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Properties

Label 72.4.a.d.1.1
Level $72$
Weight $4$
Character 72.1
Self dual yes
Analytic conductor $4.248$
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] = [72,4,Mod(1,72)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(72, 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("72.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 72.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: \(4.24813752041\)
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 \(\chi\) \(=\) 72.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+16.0000 q^{5} -12.0000 q^{7} +O(q^{10})\) \(q+16.0000 q^{5} -12.0000 q^{7} +64.0000 q^{11} +58.0000 q^{13} +32.0000 q^{17} -136.000 q^{19} -128.000 q^{23} +131.000 q^{25} -144.000 q^{29} +20.0000 q^{31} -192.000 q^{35} -18.0000 q^{37} -288.000 q^{41} -200.000 q^{43} +384.000 q^{47} -199.000 q^{49} +496.000 q^{53} +1024.00 q^{55} -128.000 q^{59} -458.000 q^{61} +928.000 q^{65} -496.000 q^{67} +512.000 q^{71} -602.000 q^{73} -768.000 q^{77} +1108.00 q^{79} +704.000 q^{83} +512.000 q^{85} -960.000 q^{89} -696.000 q^{91} -2176.00 q^{95} +206.000 q^{97} +O(q^{100})\)

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
\(5\) 16.0000 1.43108 0.715542 − 0.698570i \(-0.246180\pi\)
0.715542 + 0.698570i \(0.246180\pi\)
\(6\) 0 0
\(7\) −12.0000 −0.647939 −0.323970 − 0.946068i \(-0.605018\pi\)
−0.323970 + 0.946068i \(0.605018\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 64.0000 1.75425 0.877124 − 0.480264i \(-0.159459\pi\)
0.877124 + 0.480264i \(0.159459\pi\)
\(12\) 0 0
\(13\) 58.0000 1.23741 0.618704 − 0.785624i \(-0.287658\pi\)
0.618704 + 0.785624i \(0.287658\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 32.0000 0.456538 0.228269 − 0.973598i \(-0.426693\pi\)
0.228269 + 0.973598i \(0.426693\pi\)
\(18\) 0 0
\(19\) −136.000 −1.64213 −0.821067 − 0.570832i \(-0.806621\pi\)
−0.821067 + 0.570832i \(0.806621\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −128.000 −1.16043 −0.580214 − 0.814464i \(-0.697031\pi\)
−0.580214 + 0.814464i \(0.697031\pi\)
\(24\) 0 0
\(25\) 131.000 1.04800
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −144.000 −0.922073 −0.461037 − 0.887381i \(-0.652522\pi\)
−0.461037 + 0.887381i \(0.652522\pi\)
\(30\) 0 0
\(31\) 20.0000 0.115874 0.0579372 − 0.998320i \(-0.481548\pi\)
0.0579372 + 0.998320i \(0.481548\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −192.000 −0.927255
\(36\) 0 0
\(37\) −18.0000 −0.0799779 −0.0399889 − 0.999200i \(-0.512732\pi\)
−0.0399889 + 0.999200i \(0.512732\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −288.000 −1.09703 −0.548513 − 0.836142i \(-0.684806\pi\)
−0.548513 + 0.836142i \(0.684806\pi\)
\(42\) 0 0
\(43\) −200.000 −0.709296 −0.354648 − 0.935000i \(-0.615399\pi\)
−0.354648 + 0.935000i \(0.615399\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 384.000 1.19175 0.595874 − 0.803078i \(-0.296806\pi\)
0.595874 + 0.803078i \(0.296806\pi\)
\(48\) 0 0
\(49\) −199.000 −0.580175
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 496.000 1.28549 0.642744 − 0.766081i \(-0.277796\pi\)
0.642744 + 0.766081i \(0.277796\pi\)
\(54\) 0 0
\(55\) 1024.00 2.51048
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −128.000 −0.282444 −0.141222 − 0.989978i \(-0.545103\pi\)
−0.141222 + 0.989978i \(0.545103\pi\)
\(60\) 0 0
\(61\) −458.000 −0.961326 −0.480663 − 0.876905i \(-0.659604\pi\)
−0.480663 + 0.876905i \(0.659604\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 928.000 1.77083
\(66\) 0 0
\(67\) −496.000 −0.904419 −0.452209 − 0.891912i \(-0.649364\pi\)
−0.452209 + 0.891912i \(0.649364\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 512.000 0.855820 0.427910 − 0.903821i \(-0.359250\pi\)
0.427910 + 0.903821i \(0.359250\pi\)
\(72\) 0 0
\(73\) −602.000 −0.965189 −0.482594 − 0.875844i \(-0.660305\pi\)
−0.482594 + 0.875844i \(0.660305\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −768.000 −1.13665
\(78\) 0 0
\(79\) 1108.00 1.57797 0.788986 − 0.614412i \(-0.210607\pi\)
0.788986 + 0.614412i \(0.210607\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 704.000 0.931013 0.465506 − 0.885045i \(-0.345872\pi\)
0.465506 + 0.885045i \(0.345872\pi\)
\(84\) 0 0
\(85\) 512.000 0.653343
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −960.000 −1.14337 −0.571684 − 0.820474i \(-0.693710\pi\)
−0.571684 + 0.820474i \(0.693710\pi\)
\(90\) 0 0
\(91\) −696.000 −0.801765
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2176.00 −2.35003
\(96\) 0 0
\(97\) 206.000 0.215630 0.107815 − 0.994171i \(-0.465615\pi\)
0.107815 + 0.994171i \(0.465615\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 432.000 0.425600 0.212800 − 0.977096i \(-0.431742\pi\)
0.212800 + 0.977096i \(0.431742\pi\)
\(102\) 0 0
\(103\) −68.0000 −0.0650509 −0.0325254 − 0.999471i \(-0.510355\pi\)
−0.0325254 + 0.999471i \(0.510355\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −384.000 −0.346941 −0.173470 − 0.984839i \(-0.555498\pi\)
−0.173470 + 0.984839i \(0.555498\pi\)
\(108\) 0 0
\(109\) −518.000 −0.455187 −0.227594 − 0.973756i \(-0.573086\pi\)
−0.227594 + 0.973756i \(0.573086\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −960.000 −0.799196 −0.399598 − 0.916690i \(-0.630850\pi\)
−0.399598 + 0.916690i \(0.630850\pi\)
\(114\) 0 0
\(115\) −2048.00 −1.66067
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −384.000 −0.295809
\(120\) 0 0
\(121\) 2765.00 2.07739
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 96.0000 0.0686920
\(126\) 0 0
\(127\) 796.000 0.556170 0.278085 − 0.960556i \(-0.410300\pi\)
0.278085 + 0.960556i \(0.410300\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 512.000 0.341478 0.170739 − 0.985316i \(-0.445384\pi\)
0.170739 + 0.985316i \(0.445384\pi\)
\(132\) 0 0
\(133\) 1632.00 1.06400
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1824.00 1.13748 0.568740 − 0.822517i \(-0.307431\pi\)
0.568740 + 0.822517i \(0.307431\pi\)
\(138\) 0 0
\(139\) −2160.00 −1.31805 −0.659024 − 0.752121i \(-0.729031\pi\)
−0.659024 + 0.752121i \(0.729031\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 3712.00 2.17072
\(144\) 0 0
\(145\) −2304.00 −1.31956
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −688.000 −0.378276 −0.189138 − 0.981950i \(-0.560569\pi\)
−0.189138 + 0.981950i \(0.560569\pi\)
\(150\) 0 0
\(151\) −844.000 −0.454859 −0.227430 − 0.973795i \(-0.573032\pi\)
−0.227430 + 0.973795i \(0.573032\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 320.000 0.165826
\(156\) 0 0
\(157\) 118.000 0.0599836 0.0299918 − 0.999550i \(-0.490452\pi\)
0.0299918 + 0.999550i \(0.490452\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1536.00 0.751887
\(162\) 0 0
\(163\) 3576.00 1.71837 0.859184 − 0.511667i \(-0.170972\pi\)
0.859184 + 0.511667i \(0.170972\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 384.000 0.177933 0.0889665 − 0.996035i \(-0.471644\pi\)
0.0889665 + 0.996035i \(0.471644\pi\)
\(168\) 0 0
\(169\) 1167.00 0.531179
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2448.00 1.07583 0.537913 − 0.843000i \(-0.319213\pi\)
0.537913 + 0.843000i \(0.319213\pi\)
\(174\) 0 0
\(175\) −1572.00 −0.679040
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −4224.00 −1.76378 −0.881890 − 0.471455i \(-0.843729\pi\)
−0.881890 + 0.471455i \(0.843729\pi\)
\(180\) 0 0
\(181\) −510.000 −0.209436 −0.104718 − 0.994502i \(-0.533394\pi\)
−0.104718 + 0.994502i \(0.533394\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −288.000 −0.114455
\(186\) 0 0
\(187\) 2048.00 0.800880
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −384.000 −0.145473 −0.0727363 − 0.997351i \(-0.523173\pi\)
−0.0727363 + 0.997351i \(0.523173\pi\)
\(192\) 0 0
\(193\) −3454.00 −1.28821 −0.644105 − 0.764937i \(-0.722770\pi\)
−0.644105 + 0.764937i \(0.722770\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −3216.00 −1.16310 −0.581550 − 0.813511i \(-0.697553\pi\)
−0.581550 + 0.813511i \(0.697553\pi\)
\(198\) 0 0
\(199\) 1708.00 0.608427 0.304213 − 0.952604i \(-0.401606\pi\)
0.304213 + 0.952604i \(0.401606\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1728.00 0.597447
\(204\) 0 0
\(205\) −4608.00 −1.56994
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −8704.00 −2.88071
\(210\) 0 0
\(211\) 2320.00 0.756945 0.378472 − 0.925613i \(-0.376449\pi\)
0.378472 + 0.925613i \(0.376449\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −3200.00 −1.01506
\(216\) 0 0
\(217\) −240.000 −0.0750795
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1856.00 0.564923
\(222\) 0 0
\(223\) −116.000 −0.0348338 −0.0174169 − 0.999848i \(-0.505544\pi\)
−0.0174169 + 0.999848i \(0.505544\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1344.00 0.392971 0.196485 − 0.980507i \(-0.437047\pi\)
0.196485 + 0.980507i \(0.437047\pi\)
\(228\) 0 0
\(229\) 4594.00 1.32568 0.662839 − 0.748762i \(-0.269352\pi\)
0.662839 + 0.748762i \(0.269352\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5056.00 1.42159 0.710793 − 0.703401i \(-0.248336\pi\)
0.710793 + 0.703401i \(0.248336\pi\)
\(234\) 0 0
\(235\) 6144.00 1.70549
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −3712.00 −1.00464 −0.502321 − 0.864681i \(-0.667520\pi\)
−0.502321 + 0.864681i \(0.667520\pi\)
\(240\) 0 0
\(241\) −978.000 −0.261405 −0.130702 − 0.991422i \(-0.541723\pi\)
−0.130702 + 0.991422i \(0.541723\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −3184.00 −0.830279
\(246\) 0 0
\(247\) −7888.00 −2.03199
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1856.00 0.466732 0.233366 − 0.972389i \(-0.425026\pi\)
0.233366 + 0.972389i \(0.425026\pi\)
\(252\) 0 0
\(253\) −8192.00 −2.03568
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 7808.00 1.89513 0.947567 − 0.319556i \(-0.103534\pi\)
0.947567 + 0.319556i \(0.103534\pi\)
\(258\) 0 0
\(259\) 216.000 0.0518208
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1024.00 0.240086 0.120043 − 0.992769i \(-0.461697\pi\)
0.120043 + 0.992769i \(0.461697\pi\)
\(264\) 0 0
\(265\) 7936.00 1.83964
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 1328.00 0.301002 0.150501 − 0.988610i \(-0.451911\pi\)
0.150501 + 0.988610i \(0.451911\pi\)
\(270\) 0 0
\(271\) −5812.00 −1.30278 −0.651391 − 0.758742i \(-0.725814\pi\)
−0.651391 + 0.758742i \(0.725814\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 8384.00 1.83845
\(276\) 0 0
\(277\) 8386.00 1.81901 0.909505 − 0.415692i \(-0.136461\pi\)
0.909505 + 0.415692i \(0.136461\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −640.000 −0.135869 −0.0679345 − 0.997690i \(-0.521641\pi\)
−0.0679345 + 0.997690i \(0.521641\pi\)
\(282\) 0 0
\(283\) 4832.00 1.01496 0.507478 − 0.861665i \(-0.330578\pi\)
0.507478 + 0.861665i \(0.330578\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3456.00 0.710806
\(288\) 0 0
\(289\) −3889.00 −0.791573
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 6384.00 1.27289 0.636446 − 0.771321i \(-0.280404\pi\)
0.636446 + 0.771321i \(0.280404\pi\)
\(294\) 0 0
\(295\) −2048.00 −0.404201
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −7424.00 −1.43592
\(300\) 0 0
\(301\) 2400.00 0.459580
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −7328.00 −1.37574
\(306\) 0 0
\(307\) −3312.00 −0.615719 −0.307860 − 0.951432i \(-0.599613\pi\)
−0.307860 + 0.951432i \(0.599613\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 9984.00 1.82039 0.910194 − 0.414182i \(-0.135932\pi\)
0.910194 + 0.414182i \(0.135932\pi\)
\(312\) 0 0
\(313\) 2586.00 0.466995 0.233497 − 0.972357i \(-0.424983\pi\)
0.233497 + 0.972357i \(0.424983\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −2832.00 −0.501770 −0.250885 − 0.968017i \(-0.580722\pi\)
−0.250885 + 0.968017i \(0.580722\pi\)
\(318\) 0 0
\(319\) −9216.00 −1.61755
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −4352.00 −0.749696
\(324\) 0 0
\(325\) 7598.00 1.29680
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −4608.00 −0.772180
\(330\) 0 0
\(331\) −5920.00 −0.983059 −0.491530 − 0.870861i \(-0.663562\pi\)
−0.491530 + 0.870861i \(0.663562\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −7936.00 −1.29430
\(336\) 0 0
\(337\) −4674.00 −0.755516 −0.377758 − 0.925904i \(-0.623305\pi\)
−0.377758 + 0.925904i \(0.623305\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 1280.00 0.203272
\(342\) 0 0
\(343\) 6504.00 1.02386
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −9024.00 −1.39606 −0.698031 − 0.716067i \(-0.745940\pi\)
−0.698031 + 0.716067i \(0.745940\pi\)
\(348\) 0 0
\(349\) −4362.00 −0.669033 −0.334516 − 0.942390i \(-0.608573\pi\)
−0.334516 + 0.942390i \(0.608573\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 8768.00 1.32202 0.661011 − 0.750376i \(-0.270128\pi\)
0.661011 + 0.750376i \(0.270128\pi\)
\(354\) 0 0
\(355\) 8192.00 1.22475
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −6144.00 −0.903253 −0.451627 − 0.892207i \(-0.649156\pi\)
−0.451627 + 0.892207i \(0.649156\pi\)
\(360\) 0 0
\(361\) 11637.0 1.69660
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −9632.00 −1.38127
\(366\) 0 0
\(367\) 4564.00 0.649152 0.324576 − 0.945860i \(-0.394778\pi\)
0.324576 + 0.945860i \(0.394778\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −5952.00 −0.832918
\(372\) 0 0
\(373\) −8770.00 −1.21741 −0.608704 − 0.793397i \(-0.708310\pi\)
−0.608704 + 0.793397i \(0.708310\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −8352.00 −1.14098
\(378\) 0 0
\(379\) −1096.00 −0.148543 −0.0742714 − 0.997238i \(-0.523663\pi\)
−0.0742714 + 0.997238i \(0.523663\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 10368.0 1.38324 0.691619 − 0.722263i \(-0.256898\pi\)
0.691619 + 0.722263i \(0.256898\pi\)
\(384\) 0 0
\(385\) −12288.0 −1.62663
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −3248.00 −0.423342 −0.211671 − 0.977341i \(-0.567891\pi\)
−0.211671 + 0.977341i \(0.567891\pi\)
\(390\) 0 0
\(391\) −4096.00 −0.529779
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 17728.0 2.25821
\(396\) 0 0
\(397\) −6106.00 −0.771918 −0.385959 − 0.922516i \(-0.626129\pi\)
−0.385959 + 0.922516i \(0.626129\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 7008.00 0.872725 0.436363 − 0.899771i \(-0.356266\pi\)
0.436363 + 0.899771i \(0.356266\pi\)
\(402\) 0 0
\(403\) 1160.00 0.143384
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1152.00 −0.140301
\(408\) 0 0
\(409\) 1590.00 0.192226 0.0961130 − 0.995370i \(-0.469359\pi\)
0.0961130 + 0.995370i \(0.469359\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 1536.00 0.183006
\(414\) 0 0
\(415\) 11264.0 1.33236
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −192.000 −0.0223862 −0.0111931 − 0.999937i \(-0.503563\pi\)
−0.0111931 + 0.999937i \(0.503563\pi\)
\(420\) 0 0
\(421\) 9074.00 1.05045 0.525225 − 0.850963i \(-0.323981\pi\)
0.525225 + 0.850963i \(0.323981\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 4192.00 0.478451
\(426\) 0 0
\(427\) 5496.00 0.622881
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −5248.00 −0.586513 −0.293257 − 0.956034i \(-0.594739\pi\)
−0.293257 + 0.956034i \(0.594739\pi\)
\(432\) 0 0
\(433\) −8222.00 −0.912527 −0.456263 − 0.889845i \(-0.650813\pi\)
−0.456263 + 0.889845i \(0.650813\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 17408.0 1.90558
\(438\) 0 0
\(439\) 16236.0 1.76515 0.882576 − 0.470169i \(-0.155807\pi\)
0.882576 + 0.470169i \(0.155807\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 14528.0 1.55812 0.779059 − 0.626951i \(-0.215697\pi\)
0.779059 + 0.626951i \(0.215697\pi\)
\(444\) 0 0
\(445\) −15360.0 −1.63626
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 6304.00 0.662593 0.331296 − 0.943527i \(-0.392514\pi\)
0.331296 + 0.943527i \(0.392514\pi\)
\(450\) 0 0
\(451\) −18432.0 −1.92445
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −11136.0 −1.14739
\(456\) 0 0
\(457\) 1958.00 0.200419 0.100209 − 0.994966i \(-0.468049\pi\)
0.100209 + 0.994966i \(0.468049\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 4048.00 0.408968 0.204484 − 0.978870i \(-0.434448\pi\)
0.204484 + 0.978870i \(0.434448\pi\)
\(462\) 0 0
\(463\) 16988.0 1.70518 0.852591 − 0.522579i \(-0.175030\pi\)
0.852591 + 0.522579i \(0.175030\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 6720.00 0.665877 0.332938 − 0.942949i \(-0.391960\pi\)
0.332938 + 0.942949i \(0.391960\pi\)
\(468\) 0 0
\(469\) 5952.00 0.586008
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −12800.0 −1.24428
\(474\) 0 0
\(475\) −17816.0 −1.72096
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 9728.00 0.927941 0.463970 − 0.885851i \(-0.346424\pi\)
0.463970 + 0.885851i \(0.346424\pi\)
\(480\) 0 0
\(481\) −1044.00 −0.0989653
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 3296.00 0.308585
\(486\) 0 0
\(487\) −8444.00 −0.785696 −0.392848 − 0.919603i \(-0.628510\pi\)
−0.392848 + 0.919603i \(0.628510\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −15360.0 −1.41179 −0.705893 − 0.708318i \(-0.749454\pi\)
−0.705893 + 0.708318i \(0.749454\pi\)
\(492\) 0 0
\(493\) −4608.00 −0.420961
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −6144.00 −0.554519
\(498\) 0 0
\(499\) 6624.00 0.594250 0.297125 − 0.954839i \(-0.403972\pi\)
0.297125 + 0.954839i \(0.403972\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −6912.00 −0.612705 −0.306353 − 0.951918i \(-0.599109\pi\)
−0.306353 + 0.951918i \(0.599109\pi\)
\(504\) 0 0
\(505\) 6912.00 0.609069
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −19920.0 −1.73465 −0.867327 − 0.497739i \(-0.834164\pi\)
−0.867327 + 0.497739i \(0.834164\pi\)
\(510\) 0 0
\(511\) 7224.00 0.625383
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1088.00 −0.0930932
\(516\) 0 0
\(517\) 24576.0 2.09062
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 3680.00 0.309451 0.154725 − 0.987958i \(-0.450551\pi\)
0.154725 + 0.987958i \(0.450551\pi\)
\(522\) 0 0
\(523\) −11720.0 −0.979885 −0.489942 − 0.871755i \(-0.662982\pi\)
−0.489942 + 0.871755i \(0.662982\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 640.000 0.0529010
\(528\) 0 0
\(529\) 4217.00 0.346593
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −16704.0 −1.35747
\(534\) 0 0
\(535\) −6144.00 −0.496501
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −12736.0 −1.01777
\(540\) 0 0
\(541\) 11754.0 0.934092 0.467046 − 0.884233i \(-0.345318\pi\)
0.467046 + 0.884233i \(0.345318\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −8288.00 −0.651411
\(546\) 0 0
\(547\) −18904.0 −1.47765 −0.738827 − 0.673895i \(-0.764620\pi\)
−0.738827 + 0.673895i \(0.764620\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 19584.0 1.51417
\(552\) 0 0
\(553\) −13296.0 −1.02243
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 3088.00 0.234906 0.117453 − 0.993078i \(-0.462527\pi\)
0.117453 + 0.993078i \(0.462527\pi\)
\(558\) 0 0
\(559\) −11600.0 −0.877688
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −21440.0 −1.60495 −0.802476 − 0.596684i \(-0.796485\pi\)
−0.802476 + 0.596684i \(0.796485\pi\)
\(564\) 0 0
\(565\) −15360.0 −1.14372
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 22624.0 1.66687 0.833434 − 0.552620i \(-0.186372\pi\)
0.833434 + 0.552620i \(0.186372\pi\)
\(570\) 0 0
\(571\) −6000.00 −0.439741 −0.219871 − 0.975529i \(-0.570564\pi\)
−0.219871 + 0.975529i \(0.570564\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −16768.0 −1.21613
\(576\) 0 0
\(577\) 19922.0 1.43737 0.718686 − 0.695335i \(-0.244744\pi\)
0.718686 + 0.695335i \(0.244744\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −8448.00 −0.603239
\(582\) 0 0
\(583\) 31744.0 2.25506
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −3584.00 −0.252006 −0.126003 − 0.992030i \(-0.540215\pi\)
−0.126003 + 0.992030i \(0.540215\pi\)
\(588\) 0 0
\(589\) −2720.00 −0.190281
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1984.00 0.137391 0.0686957 − 0.997638i \(-0.478116\pi\)
0.0686957 + 0.997638i \(0.478116\pi\)
\(594\) 0 0
\(595\) −6144.00 −0.423327
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 14976.0 1.02154 0.510770 − 0.859717i \(-0.329360\pi\)
0.510770 + 0.859717i \(0.329360\pi\)
\(600\) 0 0
\(601\) 25738.0 1.74688 0.873440 − 0.486932i \(-0.161884\pi\)
0.873440 + 0.486932i \(0.161884\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 44240.0 2.97291
\(606\) 0 0
\(607\) −8548.00 −0.571586 −0.285793 − 0.958291i \(-0.592257\pi\)
−0.285793 + 0.958291i \(0.592257\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 22272.0 1.47468
\(612\) 0 0
\(613\) 8558.00 0.563873 0.281937 − 0.959433i \(-0.409023\pi\)
0.281937 + 0.959433i \(0.409023\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −10368.0 −0.676499 −0.338250 − 0.941056i \(-0.609835\pi\)
−0.338250 + 0.941056i \(0.609835\pi\)
\(618\) 0 0
\(619\) −13088.0 −0.849840 −0.424920 − 0.905231i \(-0.639698\pi\)
−0.424920 + 0.905231i \(0.639698\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 11520.0 0.740833
\(624\) 0 0
\(625\) −14839.0 −0.949696
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −576.000 −0.0365129
\(630\) 0 0
\(631\) −4412.00 −0.278350 −0.139175 − 0.990268i \(-0.544445\pi\)
−0.139175 + 0.990268i \(0.544445\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 12736.0 0.795926
\(636\) 0 0
\(637\) −11542.0 −0.717913
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −30176.0 −1.85941 −0.929704 − 0.368308i \(-0.879937\pi\)
−0.929704 + 0.368308i \(0.879937\pi\)
\(642\) 0 0
\(643\) 21288.0 1.30562 0.652812 − 0.757520i \(-0.273589\pi\)
0.652812 + 0.757520i \(0.273589\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 17024.0 1.03444 0.517220 − 0.855853i \(-0.326967\pi\)
0.517220 + 0.855853i \(0.326967\pi\)
\(648\) 0 0
\(649\) −8192.00 −0.495476
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 2256.00 0.135198 0.0675989 − 0.997713i \(-0.478466\pi\)
0.0675989 + 0.997713i \(0.478466\pi\)
\(654\) 0 0
\(655\) 8192.00 0.488684
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 23808.0 1.40733 0.703663 − 0.710534i \(-0.251546\pi\)
0.703663 + 0.710534i \(0.251546\pi\)
\(660\) 0 0
\(661\) −26242.0 −1.54417 −0.772084 − 0.635520i \(-0.780786\pi\)
−0.772084 + 0.635520i \(0.780786\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 26112.0 1.52268
\(666\) 0 0
\(667\) 18432.0 1.07000
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −29312.0 −1.68640
\(672\) 0 0
\(673\) −24590.0 −1.40843 −0.704216 − 0.709986i \(-0.748701\pi\)
−0.704216 + 0.709986i \(0.748701\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −2864.00 −0.162589 −0.0812943 − 0.996690i \(-0.525905\pi\)
−0.0812943 + 0.996690i \(0.525905\pi\)
\(678\) 0 0
\(679\) −2472.00 −0.139715
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −7616.00 −0.426674 −0.213337 − 0.976979i \(-0.568433\pi\)
−0.213337 + 0.976979i \(0.568433\pi\)
\(684\) 0 0
\(685\) 29184.0 1.62783
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 28768.0 1.59067
\(690\) 0 0
\(691\) 2168.00 0.119355 0.0596777 − 0.998218i \(-0.480993\pi\)
0.0596777 + 0.998218i \(0.480993\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −34560.0 −1.88624
\(696\) 0 0
\(697\) −9216.00 −0.500833
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −18000.0 −0.969830 −0.484915 − 0.874561i \(-0.661149\pi\)
−0.484915 + 0.874561i \(0.661149\pi\)
\(702\) 0 0
\(703\) 2448.00 0.131334
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −5184.00 −0.275763
\(708\) 0 0
\(709\) 3506.00 0.185713 0.0928566 − 0.995679i \(-0.470400\pi\)
0.0928566 + 0.995679i \(0.470400\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −2560.00 −0.134464
\(714\) 0 0
\(715\) 59392.0 3.10648
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 15616.0 0.809984 0.404992 − 0.914320i \(-0.367274\pi\)
0.404992 + 0.914320i \(0.367274\pi\)
\(720\) 0 0
\(721\) 816.000 0.0421490
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −18864.0 −0.966333
\(726\) 0 0
\(727\) −15036.0 −0.767062 −0.383531 − 0.923528i \(-0.625292\pi\)
−0.383531 + 0.923528i \(0.625292\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −6400.00 −0.323820
\(732\) 0 0
\(733\) −19126.0 −0.963758 −0.481879 − 0.876238i \(-0.660046\pi\)
−0.481879 + 0.876238i \(0.660046\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −31744.0 −1.58657
\(738\) 0 0
\(739\) −17392.0 −0.865731 −0.432865 − 0.901459i \(-0.642497\pi\)
−0.432865 + 0.901459i \(0.642497\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −32384.0 −1.59900 −0.799498 − 0.600669i \(-0.794901\pi\)
−0.799498 + 0.600669i \(0.794901\pi\)
\(744\) 0 0
\(745\) −11008.0 −0.541345
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 4608.00 0.224797
\(750\) 0 0
\(751\) −27708.0 −1.34631 −0.673155 − 0.739501i \(-0.735061\pi\)
−0.673155 + 0.739501i \(0.735061\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −13504.0 −0.650942
\(756\) 0 0
\(757\) −37246.0 −1.78828 −0.894141 − 0.447786i \(-0.852213\pi\)
−0.894141 + 0.447786i \(0.852213\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −4192.00 −0.199684 −0.0998422 − 0.995003i \(-0.531834\pi\)
−0.0998422 + 0.995003i \(0.531834\pi\)
\(762\) 0 0
\(763\) 6216.00 0.294934
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −7424.00 −0.349498
\(768\) 0 0
\(769\) 26882.0 1.26058 0.630292 − 0.776358i \(-0.282935\pi\)
0.630292 + 0.776358i \(0.282935\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −17232.0 −0.801801 −0.400900 − 0.916122i \(-0.631303\pi\)
−0.400900 + 0.916122i \(0.631303\pi\)
\(774\) 0 0
\(775\) 2620.00 0.121436
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 39168.0 1.80146
\(780\) 0 0
\(781\) 32768.0 1.50132
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1888.00 0.0858415
\(786\) 0 0
\(787\) 31816.0 1.44106 0.720532 − 0.693421i \(-0.243898\pi\)
0.720532 + 0.693421i \(0.243898\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 11520.0 0.517831
\(792\) 0 0
\(793\) −26564.0 −1.18955
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 8272.00 0.367640 0.183820 − 0.982960i \(-0.441154\pi\)
0.183820 + 0.982960i \(0.441154\pi\)
\(798\) 0 0
\(799\) 12288.0 0.544078
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −38528.0 −1.69318
\(804\) 0 0
\(805\) 24576.0 1.07601
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −9184.00 −0.399125 −0.199563 − 0.979885i \(-0.563952\pi\)
−0.199563 + 0.979885i \(0.563952\pi\)
\(810\) 0 0
\(811\) −19832.0 −0.858688 −0.429344 − 0.903141i \(-0.641255\pi\)
−0.429344 + 0.903141i \(0.641255\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 57216.0 2.45913
\(816\) 0 0
\(817\) 27200.0 1.16476
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 15216.0 0.646823 0.323412 − 0.946258i \(-0.395170\pi\)
0.323412 + 0.946258i \(0.395170\pi\)
\(822\) 0 0
\(823\) −39772.0 −1.68453 −0.842263 − 0.539067i \(-0.818777\pi\)
−0.842263 + 0.539067i \(0.818777\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −18304.0 −0.769640 −0.384820 − 0.922992i \(-0.625737\pi\)
−0.384820 + 0.922992i \(0.625737\pi\)
\(828\) 0 0
\(829\) 4906.00 0.205540 0.102770 − 0.994705i \(-0.467229\pi\)
0.102770 + 0.994705i \(0.467229\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −6368.00 −0.264872
\(834\) 0 0
\(835\) 6144.00 0.254637
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 15360.0 0.632045 0.316023 − 0.948752i \(-0.397652\pi\)
0.316023 + 0.948752i \(0.397652\pi\)
\(840\) 0 0
\(841\) −3653.00 −0.149781
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 18672.0 0.760161
\(846\) 0 0
\(847\) −33180.0 −1.34602
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 2304.00 0.0928086
\(852\) 0 0
\(853\) −24802.0 −0.995550 −0.497775 − 0.867306i \(-0.665850\pi\)
−0.497775 + 0.867306i \(0.665850\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 15072.0 0.600758 0.300379 − 0.953820i \(-0.402887\pi\)
0.300379 + 0.953820i \(0.402887\pi\)
\(858\) 0 0
\(859\) 1800.00 0.0714962 0.0357481 − 0.999361i \(-0.488619\pi\)
0.0357481 + 0.999361i \(0.488619\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 7552.00 0.297883 0.148942 − 0.988846i \(-0.452413\pi\)
0.148942 + 0.988846i \(0.452413\pi\)
\(864\) 0 0
\(865\) 39168.0 1.53960
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 70912.0 2.76815
\(870\) 0 0
\(871\) −28768.0 −1.11913
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −1152.00 −0.0445082
\(876\) 0 0
\(877\) 20838.0 0.802337 0.401168 − 0.916004i \(-0.368604\pi\)
0.401168 + 0.916004i \(0.368604\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 47744.0 1.82581 0.912904 − 0.408175i \(-0.133835\pi\)
0.912904 + 0.408175i \(0.133835\pi\)
\(882\) 0 0
\(883\) 28280.0 1.07780 0.538900 − 0.842370i \(-0.318840\pi\)
0.538900 + 0.842370i \(0.318840\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 7424.00 0.281030 0.140515 − 0.990079i \(-0.455124\pi\)
0.140515 + 0.990079i \(0.455124\pi\)
\(888\) 0 0
\(889\) −9552.00 −0.360364
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −52224.0 −1.95701
\(894\) 0 0
\(895\) −67584.0 −2.52412
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −2880.00 −0.106845
\(900\) 0 0
\(901\) 15872.0 0.586873
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −8160.00 −0.299721
\(906\) 0 0
\(907\) −5912.00 −0.216433 −0.108217 − 0.994127i \(-0.534514\pi\)
−0.108217 + 0.994127i \(0.534514\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −26240.0 −0.954303 −0.477151 − 0.878821i \(-0.658331\pi\)
−0.477151 + 0.878821i \(0.658331\pi\)
\(912\) 0 0
\(913\) 45056.0 1.63323
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −6144.00 −0.221257
\(918\) 0 0
\(919\) 35620.0 1.27856 0.639279 − 0.768975i \(-0.279233\pi\)
0.639279 + 0.768975i \(0.279233\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 29696.0 1.05900
\(924\) 0 0
\(925\) −2358.00 −0.0838168
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −3232.00 −0.114143 −0.0570713 − 0.998370i \(-0.518176\pi\)
−0.0570713 + 0.998370i \(0.518176\pi\)
\(930\) 0 0
\(931\) 27064.0 0.952725
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 32768.0 1.14613
\(936\) 0 0
\(937\) −11478.0 −0.400181 −0.200091 − 0.979777i \(-0.564124\pi\)
−0.200091 + 0.979777i \(0.564124\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −39984.0 −1.38517 −0.692583 − 0.721338i \(-0.743527\pi\)
−0.692583 + 0.721338i \(0.743527\pi\)
\(942\) 0 0
\(943\) 36864.0 1.27302
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 24192.0 0.830131 0.415066 − 0.909791i \(-0.363759\pi\)
0.415066 + 0.909791i \(0.363759\pi\)
\(948\) 0 0
\(949\) −34916.0 −1.19433
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −39456.0 −1.34114 −0.670569 − 0.741847i \(-0.733950\pi\)
−0.670569 + 0.741847i \(0.733950\pi\)
\(954\) 0 0
\(955\) −6144.00 −0.208183
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −21888.0 −0.737018
\(960\) 0 0
\(961\) −29391.0 −0.986573
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −55264.0 −1.84353
\(966\) 0 0
\(967\) −41668.0 −1.38568 −0.692840 − 0.721091i \(-0.743641\pi\)
−0.692840 + 0.721091i \(0.743641\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −51648.0 −1.70697 −0.853483 − 0.521121i \(-0.825514\pi\)
−0.853483 + 0.521121i \(0.825514\pi\)
\(972\) 0 0
\(973\) 25920.0 0.854015
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 55776.0 1.82644 0.913220 − 0.407466i \(-0.133588\pi\)
0.913220 + 0.407466i \(0.133588\pi\)
\(978\) 0 0
\(979\) −61440.0 −2.00575
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −36096.0 −1.17119 −0.585597 − 0.810602i \(-0.699140\pi\)
−0.585597 + 0.810602i \(0.699140\pi\)
\(984\) 0 0
\(985\) −51456.0 −1.66449
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 25600.0 0.823087
\(990\) 0 0
\(991\) 42532.0 1.36334 0.681672 − 0.731658i \(-0.261253\pi\)
0.681672 + 0.731658i \(0.261253\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 27328.0 0.870709
\(996\) 0 0
\(997\) 29806.0 0.946806 0.473403 − 0.880846i \(-0.343025\pi\)
0.473403 + 0.880846i \(0.343025\pi\)
\(998\) 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 72.4.a.d.1.1 yes 1
3.2 odd 2 72.4.a.a.1.1 ✓ 1
4.3 odd 2 144.4.a.f.1.1 1
5.2 odd 4 1800.4.f.x.649.1 2
5.3 odd 4 1800.4.f.x.649.2 2
5.4 even 2 1800.4.a.ba.1.1 1
8.3 odd 2 576.4.a.d.1.1 1
8.5 even 2 576.4.a.c.1.1 1
9.2 odd 6 648.4.i.l.433.1 2
9.4 even 3 648.4.i.a.217.1 2
9.5 odd 6 648.4.i.l.217.1 2
9.7 even 3 648.4.i.a.433.1 2
12.11 even 2 144.4.a.a.1.1 1
15.2 even 4 1800.4.f.b.649.1 2
15.8 even 4 1800.4.f.b.649.2 2
15.14 odd 2 1800.4.a.z.1.1 1
24.5 odd 2 576.4.a.w.1.1 1
24.11 even 2 576.4.a.x.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.4.a.a.1.1 ✓ 1 3.2 odd 2
72.4.a.d.1.1 yes 1 1.1 even 1 trivial
144.4.a.a.1.1 1 12.11 even 2
144.4.a.f.1.1 1 4.3 odd 2
576.4.a.c.1.1 1 8.5 even 2
576.4.a.d.1.1 1 8.3 odd 2
576.4.a.w.1.1 1 24.5 odd 2
576.4.a.x.1.1 1 24.11 even 2
648.4.i.a.217.1 2 9.4 even 3
648.4.i.a.433.1 2 9.7 even 3
648.4.i.l.217.1 2 9.5 odd 6
648.4.i.l.433.1 2 9.2 odd 6
1800.4.a.z.1.1 1 15.14 odd 2
1800.4.a.ba.1.1 1 5.4 even 2
1800.4.f.b.649.1 2 15.2 even 4
1800.4.f.b.649.2 2 15.8 even 4
1800.4.f.x.649.1 2 5.2 odd 4
1800.4.f.x.649.2 2 5.3 odd 4