Properties

Label 3549.2.a.a.1.1
Level $3549$
Weight $2$
Character 3549.1
Self dual yes
Analytic conductor $28.339$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3549 = 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3549.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.3389076774\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 273)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3549.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -3.00000 q^{5} -2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -3.00000 q^{5} -2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{9} +6.00000 q^{10} +2.00000 q^{12} +2.00000 q^{14} -3.00000 q^{15} -4.00000 q^{16} +2.00000 q^{17} -2.00000 q^{18} -1.00000 q^{19} -6.00000 q^{20} -1.00000 q^{21} +1.00000 q^{23} +4.00000 q^{25} +1.00000 q^{27} -2.00000 q^{28} +5.00000 q^{29} +6.00000 q^{30} -5.00000 q^{31} +8.00000 q^{32} -4.00000 q^{34} +3.00000 q^{35} +2.00000 q^{36} -8.00000 q^{37} +2.00000 q^{38} +10.0000 q^{41} +2.00000 q^{42} -9.00000 q^{43} -3.00000 q^{45} -2.00000 q^{46} +7.00000 q^{47} -4.00000 q^{48} +1.00000 q^{49} -8.00000 q^{50} +2.00000 q^{51} +9.00000 q^{53} -2.00000 q^{54} -1.00000 q^{57} -10.0000 q^{58} -4.00000 q^{59} -6.00000 q^{60} -8.00000 q^{61} +10.0000 q^{62} -1.00000 q^{63} -8.00000 q^{64} -2.00000 q^{67} +4.00000 q^{68} +1.00000 q^{69} -6.00000 q^{70} +9.00000 q^{73} +16.0000 q^{74} +4.00000 q^{75} -2.00000 q^{76} +15.0000 q^{79} +12.0000 q^{80} +1.00000 q^{81} -20.0000 q^{82} -9.00000 q^{83} -2.00000 q^{84} -6.00000 q^{85} +18.0000 q^{86} +5.00000 q^{87} -9.00000 q^{89} +6.00000 q^{90} +2.00000 q^{92} -5.00000 q^{93} -14.0000 q^{94} +3.00000 q^{95} +8.00000 q^{96} +13.0000 q^{97} -2.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 1.00000 0.577350
\(4\) 2.00000 1.00000
\(5\) −3.00000 −1.34164 −0.670820 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) −2.00000 −0.816497
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 6.00000 1.89737
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 2.00000 0.577350
\(13\) 0 0
\(14\) 2.00000 0.534522
\(15\) −3.00000 −0.774597
\(16\) −4.00000 −1.00000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) −2.00000 −0.471405
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) −6.00000 −1.34164
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514 0.104257 0.994550i \(-0.466753\pi\)
0.104257 + 0.994550i \(0.466753\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −2.00000 −0.377964
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 6.00000 1.09545
\(31\) −5.00000 −0.898027 −0.449013 0.893525i \(-0.648224\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) 8.00000 1.41421
\(33\) 0 0
\(34\) −4.00000 −0.685994
\(35\) 3.00000 0.507093
\(36\) 2.00000 0.333333
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 2.00000 0.324443
\(39\) 0 0
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 2.00000 0.308607
\(43\) −9.00000 −1.37249 −0.686244 0.727372i \(-0.740742\pi\)
−0.686244 + 0.727372i \(0.740742\pi\)
\(44\) 0 0
\(45\) −3.00000 −0.447214
\(46\) −2.00000 −0.294884
\(47\) 7.00000 1.02105 0.510527 0.859861i \(-0.329450\pi\)
0.510527 + 0.859861i \(0.329450\pi\)
\(48\) −4.00000 −0.577350
\(49\) 1.00000 0.142857
\(50\) −8.00000 −1.13137
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) 9.00000 1.23625 0.618123 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) −2.00000 −0.272166
\(55\) 0 0
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) −10.0000 −1.31306
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) −6.00000 −0.774597
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 10.0000 1.27000
\(63\) −1.00000 −0.125988
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) 4.00000 0.485071
\(69\) 1.00000 0.120386
\(70\) −6.00000 −0.717137
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 9.00000 1.05337 0.526685 0.850060i \(-0.323435\pi\)
0.526685 + 0.850060i \(0.323435\pi\)
\(74\) 16.0000 1.85996
\(75\) 4.00000 0.461880
\(76\) −2.00000 −0.229416
\(77\) 0 0
\(78\) 0 0
\(79\) 15.0000 1.68763 0.843816 0.536633i \(-0.180304\pi\)
0.843816 + 0.536633i \(0.180304\pi\)
\(80\) 12.0000 1.34164
\(81\) 1.00000 0.111111
\(82\) −20.0000 −2.20863
\(83\) −9.00000 −0.987878 −0.493939 0.869496i \(-0.664443\pi\)
−0.493939 + 0.869496i \(0.664443\pi\)
\(84\) −2.00000 −0.218218
\(85\) −6.00000 −0.650791
\(86\) 18.0000 1.94099
\(87\) 5.00000 0.536056
\(88\) 0 0
\(89\) −9.00000 −0.953998 −0.476999 0.878904i \(-0.658275\pi\)
−0.476999 + 0.878904i \(0.658275\pi\)
\(90\) 6.00000 0.632456
\(91\) 0 0
\(92\) 2.00000 0.208514
\(93\) −5.00000 −0.518476
\(94\) −14.0000 −1.44399
\(95\) 3.00000 0.307794
\(96\) 8.00000 0.816497
\(97\) 13.0000 1.31995 0.659975 0.751288i \(-0.270567\pi\)
0.659975 + 0.751288i \(0.270567\pi\)
\(98\) −2.00000 −0.202031
\(99\) 0 0
\(100\) 8.00000 0.800000
\(101\) 8.00000 0.796030 0.398015 0.917379i \(-0.369699\pi\)
0.398015 + 0.917379i \(0.369699\pi\)
\(102\) −4.00000 −0.396059
\(103\) 16.0000 1.57653 0.788263 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 0 0
\(105\) 3.00000 0.292770
\(106\) −18.0000 −1.74831
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 2.00000 0.192450
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) 4.00000 0.377964
\(113\) −21.0000 −1.97551 −0.987757 0.156001i \(-0.950140\pi\)
−0.987757 + 0.156001i \(0.950140\pi\)
\(114\) 2.00000 0.187317
\(115\) −3.00000 −0.279751
\(116\) 10.0000 0.928477
\(117\) 0 0
\(118\) 8.00000 0.736460
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 16.0000 1.44857
\(123\) 10.0000 0.901670
\(124\) −10.0000 −0.898027
\(125\) 3.00000 0.268328
\(126\) 2.00000 0.178174
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) −9.00000 −0.792406
\(130\) 0 0
\(131\) −18.0000 −1.57267 −0.786334 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) 0 0
\(133\) 1.00000 0.0867110
\(134\) 4.00000 0.345547
\(135\) −3.00000 −0.258199
\(136\) 0 0
\(137\) 12.0000 1.02523 0.512615 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) −2.00000 −0.170251
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 6.00000 0.507093
\(141\) 7.00000 0.589506
\(142\) 0 0
\(143\) 0 0
\(144\) −4.00000 −0.333333
\(145\) −15.0000 −1.24568
\(146\) −18.0000 −1.48969
\(147\) 1.00000 0.0824786
\(148\) −16.0000 −1.31519
\(149\) 4.00000 0.327693 0.163846 0.986486i \(-0.447610\pi\)
0.163846 + 0.986486i \(0.447610\pi\)
\(150\) −8.00000 −0.653197
\(151\) −20.0000 −1.62758 −0.813788 0.581161i \(-0.802599\pi\)
−0.813788 + 0.581161i \(0.802599\pi\)
\(152\) 0 0
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 15.0000 1.20483
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) −30.0000 −2.38667
\(159\) 9.00000 0.713746
\(160\) −24.0000 −1.89737
\(161\) −1.00000 −0.0788110
\(162\) −2.00000 −0.157135
\(163\) −6.00000 −0.469956 −0.234978 0.972001i \(-0.575502\pi\)
−0.234978 + 0.972001i \(0.575502\pi\)
\(164\) 20.0000 1.56174
\(165\) 0 0
\(166\) 18.0000 1.39707
\(167\) −3.00000 −0.232147 −0.116073 0.993241i \(-0.537031\pi\)
−0.116073 + 0.993241i \(0.537031\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 12.0000 0.920358
\(171\) −1.00000 −0.0764719
\(172\) −18.0000 −1.37249
\(173\) −24.0000 −1.82469 −0.912343 0.409426i \(-0.865729\pi\)
−0.912343 + 0.409426i \(0.865729\pi\)
\(174\) −10.0000 −0.758098
\(175\) −4.00000 −0.302372
\(176\) 0 0
\(177\) −4.00000 −0.300658
\(178\) 18.0000 1.34916
\(179\) 5.00000 0.373718 0.186859 0.982387i \(-0.440169\pi\)
0.186859 + 0.982387i \(0.440169\pi\)
\(180\) −6.00000 −0.447214
\(181\) 8.00000 0.594635 0.297318 0.954779i \(-0.403908\pi\)
0.297318 + 0.954779i \(0.403908\pi\)
\(182\) 0 0
\(183\) −8.00000 −0.591377
\(184\) 0 0
\(185\) 24.0000 1.76452
\(186\) 10.0000 0.733236
\(187\) 0 0
\(188\) 14.0000 1.02105
\(189\) −1.00000 −0.0727393
\(190\) −6.00000 −0.435286
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) −8.00000 −0.577350
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) −26.0000 −1.86669
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) −12.0000 −0.854965 −0.427482 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) 0 0
\(199\) −10.0000 −0.708881 −0.354441 0.935079i \(-0.615329\pi\)
−0.354441 + 0.935079i \(0.615329\pi\)
\(200\) 0 0
\(201\) −2.00000 −0.141069
\(202\) −16.0000 −1.12576
\(203\) −5.00000 −0.350931
\(204\) 4.00000 0.280056
\(205\) −30.0000 −2.09529
\(206\) −32.0000 −2.22955
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 0 0
\(210\) −6.00000 −0.414039
\(211\) −13.0000 −0.894957 −0.447478 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) 18.0000 1.23625
\(213\) 0 0
\(214\) 24.0000 1.64061
\(215\) 27.0000 1.84138
\(216\) 0 0
\(217\) 5.00000 0.339422
\(218\) −28.0000 −1.89640
\(219\) 9.00000 0.608164
\(220\) 0 0
\(221\) 0 0
\(222\) 16.0000 1.07385
\(223\) −19.0000 −1.27233 −0.636167 0.771551i \(-0.719481\pi\)
−0.636167 + 0.771551i \(0.719481\pi\)
\(224\) −8.00000 −0.534522
\(225\) 4.00000 0.266667
\(226\) 42.0000 2.79380
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) −2.00000 −0.132453
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 6.00000 0.395628
\(231\) 0 0
\(232\) 0 0
\(233\) 1.00000 0.0655122 0.0327561 0.999463i \(-0.489572\pi\)
0.0327561 + 0.999463i \(0.489572\pi\)
\(234\) 0 0
\(235\) −21.0000 −1.36989
\(236\) −8.00000 −0.520756
\(237\) 15.0000 0.974355
\(238\) 4.00000 0.259281
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 12.0000 0.774597
\(241\) −15.0000 −0.966235 −0.483117 0.875556i \(-0.660496\pi\)
−0.483117 + 0.875556i \(0.660496\pi\)
\(242\) 22.0000 1.41421
\(243\) 1.00000 0.0641500
\(244\) −16.0000 −1.02430
\(245\) −3.00000 −0.191663
\(246\) −20.0000 −1.27515
\(247\) 0 0
\(248\) 0 0
\(249\) −9.00000 −0.570352
\(250\) −6.00000 −0.379473
\(251\) 28.0000 1.76734 0.883672 0.468106i \(-0.155064\pi\)
0.883672 + 0.468106i \(0.155064\pi\)
\(252\) −2.00000 −0.125988
\(253\) 0 0
\(254\) 16.0000 1.00393
\(255\) −6.00000 −0.375735
\(256\) 16.0000 1.00000
\(257\) 12.0000 0.748539 0.374270 0.927320i \(-0.377893\pi\)
0.374270 + 0.927320i \(0.377893\pi\)
\(258\) 18.0000 1.12063
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) 5.00000 0.309492
\(262\) 36.0000 2.22409
\(263\) 19.0000 1.17159 0.585795 0.810459i \(-0.300782\pi\)
0.585795 + 0.810459i \(0.300782\pi\)
\(264\) 0 0
\(265\) −27.0000 −1.65860
\(266\) −2.00000 −0.122628
\(267\) −9.00000 −0.550791
\(268\) −4.00000 −0.244339
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 6.00000 0.365148
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) −8.00000 −0.485071
\(273\) 0 0
\(274\) −24.0000 −1.44989
\(275\) 0 0
\(276\) 2.00000 0.120386
\(277\) −3.00000 −0.180253 −0.0901263 0.995930i \(-0.528727\pi\)
−0.0901263 + 0.995930i \(0.528727\pi\)
\(278\) 40.0000 2.39904
\(279\) −5.00000 −0.299342
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) −14.0000 −0.833688
\(283\) −24.0000 −1.42665 −0.713326 0.700832i \(-0.752812\pi\)
−0.713326 + 0.700832i \(0.752812\pi\)
\(284\) 0 0
\(285\) 3.00000 0.177705
\(286\) 0 0
\(287\) −10.0000 −0.590281
\(288\) 8.00000 0.471405
\(289\) −13.0000 −0.764706
\(290\) 30.0000 1.76166
\(291\) 13.0000 0.762073
\(292\) 18.0000 1.05337
\(293\) −31.0000 −1.81104 −0.905520 0.424304i \(-0.860519\pi\)
−0.905520 + 0.424304i \(0.860519\pi\)
\(294\) −2.00000 −0.116642
\(295\) 12.0000 0.698667
\(296\) 0 0
\(297\) 0 0
\(298\) −8.00000 −0.463428
\(299\) 0 0
\(300\) 8.00000 0.461880
\(301\) 9.00000 0.518751
\(302\) 40.0000 2.30174
\(303\) 8.00000 0.459588
\(304\) 4.00000 0.229416
\(305\) 24.0000 1.37424
\(306\) −4.00000 −0.228665
\(307\) 17.0000 0.970241 0.485121 0.874447i \(-0.338776\pi\)
0.485121 + 0.874447i \(0.338776\pi\)
\(308\) 0 0
\(309\) 16.0000 0.910208
\(310\) −30.0000 −1.70389
\(311\) −2.00000 −0.113410 −0.0567048 0.998391i \(-0.518059\pi\)
−0.0567048 + 0.998391i \(0.518059\pi\)
\(312\) 0 0
\(313\) 4.00000 0.226093 0.113047 0.993590i \(-0.463939\pi\)
0.113047 + 0.993590i \(0.463939\pi\)
\(314\) 4.00000 0.225733
\(315\) 3.00000 0.169031
\(316\) 30.0000 1.68763
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) −18.0000 −1.00939
\(319\) 0 0
\(320\) 24.0000 1.34164
\(321\) −12.0000 −0.669775
\(322\) 2.00000 0.111456
\(323\) −2.00000 −0.111283
\(324\) 2.00000 0.111111
\(325\) 0 0
\(326\) 12.0000 0.664619
\(327\) 14.0000 0.774202
\(328\) 0 0
\(329\) −7.00000 −0.385922
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) −18.0000 −0.987878
\(333\) −8.00000 −0.438397
\(334\) 6.00000 0.328305
\(335\) 6.00000 0.327815
\(336\) 4.00000 0.218218
\(337\) −23.0000 −1.25289 −0.626445 0.779466i \(-0.715491\pi\)
−0.626445 + 0.779466i \(0.715491\pi\)
\(338\) 0 0
\(339\) −21.0000 −1.14056
\(340\) −12.0000 −0.650791
\(341\) 0 0
\(342\) 2.00000 0.108148
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −3.00000 −0.161515
\(346\) 48.0000 2.58050
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 10.0000 0.536056
\(349\) 1.00000 0.0535288 0.0267644 0.999642i \(-0.491480\pi\)
0.0267644 + 0.999642i \(0.491480\pi\)
\(350\) 8.00000 0.427618
\(351\) 0 0
\(352\) 0 0
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 8.00000 0.425195
\(355\) 0 0
\(356\) −18.0000 −0.953998
\(357\) −2.00000 −0.105851
\(358\) −10.0000 −0.528516
\(359\) −14.0000 −0.738892 −0.369446 0.929252i \(-0.620452\pi\)
−0.369446 + 0.929252i \(0.620452\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) −16.0000 −0.840941
\(363\) −11.0000 −0.577350
\(364\) 0 0
\(365\) −27.0000 −1.41324
\(366\) 16.0000 0.836333
\(367\) 18.0000 0.939592 0.469796 0.882775i \(-0.344327\pi\)
0.469796 + 0.882775i \(0.344327\pi\)
\(368\) −4.00000 −0.208514
\(369\) 10.0000 0.520579
\(370\) −48.0000 −2.49540
\(371\) −9.00000 −0.467257
\(372\) −10.0000 −0.518476
\(373\) −26.0000 −1.34623 −0.673114 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) 0 0
\(375\) 3.00000 0.154919
\(376\) 0 0
\(377\) 0 0
\(378\) 2.00000 0.102869
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 6.00000 0.307794
\(381\) −8.00000 −0.409852
\(382\) 16.0000 0.818631
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −28.0000 −1.42516
\(387\) −9.00000 −0.457496
\(388\) 26.0000 1.31995
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) 2.00000 0.101144
\(392\) 0 0
\(393\) −18.0000 −0.907980
\(394\) 24.0000 1.20910
\(395\) −45.0000 −2.26420
\(396\) 0 0
\(397\) 17.0000 0.853206 0.426603 0.904439i \(-0.359710\pi\)
0.426603 + 0.904439i \(0.359710\pi\)
\(398\) 20.0000 1.00251
\(399\) 1.00000 0.0500626
\(400\) −16.0000 −0.800000
\(401\) −10.0000 −0.499376 −0.249688 0.968326i \(-0.580328\pi\)
−0.249688 + 0.968326i \(0.580328\pi\)
\(402\) 4.00000 0.199502
\(403\) 0 0
\(404\) 16.0000 0.796030
\(405\) −3.00000 −0.149071
\(406\) 10.0000 0.496292
\(407\) 0 0
\(408\) 0 0
\(409\) −31.0000 −1.53285 −0.766426 0.642333i \(-0.777967\pi\)
−0.766426 + 0.642333i \(0.777967\pi\)
\(410\) 60.0000 2.96319
\(411\) 12.0000 0.591916
\(412\) 32.0000 1.57653
\(413\) 4.00000 0.196827
\(414\) −2.00000 −0.0982946
\(415\) 27.0000 1.32538
\(416\) 0 0
\(417\) −20.0000 −0.979404
\(418\) 0 0
\(419\) −20.0000 −0.977064 −0.488532 0.872546i \(-0.662467\pi\)
−0.488532 + 0.872546i \(0.662467\pi\)
\(420\) 6.00000 0.292770
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 26.0000 1.26566
\(423\) 7.00000 0.340352
\(424\) 0 0
\(425\) 8.00000 0.388057
\(426\) 0 0
\(427\) 8.00000 0.387147
\(428\) −24.0000 −1.16008
\(429\) 0 0
\(430\) −54.0000 −2.60411
\(431\) −20.0000 −0.963366 −0.481683 0.876346i \(-0.659974\pi\)
−0.481683 + 0.876346i \(0.659974\pi\)
\(432\) −4.00000 −0.192450
\(433\) 6.00000 0.288342 0.144171 0.989553i \(-0.453949\pi\)
0.144171 + 0.989553i \(0.453949\pi\)
\(434\) −10.0000 −0.480015
\(435\) −15.0000 −0.719195
\(436\) 28.0000 1.34096
\(437\) −1.00000 −0.0478365
\(438\) −18.0000 −0.860073
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −31.0000 −1.47285 −0.736427 0.676517i \(-0.763489\pi\)
−0.736427 + 0.676517i \(0.763489\pi\)
\(444\) −16.0000 −0.759326
\(445\) 27.0000 1.27992
\(446\) 38.0000 1.79935
\(447\) 4.00000 0.189194
\(448\) 8.00000 0.377964
\(449\) −34.0000 −1.60456 −0.802280 0.596948i \(-0.796380\pi\)
−0.802280 + 0.596948i \(0.796380\pi\)
\(450\) −8.00000 −0.377124
\(451\) 0 0
\(452\) −42.0000 −1.97551
\(453\) −20.0000 −0.939682
\(454\) 24.0000 1.12638
\(455\) 0 0
\(456\) 0 0
\(457\) −42.0000 −1.96468 −0.982339 0.187112i \(-0.940087\pi\)
−0.982339 + 0.187112i \(0.940087\pi\)
\(458\) 28.0000 1.30835
\(459\) 2.00000 0.0933520
\(460\) −6.00000 −0.279751
\(461\) 10.0000 0.465746 0.232873 0.972507i \(-0.425187\pi\)
0.232873 + 0.972507i \(0.425187\pi\)
\(462\) 0 0
\(463\) −26.0000 −1.20832 −0.604161 0.796862i \(-0.706492\pi\)
−0.604161 + 0.796862i \(0.706492\pi\)
\(464\) −20.0000 −0.928477
\(465\) 15.0000 0.695608
\(466\) −2.00000 −0.0926482
\(467\) −18.0000 −0.832941 −0.416470 0.909149i \(-0.636733\pi\)
−0.416470 + 0.909149i \(0.636733\pi\)
\(468\) 0 0
\(469\) 2.00000 0.0923514
\(470\) 42.0000 1.93732
\(471\) −2.00000 −0.0921551
\(472\) 0 0
\(473\) 0 0
\(474\) −30.0000 −1.37795
\(475\) −4.00000 −0.183533
\(476\) −4.00000 −0.183340
\(477\) 9.00000 0.412082
\(478\) 12.0000 0.548867
\(479\) 21.0000 0.959514 0.479757 0.877401i \(-0.340725\pi\)
0.479757 + 0.877401i \(0.340725\pi\)
\(480\) −24.0000 −1.09545
\(481\) 0 0
\(482\) 30.0000 1.36646
\(483\) −1.00000 −0.0455016
\(484\) −22.0000 −1.00000
\(485\) −39.0000 −1.77090
\(486\) −2.00000 −0.0907218
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 0 0
\(489\) −6.00000 −0.271329
\(490\) 6.00000 0.271052
\(491\) −32.0000 −1.44414 −0.722070 0.691820i \(-0.756809\pi\)
−0.722070 + 0.691820i \(0.756809\pi\)
\(492\) 20.0000 0.901670
\(493\) 10.0000 0.450377
\(494\) 0 0
\(495\) 0 0
\(496\) 20.0000 0.898027
\(497\) 0 0
\(498\) 18.0000 0.806599
\(499\) 14.0000 0.626726 0.313363 0.949633i \(-0.398544\pi\)
0.313363 + 0.949633i \(0.398544\pi\)
\(500\) 6.00000 0.268328
\(501\) −3.00000 −0.134030
\(502\) −56.0000 −2.49940
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 0 0
\(505\) −24.0000 −1.06799
\(506\) 0 0
\(507\) 0 0
\(508\) −16.0000 −0.709885
\(509\) 9.00000 0.398918 0.199459 0.979906i \(-0.436082\pi\)
0.199459 + 0.979906i \(0.436082\pi\)
\(510\) 12.0000 0.531369
\(511\) −9.00000 −0.398137
\(512\) −32.0000 −1.41421
\(513\) −1.00000 −0.0441511
\(514\) −24.0000 −1.05859
\(515\) −48.0000 −2.11513
\(516\) −18.0000 −0.792406
\(517\) 0 0
\(518\) −16.0000 −0.703000
\(519\) −24.0000 −1.05348
\(520\) 0 0
\(521\) 12.0000 0.525730 0.262865 0.964833i \(-0.415333\pi\)
0.262865 + 0.964833i \(0.415333\pi\)
\(522\) −10.0000 −0.437688
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) −36.0000 −1.57267
\(525\) −4.00000 −0.174574
\(526\) −38.0000 −1.65688
\(527\) −10.0000 −0.435607
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) 54.0000 2.34561
\(531\) −4.00000 −0.173585
\(532\) 2.00000 0.0867110
\(533\) 0 0
\(534\) 18.0000 0.778936
\(535\) 36.0000 1.55642
\(536\) 0 0
\(537\) 5.00000 0.215766
\(538\) −20.0000 −0.862261
\(539\) 0 0
\(540\) −6.00000 −0.258199
\(541\) 20.0000 0.859867 0.429934 0.902861i \(-0.358537\pi\)
0.429934 + 0.902861i \(0.358537\pi\)
\(542\) −40.0000 −1.71815
\(543\) 8.00000 0.343313
\(544\) 16.0000 0.685994
\(545\) −42.0000 −1.79908
\(546\) 0 0
\(547\) 13.0000 0.555840 0.277920 0.960604i \(-0.410355\pi\)
0.277920 + 0.960604i \(0.410355\pi\)
\(548\) 24.0000 1.02523
\(549\) −8.00000 −0.341432
\(550\) 0 0
\(551\) −5.00000 −0.213007
\(552\) 0 0
\(553\) −15.0000 −0.637865
\(554\) 6.00000 0.254916
\(555\) 24.0000 1.01874
\(556\) −40.0000 −1.69638
\(557\) 32.0000 1.35588 0.677942 0.735116i \(-0.262872\pi\)
0.677942 + 0.735116i \(0.262872\pi\)
\(558\) 10.0000 0.423334
\(559\) 0 0
\(560\) −12.0000 −0.507093
\(561\) 0 0
\(562\) −20.0000 −0.843649
\(563\) 26.0000 1.09577 0.547885 0.836554i \(-0.315433\pi\)
0.547885 + 0.836554i \(0.315433\pi\)
\(564\) 14.0000 0.589506
\(565\) 63.0000 2.65043
\(566\) 48.0000 2.01759
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 5.00000 0.209611 0.104805 0.994493i \(-0.466578\pi\)
0.104805 + 0.994493i \(0.466578\pi\)
\(570\) −6.00000 −0.251312
\(571\) 23.0000 0.962520 0.481260 0.876578i \(-0.340179\pi\)
0.481260 + 0.876578i \(0.340179\pi\)
\(572\) 0 0
\(573\) −8.00000 −0.334205
\(574\) 20.0000 0.834784
\(575\) 4.00000 0.166812
\(576\) −8.00000 −0.333333
\(577\) −22.0000 −0.915872 −0.457936 0.888985i \(-0.651411\pi\)
−0.457936 + 0.888985i \(0.651411\pi\)
\(578\) 26.0000 1.08146
\(579\) 14.0000 0.581820
\(580\) −30.0000 −1.24568
\(581\) 9.00000 0.373383
\(582\) −26.0000 −1.07773
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 62.0000 2.56120
\(587\) 3.00000 0.123823 0.0619116 0.998082i \(-0.480280\pi\)
0.0619116 + 0.998082i \(0.480280\pi\)
\(588\) 2.00000 0.0824786
\(589\) 5.00000 0.206021
\(590\) −24.0000 −0.988064
\(591\) −12.0000 −0.493614
\(592\) 32.0000 1.31519
\(593\) 9.00000 0.369586 0.184793 0.982777i \(-0.440839\pi\)
0.184793 + 0.982777i \(0.440839\pi\)
\(594\) 0 0
\(595\) 6.00000 0.245976
\(596\) 8.00000 0.327693
\(597\) −10.0000 −0.409273
\(598\) 0 0
\(599\) −35.0000 −1.43006 −0.715031 0.699093i \(-0.753587\pi\)
−0.715031 + 0.699093i \(0.753587\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) −18.0000 −0.733625
\(603\) −2.00000 −0.0814463
\(604\) −40.0000 −1.62758
\(605\) 33.0000 1.34164
\(606\) −16.0000 −0.649956
\(607\) 28.0000 1.13648 0.568242 0.822861i \(-0.307624\pi\)
0.568242 + 0.822861i \(0.307624\pi\)
\(608\) −8.00000 −0.324443
\(609\) −5.00000 −0.202610
\(610\) −48.0000 −1.94346
\(611\) 0 0
\(612\) 4.00000 0.161690
\(613\) 26.0000 1.05013 0.525065 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(614\) −34.0000 −1.37213
\(615\) −30.0000 −1.20972
\(616\) 0 0
\(617\) 48.0000 1.93241 0.966204 0.257780i \(-0.0829910\pi\)
0.966204 + 0.257780i \(0.0829910\pi\)
\(618\) −32.0000 −1.28723
\(619\) −44.0000 −1.76851 −0.884255 0.467005i \(-0.845333\pi\)
−0.884255 + 0.467005i \(0.845333\pi\)
\(620\) 30.0000 1.20483
\(621\) 1.00000 0.0401286
\(622\) 4.00000 0.160385
\(623\) 9.00000 0.360577
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) −8.00000 −0.319744
\(627\) 0 0
\(628\) −4.00000 −0.159617
\(629\) −16.0000 −0.637962
\(630\) −6.00000 −0.239046
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) −13.0000 −0.516704
\(634\) 4.00000 0.158860
\(635\) 24.0000 0.952411
\(636\) 18.0000 0.713746
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −17.0000 −0.671460 −0.335730 0.941958i \(-0.608983\pi\)
−0.335730 + 0.941958i \(0.608983\pi\)
\(642\) 24.0000 0.947204
\(643\) −44.0000 −1.73519 −0.867595 0.497271i \(-0.834335\pi\)
−0.867595 + 0.497271i \(0.834335\pi\)
\(644\) −2.00000 −0.0788110
\(645\) 27.0000 1.06312
\(646\) 4.00000 0.157378
\(647\) 2.00000 0.0786281 0.0393141 0.999227i \(-0.487483\pi\)
0.0393141 + 0.999227i \(0.487483\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 5.00000 0.195965
\(652\) −12.0000 −0.469956
\(653\) 14.0000 0.547862 0.273931 0.961749i \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(654\) −28.0000 −1.09489
\(655\) 54.0000 2.10995
\(656\) −40.0000 −1.56174
\(657\) 9.00000 0.351123
\(658\) 14.0000 0.545777
\(659\) 15.0000 0.584317 0.292159 0.956370i \(-0.405627\pi\)
0.292159 + 0.956370i \(0.405627\pi\)
\(660\) 0 0
\(661\) −15.0000 −0.583432 −0.291716 0.956505i \(-0.594226\pi\)
−0.291716 + 0.956505i \(0.594226\pi\)
\(662\) 40.0000 1.55464
\(663\) 0 0
\(664\) 0 0
\(665\) −3.00000 −0.116335
\(666\) 16.0000 0.619987
\(667\) 5.00000 0.193601
\(668\) −6.00000 −0.232147
\(669\) −19.0000 −0.734582
\(670\) −12.0000 −0.463600
\(671\) 0 0
\(672\) −8.00000 −0.308607
\(673\) −19.0000 −0.732396 −0.366198 0.930537i \(-0.619341\pi\)
−0.366198 + 0.930537i \(0.619341\pi\)
\(674\) 46.0000 1.77185
\(675\) 4.00000 0.153960
\(676\) 0 0
\(677\) 18.0000 0.691796 0.345898 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) 42.0000 1.61300
\(679\) −13.0000 −0.498894
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) 24.0000 0.918334 0.459167 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) −2.00000 −0.0764719
\(685\) −36.0000 −1.37549
\(686\) 2.00000 0.0763604
\(687\) −14.0000 −0.534133
\(688\) 36.0000 1.37249
\(689\) 0 0
\(690\) 6.00000 0.228416
\(691\) −35.0000 −1.33146 −0.665731 0.746191i \(-0.731880\pi\)
−0.665731 + 0.746191i \(0.731880\pi\)
\(692\) −48.0000 −1.82469
\(693\) 0 0
\(694\) 24.0000 0.911028
\(695\) 60.0000 2.27593
\(696\) 0 0
\(697\) 20.0000 0.757554
\(698\) −2.00000 −0.0757011
\(699\) 1.00000 0.0378235
\(700\) −8.00000 −0.302372
\(701\) 23.0000 0.868698 0.434349 0.900745i \(-0.356978\pi\)
0.434349 + 0.900745i \(0.356978\pi\)
\(702\) 0 0
\(703\) 8.00000 0.301726
\(704\) 0 0
\(705\) −21.0000 −0.790906
\(706\) −12.0000 −0.451626
\(707\) −8.00000 −0.300871
\(708\) −8.00000 −0.300658
\(709\) 16.0000 0.600893 0.300446 0.953799i \(-0.402864\pi\)
0.300446 + 0.953799i \(0.402864\pi\)
\(710\) 0 0
\(711\) 15.0000 0.562544
\(712\) 0 0
\(713\) −5.00000 −0.187251
\(714\) 4.00000 0.149696
\(715\) 0 0
\(716\) 10.0000 0.373718
\(717\) −6.00000 −0.224074
\(718\) 28.0000 1.04495
\(719\) −50.0000 −1.86469 −0.932343 0.361576i \(-0.882239\pi\)
−0.932343 + 0.361576i \(0.882239\pi\)
\(720\) 12.0000 0.447214
\(721\) −16.0000 −0.595871
\(722\) 36.0000 1.33978
\(723\) −15.0000 −0.557856
\(724\) 16.0000 0.594635
\(725\) 20.0000 0.742781
\(726\) 22.0000 0.816497
\(727\) 2.00000 0.0741759 0.0370879 0.999312i \(-0.488192\pi\)
0.0370879 + 0.999312i \(0.488192\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 54.0000 1.99863
\(731\) −18.0000 −0.665754
\(732\) −16.0000 −0.591377
\(733\) −19.0000 −0.701781 −0.350891 0.936416i \(-0.614121\pi\)
−0.350891 + 0.936416i \(0.614121\pi\)
\(734\) −36.0000 −1.32878
\(735\) −3.00000 −0.110657
\(736\) 8.00000 0.294884
\(737\) 0 0
\(738\) −20.0000 −0.736210
\(739\) −4.00000 −0.147142 −0.0735712 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) 48.0000 1.76452
\(741\) 0 0
\(742\) 18.0000 0.660801
\(743\) 26.0000 0.953847 0.476924 0.878945i \(-0.341752\pi\)
0.476924 + 0.878945i \(0.341752\pi\)
\(744\) 0 0
\(745\) −12.0000 −0.439646
\(746\) 52.0000 1.90386
\(747\) −9.00000 −0.329293
\(748\) 0 0
\(749\) 12.0000 0.438470
\(750\) −6.00000 −0.219089
\(751\) 23.0000 0.839282 0.419641 0.907690i \(-0.362156\pi\)
0.419641 + 0.907690i \(0.362156\pi\)
\(752\) −28.0000 −1.02105
\(753\) 28.0000 1.02038
\(754\) 0 0
\(755\) 60.0000 2.18362
\(756\) −2.00000 −0.0727393
\(757\) 13.0000 0.472493 0.236247 0.971693i \(-0.424083\pi\)
0.236247 + 0.971693i \(0.424083\pi\)
\(758\) −8.00000 −0.290573
\(759\) 0 0
\(760\) 0 0
\(761\) 45.0000 1.63125 0.815624 0.578582i \(-0.196394\pi\)
0.815624 + 0.578582i \(0.196394\pi\)
\(762\) 16.0000 0.579619
\(763\) −14.0000 −0.506834
\(764\) −16.0000 −0.578860
\(765\) −6.00000 −0.216930
\(766\) −32.0000 −1.15621
\(767\) 0 0
\(768\) 16.0000 0.577350
\(769\) −1.00000 −0.0360609 −0.0180305 0.999837i \(-0.505740\pi\)
−0.0180305 + 0.999837i \(0.505740\pi\)
\(770\) 0 0
\(771\) 12.0000 0.432169
\(772\) 28.0000 1.00774
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 18.0000 0.646997
\(775\) −20.0000 −0.718421
\(776\) 0 0
\(777\) 8.00000 0.286998
\(778\) 60.0000 2.15110
\(779\) −10.0000 −0.358287
\(780\) 0 0
\(781\) 0 0
\(782\) −4.00000 −0.143040
\(783\) 5.00000 0.178685
\(784\) −4.00000 −0.142857
\(785\) 6.00000 0.214149
\(786\) 36.0000 1.28408
\(787\) 7.00000 0.249523 0.124762 0.992187i \(-0.460183\pi\)
0.124762 + 0.992187i \(0.460183\pi\)
\(788\) −24.0000 −0.854965
\(789\) 19.0000 0.676418
\(790\) 90.0000 3.20206
\(791\) 21.0000 0.746674
\(792\) 0 0
\(793\) 0 0
\(794\) −34.0000 −1.20661
\(795\) −27.0000 −0.957591
\(796\) −20.0000 −0.708881
\(797\) 42.0000 1.48772 0.743858 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) −2.00000 −0.0707992
\(799\) 14.0000 0.495284
\(800\) 32.0000 1.13137
\(801\) −9.00000 −0.317999
\(802\) 20.0000 0.706225
\(803\) 0 0
\(804\) −4.00000 −0.141069
\(805\) 3.00000 0.105736
\(806\) 0 0
\(807\) 10.0000 0.352017
\(808\) 0 0
\(809\) −45.0000 −1.58212 −0.791058 0.611741i \(-0.790469\pi\)
−0.791058 + 0.611741i \(0.790469\pi\)
\(810\) 6.00000 0.210819
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) −10.0000 −0.350931
\(813\) 20.0000 0.701431
\(814\) 0 0
\(815\) 18.0000 0.630512
\(816\) −8.00000 −0.280056
\(817\) 9.00000 0.314870
\(818\) 62.0000 2.16778
\(819\) 0 0
\(820\) −60.0000 −2.09529
\(821\) −30.0000 −1.04701 −0.523504 0.852023i \(-0.675375\pi\)
−0.523504 + 0.852023i \(0.675375\pi\)
\(822\) −24.0000 −0.837096
\(823\) 16.0000 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −8.00000 −0.278356
\(827\) 42.0000 1.46048 0.730242 0.683189i \(-0.239408\pi\)
0.730242 + 0.683189i \(0.239408\pi\)
\(828\) 2.00000 0.0695048
\(829\) 10.0000 0.347314 0.173657 0.984806i \(-0.444442\pi\)
0.173657 + 0.984806i \(0.444442\pi\)
\(830\) −54.0000 −1.87437
\(831\) −3.00000 −0.104069
\(832\) 0 0
\(833\) 2.00000 0.0692959
\(834\) 40.0000 1.38509
\(835\) 9.00000 0.311458
\(836\) 0 0
\(837\) −5.00000 −0.172825
\(838\) 40.0000 1.38178
\(839\) −24.0000 −0.828572 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 20.0000 0.689246
\(843\) 10.0000 0.344418
\(844\) −26.0000 −0.894957
\(845\) 0 0
\(846\) −14.0000 −0.481330
\(847\) 11.0000 0.377964
\(848\) −36.0000 −1.23625
\(849\) −24.0000 −0.823678
\(850\) −16.0000 −0.548795
\(851\) −8.00000 −0.274236
\(852\) 0 0
\(853\) −1.00000 −0.0342393 −0.0171197 0.999853i \(-0.505450\pi\)
−0.0171197 + 0.999853i \(0.505450\pi\)
\(854\) −16.0000 −0.547509
\(855\) 3.00000 0.102598
\(856\) 0 0
\(857\) −8.00000 −0.273275 −0.136637 0.990621i \(-0.543630\pi\)
−0.136637 + 0.990621i \(0.543630\pi\)
\(858\) 0 0
\(859\) 30.0000 1.02359 0.511793 0.859109i \(-0.328981\pi\)
0.511793 + 0.859109i \(0.328981\pi\)
\(860\) 54.0000 1.84138
\(861\) −10.0000 −0.340799
\(862\) 40.0000 1.36241
\(863\) −4.00000 −0.136162 −0.0680808 0.997680i \(-0.521688\pi\)
−0.0680808 + 0.997680i \(0.521688\pi\)
\(864\) 8.00000 0.272166
\(865\) 72.0000 2.44807
\(866\) −12.0000 −0.407777
\(867\) −13.0000 −0.441503
\(868\) 10.0000 0.339422
\(869\) 0 0
\(870\) 30.0000 1.01710
\(871\) 0 0
\(872\) 0 0
\(873\) 13.0000 0.439983
\(874\) 2.00000 0.0676510
\(875\) −3.00000 −0.101419
\(876\) 18.0000 0.608164
\(877\) −22.0000 −0.742887 −0.371444 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) 0 0
\(879\) −31.0000 −1.04560
\(880\) 0 0
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) −2.00000 −0.0673435
\(883\) 16.0000 0.538443 0.269221 0.963078i \(-0.413234\pi\)
0.269221 + 0.963078i \(0.413234\pi\)
\(884\) 0 0
\(885\) 12.0000 0.403376
\(886\) 62.0000 2.08293
\(887\) 18.0000 0.604381 0.302190 0.953248i \(-0.402282\pi\)
0.302190 + 0.953248i \(0.402282\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) −54.0000 −1.81008
\(891\) 0 0
\(892\) −38.0000 −1.27233
\(893\) −7.00000 −0.234246
\(894\) −8.00000 −0.267560
\(895\) −15.0000 −0.501395
\(896\) 0 0
\(897\) 0 0
\(898\) 68.0000 2.26919
\(899\) −25.0000 −0.833797
\(900\) 8.00000 0.266667
\(901\) 18.0000 0.599667
\(902\) 0 0
\(903\) 9.00000 0.299501
\(904\) 0 0
\(905\) −24.0000 −0.797787
\(906\) 40.0000 1.32891
\(907\) 27.0000 0.896520 0.448260 0.893903i \(-0.352044\pi\)
0.448260 + 0.893903i \(0.352044\pi\)
\(908\) −24.0000 −0.796468
\(909\) 8.00000 0.265343
\(910\) 0 0
\(911\) −33.0000 −1.09334 −0.546669 0.837349i \(-0.684105\pi\)
−0.546669 + 0.837349i \(0.684105\pi\)
\(912\) 4.00000 0.132453
\(913\) 0 0
\(914\) 84.0000 2.77847
\(915\) 24.0000 0.793416
\(916\) −28.0000 −0.925146
\(917\) 18.0000 0.594412
\(918\) −4.00000 −0.132020
\(919\) 40.0000 1.31948 0.659739 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) 0 0
\(921\) 17.0000 0.560169
\(922\) −20.0000 −0.658665
\(923\) 0 0
\(924\) 0 0
\(925\) −32.0000 −1.05215
\(926\) 52.0000 1.70883
\(927\) 16.0000 0.525509
\(928\) 40.0000 1.31306
\(929\) 29.0000 0.951459 0.475730 0.879592i \(-0.342184\pi\)
0.475730 + 0.879592i \(0.342184\pi\)
\(930\) −30.0000 −0.983739
\(931\) −1.00000 −0.0327737
\(932\) 2.00000 0.0655122
\(933\) −2.00000 −0.0654771
\(934\) 36.0000 1.17796
\(935\) 0 0
\(936\) 0 0
\(937\) 28.0000 0.914720 0.457360 0.889282i \(-0.348795\pi\)
0.457360 + 0.889282i \(0.348795\pi\)
\(938\) −4.00000 −0.130605
\(939\) 4.00000 0.130535
\(940\) −42.0000 −1.36989
\(941\) −5.00000 −0.162995 −0.0814977 0.996674i \(-0.525970\pi\)
−0.0814977 + 0.996674i \(0.525970\pi\)
\(942\) 4.00000 0.130327
\(943\) 10.0000 0.325645
\(944\) 16.0000 0.520756
\(945\) 3.00000 0.0975900
\(946\) 0 0
\(947\) 2.00000 0.0649913 0.0324956 0.999472i \(-0.489654\pi\)
0.0324956 + 0.999472i \(0.489654\pi\)
\(948\) 30.0000 0.974355
\(949\) 0 0
\(950\) 8.00000 0.259554
\(951\) −2.00000 −0.0648544
\(952\) 0 0
\(953\) −29.0000 −0.939402 −0.469701 0.882826i \(-0.655638\pi\)
−0.469701 + 0.882826i \(0.655638\pi\)
\(954\) −18.0000 −0.582772
\(955\) 24.0000 0.776622
\(956\) −12.0000 −0.388108
\(957\) 0 0
\(958\) −42.0000 −1.35696
\(959\) −12.0000 −0.387500
\(960\) 24.0000 0.774597
\(961\) −6.00000 −0.193548
\(962\) 0 0
\(963\) −12.0000 −0.386695
\(964\) −30.0000 −0.966235
\(965\) −42.0000 −1.35203
\(966\) 2.00000 0.0643489
\(967\) −52.0000 −1.67221 −0.836104 0.548572i \(-0.815172\pi\)
−0.836104 + 0.548572i \(0.815172\pi\)
\(968\) 0 0
\(969\) −2.00000 −0.0642493
\(970\) 78.0000 2.50443
\(971\) −28.0000 −0.898563 −0.449281 0.893390i \(-0.648320\pi\)
−0.449281 + 0.893390i \(0.648320\pi\)
\(972\) 2.00000 0.0641500
\(973\) 20.0000 0.641171
\(974\) −16.0000 −0.512673
\(975\) 0 0
\(976\) 32.0000 1.02430
\(977\) −52.0000 −1.66363 −0.831814 0.555055i \(-0.812697\pi\)
−0.831814 + 0.555055i \(0.812697\pi\)
\(978\) 12.0000 0.383718
\(979\) 0 0
\(980\) −6.00000 −0.191663
\(981\) 14.0000 0.446986
\(982\) 64.0000 2.04232
\(983\) −31.0000 −0.988746 −0.494373 0.869250i \(-0.664602\pi\)
−0.494373 + 0.869250i \(0.664602\pi\)
\(984\) 0 0
\(985\) 36.0000 1.14706
\(986\) −20.0000 −0.636930
\(987\) −7.00000 −0.222812
\(988\) 0 0
\(989\) −9.00000 −0.286183
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) −40.0000 −1.27000
\(993\) −20.0000 −0.634681
\(994\) 0 0
\(995\) 30.0000 0.951064
\(996\) −18.0000 −0.570352
\(997\) −32.0000 −1.01345 −0.506725 0.862108i \(-0.669144\pi\)
−0.506725 + 0.862108i \(0.669144\pi\)
\(998\) −28.0000 −0.886325
\(999\) −8.00000 −0.253109
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 3549.2.a.a.1.1 1
13.5 odd 4 273.2.c.a.64.2 yes 2
13.8 odd 4 273.2.c.a.64.1 2
13.12 even 2 3549.2.a.e.1.1 1
39.5 even 4 819.2.c.a.64.1 2
39.8 even 4 819.2.c.a.64.2 2
52.31 even 4 4368.2.h.e.337.2 2
52.47 even 4 4368.2.h.e.337.1 2
91.34 even 4 1911.2.c.a.883.1 2
91.83 even 4 1911.2.c.a.883.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
273.2.c.a.64.1 2 13.8 odd 4
273.2.c.a.64.2 yes 2 13.5 odd 4
819.2.c.a.64.1 2 39.5 even 4
819.2.c.a.64.2 2 39.8 even 4
1911.2.c.a.883.1 2 91.34 even 4
1911.2.c.a.883.2 2 91.83 even 4
3549.2.a.a.1.1 1 1.1 even 1 trivial
3549.2.a.e.1.1 1 13.12 even 2
4368.2.h.e.337.1 2 52.47 even 4
4368.2.h.e.337.2 2 52.31 even 4