Properties

Label 1728.4.a.g.1.1
Level $1728$
Weight $4$
Character 1728.1
Self dual yes
Analytic conductor $101.955$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1728,4,Mod(1,1728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1728, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1728.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1728.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(101.955300490\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 108)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1728.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-9.00000 q^{5} -1.00000 q^{7} +O(q^{10})\) \(q-9.00000 q^{5} -1.00000 q^{7} -63.0000 q^{11} +28.0000 q^{13} +72.0000 q^{17} -98.0000 q^{19} +126.000 q^{23} -44.0000 q^{25} +126.000 q^{29} -259.000 q^{31} +9.00000 q^{35} -386.000 q^{37} -450.000 q^{41} +34.0000 q^{43} -54.0000 q^{47} -342.000 q^{49} +693.000 q^{53} +567.000 q^{55} -180.000 q^{59} +280.000 q^{61} -252.000 q^{65} +586.000 q^{67} +504.000 q^{71} +161.000 q^{73} +63.0000 q^{77} +440.000 q^{79} -999.000 q^{83} -648.000 q^{85} +882.000 q^{89} -28.0000 q^{91} +882.000 q^{95} -721.000 q^{97} +O(q^{100})\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −9.00000 −0.804984 −0.402492 0.915423i \(-0.631856\pi\)
−0.402492 + 0.915423i \(0.631856\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.0539949 −0.0269975 0.999636i \(-0.508595\pi\)
−0.0269975 + 0.999636i \(0.508595\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −63.0000 −1.72684 −0.863419 0.504488i \(-0.831681\pi\)
−0.863419 + 0.504488i \(0.831681\pi\)
\(12\) 0 0
\(13\) 28.0000 0.597369 0.298685 0.954352i \(-0.403452\pi\)
0.298685 + 0.954352i \(0.403452\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 72.0000 1.02721 0.513605 0.858027i \(-0.328310\pi\)
0.513605 + 0.858027i \(0.328310\pi\)
\(18\) 0 0
\(19\) −98.0000 −1.18330 −0.591651 0.806194i \(-0.701524\pi\)
−0.591651 + 0.806194i \(0.701524\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 126.000 1.14230 0.571148 0.820847i \(-0.306498\pi\)
0.571148 + 0.820847i \(0.306498\pi\)
\(24\) 0 0
\(25\) −44.0000 −0.352000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 126.000 0.806814 0.403407 0.915021i \(-0.367826\pi\)
0.403407 + 0.915021i \(0.367826\pi\)
\(30\) 0 0
\(31\) −259.000 −1.50057 −0.750287 0.661113i \(-0.770085\pi\)
−0.750287 + 0.661113i \(0.770085\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 9.00000 0.0434651
\(36\) 0 0
\(37\) −386.000 −1.71508 −0.857541 0.514416i \(-0.828009\pi\)
−0.857541 + 0.514416i \(0.828009\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −450.000 −1.71410 −0.857051 0.515231i \(-0.827706\pi\)
−0.857051 + 0.515231i \(0.827706\pi\)
\(42\) 0 0
\(43\) 34.0000 0.120580 0.0602901 0.998181i \(-0.480797\pi\)
0.0602901 + 0.998181i \(0.480797\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −54.0000 −0.167590 −0.0837948 0.996483i \(-0.526704\pi\)
−0.0837948 + 0.996483i \(0.526704\pi\)
\(48\) 0 0
\(49\) −342.000 −0.997085
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 693.000 1.79605 0.898027 0.439940i \(-0.145000\pi\)
0.898027 + 0.439940i \(0.145000\pi\)
\(54\) 0 0
\(55\) 567.000 1.39008
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −180.000 −0.397187 −0.198593 0.980082i \(-0.563637\pi\)
−0.198593 + 0.980082i \(0.563637\pi\)
\(60\) 0 0
\(61\) 280.000 0.587710 0.293855 0.955850i \(-0.405062\pi\)
0.293855 + 0.955850i \(0.405062\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −252.000 −0.480873
\(66\) 0 0
\(67\) 586.000 1.06853 0.534263 0.845318i \(-0.320589\pi\)
0.534263 + 0.845318i \(0.320589\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 504.000 0.842448 0.421224 0.906957i \(-0.361601\pi\)
0.421224 + 0.906957i \(0.361601\pi\)
\(72\) 0 0
\(73\) 161.000 0.258132 0.129066 0.991636i \(-0.458802\pi\)
0.129066 + 0.991636i \(0.458802\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 63.0000 0.0932405
\(78\) 0 0
\(79\) 440.000 0.626631 0.313316 0.949649i \(-0.398560\pi\)
0.313316 + 0.949649i \(0.398560\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −999.000 −1.32114 −0.660569 0.750765i \(-0.729685\pi\)
−0.660569 + 0.750765i \(0.729685\pi\)
\(84\) 0 0
\(85\) −648.000 −0.826888
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 882.000 1.05047 0.525235 0.850957i \(-0.323977\pi\)
0.525235 + 0.850957i \(0.323977\pi\)
\(90\) 0 0
\(91\) −28.0000 −0.0322549
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 882.000 0.952540
\(96\) 0 0
\(97\) −721.000 −0.754706 −0.377353 0.926070i \(-0.623166\pi\)
−0.377353 + 0.926070i \(0.623166\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 441.000 0.434467 0.217233 0.976120i \(-0.430297\pi\)
0.217233 + 0.976120i \(0.430297\pi\)
\(102\) 0 0
\(103\) −532.000 −0.508927 −0.254464 0.967082i \(-0.581899\pi\)
−0.254464 + 0.967082i \(0.581899\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −819.000 −0.739960 −0.369980 0.929040i \(-0.620635\pi\)
−0.369980 + 0.929040i \(0.620635\pi\)
\(108\) 0 0
\(109\) 1294.00 1.13709 0.568545 0.822652i \(-0.307507\pi\)
0.568545 + 0.822652i \(0.307507\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1134.00 0.944051 0.472025 0.881585i \(-0.343523\pi\)
0.472025 + 0.881585i \(0.343523\pi\)
\(114\) 0 0
\(115\) −1134.00 −0.919531
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −72.0000 −0.0554641
\(120\) 0 0
\(121\) 2638.00 1.98197
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1521.00 1.08834
\(126\) 0 0
\(127\) −1807.00 −1.26256 −0.631281 0.775554i \(-0.717470\pi\)
−0.631281 + 0.775554i \(0.717470\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 2205.00 1.47062 0.735312 0.677729i \(-0.237036\pi\)
0.735312 + 0.677729i \(0.237036\pi\)
\(132\) 0 0
\(133\) 98.0000 0.0638923
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1386.00 0.864336 0.432168 0.901793i \(-0.357749\pi\)
0.432168 + 0.901793i \(0.357749\pi\)
\(138\) 0 0
\(139\) −476.000 −0.290459 −0.145229 0.989398i \(-0.546392\pi\)
−0.145229 + 0.989398i \(0.546392\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −1764.00 −1.03156
\(144\) 0 0
\(145\) −1134.00 −0.649473
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1575.00 0.865967 0.432983 0.901402i \(-0.357461\pi\)
0.432983 + 0.901402i \(0.357461\pi\)
\(150\) 0 0
\(151\) 449.000 0.241981 0.120990 0.992654i \(-0.461393\pi\)
0.120990 + 0.992654i \(0.461393\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2331.00 1.20794
\(156\) 0 0
\(157\) −1820.00 −0.925171 −0.462585 0.886575i \(-0.653078\pi\)
−0.462585 + 0.886575i \(0.653078\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −126.000 −0.0616782
\(162\) 0 0
\(163\) 1828.00 0.878405 0.439202 0.898388i \(-0.355261\pi\)
0.439202 + 0.898388i \(0.355261\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −810.000 −0.375327 −0.187664 0.982233i \(-0.560092\pi\)
−0.187664 + 0.982233i \(0.560092\pi\)
\(168\) 0 0
\(169\) −1413.00 −0.643150
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1323.00 0.581421 0.290710 0.956811i \(-0.406108\pi\)
0.290710 + 0.956811i \(0.406108\pi\)
\(174\) 0 0
\(175\) 44.0000 0.0190062
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 315.000 0.131532 0.0657659 0.997835i \(-0.479051\pi\)
0.0657659 + 0.997835i \(0.479051\pi\)
\(180\) 0 0
\(181\) 2800.00 1.14985 0.574924 0.818207i \(-0.305032\pi\)
0.574924 + 0.818207i \(0.305032\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 3474.00 1.38061
\(186\) 0 0
\(187\) −4536.00 −1.77382
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −3276.00 −1.24106 −0.620532 0.784182i \(-0.713083\pi\)
−0.620532 + 0.784182i \(0.713083\pi\)
\(192\) 0 0
\(193\) 3221.00 1.20131 0.600655 0.799509i \(-0.294907\pi\)
0.600655 + 0.799509i \(0.294907\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −3339.00 −1.20758 −0.603792 0.797142i \(-0.706344\pi\)
−0.603792 + 0.797142i \(0.706344\pi\)
\(198\) 0 0
\(199\) 3689.00 1.31410 0.657051 0.753846i \(-0.271804\pi\)
0.657051 + 0.753846i \(0.271804\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −126.000 −0.0435639
\(204\) 0 0
\(205\) 4050.00 1.37983
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 6174.00 2.04337
\(210\) 0 0
\(211\) 6022.00 1.96479 0.982397 0.186805i \(-0.0598131\pi\)
0.982397 + 0.186805i \(0.0598131\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −306.000 −0.0970652
\(216\) 0 0
\(217\) 259.000 0.0810233
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2016.00 0.613624
\(222\) 0 0
\(223\) −952.000 −0.285877 −0.142939 0.989732i \(-0.545655\pi\)
−0.142939 + 0.989732i \(0.545655\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −5292.00 −1.54732 −0.773662 0.633599i \(-0.781577\pi\)
−0.773662 + 0.633599i \(0.781577\pi\)
\(228\) 0 0
\(229\) −2198.00 −0.634270 −0.317135 0.948380i \(-0.602721\pi\)
−0.317135 + 0.948380i \(0.602721\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5166.00 1.45251 0.726257 0.687423i \(-0.241258\pi\)
0.726257 + 0.687423i \(0.241258\pi\)
\(234\) 0 0
\(235\) 486.000 0.134907
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 3402.00 0.920741 0.460370 0.887727i \(-0.347717\pi\)
0.460370 + 0.887727i \(0.347717\pi\)
\(240\) 0 0
\(241\) 1862.00 0.497684 0.248842 0.968544i \(-0.419950\pi\)
0.248842 + 0.968544i \(0.419950\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 3078.00 0.802638
\(246\) 0 0
\(247\) −2744.00 −0.706869
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 5472.00 1.37605 0.688027 0.725685i \(-0.258477\pi\)
0.688027 + 0.725685i \(0.258477\pi\)
\(252\) 0 0
\(253\) −7938.00 −1.97256
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 5292.00 1.28446 0.642229 0.766513i \(-0.278010\pi\)
0.642229 + 0.766513i \(0.278010\pi\)
\(258\) 0 0
\(259\) 386.000 0.0926057
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1638.00 0.384043 0.192022 0.981391i \(-0.438496\pi\)
0.192022 + 0.981391i \(0.438496\pi\)
\(264\) 0 0
\(265\) −6237.00 −1.44580
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 1206.00 0.273350 0.136675 0.990616i \(-0.456358\pi\)
0.136675 + 0.990616i \(0.456358\pi\)
\(270\) 0 0
\(271\) 4319.00 0.968120 0.484060 0.875035i \(-0.339162\pi\)
0.484060 + 0.875035i \(0.339162\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2772.00 0.607847
\(276\) 0 0
\(277\) 2248.00 0.487615 0.243807 0.969824i \(-0.421604\pi\)
0.243807 + 0.969824i \(0.421604\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −2016.00 −0.427987 −0.213994 0.976835i \(-0.568647\pi\)
−0.213994 + 0.976835i \(0.568647\pi\)
\(282\) 0 0
\(283\) 2338.00 0.491094 0.245547 0.969385i \(-0.421032\pi\)
0.245547 + 0.969385i \(0.421032\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 450.000 0.0925528
\(288\) 0 0
\(289\) 271.000 0.0551598
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 8514.00 1.69759 0.848794 0.528724i \(-0.177329\pi\)
0.848794 + 0.528724i \(0.177329\pi\)
\(294\) 0 0
\(295\) 1620.00 0.319729
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 3528.00 0.682373
\(300\) 0 0
\(301\) −34.0000 −0.00651072
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −2520.00 −0.473098
\(306\) 0 0
\(307\) −6104.00 −1.13477 −0.567384 0.823453i \(-0.692044\pi\)
−0.567384 + 0.823453i \(0.692044\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −4338.00 −0.790950 −0.395475 0.918477i \(-0.629420\pi\)
−0.395475 + 0.918477i \(0.629420\pi\)
\(312\) 0 0
\(313\) −8155.00 −1.47268 −0.736338 0.676613i \(-0.763447\pi\)
−0.736338 + 0.676613i \(0.763447\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 4977.00 0.881818 0.440909 0.897552i \(-0.354656\pi\)
0.440909 + 0.897552i \(0.354656\pi\)
\(318\) 0 0
\(319\) −7938.00 −1.39324
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −7056.00 −1.21550
\(324\) 0 0
\(325\) −1232.00 −0.210274
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 54.0000 0.00904899
\(330\) 0 0
\(331\) −5678.00 −0.942873 −0.471437 0.881900i \(-0.656264\pi\)
−0.471437 + 0.881900i \(0.656264\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −5274.00 −0.860147
\(336\) 0 0
\(337\) 2906.00 0.469733 0.234866 0.972028i \(-0.424535\pi\)
0.234866 + 0.972028i \(0.424535\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 16317.0 2.59125
\(342\) 0 0
\(343\) 685.000 0.107832
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −6993.00 −1.08186 −0.540928 0.841069i \(-0.681927\pi\)
−0.540928 + 0.841069i \(0.681927\pi\)
\(348\) 0 0
\(349\) −7910.00 −1.21322 −0.606608 0.795001i \(-0.707470\pi\)
−0.606608 + 0.795001i \(0.707470\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 2466.00 0.371819 0.185909 0.982567i \(-0.440477\pi\)
0.185909 + 0.982567i \(0.440477\pi\)
\(354\) 0 0
\(355\) −4536.00 −0.678157
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 7182.00 1.05585 0.527927 0.849290i \(-0.322970\pi\)
0.527927 + 0.849290i \(0.322970\pi\)
\(360\) 0 0
\(361\) 2745.00 0.400204
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −1449.00 −0.207792
\(366\) 0 0
\(367\) −11431.0 −1.62587 −0.812934 0.582356i \(-0.802131\pi\)
−0.812934 + 0.582356i \(0.802131\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −693.000 −0.0969778
\(372\) 0 0
\(373\) 6616.00 0.918401 0.459200 0.888333i \(-0.348136\pi\)
0.459200 + 0.888333i \(0.348136\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3528.00 0.481966
\(378\) 0 0
\(379\) 9820.00 1.33092 0.665461 0.746433i \(-0.268235\pi\)
0.665461 + 0.746433i \(0.268235\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1440.00 0.192116 0.0960582 0.995376i \(-0.469377\pi\)
0.0960582 + 0.995376i \(0.469377\pi\)
\(384\) 0 0
\(385\) −567.000 −0.0750571
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −5985.00 −0.780081 −0.390041 0.920798i \(-0.627539\pi\)
−0.390041 + 0.920798i \(0.627539\pi\)
\(390\) 0 0
\(391\) 9072.00 1.17338
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −3960.00 −0.504428
\(396\) 0 0
\(397\) 11284.0 1.42652 0.713259 0.700900i \(-0.247218\pi\)
0.713259 + 0.700900i \(0.247218\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −7308.00 −0.910085 −0.455043 0.890470i \(-0.650376\pi\)
−0.455043 + 0.890470i \(0.650376\pi\)
\(402\) 0 0
\(403\) −7252.00 −0.896397
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 24318.0 2.96167
\(408\) 0 0
\(409\) 6335.00 0.765882 0.382941 0.923773i \(-0.374911\pi\)
0.382941 + 0.923773i \(0.374911\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 180.000 0.0214461
\(414\) 0 0
\(415\) 8991.00 1.06350
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −6372.00 −0.742942 −0.371471 0.928445i \(-0.621146\pi\)
−0.371471 + 0.928445i \(0.621146\pi\)
\(420\) 0 0
\(421\) −3320.00 −0.384339 −0.192170 0.981362i \(-0.561552\pi\)
−0.192170 + 0.981362i \(0.561552\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3168.00 −0.361578
\(426\) 0 0
\(427\) −280.000 −0.0317334
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −2898.00 −0.323879 −0.161939 0.986801i \(-0.551775\pi\)
−0.161939 + 0.986801i \(0.551775\pi\)
\(432\) 0 0
\(433\) −4291.00 −0.476241 −0.238120 0.971236i \(-0.576531\pi\)
−0.238120 + 0.971236i \(0.576531\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −12348.0 −1.35168
\(438\) 0 0
\(439\) −8323.00 −0.904864 −0.452432 0.891799i \(-0.649443\pi\)
−0.452432 + 0.891799i \(0.649443\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 3780.00 0.405402 0.202701 0.979241i \(-0.435028\pi\)
0.202701 + 0.979241i \(0.435028\pi\)
\(444\) 0 0
\(445\) −7938.00 −0.845612
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 12474.0 1.31110 0.655551 0.755151i \(-0.272437\pi\)
0.655551 + 0.755151i \(0.272437\pi\)
\(450\) 0 0
\(451\) 28350.0 2.95998
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 252.000 0.0259647
\(456\) 0 0
\(457\) 16679.0 1.70724 0.853622 0.520893i \(-0.174401\pi\)
0.853622 + 0.520893i \(0.174401\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 17271.0 1.74488 0.872441 0.488719i \(-0.162536\pi\)
0.872441 + 0.488719i \(0.162536\pi\)
\(462\) 0 0
\(463\) 17387.0 1.74523 0.872616 0.488407i \(-0.162422\pi\)
0.872616 + 0.488407i \(0.162422\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −3087.00 −0.305887 −0.152944 0.988235i \(-0.548875\pi\)
−0.152944 + 0.988235i \(0.548875\pi\)
\(468\) 0 0
\(469\) −586.000 −0.0576950
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −2142.00 −0.208223
\(474\) 0 0
\(475\) 4312.00 0.416522
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −5238.00 −0.499646 −0.249823 0.968292i \(-0.580372\pi\)
−0.249823 + 0.968292i \(0.580372\pi\)
\(480\) 0 0
\(481\) −10808.0 −1.02454
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 6489.00 0.607526
\(486\) 0 0
\(487\) −4384.00 −0.407922 −0.203961 0.978979i \(-0.565382\pi\)
−0.203961 + 0.978979i \(0.565382\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 3843.00 0.353222 0.176611 0.984281i \(-0.443486\pi\)
0.176611 + 0.984281i \(0.443486\pi\)
\(492\) 0 0
\(493\) 9072.00 0.828767
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −504.000 −0.0454879
\(498\) 0 0
\(499\) −12566.0 −1.12732 −0.563659 0.826008i \(-0.690607\pi\)
−0.563659 + 0.826008i \(0.690607\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −5238.00 −0.464316 −0.232158 0.972678i \(-0.574579\pi\)
−0.232158 + 0.972678i \(0.574579\pi\)
\(504\) 0 0
\(505\) −3969.00 −0.349739
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 6309.00 0.549394 0.274697 0.961531i \(-0.411422\pi\)
0.274697 + 0.961531i \(0.411422\pi\)
\(510\) 0 0
\(511\) −161.000 −0.0139378
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 4788.00 0.409679
\(516\) 0 0
\(517\) 3402.00 0.289400
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −5868.00 −0.493439 −0.246720 0.969087i \(-0.579353\pi\)
−0.246720 + 0.969087i \(0.579353\pi\)
\(522\) 0 0
\(523\) −6776.00 −0.566527 −0.283264 0.959042i \(-0.591417\pi\)
−0.283264 + 0.959042i \(0.591417\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −18648.0 −1.54140
\(528\) 0 0
\(529\) 3709.00 0.304841
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −12600.0 −1.02395
\(534\) 0 0
\(535\) 7371.00 0.595656
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 21546.0 1.72180
\(540\) 0 0
\(541\) 9388.00 0.746066 0.373033 0.927818i \(-0.378318\pi\)
0.373033 + 0.927818i \(0.378318\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −11646.0 −0.915339
\(546\) 0 0
\(547\) −15056.0 −1.17687 −0.588435 0.808544i \(-0.700256\pi\)
−0.588435 + 0.808544i \(0.700256\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −12348.0 −0.954705
\(552\) 0 0
\(553\) −440.000 −0.0338349
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −16569.0 −1.26041 −0.630207 0.776427i \(-0.717030\pi\)
−0.630207 + 0.776427i \(0.717030\pi\)
\(558\) 0 0
\(559\) 952.000 0.0720310
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −14553.0 −1.08941 −0.544703 0.838629i \(-0.683358\pi\)
−0.544703 + 0.838629i \(0.683358\pi\)
\(564\) 0 0
\(565\) −10206.0 −0.759946
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 4284.00 0.315632 0.157816 0.987469i \(-0.449555\pi\)
0.157816 + 0.987469i \(0.449555\pi\)
\(570\) 0 0
\(571\) −692.000 −0.0507168 −0.0253584 0.999678i \(-0.508073\pi\)
−0.0253584 + 0.999678i \(0.508073\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −5544.00 −0.402088
\(576\) 0 0
\(577\) 4886.00 0.352525 0.176262 0.984343i \(-0.443599\pi\)
0.176262 + 0.984343i \(0.443599\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 999.000 0.0713348
\(582\) 0 0
\(583\) −43659.0 −3.10149
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 11025.0 0.775214 0.387607 0.921825i \(-0.373302\pi\)
0.387607 + 0.921825i \(0.373302\pi\)
\(588\) 0 0
\(589\) 25382.0 1.77563
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −14562.0 −1.00841 −0.504207 0.863583i \(-0.668215\pi\)
−0.504207 + 0.863583i \(0.668215\pi\)
\(594\) 0 0
\(595\) 648.000 0.0446477
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 11466.0 0.782117 0.391058 0.920366i \(-0.372109\pi\)
0.391058 + 0.920366i \(0.372109\pi\)
\(600\) 0 0
\(601\) 10955.0 0.743534 0.371767 0.928326i \(-0.378752\pi\)
0.371767 + 0.928326i \(0.378752\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −23742.0 −1.59545
\(606\) 0 0
\(607\) 1232.00 0.0823811 0.0411906 0.999151i \(-0.486885\pi\)
0.0411906 + 0.999151i \(0.486885\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −1512.00 −0.100113
\(612\) 0 0
\(613\) −25622.0 −1.68819 −0.844097 0.536191i \(-0.819863\pi\)
−0.844097 + 0.536191i \(0.819863\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 21042.0 1.37296 0.686482 0.727147i \(-0.259154\pi\)
0.686482 + 0.727147i \(0.259154\pi\)
\(618\) 0 0
\(619\) −6524.00 −0.423621 −0.211811 0.977311i \(-0.567936\pi\)
−0.211811 + 0.977311i \(0.567936\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −882.000 −0.0567200
\(624\) 0 0
\(625\) −8189.00 −0.524096
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −27792.0 −1.76175
\(630\) 0 0
\(631\) 20663.0 1.30361 0.651807 0.758384i \(-0.274011\pi\)
0.651807 + 0.758384i \(0.274011\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 16263.0 1.01634
\(636\) 0 0
\(637\) −9576.00 −0.595628
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −882.000 −0.0543477 −0.0271739 0.999631i \(-0.508651\pi\)
−0.0271739 + 0.999631i \(0.508651\pi\)
\(642\) 0 0
\(643\) 7252.00 0.444776 0.222388 0.974958i \(-0.428615\pi\)
0.222388 + 0.974958i \(0.428615\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 21168.0 1.28624 0.643122 0.765764i \(-0.277639\pi\)
0.643122 + 0.765764i \(0.277639\pi\)
\(648\) 0 0
\(649\) 11340.0 0.685877
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −14301.0 −0.857031 −0.428516 0.903534i \(-0.640963\pi\)
−0.428516 + 0.903534i \(0.640963\pi\)
\(654\) 0 0
\(655\) −19845.0 −1.18383
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −15057.0 −0.890042 −0.445021 0.895520i \(-0.646804\pi\)
−0.445021 + 0.895520i \(0.646804\pi\)
\(660\) 0 0
\(661\) 4690.00 0.275976 0.137988 0.990434i \(-0.455937\pi\)
0.137988 + 0.990434i \(0.455937\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −882.000 −0.0514323
\(666\) 0 0
\(667\) 15876.0 0.921621
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −17640.0 −1.01488
\(672\) 0 0
\(673\) −5203.00 −0.298010 −0.149005 0.988836i \(-0.547607\pi\)
−0.149005 + 0.988836i \(0.547607\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −6174.00 −0.350496 −0.175248 0.984524i \(-0.556073\pi\)
−0.175248 + 0.984524i \(0.556073\pi\)
\(678\) 0 0
\(679\) 721.000 0.0407503
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −756.000 −0.0423536 −0.0211768 0.999776i \(-0.506741\pi\)
−0.0211768 + 0.999776i \(0.506741\pi\)
\(684\) 0 0
\(685\) −12474.0 −0.695777
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 19404.0 1.07291
\(690\) 0 0
\(691\) −17948.0 −0.988096 −0.494048 0.869435i \(-0.664483\pi\)
−0.494048 + 0.869435i \(0.664483\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 4284.00 0.233815
\(696\) 0 0
\(697\) −32400.0 −1.76074
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −34083.0 −1.83637 −0.918186 0.396149i \(-0.870346\pi\)
−0.918186 + 0.396149i \(0.870346\pi\)
\(702\) 0 0
\(703\) 37828.0 2.02946
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −441.000 −0.0234590
\(708\) 0 0
\(709\) 5284.00 0.279894 0.139947 0.990159i \(-0.455307\pi\)
0.139947 + 0.990159i \(0.455307\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −32634.0 −1.71410
\(714\) 0 0
\(715\) 15876.0 0.830390
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −24516.0 −1.27162 −0.635808 0.771847i \(-0.719333\pi\)
−0.635808 + 0.771847i \(0.719333\pi\)
\(720\) 0 0
\(721\) 532.000 0.0274795
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −5544.00 −0.283999
\(726\) 0 0
\(727\) −12481.0 −0.636719 −0.318359 0.947970i \(-0.603132\pi\)
−0.318359 + 0.947970i \(0.603132\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 2448.00 0.123861
\(732\) 0 0
\(733\) 3094.00 0.155907 0.0779533 0.996957i \(-0.475162\pi\)
0.0779533 + 0.996957i \(0.475162\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −36918.0 −1.84517
\(738\) 0 0
\(739\) 376.000 0.0187164 0.00935818 0.999956i \(-0.497021\pi\)
0.00935818 + 0.999956i \(0.497021\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 29484.0 1.45580 0.727902 0.685681i \(-0.240495\pi\)
0.727902 + 0.685681i \(0.240495\pi\)
\(744\) 0 0
\(745\) −14175.0 −0.697090
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 819.000 0.0399541
\(750\) 0 0
\(751\) 11681.0 0.567571 0.283785 0.958888i \(-0.408410\pi\)
0.283785 + 0.958888i \(0.408410\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −4041.00 −0.194791
\(756\) 0 0
\(757\) −890.000 −0.0427313 −0.0213657 0.999772i \(-0.506801\pi\)
−0.0213657 + 0.999772i \(0.506801\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −32832.0 −1.56394 −0.781970 0.623315i \(-0.785785\pi\)
−0.781970 + 0.623315i \(0.785785\pi\)
\(762\) 0 0
\(763\) −1294.00 −0.0613970
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −5040.00 −0.237267
\(768\) 0 0
\(769\) 3185.00 0.149355 0.0746775 0.997208i \(-0.476207\pi\)
0.0746775 + 0.997208i \(0.476207\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 21222.0 0.987454 0.493727 0.869617i \(-0.335634\pi\)
0.493727 + 0.869617i \(0.335634\pi\)
\(774\) 0 0
\(775\) 11396.0 0.528202
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 44100.0 2.02830
\(780\) 0 0
\(781\) −31752.0 −1.45477
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 16380.0 0.744748
\(786\) 0 0
\(787\) −12320.0 −0.558019 −0.279009 0.960288i \(-0.590006\pi\)
−0.279009 + 0.960288i \(0.590006\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −1134.00 −0.0509740
\(792\) 0 0
\(793\) 7840.00 0.351080
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 11907.0 0.529194 0.264597 0.964359i \(-0.414761\pi\)
0.264597 + 0.964359i \(0.414761\pi\)
\(798\) 0 0
\(799\) −3888.00 −0.172150
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −10143.0 −0.445752
\(804\) 0 0
\(805\) 1134.00 0.0496500
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −14994.0 −0.651620 −0.325810 0.945435i \(-0.605637\pi\)
−0.325810 + 0.945435i \(0.605637\pi\)
\(810\) 0 0
\(811\) 38878.0 1.68334 0.841672 0.539990i \(-0.181572\pi\)
0.841672 + 0.539990i \(0.181572\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −16452.0 −0.707102
\(816\) 0 0
\(817\) −3332.00 −0.142683
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −3906.00 −0.166042 −0.0830209 0.996548i \(-0.526457\pi\)
−0.0830209 + 0.996548i \(0.526457\pi\)
\(822\) 0 0
\(823\) −10207.0 −0.432313 −0.216157 0.976359i \(-0.569352\pi\)
−0.216157 + 0.976359i \(0.569352\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −8064.00 −0.339072 −0.169536 0.985524i \(-0.554227\pi\)
−0.169536 + 0.985524i \(0.554227\pi\)
\(828\) 0 0
\(829\) 10486.0 0.439317 0.219659 0.975577i \(-0.429506\pi\)
0.219659 + 0.975577i \(0.429506\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −24624.0 −1.02421
\(834\) 0 0
\(835\) 7290.00 0.302133
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −33696.0 −1.38655 −0.693275 0.720673i \(-0.743833\pi\)
−0.693275 + 0.720673i \(0.743833\pi\)
\(840\) 0 0
\(841\) −8513.00 −0.349051
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 12717.0 0.517726
\(846\) 0 0
\(847\) −2638.00 −0.107016
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −48636.0 −1.95913
\(852\) 0 0
\(853\) 15778.0 0.633328 0.316664 0.948538i \(-0.397437\pi\)
0.316664 + 0.948538i \(0.397437\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −26136.0 −1.04176 −0.520880 0.853630i \(-0.674396\pi\)
−0.520880 + 0.853630i \(0.674396\pi\)
\(858\) 0 0
\(859\) −23912.0 −0.949787 −0.474893 0.880043i \(-0.657513\pi\)
−0.474893 + 0.880043i \(0.657513\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 17892.0 0.705737 0.352868 0.935673i \(-0.385206\pi\)
0.352868 + 0.935673i \(0.385206\pi\)
\(864\) 0 0
\(865\) −11907.0 −0.468035
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −27720.0 −1.08209
\(870\) 0 0
\(871\) 16408.0 0.638305
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −1521.00 −0.0587648
\(876\) 0 0
\(877\) −44162.0 −1.70039 −0.850197 0.526465i \(-0.823517\pi\)
−0.850197 + 0.526465i \(0.823517\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −8820.00 −0.337291 −0.168645 0.985677i \(-0.553939\pi\)
−0.168645 + 0.985677i \(0.553939\pi\)
\(882\) 0 0
\(883\) 4654.00 0.177372 0.0886861 0.996060i \(-0.471733\pi\)
0.0886861 + 0.996060i \(0.471733\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 33084.0 1.25237 0.626185 0.779675i \(-0.284616\pi\)
0.626185 + 0.779675i \(0.284616\pi\)
\(888\) 0 0
\(889\) 1807.00 0.0681719
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 5292.00 0.198309
\(894\) 0 0
\(895\) −2835.00 −0.105881
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −32634.0 −1.21068
\(900\) 0 0
\(901\) 49896.0 1.84492
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −25200.0 −0.925609
\(906\) 0 0
\(907\) −47258.0 −1.73007 −0.865036 0.501709i \(-0.832705\pi\)
−0.865036 + 0.501709i \(0.832705\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 36666.0 1.33348 0.666739 0.745291i \(-0.267690\pi\)
0.666739 + 0.745291i \(0.267690\pi\)
\(912\) 0 0
\(913\) 62937.0 2.28139
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −2205.00 −0.0794062
\(918\) 0 0
\(919\) −2995.00 −0.107504 −0.0537519 0.998554i \(-0.517118\pi\)
−0.0537519 + 0.998554i \(0.517118\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 14112.0 0.503253
\(924\) 0 0
\(925\) 16984.0 0.603709
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −13284.0 −0.469143 −0.234572 0.972099i \(-0.575369\pi\)
−0.234572 + 0.972099i \(0.575369\pi\)
\(930\) 0 0
\(931\) 33516.0 1.17985
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 40824.0 1.42790
\(936\) 0 0
\(937\) −2989.00 −0.104212 −0.0521059 0.998642i \(-0.516593\pi\)
−0.0521059 + 0.998642i \(0.516593\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −46485.0 −1.61038 −0.805190 0.593017i \(-0.797937\pi\)
−0.805190 + 0.593017i \(0.797937\pi\)
\(942\) 0 0
\(943\) −56700.0 −1.95801
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −11529.0 −0.395609 −0.197805 0.980241i \(-0.563381\pi\)
−0.197805 + 0.980241i \(0.563381\pi\)
\(948\) 0 0
\(949\) 4508.00 0.154200
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 37044.0 1.25915 0.629577 0.776938i \(-0.283228\pi\)
0.629577 + 0.776938i \(0.283228\pi\)
\(954\) 0 0
\(955\) 29484.0 0.999036
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −1386.00 −0.0466697
\(960\) 0 0
\(961\) 37290.0 1.25172
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −28989.0 −0.967035
\(966\) 0 0
\(967\) 12455.0 0.414194 0.207097 0.978320i \(-0.433598\pi\)
0.207097 + 0.978320i \(0.433598\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 20169.0 0.666585 0.333292 0.942823i \(-0.391840\pi\)
0.333292 + 0.942823i \(0.391840\pi\)
\(972\) 0 0
\(973\) 476.000 0.0156833
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −39942.0 −1.30794 −0.653970 0.756520i \(-0.726898\pi\)
−0.653970 + 0.756520i \(0.726898\pi\)
\(978\) 0 0
\(979\) −55566.0 −1.81399
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 54234.0 1.75971 0.879856 0.475241i \(-0.157639\pi\)
0.879856 + 0.475241i \(0.157639\pi\)
\(984\) 0 0
\(985\) 30051.0 0.972086
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 4284.00 0.137738
\(990\) 0 0
\(991\) −26137.0 −0.837809 −0.418905 0.908030i \(-0.637586\pi\)
−0.418905 + 0.908030i \(0.637586\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −33201.0 −1.05783
\(996\) 0 0
\(997\) 33460.0 1.06288 0.531439 0.847097i \(-0.321652\pi\)
0.531439 + 0.847097i \(0.321652\pi\)
\(998\) 0 0
\(999\) 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 1728.4.a.g.1.1 1
3.2 odd 2 1728.4.a.y.1.1 1
4.3 odd 2 1728.4.a.h.1.1 1
8.3 odd 2 432.4.a.l.1.1 1
8.5 even 2 108.4.a.d.1.1 yes 1
12.11 even 2 1728.4.a.z.1.1 1
24.5 odd 2 108.4.a.a.1.1 1
24.11 even 2 432.4.a.c.1.1 1
72.5 odd 6 324.4.e.g.217.1 2
72.13 even 6 324.4.e.b.217.1 2
72.29 odd 6 324.4.e.g.109.1 2
72.61 even 6 324.4.e.b.109.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
108.4.a.a.1.1 1 24.5 odd 2
108.4.a.d.1.1 yes 1 8.5 even 2
324.4.e.b.109.1 2 72.61 even 6
324.4.e.b.217.1 2 72.13 even 6
324.4.e.g.109.1 2 72.29 odd 6
324.4.e.g.217.1 2 72.5 odd 6
432.4.a.c.1.1 1 24.11 even 2
432.4.a.l.1.1 1 8.3 odd 2
1728.4.a.g.1.1 1 1.1 even 1 trivial
1728.4.a.h.1.1 1 4.3 odd 2
1728.4.a.y.1.1 1 3.2 odd 2
1728.4.a.z.1.1 1 12.11 even 2