Properties

Label 2400.2.d.e.49.3
Level 2400
Weight 2
Character 2400.49
Analytic conductor 19.164
Analytic rank 0
Dimension 6
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2400.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.1640964851\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.399424.1
Defining polynomial: \(x^{6} - 2 x^{5} + 3 x^{4} - 6 x^{3} + 6 x^{2} - 8 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.3
Root \(0.264658 + 1.38923i\) of defining polynomial
Character \(\chi\) \(=\) 2400.49
Dual form 2400.2.d.e.49.4

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -0.941367i q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -0.941367i q^{7} +1.00000 q^{9} -4.49828i q^{11} -5.55691 q^{13} +7.55691i q^{17} +1.05863i q^{19} +0.941367i q^{21} -1.05863i q^{23} -1.00000 q^{27} +2.00000i q^{29} -3.55691 q^{31} +4.49828i q^{33} +7.43965 q^{37} +5.55691 q^{39} -3.88273 q^{41} +1.88273 q^{43} +10.0552i q^{47} +6.11383 q^{49} -7.55691i q^{51} +2.00000 q^{53} -1.05863i q^{57} -8.49828i q^{59} +8.99656i q^{61} -0.941367i q^{63} -4.00000 q^{67} +1.05863i q^{69} +12.9966 q^{71} +6.00000i q^{73} -4.23453 q^{77} +11.5569 q^{79} +1.00000 q^{81} -5.88273 q^{83} -2.00000i q^{87} +4.11727 q^{89} +5.23109i q^{91} +3.55691 q^{93} +17.1138i q^{97} -4.49828i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{3} + 6q^{9} + O(q^{10}) \) \( 6q - 6q^{3} + 6q^{9} - 6q^{27} + 12q^{31} + 8q^{37} - 20q^{41} + 8q^{43} - 30q^{49} + 12q^{53} - 24q^{67} + 8q^{71} - 32q^{77} + 36q^{79} + 6q^{81} - 32q^{83} + 28q^{89} - 12q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1601\) \(1951\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 0.941367i − 0.355803i −0.984048 0.177902i \(-0.943069\pi\)
0.984048 0.177902i \(-0.0569309\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) − 4.49828i − 1.35628i −0.734931 0.678141i \(-0.762786\pi\)
0.734931 0.678141i \(-0.237214\pi\)
\(12\) 0 0
\(13\) −5.55691 −1.54121 −0.770605 0.637313i \(-0.780046\pi\)
−0.770605 + 0.637313i \(0.780046\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7.55691i 1.83282i 0.400240 + 0.916410i \(0.368927\pi\)
−0.400240 + 0.916410i \(0.631073\pi\)
\(18\) 0 0
\(19\) 1.05863i 0.242867i 0.992600 + 0.121434i \(0.0387491\pi\)
−0.992600 + 0.121434i \(0.961251\pi\)
\(20\) 0 0
\(21\) 0.941367i 0.205423i
\(22\) 0 0
\(23\) − 1.05863i − 0.220740i −0.993891 0.110370i \(-0.964796\pi\)
0.993891 0.110370i \(-0.0352036\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 2.00000i 0.371391i 0.982607 + 0.185695i \(0.0594537\pi\)
−0.982607 + 0.185695i \(0.940546\pi\)
\(30\) 0 0
\(31\) −3.55691 −0.638841 −0.319420 0.947613i \(-0.603488\pi\)
−0.319420 + 0.947613i \(0.603488\pi\)
\(32\) 0 0
\(33\) 4.49828i 0.783050i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 7.43965 1.22307 0.611535 0.791217i \(-0.290552\pi\)
0.611535 + 0.791217i \(0.290552\pi\)
\(38\) 0 0
\(39\) 5.55691 0.889818
\(40\) 0 0
\(41\) −3.88273 −0.606381 −0.303191 0.952930i \(-0.598052\pi\)
−0.303191 + 0.952930i \(0.598052\pi\)
\(42\) 0 0
\(43\) 1.88273 0.287114 0.143557 0.989642i \(-0.454146\pi\)
0.143557 + 0.989642i \(0.454146\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 10.0552i 1.46670i 0.679851 + 0.733350i \(0.262045\pi\)
−0.679851 + 0.733350i \(0.737955\pi\)
\(48\) 0 0
\(49\) 6.11383 0.873404
\(50\) 0 0
\(51\) − 7.55691i − 1.05818i
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 1.05863i − 0.140219i
\(58\) 0 0
\(59\) − 8.49828i − 1.10638i −0.833054 0.553191i \(-0.813410\pi\)
0.833054 0.553191i \(-0.186590\pi\)
\(60\) 0 0
\(61\) 8.99656i 1.15189i 0.817488 + 0.575946i \(0.195366\pi\)
−0.817488 + 0.575946i \(0.804634\pi\)
\(62\) 0 0
\(63\) − 0.941367i − 0.118601i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) 1.05863i 0.127444i
\(70\) 0 0
\(71\) 12.9966 1.54241 0.771204 0.636588i \(-0.219655\pi\)
0.771204 + 0.636588i \(0.219655\pi\)
\(72\) 0 0
\(73\) 6.00000i 0.702247i 0.936329 + 0.351123i \(0.114200\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.23453 −0.482570
\(78\) 0 0
\(79\) 11.5569 1.30025 0.650127 0.759825i \(-0.274716\pi\)
0.650127 + 0.759825i \(0.274716\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −5.88273 −0.645714 −0.322857 0.946448i \(-0.604643\pi\)
−0.322857 + 0.946448i \(0.604643\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 2.00000i − 0.214423i
\(88\) 0 0
\(89\) 4.11727 0.436429 0.218215 0.975901i \(-0.429977\pi\)
0.218215 + 0.975901i \(0.429977\pi\)
\(90\) 0 0
\(91\) 5.23109i 0.548368i
\(92\) 0 0
\(93\) 3.55691 0.368835
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 17.1138i 1.73765i 0.495123 + 0.868823i \(0.335123\pi\)
−0.495123 + 0.868823i \(0.664877\pi\)
\(98\) 0 0
\(99\) − 4.49828i − 0.452094i
\(100\) 0 0
\(101\) 2.00000i 0.199007i 0.995037 + 0.0995037i \(0.0317255\pi\)
−0.995037 + 0.0995037i \(0.968274\pi\)
\(102\) 0 0
\(103\) 10.1725i 1.00232i 0.865354 + 0.501161i \(0.167094\pi\)
−0.865354 + 0.501161i \(0.832906\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −17.2311 −1.66579 −0.832896 0.553429i \(-0.813319\pi\)
−0.832896 + 0.553429i \(0.813319\pi\)
\(108\) 0 0
\(109\) − 1.88273i − 0.180333i −0.995927 0.0901666i \(-0.971260\pi\)
0.995927 0.0901666i \(-0.0287399\pi\)
\(110\) 0 0
\(111\) −7.43965 −0.706140
\(112\) 0 0
\(113\) − 15.3224i − 1.44141i −0.693243 0.720704i \(-0.743819\pi\)
0.693243 0.720704i \(-0.256181\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −5.55691 −0.513737
\(118\) 0 0
\(119\) 7.11383 0.652124
\(120\) 0 0
\(121\) −9.23453 −0.839503
\(122\) 0 0
\(123\) 3.88273 0.350094
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 18.1725i 1.61255i 0.591544 + 0.806273i \(0.298519\pi\)
−0.591544 + 0.806273i \(0.701481\pi\)
\(128\) 0 0
\(129\) −1.88273 −0.165765
\(130\) 0 0
\(131\) 6.38101i 0.557512i 0.960362 + 0.278756i \(0.0899220\pi\)
−0.960362 + 0.278756i \(0.910078\pi\)
\(132\) 0 0
\(133\) 0.996562 0.0864129
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 4.44309i − 0.379598i −0.981823 0.189799i \(-0.939216\pi\)
0.981823 0.189799i \(-0.0607837\pi\)
\(138\) 0 0
\(139\) 20.1725i 1.71101i 0.517798 + 0.855503i \(0.326752\pi\)
−0.517798 + 0.855503i \(0.673248\pi\)
\(140\) 0 0
\(141\) − 10.0552i − 0.846800i
\(142\) 0 0
\(143\) 24.9966i 2.09032i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −6.11383 −0.504260
\(148\) 0 0
\(149\) 2.00000i 0.163846i 0.996639 + 0.0819232i \(0.0261062\pi\)
−0.996639 + 0.0819232i \(0.973894\pi\)
\(150\) 0 0
\(151\) −9.67418 −0.787274 −0.393637 0.919266i \(-0.628783\pi\)
−0.393637 + 0.919266i \(0.628783\pi\)
\(152\) 0 0
\(153\) 7.55691i 0.610940i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −4.32582 −0.345238 −0.172619 0.984989i \(-0.555223\pi\)
−0.172619 + 0.984989i \(0.555223\pi\)
\(158\) 0 0
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) −0.996562 −0.0785401
\(162\) 0 0
\(163\) 6.11727 0.479141 0.239571 0.970879i \(-0.422993\pi\)
0.239571 + 0.970879i \(0.422993\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.05520i 0.468565i 0.972169 + 0.234283i \(0.0752741\pi\)
−0.972169 + 0.234283i \(0.924726\pi\)
\(168\) 0 0
\(169\) 17.8793 1.37533
\(170\) 0 0
\(171\) 1.05863i 0.0809557i
\(172\) 0 0
\(173\) 16.8793 1.28331 0.641655 0.766994i \(-0.278248\pi\)
0.641655 + 0.766994i \(0.278248\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 8.49828i 0.638770i
\(178\) 0 0
\(179\) 10.6155i 0.793443i 0.917939 + 0.396722i \(0.129852\pi\)
−0.917939 + 0.396722i \(0.870148\pi\)
\(180\) 0 0
\(181\) 14.1173i 1.04933i 0.851309 + 0.524664i \(0.175809\pi\)
−0.851309 + 0.524664i \(0.824191\pi\)
\(182\) 0 0
\(183\) − 8.99656i − 0.665045i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 33.9931 2.48582
\(188\) 0 0
\(189\) 0.941367i 0.0684744i
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) 4.87930i 0.351219i 0.984460 + 0.175610i \(0.0561897\pi\)
−0.984460 + 0.175610i \(0.943810\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.88617 0.205631 0.102816 0.994700i \(-0.467215\pi\)
0.102816 + 0.994700i \(0.467215\pi\)
\(198\) 0 0
\(199\) −17.6742 −1.25289 −0.626445 0.779466i \(-0.715491\pi\)
−0.626445 + 0.779466i \(0.715491\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) 1.88273 0.132142
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 1.05863i − 0.0735801i
\(208\) 0 0
\(209\) 4.76203 0.329396
\(210\) 0 0
\(211\) 23.9379i 1.64795i 0.566623 + 0.823977i \(0.308250\pi\)
−0.566623 + 0.823977i \(0.691750\pi\)
\(212\) 0 0
\(213\) −12.9966 −0.890510
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 3.34836i 0.227302i
\(218\) 0 0
\(219\) − 6.00000i − 0.405442i
\(220\) 0 0
\(221\) − 41.9931i − 2.82476i
\(222\) 0 0
\(223\) − 24.0552i − 1.61086i −0.592694 0.805428i \(-0.701936\pi\)
0.592694 0.805428i \(-0.298064\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 11.1138 0.737651 0.368825 0.929499i \(-0.379760\pi\)
0.368825 + 0.929499i \(0.379760\pi\)
\(228\) 0 0
\(229\) 17.2311i 1.13866i 0.822108 + 0.569331i \(0.192798\pi\)
−0.822108 + 0.569331i \(0.807202\pi\)
\(230\) 0 0
\(231\) 4.23453 0.278612
\(232\) 0 0
\(233\) 8.44309i 0.553125i 0.960996 + 0.276562i \(0.0891953\pi\)
−0.960996 + 0.276562i \(0.910805\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −11.5569 −0.750702
\(238\) 0 0
\(239\) 10.1173 0.654432 0.327216 0.944950i \(-0.393890\pi\)
0.327216 + 0.944950i \(0.393890\pi\)
\(240\) 0 0
\(241\) 16.8793 1.08729 0.543646 0.839315i \(-0.317044\pi\)
0.543646 + 0.839315i \(0.317044\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 5.88273i − 0.374309i
\(248\) 0 0
\(249\) 5.88273 0.372803
\(250\) 0 0
\(251\) − 11.8466i − 0.747753i −0.927478 0.373877i \(-0.878028\pi\)
0.927478 0.373877i \(-0.121972\pi\)
\(252\) 0 0
\(253\) −4.76203 −0.299386
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 10.6707i − 0.665623i −0.942993 0.332811i \(-0.892003\pi\)
0.942993 0.332811i \(-0.107997\pi\)
\(258\) 0 0
\(259\) − 7.00344i − 0.435172i
\(260\) 0 0
\(261\) 2.00000i 0.123797i
\(262\) 0 0
\(263\) − 1.94480i − 0.119922i −0.998201 0.0599609i \(-0.980902\pi\)
0.998201 0.0599609i \(-0.0190976\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −4.11727 −0.251973
\(268\) 0 0
\(269\) − 9.76547i − 0.595411i −0.954658 0.297706i \(-0.903779\pi\)
0.954658 0.297706i \(-0.0962214\pi\)
\(270\) 0 0
\(271\) −3.44652 −0.209361 −0.104681 0.994506i \(-0.533382\pi\)
−0.104681 + 0.994506i \(0.533382\pi\)
\(272\) 0 0
\(273\) − 5.23109i − 0.316600i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 18.7880 1.12886 0.564431 0.825480i \(-0.309096\pi\)
0.564431 + 0.825480i \(0.309096\pi\)
\(278\) 0 0
\(279\) −3.55691 −0.212947
\(280\) 0 0
\(281\) −16.8793 −1.00693 −0.503467 0.864014i \(-0.667943\pi\)
−0.503467 + 0.864014i \(0.667943\pi\)
\(282\) 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
\(287\) 3.65508i 0.215752i
\(288\) 0 0
\(289\) −40.1070 −2.35923
\(290\) 0 0
\(291\) − 17.1138i − 1.00323i
\(292\) 0 0
\(293\) −20.2277 −1.18171 −0.590856 0.806777i \(-0.701210\pi\)
−0.590856 + 0.806777i \(0.701210\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 4.49828i 0.261017i
\(298\) 0 0
\(299\) 5.88273i 0.340207i
\(300\) 0 0
\(301\) − 1.77234i − 0.102156i
\(302\) 0 0
\(303\) − 2.00000i − 0.114897i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 8.11039 0.462884 0.231442 0.972849i \(-0.425656\pi\)
0.231442 + 0.972849i \(0.425656\pi\)
\(308\) 0 0
\(309\) − 10.1725i − 0.578691i
\(310\) 0 0
\(311\) −31.8759 −1.80751 −0.903757 0.428046i \(-0.859202\pi\)
−0.903757 + 0.428046i \(0.859202\pi\)
\(312\) 0 0
\(313\) 5.11383i 0.289051i 0.989501 + 0.144525i \(0.0461655\pi\)
−0.989501 + 0.144525i \(0.953834\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −24.6448 −1.38419 −0.692094 0.721807i \(-0.743312\pi\)
−0.692094 + 0.721807i \(0.743312\pi\)
\(318\) 0 0
\(319\) 8.99656 0.503711
\(320\) 0 0
\(321\) 17.2311 0.961746
\(322\) 0 0
\(323\) −8.00000 −0.445132
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 1.88273i 0.104115i
\(328\) 0 0
\(329\) 9.46563 0.521857
\(330\) 0 0
\(331\) 11.0518i 0.607460i 0.952758 + 0.303730i \(0.0982320\pi\)
−0.952758 + 0.303730i \(0.901768\pi\)
\(332\) 0 0
\(333\) 7.43965 0.407690
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 19.9931i − 1.08909i −0.838730 0.544547i \(-0.816701\pi\)
0.838730 0.544547i \(-0.183299\pi\)
\(338\) 0 0
\(339\) 15.3224i 0.832198i
\(340\) 0 0
\(341\) 16.0000i 0.866449i
\(342\) 0 0
\(343\) − 12.3449i − 0.666563i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −6.87930 −0.369300 −0.184650 0.982804i \(-0.559115\pi\)
−0.184650 + 0.982804i \(0.559115\pi\)
\(348\) 0 0
\(349\) 4.76203i 0.254906i 0.991845 + 0.127453i \(0.0406801\pi\)
−0.991845 + 0.127453i \(0.959320\pi\)
\(350\) 0 0
\(351\) 5.55691 0.296606
\(352\) 0 0
\(353\) 3.79145i 0.201798i 0.994897 + 0.100899i \(0.0321720\pi\)
−0.994897 + 0.100899i \(0.967828\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −7.11383 −0.376504
\(358\) 0 0
\(359\) −12.9966 −0.685932 −0.342966 0.939348i \(-0.611432\pi\)
−0.342966 + 0.939348i \(0.611432\pi\)
\(360\) 0 0
\(361\) 17.8793 0.941016
\(362\) 0 0
\(363\) 9.23453 0.484687
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 22.9345i − 1.19717i −0.801059 0.598585i \(-0.795730\pi\)
0.801059 0.598585i \(-0.204270\pi\)
\(368\) 0 0
\(369\) −3.88273 −0.202127
\(370\) 0 0
\(371\) − 1.88273i − 0.0977467i
\(372\) 0 0
\(373\) 15.4396 0.799435 0.399717 0.916638i \(-0.369108\pi\)
0.399717 + 0.916638i \(0.369108\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 11.1138i − 0.572391i
\(378\) 0 0
\(379\) − 6.28973i − 0.323082i −0.986866 0.161541i \(-0.948354\pi\)
0.986866 0.161541i \(-0.0516463\pi\)
\(380\) 0 0
\(381\) − 18.1725i − 0.931003i
\(382\) 0 0
\(383\) 2.94137i 0.150297i 0.997172 + 0.0751484i \(0.0239431\pi\)
−0.997172 + 0.0751484i \(0.976057\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1.88273 0.0957047
\(388\) 0 0
\(389\) − 12.2277i − 0.619967i −0.950742 0.309983i \(-0.899676\pi\)
0.950742 0.309983i \(-0.100324\pi\)
\(390\) 0 0
\(391\) 8.00000 0.404577
\(392\) 0 0
\(393\) − 6.38101i − 0.321880i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 5.32238 0.267123 0.133561 0.991041i \(-0.457359\pi\)
0.133561 + 0.991041i \(0.457359\pi\)
\(398\) 0 0
\(399\) −0.996562 −0.0498905
\(400\) 0 0
\(401\) 6.99656 0.349392 0.174696 0.984622i \(-0.444106\pi\)
0.174696 + 0.984622i \(0.444106\pi\)
\(402\) 0 0
\(403\) 19.7655 0.984588
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 33.4656i − 1.65883i
\(408\) 0 0
\(409\) −16.2277 −0.802406 −0.401203 0.915989i \(-0.631408\pi\)
−0.401203 + 0.915989i \(0.631408\pi\)
\(410\) 0 0
\(411\) 4.44309i 0.219161i
\(412\) 0 0
\(413\) −8.00000 −0.393654
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 20.1725i − 0.987850i
\(418\) 0 0
\(419\) − 15.6121i − 0.762701i −0.924430 0.381351i \(-0.875459\pi\)
0.924430 0.381351i \(-0.124541\pi\)
\(420\) 0 0
\(421\) 33.2311i 1.61958i 0.586717 + 0.809792i \(0.300420\pi\)
−0.586717 + 0.809792i \(0.699580\pi\)
\(422\) 0 0
\(423\) 10.0552i 0.488900i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 8.46907 0.409847
\(428\) 0 0
\(429\) − 24.9966i − 1.20685i
\(430\) 0 0
\(431\) 12.9966 0.626022 0.313011 0.949749i \(-0.398662\pi\)
0.313011 + 0.949749i \(0.398662\pi\)
\(432\) 0 0
\(433\) − 20.2277i − 0.972079i −0.873937 0.486040i \(-0.838441\pi\)
0.873937 0.486040i \(-0.161559\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.12070 0.0536106
\(438\) 0 0
\(439\) −5.43965 −0.259620 −0.129810 0.991539i \(-0.541437\pi\)
−0.129810 + 0.991539i \(0.541437\pi\)
\(440\) 0 0
\(441\) 6.11383 0.291135
\(442\) 0 0
\(443\) 15.3484 0.729223 0.364611 0.931160i \(-0.381202\pi\)
0.364611 + 0.931160i \(0.381202\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 2.00000i − 0.0945968i
\(448\) 0 0
\(449\) −4.22766 −0.199515 −0.0997577 0.995012i \(-0.531807\pi\)
−0.0997577 + 0.995012i \(0.531807\pi\)
\(450\) 0 0
\(451\) 17.4656i 0.822424i
\(452\) 0 0
\(453\) 9.67418 0.454533
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 2.65164i 0.124038i 0.998075 + 0.0620192i \(0.0197540\pi\)
−0.998075 + 0.0620192i \(0.980246\pi\)
\(458\) 0 0
\(459\) − 7.55691i − 0.352727i
\(460\) 0 0
\(461\) − 10.2345i − 0.476670i −0.971183 0.238335i \(-0.923398\pi\)
0.971183 0.238335i \(-0.0766016\pi\)
\(462\) 0 0
\(463\) 19.0586i 0.885730i 0.896588 + 0.442865i \(0.146038\pi\)
−0.896588 + 0.442865i \(0.853962\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −4.11039 −0.190206 −0.0951031 0.995467i \(-0.530318\pi\)
−0.0951031 + 0.995467i \(0.530318\pi\)
\(468\) 0 0
\(469\) 3.76547i 0.173873i
\(470\) 0 0
\(471\) 4.32582 0.199323
\(472\) 0 0
\(473\) − 8.46907i − 0.389408i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 2.00000 0.0915737
\(478\) 0 0
\(479\) −25.2311 −1.15284 −0.576419 0.817154i \(-0.695550\pi\)
−0.576419 + 0.817154i \(0.695550\pi\)
\(480\) 0 0
\(481\) −41.3415 −1.88501
\(482\) 0 0
\(483\) 0.996562 0.0453451
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 21.9379i 0.994102i 0.867721 + 0.497051i \(0.165584\pi\)
−0.867721 + 0.497051i \(0.834416\pi\)
\(488\) 0 0
\(489\) −6.11727 −0.276632
\(490\) 0 0
\(491\) 7.50172i 0.338548i 0.985569 + 0.169274i \(0.0541423\pi\)
−0.985569 + 0.169274i \(0.945858\pi\)
\(492\) 0 0
\(493\) −15.1138 −0.680693
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 12.2345i − 0.548794i
\(498\) 0 0
\(499\) − 29.1690i − 1.30578i −0.757451 0.652892i \(-0.773555\pi\)
0.757451 0.652892i \(-0.226445\pi\)
\(500\) 0 0
\(501\) − 6.05520i − 0.270526i
\(502\) 0 0
\(503\) − 23.9379i − 1.06734i −0.845693 0.533670i \(-0.820813\pi\)
0.845693 0.533670i \(-0.179187\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −17.8793 −0.794047
\(508\) 0 0
\(509\) − 28.6967i − 1.27196i −0.771706 0.635980i \(-0.780596\pi\)
0.771706 0.635980i \(-0.219404\pi\)
\(510\) 0 0
\(511\) 5.64820 0.249862
\(512\) 0 0
\(513\) − 1.05863i − 0.0467398i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 45.2311 1.98926
\(518\) 0 0
\(519\) −16.8793 −0.740919
\(520\) 0 0
\(521\) 10.0000 0.438108 0.219054 0.975713i \(-0.429703\pi\)
0.219054 + 0.975713i \(0.429703\pi\)
\(522\) 0 0
\(523\) −25.7586 −1.12634 −0.563172 0.826340i \(-0.690419\pi\)
−0.563172 + 0.826340i \(0.690419\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 26.8793i − 1.17088i
\(528\) 0 0
\(529\) 21.8793 0.951274
\(530\) 0 0
\(531\) − 8.49828i − 0.368794i
\(532\) 0 0
\(533\) 21.5760 0.934561
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 10.6155i − 0.458095i
\(538\) 0 0
\(539\) − 27.5017i − 1.18458i
\(540\) 0 0
\(541\) 12.3449i 0.530750i 0.964145 + 0.265375i \(0.0854957\pi\)
−0.964145 + 0.265375i \(0.914504\pi\)
\(542\) 0 0
\(543\) − 14.1173i − 0.605830i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −19.8759 −0.849830 −0.424915 0.905233i \(-0.639696\pi\)
−0.424915 + 0.905233i \(0.639696\pi\)
\(548\) 0 0
\(549\) 8.99656i 0.383964i
\(550\) 0 0
\(551\) −2.11727 −0.0901986
\(552\) 0 0
\(553\) − 10.8793i − 0.462635i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −3.12070 −0.132228 −0.0661142 0.997812i \(-0.521060\pi\)
−0.0661142 + 0.997812i \(0.521060\pi\)
\(558\) 0 0
\(559\) −10.4622 −0.442503
\(560\) 0 0
\(561\) −33.9931 −1.43519
\(562\) 0 0
\(563\) 0.651639 0.0274633 0.0137317 0.999906i \(-0.495629\pi\)
0.0137317 + 0.999906i \(0.495629\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 0.941367i − 0.0395337i
\(568\) 0 0
\(569\) −26.9966 −1.13175 −0.565877 0.824489i \(-0.691462\pi\)
−0.565877 + 0.824489i \(0.691462\pi\)
\(570\) 0 0
\(571\) − 14.9414i − 0.625277i −0.949872 0.312638i \(-0.898787\pi\)
0.949872 0.312638i \(-0.101213\pi\)
\(572\) 0 0
\(573\) −8.00000 −0.334205
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 8.87930i − 0.369650i −0.982771 0.184825i \(-0.940828\pi\)
0.982771 0.184825i \(-0.0591718\pi\)
\(578\) 0 0
\(579\) − 4.87930i − 0.202777i
\(580\) 0 0
\(581\) 5.53781i 0.229747i
\(582\) 0 0
\(583\) − 8.99656i − 0.372600i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.23109 0.0508127 0.0254064 0.999677i \(-0.491912\pi\)
0.0254064 + 0.999677i \(0.491912\pi\)
\(588\) 0 0
\(589\) − 3.76547i − 0.155153i
\(590\) 0 0
\(591\) −2.88617 −0.118721
\(592\) 0 0
\(593\) − 3.55691i − 0.146065i −0.997330 0.0730325i \(-0.976732\pi\)
0.997330 0.0730325i \(-0.0232677\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 17.6742 0.723356
\(598\) 0 0
\(599\) −19.2242 −0.785480 −0.392740 0.919649i \(-0.628473\pi\)
−0.392740 + 0.919649i \(0.628473\pi\)
\(600\) 0 0
\(601\) −27.7586 −1.13230 −0.566148 0.824303i \(-0.691567\pi\)
−0.566148 + 0.824303i \(0.691567\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 7.16902i − 0.290982i −0.989360 0.145491i \(-0.953524\pi\)
0.989360 0.145491i \(-0.0464761\pi\)
\(608\) 0 0
\(609\) −1.88273 −0.0762922
\(610\) 0 0
\(611\) − 55.8759i − 2.26050i
\(612\) 0 0
\(613\) 9.55691 0.386000 0.193000 0.981199i \(-0.438178\pi\)
0.193000 + 0.981199i \(0.438178\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.32926i 0.0535139i 0.999642 + 0.0267569i \(0.00851802\pi\)
−0.999642 + 0.0267569i \(0.991482\pi\)
\(618\) 0 0
\(619\) 28.1725i 1.13235i 0.824286 + 0.566173i \(0.191577\pi\)
−0.824286 + 0.566173i \(0.808423\pi\)
\(620\) 0 0
\(621\) 1.05863i 0.0424815i
\(622\) 0 0
\(623\) − 3.87586i − 0.155283i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −4.76203 −0.190177
\(628\) 0 0
\(629\) 56.2208i 2.24167i
\(630\) 0 0
\(631\) 23.3224 0.928449 0.464225 0.885717i \(-0.346333\pi\)
0.464225 + 0.885717i \(0.346333\pi\)
\(632\) 0 0
\(633\) − 23.9379i − 0.951447i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −33.9740 −1.34610
\(638\) 0 0
\(639\) 12.9966 0.514136
\(640\) 0 0
\(641\) 27.1070 1.07066 0.535330 0.844643i \(-0.320187\pi\)
0.535330 + 0.844643i \(0.320187\pi\)
\(642\) 0 0
\(643\) −20.3449 −0.802325 −0.401163 0.916007i \(-0.631394\pi\)
−0.401163 + 0.916007i \(0.631394\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 37.6965i − 1.48200i −0.671503 0.741002i \(-0.734351\pi\)
0.671503 0.741002i \(-0.265649\pi\)
\(648\) 0 0
\(649\) −38.2277 −1.50057
\(650\) 0 0
\(651\) − 3.34836i − 0.131233i
\(652\) 0 0
\(653\) −8.64476 −0.338296 −0.169148 0.985591i \(-0.554102\pi\)
−0.169148 + 0.985591i \(0.554102\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 6.00000i 0.234082i
\(658\) 0 0
\(659\) 29.2603i 1.13982i 0.821707 + 0.569910i \(0.193022\pi\)
−0.821707 + 0.569910i \(0.806978\pi\)
\(660\) 0 0
\(661\) 28.7620i 1.11871i 0.828927 + 0.559357i \(0.188952\pi\)
−0.828927 + 0.559357i \(0.811048\pi\)
\(662\) 0 0
\(663\) 41.9931i 1.63088i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 2.11727 0.0819809
\(668\) 0 0
\(669\) 24.0552i 0.930028i
\(670\) 0 0
\(671\) 40.4691 1.56229
\(672\) 0 0
\(673\) 18.0000i 0.693849i 0.937893 + 0.346925i \(0.112774\pi\)
−0.937893 + 0.346925i \(0.887226\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 42.8724 1.64772 0.823860 0.566793i \(-0.191816\pi\)
0.823860 + 0.566793i \(0.191816\pi\)
\(678\) 0 0
\(679\) 16.1104 0.618260
\(680\) 0 0
\(681\) −11.1138 −0.425883
\(682\) 0 0
\(683\) 26.1173 0.999349 0.499675 0.866213i \(-0.333453\pi\)
0.499675 + 0.866213i \(0.333453\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 17.2311i − 0.657407i
\(688\) 0 0
\(689\) −11.1138 −0.423403
\(690\) 0 0
\(691\) 5.29317i 0.201362i 0.994919 + 0.100681i \(0.0321021\pi\)
−0.994919 + 0.100681i \(0.967898\pi\)
\(692\) 0 0
\(693\) −4.23453 −0.160857
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 29.3415i − 1.11139i
\(698\) 0 0
\(699\) − 8.44309i − 0.319347i
\(700\) 0 0
\(701\) 7.99312i 0.301896i 0.988542 + 0.150948i \(0.0482326\pi\)
−0.988542 + 0.150948i \(0.951767\pi\)
\(702\) 0 0
\(703\) 7.87586i 0.297044i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1.88273 0.0708075
\(708\) 0 0
\(709\) 28.9966i 1.08899i 0.838764 + 0.544494i \(0.183278\pi\)
−0.838764 + 0.544494i \(0.816722\pi\)
\(710\) 0 0
\(711\) 11.5569 0.433418
\(712\) 0 0
\(713\) 3.76547i 0.141018i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −10.1173 −0.377836
\(718\) 0 0
\(719\) 26.8793 1.00243 0.501214 0.865323i \(-0.332887\pi\)
0.501214 + 0.865323i \(0.332887\pi\)
\(720\) 0 0
\(721\) 9.57602 0.356630
\(722\) 0 0
\(723\) −16.8793 −0.627748
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 41.8138i − 1.55079i −0.631478 0.775394i \(-0.717551\pi\)
0.631478 0.775394i \(-0.282449\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 14.2277i 0.526229i
\(732\) 0 0
\(733\) 30.0844 1.11119 0.555597 0.831452i \(-0.312490\pi\)
0.555597 + 0.831452i \(0.312490\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 17.9931i 0.662785i
\(738\) 0 0
\(739\) 29.0449i 1.06843i 0.845348 + 0.534217i \(0.179393\pi\)
−0.845348 + 0.534217i \(0.820607\pi\)
\(740\) 0 0
\(741\) 5.88273i 0.216108i
\(742\) 0 0
\(743\) − 43.2863i − 1.58802i −0.607905 0.794010i \(-0.707990\pi\)
0.607905 0.794010i \(-0.292010\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −5.88273 −0.215238
\(748\) 0 0
\(749\) 16.2208i 0.592694i
\(750\) 0 0
\(751\) −41.7846 −1.52474 −0.762370 0.647141i \(-0.775964\pi\)
−0.762370 + 0.647141i \(0.775964\pi\)
\(752\) 0 0
\(753\) 11.8466i 0.431716i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 16.3258 0.593372 0.296686 0.954975i \(-0.404119\pi\)
0.296686 + 0.954975i \(0.404119\pi\)
\(758\) 0 0
\(759\) 4.76203 0.172851
\(760\) 0 0
\(761\) 50.2208 1.82050 0.910251 0.414057i \(-0.135889\pi\)
0.910251 + 0.414057i \(0.135889\pi\)
\(762\) 0 0
\(763\) −1.77234 −0.0641631
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 47.2242i 1.70517i
\(768\) 0 0
\(769\) 31.3415 1.13020 0.565101 0.825021i \(-0.308837\pi\)
0.565101 + 0.825021i \(0.308837\pi\)
\(770\) 0 0
\(771\) 10.6707i 0.384297i
\(772\) 0 0
\(773\) −9.11383 −0.327802 −0.163901 0.986477i \(-0.552408\pi\)
−0.163901 + 0.986477i \(0.552408\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 7.00344i 0.251247i
\(778\) 0 0
\(779\) − 4.11039i − 0.147270i
\(780\) 0 0
\(781\) − 58.4622i − 2.09194i
\(782\) 0 0
\(783\) − 2.00000i − 0.0714742i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 36.2208 1.29113 0.645566 0.763705i \(-0.276622\pi\)
0.645566 + 0.763705i \(0.276622\pi\)
\(788\) 0 0
\(789\) 1.94480i 0.0692369i
\(790\) 0 0
\(791\) −14.4240 −0.512858
\(792\) 0 0
\(793\) − 49.9931i − 1.77531i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −10.0000 −0.354218 −0.177109 0.984191i \(-0.556675\pi\)
−0.177109 + 0.984191i \(0.556675\pi\)
\(798\) 0 0
\(799\) −75.9862 −2.68820
\(800\) 0 0
\(801\) 4.11727 0.145476
\(802\) 0 0
\(803\) 26.9897 0.952445
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 9.76547i 0.343761i
\(808\) 0 0
\(809\) −47.5760 −1.67268 −0.836342 0.548208i \(-0.815310\pi\)
−0.836342 + 0.548208i \(0.815310\pi\)
\(810\) 0 0
\(811\) − 20.5174i − 0.720463i −0.932863 0.360231i \(-0.882698\pi\)
0.932863 0.360231i \(-0.117302\pi\)
\(812\) 0 0
\(813\) 3.44652 0.120875
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 1.99312i 0.0697306i
\(818\) 0 0
\(819\) 5.23109i 0.182789i
\(820\) 0 0
\(821\) 44.4622i 1.55174i 0.630892 + 0.775871i \(0.282689\pi\)
−0.630892 + 0.775871i \(0.717311\pi\)
\(822\) 0 0
\(823\) 32.1656i 1.12122i 0.828079 + 0.560611i \(0.189434\pi\)
−0.828079 + 0.560611i \(0.810566\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −20.0000 −0.695468 −0.347734 0.937593i \(-0.613049\pi\)
−0.347734 + 0.937593i \(0.613049\pi\)
\(828\) 0 0
\(829\) − 33.8827i − 1.17680i −0.808571 0.588398i \(-0.799759\pi\)
0.808571 0.588398i \(-0.200241\pi\)
\(830\) 0 0
\(831\) −18.7880 −0.651749
\(832\) 0 0
\(833\) 46.2017i 1.60079i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 3.55691 0.122945
\(838\) 0 0
\(839\) 4.52750 0.156307 0.0781533 0.996941i \(-0.475098\pi\)
0.0781533 + 0.996941i \(0.475098\pi\)
\(840\) 0 0
\(841\) 25.0000 0.862069
\(842\) 0 0
\(843\) 16.8793 0.581354
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 8.69308i 0.298698i
\(848\) 0 0
\(849\) 20.0000 0.686398
\(850\) 0 0
\(851\) − 7.87586i − 0.269981i
\(852\) 0 0
\(853\) −50.4293 −1.72667 −0.863334 0.504633i \(-0.831628\pi\)
−0.863334 + 0.504633i \(0.831628\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 26.4362i 0.903044i 0.892260 + 0.451522i \(0.149119\pi\)
−0.892260 + 0.451522i \(0.850881\pi\)
\(858\) 0 0
\(859\) 0.406994i 0.0138865i 0.999976 + 0.00694323i \(0.00221012\pi\)
−0.999976 + 0.00694323i \(0.997790\pi\)
\(860\) 0 0
\(861\) − 3.65508i − 0.124565i
\(862\) 0 0
\(863\) − 29.9311i − 1.01886i −0.860511 0.509432i \(-0.829855\pi\)
0.860511 0.509432i \(-0.170145\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 40.1070 1.36210
\(868\) 0 0
\(869\) − 51.9862i − 1.76351i
\(870\) 0 0
\(871\) 22.2277 0.753155
\(872\) 0 0
\(873\) 17.1138i 0.579215i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 11.2051 0.378370 0.189185 0.981941i \(-0.439415\pi\)
0.189185 + 0.981941i \(0.439415\pi\)
\(878\) 0 0
\(879\) 20.2277 0.682262
\(880\) 0 0
\(881\) −48.3380 −1.62855 −0.814275 0.580479i \(-0.802865\pi\)
−0.814275 + 0.580479i \(0.802865\pi\)
\(882\) 0 0
\(883\) 50.5726 1.70190 0.850951 0.525244i \(-0.176026\pi\)
0.850951 + 0.525244i \(0.176026\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 48.0483i 1.61330i 0.591026 + 0.806652i \(0.298723\pi\)
−0.591026 + 0.806652i \(0.701277\pi\)
\(888\) 0 0
\(889\) 17.1070 0.573749
\(890\) 0 0
\(891\) − 4.49828i − 0.150698i
\(892\) 0 0
\(893\) −10.6448 −0.356213
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 5.88273i − 0.196419i
\(898\) 0 0
\(899\) − 7.11383i − 0.237259i
\(900\) 0 0
\(901\) 15.1138i 0.503515i
\(902\) 0 0
\(903\) 1.77234i 0.0589799i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −6.46219 −0.214573 −0.107287 0.994228i \(-0.534216\pi\)
−0.107287 + 0.994228i \(0.534216\pi\)
\(908\) 0 0
\(909\) 2.00000i 0.0663358i
\(910\) 0 0
\(911\) −50.3380 −1.66777 −0.833887 0.551935i \(-0.813890\pi\)
−0.833887 + 0.551935i \(0.813890\pi\)
\(912\) 0 0
\(913\) 26.4622i 0.875771i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 6.00688 0.198365
\(918\) 0 0
\(919\) 46.4362 1.53179 0.765895 0.642966i \(-0.222296\pi\)
0.765895 + 0.642966i \(0.222296\pi\)
\(920\) 0 0
\(921\) −8.11039 −0.267246
\(922\) 0 0
\(923\) −72.2208 −2.37718
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 10.1725i 0.334107i
\(928\) 0 0
\(929\) 35.9931 1.18090 0.590448 0.807076i \(-0.298951\pi\)
0.590448 + 0.807076i \(0.298951\pi\)
\(930\) 0 0
\(931\) 6.47230i 0.212121i
\(932\) 0 0
\(933\) 31.8759 1.04357
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 2.70360i 0.0883227i 0.999024 + 0.0441613i \(0.0140616\pi\)
−0.999024 + 0.0441613i \(0.985938\pi\)
\(938\) 0 0
\(939\) − 5.11383i − 0.166883i
\(940\) 0 0
\(941\) − 17.7655i − 0.579138i −0.957157 0.289569i \(-0.906488\pi\)
0.957157 0.289569i \(-0.0935119\pi\)
\(942\) 0 0
\(943\) 4.11039i 0.133853i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −26.2277 −0.852284 −0.426142 0.904656i \(-0.640128\pi\)
−0.426142 + 0.904656i \(0.640128\pi\)
\(948\) 0 0
\(949\) − 33.3415i − 1.08231i
\(950\) 0 0
\(951\) 24.6448 0.799161
\(952\) 0 0
\(953\) − 9.09472i − 0.294607i −0.989091 0.147304i \(-0.952941\pi\)
0.989091 0.147304i \(-0.0470594\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −8.99656 −0.290818
\(958\) 0 0
\(959\) −4.18257 −0.135062
\(960\) 0 0
\(961\) −18.3484 −0.591883
\(962\) 0 0
\(963\) −17.2311 −0.555264
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 7.47574i 0.240404i 0.992749 + 0.120202i \(0.0383542\pi\)
−0.992749 + 0.120202i \(0.961646\pi\)
\(968\) 0 0
\(969\) 8.00000 0.256997
\(970\) 0 0
\(971\) 41.0777i 1.31825i 0.752035 + 0.659124i \(0.229073\pi\)
−0.752035 + 0.659124i \(0.770927\pi\)
\(972\) 0 0
\(973\) 18.9897 0.608781
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 4.20855i − 0.134644i −0.997731 0.0673218i \(-0.978555\pi\)
0.997731 0.0673218i \(-0.0214454\pi\)
\(978\) 0 0
\(979\) − 18.5206i − 0.591922i
\(980\) 0 0
\(981\) − 1.88273i − 0.0601111i
\(982\) 0 0
\(983\) − 8.35504i − 0.266484i −0.991084 0.133242i \(-0.957461\pi\)
0.991084 0.133242i \(-0.0425388\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −9.46563 −0.301294
\(988\) 0 0
\(989\) − 1.99312i − 0.0633777i
\(990\) 0 0
\(991\) 13.9087 0.441825 0.220912 0.975294i \(-0.429097\pi\)
0.220912 + 0.975294i \(0.429097\pi\)
\(992\) 0 0
\(993\) − 11.0518i − 0.350717i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −34.8984 −1.10524 −0.552622 0.833432i \(-0.686373\pi\)
−0.552622 + 0.833432i \(0.686373\pi\)
\(998\) 0 0
\(999\) −7.43965 −0.235380
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 2400.2.d.e.49.3 6
3.2 odd 2 7200.2.d.r.2449.3 6
4.3 odd 2 600.2.d.f.349.2 6
5.2 odd 4 2400.2.k.c.1201.5 6
5.3 odd 4 480.2.k.b.241.2 6
5.4 even 2 2400.2.d.f.49.4 6
8.3 odd 2 600.2.d.e.349.6 6
8.5 even 2 2400.2.d.f.49.3 6
12.11 even 2 1800.2.d.r.1549.5 6
15.2 even 4 7200.2.k.p.3601.4 6
15.8 even 4 1440.2.k.f.721.2 6
15.14 odd 2 7200.2.d.q.2449.4 6
20.3 even 4 120.2.k.b.61.4 yes 6
20.7 even 4 600.2.k.c.301.3 6
20.19 odd 2 600.2.d.e.349.5 6
24.5 odd 2 7200.2.d.q.2449.3 6
24.11 even 2 1800.2.d.q.1549.1 6
40.3 even 4 120.2.k.b.61.3 6
40.13 odd 4 480.2.k.b.241.5 6
40.19 odd 2 600.2.d.f.349.1 6
40.27 even 4 600.2.k.c.301.4 6
40.29 even 2 inner 2400.2.d.e.49.4 6
40.37 odd 4 2400.2.k.c.1201.2 6
60.23 odd 4 360.2.k.f.181.3 6
60.47 odd 4 1800.2.k.p.901.4 6
60.59 even 2 1800.2.d.q.1549.2 6
80.3 even 4 3840.2.a.bp.1.2 3
80.13 odd 4 3840.2.a.br.1.2 3
80.43 even 4 3840.2.a.bq.1.2 3
80.53 odd 4 3840.2.a.bo.1.2 3
120.29 odd 2 7200.2.d.r.2449.4 6
120.53 even 4 1440.2.k.f.721.5 6
120.59 even 2 1800.2.d.r.1549.6 6
120.77 even 4 7200.2.k.p.3601.3 6
120.83 odd 4 360.2.k.f.181.4 6
120.107 odd 4 1800.2.k.p.901.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.2.k.b.61.3 6 40.3 even 4
120.2.k.b.61.4 yes 6 20.3 even 4
360.2.k.f.181.3 6 60.23 odd 4
360.2.k.f.181.4 6 120.83 odd 4
480.2.k.b.241.2 6 5.3 odd 4
480.2.k.b.241.5 6 40.13 odd 4
600.2.d.e.349.5 6 20.19 odd 2
600.2.d.e.349.6 6 8.3 odd 2
600.2.d.f.349.1 6 40.19 odd 2
600.2.d.f.349.2 6 4.3 odd 2
600.2.k.c.301.3 6 20.7 even 4
600.2.k.c.301.4 6 40.27 even 4
1440.2.k.f.721.2 6 15.8 even 4
1440.2.k.f.721.5 6 120.53 even 4
1800.2.d.q.1549.1 6 24.11 even 2
1800.2.d.q.1549.2 6 60.59 even 2
1800.2.d.r.1549.5 6 12.11 even 2
1800.2.d.r.1549.6 6 120.59 even 2
1800.2.k.p.901.3 6 120.107 odd 4
1800.2.k.p.901.4 6 60.47 odd 4
2400.2.d.e.49.3 6 1.1 even 1 trivial
2400.2.d.e.49.4 6 40.29 even 2 inner
2400.2.d.f.49.3 6 8.5 even 2
2400.2.d.f.49.4 6 5.4 even 2
2400.2.k.c.1201.2 6 40.37 odd 4
2400.2.k.c.1201.5 6 5.2 odd 4
3840.2.a.bo.1.2 3 80.53 odd 4
3840.2.a.bp.1.2 3 80.3 even 4
3840.2.a.bq.1.2 3 80.43 even 4
3840.2.a.br.1.2 3 80.13 odd 4
7200.2.d.q.2449.3 6 24.5 odd 2
7200.2.d.q.2449.4 6 15.14 odd 2
7200.2.d.r.2449.3 6 3.2 odd 2
7200.2.d.r.2449.4 6 120.29 odd 2
7200.2.k.p.3601.3 6 120.77 even 4
7200.2.k.p.3601.4 6 15.2 even 4