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Properties

Label 1872.4.a.g.1.1
Level $1872$
Weight $4$
Character 1872.1
Self dual yes
Analytic conductor $110.452$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1872,4,Mod(1,1872)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1872.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1872, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1872.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,-2,0,26] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(110.451575531\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 234)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1872.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{5} +26.0000 q^{7} +52.0000 q^{11} -13.0000 q^{13} -48.0000 q^{17} -18.0000 q^{19} -52.0000 q^{23} -121.000 q^{25} -224.000 q^{29} -310.000 q^{31} -52.0000 q^{35} -18.0000 q^{37} -330.000 q^{41} -328.000 q^{43} +616.000 q^{47} +333.000 q^{49} +324.000 q^{53} -104.000 q^{55} +188.000 q^{59} -110.000 q^{61} +26.0000 q^{65} -118.000 q^{67} -656.000 q^{71} -178.000 q^{73} +1352.00 q^{77} -836.000 q^{79} +60.0000 q^{83} +96.0000 q^{85} -870.000 q^{89} -338.000 q^{91} +36.0000 q^{95} +1238.00 q^{97} +O(q^{100})\)

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
\(5\) −2.00000 −0.178885 −0.0894427 − 0.995992i \(-0.528509\pi\)
−0.0894427 + 0.995992i \(0.528509\pi\)
\(6\) 0 0
\(7\) 26.0000 1.40387 0.701934 − 0.712242i \(-0.252320\pi\)
0.701934 + 0.712242i \(0.252320\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 52.0000 1.42533 0.712663 − 0.701506i \(-0.247489\pi\)
0.712663 + 0.701506i \(0.247489\pi\)
\(12\) 0 0
\(13\) −13.0000 −0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −48.0000 −0.684806 −0.342403 − 0.939553i \(-0.611241\pi\)
−0.342403 + 0.939553i \(0.611241\pi\)
\(18\) 0 0
\(19\) −18.0000 −0.217341 −0.108671 − 0.994078i \(-0.534659\pi\)
−0.108671 + 0.994078i \(0.534659\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −52.0000 −0.471424 −0.235712 − 0.971823i \(-0.575742\pi\)
−0.235712 + 0.971823i \(0.575742\pi\)
\(24\) 0 0
\(25\) −121.000 −0.968000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −224.000 −1.43434 −0.717168 − 0.696900i \(-0.754562\pi\)
−0.717168 + 0.696900i \(0.754562\pi\)
\(30\) 0 0
\(31\) −310.000 −1.79605 −0.898027 − 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −52.0000 −0.251132
\(36\) 0 0
\(37\) −18.0000 −0.0799779 −0.0399889 − 0.999200i \(-0.512732\pi\)
−0.0399889 + 0.999200i \(0.512732\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −330.000 −1.25701 −0.628504 − 0.777806i \(-0.716332\pi\)
−0.628504 + 0.777806i \(0.716332\pi\)
\(42\) 0 0
\(43\) −328.000 −1.16324 −0.581622 − 0.813459i \(-0.697582\pi\)
−0.581622 + 0.813459i \(0.697582\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 616.000 1.91176 0.955881 − 0.293753i \(-0.0949045\pi\)
0.955881 + 0.293753i \(0.0949045\pi\)
\(48\) 0 0
\(49\) 333.000 0.970845
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 324.000 0.839714 0.419857 − 0.907590i \(-0.362080\pi\)
0.419857 + 0.907590i \(0.362080\pi\)
\(54\) 0 0
\(55\) −104.000 −0.254970
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 188.000 0.414839 0.207420 − 0.978252i \(-0.433493\pi\)
0.207420 + 0.978252i \(0.433493\pi\)
\(60\) 0 0
\(61\) −110.000 −0.230886 −0.115443 − 0.993314i \(-0.536829\pi\)
−0.115443 + 0.993314i \(0.536829\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 26.0000 0.0496139
\(66\) 0 0
\(67\) −118.000 −0.215164 −0.107582 − 0.994196i \(-0.534311\pi\)
−0.107582 + 0.994196i \(0.534311\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −656.000 −1.09652 −0.548260 − 0.836308i \(-0.684709\pi\)
−0.548260 + 0.836308i \(0.684709\pi\)
\(72\) 0 0
\(73\) −178.000 −0.285388 −0.142694 − 0.989767i \(-0.545576\pi\)
−0.142694 + 0.989767i \(0.545576\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1352.00 2.00097
\(78\) 0 0
\(79\) −836.000 −1.19060 −0.595300 − 0.803504i \(-0.702967\pi\)
−0.595300 + 0.803504i \(0.702967\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 60.0000 0.0793477 0.0396738 − 0.999213i \(-0.487368\pi\)
0.0396738 + 0.999213i \(0.487368\pi\)
\(84\) 0 0
\(85\) 96.0000 0.122502
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −870.000 −1.03618 −0.518089 − 0.855327i \(-0.673356\pi\)
−0.518089 + 0.855327i \(0.673356\pi\)
\(90\) 0 0
\(91\) −338.000 −0.389363
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 36.0000 0.0388792
\(96\) 0 0
\(97\) 1238.00 1.29587 0.647937 − 0.761694i \(-0.275632\pi\)
0.647937 + 0.761694i \(0.275632\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 920.000 0.906371 0.453185 − 0.891416i \(-0.350288\pi\)
0.453185 + 0.891416i \(0.350288\pi\)
\(102\) 0 0
\(103\) 836.000 0.799743 0.399871 − 0.916571i \(-0.369055\pi\)
0.399871 + 0.916571i \(0.369055\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 260.000 0.234908 0.117454 − 0.993078i \(-0.462527\pi\)
0.117454 + 0.993078i \(0.462527\pi\)
\(108\) 0 0
\(109\) −2078.00 −1.82602 −0.913011 − 0.407936i \(-0.866249\pi\)
−0.913011 + 0.407936i \(0.866249\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1156.00 0.962366 0.481183 − 0.876620i \(-0.340207\pi\)
0.481183 + 0.876620i \(0.340207\pi\)
\(114\) 0 0
\(115\) 104.000 0.0843309
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −1248.00 −0.961378
\(120\) 0 0
\(121\) 1373.00 1.03156
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 492.000 0.352047
\(126\) 0 0
\(127\) 752.000 0.525427 0.262713 − 0.964874i \(-0.415383\pi\)
0.262713 + 0.964874i \(0.415383\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1972.00 −1.31522 −0.657612 − 0.753356i \(-0.728434\pi\)
−0.657612 + 0.753356i \(0.728434\pi\)
\(132\) 0 0
\(133\) −468.000 −0.305118
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −258.000 −0.160894 −0.0804468 − 0.996759i \(-0.525635\pi\)
−0.0804468 + 0.996759i \(0.525635\pi\)
\(138\) 0 0
\(139\) −1468.00 −0.895785 −0.447893 − 0.894087i \(-0.647825\pi\)
−0.447893 + 0.894087i \(0.647825\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −676.000 −0.395314
\(144\) 0 0
\(145\) 448.000 0.256582
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −3414.00 −1.87709 −0.938543 − 0.345163i \(-0.887824\pi\)
−0.938543 + 0.345163i \(0.887824\pi\)
\(150\) 0 0
\(151\) 2214.00 1.19320 0.596599 − 0.802540i \(-0.296518\pi\)
0.596599 + 0.802540i \(0.296518\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 620.000 0.321288
\(156\) 0 0
\(157\) −2230.00 −1.13359 −0.566794 − 0.823859i \(-0.691817\pi\)
−0.566794 + 0.823859i \(0.691817\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −1352.00 −0.661817
\(162\) 0 0
\(163\) −1118.00 −0.537230 −0.268615 − 0.963248i \(-0.586566\pi\)
−0.268615 + 0.963248i \(0.586566\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −288.000 −0.133450 −0.0667249 − 0.997771i \(-0.521255\pi\)
−0.0667249 + 0.997771i \(0.521255\pi\)
\(168\) 0 0
\(169\) 169.000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 328.000 0.144147 0.0720733 − 0.997399i \(-0.477038\pi\)
0.0720733 + 0.997399i \(0.477038\pi\)
\(174\) 0 0
\(175\) −3146.00 −1.35894
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 2820.00 1.17752 0.588762 − 0.808307i \(-0.299616\pi\)
0.588762 + 0.808307i \(0.299616\pi\)
\(180\) 0 0
\(181\) −2250.00 −0.923984 −0.461992 − 0.886884i \(-0.652865\pi\)
−0.461992 + 0.886884i \(0.652865\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 36.0000 0.0143069
\(186\) 0 0
\(187\) −2496.00 −0.976073
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3604.00 1.36532 0.682660 − 0.730736i \(-0.260823\pi\)
0.682660 + 0.730736i \(0.260823\pi\)
\(192\) 0 0
\(193\) −1198.00 −0.446808 −0.223404 − 0.974726i \(-0.571717\pi\)
−0.223404 + 0.974726i \(0.571717\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −1026.00 −0.371063 −0.185532 − 0.982638i \(-0.559401\pi\)
−0.185532 + 0.982638i \(0.559401\pi\)
\(198\) 0 0
\(199\) −2528.00 −0.900528 −0.450264 − 0.892895i \(-0.648670\pi\)
−0.450264 + 0.892895i \(0.648670\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −5824.00 −2.01362
\(204\) 0 0
\(205\) 660.000 0.224860
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −936.000 −0.309782
\(210\) 0 0
\(211\) 3160.00 1.03101 0.515506 − 0.856886i \(-0.327604\pi\)
0.515506 + 0.856886i \(0.327604\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 656.000 0.208088
\(216\) 0 0
\(217\) −8060.00 −2.52142
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 624.000 0.189931
\(222\) 0 0
\(223\) −2670.00 −0.801778 −0.400889 − 0.916127i \(-0.631299\pi\)
−0.400889 + 0.916127i \(0.631299\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 5220.00 1.52627 0.763136 − 0.646238i \(-0.223659\pi\)
0.763136 + 0.646238i \(0.223659\pi\)
\(228\) 0 0
\(229\) −6246.00 −1.80239 −0.901195 − 0.433414i \(-0.857309\pi\)
−0.901195 + 0.433414i \(0.857309\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −2860.00 −0.804141 −0.402070 − 0.915609i \(-0.631709\pi\)
−0.402070 + 0.915609i \(0.631709\pi\)
\(234\) 0 0
\(235\) −1232.00 −0.341986
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 7176.00 1.94216 0.971081 − 0.238749i \(-0.0767374\pi\)
0.971081 + 0.238749i \(0.0767374\pi\)
\(240\) 0 0
\(241\) −2922.00 −0.781006 −0.390503 − 0.920602i \(-0.627699\pi\)
−0.390503 + 0.920602i \(0.627699\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −666.000 −0.173670
\(246\) 0 0
\(247\) 234.000 0.0602796
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −1440.00 −0.362119 −0.181060 − 0.983472i \(-0.557953\pi\)
−0.181060 + 0.983472i \(0.557953\pi\)
\(252\) 0 0
\(253\) −2704.00 −0.671933
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3852.00 0.934946 0.467473 − 0.884007i \(-0.345164\pi\)
0.467473 + 0.884007i \(0.345164\pi\)
\(258\) 0 0
\(259\) −468.000 −0.112278
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 7008.00 1.64309 0.821543 − 0.570146i \(-0.193113\pi\)
0.821543 + 0.570146i \(0.193113\pi\)
\(264\) 0 0
\(265\) −648.000 −0.150213
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 1992.00 0.451503 0.225752 − 0.974185i \(-0.427516\pi\)
0.225752 + 0.974185i \(0.427516\pi\)
\(270\) 0 0
\(271\) 3362.00 0.753605 0.376803 − 0.926294i \(-0.377024\pi\)
0.376803 + 0.926294i \(0.377024\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −6292.00 −1.37972
\(276\) 0 0
\(277\) 1506.00 0.326667 0.163334 − 0.986571i \(-0.447775\pi\)
0.163334 + 0.986571i \(0.447775\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 3638.00 0.772331 0.386165 − 0.922430i \(-0.373799\pi\)
0.386165 + 0.922430i \(0.373799\pi\)
\(282\) 0 0
\(283\) −5404.00 −1.13510 −0.567552 − 0.823338i \(-0.692109\pi\)
−0.567552 + 0.823338i \(0.692109\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −8580.00 −1.76467
\(288\) 0 0
\(289\) −2609.00 −0.531040
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 4994.00 0.995743 0.497871 − 0.867251i \(-0.334115\pi\)
0.497871 + 0.867251i \(0.334115\pi\)
\(294\) 0 0
\(295\) −376.000 −0.0742087
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 676.000 0.130749
\(300\) 0 0
\(301\) −8528.00 −1.63304
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 220.000 0.0413022
\(306\) 0 0
\(307\) −2474.00 −0.459930 −0.229965 − 0.973199i \(-0.573861\pi\)
−0.229965 + 0.973199i \(0.573861\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −9156.00 −1.66942 −0.834709 − 0.550691i \(-0.814364\pi\)
−0.834709 + 0.550691i \(0.814364\pi\)
\(312\) 0 0
\(313\) 3818.00 0.689476 0.344738 − 0.938699i \(-0.387968\pi\)
0.344738 + 0.938699i \(0.387968\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −6234.00 −1.10453 −0.552265 − 0.833668i \(-0.686237\pi\)
−0.552265 + 0.833668i \(0.686237\pi\)
\(318\) 0 0
\(319\) −11648.0 −2.04440
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 864.000 0.148837
\(324\) 0 0
\(325\) 1573.00 0.268475
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 16016.0 2.68386
\(330\) 0 0
\(331\) −5054.00 −0.839254 −0.419627 − 0.907697i \(-0.637839\pi\)
−0.419627 + 0.907697i \(0.637839\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 236.000 0.0384897
\(336\) 0 0
\(337\) 10238.0 1.65489 0.827447 − 0.561544i \(-0.189792\pi\)
0.827447 + 0.561544i \(0.189792\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −16120.0 −2.55996
\(342\) 0 0
\(343\) −260.000 −0.0409291
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −2184.00 −0.337877 −0.168938 − 0.985627i \(-0.554034\pi\)
−0.168938 + 0.985627i \(0.554034\pi\)
\(348\) 0 0
\(349\) 6766.00 1.03775 0.518876 − 0.854849i \(-0.326351\pi\)
0.518876 + 0.854849i \(0.326351\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −8290.00 −1.24995 −0.624975 − 0.780645i \(-0.714891\pi\)
−0.624975 + 0.780645i \(0.714891\pi\)
\(354\) 0 0
\(355\) 1312.00 0.196151
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −9456.00 −1.39016 −0.695082 − 0.718931i \(-0.744632\pi\)
−0.695082 + 0.718931i \(0.744632\pi\)
\(360\) 0 0
\(361\) −6535.00 −0.952763
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 356.000 0.0510518
\(366\) 0 0
\(367\) 10096.0 1.43599 0.717993 − 0.696050i \(-0.245061\pi\)
0.717993 + 0.696050i \(0.245061\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 8424.00 1.17885
\(372\) 0 0
\(373\) −1234.00 −0.171298 −0.0856489 − 0.996325i \(-0.527296\pi\)
−0.0856489 + 0.996325i \(0.527296\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2912.00 0.397813
\(378\) 0 0
\(379\) −11218.0 −1.52040 −0.760198 − 0.649692i \(-0.774898\pi\)
−0.760198 + 0.649692i \(0.774898\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −7488.00 −0.999005 −0.499503 − 0.866312i \(-0.666484\pi\)
−0.499503 + 0.866312i \(0.666484\pi\)
\(384\) 0 0
\(385\) −2704.00 −0.357944
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 14364.0 1.87219 0.936097 − 0.351741i \(-0.114410\pi\)
0.936097 + 0.351741i \(0.114410\pi\)
\(390\) 0 0
\(391\) 2496.00 0.322834
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1672.00 0.212981
\(396\) 0 0
\(397\) 7454.00 0.942331 0.471166 − 0.882045i \(-0.343833\pi\)
0.471166 + 0.882045i \(0.343833\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −5766.00 −0.718056 −0.359028 − 0.933327i \(-0.616892\pi\)
−0.359028 + 0.933327i \(0.616892\pi\)
\(402\) 0 0
\(403\) 4030.00 0.498135
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −936.000 −0.113995
\(408\) 0 0
\(409\) 11926.0 1.44182 0.720908 − 0.693031i \(-0.243725\pi\)
0.720908 + 0.693031i \(0.243725\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 4888.00 0.582380
\(414\) 0 0
\(415\) −120.000 −0.0141941
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 9996.00 1.16548 0.582741 − 0.812658i \(-0.301980\pi\)
0.582741 + 0.812658i \(0.301980\pi\)
\(420\) 0 0
\(421\) −2030.00 −0.235003 −0.117501 − 0.993073i \(-0.537488\pi\)
−0.117501 + 0.993073i \(0.537488\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 5808.00 0.662893
\(426\) 0 0
\(427\) −2860.00 −0.324134
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 15888.0 1.77563 0.887817 − 0.460197i \(-0.152221\pi\)
0.887817 + 0.460197i \(0.152221\pi\)
\(432\) 0 0
\(433\) −14402.0 −1.59842 −0.799210 − 0.601052i \(-0.794749\pi\)
−0.799210 + 0.601052i \(0.794749\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 936.000 0.102460
\(438\) 0 0
\(439\) −404.000 −0.0439223 −0.0219611 − 0.999759i \(-0.506991\pi\)
−0.0219611 + 0.999759i \(0.506991\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −16692.0 −1.79021 −0.895103 − 0.445860i \(-0.852898\pi\)
−0.895103 + 0.445860i \(0.852898\pi\)
\(444\) 0 0
\(445\) 1740.00 0.185357
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 4058.00 0.426523 0.213262 − 0.976995i \(-0.431591\pi\)
0.213262 + 0.976995i \(0.431591\pi\)
\(450\) 0 0
\(451\) −17160.0 −1.79165
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 676.000 0.0696514
\(456\) 0 0
\(457\) −14642.0 −1.49874 −0.749370 − 0.662152i \(-0.769643\pi\)
−0.749370 + 0.662152i \(0.769643\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −17970.0 −1.81550 −0.907751 − 0.419510i \(-0.862202\pi\)
−0.907751 + 0.419510i \(0.862202\pi\)
\(462\) 0 0
\(463\) 6242.00 0.626545 0.313273 − 0.949663i \(-0.398575\pi\)
0.313273 + 0.949663i \(0.398575\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 5000.00 0.495444 0.247722 − 0.968831i \(-0.420318\pi\)
0.247722 + 0.968831i \(0.420318\pi\)
\(468\) 0 0
\(469\) −3068.00 −0.302062
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −17056.0 −1.65800
\(474\) 0 0
\(475\) 2178.00 0.210386
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 9408.00 0.897416 0.448708 − 0.893678i \(-0.351884\pi\)
0.448708 + 0.893678i \(0.351884\pi\)
\(480\) 0 0
\(481\) 234.000 0.0221819
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −2476.00 −0.231813
\(486\) 0 0
\(487\) 7090.00 0.659710 0.329855 − 0.944032i \(-0.393000\pi\)
0.329855 + 0.944032i \(0.393000\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 8176.00 0.751482 0.375741 − 0.926725i \(-0.377388\pi\)
0.375741 + 0.926725i \(0.377388\pi\)
\(492\) 0 0
\(493\) 10752.0 0.982243
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −17056.0 −1.53937
\(498\) 0 0
\(499\) −9646.00 −0.865359 −0.432680 − 0.901548i \(-0.642432\pi\)
−0.432680 + 0.901548i \(0.642432\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −4352.00 −0.385778 −0.192889 − 0.981221i \(-0.561786\pi\)
−0.192889 + 0.981221i \(0.561786\pi\)
\(504\) 0 0
\(505\) −1840.00 −0.162136
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 14582.0 1.26981 0.634907 − 0.772588i \(-0.281038\pi\)
0.634907 + 0.772588i \(0.281038\pi\)
\(510\) 0 0
\(511\) −4628.00 −0.400647
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1672.00 −0.143062
\(516\) 0 0
\(517\) 32032.0 2.72489
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 11820.0 0.993942 0.496971 − 0.867767i \(-0.334446\pi\)
0.496971 + 0.867767i \(0.334446\pi\)
\(522\) 0 0
\(523\) −2676.00 −0.223735 −0.111867 − 0.993723i \(-0.535683\pi\)
−0.111867 + 0.993723i \(0.535683\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 14880.0 1.22995
\(528\) 0 0
\(529\) −9463.00 −0.777760
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 4290.00 0.348631
\(534\) 0 0
\(535\) −520.000 −0.0420216
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 17316.0 1.38377
\(540\) 0 0
\(541\) 842.000 0.0669139 0.0334569 − 0.999440i \(-0.489348\pi\)
0.0334569 + 0.999440i \(0.489348\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 4156.00 0.326649
\(546\) 0 0
\(547\) −1036.00 −0.0809802 −0.0404901 − 0.999180i \(-0.512892\pi\)
−0.0404901 + 0.999180i \(0.512892\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 4032.00 0.311740
\(552\) 0 0
\(553\) −21736.0 −1.67144
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −3342.00 −0.254228 −0.127114 − 0.991888i \(-0.540571\pi\)
−0.127114 + 0.991888i \(0.540571\pi\)
\(558\) 0 0
\(559\) 4264.00 0.322626
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −12412.0 −0.929136 −0.464568 − 0.885538i \(-0.653790\pi\)
−0.464568 + 0.885538i \(0.653790\pi\)
\(564\) 0 0
\(565\) −2312.00 −0.172153
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −21556.0 −1.58818 −0.794090 − 0.607800i \(-0.792052\pi\)
−0.794090 + 0.607800i \(0.792052\pi\)
\(570\) 0 0
\(571\) −9012.00 −0.660491 −0.330246 − 0.943895i \(-0.607132\pi\)
−0.330246 + 0.943895i \(0.607132\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 6292.00 0.456338
\(576\) 0 0
\(577\) −10446.0 −0.753679 −0.376839 − 0.926279i \(-0.622989\pi\)
−0.376839 + 0.926279i \(0.622989\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1560.00 0.111394
\(582\) 0 0
\(583\) 16848.0 1.19687
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −23452.0 −1.64901 −0.824504 − 0.565856i \(-0.808546\pi\)
−0.824504 + 0.565856i \(0.808546\pi\)
\(588\) 0 0
\(589\) 5580.00 0.390356
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 23914.0 1.65604 0.828019 − 0.560700i \(-0.189468\pi\)
0.828019 + 0.560700i \(0.189468\pi\)
\(594\) 0 0
\(595\) 2496.00 0.171977
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −2552.00 −0.174077 −0.0870383 − 0.996205i \(-0.527740\pi\)
−0.0870383 + 0.996205i \(0.527740\pi\)
\(600\) 0 0
\(601\) 3418.00 0.231985 0.115993 − 0.993250i \(-0.462995\pi\)
0.115993 + 0.993250i \(0.462995\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −2746.00 −0.184530
\(606\) 0 0
\(607\) −12336.0 −0.824881 −0.412441 − 0.910985i \(-0.635324\pi\)
−0.412441 + 0.910985i \(0.635324\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −8008.00 −0.530228
\(612\) 0 0
\(613\) −21946.0 −1.44599 −0.722994 − 0.690854i \(-0.757235\pi\)
−0.722994 + 0.690854i \(0.757235\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −23890.0 −1.55879 −0.779396 − 0.626531i \(-0.784474\pi\)
−0.779396 + 0.626531i \(0.784474\pi\)
\(618\) 0 0
\(619\) −10042.0 −0.652055 −0.326028 − 0.945360i \(-0.605710\pi\)
−0.326028 + 0.945360i \(0.605710\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −22620.0 −1.45466
\(624\) 0 0
\(625\) 14141.0 0.905024
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 864.000 0.0547694
\(630\) 0 0
\(631\) 21830.0 1.37724 0.688620 − 0.725122i \(-0.258217\pi\)
0.688620 + 0.725122i \(0.258217\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1504.00 −0.0939912
\(636\) 0 0
\(637\) −4329.00 −0.269264
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 776.000 0.0478162 0.0239081 − 0.999714i \(-0.492389\pi\)
0.0239081 + 0.999714i \(0.492389\pi\)
\(642\) 0 0
\(643\) −4094.00 −0.251091 −0.125546 − 0.992088i \(-0.540068\pi\)
−0.125546 + 0.992088i \(0.540068\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −13064.0 −0.793816 −0.396908 − 0.917858i \(-0.629917\pi\)
−0.396908 + 0.917858i \(0.629917\pi\)
\(648\) 0 0
\(649\) 9776.00 0.591281
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 24504.0 1.46848 0.734239 − 0.678891i \(-0.237539\pi\)
0.734239 + 0.678891i \(0.237539\pi\)
\(654\) 0 0
\(655\) 3944.00 0.235275
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 4752.00 0.280898 0.140449 − 0.990088i \(-0.455145\pi\)
0.140449 + 0.990088i \(0.455145\pi\)
\(660\) 0 0
\(661\) −9514.00 −0.559836 −0.279918 − 0.960024i \(-0.590307\pi\)
−0.279918 + 0.960024i \(0.590307\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 936.000 0.0545812
\(666\) 0 0
\(667\) 11648.0 0.676180
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −5720.00 −0.329088
\(672\) 0 0
\(673\) 8374.00 0.479634 0.239817 − 0.970818i \(-0.422912\pi\)
0.239817 + 0.970818i \(0.422912\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 14556.0 0.826341 0.413170 − 0.910654i \(-0.364421\pi\)
0.413170 + 0.910654i \(0.364421\pi\)
\(678\) 0 0
\(679\) 32188.0 1.81924
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −23276.0 −1.30400 −0.652000 − 0.758219i \(-0.726070\pi\)
−0.652000 + 0.758219i \(0.726070\pi\)
\(684\) 0 0
\(685\) 516.000 0.0287815
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −4212.00 −0.232895
\(690\) 0 0
\(691\) 29522.0 1.62528 0.812641 − 0.582765i \(-0.198029\pi\)
0.812641 + 0.582765i \(0.198029\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 2936.00 0.160243
\(696\) 0 0
\(697\) 15840.0 0.860807
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −27252.0 −1.46832 −0.734161 − 0.678975i \(-0.762424\pi\)
−0.734161 + 0.678975i \(0.762424\pi\)
\(702\) 0 0
\(703\) 324.000 0.0173825
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 23920.0 1.27242
\(708\) 0 0
\(709\) 2162.00 0.114521 0.0572607 − 0.998359i \(-0.481763\pi\)
0.0572607 + 0.998359i \(0.481763\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 16120.0 0.846702
\(714\) 0 0
\(715\) 1352.00 0.0707160
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −8044.00 −0.417233 −0.208617 − 0.977998i \(-0.566896\pi\)
−0.208617 + 0.977998i \(0.566896\pi\)
\(720\) 0 0
\(721\) 21736.0 1.12273
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 27104.0 1.38844
\(726\) 0 0
\(727\) 22496.0 1.14763 0.573817 − 0.818983i \(-0.305462\pi\)
0.573817 + 0.818983i \(0.305462\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 15744.0 0.796598
\(732\) 0 0
\(733\) −15934.0 −0.802914 −0.401457 − 0.915878i \(-0.631496\pi\)
−0.401457 + 0.915878i \(0.631496\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −6136.00 −0.306679
\(738\) 0 0
\(739\) 29838.0 1.48526 0.742631 − 0.669701i \(-0.233578\pi\)
0.742631 + 0.669701i \(0.233578\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −10760.0 −0.531287 −0.265643 − 0.964071i \(-0.585584\pi\)
−0.265643 + 0.964071i \(0.585584\pi\)
\(744\) 0 0
\(745\) 6828.00 0.335783
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 6760.00 0.329780
\(750\) 0 0
\(751\) 14916.0 0.724757 0.362379 − 0.932031i \(-0.381965\pi\)
0.362379 + 0.932031i \(0.381965\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −4428.00 −0.213446
\(756\) 0 0
\(757\) 29046.0 1.39458 0.697289 − 0.716791i \(-0.254390\pi\)
0.697289 + 0.716791i \(0.254390\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 2766.00 0.131757 0.0658787 − 0.997828i \(-0.479015\pi\)
0.0658787 + 0.997828i \(0.479015\pi\)
\(762\) 0 0
\(763\) −54028.0 −2.56349
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −2444.00 −0.115056
\(768\) 0 0
\(769\) 5394.00 0.252942 0.126471 − 0.991970i \(-0.459635\pi\)
0.126471 + 0.991970i \(0.459635\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 19378.0 0.901654 0.450827 − 0.892611i \(-0.351129\pi\)
0.450827 + 0.892611i \(0.351129\pi\)
\(774\) 0 0
\(775\) 37510.0 1.73858
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 5940.00 0.273200
\(780\) 0 0
\(781\) −34112.0 −1.56290
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 4460.00 0.202782
\(786\) 0 0
\(787\) 8350.00 0.378202 0.189101 − 0.981958i \(-0.439443\pi\)
0.189101 + 0.981958i \(0.439443\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 30056.0 1.35103
\(792\) 0 0
\(793\) 1430.00 0.0640363
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −22116.0 −0.982922 −0.491461 − 0.870900i \(-0.663537\pi\)
−0.491461 + 0.870900i \(0.663537\pi\)
\(798\) 0 0
\(799\) −29568.0 −1.30919
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −9256.00 −0.406771
\(804\) 0 0
\(805\) 2704.00 0.118389
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −6444.00 −0.280048 −0.140024 − 0.990148i \(-0.544718\pi\)
−0.140024 + 0.990148i \(0.544718\pi\)
\(810\) 0 0
\(811\) 8998.00 0.389596 0.194798 − 0.980843i \(-0.437595\pi\)
0.194798 + 0.980843i \(0.437595\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 2236.00 0.0961027
\(816\) 0 0
\(817\) 5904.00 0.252821
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 11938.0 0.507478 0.253739 − 0.967273i \(-0.418340\pi\)
0.253739 + 0.967273i \(0.418340\pi\)
\(822\) 0 0
\(823\) 41028.0 1.73772 0.868862 − 0.495055i \(-0.164852\pi\)
0.868862 + 0.495055i \(0.164852\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 20460.0 0.860295 0.430147 − 0.902759i \(-0.358462\pi\)
0.430147 + 0.902759i \(0.358462\pi\)
\(828\) 0 0
\(829\) 27814.0 1.16528 0.582642 − 0.812729i \(-0.302019\pi\)
0.582642 + 0.812729i \(0.302019\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −15984.0 −0.664841
\(834\) 0 0
\(835\) 576.000 0.0238722
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −24880.0 −1.02378 −0.511891 − 0.859050i \(-0.671055\pi\)
−0.511891 + 0.859050i \(0.671055\pi\)
\(840\) 0 0
\(841\) 25787.0 1.05732
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −338.000 −0.0137604
\(846\) 0 0
\(847\) 35698.0 1.44817
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 936.000 0.0377035
\(852\) 0 0
\(853\) 37678.0 1.51239 0.756196 − 0.654345i \(-0.227056\pi\)
0.756196 + 0.654345i \(0.227056\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 14708.0 0.586249 0.293125 − 0.956074i \(-0.405305\pi\)
0.293125 + 0.956074i \(0.405305\pi\)
\(858\) 0 0
\(859\) −16472.0 −0.654269 −0.327135 − 0.944978i \(-0.606083\pi\)
−0.327135 + 0.944978i \(0.606083\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −13640.0 −0.538020 −0.269010 − 0.963137i \(-0.586696\pi\)
−0.269010 + 0.963137i \(0.586696\pi\)
\(864\) 0 0
\(865\) −656.000 −0.0257857
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −43472.0 −1.69699
\(870\) 0 0
\(871\) 1534.00 0.0596758
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 12792.0 0.494227
\(876\) 0 0
\(877\) 44430.0 1.71071 0.855356 − 0.518041i \(-0.173338\pi\)
0.855356 + 0.518041i \(0.173338\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −6684.00 −0.255607 −0.127803 − 0.991800i \(-0.540793\pi\)
−0.127803 + 0.991800i \(0.540793\pi\)
\(882\) 0 0
\(883\) −9196.00 −0.350476 −0.175238 − 0.984526i \(-0.556069\pi\)
−0.175238 + 0.984526i \(0.556069\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 34852.0 1.31930 0.659648 − 0.751575i \(-0.270705\pi\)
0.659648 + 0.751575i \(0.270705\pi\)
\(888\) 0 0
\(889\) 19552.0 0.737630
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −11088.0 −0.415505
\(894\) 0 0
\(895\) −5640.00 −0.210642
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 69440.0 2.57614
\(900\) 0 0
\(901\) −15552.0 −0.575041
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 4500.00 0.165287
\(906\) 0 0
\(907\) −36780.0 −1.34648 −0.673241 − 0.739423i \(-0.735099\pi\)
−0.673241 + 0.739423i \(0.735099\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 22980.0 0.835742 0.417871 − 0.908506i \(-0.362776\pi\)
0.417871 + 0.908506i \(0.362776\pi\)
\(912\) 0 0
\(913\) 3120.00 0.113096
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −51272.0 −1.84640
\(918\) 0 0
\(919\) 19980.0 0.717170 0.358585 − 0.933497i \(-0.383259\pi\)
0.358585 + 0.933497i \(0.383259\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 8528.00 0.304120
\(924\) 0 0
\(925\) 2178.00 0.0774186
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 51486.0 1.81830 0.909150 − 0.416469i \(-0.136732\pi\)
0.909150 + 0.416469i \(0.136732\pi\)
\(930\) 0 0
\(931\) −5994.00 −0.211005
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 4992.00 0.174605
\(936\) 0 0
\(937\) 21774.0 0.759152 0.379576 − 0.925161i \(-0.376070\pi\)
0.379576 + 0.925161i \(0.376070\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −29158.0 −1.01012 −0.505060 − 0.863084i \(-0.668530\pi\)
−0.505060 + 0.863084i \(0.668530\pi\)
\(942\) 0 0
\(943\) 17160.0 0.592584
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −47844.0 −1.64173 −0.820866 − 0.571120i \(-0.806509\pi\)
−0.820866 + 0.571120i \(0.806509\pi\)
\(948\) 0 0
\(949\) 2314.00 0.0791524
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 20716.0 0.704152 0.352076 − 0.935971i \(-0.385476\pi\)
0.352076 + 0.935971i \(0.385476\pi\)
\(954\) 0 0
\(955\) −7208.00 −0.244236
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −6708.00 −0.225873
\(960\) 0 0
\(961\) 66309.0 2.22581
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 2396.00 0.0799274
\(966\) 0 0
\(967\) −1406.00 −0.0467569 −0.0233784 − 0.999727i \(-0.507442\pi\)
−0.0233784 + 0.999727i \(0.507442\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −40176.0 −1.32782 −0.663908 − 0.747814i \(-0.731103\pi\)
−0.663908 + 0.747814i \(0.731103\pi\)
\(972\) 0 0
\(973\) −38168.0 −1.25756
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −20910.0 −0.684719 −0.342359 − 0.939569i \(-0.611226\pi\)
−0.342359 + 0.939569i \(0.611226\pi\)
\(978\) 0 0
\(979\) −45240.0 −1.47689
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 26288.0 0.852957 0.426479 − 0.904498i \(-0.359754\pi\)
0.426479 + 0.904498i \(0.359754\pi\)
\(984\) 0 0
\(985\) 2052.00 0.0663778
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 17056.0 0.548381
\(990\) 0 0
\(991\) 25608.0 0.820853 0.410426 − 0.911894i \(-0.365380\pi\)
0.410426 + 0.911894i \(0.365380\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 5056.00 0.161091
\(996\) 0 0
\(997\) −20098.0 −0.638425 −0.319213 − 0.947683i \(-0.603418\pi\)
−0.319213 + 0.947683i \(0.603418\pi\)
\(998\) 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 1872.4.a.g.1.1 1
3.2 odd 2 1872.4.a.h.1.1 1
4.3 odd 2 234.4.a.i.1.1 yes 1
12.11 even 2 234.4.a.d.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
234.4.a.d.1.1 ✓ 1 12.11 even 2
234.4.a.i.1.1 yes 1 4.3 odd 2
1872.4.a.g.1.1 1 1.1 even 1 trivial
1872.4.a.h.1.1 1 3.2 odd 2