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Properties

Label 1296.4.a.a.1.1
Level $1296$
Weight $4$
Character 1296.1
Self dual yes
Analytic conductor $76.466$
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] = [1296,4,Mod(1,1296)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1296, 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("1296.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1296.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: \(76.4664753674\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 162)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1296.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-21.0000 q^{5} -8.00000 q^{7} +O(q^{10})\) \(q-21.0000 q^{5} -8.00000 q^{7} +36.0000 q^{11} -49.0000 q^{13} -21.0000 q^{17} +112.000 q^{19} +180.000 q^{23} +316.000 q^{25} +135.000 q^{29} -308.000 q^{31} +168.000 q^{35} -1.00000 q^{37} +42.0000 q^{41} -20.0000 q^{43} +84.0000 q^{47} -279.000 q^{49} +174.000 q^{53} -756.000 q^{55} +504.000 q^{59} -385.000 q^{61} +1029.00 q^{65} -272.000 q^{67} -888.000 q^{71} +371.000 q^{73} -288.000 q^{77} +652.000 q^{79} +84.0000 q^{83} +441.000 q^{85} -21.0000 q^{89} +392.000 q^{91} -2352.00 q^{95} -1246.00 q^{97} +O(q^{100})\)

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
\(5\) −21.0000 −1.87830 −0.939149 − 0.343511i \(-0.888384\pi\)
−0.939149 + 0.343511i \(0.888384\pi\)
\(6\) 0 0
\(7\) −8.00000 −0.431959 −0.215980 − 0.976398i \(-0.569295\pi\)
−0.215980 + 0.976398i \(0.569295\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 36.0000 0.986764 0.493382 − 0.869813i \(-0.335760\pi\)
0.493382 + 0.869813i \(0.335760\pi\)
\(12\) 0 0
\(13\) −49.0000 −1.04540 −0.522698 − 0.852518i \(-0.675075\pi\)
−0.522698 + 0.852518i \(0.675075\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −21.0000 −0.299603 −0.149801 − 0.988716i \(-0.547863\pi\)
−0.149801 + 0.988716i \(0.547863\pi\)
\(18\) 0 0
\(19\) 112.000 1.35235 0.676173 − 0.736743i \(-0.263637\pi\)
0.676173 + 0.736743i \(0.263637\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 180.000 1.63185 0.815926 − 0.578156i \(-0.196228\pi\)
0.815926 + 0.578156i \(0.196228\pi\)
\(24\) 0 0
\(25\) 316.000 2.52800
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 135.000 0.864444 0.432222 − 0.901767i \(-0.357730\pi\)
0.432222 + 0.901767i \(0.357730\pi\)
\(30\) 0 0
\(31\) −308.000 −1.78447 −0.892233 − 0.451576i \(-0.850862\pi\)
−0.892233 + 0.451576i \(0.850862\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 168.000 0.811348
\(36\) 0 0
\(37\) −1.00000 −0.00444322 −0.00222161 − 0.999998i \(-0.500707\pi\)
−0.00222161 + 0.999998i \(0.500707\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 42.0000 0.159983 0.0799914 − 0.996796i \(-0.474511\pi\)
0.0799914 + 0.996796i \(0.474511\pi\)
\(42\) 0 0
\(43\) −20.0000 −0.0709296 −0.0354648 − 0.999371i \(-0.511291\pi\)
−0.0354648 + 0.999371i \(0.511291\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 84.0000 0.260695 0.130347 − 0.991468i \(-0.458391\pi\)
0.130347 + 0.991468i \(0.458391\pi\)
\(48\) 0 0
\(49\) −279.000 −0.813411
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 174.000 0.450957 0.225479 − 0.974248i \(-0.427605\pi\)
0.225479 + 0.974248i \(0.427605\pi\)
\(54\) 0 0
\(55\) −756.000 −1.85344
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 504.000 1.11212 0.556061 − 0.831141i \(-0.312312\pi\)
0.556061 + 0.831141i \(0.312312\pi\)
\(60\) 0 0
\(61\) −385.000 −0.808102 −0.404051 − 0.914737i \(-0.632398\pi\)
−0.404051 + 0.914737i \(0.632398\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1029.00 1.96357
\(66\) 0 0
\(67\) −272.000 −0.495971 −0.247986 − 0.968764i \(-0.579769\pi\)
−0.247986 + 0.968764i \(0.579769\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −888.000 −1.48431 −0.742156 − 0.670227i \(-0.766197\pi\)
−0.742156 + 0.670227i \(0.766197\pi\)
\(72\) 0 0
\(73\) 371.000 0.594826 0.297413 − 0.954749i \(-0.403876\pi\)
0.297413 + 0.954749i \(0.403876\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −288.000 −0.426242
\(78\) 0 0
\(79\) 652.000 0.928554 0.464277 − 0.885690i \(-0.346314\pi\)
0.464277 + 0.885690i \(0.346314\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 84.0000 0.111087 0.0555434 − 0.998456i \(-0.482311\pi\)
0.0555434 + 0.998456i \(0.482311\pi\)
\(84\) 0 0
\(85\) 441.000 0.562743
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −21.0000 −0.0250112 −0.0125056 − 0.999922i \(-0.503981\pi\)
−0.0125056 + 0.999922i \(0.503981\pi\)
\(90\) 0 0
\(91\) 392.000 0.451569
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2352.00 −2.54011
\(96\) 0 0
\(97\) −1246.00 −1.30425 −0.652124 − 0.758112i \(-0.726122\pi\)
−0.652124 + 0.758112i \(0.726122\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −546.000 −0.537911 −0.268956 − 0.963153i \(-0.586678\pi\)
−0.268956 + 0.963153i \(0.586678\pi\)
\(102\) 0 0
\(103\) 196.000 0.187500 0.0937498 − 0.995596i \(-0.470115\pi\)
0.0937498 + 0.995596i \(0.470115\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −300.000 −0.271048 −0.135524 − 0.990774i \(-0.543272\pi\)
−0.135524 + 0.990774i \(0.543272\pi\)
\(108\) 0 0
\(109\) −1069.00 −0.939373 −0.469686 − 0.882833i \(-0.655633\pi\)
−0.469686 + 0.882833i \(0.655633\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −897.000 −0.746749 −0.373375 − 0.927681i \(-0.621799\pi\)
−0.373375 + 0.927681i \(0.621799\pi\)
\(114\) 0 0
\(115\) −3780.00 −3.06510
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 168.000 0.129416
\(120\) 0 0
\(121\) −35.0000 −0.0262960
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −4011.00 −2.87004
\(126\) 0 0
\(127\) −1532.00 −1.07042 −0.535209 − 0.844720i \(-0.679767\pi\)
−0.535209 + 0.844720i \(0.679767\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 840.000 0.560238 0.280119 − 0.959965i \(-0.409626\pi\)
0.280119 + 0.959965i \(0.409626\pi\)
\(132\) 0 0
\(133\) −896.000 −0.584158
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −729.000 −0.454618 −0.227309 − 0.973823i \(-0.572993\pi\)
−0.227309 + 0.973823i \(0.572993\pi\)
\(138\) 0 0
\(139\) 2044.00 1.24726 0.623632 − 0.781718i \(-0.285656\pi\)
0.623632 + 0.781718i \(0.285656\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −1764.00 −1.03156
\(144\) 0 0
\(145\) −2835.00 −1.62368
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1287.00 0.707618 0.353809 − 0.935318i \(-0.384886\pi\)
0.353809 + 0.935318i \(0.384886\pi\)
\(150\) 0 0
\(151\) 736.000 0.396655 0.198327 − 0.980136i \(-0.436449\pi\)
0.198327 + 0.980136i \(0.436449\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 6468.00 3.35176
\(156\) 0 0
\(157\) −2149.00 −1.09241 −0.546207 − 0.837650i \(-0.683929\pi\)
−0.546207 + 0.837650i \(0.683929\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −1440.00 −0.704894
\(162\) 0 0
\(163\) 3088.00 1.48387 0.741935 − 0.670472i \(-0.233908\pi\)
0.741935 + 0.670472i \(0.233908\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 168.000 0.0778457 0.0389228 − 0.999242i \(-0.487607\pi\)
0.0389228 + 0.999242i \(0.487607\pi\)
\(168\) 0 0
\(169\) 204.000 0.0928539
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 3003.00 1.31973 0.659867 − 0.751383i \(-0.270613\pi\)
0.659867 + 0.751383i \(0.270613\pi\)
\(174\) 0 0
\(175\) −2528.00 −1.09199
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −1164.00 −0.486042 −0.243021 − 0.970021i \(-0.578138\pi\)
−0.243021 + 0.970021i \(0.578138\pi\)
\(180\) 0 0
\(181\) −1666.00 −0.684159 −0.342080 − 0.939671i \(-0.611131\pi\)
−0.342080 + 0.939671i \(0.611131\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 21.0000 0.00834568
\(186\) 0 0
\(187\) −756.000 −0.295637
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2064.00 −0.781915 −0.390958 − 0.920409i \(-0.627856\pi\)
−0.390958 + 0.920409i \(0.627856\pi\)
\(192\) 0 0
\(193\) −565.000 −0.210723 −0.105362 − 0.994434i \(-0.533600\pi\)
−0.105362 + 0.994434i \(0.533600\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 4731.00 1.71101 0.855507 − 0.517791i \(-0.173246\pi\)
0.855507 + 0.517791i \(0.173246\pi\)
\(198\) 0 0
\(199\) −4676.00 −1.66569 −0.832846 − 0.553504i \(-0.813290\pi\)
−0.832846 + 0.553504i \(0.813290\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1080.00 −0.373405
\(204\) 0 0
\(205\) −882.000 −0.300495
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 4032.00 1.33445
\(210\) 0 0
\(211\) −3380.00 −1.10279 −0.551395 − 0.834244i \(-0.685904\pi\)
−0.551395 + 0.834244i \(0.685904\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 420.000 0.133227
\(216\) 0 0
\(217\) 2464.00 0.770817
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1029.00 0.313204
\(222\) 0 0
\(223\) 5236.00 1.57233 0.786163 − 0.618020i \(-0.212065\pi\)
0.786163 + 0.618020i \(0.212065\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3864.00 1.12979 0.564896 − 0.825162i \(-0.308916\pi\)
0.564896 + 0.825162i \(0.308916\pi\)
\(228\) 0 0
\(229\) −3913.00 −1.12916 −0.564581 − 0.825377i \(-0.690962\pi\)
−0.564581 + 0.825377i \(0.690962\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6333.00 −1.78064 −0.890319 − 0.455337i \(-0.849519\pi\)
−0.890319 + 0.455337i \(0.849519\pi\)
\(234\) 0 0
\(235\) −1764.00 −0.489662
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −3828.00 −1.03604 −0.518018 − 0.855370i \(-0.673330\pi\)
−0.518018 + 0.855370i \(0.673330\pi\)
\(240\) 0 0
\(241\) −1477.00 −0.394780 −0.197390 − 0.980325i \(-0.563246\pi\)
−0.197390 + 0.980325i \(0.563246\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 5859.00 1.52783
\(246\) 0 0
\(247\) −5488.00 −1.41374
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −3612.00 −0.908316 −0.454158 − 0.890921i \(-0.650060\pi\)
−0.454158 + 0.890921i \(0.650060\pi\)
\(252\) 0 0
\(253\) 6480.00 1.61025
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 399.000 0.0968441 0.0484221 − 0.998827i \(-0.484581\pi\)
0.0484221 + 0.998827i \(0.484581\pi\)
\(258\) 0 0
\(259\) 8.00000 0.00191929
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 3228.00 0.756833 0.378416 − 0.925635i \(-0.376469\pi\)
0.378416 + 0.925635i \(0.376469\pi\)
\(264\) 0 0
\(265\) −3654.00 −0.847032
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 147.000 0.0333188 0.0166594 − 0.999861i \(-0.494697\pi\)
0.0166594 + 0.999861i \(0.494697\pi\)
\(270\) 0 0
\(271\) −3332.00 −0.746880 −0.373440 − 0.927654i \(-0.621822\pi\)
−0.373440 + 0.927654i \(0.621822\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 11376.0 2.49454
\(276\) 0 0
\(277\) 2414.00 0.523622 0.261811 − 0.965119i \(-0.415680\pi\)
0.261811 + 0.965119i \(0.415680\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 3555.00 0.754710 0.377355 − 0.926069i \(-0.376834\pi\)
0.377355 + 0.926069i \(0.376834\pi\)
\(282\) 0 0
\(283\) −5348.00 −1.12334 −0.561671 − 0.827361i \(-0.689841\pi\)
−0.561671 + 0.827361i \(0.689841\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −336.000 −0.0691061
\(288\) 0 0
\(289\) −4472.00 −0.910238
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −6489.00 −1.29383 −0.646914 − 0.762563i \(-0.723941\pi\)
−0.646914 + 0.762563i \(0.723941\pi\)
\(294\) 0 0
\(295\) −10584.0 −2.08890
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −8820.00 −1.70593
\(300\) 0 0
\(301\) 160.000 0.0306387
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 8085.00 1.51785
\(306\) 0 0
\(307\) 1204.00 0.223830 0.111915 − 0.993718i \(-0.464302\pi\)
0.111915 + 0.993718i \(0.464302\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −3192.00 −0.581999 −0.291000 − 0.956723i \(-0.593988\pi\)
−0.291000 + 0.956723i \(0.593988\pi\)
\(312\) 0 0
\(313\) −3241.00 −0.585278 −0.292639 − 0.956223i \(-0.594533\pi\)
−0.292639 + 0.956223i \(0.594533\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −5325.00 −0.943476 −0.471738 − 0.881739i \(-0.656373\pi\)
−0.471738 + 0.881739i \(0.656373\pi\)
\(318\) 0 0
\(319\) 4860.00 0.853002
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −2352.00 −0.405167
\(324\) 0 0
\(325\) −15484.0 −2.64276
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −672.000 −0.112610
\(330\) 0 0
\(331\) −968.000 −0.160743 −0.0803717 − 0.996765i \(-0.525611\pi\)
−0.0803717 + 0.996765i \(0.525611\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 5712.00 0.931582
\(336\) 0 0
\(337\) 9890.00 1.59864 0.799321 − 0.600904i \(-0.205192\pi\)
0.799321 + 0.600904i \(0.205192\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −11088.0 −1.76085
\(342\) 0 0
\(343\) 4976.00 0.783320
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −1560.00 −0.241341 −0.120670 − 0.992693i \(-0.538504\pi\)
−0.120670 + 0.992693i \(0.538504\pi\)
\(348\) 0 0
\(349\) 2870.00 0.440194 0.220097 − 0.975478i \(-0.429363\pi\)
0.220097 + 0.975478i \(0.429363\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −7182.00 −1.08289 −0.541444 − 0.840737i \(-0.682122\pi\)
−0.541444 + 0.840737i \(0.682122\pi\)
\(354\) 0 0
\(355\) 18648.0 2.78798
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 8100.00 1.19081 0.595406 − 0.803425i \(-0.296991\pi\)
0.595406 + 0.803425i \(0.296991\pi\)
\(360\) 0 0
\(361\) 5685.00 0.828838
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −7791.00 −1.11726
\(366\) 0 0
\(367\) −11144.0 −1.58505 −0.792523 − 0.609842i \(-0.791233\pi\)
−0.792523 + 0.609842i \(0.791233\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −1392.00 −0.194795
\(372\) 0 0
\(373\) 13838.0 1.92092 0.960462 − 0.278412i \(-0.0898080\pi\)
0.960462 + 0.278412i \(0.0898080\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −6615.00 −0.903687
\(378\) 0 0
\(379\) −1196.00 −0.162096 −0.0810480 − 0.996710i \(-0.525827\pi\)
−0.0810480 + 0.996710i \(0.525827\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 3864.00 0.515512 0.257756 − 0.966210i \(-0.417017\pi\)
0.257756 + 0.966210i \(0.417017\pi\)
\(384\) 0 0
\(385\) 6048.00 0.800609
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 5070.00 0.660821 0.330410 − 0.943837i \(-0.392813\pi\)
0.330410 + 0.943837i \(0.392813\pi\)
\(390\) 0 0
\(391\) −3780.00 −0.488907
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −13692.0 −1.74410
\(396\) 0 0
\(397\) 15239.0 1.92651 0.963254 − 0.268593i \(-0.0865587\pi\)
0.963254 + 0.268593i \(0.0865587\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1707.00 0.212577 0.106289 − 0.994335i \(-0.466103\pi\)
0.106289 + 0.994335i \(0.466103\pi\)
\(402\) 0 0
\(403\) 15092.0 1.86547
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −36.0000 −0.00438441
\(408\) 0 0
\(409\) −13321.0 −1.61047 −0.805234 − 0.592958i \(-0.797960\pi\)
−0.805234 + 0.592958i \(0.797960\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −4032.00 −0.480392
\(414\) 0 0
\(415\) −1764.00 −0.208654
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −13944.0 −1.62580 −0.812899 − 0.582405i \(-0.802112\pi\)
−0.812899 + 0.582405i \(0.802112\pi\)
\(420\) 0 0
\(421\) −10837.0 −1.25454 −0.627272 − 0.778800i \(-0.715829\pi\)
−0.627272 + 0.778800i \(0.715829\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −6636.00 −0.757396
\(426\) 0 0
\(427\) 3080.00 0.349067
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 12612.0 1.40951 0.704755 − 0.709451i \(-0.251057\pi\)
0.704755 + 0.709451i \(0.251057\pi\)
\(432\) 0 0
\(433\) −9709.00 −1.07756 −0.538781 − 0.842446i \(-0.681115\pi\)
−0.538781 + 0.842446i \(0.681115\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 20160.0 2.20683
\(438\) 0 0
\(439\) −10388.0 −1.12937 −0.564684 − 0.825307i \(-0.691002\pi\)
−0.564684 + 0.825307i \(0.691002\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −2508.00 −0.268981 −0.134491 − 0.990915i \(-0.542940\pi\)
−0.134491 + 0.990915i \(0.542940\pi\)
\(444\) 0 0
\(445\) 441.000 0.0469784
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 13698.0 1.43975 0.719876 − 0.694103i \(-0.244199\pi\)
0.719876 + 0.694103i \(0.244199\pi\)
\(450\) 0 0
\(451\) 1512.00 0.157865
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −8232.00 −0.848180
\(456\) 0 0
\(457\) −9745.00 −0.997488 −0.498744 − 0.866749i \(-0.666205\pi\)
−0.498744 + 0.866749i \(0.666205\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −17514.0 −1.76943 −0.884716 − 0.466130i \(-0.845648\pi\)
−0.884716 + 0.466130i \(0.845648\pi\)
\(462\) 0 0
\(463\) −4640.00 −0.465743 −0.232872 − 0.972507i \(-0.574812\pi\)
−0.232872 + 0.972507i \(0.574812\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −4368.00 −0.432820 −0.216410 − 0.976303i \(-0.569435\pi\)
−0.216410 + 0.976303i \(0.569435\pi\)
\(468\) 0 0
\(469\) 2176.00 0.214240
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −720.000 −0.0699908
\(474\) 0 0
\(475\) 35392.0 3.41873
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −18816.0 −1.79483 −0.897416 − 0.441184i \(-0.854558\pi\)
−0.897416 + 0.441184i \(0.854558\pi\)
\(480\) 0 0
\(481\) 49.0000 0.00464492
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 26166.0 2.44977
\(486\) 0 0
\(487\) 13756.0 1.27997 0.639983 − 0.768389i \(-0.278941\pi\)
0.639983 + 0.768389i \(0.278941\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 7740.00 0.711408 0.355704 − 0.934599i \(-0.384241\pi\)
0.355704 + 0.934599i \(0.384241\pi\)
\(492\) 0 0
\(493\) −2835.00 −0.258990
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 7104.00 0.641163
\(498\) 0 0
\(499\) −2396.00 −0.214949 −0.107475 − 0.994208i \(-0.534276\pi\)
−0.107475 + 0.994208i \(0.534276\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 12096.0 1.07223 0.536117 − 0.844144i \(-0.319890\pi\)
0.536117 + 0.844144i \(0.319890\pi\)
\(504\) 0 0
\(505\) 11466.0 1.01036
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −1722.00 −0.149953 −0.0749767 − 0.997185i \(-0.523888\pi\)
−0.0749767 + 0.997185i \(0.523888\pi\)
\(510\) 0 0
\(511\) −2968.00 −0.256940
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −4116.00 −0.352180
\(516\) 0 0
\(517\) 3024.00 0.257244
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −2982.00 −0.250756 −0.125378 − 0.992109i \(-0.540014\pi\)
−0.125378 + 0.992109i \(0.540014\pi\)
\(522\) 0 0
\(523\) −812.000 −0.0678896 −0.0339448 − 0.999424i \(-0.510807\pi\)
−0.0339448 + 0.999424i \(0.510807\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6468.00 0.534631
\(528\) 0 0
\(529\) 20233.0 1.66294
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −2058.00 −0.167246
\(534\) 0 0
\(535\) 6300.00 0.509108
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −10044.0 −0.802645
\(540\) 0 0
\(541\) 7055.00 0.560662 0.280331 − 0.959903i \(-0.409556\pi\)
0.280331 + 0.959903i \(0.409556\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 22449.0 1.76442
\(546\) 0 0
\(547\) 14596.0 1.14091 0.570457 − 0.821328i \(-0.306766\pi\)
0.570457 + 0.821328i \(0.306766\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 15120.0 1.16903
\(552\) 0 0
\(553\) −5216.00 −0.401097
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 7755.00 0.589928 0.294964 − 0.955508i \(-0.404692\pi\)
0.294964 + 0.955508i \(0.404692\pi\)
\(558\) 0 0
\(559\) 980.000 0.0741495
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 16044.0 1.20102 0.600510 − 0.799617i \(-0.294964\pi\)
0.600510 + 0.799617i \(0.294964\pi\)
\(564\) 0 0
\(565\) 18837.0 1.40262
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −17025.0 −1.25435 −0.627175 − 0.778878i \(-0.715789\pi\)
−0.627175 + 0.778878i \(0.715789\pi\)
\(570\) 0 0
\(571\) −3320.00 −0.243323 −0.121662 − 0.992572i \(-0.538822\pi\)
−0.121662 + 0.992572i \(0.538822\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 56880.0 4.12532
\(576\) 0 0
\(577\) 1127.00 0.0813130 0.0406565 − 0.999173i \(-0.487055\pi\)
0.0406565 + 0.999173i \(0.487055\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −672.000 −0.0479850
\(582\) 0 0
\(583\) 6264.00 0.444989
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −84.0000 −0.00590639 −0.00295320 − 0.999996i \(-0.500940\pi\)
−0.00295320 + 0.999996i \(0.500940\pi\)
\(588\) 0 0
\(589\) −34496.0 −2.41321
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1743.00 0.120702 0.0603511 − 0.998177i \(-0.480778\pi\)
0.0603511 + 0.998177i \(0.480778\pi\)
\(594\) 0 0
\(595\) −3528.00 −0.243082
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −16092.0 −1.09766 −0.548832 − 0.835932i \(-0.684928\pi\)
−0.548832 + 0.835932i \(0.684928\pi\)
\(600\) 0 0
\(601\) 21035.0 1.42768 0.713840 − 0.700309i \(-0.246954\pi\)
0.713840 + 0.700309i \(0.246954\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 735.000 0.0493917
\(606\) 0 0
\(607\) −6776.00 −0.453096 −0.226548 − 0.974000i \(-0.572744\pi\)
−0.226548 + 0.974000i \(0.572744\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −4116.00 −0.272530
\(612\) 0 0
\(613\) −23794.0 −1.56775 −0.783875 − 0.620919i \(-0.786760\pi\)
−0.783875 + 0.620919i \(0.786760\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −21621.0 −1.41074 −0.705372 − 0.708838i \(-0.749220\pi\)
−0.705372 + 0.708838i \(0.749220\pi\)
\(618\) 0 0
\(619\) −22232.0 −1.44359 −0.721793 − 0.692109i \(-0.756682\pi\)
−0.721793 + 0.692109i \(0.756682\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 168.000 0.0108038
\(624\) 0 0
\(625\) 44731.0 2.86278
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 21.0000 0.00133120
\(630\) 0 0
\(631\) 9280.00 0.585469 0.292735 − 0.956194i \(-0.405435\pi\)
0.292735 + 0.956194i \(0.405435\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 32172.0 2.01056
\(636\) 0 0
\(637\) 13671.0 0.850337
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 19179.0 1.18179 0.590893 − 0.806750i \(-0.298775\pi\)
0.590893 + 0.806750i \(0.298775\pi\)
\(642\) 0 0
\(643\) 3220.00 0.197487 0.0987437 − 0.995113i \(-0.468518\pi\)
0.0987437 + 0.995113i \(0.468518\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 14112.0 0.857496 0.428748 − 0.903424i \(-0.358955\pi\)
0.428748 + 0.903424i \(0.358955\pi\)
\(648\) 0 0
\(649\) 18144.0 1.09740
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −22842.0 −1.36888 −0.684438 − 0.729071i \(-0.739953\pi\)
−0.684438 + 0.729071i \(0.739953\pi\)
\(654\) 0 0
\(655\) −17640.0 −1.05229
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −21720.0 −1.28390 −0.641951 − 0.766746i \(-0.721875\pi\)
−0.641951 + 0.766746i \(0.721875\pi\)
\(660\) 0 0
\(661\) 26327.0 1.54917 0.774585 − 0.632470i \(-0.217959\pi\)
0.774585 + 0.632470i \(0.217959\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 18816.0 1.09722
\(666\) 0 0
\(667\) 24300.0 1.41064
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −13860.0 −0.797406
\(672\) 0 0
\(673\) −19741.0 −1.13070 −0.565349 − 0.824852i \(-0.691258\pi\)
−0.565349 + 0.824852i \(0.691258\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −12642.0 −0.717683 −0.358842 − 0.933398i \(-0.616828\pi\)
−0.358842 + 0.933398i \(0.616828\pi\)
\(678\) 0 0
\(679\) 9968.00 0.563383
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −26172.0 −1.46624 −0.733121 − 0.680098i \(-0.761937\pi\)
−0.733121 + 0.680098i \(0.761937\pi\)
\(684\) 0 0
\(685\) 15309.0 0.853908
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −8526.00 −0.471429
\(690\) 0 0
\(691\) 9520.00 0.524107 0.262053 − 0.965053i \(-0.415600\pi\)
0.262053 + 0.965053i \(0.415600\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −42924.0 −2.34273
\(696\) 0 0
\(697\) −882.000 −0.0479313
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −16773.0 −0.903720 −0.451860 − 0.892089i \(-0.649239\pi\)
−0.451860 + 0.892089i \(0.649239\pi\)
\(702\) 0 0
\(703\) −112.000 −0.00600876
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 4368.00 0.232356
\(708\) 0 0
\(709\) 12767.0 0.676269 0.338135 − 0.941098i \(-0.390204\pi\)
0.338135 + 0.941098i \(0.390204\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −55440.0 −2.91198
\(714\) 0 0
\(715\) 37044.0 1.93758
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −24948.0 −1.29402 −0.647012 − 0.762480i \(-0.723982\pi\)
−0.647012 + 0.762480i \(0.723982\pi\)
\(720\) 0 0
\(721\) −1568.00 −0.0809922
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 42660.0 2.18531
\(726\) 0 0
\(727\) −56.0000 −0.00285684 −0.00142842 − 0.999999i \(-0.500455\pi\)
−0.00142842 + 0.999999i \(0.500455\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 420.000 0.0212507
\(732\) 0 0
\(733\) 1190.00 0.0599641 0.0299820 − 0.999550i \(-0.490455\pi\)
0.0299820 + 0.999550i \(0.490455\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −9792.00 −0.489407
\(738\) 0 0
\(739\) 26692.0 1.32866 0.664331 − 0.747439i \(-0.268717\pi\)
0.664331 + 0.747439i \(0.268717\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 18852.0 0.930838 0.465419 − 0.885090i \(-0.345904\pi\)
0.465419 + 0.885090i \(0.345904\pi\)
\(744\) 0 0
\(745\) −27027.0 −1.32912
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 2400.00 0.117082
\(750\) 0 0
\(751\) −22616.0 −1.09889 −0.549447 − 0.835528i \(-0.685162\pi\)
−0.549447 + 0.835528i \(0.685162\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −15456.0 −0.745035
\(756\) 0 0
\(757\) 9326.00 0.447766 0.223883 − 0.974616i \(-0.428127\pi\)
0.223883 + 0.974616i \(0.428127\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −20769.0 −0.989324 −0.494662 − 0.869085i \(-0.664708\pi\)
−0.494662 + 0.869085i \(0.664708\pi\)
\(762\) 0 0
\(763\) 8552.00 0.405771
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −24696.0 −1.16261
\(768\) 0 0
\(769\) −301.000 −0.0141149 −0.00705744 − 0.999975i \(-0.502246\pi\)
−0.00705744 + 0.999975i \(0.502246\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 17955.0 0.835442 0.417721 − 0.908575i \(-0.362829\pi\)
0.417721 + 0.908575i \(0.362829\pi\)
\(774\) 0 0
\(775\) −97328.0 −4.51113
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 4704.00 0.216352
\(780\) 0 0
\(781\) −31968.0 −1.46467
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 45129.0 2.05188
\(786\) 0 0
\(787\) 27412.0 1.24159 0.620796 − 0.783972i \(-0.286810\pi\)
0.620796 + 0.783972i \(0.286810\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 7176.00 0.322565
\(792\) 0 0
\(793\) 18865.0 0.844787
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −22533.0 −1.00146 −0.500728 − 0.865605i \(-0.666934\pi\)
−0.500728 + 0.865605i \(0.666934\pi\)
\(798\) 0 0
\(799\) −1764.00 −0.0781049
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 13356.0 0.586953
\(804\) 0 0
\(805\) 30240.0 1.32400
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −6105.00 −0.265316 −0.132658 − 0.991162i \(-0.542351\pi\)
−0.132658 + 0.991162i \(0.542351\pi\)
\(810\) 0 0
\(811\) 3472.00 0.150331 0.0751655 − 0.997171i \(-0.476051\pi\)
0.0751655 + 0.997171i \(0.476051\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −64848.0 −2.78715
\(816\) 0 0
\(817\) −2240.00 −0.0959213
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −4929.00 −0.209529 −0.104764 − 0.994497i \(-0.533409\pi\)
−0.104764 + 0.994497i \(0.533409\pi\)
\(822\) 0 0
\(823\) −39524.0 −1.67402 −0.837011 − 0.547186i \(-0.815699\pi\)
−0.837011 + 0.547186i \(0.815699\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 38676.0 1.62623 0.813117 − 0.582100i \(-0.197769\pi\)
0.813117 + 0.582100i \(0.197769\pi\)
\(828\) 0 0
\(829\) 16646.0 0.697394 0.348697 − 0.937236i \(-0.386624\pi\)
0.348697 + 0.937236i \(0.386624\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 5859.00 0.243700
\(834\) 0 0
\(835\) −3528.00 −0.146217
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 6636.00 0.273063 0.136532 − 0.990636i \(-0.456404\pi\)
0.136532 + 0.990636i \(0.456404\pi\)
\(840\) 0 0
\(841\) −6164.00 −0.252737
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −4284.00 −0.174407
\(846\) 0 0
\(847\) 280.000 0.0113588
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −180.000 −0.00725067
\(852\) 0 0
\(853\) −9394.00 −0.377074 −0.188537 − 0.982066i \(-0.560375\pi\)
−0.188537 + 0.982066i \(0.560375\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 7287.00 0.290454 0.145227 − 0.989398i \(-0.453609\pi\)
0.145227 + 0.989398i \(0.453609\pi\)
\(858\) 0 0
\(859\) 25732.0 1.02208 0.511039 − 0.859558i \(-0.329261\pi\)
0.511039 + 0.859558i \(0.329261\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −7188.00 −0.283525 −0.141763 − 0.989901i \(-0.545277\pi\)
−0.141763 + 0.989901i \(0.545277\pi\)
\(864\) 0 0
\(865\) −63063.0 −2.47885
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 23472.0 0.916264
\(870\) 0 0
\(871\) 13328.0 0.518487
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 32088.0 1.23974
\(876\) 0 0
\(877\) −12601.0 −0.485183 −0.242592 − 0.970129i \(-0.577997\pi\)
−0.242592 + 0.970129i \(0.577997\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 210.000 0.00803074 0.00401537 − 0.999992i \(-0.498722\pi\)
0.00401537 + 0.999992i \(0.498722\pi\)
\(882\) 0 0
\(883\) 47524.0 1.81122 0.905612 − 0.424108i \(-0.139412\pi\)
0.905612 + 0.424108i \(0.139412\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −41748.0 −1.58034 −0.790169 − 0.612888i \(-0.790008\pi\)
−0.790169 + 0.612888i \(0.790008\pi\)
\(888\) 0 0
\(889\) 12256.0 0.462377
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 9408.00 0.352550
\(894\) 0 0
\(895\) 24444.0 0.912931
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −41580.0 −1.54257
\(900\) 0 0
\(901\) −3654.00 −0.135108
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 34986.0 1.28505
\(906\) 0 0
\(907\) −14720.0 −0.538886 −0.269443 − 0.963016i \(-0.586840\pi\)
−0.269443 + 0.963016i \(0.586840\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 17652.0 0.641972 0.320986 − 0.947084i \(-0.395986\pi\)
0.320986 + 0.947084i \(0.395986\pi\)
\(912\) 0 0
\(913\) 3024.00 0.109616
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −6720.00 −0.242000
\(918\) 0 0
\(919\) −17156.0 −0.615804 −0.307902 − 0.951418i \(-0.599627\pi\)
−0.307902 + 0.951418i \(0.599627\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 43512.0 1.55170
\(924\) 0 0
\(925\) −316.000 −0.0112324
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −24213.0 −0.855116 −0.427558 − 0.903988i \(-0.640626\pi\)
−0.427558 + 0.903988i \(0.640626\pi\)
\(930\) 0 0
\(931\) −31248.0 −1.10001
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 15876.0 0.555295
\(936\) 0 0
\(937\) −15085.0 −0.525940 −0.262970 − 0.964804i \(-0.584702\pi\)
−0.262970 + 0.964804i \(0.584702\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −5229.00 −0.181148 −0.0905741 − 0.995890i \(-0.528870\pi\)
−0.0905741 + 0.995890i \(0.528870\pi\)
\(942\) 0 0
\(943\) 7560.00 0.261068
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 28488.0 0.977546 0.488773 − 0.872411i \(-0.337445\pi\)
0.488773 + 0.872411i \(0.337445\pi\)
\(948\) 0 0
\(949\) −18179.0 −0.621829
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1143.00 0.0388514 0.0194257 − 0.999811i \(-0.493816\pi\)
0.0194257 + 0.999811i \(0.493816\pi\)
\(954\) 0 0
\(955\) 43344.0 1.46867
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 5832.00 0.196377
\(960\) 0 0
\(961\) 65073.0 2.18432
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 11865.0 0.395801
\(966\) 0 0
\(967\) −36416.0 −1.21102 −0.605512 − 0.795836i \(-0.707032\pi\)
−0.605512 + 0.795836i \(0.707032\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 25200.0 0.832859 0.416430 − 0.909168i \(-0.363281\pi\)
0.416430 + 0.909168i \(0.363281\pi\)
\(972\) 0 0
\(973\) −16352.0 −0.538768
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −23742.0 −0.777455 −0.388728 − 0.921353i \(-0.627085\pi\)
−0.388728 + 0.921353i \(0.627085\pi\)
\(978\) 0 0
\(979\) −756.000 −0.0246801
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 4704.00 0.152629 0.0763145 − 0.997084i \(-0.475685\pi\)
0.0763145 + 0.997084i \(0.475685\pi\)
\(984\) 0 0
\(985\) −99351.0 −3.21379
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −3600.00 −0.115747
\(990\) 0 0
\(991\) 6280.00 0.201302 0.100651 − 0.994922i \(-0.467907\pi\)
0.100651 + 0.994922i \(0.467907\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 98196.0 3.12867
\(996\) 0 0
\(997\) −8701.00 −0.276393 −0.138196 − 0.990405i \(-0.544130\pi\)
−0.138196 + 0.990405i \(0.544130\pi\)
\(998\) 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 1296.4.a.a.1.1 1
3.2 odd 2 1296.4.a.h.1.1 1
4.3 odd 2 162.4.a.c.1.1 yes 1
12.11 even 2 162.4.a.b.1.1 ✓ 1
36.7 odd 6 162.4.c.d.109.1 2
36.11 even 6 162.4.c.e.109.1 2
36.23 even 6 162.4.c.e.55.1 2
36.31 odd 6 162.4.c.d.55.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
162.4.a.b.1.1 ✓ 1 12.11 even 2
162.4.a.c.1.1 yes 1 4.3 odd 2
162.4.c.d.55.1 2 36.31 odd 6
162.4.c.d.109.1 2 36.7 odd 6
162.4.c.e.55.1 2 36.23 even 6
162.4.c.e.109.1 2 36.11 even 6
1296.4.a.a.1.1 1 1.1 even 1 trivial
1296.4.a.h.1.1 1 3.2 odd 2