Properties

Label 8470.2.a.v.1.1
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -2.00000 q^{12} +1.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} +1.00000 q^{18} +1.00000 q^{20} -2.00000 q^{21} -4.00000 q^{23} -2.00000 q^{24} +1.00000 q^{25} +4.00000 q^{27} +1.00000 q^{28} -2.00000 q^{29} -2.00000 q^{30} -2.00000 q^{31} +1.00000 q^{32} +1.00000 q^{35} +1.00000 q^{36} -6.00000 q^{37} +1.00000 q^{40} -8.00000 q^{41} -2.00000 q^{42} +12.0000 q^{43} +1.00000 q^{45} -4.00000 q^{46} -6.00000 q^{47} -2.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -6.00000 q^{53} +4.00000 q^{54} +1.00000 q^{56} -2.00000 q^{58} -10.0000 q^{59} -2.00000 q^{60} +4.00000 q^{61} -2.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -8.00000 q^{67} +8.00000 q^{69} +1.00000 q^{70} -4.00000 q^{71} +1.00000 q^{72} +4.00000 q^{73} -6.00000 q^{74} -2.00000 q^{75} +16.0000 q^{79} +1.00000 q^{80} -11.0000 q^{81} -8.00000 q^{82} -2.00000 q^{84} +12.0000 q^{86} +4.00000 q^{87} -6.00000 q^{89} +1.00000 q^{90} -4.00000 q^{92} +4.00000 q^{93} -6.00000 q^{94} -2.00000 q^{96} +14.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −2.00000 −0.816497
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) −2.00000 −0.577350
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 1.00000 0.267261
\(15\) −2.00000 −0.516398
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 1.00000 0.235702
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 1.00000 0.223607
\(21\) −2.00000 −0.436436
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) −2.00000 −0.408248
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 1.00000 0.188982
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) −2.00000 −0.365148
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 1.00000 0.166667
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) −8.00000 −1.24939 −0.624695 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) −2.00000 −0.308607
\(43\) 12.0000 1.82998 0.914991 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) −4.00000 −0.589768
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) −2.00000 −0.288675
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 4.00000 0.544331
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −2.00000 −0.262613
\(59\) −10.0000 −1.30189 −0.650945 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) −2.00000 −0.258199
\(61\) 4.00000 0.512148 0.256074 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) −2.00000 −0.254000
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) 8.00000 0.963087
\(70\) 1.00000 0.119523
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 1.00000 0.117851
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) −6.00000 −0.697486
\(75\) −2.00000 −0.230940
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 16.0000 1.80014 0.900070 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) 1.00000 0.111803
\(81\) −11.0000 −1.22222
\(82\) −8.00000 −0.883452
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) 12.0000 1.29399
\(87\) 4.00000 0.428845
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) −4.00000 −0.417029
\(93\) 4.00000 0.414781
\(94\) −6.00000 −0.618853
\(95\) 0 0
\(96\) −2.00000 −0.204124
\(97\) 14.0000 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) −14.0000 −1.37946 −0.689730 0.724066i \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) 0 0
\(105\) −2.00000 −0.195180
\(106\) −6.00000 −0.582772
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 4.00000 0.384900
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 0 0
\(111\) 12.0000 1.13899
\(112\) 1.00000 0.0944911
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 0 0
\(115\) −4.00000 −0.373002
\(116\) −2.00000 −0.185695
\(117\) 0 0
\(118\) −10.0000 −0.920575
\(119\) 0 0
\(120\) −2.00000 −0.182574
\(121\) 0 0
\(122\) 4.00000 0.362143
\(123\) 16.0000 1.44267
\(124\) −2.00000 −0.179605
\(125\) 1.00000 0.0894427
\(126\) 1.00000 0.0890871
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 1.00000 0.0883883
\(129\) −24.0000 −2.11308
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −8.00000 −0.691095
\(135\) 4.00000 0.344265
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 8.00000 0.681005
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 1.00000 0.0845154
\(141\) 12.0000 1.01058
\(142\) −4.00000 −0.335673
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −2.00000 −0.166091
\(146\) 4.00000 0.331042
\(147\) −2.00000 −0.164957
\(148\) −6.00000 −0.493197
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) −2.00000 −0.163299
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2.00000 −0.160644
\(156\) 0 0
\(157\) 6.00000 0.478852 0.239426 0.970915i \(-0.423041\pi\)
0.239426 + 0.970915i \(0.423041\pi\)
\(158\) 16.0000 1.27289
\(159\) 12.0000 0.951662
\(160\) 1.00000 0.0790569
\(161\) −4.00000 −0.315244
\(162\) −11.0000 −0.864242
\(163\) −8.00000 −0.626608 −0.313304 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) −8.00000 −0.624695
\(165\) 0 0
\(166\) 0 0
\(167\) −16.0000 −1.23812 −0.619059 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) −2.00000 −0.154303
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 12.0000 0.914991
\(173\) −24.0000 −1.82469 −0.912343 0.409426i \(-0.865729\pi\)
−0.912343 + 0.409426i \(0.865729\pi\)
\(174\) 4.00000 0.303239
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 20.0000 1.50329
\(178\) −6.00000 −0.449719
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 1.00000 0.0745356
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) −8.00000 −0.591377
\(184\) −4.00000 −0.294884
\(185\) −6.00000 −0.441129
\(186\) 4.00000 0.293294
\(187\) 0 0
\(188\) −6.00000 −0.437595
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) −2.00000 −0.144338
\(193\) 22.0000 1.58359 0.791797 0.610784i \(-0.209146\pi\)
0.791797 + 0.610784i \(0.209146\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −14.0000 −0.997459 −0.498729 0.866758i \(-0.666200\pi\)
−0.498729 + 0.866758i \(0.666200\pi\)
\(198\) 0 0
\(199\) −6.00000 −0.425329 −0.212664 0.977125i \(-0.568214\pi\)
−0.212664 + 0.977125i \(0.568214\pi\)
\(200\) 1.00000 0.0707107
\(201\) 16.0000 1.12855
\(202\) 0 0
\(203\) −2.00000 −0.140372
\(204\) 0 0
\(205\) −8.00000 −0.558744
\(206\) −14.0000 −0.975426
\(207\) −4.00000 −0.278019
\(208\) 0 0
\(209\) 0 0
\(210\) −2.00000 −0.138013
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) −6.00000 −0.412082
\(213\) 8.00000 0.548151
\(214\) −12.0000 −0.820303
\(215\) 12.0000 0.818393
\(216\) 4.00000 0.272166
\(217\) −2.00000 −0.135769
\(218\) 6.00000 0.406371
\(219\) −8.00000 −0.540590
\(220\) 0 0
\(221\) 0 0
\(222\) 12.0000 0.805387
\(223\) −6.00000 −0.401790 −0.200895 0.979613i \(-0.564385\pi\)
−0.200895 + 0.979613i \(0.564385\pi\)
\(224\) 1.00000 0.0668153
\(225\) 1.00000 0.0666667
\(226\) 2.00000 0.133038
\(227\) −4.00000 −0.265489 −0.132745 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) 0 0
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) −4.00000 −0.263752
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) −6.00000 −0.391397
\(236\) −10.0000 −0.650945
\(237\) −32.0000 −2.07862
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) −2.00000 −0.129099
\(241\) 16.0000 1.03065 0.515325 0.856995i \(-0.327671\pi\)
0.515325 + 0.856995i \(0.327671\pi\)
\(242\) 0 0
\(243\) 10.0000 0.641500
\(244\) 4.00000 0.256074
\(245\) 1.00000 0.0638877
\(246\) 16.0000 1.02012
\(247\) 0 0
\(248\) −2.00000 −0.127000
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 10.0000 0.631194 0.315597 0.948893i \(-0.397795\pi\)
0.315597 + 0.948893i \(0.397795\pi\)
\(252\) 1.00000 0.0629941
\(253\) 0 0
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) −24.0000 −1.49417
\(259\) −6.00000 −0.372822
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) −4.00000 −0.247121
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) 0 0
\(267\) 12.0000 0.734388
\(268\) −8.00000 −0.488678
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 4.00000 0.243432
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 8.00000 0.481543
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) −8.00000 −0.479808
\(279\) −2.00000 −0.119737
\(280\) 1.00000 0.0597614
\(281\) 2.00000 0.119310 0.0596550 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) 12.0000 0.714590
\(283\) −28.0000 −1.66443 −0.832214 0.554455i \(-0.812927\pi\)
−0.832214 + 0.554455i \(0.812927\pi\)
\(284\) −4.00000 −0.237356
\(285\) 0 0
\(286\) 0 0
\(287\) −8.00000 −0.472225
\(288\) 1.00000 0.0589256
\(289\) −17.0000 −1.00000
\(290\) −2.00000 −0.117444
\(291\) −28.0000 −1.64139
\(292\) 4.00000 0.234082
\(293\) 4.00000 0.233682 0.116841 0.993151i \(-0.462723\pi\)
0.116841 + 0.993151i \(0.462723\pi\)
\(294\) −2.00000 −0.116642
\(295\) −10.0000 −0.582223
\(296\) −6.00000 −0.348743
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) 0 0
\(300\) −2.00000 −0.115470
\(301\) 12.0000 0.691669
\(302\) 8.00000 0.460348
\(303\) 0 0
\(304\) 0 0
\(305\) 4.00000 0.229039
\(306\) 0 0
\(307\) −28.0000 −1.59804 −0.799022 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) 0 0
\(309\) 28.0000 1.59286
\(310\) −2.00000 −0.113592
\(311\) −6.00000 −0.340229 −0.170114 0.985424i \(-0.554414\pi\)
−0.170114 + 0.985424i \(0.554414\pi\)
\(312\) 0 0
\(313\) 2.00000 0.113047 0.0565233 0.998401i \(-0.481998\pi\)
0.0565233 + 0.998401i \(0.481998\pi\)
\(314\) 6.00000 0.338600
\(315\) 1.00000 0.0563436
\(316\) 16.0000 0.900070
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) 12.0000 0.672927
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) 24.0000 1.33955
\(322\) −4.00000 −0.222911
\(323\) 0 0
\(324\) −11.0000 −0.611111
\(325\) 0 0
\(326\) −8.00000 −0.443079
\(327\) −12.0000 −0.663602
\(328\) −8.00000 −0.441726
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) 0 0
\(333\) −6.00000 −0.328798
\(334\) −16.0000 −0.875481
\(335\) −8.00000 −0.437087
\(336\) −2.00000 −0.109109
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) −13.0000 −0.707107
\(339\) −4.00000 −0.217250
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 12.0000 0.646997
\(345\) 8.00000 0.430706
\(346\) −24.0000 −1.29025
\(347\) −28.0000 −1.50312 −0.751559 0.659665i \(-0.770698\pi\)
−0.751559 + 0.659665i \(0.770698\pi\)
\(348\) 4.00000 0.214423
\(349\) 20.0000 1.07058 0.535288 0.844670i \(-0.320203\pi\)
0.535288 + 0.844670i \(0.320203\pi\)
\(350\) 1.00000 0.0534522
\(351\) 0 0
\(352\) 0 0
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 20.0000 1.06299
\(355\) −4.00000 −0.212298
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) 8.00000 0.422224 0.211112 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) 1.00000 0.0527046
\(361\) −19.0000 −1.00000
\(362\) −6.00000 −0.315353
\(363\) 0 0
\(364\) 0 0
\(365\) 4.00000 0.209370
\(366\) −8.00000 −0.418167
\(367\) −22.0000 −1.14839 −0.574195 0.818718i \(-0.694685\pi\)
−0.574195 + 0.818718i \(0.694685\pi\)
\(368\) −4.00000 −0.208514
\(369\) −8.00000 −0.416463
\(370\) −6.00000 −0.311925
\(371\) −6.00000 −0.311504
\(372\) 4.00000 0.207390
\(373\) −14.0000 −0.724893 −0.362446 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) 0 0
\(375\) −2.00000 −0.103280
\(376\) −6.00000 −0.309426
\(377\) 0 0
\(378\) 4.00000 0.205738
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) 16.0000 0.819705
\(382\) −8.00000 −0.409316
\(383\) −2.00000 −0.102195 −0.0510976 0.998694i \(-0.516272\pi\)
−0.0510976 + 0.998694i \(0.516272\pi\)
\(384\) −2.00000 −0.102062
\(385\) 0 0
\(386\) 22.0000 1.11977
\(387\) 12.0000 0.609994
\(388\) 14.0000 0.710742
\(389\) 22.0000 1.11544 0.557722 0.830028i \(-0.311675\pi\)
0.557722 + 0.830028i \(0.311675\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 1.00000 0.0505076
\(393\) 8.00000 0.403547
\(394\) −14.0000 −0.705310
\(395\) 16.0000 0.805047
\(396\) 0 0
\(397\) −34.0000 −1.70641 −0.853206 0.521575i \(-0.825345\pi\)
−0.853206 + 0.521575i \(0.825345\pi\)
\(398\) −6.00000 −0.300753
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 16.0000 0.798007
\(403\) 0 0
\(404\) 0 0
\(405\) −11.0000 −0.546594
\(406\) −2.00000 −0.0992583
\(407\) 0 0
\(408\) 0 0
\(409\) −4.00000 −0.197787 −0.0988936 0.995098i \(-0.531530\pi\)
−0.0988936 + 0.995098i \(0.531530\pi\)
\(410\) −8.00000 −0.395092
\(411\) −12.0000 −0.591916
\(412\) −14.0000 −0.689730
\(413\) −10.0000 −0.492068
\(414\) −4.00000 −0.196589
\(415\) 0 0
\(416\) 0 0
\(417\) 16.0000 0.783523
\(418\) 0 0
\(419\) −26.0000 −1.27018 −0.635092 0.772437i \(-0.719038\pi\)
−0.635092 + 0.772437i \(0.719038\pi\)
\(420\) −2.00000 −0.0975900
\(421\) 18.0000 0.877266 0.438633 0.898666i \(-0.355463\pi\)
0.438633 + 0.898666i \(0.355463\pi\)
\(422\) 12.0000 0.584151
\(423\) −6.00000 −0.291730
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 8.00000 0.387601
\(427\) 4.00000 0.193574
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 12.0000 0.578691
\(431\) −8.00000 −0.385346 −0.192673 0.981263i \(-0.561716\pi\)
−0.192673 + 0.981263i \(0.561716\pi\)
\(432\) 4.00000 0.192450
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) −2.00000 −0.0960031
\(435\) 4.00000 0.191785
\(436\) 6.00000 0.287348
\(437\) 0 0
\(438\) −8.00000 −0.382255
\(439\) −4.00000 −0.190910 −0.0954548 0.995434i \(-0.530431\pi\)
−0.0954548 + 0.995434i \(0.530431\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −20.0000 −0.950229 −0.475114 0.879924i \(-0.657593\pi\)
−0.475114 + 0.879924i \(0.657593\pi\)
\(444\) 12.0000 0.569495
\(445\) −6.00000 −0.284427
\(446\) −6.00000 −0.284108
\(447\) 12.0000 0.567581
\(448\) 1.00000 0.0472456
\(449\) −10.0000 −0.471929 −0.235965 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0 0
\(452\) 2.00000 0.0940721
\(453\) −16.0000 −0.751746
\(454\) −4.00000 −0.187729
\(455\) 0 0
\(456\) 0 0
\(457\) −26.0000 −1.21623 −0.608114 0.793849i \(-0.708074\pi\)
−0.608114 + 0.793849i \(0.708074\pi\)
\(458\) 2.00000 0.0934539
\(459\) 0 0
\(460\) −4.00000 −0.186501
\(461\) 20.0000 0.931493 0.465746 0.884918i \(-0.345786\pi\)
0.465746 + 0.884918i \(0.345786\pi\)
\(462\) 0 0
\(463\) 4.00000 0.185896 0.0929479 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 4.00000 0.185496
\(466\) 18.0000 0.833834
\(467\) −30.0000 −1.38823 −0.694117 0.719862i \(-0.744205\pi\)
−0.694117 + 0.719862i \(0.744205\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) −6.00000 −0.276759
\(471\) −12.0000 −0.552931
\(472\) −10.0000 −0.460287
\(473\) 0 0
\(474\) −32.0000 −1.46981
\(475\) 0 0
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) 0 0
\(479\) −20.0000 −0.913823 −0.456912 0.889512i \(-0.651044\pi\)
−0.456912 + 0.889512i \(0.651044\pi\)
\(480\) −2.00000 −0.0912871
\(481\) 0 0
\(482\) 16.0000 0.728780
\(483\) 8.00000 0.364013
\(484\) 0 0
\(485\) 14.0000 0.635707
\(486\) 10.0000 0.453609
\(487\) 20.0000 0.906287 0.453143 0.891438i \(-0.350303\pi\)
0.453143 + 0.891438i \(0.350303\pi\)
\(488\) 4.00000 0.181071
\(489\) 16.0000 0.723545
\(490\) 1.00000 0.0451754
\(491\) −36.0000 −1.62466 −0.812329 0.583200i \(-0.801800\pi\)
−0.812329 + 0.583200i \(0.801800\pi\)
\(492\) 16.0000 0.721336
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −2.00000 −0.0898027
\(497\) −4.00000 −0.179425
\(498\) 0 0
\(499\) −40.0000 −1.79065 −0.895323 0.445418i \(-0.853055\pi\)
−0.895323 + 0.445418i \(0.853055\pi\)
\(500\) 1.00000 0.0447214
\(501\) 32.0000 1.42965
\(502\) 10.0000 0.446322
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 1.00000 0.0445435
\(505\) 0 0
\(506\) 0 0
\(507\) 26.0000 1.15470
\(508\) −8.00000 −0.354943
\(509\) 18.0000 0.797836 0.398918 0.916987i \(-0.369386\pi\)
0.398918 + 0.916987i \(0.369386\pi\)
\(510\) 0 0
\(511\) 4.00000 0.176950
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −14.0000 −0.617514
\(515\) −14.0000 −0.616914
\(516\) −24.0000 −1.05654
\(517\) 0 0
\(518\) −6.00000 −0.263625
\(519\) 48.0000 2.10697
\(520\) 0 0
\(521\) −34.0000 −1.48957 −0.744784 0.667306i \(-0.767447\pi\)
−0.744784 + 0.667306i \(0.767447\pi\)
\(522\) −2.00000 −0.0875376
\(523\) −28.0000 −1.22435 −0.612177 0.790721i \(-0.709706\pi\)
−0.612177 + 0.790721i \(0.709706\pi\)
\(524\) −4.00000 −0.174741
\(525\) −2.00000 −0.0872872
\(526\) 24.0000 1.04645
\(527\) 0 0
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) −6.00000 −0.260623
\(531\) −10.0000 −0.433963
\(532\) 0 0
\(533\) 0 0
\(534\) 12.0000 0.519291
\(535\) −12.0000 −0.518805
\(536\) −8.00000 −0.345547
\(537\) −24.0000 −1.03568
\(538\) 10.0000 0.431131
\(539\) 0 0
\(540\) 4.00000 0.172133
\(541\) 10.0000 0.429934 0.214967 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) −20.0000 −0.859074
\(543\) 12.0000 0.514969
\(544\) 0 0
\(545\) 6.00000 0.257012
\(546\) 0 0
\(547\) 44.0000 1.88130 0.940652 0.339372i \(-0.110215\pi\)
0.940652 + 0.339372i \(0.110215\pi\)
\(548\) 6.00000 0.256307
\(549\) 4.00000 0.170716
\(550\) 0 0
\(551\) 0 0
\(552\) 8.00000 0.340503
\(553\) 16.0000 0.680389
\(554\) 2.00000 0.0849719
\(555\) 12.0000 0.509372
\(556\) −8.00000 −0.339276
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) −2.00000 −0.0846668
\(559\) 0 0
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 2.00000 0.0843649
\(563\) −32.0000 −1.34864 −0.674320 0.738440i \(-0.735563\pi\)
−0.674320 + 0.738440i \(0.735563\pi\)
\(564\) 12.0000 0.505291
\(565\) 2.00000 0.0841406
\(566\) −28.0000 −1.17693
\(567\) −11.0000 −0.461957
\(568\) −4.00000 −0.167836
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 0 0
\(573\) 16.0000 0.668410
\(574\) −8.00000 −0.333914
\(575\) −4.00000 −0.166812
\(576\) 1.00000 0.0416667
\(577\) −26.0000 −1.08239 −0.541197 0.840896i \(-0.682029\pi\)
−0.541197 + 0.840896i \(0.682029\pi\)
\(578\) −17.0000 −0.707107
\(579\) −44.0000 −1.82858
\(580\) −2.00000 −0.0830455
\(581\) 0 0
\(582\) −28.0000 −1.16064
\(583\) 0 0
\(584\) 4.00000 0.165521
\(585\) 0 0
\(586\) 4.00000 0.165238
\(587\) 2.00000 0.0825488 0.0412744 0.999148i \(-0.486858\pi\)
0.0412744 + 0.999148i \(0.486858\pi\)
\(588\) −2.00000 −0.0824786
\(589\) 0 0
\(590\) −10.0000 −0.411693
\(591\) 28.0000 1.15177
\(592\) −6.00000 −0.246598
\(593\) 12.0000 0.492781 0.246390 0.969171i \(-0.420755\pi\)
0.246390 + 0.969171i \(0.420755\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) 12.0000 0.491127
\(598\) 0 0
\(599\) 20.0000 0.817178 0.408589 0.912719i \(-0.366021\pi\)
0.408589 + 0.912719i \(0.366021\pi\)
\(600\) −2.00000 −0.0816497
\(601\) 8.00000 0.326327 0.163163 0.986599i \(-0.447830\pi\)
0.163163 + 0.986599i \(0.447830\pi\)
\(602\) 12.0000 0.489083
\(603\) −8.00000 −0.325785
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) 0 0
\(607\) −24.0000 −0.974130 −0.487065 0.873366i \(-0.661933\pi\)
−0.487065 + 0.873366i \(0.661933\pi\)
\(608\) 0 0
\(609\) 4.00000 0.162088
\(610\) 4.00000 0.161955
\(611\) 0 0
\(612\) 0 0
\(613\) 6.00000 0.242338 0.121169 0.992632i \(-0.461336\pi\)
0.121169 + 0.992632i \(0.461336\pi\)
\(614\) −28.0000 −1.12999
\(615\) 16.0000 0.645182
\(616\) 0 0
\(617\) −42.0000 −1.69086 −0.845428 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) 28.0000 1.12633
\(619\) −22.0000 −0.884255 −0.442127 0.896952i \(-0.645776\pi\)
−0.442127 + 0.896952i \(0.645776\pi\)
\(620\) −2.00000 −0.0803219
\(621\) −16.0000 −0.642058
\(622\) −6.00000 −0.240578
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 2.00000 0.0799361
\(627\) 0 0
\(628\) 6.00000 0.239426
\(629\) 0 0
\(630\) 1.00000 0.0398410
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 16.0000 0.636446
\(633\) −24.0000 −0.953914
\(634\) −2.00000 −0.0794301
\(635\) −8.00000 −0.317470
\(636\) 12.0000 0.475831
\(637\) 0 0
\(638\) 0 0
\(639\) −4.00000 −0.158238
\(640\) 1.00000 0.0395285
\(641\) 46.0000 1.81689 0.908445 0.418004i \(-0.137270\pi\)
0.908445 + 0.418004i \(0.137270\pi\)
\(642\) 24.0000 0.947204
\(643\) 34.0000 1.34083 0.670415 0.741987i \(-0.266116\pi\)
0.670415 + 0.741987i \(0.266116\pi\)
\(644\) −4.00000 −0.157622
\(645\) −24.0000 −0.944999
\(646\) 0 0
\(647\) −42.0000 −1.65119 −0.825595 0.564263i \(-0.809160\pi\)
−0.825595 + 0.564263i \(0.809160\pi\)
\(648\) −11.0000 −0.432121
\(649\) 0 0
\(650\) 0 0
\(651\) 4.00000 0.156772
\(652\) −8.00000 −0.313304
\(653\) 22.0000 0.860927 0.430463 0.902608i \(-0.358350\pi\)
0.430463 + 0.902608i \(0.358350\pi\)
\(654\) −12.0000 −0.469237
\(655\) −4.00000 −0.156293
\(656\) −8.00000 −0.312348
\(657\) 4.00000 0.156055
\(658\) −6.00000 −0.233904
\(659\) 20.0000 0.779089 0.389545 0.921008i \(-0.372632\pi\)
0.389545 + 0.921008i \(0.372632\pi\)
\(660\) 0 0
\(661\) 46.0000 1.78919 0.894596 0.446875i \(-0.147463\pi\)
0.894596 + 0.446875i \(0.147463\pi\)
\(662\) 28.0000 1.08825
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) 8.00000 0.309761
\(668\) −16.0000 −0.619059
\(669\) 12.0000 0.463947
\(670\) −8.00000 −0.309067
\(671\) 0 0
\(672\) −2.00000 −0.0771517
\(673\) −14.0000 −0.539660 −0.269830 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) −14.0000 −0.539260
\(675\) 4.00000 0.153960
\(676\) −13.0000 −0.500000
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) −4.00000 −0.153619
\(679\) 14.0000 0.537271
\(680\) 0 0
\(681\) 8.00000 0.306561
\(682\) 0 0
\(683\) 20.0000 0.765279 0.382639 0.923898i \(-0.375015\pi\)
0.382639 + 0.923898i \(0.375015\pi\)
\(684\) 0 0
\(685\) 6.00000 0.229248
\(686\) 1.00000 0.0381802
\(687\) −4.00000 −0.152610
\(688\) 12.0000 0.457496
\(689\) 0 0
\(690\) 8.00000 0.304555
\(691\) 6.00000 0.228251 0.114125 0.993466i \(-0.463593\pi\)
0.114125 + 0.993466i \(0.463593\pi\)
\(692\) −24.0000 −0.912343
\(693\) 0 0
\(694\) −28.0000 −1.06287
\(695\) −8.00000 −0.303457
\(696\) 4.00000 0.151620
\(697\) 0 0
\(698\) 20.0000 0.757011
\(699\) −36.0000 −1.36165
\(700\) 1.00000 0.0377964
\(701\) 14.0000 0.528773 0.264386 0.964417i \(-0.414831\pi\)
0.264386 + 0.964417i \(0.414831\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 12.0000 0.451946
\(706\) 18.0000 0.677439
\(707\) 0 0
\(708\) 20.0000 0.751646
\(709\) 30.0000 1.12667 0.563337 0.826227i \(-0.309517\pi\)
0.563337 + 0.826227i \(0.309517\pi\)
\(710\) −4.00000 −0.150117
\(711\) 16.0000 0.600047
\(712\) −6.00000 −0.224860
\(713\) 8.00000 0.299602
\(714\) 0 0
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) 8.00000 0.298557
\(719\) −34.0000 −1.26799 −0.633993 0.773339i \(-0.718585\pi\)
−0.633993 + 0.773339i \(0.718585\pi\)
\(720\) 1.00000 0.0372678
\(721\) −14.0000 −0.521387
\(722\) −19.0000 −0.707107
\(723\) −32.0000 −1.19009
\(724\) −6.00000 −0.222988
\(725\) −2.00000 −0.0742781
\(726\) 0 0
\(727\) 14.0000 0.519231 0.259616 0.965712i \(-0.416404\pi\)
0.259616 + 0.965712i \(0.416404\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 4.00000 0.148047
\(731\) 0 0
\(732\) −8.00000 −0.295689
\(733\) −4.00000 −0.147743 −0.0738717 0.997268i \(-0.523536\pi\)
−0.0738717 + 0.997268i \(0.523536\pi\)
\(734\) −22.0000 −0.812035
\(735\) −2.00000 −0.0737711
\(736\) −4.00000 −0.147442
\(737\) 0 0
\(738\) −8.00000 −0.294484
\(739\) −12.0000 −0.441427 −0.220714 0.975339i \(-0.570839\pi\)
−0.220714 + 0.975339i \(0.570839\pi\)
\(740\) −6.00000 −0.220564
\(741\) 0 0
\(742\) −6.00000 −0.220267
\(743\) 8.00000 0.293492 0.146746 0.989174i \(-0.453120\pi\)
0.146746 + 0.989174i \(0.453120\pi\)
\(744\) 4.00000 0.146647
\(745\) −6.00000 −0.219823
\(746\) −14.0000 −0.512576
\(747\) 0 0
\(748\) 0 0
\(749\) −12.0000 −0.438470
\(750\) −2.00000 −0.0730297
\(751\) 52.0000 1.89751 0.948753 0.316017i \(-0.102346\pi\)
0.948753 + 0.316017i \(0.102346\pi\)
\(752\) −6.00000 −0.218797
\(753\) −20.0000 −0.728841
\(754\) 0 0
\(755\) 8.00000 0.291150
\(756\) 4.00000 0.145479
\(757\) −26.0000 −0.944986 −0.472493 0.881334i \(-0.656646\pi\)
−0.472493 + 0.881334i \(0.656646\pi\)
\(758\) −16.0000 −0.581146
\(759\) 0 0
\(760\) 0 0
\(761\) −12.0000 −0.435000 −0.217500 0.976060i \(-0.569790\pi\)
−0.217500 + 0.976060i \(0.569790\pi\)
\(762\) 16.0000 0.579619
\(763\) 6.00000 0.217215
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) −2.00000 −0.0722629
\(767\) 0 0
\(768\) −2.00000 −0.0721688
\(769\) −12.0000 −0.432731 −0.216366 0.976312i \(-0.569420\pi\)
−0.216366 + 0.976312i \(0.569420\pi\)
\(770\) 0 0
\(771\) 28.0000 1.00840
\(772\) 22.0000 0.791797
\(773\) 18.0000 0.647415 0.323708 0.946157i \(-0.395071\pi\)
0.323708 + 0.946157i \(0.395071\pi\)
\(774\) 12.0000 0.431331
\(775\) −2.00000 −0.0718421
\(776\) 14.0000 0.502571
\(777\) 12.0000 0.430498
\(778\) 22.0000 0.788738
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −8.00000 −0.285897
\(784\) 1.00000 0.0357143
\(785\) 6.00000 0.214149
\(786\) 8.00000 0.285351
\(787\) −48.0000 −1.71102 −0.855508 0.517790i \(-0.826755\pi\)
−0.855508 + 0.517790i \(0.826755\pi\)
\(788\) −14.0000 −0.498729
\(789\) −48.0000 −1.70885
\(790\) 16.0000 0.569254
\(791\) 2.00000 0.0711118
\(792\) 0 0
\(793\) 0 0
\(794\) −34.0000 −1.20661
\(795\) 12.0000 0.425596
\(796\) −6.00000 −0.212664
\(797\) 6.00000 0.212531 0.106265 0.994338i \(-0.466111\pi\)
0.106265 + 0.994338i \(0.466111\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) −6.00000 −0.212000
\(802\) −6.00000 −0.211867
\(803\) 0 0
\(804\) 16.0000 0.564276
\(805\) −4.00000 −0.140981
\(806\) 0 0
\(807\) −20.0000 −0.704033
\(808\) 0 0
\(809\) −18.0000 −0.632846 −0.316423 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) −11.0000 −0.386501
\(811\) 44.0000 1.54505 0.772524 0.634985i \(-0.218994\pi\)
0.772524 + 0.634985i \(0.218994\pi\)
\(812\) −2.00000 −0.0701862
\(813\) 40.0000 1.40286
\(814\) 0 0
\(815\) −8.00000 −0.280228
\(816\) 0 0
\(817\) 0 0
\(818\) −4.00000 −0.139857
\(819\) 0 0
\(820\) −8.00000 −0.279372
\(821\) 34.0000 1.18661 0.593304 0.804978i \(-0.297823\pi\)
0.593304 + 0.804978i \(0.297823\pi\)
\(822\) −12.0000 −0.418548
\(823\) 32.0000 1.11545 0.557725 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) −14.0000 −0.487713
\(825\) 0 0
\(826\) −10.0000 −0.347945
\(827\) 36.0000 1.25184 0.625921 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) −4.00000 −0.139010
\(829\) −10.0000 −0.347314 −0.173657 0.984806i \(-0.555558\pi\)
−0.173657 + 0.984806i \(0.555558\pi\)
\(830\) 0 0
\(831\) −4.00000 −0.138758
\(832\) 0 0
\(833\) 0 0
\(834\) 16.0000 0.554035
\(835\) −16.0000 −0.553703
\(836\) 0 0
\(837\) −8.00000 −0.276520
\(838\) −26.0000 −0.898155
\(839\) −10.0000 −0.345238 −0.172619 0.984989i \(-0.555223\pi\)
−0.172619 + 0.984989i \(0.555223\pi\)
\(840\) −2.00000 −0.0690066
\(841\) −25.0000 −0.862069
\(842\) 18.0000 0.620321
\(843\) −4.00000 −0.137767
\(844\) 12.0000 0.413057
\(845\) −13.0000 −0.447214
\(846\) −6.00000 −0.206284
\(847\) 0 0
\(848\) −6.00000 −0.206041
\(849\) 56.0000 1.92192
\(850\) 0 0
\(851\) 24.0000 0.822709
\(852\) 8.00000 0.274075
\(853\) 8.00000 0.273915 0.136957 0.990577i \(-0.456268\pi\)
0.136957 + 0.990577i \(0.456268\pi\)
\(854\) 4.00000 0.136877
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) −12.0000 −0.409912 −0.204956 0.978771i \(-0.565705\pi\)
−0.204956 + 0.978771i \(0.565705\pi\)
\(858\) 0 0
\(859\) 2.00000 0.0682391 0.0341196 0.999418i \(-0.489137\pi\)
0.0341196 + 0.999418i \(0.489137\pi\)
\(860\) 12.0000 0.409197
\(861\) 16.0000 0.545279
\(862\) −8.00000 −0.272481
\(863\) −32.0000 −1.08929 −0.544646 0.838666i \(-0.683336\pi\)
−0.544646 + 0.838666i \(0.683336\pi\)
\(864\) 4.00000 0.136083
\(865\) −24.0000 −0.816024
\(866\) 14.0000 0.475739
\(867\) 34.0000 1.15470
\(868\) −2.00000 −0.0678844
\(869\) 0 0
\(870\) 4.00000 0.135613
\(871\) 0 0
\(872\) 6.00000 0.203186
\(873\) 14.0000 0.473828
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) −8.00000 −0.270295
\(877\) −10.0000 −0.337676 −0.168838 0.985644i \(-0.554001\pi\)
−0.168838 + 0.985644i \(0.554001\pi\)
\(878\) −4.00000 −0.134993
\(879\) −8.00000 −0.269833
\(880\) 0 0
\(881\) −26.0000 −0.875962 −0.437981 0.898984i \(-0.644306\pi\)
−0.437981 + 0.898984i \(0.644306\pi\)
\(882\) 1.00000 0.0336718
\(883\) −44.0000 −1.48072 −0.740359 0.672212i \(-0.765344\pi\)
−0.740359 + 0.672212i \(0.765344\pi\)
\(884\) 0 0
\(885\) 20.0000 0.672293
\(886\) −20.0000 −0.671913
\(887\) 52.0000 1.74599 0.872995 0.487730i \(-0.162175\pi\)
0.872995 + 0.487730i \(0.162175\pi\)
\(888\) 12.0000 0.402694
\(889\) −8.00000 −0.268311
\(890\) −6.00000 −0.201120
\(891\) 0 0
\(892\) −6.00000 −0.200895
\(893\) 0 0
\(894\) 12.0000 0.401340
\(895\) 12.0000 0.401116
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −10.0000 −0.333704
\(899\) 4.00000 0.133407
\(900\) 1.00000 0.0333333
\(901\) 0 0
\(902\) 0 0
\(903\) −24.0000 −0.798670
\(904\) 2.00000 0.0665190
\(905\) −6.00000 −0.199447
\(906\) −16.0000 −0.531564
\(907\) 48.0000 1.59381 0.796907 0.604102i \(-0.206468\pi\)
0.796907 + 0.604102i \(0.206468\pi\)
\(908\) −4.00000 −0.132745
\(909\) 0 0
\(910\) 0 0
\(911\) 28.0000 0.927681 0.463841 0.885919i \(-0.346471\pi\)
0.463841 + 0.885919i \(0.346471\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −26.0000 −0.860004
\(915\) −8.00000 −0.264472
\(916\) 2.00000 0.0660819
\(917\) −4.00000 −0.132092
\(918\) 0 0
\(919\) 8.00000 0.263896 0.131948 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) −4.00000 −0.131876
\(921\) 56.0000 1.84526
\(922\) 20.0000 0.658665
\(923\) 0 0
\(924\) 0 0
\(925\) −6.00000 −0.197279
\(926\) 4.00000 0.131448
\(927\) −14.0000 −0.459820
\(928\) −2.00000 −0.0656532
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 4.00000 0.131165
\(931\) 0 0
\(932\) 18.0000 0.589610
\(933\) 12.0000 0.392862
\(934\) −30.0000 −0.981630
\(935\) 0 0
\(936\) 0 0
\(937\) −36.0000 −1.17607 −0.588034 0.808836i \(-0.700098\pi\)
−0.588034 + 0.808836i \(0.700098\pi\)
\(938\) −8.00000 −0.261209
\(939\) −4.00000 −0.130535
\(940\) −6.00000 −0.195698
\(941\) 60.0000 1.95594 0.977972 0.208736i \(-0.0669349\pi\)
0.977972 + 0.208736i \(0.0669349\pi\)
\(942\) −12.0000 −0.390981
\(943\) 32.0000 1.04206
\(944\) −10.0000 −0.325472
\(945\) 4.00000 0.130120
\(946\) 0 0
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) −32.0000 −1.03931
\(949\) 0 0
\(950\) 0 0
\(951\) 4.00000 0.129709
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) −6.00000 −0.194257
\(955\) −8.00000 −0.258874
\(956\) 0 0
\(957\) 0 0
\(958\) −20.0000 −0.646171
\(959\) 6.00000 0.193750
\(960\) −2.00000 −0.0645497
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) −12.0000 −0.386695
\(964\) 16.0000 0.515325
\(965\) 22.0000 0.708205
\(966\) 8.00000 0.257396
\(967\) 48.0000 1.54358 0.771788 0.635880i \(-0.219363\pi\)
0.771788 + 0.635880i \(0.219363\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 14.0000 0.449513
\(971\) 26.0000 0.834380 0.417190 0.908819i \(-0.363015\pi\)
0.417190 + 0.908819i \(0.363015\pi\)
\(972\) 10.0000 0.320750
\(973\) −8.00000 −0.256468
\(974\) 20.0000 0.640841
\(975\) 0 0
\(976\) 4.00000 0.128037
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 16.0000 0.511624
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) 6.00000 0.191565
\(982\) −36.0000 −1.14881
\(983\) −22.0000 −0.701691 −0.350846 0.936433i \(-0.614106\pi\)
−0.350846 + 0.936433i \(0.614106\pi\)
\(984\) 16.0000 0.510061
\(985\) −14.0000 −0.446077
\(986\) 0 0
\(987\) 12.0000 0.381964
\(988\) 0 0
\(989\) −48.0000 −1.52631
\(990\) 0 0
\(991\) −4.00000 −0.127064 −0.0635321 0.997980i \(-0.520237\pi\)
−0.0635321 + 0.997980i \(0.520237\pi\)
\(992\) −2.00000 −0.0635001
\(993\) −56.0000 −1.77711
\(994\) −4.00000 −0.126872
\(995\) −6.00000 −0.190213
\(996\) 0 0
\(997\) 48.0000 1.52018 0.760088 0.649821i \(-0.225156\pi\)
0.760088 + 0.649821i \(0.225156\pi\)
\(998\) −40.0000 −1.26618
\(999\) −24.0000 −0.759326
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 8470.2.a.v.1.1 1
11.10 odd 2 770.2.a.b.1.1 1
33.32 even 2 6930.2.a.s.1.1 1
44.43 even 2 6160.2.a.p.1.1 1
55.32 even 4 3850.2.c.p.1849.1 2
55.43 even 4 3850.2.c.p.1849.2 2
55.54 odd 2 3850.2.a.bb.1.1 1
77.76 even 2 5390.2.a.q.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.b.1.1 1 11.10 odd 2
3850.2.a.bb.1.1 1 55.54 odd 2
3850.2.c.p.1849.1 2 55.32 even 4
3850.2.c.p.1849.2 2 55.43 even 4
5390.2.a.q.1.1 1 77.76 even 2
6160.2.a.p.1.1 1 44.43 even 2
6930.2.a.s.1.1 1 33.32 even 2
8470.2.a.v.1.1 1 1.1 even 1 trivial