Properties

Label 171.2.a.a.1.1
Level $171$
Weight $2$
Character 171.1
Self dual yes
Analytic conductor $1.365$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 171.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.36544187456\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 57)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 171.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{4} +2.00000 q^{5} +3.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{4} +2.00000 q^{5} +3.00000 q^{8} -2.00000 q^{10} +6.00000 q^{13} -1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{19} -2.00000 q^{20} -4.00000 q^{23} -1.00000 q^{25} -6.00000 q^{26} -2.00000 q^{29} +8.00000 q^{31} -5.00000 q^{32} -6.00000 q^{34} -10.0000 q^{37} +1.00000 q^{38} +6.00000 q^{40} +2.00000 q^{41} -4.00000 q^{43} +4.00000 q^{46} -12.0000 q^{47} -7.00000 q^{49} +1.00000 q^{50} -6.00000 q^{52} +6.00000 q^{53} +2.00000 q^{58} +12.0000 q^{59} -2.00000 q^{61} -8.00000 q^{62} +7.00000 q^{64} +12.0000 q^{65} -4.00000 q^{67} -6.00000 q^{68} +10.0000 q^{73} +10.0000 q^{74} +1.00000 q^{76} -2.00000 q^{80} -2.00000 q^{82} -16.0000 q^{83} +12.0000 q^{85} +4.00000 q^{86} +2.00000 q^{89} +4.00000 q^{92} +12.0000 q^{94} -2.00000 q^{95} +10.0000 q^{97} +7.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107 −0.353553 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 3.00000 1.06066
\(9\) 0 0
\(10\) −2.00000 −0.632456
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) −6.00000 −1.17670
\(27\) 0 0
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) −5.00000 −0.883883
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) 6.00000 0.948683
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) −12.0000 −1.75038 −0.875190 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −6.00000 −0.832050
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 2.00000 0.262613
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −8.00000 −1.01600
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) 12.0000 1.48842
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 10.0000 1.16248
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −2.00000 −0.223607
\(81\) 0 0
\(82\) −2.00000 −0.220863
\(83\) −16.0000 −1.75623 −0.878114 0.478451i \(-0.841198\pi\)
−0.878114 + 0.478451i \(0.841198\pi\)
\(84\) 0 0
\(85\) 12.0000 1.30158
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) 0 0
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 4.00000 0.417029
\(93\) 0 0
\(94\) 12.0000 1.23771
\(95\) −2.00000 −0.205196
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 7.00000 0.707107
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 18.0000 1.76505
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) −8.00000 −0.746004
\(116\) 2.00000 0.185695
\(117\) 0 0
\(118\) −12.0000 −1.10469
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 2.00000 0.181071
\(123\) 0 0
\(124\) −8.00000 −0.718421
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 3.00000 0.265165
\(129\) 0 0
\(130\) −12.0000 −1.05247
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) 18.0000 1.54349
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −4.00000 −0.332182
\(146\) −10.0000 −0.827606
\(147\) 0 0
\(148\) 10.0000 0.821995
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) −3.00000 −0.243332
\(153\) 0 0
\(154\) 0 0
\(155\) 16.0000 1.28515
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −10.0000 −0.790569
\(161\) 0 0
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) 16.0000 1.24184
\(167\) −24.0000 −1.85718 −0.928588 0.371113i \(-0.878976\pi\)
−0.928588 + 0.371113i \(0.878976\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) −12.0000 −0.920358
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) 22.0000 1.67263 0.836315 0.548250i \(-0.184706\pi\)
0.836315 + 0.548250i \(0.184706\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) −2.00000 −0.149906
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −12.0000 −0.884652
\(185\) −20.0000 −1.47043
\(186\) 0 0
\(187\) 0 0
\(188\) 12.0000 0.875190
\(189\) 0 0
\(190\) 2.00000 0.145095
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) 7.00000 0.500000
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) −3.00000 −0.212132
\(201\) 0 0
\(202\) −10.0000 −0.703598
\(203\) 0 0
\(204\) 0 0
\(205\) 4.00000 0.279372
\(206\) −8.00000 −0.557386
\(207\) 0 0
\(208\) −6.00000 −0.416025
\(209\) 0 0
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) 4.00000 0.273434
\(215\) −8.00000 −0.545595
\(216\) 0 0
\(217\) 0 0
\(218\) 10.0000 0.677285
\(219\) 0 0
\(220\) 0 0
\(221\) 36.0000 2.42162
\(222\) 0 0
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 8.00000 0.527504
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) 0 0
\(235\) −24.0000 −1.56559
\(236\) −12.0000 −0.781133
\(237\) 0 0
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) −6.00000 −0.386494 −0.193247 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) 11.0000 0.707107
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) −14.0000 −0.894427
\(246\) 0 0
\(247\) −6.00000 −0.381771
\(248\) 24.0000 1.52400
\(249\) 0 0
\(250\) 12.0000 0.758947
\(251\) 24.0000 1.51487 0.757433 0.652913i \(-0.226453\pi\)
0.757433 + 0.652913i \(0.226453\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −12.0000 −0.744208
\(261\) 0 0
\(262\) 8.00000 0.494242
\(263\) −12.0000 −0.739952 −0.369976 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 0 0
\(265\) 12.0000 0.737154
\(266\) 0 0
\(267\) 0 0
\(268\) 4.00000 0.244339
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) −6.00000 −0.363803
\(273\) 0 0
\(274\) 18.0000 1.08742
\(275\) 0 0
\(276\) 0 0
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) −4.00000 −0.239904
\(279\) 0 0
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) −20.0000 −1.18888 −0.594438 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 4.00000 0.234888
\(291\) 0 0
\(292\) −10.0000 −0.585206
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 0 0
\(295\) 24.0000 1.39733
\(296\) −30.0000 −1.74371
\(297\) 0 0
\(298\) 6.00000 0.347571
\(299\) −24.0000 −1.38796
\(300\) 0 0
\(301\) 0 0
\(302\) 8.00000 0.460348
\(303\) 0 0
\(304\) 1.00000 0.0573539
\(305\) −4.00000 −0.229039
\(306\) 0 0
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −16.0000 −0.908739
\(311\) −4.00000 −0.226819 −0.113410 0.993548i \(-0.536177\pi\)
−0.113410 + 0.993548i \(0.536177\pi\)
\(312\) 0 0
\(313\) −22.0000 −1.24351 −0.621757 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) 2.00000 0.112867
\(315\) 0 0
\(316\) 0 0
\(317\) 6.00000 0.336994 0.168497 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 14.0000 0.782624
\(321\) 0 0
\(322\) 0 0
\(323\) −6.00000 −0.333849
\(324\) 0 0
\(325\) −6.00000 −0.332820
\(326\) 4.00000 0.221540
\(327\) 0 0
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) 16.0000 0.878114
\(333\) 0 0
\(334\) 24.0000 1.31322
\(335\) −8.00000 −0.437087
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) −23.0000 −1.25104
\(339\) 0 0
\(340\) −12.0000 −0.650791
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) −12.0000 −0.646997
\(345\) 0 0
\(346\) −22.0000 −1.18273
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 22.0000 1.17094 0.585471 0.810693i \(-0.300910\pi\)
0.585471 + 0.810693i \(0.300910\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −2.00000 −0.106000
\(357\) 0 0
\(358\) −4.00000 −0.211407
\(359\) 20.0000 1.05556 0.527780 0.849381i \(-0.323025\pi\)
0.527780 + 0.849381i \(0.323025\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −14.0000 −0.735824
\(363\) 0 0
\(364\) 0 0
\(365\) 20.0000 1.04685
\(366\) 0 0
\(367\) 32.0000 1.67039 0.835193 0.549957i \(-0.185356\pi\)
0.835193 + 0.549957i \(0.185356\pi\)
\(368\) 4.00000 0.208514
\(369\) 0 0
\(370\) 20.0000 1.03975
\(371\) 0 0
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −36.0000 −1.85656
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) 12.0000 0.616399 0.308199 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) 2.00000 0.102598
\(381\) 0 0
\(382\) −12.0000 −0.613973
\(383\) −8.00000 −0.408781 −0.204390 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 14.0000 0.712581
\(387\) 0 0
\(388\) −10.0000 −0.507673
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) −24.0000 −1.21373
\(392\) −21.0000 −1.06066
\(393\) 0 0
\(394\) −2.00000 −0.100759
\(395\) 0 0
\(396\) 0 0
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 8.00000 0.401004
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −38.0000 −1.89763 −0.948815 0.315833i \(-0.897716\pi\)
−0.948815 + 0.315833i \(0.897716\pi\)
\(402\) 0 0
\(403\) 48.0000 2.39105
\(404\) −10.0000 −0.497519
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) −4.00000 −0.197546
\(411\) 0 0
\(412\) −8.00000 −0.394132
\(413\) 0 0
\(414\) 0 0
\(415\) −32.0000 −1.57082
\(416\) −30.0000 −1.47087
\(417\) 0 0
\(418\) 0 0
\(419\) −8.00000 −0.390826 −0.195413 0.980721i \(-0.562605\pi\)
−0.195413 + 0.980721i \(0.562605\pi\)
\(420\) 0 0
\(421\) 14.0000 0.682318 0.341159 0.940006i \(-0.389181\pi\)
0.341159 + 0.940006i \(0.389181\pi\)
\(422\) 4.00000 0.194717
\(423\) 0 0
\(424\) 18.0000 0.874157
\(425\) −6.00000 −0.291043
\(426\) 0 0
\(427\) 0 0
\(428\) 4.00000 0.193347
\(429\) 0 0
\(430\) 8.00000 0.385794
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 0 0
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 10.0000 0.478913
\(437\) 4.00000 0.191346
\(438\) 0 0
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −36.0000 −1.71235
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 4.00000 0.189618
\(446\) −16.0000 −0.757622
\(447\) 0 0
\(448\) 0 0
\(449\) 2.00000 0.0943858 0.0471929 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 6.00000 0.282216
\(453\) 0 0
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) 0 0
\(457\) −6.00000 −0.280668 −0.140334 0.990104i \(-0.544818\pi\)
−0.140334 + 0.990104i \(0.544818\pi\)
\(458\) −6.00000 −0.280362
\(459\) 0 0
\(460\) 8.00000 0.373002
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) 32.0000 1.48717 0.743583 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) 10.0000 0.463241
\(467\) 32.0000 1.48078 0.740392 0.672176i \(-0.234640\pi\)
0.740392 + 0.672176i \(0.234640\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 24.0000 1.10704
\(471\) 0 0
\(472\) 36.0000 1.65703
\(473\) 0 0
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) 0 0
\(478\) −12.0000 −0.548867
\(479\) −20.0000 −0.913823 −0.456912 0.889512i \(-0.651044\pi\)
−0.456912 + 0.889512i \(0.651044\pi\)
\(480\) 0 0
\(481\) −60.0000 −2.73576
\(482\) 6.00000 0.273293
\(483\) 0 0
\(484\) 11.0000 0.500000
\(485\) 20.0000 0.908153
\(486\) 0 0
\(487\) 32.0000 1.45006 0.725029 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) −6.00000 −0.271607
\(489\) 0 0
\(490\) 14.0000 0.632456
\(491\) 32.0000 1.44414 0.722070 0.691820i \(-0.243191\pi\)
0.722070 + 0.691820i \(0.243191\pi\)
\(492\) 0 0
\(493\) −12.0000 −0.540453
\(494\) 6.00000 0.269953
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) 0 0
\(498\) 0 0
\(499\) 28.0000 1.25345 0.626726 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(500\) 12.0000 0.536656
\(501\) 0 0
\(502\) −24.0000 −1.07117
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 0 0
\(505\) 20.0000 0.889988
\(506\) 0 0
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) 22.0000 0.975133 0.487566 0.873086i \(-0.337885\pi\)
0.487566 + 0.873086i \(0.337885\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 11.0000 0.486136
\(513\) 0 0
\(514\) 14.0000 0.617514
\(515\) 16.0000 0.705044
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 36.0000 1.57870
\(521\) −14.0000 −0.613351 −0.306676 0.951814i \(-0.599217\pi\)
−0.306676 + 0.951814i \(0.599217\pi\)
\(522\) 0 0
\(523\) −28.0000 −1.22435 −0.612177 0.790721i \(-0.709706\pi\)
−0.612177 + 0.790721i \(0.709706\pi\)
\(524\) 8.00000 0.349482
\(525\) 0 0
\(526\) 12.0000 0.523225
\(527\) 48.0000 2.09091
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) −12.0000 −0.521247
\(531\) 0 0
\(532\) 0 0
\(533\) 12.0000 0.519778
\(534\) 0 0
\(535\) −8.00000 −0.345870
\(536\) −12.0000 −0.518321
\(537\) 0 0
\(538\) −6.00000 −0.258678
\(539\) 0 0
\(540\) 0 0
\(541\) 14.0000 0.601907 0.300954 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −30.0000 −1.28624
\(545\) −20.0000 −0.856706
\(546\) 0 0
\(547\) 4.00000 0.171028 0.0855138 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) 18.0000 0.768922
\(549\) 0 0
\(550\) 0 0
\(551\) 2.00000 0.0852029
\(552\) 0 0
\(553\) 0 0
\(554\) −22.0000 −0.934690
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) −30.0000 −1.27114 −0.635570 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) 0 0
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) 0 0
\(562\) −10.0000 −0.421825
\(563\) −20.0000 −0.842900 −0.421450 0.906852i \(-0.638479\pi\)
−0.421450 + 0.906852i \(0.638479\pi\)
\(564\) 0 0
\(565\) −12.0000 −0.504844
\(566\) 20.0000 0.840663
\(567\) 0 0
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) 44.0000 1.84134 0.920671 0.390339i \(-0.127642\pi\)
0.920671 + 0.390339i \(0.127642\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) 18.0000 0.749350 0.374675 0.927156i \(-0.377754\pi\)
0.374675 + 0.927156i \(0.377754\pi\)
\(578\) −19.0000 −0.790296
\(579\) 0 0
\(580\) 4.00000 0.166091
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 30.0000 1.24141
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) −8.00000 −0.330195 −0.165098 0.986277i \(-0.552794\pi\)
−0.165098 + 0.986277i \(0.552794\pi\)
\(588\) 0 0
\(589\) −8.00000 −0.329634
\(590\) −24.0000 −0.988064
\(591\) 0 0
\(592\) 10.0000 0.410997
\(593\) 30.0000 1.23195 0.615976 0.787765i \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) 0 0
\(598\) 24.0000 0.981433
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 8.00000 0.325515
\(605\) −22.0000 −0.894427
\(606\) 0 0
\(607\) 24.0000 0.974130 0.487065 0.873366i \(-0.338067\pi\)
0.487065 + 0.873366i \(0.338067\pi\)
\(608\) 5.00000 0.202777
\(609\) 0 0
\(610\) 4.00000 0.161955
\(611\) −72.0000 −2.91281
\(612\) 0 0
\(613\) 6.00000 0.242338 0.121169 0.992632i \(-0.461336\pi\)
0.121169 + 0.992632i \(0.461336\pi\)
\(614\) 12.0000 0.484281
\(615\) 0 0
\(616\) 0 0
\(617\) −2.00000 −0.0805170 −0.0402585 0.999189i \(-0.512818\pi\)
−0.0402585 + 0.999189i \(0.512818\pi\)
\(618\) 0 0
\(619\) −28.0000 −1.12542 −0.562708 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) −16.0000 −0.642575
\(621\) 0 0
\(622\) 4.00000 0.160385
\(623\) 0 0
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 22.0000 0.879297
\(627\) 0 0
\(628\) 2.00000 0.0798087
\(629\) −60.0000 −2.39236
\(630\) 0 0
\(631\) −32.0000 −1.27390 −0.636950 0.770905i \(-0.719804\pi\)
−0.636950 + 0.770905i \(0.719804\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −6.00000 −0.238290
\(635\) −16.0000 −0.634941
\(636\) 0 0
\(637\) −42.0000 −1.66410
\(638\) 0 0
\(639\) 0 0
\(640\) 6.00000 0.237171
\(641\) −38.0000 −1.50091 −0.750455 0.660922i \(-0.770166\pi\)
−0.750455 + 0.660922i \(0.770166\pi\)
\(642\) 0 0
\(643\) 20.0000 0.788723 0.394362 0.918955i \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 6.00000 0.236067
\(647\) −36.0000 −1.41531 −0.707653 0.706560i \(-0.750246\pi\)
−0.707653 + 0.706560i \(0.750246\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 6.00000 0.235339
\(651\) 0 0
\(652\) 4.00000 0.156652
\(653\) −14.0000 −0.547862 −0.273931 0.961749i \(-0.588324\pi\)
−0.273931 + 0.961749i \(0.588324\pi\)
\(654\) 0 0
\(655\) −16.0000 −0.625172
\(656\) −2.00000 −0.0780869
\(657\) 0 0
\(658\) 0 0
\(659\) 44.0000 1.71400 0.856998 0.515319i \(-0.172327\pi\)
0.856998 + 0.515319i \(0.172327\pi\)
\(660\) 0 0
\(661\) 6.00000 0.233373 0.116686 0.993169i \(-0.462773\pi\)
0.116686 + 0.993169i \(0.462773\pi\)
\(662\) −12.0000 −0.466393
\(663\) 0 0
\(664\) −48.0000 −1.86276
\(665\) 0 0
\(666\) 0 0
\(667\) 8.00000 0.309761
\(668\) 24.0000 0.928588
\(669\) 0 0
\(670\) 8.00000 0.309067
\(671\) 0 0
\(672\) 0 0
\(673\) −46.0000 −1.77317 −0.886585 0.462566i \(-0.846929\pi\)
−0.886585 + 0.462566i \(0.846929\pi\)
\(674\) 22.0000 0.847408
\(675\) 0 0
\(676\) −23.0000 −0.884615
\(677\) 22.0000 0.845529 0.422764 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 36.0000 1.38054
\(681\) 0 0
\(682\) 0 0
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 0 0
\(685\) −36.0000 −1.37549
\(686\) 0 0
\(687\) 0 0
\(688\) 4.00000 0.152499
\(689\) 36.0000 1.37149
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) −22.0000 −0.836315
\(693\) 0 0
\(694\) 0 0
\(695\) 8.00000 0.303457
\(696\) 0 0
\(697\) 12.0000 0.454532
\(698\) 2.00000 0.0757011
\(699\) 0 0
\(700\) 0 0
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 0 0
\(703\) 10.0000 0.377157
\(704\) 0 0
\(705\) 0 0
\(706\) −22.0000 −0.827981
\(707\) 0 0
\(708\) 0 0
\(709\) 38.0000 1.42712 0.713560 0.700594i \(-0.247082\pi\)
0.713560 + 0.700594i \(0.247082\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 6.00000 0.224860
\(713\) −32.0000 −1.19841
\(714\) 0 0
\(715\) 0 0
\(716\) −4.00000 −0.149487
\(717\) 0 0
\(718\) −20.0000 −0.746393
\(719\) 20.0000 0.745874 0.372937 0.927857i \(-0.378351\pi\)
0.372937 + 0.927857i \(0.378351\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −1.00000 −0.0372161
\(723\) 0 0
\(724\) −14.0000 −0.520306
\(725\) 2.00000 0.0742781
\(726\) 0 0
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −20.0000 −0.740233
\(731\) −24.0000 −0.887672
\(732\) 0 0
\(733\) 46.0000 1.69905 0.849524 0.527549i \(-0.176889\pi\)
0.849524 + 0.527549i \(0.176889\pi\)
\(734\) −32.0000 −1.18114
\(735\) 0 0
\(736\) 20.0000 0.737210
\(737\) 0 0
\(738\) 0 0
\(739\) −36.0000 −1.32428 −0.662141 0.749380i \(-0.730352\pi\)
−0.662141 + 0.749380i \(0.730352\pi\)
\(740\) 20.0000 0.735215
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) −12.0000 −0.439646
\(746\) 10.0000 0.366126
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 16.0000 0.583848 0.291924 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) 12.0000 0.437595
\(753\) 0 0
\(754\) 12.0000 0.437014
\(755\) −16.0000 −0.582300
\(756\) 0 0
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) −12.0000 −0.435860
\(759\) 0 0
\(760\) −6.00000 −0.217643
\(761\) −50.0000 −1.81250 −0.906249 0.422744i \(-0.861067\pi\)
−0.906249 + 0.422744i \(0.861067\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) 8.00000 0.289052
\(767\) 72.0000 2.59977
\(768\) 0 0
\(769\) 18.0000 0.649097 0.324548 0.945869i \(-0.394788\pi\)
0.324548 + 0.945869i \(0.394788\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 14.0000 0.503871
\(773\) −18.0000 −0.647415 −0.323708 0.946157i \(-0.604929\pi\)
−0.323708 + 0.946157i \(0.604929\pi\)
\(774\) 0 0
\(775\) −8.00000 −0.287368
\(776\) 30.0000 1.07694
\(777\) 0 0
\(778\) 30.0000 1.07555
\(779\) −2.00000 −0.0716574
\(780\) 0 0
\(781\) 0 0
\(782\) 24.0000 0.858238
\(783\) 0 0
\(784\) 7.00000 0.250000
\(785\) −4.00000 −0.142766
\(786\) 0 0
\(787\) 44.0000 1.56843 0.784215 0.620489i \(-0.213066\pi\)
0.784215 + 0.620489i \(0.213066\pi\)
\(788\) −2.00000 −0.0712470
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −12.0000 −0.426132
\(794\) −14.0000 −0.496841
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) 6.00000 0.212531 0.106265 0.994338i \(-0.466111\pi\)
0.106265 + 0.994338i \(0.466111\pi\)
\(798\) 0 0
\(799\) −72.0000 −2.54718
\(800\) 5.00000 0.176777
\(801\) 0 0
\(802\) 38.0000 1.34183
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) −48.0000 −1.69073
\(807\) 0 0
\(808\) 30.0000 1.05540
\(809\) −26.0000 −0.914111 −0.457056 0.889438i \(-0.651096\pi\)
−0.457056 + 0.889438i \(0.651096\pi\)
\(810\) 0 0
\(811\) 44.0000 1.54505 0.772524 0.634985i \(-0.218994\pi\)
0.772524 + 0.634985i \(0.218994\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −8.00000 −0.280228
\(816\) 0 0
\(817\) 4.00000 0.139942
\(818\) 14.0000 0.489499
\(819\) 0 0
\(820\) −4.00000 −0.139686
\(821\) 42.0000 1.46581 0.732905 0.680331i \(-0.238164\pi\)
0.732905 + 0.680331i \(0.238164\pi\)
\(822\) 0 0
\(823\) −32.0000 −1.11545 −0.557725 0.830026i \(-0.688326\pi\)
−0.557725 + 0.830026i \(0.688326\pi\)
\(824\) 24.0000 0.836080
\(825\) 0 0
\(826\) 0 0
\(827\) 28.0000 0.973655 0.486828 0.873498i \(-0.338154\pi\)
0.486828 + 0.873498i \(0.338154\pi\)
\(828\) 0 0
\(829\) −10.0000 −0.347314 −0.173657 0.984806i \(-0.555558\pi\)
−0.173657 + 0.984806i \(0.555558\pi\)
\(830\) 32.0000 1.11074
\(831\) 0 0
\(832\) 42.0000 1.45609
\(833\) −42.0000 −1.45521
\(834\) 0 0
\(835\) −48.0000 −1.66111
\(836\) 0 0
\(837\) 0 0
\(838\) 8.00000 0.276355
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −14.0000 −0.482472
\(843\) 0 0
\(844\) 4.00000 0.137686
\(845\) 46.0000 1.58245
\(846\) 0 0
\(847\) 0 0
\(848\) −6.00000 −0.206041
\(849\) 0 0
\(850\) 6.00000 0.205798
\(851\) 40.0000 1.37118
\(852\) 0 0
\(853\) 22.0000 0.753266 0.376633 0.926363i \(-0.377082\pi\)
0.376633 + 0.926363i \(0.377082\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) −30.0000 −1.02478 −0.512390 0.858753i \(-0.671240\pi\)
−0.512390 + 0.858753i \(0.671240\pi\)
\(858\) 0 0
\(859\) −28.0000 −0.955348 −0.477674 0.878537i \(-0.658520\pi\)
−0.477674 + 0.878537i \(0.658520\pi\)
\(860\) 8.00000 0.272798
\(861\) 0 0
\(862\) −24.0000 −0.817443
\(863\) 40.0000 1.36162 0.680808 0.732462i \(-0.261629\pi\)
0.680808 + 0.732462i \(0.261629\pi\)
\(864\) 0 0
\(865\) 44.0000 1.49604
\(866\) 14.0000 0.475739
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −24.0000 −0.813209
\(872\) −30.0000 −1.01593
\(873\) 0 0
\(874\) −4.00000 −0.135302
\(875\) 0 0
\(876\) 0 0
\(877\) −34.0000 −1.14810 −0.574049 0.818821i \(-0.694628\pi\)
−0.574049 + 0.818821i \(0.694628\pi\)
\(878\) 8.00000 0.269987
\(879\) 0 0
\(880\) 0 0
\(881\) 38.0000 1.28025 0.640126 0.768270i \(-0.278882\pi\)
0.640126 + 0.768270i \(0.278882\pi\)
\(882\) 0 0
\(883\) −36.0000 −1.21150 −0.605748 0.795656i \(-0.707126\pi\)
−0.605748 + 0.795656i \(0.707126\pi\)
\(884\) −36.0000 −1.21081
\(885\) 0 0
\(886\) 0 0
\(887\) 40.0000 1.34307 0.671534 0.740973i \(-0.265636\pi\)
0.671534 + 0.740973i \(0.265636\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −4.00000 −0.134080
\(891\) 0 0
\(892\) −16.0000 −0.535720
\(893\) 12.0000 0.401565
\(894\) 0 0
\(895\) 8.00000 0.267411
\(896\) 0 0
\(897\) 0 0
\(898\) −2.00000 −0.0667409
\(899\) −16.0000 −0.533630
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) 0 0
\(903\) 0 0
\(904\) −18.0000 −0.598671
\(905\) 28.0000 0.930751
\(906\) 0 0
\(907\) 4.00000 0.132818 0.0664089 0.997792i \(-0.478846\pi\)
0.0664089 + 0.997792i \(0.478846\pi\)
\(908\) −12.0000 −0.398234
\(909\) 0 0
\(910\) 0 0
\(911\) 16.0000 0.530104 0.265052 0.964234i \(-0.414611\pi\)
0.265052 + 0.964234i \(0.414611\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 6.00000 0.198462
\(915\) 0 0
\(916\) −6.00000 −0.198246
\(917\) 0 0
\(918\) 0 0
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) −24.0000 −0.791257
\(921\) 0 0
\(922\) −18.0000 −0.592798
\(923\) 0 0
\(924\) 0 0
\(925\) 10.0000 0.328798
\(926\) −32.0000 −1.05159
\(927\) 0 0
\(928\) 10.0000 0.328266
\(929\) −34.0000 −1.11550 −0.557752 0.830008i \(-0.688336\pi\)
−0.557752 + 0.830008i \(0.688336\pi\)
\(930\) 0 0
\(931\) 7.00000 0.229416
\(932\) 10.0000 0.327561
\(933\) 0 0
\(934\) −32.0000 −1.04707
\(935\) 0 0
\(936\) 0 0
\(937\) −22.0000 −0.718709 −0.359354 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 24.0000 0.782794
\(941\) 22.0000 0.717180 0.358590 0.933495i \(-0.383258\pi\)
0.358590 + 0.933495i \(0.383258\pi\)
\(942\) 0 0
\(943\) −8.00000 −0.260516
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) 0 0
\(947\) 8.00000 0.259965 0.129983 0.991516i \(-0.458508\pi\)
0.129983 + 0.991516i \(0.458508\pi\)
\(948\) 0 0
\(949\) 60.0000 1.94768
\(950\) −1.00000 −0.0324443
\(951\) 0 0
\(952\) 0 0
\(953\) −38.0000 −1.23094 −0.615470 0.788160i \(-0.711034\pi\)
−0.615470 + 0.788160i \(0.711034\pi\)
\(954\) 0 0
\(955\) 24.0000 0.776622
\(956\) −12.0000 −0.388108
\(957\) 0 0
\(958\) 20.0000 0.646171
\(959\) 0 0
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 60.0000 1.93448
\(963\) 0 0
\(964\) 6.00000 0.193247
\(965\) −28.0000 −0.901352
\(966\) 0 0
\(967\) −32.0000 −1.02905 −0.514525 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) −33.0000 −1.06066
\(969\) 0 0
\(970\) −20.0000 −0.642161
\(971\) 36.0000 1.15529 0.577647 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −32.0000 −1.02535
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 14.0000 0.447214
\(981\) 0 0
\(982\) −32.0000 −1.02116
\(983\) 8.00000 0.255160 0.127580 0.991828i \(-0.459279\pi\)
0.127580 + 0.991828i \(0.459279\pi\)
\(984\) 0 0
\(985\) 4.00000 0.127451
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) 6.00000 0.190885
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) 40.0000 1.27064 0.635321 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) −40.0000 −1.27000
\(993\) 0 0
\(994\) 0 0
\(995\) −16.0000 −0.507234
\(996\) 0 0
\(997\) −58.0000 −1.83688 −0.918439 0.395562i \(-0.870550\pi\)
−0.918439 + 0.395562i \(0.870550\pi\)
\(998\) −28.0000 −0.886325
\(999\) 0 0
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 171.2.a.a.1.1 1
3.2 odd 2 57.2.a.c.1.1 1
4.3 odd 2 2736.2.a.s.1.1 1
5.4 even 2 4275.2.a.m.1.1 1
7.6 odd 2 8379.2.a.e.1.1 1
12.11 even 2 912.2.a.b.1.1 1
15.2 even 4 1425.2.c.g.799.2 2
15.8 even 4 1425.2.c.g.799.1 2
15.14 odd 2 1425.2.a.a.1.1 1
19.18 odd 2 3249.2.a.g.1.1 1
21.20 even 2 2793.2.a.i.1.1 1
24.5 odd 2 3648.2.a.o.1.1 1
24.11 even 2 3648.2.a.bf.1.1 1
33.32 even 2 6897.2.a.a.1.1 1
39.38 odd 2 9633.2.a.h.1.1 1
57.56 even 2 1083.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.a.c.1.1 1 3.2 odd 2
171.2.a.a.1.1 1 1.1 even 1 trivial
912.2.a.b.1.1 1 12.11 even 2
1083.2.a.a.1.1 1 57.56 even 2
1425.2.a.a.1.1 1 15.14 odd 2
1425.2.c.g.799.1 2 15.8 even 4
1425.2.c.g.799.2 2 15.2 even 4
2736.2.a.s.1.1 1 4.3 odd 2
2793.2.a.i.1.1 1 21.20 even 2
3249.2.a.g.1.1 1 19.18 odd 2
3648.2.a.o.1.1 1 24.5 odd 2
3648.2.a.bf.1.1 1 24.11 even 2
4275.2.a.m.1.1 1 5.4 even 2
6897.2.a.a.1.1 1 33.32 even 2
8379.2.a.e.1.1 1 7.6 odd 2
9633.2.a.h.1.1 1 39.38 odd 2