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Properties

Label 3536.2.a.k.1.1
Level $3536$
Weight $2$
Character 3536.1
Self dual yes
Analytic conductor $28.235$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3536,2,Mod(1,3536)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3536, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3536.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3536 = 2^{4} \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3536.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: \(28.2351021547\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1768)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3536.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{3} +2.00000 q^{5} -2.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{3} +2.00000 q^{5} -2.00000 q^{7} +1.00000 q^{9} -4.00000 q^{11} -1.00000 q^{13} +4.00000 q^{15} -1.00000 q^{17} -4.00000 q^{19} -4.00000 q^{21} -4.00000 q^{23} -1.00000 q^{25} -4.00000 q^{27} -2.00000 q^{31} -8.00000 q^{33} -4.00000 q^{35} +2.00000 q^{37} -2.00000 q^{39} -4.00000 q^{41} +4.00000 q^{43} +2.00000 q^{45} -3.00000 q^{49} -2.00000 q^{51} -6.00000 q^{53} -8.00000 q^{55} -8.00000 q^{57} +12.0000 q^{59} -2.00000 q^{63} -2.00000 q^{65} -12.0000 q^{67} -8.00000 q^{69} -10.0000 q^{71} +16.0000 q^{73} -2.00000 q^{75} +8.00000 q^{77} -11.0000 q^{81} +4.00000 q^{83} -2.00000 q^{85} +14.0000 q^{89} +2.00000 q^{91} -4.00000 q^{93} -8.00000 q^{95} +8.00000 q^{97} -4.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.00000 1.15470 0.577350 − 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 0 0
\(5\) 2.00000 0.894427 0.447214 − 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 4.00000 1.03280
\(16\) 0 0
\(17\) −1.00000 −0.242536
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 − 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 − 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) −8.00000 −1.39262
\(34\) 0 0
\(35\) −4.00000 −0.676123
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −4.00000 −0.624695 −0.312348 − 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 2.00000 0.298142
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) −8.00000 −1.07872
\(56\) 0 0
\(57\) −8.00000 −1.05963
\(58\) 0 0
\(59\) 12.0000 1.56227 0.781133 − 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) −2.00000 −0.251976
\(64\) 0 0
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 − 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 0 0
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) −10.0000 −1.18678 −0.593391 − 0.804914i \(-0.702211\pi\)
−0.593391 + 0.804914i \(0.702211\pi\)
\(72\) 0 0
\(73\) 16.0000 1.87266 0.936329 − 0.351123i \(-0.114200\pi\)
0.936329 + 0.351123i \(0.114200\pi\)
\(74\) 0 0
\(75\) −2.00000 −0.230940
\(76\) 0 0
\(77\) 8.00000 0.911685
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 4.00000 0.439057 0.219529 − 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 14.0000 1.48400 0.741999 − 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) −4.00000 −0.414781
\(94\) 0 0
\(95\) −8.00000 −0.820783
\(96\) 0 0
\(97\) 8.00000 0.812277 0.406138 − 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) 10.0000 0.995037 0.497519 − 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 − 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 0 0
\(105\) −8.00000 −0.780720
\(106\) 0 0
\(107\) −14.0000 −1.35343 −0.676716 − 0.736245i \(-0.736597\pi\)
−0.676716 + 0.736245i \(0.736597\pi\)
\(108\) 0 0
\(109\) −14.0000 −1.34096 −0.670478 − 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) 4.00000 0.379663
\(112\) 0 0
\(113\) 14.0000 1.31701 0.658505 − 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 0 0
\(115\) −8.00000 −0.746004
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) 2.00000 0.183340
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) −8.00000 −0.721336
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 − 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 0 0
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) −18.0000 −1.57267 −0.786334 − 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) 0 0
\(133\) 8.00000 0.693688
\(134\) 0 0
\(135\) −8.00000 −0.688530
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 − 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) 18.0000 1.52674 0.763370 − 0.645961i \(-0.223543\pi\)
0.763370 + 0.645961i \(0.223543\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −6.00000 −0.494872
\(148\) 0 0
\(149\) −10.0000 −0.819232 −0.409616 − 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) 20.0000 1.62758 0.813788 − 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) 0 0
\(153\) −1.00000 −0.0808452
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 − 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 0 0
\(159\) −12.0000 −0.951662
\(160\) 0 0
\(161\) 8.00000 0.630488
\(162\) 0 0
\(163\) 20.0000 1.56652 0.783260 − 0.621694i \(-0.213555\pi\)
0.783260 + 0.621694i \(0.213555\pi\)
\(164\) 0 0
\(165\) −16.0000 −1.24560
\(166\) 0 0
\(167\) 10.0000 0.773823 0.386912 − 0.922117i \(-0.373542\pi\)
0.386912 + 0.922117i \(0.373542\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 0 0
\(173\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 2.00000 0.151186
\(176\) 0 0
\(177\) 24.0000 1.80395
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 − 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 16.0000 1.18927 0.594635 − 0.803996i \(-0.297296\pi\)
0.594635 + 0.803996i \(0.297296\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 4.00000 0.294086
\(186\) 0 0
\(187\) 4.00000 0.292509
\(188\) 0 0
\(189\) 8.00000 0.581914
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 − 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) 4.00000 0.287926 0.143963 − 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) 0 0
\(195\) −4.00000 −0.286446
\(196\) 0 0
\(197\) −10.0000 −0.712470 −0.356235 − 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) 0 0
\(199\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) −24.0000 −1.69283
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −8.00000 −0.558744
\(206\) 0 0
\(207\) −4.00000 −0.278019
\(208\) 0 0
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) −10.0000 −0.688428 −0.344214 − 0.938891i \(-0.611855\pi\)
−0.344214 + 0.938891i \(0.611855\pi\)
\(212\) 0 0
\(213\) −20.0000 −1.37038
\(214\) 0 0
\(215\) 8.00000 0.545595
\(216\) 0 0
\(217\) 4.00000 0.271538
\(218\) 0 0
\(219\) 32.0000 2.16236
\(220\) 0 0
\(221\) 1.00000 0.0672673
\(222\) 0 0
\(223\) 8.00000 0.535720 0.267860 − 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 0 0
\(227\) −8.00000 −0.530979 −0.265489 − 0.964114i \(-0.585534\pi\)
−0.265489 + 0.964114i \(0.585534\pi\)
\(228\) 0 0
\(229\) 6.00000 0.396491 0.198246 − 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 0 0
\(231\) 16.0000 1.05272
\(232\) 0 0
\(233\) −2.00000 −0.131024 −0.0655122 − 0.997852i \(-0.520868\pi\)
−0.0655122 + 0.997852i \(0.520868\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 − 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 − 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) 0 0
\(243\) −10.0000 −0.641500
\(244\) 0 0
\(245\) −6.00000 −0.383326
\(246\) 0 0
\(247\) 4.00000 0.254514
\(248\) 0 0
\(249\) 8.00000 0.506979
\(250\) 0 0
\(251\) 16.0000 1.00991 0.504956 − 0.863145i \(-0.331509\pi\)
0.504956 + 0.863145i \(0.331509\pi\)
\(252\) 0 0
\(253\) 16.0000 1.00591
\(254\) 0 0
\(255\) −4.00000 −0.250490
\(256\) 0 0
\(257\) 2.00000 0.124757 0.0623783 − 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 0 0
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −8.00000 −0.493301 −0.246651 − 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) 0 0
\(265\) −12.0000 −0.737154
\(266\) 0 0
\(267\) 28.0000 1.71357
\(268\) 0 0
\(269\) −8.00000 −0.487769 −0.243884 − 0.969804i \(-0.578422\pi\)
−0.243884 + 0.969804i \(0.578422\pi\)
\(270\) 0 0
\(271\) −20.0000 −1.21491 −0.607457 − 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) 0 0
\(273\) 4.00000 0.242091
\(274\) 0 0
\(275\) 4.00000 0.241209
\(276\) 0 0
\(277\) −24.0000 −1.44202 −0.721010 − 0.692925i \(-0.756322\pi\)
−0.721010 + 0.692925i \(0.756322\pi\)
\(278\) 0 0
\(279\) −2.00000 −0.119737
\(280\) 0 0
\(281\) −10.0000 −0.596550 −0.298275 − 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 0 0
\(283\) 2.00000 0.118888 0.0594438 − 0.998232i \(-0.481067\pi\)
0.0594438 + 0.998232i \(0.481067\pi\)
\(284\) 0 0
\(285\) −16.0000 −0.947758
\(286\) 0 0
\(287\) 8.00000 0.472225
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 16.0000 0.937937
\(292\) 0 0
\(293\) −6.00000 −0.350524 −0.175262 − 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) 24.0000 1.39733
\(296\) 0 0
\(297\) 16.0000 0.928414
\(298\) 0 0
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) 0 0
\(303\) 20.0000 1.14897
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −4.00000 −0.228292 −0.114146 − 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) 0 0
\(309\) −32.0000 −1.82042
\(310\) 0 0
\(311\) 4.00000 0.226819 0.113410 − 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 − 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 0 0
\(315\) −4.00000 −0.225374
\(316\) 0 0
\(317\) 22.0000 1.23564 0.617822 − 0.786318i \(-0.288015\pi\)
0.617822 + 0.786318i \(0.288015\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −28.0000 −1.56281
\(322\) 0 0
\(323\) 4.00000 0.222566
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) 0 0
\(327\) −28.0000 −1.54840
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 − 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) 0 0
\(333\) 2.00000 0.109599
\(334\) 0 0
\(335\) −24.0000 −1.31126
\(336\) 0 0
\(337\) −6.00000 −0.326841 −0.163420 − 0.986557i \(-0.552253\pi\)
−0.163420 + 0.986557i \(0.552253\pi\)
\(338\) 0 0
\(339\) 28.0000 1.52075
\(340\) 0 0
\(341\) 8.00000 0.433224
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 0 0
\(345\) −16.0000 −0.861411
\(346\) 0 0
\(347\) 18.0000 0.966291 0.483145 − 0.875540i \(-0.339494\pi\)
0.483145 + 0.875540i \(0.339494\pi\)
\(348\) 0 0
\(349\) −30.0000 −1.60586 −0.802932 − 0.596071i \(-0.796728\pi\)
−0.802932 + 0.596071i \(0.796728\pi\)
\(350\) 0 0
\(351\) 4.00000 0.213504
\(352\) 0 0
\(353\) −22.0000 −1.17094 −0.585471 − 0.810693i \(-0.699090\pi\)
−0.585471 + 0.810693i \(0.699090\pi\)
\(354\) 0 0
\(355\) −20.0000 −1.06149
\(356\) 0 0
\(357\) 4.00000 0.211702
\(358\) 0 0
\(359\) −12.0000 −0.633336 −0.316668 − 0.948536i \(-0.602564\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 10.0000 0.524864
\(364\) 0 0
\(365\) 32.0000 1.67496
\(366\) 0 0
\(367\) −12.0000 −0.626395 −0.313197 − 0.949688i \(-0.601400\pi\)
−0.313197 + 0.949688i \(0.601400\pi\)
\(368\) 0 0
\(369\) −4.00000 −0.208232
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) 0 0
\(373\) −22.0000 −1.13912 −0.569558 − 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) 0 0
\(375\) −24.0000 −1.23935
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 36.0000 1.84920 0.924598 − 0.380945i \(-0.124401\pi\)
0.924598 + 0.380945i \(0.124401\pi\)
\(380\) 0 0
\(381\) −32.0000 −1.63941
\(382\) 0 0
\(383\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 16.0000 0.815436
\(386\) 0 0
\(387\) 4.00000 0.203331
\(388\) 0 0
\(389\) 30.0000 1.52106 0.760530 − 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) 0 0
\(393\) −36.0000 −1.81596
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −14.0000 −0.702640 −0.351320 − 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 0 0
\(399\) 16.0000 0.801002
\(400\) 0 0
\(401\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 2.00000 0.0996271
\(404\) 0 0
\(405\) −22.0000 −1.09319
\(406\) 0 0
\(407\) −8.00000 −0.396545
\(408\) 0 0
\(409\) 2.00000 0.0988936 0.0494468 − 0.998777i \(-0.484254\pi\)
0.0494468 + 0.998777i \(0.484254\pi\)
\(410\) 0 0
\(411\) 12.0000 0.591916
\(412\) 0 0
\(413\) −24.0000 −1.18096
\(414\) 0 0
\(415\) 8.00000 0.392705
\(416\) 0 0
\(417\) 36.0000 1.76293
\(418\) 0 0
\(419\) −30.0000 −1.46560 −0.732798 − 0.680446i \(-0.761786\pi\)
−0.732798 + 0.680446i \(0.761786\pi\)
\(420\) 0 0
\(421\) −34.0000 −1.65706 −0.828529 − 0.559946i \(-0.810822\pi\)
−0.828529 + 0.559946i \(0.810822\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.00000 0.0485071
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 8.00000 0.386244
\(430\) 0 0
\(431\) −10.0000 −0.481683 −0.240842 − 0.970564i \(-0.577423\pi\)
−0.240842 + 0.970564i \(0.577423\pi\)
\(432\) 0 0
\(433\) −14.0000 −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
\(437\) 16.0000 0.765384
\(438\) 0 0
\(439\) −4.00000 −0.190910 −0.0954548 − 0.995434i \(-0.530431\pi\)
−0.0954548 + 0.995434i \(0.530431\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) −36.0000 −1.71041 −0.855206 − 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) 0 0
\(445\) 28.0000 1.32733
\(446\) 0 0
\(447\) −20.0000 −0.945968
\(448\) 0 0
\(449\) −24.0000 −1.13263 −0.566315 − 0.824189i \(-0.691631\pi\)
−0.566315 + 0.824189i \(0.691631\pi\)
\(450\) 0 0
\(451\) 16.0000 0.753411
\(452\) 0 0
\(453\) 40.0000 1.87936
\(454\) 0 0
\(455\) 4.00000 0.187523
\(456\) 0 0
\(457\) 10.0000 0.467780 0.233890 − 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 0 0
\(459\) 4.00000 0.186704
\(460\) 0 0
\(461\) 6.00000 0.279448 0.139724 − 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 0 0
\(463\) −12.0000 −0.557687 −0.278844 − 0.960337i \(-0.589951\pi\)
−0.278844 + 0.960337i \(0.589951\pi\)
\(464\) 0 0
\(465\) −8.00000 −0.370991
\(466\) 0 0
\(467\) 8.00000 0.370196 0.185098 − 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 0 0
\(469\) 24.0000 1.10822
\(470\) 0 0
\(471\) −4.00000 −0.184310
\(472\) 0 0
\(473\) −16.0000 −0.735681
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) 0 0
\(479\) 6.00000 0.274147 0.137073 − 0.990561i \(-0.456230\pi\)
0.137073 + 0.990561i \(0.456230\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) 0 0
\(483\) 16.0000 0.728025
\(484\) 0 0
\(485\) 16.0000 0.726523
\(486\) 0 0
\(487\) −2.00000 −0.0906287 −0.0453143 − 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) 0 0
\(489\) 40.0000 1.80886
\(490\) 0 0
\(491\) 24.0000 1.08310 0.541552 − 0.840667i \(-0.317837\pi\)
0.541552 + 0.840667i \(0.317837\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −8.00000 −0.359573
\(496\) 0 0
\(497\) 20.0000 0.897123
\(498\) 0 0
\(499\) 24.0000 1.07439 0.537194 − 0.843459i \(-0.319484\pi\)
0.537194 + 0.843459i \(0.319484\pi\)
\(500\) 0 0
\(501\) 20.0000 0.893534
\(502\) 0 0
\(503\) 4.00000 0.178351 0.0891756 − 0.996016i \(-0.471577\pi\)
0.0891756 + 0.996016i \(0.471577\pi\)
\(504\) 0 0
\(505\) 20.0000 0.889988
\(506\) 0 0
\(507\) 2.00000 0.0888231
\(508\) 0 0
\(509\) 26.0000 1.15243 0.576215 − 0.817298i \(-0.304529\pi\)
0.576215 + 0.817298i \(0.304529\pi\)
\(510\) 0 0
\(511\) −32.0000 −1.41560
\(512\) 0 0
\(513\) 16.0000 0.706417
\(514\) 0 0
\(515\) −32.0000 −1.41009
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 − 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) 4.00000 0.174908 0.0874539 − 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) 0 0
\(525\) 4.00000 0.174574
\(526\) 0 0
\(527\) 2.00000 0.0871214
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) 4.00000 0.173259
\(534\) 0 0
\(535\) −28.0000 −1.21055
\(536\) 0 0
\(537\) −24.0000 −1.03568
\(538\) 0 0
\(539\) 12.0000 0.516877
\(540\) 0 0
\(541\) −42.0000 −1.80572 −0.902861 − 0.429934i \(-0.858537\pi\)
−0.902861 + 0.429934i \(0.858537\pi\)
\(542\) 0 0
\(543\) 32.0000 1.37325
\(544\) 0 0
\(545\) −28.0000 −1.19939
\(546\) 0 0
\(547\) −42.0000 −1.79579 −0.897895 − 0.440209i \(-0.854904\pi\)
−0.897895 + 0.440209i \(0.854904\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 8.00000 0.339581
\(556\) 0 0
\(557\) 34.0000 1.44063 0.720313 − 0.693649i \(-0.243998\pi\)
0.720313 + 0.693649i \(0.243998\pi\)
\(558\) 0 0
\(559\) −4.00000 −0.169182
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) 0 0
\(563\) 24.0000 1.01148 0.505740 − 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 0 0
\(565\) 28.0000 1.17797
\(566\) 0 0
\(567\) 22.0000 0.923913
\(568\) 0 0
\(569\) −26.0000 −1.08998 −0.544988 − 0.838444i \(-0.683466\pi\)
−0.544988 + 0.838444i \(0.683466\pi\)
\(570\) 0 0
\(571\) 2.00000 0.0836974 0.0418487 − 0.999124i \(-0.486675\pi\)
0.0418487 + 0.999124i \(0.486675\pi\)
\(572\) 0 0
\(573\) 16.0000 0.668410
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) −2.00000 −0.0832611 −0.0416305 − 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) 0 0
\(579\) 8.00000 0.332469
\(580\) 0 0
\(581\) −8.00000 −0.331896
\(582\) 0 0
\(583\) 24.0000 0.993978
\(584\) 0 0
\(585\) −2.00000 −0.0826898
\(586\) 0 0
\(587\) −12.0000 −0.495293 −0.247647 − 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) −20.0000 −0.822690
\(592\) 0 0
\(593\) 14.0000 0.574911 0.287456 − 0.957794i \(-0.407191\pi\)
0.287456 + 0.957794i \(0.407191\pi\)
\(594\) 0 0
\(595\) 4.00000 0.163984
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 8.00000 0.326871 0.163436 − 0.986554i \(-0.447742\pi\)
0.163436 + 0.986554i \(0.447742\pi\)
\(600\) 0 0
\(601\) −42.0000 −1.71322 −0.856608 − 0.515968i \(-0.827432\pi\)
−0.856608 + 0.515968i \(0.827432\pi\)
\(602\) 0 0
\(603\) −12.0000 −0.488678
\(604\) 0 0
\(605\) 10.0000 0.406558
\(606\) 0 0
\(607\) −20.0000 −0.811775 −0.405887 − 0.913923i \(-0.633038\pi\)
−0.405887 + 0.913923i \(0.633038\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −30.0000 −1.21169 −0.605844 − 0.795583i \(-0.707165\pi\)
−0.605844 + 0.795583i \(0.707165\pi\)
\(614\) 0 0
\(615\) −16.0000 −0.645182
\(616\) 0 0
\(617\) 8.00000 0.322068 0.161034 − 0.986949i \(-0.448517\pi\)
0.161034 + 0.986949i \(0.448517\pi\)
\(618\) 0 0
\(619\) 32.0000 1.28619 0.643094 − 0.765787i \(-0.277650\pi\)
0.643094 + 0.765787i \(0.277650\pi\)
\(620\) 0 0
\(621\) 16.0000 0.642058
\(622\) 0 0
\(623\) −28.0000 −1.12180
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 32.0000 1.27796
\(628\) 0 0
\(629\) −2.00000 −0.0797452
\(630\) 0 0
\(631\) −12.0000 −0.477712 −0.238856 − 0.971055i \(-0.576772\pi\)
−0.238856 + 0.971055i \(0.576772\pi\)
\(632\) 0 0
\(633\) −20.0000 −0.794929
\(634\) 0 0
\(635\) −32.0000 −1.26988
\(636\) 0 0
\(637\) 3.00000 0.118864
\(638\) 0 0
\(639\) −10.0000 −0.395594
\(640\) 0 0
\(641\) 14.0000 0.552967 0.276483 − 0.961019i \(-0.410831\pi\)
0.276483 + 0.961019i \(0.410831\pi\)
\(642\) 0 0
\(643\) 20.0000 0.788723 0.394362 − 0.918955i \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(644\) 0 0
\(645\) 16.0000 0.629999
\(646\) 0 0
\(647\) −24.0000 −0.943537 −0.471769 − 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 0 0
\(649\) −48.0000 −1.88416
\(650\) 0 0
\(651\) 8.00000 0.313545
\(652\) 0 0
\(653\) −24.0000 −0.939193 −0.469596 − 0.882881i \(-0.655601\pi\)
−0.469596 + 0.882881i \(0.655601\pi\)
\(654\) 0 0
\(655\) −36.0000 −1.40664
\(656\) 0 0
\(657\) 16.0000 0.624219
\(658\) 0 0
\(659\) 8.00000 0.311636 0.155818 − 0.987786i \(-0.450199\pi\)
0.155818 + 0.987786i \(0.450199\pi\)
\(660\) 0 0
\(661\) 38.0000 1.47803 0.739014 − 0.673690i \(-0.235292\pi\)
0.739014 + 0.673690i \(0.235292\pi\)
\(662\) 0 0
\(663\) 2.00000 0.0776736
\(664\) 0 0
\(665\) 16.0000 0.620453
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 6.00000 0.231283 0.115642 − 0.993291i \(-0.463108\pi\)
0.115642 + 0.993291i \(0.463108\pi\)
\(674\) 0 0
\(675\) 4.00000 0.153960
\(676\) 0 0
\(677\) 24.0000 0.922395 0.461197 − 0.887298i \(-0.347420\pi\)
0.461197 + 0.887298i \(0.347420\pi\)
\(678\) 0 0
\(679\) −16.0000 −0.614024
\(680\) 0 0
\(681\) −16.0000 −0.613121
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 − 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) 12.0000 0.458496
\(686\) 0 0
\(687\) 12.0000 0.457829
\(688\) 0 0
\(689\) 6.00000 0.228582
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 − 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) 0 0
\(693\) 8.00000 0.303895
\(694\) 0 0
\(695\) 36.0000 1.36556
\(696\) 0 0
\(697\) 4.00000 0.151511
\(698\) 0 0
\(699\) −4.00000 −0.151294
\(700\) 0 0
\(701\) −22.0000 −0.830929 −0.415464 − 0.909610i \(-0.636381\pi\)
−0.415464 + 0.909610i \(0.636381\pi\)
\(702\) 0 0
\(703\) −8.00000 −0.301726
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −20.0000 −0.752177
\(708\) 0 0
\(709\) −26.0000 −0.976450 −0.488225 − 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 8.00000 0.299602
\(714\) 0 0
\(715\) 8.00000 0.299183
\(716\) 0 0
\(717\) −24.0000 −0.896296
\(718\) 0 0
\(719\) 36.0000 1.34257 0.671287 − 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) 0 0
\(721\) 32.0000 1.19174
\(722\) 0 0
\(723\) 16.0000 0.595046
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −16.0000 −0.593407 −0.296704 − 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −4.00000 −0.147945
\(732\) 0 0
\(733\) −14.0000 −0.517102 −0.258551 − 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) 0 0
\(735\) −12.0000 −0.442627
\(736\) 0 0
\(737\) 48.0000 1.76810
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 − 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) 8.00000 0.293887
\(742\) 0 0
\(743\) −26.0000 −0.953847 −0.476924 − 0.878945i \(-0.658248\pi\)
−0.476924 + 0.878945i \(0.658248\pi\)
\(744\) 0 0
\(745\) −20.0000 −0.732743
\(746\) 0 0
\(747\) 4.00000 0.146352
\(748\) 0 0
\(749\) 28.0000 1.02310
\(750\) 0 0
\(751\) 16.0000 0.583848 0.291924 − 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) 0 0
\(753\) 32.0000 1.16614
\(754\) 0 0
\(755\) 40.0000 1.45575
\(756\) 0 0
\(757\) −10.0000 −0.363456 −0.181728 − 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 0 0
\(759\) 32.0000 1.16153
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 − 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 0 0
\(763\) 28.0000 1.01367
\(764\) 0 0
\(765\) −2.00000 −0.0723102
\(766\) 0 0
\(767\) −12.0000 −0.433295
\(768\) 0 0
\(769\) 46.0000 1.65880 0.829401 − 0.558653i \(-0.188682\pi\)
0.829401 + 0.558653i \(0.188682\pi\)
\(770\) 0 0
\(771\) 4.00000 0.144056
\(772\) 0 0
\(773\) 42.0000 1.51064 0.755318 − 0.655359i \(-0.227483\pi\)
0.755318 + 0.655359i \(0.227483\pi\)
\(774\) 0 0
\(775\) 2.00000 0.0718421
\(776\) 0 0
\(777\) −8.00000 −0.286998
\(778\) 0 0
\(779\) 16.0000 0.573259
\(780\) 0 0
\(781\) 40.0000 1.43131
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −4.00000 −0.142766
\(786\) 0 0
\(787\) 12.0000 0.427754 0.213877 − 0.976861i \(-0.431391\pi\)
0.213877 + 0.976861i \(0.431391\pi\)
\(788\) 0 0
\(789\) −16.0000 −0.569615
\(790\) 0 0
\(791\) −28.0000 −0.995565
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −24.0000 −0.851192
\(796\) 0 0
\(797\) 2.00000 0.0708436 0.0354218 − 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 14.0000 0.494666
\(802\) 0 0
\(803\) −64.0000 −2.25851
\(804\) 0 0
\(805\) 16.0000 0.563926
\(806\) 0 0
\(807\) −16.0000 −0.563227
\(808\) 0 0
\(809\) −30.0000 −1.05474 −0.527372 − 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) 16.0000 0.561836 0.280918 − 0.959732i \(-0.409361\pi\)
0.280918 + 0.959732i \(0.409361\pi\)
\(812\) 0 0
\(813\) −40.0000 −1.40286
\(814\) 0 0
\(815\) 40.0000 1.40114
\(816\) 0 0
\(817\) −16.0000 −0.559769
\(818\) 0 0
\(819\) 2.00000 0.0698857
\(820\) 0 0
\(821\) −38.0000 −1.32621 −0.663105 − 0.748527i \(-0.730762\pi\)
−0.663105 + 0.748527i \(0.730762\pi\)
\(822\) 0 0
\(823\) 8.00000 0.278862 0.139431 − 0.990232i \(-0.455473\pi\)
0.139431 + 0.990232i \(0.455473\pi\)
\(824\) 0 0
\(825\) 8.00000 0.278524
\(826\) 0 0
\(827\) 28.0000 0.973655 0.486828 − 0.873498i \(-0.338154\pi\)
0.486828 + 0.873498i \(0.338154\pi\)
\(828\) 0 0
\(829\) 30.0000 1.04194 0.520972 − 0.853574i \(-0.325570\pi\)
0.520972 + 0.853574i \(0.325570\pi\)
\(830\) 0 0
\(831\) −48.0000 −1.66510
\(832\) 0 0
\(833\) 3.00000 0.103944
\(834\) 0 0
\(835\) 20.0000 0.692129
\(836\) 0 0
\(837\) 8.00000 0.276520
\(838\) 0 0
\(839\) 42.0000 1.45000 0.725001 − 0.688748i \(-0.241839\pi\)
0.725001 + 0.688748i \(0.241839\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) −20.0000 −0.688837
\(844\) 0 0
\(845\) 2.00000 0.0688021
\(846\) 0 0
\(847\) −10.0000 −0.343604
\(848\) 0 0
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) −8.00000 −0.274236
\(852\) 0 0
\(853\) −50.0000 −1.71197 −0.855984 − 0.517003i \(-0.827048\pi\)
−0.855984 + 0.517003i \(0.827048\pi\)
\(854\) 0 0
\(855\) −8.00000 −0.273594
\(856\) 0 0
\(857\) 18.0000 0.614868 0.307434 − 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) 28.0000 0.955348 0.477674 − 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) 0 0
\(861\) 16.0000 0.545279
\(862\) 0 0
\(863\) −32.0000 −1.08929 −0.544646 − 0.838666i \(-0.683336\pi\)
−0.544646 + 0.838666i \(0.683336\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 2.00000 0.0679236
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 12.0000 0.406604
\(872\) 0 0
\(873\) 8.00000 0.270759
\(874\) 0 0
\(875\) 24.0000 0.811348
\(876\) 0 0
\(877\) 38.0000 1.28317 0.641584 − 0.767052i \(-0.278277\pi\)
0.641584 + 0.767052i \(0.278277\pi\)
\(878\) 0 0
\(879\) −12.0000 −0.404750
\(880\) 0 0
\(881\) −18.0000 −0.606435 −0.303218 − 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) 36.0000 1.21150 0.605748 − 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) 0 0
\(885\) 48.0000 1.61350
\(886\) 0 0
\(887\) 36.0000 1.20876 0.604381 − 0.796696i \(-0.293421\pi\)
0.604381 + 0.796696i \(0.293421\pi\)
\(888\) 0 0
\(889\) 32.0000 1.07325
\(890\) 0 0
\(891\) 44.0000 1.47406
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −24.0000 −0.802232
\(896\) 0 0
\(897\) 8.00000 0.267112
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 6.00000 0.199889
\(902\) 0 0
\(903\) −16.0000 −0.532447
\(904\) 0 0
\(905\) 32.0000 1.06372
\(906\) 0 0
\(907\) 50.0000 1.66022 0.830111 − 0.557598i \(-0.188277\pi\)
0.830111 + 0.557598i \(0.188277\pi\)
\(908\) 0 0
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) −32.0000 −1.06021 −0.530104 − 0.847933i \(-0.677847\pi\)
−0.530104 + 0.847933i \(0.677847\pi\)
\(912\) 0 0
\(913\) −16.0000 −0.529523
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 36.0000 1.18882
\(918\) 0 0
\(919\) 24.0000 0.791687 0.395843 − 0.918318i \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(920\) 0 0
\(921\) −8.00000 −0.263609
\(922\) 0 0
\(923\) 10.0000 0.329154
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) 0 0
\(927\) −16.0000 −0.525509
\(928\) 0 0
\(929\) 40.0000 1.31236 0.656179 − 0.754606i \(-0.272172\pi\)
0.656179 + 0.754606i \(0.272172\pi\)
\(930\) 0 0
\(931\) 12.0000 0.393284
\(932\) 0 0
\(933\) 8.00000 0.261908
\(934\) 0 0
\(935\) 8.00000 0.261628
\(936\) 0 0
\(937\) 22.0000 0.718709 0.359354 − 0.933201i \(-0.382997\pi\)
0.359354 + 0.933201i \(0.382997\pi\)
\(938\) 0 0
\(939\) −12.0000 −0.391605
\(940\) 0 0
\(941\) 14.0000 0.456387 0.228193 − 0.973616i \(-0.426718\pi\)
0.228193 + 0.973616i \(0.426718\pi\)
\(942\) 0 0
\(943\) 16.0000 0.521032
\(944\) 0 0
\(945\) 16.0000 0.520480
\(946\) 0 0
\(947\) 36.0000 1.16984 0.584921 − 0.811090i \(-0.301125\pi\)
0.584921 + 0.811090i \(0.301125\pi\)
\(948\) 0 0
\(949\) −16.0000 −0.519382
\(950\) 0 0
\(951\) 44.0000 1.42680
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 − 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 0 0
\(955\) 16.0000 0.517748
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −12.0000 −0.387500
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) −14.0000 −0.451144
\(964\) 0 0
\(965\) 8.00000 0.257529
\(966\) 0 0
\(967\) 4.00000 0.128631 0.0643157 − 0.997930i \(-0.479514\pi\)
0.0643157 + 0.997930i \(0.479514\pi\)
\(968\) 0 0
\(969\) 8.00000 0.256997
\(970\) 0 0
\(971\) 48.0000 1.54039 0.770197 − 0.637806i \(-0.220158\pi\)
0.770197 + 0.637806i \(0.220158\pi\)
\(972\) 0 0
\(973\) −36.0000 −1.15411
\(974\) 0 0
\(975\) 2.00000 0.0640513
\(976\) 0 0
\(977\) −18.0000 −0.575871 −0.287936 − 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 0 0
\(979\) −56.0000 −1.78977
\(980\) 0 0
\(981\) −14.0000 −0.446986
\(982\) 0 0
\(983\) 34.0000 1.08443 0.542216 − 0.840239i \(-0.317586\pi\)
0.542216 + 0.840239i \(0.317586\pi\)
\(984\) 0 0
\(985\) −20.0000 −0.637253
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −16.0000 −0.508770
\(990\) 0 0
\(991\) −40.0000 −1.27064 −0.635321 − 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) 0 0
\(993\) 24.0000 0.761617
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 8.00000 0.253363 0.126681 − 0.991943i \(-0.459567\pi\)
0.126681 + 0.991943i \(0.459567\pi\)
\(998\) 0 0
\(999\) −8.00000 −0.253109
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 3536.2.a.k.1.1 1
4.3 odd 2 1768.2.a.b.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1768.2.a.b.1.1 ✓ 1 4.3 odd 2
3536.2.a.k.1.1 1 1.1 even 1 trivial