Properties

Label 128.4.a.b.1.1
Level $128$
Weight $4$
Character 128.1
Self dual yes
Analytic conductor $7.552$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 128.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.55224448073\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 128.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{3} +6.00000 q^{5} -20.0000 q^{7} -23.0000 q^{9} +O(q^{10})\) \(q-2.00000 q^{3} +6.00000 q^{5} -20.0000 q^{7} -23.0000 q^{9} -14.0000 q^{11} +54.0000 q^{13} -12.0000 q^{15} -66.0000 q^{17} -162.000 q^{19} +40.0000 q^{21} -172.000 q^{23} -89.0000 q^{25} +100.000 q^{27} -2.00000 q^{29} +128.000 q^{31} +28.0000 q^{33} -120.000 q^{35} +158.000 q^{37} -108.000 q^{39} +202.000 q^{41} +298.000 q^{43} -138.000 q^{45} +408.000 q^{47} +57.0000 q^{49} +132.000 q^{51} -690.000 q^{53} -84.0000 q^{55} +324.000 q^{57} +322.000 q^{59} -298.000 q^{61} +460.000 q^{63} +324.000 q^{65} -202.000 q^{67} +344.000 q^{69} +700.000 q^{71} -418.000 q^{73} +178.000 q^{75} +280.000 q^{77} -744.000 q^{79} +421.000 q^{81} +678.000 q^{83} -396.000 q^{85} +4.00000 q^{87} -82.0000 q^{89} -1080.00 q^{91} -256.000 q^{93} -972.000 q^{95} -1122.00 q^{97} +322.000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −2.00000 −0.384900 −0.192450 0.981307i \(-0.561643\pi\)
−0.192450 + 0.981307i \(0.561643\pi\)
\(4\) 0 0
\(5\) 6.00000 0.536656 0.268328 0.963328i \(-0.413529\pi\)
0.268328 + 0.963328i \(0.413529\pi\)
\(6\) 0 0
\(7\) −20.0000 −1.07990 −0.539949 0.841698i \(-0.681557\pi\)
−0.539949 + 0.841698i \(0.681557\pi\)
\(8\) 0 0
\(9\) −23.0000 −0.851852
\(10\) 0 0
\(11\) −14.0000 −0.383742 −0.191871 0.981420i \(-0.561455\pi\)
−0.191871 + 0.981420i \(0.561455\pi\)
\(12\) 0 0
\(13\) 54.0000 1.15207 0.576035 0.817425i \(-0.304599\pi\)
0.576035 + 0.817425i \(0.304599\pi\)
\(14\) 0 0
\(15\) −12.0000 −0.206559
\(16\) 0 0
\(17\) −66.0000 −0.941609 −0.470804 0.882238i \(-0.656036\pi\)
−0.470804 + 0.882238i \(0.656036\pi\)
\(18\) 0 0
\(19\) −162.000 −1.95607 −0.978035 0.208438i \(-0.933162\pi\)
−0.978035 + 0.208438i \(0.933162\pi\)
\(20\) 0 0
\(21\) 40.0000 0.415653
\(22\) 0 0
\(23\) −172.000 −1.55933 −0.779663 0.626200i \(-0.784609\pi\)
−0.779663 + 0.626200i \(0.784609\pi\)
\(24\) 0 0
\(25\) −89.0000 −0.712000
\(26\) 0 0
\(27\) 100.000 0.712778
\(28\) 0 0
\(29\) −2.00000 −0.0128066 −0.00640329 0.999979i \(-0.502038\pi\)
−0.00640329 + 0.999979i \(0.502038\pi\)
\(30\) 0 0
\(31\) 128.000 0.741596 0.370798 0.928714i \(-0.379084\pi\)
0.370798 + 0.928714i \(0.379084\pi\)
\(32\) 0 0
\(33\) 28.0000 0.147702
\(34\) 0 0
\(35\) −120.000 −0.579534
\(36\) 0 0
\(37\) 158.000 0.702028 0.351014 0.936370i \(-0.385837\pi\)
0.351014 + 0.936370i \(0.385837\pi\)
\(38\) 0 0
\(39\) −108.000 −0.443432
\(40\) 0 0
\(41\) 202.000 0.769441 0.384721 0.923033i \(-0.374298\pi\)
0.384721 + 0.923033i \(0.374298\pi\)
\(42\) 0 0
\(43\) 298.000 1.05685 0.528425 0.848980i \(-0.322783\pi\)
0.528425 + 0.848980i \(0.322783\pi\)
\(44\) 0 0
\(45\) −138.000 −0.457152
\(46\) 0 0
\(47\) 408.000 1.26623 0.633116 0.774057i \(-0.281776\pi\)
0.633116 + 0.774057i \(0.281776\pi\)
\(48\) 0 0
\(49\) 57.0000 0.166181
\(50\) 0 0
\(51\) 132.000 0.362425
\(52\) 0 0
\(53\) −690.000 −1.78828 −0.894140 0.447788i \(-0.852212\pi\)
−0.894140 + 0.447788i \(0.852212\pi\)
\(54\) 0 0
\(55\) −84.0000 −0.205937
\(56\) 0 0
\(57\) 324.000 0.752892
\(58\) 0 0
\(59\) 322.000 0.710523 0.355261 0.934767i \(-0.384392\pi\)
0.355261 + 0.934767i \(0.384392\pi\)
\(60\) 0 0
\(61\) −298.000 −0.625492 −0.312746 0.949837i \(-0.601249\pi\)
−0.312746 + 0.949837i \(0.601249\pi\)
\(62\) 0 0
\(63\) 460.000 0.919914
\(64\) 0 0
\(65\) 324.000 0.618265
\(66\) 0 0
\(67\) −202.000 −0.368332 −0.184166 0.982895i \(-0.558958\pi\)
−0.184166 + 0.982895i \(0.558958\pi\)
\(68\) 0 0
\(69\) 344.000 0.600185
\(70\) 0 0
\(71\) 700.000 1.17007 0.585033 0.811009i \(-0.301081\pi\)
0.585033 + 0.811009i \(0.301081\pi\)
\(72\) 0 0
\(73\) −418.000 −0.670181 −0.335090 0.942186i \(-0.608767\pi\)
−0.335090 + 0.942186i \(0.608767\pi\)
\(74\) 0 0
\(75\) 178.000 0.274049
\(76\) 0 0
\(77\) 280.000 0.414402
\(78\) 0 0
\(79\) −744.000 −1.05958 −0.529788 0.848130i \(-0.677729\pi\)
−0.529788 + 0.848130i \(0.677729\pi\)
\(80\) 0 0
\(81\) 421.000 0.577503
\(82\) 0 0
\(83\) 678.000 0.896629 0.448314 0.893876i \(-0.352025\pi\)
0.448314 + 0.893876i \(0.352025\pi\)
\(84\) 0 0
\(85\) −396.000 −0.505320
\(86\) 0 0
\(87\) 4.00000 0.00492925
\(88\) 0 0
\(89\) −82.0000 −0.0976627 −0.0488314 0.998807i \(-0.515550\pi\)
−0.0488314 + 0.998807i \(0.515550\pi\)
\(90\) 0 0
\(91\) −1080.00 −1.24412
\(92\) 0 0
\(93\) −256.000 −0.285440
\(94\) 0 0
\(95\) −972.000 −1.04974
\(96\) 0 0
\(97\) −1122.00 −1.17445 −0.587226 0.809423i \(-0.699780\pi\)
−0.587226 + 0.809423i \(0.699780\pi\)
\(98\) 0 0
\(99\) 322.000 0.326891
\(100\) 0 0
\(101\) 1390.00 1.36941 0.684704 0.728821i \(-0.259932\pi\)
0.684704 + 0.728821i \(0.259932\pi\)
\(102\) 0 0
\(103\) −788.000 −0.753825 −0.376912 0.926249i \(-0.623014\pi\)
−0.376912 + 0.926249i \(0.623014\pi\)
\(104\) 0 0
\(105\) 240.000 0.223063
\(106\) 0 0
\(107\) −1614.00 −1.45824 −0.729118 0.684388i \(-0.760070\pi\)
−0.729118 + 0.684388i \(0.760070\pi\)
\(108\) 0 0
\(109\) 2014.00 1.76978 0.884891 0.465798i \(-0.154233\pi\)
0.884891 + 0.465798i \(0.154233\pi\)
\(110\) 0 0
\(111\) −316.000 −0.270211
\(112\) 0 0
\(113\) −542.000 −0.451213 −0.225607 0.974219i \(-0.572436\pi\)
−0.225607 + 0.974219i \(0.572436\pi\)
\(114\) 0 0
\(115\) −1032.00 −0.836822
\(116\) 0 0
\(117\) −1242.00 −0.981393
\(118\) 0 0
\(119\) 1320.00 1.01684
\(120\) 0 0
\(121\) −1135.00 −0.852742
\(122\) 0 0
\(123\) −404.000 −0.296158
\(124\) 0 0
\(125\) −1284.00 −0.918756
\(126\) 0 0
\(127\) −1712.00 −1.19618 −0.598092 0.801427i \(-0.704074\pi\)
−0.598092 + 0.801427i \(0.704074\pi\)
\(128\) 0 0
\(129\) −596.000 −0.406782
\(130\) 0 0
\(131\) 2118.00 1.41260 0.706300 0.707913i \(-0.250363\pi\)
0.706300 + 0.707913i \(0.250363\pi\)
\(132\) 0 0
\(133\) 3240.00 2.11236
\(134\) 0 0
\(135\) 600.000 0.382517
\(136\) 0 0
\(137\) −486.000 −0.303079 −0.151539 0.988451i \(-0.548423\pi\)
−0.151539 + 0.988451i \(0.548423\pi\)
\(138\) 0 0
\(139\) −1286.00 −0.784727 −0.392364 0.919810i \(-0.628343\pi\)
−0.392364 + 0.919810i \(0.628343\pi\)
\(140\) 0 0
\(141\) −816.000 −0.487373
\(142\) 0 0
\(143\) −756.000 −0.442097
\(144\) 0 0
\(145\) −12.0000 −0.00687273
\(146\) 0 0
\(147\) −114.000 −0.0639630
\(148\) 0 0
\(149\) −2666.00 −1.46582 −0.732910 0.680325i \(-0.761838\pi\)
−0.732910 + 0.680325i \(0.761838\pi\)
\(150\) 0 0
\(151\) −172.000 −0.0926964 −0.0463482 0.998925i \(-0.514758\pi\)
−0.0463482 + 0.998925i \(0.514758\pi\)
\(152\) 0 0
\(153\) 1518.00 0.802111
\(154\) 0 0
\(155\) 768.000 0.397982
\(156\) 0 0
\(157\) 838.000 0.425985 0.212993 0.977054i \(-0.431679\pi\)
0.212993 + 0.977054i \(0.431679\pi\)
\(158\) 0 0
\(159\) 1380.00 0.688309
\(160\) 0 0
\(161\) 3440.00 1.68391
\(162\) 0 0
\(163\) −1346.00 −0.646791 −0.323395 0.946264i \(-0.604824\pi\)
−0.323395 + 0.946264i \(0.604824\pi\)
\(164\) 0 0
\(165\) 168.000 0.0792653
\(166\) 0 0
\(167\) 1052.00 0.487462 0.243731 0.969843i \(-0.421629\pi\)
0.243731 + 0.969843i \(0.421629\pi\)
\(168\) 0 0
\(169\) 719.000 0.327264
\(170\) 0 0
\(171\) 3726.00 1.66628
\(172\) 0 0
\(173\) 38.0000 0.0166999 0.00834996 0.999965i \(-0.497342\pi\)
0.00834996 + 0.999965i \(0.497342\pi\)
\(174\) 0 0
\(175\) 1780.00 0.768888
\(176\) 0 0
\(177\) −644.000 −0.273480
\(178\) 0 0
\(179\) 2790.00 1.16500 0.582498 0.812832i \(-0.302075\pi\)
0.582498 + 0.812832i \(0.302075\pi\)
\(180\) 0 0
\(181\) −3418.00 −1.40364 −0.701818 0.712357i \(-0.747628\pi\)
−0.701818 + 0.712357i \(0.747628\pi\)
\(182\) 0 0
\(183\) 596.000 0.240752
\(184\) 0 0
\(185\) 948.000 0.376748
\(186\) 0 0
\(187\) 924.000 0.361335
\(188\) 0 0
\(189\) −2000.00 −0.769728
\(190\) 0 0
\(191\) 1968.00 0.745547 0.372774 0.927922i \(-0.378407\pi\)
0.372774 + 0.927922i \(0.378407\pi\)
\(192\) 0 0
\(193\) −1058.00 −0.394593 −0.197297 0.980344i \(-0.563216\pi\)
−0.197297 + 0.980344i \(0.563216\pi\)
\(194\) 0 0
\(195\) −648.000 −0.237970
\(196\) 0 0
\(197\) 726.000 0.262565 0.131283 0.991345i \(-0.458090\pi\)
0.131283 + 0.991345i \(0.458090\pi\)
\(198\) 0 0
\(199\) −4116.00 −1.46621 −0.733104 0.680116i \(-0.761929\pi\)
−0.733104 + 0.680116i \(0.761929\pi\)
\(200\) 0 0
\(201\) 404.000 0.141771
\(202\) 0 0
\(203\) 40.0000 0.0138298
\(204\) 0 0
\(205\) 1212.00 0.412926
\(206\) 0 0
\(207\) 3956.00 1.32831
\(208\) 0 0
\(209\) 2268.00 0.750626
\(210\) 0 0
\(211\) −1482.00 −0.483531 −0.241766 0.970335i \(-0.577727\pi\)
−0.241766 + 0.970335i \(0.577727\pi\)
\(212\) 0 0
\(213\) −1400.00 −0.450359
\(214\) 0 0
\(215\) 1788.00 0.567166
\(216\) 0 0
\(217\) −2560.00 −0.800848
\(218\) 0 0
\(219\) 836.000 0.257953
\(220\) 0 0
\(221\) −3564.00 −1.08480
\(222\) 0 0
\(223\) 896.000 0.269061 0.134530 0.990909i \(-0.457047\pi\)
0.134530 + 0.990909i \(0.457047\pi\)
\(224\) 0 0
\(225\) 2047.00 0.606519
\(226\) 0 0
\(227\) −3410.00 −0.997047 −0.498523 0.866876i \(-0.666124\pi\)
−0.498523 + 0.866876i \(0.666124\pi\)
\(228\) 0 0
\(229\) 4502.00 1.29913 0.649564 0.760307i \(-0.274951\pi\)
0.649564 + 0.760307i \(0.274951\pi\)
\(230\) 0 0
\(231\) −560.000 −0.159503
\(232\) 0 0
\(233\) 2302.00 0.647249 0.323625 0.946186i \(-0.395099\pi\)
0.323625 + 0.946186i \(0.395099\pi\)
\(234\) 0 0
\(235\) 2448.00 0.679532
\(236\) 0 0
\(237\) 1488.00 0.407831
\(238\) 0 0
\(239\) −4024.00 −1.08908 −0.544542 0.838734i \(-0.683296\pi\)
−0.544542 + 0.838734i \(0.683296\pi\)
\(240\) 0 0
\(241\) −3586.00 −0.958484 −0.479242 0.877683i \(-0.659088\pi\)
−0.479242 + 0.877683i \(0.659088\pi\)
\(242\) 0 0
\(243\) −3542.00 −0.935059
\(244\) 0 0
\(245\) 342.000 0.0891820
\(246\) 0 0
\(247\) −8748.00 −2.25353
\(248\) 0 0
\(249\) −1356.00 −0.345112
\(250\) 0 0
\(251\) 1250.00 0.314340 0.157170 0.987572i \(-0.449763\pi\)
0.157170 + 0.987572i \(0.449763\pi\)
\(252\) 0 0
\(253\) 2408.00 0.598378
\(254\) 0 0
\(255\) 792.000 0.194498
\(256\) 0 0
\(257\) −6638.00 −1.61116 −0.805578 0.592490i \(-0.798145\pi\)
−0.805578 + 0.592490i \(0.798145\pi\)
\(258\) 0 0
\(259\) −3160.00 −0.758119
\(260\) 0 0
\(261\) 46.0000 0.0109093
\(262\) 0 0
\(263\) 1724.00 0.404207 0.202103 0.979364i \(-0.435222\pi\)
0.202103 + 0.979364i \(0.435222\pi\)
\(264\) 0 0
\(265\) −4140.00 −0.959691
\(266\) 0 0
\(267\) 164.000 0.0375904
\(268\) 0 0
\(269\) 4814.00 1.09113 0.545566 0.838068i \(-0.316315\pi\)
0.545566 + 0.838068i \(0.316315\pi\)
\(270\) 0 0
\(271\) 1640.00 0.367612 0.183806 0.982963i \(-0.441158\pi\)
0.183806 + 0.982963i \(0.441158\pi\)
\(272\) 0 0
\(273\) 2160.00 0.478861
\(274\) 0 0
\(275\) 1246.00 0.273224
\(276\) 0 0
\(277\) 3982.00 0.863737 0.431869 0.901937i \(-0.357854\pi\)
0.431869 + 0.901937i \(0.357854\pi\)
\(278\) 0 0
\(279\) −2944.00 −0.631730
\(280\) 0 0
\(281\) 4126.00 0.875931 0.437965 0.898992i \(-0.355699\pi\)
0.437965 + 0.898992i \(0.355699\pi\)
\(282\) 0 0
\(283\) −3446.00 −0.723828 −0.361914 0.932211i \(-0.617877\pi\)
−0.361914 + 0.932211i \(0.617877\pi\)
\(284\) 0 0
\(285\) 1944.00 0.404044
\(286\) 0 0
\(287\) −4040.00 −0.830919
\(288\) 0 0
\(289\) −557.000 −0.113373
\(290\) 0 0
\(291\) 2244.00 0.452047
\(292\) 0 0
\(293\) −1514.00 −0.301873 −0.150937 0.988543i \(-0.548229\pi\)
−0.150937 + 0.988543i \(0.548229\pi\)
\(294\) 0 0
\(295\) 1932.00 0.381306
\(296\) 0 0
\(297\) −1400.00 −0.273523
\(298\) 0 0
\(299\) −9288.00 −1.79645
\(300\) 0 0
\(301\) −5960.00 −1.14129
\(302\) 0 0
\(303\) −2780.00 −0.527085
\(304\) 0 0
\(305\) −1788.00 −0.335674
\(306\) 0 0
\(307\) −5490.00 −1.02062 −0.510311 0.859990i \(-0.670470\pi\)
−0.510311 + 0.859990i \(0.670470\pi\)
\(308\) 0 0
\(309\) 1576.00 0.290147
\(310\) 0 0
\(311\) 5556.00 1.01303 0.506514 0.862232i \(-0.330934\pi\)
0.506514 + 0.862232i \(0.330934\pi\)
\(312\) 0 0
\(313\) −2054.00 −0.370923 −0.185462 0.982652i \(-0.559378\pi\)
−0.185462 + 0.982652i \(0.559378\pi\)
\(314\) 0 0
\(315\) 2760.00 0.493677
\(316\) 0 0
\(317\) 2494.00 0.441883 0.220942 0.975287i \(-0.429087\pi\)
0.220942 + 0.975287i \(0.429087\pi\)
\(318\) 0 0
\(319\) 28.0000 0.00491442
\(320\) 0 0
\(321\) 3228.00 0.561275
\(322\) 0 0
\(323\) 10692.0 1.84185
\(324\) 0 0
\(325\) −4806.00 −0.820274
\(326\) 0 0
\(327\) −4028.00 −0.681189
\(328\) 0 0
\(329\) −8160.00 −1.36740
\(330\) 0 0
\(331\) 2914.00 0.483891 0.241946 0.970290i \(-0.422214\pi\)
0.241946 + 0.970290i \(0.422214\pi\)
\(332\) 0 0
\(333\) −3634.00 −0.598024
\(334\) 0 0
\(335\) −1212.00 −0.197668
\(336\) 0 0
\(337\) 7186.00 1.16156 0.580781 0.814060i \(-0.302747\pi\)
0.580781 + 0.814060i \(0.302747\pi\)
\(338\) 0 0
\(339\) 1084.00 0.173672
\(340\) 0 0
\(341\) −1792.00 −0.284581
\(342\) 0 0
\(343\) 5720.00 0.900440
\(344\) 0 0
\(345\) 2064.00 0.322093
\(346\) 0 0
\(347\) −4446.00 −0.687821 −0.343910 0.939002i \(-0.611752\pi\)
−0.343910 + 0.939002i \(0.611752\pi\)
\(348\) 0 0
\(349\) 430.000 0.0659524 0.0329762 0.999456i \(-0.489501\pi\)
0.0329762 + 0.999456i \(0.489501\pi\)
\(350\) 0 0
\(351\) 5400.00 0.821170
\(352\) 0 0
\(353\) 850.000 0.128161 0.0640806 0.997945i \(-0.479589\pi\)
0.0640806 + 0.997945i \(0.479589\pi\)
\(354\) 0 0
\(355\) 4200.00 0.627924
\(356\) 0 0
\(357\) −2640.00 −0.391383
\(358\) 0 0
\(359\) 10988.0 1.61539 0.807694 0.589602i \(-0.200715\pi\)
0.807694 + 0.589602i \(0.200715\pi\)
\(360\) 0 0
\(361\) 19385.0 2.82621
\(362\) 0 0
\(363\) 2270.00 0.328221
\(364\) 0 0
\(365\) −2508.00 −0.359657
\(366\) 0 0
\(367\) −872.000 −0.124027 −0.0620137 0.998075i \(-0.519752\pi\)
−0.0620137 + 0.998075i \(0.519752\pi\)
\(368\) 0 0
\(369\) −4646.00 −0.655450
\(370\) 0 0
\(371\) 13800.0 1.93116
\(372\) 0 0
\(373\) 3454.00 0.479467 0.239734 0.970839i \(-0.422940\pi\)
0.239734 + 0.970839i \(0.422940\pi\)
\(374\) 0 0
\(375\) 2568.00 0.353629
\(376\) 0 0
\(377\) −108.000 −0.0147541
\(378\) 0 0
\(379\) 1490.00 0.201942 0.100971 0.994889i \(-0.467805\pi\)
0.100971 + 0.994889i \(0.467805\pi\)
\(380\) 0 0
\(381\) 3424.00 0.460412
\(382\) 0 0
\(383\) −10240.0 −1.36616 −0.683080 0.730343i \(-0.739360\pi\)
−0.683080 + 0.730343i \(0.739360\pi\)
\(384\) 0 0
\(385\) 1680.00 0.222392
\(386\) 0 0
\(387\) −6854.00 −0.900280
\(388\) 0 0
\(389\) −7458.00 −0.972071 −0.486035 0.873939i \(-0.661557\pi\)
−0.486035 + 0.873939i \(0.661557\pi\)
\(390\) 0 0
\(391\) 11352.0 1.46827
\(392\) 0 0
\(393\) −4236.00 −0.543710
\(394\) 0 0
\(395\) −4464.00 −0.568628
\(396\) 0 0
\(397\) −4706.00 −0.594930 −0.297465 0.954733i \(-0.596141\pi\)
−0.297465 + 0.954733i \(0.596141\pi\)
\(398\) 0 0
\(399\) −6480.00 −0.813047
\(400\) 0 0
\(401\) 5598.00 0.697134 0.348567 0.937284i \(-0.386668\pi\)
0.348567 + 0.937284i \(0.386668\pi\)
\(402\) 0 0
\(403\) 6912.00 0.854370
\(404\) 0 0
\(405\) 2526.00 0.309921
\(406\) 0 0
\(407\) −2212.00 −0.269397
\(408\) 0 0
\(409\) −838.000 −0.101312 −0.0506558 0.998716i \(-0.516131\pi\)
−0.0506558 + 0.998716i \(0.516131\pi\)
\(410\) 0 0
\(411\) 972.000 0.116655
\(412\) 0 0
\(413\) −6440.00 −0.767292
\(414\) 0 0
\(415\) 4068.00 0.481181
\(416\) 0 0
\(417\) 2572.00 0.302042
\(418\) 0 0
\(419\) −26.0000 −0.00303146 −0.00151573 0.999999i \(-0.500482\pi\)
−0.00151573 + 0.999999i \(0.500482\pi\)
\(420\) 0 0
\(421\) −15578.0 −1.80339 −0.901693 0.432377i \(-0.857675\pi\)
−0.901693 + 0.432377i \(0.857675\pi\)
\(422\) 0 0
\(423\) −9384.00 −1.07864
\(424\) 0 0
\(425\) 5874.00 0.670426
\(426\) 0 0
\(427\) 5960.00 0.675467
\(428\) 0 0
\(429\) 1512.00 0.170163
\(430\) 0 0
\(431\) 6792.00 0.759070 0.379535 0.925177i \(-0.376084\pi\)
0.379535 + 0.925177i \(0.376084\pi\)
\(432\) 0 0
\(433\) −9314.00 −1.03372 −0.516862 0.856069i \(-0.672900\pi\)
−0.516862 + 0.856069i \(0.672900\pi\)
\(434\) 0 0
\(435\) 24.0000 0.00264531
\(436\) 0 0
\(437\) 27864.0 3.05015
\(438\) 0 0
\(439\) 3828.00 0.416174 0.208087 0.978110i \(-0.433276\pi\)
0.208087 + 0.978110i \(0.433276\pi\)
\(440\) 0 0
\(441\) −1311.00 −0.141561
\(442\) 0 0
\(443\) −15414.0 −1.65314 −0.826570 0.562834i \(-0.809711\pi\)
−0.826570 + 0.562834i \(0.809711\pi\)
\(444\) 0 0
\(445\) −492.000 −0.0524113
\(446\) 0 0
\(447\) 5332.00 0.564195
\(448\) 0 0
\(449\) −3650.00 −0.383640 −0.191820 0.981430i \(-0.561439\pi\)
−0.191820 + 0.981430i \(0.561439\pi\)
\(450\) 0 0
\(451\) −2828.00 −0.295267
\(452\) 0 0
\(453\) 344.000 0.0356789
\(454\) 0 0
\(455\) −6480.00 −0.667664
\(456\) 0 0
\(457\) −15862.0 −1.62362 −0.811809 0.583924i \(-0.801517\pi\)
−0.811809 + 0.583924i \(0.801517\pi\)
\(458\) 0 0
\(459\) −6600.00 −0.671158
\(460\) 0 0
\(461\) 78.0000 0.00788031 0.00394015 0.999992i \(-0.498746\pi\)
0.00394015 + 0.999992i \(0.498746\pi\)
\(462\) 0 0
\(463\) 4376.00 0.439244 0.219622 0.975585i \(-0.429518\pi\)
0.219622 + 0.975585i \(0.429518\pi\)
\(464\) 0 0
\(465\) −1536.00 −0.153183
\(466\) 0 0
\(467\) −13714.0 −1.35890 −0.679452 0.733720i \(-0.737782\pi\)
−0.679452 + 0.733720i \(0.737782\pi\)
\(468\) 0 0
\(469\) 4040.00 0.397761
\(470\) 0 0
\(471\) −1676.00 −0.163962
\(472\) 0 0
\(473\) −4172.00 −0.405558
\(474\) 0 0
\(475\) 14418.0 1.39272
\(476\) 0 0
\(477\) 15870.0 1.52335
\(478\) 0 0
\(479\) −19072.0 −1.81925 −0.909626 0.415428i \(-0.863632\pi\)
−0.909626 + 0.415428i \(0.863632\pi\)
\(480\) 0 0
\(481\) 8532.00 0.808785
\(482\) 0 0
\(483\) −6880.00 −0.648138
\(484\) 0 0
\(485\) −6732.00 −0.630277
\(486\) 0 0
\(487\) −19428.0 −1.80773 −0.903867 0.427813i \(-0.859284\pi\)
−0.903867 + 0.427813i \(0.859284\pi\)
\(488\) 0 0
\(489\) 2692.00 0.248950
\(490\) 0 0
\(491\) 2490.00 0.228864 0.114432 0.993431i \(-0.463495\pi\)
0.114432 + 0.993431i \(0.463495\pi\)
\(492\) 0 0
\(493\) 132.000 0.0120588
\(494\) 0 0
\(495\) 1932.00 0.175428
\(496\) 0 0
\(497\) −14000.0 −1.26355
\(498\) 0 0
\(499\) −2826.00 −0.253525 −0.126763 0.991933i \(-0.540459\pi\)
−0.126763 + 0.991933i \(0.540459\pi\)
\(500\) 0 0
\(501\) −2104.00 −0.187624
\(502\) 0 0
\(503\) −2268.00 −0.201044 −0.100522 0.994935i \(-0.532051\pi\)
−0.100522 + 0.994935i \(0.532051\pi\)
\(504\) 0 0
\(505\) 8340.00 0.734901
\(506\) 0 0
\(507\) −1438.00 −0.125964
\(508\) 0 0
\(509\) 10534.0 0.917311 0.458656 0.888614i \(-0.348331\pi\)
0.458656 + 0.888614i \(0.348331\pi\)
\(510\) 0 0
\(511\) 8360.00 0.723727
\(512\) 0 0
\(513\) −16200.0 −1.39424
\(514\) 0 0
\(515\) −4728.00 −0.404545
\(516\) 0 0
\(517\) −5712.00 −0.485906
\(518\) 0 0
\(519\) −76.0000 −0.00642780
\(520\) 0 0
\(521\) −9478.00 −0.797003 −0.398502 0.917168i \(-0.630470\pi\)
−0.398502 + 0.917168i \(0.630470\pi\)
\(522\) 0 0
\(523\) 5858.00 0.489775 0.244888 0.969551i \(-0.421249\pi\)
0.244888 + 0.969551i \(0.421249\pi\)
\(524\) 0 0
\(525\) −3560.00 −0.295945
\(526\) 0 0
\(527\) −8448.00 −0.698293
\(528\) 0 0
\(529\) 17417.0 1.43150
\(530\) 0 0
\(531\) −7406.00 −0.605260
\(532\) 0 0
\(533\) 10908.0 0.886450
\(534\) 0 0
\(535\) −9684.00 −0.782572
\(536\) 0 0
\(537\) −5580.00 −0.448407
\(538\) 0 0
\(539\) −798.000 −0.0637705
\(540\) 0 0
\(541\) 1910.00 0.151788 0.0758940 0.997116i \(-0.475819\pi\)
0.0758940 + 0.997116i \(0.475819\pi\)
\(542\) 0 0
\(543\) 6836.00 0.540259
\(544\) 0 0
\(545\) 12084.0 0.949765
\(546\) 0 0
\(547\) −1754.00 −0.137104 −0.0685518 0.997648i \(-0.521838\pi\)
−0.0685518 + 0.997648i \(0.521838\pi\)
\(548\) 0 0
\(549\) 6854.00 0.532826
\(550\) 0 0
\(551\) 324.000 0.0250506
\(552\) 0 0
\(553\) 14880.0 1.14424
\(554\) 0 0
\(555\) −1896.00 −0.145010
\(556\) 0 0
\(557\) 2294.00 0.174506 0.0872531 0.996186i \(-0.472191\pi\)
0.0872531 + 0.996186i \(0.472191\pi\)
\(558\) 0 0
\(559\) 16092.0 1.21757
\(560\) 0 0
\(561\) −1848.00 −0.139078
\(562\) 0 0
\(563\) −16242.0 −1.21584 −0.607921 0.793998i \(-0.707996\pi\)
−0.607921 + 0.793998i \(0.707996\pi\)
\(564\) 0 0
\(565\) −3252.00 −0.242146
\(566\) 0 0
\(567\) −8420.00 −0.623645
\(568\) 0 0
\(569\) −15990.0 −1.17809 −0.589047 0.808099i \(-0.700497\pi\)
−0.589047 + 0.808099i \(0.700497\pi\)
\(570\) 0 0
\(571\) 21674.0 1.58849 0.794246 0.607597i \(-0.207866\pi\)
0.794246 + 0.607597i \(0.207866\pi\)
\(572\) 0 0
\(573\) −3936.00 −0.286961
\(574\) 0 0
\(575\) 15308.0 1.11024
\(576\) 0 0
\(577\) −12542.0 −0.904905 −0.452453 0.891788i \(-0.649451\pi\)
−0.452453 + 0.891788i \(0.649451\pi\)
\(578\) 0 0
\(579\) 2116.00 0.151879
\(580\) 0 0
\(581\) −13560.0 −0.968268
\(582\) 0 0
\(583\) 9660.00 0.686237
\(584\) 0 0
\(585\) −7452.00 −0.526671
\(586\) 0 0
\(587\) 18578.0 1.30630 0.653148 0.757230i \(-0.273448\pi\)
0.653148 + 0.757230i \(0.273448\pi\)
\(588\) 0 0
\(589\) −20736.0 −1.45061
\(590\) 0 0
\(591\) −1452.00 −0.101061
\(592\) 0 0
\(593\) 14514.0 1.00509 0.502545 0.864551i \(-0.332397\pi\)
0.502545 + 0.864551i \(0.332397\pi\)
\(594\) 0 0
\(595\) 7920.00 0.545695
\(596\) 0 0
\(597\) 8232.00 0.564344
\(598\) 0 0
\(599\) 3460.00 0.236013 0.118006 0.993013i \(-0.462350\pi\)
0.118006 + 0.993013i \(0.462350\pi\)
\(600\) 0 0
\(601\) 20686.0 1.40399 0.701996 0.712181i \(-0.252292\pi\)
0.701996 + 0.712181i \(0.252292\pi\)
\(602\) 0 0
\(603\) 4646.00 0.313764
\(604\) 0 0
\(605\) −6810.00 −0.457630
\(606\) 0 0
\(607\) 9776.00 0.653700 0.326850 0.945076i \(-0.394013\pi\)
0.326850 + 0.945076i \(0.394013\pi\)
\(608\) 0 0
\(609\) −80.0000 −0.00532309
\(610\) 0 0
\(611\) 22032.0 1.45879
\(612\) 0 0
\(613\) −8794.00 −0.579423 −0.289712 0.957114i \(-0.593559\pi\)
−0.289712 + 0.957114i \(0.593559\pi\)
\(614\) 0 0
\(615\) −2424.00 −0.158935
\(616\) 0 0
\(617\) 16398.0 1.06995 0.534975 0.844868i \(-0.320321\pi\)
0.534975 + 0.844868i \(0.320321\pi\)
\(618\) 0 0
\(619\) −21374.0 −1.38787 −0.693937 0.720036i \(-0.744125\pi\)
−0.693937 + 0.720036i \(0.744125\pi\)
\(620\) 0 0
\(621\) −17200.0 −1.11145
\(622\) 0 0
\(623\) 1640.00 0.105466
\(624\) 0 0
\(625\) 3421.00 0.218944
\(626\) 0 0
\(627\) −4536.00 −0.288916
\(628\) 0 0
\(629\) −10428.0 −0.661036
\(630\) 0 0
\(631\) 18916.0 1.19340 0.596699 0.802465i \(-0.296479\pi\)
0.596699 + 0.802465i \(0.296479\pi\)
\(632\) 0 0
\(633\) 2964.00 0.186111
\(634\) 0 0
\(635\) −10272.0 −0.641940
\(636\) 0 0
\(637\) 3078.00 0.191452
\(638\) 0 0
\(639\) −16100.0 −0.996723
\(640\) 0 0
\(641\) −5250.00 −0.323498 −0.161749 0.986832i \(-0.551714\pi\)
−0.161749 + 0.986832i \(0.551714\pi\)
\(642\) 0 0
\(643\) 4502.00 0.276114 0.138057 0.990424i \(-0.455914\pi\)
0.138057 + 0.990424i \(0.455914\pi\)
\(644\) 0 0
\(645\) −3576.00 −0.218302
\(646\) 0 0
\(647\) −11076.0 −0.673018 −0.336509 0.941680i \(-0.609246\pi\)
−0.336509 + 0.941680i \(0.609246\pi\)
\(648\) 0 0
\(649\) −4508.00 −0.272657
\(650\) 0 0
\(651\) 5120.00 0.308247
\(652\) 0 0
\(653\) 30766.0 1.84375 0.921873 0.387491i \(-0.126658\pi\)
0.921873 + 0.387491i \(0.126658\pi\)
\(654\) 0 0
\(655\) 12708.0 0.758080
\(656\) 0 0
\(657\) 9614.00 0.570895
\(658\) 0 0
\(659\) 20518.0 1.21285 0.606425 0.795141i \(-0.292603\pi\)
0.606425 + 0.795141i \(0.292603\pi\)
\(660\) 0 0
\(661\) 70.0000 0.00411904 0.00205952 0.999998i \(-0.499344\pi\)
0.00205952 + 0.999998i \(0.499344\pi\)
\(662\) 0 0
\(663\) 7128.00 0.417539
\(664\) 0 0
\(665\) 19440.0 1.13361
\(666\) 0 0
\(667\) 344.000 0.0199696
\(668\) 0 0
\(669\) −1792.00 −0.103562
\(670\) 0 0
\(671\) 4172.00 0.240027
\(672\) 0 0
\(673\) 23070.0 1.32137 0.660686 0.750663i \(-0.270266\pi\)
0.660686 + 0.750663i \(0.270266\pi\)
\(674\) 0 0
\(675\) −8900.00 −0.507498
\(676\) 0 0
\(677\) 2622.00 0.148850 0.0744251 0.997227i \(-0.476288\pi\)
0.0744251 + 0.997227i \(0.476288\pi\)
\(678\) 0 0
\(679\) 22440.0 1.26829
\(680\) 0 0
\(681\) 6820.00 0.383764
\(682\) 0 0
\(683\) 14682.0 0.822535 0.411267 0.911515i \(-0.365086\pi\)
0.411267 + 0.911515i \(0.365086\pi\)
\(684\) 0 0
\(685\) −2916.00 −0.162649
\(686\) 0 0
\(687\) −9004.00 −0.500035
\(688\) 0 0
\(689\) −37260.0 −2.06022
\(690\) 0 0
\(691\) 23270.0 1.28109 0.640545 0.767921i \(-0.278709\pi\)
0.640545 + 0.767921i \(0.278709\pi\)
\(692\) 0 0
\(693\) −6440.00 −0.353009
\(694\) 0 0
\(695\) −7716.00 −0.421129
\(696\) 0 0
\(697\) −13332.0 −0.724513
\(698\) 0 0
\(699\) −4604.00 −0.249126
\(700\) 0 0
\(701\) −31530.0 −1.69882 −0.849409 0.527735i \(-0.823041\pi\)
−0.849409 + 0.527735i \(0.823041\pi\)
\(702\) 0 0
\(703\) −25596.0 −1.37322
\(704\) 0 0
\(705\) −4896.00 −0.261552
\(706\) 0 0
\(707\) −27800.0 −1.47882
\(708\) 0 0
\(709\) 22014.0 1.16608 0.583042 0.812442i \(-0.301862\pi\)
0.583042 + 0.812442i \(0.301862\pi\)
\(710\) 0 0
\(711\) 17112.0 0.902602
\(712\) 0 0
\(713\) −22016.0 −1.15639
\(714\) 0 0
\(715\) −4536.00 −0.237254
\(716\) 0 0
\(717\) 8048.00 0.419188
\(718\) 0 0
\(719\) −19016.0 −0.986338 −0.493169 0.869933i \(-0.664162\pi\)
−0.493169 + 0.869933i \(0.664162\pi\)
\(720\) 0 0
\(721\) 15760.0 0.814054
\(722\) 0 0
\(723\) 7172.00 0.368921
\(724\) 0 0
\(725\) 178.000 0.00911828
\(726\) 0 0
\(727\) −15996.0 −0.816037 −0.408018 0.912974i \(-0.633780\pi\)
−0.408018 + 0.912974i \(0.633780\pi\)
\(728\) 0 0
\(729\) −4283.00 −0.217599
\(730\) 0 0
\(731\) −19668.0 −0.995140
\(732\) 0 0
\(733\) 10622.0 0.535242 0.267621 0.963524i \(-0.413762\pi\)
0.267621 + 0.963524i \(0.413762\pi\)
\(734\) 0 0
\(735\) −684.000 −0.0343261
\(736\) 0 0
\(737\) 2828.00 0.141344
\(738\) 0 0
\(739\) 8878.00 0.441925 0.220962 0.975282i \(-0.429080\pi\)
0.220962 + 0.975282i \(0.429080\pi\)
\(740\) 0 0
\(741\) 17496.0 0.867384
\(742\) 0 0
\(743\) 21852.0 1.07897 0.539483 0.841996i \(-0.318620\pi\)
0.539483 + 0.841996i \(0.318620\pi\)
\(744\) 0 0
\(745\) −15996.0 −0.786642
\(746\) 0 0
\(747\) −15594.0 −0.763795
\(748\) 0 0
\(749\) 32280.0 1.57475
\(750\) 0 0
\(751\) −32024.0 −1.55602 −0.778011 0.628251i \(-0.783771\pi\)
−0.778011 + 0.628251i \(0.783771\pi\)
\(752\) 0 0
\(753\) −2500.00 −0.120989
\(754\) 0 0
\(755\) −1032.00 −0.0497461
\(756\) 0 0
\(757\) −26602.0 −1.27723 −0.638617 0.769525i \(-0.720493\pi\)
−0.638617 + 0.769525i \(0.720493\pi\)
\(758\) 0 0
\(759\) −4816.00 −0.230316
\(760\) 0 0
\(761\) −13958.0 −0.664885 −0.332442 0.943124i \(-0.607873\pi\)
−0.332442 + 0.943124i \(0.607873\pi\)
\(762\) 0 0
\(763\) −40280.0 −1.91118
\(764\) 0 0
\(765\) 9108.00 0.430458
\(766\) 0 0
\(767\) 17388.0 0.818571
\(768\) 0 0
\(769\) −11970.0 −0.561312 −0.280656 0.959808i \(-0.590552\pi\)
−0.280656 + 0.959808i \(0.590552\pi\)
\(770\) 0 0
\(771\) 13276.0 0.620134
\(772\) 0 0
\(773\) 12318.0 0.573154 0.286577 0.958057i \(-0.407483\pi\)
0.286577 + 0.958057i \(0.407483\pi\)
\(774\) 0 0
\(775\) −11392.0 −0.528016
\(776\) 0 0
\(777\) 6320.00 0.291800
\(778\) 0 0
\(779\) −32724.0 −1.50508
\(780\) 0 0
\(781\) −9800.00 −0.449003
\(782\) 0 0
\(783\) −200.000 −0.00912825
\(784\) 0 0
\(785\) 5028.00 0.228608
\(786\) 0 0
\(787\) −26698.0 −1.20925 −0.604626 0.796510i \(-0.706677\pi\)
−0.604626 + 0.796510i \(0.706677\pi\)
\(788\) 0 0
\(789\) −3448.00 −0.155579
\(790\) 0 0
\(791\) 10840.0 0.487264
\(792\) 0 0
\(793\) −16092.0 −0.720610
\(794\) 0 0
\(795\) 8280.00 0.369385
\(796\) 0 0
\(797\) −4794.00 −0.213064 −0.106532 0.994309i \(-0.533975\pi\)
−0.106532 + 0.994309i \(0.533975\pi\)
\(798\) 0 0
\(799\) −26928.0 −1.19230
\(800\) 0 0
\(801\) 1886.00 0.0831942
\(802\) 0 0
\(803\) 5852.00 0.257176
\(804\) 0 0
\(805\) 20640.0 0.903683
\(806\) 0 0
\(807\) −9628.00 −0.419977
\(808\) 0 0
\(809\) 39114.0 1.69985 0.849923 0.526907i \(-0.176649\pi\)
0.849923 + 0.526907i \(0.176649\pi\)
\(810\) 0 0
\(811\) 1090.00 0.0471949 0.0235975 0.999722i \(-0.492488\pi\)
0.0235975 + 0.999722i \(0.492488\pi\)
\(812\) 0 0
\(813\) −3280.00 −0.141494
\(814\) 0 0
\(815\) −8076.00 −0.347104
\(816\) 0 0
\(817\) −48276.0 −2.06727
\(818\) 0 0
\(819\) 24840.0 1.05980
\(820\) 0 0
\(821\) −9730.00 −0.413617 −0.206808 0.978381i \(-0.566308\pi\)
−0.206808 + 0.978381i \(0.566308\pi\)
\(822\) 0 0
\(823\) −1100.00 −0.0465900 −0.0232950 0.999729i \(-0.507416\pi\)
−0.0232950 + 0.999729i \(0.507416\pi\)
\(824\) 0 0
\(825\) −2492.00 −0.105164
\(826\) 0 0
\(827\) 38074.0 1.60092 0.800461 0.599385i \(-0.204588\pi\)
0.800461 + 0.599385i \(0.204588\pi\)
\(828\) 0 0
\(829\) 24230.0 1.01513 0.507565 0.861613i \(-0.330546\pi\)
0.507565 + 0.861613i \(0.330546\pi\)
\(830\) 0 0
\(831\) −7964.00 −0.332453
\(832\) 0 0
\(833\) −3762.00 −0.156477
\(834\) 0 0
\(835\) 6312.00 0.261600
\(836\) 0 0
\(837\) 12800.0 0.528593
\(838\) 0 0
\(839\) −16820.0 −0.692123 −0.346061 0.938212i \(-0.612481\pi\)
−0.346061 + 0.938212i \(0.612481\pi\)
\(840\) 0 0
\(841\) −24385.0 −0.999836
\(842\) 0 0
\(843\) −8252.00 −0.337146
\(844\) 0 0
\(845\) 4314.00 0.175629
\(846\) 0 0
\(847\) 22700.0 0.920875
\(848\) 0 0
\(849\) 6892.00 0.278602
\(850\) 0 0
\(851\) −27176.0 −1.09469
\(852\) 0 0
\(853\) −22162.0 −0.889581 −0.444790 0.895635i \(-0.646722\pi\)
−0.444790 + 0.895635i \(0.646722\pi\)
\(854\) 0 0
\(855\) 22356.0 0.894221
\(856\) 0 0
\(857\) −8790.00 −0.350363 −0.175181 0.984536i \(-0.556051\pi\)
−0.175181 + 0.984536i \(0.556051\pi\)
\(858\) 0 0
\(859\) −10558.0 −0.419365 −0.209682 0.977770i \(-0.567243\pi\)
−0.209682 + 0.977770i \(0.567243\pi\)
\(860\) 0 0
\(861\) 8080.00 0.319821
\(862\) 0 0
\(863\) 7392.00 0.291572 0.145786 0.989316i \(-0.453429\pi\)
0.145786 + 0.989316i \(0.453429\pi\)
\(864\) 0 0
\(865\) 228.000 0.00896212
\(866\) 0 0
\(867\) 1114.00 0.0436372
\(868\) 0 0
\(869\) 10416.0 0.406604
\(870\) 0 0
\(871\) −10908.0 −0.424344
\(872\) 0 0
\(873\) 25806.0 1.00046
\(874\) 0 0
\(875\) 25680.0 0.992163
\(876\) 0 0
\(877\) 6574.00 0.253122 0.126561 0.991959i \(-0.459606\pi\)
0.126561 + 0.991959i \(0.459606\pi\)
\(878\) 0 0
\(879\) 3028.00 0.116191
\(880\) 0 0
\(881\) 47154.0 1.80324 0.901622 0.432524i \(-0.142377\pi\)
0.901622 + 0.432524i \(0.142377\pi\)
\(882\) 0 0
\(883\) −642.000 −0.0244677 −0.0122339 0.999925i \(-0.503894\pi\)
−0.0122339 + 0.999925i \(0.503894\pi\)
\(884\) 0 0
\(885\) −3864.00 −0.146765
\(886\) 0 0
\(887\) −23308.0 −0.882307 −0.441153 0.897432i \(-0.645431\pi\)
−0.441153 + 0.897432i \(0.645431\pi\)
\(888\) 0 0
\(889\) 34240.0 1.29176
\(890\) 0 0
\(891\) −5894.00 −0.221612
\(892\) 0 0
\(893\) −66096.0 −2.47684
\(894\) 0 0
\(895\) 16740.0 0.625203
\(896\) 0 0
\(897\) 18576.0 0.691454
\(898\) 0 0
\(899\) −256.000 −0.00949731
\(900\) 0 0
\(901\) 45540.0 1.68386
\(902\) 0 0
\(903\) 11920.0 0.439283
\(904\) 0 0
\(905\) −20508.0 −0.753270
\(906\) 0 0
\(907\) 21450.0 0.785265 0.392633 0.919695i \(-0.371564\pi\)
0.392633 + 0.919695i \(0.371564\pi\)
\(908\) 0 0
\(909\) −31970.0 −1.16653
\(910\) 0 0
\(911\) 40904.0 1.48761 0.743804 0.668398i \(-0.233020\pi\)
0.743804 + 0.668398i \(0.233020\pi\)
\(912\) 0 0
\(913\) −9492.00 −0.344074
\(914\) 0 0
\(915\) 3576.00 0.129201
\(916\) 0 0
\(917\) −42360.0 −1.52546
\(918\) 0 0
\(919\) 27380.0 0.982789 0.491394 0.870937i \(-0.336487\pi\)
0.491394 + 0.870937i \(0.336487\pi\)
\(920\) 0 0
\(921\) 10980.0 0.392837
\(922\) 0 0
\(923\) 37800.0 1.34800
\(924\) 0 0
\(925\) −14062.0 −0.499844
\(926\) 0 0
\(927\) 18124.0 0.642147
\(928\) 0 0
\(929\) 10302.0 0.363830 0.181915 0.983314i \(-0.441770\pi\)
0.181915 + 0.983314i \(0.441770\pi\)
\(930\) 0 0
\(931\) −9234.00 −0.325061
\(932\) 0 0
\(933\) −11112.0 −0.389915
\(934\) 0 0
\(935\) 5544.00 0.193912
\(936\) 0 0
\(937\) 5054.00 0.176208 0.0881040 0.996111i \(-0.471919\pi\)
0.0881040 + 0.996111i \(0.471919\pi\)
\(938\) 0 0
\(939\) 4108.00 0.142768
\(940\) 0 0
\(941\) 30462.0 1.05530 0.527648 0.849463i \(-0.323074\pi\)
0.527648 + 0.849463i \(0.323074\pi\)
\(942\) 0 0
\(943\) −34744.0 −1.19981
\(944\) 0 0
\(945\) −12000.0 −0.413079
\(946\) 0 0
\(947\) −32082.0 −1.10087 −0.550436 0.834878i \(-0.685538\pi\)
−0.550436 + 0.834878i \(0.685538\pi\)
\(948\) 0 0
\(949\) −22572.0 −0.772095
\(950\) 0 0
\(951\) −4988.00 −0.170081
\(952\) 0 0
\(953\) 12970.0 0.440860 0.220430 0.975403i \(-0.429254\pi\)
0.220430 + 0.975403i \(0.429254\pi\)
\(954\) 0 0
\(955\) 11808.0 0.400103
\(956\) 0 0
\(957\) −56.0000 −0.00189156
\(958\) 0 0
\(959\) 9720.00 0.327294
\(960\) 0 0
\(961\) −13407.0 −0.450035
\(962\) 0 0
\(963\) 37122.0 1.24220
\(964\) 0 0
\(965\) −6348.00 −0.211761
\(966\) 0 0
\(967\) −1652.00 −0.0549377 −0.0274688 0.999623i \(-0.508745\pi\)
−0.0274688 + 0.999623i \(0.508745\pi\)
\(968\) 0 0
\(969\) −21384.0 −0.708930
\(970\) 0 0
\(971\) 24650.0 0.814682 0.407341 0.913276i \(-0.366456\pi\)
0.407341 + 0.913276i \(0.366456\pi\)
\(972\) 0 0
\(973\) 25720.0 0.847426
\(974\) 0 0
\(975\) 9612.00 0.315723
\(976\) 0 0
\(977\) 23646.0 0.774312 0.387156 0.922014i \(-0.373458\pi\)
0.387156 + 0.922014i \(0.373458\pi\)
\(978\) 0 0
\(979\) 1148.00 0.0374773
\(980\) 0 0
\(981\) −46322.0 −1.50759
\(982\) 0 0
\(983\) −38108.0 −1.23648 −0.618238 0.785991i \(-0.712153\pi\)
−0.618238 + 0.785991i \(0.712153\pi\)
\(984\) 0 0
\(985\) 4356.00 0.140907
\(986\) 0 0
\(987\) 16320.0 0.526313
\(988\) 0 0
\(989\) −51256.0 −1.64797
\(990\) 0 0
\(991\) 18640.0 0.597497 0.298748 0.954332i \(-0.403431\pi\)
0.298748 + 0.954332i \(0.403431\pi\)
\(992\) 0 0
\(993\) −5828.00 −0.186250
\(994\) 0 0
\(995\) −24696.0 −0.786850
\(996\) 0 0
\(997\) 29022.0 0.921902 0.460951 0.887426i \(-0.347508\pi\)
0.460951 + 0.887426i \(0.347508\pi\)
\(998\) 0 0
\(999\) 15800.0 0.500390
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 128.4.a.b.1.1 yes 1
3.2 odd 2 1152.4.a.c.1.1 1
4.3 odd 2 128.4.a.d.1.1 yes 1
8.3 odd 2 128.4.a.a.1.1 1
8.5 even 2 128.4.a.c.1.1 yes 1
12.11 even 2 1152.4.a.d.1.1 1
16.3 odd 4 256.4.b.b.129.2 2
16.5 even 4 256.4.b.f.129.2 2
16.11 odd 4 256.4.b.b.129.1 2
16.13 even 4 256.4.b.f.129.1 2
24.5 odd 2 1152.4.a.i.1.1 1
24.11 even 2 1152.4.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.4.a.a.1.1 1 8.3 odd 2
128.4.a.b.1.1 yes 1 1.1 even 1 trivial
128.4.a.c.1.1 yes 1 8.5 even 2
128.4.a.d.1.1 yes 1 4.3 odd 2
256.4.b.b.129.1 2 16.11 odd 4
256.4.b.b.129.2 2 16.3 odd 4
256.4.b.f.129.1 2 16.13 even 4
256.4.b.f.129.2 2 16.5 even 4
1152.4.a.c.1.1 1 3.2 odd 2
1152.4.a.d.1.1 1 12.11 even 2
1152.4.a.i.1.1 1 24.5 odd 2
1152.4.a.j.1.1 1 24.11 even 2