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Properties

Label 2800.2.a.x.1.1
Level $2800$
Weight $2$
Character 2800.1
Self dual yes
Analytic conductor $22.358$
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] = [2800,2,Mod(1,2800)] 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("2800.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2800.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,1,0,0,0,1,0,-2,0,-3,0,-2] 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: \(22.3581125660\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 350)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2800.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{7} -2.00000 q^{9} -3.00000 q^{11} -2.00000 q^{13} -3.00000 q^{17} +7.00000 q^{19} +1.00000 q^{21} -5.00000 q^{27} -6.00000 q^{29} +4.00000 q^{31} -3.00000 q^{33} -8.00000 q^{37} -2.00000 q^{39} -9.00000 q^{41} +8.00000 q^{43} -6.00000 q^{47} +1.00000 q^{49} -3.00000 q^{51} +12.0000 q^{53} +7.00000 q^{57} -12.0000 q^{59} -10.0000 q^{61} -2.00000 q^{63} -7.00000 q^{67} -6.00000 q^{71} -5.00000 q^{73} -3.00000 q^{77} -14.0000 q^{79} +1.00000 q^{81} -9.00000 q^{83} -6.00000 q^{87} -15.0000 q^{89} -2.00000 q^{91} +4.00000 q^{93} +10.0000 q^{97} +6.00000 q^{99} +O(q^{100})\)

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
\(3\) 1.00000 0.577350 0.288675 − 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 − 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.00000 −0.727607 −0.363803 − 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) 7.00000 1.60591 0.802955 − 0.596040i \(-0.203260\pi\)
0.802955 + 0.596040i \(0.203260\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) −3.00000 −0.522233
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −8.00000 −1.31519 −0.657596 − 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −9.00000 −1.40556 −0.702782 − 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.00000 −0.875190 −0.437595 − 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −3.00000 −0.420084
\(52\) 0 0
\(53\) 12.0000 1.64833 0.824163 − 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 7.00000 0.927173
\(58\) 0 0
\(59\) −12.0000 −1.56227 −0.781133 − 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) −2.00000 −0.251976
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −7.00000 −0.855186 −0.427593 − 0.903971i \(-0.640638\pi\)
−0.427593 + 0.903971i \(0.640638\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 − 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) −5.00000 −0.585206 −0.292603 − 0.956234i \(-0.594521\pi\)
−0.292603 + 0.956234i \(0.594521\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.00000 −0.341882
\(78\) 0 0
\(79\) −14.0000 −1.57512 −0.787562 − 0.616236i \(-0.788657\pi\)
−0.787562 + 0.616236i \(0.788657\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −9.00000 −0.987878 −0.493939 − 0.869496i \(-0.664443\pi\)
−0.493939 + 0.869496i \(0.664443\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) −15.0000 −1.59000 −0.794998 − 0.606612i \(-0.792528\pi\)
−0.794998 + 0.606612i \(0.792528\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 0 0
\(93\) 4.00000 0.414781
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 − 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 0 0
\(99\) 6.00000 0.603023
\(100\) 0 0
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 20.0000 1.97066 0.985329 − 0.170664i \(-0.0545913\pi\)
0.985329 + 0.170664i \(0.0545913\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3.00000 0.290021 0.145010 − 0.989430i \(-0.453678\pi\)
0.145010 + 0.989430i \(0.453678\pi\)
\(108\) 0 0
\(109\) 14.0000 1.34096 0.670478 − 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) 0 0
\(113\) 9.00000 0.846649 0.423324 − 0.905978i \(-0.360863\pi\)
0.423324 + 0.905978i \(0.360863\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 4.00000 0.369800
\(118\) 0 0
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) −9.00000 −0.811503
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 − 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 0 0
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 7.00000 0.606977
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −21.0000 −1.79415 −0.897076 − 0.441877i \(-0.854313\pi\)
−0.897076 + 0.441877i \(0.854313\pi\)
\(138\) 0 0
\(139\) 7.00000 0.593732 0.296866 − 0.954919i \(-0.404058\pi\)
0.296866 + 0.954919i \(0.404058\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 0 0
\(143\) 6.00000 0.501745
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 12.0000 0.983078 0.491539 − 0.870855i \(-0.336434\pi\)
0.491539 + 0.870855i \(0.336434\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 − 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −20.0000 −1.59617 −0.798087 − 0.602542i \(-0.794154\pi\)
−0.798087 + 0.602542i \(0.794154\pi\)
\(158\) 0 0
\(159\) 12.0000 0.951662
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 5.00000 0.391630 0.195815 − 0.980641i \(-0.437265\pi\)
0.195815 + 0.980641i \(0.437265\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.0000 0.928588 0.464294 − 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −14.0000 −1.07061
\(172\) 0 0
\(173\) 6.00000 0.456172 0.228086 − 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −12.0000 −0.901975
\(178\) 0 0
\(179\) −3.00000 −0.224231 −0.112115 − 0.993695i \(-0.535763\pi\)
−0.112115 + 0.993695i \(0.535763\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 − 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) −10.0000 −0.739221
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 9.00000 0.658145
\(188\) 0 0
\(189\) −5.00000 −0.363696
\(190\) 0 0
\(191\) 6.00000 0.434145 0.217072 − 0.976156i \(-0.430349\pi\)
0.217072 + 0.976156i \(0.430349\pi\)
\(192\) 0 0
\(193\) −5.00000 −0.359908 −0.179954 − 0.983675i \(-0.557595\pi\)
−0.179954 + 0.983675i \(0.557595\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 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 \(-0.665278\pi\)
−0.496217 + 0.868199i \(0.665278\pi\)
\(200\) 0 0
\(201\) −7.00000 −0.493742
\(202\) 0 0
\(203\) −6.00000 −0.421117
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −21.0000 −1.45260
\(210\) 0 0
\(211\) −17.0000 −1.17033 −0.585164 − 0.810915i \(-0.698970\pi\)
−0.585164 + 0.810915i \(0.698970\pi\)
\(212\) 0 0
\(213\) −6.00000 −0.411113
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 4.00000 0.271538
\(218\) 0 0
\(219\) −5.00000 −0.337869
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) 0 0
\(223\) 14.0000 0.937509 0.468755 − 0.883328i \(-0.344703\pi\)
0.468755 + 0.883328i \(0.344703\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 12.0000 0.796468 0.398234 − 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) 26.0000 1.71813 0.859064 − 0.511868i \(-0.171046\pi\)
0.859064 + 0.511868i \(0.171046\pi\)
\(230\) 0 0
\(231\) −3.00000 −0.197386
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 − 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −14.0000 −0.909398
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 − 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) −25.0000 −1.61039 −0.805196 − 0.593009i \(-0.797940\pi\)
−0.805196 + 0.593009i \(0.797940\pi\)
\(242\) 0 0
\(243\) 16.0000 1.02640
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −14.0000 −0.890799
\(248\) 0 0
\(249\) −9.00000 −0.570352
\(250\) 0 0
\(251\) −15.0000 −0.946792 −0.473396 − 0.880850i \(-0.656972\pi\)
−0.473396 + 0.880850i \(0.656972\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 30.0000 1.87135 0.935674 − 0.352865i \(-0.114792\pi\)
0.935674 + 0.352865i \(0.114792\pi\)
\(258\) 0 0
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) 12.0000 0.742781
\(262\) 0 0
\(263\) 6.00000 0.369976 0.184988 − 0.982741i \(-0.440775\pi\)
0.184988 + 0.982741i \(0.440775\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −15.0000 −0.917985
\(268\) 0 0
\(269\) −6.00000 −0.365826 −0.182913 − 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) −2.00000 −0.121491 −0.0607457 − 0.998153i \(-0.519348\pi\)
−0.0607457 + 0.998153i \(0.519348\pi\)
\(272\) 0 0
\(273\) −2.00000 −0.121046
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 − 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 0 0
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 − 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) −1.00000 −0.0594438 −0.0297219 − 0.999558i \(-0.509462\pi\)
−0.0297219 + 0.999558i \(0.509462\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −9.00000 −0.531253
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 10.0000 0.586210
\(292\) 0 0
\(293\) 6.00000 0.350524 0.175262 − 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 15.0000 0.870388
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −7.00000 −0.399511 −0.199756 − 0.979846i \(-0.564015\pi\)
−0.199756 + 0.979846i \(0.564015\pi\)
\(308\) 0 0
\(309\) 20.0000 1.13776
\(310\) 0 0
\(311\) 18.0000 1.02069 0.510343 − 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 0 0
\(313\) 10.0000 0.565233 0.282617 − 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 12.0000 0.673987 0.336994 − 0.941507i \(-0.390590\pi\)
0.336994 + 0.941507i \(0.390590\pi\)
\(318\) 0 0
\(319\) 18.0000 1.00781
\(320\) 0 0
\(321\) 3.00000 0.167444
\(322\) 0 0
\(323\) −21.0000 −1.16847
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 14.0000 0.774202
\(328\) 0 0
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) 25.0000 1.37412 0.687062 − 0.726599i \(-0.258900\pi\)
0.687062 + 0.726599i \(0.258900\pi\)
\(332\) 0 0
\(333\) 16.0000 0.876795
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 13.0000 0.708155 0.354078 − 0.935216i \(-0.384795\pi\)
0.354078 + 0.935216i \(0.384795\pi\)
\(338\) 0 0
\(339\) 9.00000 0.488813
\(340\) 0 0
\(341\) −12.0000 −0.649836
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −21.0000 −1.12734 −0.563670 − 0.826000i \(-0.690611\pi\)
−0.563670 + 0.826000i \(0.690611\pi\)
\(348\) 0 0
\(349\) 8.00000 0.428230 0.214115 − 0.976808i \(-0.431313\pi\)
0.214115 + 0.976808i \(0.431313\pi\)
\(350\) 0 0
\(351\) 10.0000 0.533761
\(352\) 0 0
\(353\) −30.0000 −1.59674 −0.798369 − 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −3.00000 −0.158777
\(358\) 0 0
\(359\) 6.00000 0.316668 0.158334 − 0.987386i \(-0.449388\pi\)
0.158334 + 0.987386i \(0.449388\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) 0 0
\(363\) −2.00000 −0.104973
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 − 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 0 0
\(369\) 18.0000 0.937043
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) 0 0
\(373\) 4.00000 0.207112 0.103556 − 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) −17.0000 −0.873231 −0.436616 − 0.899648i \(-0.643823\pi\)
−0.436616 + 0.899648i \(0.643823\pi\)
\(380\) 0 0
\(381\) 2.00000 0.102463
\(382\) 0 0
\(383\) −30.0000 −1.53293 −0.766464 − 0.642287i \(-0.777986\pi\)
−0.766464 + 0.642287i \(0.777986\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −16.0000 −0.813326
\(388\) 0 0
\(389\) −24.0000 −1.21685 −0.608424 − 0.793612i \(-0.708198\pi\)
−0.608424 + 0.793612i \(0.708198\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −2.00000 −0.100377 −0.0501886 − 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 0 0
\(399\) 7.00000 0.350438
\(400\) 0 0
\(401\) −27.0000 −1.34832 −0.674158 − 0.738587i \(-0.735493\pi\)
−0.674158 + 0.738587i \(0.735493\pi\)
\(402\) 0 0
\(403\) −8.00000 −0.398508
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 24.0000 1.18964
\(408\) 0 0
\(409\) −25.0000 −1.23617 −0.618085 − 0.786111i \(-0.712091\pi\)
−0.618085 + 0.786111i \(0.712091\pi\)
\(410\) 0 0
\(411\) −21.0000 −1.03585
\(412\) 0 0
\(413\) −12.0000 −0.590481
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 7.00000 0.342791
\(418\) 0 0
\(419\) 3.00000 0.146560 0.0732798 − 0.997311i \(-0.476653\pi\)
0.0732798 + 0.997311i \(0.476653\pi\)
\(420\) 0 0
\(421\) 20.0000 0.974740 0.487370 − 0.873195i \(-0.337956\pi\)
0.487370 + 0.873195i \(0.337956\pi\)
\(422\) 0 0
\(423\) 12.0000 0.583460
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −10.0000 −0.483934
\(428\) 0 0
\(429\) 6.00000 0.289683
\(430\) 0 0
\(431\) 36.0000 1.73406 0.867029 − 0.498257i \(-0.166026\pi\)
0.867029 + 0.498257i \(0.166026\pi\)
\(432\) 0 0
\(433\) −11.0000 −0.528626 −0.264313 − 0.964437i \(-0.585145\pi\)
−0.264313 + 0.964437i \(0.585145\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 4.00000 0.190910 0.0954548 − 0.995434i \(-0.469569\pi\)
0.0954548 + 0.995434i \(0.469569\pi\)
\(440\) 0 0
\(441\) −2.00000 −0.0952381
\(442\) 0 0
\(443\) 21.0000 0.997740 0.498870 − 0.866677i \(-0.333748\pi\)
0.498870 + 0.866677i \(0.333748\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 12.0000 0.567581
\(448\) 0 0
\(449\) −15.0000 −0.707894 −0.353947 − 0.935266i \(-0.615161\pi\)
−0.353947 + 0.935266i \(0.615161\pi\)
\(450\) 0 0
\(451\) 27.0000 1.27138
\(452\) 0 0
\(453\) −8.00000 −0.375873
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −17.0000 −0.795226 −0.397613 − 0.917553i \(-0.630161\pi\)
−0.397613 + 0.917553i \(0.630161\pi\)
\(458\) 0 0
\(459\) 15.0000 0.700140
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 − 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 − 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 36.0000 1.66588 0.832941 − 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) 0 0
\(469\) −7.00000 −0.323230
\(470\) 0 0
\(471\) −20.0000 −0.921551
\(472\) 0 0
\(473\) −24.0000 −1.10352
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −24.0000 −1.09888
\(478\) 0 0
\(479\) −18.0000 −0.822441 −0.411220 − 0.911536i \(-0.634897\pi\)
−0.411220 + 0.911536i \(0.634897\pi\)
\(480\) 0 0
\(481\) 16.0000 0.729537
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −34.0000 −1.54069 −0.770344 − 0.637629i \(-0.779915\pi\)
−0.770344 + 0.637629i \(0.779915\pi\)
\(488\) 0 0
\(489\) 5.00000 0.226108
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 − 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) 18.0000 0.810679
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −6.00000 −0.269137
\(498\) 0 0
\(499\) 28.0000 1.25345 0.626726 − 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 0 0
\(503\) −6.00000 −0.267527 −0.133763 − 0.991013i \(-0.542706\pi\)
−0.133763 + 0.991013i \(0.542706\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) 0 0
\(509\) 42.0000 1.86162 0.930809 − 0.365507i \(-0.119104\pi\)
0.930809 + 0.365507i \(0.119104\pi\)
\(510\) 0 0
\(511\) −5.00000 −0.221187
\(512\) 0 0
\(513\) −35.0000 −1.54529
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 18.0000 0.791639
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) 39.0000 1.70862 0.854311 − 0.519763i \(-0.173980\pi\)
0.854311 + 0.519763i \(0.173980\pi\)
\(522\) 0 0
\(523\) −7.00000 −0.306089 −0.153044 − 0.988219i \(-0.548908\pi\)
−0.153044 + 0.988219i \(0.548908\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −12.0000 −0.522728
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 24.0000 1.04151
\(532\) 0 0
\(533\) 18.0000 0.779667
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −3.00000 −0.129460
\(538\) 0 0
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 − 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) 0 0
\(543\) 2.00000 0.0858282
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 35.0000 1.49649 0.748246 − 0.663421i \(-0.230896\pi\)
0.748246 + 0.663421i \(0.230896\pi\)
\(548\) 0 0
\(549\) 20.0000 0.853579
\(550\) 0 0
\(551\) −42.0000 −1.78926
\(552\) 0 0
\(553\) −14.0000 −0.595341
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −36.0000 −1.52537 −0.762684 − 0.646771i \(-0.776119\pi\)
−0.762684 + 0.646771i \(0.776119\pi\)
\(558\) 0 0
\(559\) −16.0000 −0.676728
\(560\) 0 0
\(561\) 9.00000 0.379980
\(562\) 0 0
\(563\) 12.0000 0.505740 0.252870 − 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −27.0000 −1.13190 −0.565949 − 0.824440i \(-0.691490\pi\)
−0.565949 + 0.824440i \(0.691490\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 − 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 0 0
\(573\) 6.00000 0.250654
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 7.00000 0.291414 0.145707 − 0.989328i \(-0.453454\pi\)
0.145707 + 0.989328i \(0.453454\pi\)
\(578\) 0 0
\(579\) −5.00000 −0.207793
\(580\) 0 0
\(581\) −9.00000 −0.373383
\(582\) 0 0
\(583\) −36.0000 −1.49097
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 39.0000 1.60970 0.804851 − 0.593477i \(-0.202245\pi\)
0.804851 + 0.593477i \(0.202245\pi\)
\(588\) 0 0
\(589\) 28.0000 1.15372
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) 0 0
\(593\) 27.0000 1.10876 0.554379 − 0.832265i \(-0.312956\pi\)
0.554379 + 0.832265i \(0.312956\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −14.0000 −0.572982
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 − 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) −7.00000 −0.285536 −0.142768 − 0.989756i \(-0.545600\pi\)
−0.142768 + 0.989756i \(0.545600\pi\)
\(602\) 0 0
\(603\) 14.0000 0.570124
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 44.0000 1.78590 0.892952 − 0.450151i \(-0.148630\pi\)
0.892952 + 0.450151i \(0.148630\pi\)
\(608\) 0 0
\(609\) −6.00000 −0.243132
\(610\) 0 0
\(611\) 12.0000 0.485468
\(612\) 0 0
\(613\) −2.00000 −0.0807792 −0.0403896 − 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 − 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 0 0
\(619\) 28.0000 1.12542 0.562708 − 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −15.0000 −0.600962
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −21.0000 −0.838659
\(628\) 0 0
\(629\) 24.0000 0.956943
\(630\) 0 0
\(631\) −32.0000 −1.27390 −0.636950 − 0.770905i \(-0.719804\pi\)
−0.636950 + 0.770905i \(0.719804\pi\)
\(632\) 0 0
\(633\) −17.0000 −0.675689
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −2.00000 −0.0792429
\(638\) 0 0
\(639\) 12.0000 0.474713
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 − 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 0 0
\(643\) −40.0000 −1.57745 −0.788723 − 0.614749i \(-0.789257\pi\)
−0.788723 + 0.614749i \(0.789257\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −24.0000 −0.943537 −0.471769 − 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 0 0
\(649\) 36.0000 1.41312
\(650\) 0 0
\(651\) 4.00000 0.156772
\(652\) 0 0
\(653\) −36.0000 −1.40879 −0.704394 − 0.709809i \(-0.748781\pi\)
−0.704394 + 0.709809i \(0.748781\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 10.0000 0.390137
\(658\) 0 0
\(659\) −9.00000 −0.350590 −0.175295 − 0.984516i \(-0.556088\pi\)
−0.175295 + 0.984516i \(0.556088\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 − 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) 0 0
\(663\) 6.00000 0.233021
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 14.0000 0.541271
\(670\) 0 0
\(671\) 30.0000 1.15814
\(672\) 0 0
\(673\) −2.00000 −0.0770943 −0.0385472 − 0.999257i \(-0.512273\pi\)
−0.0385472 + 0.999257i \(0.512273\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 10.0000 0.383765
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) 3.00000 0.114792 0.0573959 − 0.998351i \(-0.481720\pi\)
0.0573959 + 0.998351i \(0.481720\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 26.0000 0.991962
\(688\) 0 0
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) 19.0000 0.722794 0.361397 − 0.932412i \(-0.382300\pi\)
0.361397 + 0.932412i \(0.382300\pi\)
\(692\) 0 0
\(693\) 6.00000 0.227921
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 27.0000 1.02270
\(698\) 0 0
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) −56.0000 −2.11208
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −28.0000 −1.05156 −0.525781 − 0.850620i \(-0.676227\pi\)
−0.525781 + 0.850620i \(0.676227\pi\)
\(710\) 0 0
\(711\) 28.0000 1.05008
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 12.0000 0.448148
\(718\) 0 0
\(719\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 20.0000 0.744839
\(722\) 0 0
\(723\) −25.0000 −0.929760
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −34.0000 −1.26099 −0.630495 − 0.776193i \(-0.717148\pi\)
−0.630495 + 0.776193i \(0.717148\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −24.0000 −0.887672
\(732\) 0 0
\(733\) 40.0000 1.47743 0.738717 − 0.674016i \(-0.235432\pi\)
0.738717 + 0.674016i \(0.235432\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 21.0000 0.773545
\(738\) 0 0
\(739\) −20.0000 −0.735712 −0.367856 − 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) −14.0000 −0.514303
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 − 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 18.0000 0.658586
\(748\) 0 0
\(749\) 3.00000 0.109618
\(750\) 0 0
\(751\) 46.0000 1.67856 0.839282 − 0.543696i \(-0.182976\pi\)
0.839282 + 0.543696i \(0.182976\pi\)
\(752\) 0 0
\(753\) −15.0000 −0.546630
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 34.0000 1.23575 0.617876 − 0.786276i \(-0.287994\pi\)
0.617876 + 0.786276i \(0.287994\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −9.00000 −0.326250 −0.163125 − 0.986605i \(-0.552157\pi\)
−0.163125 + 0.986605i \(0.552157\pi\)
\(762\) 0 0
\(763\) 14.0000 0.506834
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 24.0000 0.866590
\(768\) 0 0
\(769\) 23.0000 0.829401 0.414701 − 0.909958i \(-0.363886\pi\)
0.414701 + 0.909958i \(0.363886\pi\)
\(770\) 0 0
\(771\) 30.0000 1.08042
\(772\) 0 0
\(773\) 12.0000 0.431610 0.215805 − 0.976436i \(-0.430762\pi\)
0.215805 + 0.976436i \(0.430762\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −8.00000 −0.286998
\(778\) 0 0
\(779\) −63.0000 −2.25721
\(780\) 0 0
\(781\) 18.0000 0.644091
\(782\) 0 0
\(783\) 30.0000 1.07211
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −4.00000 −0.142585 −0.0712923 − 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) 0 0
\(789\) 6.00000 0.213606
\(790\) 0 0
\(791\) 9.00000 0.320003
\(792\) 0 0
\(793\) 20.0000 0.710221
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 30.0000 1.06265 0.531327 − 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 0 0
\(799\) 18.0000 0.636794
\(800\) 0 0
\(801\) 30.0000 1.06000
\(802\) 0 0
\(803\) 15.0000 0.529339
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −6.00000 −0.211210
\(808\) 0 0
\(809\) 6.00000 0.210949 0.105474 − 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) −44.0000 −1.54505 −0.772524 − 0.634985i \(-0.781006\pi\)
−0.772524 + 0.634985i \(0.781006\pi\)
\(812\) 0 0
\(813\) −2.00000 −0.0701431
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 56.0000 1.95919
\(818\) 0 0
\(819\) 4.00000 0.139771
\(820\) 0 0
\(821\) 24.0000 0.837606 0.418803 − 0.908077i \(-0.362450\pi\)
0.418803 + 0.908077i \(0.362450\pi\)
\(822\) 0 0
\(823\) 26.0000 0.906303 0.453152 − 0.891434i \(-0.350300\pi\)
0.453152 + 0.891434i \(0.350300\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 9.00000 0.312961 0.156480 − 0.987681i \(-0.449985\pi\)
0.156480 + 0.987681i \(0.449985\pi\)
\(828\) 0 0
\(829\) −4.00000 −0.138926 −0.0694629 − 0.997585i \(-0.522129\pi\)
−0.0694629 + 0.997585i \(0.522129\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) 0 0
\(833\) −3.00000 −0.103944
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −20.0000 −0.691301
\(838\) 0 0
\(839\) 6.00000 0.207143 0.103572 − 0.994622i \(-0.466973\pi\)
0.103572 + 0.994622i \(0.466973\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) −18.0000 −0.619953
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −2.00000 −0.0687208
\(848\) 0 0
\(849\) −1.00000 −0.0343199
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 10.0000 0.342393 0.171197 − 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −15.0000 −0.512390 −0.256195 − 0.966625i \(-0.582469\pi\)
−0.256195 + 0.966625i \(0.582469\pi\)
\(858\) 0 0
\(859\) 31.0000 1.05771 0.528853 − 0.848713i \(-0.322622\pi\)
0.528853 + 0.848713i \(0.322622\pi\)
\(860\) 0 0
\(861\) −9.00000 −0.306719
\(862\) 0 0
\(863\) −12.0000 −0.408485 −0.204242 − 0.978920i \(-0.565473\pi\)
−0.204242 + 0.978920i \(0.565473\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −8.00000 −0.271694
\(868\) 0 0
\(869\) 42.0000 1.42475
\(870\) 0 0
\(871\) 14.0000 0.474372
\(872\) 0 0
\(873\) −20.0000 −0.676897
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −32.0000 −1.08056 −0.540282 − 0.841484i \(-0.681682\pi\)
−0.540282 + 0.841484i \(0.681682\pi\)
\(878\) 0 0
\(879\) 6.00000 0.202375
\(880\) 0 0
\(881\) −6.00000 −0.202145 −0.101073 − 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) 0 0
\(883\) 47.0000 1.58168 0.790838 − 0.612026i \(-0.209645\pi\)
0.790838 + 0.612026i \(0.209645\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 6.00000 0.201460 0.100730 − 0.994914i \(-0.467882\pi\)
0.100730 + 0.994914i \(0.467882\pi\)
\(888\) 0 0
\(889\) 2.00000 0.0670778
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) 0 0
\(893\) −42.0000 −1.40548
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −24.0000 −0.800445
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) 8.00000 0.266223
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −4.00000 −0.132818 −0.0664089 − 0.997792i \(-0.521154\pi\)
−0.0664089 + 0.997792i \(0.521154\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −36.0000 −1.19273 −0.596367 − 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) 0 0
\(913\) 27.0000 0.893570
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 34.0000 1.12156 0.560778 − 0.827966i \(-0.310502\pi\)
0.560778 + 0.827966i \(0.310502\pi\)
\(920\) 0 0
\(921\) −7.00000 −0.230658
\(922\) 0 0
\(923\) 12.0000 0.394985
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −40.0000 −1.31377
\(928\) 0 0
\(929\) −18.0000 −0.590561 −0.295280 − 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 0 0
\(931\) 7.00000 0.229416
\(932\) 0 0
\(933\) 18.0000 0.589294
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −29.0000 −0.947389 −0.473694 − 0.880689i \(-0.657080\pi\)
−0.473694 + 0.880689i \(0.657080\pi\)
\(938\) 0 0
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) 24.0000 0.782378 0.391189 − 0.920310i \(-0.372064\pi\)
0.391189 + 0.920310i \(0.372064\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 12.0000 0.389948 0.194974 − 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 0 0
\(949\) 10.0000 0.324614
\(950\) 0 0
\(951\) 12.0000 0.389127
\(952\) 0 0
\(953\) −57.0000 −1.84641 −0.923206 − 0.384307i \(-0.874441\pi\)
−0.923206 + 0.384307i \(0.874441\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 18.0000 0.581857
\(958\) 0 0
\(959\) −21.0000 −0.678125
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) −6.00000 −0.193347
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −34.0000 −1.09337 −0.546683 − 0.837340i \(-0.684110\pi\)
−0.546683 + 0.837340i \(0.684110\pi\)
\(968\) 0 0
\(969\) −21.0000 −0.674617
\(970\) 0 0
\(971\) −9.00000 −0.288824 −0.144412 − 0.989518i \(-0.546129\pi\)
−0.144412 + 0.989518i \(0.546129\pi\)
\(972\) 0 0
\(973\) 7.00000 0.224410
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 3.00000 0.0959785 0.0479893 − 0.998848i \(-0.484719\pi\)
0.0479893 + 0.998848i \(0.484719\pi\)
\(978\) 0 0
\(979\) 45.0000 1.43821
\(980\) 0 0
\(981\) −28.0000 −0.893971
\(982\) 0 0
\(983\) 12.0000 0.382741 0.191370 − 0.981518i \(-0.438707\pi\)
0.191370 + 0.981518i \(0.438707\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −6.00000 −0.190982
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −20.0000 −0.635321 −0.317660 − 0.948205i \(-0.602897\pi\)
−0.317660 + 0.948205i \(0.602897\pi\)
\(992\) 0 0
\(993\) 25.0000 0.793351
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −62.0000 −1.96356 −0.981780 − 0.190022i \(-0.939144\pi\)
−0.981780 + 0.190022i \(0.939144\pi\)
\(998\) 0 0
\(999\) 40.0000 1.26554
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 2800.2.a.x.1.1 1
4.3 odd 2 350.2.a.a.1.1 ✓ 1
5.2 odd 4 2800.2.g.i.449.1 2
5.3 odd 4 2800.2.g.i.449.2 2
5.4 even 2 2800.2.a.h.1.1 1
12.11 even 2 3150.2.a.x.1.1 1
20.3 even 4 350.2.c.c.99.2 2
20.7 even 4 350.2.c.c.99.1 2
20.19 odd 2 350.2.a.e.1.1 yes 1
28.27 even 2 2450.2.a.m.1.1 1
60.23 odd 4 3150.2.g.f.2899.1 2
60.47 odd 4 3150.2.g.f.2899.2 2
60.59 even 2 3150.2.a.m.1.1 1
140.27 odd 4 2450.2.c.h.99.1 2
140.83 odd 4 2450.2.c.h.99.2 2
140.139 even 2 2450.2.a.x.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
350.2.a.a.1.1 ✓ 1 4.3 odd 2
350.2.a.e.1.1 yes 1 20.19 odd 2
350.2.c.c.99.1 2 20.7 even 4
350.2.c.c.99.2 2 20.3 even 4
2450.2.a.m.1.1 1 28.27 even 2
2450.2.a.x.1.1 1 140.139 even 2
2450.2.c.h.99.1 2 140.27 odd 4
2450.2.c.h.99.2 2 140.83 odd 4
2800.2.a.h.1.1 1 5.4 even 2
2800.2.a.x.1.1 1 1.1 even 1 trivial
2800.2.g.i.449.1 2 5.2 odd 4
2800.2.g.i.449.2 2 5.3 odd 4
3150.2.a.m.1.1 1 60.59 even 2
3150.2.a.x.1.1 1 12.11 even 2
3150.2.g.f.2899.1 2 60.23 odd 4
3150.2.g.f.2899.2 2 60.47 odd 4