Properties

Label 1456.2.a.m.1.1
Level $1456$
Weight $2$
Character 1456.1
Self dual yes
Analytic conductor $11.626$
Analytic rank $0$
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] = [1456,2,Mod(1,1456)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1456, 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("1456.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1456.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: \(11.6262185343\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 364)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1456.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +4.00000 q^{11} +1.00000 q^{13} +2.00000 q^{15} -2.00000 q^{17} +1.00000 q^{19} +2.00000 q^{21} +7.00000 q^{23} -4.00000 q^{25} -4.00000 q^{27} -5.00000 q^{29} +9.00000 q^{31} +8.00000 q^{33} +1.00000 q^{35} -2.00000 q^{37} +2.00000 q^{39} +2.00000 q^{41} -1.00000 q^{43} +1.00000 q^{45} -9.00000 q^{47} +1.00000 q^{49} -4.00000 q^{51} +3.00000 q^{53} +4.00000 q^{55} +2.00000 q^{57} +14.0000 q^{61} +1.00000 q^{63} +1.00000 q^{65} -10.0000 q^{67} +14.0000 q^{69} +14.0000 q^{71} +3.00000 q^{73} -8.00000 q^{75} +4.00000 q^{77} -5.00000 q^{79} -11.0000 q^{81} -5.00000 q^{83} -2.00000 q^{85} -10.0000 q^{87} -9.00000 q^{89} +1.00000 q^{91} +18.0000 q^{93} +1.00000 q^{95} -1.00000 q^{97} +4.00000 q^{99} +O(q^{100})\)

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
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 2.00000 0.516398
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416 0.114708 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) 7.00000 1.45960 0.729800 0.683660i \(-0.239613\pi\)
0.729800 + 0.683660i \(0.239613\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) 9.00000 1.61645 0.808224 0.588875i \(-0.200429\pi\)
0.808224 + 0.588875i \(0.200429\pi\)
\(32\) 0 0
\(33\) 8.00000 1.39262
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −9.00000 −1.31278 −0.656392 0.754420i \(-0.727918\pi\)
−0.656392 + 0.754420i \(0.727918\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) 3.00000 0.412082 0.206041 0.978543i \(-0.433942\pi\)
0.206041 + 0.978543i \(0.433942\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 14.0000 1.79252 0.896258 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) −10.0000 −1.22169 −0.610847 0.791748i \(-0.709171\pi\)
−0.610847 + 0.791748i \(0.709171\pi\)
\(68\) 0 0
\(69\) 14.0000 1.68540
\(70\) 0 0
\(71\) 14.0000 1.66149 0.830747 0.556650i \(-0.187914\pi\)
0.830747 + 0.556650i \(0.187914\pi\)
\(72\) 0 0
\(73\) 3.00000 0.351123 0.175562 0.984468i \(-0.443826\pi\)
0.175562 + 0.984468i \(0.443826\pi\)
\(74\) 0 0
\(75\) −8.00000 −0.923760
\(76\) 0 0
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) −5.00000 −0.562544 −0.281272 0.959628i \(-0.590756\pi\)
−0.281272 + 0.959628i \(0.590756\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) −5.00000 −0.548821 −0.274411 0.961613i \(-0.588483\pi\)
−0.274411 + 0.961613i \(0.588483\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) 0 0
\(87\) −10.0000 −1.07211
\(88\) 0 0
\(89\) −9.00000 −0.953998 −0.476999 0.878904i \(-0.658275\pi\)
−0.476999 + 0.878904i \(0.658275\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) 18.0000 1.86651
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) −1.00000 −0.101535 −0.0507673 0.998711i \(-0.516167\pi\)
−0.0507673 + 0.998711i \(0.516167\pi\)
\(98\) 0 0
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) 2.00000 0.195180
\(106\) 0 0
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) 0 0
\(113\) 1.00000 0.0940721 0.0470360 0.998893i \(-0.485022\pi\)
0.0470360 + 0.998893i \(0.485022\pi\)
\(114\) 0 0
\(115\) 7.00000 0.652753
\(116\) 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
\(123\) 4.00000 0.360668
\(124\) 0 0
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 0 0
\(129\) −2.00000 −0.176090
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 1.00000 0.0867110
\(134\) 0 0
\(135\) −4.00000 −0.344265
\(136\) 0 0
\(137\) −14.0000 −1.19610 −0.598050 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) 0 0
\(139\) −16.0000 −1.35710 −0.678551 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 0 0
\(141\) −18.0000 −1.51587
\(142\) 0 0
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) −5.00000 −0.415227
\(146\) 0 0
\(147\) 2.00000 0.164957
\(148\) 0 0
\(149\) 12.0000 0.983078 0.491539 0.870855i \(-0.336434\pi\)
0.491539 + 0.870855i \(0.336434\pi\)
\(150\) 0 0
\(151\) −14.0000 −1.13930 −0.569652 0.821886i \(-0.692922\pi\)
−0.569652 + 0.821886i \(0.692922\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) 9.00000 0.722897
\(156\) 0 0
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 7.00000 0.551677
\(162\) 0 0
\(163\) −24.0000 −1.87983 −0.939913 0.341415i \(-0.889094\pi\)
−0.939913 + 0.341415i \(0.889094\pi\)
\(164\) 0 0
\(165\) 8.00000 0.622799
\(166\) 0 0
\(167\) −3.00000 −0.232147 −0.116073 0.993241i \(-0.537031\pi\)
−0.116073 + 0.993241i \(0.537031\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 0 0
\(173\) 2.00000 0.152057 0.0760286 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −9.00000 −0.672692 −0.336346 0.941739i \(-0.609191\pi\)
−0.336346 + 0.941739i \(0.609191\pi\)
\(180\) 0 0
\(181\) −20.0000 −1.48659 −0.743294 0.668965i \(-0.766738\pi\)
−0.743294 + 0.668965i \(0.766738\pi\)
\(182\) 0 0
\(183\) 28.0000 2.06982
\(184\) 0 0
\(185\) −2.00000 −0.147043
\(186\) 0 0
\(187\) −8.00000 −0.585018
\(188\) 0 0
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) 24.0000 1.73658 0.868290 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) 0 0
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 0 0
\(195\) 2.00000 0.143223
\(196\) 0 0
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) 14.0000 0.992434 0.496217 0.868199i \(-0.334722\pi\)
0.496217 + 0.868199i \(0.334722\pi\)
\(200\) 0 0
\(201\) −20.0000 −1.41069
\(202\) 0 0
\(203\) −5.00000 −0.350931
\(204\) 0 0
\(205\) 2.00000 0.139686
\(206\) 0 0
\(207\) 7.00000 0.486534
\(208\) 0 0
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) 23.0000 1.58339 0.791693 0.610920i \(-0.209200\pi\)
0.791693 + 0.610920i \(0.209200\pi\)
\(212\) 0 0
\(213\) 28.0000 1.91853
\(214\) 0 0
\(215\) −1.00000 −0.0681994
\(216\) 0 0
\(217\) 9.00000 0.610960
\(218\) 0 0
\(219\) 6.00000 0.405442
\(220\) 0 0
\(221\) −2.00000 −0.134535
\(222\) 0 0
\(223\) −21.0000 −1.40626 −0.703132 0.711059i \(-0.748216\pi\)
−0.703132 + 0.711059i \(0.748216\pi\)
\(224\) 0 0
\(225\) −4.00000 −0.266667
\(226\) 0 0
\(227\) 8.00000 0.530979 0.265489 0.964114i \(-0.414466\pi\)
0.265489 + 0.964114i \(0.414466\pi\)
\(228\) 0 0
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 0 0
\(231\) 8.00000 0.526361
\(232\) 0 0
\(233\) −25.0000 −1.63780 −0.818902 0.573933i \(-0.805417\pi\)
−0.818902 + 0.573933i \(0.805417\pi\)
\(234\) 0 0
\(235\) −9.00000 −0.587095
\(236\) 0 0
\(237\) −10.0000 −0.649570
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) −25.0000 −1.61039 −0.805196 0.593009i \(-0.797940\pi\)
−0.805196 + 0.593009i \(0.797940\pi\)
\(242\) 0 0
\(243\) −10.0000 −0.641500
\(244\) 0 0
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 1.00000 0.0636285
\(248\) 0 0
\(249\) −10.0000 −0.633724
\(250\) 0 0
\(251\) 16.0000 1.00991 0.504956 0.863145i \(-0.331509\pi\)
0.504956 + 0.863145i \(0.331509\pi\)
\(252\) 0 0
\(253\) 28.0000 1.76034
\(254\) 0 0
\(255\) −4.00000 −0.250490
\(256\) 0 0
\(257\) −12.0000 −0.748539 −0.374270 0.927320i \(-0.622107\pi\)
−0.374270 + 0.927320i \(0.622107\pi\)
\(258\) 0 0
\(259\) −2.00000 −0.124274
\(260\) 0 0
\(261\) −5.00000 −0.309492
\(262\) 0 0
\(263\) −19.0000 −1.17159 −0.585795 0.810459i \(-0.699218\pi\)
−0.585795 + 0.810459i \(0.699218\pi\)
\(264\) 0 0
\(265\) 3.00000 0.184289
\(266\) 0 0
\(267\) −18.0000 −1.10158
\(268\) 0 0
\(269\) 24.0000 1.46331 0.731653 0.681677i \(-0.238749\pi\)
0.731653 + 0.681677i \(0.238749\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 2.00000 0.121046
\(274\) 0 0
\(275\) −16.0000 −0.964836
\(276\) 0 0
\(277\) −23.0000 −1.38194 −0.690968 0.722885i \(-0.742815\pi\)
−0.690968 + 0.722885i \(0.742815\pi\)
\(278\) 0 0
\(279\) 9.00000 0.538816
\(280\) 0 0
\(281\) −8.00000 −0.477240 −0.238620 0.971113i \(-0.576695\pi\)
−0.238620 + 0.971113i \(0.576695\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 0 0
\(285\) 2.00000 0.118470
\(286\) 0 0
\(287\) 2.00000 0.118056
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) 0 0
\(293\) 5.00000 0.292103 0.146052 0.989277i \(-0.453343\pi\)
0.146052 + 0.989277i \(0.453343\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −16.0000 −0.928414
\(298\) 0 0
\(299\) 7.00000 0.404820
\(300\) 0 0
\(301\) −1.00000 −0.0576390
\(302\) 0 0
\(303\) −24.0000 −1.37876
\(304\) 0 0
\(305\) 14.0000 0.801638
\(306\) 0 0
\(307\) 27.0000 1.54097 0.770486 0.637457i \(-0.220014\pi\)
0.770486 + 0.637457i \(0.220014\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −12.0000 −0.678280 −0.339140 0.940736i \(-0.610136\pi\)
−0.339140 + 0.940736i \(0.610136\pi\)
\(314\) 0 0
\(315\) 1.00000 0.0563436
\(316\) 0 0
\(317\) 20.0000 1.12331 0.561656 0.827371i \(-0.310164\pi\)
0.561656 + 0.827371i \(0.310164\pi\)
\(318\) 0 0
\(319\) −20.0000 −1.11979
\(320\) 0 0
\(321\) 24.0000 1.33955
\(322\) 0 0
\(323\) −2.00000 −0.111283
\(324\) 0 0
\(325\) −4.00000 −0.221880
\(326\) 0 0
\(327\) −8.00000 −0.442401
\(328\) 0 0
\(329\) −9.00000 −0.496186
\(330\) 0 0
\(331\) −30.0000 −1.64895 −0.824475 0.565899i \(-0.808529\pi\)
−0.824475 + 0.565899i \(0.808529\pi\)
\(332\) 0 0
\(333\) −2.00000 −0.109599
\(334\) 0 0
\(335\) −10.0000 −0.546358
\(336\) 0 0
\(337\) 21.0000 1.14394 0.571971 0.820274i \(-0.306179\pi\)
0.571971 + 0.820274i \(0.306179\pi\)
\(338\) 0 0
\(339\) 2.00000 0.108625
\(340\) 0 0
\(341\) 36.0000 1.94951
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 14.0000 0.753735
\(346\) 0 0
\(347\) −16.0000 −0.858925 −0.429463 0.903085i \(-0.641297\pi\)
−0.429463 + 0.903085i \(0.641297\pi\)
\(348\) 0 0
\(349\) 35.0000 1.87351 0.936754 0.349990i \(-0.113815\pi\)
0.936754 + 0.349990i \(0.113815\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) 0 0
\(353\) −2.00000 −0.106449 −0.0532246 0.998583i \(-0.516950\pi\)
−0.0532246 + 0.998583i \(0.516950\pi\)
\(354\) 0 0
\(355\) 14.0000 0.743043
\(356\) 0 0
\(357\) −4.00000 −0.211702
\(358\) 0 0
\(359\) 30.0000 1.58334 0.791670 0.610949i \(-0.209212\pi\)
0.791670 + 0.610949i \(0.209212\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) 10.0000 0.524864
\(364\) 0 0
\(365\) 3.00000 0.157027
\(366\) 0 0
\(367\) −24.0000 −1.25279 −0.626395 0.779506i \(-0.715470\pi\)
−0.626395 + 0.779506i \(0.715470\pi\)
\(368\) 0 0
\(369\) 2.00000 0.104116
\(370\) 0 0
\(371\) 3.00000 0.155752
\(372\) 0 0
\(373\) −26.0000 −1.34623 −0.673114 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) 0 0
\(375\) −18.0000 −0.929516
\(376\) 0 0
\(377\) −5.00000 −0.257513
\(378\) 0 0
\(379\) −28.0000 −1.43826 −0.719132 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) 0 0
\(381\) 16.0000 0.819705
\(382\) 0 0
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 0 0
\(385\) 4.00000 0.203859
\(386\) 0 0
\(387\) −1.00000 −0.0508329
\(388\) 0 0
\(389\) −22.0000 −1.11544 −0.557722 0.830028i \(-0.688325\pi\)
−0.557722 + 0.830028i \(0.688325\pi\)
\(390\) 0 0
\(391\) −14.0000 −0.708010
\(392\) 0 0
\(393\) 24.0000 1.21064
\(394\) 0 0
\(395\) −5.00000 −0.251577
\(396\) 0 0
\(397\) −25.0000 −1.25471 −0.627357 0.778732i \(-0.715863\pi\)
−0.627357 + 0.778732i \(0.715863\pi\)
\(398\) 0 0
\(399\) 2.00000 0.100125
\(400\) 0 0
\(401\) −28.0000 −1.39825 −0.699127 0.714998i \(-0.746428\pi\)
−0.699127 + 0.714998i \(0.746428\pi\)
\(402\) 0 0
\(403\) 9.00000 0.448322
\(404\) 0 0
\(405\) −11.0000 −0.546594
\(406\) 0 0
\(407\) −8.00000 −0.396545
\(408\) 0 0
\(409\) −25.0000 −1.23617 −0.618085 0.786111i \(-0.712091\pi\)
−0.618085 + 0.786111i \(0.712091\pi\)
\(410\) 0 0
\(411\) −28.0000 −1.38114
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −5.00000 −0.245440
\(416\) 0 0
\(417\) −32.0000 −1.56705
\(418\) 0 0
\(419\) 16.0000 0.781651 0.390826 0.920465i \(-0.372190\pi\)
0.390826 + 0.920465i \(0.372190\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 0 0
\(423\) −9.00000 −0.437595
\(424\) 0 0
\(425\) 8.00000 0.388057
\(426\) 0 0
\(427\) 14.0000 0.677507
\(428\) 0 0
\(429\) 8.00000 0.386244
\(430\) 0 0
\(431\) −14.0000 −0.674356 −0.337178 0.941441i \(-0.609472\pi\)
−0.337178 + 0.941441i \(0.609472\pi\)
\(432\) 0 0
\(433\) 24.0000 1.15337 0.576683 0.816968i \(-0.304347\pi\)
0.576683 + 0.816968i \(0.304347\pi\)
\(434\) 0 0
\(435\) −10.0000 −0.479463
\(436\) 0 0
\(437\) 7.00000 0.334855
\(438\) 0 0
\(439\) −10.0000 −0.477274 −0.238637 0.971109i \(-0.576701\pi\)
−0.238637 + 0.971109i \(0.576701\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 39.0000 1.85295 0.926473 0.376361i \(-0.122825\pi\)
0.926473 + 0.376361i \(0.122825\pi\)
\(444\) 0 0
\(445\) −9.00000 −0.426641
\(446\) 0 0
\(447\) 24.0000 1.13516
\(448\) 0 0
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 8.00000 0.376705
\(452\) 0 0
\(453\) −28.0000 −1.31555
\(454\) 0 0
\(455\) 1.00000 0.0468807
\(456\) 0 0
\(457\) −6.00000 −0.280668 −0.140334 0.990104i \(-0.544818\pi\)
−0.140334 + 0.990104i \(0.544818\pi\)
\(458\) 0 0
\(459\) 8.00000 0.373408
\(460\) 0 0
\(461\) 10.0000 0.465746 0.232873 0.972507i \(-0.425187\pi\)
0.232873 + 0.972507i \(0.425187\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) 0 0
\(465\) 18.0000 0.834730
\(466\) 0 0
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 0 0
\(469\) −10.0000 −0.461757
\(470\) 0 0
\(471\) 20.0000 0.921551
\(472\) 0 0
\(473\) −4.00000 −0.183920
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) 3.00000 0.137361
\(478\) 0 0
\(479\) −27.0000 −1.23366 −0.616831 0.787096i \(-0.711584\pi\)
−0.616831 + 0.787096i \(0.711584\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) 0 0
\(483\) 14.0000 0.637022
\(484\) 0 0
\(485\) −1.00000 −0.0454077
\(486\) 0 0
\(487\) 34.0000 1.54069 0.770344 0.637629i \(-0.220085\pi\)
0.770344 + 0.637629i \(0.220085\pi\)
\(488\) 0 0
\(489\) −48.0000 −2.17064
\(490\) 0 0
\(491\) 8.00000 0.361035 0.180517 0.983572i \(-0.442223\pi\)
0.180517 + 0.983572i \(0.442223\pi\)
\(492\) 0 0
\(493\) 10.0000 0.450377
\(494\) 0 0
\(495\) 4.00000 0.179787
\(496\) 0 0
\(497\) 14.0000 0.627986
\(498\) 0 0
\(499\) 2.00000 0.0895323 0.0447661 0.998997i \(-0.485746\pi\)
0.0447661 + 0.998997i \(0.485746\pi\)
\(500\) 0 0
\(501\) −6.00000 −0.268060
\(502\) 0 0
\(503\) 34.0000 1.51599 0.757993 0.652263i \(-0.226180\pi\)
0.757993 + 0.652263i \(0.226180\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) 0 0
\(507\) 2.00000 0.0888231
\(508\) 0 0
\(509\) 9.00000 0.398918 0.199459 0.979906i \(-0.436082\pi\)
0.199459 + 0.979906i \(0.436082\pi\)
\(510\) 0 0
\(511\) 3.00000 0.132712
\(512\) 0 0
\(513\) −4.00000 −0.176604
\(514\) 0 0
\(515\) −4.00000 −0.176261
\(516\) 0 0
\(517\) −36.0000 −1.58328
\(518\) 0 0
\(519\) 4.00000 0.175581
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) 18.0000 0.787085 0.393543 0.919306i \(-0.371249\pi\)
0.393543 + 0.919306i \(0.371249\pi\)
\(524\) 0 0
\(525\) −8.00000 −0.349149
\(526\) 0 0
\(527\) −18.0000 −0.784092
\(528\) 0 0
\(529\) 26.0000 1.13043
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 2.00000 0.0866296
\(534\) 0 0
\(535\) 12.0000 0.518805
\(536\) 0 0
\(537\) −18.0000 −0.776757
\(538\) 0 0
\(539\) 4.00000 0.172292
\(540\) 0 0
\(541\) −38.0000 −1.63375 −0.816874 0.576816i \(-0.804295\pi\)
−0.816874 + 0.576816i \(0.804295\pi\)
\(542\) 0 0
\(543\) −40.0000 −1.71656
\(544\) 0 0
\(545\) −4.00000 −0.171341
\(546\) 0 0
\(547\) −23.0000 −0.983409 −0.491704 0.870762i \(-0.663626\pi\)
−0.491704 + 0.870762i \(0.663626\pi\)
\(548\) 0 0
\(549\) 14.0000 0.597505
\(550\) 0 0
\(551\) −5.00000 −0.213007
\(552\) 0 0
\(553\) −5.00000 −0.212622
\(554\) 0 0
\(555\) −4.00000 −0.169791
\(556\) 0 0
\(557\) 4.00000 0.169485 0.0847427 0.996403i \(-0.472993\pi\)
0.0847427 + 0.996403i \(0.472993\pi\)
\(558\) 0 0
\(559\) −1.00000 −0.0422955
\(560\) 0 0
\(561\) −16.0000 −0.675521
\(562\) 0 0
\(563\) 40.0000 1.68580 0.842900 0.538071i \(-0.180847\pi\)
0.842900 + 0.538071i \(0.180847\pi\)
\(564\) 0 0
\(565\) 1.00000 0.0420703
\(566\) 0 0
\(567\) −11.0000 −0.461957
\(568\) 0 0
\(569\) −37.0000 −1.55112 −0.775560 0.631273i \(-0.782533\pi\)
−0.775560 + 0.631273i \(0.782533\pi\)
\(570\) 0 0
\(571\) 7.00000 0.292941 0.146470 0.989215i \(-0.453209\pi\)
0.146470 + 0.989215i \(0.453209\pi\)
\(572\) 0 0
\(573\) 48.0000 2.00523
\(574\) 0 0
\(575\) −28.0000 −1.16768
\(576\) 0 0
\(577\) −18.0000 −0.749350 −0.374675 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(578\) 0 0
\(579\) 4.00000 0.166234
\(580\) 0 0
\(581\) −5.00000 −0.207435
\(582\) 0 0
\(583\) 12.0000 0.496989
\(584\) 0 0
\(585\) 1.00000 0.0413449
\(586\) 0 0
\(587\) 15.0000 0.619116 0.309558 0.950881i \(-0.399819\pi\)
0.309558 + 0.950881i \(0.399819\pi\)
\(588\) 0 0
\(589\) 9.00000 0.370839
\(590\) 0 0
\(591\) 12.0000 0.493614
\(592\) 0 0
\(593\) 41.0000 1.68367 0.841834 0.539736i \(-0.181476\pi\)
0.841834 + 0.539736i \(0.181476\pi\)
\(594\) 0 0
\(595\) −2.00000 −0.0819920
\(596\) 0 0
\(597\) 28.0000 1.14596
\(598\) 0 0
\(599\) −21.0000 −0.858037 −0.429018 0.903296i \(-0.641140\pi\)
−0.429018 + 0.903296i \(0.641140\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 0 0
\(603\) −10.0000 −0.407231
\(604\) 0 0
\(605\) 5.00000 0.203279
\(606\) 0 0
\(607\) 40.0000 1.62355 0.811775 0.583970i \(-0.198502\pi\)
0.811775 + 0.583970i \(0.198502\pi\)
\(608\) 0 0
\(609\) −10.0000 −0.405220
\(610\) 0 0
\(611\) −9.00000 −0.364101
\(612\) 0 0
\(613\) −16.0000 −0.646234 −0.323117 0.946359i \(-0.604731\pi\)
−0.323117 + 0.946359i \(0.604731\pi\)
\(614\) 0 0
\(615\) 4.00000 0.161296
\(616\) 0 0
\(617\) 34.0000 1.36879 0.684394 0.729112i \(-0.260067\pi\)
0.684394 + 0.729112i \(0.260067\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) −28.0000 −1.12360
\(622\) 0 0
\(623\) −9.00000 −0.360577
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) 8.00000 0.319489
\(628\) 0 0
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) 2.00000 0.0796187 0.0398094 0.999207i \(-0.487325\pi\)
0.0398094 + 0.999207i \(0.487325\pi\)
\(632\) 0 0
\(633\) 46.0000 1.82834
\(634\) 0 0
\(635\) 8.00000 0.317470
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 0 0
\(639\) 14.0000 0.553831
\(640\) 0 0
\(641\) 41.0000 1.61940 0.809701 0.586842i \(-0.199629\pi\)
0.809701 + 0.586842i \(0.199629\pi\)
\(642\) 0 0
\(643\) 28.0000 1.10421 0.552106 0.833774i \(-0.313824\pi\)
0.552106 + 0.833774i \(0.313824\pi\)
\(644\) 0 0
\(645\) −2.00000 −0.0787499
\(646\) 0 0
\(647\) −12.0000 −0.471769 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 18.0000 0.705476
\(652\) 0 0
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 0 0
\(655\) 12.0000 0.468879
\(656\) 0 0
\(657\) 3.00000 0.117041
\(658\) 0 0
\(659\) 45.0000 1.75295 0.876476 0.481446i \(-0.159888\pi\)
0.876476 + 0.481446i \(0.159888\pi\)
\(660\) 0 0
\(661\) 31.0000 1.20576 0.602880 0.797832i \(-0.294020\pi\)
0.602880 + 0.797832i \(0.294020\pi\)
\(662\) 0 0
\(663\) −4.00000 −0.155347
\(664\) 0 0
\(665\) 1.00000 0.0387783
\(666\) 0 0
\(667\) −35.0000 −1.35521
\(668\) 0 0
\(669\) −42.0000 −1.62381
\(670\) 0 0
\(671\) 56.0000 2.16186
\(672\) 0 0
\(673\) 21.0000 0.809491 0.404745 0.914429i \(-0.367360\pi\)
0.404745 + 0.914429i \(0.367360\pi\)
\(674\) 0 0
\(675\) 16.0000 0.615840
\(676\) 0 0
\(677\) 16.0000 0.614930 0.307465 0.951559i \(-0.400519\pi\)
0.307465 + 0.951559i \(0.400519\pi\)
\(678\) 0 0
\(679\) −1.00000 −0.0383765
\(680\) 0 0
\(681\) 16.0000 0.613121
\(682\) 0 0
\(683\) −4.00000 −0.153056 −0.0765279 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) 0 0
\(685\) −14.0000 −0.534913
\(686\) 0 0
\(687\) −4.00000 −0.152610
\(688\) 0 0
\(689\) 3.00000 0.114291
\(690\) 0 0
\(691\) 35.0000 1.33146 0.665731 0.746191i \(-0.268120\pi\)
0.665731 + 0.746191i \(0.268120\pi\)
\(692\) 0 0
\(693\) 4.00000 0.151947
\(694\) 0 0
\(695\) −16.0000 −0.606915
\(696\) 0 0
\(697\) −4.00000 −0.151511
\(698\) 0 0
\(699\) −50.0000 −1.89117
\(700\) 0 0
\(701\) 45.0000 1.69963 0.849813 0.527084i \(-0.176715\pi\)
0.849813 + 0.527084i \(0.176715\pi\)
\(702\) 0 0
\(703\) −2.00000 −0.0754314
\(704\) 0 0
\(705\) −18.0000 −0.677919
\(706\) 0 0
\(707\) −12.0000 −0.451306
\(708\) 0 0
\(709\) −12.0000 −0.450669 −0.225335 0.974281i \(-0.572348\pi\)
−0.225335 + 0.974281i \(0.572348\pi\)
\(710\) 0 0
\(711\) −5.00000 −0.187515
\(712\) 0 0
\(713\) 63.0000 2.35937
\(714\) 0 0
\(715\) 4.00000 0.149592
\(716\) 0 0
\(717\) −24.0000 −0.896296
\(718\) 0 0
\(719\) 40.0000 1.49175 0.745874 0.666087i \(-0.232032\pi\)
0.745874 + 0.666087i \(0.232032\pi\)
\(720\) 0 0
\(721\) −4.00000 −0.148968
\(722\) 0 0
\(723\) −50.0000 −1.85952
\(724\) 0 0
\(725\) 20.0000 0.742781
\(726\) 0 0
\(727\) 42.0000 1.55769 0.778847 0.627214i \(-0.215805\pi\)
0.778847 + 0.627214i \(0.215805\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 2.00000 0.0739727
\(732\) 0 0
\(733\) −21.0000 −0.775653 −0.387826 0.921732i \(-0.626774\pi\)
−0.387826 + 0.921732i \(0.626774\pi\)
\(734\) 0 0
\(735\) 2.00000 0.0737711
\(736\) 0 0
\(737\) −40.0000 −1.47342
\(738\) 0 0
\(739\) 8.00000 0.294285 0.147142 0.989115i \(-0.452992\pi\)
0.147142 + 0.989115i \(0.452992\pi\)
\(740\) 0 0
\(741\) 2.00000 0.0734718
\(742\) 0 0
\(743\) 8.00000 0.293492 0.146746 0.989174i \(-0.453120\pi\)
0.146746 + 0.989174i \(0.453120\pi\)
\(744\) 0 0
\(745\) 12.0000 0.439646
\(746\) 0 0
\(747\) −5.00000 −0.182940
\(748\) 0 0
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) −25.0000 −0.912263 −0.456131 0.889912i \(-0.650765\pi\)
−0.456131 + 0.889912i \(0.650765\pi\)
\(752\) 0 0
\(753\) 32.0000 1.16614
\(754\) 0 0
\(755\) −14.0000 −0.509512
\(756\) 0 0
\(757\) −47.0000 −1.70824 −0.854122 0.520073i \(-0.825905\pi\)
−0.854122 + 0.520073i \(0.825905\pi\)
\(758\) 0 0
\(759\) 56.0000 2.03267
\(760\) 0 0
\(761\) −35.0000 −1.26875 −0.634375 0.773026i \(-0.718742\pi\)
−0.634375 + 0.773026i \(0.718742\pi\)
\(762\) 0 0
\(763\) −4.00000 −0.144810
\(764\) 0 0
\(765\) −2.00000 −0.0723102
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 37.0000 1.33425 0.667127 0.744944i \(-0.267524\pi\)
0.667127 + 0.744944i \(0.267524\pi\)
\(770\) 0 0
\(771\) −24.0000 −0.864339
\(772\) 0 0
\(773\) 38.0000 1.36677 0.683383 0.730061i \(-0.260508\pi\)
0.683383 + 0.730061i \(0.260508\pi\)
\(774\) 0 0
\(775\) −36.0000 −1.29316
\(776\) 0 0
\(777\) −4.00000 −0.143499
\(778\) 0 0
\(779\) 2.00000 0.0716574
\(780\) 0 0
\(781\) 56.0000 2.00384
\(782\) 0 0
\(783\) 20.0000 0.714742
\(784\) 0 0
\(785\) 10.0000 0.356915
\(786\) 0 0
\(787\) 37.0000 1.31891 0.659454 0.751745i \(-0.270788\pi\)
0.659454 + 0.751745i \(0.270788\pi\)
\(788\) 0 0
\(789\) −38.0000 −1.35284
\(790\) 0 0
\(791\) 1.00000 0.0355559
\(792\) 0 0
\(793\) 14.0000 0.497155
\(794\) 0 0
\(795\) 6.00000 0.212798
\(796\) 0 0
\(797\) 6.00000 0.212531 0.106265 0.994338i \(-0.466111\pi\)
0.106265 + 0.994338i \(0.466111\pi\)
\(798\) 0 0
\(799\) 18.0000 0.636794
\(800\) 0 0
\(801\) −9.00000 −0.317999
\(802\) 0 0
\(803\) 12.0000 0.423471
\(804\) 0 0
\(805\) 7.00000 0.246718
\(806\) 0 0
\(807\) 48.0000 1.68968
\(808\) 0 0
\(809\) 45.0000 1.58212 0.791058 0.611741i \(-0.209531\pi\)
0.791058 + 0.611741i \(0.209531\pi\)
\(810\) 0 0
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −24.0000 −0.840683
\(816\) 0 0
\(817\) −1.00000 −0.0349856
\(818\) 0 0
\(819\) 1.00000 0.0349428
\(820\) 0 0
\(821\) 34.0000 1.18661 0.593304 0.804978i \(-0.297823\pi\)
0.593304 + 0.804978i \(0.297823\pi\)
\(822\) 0 0
\(823\) −16.0000 −0.557725 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) 0 0
\(825\) −32.0000 −1.11410
\(826\) 0 0
\(827\) −40.0000 −1.39094 −0.695468 0.718557i \(-0.744803\pi\)
−0.695468 + 0.718557i \(0.744803\pi\)
\(828\) 0 0
\(829\) 24.0000 0.833554 0.416777 0.909009i \(-0.363160\pi\)
0.416777 + 0.909009i \(0.363160\pi\)
\(830\) 0 0
\(831\) −46.0000 −1.59572
\(832\) 0 0
\(833\) −2.00000 −0.0692959
\(834\) 0 0
\(835\) −3.00000 −0.103819
\(836\) 0 0
\(837\) −36.0000 −1.24434
\(838\) 0 0
\(839\) 12.0000 0.414286 0.207143 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 0 0
\(843\) −16.0000 −0.551069
\(844\) 0 0
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) 5.00000 0.171802
\(848\) 0 0
\(849\) 8.00000 0.274559
\(850\) 0 0
\(851\) −14.0000 −0.479914
\(852\) 0 0
\(853\) 37.0000 1.26686 0.633428 0.773802i \(-0.281647\pi\)
0.633428 + 0.773802i \(0.281647\pi\)
\(854\) 0 0
\(855\) 1.00000 0.0341993
\(856\) 0 0
\(857\) 30.0000 1.02478 0.512390 0.858753i \(-0.328760\pi\)
0.512390 + 0.858753i \(0.328760\pi\)
\(858\) 0 0
\(859\) −22.0000 −0.750630 −0.375315 0.926897i \(-0.622466\pi\)
−0.375315 + 0.926897i \(0.622466\pi\)
\(860\) 0 0
\(861\) 4.00000 0.136320
\(862\) 0 0
\(863\) 48.0000 1.63394 0.816970 0.576681i \(-0.195652\pi\)
0.816970 + 0.576681i \(0.195652\pi\)
\(864\) 0 0
\(865\) 2.00000 0.0680020
\(866\) 0 0
\(867\) −26.0000 −0.883006
\(868\) 0 0
\(869\) −20.0000 −0.678454
\(870\) 0 0
\(871\) −10.0000 −0.338837
\(872\) 0 0
\(873\) −1.00000 −0.0338449
\(874\) 0 0
\(875\) −9.00000 −0.304256
\(876\) 0 0
\(877\) 14.0000 0.472746 0.236373 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) 0 0
\(879\) 10.0000 0.337292
\(880\) 0 0
\(881\) −2.00000 −0.0673817 −0.0336909 0.999432i \(-0.510726\pi\)
−0.0336909 + 0.999432i \(0.510726\pi\)
\(882\) 0 0
\(883\) 16.0000 0.538443 0.269221 0.963078i \(-0.413234\pi\)
0.269221 + 0.963078i \(0.413234\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −32.0000 −1.07445 −0.537227 0.843437i \(-0.680528\pi\)
−0.537227 + 0.843437i \(0.680528\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) −44.0000 −1.47406
\(892\) 0 0
\(893\) −9.00000 −0.301174
\(894\) 0 0
\(895\) −9.00000 −0.300837
\(896\) 0 0
\(897\) 14.0000 0.467446
\(898\) 0 0
\(899\) −45.0000 −1.50083
\(900\) 0 0
\(901\) −6.00000 −0.199889
\(902\) 0 0
\(903\) −2.00000 −0.0665558
\(904\) 0 0
\(905\) −20.0000 −0.664822
\(906\) 0 0
\(907\) 3.00000 0.0996134 0.0498067 0.998759i \(-0.484139\pi\)
0.0498067 + 0.998759i \(0.484139\pi\)
\(908\) 0 0
\(909\) −12.0000 −0.398015
\(910\) 0 0
\(911\) 21.0000 0.695761 0.347881 0.937539i \(-0.386901\pi\)
0.347881 + 0.937539i \(0.386901\pi\)
\(912\) 0 0
\(913\) −20.0000 −0.661903
\(914\) 0 0
\(915\) 28.0000 0.925651
\(916\) 0 0
\(917\) 12.0000 0.396275
\(918\) 0 0
\(919\) 40.0000 1.31948 0.659739 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) 0 0
\(921\) 54.0000 1.77936
\(922\) 0 0
\(923\) 14.0000 0.460816
\(924\) 0 0
\(925\) 8.00000 0.263038
\(926\) 0 0
\(927\) −4.00000 −0.131377
\(928\) 0 0
\(929\) −3.00000 −0.0984268 −0.0492134 0.998788i \(-0.515671\pi\)
−0.0492134 + 0.998788i \(0.515671\pi\)
\(930\) 0 0
\(931\) 1.00000 0.0327737
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −8.00000 −0.261628
\(936\) 0 0
\(937\) 30.0000 0.980057 0.490029 0.871706i \(-0.336986\pi\)
0.490029 + 0.871706i \(0.336986\pi\)
\(938\) 0 0
\(939\) −24.0000 −0.783210
\(940\) 0 0
\(941\) 35.0000 1.14097 0.570484 0.821309i \(-0.306756\pi\)
0.570484 + 0.821309i \(0.306756\pi\)
\(942\) 0 0
\(943\) 14.0000 0.455903
\(944\) 0 0
\(945\) −4.00000 −0.130120
\(946\) 0 0
\(947\) −10.0000 −0.324956 −0.162478 0.986712i \(-0.551949\pi\)
−0.162478 + 0.986712i \(0.551949\pi\)
\(948\) 0 0
\(949\) 3.00000 0.0973841
\(950\) 0 0
\(951\) 40.0000 1.29709
\(952\) 0 0
\(953\) −19.0000 −0.615470 −0.307735 0.951472i \(-0.599571\pi\)
−0.307735 + 0.951472i \(0.599571\pi\)
\(954\) 0 0
\(955\) 24.0000 0.776622
\(956\) 0 0
\(957\) −40.0000 −1.29302
\(958\) 0 0
\(959\) −14.0000 −0.452084
\(960\) 0 0
\(961\) 50.0000 1.61290
\(962\) 0 0
\(963\) 12.0000 0.386695
\(964\) 0 0
\(965\) 2.00000 0.0643823
\(966\) 0 0
\(967\) 12.0000 0.385894 0.192947 0.981209i \(-0.438195\pi\)
0.192947 + 0.981209i \(0.438195\pi\)
\(968\) 0 0
\(969\) −4.00000 −0.128499
\(970\) 0 0
\(971\) −30.0000 −0.962746 −0.481373 0.876516i \(-0.659862\pi\)
−0.481373 + 0.876516i \(0.659862\pi\)
\(972\) 0 0
\(973\) −16.0000 −0.512936
\(974\) 0 0
\(975\) −8.00000 −0.256205
\(976\) 0 0
\(977\) −60.0000 −1.91957 −0.959785 0.280736i \(-0.909421\pi\)
−0.959785 + 0.280736i \(0.909421\pi\)
\(978\) 0 0
\(979\) −36.0000 −1.15056
\(980\) 0 0
\(981\) −4.00000 −0.127710
\(982\) 0 0
\(983\) −11.0000 −0.350846 −0.175423 0.984493i \(-0.556129\pi\)
−0.175423 + 0.984493i \(0.556129\pi\)
\(984\) 0 0
\(985\) 6.00000 0.191176
\(986\) 0 0
\(987\) −18.0000 −0.572946
\(988\) 0 0
\(989\) −7.00000 −0.222587
\(990\) 0 0
\(991\) −24.0000 −0.762385 −0.381193 0.924496i \(-0.624487\pi\)
−0.381193 + 0.924496i \(0.624487\pi\)
\(992\) 0 0
\(993\) −60.0000 −1.90404
\(994\) 0 0
\(995\) 14.0000 0.443830
\(996\) 0 0
\(997\) −32.0000 −1.01345 −0.506725 0.862108i \(-0.669144\pi\)
−0.506725 + 0.862108i \(0.669144\pi\)
\(998\) 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 1456.2.a.m.1.1 1
4.3 odd 2 364.2.a.a.1.1 1
8.3 odd 2 5824.2.a.bb.1.1 1
8.5 even 2 5824.2.a.d.1.1 1
12.11 even 2 3276.2.a.b.1.1 1
20.19 odd 2 9100.2.a.l.1.1 1
28.3 even 6 2548.2.j.c.1353.1 2
28.11 odd 6 2548.2.j.j.1353.1 2
28.19 even 6 2548.2.j.c.1145.1 2
28.23 odd 6 2548.2.j.j.1145.1 2
28.27 even 2 2548.2.a.i.1.1 1
52.31 even 4 4732.2.g.a.337.1 2
52.47 even 4 4732.2.g.a.337.2 2
52.51 odd 2 4732.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
364.2.a.a.1.1 1 4.3 odd 2
1456.2.a.m.1.1 1 1.1 even 1 trivial
2548.2.a.i.1.1 1 28.27 even 2
2548.2.j.c.1145.1 2 28.19 even 6
2548.2.j.c.1353.1 2 28.3 even 6
2548.2.j.j.1145.1 2 28.23 odd 6
2548.2.j.j.1353.1 2 28.11 odd 6
3276.2.a.b.1.1 1 12.11 even 2
4732.2.a.a.1.1 1 52.51 odd 2
4732.2.g.a.337.1 2 52.31 even 4
4732.2.g.a.337.2 2 52.47 even 4
5824.2.a.d.1.1 1 8.5 even 2
5824.2.a.bb.1.1 1 8.3 odd 2
9100.2.a.l.1.1 1 20.19 odd 2