↑

Properties

Label 1568.4.a.b.1.1
Level $1568$
Weight $4$
Character 1568.1
Self dual yes
Analytic conductor $92.515$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1568,4,Mod(1,1568)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1568, 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("1568.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1568 = 2^{5} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1568.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: \(92.5149948890\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 1568.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.00000 q^{3} +4.00000 q^{5} +37.0000 q^{9} +O(q^{10})\) \(q-8.00000 q^{3} +4.00000 q^{5} +37.0000 q^{9} +40.0000 q^{11} -36.0000 q^{13} -32.0000 q^{15} +40.0000 q^{17} +72.0000 q^{19} +176.000 q^{23} -109.000 q^{25} -80.0000 q^{27} +162.000 q^{29} +16.0000 q^{31} -320.000 q^{33} -54.0000 q^{37} +288.000 q^{39} -472.000 q^{41} +72.0000 q^{43} +148.000 q^{45} -144.000 q^{47} -320.000 q^{51} +486.000 q^{53} +160.000 q^{55} -576.000 q^{57} +648.000 q^{59} -684.000 q^{61} -144.000 q^{65} -216.000 q^{67} -1408.00 q^{69} +608.000 q^{71} +1008.00 q^{73} +872.000 q^{75} -1008.00 q^{79} -359.000 q^{81} -216.000 q^{83} +160.000 q^{85} -1296.00 q^{87} -1040.00 q^{89} -128.000 q^{93} +288.000 q^{95} -936.000 q^{97} +1480.00 q^{99} +O(q^{100})\)

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
\(3\) −8.00000 −1.53960 −0.769800 − 0.638285i \(-0.779644\pi\)
−0.769800 + 0.638285i \(0.779644\pi\)
\(4\) 0 0
\(5\) 4.00000 0.357771 0.178885 − 0.983870i \(-0.442751\pi\)
0.178885 + 0.983870i \(0.442751\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 37.0000 1.37037
\(10\) 0 0
\(11\) 40.0000 1.09640 0.548202 − 0.836346i \(-0.315312\pi\)
0.548202 + 0.836346i \(0.315312\pi\)
\(12\) 0 0
\(13\) −36.0000 −0.768046 −0.384023 − 0.923323i \(-0.625462\pi\)
−0.384023 + 0.923323i \(0.625462\pi\)
\(14\) 0 0
\(15\) −32.0000 −0.550824
\(16\) 0 0
\(17\) 40.0000 0.570672 0.285336 − 0.958428i \(-0.407895\pi\)
0.285336 + 0.958428i \(0.407895\pi\)
\(18\) 0 0
\(19\) 72.0000 0.869365 0.434682 − 0.900584i \(-0.356861\pi\)
0.434682 + 0.900584i \(0.356861\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 176.000 1.59559 0.797794 − 0.602930i \(-0.206000\pi\)
0.797794 + 0.602930i \(0.206000\pi\)
\(24\) 0 0
\(25\) −109.000 −0.872000
\(26\) 0 0
\(27\) −80.0000 −0.570222
\(28\) 0 0
\(29\) 162.000 1.03733 0.518666 − 0.854977i \(-0.326429\pi\)
0.518666 + 0.854977i \(0.326429\pi\)
\(30\) 0 0
\(31\) 16.0000 0.0926995 0.0463498 − 0.998925i \(-0.485241\pi\)
0.0463498 + 0.998925i \(0.485241\pi\)
\(32\) 0 0
\(33\) −320.000 −1.68803
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −54.0000 −0.239934 −0.119967 − 0.992778i \(-0.538279\pi\)
−0.119967 + 0.992778i \(0.538279\pi\)
\(38\) 0 0
\(39\) 288.000 1.18248
\(40\) 0 0
\(41\) −472.000 −1.79790 −0.898951 − 0.438048i \(-0.855670\pi\)
−0.898951 + 0.438048i \(0.855670\pi\)
\(42\) 0 0
\(43\) 72.0000 0.255346 0.127673 − 0.991816i \(-0.459249\pi\)
0.127673 + 0.991816i \(0.459249\pi\)
\(44\) 0 0
\(45\) 148.000 0.490279
\(46\) 0 0
\(47\) −144.000 −0.446906 −0.223453 − 0.974715i \(-0.571733\pi\)
−0.223453 + 0.974715i \(0.571733\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −320.000 −0.878607
\(52\) 0 0
\(53\) 486.000 1.25957 0.629785 − 0.776769i \(-0.283143\pi\)
0.629785 + 0.776769i \(0.283143\pi\)
\(54\) 0 0
\(55\) 160.000 0.392262
\(56\) 0 0
\(57\) −576.000 −1.33847
\(58\) 0 0
\(59\) 648.000 1.42987 0.714936 − 0.699190i \(-0.246456\pi\)
0.714936 + 0.699190i \(0.246456\pi\)
\(60\) 0 0
\(61\) −684.000 −1.43569 −0.717846 − 0.696202i \(-0.754872\pi\)
−0.717846 + 0.696202i \(0.754872\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −144.000 −0.274785
\(66\) 0 0
\(67\) −216.000 −0.393860 −0.196930 − 0.980418i \(-0.563097\pi\)
−0.196930 + 0.980418i \(0.563097\pi\)
\(68\) 0 0
\(69\) −1408.00 −2.45657
\(70\) 0 0
\(71\) 608.000 1.01629 0.508143 − 0.861273i \(-0.330332\pi\)
0.508143 + 0.861273i \(0.330332\pi\)
\(72\) 0 0
\(73\) 1008.00 1.61613 0.808065 − 0.589093i \(-0.200515\pi\)
0.808065 + 0.589093i \(0.200515\pi\)
\(74\) 0 0
\(75\) 872.000 1.34253
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −1008.00 −1.43556 −0.717778 − 0.696272i \(-0.754841\pi\)
−0.717778 + 0.696272i \(0.754841\pi\)
\(80\) 0 0
\(81\) −359.000 −0.492455
\(82\) 0 0
\(83\) −216.000 −0.285652 −0.142826 − 0.989748i \(-0.545619\pi\)
−0.142826 + 0.989748i \(0.545619\pi\)
\(84\) 0 0
\(85\) 160.000 0.204170
\(86\) 0 0
\(87\) −1296.00 −1.59708
\(88\) 0 0
\(89\) −1040.00 −1.23865 −0.619325 − 0.785135i \(-0.712594\pi\)
−0.619325 + 0.785135i \(0.712594\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −128.000 −0.142720
\(94\) 0 0
\(95\) 288.000 0.311033
\(96\) 0 0
\(97\) −936.000 −0.979757 −0.489878 − 0.871791i \(-0.662959\pi\)
−0.489878 + 0.871791i \(0.662959\pi\)
\(98\) 0 0
\(99\) 1480.00 1.50248
\(100\) 0 0
\(101\) 148.000 0.145807 0.0729037 − 0.997339i \(-0.476773\pi\)
0.0729037 + 0.997339i \(0.476773\pi\)
\(102\) 0 0
\(103\) −1744.00 −1.66836 −0.834182 − 0.551490i \(-0.814060\pi\)
−0.834182 + 0.551490i \(0.814060\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −328.000 −0.296345 −0.148173 − 0.988962i \(-0.547339\pi\)
−0.148173 + 0.988962i \(0.547339\pi\)
\(108\) 0 0
\(109\) −1294.00 −1.13709 −0.568545 − 0.822652i \(-0.692493\pi\)
−0.568545 + 0.822652i \(0.692493\pi\)
\(110\) 0 0
\(111\) 432.000 0.369402
\(112\) 0 0
\(113\) 594.000 0.494503 0.247251 − 0.968951i \(-0.420473\pi\)
0.247251 + 0.968951i \(0.420473\pi\)
\(114\) 0 0
\(115\) 704.000 0.570855
\(116\) 0 0
\(117\) −1332.00 −1.05251
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 269.000 0.202104
\(122\) 0 0
\(123\) 3776.00 2.76805
\(124\) 0 0
\(125\) −936.000 −0.669747
\(126\) 0 0
\(127\) 576.000 0.402455 0.201227 − 0.979545i \(-0.435507\pi\)
0.201227 + 0.979545i \(0.435507\pi\)
\(128\) 0 0
\(129\) −576.000 −0.393132
\(130\) 0 0
\(131\) 2088.00 1.39259 0.696295 − 0.717755i \(-0.254830\pi\)
0.696295 + 0.717755i \(0.254830\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −320.000 −0.204009
\(136\) 0 0
\(137\) 2502.00 1.56029 0.780147 − 0.625596i \(-0.215144\pi\)
0.780147 + 0.625596i \(0.215144\pi\)
\(138\) 0 0
\(139\) −1960.00 −1.19601 −0.598004 − 0.801493i \(-0.704039\pi\)
−0.598004 + 0.801493i \(0.704039\pi\)
\(140\) 0 0
\(141\) 1152.00 0.688056
\(142\) 0 0
\(143\) −1440.00 −0.842090
\(144\) 0 0
\(145\) 648.000 0.371127
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3366.00 1.85069 0.925347 − 0.379121i \(-0.123774\pi\)
0.925347 + 0.379121i \(0.123774\pi\)
\(150\) 0 0
\(151\) 2736.00 1.47452 0.737260 − 0.675609i \(-0.236119\pi\)
0.737260 + 0.675609i \(0.236119\pi\)
\(152\) 0 0
\(153\) 1480.00 0.782032
\(154\) 0 0
\(155\) 64.0000 0.0331652
\(156\) 0 0
\(157\) 684.000 0.347702 0.173851 − 0.984772i \(-0.444379\pi\)
0.173851 + 0.984772i \(0.444379\pi\)
\(158\) 0 0
\(159\) −3888.00 −1.93924
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1656.00 0.795754 0.397877 − 0.917439i \(-0.369747\pi\)
0.397877 + 0.917439i \(0.369747\pi\)
\(164\) 0 0
\(165\) −1280.00 −0.603926
\(166\) 0 0
\(167\) 3600.00 1.66812 0.834061 − 0.551672i \(-0.186010\pi\)
0.834061 + 0.551672i \(0.186010\pi\)
\(168\) 0 0
\(169\) −901.000 −0.410105
\(170\) 0 0
\(171\) 2664.00 1.19135
\(172\) 0 0
\(173\) 3676.00 1.61550 0.807749 − 0.589527i \(-0.200686\pi\)
0.807749 + 0.589527i \(0.200686\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −5184.00 −2.20143
\(178\) 0 0
\(179\) 2984.00 1.24600 0.623002 − 0.782220i \(-0.285913\pi\)
0.623002 + 0.782220i \(0.285913\pi\)
\(180\) 0 0
\(181\) 2988.00 1.22705 0.613526 − 0.789675i \(-0.289751\pi\)
0.613526 + 0.789675i \(0.289751\pi\)
\(182\) 0 0
\(183\) 5472.00 2.21039
\(184\) 0 0
\(185\) −216.000 −0.0858413
\(186\) 0 0
\(187\) 1600.00 0.625688
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1328.00 −0.503093 −0.251546 − 0.967845i \(-0.580939\pi\)
−0.251546 + 0.967845i \(0.580939\pi\)
\(192\) 0 0
\(193\) 306.000 0.114126 0.0570631 − 0.998371i \(-0.481826\pi\)
0.0570631 + 0.998371i \(0.481826\pi\)
\(194\) 0 0
\(195\) 1152.00 0.423059
\(196\) 0 0
\(197\) −1674.00 −0.605419 −0.302710 − 0.953083i \(-0.597891\pi\)
−0.302710 + 0.953083i \(0.597891\pi\)
\(198\) 0 0
\(199\) −2736.00 −0.974623 −0.487311 − 0.873228i \(-0.662022\pi\)
−0.487311 + 0.873228i \(0.662022\pi\)
\(200\) 0 0
\(201\) 1728.00 0.606387
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −1888.00 −0.643237
\(206\) 0 0
\(207\) 6512.00 2.18655
\(208\) 0 0
\(209\) 2880.00 0.953176
\(210\) 0 0
\(211\) 4104.00 1.33901 0.669505 − 0.742808i \(-0.266506\pi\)
0.669505 + 0.742808i \(0.266506\pi\)
\(212\) 0 0
\(213\) −4864.00 −1.56468
\(214\) 0 0
\(215\) 288.000 0.0913555
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −8064.00 −2.48819
\(220\) 0 0
\(221\) −1440.00 −0.438303
\(222\) 0 0
\(223\) −3168.00 −0.951323 −0.475661 − 0.879628i \(-0.657791\pi\)
−0.475661 + 0.879628i \(0.657791\pi\)
\(224\) 0 0
\(225\) −4033.00 −1.19496
\(226\) 0 0
\(227\) 2088.00 0.610508 0.305254 − 0.952271i \(-0.401259\pi\)
0.305254 + 0.952271i \(0.401259\pi\)
\(228\) 0 0
\(229\) 3708.00 1.07001 0.535003 − 0.844850i \(-0.320310\pi\)
0.535003 + 0.844850i \(0.320310\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2070.00 0.582018 0.291009 − 0.956720i \(-0.406009\pi\)
0.291009 + 0.956720i \(0.406009\pi\)
\(234\) 0 0
\(235\) −576.000 −0.159890
\(236\) 0 0
\(237\) 8064.00 2.21018
\(238\) 0 0
\(239\) 896.000 0.242500 0.121250 − 0.992622i \(-0.461310\pi\)
0.121250 + 0.992622i \(0.461310\pi\)
\(240\) 0 0
\(241\) −3240.00 −0.866003 −0.433002 − 0.901393i \(-0.642546\pi\)
−0.433002 + 0.901393i \(0.642546\pi\)
\(242\) 0 0
\(243\) 5032.00 1.32841
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −2592.00 −0.667713
\(248\) 0 0
\(249\) 1728.00 0.439789
\(250\) 0 0
\(251\) −3816.00 −0.959617 −0.479808 − 0.877373i \(-0.659294\pi\)
−0.479808 + 0.877373i \(0.659294\pi\)
\(252\) 0 0
\(253\) 7040.00 1.74941
\(254\) 0 0
\(255\) −1280.00 −0.314340
\(256\) 0 0
\(257\) 4064.00 0.986402 0.493201 − 0.869915i \(-0.335827\pi\)
0.493201 + 0.869915i \(0.335827\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 5994.00 1.42153
\(262\) 0 0
\(263\) 256.000 0.0600214 0.0300107 − 0.999550i \(-0.490446\pi\)
0.0300107 + 0.999550i \(0.490446\pi\)
\(264\) 0 0
\(265\) 1944.00 0.450638
\(266\) 0 0
\(267\) 8320.00 1.90703
\(268\) 0 0
\(269\) 1868.00 0.423398 0.211699 − 0.977335i \(-0.432100\pi\)
0.211699 + 0.977335i \(0.432100\pi\)
\(270\) 0 0
\(271\) −2432.00 −0.545142 −0.272571 − 0.962136i \(-0.587874\pi\)
−0.272571 + 0.962136i \(0.587874\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −4360.00 −0.956065
\(276\) 0 0
\(277\) 1926.00 0.417769 0.208885 − 0.977940i \(-0.433017\pi\)
0.208885 + 0.977940i \(0.433017\pi\)
\(278\) 0 0
\(279\) 592.000 0.127033
\(280\) 0 0
\(281\) −6714.00 −1.42535 −0.712676 − 0.701494i \(-0.752517\pi\)
−0.712676 + 0.701494i \(0.752517\pi\)
\(282\) 0 0
\(283\) 72.0000 0.0151235 0.00756176 − 0.999971i \(-0.497593\pi\)
0.00756176 + 0.999971i \(0.497593\pi\)
\(284\) 0 0
\(285\) −2304.00 −0.478867
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −3313.00 −0.674333
\(290\) 0 0
\(291\) 7488.00 1.50843
\(292\) 0 0
\(293\) 2308.00 0.460187 0.230094 − 0.973169i \(-0.426097\pi\)
0.230094 + 0.973169i \(0.426097\pi\)
\(294\) 0 0
\(295\) 2592.00 0.511566
\(296\) 0 0
\(297\) −3200.00 −0.625195
\(298\) 0 0
\(299\) −6336.00 −1.22549
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −1184.00 −0.224485
\(304\) 0 0
\(305\) −2736.00 −0.513649
\(306\) 0 0
\(307\) 3816.00 0.709416 0.354708 − 0.934977i \(-0.384580\pi\)
0.354708 + 0.934977i \(0.384580\pi\)
\(308\) 0 0
\(309\) 13952.0 2.56861
\(310\) 0 0
\(311\) 4032.00 0.735157 0.367578 − 0.929993i \(-0.380187\pi\)
0.367578 + 0.929993i \(0.380187\pi\)
\(312\) 0 0
\(313\) 8712.00 1.57326 0.786632 − 0.617423i \(-0.211823\pi\)
0.786632 + 0.617423i \(0.211823\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1854.00 0.328489 0.164245 − 0.986420i \(-0.447481\pi\)
0.164245 + 0.986420i \(0.447481\pi\)
\(318\) 0 0
\(319\) 6480.00 1.13734
\(320\) 0 0
\(321\) 2624.00 0.456254
\(322\) 0 0
\(323\) 2880.00 0.496122
\(324\) 0 0
\(325\) 3924.00 0.669736
\(326\) 0 0
\(327\) 10352.0 1.75066
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 6696.00 1.11192 0.555960 − 0.831209i \(-0.312351\pi\)
0.555960 + 0.831209i \(0.312351\pi\)
\(332\) 0 0
\(333\) −1998.00 −0.328798
\(334\) 0 0
\(335\) −864.000 −0.140912
\(336\) 0 0
\(337\) −3634.00 −0.587408 −0.293704 − 0.955896i \(-0.594888\pi\)
−0.293704 + 0.955896i \(0.594888\pi\)
\(338\) 0 0
\(339\) −4752.00 −0.761337
\(340\) 0 0
\(341\) 640.000 0.101636
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −5632.00 −0.878889
\(346\) 0 0
\(347\) 7016.00 1.08541 0.542707 − 0.839922i \(-0.317399\pi\)
0.542707 + 0.839922i \(0.317399\pi\)
\(348\) 0 0
\(349\) −11268.0 −1.72826 −0.864129 − 0.503270i \(-0.832130\pi\)
−0.864129 + 0.503270i \(0.832130\pi\)
\(350\) 0 0
\(351\) 2880.00 0.437957
\(352\) 0 0
\(353\) 2768.00 0.417353 0.208677 − 0.977985i \(-0.433084\pi\)
0.208677 + 0.977985i \(0.433084\pi\)
\(354\) 0 0
\(355\) 2432.00 0.363598
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 176.000 0.0258744 0.0129372 − 0.999916i \(-0.495882\pi\)
0.0129372 + 0.999916i \(0.495882\pi\)
\(360\) 0 0
\(361\) −1675.00 −0.244205
\(362\) 0 0
\(363\) −2152.00 −0.311159
\(364\) 0 0
\(365\) 4032.00 0.578204
\(366\) 0 0
\(367\) 10656.0 1.51564 0.757818 − 0.652466i \(-0.226265\pi\)
0.757818 + 0.652466i \(0.226265\pi\)
\(368\) 0 0
\(369\) −17464.0 −2.46379
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 5222.00 0.724893 0.362446 − 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 0 0
\(375\) 7488.00 1.03114
\(376\) 0 0
\(377\) −5832.00 −0.796720
\(378\) 0 0
\(379\) −7992.00 −1.08317 −0.541585 − 0.840646i \(-0.682176\pi\)
−0.541585 + 0.840646i \(0.682176\pi\)
\(380\) 0 0
\(381\) −4608.00 −0.619619
\(382\) 0 0
\(383\) −4176.00 −0.557137 −0.278569 − 0.960416i \(-0.589860\pi\)
−0.278569 + 0.960416i \(0.589860\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 2664.00 0.349919
\(388\) 0 0
\(389\) −1494.00 −0.194727 −0.0973635 − 0.995249i \(-0.531041\pi\)
−0.0973635 + 0.995249i \(0.531041\pi\)
\(390\) 0 0
\(391\) 7040.00 0.910558
\(392\) 0 0
\(393\) −16704.0 −2.14403
\(394\) 0 0
\(395\) −4032.00 −0.513600
\(396\) 0 0
\(397\) −3348.00 −0.423253 −0.211626 − 0.977351i \(-0.567876\pi\)
−0.211626 + 0.977351i \(0.567876\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −7650.00 −0.952675 −0.476338 − 0.879262i \(-0.658036\pi\)
−0.476338 + 0.879262i \(0.658036\pi\)
\(402\) 0 0
\(403\) −576.000 −0.0711975
\(404\) 0 0
\(405\) −1436.00 −0.176186
\(406\) 0 0
\(407\) −2160.00 −0.263064
\(408\) 0 0
\(409\) 4680.00 0.565797 0.282899 − 0.959150i \(-0.408704\pi\)
0.282899 + 0.959150i \(0.408704\pi\)
\(410\) 0 0
\(411\) −20016.0 −2.40223
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −864.000 −0.102198
\(416\) 0 0
\(417\) 15680.0 1.84137
\(418\) 0 0
\(419\) 13176.0 1.53625 0.768126 − 0.640299i \(-0.221189\pi\)
0.768126 + 0.640299i \(0.221189\pi\)
\(420\) 0 0
\(421\) −8874.00 −1.02730 −0.513649 − 0.858001i \(-0.671707\pi\)
−0.513649 + 0.858001i \(0.671707\pi\)
\(422\) 0 0
\(423\) −5328.00 −0.612426
\(424\) 0 0
\(425\) −4360.00 −0.497626
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 11520.0 1.29648
\(430\) 0 0
\(431\) −8096.00 −0.904804 −0.452402 − 0.891814i \(-0.649433\pi\)
−0.452402 + 0.891814i \(0.649433\pi\)
\(432\) 0 0
\(433\) −11016.0 −1.22262 −0.611311 − 0.791391i \(-0.709357\pi\)
−0.611311 + 0.791391i \(0.709357\pi\)
\(434\) 0 0
\(435\) −5184.00 −0.571388
\(436\) 0 0
\(437\) 12672.0 1.38715
\(438\) 0 0
\(439\) 16128.0 1.75341 0.876706 − 0.481027i \(-0.159736\pi\)
0.876706 + 0.481027i \(0.159736\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 5368.00 0.575714 0.287857 − 0.957673i \(-0.407057\pi\)
0.287857 + 0.957673i \(0.407057\pi\)
\(444\) 0 0
\(445\) −4160.00 −0.443153
\(446\) 0 0
\(447\) −26928.0 −2.84933
\(448\) 0 0
\(449\) 306.000 0.0321627 0.0160813 − 0.999871i \(-0.494881\pi\)
0.0160813 + 0.999871i \(0.494881\pi\)
\(450\) 0 0
\(451\) −18880.0 −1.97123
\(452\) 0 0
\(453\) −21888.0 −2.27017
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −10118.0 −1.03567 −0.517834 − 0.855481i \(-0.673261\pi\)
−0.517834 + 0.855481i \(0.673261\pi\)
\(458\) 0 0
\(459\) −3200.00 −0.325410
\(460\) 0 0
\(461\) −15412.0 −1.55707 −0.778534 − 0.627602i \(-0.784036\pi\)
−0.778534 + 0.627602i \(0.784036\pi\)
\(462\) 0 0
\(463\) 18288.0 1.83567 0.917835 − 0.396962i \(-0.129935\pi\)
0.917835 + 0.396962i \(0.129935\pi\)
\(464\) 0 0
\(465\) −512.000 −0.0510611
\(466\) 0 0
\(467\) −3672.00 −0.363854 −0.181927 − 0.983312i \(-0.558233\pi\)
−0.181927 + 0.983312i \(0.558233\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −5472.00 −0.535322
\(472\) 0 0
\(473\) 2880.00 0.279963
\(474\) 0 0
\(475\) −7848.00 −0.758086
\(476\) 0 0
\(477\) 17982.0 1.72608
\(478\) 0 0
\(479\) 12240.0 1.16756 0.583779 − 0.811913i \(-0.301574\pi\)
0.583779 + 0.811913i \(0.301574\pi\)
\(480\) 0 0
\(481\) 1944.00 0.184280
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −3744.00 −0.350528
\(486\) 0 0
\(487\) −12816.0 −1.19250 −0.596251 − 0.802798i \(-0.703344\pi\)
−0.596251 + 0.802798i \(0.703344\pi\)
\(488\) 0 0
\(489\) −13248.0 −1.22514
\(490\) 0 0
\(491\) 6520.00 0.599274 0.299637 − 0.954053i \(-0.403134\pi\)
0.299637 + 0.954053i \(0.403134\pi\)
\(492\) 0 0
\(493\) 6480.00 0.591977
\(494\) 0 0
\(495\) 5920.00 0.537544
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −15192.0 −1.36290 −0.681450 − 0.731864i \(-0.738650\pi\)
−0.681450 + 0.731864i \(0.738650\pi\)
\(500\) 0 0
\(501\) −28800.0 −2.56824
\(502\) 0 0
\(503\) 8928.00 0.791411 0.395706 − 0.918377i \(-0.370500\pi\)
0.395706 + 0.918377i \(0.370500\pi\)
\(504\) 0 0
\(505\) 592.000 0.0521657
\(506\) 0 0
\(507\) 7208.00 0.631397
\(508\) 0 0
\(509\) 13388.0 1.16584 0.582920 − 0.812529i \(-0.301910\pi\)
0.582920 + 0.812529i \(0.301910\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −5760.00 −0.495731
\(514\) 0 0
\(515\) −6976.00 −0.596892
\(516\) 0 0
\(517\) −5760.00 −0.489989
\(518\) 0 0
\(519\) −29408.0 −2.48722
\(520\) 0 0
\(521\) 13496.0 1.13488 0.567438 − 0.823416i \(-0.307935\pi\)
0.567438 + 0.823416i \(0.307935\pi\)
\(522\) 0 0
\(523\) −13880.0 −1.16048 −0.580239 − 0.814446i \(-0.697041\pi\)
−0.580239 + 0.814446i \(0.697041\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 640.000 0.0529010
\(528\) 0 0
\(529\) 18809.0 1.54590
\(530\) 0 0
\(531\) 23976.0 1.95945
\(532\) 0 0
\(533\) 16992.0 1.38087
\(534\) 0 0
\(535\) −1312.00 −0.106024
\(536\) 0 0
\(537\) −23872.0 −1.91835
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 23742.0 1.88678 0.943390 − 0.331685i \(-0.107617\pi\)
0.943390 + 0.331685i \(0.107617\pi\)
\(542\) 0 0
\(543\) −23904.0 −1.88917
\(544\) 0 0
\(545\) −5176.00 −0.406817
\(546\) 0 0
\(547\) −2952.00 −0.230747 −0.115373 − 0.993322i \(-0.536806\pi\)
−0.115373 + 0.993322i \(0.536806\pi\)
\(548\) 0 0
\(549\) −25308.0 −1.96743
\(550\) 0 0
\(551\) 11664.0 0.901821
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 1728.00 0.132161
\(556\) 0 0
\(557\) 8622.00 0.655881 0.327941 − 0.944698i \(-0.393645\pi\)
0.327941 + 0.944698i \(0.393645\pi\)
\(558\) 0 0
\(559\) −2592.00 −0.196118
\(560\) 0 0
\(561\) −12800.0 −0.963309
\(562\) 0 0
\(563\) −12744.0 −0.953989 −0.476994 − 0.878906i \(-0.658274\pi\)
−0.476994 + 0.878906i \(0.658274\pi\)
\(564\) 0 0
\(565\) 2376.00 0.176919
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −14922.0 −1.09941 −0.549704 − 0.835360i \(-0.685259\pi\)
−0.549704 + 0.835360i \(0.685259\pi\)
\(570\) 0 0
\(571\) −21528.0 −1.57779 −0.788896 − 0.614527i \(-0.789347\pi\)
−0.788896 + 0.614527i \(0.789347\pi\)
\(572\) 0 0
\(573\) 10624.0 0.774562
\(574\) 0 0
\(575\) −19184.0 −1.39135
\(576\) 0 0
\(577\) 19728.0 1.42338 0.711688 − 0.702496i \(-0.247931\pi\)
0.711688 + 0.702496i \(0.247931\pi\)
\(578\) 0 0
\(579\) −2448.00 −0.175709
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 19440.0 1.38100
\(584\) 0 0
\(585\) −5328.00 −0.376557
\(586\) 0 0
\(587\) −17928.0 −1.26059 −0.630296 − 0.776355i \(-0.717067\pi\)
−0.630296 + 0.776355i \(0.717067\pi\)
\(588\) 0 0
\(589\) 1152.00 0.0805897
\(590\) 0 0
\(591\) 13392.0 0.932104
\(592\) 0 0
\(593\) 8752.00 0.606073 0.303037 − 0.952979i \(-0.402000\pi\)
0.303037 + 0.952979i \(0.402000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 21888.0 1.50053
\(598\) 0 0
\(599\) 16448.0 1.12195 0.560974 − 0.827833i \(-0.310427\pi\)
0.560974 + 0.827833i \(0.310427\pi\)
\(600\) 0 0
\(601\) 11088.0 0.752561 0.376280 − 0.926506i \(-0.377203\pi\)
0.376280 + 0.926506i \(0.377203\pi\)
\(602\) 0 0
\(603\) −7992.00 −0.539734
\(604\) 0 0
\(605\) 1076.00 0.0723068
\(606\) 0 0
\(607\) −4032.00 −0.269611 −0.134805 − 0.990872i \(-0.543041\pi\)
−0.134805 + 0.990872i \(0.543041\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 5184.00 0.343244
\(612\) 0 0
\(613\) −8854.00 −0.583376 −0.291688 − 0.956513i \(-0.594217\pi\)
−0.291688 + 0.956513i \(0.594217\pi\)
\(614\) 0 0
\(615\) 15104.0 0.990329
\(616\) 0 0
\(617\) −1962.00 −0.128018 −0.0640090 − 0.997949i \(-0.520389\pi\)
−0.0640090 + 0.997949i \(0.520389\pi\)
\(618\) 0 0
\(619\) 1384.00 0.0898670 0.0449335 − 0.998990i \(-0.485692\pi\)
0.0449335 + 0.998990i \(0.485692\pi\)
\(620\) 0 0
\(621\) −14080.0 −0.909840
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 9881.00 0.632384
\(626\) 0 0
\(627\) −23040.0 −1.46751
\(628\) 0 0
\(629\) −2160.00 −0.136923
\(630\) 0 0
\(631\) 8352.00 0.526922 0.263461 − 0.964670i \(-0.415136\pi\)
0.263461 + 0.964670i \(0.415136\pi\)
\(632\) 0 0
\(633\) −32832.0 −2.06154
\(634\) 0 0
\(635\) 2304.00 0.143987
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 22496.0 1.39269
\(640\) 0 0
\(641\) −4338.00 −0.267302 −0.133651 − 0.991028i \(-0.542670\pi\)
−0.133651 + 0.991028i \(0.542670\pi\)
\(642\) 0 0
\(643\) 6264.00 0.384180 0.192090 − 0.981377i \(-0.438473\pi\)
0.192090 + 0.981377i \(0.438473\pi\)
\(644\) 0 0
\(645\) −2304.00 −0.140651
\(646\) 0 0
\(647\) 13392.0 0.813746 0.406873 − 0.913485i \(-0.366619\pi\)
0.406873 + 0.913485i \(0.366619\pi\)
\(648\) 0 0
\(649\) 25920.0 1.56772
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −13230.0 −0.792848 −0.396424 − 0.918067i \(-0.629749\pi\)
−0.396424 + 0.918067i \(0.629749\pi\)
\(654\) 0 0
\(655\) 8352.00 0.498228
\(656\) 0 0
\(657\) 37296.0 2.21470
\(658\) 0 0
\(659\) −29704.0 −1.75585 −0.877924 − 0.478800i \(-0.841072\pi\)
−0.877924 + 0.478800i \(0.841072\pi\)
\(660\) 0 0
\(661\) 10548.0 0.620680 0.310340 − 0.950626i \(-0.399557\pi\)
0.310340 + 0.950626i \(0.399557\pi\)
\(662\) 0 0
\(663\) 11520.0 0.674811
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 28512.0 1.65516
\(668\) 0 0
\(669\) 25344.0 1.46466
\(670\) 0 0
\(671\) −27360.0 −1.57410
\(672\) 0 0
\(673\) −20178.0 −1.15573 −0.577864 − 0.816133i \(-0.696114\pi\)
−0.577864 + 0.816133i \(0.696114\pi\)
\(674\) 0 0
\(675\) 8720.00 0.497234
\(676\) 0 0
\(677\) 24908.0 1.41402 0.707010 − 0.707203i \(-0.250043\pi\)
0.707010 + 0.707203i \(0.250043\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −16704.0 −0.939939
\(682\) 0 0
\(683\) −8456.00 −0.473733 −0.236867 − 0.971542i \(-0.576120\pi\)
−0.236867 + 0.971542i \(0.576120\pi\)
\(684\) 0 0
\(685\) 10008.0 0.558228
\(686\) 0 0
\(687\) −29664.0 −1.64738
\(688\) 0 0
\(689\) −17496.0 −0.967409
\(690\) 0 0
\(691\) 24104.0 1.32700 0.663502 − 0.748175i \(-0.269070\pi\)
0.663502 + 0.748175i \(0.269070\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −7840.00 −0.427897
\(696\) 0 0
\(697\) −18880.0 −1.02601
\(698\) 0 0
\(699\) −16560.0 −0.896075
\(700\) 0 0
\(701\) 9954.00 0.536316 0.268158 − 0.963375i \(-0.413585\pi\)
0.268158 + 0.963375i \(0.413585\pi\)
\(702\) 0 0
\(703\) −3888.00 −0.208590
\(704\) 0 0
\(705\) 4608.00 0.246166
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −28694.0 −1.51992 −0.759962 − 0.649968i \(-0.774782\pi\)
−0.759962 + 0.649968i \(0.774782\pi\)
\(710\) 0 0
\(711\) −37296.0 −1.96724
\(712\) 0 0
\(713\) 2816.00 0.147910
\(714\) 0 0
\(715\) −5760.00 −0.301275
\(716\) 0 0
\(717\) −7168.00 −0.373353
\(718\) 0 0
\(719\) −28080.0 −1.45648 −0.728239 − 0.685324i \(-0.759661\pi\)
−0.728239 + 0.685324i \(0.759661\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 25920.0 1.33330
\(724\) 0 0
\(725\) −17658.0 −0.904554
\(726\) 0 0
\(727\) 17584.0 0.897049 0.448524 − 0.893771i \(-0.351950\pi\)
0.448524 + 0.893771i \(0.351950\pi\)
\(728\) 0 0
\(729\) −30563.0 −1.55276
\(730\) 0 0
\(731\) 2880.00 0.145719
\(732\) 0 0
\(733\) 28260.0 1.42402 0.712010 − 0.702169i \(-0.247785\pi\)
0.712010 + 0.702169i \(0.247785\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −8640.00 −0.431830
\(738\) 0 0
\(739\) −1224.00 −0.0609277 −0.0304638 − 0.999536i \(-0.509698\pi\)
−0.0304638 + 0.999536i \(0.509698\pi\)
\(740\) 0 0
\(741\) 20736.0 1.02801
\(742\) 0 0
\(743\) −8240.00 −0.406859 −0.203430 − 0.979090i \(-0.565209\pi\)
−0.203430 + 0.979090i \(0.565209\pi\)
\(744\) 0 0
\(745\) 13464.0 0.662125
\(746\) 0 0
\(747\) −7992.00 −0.391448
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 16704.0 0.811635 0.405817 − 0.913954i \(-0.366987\pi\)
0.405817 + 0.913954i \(0.366987\pi\)
\(752\) 0 0
\(753\) 30528.0 1.47743
\(754\) 0 0
\(755\) 10944.0 0.527540
\(756\) 0 0
\(757\) 30778.0 1.47774 0.738868 − 0.673851i \(-0.235361\pi\)
0.738868 + 0.673851i \(0.235361\pi\)
\(758\) 0 0
\(759\) −56320.0 −2.69339
\(760\) 0 0
\(761\) 29624.0 1.41113 0.705564 − 0.708646i \(-0.250694\pi\)
0.705564 + 0.708646i \(0.250694\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 5920.00 0.279788
\(766\) 0 0
\(767\) −23328.0 −1.09821
\(768\) 0 0
\(769\) 11016.0 0.516576 0.258288 − 0.966068i \(-0.416842\pi\)
0.258288 + 0.966068i \(0.416842\pi\)
\(770\) 0 0
\(771\) −32512.0 −1.51867
\(772\) 0 0
\(773\) −940.000 −0.0437380 −0.0218690 − 0.999761i \(-0.506962\pi\)
−0.0218690 + 0.999761i \(0.506962\pi\)
\(774\) 0 0
\(775\) −1744.00 −0.0808340
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −33984.0 −1.56303
\(780\) 0 0
\(781\) 24320.0 1.11426
\(782\) 0 0
\(783\) −12960.0 −0.591510
\(784\) 0 0
\(785\) 2736.00 0.124397
\(786\) 0 0
\(787\) 1960.00 0.0887757 0.0443878 − 0.999014i \(-0.485866\pi\)
0.0443878 + 0.999014i \(0.485866\pi\)
\(788\) 0 0
\(789\) −2048.00 −0.0924090
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 24624.0 1.10268
\(794\) 0 0
\(795\) −15552.0 −0.693802
\(796\) 0 0
\(797\) 37868.0 1.68300 0.841501 − 0.540255i \(-0.181672\pi\)
0.841501 + 0.540255i \(0.181672\pi\)
\(798\) 0 0
\(799\) −5760.00 −0.255036
\(800\) 0 0
\(801\) −38480.0 −1.69741
\(802\) 0 0
\(803\) 40320.0 1.77193
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −14944.0 −0.651863
\(808\) 0 0
\(809\) −28134.0 −1.22267 −0.611334 − 0.791373i \(-0.709367\pi\)
−0.611334 + 0.791373i \(0.709367\pi\)
\(810\) 0 0
\(811\) −25112.0 −1.08730 −0.543651 − 0.839312i \(-0.682958\pi\)
−0.543651 + 0.839312i \(0.682958\pi\)
\(812\) 0 0
\(813\) 19456.0 0.839301
\(814\) 0 0
\(815\) 6624.00 0.284698
\(816\) 0 0
\(817\) 5184.00 0.221989
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 43686.0 1.85707 0.928533 − 0.371249i \(-0.121070\pi\)
0.928533 + 0.371249i \(0.121070\pi\)
\(822\) 0 0
\(823\) −43776.0 −1.85411 −0.927057 − 0.374921i \(-0.877670\pi\)
−0.927057 + 0.374921i \(0.877670\pi\)
\(824\) 0 0
\(825\) 34880.0 1.47196
\(826\) 0 0
\(827\) 1336.00 0.0561757 0.0280878 − 0.999605i \(-0.491058\pi\)
0.0280878 + 0.999605i \(0.491058\pi\)
\(828\) 0 0
\(829\) −15372.0 −0.644019 −0.322009 − 0.946736i \(-0.604358\pi\)
−0.322009 + 0.946736i \(0.604358\pi\)
\(830\) 0 0
\(831\) −15408.0 −0.643198
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 14400.0 0.596805
\(836\) 0 0
\(837\) −1280.00 −0.0528593
\(838\) 0 0
\(839\) 40752.0 1.67690 0.838448 − 0.544982i \(-0.183464\pi\)
0.838448 + 0.544982i \(0.183464\pi\)
\(840\) 0 0
\(841\) 1855.00 0.0760589
\(842\) 0 0
\(843\) 53712.0 2.19447
\(844\) 0 0
\(845\) −3604.00 −0.146724
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −576.000 −0.0232842
\(850\) 0 0
\(851\) −9504.00 −0.382835
\(852\) 0 0
\(853\) −12132.0 −0.486977 −0.243489 − 0.969904i \(-0.578292\pi\)
−0.243489 + 0.969904i \(0.578292\pi\)
\(854\) 0 0
\(855\) 10656.0 0.426231
\(856\) 0 0
\(857\) 36392.0 1.45056 0.725278 − 0.688456i \(-0.241711\pi\)
0.725278 + 0.688456i \(0.241711\pi\)
\(858\) 0 0
\(859\) −10440.0 −0.414678 −0.207339 − 0.978269i \(-0.566480\pi\)
−0.207339 + 0.978269i \(0.566480\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −13360.0 −0.526975 −0.263488 − 0.964663i \(-0.584873\pi\)
−0.263488 + 0.964663i \(0.584873\pi\)
\(864\) 0 0
\(865\) 14704.0 0.577978
\(866\) 0 0
\(867\) 26504.0 1.03820
\(868\) 0 0
\(869\) −40320.0 −1.57395
\(870\) 0 0
\(871\) 7776.00 0.302503
\(872\) 0 0
\(873\) −34632.0 −1.34263
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −11086.0 −0.426850 −0.213425 − 0.976959i \(-0.568462\pi\)
−0.213425 + 0.976959i \(0.568462\pi\)
\(878\) 0 0
\(879\) −18464.0 −0.708504
\(880\) 0 0
\(881\) −36832.0 −1.40851 −0.704257 − 0.709945i \(-0.748720\pi\)
−0.704257 + 0.709945i \(0.748720\pi\)
\(882\) 0 0
\(883\) 49608.0 1.89065 0.945324 − 0.326133i \(-0.105746\pi\)
0.945324 + 0.326133i \(0.105746\pi\)
\(884\) 0 0
\(885\) −20736.0 −0.787608
\(886\) 0 0
\(887\) −15696.0 −0.594160 −0.297080 − 0.954853i \(-0.596013\pi\)
−0.297080 + 0.954853i \(0.596013\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −14360.0 −0.539931
\(892\) 0 0
\(893\) −10368.0 −0.388524
\(894\) 0 0
\(895\) 11936.0 0.445784
\(896\) 0 0
\(897\) 50688.0 1.88676
\(898\) 0 0
\(899\) 2592.00 0.0961602
\(900\) 0 0
\(901\) 19440.0 0.718802
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 11952.0 0.439003
\(906\) 0 0
\(907\) −2664.00 −0.0975266 −0.0487633 − 0.998810i \(-0.515528\pi\)
−0.0487633 + 0.998810i \(0.515528\pi\)
\(908\) 0 0
\(909\) 5476.00 0.199810
\(910\) 0 0
\(911\) −4864.00 −0.176895 −0.0884476 − 0.996081i \(-0.528191\pi\)
−0.0884476 + 0.996081i \(0.528191\pi\)
\(912\) 0 0
\(913\) −8640.00 −0.313190
\(914\) 0 0
\(915\) 21888.0 0.790814
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 12384.0 0.444516 0.222258 − 0.974988i \(-0.428657\pi\)
0.222258 + 0.974988i \(0.428657\pi\)
\(920\) 0 0
\(921\) −30528.0 −1.09222
\(922\) 0 0
\(923\) −21888.0 −0.780555
\(924\) 0 0
\(925\) 5886.00 0.209222
\(926\) 0 0
\(927\) −64528.0 −2.28628
\(928\) 0 0
\(929\) 33224.0 1.17335 0.586676 − 0.809822i \(-0.300436\pi\)
0.586676 + 0.809822i \(0.300436\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −32256.0 −1.13185
\(934\) 0 0
\(935\) 6400.00 0.223853
\(936\) 0 0
\(937\) −11088.0 −0.386584 −0.193292 − 0.981141i \(-0.561916\pi\)
−0.193292 + 0.981141i \(0.561916\pi\)
\(938\) 0 0
\(939\) −69696.0 −2.42220
\(940\) 0 0
\(941\) 22324.0 0.773370 0.386685 − 0.922212i \(-0.373620\pi\)
0.386685 + 0.922212i \(0.373620\pi\)
\(942\) 0 0
\(943\) −83072.0 −2.86871
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 23800.0 0.816680 0.408340 − 0.912830i \(-0.366108\pi\)
0.408340 + 0.912830i \(0.366108\pi\)
\(948\) 0 0
\(949\) −36288.0 −1.24126
\(950\) 0 0
\(951\) −14832.0 −0.505742
\(952\) 0 0
\(953\) −2358.00 −0.0801502 −0.0400751 − 0.999197i \(-0.512760\pi\)
−0.0400751 + 0.999197i \(0.512760\pi\)
\(954\) 0 0
\(955\) −5312.00 −0.179992
\(956\) 0 0
\(957\) −51840.0 −1.75104
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −29535.0 −0.991407
\(962\) 0 0
\(963\) −12136.0 −0.406103
\(964\) 0 0
\(965\) 1224.00 0.0408310
\(966\) 0 0
\(967\) −8496.00 −0.282537 −0.141268 − 0.989971i \(-0.545118\pi\)
−0.141268 + 0.989971i \(0.545118\pi\)
\(968\) 0 0
\(969\) −23040.0 −0.763830
\(970\) 0 0
\(971\) 16200.0 0.535410 0.267705 − 0.963501i \(-0.413735\pi\)
0.267705 + 0.963501i \(0.413735\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −31392.0 −1.03113
\(976\) 0 0
\(977\) 21726.0 0.711439 0.355720 − 0.934593i \(-0.384236\pi\)
0.355720 + 0.934593i \(0.384236\pi\)
\(978\) 0 0
\(979\) −41600.0 −1.35806
\(980\) 0 0
\(981\) −47878.0 −1.55823
\(982\) 0 0
\(983\) −8208.00 −0.266322 −0.133161 − 0.991094i \(-0.542513\pi\)
−0.133161 + 0.991094i \(0.542513\pi\)
\(984\) 0 0
\(985\) −6696.00 −0.216601
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 12672.0 0.407428
\(990\) 0 0
\(991\) 18000.0 0.576982 0.288491 − 0.957483i \(-0.406847\pi\)
0.288491 + 0.957483i \(0.406847\pi\)
\(992\) 0 0
\(993\) −53568.0 −1.71191
\(994\) 0 0
\(995\) −10944.0 −0.348692
\(996\) 0 0
\(997\) 57204.0 1.81712 0.908560 − 0.417754i \(-0.137183\pi\)
0.908560 + 0.417754i \(0.137183\pi\)
\(998\) 0 0
\(999\) 4320.00 0.136816
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 1568.4.a.b.1.1 yes 1
4.3 odd 2 1568.4.a.n.1.1 yes 1
7.6 odd 2 1568.4.a.m.1.1 yes 1
28.27 even 2 1568.4.a.a.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1568.4.a.a.1.1 ✓ 1 28.27 even 2
1568.4.a.b.1.1 yes 1 1.1 even 1 trivial
1568.4.a.m.1.1 yes 1 7.6 odd 2
1568.4.a.n.1.1 yes 1 4.3 odd 2