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Properties

Label 7744.2.a.a.1.1
Level $7744$
Weight $2$
Character 7744.1
Self dual yes
Analytic conductor $61.836$
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] = [7744,2,Mod(1,7744)] 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("7744.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7744, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7744 = 2^{6} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7744.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-3,0,-1,0,0,0,6,0,0,0,-6] 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: \(61.8361513253\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 352)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 7744.1

$q$-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)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -1.00000 q^{5} +6.00000 q^{9} -6.00000 q^{13} +3.00000 q^{15} +4.00000 q^{17} +6.00000 q^{19} +3.00000 q^{23} -4.00000 q^{25} -9.00000 q^{27} -4.00000 q^{29} -9.00000 q^{31} -7.00000 q^{37} +18.0000 q^{39} +2.00000 q^{41} +6.00000 q^{43} -6.00000 q^{45} +12.0000 q^{47} -7.00000 q^{49} -12.0000 q^{51} -2.00000 q^{53} -18.0000 q^{57} -9.00000 q^{59} +8.00000 q^{61} +6.00000 q^{65} +15.0000 q^{67} -9.00000 q^{69} -3.00000 q^{71} +6.00000 q^{73} +12.0000 q^{75} +6.00000 q^{79} +9.00000 q^{81} -6.00000 q^{83} -4.00000 q^{85} +12.0000 q^{87} -5.00000 q^{89} +27.0000 q^{93} -6.00000 q^{95} -3.00000 q^{97} +O(q^{100})\)

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
\(3\) −3.00000 −1.73205 −0.866025 − 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214 −0.223607 − 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −6.00000 −1.66410 −0.832050 − 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 3.00000 0.774597
\(16\) 0 0
\(17\) 4.00000 0.970143 0.485071 − 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 − 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.00000 0.625543 0.312772 − 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) −9.00000 −1.73205
\(28\) 0 0
\(29\) −4.00000 −0.742781 −0.371391 − 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) −9.00000 −1.61645 −0.808224 − 0.588875i \(-0.799571\pi\)
−0.808224 + 0.588875i \(0.799571\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −7.00000 −1.15079 −0.575396 − 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) 0 0
\(39\) 18.0000 2.88231
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 − 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 0 0
\(45\) −6.00000 −0.894427
\(46\) 0 0
\(47\) 12.0000 1.75038 0.875190 − 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) −12.0000 −1.68034
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 − 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −18.0000 −2.38416
\(58\) 0 0
\(59\) −9.00000 −1.17170 −0.585850 − 0.810419i \(-0.699239\pi\)
−0.585850 + 0.810419i \(0.699239\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 − 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.00000 0.744208
\(66\) 0 0
\(67\) 15.0000 1.83254 0.916271 − 0.400559i \(-0.131184\pi\)
0.916271 + 0.400559i \(0.131184\pi\)
\(68\) 0 0
\(69\) −9.00000 −1.08347
\(70\) 0 0
\(71\) −3.00000 −0.356034 −0.178017 − 0.984027i \(-0.556968\pi\)
−0.178017 + 0.984027i \(0.556968\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 − 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) 12.0000 1.38564
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 6.00000 0.675053 0.337526 − 0.941316i \(-0.390410\pi\)
0.337526 + 0.941316i \(0.390410\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) −6.00000 −0.658586 −0.329293 − 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) 0 0
\(87\) 12.0000 1.28654
\(88\) 0 0
\(89\) −5.00000 −0.529999 −0.264999 − 0.964249i \(-0.585372\pi\)
−0.264999 + 0.964249i \(0.585372\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 27.0000 2.79977
\(94\) 0 0
\(95\) −6.00000 −0.615587
\(96\) 0 0
\(97\) −3.00000 −0.304604 −0.152302 − 0.988334i \(-0.548669\pi\)
−0.152302 + 0.988334i \(0.548669\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 16.0000 1.59206 0.796030 − 0.605257i \(-0.206930\pi\)
0.796030 + 0.605257i \(0.206930\pi\)
\(102\) 0 0
\(103\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0000 1.16008 0.580042 − 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) −6.00000 −0.574696 −0.287348 − 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) 0 0
\(111\) 21.0000 1.99323
\(112\) 0 0
\(113\) 1.00000 0.0940721 0.0470360 − 0.998893i \(-0.485022\pi\)
0.0470360 + 0.998893i \(0.485022\pi\)
\(114\) 0 0
\(115\) −3.00000 −0.279751
\(116\) 0 0
\(117\) −36.0000 −3.32820
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) −6.00000 −0.541002
\(124\) 0 0
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) 6.00000 0.532414 0.266207 − 0.963916i \(-0.414230\pi\)
0.266207 + 0.963916i \(0.414230\pi\)
\(128\) 0 0
\(129\) −18.0000 −1.58481
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 − 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 9.00000 0.774597
\(136\) 0 0
\(137\) 5.00000 0.427179 0.213589 − 0.976924i \(-0.431485\pi\)
0.213589 + 0.976924i \(0.431485\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 − 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) −36.0000 −3.03175
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 4.00000 0.332182
\(146\) 0 0
\(147\) 21.0000 1.73205
\(148\) 0 0
\(149\) −16.0000 −1.31077 −0.655386 − 0.755295i \(-0.727494\pi\)
−0.655386 + 0.755295i \(0.727494\pi\)
\(150\) 0 0
\(151\) 6.00000 0.488273 0.244137 − 0.969741i \(-0.421495\pi\)
0.244137 + 0.969741i \(0.421495\pi\)
\(152\) 0 0
\(153\) 24.0000 1.94029
\(154\) 0 0
\(155\) 9.00000 0.722897
\(156\) 0 0
\(157\) −13.0000 −1.03751 −0.518756 − 0.854922i \(-0.673605\pi\)
−0.518756 + 0.854922i \(0.673605\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −12.0000 −0.939913 −0.469956 − 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −18.0000 −1.39288 −0.696441 − 0.717614i \(-0.745234\pi\)
−0.696441 + 0.717614i \(0.745234\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) 36.0000 2.75299
\(172\) 0 0
\(173\) 10.0000 0.760286 0.380143 − 0.924928i \(-0.375875\pi\)
0.380143 + 0.924928i \(0.375875\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 27.0000 2.02944
\(178\) 0 0
\(179\) 15.0000 1.12115 0.560576 − 0.828103i \(-0.310580\pi\)
0.560576 + 0.828103i \(0.310580\pi\)
\(180\) 0 0
\(181\) −3.00000 −0.222988 −0.111494 − 0.993765i \(-0.535564\pi\)
−0.111494 + 0.993765i \(0.535564\pi\)
\(182\) 0 0
\(183\) −24.0000 −1.77413
\(184\) 0 0
\(185\) 7.00000 0.514650
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 21.0000 1.51951 0.759753 − 0.650211i \(-0.225320\pi\)
0.759753 + 0.650211i \(0.225320\pi\)
\(192\) 0 0
\(193\) −4.00000 −0.287926 −0.143963 − 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 0 0
\(195\) −18.0000 −1.28901
\(196\) 0 0
\(197\) 22.0000 1.56744 0.783718 − 0.621117i \(-0.213321\pi\)
0.783718 + 0.621117i \(0.213321\pi\)
\(198\) 0 0
\(199\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) −45.0000 −3.17406
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −2.00000 −0.139686
\(206\) 0 0
\(207\) 18.0000 1.25109
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 − 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 0 0
\(213\) 9.00000 0.616670
\(214\) 0 0
\(215\) −6.00000 −0.409197
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −18.0000 −1.21633
\(220\) 0 0
\(221\) −24.0000 −1.61441
\(222\) 0 0
\(223\) −21.0000 −1.40626 −0.703132 − 0.711059i \(-0.748216\pi\)
−0.703132 + 0.711059i \(0.748216\pi\)
\(224\) 0 0
\(225\) −24.0000 −1.60000
\(226\) 0 0
\(227\) 12.0000 0.796468 0.398234 − 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) −15.0000 −0.991228 −0.495614 − 0.868543i \(-0.665057\pi\)
−0.495614 + 0.868543i \(0.665057\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 4.00000 0.262049 0.131024 − 0.991379i \(-0.458173\pi\)
0.131024 + 0.991379i \(0.458173\pi\)
\(234\) 0 0
\(235\) −12.0000 −0.782794
\(236\) 0 0
\(237\) −18.0000 −1.16923
\(238\) 0 0
\(239\) 6.00000 0.388108 0.194054 − 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) 0 0
\(241\) 12.0000 0.772988 0.386494 − 0.922292i \(-0.373686\pi\)
0.386494 + 0.922292i \(0.373686\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 7.00000 0.447214
\(246\) 0 0
\(247\) −36.0000 −2.29063
\(248\) 0 0
\(249\) 18.0000 1.14070
\(250\) 0 0
\(251\) −21.0000 −1.32551 −0.662754 − 0.748837i \(-0.730613\pi\)
−0.662754 + 0.748837i \(0.730613\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 12.0000 0.751469
\(256\) 0 0
\(257\) 14.0000 0.873296 0.436648 − 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −24.0000 −1.48556
\(262\) 0 0
\(263\) −24.0000 −1.47990 −0.739952 − 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) 2.00000 0.122859
\(266\) 0 0
\(267\) 15.0000 0.917985
\(268\) 0 0
\(269\) 2.00000 0.121942 0.0609711 − 0.998140i \(-0.480580\pi\)
0.0609711 + 0.998140i \(0.480580\pi\)
\(270\) 0 0
\(271\) 6.00000 0.364474 0.182237 − 0.983255i \(-0.441666\pi\)
0.182237 + 0.983255i \(0.441666\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −6.00000 −0.360505 −0.180253 − 0.983620i \(-0.557691\pi\)
−0.180253 + 0.983620i \(0.557691\pi\)
\(278\) 0 0
\(279\) −54.0000 −3.23290
\(280\) 0 0
\(281\) −10.0000 −0.596550 −0.298275 − 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 0 0
\(283\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) 18.0000 1.06623
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 9.00000 0.527589
\(292\) 0 0
\(293\) −22.0000 −1.28525 −0.642627 − 0.766179i \(-0.722155\pi\)
−0.642627 + 0.766179i \(0.722155\pi\)
\(294\) 0 0
\(295\) 9.00000 0.524000
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −18.0000 −1.04097
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −48.0000 −2.75753
\(304\) 0 0
\(305\) −8.00000 −0.458079
\(306\) 0 0
\(307\) 6.00000 0.342438 0.171219 − 0.985233i \(-0.445229\pi\)
0.171219 + 0.985233i \(0.445229\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 − 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) −1.00000 −0.0565233 −0.0282617 − 0.999601i \(-0.508997\pi\)
−0.0282617 + 0.999601i \(0.508997\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 23.0000 1.29181 0.645904 − 0.763418i \(-0.276480\pi\)
0.645904 + 0.763418i \(0.276480\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −36.0000 −2.00932
\(322\) 0 0
\(323\) 24.0000 1.33540
\(324\) 0 0
\(325\) 24.0000 1.33128
\(326\) 0 0
\(327\) 18.0000 0.995402
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −15.0000 −0.824475 −0.412237 − 0.911077i \(-0.635253\pi\)
−0.412237 + 0.911077i \(0.635253\pi\)
\(332\) 0 0
\(333\) −42.0000 −2.30159
\(334\) 0 0
\(335\) −15.0000 −0.819538
\(336\) 0 0
\(337\) −24.0000 −1.30736 −0.653682 − 0.756770i \(-0.726776\pi\)
−0.653682 + 0.756770i \(0.726776\pi\)
\(338\) 0 0
\(339\) −3.00000 −0.162938
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 9.00000 0.484544
\(346\) 0 0
\(347\) 6.00000 0.322097 0.161048 − 0.986947i \(-0.448512\pi\)
0.161048 + 0.986947i \(0.448512\pi\)
\(348\) 0 0
\(349\) −28.0000 −1.49881 −0.749403 − 0.662114i \(-0.769659\pi\)
−0.749403 + 0.662114i \(0.769659\pi\)
\(350\) 0 0
\(351\) 54.0000 2.88231
\(352\) 0 0
\(353\) −5.00000 −0.266123 −0.133062 − 0.991108i \(-0.542481\pi\)
−0.133062 + 0.991108i \(0.542481\pi\)
\(354\) 0 0
\(355\) 3.00000 0.159223
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 6.00000 0.316668 0.158334 − 0.987386i \(-0.449388\pi\)
0.158334 + 0.987386i \(0.449388\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −6.00000 −0.314054
\(366\) 0 0
\(367\) 15.0000 0.782994 0.391497 − 0.920179i \(-0.371957\pi\)
0.391497 + 0.920179i \(0.371957\pi\)
\(368\) 0 0
\(369\) 12.0000 0.624695
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −4.00000 −0.207112 −0.103556 − 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) 0 0
\(375\) −27.0000 −1.39427
\(376\) 0 0
\(377\) 24.0000 1.23606
\(378\) 0 0
\(379\) 21.0000 1.07870 0.539349 − 0.842082i \(-0.318670\pi\)
0.539349 + 0.842082i \(0.318670\pi\)
\(380\) 0 0
\(381\) −18.0000 −0.922168
\(382\) 0 0
\(383\) −21.0000 −1.07305 −0.536525 − 0.843884i \(-0.680263\pi\)
−0.536525 + 0.843884i \(0.680263\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 36.0000 1.82998
\(388\) 0 0
\(389\) −5.00000 −0.253510 −0.126755 − 0.991934i \(-0.540456\pi\)
−0.126755 + 0.991934i \(0.540456\pi\)
\(390\) 0 0
\(391\) 12.0000 0.606866
\(392\) 0 0
\(393\) −36.0000 −1.81596
\(394\) 0 0
\(395\) −6.00000 −0.301893
\(396\) 0 0
\(397\) 34.0000 1.70641 0.853206 − 0.521575i \(-0.174655\pi\)
0.853206 + 0.521575i \(0.174655\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −22.0000 −1.09863 −0.549314 − 0.835616i \(-0.685111\pi\)
−0.549314 + 0.835616i \(0.685111\pi\)
\(402\) 0 0
\(403\) 54.0000 2.68993
\(404\) 0 0
\(405\) −9.00000 −0.447214
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −18.0000 −0.890043 −0.445021 − 0.895520i \(-0.646804\pi\)
−0.445021 + 0.895520i \(0.646804\pi\)
\(410\) 0 0
\(411\) −15.0000 −0.739895
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 6.00000 0.294528
\(416\) 0 0
\(417\) 36.0000 1.76293
\(418\) 0 0
\(419\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 − 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 0 0
\(423\) 72.0000 3.50076
\(424\) 0 0
\(425\) −16.0000 −0.776114
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 12.0000 0.578020 0.289010 − 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 0 0
\(433\) −11.0000 −0.528626 −0.264313 − 0.964437i \(-0.585145\pi\)
−0.264313 + 0.964437i \(0.585145\pi\)
\(434\) 0 0
\(435\) −12.0000 −0.575356
\(436\) 0 0
\(437\) 18.0000 0.861057
\(438\) 0 0
\(439\) −36.0000 −1.71819 −0.859093 − 0.511819i \(-0.828972\pi\)
−0.859093 + 0.511819i \(0.828972\pi\)
\(440\) 0 0
\(441\) −42.0000 −2.00000
\(442\) 0 0
\(443\) −9.00000 −0.427603 −0.213801 − 0.976877i \(-0.568585\pi\)
−0.213801 + 0.976877i \(0.568585\pi\)
\(444\) 0 0
\(445\) 5.00000 0.237023
\(446\) 0 0
\(447\) 48.0000 2.27032
\(448\) 0 0
\(449\) −17.0000 −0.802280 −0.401140 − 0.916017i \(-0.631386\pi\)
−0.401140 + 0.916017i \(0.631386\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −18.0000 −0.845714
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 24.0000 1.12267 0.561336 − 0.827588i \(-0.310287\pi\)
0.561336 + 0.827588i \(0.310287\pi\)
\(458\) 0 0
\(459\) −36.0000 −1.68034
\(460\) 0 0
\(461\) −40.0000 −1.86299 −0.931493 − 0.363760i \(-0.881493\pi\)
−0.931493 + 0.363760i \(0.881493\pi\)
\(462\) 0 0
\(463\) −15.0000 −0.697109 −0.348555 − 0.937288i \(-0.613327\pi\)
−0.348555 + 0.937288i \(0.613327\pi\)
\(464\) 0 0
\(465\) −27.0000 −1.25210
\(466\) 0 0
\(467\) 15.0000 0.694117 0.347059 − 0.937843i \(-0.387180\pi\)
0.347059 + 0.937843i \(0.387180\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 39.0000 1.79703
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −24.0000 −1.10120
\(476\) 0 0
\(477\) −12.0000 −0.549442
\(478\) 0 0
\(479\) −6.00000 −0.274147 −0.137073 − 0.990561i \(-0.543770\pi\)
−0.137073 + 0.990561i \(0.543770\pi\)
\(480\) 0 0
\(481\) 42.0000 1.91504
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 3.00000 0.136223
\(486\) 0 0
\(487\) 3.00000 0.135943 0.0679715 − 0.997687i \(-0.478347\pi\)
0.0679715 + 0.997687i \(0.478347\pi\)
\(488\) 0 0
\(489\) 36.0000 1.62798
\(490\) 0 0
\(491\) −24.0000 −1.08310 −0.541552 − 0.840667i \(-0.682163\pi\)
−0.541552 + 0.840667i \(0.682163\pi\)
\(492\) 0 0
\(493\) −16.0000 −0.720604
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −12.0000 −0.537194 −0.268597 − 0.963253i \(-0.586560\pi\)
−0.268597 + 0.963253i \(0.586560\pi\)
\(500\) 0 0
\(501\) 54.0000 2.41254
\(502\) 0 0
\(503\) −42.0000 −1.87269 −0.936344 − 0.351085i \(-0.885813\pi\)
−0.936344 + 0.351085i \(0.885813\pi\)
\(504\) 0 0
\(505\) −16.0000 −0.711991
\(506\) 0 0
\(507\) −69.0000 −3.06440
\(508\) 0 0
\(509\) 1.00000 0.0443242 0.0221621 − 0.999754i \(-0.492945\pi\)
0.0221621 + 0.999754i \(0.492945\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −54.0000 −2.38416
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −30.0000 −1.31685
\(520\) 0 0
\(521\) −31.0000 −1.35813 −0.679067 − 0.734076i \(-0.737616\pi\)
−0.679067 + 0.734076i \(0.737616\pi\)
\(522\) 0 0
\(523\) −30.0000 −1.31181 −0.655904 − 0.754844i \(-0.727712\pi\)
−0.655904 + 0.754844i \(0.727712\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −36.0000 −1.56818
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) −54.0000 −2.34340
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) 0 0
\(535\) −12.0000 −0.518805
\(536\) 0 0
\(537\) −45.0000 −1.94189
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −6.00000 −0.257960 −0.128980 − 0.991647i \(-0.541170\pi\)
−0.128980 + 0.991647i \(0.541170\pi\)
\(542\) 0 0
\(543\) 9.00000 0.386227
\(544\) 0 0
\(545\) 6.00000 0.257012
\(546\) 0 0
\(547\) 36.0000 1.53925 0.769624 − 0.638497i \(-0.220443\pi\)
0.769624 + 0.638497i \(0.220443\pi\)
\(548\) 0 0
\(549\) 48.0000 2.04859
\(550\) 0 0
\(551\) −24.0000 −1.02243
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −21.0000 −0.891400
\(556\) 0 0
\(557\) −16.0000 −0.677942 −0.338971 − 0.940797i \(-0.610079\pi\)
−0.338971 + 0.940797i \(0.610079\pi\)
\(558\) 0 0
\(559\) −36.0000 −1.52264
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) −1.00000 −0.0420703
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −44.0000 −1.84458 −0.922288 − 0.386503i \(-0.873683\pi\)
−0.922288 + 0.386503i \(0.873683\pi\)
\(570\) 0 0
\(571\) −12.0000 −0.502184 −0.251092 − 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 0 0
\(573\) −63.0000 −2.63186
\(574\) 0 0
\(575\) −12.0000 −0.500435
\(576\) 0 0
\(577\) 29.0000 1.20729 0.603643 − 0.797255i \(-0.293715\pi\)
0.603643 + 0.797255i \(0.293715\pi\)
\(578\) 0 0
\(579\) 12.0000 0.498703
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 36.0000 1.48842
\(586\) 0 0
\(587\) −36.0000 −1.48588 −0.742940 − 0.669359i \(-0.766569\pi\)
−0.742940 + 0.669359i \(0.766569\pi\)
\(588\) 0 0
\(589\) −54.0000 −2.22503
\(590\) 0 0
\(591\) −66.0000 −2.71488
\(592\) 0 0
\(593\) −20.0000 −0.821302 −0.410651 − 0.911793i \(-0.634698\pi\)
−0.410651 + 0.911793i \(0.634698\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 − 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) −28.0000 −1.14214 −0.571072 − 0.820900i \(-0.693472\pi\)
−0.571072 + 0.820900i \(0.693472\pi\)
\(602\) 0 0
\(603\) 90.0000 3.66508
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 18.0000 0.730597 0.365299 − 0.930890i \(-0.380967\pi\)
0.365299 + 0.930890i \(0.380967\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −72.0000 −2.91281
\(612\) 0 0
\(613\) −16.0000 −0.646234 −0.323117 − 0.946359i \(-0.604731\pi\)
−0.323117 + 0.946359i \(0.604731\pi\)
\(614\) 0 0
\(615\) 6.00000 0.241943
\(616\) 0 0
\(617\) 22.0000 0.885687 0.442843 − 0.896599i \(-0.353970\pi\)
0.442843 + 0.896599i \(0.353970\pi\)
\(618\) 0 0
\(619\) −15.0000 −0.602901 −0.301450 − 0.953482i \(-0.597471\pi\)
−0.301450 + 0.953482i \(0.597471\pi\)
\(620\) 0 0
\(621\) −27.0000 −1.08347
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −28.0000 −1.11643
\(630\) 0 0
\(631\) 15.0000 0.597141 0.298570 − 0.954388i \(-0.403490\pi\)
0.298570 + 0.954388i \(0.403490\pi\)
\(632\) 0 0
\(633\) −36.0000 −1.43087
\(634\) 0 0
\(635\) −6.00000 −0.238103
\(636\) 0 0
\(637\) 42.0000 1.66410
\(638\) 0 0
\(639\) −18.0000 −0.712069
\(640\) 0 0
\(641\) −37.0000 −1.46141 −0.730706 − 0.682692i \(-0.760809\pi\)
−0.730706 + 0.682692i \(0.760809\pi\)
\(642\) 0 0
\(643\) −21.0000 −0.828159 −0.414080 − 0.910241i \(-0.635896\pi\)
−0.414080 + 0.910241i \(0.635896\pi\)
\(644\) 0 0
\(645\) 18.0000 0.708749
\(646\) 0 0
\(647\) −3.00000 −0.117942 −0.0589711 − 0.998260i \(-0.518782\pi\)
−0.0589711 + 0.998260i \(0.518782\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 17.0000 0.665261 0.332631 − 0.943057i \(-0.392064\pi\)
0.332631 + 0.943057i \(0.392064\pi\)
\(654\) 0 0
\(655\) −12.0000 −0.468879
\(656\) 0 0
\(657\) 36.0000 1.40449
\(658\) 0 0
\(659\) −18.0000 −0.701180 −0.350590 − 0.936529i \(-0.614019\pi\)
−0.350590 + 0.936529i \(0.614019\pi\)
\(660\) 0 0
\(661\) 23.0000 0.894596 0.447298 − 0.894385i \(-0.352386\pi\)
0.447298 + 0.894385i \(0.352386\pi\)
\(662\) 0 0
\(663\) 72.0000 2.79625
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −12.0000 −0.464642
\(668\) 0 0
\(669\) 63.0000 2.43572
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 10.0000 0.385472 0.192736 − 0.981251i \(-0.438264\pi\)
0.192736 + 0.981251i \(0.438264\pi\)
\(674\) 0 0
\(675\) 36.0000 1.38564
\(676\) 0 0
\(677\) −8.00000 −0.307465 −0.153732 − 0.988113i \(-0.549129\pi\)
−0.153732 + 0.988113i \(0.549129\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −36.0000 −1.37952
\(682\) 0 0
\(683\) 36.0000 1.37750 0.688751 − 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 0 0
\(685\) −5.00000 −0.191040
\(686\) 0 0
\(687\) 45.0000 1.71686
\(688\) 0 0
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) −33.0000 −1.25538 −0.627690 − 0.778464i \(-0.715999\pi\)
−0.627690 + 0.778464i \(0.715999\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 12.0000 0.455186
\(696\) 0 0
\(697\) 8.00000 0.303022
\(698\) 0 0
\(699\) −12.0000 −0.453882
\(700\) 0 0
\(701\) −40.0000 −1.51078 −0.755390 − 0.655276i \(-0.772552\pi\)
−0.755390 + 0.655276i \(0.772552\pi\)
\(702\) 0 0
\(703\) −42.0000 −1.58406
\(704\) 0 0
\(705\) 36.0000 1.35584
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 33.0000 1.23934 0.619671 − 0.784862i \(-0.287266\pi\)
0.619671 + 0.784862i \(0.287266\pi\)
\(710\) 0 0
\(711\) 36.0000 1.35011
\(712\) 0 0
\(713\) −27.0000 −1.01116
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −18.0000 −0.672222
\(718\) 0 0
\(719\) 27.0000 1.00693 0.503465 − 0.864016i \(-0.332058\pi\)
0.503465 + 0.864016i \(0.332058\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −36.0000 −1.33885
\(724\) 0 0
\(725\) 16.0000 0.594225
\(726\) 0 0
\(727\) 3.00000 0.111264 0.0556319 − 0.998451i \(-0.482283\pi\)
0.0556319 + 0.998451i \(0.482283\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) 0 0
\(733\) 48.0000 1.77292 0.886460 − 0.462805i \(-0.153157\pi\)
0.886460 + 0.462805i \(0.153157\pi\)
\(734\) 0 0
\(735\) −21.0000 −0.774597
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −6.00000 −0.220714 −0.110357 − 0.993892i \(-0.535199\pi\)
−0.110357 + 0.993892i \(0.535199\pi\)
\(740\) 0 0
\(741\) 108.000 3.96748
\(742\) 0 0
\(743\) 36.0000 1.32071 0.660356 − 0.750953i \(-0.270405\pi\)
0.660356 + 0.750953i \(0.270405\pi\)
\(744\) 0 0
\(745\) 16.0000 0.586195
\(746\) 0 0
\(747\) −36.0000 −1.31717
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −3.00000 −0.109472 −0.0547358 − 0.998501i \(-0.517432\pi\)
−0.0547358 + 0.998501i \(0.517432\pi\)
\(752\) 0 0
\(753\) 63.0000 2.29585
\(754\) 0 0
\(755\) −6.00000 −0.218362
\(756\) 0 0
\(757\) −6.00000 −0.218074 −0.109037 − 0.994038i \(-0.534777\pi\)
−0.109037 + 0.994038i \(0.534777\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 26.0000 0.942499 0.471250 − 0.882000i \(-0.343803\pi\)
0.471250 + 0.882000i \(0.343803\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −24.0000 −0.867722
\(766\) 0 0
\(767\) 54.0000 1.94983
\(768\) 0 0
\(769\) 22.0000 0.793340 0.396670 − 0.917961i \(-0.370166\pi\)
0.396670 + 0.917961i \(0.370166\pi\)
\(770\) 0 0
\(771\) −42.0000 −1.51259
\(772\) 0 0
\(773\) −2.00000 −0.0719350 −0.0359675 − 0.999353i \(-0.511451\pi\)
−0.0359675 + 0.999353i \(0.511451\pi\)
\(774\) 0 0
\(775\) 36.0000 1.29316
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 12.0000 0.429945
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 36.0000 1.28654
\(784\) 0 0
\(785\) 13.0000 0.463990
\(786\) 0 0
\(787\) −36.0000 −1.28326 −0.641631 − 0.767014i \(-0.721742\pi\)
−0.641631 + 0.767014i \(0.721742\pi\)
\(788\) 0 0
\(789\) 72.0000 2.56327
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −48.0000 −1.70453
\(794\) 0 0
\(795\) −6.00000 −0.212798
\(796\) 0 0
\(797\) −49.0000 −1.73567 −0.867835 − 0.496853i \(-0.834489\pi\)
−0.867835 + 0.496853i \(0.834489\pi\)
\(798\) 0 0
\(799\) 48.0000 1.69812
\(800\) 0 0
\(801\) −30.0000 −1.06000
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −6.00000 −0.211210
\(808\) 0 0
\(809\) −46.0000 −1.61727 −0.808637 − 0.588308i \(-0.799794\pi\)
−0.808637 + 0.588308i \(0.799794\pi\)
\(810\) 0 0
\(811\) 30.0000 1.05344 0.526721 − 0.850038i \(-0.323421\pi\)
0.526721 + 0.850038i \(0.323421\pi\)
\(812\) 0 0
\(813\) −18.0000 −0.631288
\(814\) 0 0
\(815\) 12.0000 0.420342
\(816\) 0 0
\(817\) 36.0000 1.25948
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 22.0000 0.767805 0.383903 − 0.923374i \(-0.374580\pi\)
0.383903 + 0.923374i \(0.374580\pi\)
\(822\) 0 0
\(823\) −33.0000 −1.15031 −0.575154 − 0.818045i \(-0.695058\pi\)
−0.575154 + 0.818045i \(0.695058\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −18.0000 −0.625921 −0.312961 − 0.949766i \(-0.601321\pi\)
−0.312961 + 0.949766i \(0.601321\pi\)
\(828\) 0 0
\(829\) −21.0000 −0.729360 −0.364680 − 0.931133i \(-0.618822\pi\)
−0.364680 + 0.931133i \(0.618822\pi\)
\(830\) 0 0
\(831\) 18.0000 0.624413
\(832\) 0 0
\(833\) −28.0000 −0.970143
\(834\) 0 0
\(835\) 18.0000 0.622916
\(836\) 0 0
\(837\) 81.0000 2.79977
\(838\) 0 0
\(839\) −33.0000 −1.13929 −0.569643 − 0.821892i \(-0.692919\pi\)
−0.569643 + 0.821892i \(0.692919\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 0 0
\(843\) 30.0000 1.03325
\(844\) 0 0
\(845\) −23.0000 −0.791224
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −21.0000 −0.719871
\(852\) 0 0
\(853\) 46.0000 1.57501 0.787505 − 0.616308i \(-0.211372\pi\)
0.787505 + 0.616308i \(0.211372\pi\)
\(854\) 0 0
\(855\) −36.0000 −1.23117
\(856\) 0 0
\(857\) 32.0000 1.09310 0.546550 − 0.837427i \(-0.315941\pi\)
0.546550 + 0.837427i \(0.315941\pi\)
\(858\) 0 0
\(859\) 27.0000 0.921228 0.460614 − 0.887601i \(-0.347629\pi\)
0.460614 + 0.887601i \(0.347629\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 24.0000 0.816970 0.408485 − 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 0 0
\(865\) −10.0000 −0.340010
\(866\) 0 0
\(867\) 3.00000 0.101885
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −90.0000 −3.04953
\(872\) 0 0
\(873\) −18.0000 −0.609208
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 14.0000 0.472746 0.236373 − 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) 0 0
\(879\) 66.0000 2.22612
\(880\) 0 0
\(881\) 29.0000 0.977035 0.488517 − 0.872554i \(-0.337538\pi\)
0.488517 + 0.872554i \(0.337538\pi\)
\(882\) 0 0
\(883\) 12.0000 0.403832 0.201916 − 0.979403i \(-0.435283\pi\)
0.201916 + 0.979403i \(0.435283\pi\)
\(884\) 0 0
\(885\) −27.0000 −0.907595
\(886\) 0 0
\(887\) −6.00000 −0.201460 −0.100730 − 0.994914i \(-0.532118\pi\)
−0.100730 + 0.994914i \(0.532118\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 72.0000 2.40939
\(894\) 0 0
\(895\) −15.0000 −0.501395
\(896\) 0 0
\(897\) 54.0000 1.80301
\(898\) 0 0
\(899\) 36.0000 1.20067
\(900\) 0 0
\(901\) −8.00000 −0.266519
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 3.00000 0.0997234
\(906\) 0 0
\(907\) 12.0000 0.398453 0.199227 − 0.979953i \(-0.436157\pi\)
0.199227 + 0.979953i \(0.436157\pi\)
\(908\) 0 0
\(909\) 96.0000 3.18412
\(910\) 0 0
\(911\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 24.0000 0.793416
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −30.0000 −0.989609 −0.494804 − 0.869004i \(-0.664760\pi\)
−0.494804 + 0.869004i \(0.664760\pi\)
\(920\) 0 0
\(921\) −18.0000 −0.593120
\(922\) 0 0
\(923\) 18.0000 0.592477
\(924\) 0 0
\(925\) 28.0000 0.920634
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −22.0000 −0.721797 −0.360898 − 0.932605i \(-0.617530\pi\)
−0.360898 + 0.932605i \(0.617530\pi\)
\(930\) 0 0
\(931\) −42.0000 −1.37649
\(932\) 0 0
\(933\) 72.0000 2.35717
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 2.00000 0.0653372 0.0326686 − 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 0 0
\(939\) 3.00000 0.0979013
\(940\) 0 0
\(941\) −10.0000 −0.325991 −0.162995 − 0.986627i \(-0.552116\pi\)
−0.162995 + 0.986627i \(0.552116\pi\)
\(942\) 0 0
\(943\) 6.00000 0.195387
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 15.0000 0.487435 0.243717 − 0.969846i \(-0.421633\pi\)
0.243717 + 0.969846i \(0.421633\pi\)
\(948\) 0 0
\(949\) −36.0000 −1.16861
\(950\) 0 0
\(951\) −69.0000 −2.23748
\(952\) 0 0
\(953\) 40.0000 1.29573 0.647864 − 0.761756i \(-0.275663\pi\)
0.647864 + 0.761756i \(0.275663\pi\)
\(954\) 0 0
\(955\) −21.0000 −0.679544
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 50.0000 1.61290
\(962\) 0 0
\(963\) 72.0000 2.32017
\(964\) 0 0
\(965\) 4.00000 0.128765
\(966\) 0 0
\(967\) −12.0000 −0.385894 −0.192947 − 0.981209i \(-0.561805\pi\)
−0.192947 + 0.981209i \(0.561805\pi\)
\(968\) 0 0
\(969\) −72.0000 −2.31297
\(970\) 0 0
\(971\) −15.0000 −0.481373 −0.240686 − 0.970603i \(-0.577373\pi\)
−0.240686 + 0.970603i \(0.577373\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −72.0000 −2.30585
\(976\) 0 0
\(977\) −23.0000 −0.735835 −0.367918 − 0.929858i \(-0.619929\pi\)
−0.367918 + 0.929858i \(0.619929\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −36.0000 −1.14939
\(982\) 0 0
\(983\) −45.0000 −1.43528 −0.717639 − 0.696416i \(-0.754777\pi\)
−0.717639 + 0.696416i \(0.754777\pi\)
\(984\) 0 0
\(985\) −22.0000 −0.700978
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 18.0000 0.572367
\(990\) 0 0
\(991\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) 45.0000 1.42803
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 8.00000 0.253363 0.126681 − 0.991943i \(-0.459567\pi\)
0.126681 + 0.991943i \(0.459567\pi\)
\(998\) 0 0
\(999\) 63.0000 1.99323
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 7744.2.a.a.1.1 1
4.3 odd 2 7744.2.a.bj.1.1 1
8.3 odd 2 3872.2.a.a.1.1 1
8.5 even 2 3872.2.a.m.1.1 1
11.10 odd 2 704.2.a.a.1.1 1
33.32 even 2 6336.2.a.bs.1.1 1
44.43 even 2 704.2.a.k.1.1 1
88.21 odd 2 352.2.a.f.1.1 yes 1
88.43 even 2 352.2.a.a.1.1 ✓ 1
132.131 odd 2 6336.2.a.bt.1.1 1
176.21 odd 4 2816.2.c.h.1409.2 2
176.43 even 4 2816.2.c.g.1409.1 2
176.109 odd 4 2816.2.c.h.1409.1 2
176.131 even 4 2816.2.c.g.1409.2 2
264.131 odd 2 3168.2.a.h.1.1 1
264.197 even 2 3168.2.a.i.1.1 1
440.109 odd 2 8800.2.a.a.1.1 1
440.219 even 2 8800.2.a.bb.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
352.2.a.a.1.1 ✓ 1 88.43 even 2
352.2.a.f.1.1 yes 1 88.21 odd 2
704.2.a.a.1.1 1 11.10 odd 2
704.2.a.k.1.1 1 44.43 even 2
2816.2.c.g.1409.1 2 176.43 even 4
2816.2.c.g.1409.2 2 176.131 even 4
2816.2.c.h.1409.1 2 176.109 odd 4
2816.2.c.h.1409.2 2 176.21 odd 4
3168.2.a.h.1.1 1 264.131 odd 2
3168.2.a.i.1.1 1 264.197 even 2
3872.2.a.a.1.1 1 8.3 odd 2
3872.2.a.m.1.1 1 8.5 even 2
6336.2.a.bs.1.1 1 33.32 even 2
6336.2.a.bt.1.1 1 132.131 odd 2
7744.2.a.a.1.1 1 1.1 even 1 trivial
7744.2.a.bj.1.1 1 4.3 odd 2
8800.2.a.a.1.1 1 440.109 odd 2
8800.2.a.bb.1.1 1 440.219 even 2