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Properties

Label 1008.4.a.e.1.1
Level $1008$
Weight $4$
Character 1008.1
Self dual yes
Analytic conductor $59.474$
Analytic rank $0$
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] = [1008,4,Mod(1,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1008.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1008.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: \(59.4739252858\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 56)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1008.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.00000 q^{5} +7.00000 q^{7} +O(q^{10})\) \(q-8.00000 q^{5} +7.00000 q^{7} +56.0000 q^{11} -28.0000 q^{13} +90.0000 q^{17} -74.0000 q^{19} -96.0000 q^{23} -61.0000 q^{25} +222.000 q^{29} +100.000 q^{31} -56.0000 q^{35} +58.0000 q^{37} -422.000 q^{41} -512.000 q^{43} +148.000 q^{47} +49.0000 q^{49} +642.000 q^{53} -448.000 q^{55} -318.000 q^{59} +720.000 q^{61} +224.000 q^{65} +412.000 q^{67} +448.000 q^{71} +994.000 q^{73} +392.000 q^{77} +296.000 q^{79} +386.000 q^{83} -720.000 q^{85} +6.00000 q^{89} -196.000 q^{91} +592.000 q^{95} -138.000 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −8.00000 −0.715542 −0.357771 − 0.933809i \(-0.616463\pi\)
−0.357771 + 0.933809i \(0.616463\pi\)
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 56.0000 1.53497 0.767483 − 0.641069i \(-0.221509\pi\)
0.767483 + 0.641069i \(0.221509\pi\)
\(12\) 0 0
\(13\) −28.0000 −0.597369 −0.298685 − 0.954352i \(-0.596548\pi\)
−0.298685 + 0.954352i \(0.596548\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 90.0000 1.28401 0.642006 − 0.766700i \(-0.278102\pi\)
0.642006 + 0.766700i \(0.278102\pi\)
\(18\) 0 0
\(19\) −74.0000 −0.893514 −0.446757 − 0.894655i \(-0.647421\pi\)
−0.446757 + 0.894655i \(0.647421\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −96.0000 −0.870321 −0.435161 − 0.900353i \(-0.643308\pi\)
−0.435161 + 0.900353i \(0.643308\pi\)
\(24\) 0 0
\(25\) −61.0000 −0.488000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 222.000 1.42153 0.710765 − 0.703430i \(-0.248349\pi\)
0.710765 + 0.703430i \(0.248349\pi\)
\(30\) 0 0
\(31\) 100.000 0.579372 0.289686 − 0.957122i \(-0.406449\pi\)
0.289686 + 0.957122i \(0.406449\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −56.0000 −0.270449
\(36\) 0 0
\(37\) 58.0000 0.257707 0.128853 − 0.991664i \(-0.458870\pi\)
0.128853 + 0.991664i \(0.458870\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −422.000 −1.60745 −0.803724 − 0.595003i \(-0.797151\pi\)
−0.803724 + 0.595003i \(0.797151\pi\)
\(42\) 0 0
\(43\) −512.000 −1.81580 −0.907898 − 0.419190i \(-0.862314\pi\)
−0.907898 + 0.419190i \(0.862314\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 148.000 0.459320 0.229660 − 0.973271i \(-0.426239\pi\)
0.229660 + 0.973271i \(0.426239\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 642.000 1.66388 0.831939 − 0.554868i \(-0.187231\pi\)
0.831939 + 0.554868i \(0.187231\pi\)
\(54\) 0 0
\(55\) −448.000 −1.09833
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −318.000 −0.701696 −0.350848 − 0.936432i \(-0.614107\pi\)
−0.350848 + 0.936432i \(0.614107\pi\)
\(60\) 0 0
\(61\) 720.000 1.51125 0.755627 − 0.655002i \(-0.227332\pi\)
0.755627 + 0.655002i \(0.227332\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 224.000 0.427443
\(66\) 0 0
\(67\) 412.000 0.751251 0.375625 − 0.926772i \(-0.377428\pi\)
0.375625 + 0.926772i \(0.377428\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 448.000 0.748843 0.374421 − 0.927259i \(-0.377841\pi\)
0.374421 + 0.927259i \(0.377841\pi\)
\(72\) 0 0
\(73\) 994.000 1.59368 0.796842 − 0.604188i \(-0.206502\pi\)
0.796842 + 0.604188i \(0.206502\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 392.000 0.580163
\(78\) 0 0
\(79\) 296.000 0.421552 0.210776 − 0.977534i \(-0.432401\pi\)
0.210776 + 0.977534i \(0.432401\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 386.000 0.510470 0.255235 − 0.966879i \(-0.417847\pi\)
0.255235 + 0.966879i \(0.417847\pi\)
\(84\) 0 0
\(85\) −720.000 −0.918764
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.00714605 0.00357303 − 0.999994i \(-0.498863\pi\)
0.00357303 + 0.999994i \(0.498863\pi\)
\(90\) 0 0
\(91\) −196.000 −0.225784
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 592.000 0.639347
\(96\) 0 0
\(97\) −138.000 −0.144451 −0.0722257 − 0.997388i \(-0.523010\pi\)
−0.0722257 + 0.997388i \(0.523010\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 664.000 0.654163 0.327082 − 0.944996i \(-0.393935\pi\)
0.327082 + 0.944996i \(0.393935\pi\)
\(102\) 0 0
\(103\) −2012.00 −1.92474 −0.962370 − 0.271742i \(-0.912400\pi\)
−0.962370 + 0.271742i \(0.912400\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 900.000 0.813143 0.406571 − 0.913619i \(-0.366724\pi\)
0.406571 + 0.913619i \(0.366724\pi\)
\(108\) 0 0
\(109\) 1706.00 1.49913 0.749565 − 0.661931i \(-0.230263\pi\)
0.749565 + 0.661931i \(0.230263\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 442.000 0.367963 0.183982 − 0.982930i \(-0.441101\pi\)
0.183982 + 0.982930i \(0.441101\pi\)
\(114\) 0 0
\(115\) 768.000 0.622751
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 630.000 0.485311
\(120\) 0 0
\(121\) 1805.00 1.35612
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1488.00 1.06473
\(126\) 0 0
\(127\) 1952.00 1.36387 0.681937 − 0.731411i \(-0.261138\pi\)
0.681937 + 0.731411i \(0.261138\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −742.000 −0.494877 −0.247438 − 0.968904i \(-0.579589\pi\)
−0.247438 + 0.968904i \(0.579589\pi\)
\(132\) 0 0
\(133\) −518.000 −0.337717
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1854.00 1.15619 0.578095 − 0.815970i \(-0.303796\pi\)
0.578095 + 0.815970i \(0.303796\pi\)
\(138\) 0 0
\(139\) 74.0000 0.0451554 0.0225777 − 0.999745i \(-0.492813\pi\)
0.0225777 + 0.999745i \(0.492813\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −1568.00 −0.916942
\(144\) 0 0
\(145\) −1776.00 −1.01716
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2502.00 −1.37565 −0.687825 − 0.725877i \(-0.741434\pi\)
−0.687825 + 0.725877i \(0.741434\pi\)
\(150\) 0 0
\(151\) 888.000 0.478572 0.239286 − 0.970949i \(-0.423087\pi\)
0.239286 + 0.970949i \(0.423087\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −800.000 −0.414565
\(156\) 0 0
\(157\) 1092.00 0.555102 0.277551 − 0.960711i \(-0.410477\pi\)
0.277551 + 0.960711i \(0.410477\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −672.000 −0.328950
\(162\) 0 0
\(163\) 1704.00 0.818820 0.409410 − 0.912351i \(-0.365735\pi\)
0.409410 + 0.912351i \(0.365735\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 436.000 0.202028 0.101014 − 0.994885i \(-0.467791\pi\)
0.101014 + 0.994885i \(0.467791\pi\)
\(168\) 0 0
\(169\) −1413.00 −0.643150
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 996.000 0.437714 0.218857 − 0.975757i \(-0.429767\pi\)
0.218857 + 0.975757i \(0.429767\pi\)
\(174\) 0 0
\(175\) −427.000 −0.184447
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1244.00 0.519447 0.259723 − 0.965683i \(-0.416369\pi\)
0.259723 + 0.965683i \(0.416369\pi\)
\(180\) 0 0
\(181\) 412.000 0.169192 0.0845959 − 0.996415i \(-0.473040\pi\)
0.0845959 + 0.996415i \(0.473040\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −464.000 −0.184400
\(186\) 0 0
\(187\) 5040.00 1.97092
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1160.00 0.439448 0.219724 − 0.975562i \(-0.429484\pi\)
0.219724 + 0.975562i \(0.429484\pi\)
\(192\) 0 0
\(193\) −3698.00 −1.37921 −0.689606 − 0.724185i \(-0.742216\pi\)
−0.689606 + 0.724185i \(0.742216\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2874.00 1.03941 0.519706 − 0.854345i \(-0.326042\pi\)
0.519706 + 0.854345i \(0.326042\pi\)
\(198\) 0 0
\(199\) −3348.00 −1.19263 −0.596315 − 0.802750i \(-0.703369\pi\)
−0.596315 + 0.802750i \(0.703369\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1554.00 0.537288
\(204\) 0 0
\(205\) 3376.00 1.15020
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −4144.00 −1.37151
\(210\) 0 0
\(211\) 2556.00 0.833945 0.416972 − 0.908919i \(-0.363091\pi\)
0.416972 + 0.908919i \(0.363091\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 4096.00 1.29928
\(216\) 0 0
\(217\) 700.000 0.218982
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −2520.00 −0.767030
\(222\) 0 0
\(223\) 312.000 0.0936909 0.0468454 − 0.998902i \(-0.485083\pi\)
0.0468454 + 0.998902i \(0.485083\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 826.000 0.241513 0.120757 − 0.992682i \(-0.461468\pi\)
0.120757 + 0.992682i \(0.461468\pi\)
\(228\) 0 0
\(229\) 4036.00 1.16466 0.582328 − 0.812954i \(-0.302142\pi\)
0.582328 + 0.812954i \(0.302142\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2902.00 0.815950 0.407975 − 0.912993i \(-0.366235\pi\)
0.407975 + 0.912993i \(0.366235\pi\)
\(234\) 0 0
\(235\) −1184.00 −0.328662
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 3176.00 0.859575 0.429787 − 0.902930i \(-0.358589\pi\)
0.429787 + 0.902930i \(0.358589\pi\)
\(240\) 0 0
\(241\) −4682.00 −1.25143 −0.625714 − 0.780053i \(-0.715192\pi\)
−0.625714 + 0.780053i \(0.715192\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −392.000 −0.102220
\(246\) 0 0
\(247\) 2072.00 0.533758
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 630.000 0.158427 0.0792136 − 0.996858i \(-0.474759\pi\)
0.0792136 + 0.996858i \(0.474759\pi\)
\(252\) 0 0
\(253\) −5376.00 −1.33591
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1870.00 0.453881 0.226940 − 0.973909i \(-0.427128\pi\)
0.226940 + 0.973909i \(0.427128\pi\)
\(258\) 0 0
\(259\) 406.000 0.0974039
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 3360.00 0.787781 0.393891 − 0.919157i \(-0.371129\pi\)
0.393891 + 0.919157i \(0.371129\pi\)
\(264\) 0 0
\(265\) −5136.00 −1.19057
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 5924.00 1.34272 0.671362 − 0.741130i \(-0.265710\pi\)
0.671362 + 0.741130i \(0.265710\pi\)
\(270\) 0 0
\(271\) −2064.00 −0.462653 −0.231327 − 0.972876i \(-0.574307\pi\)
−0.231327 + 0.972876i \(0.574307\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −3416.00 −0.749064
\(276\) 0 0
\(277\) −3314.00 −0.718841 −0.359421 − 0.933176i \(-0.617026\pi\)
−0.359421 + 0.933176i \(0.617026\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −3018.00 −0.640707 −0.320354 − 0.947298i \(-0.603802\pi\)
−0.320354 + 0.947298i \(0.603802\pi\)
\(282\) 0 0
\(283\) −6802.00 −1.42875 −0.714376 − 0.699762i \(-0.753289\pi\)
−0.714376 + 0.699762i \(0.753289\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2954.00 −0.607558
\(288\) 0 0
\(289\) 3187.00 0.648687
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −2784.00 −0.555096 −0.277548 − 0.960712i \(-0.589522\pi\)
−0.277548 + 0.960712i \(0.589522\pi\)
\(294\) 0 0
\(295\) 2544.00 0.502093
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 2688.00 0.519903
\(300\) 0 0
\(301\) −3584.00 −0.686307
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −5760.00 −1.08137
\(306\) 0 0
\(307\) 5654.00 1.05111 0.525555 − 0.850760i \(-0.323858\pi\)
0.525555 + 0.850760i \(0.323858\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 2504.00 0.456556 0.228278 − 0.973596i \(-0.426691\pi\)
0.228278 + 0.973596i \(0.426691\pi\)
\(312\) 0 0
\(313\) 6454.00 1.16550 0.582750 − 0.812651i \(-0.301977\pi\)
0.582750 + 0.812651i \(0.301977\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 4146.00 0.734582 0.367291 − 0.930106i \(-0.380285\pi\)
0.367291 + 0.930106i \(0.380285\pi\)
\(318\) 0 0
\(319\) 12432.0 2.18200
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −6660.00 −1.14728
\(324\) 0 0
\(325\) 1708.00 0.291516
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 1036.00 0.173606
\(330\) 0 0
\(331\) −10208.0 −1.69511 −0.847557 − 0.530705i \(-0.821927\pi\)
−0.847557 + 0.530705i \(0.821927\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −3296.00 −0.537551
\(336\) 0 0
\(337\) 8078.00 1.30575 0.652873 − 0.757467i \(-0.273563\pi\)
0.652873 + 0.757467i \(0.273563\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 5600.00 0.889317
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −3456.00 −0.534662 −0.267331 − 0.963605i \(-0.586142\pi\)
−0.267331 + 0.963605i \(0.586142\pi\)
\(348\) 0 0
\(349\) 6348.00 0.973641 0.486820 − 0.873502i \(-0.338157\pi\)
0.486820 + 0.873502i \(0.338157\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −2290.00 −0.345282 −0.172641 − 0.984985i \(-0.555230\pi\)
−0.172641 + 0.984985i \(0.555230\pi\)
\(354\) 0 0
\(355\) −3584.00 −0.535828
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −9912.00 −1.45720 −0.728601 − 0.684939i \(-0.759829\pi\)
−0.728601 + 0.684939i \(0.759829\pi\)
\(360\) 0 0
\(361\) −1383.00 −0.201633
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −7952.00 −1.14035
\(366\) 0 0
\(367\) −4840.00 −0.688409 −0.344204 − 0.938895i \(-0.611851\pi\)
−0.344204 + 0.938895i \(0.611851\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 4494.00 0.628886
\(372\) 0 0
\(373\) −8090.00 −1.12301 −0.561507 − 0.827472i \(-0.689778\pi\)
−0.561507 + 0.827472i \(0.689778\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −6216.00 −0.849178
\(378\) 0 0
\(379\) 7000.00 0.948723 0.474361 − 0.880330i \(-0.342679\pi\)
0.474361 + 0.880330i \(0.342679\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 10308.0 1.37523 0.687616 − 0.726074i \(-0.258657\pi\)
0.687616 + 0.726074i \(0.258657\pi\)
\(384\) 0 0
\(385\) −3136.00 −0.415131
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 11510.0 1.50021 0.750103 − 0.661321i \(-0.230004\pi\)
0.750103 + 0.661321i \(0.230004\pi\)
\(390\) 0 0
\(391\) −8640.00 −1.11750
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −2368.00 −0.301638
\(396\) 0 0
\(397\) −8132.00 −1.02804 −0.514022 − 0.857777i \(-0.671845\pi\)
−0.514022 + 0.857777i \(0.671845\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1074.00 0.133748 0.0668741 − 0.997761i \(-0.478697\pi\)
0.0668741 + 0.997761i \(0.478697\pi\)
\(402\) 0 0
\(403\) −2800.00 −0.346099
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 3248.00 0.395571
\(408\) 0 0
\(409\) 3406.00 0.411775 0.205887 − 0.978576i \(-0.433992\pi\)
0.205887 + 0.978576i \(0.433992\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −2226.00 −0.265216
\(414\) 0 0
\(415\) −3088.00 −0.365263
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −13522.0 −1.57659 −0.788297 − 0.615295i \(-0.789037\pi\)
−0.788297 + 0.615295i \(0.789037\pi\)
\(420\) 0 0
\(421\) 1198.00 0.138686 0.0693432 − 0.997593i \(-0.477910\pi\)
0.0693432 + 0.997593i \(0.477910\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −5490.00 −0.626598
\(426\) 0 0
\(427\) 5040.00 0.571201
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −12824.0 −1.43320 −0.716601 − 0.697483i \(-0.754303\pi\)
−0.716601 + 0.697483i \(0.754303\pi\)
\(432\) 0 0
\(433\) −12770.0 −1.41729 −0.708646 − 0.705565i \(-0.750693\pi\)
−0.708646 + 0.705565i \(0.750693\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 7104.00 0.777644
\(438\) 0 0
\(439\) −13224.0 −1.43769 −0.718846 − 0.695169i \(-0.755330\pi\)
−0.718846 + 0.695169i \(0.755330\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 972.000 0.104246 0.0521232 − 0.998641i \(-0.483401\pi\)
0.0521232 + 0.998641i \(0.483401\pi\)
\(444\) 0 0
\(445\) −48.0000 −0.00511330
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −6034.00 −0.634214 −0.317107 − 0.948390i \(-0.602711\pi\)
−0.317107 + 0.948390i \(0.602711\pi\)
\(450\) 0 0
\(451\) −23632.0 −2.46738
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1568.00 0.161558
\(456\) 0 0
\(457\) 12742.0 1.30426 0.652129 − 0.758108i \(-0.273876\pi\)
0.652129 + 0.758108i \(0.273876\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −6132.00 −0.619513 −0.309757 − 0.950816i \(-0.600248\pi\)
−0.309757 + 0.950816i \(0.600248\pi\)
\(462\) 0 0
\(463\) −2672.00 −0.268204 −0.134102 − 0.990968i \(-0.542815\pi\)
−0.134102 + 0.990968i \(0.542815\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −17246.0 −1.70889 −0.854443 − 0.519545i \(-0.826101\pi\)
−0.854443 + 0.519545i \(0.826101\pi\)
\(468\) 0 0
\(469\) 2884.00 0.283946
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −28672.0 −2.78719
\(474\) 0 0
\(475\) 4514.00 0.436035
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −19812.0 −1.88984 −0.944920 − 0.327301i \(-0.893861\pi\)
−0.944920 + 0.327301i \(0.893861\pi\)
\(480\) 0 0
\(481\) −1624.00 −0.153946
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1104.00 0.103361
\(486\) 0 0
\(487\) −14864.0 −1.38306 −0.691532 − 0.722346i \(-0.743064\pi\)
−0.691532 + 0.722346i \(0.743064\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 15756.0 1.44818 0.724092 − 0.689703i \(-0.242259\pi\)
0.724092 + 0.689703i \(0.242259\pi\)
\(492\) 0 0
\(493\) 19980.0 1.82526
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 3136.00 0.283036
\(498\) 0 0
\(499\) −1604.00 −0.143898 −0.0719488 − 0.997408i \(-0.522922\pi\)
−0.0719488 + 0.997408i \(0.522922\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −19232.0 −1.70480 −0.852398 − 0.522893i \(-0.824853\pi\)
−0.852398 + 0.522893i \(0.824853\pi\)
\(504\) 0 0
\(505\) −5312.00 −0.468081
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −76.0000 −0.00661815 −0.00330908 − 0.999995i \(-0.501053\pi\)
−0.00330908 + 0.999995i \(0.501053\pi\)
\(510\) 0 0
\(511\) 6958.00 0.602356
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 16096.0 1.37723
\(516\) 0 0
\(517\) 8288.00 0.705040
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 12570.0 1.05701 0.528505 − 0.848930i \(-0.322753\pi\)
0.528505 + 0.848930i \(0.322753\pi\)
\(522\) 0 0
\(523\) 11510.0 0.962327 0.481164 − 0.876631i \(-0.340214\pi\)
0.481164 + 0.876631i \(0.340214\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 9000.00 0.743921
\(528\) 0 0
\(529\) −2951.00 −0.242541
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 11816.0 0.960240
\(534\) 0 0
\(535\) −7200.00 −0.581838
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 2744.00 0.219281
\(540\) 0 0
\(541\) 23438.0 1.86262 0.931311 − 0.364225i \(-0.118666\pi\)
0.931311 + 0.364225i \(0.118666\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −13648.0 −1.07269
\(546\) 0 0
\(547\) 5328.00 0.416470 0.208235 − 0.978079i \(-0.433228\pi\)
0.208235 + 0.978079i \(0.433228\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −16428.0 −1.27016
\(552\) 0 0
\(553\) 2072.00 0.159332
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 2386.00 0.181505 0.0907523 − 0.995873i \(-0.471073\pi\)
0.0907523 + 0.995873i \(0.471073\pi\)
\(558\) 0 0
\(559\) 14336.0 1.08470
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 13094.0 0.980189 0.490094 − 0.871669i \(-0.336962\pi\)
0.490094 + 0.871669i \(0.336962\pi\)
\(564\) 0 0
\(565\) −3536.00 −0.263293
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −15638.0 −1.15216 −0.576080 − 0.817393i \(-0.695418\pi\)
−0.576080 + 0.817393i \(0.695418\pi\)
\(570\) 0 0
\(571\) 4040.00 0.296092 0.148046 − 0.988980i \(-0.452702\pi\)
0.148046 + 0.988980i \(0.452702\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 5856.00 0.424717
\(576\) 0 0
\(577\) 9154.00 0.660461 0.330231 − 0.943900i \(-0.392874\pi\)
0.330231 + 0.943900i \(0.392874\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 2702.00 0.192939
\(582\) 0 0
\(583\) 35952.0 2.55400
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −2754.00 −0.193645 −0.0968226 − 0.995302i \(-0.530868\pi\)
−0.0968226 + 0.995302i \(0.530868\pi\)
\(588\) 0 0
\(589\) −7400.00 −0.517677
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 20678.0 1.43195 0.715973 − 0.698128i \(-0.245983\pi\)
0.715973 + 0.698128i \(0.245983\pi\)
\(594\) 0 0
\(595\) −5040.00 −0.347260
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −10200.0 −0.695761 −0.347880 − 0.937539i \(-0.613098\pi\)
−0.347880 + 0.937539i \(0.613098\pi\)
\(600\) 0 0
\(601\) −23782.0 −1.61412 −0.807062 − 0.590467i \(-0.798943\pi\)
−0.807062 + 0.590467i \(0.798943\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −14440.0 −0.970363
\(606\) 0 0
\(607\) −9920.00 −0.663328 −0.331664 − 0.943397i \(-0.607610\pi\)
−0.331664 + 0.943397i \(0.607610\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −4144.00 −0.274383
\(612\) 0 0
\(613\) −21030.0 −1.38563 −0.692817 − 0.721113i \(-0.743631\pi\)
−0.692817 + 0.721113i \(0.743631\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −8854.00 −0.577713 −0.288856 − 0.957372i \(-0.593275\pi\)
−0.288856 + 0.957372i \(0.593275\pi\)
\(618\) 0 0
\(619\) 1214.00 0.0788284 0.0394142 − 0.999223i \(-0.487451\pi\)
0.0394142 + 0.999223i \(0.487451\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 42.0000 0.00270095
\(624\) 0 0
\(625\) −4279.00 −0.273856
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 5220.00 0.330898
\(630\) 0 0
\(631\) 1200.00 0.0757072 0.0378536 − 0.999283i \(-0.487948\pi\)
0.0378536 + 0.999283i \(0.487948\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −15616.0 −0.975909
\(636\) 0 0
\(637\) −1372.00 −0.0853385
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −6894.00 −0.424800 −0.212400 − 0.977183i \(-0.568128\pi\)
−0.212400 + 0.977183i \(0.568128\pi\)
\(642\) 0 0
\(643\) −13174.0 −0.807981 −0.403991 − 0.914763i \(-0.632377\pi\)
−0.403991 + 0.914763i \(0.632377\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1684.00 0.102326 0.0511630 − 0.998690i \(-0.483707\pi\)
0.0511630 + 0.998690i \(0.483707\pi\)
\(648\) 0 0
\(649\) −17808.0 −1.07708
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 20582.0 1.23344 0.616720 − 0.787183i \(-0.288461\pi\)
0.616720 + 0.787183i \(0.288461\pi\)
\(654\) 0 0
\(655\) 5936.00 0.354105
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 22416.0 1.32504 0.662522 − 0.749043i \(-0.269486\pi\)
0.662522 + 0.749043i \(0.269486\pi\)
\(660\) 0 0
\(661\) 20864.0 1.22771 0.613854 − 0.789419i \(-0.289618\pi\)
0.613854 + 0.789419i \(0.289618\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 4144.00 0.241650
\(666\) 0 0
\(667\) −21312.0 −1.23719
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 40320.0 2.31973
\(672\) 0 0
\(673\) −14438.0 −0.826960 −0.413480 − 0.910513i \(-0.635687\pi\)
−0.413480 + 0.910513i \(0.635687\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 2724.00 0.154641 0.0773204 − 0.997006i \(-0.475364\pi\)
0.0773204 + 0.997006i \(0.475364\pi\)
\(678\) 0 0
\(679\) −966.000 −0.0545975
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 5676.00 0.317988 0.158994 − 0.987280i \(-0.449175\pi\)
0.158994 + 0.987280i \(0.449175\pi\)
\(684\) 0 0
\(685\) −14832.0 −0.827302
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −17976.0 −0.993949
\(690\) 0 0
\(691\) 2534.00 0.139505 0.0697525 − 0.997564i \(-0.477779\pi\)
0.0697525 + 0.997564i \(0.477779\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −592.000 −0.0323106
\(696\) 0 0
\(697\) −37980.0 −2.06398
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 15558.0 0.838256 0.419128 − 0.907927i \(-0.362336\pi\)
0.419128 + 0.907927i \(0.362336\pi\)
\(702\) 0 0
\(703\) −4292.00 −0.230264
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 4648.00 0.247250
\(708\) 0 0
\(709\) 14538.0 0.770079 0.385040 − 0.922900i \(-0.374188\pi\)
0.385040 + 0.922900i \(0.374188\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −9600.00 −0.504240
\(714\) 0 0
\(715\) 12544.0 0.656110
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 16364.0 0.848782 0.424391 − 0.905479i \(-0.360488\pi\)
0.424391 + 0.905479i \(0.360488\pi\)
\(720\) 0 0
\(721\) −14084.0 −0.727483
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −13542.0 −0.693707
\(726\) 0 0
\(727\) 24884.0 1.26946 0.634729 − 0.772735i \(-0.281112\pi\)
0.634729 + 0.772735i \(0.281112\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −46080.0 −2.33151
\(732\) 0 0
\(733\) 9248.00 0.466006 0.233003 − 0.972476i \(-0.425145\pi\)
0.233003 + 0.972476i \(0.425145\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 23072.0 1.15315
\(738\) 0 0
\(739\) −3928.00 −0.195526 −0.0977631 − 0.995210i \(-0.531169\pi\)
−0.0977631 + 0.995210i \(0.531169\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 2840.00 0.140228 0.0701141 − 0.997539i \(-0.477664\pi\)
0.0701141 + 0.997539i \(0.477664\pi\)
\(744\) 0 0
\(745\) 20016.0 0.984335
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 6300.00 0.307339
\(750\) 0 0
\(751\) −904.000 −0.0439247 −0.0219623 − 0.999759i \(-0.506991\pi\)
−0.0219623 + 0.999759i \(0.506991\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −7104.00 −0.342438
\(756\) 0 0
\(757\) −326.000 −0.0156521 −0.00782607 − 0.999969i \(-0.502491\pi\)
−0.00782607 + 0.999969i \(0.502491\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −13902.0 −0.662217 −0.331108 − 0.943593i \(-0.607423\pi\)
−0.331108 + 0.943593i \(0.607423\pi\)
\(762\) 0 0
\(763\) 11942.0 0.566618
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 8904.00 0.419172
\(768\) 0 0
\(769\) −19378.0 −0.908698 −0.454349 − 0.890824i \(-0.650128\pi\)
−0.454349 + 0.890824i \(0.650128\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 33280.0 1.54851 0.774255 − 0.632874i \(-0.218125\pi\)
0.774255 + 0.632874i \(0.218125\pi\)
\(774\) 0 0
\(775\) −6100.00 −0.282734
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 31228.0 1.43628
\(780\) 0 0
\(781\) 25088.0 1.14945
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −8736.00 −0.397199
\(786\) 0 0
\(787\) 32974.0 1.49351 0.746757 − 0.665097i \(-0.231610\pi\)
0.746757 + 0.665097i \(0.231610\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 3094.00 0.139077
\(792\) 0 0
\(793\) −20160.0 −0.902778
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −17244.0 −0.766391 −0.383196 − 0.923667i \(-0.625176\pi\)
−0.383196 + 0.923667i \(0.625176\pi\)
\(798\) 0 0
\(799\) 13320.0 0.589772
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 55664.0 2.44625
\(804\) 0 0
\(805\) 5376.00 0.235378
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 27178.0 1.18112 0.590561 − 0.806993i \(-0.298907\pi\)
0.590561 + 0.806993i \(0.298907\pi\)
\(810\) 0 0
\(811\) −33514.0 −1.45109 −0.725546 − 0.688174i \(-0.758413\pi\)
−0.725546 + 0.688174i \(0.758413\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −13632.0 −0.585900
\(816\) 0 0
\(817\) 37888.0 1.62244
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 10978.0 0.466669 0.233334 − 0.972397i \(-0.425036\pi\)
0.233334 + 0.972397i \(0.425036\pi\)
\(822\) 0 0
\(823\) 2376.00 0.100634 0.0503172 − 0.998733i \(-0.483977\pi\)
0.0503172 + 0.998733i \(0.483977\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −12844.0 −0.540060 −0.270030 − 0.962852i \(-0.587034\pi\)
−0.270030 + 0.962852i \(0.587034\pi\)
\(828\) 0 0
\(829\) −30880.0 −1.29374 −0.646868 − 0.762602i \(-0.723921\pi\)
−0.646868 + 0.762602i \(0.723921\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 4410.00 0.183430
\(834\) 0 0
\(835\) −3488.00 −0.144560
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −27924.0 −1.14904 −0.574519 − 0.818491i \(-0.694811\pi\)
−0.574519 + 0.818491i \(0.694811\pi\)
\(840\) 0 0
\(841\) 24895.0 1.02075
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 11304.0 0.460200
\(846\) 0 0
\(847\) 12635.0 0.512566
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −5568.00 −0.224287
\(852\) 0 0
\(853\) −10636.0 −0.426928 −0.213464 − 0.976951i \(-0.568475\pi\)
−0.213464 + 0.976951i \(0.568475\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −41934.0 −1.67146 −0.835728 − 0.549143i \(-0.814954\pi\)
−0.835728 + 0.549143i \(0.814954\pi\)
\(858\) 0 0
\(859\) 15946.0 0.633377 0.316688 − 0.948530i \(-0.397429\pi\)
0.316688 + 0.948530i \(0.397429\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 16984.0 0.669921 0.334961 − 0.942232i \(-0.391277\pi\)
0.334961 + 0.942232i \(0.391277\pi\)
\(864\) 0 0
\(865\) −7968.00 −0.313202
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 16576.0 0.647068
\(870\) 0 0
\(871\) −11536.0 −0.448774
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 10416.0 0.402429
\(876\) 0 0
\(877\) −6702.00 −0.258051 −0.129025 − 0.991641i \(-0.541185\pi\)
−0.129025 + 0.991641i \(0.541185\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −36810.0 −1.40767 −0.703837 − 0.710362i \(-0.748531\pi\)
−0.703837 + 0.710362i \(0.748531\pi\)
\(882\) 0 0
\(883\) 5060.00 0.192845 0.0964227 − 0.995340i \(-0.469260\pi\)
0.0964227 + 0.995340i \(0.469260\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 32156.0 1.21724 0.608621 − 0.793461i \(-0.291723\pi\)
0.608621 + 0.793461i \(0.291723\pi\)
\(888\) 0 0
\(889\) 13664.0 0.515496
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −10952.0 −0.410408
\(894\) 0 0
\(895\) −9952.00 −0.371686
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 22200.0 0.823594
\(900\) 0 0
\(901\) 57780.0 2.13644
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −3296.00 −0.121064
\(906\) 0 0
\(907\) 2716.00 0.0994303 0.0497152 − 0.998763i \(-0.484169\pi\)
0.0497152 + 0.998763i \(0.484169\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 42048.0 1.52921 0.764606 − 0.644498i \(-0.222934\pi\)
0.764606 + 0.644498i \(0.222934\pi\)
\(912\) 0 0
\(913\) 21616.0 0.783554
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −5194.00 −0.187046
\(918\) 0 0
\(919\) 7816.00 0.280551 0.140275 − 0.990113i \(-0.455201\pi\)
0.140275 + 0.990113i \(0.455201\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −12544.0 −0.447336
\(924\) 0 0
\(925\) −3538.00 −0.125761
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 15578.0 0.550159 0.275079 − 0.961422i \(-0.411296\pi\)
0.275079 + 0.961422i \(0.411296\pi\)
\(930\) 0 0
\(931\) −3626.00 −0.127645
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −40320.0 −1.41027
\(936\) 0 0
\(937\) 46210.0 1.61112 0.805558 − 0.592518i \(-0.201866\pi\)
0.805558 + 0.592518i \(0.201866\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 24816.0 0.859701 0.429850 − 0.902900i \(-0.358566\pi\)
0.429850 + 0.902900i \(0.358566\pi\)
\(942\) 0 0
\(943\) 40512.0 1.39899
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −12800.0 −0.439223 −0.219611 − 0.975587i \(-0.570479\pi\)
−0.219611 + 0.975587i \(0.570479\pi\)
\(948\) 0 0
\(949\) −27832.0 −0.952018
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 16902.0 0.574512 0.287256 − 0.957854i \(-0.407257\pi\)
0.287256 + 0.957854i \(0.407257\pi\)
\(954\) 0 0
\(955\) −9280.00 −0.314444
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 12978.0 0.436998
\(960\) 0 0
\(961\) −19791.0 −0.664328
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 29584.0 0.986884
\(966\) 0 0
\(967\) 4000.00 0.133021 0.0665105 − 0.997786i \(-0.478813\pi\)
0.0665105 + 0.997786i \(0.478813\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −27486.0 −0.908412 −0.454206 − 0.890897i \(-0.650077\pi\)
−0.454206 + 0.890897i \(0.650077\pi\)
\(972\) 0 0
\(973\) 518.000 0.0170671
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1126.00 0.0368720 0.0184360 − 0.999830i \(-0.494131\pi\)
0.0184360 + 0.999830i \(0.494131\pi\)
\(978\) 0 0
\(979\) 336.000 0.0109690
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 4668.00 0.151461 0.0757305 − 0.997128i \(-0.475871\pi\)
0.0757305 + 0.997128i \(0.475871\pi\)
\(984\) 0 0
\(985\) −22992.0 −0.743742
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 49152.0 1.58033
\(990\) 0 0
\(991\) −2048.00 −0.0656477 −0.0328238 − 0.999461i \(-0.510450\pi\)
−0.0328238 + 0.999461i \(0.510450\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 26784.0 0.853377
\(996\) 0 0
\(997\) −26864.0 −0.853351 −0.426676 − 0.904405i \(-0.640315\pi\)
−0.426676 + 0.904405i \(0.640315\pi\)
\(998\) 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 1008.4.a.e.1.1 1
3.2 odd 2 112.4.a.b.1.1 1
4.3 odd 2 504.4.a.a.1.1 1
12.11 even 2 56.4.a.b.1.1 ✓ 1
21.20 even 2 784.4.a.q.1.1 1
24.5 odd 2 448.4.a.n.1.1 1
24.11 even 2 448.4.a.c.1.1 1
60.23 odd 4 1400.4.g.b.449.2 2
60.47 odd 4 1400.4.g.b.449.1 2
60.59 even 2 1400.4.a.b.1.1 1
84.11 even 6 392.4.i.a.177.1 2
84.23 even 6 392.4.i.a.361.1 2
84.47 odd 6 392.4.i.h.361.1 2
84.59 odd 6 392.4.i.h.177.1 2
84.83 odd 2 392.4.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.4.a.b.1.1 ✓ 1 12.11 even 2
112.4.a.b.1.1 1 3.2 odd 2
392.4.a.a.1.1 1 84.83 odd 2
392.4.i.a.177.1 2 84.11 even 6
392.4.i.a.361.1 2 84.23 even 6
392.4.i.h.177.1 2 84.59 odd 6
392.4.i.h.361.1 2 84.47 odd 6
448.4.a.c.1.1 1 24.11 even 2
448.4.a.n.1.1 1 24.5 odd 2
504.4.a.a.1.1 1 4.3 odd 2
784.4.a.q.1.1 1 21.20 even 2
1008.4.a.e.1.1 1 1.1 even 1 trivial
1400.4.a.b.1.1 1 60.59 even 2
1400.4.g.b.449.1 2 60.47 odd 4
1400.4.g.b.449.2 2 60.23 odd 4