Properties

Label 2640.2.a.x.1.2
Level $2640$
Weight $2$
Character 2640.1
Self dual yes
Analytic conductor $21.081$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2640.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(21.0805061336\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 2640.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} -2.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} -2.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} +5.46410 q^{13} +1.00000 q^{15} -5.46410 q^{19} +2.00000 q^{21} -6.92820 q^{23} +1.00000 q^{25} -1.00000 q^{27} -3.46410 q^{29} +10.9282 q^{31} -1.00000 q^{33} +2.00000 q^{35} -4.92820 q^{37} -5.46410 q^{39} +3.46410 q^{41} +4.92820 q^{43} -1.00000 q^{45} +6.92820 q^{47} -3.00000 q^{49} +0.928203 q^{53} -1.00000 q^{55} +5.46410 q^{57} +6.92820 q^{59} +2.00000 q^{61} -2.00000 q^{63} -5.46410 q^{65} -8.00000 q^{67} +6.92820 q^{69} -13.8564 q^{71} -8.39230 q^{73} -1.00000 q^{75} -2.00000 q^{77} +6.53590 q^{79} +1.00000 q^{81} -8.53590 q^{83} +3.46410 q^{87} +0.928203 q^{89} -10.9282 q^{91} -10.9282 q^{93} +5.46410 q^{95} -10.0000 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 2q^{5} - 4q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 2q^{5} - 4q^{7} + 2q^{9} + 2q^{11} + 4q^{13} + 2q^{15} - 4q^{19} + 4q^{21} + 2q^{25} - 2q^{27} + 8q^{31} - 2q^{33} + 4q^{35} + 4q^{37} - 4q^{39} - 4q^{43} - 2q^{45} - 6q^{49} - 12q^{53} - 2q^{55} + 4q^{57} + 4q^{61} - 4q^{63} - 4q^{65} - 16q^{67} + 4q^{73} - 2q^{75} - 4q^{77} + 20q^{79} + 2q^{81} - 24q^{83} - 12q^{89} - 8q^{91} - 8q^{93} + 4q^{95} - 20q^{97} + 2q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 5.46410 1.51547 0.757735 0.652563i \(-0.226306\pi\)
0.757735 + 0.652563i \(0.226306\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −5.46410 −1.25355 −0.626775 0.779200i \(-0.715626\pi\)
−0.626775 + 0.779200i \(0.715626\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) −6.92820 −1.44463 −0.722315 0.691564i \(-0.756922\pi\)
−0.722315 + 0.691564i \(0.756922\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −3.46410 −0.643268 −0.321634 0.946864i \(-0.604232\pi\)
−0.321634 + 0.946864i \(0.604232\pi\)
\(30\) 0 0
\(31\) 10.9282 1.96276 0.981382 0.192068i \(-0.0615194\pi\)
0.981382 + 0.192068i \(0.0615194\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) −4.92820 −0.810192 −0.405096 0.914274i \(-0.632762\pi\)
−0.405096 + 0.914274i \(0.632762\pi\)
\(38\) 0 0
\(39\) −5.46410 −0.874957
\(40\) 0 0
\(41\) 3.46410 0.541002 0.270501 0.962720i \(-0.412811\pi\)
0.270501 + 0.962720i \(0.412811\pi\)
\(42\) 0 0
\(43\) 4.92820 0.751544 0.375772 0.926712i \(-0.377378\pi\)
0.375772 + 0.926712i \(0.377378\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 6.92820 1.01058 0.505291 0.862949i \(-0.331385\pi\)
0.505291 + 0.862949i \(0.331385\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.928203 0.127499 0.0637493 0.997966i \(-0.479694\pi\)
0.0637493 + 0.997966i \(0.479694\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) 5.46410 0.723738
\(58\) 0 0
\(59\) 6.92820 0.901975 0.450988 0.892530i \(-0.351072\pi\)
0.450988 + 0.892530i \(0.351072\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) −2.00000 −0.251976
\(64\) 0 0
\(65\) −5.46410 −0.677738
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) 6.92820 0.834058
\(70\) 0 0
\(71\) −13.8564 −1.64445 −0.822226 0.569160i \(-0.807268\pi\)
−0.822226 + 0.569160i \(0.807268\pi\)
\(72\) 0 0
\(73\) −8.39230 −0.982245 −0.491122 0.871091i \(-0.663413\pi\)
−0.491122 + 0.871091i \(0.663413\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −2.00000 −0.227921
\(78\) 0 0
\(79\) 6.53590 0.735346 0.367673 0.929955i \(-0.380155\pi\)
0.367673 + 0.929955i \(0.380155\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −8.53590 −0.936937 −0.468468 0.883480i \(-0.655194\pi\)
−0.468468 + 0.883480i \(0.655194\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 3.46410 0.371391
\(88\) 0 0
\(89\) 0.928203 0.0983893 0.0491947 0.998789i \(-0.484335\pi\)
0.0491947 + 0.998789i \(0.484335\pi\)
\(90\) 0 0
\(91\) −10.9282 −1.14559
\(92\) 0 0
\(93\) −10.9282 −1.13320
\(94\) 0 0
\(95\) 5.46410 0.560605
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) −10.3923 −1.03407 −0.517036 0.855963i \(-0.672965\pi\)
−0.517036 + 0.855963i \(0.672965\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 0 0
\(105\) −2.00000 −0.195180
\(106\) 0 0
\(107\) −8.53590 −0.825196 −0.412598 0.910913i \(-0.635379\pi\)
−0.412598 + 0.910913i \(0.635379\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 4.92820 0.467764
\(112\) 0 0
\(113\) −12.9282 −1.21618 −0.608092 0.793867i \(-0.708065\pi\)
−0.608092 + 0.793867i \(0.708065\pi\)
\(114\) 0 0
\(115\) 6.92820 0.646058
\(116\) 0 0
\(117\) 5.46410 0.505156
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −3.46410 −0.312348
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −8.92820 −0.792250 −0.396125 0.918197i \(-0.629645\pi\)
−0.396125 + 0.918197i \(0.629645\pi\)
\(128\) 0 0
\(129\) −4.92820 −0.433904
\(130\) 0 0
\(131\) −18.9282 −1.65376 −0.826882 0.562375i \(-0.809888\pi\)
−0.826882 + 0.562375i \(0.809888\pi\)
\(132\) 0 0
\(133\) 10.9282 0.947595
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) −12.3923 −1.05110 −0.525551 0.850762i \(-0.676141\pi\)
−0.525551 + 0.850762i \(0.676141\pi\)
\(140\) 0 0
\(141\) −6.92820 −0.583460
\(142\) 0 0
\(143\) 5.46410 0.456931
\(144\) 0 0
\(145\) 3.46410 0.287678
\(146\) 0 0
\(147\) 3.00000 0.247436
\(148\) 0 0
\(149\) −15.4641 −1.26687 −0.633434 0.773796i \(-0.718355\pi\)
−0.633434 + 0.773796i \(0.718355\pi\)
\(150\) 0 0
\(151\) 20.3923 1.65950 0.829751 0.558134i \(-0.188482\pi\)
0.829751 + 0.558134i \(0.188482\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −10.9282 −0.877774
\(156\) 0 0
\(157\) −3.07180 −0.245156 −0.122578 0.992459i \(-0.539116\pi\)
−0.122578 + 0.992459i \(0.539116\pi\)
\(158\) 0 0
\(159\) −0.928203 −0.0736113
\(160\) 0 0
\(161\) 13.8564 1.09204
\(162\) 0 0
\(163\) −9.85641 −0.772013 −0.386007 0.922496i \(-0.626146\pi\)
−0.386007 + 0.922496i \(0.626146\pi\)
\(164\) 0 0
\(165\) 1.00000 0.0778499
\(166\) 0 0
\(167\) 10.3923 0.804181 0.402090 0.915600i \(-0.368284\pi\)
0.402090 + 0.915600i \(0.368284\pi\)
\(168\) 0 0
\(169\) 16.8564 1.29665
\(170\) 0 0
\(171\) −5.46410 −0.417850
\(172\) 0 0
\(173\) −12.0000 −0.912343 −0.456172 0.889892i \(-0.650780\pi\)
−0.456172 + 0.889892i \(0.650780\pi\)
\(174\) 0 0
\(175\) −2.00000 −0.151186
\(176\) 0 0
\(177\) −6.92820 −0.520756
\(178\) 0 0
\(179\) −6.92820 −0.517838 −0.258919 0.965899i \(-0.583366\pi\)
−0.258919 + 0.965899i \(0.583366\pi\)
\(180\) 0 0
\(181\) 15.8564 1.17860 0.589299 0.807915i \(-0.299404\pi\)
0.589299 + 0.807915i \(0.299404\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) 0 0
\(185\) 4.92820 0.362329
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) 18.9282 1.36960 0.684798 0.728733i \(-0.259890\pi\)
0.684798 + 0.728733i \(0.259890\pi\)
\(192\) 0 0
\(193\) 24.3923 1.75580 0.877898 0.478847i \(-0.158945\pi\)
0.877898 + 0.478847i \(0.158945\pi\)
\(194\) 0 0
\(195\) 5.46410 0.391292
\(196\) 0 0
\(197\) −12.0000 −0.854965 −0.427482 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) 0 0
\(199\) 24.7846 1.75693 0.878467 0.477803i \(-0.158567\pi\)
0.878467 + 0.477803i \(0.158567\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) 0 0
\(203\) 6.92820 0.486265
\(204\) 0 0
\(205\) −3.46410 −0.241943
\(206\) 0 0
\(207\) −6.92820 −0.481543
\(208\) 0 0
\(209\) −5.46410 −0.377960
\(210\) 0 0
\(211\) 8.39230 0.577750 0.288875 0.957367i \(-0.406719\pi\)
0.288875 + 0.957367i \(0.406719\pi\)
\(212\) 0 0
\(213\) 13.8564 0.949425
\(214\) 0 0
\(215\) −4.92820 −0.336101
\(216\) 0 0
\(217\) −21.8564 −1.48371
\(218\) 0 0
\(219\) 8.39230 0.567099
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −9.85641 −0.660034 −0.330017 0.943975i \(-0.607054\pi\)
−0.330017 + 0.943975i \(0.607054\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −15.4641 −1.02639 −0.513194 0.858272i \(-0.671538\pi\)
−0.513194 + 0.858272i \(0.671538\pi\)
\(228\) 0 0
\(229\) −23.8564 −1.57648 −0.788238 0.615371i \(-0.789006\pi\)
−0.788238 + 0.615371i \(0.789006\pi\)
\(230\) 0 0
\(231\) 2.00000 0.131590
\(232\) 0 0
\(233\) 12.0000 0.786146 0.393073 0.919507i \(-0.371412\pi\)
0.393073 + 0.919507i \(0.371412\pi\)
\(234\) 0 0
\(235\) −6.92820 −0.451946
\(236\) 0 0
\(237\) −6.53590 −0.424552
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) 0.143594 0.00924967 0.00462484 0.999989i \(-0.498528\pi\)
0.00462484 + 0.999989i \(0.498528\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 3.00000 0.191663
\(246\) 0 0
\(247\) −29.8564 −1.89972
\(248\) 0 0
\(249\) 8.53590 0.540941
\(250\) 0 0
\(251\) 1.85641 0.117175 0.0585877 0.998282i \(-0.481340\pi\)
0.0585877 + 0.998282i \(0.481340\pi\)
\(252\) 0 0
\(253\) −6.92820 −0.435572
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −19.8564 −1.23861 −0.619304 0.785151i \(-0.712585\pi\)
−0.619304 + 0.785151i \(0.712585\pi\)
\(258\) 0 0
\(259\) 9.85641 0.612447
\(260\) 0 0
\(261\) −3.46410 −0.214423
\(262\) 0 0
\(263\) −20.5359 −1.26630 −0.633149 0.774030i \(-0.718238\pi\)
−0.633149 + 0.774030i \(0.718238\pi\)
\(264\) 0 0
\(265\) −0.928203 −0.0570191
\(266\) 0 0
\(267\) −0.928203 −0.0568051
\(268\) 0 0
\(269\) 19.8564 1.21067 0.605333 0.795972i \(-0.293040\pi\)
0.605333 + 0.795972i \(0.293040\pi\)
\(270\) 0 0
\(271\) 11.6077 0.705117 0.352559 0.935790i \(-0.385312\pi\)
0.352559 + 0.935790i \(0.385312\pi\)
\(272\) 0 0
\(273\) 10.9282 0.661405
\(274\) 0 0
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) 29.4641 1.77033 0.885163 0.465281i \(-0.154047\pi\)
0.885163 + 0.465281i \(0.154047\pi\)
\(278\) 0 0
\(279\) 10.9282 0.654254
\(280\) 0 0
\(281\) 3.46410 0.206651 0.103325 0.994648i \(-0.467052\pi\)
0.103325 + 0.994648i \(0.467052\pi\)
\(282\) 0 0
\(283\) 4.92820 0.292951 0.146476 0.989214i \(-0.453207\pi\)
0.146476 + 0.989214i \(0.453207\pi\)
\(284\) 0 0
\(285\) −5.46410 −0.323665
\(286\) 0 0
\(287\) −6.92820 −0.408959
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 10.0000 0.586210
\(292\) 0 0
\(293\) −13.8564 −0.809500 −0.404750 0.914427i \(-0.632641\pi\)
−0.404750 + 0.914427i \(0.632641\pi\)
\(294\) 0 0
\(295\) −6.92820 −0.403376
\(296\) 0 0
\(297\) −1.00000 −0.0580259
\(298\) 0 0
\(299\) −37.8564 −2.18929
\(300\) 0 0
\(301\) −9.85641 −0.568114
\(302\) 0 0
\(303\) 10.3923 0.597022
\(304\) 0 0
\(305\) −2.00000 −0.114520
\(306\) 0 0
\(307\) −14.0000 −0.799022 −0.399511 0.916728i \(-0.630820\pi\)
−0.399511 + 0.916728i \(0.630820\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −5.07180 −0.287595 −0.143798 0.989607i \(-0.545931\pi\)
−0.143798 + 0.989607i \(0.545931\pi\)
\(312\) 0 0
\(313\) 20.9282 1.18293 0.591466 0.806330i \(-0.298549\pi\)
0.591466 + 0.806330i \(0.298549\pi\)
\(314\) 0 0
\(315\) 2.00000 0.112687
\(316\) 0 0
\(317\) −24.9282 −1.40011 −0.700054 0.714090i \(-0.746841\pi\)
−0.700054 + 0.714090i \(0.746841\pi\)
\(318\) 0 0
\(319\) −3.46410 −0.193952
\(320\) 0 0
\(321\) 8.53590 0.476427
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 5.46410 0.303094
\(326\) 0 0
\(327\) 10.0000 0.553001
\(328\) 0 0
\(329\) −13.8564 −0.763928
\(330\) 0 0
\(331\) −9.85641 −0.541757 −0.270879 0.962614i \(-0.587314\pi\)
−0.270879 + 0.962614i \(0.587314\pi\)
\(332\) 0 0
\(333\) −4.92820 −0.270064
\(334\) 0 0
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) 33.1769 1.80726 0.903631 0.428312i \(-0.140892\pi\)
0.903631 + 0.428312i \(0.140892\pi\)
\(338\) 0 0
\(339\) 12.9282 0.702164
\(340\) 0 0
\(341\) 10.9282 0.591795
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 0 0
\(345\) −6.92820 −0.373002
\(346\) 0 0
\(347\) −22.3923 −1.20208 −0.601041 0.799218i \(-0.705247\pi\)
−0.601041 + 0.799218i \(0.705247\pi\)
\(348\) 0 0
\(349\) −8.14359 −0.435917 −0.217958 0.975958i \(-0.569940\pi\)
−0.217958 + 0.975958i \(0.569940\pi\)
\(350\) 0 0
\(351\) −5.46410 −0.291652
\(352\) 0 0
\(353\) 12.9282 0.688099 0.344049 0.938952i \(-0.388201\pi\)
0.344049 + 0.938952i \(0.388201\pi\)
\(354\) 0 0
\(355\) 13.8564 0.735422
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 20.7846 1.09697 0.548485 0.836160i \(-0.315205\pi\)
0.548485 + 0.836160i \(0.315205\pi\)
\(360\) 0 0
\(361\) 10.8564 0.571390
\(362\) 0 0
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) 8.39230 0.439273
\(366\) 0 0
\(367\) −20.0000 −1.04399 −0.521996 0.852948i \(-0.674812\pi\)
−0.521996 + 0.852948i \(0.674812\pi\)
\(368\) 0 0
\(369\) 3.46410 0.180334
\(370\) 0 0
\(371\) −1.85641 −0.0963798
\(372\) 0 0
\(373\) −20.3923 −1.05587 −0.527937 0.849284i \(-0.677034\pi\)
−0.527937 + 0.849284i \(0.677034\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −18.9282 −0.974852
\(378\) 0 0
\(379\) 17.8564 0.917222 0.458611 0.888637i \(-0.348347\pi\)
0.458611 + 0.888637i \(0.348347\pi\)
\(380\) 0 0
\(381\) 8.92820 0.457406
\(382\) 0 0
\(383\) 13.8564 0.708029 0.354015 0.935240i \(-0.384816\pi\)
0.354015 + 0.935240i \(0.384816\pi\)
\(384\) 0 0
\(385\) 2.00000 0.101929
\(386\) 0 0
\(387\) 4.92820 0.250515
\(388\) 0 0
\(389\) 11.0718 0.561362 0.280681 0.959801i \(-0.409440\pi\)
0.280681 + 0.959801i \(0.409440\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 18.9282 0.954802
\(394\) 0 0
\(395\) −6.53590 −0.328857
\(396\) 0 0
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 0 0
\(399\) −10.9282 −0.547094
\(400\) 0 0
\(401\) −7.85641 −0.392330 −0.196165 0.980571i \(-0.562849\pi\)
−0.196165 + 0.980571i \(0.562849\pi\)
\(402\) 0 0
\(403\) 59.7128 2.97451
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −4.92820 −0.244282
\(408\) 0 0
\(409\) −6.78461 −0.335477 −0.167739 0.985831i \(-0.553646\pi\)
−0.167739 + 0.985831i \(0.553646\pi\)
\(410\) 0 0
\(411\) 18.0000 0.887875
\(412\) 0 0
\(413\) −13.8564 −0.681829
\(414\) 0 0
\(415\) 8.53590 0.419011
\(416\) 0 0
\(417\) 12.3923 0.606854
\(418\) 0 0
\(419\) −30.9282 −1.51094 −0.755471 0.655182i \(-0.772592\pi\)
−0.755471 + 0.655182i \(0.772592\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) 0 0
\(423\) 6.92820 0.336861
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −4.00000 −0.193574
\(428\) 0 0
\(429\) −5.46410 −0.263809
\(430\) 0 0
\(431\) −8.78461 −0.423140 −0.211570 0.977363i \(-0.567858\pi\)
−0.211570 + 0.977363i \(0.567858\pi\)
\(432\) 0 0
\(433\) 0.143594 0.00690067 0.00345033 0.999994i \(-0.498902\pi\)
0.00345033 + 0.999994i \(0.498902\pi\)
\(434\) 0 0
\(435\) −3.46410 −0.166091
\(436\) 0 0
\(437\) 37.8564 1.81092
\(438\) 0 0
\(439\) −33.1769 −1.58345 −0.791724 0.610879i \(-0.790816\pi\)
−0.791724 + 0.610879i \(0.790816\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 0 0
\(445\) −0.928203 −0.0440011
\(446\) 0 0
\(447\) 15.4641 0.731427
\(448\) 0 0
\(449\) −26.7846 −1.26404 −0.632022 0.774950i \(-0.717775\pi\)
−0.632022 + 0.774950i \(0.717775\pi\)
\(450\) 0 0
\(451\) 3.46410 0.163118
\(452\) 0 0
\(453\) −20.3923 −0.958114
\(454\) 0 0
\(455\) 10.9282 0.512322
\(456\) 0 0
\(457\) 12.3923 0.579688 0.289844 0.957074i \(-0.406397\pi\)
0.289844 + 0.957074i \(0.406397\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 36.2487 1.68827 0.844135 0.536130i \(-0.180114\pi\)
0.844135 + 0.536130i \(0.180114\pi\)
\(462\) 0 0
\(463\) 28.0000 1.30127 0.650635 0.759390i \(-0.274503\pi\)
0.650635 + 0.759390i \(0.274503\pi\)
\(464\) 0 0
\(465\) 10.9282 0.506783
\(466\) 0 0
\(467\) −5.07180 −0.234695 −0.117347 0.993091i \(-0.537439\pi\)
−0.117347 + 0.993091i \(0.537439\pi\)
\(468\) 0 0
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) 3.07180 0.141541
\(472\) 0 0
\(473\) 4.92820 0.226599
\(474\) 0 0
\(475\) −5.46410 −0.250710
\(476\) 0 0
\(477\) 0.928203 0.0424995
\(478\) 0 0
\(479\) −12.0000 −0.548294 −0.274147 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) 0 0
\(481\) −26.9282 −1.22782
\(482\) 0 0
\(483\) −13.8564 −0.630488
\(484\) 0 0
\(485\) 10.0000 0.454077
\(486\) 0 0
\(487\) 31.7128 1.43704 0.718522 0.695504i \(-0.244819\pi\)
0.718522 + 0.695504i \(0.244819\pi\)
\(488\) 0 0
\(489\) 9.85641 0.445722
\(490\) 0 0
\(491\) 30.9282 1.39577 0.697885 0.716210i \(-0.254125\pi\)
0.697885 + 0.716210i \(0.254125\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −1.00000 −0.0449467
\(496\) 0 0
\(497\) 27.7128 1.24309
\(498\) 0 0
\(499\) −28.7846 −1.28858 −0.644288 0.764783i \(-0.722846\pi\)
−0.644288 + 0.764783i \(0.722846\pi\)
\(500\) 0 0
\(501\) −10.3923 −0.464294
\(502\) 0 0
\(503\) −31.1769 −1.39011 −0.695055 0.718957i \(-0.744620\pi\)
−0.695055 + 0.718957i \(0.744620\pi\)
\(504\) 0 0
\(505\) 10.3923 0.462451
\(506\) 0 0
\(507\) −16.8564 −0.748619
\(508\) 0 0
\(509\) 19.8564 0.880120 0.440060 0.897968i \(-0.354957\pi\)
0.440060 + 0.897968i \(0.354957\pi\)
\(510\) 0 0
\(511\) 16.7846 0.742507
\(512\) 0 0
\(513\) 5.46410 0.241246
\(514\) 0 0
\(515\) 8.00000 0.352522
\(516\) 0 0
\(517\) 6.92820 0.304702
\(518\) 0 0
\(519\) 12.0000 0.526742
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 0 0
\(523\) 22.0000 0.961993 0.480996 0.876723i \(-0.340275\pi\)
0.480996 + 0.876723i \(0.340275\pi\)
\(524\) 0 0
\(525\) 2.00000 0.0872872
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 25.0000 1.08696
\(530\) 0 0
\(531\) 6.92820 0.300658
\(532\) 0 0
\(533\) 18.9282 0.819871
\(534\) 0 0
\(535\) 8.53590 0.369039
\(536\) 0 0
\(537\) 6.92820 0.298974
\(538\) 0 0
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) 27.8564 1.19764 0.598820 0.800883i \(-0.295636\pi\)
0.598820 + 0.800883i \(0.295636\pi\)
\(542\) 0 0
\(543\) −15.8564 −0.680464
\(544\) 0 0
\(545\) 10.0000 0.428353
\(546\) 0 0
\(547\) −2.00000 −0.0855138 −0.0427569 0.999086i \(-0.513614\pi\)
−0.0427569 + 0.999086i \(0.513614\pi\)
\(548\) 0 0
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) 18.9282 0.806369
\(552\) 0 0
\(553\) −13.0718 −0.555869
\(554\) 0 0
\(555\) −4.92820 −0.209191
\(556\) 0 0
\(557\) 3.21539 0.136240 0.0681202 0.997677i \(-0.478300\pi\)
0.0681202 + 0.997677i \(0.478300\pi\)
\(558\) 0 0
\(559\) 26.9282 1.13894
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −10.3923 −0.437983 −0.218992 0.975727i \(-0.570277\pi\)
−0.218992 + 0.975727i \(0.570277\pi\)
\(564\) 0 0
\(565\) 12.9282 0.543894
\(566\) 0 0
\(567\) −2.00000 −0.0839921
\(568\) 0 0
\(569\) 5.32051 0.223047 0.111524 0.993762i \(-0.464427\pi\)
0.111524 + 0.993762i \(0.464427\pi\)
\(570\) 0 0
\(571\) −3.60770 −0.150977 −0.0754887 0.997147i \(-0.524052\pi\)
−0.0754887 + 0.997147i \(0.524052\pi\)
\(572\) 0 0
\(573\) −18.9282 −0.790737
\(574\) 0 0
\(575\) −6.92820 −0.288926
\(576\) 0 0
\(577\) −18.7846 −0.782014 −0.391007 0.920388i \(-0.627873\pi\)
−0.391007 + 0.920388i \(0.627873\pi\)
\(578\) 0 0
\(579\) −24.3923 −1.01371
\(580\) 0 0
\(581\) 17.0718 0.708257
\(582\) 0 0
\(583\) 0.928203 0.0384422
\(584\) 0 0
\(585\) −5.46410 −0.225913
\(586\) 0 0
\(587\) 18.9282 0.781251 0.390625 0.920550i \(-0.372259\pi\)
0.390625 + 0.920550i \(0.372259\pi\)
\(588\) 0 0
\(589\) −59.7128 −2.46042
\(590\) 0 0
\(591\) 12.0000 0.493614
\(592\) 0 0
\(593\) 8.78461 0.360741 0.180370 0.983599i \(-0.442270\pi\)
0.180370 + 0.983599i \(0.442270\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −24.7846 −1.01437
\(598\) 0 0
\(599\) 37.8564 1.54677 0.773385 0.633936i \(-0.218562\pi\)
0.773385 + 0.633936i \(0.218562\pi\)
\(600\) 0 0
\(601\) −32.6410 −1.33145 −0.665727 0.746195i \(-0.731879\pi\)
−0.665727 + 0.746195i \(0.731879\pi\)
\(602\) 0 0
\(603\) −8.00000 −0.325785
\(604\) 0 0
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) 18.7846 0.762444 0.381222 0.924484i \(-0.375503\pi\)
0.381222 + 0.924484i \(0.375503\pi\)
\(608\) 0 0
\(609\) −6.92820 −0.280745
\(610\) 0 0
\(611\) 37.8564 1.53151
\(612\) 0 0
\(613\) −20.3923 −0.823637 −0.411819 0.911266i \(-0.635106\pi\)
−0.411819 + 0.911266i \(0.635106\pi\)
\(614\) 0 0
\(615\) 3.46410 0.139686
\(616\) 0 0
\(617\) −36.9282 −1.48667 −0.743337 0.668917i \(-0.766758\pi\)
−0.743337 + 0.668917i \(0.766758\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) 6.92820 0.278019
\(622\) 0 0
\(623\) −1.85641 −0.0743754
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 5.46410 0.218215
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 21.0718 0.838855 0.419427 0.907789i \(-0.362231\pi\)
0.419427 + 0.907789i \(0.362231\pi\)
\(632\) 0 0
\(633\) −8.39230 −0.333564
\(634\) 0 0
\(635\) 8.92820 0.354305
\(636\) 0 0
\(637\) −16.3923 −0.649487
\(638\) 0 0
\(639\) −13.8564 −0.548151
\(640\) 0 0
\(641\) −12.9282 −0.510633 −0.255317 0.966857i \(-0.582180\pi\)
−0.255317 + 0.966857i \(0.582180\pi\)
\(642\) 0 0
\(643\) −37.5692 −1.48159 −0.740793 0.671734i \(-0.765550\pi\)
−0.740793 + 0.671734i \(0.765550\pi\)
\(644\) 0 0
\(645\) 4.92820 0.194048
\(646\) 0 0
\(647\) −27.7128 −1.08950 −0.544752 0.838597i \(-0.683376\pi\)
−0.544752 + 0.838597i \(0.683376\pi\)
\(648\) 0 0
\(649\) 6.92820 0.271956
\(650\) 0 0
\(651\) 21.8564 0.856620
\(652\) 0 0
\(653\) 19.8564 0.777041 0.388521 0.921440i \(-0.372986\pi\)
0.388521 + 0.921440i \(0.372986\pi\)
\(654\) 0 0
\(655\) 18.9282 0.739586
\(656\) 0 0
\(657\) −8.39230 −0.327415
\(658\) 0 0
\(659\) 15.7128 0.612084 0.306042 0.952018i \(-0.400995\pi\)
0.306042 + 0.952018i \(0.400995\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −10.9282 −0.423778
\(666\) 0 0
\(667\) 24.0000 0.929284
\(668\) 0 0
\(669\) 9.85641 0.381071
\(670\) 0 0
\(671\) 2.00000 0.0772091
\(672\) 0 0
\(673\) −3.32051 −0.127996 −0.0639981 0.997950i \(-0.520385\pi\)
−0.0639981 + 0.997950i \(0.520385\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 8.78461 0.337620 0.168810 0.985649i \(-0.446008\pi\)
0.168810 + 0.985649i \(0.446008\pi\)
\(678\) 0 0
\(679\) 20.0000 0.767530
\(680\) 0 0
\(681\) 15.4641 0.592586
\(682\) 0 0
\(683\) 32.7846 1.25447 0.627234 0.778831i \(-0.284187\pi\)
0.627234 + 0.778831i \(0.284187\pi\)
\(684\) 0 0
\(685\) 18.0000 0.687745
\(686\) 0 0
\(687\) 23.8564 0.910179
\(688\) 0 0
\(689\) 5.07180 0.193220
\(690\) 0 0
\(691\) −47.7128 −1.81508 −0.907540 0.419965i \(-0.862042\pi\)
−0.907540 + 0.419965i \(0.862042\pi\)
\(692\) 0 0
\(693\) −2.00000 −0.0759737
\(694\) 0 0
\(695\) 12.3923 0.470067
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −12.0000 −0.453882
\(700\) 0 0
\(701\) 39.4641 1.49054 0.745269 0.666764i \(-0.232321\pi\)
0.745269 + 0.666764i \(0.232321\pi\)
\(702\) 0 0
\(703\) 26.9282 1.01562
\(704\) 0 0
\(705\) 6.92820 0.260931
\(706\) 0 0
\(707\) 20.7846 0.781686
\(708\) 0 0
\(709\) −11.8564 −0.445277 −0.222638 0.974901i \(-0.571467\pi\)
−0.222638 + 0.974901i \(0.571467\pi\)
\(710\) 0 0
\(711\) 6.53590 0.245115
\(712\) 0 0
\(713\) −75.7128 −2.83547
\(714\) 0 0
\(715\) −5.46410 −0.204346
\(716\) 0 0
\(717\) −12.0000 −0.448148
\(718\) 0 0
\(719\) 5.07180 0.189146 0.0945731 0.995518i \(-0.469851\pi\)
0.0945731 + 0.995518i \(0.469851\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 0 0
\(723\) −0.143594 −0.00534030
\(724\) 0 0
\(725\) −3.46410 −0.128654
\(726\) 0 0
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 53.9615 1.99311 0.996557 0.0829082i \(-0.0264208\pi\)
0.996557 + 0.0829082i \(0.0264208\pi\)
\(734\) 0 0
\(735\) −3.00000 −0.110657
\(736\) 0 0
\(737\) −8.00000 −0.294684
\(738\) 0 0
\(739\) −17.4641 −0.642427 −0.321214 0.947007i \(-0.604091\pi\)
−0.321214 + 0.947007i \(0.604091\pi\)
\(740\) 0 0
\(741\) 29.8564 1.09680
\(742\) 0 0
\(743\) −25.6077 −0.939455 −0.469728 0.882811i \(-0.655648\pi\)
−0.469728 + 0.882811i \(0.655648\pi\)
\(744\) 0 0
\(745\) 15.4641 0.566561
\(746\) 0 0
\(747\) −8.53590 −0.312312
\(748\) 0 0
\(749\) 17.0718 0.623790
\(750\) 0 0
\(751\) −26.9282 −0.982624 −0.491312 0.870984i \(-0.663483\pi\)
−0.491312 + 0.870984i \(0.663483\pi\)
\(752\) 0 0
\(753\) −1.85641 −0.0676512
\(754\) 0 0
\(755\) −20.3923 −0.742152
\(756\) 0 0
\(757\) 34.7846 1.26427 0.632134 0.774859i \(-0.282179\pi\)
0.632134 + 0.774859i \(0.282179\pi\)
\(758\) 0 0
\(759\) 6.92820 0.251478
\(760\) 0 0
\(761\) −32.5359 −1.17943 −0.589713 0.807613i \(-0.700759\pi\)
−0.589713 + 0.807613i \(0.700759\pi\)
\(762\) 0 0
\(763\) 20.0000 0.724049
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 37.8564 1.36692
\(768\) 0 0
\(769\) 50.4974 1.82098 0.910492 0.413527i \(-0.135703\pi\)
0.910492 + 0.413527i \(0.135703\pi\)
\(770\) 0 0
\(771\) 19.8564 0.715111
\(772\) 0 0
\(773\) −4.14359 −0.149035 −0.0745174 0.997220i \(-0.523742\pi\)
−0.0745174 + 0.997220i \(0.523742\pi\)
\(774\) 0 0
\(775\) 10.9282 0.392553
\(776\) 0 0
\(777\) −9.85641 −0.353597
\(778\) 0 0
\(779\) −18.9282 −0.678173
\(780\) 0 0
\(781\) −13.8564 −0.495821
\(782\) 0 0
\(783\) 3.46410 0.123797
\(784\) 0 0
\(785\) 3.07180 0.109637
\(786\) 0 0
\(787\) −22.7846 −0.812184 −0.406092 0.913832i \(-0.633109\pi\)
−0.406092 + 0.913832i \(0.633109\pi\)
\(788\) 0 0
\(789\) 20.5359 0.731097
\(790\) 0 0
\(791\) 25.8564 0.919348
\(792\) 0 0
\(793\) 10.9282 0.388072
\(794\) 0 0
\(795\) 0.928203 0.0329200
\(796\) 0 0
\(797\) −52.6410 −1.86464 −0.932320 0.361634i \(-0.882219\pi\)
−0.932320 + 0.361634i \(0.882219\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0.928203 0.0327964
\(802\) 0 0
\(803\) −8.39230 −0.296158
\(804\) 0 0
\(805\) −13.8564 −0.488374
\(806\) 0 0
\(807\) −19.8564 −0.698979
\(808\) 0 0
\(809\) −15.4641 −0.543689 −0.271844 0.962341i \(-0.587634\pi\)
−0.271844 + 0.962341i \(0.587634\pi\)
\(810\) 0 0
\(811\) −12.3923 −0.435153 −0.217576 0.976043i \(-0.569815\pi\)
−0.217576 + 0.976043i \(0.569815\pi\)
\(812\) 0 0
\(813\) −11.6077 −0.407100
\(814\) 0 0
\(815\) 9.85641 0.345255
\(816\) 0 0
\(817\) −26.9282 −0.942099
\(818\) 0 0
\(819\) −10.9282 −0.381862
\(820\) 0 0
\(821\) 20.5359 0.716708 0.358354 0.933586i \(-0.383338\pi\)
0.358354 + 0.933586i \(0.383338\pi\)
\(822\) 0 0
\(823\) 33.5692 1.17015 0.585075 0.810979i \(-0.301065\pi\)
0.585075 + 0.810979i \(0.301065\pi\)
\(824\) 0 0
\(825\) −1.00000 −0.0348155
\(826\) 0 0
\(827\) −22.3923 −0.778657 −0.389328 0.921099i \(-0.627293\pi\)
−0.389328 + 0.921099i \(0.627293\pi\)
\(828\) 0 0
\(829\) 29.7128 1.03197 0.515984 0.856598i \(-0.327426\pi\)
0.515984 + 0.856598i \(0.327426\pi\)
\(830\) 0 0
\(831\) −29.4641 −1.02210
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −10.3923 −0.359641
\(836\) 0 0
\(837\) −10.9282 −0.377734
\(838\) 0 0
\(839\) −56.7846 −1.96042 −0.980211 0.197954i \(-0.936570\pi\)
−0.980211 + 0.197954i \(0.936570\pi\)
\(840\) 0 0
\(841\) −17.0000 −0.586207
\(842\) 0 0
\(843\) −3.46410 −0.119310
\(844\) 0 0
\(845\) −16.8564 −0.579878
\(846\) 0 0
\(847\) −2.00000 −0.0687208
\(848\) 0 0
\(849\) −4.92820 −0.169135
\(850\) 0 0
\(851\) 34.1436 1.17043
\(852\) 0 0
\(853\) 3.60770 0.123525 0.0617626 0.998091i \(-0.480328\pi\)
0.0617626 + 0.998091i \(0.480328\pi\)
\(854\) 0 0
\(855\) 5.46410 0.186868
\(856\) 0 0
\(857\) −37.8564 −1.29315 −0.646575 0.762850i \(-0.723799\pi\)
−0.646575 + 0.762850i \(0.723799\pi\)
\(858\) 0 0
\(859\) 7.71281 0.263158 0.131579 0.991306i \(-0.457995\pi\)
0.131579 + 0.991306i \(0.457995\pi\)
\(860\) 0 0
\(861\) 6.92820 0.236113
\(862\) 0 0
\(863\) 37.8564 1.28865 0.644324 0.764753i \(-0.277139\pi\)
0.644324 + 0.764753i \(0.277139\pi\)
\(864\) 0 0
\(865\) 12.0000 0.408012
\(866\) 0 0
\(867\) 17.0000 0.577350
\(868\) 0 0
\(869\) 6.53590 0.221715
\(870\) 0 0
\(871\) −43.7128 −1.48115
\(872\) 0 0
\(873\) −10.0000 −0.338449
\(874\) 0 0
\(875\) 2.00000 0.0676123
\(876\) 0 0
\(877\) −34.2487 −1.15650 −0.578248 0.815861i \(-0.696264\pi\)
−0.578248 + 0.815861i \(0.696264\pi\)
\(878\) 0 0
\(879\) 13.8564 0.467365
\(880\) 0 0
\(881\) −0.928203 −0.0312720 −0.0156360 0.999878i \(-0.504977\pi\)
−0.0156360 + 0.999878i \(0.504977\pi\)
\(882\) 0 0
\(883\) 4.00000 0.134611 0.0673054 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) 0 0
\(885\) 6.92820 0.232889
\(886\) 0 0
\(887\) −12.2487 −0.411271 −0.205636 0.978629i \(-0.565926\pi\)
−0.205636 + 0.978629i \(0.565926\pi\)
\(888\) 0 0
\(889\) 17.8564 0.598885
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 0 0
\(893\) −37.8564 −1.26682
\(894\) 0 0
\(895\) 6.92820 0.231584
\(896\) 0 0
\(897\) 37.8564 1.26399
\(898\) 0 0
\(899\) −37.8564 −1.26258
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 9.85641 0.328001
\(904\) 0 0
\(905\) −15.8564 −0.527085
\(906\) 0 0
\(907\) −18.1436 −0.602448 −0.301224 0.953553i \(-0.597395\pi\)
−0.301224 + 0.953553i \(0.597395\pi\)
\(908\) 0 0
\(909\) −10.3923 −0.344691
\(910\) 0 0
\(911\) −18.9282 −0.627119 −0.313560 0.949568i \(-0.601522\pi\)
−0.313560 + 0.949568i \(0.601522\pi\)
\(912\) 0 0
\(913\) −8.53590 −0.282497
\(914\) 0 0
\(915\) 2.00000 0.0661180
\(916\) 0 0
\(917\) 37.8564 1.25013
\(918\) 0 0
\(919\) 32.3923 1.06852 0.534262 0.845319i \(-0.320590\pi\)
0.534262 + 0.845319i \(0.320590\pi\)
\(920\) 0 0
\(921\) 14.0000 0.461316
\(922\) 0 0
\(923\) −75.7128 −2.49212
\(924\) 0 0
\(925\) −4.92820 −0.162038
\(926\) 0 0
\(927\) −8.00000 −0.262754
\(928\) 0 0
\(929\) 2.78461 0.0913601 0.0456800 0.998956i \(-0.485455\pi\)
0.0456800 + 0.998956i \(0.485455\pi\)
\(930\) 0 0
\(931\) 16.3923 0.537236
\(932\) 0 0
\(933\) 5.07180 0.166043
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −20.3923 −0.666188 −0.333094 0.942894i \(-0.608093\pi\)
−0.333094 + 0.942894i \(0.608093\pi\)
\(938\) 0 0
\(939\) −20.9282 −0.682966
\(940\) 0 0
\(941\) 27.4641 0.895304 0.447652 0.894208i \(-0.352260\pi\)
0.447652 + 0.894208i \(0.352260\pi\)
\(942\) 0 0
\(943\) −24.0000 −0.781548
\(944\) 0 0
\(945\) −2.00000 −0.0650600
\(946\) 0 0
\(947\) 18.9282 0.615084 0.307542 0.951535i \(-0.400494\pi\)
0.307542 + 0.951535i \(0.400494\pi\)
\(948\) 0 0
\(949\) −45.8564 −1.48856
\(950\) 0 0
\(951\) 24.9282 0.808352
\(952\) 0 0
\(953\) −3.21539 −0.104157 −0.0520784 0.998643i \(-0.516585\pi\)
−0.0520784 + 0.998643i \(0.516585\pi\)
\(954\) 0 0
\(955\) −18.9282 −0.612502
\(956\) 0 0
\(957\) 3.46410 0.111979
\(958\) 0 0
\(959\) 36.0000 1.16250
\(960\) 0 0
\(961\) 88.4256 2.85244
\(962\) 0 0
\(963\) −8.53590 −0.275065
\(964\) 0 0
\(965\) −24.3923 −0.785216
\(966\) 0 0
\(967\) −22.7846 −0.732704 −0.366352 0.930476i \(-0.619393\pi\)
−0.366352 + 0.930476i \(0.619393\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −25.8564 −0.829772 −0.414886 0.909873i \(-0.636178\pi\)
−0.414886 + 0.909873i \(0.636178\pi\)
\(972\) 0 0
\(973\) 24.7846 0.794558
\(974\) 0 0
\(975\) −5.46410 −0.174991
\(976\) 0 0
\(977\) 47.5692 1.52187 0.760937 0.648826i \(-0.224740\pi\)
0.760937 + 0.648826i \(0.224740\pi\)
\(978\) 0 0
\(979\) 0.928203 0.0296655
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) 0 0
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 0 0
\(985\) 12.0000 0.382352
\(986\) 0 0
\(987\) 13.8564 0.441054
\(988\) 0 0
\(989\) −34.1436 −1.08570
\(990\) 0 0
\(991\) 7.21539 0.229204 0.114602 0.993411i \(-0.463441\pi\)
0.114602 + 0.993411i \(0.463441\pi\)
\(992\) 0 0
\(993\) 9.85641 0.312784
\(994\) 0 0
\(995\) −24.7846 −0.785725
\(996\) 0 0
\(997\) 27.6077 0.874344 0.437172 0.899378i \(-0.355980\pi\)
0.437172 + 0.899378i \(0.355980\pi\)
\(998\) 0 0
\(999\) 4.92820 0.155921
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 2640.2.a.x.1.2 2
3.2 odd 2 7920.2.a.bz.1.2 2
4.3 odd 2 165.2.a.b.1.1 2
12.11 even 2 495.2.a.c.1.2 2
20.3 even 4 825.2.c.c.199.4 4
20.7 even 4 825.2.c.c.199.1 4
20.19 odd 2 825.2.a.e.1.2 2
28.27 even 2 8085.2.a.bd.1.1 2
44.43 even 2 1815.2.a.i.1.2 2
60.23 odd 4 2475.2.c.n.199.1 4
60.47 odd 4 2475.2.c.n.199.4 4
60.59 even 2 2475.2.a.r.1.1 2
132.131 odd 2 5445.2.a.s.1.1 2
220.219 even 2 9075.2.a.bh.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.b.1.1 2 4.3 odd 2
495.2.a.c.1.2 2 12.11 even 2
825.2.a.e.1.2 2 20.19 odd 2
825.2.c.c.199.1 4 20.7 even 4
825.2.c.c.199.4 4 20.3 even 4
1815.2.a.i.1.2 2 44.43 even 2
2475.2.a.r.1.1 2 60.59 even 2
2475.2.c.n.199.1 4 60.23 odd 4
2475.2.c.n.199.4 4 60.47 odd 4
2640.2.a.x.1.2 2 1.1 even 1 trivial
5445.2.a.s.1.1 2 132.131 odd 2
7920.2.a.bz.1.2 2 3.2 odd 2
8085.2.a.bd.1.1 2 28.27 even 2
9075.2.a.bh.1.1 2 220.219 even 2