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Properties

Label 2960.2.a.c.1.1
Level $2960$
Weight $2$
Character 2960.1
Self dual yes
Analytic conductor $23.636$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2960,2,Mod(1,2960)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2960.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2960, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2960.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-2,0,1,0,-1,0,1,0,-3,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(23.6357189983\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 370)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2960.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -3.00000 q^{11} -2.00000 q^{15} +3.00000 q^{17} +6.00000 q^{19} +2.00000 q^{21} -2.00000 q^{23} +1.00000 q^{25} +4.00000 q^{27} -3.00000 q^{29} -3.00000 q^{31} +6.00000 q^{33} -1.00000 q^{35} -1.00000 q^{37} +3.00000 q^{41} +1.00000 q^{43} +1.00000 q^{45} -4.00000 q^{47} -6.00000 q^{49} -6.00000 q^{51} +13.0000 q^{53} -3.00000 q^{55} -12.0000 q^{57} -15.0000 q^{61} -1.00000 q^{63} +4.00000 q^{69} +2.00000 q^{71} -2.00000 q^{75} +3.00000 q^{77} +8.00000 q^{79} -11.0000 q^{81} +4.00000 q^{83} +3.00000 q^{85} +6.00000 q^{87} -18.0000 q^{89} +6.00000 q^{93} +6.00000 q^{95} -7.00000 q^{97} -3.00000 q^{99} +O(q^{100})\)

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 \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964 −0.188982 − 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 − 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 0 0
\(17\) 3.00000 0.727607 0.363803 − 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 − 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) −2.00000 −0.417029 −0.208514 − 0.978019i \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) −3.00000 −0.557086 −0.278543 − 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) −3.00000 −0.538816 −0.269408 − 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) 0 0
\(33\) 6.00000 1.04447
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.00000 0.468521 0.234261 − 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) 1.00000 0.152499 0.0762493 − 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −4.00000 −0.583460 −0.291730 − 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 0 0
\(53\) 13.0000 1.78569 0.892844 − 0.450367i \(-0.148707\pi\)
0.892844 + 0.450367i \(0.148707\pi\)
\(54\) 0 0
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) −12.0000 −1.58944
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −15.0000 −1.92055 −0.960277 − 0.279050i \(-0.909981\pi\)
−0.960277 + 0.279050i \(0.909981\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 − 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 0 0
\(73\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) −2.00000 −0.230940
\(76\) 0 0
\(77\) 3.00000 0.341882
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 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
\(85\) 3.00000 0.325396
\(86\) 0 0
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) −18.0000 −1.90800 −0.953998 − 0.299813i \(-0.903076\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 6.00000 0.622171
\(94\) 0 0
\(95\) 6.00000 0.615587
\(96\) 0 0
\(97\) −7.00000 −0.710742 −0.355371 − 0.934725i \(-0.615646\pi\)
−0.355371 + 0.934725i \(0.615646\pi\)
\(98\) 0 0
\(99\) −3.00000 −0.301511
\(100\) 0 0
\(101\) −10.0000 −0.995037 −0.497519 − 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 − 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 0 0
\(105\) 2.00000 0.195180
\(106\) 0 0
\(107\) −2.00000 −0.193347 −0.0966736 − 0.995316i \(-0.530820\pi\)
−0.0966736 + 0.995316i \(0.530820\pi\)
\(108\) 0 0
\(109\) −3.00000 −0.287348 −0.143674 − 0.989625i \(-0.545892\pi\)
−0.143674 + 0.989625i \(0.545892\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) −7.00000 −0.658505 −0.329252 − 0.944242i \(-0.606797\pi\)
−0.329252 + 0.944242i \(0.606797\pi\)
\(114\) 0 0
\(115\) −2.00000 −0.186501
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) −6.00000 −0.541002
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 4.00000 0.354943 0.177471 − 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 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
\(133\) −6.00000 −0.520266
\(134\) 0 0
\(135\) 4.00000 0.344265
\(136\) 0 0
\(137\) 8.00000 0.683486 0.341743 − 0.939793i \(-0.388983\pi\)
0.341743 + 0.939793i \(0.388983\pi\)
\(138\) 0 0
\(139\) −3.00000 −0.254457 −0.127228 − 0.991873i \(-0.540608\pi\)
−0.127228 + 0.991873i \(0.540608\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −3.00000 −0.249136
\(146\) 0 0
\(147\) 12.0000 0.989743
\(148\) 0 0
\(149\) 18.0000 1.47462 0.737309 − 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 − 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 0 0
\(153\) 3.00000 0.242536
\(154\) 0 0
\(155\) −3.00000 −0.240966
\(156\) 0 0
\(157\) −3.00000 −0.239426 −0.119713 − 0.992809i \(-0.538197\pi\)
−0.119713 + 0.992809i \(0.538197\pi\)
\(158\) 0 0
\(159\) −26.0000 −2.06193
\(160\) 0 0
\(161\) 2.00000 0.157622
\(162\) 0 0
\(163\) 5.00000 0.391630 0.195815 − 0.980641i \(-0.437265\pi\)
0.195815 + 0.980641i \(0.437265\pi\)
\(164\) 0 0
\(165\) 6.00000 0.467099
\(166\) 0 0
\(167\) −12.0000 −0.928588 −0.464294 − 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 6.00000 0.458831
\(172\) 0 0
\(173\) 9.00000 0.684257 0.342129 − 0.939653i \(-0.388852\pi\)
0.342129 + 0.939653i \(0.388852\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −24.0000 −1.79384 −0.896922 − 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 − 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 30.0000 2.21766
\(184\) 0 0
\(185\) −1.00000 −0.0735215
\(186\) 0 0
\(187\) −9.00000 −0.658145
\(188\) 0 0
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) −21.0000 −1.51951 −0.759753 − 0.650211i \(-0.774680\pi\)
−0.759753 + 0.650211i \(0.774680\pi\)
\(192\) 0 0
\(193\) −10.0000 −0.719816 −0.359908 − 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 − 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 − 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 3.00000 0.210559
\(204\) 0 0
\(205\) 3.00000 0.209529
\(206\) 0 0
\(207\) −2.00000 −0.139010
\(208\) 0 0
\(209\) −18.0000 −1.24509
\(210\) 0 0
\(211\) 3.00000 0.206529 0.103264 − 0.994654i \(-0.467071\pi\)
0.103264 + 0.994654i \(0.467071\pi\)
\(212\) 0 0
\(213\) −4.00000 −0.274075
\(214\) 0 0
\(215\) 1.00000 0.0681994
\(216\) 0 0
\(217\) 3.00000 0.203653
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −23.0000 −1.54019 −0.770097 − 0.637927i \(-0.779792\pi\)
−0.770097 + 0.637927i \(0.779792\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −13.0000 −0.862840 −0.431420 − 0.902151i \(-0.641987\pi\)
−0.431420 + 0.902151i \(0.641987\pi\)
\(228\) 0 0
\(229\) −6.00000 −0.396491 −0.198246 − 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 0 0
\(231\) −6.00000 −0.394771
\(232\) 0 0
\(233\) 4.00000 0.262049 0.131024 − 0.991379i \(-0.458173\pi\)
0.131024 + 0.991379i \(0.458173\pi\)
\(234\) 0 0
\(235\) −4.00000 −0.260931
\(236\) 0 0
\(237\) −16.0000 −1.03931
\(238\) 0 0
\(239\) −9.00000 −0.582162 −0.291081 − 0.956698i \(-0.594015\pi\)
−0.291081 + 0.956698i \(0.594015\pi\)
\(240\) 0 0
\(241\) 24.0000 1.54598 0.772988 − 0.634421i \(-0.218761\pi\)
0.772988 + 0.634421i \(0.218761\pi\)
\(242\) 0 0
\(243\) 10.0000 0.641500
\(244\) 0 0
\(245\) −6.00000 −0.383326
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −8.00000 −0.506979
\(250\) 0 0
\(251\) −6.00000 −0.378717 −0.189358 − 0.981908i \(-0.560641\pi\)
−0.189358 + 0.981908i \(0.560641\pi\)
\(252\) 0 0
\(253\) 6.00000 0.377217
\(254\) 0 0
\(255\) −6.00000 −0.375735
\(256\) 0 0
\(257\) −22.0000 −1.37232 −0.686161 − 0.727450i \(-0.740706\pi\)
−0.686161 + 0.727450i \(0.740706\pi\)
\(258\) 0 0
\(259\) 1.00000 0.0621370
\(260\) 0 0
\(261\) −3.00000 −0.185695
\(262\) 0 0
\(263\) 1.00000 0.0616626 0.0308313 − 0.999525i \(-0.490185\pi\)
0.0308313 + 0.999525i \(0.490185\pi\)
\(264\) 0 0
\(265\) 13.0000 0.798584
\(266\) 0 0
\(267\) 36.0000 2.20316
\(268\) 0 0
\(269\) −10.0000 −0.609711 −0.304855 − 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 0 0
\(271\) 10.0000 0.607457 0.303728 − 0.952759i \(-0.401768\pi\)
0.303728 + 0.952759i \(0.401768\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −3.00000 −0.180907
\(276\) 0 0
\(277\) −20.0000 −1.20168 −0.600842 − 0.799368i \(-0.705168\pi\)
−0.600842 + 0.799368i \(0.705168\pi\)
\(278\) 0 0
\(279\) −3.00000 −0.179605
\(280\) 0 0
\(281\) −24.0000 −1.43172 −0.715860 − 0.698244i \(-0.753965\pi\)
−0.715860 + 0.698244i \(0.753965\pi\)
\(282\) 0 0
\(283\) 20.0000 1.18888 0.594438 − 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) 0 0
\(285\) −12.0000 −0.710819
\(286\) 0 0
\(287\) −3.00000 −0.177084
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 14.0000 0.820695
\(292\) 0 0
\(293\) 3.00000 0.175262 0.0876309 − 0.996153i \(-0.472070\pi\)
0.0876309 + 0.996153i \(0.472070\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −12.0000 −0.696311
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −1.00000 −0.0576390
\(302\) 0 0
\(303\) 20.0000 1.14897
\(304\) 0 0
\(305\) −15.0000 −0.858898
\(306\) 0 0
\(307\) −34.0000 −1.94048 −0.970241 − 0.242140i \(-0.922151\pi\)
−0.970241 + 0.242140i \(0.922151\pi\)
\(308\) 0 0
\(309\) 16.0000 0.910208
\(310\) 0 0
\(311\) −25.0000 −1.41762 −0.708810 − 0.705399i \(-0.750768\pi\)
−0.708810 + 0.705399i \(0.750768\pi\)
\(312\) 0 0
\(313\) 26.0000 1.46961 0.734803 − 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) 0 0
\(315\) −1.00000 −0.0563436
\(316\) 0 0
\(317\) 21.0000 1.17948 0.589739 − 0.807594i \(-0.299231\pi\)
0.589739 + 0.807594i \(0.299231\pi\)
\(318\) 0 0
\(319\) 9.00000 0.503903
\(320\) 0 0
\(321\) 4.00000 0.223258
\(322\) 0 0
\(323\) 18.0000 1.00155
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 6.00000 0.331801
\(328\) 0 0
\(329\) 4.00000 0.220527
\(330\) 0 0
\(331\) −30.0000 −1.64895 −0.824475 − 0.565899i \(-0.808529\pi\)
−0.824475 + 0.565899i \(0.808529\pi\)
\(332\) 0 0
\(333\) −1.00000 −0.0547997
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 12.0000 0.653682 0.326841 − 0.945079i \(-0.394016\pi\)
0.326841 + 0.945079i \(0.394016\pi\)
\(338\) 0 0
\(339\) 14.0000 0.760376
\(340\) 0 0
\(341\) 9.00000 0.487377
\(342\) 0 0
\(343\) 13.0000 0.701934
\(344\) 0 0
\(345\) 4.00000 0.215353
\(346\) 0 0
\(347\) 28.0000 1.50312 0.751559 − 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) 0 0
\(349\) −22.0000 −1.17763 −0.588817 − 0.808267i \(-0.700406\pi\)
−0.588817 + 0.808267i \(0.700406\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 19.0000 1.01127 0.505634 − 0.862748i \(-0.331259\pi\)
0.505634 + 0.862748i \(0.331259\pi\)
\(354\) 0 0
\(355\) 2.00000 0.106149
\(356\) 0 0
\(357\) 6.00000 0.317554
\(358\) 0 0
\(359\) −16.0000 −0.844448 −0.422224 − 0.906492i \(-0.638750\pi\)
−0.422224 + 0.906492i \(0.638750\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) 4.00000 0.209946
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −13.0000 −0.678594 −0.339297 − 0.940679i \(-0.610189\pi\)
−0.339297 + 0.940679i \(0.610189\pi\)
\(368\) 0 0
\(369\) 3.00000 0.156174
\(370\) 0 0
\(371\) −13.0000 −0.674926
\(372\) 0 0
\(373\) 22.0000 1.13912 0.569558 − 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) 0 0
\(375\) −2.00000 −0.103280
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 − 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 0 0
\(383\) 12.0000 0.613171 0.306586 − 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) 0 0
\(385\) 3.00000 0.152894
\(386\) 0 0
\(387\) 1.00000 0.0508329
\(388\) 0 0
\(389\) 15.0000 0.760530 0.380265 − 0.924878i \(-0.375833\pi\)
0.380265 + 0.924878i \(0.375833\pi\)
\(390\) 0 0
\(391\) −6.00000 −0.303433
\(392\) 0 0
\(393\) −24.0000 −1.21064
\(394\) 0 0
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) 18.0000 0.903394 0.451697 − 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) 0 0
\(399\) 12.0000 0.600751
\(400\) 0 0
\(401\) 34.0000 1.69788 0.848939 − 0.528490i \(-0.177242\pi\)
0.848939 + 0.528490i \(0.177242\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −11.0000 −0.546594
\(406\) 0 0
\(407\) 3.00000 0.148704
\(408\) 0 0
\(409\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) −16.0000 −0.789222
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 4.00000 0.196352
\(416\) 0 0
\(417\) 6.00000 0.293821
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 − 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 34.0000 1.65706 0.828529 − 0.559946i \(-0.189178\pi\)
0.828529 + 0.559946i \(0.189178\pi\)
\(422\) 0 0
\(423\) −4.00000 −0.194487
\(424\) 0 0
\(425\) 3.00000 0.145521
\(426\) 0 0
\(427\) 15.0000 0.725901
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 31.0000 1.49322 0.746609 − 0.665263i \(-0.231681\pi\)
0.746609 + 0.665263i \(0.231681\pi\)
\(432\) 0 0
\(433\) −22.0000 −1.05725 −0.528626 − 0.848855i \(-0.677293\pi\)
−0.528626 + 0.848855i \(0.677293\pi\)
\(434\) 0 0
\(435\) 6.00000 0.287678
\(436\) 0 0
\(437\) −12.0000 −0.574038
\(438\) 0 0
\(439\) −37.0000 −1.76591 −0.882957 − 0.469454i \(-0.844451\pi\)
−0.882957 + 0.469454i \(0.844451\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 0 0
\(443\) 34.0000 1.61539 0.807694 − 0.589601i \(-0.200715\pi\)
0.807694 + 0.589601i \(0.200715\pi\)
\(444\) 0 0
\(445\) −18.0000 −0.853282
\(446\) 0 0
\(447\) −36.0000 −1.70274
\(448\) 0 0
\(449\) −10.0000 −0.471929 −0.235965 − 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) 0 0
\(451\) −9.00000 −0.423793
\(452\) 0 0
\(453\) 32.0000 1.50349
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 9.00000 0.421002 0.210501 − 0.977594i \(-0.432490\pi\)
0.210501 + 0.977594i \(0.432490\pi\)
\(458\) 0 0
\(459\) 12.0000 0.560112
\(460\) 0 0
\(461\) −1.00000 −0.0465746 −0.0232873 − 0.999729i \(-0.507413\pi\)
−0.0232873 + 0.999729i \(0.507413\pi\)
\(462\) 0 0
\(463\) −12.0000 −0.557687 −0.278844 − 0.960337i \(-0.589951\pi\)
−0.278844 + 0.960337i \(0.589951\pi\)
\(464\) 0 0
\(465\) 6.00000 0.278243
\(466\) 0 0
\(467\) 37.0000 1.71216 0.856078 − 0.516847i \(-0.172894\pi\)
0.856078 + 0.516847i \(0.172894\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 6.00000 0.276465
\(472\) 0 0
\(473\) −3.00000 −0.137940
\(474\) 0 0
\(475\) 6.00000 0.275299
\(476\) 0 0
\(477\) 13.0000 0.595229
\(478\) 0 0
\(479\) 8.00000 0.365529 0.182765 − 0.983157i \(-0.441495\pi\)
0.182765 + 0.983157i \(0.441495\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −4.00000 −0.182006
\(484\) 0 0
\(485\) −7.00000 −0.317854
\(486\) 0 0
\(487\) −34.0000 −1.54069 −0.770344 − 0.637629i \(-0.779915\pi\)
−0.770344 + 0.637629i \(0.779915\pi\)
\(488\) 0 0
\(489\) −10.0000 −0.452216
\(490\) 0 0
\(491\) 20.0000 0.902587 0.451294 − 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 0 0
\(493\) −9.00000 −0.405340
\(494\) 0 0
\(495\) −3.00000 −0.134840
\(496\) 0 0
\(497\) −2.00000 −0.0897123
\(498\) 0 0
\(499\) 22.0000 0.984855 0.492428 − 0.870353i \(-0.336110\pi\)
0.492428 + 0.870353i \(0.336110\pi\)
\(500\) 0 0
\(501\) 24.0000 1.07224
\(502\) 0 0
\(503\) 24.0000 1.07011 0.535054 − 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 0 0
\(505\) −10.0000 −0.444994
\(506\) 0 0
\(507\) 26.0000 1.15470
\(508\) 0 0
\(509\) 36.0000 1.59567 0.797836 − 0.602875i \(-0.205978\pi\)
0.797836 + 0.602875i \(0.205978\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 24.0000 1.05963
\(514\) 0 0
\(515\) −8.00000 −0.352522
\(516\) 0 0
\(517\) 12.0000 0.527759
\(518\) 0 0
\(519\) −18.0000 −0.790112
\(520\) 0 0
\(521\) −23.0000 −1.00765 −0.503824 − 0.863806i \(-0.668074\pi\)
−0.503824 + 0.863806i \(0.668074\pi\)
\(522\) 0 0
\(523\) 44.0000 1.92399 0.961993 − 0.273075i \(-0.0880406\pi\)
0.961993 + 0.273075i \(0.0880406\pi\)
\(524\) 0 0
\(525\) 2.00000 0.0872872
\(526\) 0 0
\(527\) −9.00000 −0.392046
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −2.00000 −0.0864675
\(536\) 0 0
\(537\) 48.0000 2.07135
\(538\) 0 0
\(539\) 18.0000 0.775315
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 − 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 0 0
\(543\) −4.00000 −0.171656
\(544\) 0 0
\(545\) −3.00000 −0.128506
\(546\) 0 0
\(547\) −23.0000 −0.983409 −0.491704 − 0.870762i \(-0.663626\pi\)
−0.491704 + 0.870762i \(0.663626\pi\)
\(548\) 0 0
\(549\) −15.0000 −0.640184
\(550\) 0 0
\(551\) −18.0000 −0.766826
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) 0 0
\(555\) 2.00000 0.0848953
\(556\) 0 0
\(557\) −40.0000 −1.69485 −0.847427 − 0.530912i \(-0.821850\pi\)
−0.847427 + 0.530912i \(0.821850\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 18.0000 0.759961
\(562\) 0 0
\(563\) 11.0000 0.463595 0.231797 − 0.972764i \(-0.425539\pi\)
0.231797 + 0.972764i \(0.425539\pi\)
\(564\) 0 0
\(565\) −7.00000 −0.294492
\(566\) 0 0
\(567\) 11.0000 0.461957
\(568\) 0 0
\(569\) 10.0000 0.419222 0.209611 − 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 0 0
\(571\) 31.0000 1.29731 0.648655 − 0.761083i \(-0.275332\pi\)
0.648655 + 0.761083i \(0.275332\pi\)
\(572\) 0 0
\(573\) 42.0000 1.75458
\(574\) 0 0
\(575\) −2.00000 −0.0834058
\(576\) 0 0
\(577\) 30.0000 1.24892 0.624458 − 0.781058i \(-0.285320\pi\)
0.624458 + 0.781058i \(0.285320\pi\)
\(578\) 0 0
\(579\) 20.0000 0.831172
\(580\) 0 0
\(581\) −4.00000 −0.165948
\(582\) 0 0
\(583\) −39.0000 −1.61521
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −35.0000 −1.44460 −0.722302 − 0.691577i \(-0.756916\pi\)
−0.722302 + 0.691577i \(0.756916\pi\)
\(588\) 0 0
\(589\) −18.0000 −0.741677
\(590\) 0 0
\(591\) 36.0000 1.48084
\(592\) 0 0
\(593\) 12.0000 0.492781 0.246390 − 0.969171i \(-0.420755\pi\)
0.246390 + 0.969171i \(0.420755\pi\)
\(594\) 0 0
\(595\) −3.00000 −0.122988
\(596\) 0 0
\(597\) 16.0000 0.654836
\(598\) 0 0
\(599\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 21.0000 0.856608 0.428304 − 0.903635i \(-0.359111\pi\)
0.428304 + 0.903635i \(0.359111\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −2.00000 −0.0813116
\(606\) 0 0
\(607\) −22.0000 −0.892952 −0.446476 − 0.894795i \(-0.647321\pi\)
−0.446476 + 0.894795i \(0.647321\pi\)
\(608\) 0 0
\(609\) −6.00000 −0.243132
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 11.0000 0.444286 0.222143 − 0.975014i \(-0.428695\pi\)
0.222143 + 0.975014i \(0.428695\pi\)
\(614\) 0 0
\(615\) −6.00000 −0.241943
\(616\) 0 0
\(617\) 36.0000 1.44931 0.724653 − 0.689114i \(-0.242000\pi\)
0.724653 + 0.689114i \(0.242000\pi\)
\(618\) 0 0
\(619\) 37.0000 1.48716 0.743578 − 0.668649i \(-0.233127\pi\)
0.743578 + 0.668649i \(0.233127\pi\)
\(620\) 0 0
\(621\) −8.00000 −0.321029
\(622\) 0 0
\(623\) 18.0000 0.721155
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 36.0000 1.43770
\(628\) 0 0
\(629\) −3.00000 −0.119618
\(630\) 0 0
\(631\) 41.0000 1.63218 0.816092 − 0.577922i \(-0.196136\pi\)
0.816092 + 0.577922i \(0.196136\pi\)
\(632\) 0 0
\(633\) −6.00000 −0.238479
\(634\) 0 0
\(635\) 4.00000 0.158735
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 2.00000 0.0791188
\(640\) 0 0
\(641\) −33.0000 −1.30342 −0.651711 − 0.758468i \(-0.725948\pi\)
−0.651711 + 0.758468i \(0.725948\pi\)
\(642\) 0 0
\(643\) −11.0000 −0.433798 −0.216899 − 0.976194i \(-0.569594\pi\)
−0.216899 + 0.976194i \(0.569594\pi\)
\(644\) 0 0
\(645\) −2.00000 −0.0787499
\(646\) 0 0
\(647\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −6.00000 −0.235159
\(652\) 0 0
\(653\) 30.0000 1.17399 0.586995 − 0.809590i \(-0.300311\pi\)
0.586995 + 0.809590i \(0.300311\pi\)
\(654\) 0 0
\(655\) 12.0000 0.468879
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 − 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) −13.0000 −0.505641 −0.252821 − 0.967513i \(-0.581358\pi\)
−0.252821 + 0.967513i \(0.581358\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −6.00000 −0.232670
\(666\) 0 0
\(667\) 6.00000 0.232321
\(668\) 0 0
\(669\) 46.0000 1.77846
\(670\) 0 0
\(671\) 45.0000 1.73721
\(672\) 0 0
\(673\) −34.0000 −1.31060 −0.655302 − 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) 0 0
\(675\) 4.00000 0.153960
\(676\) 0 0
\(677\) 34.0000 1.30673 0.653363 − 0.757045i \(-0.273358\pi\)
0.653363 + 0.757045i \(0.273358\pi\)
\(678\) 0 0
\(679\) 7.00000 0.268635
\(680\) 0 0
\(681\) 26.0000 0.996322
\(682\) 0 0
\(683\) 31.0000 1.18618 0.593091 − 0.805135i \(-0.297907\pi\)
0.593091 + 0.805135i \(0.297907\pi\)
\(684\) 0 0
\(685\) 8.00000 0.305664
\(686\) 0 0
\(687\) 12.0000 0.457829
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −5.00000 −0.190209 −0.0951045 − 0.995467i \(-0.530319\pi\)
−0.0951045 + 0.995467i \(0.530319\pi\)
\(692\) 0 0
\(693\) 3.00000 0.113961
\(694\) 0 0
\(695\) −3.00000 −0.113796
\(696\) 0 0
\(697\) 9.00000 0.340899
\(698\) 0 0
\(699\) −8.00000 −0.302588
\(700\) 0 0
\(701\) −22.0000 −0.830929 −0.415464 − 0.909610i \(-0.636381\pi\)
−0.415464 + 0.909610i \(0.636381\pi\)
\(702\) 0 0
\(703\) −6.00000 −0.226294
\(704\) 0 0
\(705\) 8.00000 0.301297
\(706\) 0 0
\(707\) 10.0000 0.376089
\(708\) 0 0
\(709\) −19.0000 −0.713560 −0.356780 − 0.934188i \(-0.616125\pi\)
−0.356780 + 0.934188i \(0.616125\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) 6.00000 0.224702
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 18.0000 0.672222
\(718\) 0 0
\(719\) 18.0000 0.671287 0.335643 − 0.941989i \(-0.391046\pi\)
0.335643 + 0.941989i \(0.391046\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) 0 0
\(723\) −48.0000 −1.78514
\(724\) 0 0
\(725\) −3.00000 −0.111417
\(726\) 0 0
\(727\) −52.0000 −1.92857 −0.964287 − 0.264861i \(-0.914674\pi\)
−0.964287 + 0.264861i \(0.914674\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 3.00000 0.110959
\(732\) 0 0
\(733\) −41.0000 −1.51437 −0.757185 − 0.653201i \(-0.773426\pi\)
−0.757185 + 0.653201i \(0.773426\pi\)
\(734\) 0 0
\(735\) 12.0000 0.442627
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −27.0000 −0.993211 −0.496606 − 0.867976i \(-0.665420\pi\)
−0.496606 + 0.867976i \(0.665420\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −3.00000 −0.110059 −0.0550297 − 0.998485i \(-0.517525\pi\)
−0.0550297 + 0.998485i \(0.517525\pi\)
\(744\) 0 0
\(745\) 18.0000 0.659469
\(746\) 0 0
\(747\) 4.00000 0.146352
\(748\) 0 0
\(749\) 2.00000 0.0730784
\(750\) 0 0
\(751\) 22.0000 0.802791 0.401396 − 0.915905i \(-0.368525\pi\)
0.401396 + 0.915905i \(0.368525\pi\)
\(752\) 0 0
\(753\) 12.0000 0.437304
\(754\) 0 0
\(755\) −16.0000 −0.582300
\(756\) 0 0
\(757\) −50.0000 −1.81728 −0.908640 − 0.417579i \(-0.862879\pi\)
−0.908640 + 0.417579i \(0.862879\pi\)
\(758\) 0 0
\(759\) −12.0000 −0.435572
\(760\) 0 0
\(761\) −21.0000 −0.761249 −0.380625 − 0.924730i \(-0.624291\pi\)
−0.380625 + 0.924730i \(0.624291\pi\)
\(762\) 0 0
\(763\) 3.00000 0.108607
\(764\) 0 0
\(765\) 3.00000 0.108465
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 8.00000 0.288487 0.144244 − 0.989542i \(-0.453925\pi\)
0.144244 + 0.989542i \(0.453925\pi\)
\(770\) 0 0
\(771\) 44.0000 1.58462
\(772\) 0 0
\(773\) −41.0000 −1.47467 −0.737334 − 0.675529i \(-0.763915\pi\)
−0.737334 + 0.675529i \(0.763915\pi\)
\(774\) 0 0
\(775\) −3.00000 −0.107763
\(776\) 0 0
\(777\) −2.00000 −0.0717496
\(778\) 0 0
\(779\) 18.0000 0.644917
\(780\) 0 0
\(781\) −6.00000 −0.214697
\(782\) 0 0
\(783\) −12.0000 −0.428845
\(784\) 0 0
\(785\) −3.00000 −0.107075
\(786\) 0 0
\(787\) 10.0000 0.356462 0.178231 − 0.983989i \(-0.442963\pi\)
0.178231 + 0.983989i \(0.442963\pi\)
\(788\) 0 0
\(789\) −2.00000 −0.0712019
\(790\) 0 0
\(791\) 7.00000 0.248891
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −26.0000 −0.922125
\(796\) 0 0
\(797\) −26.0000 −0.920967 −0.460484 − 0.887668i \(-0.652324\pi\)
−0.460484 + 0.887668i \(0.652324\pi\)
\(798\) 0 0
\(799\) −12.0000 −0.424529
\(800\) 0 0
\(801\) −18.0000 −0.635999
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 2.00000 0.0704907
\(806\) 0 0
\(807\) 20.0000 0.704033
\(808\) 0 0
\(809\) 34.0000 1.19538 0.597688 − 0.801729i \(-0.296086\pi\)
0.597688 + 0.801729i \(0.296086\pi\)
\(810\) 0 0
\(811\) 16.0000 0.561836 0.280918 − 0.959732i \(-0.409361\pi\)
0.280918 + 0.959732i \(0.409361\pi\)
\(812\) 0 0
\(813\) −20.0000 −0.701431
\(814\) 0 0
\(815\) 5.00000 0.175142
\(816\) 0 0
\(817\) 6.00000 0.209913
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 8.00000 0.279202 0.139601 − 0.990208i \(-0.455418\pi\)
0.139601 + 0.990208i \(0.455418\pi\)
\(822\) 0 0
\(823\) −24.0000 −0.836587 −0.418294 − 0.908312i \(-0.637372\pi\)
−0.418294 + 0.908312i \(0.637372\pi\)
\(824\) 0 0
\(825\) 6.00000 0.208893
\(826\) 0 0
\(827\) 11.0000 0.382507 0.191254 − 0.981541i \(-0.438745\pi\)
0.191254 + 0.981541i \(0.438745\pi\)
\(828\) 0 0
\(829\) −39.0000 −1.35453 −0.677263 − 0.735741i \(-0.736834\pi\)
−0.677263 + 0.735741i \(0.736834\pi\)
\(830\) 0 0
\(831\) 40.0000 1.38758
\(832\) 0 0
\(833\) −18.0000 −0.623663
\(834\) 0 0
\(835\) −12.0000 −0.415277
\(836\) 0 0
\(837\) −12.0000 −0.414781
\(838\) 0 0
\(839\) −36.0000 −1.24286 −0.621429 − 0.783470i \(-0.713448\pi\)
−0.621429 + 0.783470i \(0.713448\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) 48.0000 1.65321
\(844\) 0 0
\(845\) −13.0000 −0.447214
\(846\) 0 0
\(847\) 2.00000 0.0687208
\(848\) 0 0
\(849\) −40.0000 −1.37280
\(850\) 0 0
\(851\) 2.00000 0.0685591
\(852\) 0 0
\(853\) 22.0000 0.753266 0.376633 − 0.926363i \(-0.377082\pi\)
0.376633 + 0.926363i \(0.377082\pi\)
\(854\) 0 0
\(855\) 6.00000 0.205196
\(856\) 0 0
\(857\) 33.0000 1.12726 0.563629 − 0.826028i \(-0.309405\pi\)
0.563629 + 0.826028i \(0.309405\pi\)
\(858\) 0 0
\(859\) 36.0000 1.22830 0.614152 − 0.789188i \(-0.289498\pi\)
0.614152 + 0.789188i \(0.289498\pi\)
\(860\) 0 0
\(861\) 6.00000 0.204479
\(862\) 0 0
\(863\) −33.0000 −1.12333 −0.561667 − 0.827364i \(-0.689840\pi\)
−0.561667 + 0.827364i \(0.689840\pi\)
\(864\) 0 0
\(865\) 9.00000 0.306009
\(866\) 0 0
\(867\) 16.0000 0.543388
\(868\) 0 0
\(869\) −24.0000 −0.814144
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −7.00000 −0.236914
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) −43.0000 −1.45201 −0.726003 − 0.687691i \(-0.758624\pi\)
−0.726003 + 0.687691i \(0.758624\pi\)
\(878\) 0 0
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) −9.00000 −0.303218 −0.151609 − 0.988441i \(-0.548445\pi\)
−0.151609 + 0.988441i \(0.548445\pi\)
\(882\) 0 0
\(883\) −29.0000 −0.975928 −0.487964 − 0.872864i \(-0.662260\pi\)
−0.487964 + 0.872864i \(0.662260\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 33.0000 1.10803 0.554016 − 0.832506i \(-0.313095\pi\)
0.554016 + 0.832506i \(0.313095\pi\)
\(888\) 0 0
\(889\) −4.00000 −0.134156
\(890\) 0 0
\(891\) 33.0000 1.10554
\(892\) 0 0
\(893\) −24.0000 −0.803129
\(894\) 0 0
\(895\) −24.0000 −0.802232
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 9.00000 0.300167
\(900\) 0 0
\(901\) 39.0000 1.29928
\(902\) 0 0
\(903\) 2.00000 0.0665558
\(904\) 0 0
\(905\) 2.00000 0.0664822
\(906\) 0 0
\(907\) 12.0000 0.398453 0.199227 − 0.979953i \(-0.436157\pi\)
0.199227 + 0.979953i \(0.436157\pi\)
\(908\) 0 0
\(909\) −10.0000 −0.331679
\(910\) 0 0
\(911\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) −12.0000 −0.397142
\(914\) 0 0
\(915\) 30.0000 0.991769
\(916\) 0 0
\(917\) −12.0000 −0.396275
\(918\) 0 0
\(919\) −32.0000 −1.05558 −0.527791 − 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 0 0
\(921\) 68.0000 2.24068
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −1.00000 −0.0328798
\(926\) 0 0
\(927\) −8.00000 −0.262754
\(928\) 0 0
\(929\) −3.00000 −0.0984268 −0.0492134 − 0.998788i \(-0.515671\pi\)
−0.0492134 + 0.998788i \(0.515671\pi\)
\(930\) 0 0
\(931\) −36.0000 −1.17985
\(932\) 0 0
\(933\) 50.0000 1.63693
\(934\) 0 0
\(935\) −9.00000 −0.294331
\(936\) 0 0
\(937\) −14.0000 −0.457360 −0.228680 − 0.973502i \(-0.573441\pi\)
−0.228680 + 0.973502i \(0.573441\pi\)
\(938\) 0 0
\(939\) −52.0000 −1.69696
\(940\) 0 0
\(941\) −6.00000 −0.195594 −0.0977972 − 0.995206i \(-0.531180\pi\)
−0.0977972 + 0.995206i \(0.531180\pi\)
\(942\) 0 0
\(943\) −6.00000 −0.195387
\(944\) 0 0
\(945\) −4.00000 −0.130120
\(946\) 0 0
\(947\) −9.00000 −0.292461 −0.146230 − 0.989251i \(-0.546714\pi\)
−0.146230 + 0.989251i \(0.546714\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −42.0000 −1.36194
\(952\) 0 0
\(953\) 36.0000 1.16615 0.583077 − 0.812417i \(-0.301849\pi\)
0.583077 + 0.812417i \(0.301849\pi\)
\(954\) 0 0
\(955\) −21.0000 −0.679544
\(956\) 0 0
\(957\) −18.0000 −0.581857
\(958\) 0 0
\(959\) −8.00000 −0.258333
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 0 0
\(963\) −2.00000 −0.0644491
\(964\) 0 0
\(965\) −10.0000 −0.321911
\(966\) 0 0
\(967\) −48.0000 −1.54358 −0.771788 − 0.635880i \(-0.780637\pi\)
−0.771788 + 0.635880i \(0.780637\pi\)
\(968\) 0 0
\(969\) −36.0000 −1.15649
\(970\) 0 0
\(971\) −9.00000 −0.288824 −0.144412 − 0.989518i \(-0.546129\pi\)
−0.144412 + 0.989518i \(0.546129\pi\)
\(972\) 0 0
\(973\) 3.00000 0.0961756
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 35.0000 1.11975 0.559875 − 0.828577i \(-0.310849\pi\)
0.559875 + 0.828577i \(0.310849\pi\)
\(978\) 0 0
\(979\) 54.0000 1.72585
\(980\) 0 0
\(981\) −3.00000 −0.0957826
\(982\) 0 0
\(983\) 27.0000 0.861166 0.430583 − 0.902551i \(-0.358308\pi\)
0.430583 + 0.902551i \(0.358308\pi\)
\(984\) 0 0
\(985\) −18.0000 −0.573528
\(986\) 0 0
\(987\) −8.00000 −0.254643
\(988\) 0 0
\(989\) −2.00000 −0.0635963
\(990\) 0 0
\(991\) −33.0000 −1.04828 −0.524140 − 0.851632i \(-0.675613\pi\)
−0.524140 + 0.851632i \(0.675613\pi\)
\(992\) 0 0
\(993\) 60.0000 1.90404
\(994\) 0 0
\(995\) −8.00000 −0.253617
\(996\) 0 0
\(997\) −12.0000 −0.380044 −0.190022 − 0.981780i \(-0.560856\pi\)
−0.190022 + 0.981780i \(0.560856\pi\)
\(998\) 0 0
\(999\) −4.00000 −0.126554
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 2960.2.a.c.1.1 1
4.3 odd 2 370.2.a.c.1.1 ✓ 1
12.11 even 2 3330.2.a.p.1.1 1
20.3 even 4 1850.2.b.c.149.2 2
20.7 even 4 1850.2.b.c.149.1 2
20.19 odd 2 1850.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.c.1.1 ✓ 1 4.3 odd 2
1850.2.a.i.1.1 1 20.19 odd 2
1850.2.b.c.149.1 2 20.7 even 4
1850.2.b.c.149.2 2 20.3 even 4
2960.2.a.c.1.1 1 1.1 even 1 trivial
3330.2.a.p.1.1 1 12.11 even 2