0.598848\pi\)
\(662\) −2647.40 −0.155429
\(663\) 0 0
\(664\) −10712.4 −0.626084
\(665\) −25246.0 −1.47218
\(666\) 0 0
\(667\) 1018.47 0.0591232
\(668\) −20060.5 −1.16192
\(669\) 0 0
\(670\) −11306.7 −0.651962
\(671\) 16550.6 0.952206
\(672\) 0 0
\(673\) −19449.6 −1.11400 −0.557002 0.830511i \(-0.688049\pi\)
−0.557002 + 0.830511i \(0.688049\pi\)
\(674\) 4444.92 0.254024
\(675\) 0 0
\(676\) 0 0
\(677\) 6629.48 0.376354 0.188177 0.982135i \(-0.439742\pi\)
0.188177 + 0.982135i \(0.439742\pi\)
\(678\) 0 0
\(679\) −1661.09 −0.0938835
\(680\) −17339.8 −0.977867
\(681\) 0 0
\(682\) −15038.4 −0.844356
\(683\) 2526.40 0.141537 0.0707687 0.997493i \(-0.477455\pi\)
0.0707687 + 0.997493i \(0.477455\pi\)
\(684\) 0 0
\(685\) 4914.13 0.274101
\(686\) −6332.57 −0.352447
\(687\) 0 0
\(688\) 2897.91 0.160584
\(689\) 0 0
\(690\) 0 0
\(691\) 3808.76 0.209685 0.104842 0.994489i \(-0.466566\pi\)
0.104842 + 0.994489i \(0.466566\pi\)
\(692\) −18049.2 −0.991515
\(693\) 0 0
\(694\) −7741.54 −0.423436
\(695\) 7311.59 0.399056
\(696\) 0 0
\(697\) −22891.1 −1.24399
\(698\) 6499.69 0.352460
\(699\) 0 0
\(700\) 4027.81 0.217482
\(701\) −33617.8 −1.81131 −0.905655 0.424015i \(-0.860620\pi\)
−0.905655 + 0.424015i \(0.860620\pi\)
\(702\) 0 0
\(703\) −18049.0 −0.968323
\(704\) 8870.43 0.474882
\(705\) 0 0
\(706\) −4973.63 −0.265134
\(707\) −13629.0 −0.724994
\(708\) 0 0
\(709\) 26606.5 1.40935 0.704675 0.709530i \(-0.251093\pi\)
0.704675 + 0.709530i \(0.251093\pi\)
\(710\) 3899.05 0.206097
\(711\) 0 0
\(712\) 11001.4 0.579066
\(713\) 14127.5 0.742046
\(714\) 0 0
\(715\) 0 0
\(716\) 15566.5 0.812494
\(717\) 0 0
\(718\) 6729.51 0.349781
\(719\) 16539.3 0.857877 0.428939 0.903334i \(-0.358888\pi\)
0.428939 + 0.903334i \(0.358888\pi\)
\(720\) 0 0
\(721\) −725.398 −0.0374691
\(722\) 10363.0 0.534171
\(723\) 0 0
\(724\) −21983.1 −1.12844
\(725\) 448.622 0.0229812
\(726\) 0 0
\(727\) −12757.5 −0.650823 −0.325411 0.945573i \(-0.605503\pi\)
−0.325411 + 0.945573i \(0.605503\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −11349.9 −0.575448
\(731\) −18250.3 −0.923408
\(732\) 0 0
\(733\) −20523.4 −1.03417 −0.517087 0.855933i \(-0.672984\pi\)
−0.517087 + 0.855933i \(0.672984\pi\)
\(734\) −2651.31 −0.133327
\(735\) 0 0
\(736\) −13504.6 −0.676338
\(737\) 38652.9 1.93188
\(738\) 0 0
\(739\) 7462.66 0.371473 0.185736 0.982600i \(-0.440533\pi\)
0.185736 + 0.982600i \(0.440533\pi\)
\(740\) −8469.84 −0.420753
\(741\) 0 0
\(742\) −23393.7 −1.15742
\(743\) 18847.8 0.930632 0.465316 0.885145i \(-0.345941\pi\)
0.465316 + 0.885145i \(0.345941\pi\)
\(744\) 0 0
\(745\) −31368.5 −1.54262
\(746\) 1792.13 0.0879549
\(747\) 0 0
\(748\) 24609.2 1.20294
\(749\) 38587.9 1.88247
\(750\) 0 0
\(751\) −20832.6 −1.01224 −0.506119 0.862464i \(-0.668920\pi\)
−0.506119 + 0.862464i \(0.668920\pi\)
\(752\) −924.050 −0.0448093
\(753\) 0 0
\(754\) 0 0
\(755\) −7681.29 −0.370266
\(756\) 0 0
\(757\) −20860.9 −1.00159 −0.500795 0.865566i \(-0.666959\pi\)
−0.500795 + 0.865566i \(0.666959\pi\)
\(758\) −10871.4 −0.520932
\(759\) 0 0
\(760\) 23526.1 1.12287
\(761\) −4464.86 −0.212682 −0.106341 0.994330i \(-0.533914\pi\)
−0.106341 + 0.994330i \(0.533914\pi\)
\(762\) 0 0
\(763\) 28479.0 1.35126
\(764\) −5271.56 −0.249631
\(765\) 0 0
\(766\) −2960.08 −0.139624
\(767\) 0 0
\(768\) 0 0
\(769\) −23797.7 −1.11595 −0.557977 0.829857i \(-0.688422\pi\)
−0.557977 + 0.829857i \(0.688422\pi\)
\(770\) −16554.8 −0.774795
\(771\) 0 0
\(772\) 12842.6 0.598725
\(773\) 29418.7 1.36885 0.684423 0.729085i \(-0.260054\pi\)
0.684423 + 0.729085i \(0.260054\pi\)
\(774\) 0 0
\(775\) 6222.99 0.288434
\(776\) 1547.93 0.0716075
\(777\) 0 0
\(778\) 11496.6 0.529787
\(779\) 31058.0 1.42846
\(780\) 0 0
\(781\) −13329.3 −0.610703
\(782\) 9449.80 0.432128
\(783\) 0 0
\(784\) 2149.56 0.0979208
\(785\) 630.435 0.0286640
\(786\) 0 0
\(787\) −23896.0 −1.08234 −0.541170 0.840913i \(-0.682018\pi\)
−0.541170 + 0.840913i \(0.682018\pi\)
\(788\) −12865.8 −0.581630
\(789\) 0 0
\(790\) −18053.4 −0.813052
\(791\) −24929.7 −1.12060
\(792\) 0 0
\(793\) 0 0
\(794\) −633.680 −0.0283230
\(795\) 0 0
\(796\) 1481.57 0.0659708
\(797\) 1034.67 0.0459847 0.0229923 0.999736i \(-0.492681\pi\)
0.0229923 + 0.999736i \(0.492681\pi\)
\(798\) 0 0
\(799\) 5819.42 0.257667
\(800\) −5948.59 −0.262893
\(801\) 0 0
\(802\) −1459.84 −0.0642751
\(803\) 38800.6 1.70516
\(804\) 0 0
\(805\) 15552.0 0.680914
\(806\) 0 0
\(807\) 0 0
\(808\) 12700.5 0.552973
\(809\) −29557.9 −1.28455 −0.642275 0.766474i \(-0.722009\pi\)
−0.642275 + 0.766474i \(0.722009\pi\)
\(810\) 0 0
\(811\) 30268.9 1.31058 0.655292 0.755376i \(-0.272546\pi\)
0.655292 + 0.755376i \(0.272546\pi\)
\(812\) 1796.46 0.0776396
\(813\) 0 0
\(814\) −11835.4 −0.509621
\(815\) 17920.2 0.770203
\(816\) 0 0
\(817\) 24761.5 1.06034
\(818\) −6528.17 −0.279037
\(819\) 0 0
\(820\) 14574.6 0.620690
\(821\) −39292.7 −1.67031 −0.835156 0.550013i \(-0.814623\pi\)
−0.835156 + 0.550013i \(0.814623\pi\)
\(822\) 0 0
\(823\) −25988.8 −1.10074 −0.550372 0.834920i \(-0.685514\pi\)
−0.550372 + 0.834920i \(0.685514\pi\)
\(824\) 675.980 0.0285787
\(825\) 0 0
\(826\) 3132.31 0.131946
\(827\) −19585.1 −0.823508 −0.411754 0.911295i \(-0.635084\pi\)
−0.411754 + 0.911295i \(0.635084\pi\)
\(828\) 0 0
\(829\) 666.472 0.0279222 0.0139611 0.999903i \(-0.495556\pi\)
0.0139611 + 0.999903i \(0.495556\pi\)
\(830\) −7563.86 −0.316320
\(831\) 0 0
\(832\) 0 0
\(833\) −13537.3 −0.563075
\(834\) 0 0
\(835\) −34118.7 −1.41404
\(836\) −33389.1 −1.38133
\(837\) 0 0
\(838\) 15157.9 0.624845
\(839\) −36379.7 −1.49698 −0.748490 0.663146i \(-0.769221\pi\)
−0.748490 + 0.663146i \(0.769221\pi\)
\(840\) 0 0
\(841\) −24188.9 −0.991796
\(842\) 574.762 0.0235245
\(843\) 0 0
\(844\) −33227.8 −1.35515
\(845\) 0 0
\(846\) 0 0
\(847\) 26827.6 1.08832
\(848\) −9391.60 −0.380317
\(849\) 0 0
\(850\) 4162.52 0.167969
\(851\) 11118.5 0.447871
\(852\) 0 0
\(853\) 39951.4 1.60364 0.801822 0.597563i \(-0.203864\pi\)
0.801822 + 0.597563i \(0.203864\pi\)
\(854\) 11210.1 0.449183
\(855\) 0 0
\(856\) −35959.1 −1.43581
\(857\) −17226.4 −0.686629 −0.343314 0.939221i \(-0.611550\pi\)
−0.343314 + 0.939221i \(0.611550\pi\)
\(858\) 0 0
\(859\) −33392.3 −1.32634 −0.663172 0.748467i \(-0.730790\pi\)
−0.663172 + 0.748467i \(0.730790\pi\)
\(860\) 11619.8 0.460735
\(861\) 0 0
\(862\) 3713.11 0.146716
\(863\) 17381.9 0.685614 0.342807 0.939406i \(-0.388622\pi\)
0.342807 + 0.939406i \(0.388622\pi\)
\(864\) 0 0
\(865\) −30697.9 −1.20666
\(866\) 17086.8 0.670476
\(867\) 0 0
\(868\) 24919.3 0.974443
\(869\) 61717.2 2.40922
\(870\) 0 0
\(871\) 0 0
\(872\) −26538.8 −1.03064
\(873\) 0 0
\(874\) −12821.2 −0.496207
\(875\) 33850.5 1.30783
\(876\) 0 0
\(877\) 14335.6 0.551970 0.275985 0.961162i \(-0.410996\pi\)
0.275985 + 0.961162i \(0.410996\pi\)
\(878\) −2796.02 −0.107473
\(879\) 0 0
\(880\) −6646.06 −0.254589
\(881\) −5436.53 −0.207901 −0.103951 0.994582i \(-0.533148\pi\)
−0.103951 + 0.994582i \(0.533148\pi\)
\(882\) 0 0
\(883\) 21185.9 0.807430 0.403715 0.914885i \(-0.367719\pi\)
0.403715 + 0.914885i \(0.367719\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −17688.6 −0.670724
\(887\) 12661.4 0.479287 0.239644 0.970861i \(-0.422969\pi\)
0.239644 + 0.970861i \(0.422969\pi\)
\(888\) 0 0
\(889\) 26255.5 0.990531
\(890\) 7767.94 0.292564
\(891\) 0 0
\(892\) 19669.0 0.738305
\(893\) −7895.63 −0.295876
\(894\) 0 0
\(895\) 26475.3 0.988794
\(896\) −27549.1 −1.02718
\(897\) 0 0
\(898\) 21462.3 0.797558
\(899\) 2775.54 0.102969
\(900\) 0 0
\(901\) 59145.8 2.18694
\(902\) 20365.9 0.751786
\(903\) 0 0
\(904\) 23231.3 0.854715
\(905\) −37388.6 −1.37330
\(906\) 0 0
\(907\) 3545.27 0.129789 0.0648946 0.997892i \(-0.479329\pi\)
0.0648946 + 0.997892i \(0.479329\pi\)
\(908\) 30263.1 1.10607
\(909\) 0 0
\(910\) 0 0
\(911\) 3913.88 0.142341 0.0711706 0.997464i \(-0.477327\pi\)
0.0711706 + 0.997464i \(0.477327\pi\)
\(912\) 0 0
\(913\) 25857.7 0.937313
\(914\) 3618.46 0.130950
\(915\) 0 0
\(916\) −27110.2 −0.977890
\(917\) 32319.5 1.16389
\(918\) 0 0
\(919\) 6917.79 0.248310 0.124155 0.992263i \(-0.460378\pi\)
0.124155 + 0.992263i \(0.460378\pi\)
\(920\) −14492.5 −0.519352
\(921\) 0 0
\(922\) 9708.18 0.346770
\(923\) 0 0
\(924\) 0 0
\(925\) 4897.57 0.174088
\(926\) −96.7050 −0.00343188
\(927\) 0 0
\(928\) −2653.15 −0.0938512
\(929\) 3753.03 0.132543 0.0662717 0.997802i \(-0.478890\pi\)
0.0662717 + 0.997802i \(0.478890\pi\)
\(930\) 0 0
\(931\) 18367.1 0.646571
\(932\) −27336.4 −0.960765
\(933\) 0 0
\(934\) 11968.7 0.419302
\(935\) 41855.1 1.46397
\(936\) 0 0
\(937\) 48189.0 1.68011 0.840056 0.542499i \(-0.182522\pi\)
0.840056 + 0.542499i \(0.182522\pi\)
\(938\) 26180.4 0.911323
\(939\) 0 0
\(940\) −3705.17 −0.128563
\(941\) −26656.4 −0.923456 −0.461728 0.887022i \(-0.652770\pi\)
−0.461728 + 0.887022i \(0.652770\pi\)
\(942\) 0 0
\(943\) −19132.3 −0.660693
\(944\) 1257.50 0.0433559
\(945\) 0 0
\(946\) 16237.1 0.558047
\(947\) −31258.9 −1.07263 −0.536314 0.844018i \(-0.680184\pi\)
−0.536314 + 0.844018i \(0.680184\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −5647.60 −0.192876
\(951\) 0 0
\(952\) 40150.0 1.36688
\(953\) −10602.4 −0.360382 −0.180191 0.983632i \(-0.557672\pi\)
−0.180191 + 0.983632i \(0.557672\pi\)
\(954\) 0 0
\(955\) −8965.81 −0.303798
\(956\) 9558.44 0.323370
\(957\) 0 0
\(958\) 19856.9 0.669674
\(959\) −11378.6 −0.383144
\(960\) 0 0
\(961\) 8709.45 0.292352
\(962\) 0 0
\(963\) 0 0
\(964\) 3775.60 0.126145
\(965\) 21842.6 0.728640
\(966\) 0 0
\(967\) −4815.93 −0.160155 −0.0800774 0.996789i \(-0.525517\pi\)
−0.0800774 + 0.996789i \(0.525517\pi\)
\(968\) −25000.0 −0.830092
\(969\) 0 0
\(970\) 1092.97 0.0361786
\(971\) −56729.5 −1.87491 −0.937454 0.348109i \(-0.886824\pi\)
−0.937454 + 0.348109i \(0.886824\pi\)
\(972\) 0 0
\(973\) −16929.9 −0.557808
\(974\) −106.493 −0.00350334
\(975\) 0 0
\(976\) 4500.40 0.147597
\(977\) 55246.2 1.80909 0.904546 0.426375i \(-0.140210\pi\)
0.904546 + 0.426375i \(0.140210\pi\)
\(978\) 0 0
\(979\) −26555.4 −0.866921
\(980\) 8619.11 0.280946
\(981\) 0 0
\(982\) 4000.34 0.129996
\(983\) 22820.6 0.740451 0.370226 0.928942i \(-0.379280\pi\)
0.370226 + 0.928942i \(0.379280\pi\)
\(984\) 0 0
\(985\) −21882.0 −0.707836
\(986\) 1856.54 0.0599637
\(987\) 0 0
\(988\) 0 0
\(989\) −15253.5 −0.490429
\(990\) 0 0
\(991\) −46490.4 −1.49023 −0.745115 0.666936i \(-0.767605\pi\)
−0.745115 + 0.666936i \(0.767605\pi\)
\(992\) −36802.8 −1.17791
\(993\) 0 0
\(994\) −9028.21 −0.288086
\(995\) 2519.84 0.0802856
\(996\) 0 0
\(997\) 24943.4 0.792344 0.396172 0.918176i \(-0.370338\pi\)
0.396172 + 0.918176i \(0.370338\pi\)
\(998\) 11626.8 0.368777
\(999\) 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.w.1.3 4
3.2 odd 2 507.4.a.l.1.2 4
13.5 odd 4 117.4.b.e.64.2 4
13.8 odd 4 117.4.b.e.64.3 4
13.12 even 2 inner 1521.4.a.w.1.2 4
39.5 even 4 39.4.b.b.25.3 yes 4
39.8 even 4 39.4.b.b.25.2 4
39.38 odd 2 507.4.a.l.1.3 4
156.47 odd 4 624.4.c.c.337.3 4
156.83 odd 4 624.4.c.c.337.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.b.b.25.2 4 39.8 even 4
39.4.b.b.25.3 yes 4 39.5 even 4
117.4.b.e.64.2 4 13.5 odd 4
117.4.b.e.64.3 4 13.8 odd 4
507.4.a.l.1.2 4 3.2 odd 2
507.4.a.l.1.3 4 39.38 odd 2
624.4.c.c.337.2 4 156.83 odd 4
624.4.c.c.337.3 4 156.47 odd 4
1521.4.a.w.1.2 4 13.12 even 2 inner
1521.4.a.w.1.3 4 1.1 even 1 trivial