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Properties

Label 1728.4.a.x.1.1
Level $1728$
Weight $4$
Character 1728.1
Self dual yes
Analytic conductor $101.955$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1728,4,Mod(1,1728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1728, 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("1728.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1728.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: \(101.955300490\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 216)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1728.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.00000 q^{5} +3.00000 q^{7} +O(q^{10})\) \(q+4.00000 q^{5} +3.00000 q^{7} +28.0000 q^{11} +11.0000 q^{13} -44.0000 q^{17} -29.0000 q^{19} -172.000 q^{23} -109.000 q^{25} +192.000 q^{29} +116.000 q^{31} +12.0000 q^{35} +69.0000 q^{37} -384.000 q^{41} -328.000 q^{43} -156.000 q^{47} -334.000 q^{49} -392.000 q^{53} +112.000 q^{55} +412.000 q^{59} +425.000 q^{61} +44.0000 q^{65} -257.000 q^{67} +1000.00 q^{71} -359.000 q^{73} +84.0000 q^{77} +877.000 q^{79} -328.000 q^{83} -176.000 q^{85} +1572.00 q^{89} +33.0000 q^{91} -116.000 q^{95} -1483.00 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 4.00000 0.357771 0.178885 − 0.983870i \(-0.442751\pi\)
0.178885 + 0.983870i \(0.442751\pi\)
\(6\) 0 0
\(7\) 3.00000 0.161985 0.0809924 − 0.996715i \(-0.474191\pi\)
0.0809924 + 0.996715i \(0.474191\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 28.0000 0.767483 0.383742 − 0.923440i \(-0.374635\pi\)
0.383742 + 0.923440i \(0.374635\pi\)
\(12\) 0 0
\(13\) 11.0000 0.234681 0.117340 − 0.993092i \(-0.462563\pi\)
0.117340 + 0.993092i \(0.462563\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −44.0000 −0.627739 −0.313870 − 0.949466i \(-0.601625\pi\)
−0.313870 + 0.949466i \(0.601625\pi\)
\(18\) 0 0
\(19\) −29.0000 −0.350161 −0.175080 − 0.984554i \(-0.556019\pi\)
−0.175080 + 0.984554i \(0.556019\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −172.000 −1.55933 −0.779663 − 0.626200i \(-0.784609\pi\)
−0.779663 + 0.626200i \(0.784609\pi\)
\(24\) 0 0
\(25\) −109.000 −0.872000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 192.000 1.22943 0.614716 − 0.788749i \(-0.289271\pi\)
0.614716 + 0.788749i \(0.289271\pi\)
\(30\) 0 0
\(31\) 116.000 0.672071 0.336036 − 0.941849i \(-0.390914\pi\)
0.336036 + 0.941849i \(0.390914\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 12.0000 0.0579534
\(36\) 0 0
\(37\) 69.0000 0.306582 0.153291 − 0.988181i \(-0.451013\pi\)
0.153291 + 0.988181i \(0.451013\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −384.000 −1.46270 −0.731350 − 0.682002i \(-0.761110\pi\)
−0.731350 + 0.682002i \(0.761110\pi\)
\(42\) 0 0
\(43\) −328.000 −1.16324 −0.581622 − 0.813459i \(-0.697582\pi\)
−0.581622 + 0.813459i \(0.697582\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −156.000 −0.484148 −0.242074 − 0.970258i \(-0.577828\pi\)
−0.242074 + 0.970258i \(0.577828\pi\)
\(48\) 0 0
\(49\) −334.000 −0.973761
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −392.000 −1.01595 −0.507975 − 0.861372i \(-0.669606\pi\)
−0.507975 + 0.861372i \(0.669606\pi\)
\(54\) 0 0
\(55\) 112.000 0.274583
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 412.000 0.909116 0.454558 − 0.890717i \(-0.349797\pi\)
0.454558 + 0.890717i \(0.349797\pi\)
\(60\) 0 0
\(61\) 425.000 0.892060 0.446030 − 0.895018i \(-0.352837\pi\)
0.446030 + 0.895018i \(0.352837\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 44.0000 0.0839620
\(66\) 0 0
\(67\) −257.000 −0.468620 −0.234310 − 0.972162i \(-0.575283\pi\)
−0.234310 + 0.972162i \(0.575283\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1000.00 1.67152 0.835762 − 0.549092i \(-0.185026\pi\)
0.835762 + 0.549092i \(0.185026\pi\)
\(72\) 0 0
\(73\) −359.000 −0.575586 −0.287793 − 0.957693i \(-0.592921\pi\)
−0.287793 + 0.957693i \(0.592921\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 84.0000 0.124321
\(78\) 0 0
\(79\) 877.000 1.24899 0.624495 − 0.781029i \(-0.285305\pi\)
0.624495 + 0.781029i \(0.285305\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −328.000 −0.433767 −0.216884 − 0.976197i \(-0.569589\pi\)
−0.216884 + 0.976197i \(0.569589\pi\)
\(84\) 0 0
\(85\) −176.000 −0.224587
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1572.00 1.87227 0.936133 − 0.351646i \(-0.114378\pi\)
0.936133 + 0.351646i \(0.114378\pi\)
\(90\) 0 0
\(91\) 33.0000 0.0380147
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −116.000 −0.125277
\(96\) 0 0
\(97\) −1483.00 −1.55233 −0.776164 − 0.630531i \(-0.782837\pi\)
−0.776164 + 0.630531i \(0.782837\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −576.000 −0.567467 −0.283733 − 0.958903i \(-0.591573\pi\)
−0.283733 + 0.958903i \(0.591573\pi\)
\(102\) 0 0
\(103\) 79.0000 0.0755738 0.0377869 − 0.999286i \(-0.487969\pi\)
0.0377869 + 0.999286i \(0.487969\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1836.00 −1.65881 −0.829406 − 0.558647i \(-0.811321\pi\)
−0.829406 + 0.558647i \(0.811321\pi\)
\(108\) 0 0
\(109\) −1450.00 −1.27417 −0.637086 − 0.770792i \(-0.719861\pi\)
−0.637086 + 0.770792i \(0.719861\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −636.000 −0.529468 −0.264734 − 0.964322i \(-0.585284\pi\)
−0.264734 + 0.964322i \(0.585284\pi\)
\(114\) 0 0
\(115\) −688.000 −0.557881
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −132.000 −0.101684
\(120\) 0 0
\(121\) −547.000 −0.410969
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −936.000 −0.669747
\(126\) 0 0
\(127\) −1460.00 −1.02011 −0.510055 − 0.860142i \(-0.670375\pi\)
−0.510055 + 0.860142i \(0.670375\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 464.000 0.309465 0.154732 − 0.987956i \(-0.450548\pi\)
0.154732 + 0.987956i \(0.450548\pi\)
\(132\) 0 0
\(133\) −87.0000 −0.0567207
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −36.0000 −0.0224503 −0.0112251 − 0.999937i \(-0.503573\pi\)
−0.0112251 + 0.999937i \(0.503573\pi\)
\(138\) 0 0
\(139\) 1149.00 0.701129 0.350564 − 0.936539i \(-0.385990\pi\)
0.350564 + 0.936539i \(0.385990\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 308.000 0.180114
\(144\) 0 0
\(145\) 768.000 0.439855
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1736.00 0.954488 0.477244 − 0.878771i \(-0.341636\pi\)
0.477244 + 0.878771i \(0.341636\pi\)
\(150\) 0 0
\(151\) 737.000 0.397193 0.198597 − 0.980081i \(-0.436362\pi\)
0.198597 + 0.980081i \(0.436362\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 464.000 0.240448
\(156\) 0 0
\(157\) −2854.00 −1.45079 −0.725395 − 0.688333i \(-0.758343\pi\)
−0.725395 + 0.688333i \(0.758343\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −516.000 −0.252587
\(162\) 0 0
\(163\) 2013.00 0.967303 0.483651 − 0.875261i \(-0.339310\pi\)
0.483651 + 0.875261i \(0.339310\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1212.00 −0.561601 −0.280801 − 0.959766i \(-0.590600\pi\)
−0.280801 + 0.959766i \(0.590600\pi\)
\(168\) 0 0
\(169\) −2076.00 −0.944925
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1920.00 0.843786 0.421893 − 0.906646i \(-0.361366\pi\)
0.421893 + 0.906646i \(0.361366\pi\)
\(174\) 0 0
\(175\) −327.000 −0.141251
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 3240.00 1.35290 0.676450 − 0.736489i \(-0.263518\pi\)
0.676450 + 0.736489i \(0.263518\pi\)
\(180\) 0 0
\(181\) −4545.00 −1.86645 −0.933224 − 0.359294i \(-0.883017\pi\)
−0.933224 + 0.359294i \(0.883017\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 276.000 0.109686
\(186\) 0 0
\(187\) −1232.00 −0.481779
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2148.00 −0.813737 −0.406869 − 0.913487i \(-0.633379\pi\)
−0.406869 + 0.913487i \(0.633379\pi\)
\(192\) 0 0
\(193\) −841.000 −0.313661 −0.156830 − 0.987626i \(-0.550128\pi\)
−0.156830 + 0.987626i \(0.550128\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1524.00 0.551170 0.275585 − 0.961277i \(-0.411128\pi\)
0.275585 + 0.961277i \(0.411128\pi\)
\(198\) 0 0
\(199\) −5321.00 −1.89546 −0.947728 − 0.319080i \(-0.896626\pi\)
−0.947728 + 0.319080i \(0.896626\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 576.000 0.199149
\(204\) 0 0
\(205\) −1536.00 −0.523312
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −812.000 −0.268743
\(210\) 0 0
\(211\) 3497.00 1.14096 0.570482 − 0.821310i \(-0.306756\pi\)
0.570482 + 0.821310i \(0.306756\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −1312.00 −0.416175
\(216\) 0 0
\(217\) 348.000 0.108865
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −484.000 −0.147318
\(222\) 0 0
\(223\) 1804.00 0.541725 0.270863 − 0.962618i \(-0.412691\pi\)
0.270863 + 0.962618i \(0.412691\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1320.00 −0.385954 −0.192977 − 0.981203i \(-0.561814\pi\)
−0.192977 + 0.981203i \(0.561814\pi\)
\(228\) 0 0
\(229\) −4114.00 −1.18716 −0.593582 − 0.804773i \(-0.702287\pi\)
−0.593582 + 0.804773i \(0.702287\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −5008.00 −1.40809 −0.704045 − 0.710155i \(-0.748625\pi\)
−0.704045 + 0.710155i \(0.748625\pi\)
\(234\) 0 0
\(235\) −624.000 −0.173214
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 2536.00 0.686361 0.343180 − 0.939270i \(-0.388496\pi\)
0.343180 + 0.939270i \(0.388496\pi\)
\(240\) 0 0
\(241\) −4287.00 −1.14585 −0.572925 − 0.819608i \(-0.694191\pi\)
−0.572925 + 0.819608i \(0.694191\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1336.00 −0.348383
\(246\) 0 0
\(247\) −319.000 −0.0821760
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1424.00 0.358096 0.179048 − 0.983840i \(-0.442698\pi\)
0.179048 + 0.983840i \(0.442698\pi\)
\(252\) 0 0
\(253\) −4816.00 −1.19676
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 5848.00 1.41941 0.709705 − 0.704499i \(-0.248828\pi\)
0.709705 + 0.704499i \(0.248828\pi\)
\(258\) 0 0
\(259\) 207.000 0.0496616
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 3800.00 0.890943 0.445472 − 0.895296i \(-0.353036\pi\)
0.445472 + 0.895296i \(0.353036\pi\)
\(264\) 0 0
\(265\) −1568.00 −0.363477
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 1820.00 0.412518 0.206259 − 0.978497i \(-0.433871\pi\)
0.206259 + 0.978497i \(0.433871\pi\)
\(270\) 0 0
\(271\) −1615.00 −0.362008 −0.181004 − 0.983482i \(-0.557935\pi\)
−0.181004 + 0.983482i \(0.557935\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −3052.00 −0.669246
\(276\) 0 0
\(277\) −754.000 −0.163550 −0.0817752 − 0.996651i \(-0.526059\pi\)
−0.0817752 + 0.996651i \(0.526059\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 544.000 0.115489 0.0577443 − 0.998331i \(-0.481609\pi\)
0.0577443 + 0.998331i \(0.481609\pi\)
\(282\) 0 0
\(283\) −848.000 −0.178121 −0.0890607 − 0.996026i \(-0.528387\pi\)
−0.0890607 + 0.996026i \(0.528387\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −1152.00 −0.236935
\(288\) 0 0
\(289\) −2977.00 −0.605943
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −6708.00 −1.33749 −0.668747 − 0.743490i \(-0.733169\pi\)
−0.668747 + 0.743490i \(0.733169\pi\)
\(294\) 0 0
\(295\) 1648.00 0.325255
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1892.00 −0.365944
\(300\) 0 0
\(301\) −984.000 −0.188428
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1700.00 0.319153
\(306\) 0 0
\(307\) −2112.00 −0.392633 −0.196316 − 0.980541i \(-0.562898\pi\)
−0.196316 + 0.980541i \(0.562898\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 5820.00 1.06116 0.530582 − 0.847634i \(-0.321973\pi\)
0.530582 + 0.847634i \(0.321973\pi\)
\(312\) 0 0
\(313\) −7413.00 −1.33868 −0.669341 − 0.742955i \(-0.733424\pi\)
−0.669341 + 0.742955i \(0.733424\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −8856.00 −1.56909 −0.784547 − 0.620070i \(-0.787104\pi\)
−0.784547 + 0.620070i \(0.787104\pi\)
\(318\) 0 0
\(319\) 5376.00 0.943568
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1276.00 0.219810
\(324\) 0 0
\(325\) −1199.00 −0.204642
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −468.000 −0.0784245
\(330\) 0 0
\(331\) −3131.00 −0.519925 −0.259963 − 0.965619i \(-0.583710\pi\)
−0.259963 + 0.965619i \(0.583710\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −1028.00 −0.167659
\(336\) 0 0
\(337\) −6819.00 −1.10224 −0.551120 − 0.834426i \(-0.685799\pi\)
−0.551120 + 0.834426i \(0.685799\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 3248.00 0.515804
\(342\) 0 0
\(343\) −2031.00 −0.319719
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −11304.0 −1.74879 −0.874396 − 0.485214i \(-0.838742\pi\)
−0.874396 + 0.485214i \(0.838742\pi\)
\(348\) 0 0
\(349\) −2235.00 −0.342799 −0.171399 − 0.985202i \(-0.554829\pi\)
−0.171399 + 0.985202i \(0.554829\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −296.000 −0.0446303 −0.0223151 − 0.999751i \(-0.507104\pi\)
−0.0223151 + 0.999751i \(0.507104\pi\)
\(354\) 0 0
\(355\) 4000.00 0.598022
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −5700.00 −0.837979 −0.418990 − 0.907991i \(-0.637616\pi\)
−0.418990 + 0.907991i \(0.637616\pi\)
\(360\) 0 0
\(361\) −6018.00 −0.877387
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −1436.00 −0.205928
\(366\) 0 0
\(367\) 6943.00 0.987525 0.493762 − 0.869597i \(-0.335621\pi\)
0.493762 + 0.869597i \(0.335621\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −1176.00 −0.164568
\(372\) 0 0
\(373\) 877.000 0.121741 0.0608704 − 0.998146i \(-0.480612\pi\)
0.0608704 + 0.998146i \(0.480612\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2112.00 0.288524
\(378\) 0 0
\(379\) −10361.0 −1.40424 −0.702122 − 0.712056i \(-0.747764\pi\)
−0.702122 + 0.712056i \(0.747764\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −5184.00 −0.691619 −0.345809 − 0.938305i \(-0.612396\pi\)
−0.345809 + 0.938305i \(0.612396\pi\)
\(384\) 0 0
\(385\) 336.000 0.0444783
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 8260.00 1.07660 0.538302 − 0.842752i \(-0.319066\pi\)
0.538302 + 0.842752i \(0.319066\pi\)
\(390\) 0 0
\(391\) 7568.00 0.978850
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 3508.00 0.446852
\(396\) 0 0
\(397\) 5098.00 0.644487 0.322243 − 0.946657i \(-0.395563\pi\)
0.322243 + 0.946657i \(0.395563\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −3192.00 −0.397508 −0.198754 − 0.980049i \(-0.563690\pi\)
−0.198754 + 0.980049i \(0.563690\pi\)
\(402\) 0 0
\(403\) 1276.00 0.157722
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1932.00 0.235297
\(408\) 0 0
\(409\) −639.000 −0.0772531 −0.0386265 − 0.999254i \(-0.512298\pi\)
−0.0386265 + 0.999254i \(0.512298\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 1236.00 0.147263
\(414\) 0 0
\(415\) −1312.00 −0.155189
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 11340.0 1.32218 0.661092 − 0.750305i \(-0.270093\pi\)
0.661092 + 0.750305i \(0.270093\pi\)
\(420\) 0 0
\(421\) 115.000 0.0133130 0.00665648 − 0.999978i \(-0.497881\pi\)
0.00665648 + 0.999978i \(0.497881\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 4796.00 0.547389
\(426\) 0 0
\(427\) 1275.00 0.144500
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 3796.00 0.424239 0.212119 − 0.977244i \(-0.431963\pi\)
0.212119 + 0.977244i \(0.431963\pi\)
\(432\) 0 0
\(433\) −6062.00 −0.672797 −0.336399 − 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4988.00 0.546015
\(438\) 0 0
\(439\) −10548.0 −1.14676 −0.573381 − 0.819289i \(-0.694369\pi\)
−0.573381 + 0.819289i \(0.694369\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 15416.0 1.65335 0.826677 − 0.562676i \(-0.190228\pi\)
0.826677 + 0.562676i \(0.190228\pi\)
\(444\) 0 0
\(445\) 6288.00 0.669842
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −12820.0 −1.34747 −0.673734 − 0.738974i \(-0.735311\pi\)
−0.673734 + 0.738974i \(0.735311\pi\)
\(450\) 0 0
\(451\) −10752.0 −1.12260
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 132.000 0.0136006
\(456\) 0 0
\(457\) 470.000 0.0481087 0.0240543 − 0.999711i \(-0.492343\pi\)
0.0240543 + 0.999711i \(0.492343\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 12796.0 1.29277 0.646387 − 0.763009i \(-0.276279\pi\)
0.646387 + 0.763009i \(0.276279\pi\)
\(462\) 0 0
\(463\) 14603.0 1.46579 0.732893 − 0.680344i \(-0.238170\pi\)
0.732893 + 0.680344i \(0.238170\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −12084.0 −1.19739 −0.598695 − 0.800977i \(-0.704314\pi\)
−0.598695 + 0.800977i \(0.704314\pi\)
\(468\) 0 0
\(469\) −771.000 −0.0759093
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −9184.00 −0.892771
\(474\) 0 0
\(475\) 3161.00 0.305340
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 1528.00 0.145754 0.0728769 − 0.997341i \(-0.476782\pi\)
0.0728769 + 0.997341i \(0.476782\pi\)
\(480\) 0 0
\(481\) 759.000 0.0719489
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −5932.00 −0.555378
\(486\) 0 0
\(487\) −8117.00 −0.755270 −0.377635 − 0.925955i \(-0.623263\pi\)
−0.377635 + 0.925955i \(0.623263\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 16596.0 1.52539 0.762696 − 0.646758i \(-0.223876\pi\)
0.762696 + 0.646758i \(0.223876\pi\)
\(492\) 0 0
\(493\) −8448.00 −0.771762
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 3000.00 0.270761
\(498\) 0 0
\(499\) 13440.0 1.20573 0.602863 − 0.797845i \(-0.294027\pi\)
0.602863 + 0.797845i \(0.294027\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −8652.00 −0.766946 −0.383473 − 0.923552i \(-0.625272\pi\)
−0.383473 + 0.923552i \(0.625272\pi\)
\(504\) 0 0
\(505\) −2304.00 −0.203023
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 4884.00 0.425304 0.212652 − 0.977128i \(-0.431790\pi\)
0.212652 + 0.977128i \(0.431790\pi\)
\(510\) 0 0
\(511\) −1077.00 −0.0932362
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 316.000 0.0270381
\(516\) 0 0
\(517\) −4368.00 −0.371575
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 18028.0 1.51597 0.757986 − 0.652271i \(-0.226184\pi\)
0.757986 + 0.652271i \(0.226184\pi\)
\(522\) 0 0
\(523\) −20593.0 −1.72174 −0.860869 − 0.508827i \(-0.830079\pi\)
−0.860869 + 0.508827i \(0.830079\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −5104.00 −0.421886
\(528\) 0 0
\(529\) 17417.0 1.43150
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −4224.00 −0.343268
\(534\) 0 0
\(535\) −7344.00 −0.593474
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −9352.00 −0.747345
\(540\) 0 0
\(541\) 7443.00 0.591496 0.295748 − 0.955266i \(-0.404431\pi\)
0.295748 + 0.955266i \(0.404431\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −5800.00 −0.455862
\(546\) 0 0
\(547\) 23653.0 1.84887 0.924433 − 0.381346i \(-0.124539\pi\)
0.924433 + 0.381346i \(0.124539\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −5568.00 −0.430499
\(552\) 0 0
\(553\) 2631.00 0.202317
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −12356.0 −0.939929 −0.469965 − 0.882685i \(-0.655733\pi\)
−0.469965 + 0.882685i \(0.655733\pi\)
\(558\) 0 0
\(559\) −3608.00 −0.272991
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −8288.00 −0.620422 −0.310211 − 0.950668i \(-0.600400\pi\)
−0.310211 + 0.950668i \(0.600400\pi\)
\(564\) 0 0
\(565\) −2544.00 −0.189428
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −13228.0 −0.974599 −0.487299 − 0.873235i \(-0.662018\pi\)
−0.487299 + 0.873235i \(0.662018\pi\)
\(570\) 0 0
\(571\) 2811.00 0.206019 0.103009 − 0.994680i \(-0.467153\pi\)
0.103009 + 0.994680i \(0.467153\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 18748.0 1.35973
\(576\) 0 0
\(577\) 8963.00 0.646680 0.323340 − 0.946283i \(-0.395194\pi\)
0.323340 + 0.946283i \(0.395194\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −984.000 −0.0702637
\(582\) 0 0
\(583\) −10976.0 −0.779725
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −22844.0 −1.60626 −0.803128 − 0.595806i \(-0.796833\pi\)
−0.803128 + 0.595806i \(0.796833\pi\)
\(588\) 0 0
\(589\) −3364.00 −0.235333
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 17528.0 1.21381 0.606904 − 0.794775i \(-0.292411\pi\)
0.606904 + 0.794775i \(0.292411\pi\)
\(594\) 0 0
\(595\) −528.000 −0.0363796
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −15168.0 −1.03464 −0.517319 − 0.855793i \(-0.673070\pi\)
−0.517319 + 0.855793i \(0.673070\pi\)
\(600\) 0 0
\(601\) 22906.0 1.55467 0.777334 − 0.629088i \(-0.216572\pi\)
0.777334 + 0.629088i \(0.216572\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −2188.00 −0.147033
\(606\) 0 0
\(607\) 17219.0 1.15140 0.575698 − 0.817662i \(-0.304730\pi\)
0.575698 + 0.817662i \(0.304730\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −1716.00 −0.113620
\(612\) 0 0
\(613\) 20569.0 1.35526 0.677630 − 0.735403i \(-0.263007\pi\)
0.677630 + 0.735403i \(0.263007\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 17772.0 1.15960 0.579800 − 0.814759i \(-0.303131\pi\)
0.579800 + 0.814759i \(0.303131\pi\)
\(618\) 0 0
\(619\) 11099.0 0.720689 0.360344 − 0.932819i \(-0.382659\pi\)
0.360344 + 0.932819i \(0.382659\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 4716.00 0.303279
\(624\) 0 0
\(625\) 9881.00 0.632384
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −3036.00 −0.192453
\(630\) 0 0
\(631\) −25895.0 −1.63370 −0.816849 − 0.576851i \(-0.804281\pi\)
−0.816849 + 0.576851i \(0.804281\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −5840.00 −0.364966
\(636\) 0 0
\(637\) −3674.00 −0.228523
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 9080.00 0.559498 0.279749 − 0.960073i \(-0.409749\pi\)
0.279749 + 0.960073i \(0.409749\pi\)
\(642\) 0 0
\(643\) 22632.0 1.38805 0.694027 − 0.719949i \(-0.255835\pi\)
0.694027 + 0.719949i \(0.255835\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 30112.0 1.82971 0.914857 − 0.403778i \(-0.132303\pi\)
0.914857 + 0.403778i \(0.132303\pi\)
\(648\) 0 0
\(649\) 11536.0 0.697731
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 25368.0 1.52026 0.760128 − 0.649774i \(-0.225136\pi\)
0.760128 + 0.649774i \(0.225136\pi\)
\(654\) 0 0
\(655\) 1856.00 0.110717
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 13080.0 0.773178 0.386589 − 0.922252i \(-0.373653\pi\)
0.386589 + 0.922252i \(0.373653\pi\)
\(660\) 0 0
\(661\) 9433.00 0.555070 0.277535 − 0.960716i \(-0.410483\pi\)
0.277535 + 0.960716i \(0.410483\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −348.000 −0.0202930
\(666\) 0 0
\(667\) −33024.0 −1.91708
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 11900.0 0.684641
\(672\) 0 0
\(673\) 14275.0 0.817624 0.408812 − 0.912619i \(-0.365943\pi\)
0.408812 + 0.912619i \(0.365943\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −28268.0 −1.60477 −0.802384 − 0.596809i \(-0.796435\pi\)
−0.802384 + 0.596809i \(0.796435\pi\)
\(678\) 0 0
\(679\) −4449.00 −0.251454
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −5840.00 −0.327176 −0.163588 − 0.986529i \(-0.552307\pi\)
−0.163588 + 0.986529i \(0.552307\pi\)
\(684\) 0 0
\(685\) −144.000 −0.00803205
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −4312.00 −0.238424
\(690\) 0 0
\(691\) 12376.0 0.681339 0.340669 − 0.940183i \(-0.389346\pi\)
0.340669 + 0.940183i \(0.389346\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 4596.00 0.250843
\(696\) 0 0
\(697\) 16896.0 0.918195
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −11076.0 −0.596769 −0.298384 − 0.954446i \(-0.596448\pi\)
−0.298384 + 0.954446i \(0.596448\pi\)
\(702\) 0 0
\(703\) −2001.00 −0.107353
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −1728.00 −0.0919210
\(708\) 0 0
\(709\) −7253.00 −0.384192 −0.192096 − 0.981376i \(-0.561528\pi\)
−0.192096 + 0.981376i \(0.561528\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −19952.0 −1.04798
\(714\) 0 0
\(715\) 1232.00 0.0644394
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −28456.0 −1.47598 −0.737990 − 0.674812i \(-0.764225\pi\)
−0.737990 + 0.674812i \(0.764225\pi\)
\(720\) 0 0
\(721\) 237.000 0.0122418
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −20928.0 −1.07206
\(726\) 0 0
\(727\) −10428.0 −0.531985 −0.265993 − 0.963975i \(-0.585700\pi\)
−0.265993 + 0.963975i \(0.585700\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 14432.0 0.730215
\(732\) 0 0
\(733\) 23206.0 1.16935 0.584675 − 0.811268i \(-0.301222\pi\)
0.584675 + 0.811268i \(0.301222\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −7196.00 −0.359658
\(738\) 0 0
\(739\) 3424.00 0.170438 0.0852191 − 0.996362i \(-0.472841\pi\)
0.0852191 + 0.996362i \(0.472841\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −11692.0 −0.577305 −0.288653 − 0.957434i \(-0.593207\pi\)
−0.288653 + 0.957434i \(0.593207\pi\)
\(744\) 0 0
\(745\) 6944.00 0.341488
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −5508.00 −0.268702
\(750\) 0 0
\(751\) 29349.0 1.42605 0.713023 − 0.701141i \(-0.247326\pi\)
0.713023 + 0.701141i \(0.247326\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 2948.00 0.142104
\(756\) 0 0
\(757\) 13555.0 0.650812 0.325406 − 0.945574i \(-0.394499\pi\)
0.325406 + 0.945574i \(0.394499\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −10316.0 −0.491399 −0.245700 − 0.969346i \(-0.579018\pi\)
−0.245700 + 0.969346i \(0.579018\pi\)
\(762\) 0 0
\(763\) −4350.00 −0.206397
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 4532.00 0.213352
\(768\) 0 0
\(769\) −145.000 −0.00679952 −0.00339976 − 0.999994i \(-0.501082\pi\)
−0.00339976 + 0.999994i \(0.501082\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 33576.0 1.56228 0.781142 − 0.624354i \(-0.214638\pi\)
0.781142 + 0.624354i \(0.214638\pi\)
\(774\) 0 0
\(775\) −12644.0 −0.586046
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 11136.0 0.512180
\(780\) 0 0
\(781\) 28000.0 1.28287
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −11416.0 −0.519050
\(786\) 0 0
\(787\) 26795.0 1.21364 0.606822 − 0.794837i \(-0.292444\pi\)
0.606822 + 0.794837i \(0.292444\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −1908.00 −0.0857657
\(792\) 0 0
\(793\) 4675.00 0.209349
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −4784.00 −0.212620 −0.106310 − 0.994333i \(-0.533904\pi\)
−0.106310 + 0.994333i \(0.533904\pi\)
\(798\) 0 0
\(799\) 6864.00 0.303918
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −10052.0 −0.441753
\(804\) 0 0
\(805\) −2064.00 −0.0903683
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −18488.0 −0.803465 −0.401733 − 0.915757i \(-0.631592\pi\)
−0.401733 + 0.915757i \(0.631592\pi\)
\(810\) 0 0
\(811\) −33304.0 −1.44200 −0.721000 − 0.692935i \(-0.756317\pi\)
−0.721000 + 0.692935i \(0.756317\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 8052.00 0.346073
\(816\) 0 0
\(817\) 9512.00 0.407323
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 43380.0 1.84406 0.922029 − 0.387120i \(-0.126530\pi\)
0.922029 + 0.387120i \(0.126530\pi\)
\(822\) 0 0
\(823\) 33785.0 1.43095 0.715475 − 0.698639i \(-0.246211\pi\)
0.715475 + 0.698639i \(0.246211\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 38732.0 1.62859 0.814295 − 0.580452i \(-0.197124\pi\)
0.814295 + 0.580452i \(0.197124\pi\)
\(828\) 0 0
\(829\) 8255.00 0.345848 0.172924 − 0.984935i \(-0.444678\pi\)
0.172924 + 0.984935i \(0.444678\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 14696.0 0.611268
\(834\) 0 0
\(835\) −4848.00 −0.200925
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −36360.0 −1.49617 −0.748085 − 0.663603i \(-0.769026\pi\)
−0.748085 + 0.663603i \(0.769026\pi\)
\(840\) 0 0
\(841\) 12475.0 0.511501
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −8304.00 −0.338067
\(846\) 0 0
\(847\) −1641.00 −0.0665708
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −11868.0 −0.478061
\(852\) 0 0
\(853\) 40525.0 1.62667 0.813335 − 0.581796i \(-0.197650\pi\)
0.813335 + 0.581796i \(0.197650\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 45192.0 1.80132 0.900659 − 0.434527i \(-0.143084\pi\)
0.900659 + 0.434527i \(0.143084\pi\)
\(858\) 0 0
\(859\) 4425.00 0.175761 0.0878807 − 0.996131i \(-0.471991\pi\)
0.0878807 + 0.996131i \(0.471991\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −32380.0 −1.27721 −0.638603 − 0.769537i \(-0.720487\pi\)
−0.638603 + 0.769537i \(0.720487\pi\)
\(864\) 0 0
\(865\) 7680.00 0.301882
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 24556.0 0.958579
\(870\) 0 0
\(871\) −2827.00 −0.109976
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −2808.00 −0.108489
\(876\) 0 0
\(877\) 32565.0 1.25387 0.626934 − 0.779073i \(-0.284310\pi\)
0.626934 + 0.779073i \(0.284310\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −40052.0 −1.53165 −0.765826 − 0.643047i \(-0.777670\pi\)
−0.765826 + 0.643047i \(0.777670\pi\)
\(882\) 0 0
\(883\) −26891.0 −1.02486 −0.512432 − 0.858728i \(-0.671255\pi\)
−0.512432 + 0.858728i \(0.671255\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 46168.0 1.74765 0.873827 − 0.486236i \(-0.161631\pi\)
0.873827 + 0.486236i \(0.161631\pi\)
\(888\) 0 0
\(889\) −4380.00 −0.165242
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 4524.00 0.169530
\(894\) 0 0
\(895\) 12960.0 0.484028
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 22272.0 0.826266
\(900\) 0 0
\(901\) 17248.0 0.637752
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −18180.0 −0.667761
\(906\) 0 0
\(907\) −47449.0 −1.73707 −0.868533 − 0.495632i \(-0.834936\pi\)
−0.868533 + 0.495632i \(0.834936\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 8528.00 0.310148 0.155074 − 0.987903i \(-0.450438\pi\)
0.155074 + 0.987903i \(0.450438\pi\)
\(912\) 0 0
\(913\) −9184.00 −0.332909
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1392.00 0.0501286
\(918\) 0 0
\(919\) −45596.0 −1.63664 −0.818321 − 0.574762i \(-0.805095\pi\)
−0.818321 + 0.574762i \(0.805095\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 11000.0 0.392275
\(924\) 0 0
\(925\) −7521.00 −0.267339
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −2744.00 −0.0969082 −0.0484541 − 0.998825i \(-0.515429\pi\)
−0.0484541 + 0.998825i \(0.515429\pi\)
\(930\) 0 0
\(931\) 9686.00 0.340973
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −4928.00 −0.172367
\(936\) 0 0
\(937\) −10389.0 −0.362213 −0.181107 − 0.983463i \(-0.557968\pi\)
−0.181107 + 0.983463i \(0.557968\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −55188.0 −1.91188 −0.955939 − 0.293565i \(-0.905158\pi\)
−0.955939 + 0.293565i \(0.905158\pi\)
\(942\) 0 0
\(943\) 66048.0 2.28083
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −22044.0 −0.756424 −0.378212 − 0.925719i \(-0.623461\pi\)
−0.378212 + 0.925719i \(0.623461\pi\)
\(948\) 0 0
\(949\) −3949.00 −0.135079
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −4140.00 −0.140722 −0.0703608 − 0.997522i \(-0.522415\pi\)
−0.0703608 + 0.997522i \(0.522415\pi\)
\(954\) 0 0
\(955\) −8592.00 −0.291132
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −108.000 −0.00363660
\(960\) 0 0
\(961\) −16335.0 −0.548320
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −3364.00 −0.112219
\(966\) 0 0
\(967\) −18139.0 −0.603217 −0.301609 − 0.953432i \(-0.597524\pi\)
−0.301609 + 0.953432i \(0.597524\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −23340.0 −0.771386 −0.385693 − 0.922627i \(-0.626038\pi\)
−0.385693 + 0.922627i \(0.626038\pi\)
\(972\) 0 0
\(973\) 3447.00 0.113572
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 39096.0 1.28024 0.640119 − 0.768276i \(-0.278885\pi\)
0.640119 + 0.768276i \(0.278885\pi\)
\(978\) 0 0
\(979\) 44016.0 1.43693
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −1092.00 −0.0354317 −0.0177159 − 0.999843i \(-0.505639\pi\)
−0.0177159 + 0.999843i \(0.505639\pi\)
\(984\) 0 0
\(985\) 6096.00 0.197193
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 56416.0 1.81388
\(990\) 0 0
\(991\) 5887.00 0.188705 0.0943525 − 0.995539i \(-0.469922\pi\)
0.0943525 + 0.995539i \(0.469922\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −21284.0 −0.678139
\(996\) 0 0
\(997\) 35426.0 1.12533 0.562664 − 0.826685i \(-0.309776\pi\)
0.562664 + 0.826685i \(0.309776\pi\)
\(998\) 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 1728.4.a.x.1.1 1
3.2 odd 2 1728.4.a.j.1.1 1
4.3 odd 2 1728.4.a.w.1.1 1
8.3 odd 2 432.4.a.d.1.1 1
8.5 even 2 216.4.a.a.1.1 ✓ 1
12.11 even 2 1728.4.a.i.1.1 1
24.5 odd 2 216.4.a.d.1.1 yes 1
24.11 even 2 432.4.a.k.1.1 1
72.5 odd 6 648.4.i.d.217.1 2
72.13 even 6 648.4.i.i.217.1 2
72.29 odd 6 648.4.i.d.433.1 2
72.61 even 6 648.4.i.i.433.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
216.4.a.a.1.1 ✓ 1 8.5 even 2
216.4.a.d.1.1 yes 1 24.5 odd 2
432.4.a.d.1.1 1 8.3 odd 2
432.4.a.k.1.1 1 24.11 even 2
648.4.i.d.217.1 2 72.5 odd 6
648.4.i.d.433.1 2 72.29 odd 6
648.4.i.i.217.1 2 72.13 even 6
648.4.i.i.433.1 2 72.61 even 6
1728.4.a.i.1.1 1 12.11 even 2
1728.4.a.j.1.1 1 3.2 odd 2
1728.4.a.w.1.1 1 4.3 odd 2
1728.4.a.x.1.1 1 1.1 even 1 trivial