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Properties

Label 108.4.a.a.1.1
Level $108$
Weight $4$
Character 108.1
Self dual yes
Analytic conductor $6.372$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [108,4,Mod(1,108)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(108, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("108.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 108.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: \(6.37220628062\)
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 \(\chi\) \(=\) 108.1

$q$-expansion

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

Display 100 coefficients 10 coefficients

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 \(-0.596548\pi\)
−0.298685 + 0.954352i \(0.596548\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −72.0000 −1.02721 −0.513605 − 0.858027i \(-0.671690\pi\)
−0.513605 + 0.858027i \(0.671690\pi\)
\(18\) 0 0
\(19\) 98.0000 1.18330 0.591651 − 0.806194i \(-0.298476\pi\)
0.591651 + 0.806194i \(0.298476\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −126.000 −1.14230 −0.571148 − 0.820847i \(-0.693502\pi\)
−0.571148 + 0.820847i \(0.693502\pi\)
\(24\) 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 \(-0.171991\pi\)
0.857541 + 0.514416i \(0.171991\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 450.000 1.71410 0.857051 − 0.515231i \(-0.172294\pi\)
0.857051 + 0.515231i \(0.172294\pi\)
\(42\) 0 0
\(43\) −34.0000 −0.120580 −0.0602901 − 0.998181i \(-0.519203\pi\)
−0.0602901 + 0.998181i \(0.519203\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 54.0000 0.167590 0.0837948 − 0.996483i \(-0.473296\pi\)
0.0837948 + 0.996483i \(0.473296\pi\)
\(48\) 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 \(-0.594938\pi\)
−0.293855 + 0.955850i \(0.594938\pi\)
\(62\) 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 \(-0.679411\pi\)
−0.534263 + 0.845318i \(0.679411\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −504.000 −0.842448 −0.421224 − 0.906957i \(-0.638399\pi\)
−0.421224 + 0.906957i \(0.638399\pi\)
\(72\) 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 \(-0.676023\pi\)
−0.525235 + 0.850957i \(0.676023\pi\)
\(90\) 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 \(-0.692493\pi\)
−0.568545 + 0.822652i \(0.692493\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1134.00 −0.944051 −0.472025 − 0.881585i \(-0.656477\pi\)
−0.472025 + 0.881585i \(0.656477\pi\)
\(114\) 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 \(-0.642251\pi\)
−0.432168 + 0.901793i \(0.642251\pi\)
\(138\) 0 0
\(139\) 476.000 0.290459 0.145229 − 0.989398i \(-0.453608\pi\)
0.145229 + 0.989398i \(0.453608\pi\)
\(140\) 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 \(-0.346922\pi\)
0.462585 + 0.886575i \(0.346922\pi\)
\(158\) 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 \(-0.644739\pi\)
−0.439202 + 0.898388i \(0.644739\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 810.000 0.375327 0.187664 − 0.982233i \(-0.439908\pi\)
0.187664 + 0.982233i \(0.439908\pi\)
\(168\) 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 \(-0.694968\pi\)
−0.574924 + 0.818207i \(0.694968\pi\)
\(182\) 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 \(-0.286917\pi\)
0.620532 + 0.784182i \(0.286917\pi\)
\(192\) 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 \(-0.940187\pi\)
−0.982397 + 0.186805i \(0.940187\pi\)
\(212\) 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 \(-0.397279\pi\)
0.317135 + 0.948380i \(0.397279\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −5166.00 −1.45251 −0.726257 − 0.687423i \(-0.758742\pi\)
−0.726257 + 0.687423i \(0.758742\pi\)
\(234\) 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 \(-0.652283\pi\)
−0.460370 + 0.887727i \(0.652283\pi\)
\(240\) 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 \(-0.721990\pi\)
−0.642229 + 0.766513i \(0.721990\pi\)
\(258\) 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 \(-0.561504\pi\)
−0.192022 + 0.981391i \(0.561504\pi\)
\(264\) 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 \(-0.578396\pi\)
−0.243807 + 0.969824i \(0.578396\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2016.00 0.427987 0.213994 − 0.976835i \(-0.431353\pi\)
0.213994 + 0.976835i \(0.431353\pi\)
\(282\) 0 0
\(283\) −2338.00 −0.491094 −0.245547 − 0.969385i \(-0.578968\pi\)
−0.245547 + 0.969385i \(0.578968\pi\)
\(284\) 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 \(-0.307956\pi\)
0.567384 + 0.823453i \(0.307956\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 4338.00 0.790950 0.395475 − 0.918477i \(-0.370580\pi\)
0.395475 + 0.918477i \(0.370580\pi\)
\(312\) 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 \(-0.343736\pi\)
0.471437 + 0.881900i \(0.343736\pi\)
\(332\) 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 \(-0.292530\pi\)
0.606608 + 0.795001i \(0.292530\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −2466.00 −0.371819 −0.185909 − 0.982567i \(-0.559523\pi\)
−0.185909 + 0.982567i \(0.559523\pi\)
\(354\) 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 \(-0.677030\pi\)
−0.527927 + 0.849290i \(0.677030\pi\)
\(360\) 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 \(-0.651864\pi\)
−0.459200 + 0.888333i \(0.651864\pi\)
\(374\) 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 \(-0.731765\pi\)
−0.665461 + 0.746433i \(0.731765\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1440.00 −0.192116 −0.0960582 − 0.995376i \(-0.530623\pi\)
−0.0960582 + 0.995376i \(0.530623\pi\)
\(384\) 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 \(-0.752782\pi\)
−0.713259 + 0.700900i \(0.752782\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 7308.00 0.910085 0.455043 − 0.890470i \(-0.349624\pi\)
0.455043 + 0.890470i \(0.349624\pi\)
\(402\) 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 \(-0.438448\pi\)
0.192170 + 0.981362i \(0.438448\pi\)
\(422\) 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 \(-0.448225\pi\)
0.161939 + 0.986801i \(0.448225\pi\)
\(432\) 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 \(-0.727563\pi\)
−0.655551 + 0.755151i \(0.727563\pi\)
\(450\) 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 \(-0.419628\pi\)
0.249823 + 0.968292i \(0.419628\pi\)
\(480\) 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 \(-0.309393\pi\)
0.563659 + 0.826008i \(0.309393\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 5238.00 0.464316 0.232158 − 0.972678i \(-0.425421\pi\)
0.232158 + 0.972678i \(0.425421\pi\)
\(504\) 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 \(-0.420647\pi\)
0.246720 + 0.969087i \(0.420647\pi\)
\(522\) 0 0
\(523\) 6776.00 0.566527 0.283264 − 0.959042i \(-0.408583\pi\)
0.283264 + 0.959042i \(0.408583\pi\)
\(524\) 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 \(-0.621682\pi\)
−0.373033 + 0.927818i \(0.621682\pi\)
\(542\) 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 \(-0.299744\pi\)
0.588435 + 0.808544i \(0.299744\pi\)
\(548\) 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 \(-0.550445\pi\)
−0.157816 + 0.987469i \(0.550445\pi\)
\(570\) 0 0
\(571\) 692.000 0.0507168 0.0253584 − 0.999678i \(-0.491927\pi\)
0.0253584 + 0.999678i \(0.491927\pi\)
\(572\) 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 \(-0.331785\pi\)
0.504207 + 0.863583i \(0.331785\pi\)
\(594\) 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 \(-0.627891\pi\)
−0.391058 + 0.920366i \(0.627891\pi\)
\(600\) 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 \(-0.180137\pi\)
0.844097 + 0.536191i \(0.180137\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −21042.0 −1.37296 −0.686482 − 0.727147i \(-0.740846\pi\)
−0.686482 + 0.727147i \(0.740846\pi\)
\(618\) 0 0
\(619\) 6524.00 0.423621 0.211811 − 0.977311i \(-0.432064\pi\)
0.211811 + 0.977311i \(0.432064\pi\)
\(620\) 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 \(-0.491349\pi\)
0.0271739 + 0.999631i \(0.491349\pi\)
\(642\) 0 0
\(643\) −7252.00 −0.444776 −0.222388 − 0.974958i \(-0.571385\pi\)
−0.222388 + 0.974958i \(0.571385\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −21168.0 −1.28624 −0.643122 − 0.765764i \(-0.722361\pi\)
−0.643122 + 0.765764i \(0.722361\pi\)
\(648\) 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 \(-0.544063\pi\)
−0.137988 + 0.990434i \(0.544063\pi\)
\(662\) 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 \(-0.335517\pi\)
0.494048 + 0.869435i \(0.335517\pi\)
\(692\) 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 \(-0.544693\pi\)
−0.139947 + 0.990159i \(0.544693\pi\)
\(710\) 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 \(-0.280667\pi\)
0.635808 + 0.771847i \(0.280667\pi\)
\(720\) 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 \(-0.524838\pi\)
−0.0779533 + 0.996957i \(0.524838\pi\)
\(734\) 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 \(-0.502979\pi\)
−0.00935818 + 0.999956i \(0.502979\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −29484.0 −1.45580 −0.727902 − 0.685681i \(-0.759505\pi\)
−0.727902 + 0.685681i \(0.759505\pi\)
\(744\) 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 \(-0.493199\pi\)
0.0213657 + 0.999772i \(0.493199\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 32832.0 1.56394 0.781970 − 0.623315i \(-0.214215\pi\)
0.781970 + 0.623315i \(0.214215\pi\)
\(762\) 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 \(-0.409994\pi\)
0.279009 + 0.960288i \(0.409994\pi\)
\(788\) 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 \(-0.394363\pi\)
0.325810 + 0.945435i \(0.394363\pi\)
\(810\) 0 0
\(811\) −38878.0 −1.68334 −0.841672 − 0.539990i \(-0.818428\pi\)
−0.841672 + 0.539990i \(0.818428\pi\)
\(812\) 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 \(-0.570494\pi\)
−0.219659 + 0.975577i \(0.570494\pi\)
\(830\) 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 \(-0.256167\pi\)
0.693275 + 0.720673i \(0.256167\pi\)
\(840\) 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 \(-0.602563\pi\)
−0.316664 + 0.948538i \(0.602563\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 26136.0 1.04176 0.520880 − 0.853630i \(-0.325604\pi\)
0.520880 + 0.853630i \(0.325604\pi\)
\(858\) 0 0
\(859\) 23912.0 0.949787 0.474893 − 0.880043i \(-0.342487\pi\)
0.474893 + 0.880043i \(0.342487\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −17892.0 −0.705737 −0.352868 − 0.935673i \(-0.614794\pi\)
−0.352868 + 0.935673i \(0.614794\pi\)
\(864\) 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 \(-0.176483\pi\)
0.850197 + 0.526465i \(0.176483\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 8820.00 0.337291 0.168645 − 0.985677i \(-0.446061\pi\)
0.168645 + 0.985677i \(0.446061\pi\)
\(882\) 0 0
\(883\) −4654.00 −0.177372 −0.0886861 − 0.996060i \(-0.528267\pi\)
−0.0886861 + 0.996060i \(0.528267\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −33084.0 −1.25237 −0.626185 − 0.779675i \(-0.715384\pi\)
−0.626185 + 0.779675i \(0.715384\pi\)
\(888\) 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 \(-0.167295\pi\)
0.865036 + 0.501709i \(0.167295\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −36666.0 −1.33348 −0.666739 − 0.745291i \(-0.732310\pi\)
−0.666739 + 0.745291i \(0.732310\pi\)
\(912\) 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 \(-0.424631\pi\)
0.234572 + 0.972099i \(0.424631\pi\)
\(930\) 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 \(-0.716772\pi\)
−0.629577 + 0.776938i \(0.716772\pi\)
\(954\) 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 \(-0.273102\pi\)
0.653970 + 0.756520i \(0.273102\pi\)
\(978\) 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 \(-0.842361\pi\)
−0.879856 + 0.475241i \(0.842361\pi\)
\(984\) 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 \(-0.678348\pi\)
−0.531439 + 0.847097i \(0.678348\pi\)
\(998\) 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 108.4.a.a.1.1 ✓ 1
3.2 odd 2 108.4.a.d.1.1 yes 1
4.3 odd 2 432.4.a.c.1.1 1
8.3 odd 2 1728.4.a.z.1.1 1
8.5 even 2 1728.4.a.y.1.1 1
9.2 odd 6 324.4.e.b.109.1 2
9.4 even 3 324.4.e.g.217.1 2
9.5 odd 6 324.4.e.b.217.1 2
9.7 even 3 324.4.e.g.109.1 2
12.11 even 2 432.4.a.l.1.1 1
24.5 odd 2 1728.4.a.g.1.1 1
24.11 even 2 1728.4.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
108.4.a.a.1.1 ✓ 1 1.1 even 1 trivial
108.4.a.d.1.1 yes 1 3.2 odd 2
324.4.e.b.109.1 2 9.2 odd 6
324.4.e.b.217.1 2 9.5 odd 6
324.4.e.g.109.1 2 9.7 even 3
324.4.e.g.217.1 2 9.4 even 3
432.4.a.c.1.1 1 4.3 odd 2
432.4.a.l.1.1 1 12.11 even 2
1728.4.a.g.1.1 1 24.5 odd 2
1728.4.a.h.1.1 1 24.11 even 2
1728.4.a.y.1.1 1 8.5 even 2
1728.4.a.z.1.1 1 8.3 odd 2