Properties

Label 2760.2.a.v.1.2
Level $2760$
Weight $2$
Character 2760.1
Self dual yes
Analytic conductor $22.039$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2760.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.0387109579\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.54764.1
Defining polynomial: \( x^{4} - x^{3} - 9x^{2} + 3x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.339102\) of defining polynomial
Character \(\chi\) \(=\) 2760.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} -0.845563 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} -0.845563 q^{7} +1.00000 q^{9} -4.55883 q^{11} +5.23704 q^{13} +1.00000 q^{15} +4.72619 q^{17} -5.54591 q^{19} +0.845563 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +7.06968 q^{29} -1.83264 q^{31} +4.55883 q^{33} +0.845563 q^{35} -9.93738 q^{37} -5.23704 q^{39} +5.71327 q^{41} +4.78294 q^{43} -1.00000 q^{45} +5.54591 q^{47} -6.28502 q^{49} -4.72619 q^{51} -14.4962 q^{53} +4.55883 q^{55} +5.54591 q^{57} +5.06968 q^{59} -6.22411 q^{61} -0.845563 q^{63} -5.23704 q^{65} -7.85849 q^{67} -1.00000 q^{69} +3.71327 q^{71} +1.23704 q^{73} -1.00000 q^{75} +3.85478 q^{77} -13.7700 q^{79} +1.00000 q^{81} +7.50913 q^{83} -4.72619 q^{85} -7.06968 q^{87} +2.98708 q^{89} -4.42825 q^{91} +1.83264 q^{93} +5.54591 q^{95} -2.33472 q^{97} -4.55883 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 4 q^{5} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} - 4 q^{5} + 4 q^{9} - 2 q^{13} + 4 q^{15} + 2 q^{17} - 6 q^{19} + 4 q^{23} + 4 q^{25} - 4 q^{27} + 4 q^{29} - 6 q^{31} - 4 q^{37} + 2 q^{39} + 8 q^{41} - 20 q^{43} - 4 q^{45} + 6 q^{47} + 10 q^{49} - 2 q^{51} - 4 q^{53} + 6 q^{57} - 4 q^{59} - 4 q^{61} + 2 q^{65} - 26 q^{67} - 4 q^{69} - 18 q^{73} - 4 q^{75} + 6 q^{77} - 18 q^{79} + 4 q^{81} - 26 q^{83} - 2 q^{85} - 4 q^{87} + 14 q^{89} - 38 q^{91} + 6 q^{93} + 6 q^{95} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(7\) −0.845563 −0.319593 −0.159796 0.987150i \(-0.551084\pi\)
−0.159796 + 0.987150i \(0.551084\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.55883 −1.37454 −0.687270 0.726402i \(-0.741191\pi\)
−0.687270 + 0.726402i \(0.741191\pi\)
\(12\) 0 0
\(13\) 5.23704 1.45249 0.726246 0.687435i \(-0.241263\pi\)
0.726246 + 0.687435i \(0.241263\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 4.72619 1.14627 0.573135 0.819461i \(-0.305727\pi\)
0.573135 + 0.819461i \(0.305727\pi\)
\(18\) 0 0
\(19\) −5.54591 −1.27232 −0.636159 0.771558i \(-0.719478\pi\)
−0.636159 + 0.771558i \(0.719478\pi\)
\(20\) 0 0
\(21\) 0.845563 0.184517
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 7.06968 1.31281 0.656403 0.754411i \(-0.272077\pi\)
0.656403 + 0.754411i \(0.272077\pi\)
\(30\) 0 0
\(31\) −1.83264 −0.329152 −0.164576 0.986364i \(-0.552626\pi\)
−0.164576 + 0.986364i \(0.552626\pi\)
\(32\) 0 0
\(33\) 4.55883 0.793591
\(34\) 0 0
\(35\) 0.845563 0.142926
\(36\) 0 0
\(37\) −9.93738 −1.63370 −0.816848 0.576854i \(-0.804280\pi\)
−0.816848 + 0.576854i \(0.804280\pi\)
\(38\) 0 0
\(39\) −5.23704 −0.838597
\(40\) 0 0
\(41\) 5.71327 0.892263 0.446131 0.894968i \(-0.352801\pi\)
0.446131 + 0.894968i \(0.352801\pi\)
\(42\) 0 0
\(43\) 4.78294 0.729392 0.364696 0.931127i \(-0.381173\pi\)
0.364696 + 0.931127i \(0.381173\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 5.54591 0.808954 0.404477 0.914548i \(-0.367454\pi\)
0.404477 + 0.914548i \(0.367454\pi\)
\(48\) 0 0
\(49\) −6.28502 −0.897860
\(50\) 0 0
\(51\) −4.72619 −0.661799
\(52\) 0 0
\(53\) −14.4962 −1.99121 −0.995604 0.0936641i \(-0.970142\pi\)
−0.995604 + 0.0936641i \(0.970142\pi\)
\(54\) 0 0
\(55\) 4.55883 0.614713
\(56\) 0 0
\(57\) 5.54591 0.734573
\(58\) 0 0
\(59\) 5.06968 0.660015 0.330008 0.943978i \(-0.392949\pi\)
0.330008 + 0.943978i \(0.392949\pi\)
\(60\) 0 0
\(61\) −6.22411 −0.796916 −0.398458 0.917187i \(-0.630455\pi\)
−0.398458 + 0.917187i \(0.630455\pi\)
\(62\) 0 0
\(63\) −0.845563 −0.106531
\(64\) 0 0
\(65\) −5.23704 −0.649574
\(66\) 0 0
\(67\) −7.85849 −0.960067 −0.480033 0.877250i \(-0.659375\pi\)
−0.480033 + 0.877250i \(0.659375\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 3.71327 0.440684 0.220342 0.975423i \(-0.429283\pi\)
0.220342 + 0.975423i \(0.429283\pi\)
\(72\) 0 0
\(73\) 1.23704 0.144784 0.0723920 0.997376i \(-0.476937\pi\)
0.0723920 + 0.997376i \(0.476937\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 3.85478 0.439293
\(78\) 0 0
\(79\) −13.7700 −1.54925 −0.774624 0.632422i \(-0.782061\pi\)
−0.774624 + 0.632422i \(0.782061\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 7.50913 0.824235 0.412117 0.911131i \(-0.364789\pi\)
0.412117 + 0.911131i \(0.364789\pi\)
\(84\) 0 0
\(85\) −4.72619 −0.512627
\(86\) 0 0
\(87\) −7.06968 −0.757949
\(88\) 0 0
\(89\) 2.98708 0.316630 0.158315 0.987389i \(-0.449394\pi\)
0.158315 + 0.987389i \(0.449394\pi\)
\(90\) 0 0
\(91\) −4.42825 −0.464206
\(92\) 0 0
\(93\) 1.83264 0.190036
\(94\) 0 0
\(95\) 5.54591 0.568998
\(96\) 0 0
\(97\) −2.33472 −0.237055 −0.118527 0.992951i \(-0.537817\pi\)
−0.118527 + 0.992951i \(0.537817\pi\)
\(98\) 0 0
\(99\) −4.55883 −0.458180
\(100\) 0 0
\(101\) −12.1873 −1.21269 −0.606343 0.795203i \(-0.707364\pi\)
−0.606343 + 0.795203i \(0.707364\pi\)
\(102\) 0 0
\(103\) −18.1047 −1.78391 −0.891956 0.452121i \(-0.850667\pi\)
−0.891956 + 0.452121i \(0.850667\pi\)
\(104\) 0 0
\(105\) −0.845563 −0.0825185
\(106\) 0 0
\(107\) −6.39147 −0.617887 −0.308943 0.951080i \(-0.599975\pi\)
−0.308943 + 0.951080i \(0.599975\pi\)
\(108\) 0 0
\(109\) 6.55883 0.628222 0.314111 0.949386i \(-0.398294\pi\)
0.314111 + 0.949386i \(0.398294\pi\)
\(110\) 0 0
\(111\) 9.93738 0.943214
\(112\) 0 0
\(113\) −14.0826 −1.32478 −0.662390 0.749159i \(-0.730458\pi\)
−0.662390 + 0.749159i \(0.730458\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) 0 0
\(117\) 5.23704 0.484164
\(118\) 0 0
\(119\) −3.99629 −0.366340
\(120\) 0 0
\(121\) 9.78294 0.889358
\(122\) 0 0
\(123\) −5.71327 −0.515148
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −13.2629 −1.17689 −0.588445 0.808537i \(-0.700260\pi\)
−0.588445 + 0.808537i \(0.700260\pi\)
\(128\) 0 0
\(129\) −4.78294 −0.421115
\(130\) 0 0
\(131\) −7.38225 −0.644991 −0.322495 0.946571i \(-0.604522\pi\)
−0.322495 + 0.946571i \(0.604522\pi\)
\(132\) 0 0
\(133\) 4.68942 0.406624
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 10.1306 0.865514 0.432757 0.901511i \(-0.357541\pi\)
0.432757 + 0.901511i \(0.357541\pi\)
\(138\) 0 0
\(139\) 11.7332 0.995201 0.497600 0.867406i \(-0.334215\pi\)
0.497600 + 0.867406i \(0.334215\pi\)
\(140\) 0 0
\(141\) −5.54591 −0.467050
\(142\) 0 0
\(143\) −23.8748 −1.99651
\(144\) 0 0
\(145\) −7.06968 −0.587105
\(146\) 0 0
\(147\) 6.28502 0.518380
\(148\) 0 0
\(149\) −10.5934 −0.867849 −0.433924 0.900949i \(-0.642871\pi\)
−0.433924 + 0.900949i \(0.642871\pi\)
\(150\) 0 0
\(151\) 8.16520 0.664474 0.332237 0.943196i \(-0.392197\pi\)
0.332237 + 0.943196i \(0.392197\pi\)
\(152\) 0 0
\(153\) 4.72619 0.382090
\(154\) 0 0
\(155\) 1.83264 0.147201
\(156\) 0 0
\(157\) −10.6068 −0.846516 −0.423258 0.906009i \(-0.639114\pi\)
−0.423258 + 0.906009i \(0.639114\pi\)
\(158\) 0 0
\(159\) 14.4962 1.14962
\(160\) 0 0
\(161\) −0.845563 −0.0666397
\(162\) 0 0
\(163\) −15.6653 −1.22700 −0.613500 0.789695i \(-0.710239\pi\)
−0.613500 + 0.789695i \(0.710239\pi\)
\(164\) 0 0
\(165\) −4.55883 −0.354905
\(166\) 0 0
\(167\) −2.16365 −0.167429 −0.0837143 0.996490i \(-0.526678\pi\)
−0.0837143 + 0.996490i \(0.526678\pi\)
\(168\) 0 0
\(169\) 14.4265 1.10973
\(170\) 0 0
\(171\) −5.54591 −0.424106
\(172\) 0 0
\(173\) −8.30887 −0.631712 −0.315856 0.948807i \(-0.602292\pi\)
−0.315856 + 0.948807i \(0.602292\pi\)
\(174\) 0 0
\(175\) −0.845563 −0.0639186
\(176\) 0 0
\(177\) −5.06968 −0.381060
\(178\) 0 0
\(179\) −10.7571 −0.804023 −0.402012 0.915635i \(-0.631689\pi\)
−0.402012 + 0.915635i \(0.631689\pi\)
\(180\) 0 0
\(181\) 19.2482 1.43071 0.715356 0.698761i \(-0.246265\pi\)
0.715356 + 0.698761i \(0.246265\pi\)
\(182\) 0 0
\(183\) 6.22411 0.460100
\(184\) 0 0
\(185\) 9.93738 0.730611
\(186\) 0 0
\(187\) −21.5459 −1.57559
\(188\) 0 0
\(189\) 0.845563 0.0615057
\(190\) 0 0
\(191\) 1.78465 0.129133 0.0645665 0.997913i \(-0.479434\pi\)
0.0645665 + 0.997913i \(0.479434\pi\)
\(192\) 0 0
\(193\) −11.4524 −0.824361 −0.412180 0.911102i \(-0.635233\pi\)
−0.412180 + 0.911102i \(0.635233\pi\)
\(194\) 0 0
\(195\) 5.23704 0.375032
\(196\) 0 0
\(197\) 2.23874 0.159504 0.0797520 0.996815i \(-0.474587\pi\)
0.0797520 + 0.996815i \(0.474587\pi\)
\(198\) 0 0
\(199\) 18.5530 1.31518 0.657592 0.753374i \(-0.271575\pi\)
0.657592 + 0.753374i \(0.271575\pi\)
\(200\) 0 0
\(201\) 7.85849 0.554295
\(202\) 0 0
\(203\) −5.97786 −0.419563
\(204\) 0 0
\(205\) −5.71327 −0.399032
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 25.2829 1.74885
\(210\) 0 0
\(211\) −25.0204 −1.72248 −0.861239 0.508201i \(-0.830311\pi\)
−0.861239 + 0.508201i \(0.830311\pi\)
\(212\) 0 0
\(213\) −3.71327 −0.254429
\(214\) 0 0
\(215\) −4.78294 −0.326194
\(216\) 0 0
\(217\) 1.54961 0.105195
\(218\) 0 0
\(219\) −1.23704 −0.0835911
\(220\) 0 0
\(221\) 24.7512 1.66495
\(222\) 0 0
\(223\) −7.71697 −0.516767 −0.258383 0.966042i \(-0.583190\pi\)
−0.258383 + 0.966042i \(0.583190\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −14.4395 −0.958381 −0.479190 0.877711i \(-0.659070\pi\)
−0.479190 + 0.877711i \(0.659070\pi\)
\(228\) 0 0
\(229\) 15.8660 1.04845 0.524227 0.851578i \(-0.324354\pi\)
0.524227 + 0.851578i \(0.324354\pi\)
\(230\) 0 0
\(231\) −3.85478 −0.253626
\(232\) 0 0
\(233\) −8.36057 −0.547719 −0.273859 0.961770i \(-0.588300\pi\)
−0.273859 + 0.961770i \(0.588300\pi\)
\(234\) 0 0
\(235\) −5.54591 −0.361775
\(236\) 0 0
\(237\) 13.7700 0.894459
\(238\) 0 0
\(239\) 11.6173 0.751460 0.375730 0.926729i \(-0.377392\pi\)
0.375730 + 0.926729i \(0.377392\pi\)
\(240\) 0 0
\(241\) −25.0675 −1.61474 −0.807370 0.590045i \(-0.799110\pi\)
−0.807370 + 0.590045i \(0.799110\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 6.28502 0.401535
\(246\) 0 0
\(247\) −29.0441 −1.84803
\(248\) 0 0
\(249\) −7.50913 −0.475872
\(250\) 0 0
\(251\) −13.4870 −0.851291 −0.425646 0.904890i \(-0.639953\pi\)
−0.425646 + 0.904890i \(0.639953\pi\)
\(252\) 0 0
\(253\) −4.55883 −0.286611
\(254\) 0 0
\(255\) 4.72619 0.295966
\(256\) 0 0
\(257\) −1.85478 −0.115698 −0.0578490 0.998325i \(-0.518424\pi\)
−0.0578490 + 0.998325i \(0.518424\pi\)
\(258\) 0 0
\(259\) 8.40269 0.522117
\(260\) 0 0
\(261\) 7.06968 0.437602
\(262\) 0 0
\(263\) 16.2132 0.999748 0.499874 0.866098i \(-0.333380\pi\)
0.499874 + 0.866098i \(0.333380\pi\)
\(264\) 0 0
\(265\) 14.4962 0.890495
\(266\) 0 0
\(267\) −2.98708 −0.182806
\(268\) 0 0
\(269\) −8.80508 −0.536855 −0.268428 0.963300i \(-0.586504\pi\)
−0.268428 + 0.963300i \(0.586504\pi\)
\(270\) 0 0
\(271\) 3.23333 0.196411 0.0982054 0.995166i \(-0.468690\pi\)
0.0982054 + 0.995166i \(0.468690\pi\)
\(272\) 0 0
\(273\) 4.42825 0.268010
\(274\) 0 0
\(275\) −4.55883 −0.274908
\(276\) 0 0
\(277\) 2.61775 0.157285 0.0786426 0.996903i \(-0.474941\pi\)
0.0786426 + 0.996903i \(0.474941\pi\)
\(278\) 0 0
\(279\) −1.83264 −0.109717
\(280\) 0 0
\(281\) −14.6031 −0.871149 −0.435574 0.900153i \(-0.643455\pi\)
−0.435574 + 0.900153i \(0.643455\pi\)
\(282\) 0 0
\(283\) −22.7808 −1.35418 −0.677088 0.735902i \(-0.736759\pi\)
−0.677088 + 0.735902i \(0.736759\pi\)
\(284\) 0 0
\(285\) −5.54591 −0.328511
\(286\) 0 0
\(287\) −4.83093 −0.285161
\(288\) 0 0
\(289\) 5.33688 0.313934
\(290\) 0 0
\(291\) 2.33472 0.136864
\(292\) 0 0
\(293\) −25.5880 −1.49487 −0.747434 0.664336i \(-0.768715\pi\)
−0.747434 + 0.664336i \(0.768715\pi\)
\(294\) 0 0
\(295\) −5.06968 −0.295168
\(296\) 0 0
\(297\) 4.55883 0.264530
\(298\) 0 0
\(299\) 5.23704 0.302866
\(300\) 0 0
\(301\) −4.04428 −0.233109
\(302\) 0 0
\(303\) 12.1873 0.700144
\(304\) 0 0
\(305\) 6.22411 0.356392
\(306\) 0 0
\(307\) 12.4507 0.710597 0.355299 0.934753i \(-0.384379\pi\)
0.355299 + 0.934753i \(0.384379\pi\)
\(308\) 0 0
\(309\) 18.1047 1.02994
\(310\) 0 0
\(311\) 10.9048 0.618352 0.309176 0.951005i \(-0.399947\pi\)
0.309176 + 0.951005i \(0.399947\pi\)
\(312\) 0 0
\(313\) −30.7549 −1.73837 −0.869186 0.494486i \(-0.835356\pi\)
−0.869186 + 0.494486i \(0.835356\pi\)
\(314\) 0 0
\(315\) 0.845563 0.0476421
\(316\) 0 0
\(317\) 24.9983 1.40404 0.702022 0.712155i \(-0.252281\pi\)
0.702022 + 0.712155i \(0.252281\pi\)
\(318\) 0 0
\(319\) −32.2295 −1.80450
\(320\) 0 0
\(321\) 6.39147 0.356737
\(322\) 0 0
\(323\) −26.2110 −1.45842
\(324\) 0 0
\(325\) 5.23704 0.290498
\(326\) 0 0
\(327\) −6.55883 −0.362704
\(328\) 0 0
\(329\) −4.68942 −0.258536
\(330\) 0 0
\(331\) −9.59390 −0.527328 −0.263664 0.964615i \(-0.584931\pi\)
−0.263664 + 0.964615i \(0.584931\pi\)
\(332\) 0 0
\(333\) −9.93738 −0.544565
\(334\) 0 0
\(335\) 7.85849 0.429355
\(336\) 0 0
\(337\) 3.97415 0.216486 0.108243 0.994124i \(-0.465478\pi\)
0.108243 + 0.994124i \(0.465478\pi\)
\(338\) 0 0
\(339\) 14.0826 0.764862
\(340\) 0 0
\(341\) 8.35470 0.452432
\(342\) 0 0
\(343\) 11.2333 0.606543
\(344\) 0 0
\(345\) 1.00000 0.0538382
\(346\) 0 0
\(347\) 13.1177 0.704193 0.352097 0.935964i \(-0.385469\pi\)
0.352097 + 0.935964i \(0.385469\pi\)
\(348\) 0 0
\(349\) 9.80679 0.524946 0.262473 0.964939i \(-0.415462\pi\)
0.262473 + 0.964939i \(0.415462\pi\)
\(350\) 0 0
\(351\) −5.23704 −0.279532
\(352\) 0 0
\(353\) −8.55007 −0.455074 −0.227537 0.973769i \(-0.573067\pi\)
−0.227537 + 0.973769i \(0.573067\pi\)
\(354\) 0 0
\(355\) −3.71327 −0.197080
\(356\) 0 0
\(357\) 3.99629 0.211506
\(358\) 0 0
\(359\) 28.3990 1.49884 0.749420 0.662094i \(-0.230332\pi\)
0.749420 + 0.662094i \(0.230332\pi\)
\(360\) 0 0
\(361\) 11.7571 0.618795
\(362\) 0 0
\(363\) −9.78294 −0.513471
\(364\) 0 0
\(365\) −1.23704 −0.0647494
\(366\) 0 0
\(367\) 24.3121 1.26908 0.634539 0.772890i \(-0.281190\pi\)
0.634539 + 0.772890i \(0.281190\pi\)
\(368\) 0 0
\(369\) 5.71327 0.297421
\(370\) 0 0
\(371\) 12.2575 0.636376
\(372\) 0 0
\(373\) −20.3435 −1.05335 −0.526673 0.850068i \(-0.676561\pi\)
−0.526673 + 0.850068i \(0.676561\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 37.0241 1.90684
\(378\) 0 0
\(379\) 7.33056 0.376546 0.188273 0.982117i \(-0.439711\pi\)
0.188273 + 0.982117i \(0.439711\pi\)
\(380\) 0 0
\(381\) 13.2629 0.679478
\(382\) 0 0
\(383\) 28.2575 1.44389 0.721945 0.691951i \(-0.243249\pi\)
0.721945 + 0.691951i \(0.243249\pi\)
\(384\) 0 0
\(385\) −3.85478 −0.196458
\(386\) 0 0
\(387\) 4.78294 0.243131
\(388\) 0 0
\(389\) 23.9966 1.21667 0.608337 0.793678i \(-0.291837\pi\)
0.608337 + 0.793678i \(0.291837\pi\)
\(390\) 0 0
\(391\) 4.72619 0.239014
\(392\) 0 0
\(393\) 7.38225 0.372385
\(394\) 0 0
\(395\) 13.7700 0.692845
\(396\) 0 0
\(397\) −3.02169 −0.151654 −0.0758271 0.997121i \(-0.524160\pi\)
−0.0758271 + 0.997121i \(0.524160\pi\)
\(398\) 0 0
\(399\) −4.68942 −0.234765
\(400\) 0 0
\(401\) −29.6263 −1.47947 −0.739735 0.672899i \(-0.765049\pi\)
−0.739735 + 0.672899i \(0.765049\pi\)
\(402\) 0 0
\(403\) −9.59760 −0.478091
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 45.3028 2.24558
\(408\) 0 0
\(409\) −7.42437 −0.367112 −0.183556 0.983009i \(-0.558761\pi\)
−0.183556 + 0.983009i \(0.558761\pi\)
\(410\) 0 0
\(411\) −10.1306 −0.499705
\(412\) 0 0
\(413\) −4.28673 −0.210936
\(414\) 0 0
\(415\) −7.50913 −0.368609
\(416\) 0 0
\(417\) −11.7332 −0.574580
\(418\) 0 0
\(419\) −2.20568 −0.107754 −0.0538772 0.998548i \(-0.517158\pi\)
−0.0538772 + 0.998548i \(0.517158\pi\)
\(420\) 0 0
\(421\) 34.0989 1.66188 0.830939 0.556364i \(-0.187804\pi\)
0.830939 + 0.556364i \(0.187804\pi\)
\(422\) 0 0
\(423\) 5.54591 0.269651
\(424\) 0 0
\(425\) 4.72619 0.229254
\(426\) 0 0
\(427\) 5.26288 0.254689
\(428\) 0 0
\(429\) 23.8748 1.15268
\(430\) 0 0
\(431\) 4.16952 0.200839 0.100419 0.994945i \(-0.467982\pi\)
0.100419 + 0.994945i \(0.467982\pi\)
\(432\) 0 0
\(433\) 7.08965 0.340707 0.170353 0.985383i \(-0.445509\pi\)
0.170353 + 0.985383i \(0.445509\pi\)
\(434\) 0 0
\(435\) 7.06968 0.338965
\(436\) 0 0
\(437\) −5.54591 −0.265297
\(438\) 0 0
\(439\) −36.7537 −1.75416 −0.877079 0.480347i \(-0.840511\pi\)
−0.877079 + 0.480347i \(0.840511\pi\)
\(440\) 0 0
\(441\) −6.28502 −0.299287
\(442\) 0 0
\(443\) 23.3505 1.10942 0.554709 0.832045i \(-0.312830\pi\)
0.554709 + 0.832045i \(0.312830\pi\)
\(444\) 0 0
\(445\) −2.98708 −0.141601
\(446\) 0 0
\(447\) 10.5934 0.501053
\(448\) 0 0
\(449\) 30.0437 1.41785 0.708924 0.705285i \(-0.249181\pi\)
0.708924 + 0.705285i \(0.249181\pi\)
\(450\) 0 0
\(451\) −26.0458 −1.22645
\(452\) 0 0
\(453\) −8.16520 −0.383634
\(454\) 0 0
\(455\) 4.42825 0.207599
\(456\) 0 0
\(457\) 19.5496 0.914492 0.457246 0.889340i \(-0.348836\pi\)
0.457246 + 0.889340i \(0.348836\pi\)
\(458\) 0 0
\(459\) −4.72619 −0.220600
\(460\) 0 0
\(461\) −22.6577 −1.05527 −0.527637 0.849470i \(-0.676922\pi\)
−0.527637 + 0.849470i \(0.676922\pi\)
\(462\) 0 0
\(463\) −21.7037 −1.00866 −0.504328 0.863512i \(-0.668260\pi\)
−0.504328 + 0.863512i \(0.668260\pi\)
\(464\) 0 0
\(465\) −1.83264 −0.0849867
\(466\) 0 0
\(467\) 25.2003 1.16613 0.583065 0.812426i \(-0.301853\pi\)
0.583065 + 0.812426i \(0.301853\pi\)
\(468\) 0 0
\(469\) 6.64485 0.306831
\(470\) 0 0
\(471\) 10.6068 0.488736
\(472\) 0 0
\(473\) −21.8046 −1.00258
\(474\) 0 0
\(475\) −5.54591 −0.254464
\(476\) 0 0
\(477\) −14.4962 −0.663736
\(478\) 0 0
\(479\) −19.1635 −0.875602 −0.437801 0.899072i \(-0.644243\pi\)
−0.437801 + 0.899072i \(0.644243\pi\)
\(480\) 0 0
\(481\) −52.0424 −2.37293
\(482\) 0 0
\(483\) 0.845563 0.0384745
\(484\) 0 0
\(485\) 2.33472 0.106014
\(486\) 0 0
\(487\) 14.3805 0.651645 0.325822 0.945431i \(-0.394359\pi\)
0.325822 + 0.945431i \(0.394359\pi\)
\(488\) 0 0
\(489\) 15.6653 0.708408
\(490\) 0 0
\(491\) −16.3267 −0.736813 −0.368407 0.929665i \(-0.620097\pi\)
−0.368407 + 0.929665i \(0.620097\pi\)
\(492\) 0 0
\(493\) 33.4126 1.50483
\(494\) 0 0
\(495\) 4.55883 0.204904
\(496\) 0 0
\(497\) −3.13980 −0.140839
\(498\) 0 0
\(499\) 29.8726 1.33728 0.668641 0.743586i \(-0.266877\pi\)
0.668641 + 0.743586i \(0.266877\pi\)
\(500\) 0 0
\(501\) 2.16365 0.0966649
\(502\) 0 0
\(503\) 33.6831 1.50186 0.750928 0.660385i \(-0.229607\pi\)
0.750928 + 0.660385i \(0.229607\pi\)
\(504\) 0 0
\(505\) 12.1873 0.542329
\(506\) 0 0
\(507\) −14.4265 −0.640705
\(508\) 0 0
\(509\) −29.9265 −1.32647 −0.663233 0.748413i \(-0.730816\pi\)
−0.663233 + 0.748413i \(0.730816\pi\)
\(510\) 0 0
\(511\) −1.04599 −0.0462719
\(512\) 0 0
\(513\) 5.54591 0.244858
\(514\) 0 0
\(515\) 18.1047 0.797790
\(516\) 0 0
\(517\) −25.2829 −1.11194
\(518\) 0 0
\(519\) 8.30887 0.364719
\(520\) 0 0
\(521\) −8.19827 −0.359173 −0.179586 0.983742i \(-0.557476\pi\)
−0.179586 + 0.983742i \(0.557476\pi\)
\(522\) 0 0
\(523\) −8.15779 −0.356715 −0.178358 0.983966i \(-0.557078\pi\)
−0.178358 + 0.983966i \(0.557078\pi\)
\(524\) 0 0
\(525\) 0.845563 0.0369034
\(526\) 0 0
\(527\) −8.66141 −0.377297
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 5.06968 0.220005
\(532\) 0 0
\(533\) 29.9206 1.29600
\(534\) 0 0
\(535\) 6.39147 0.276327
\(536\) 0 0
\(537\) 10.7571 0.464203
\(538\) 0 0
\(539\) 28.6524 1.23414
\(540\) 0 0
\(541\) −43.4406 −1.86766 −0.933830 0.357718i \(-0.883555\pi\)
−0.933830 + 0.357718i \(0.883555\pi\)
\(542\) 0 0
\(543\) −19.2482 −0.826021
\(544\) 0 0
\(545\) −6.55883 −0.280949
\(546\) 0 0
\(547\) −37.5659 −1.60620 −0.803101 0.595843i \(-0.796818\pi\)
−0.803101 + 0.595843i \(0.796818\pi\)
\(548\) 0 0
\(549\) −6.22411 −0.265639
\(550\) 0 0
\(551\) −39.2078 −1.67031
\(552\) 0 0
\(553\) 11.6434 0.495129
\(554\) 0 0
\(555\) −9.93738 −0.421818
\(556\) 0 0
\(557\) 13.0438 0.552685 0.276342 0.961059i \(-0.410878\pi\)
0.276342 + 0.961059i \(0.410878\pi\)
\(558\) 0 0
\(559\) 25.0484 1.05944
\(560\) 0 0
\(561\) 21.5459 0.909669
\(562\) 0 0
\(563\) −4.25212 −0.179206 −0.0896028 0.995978i \(-0.528560\pi\)
−0.0896028 + 0.995978i \(0.528560\pi\)
\(564\) 0 0
\(565\) 14.0826 0.592459
\(566\) 0 0
\(567\) −0.845563 −0.0355103
\(568\) 0 0
\(569\) 45.8358 1.92154 0.960769 0.277350i \(-0.0894563\pi\)
0.960769 + 0.277350i \(0.0894563\pi\)
\(570\) 0 0
\(571\) 22.7595 0.952457 0.476229 0.879321i \(-0.342003\pi\)
0.476229 + 0.879321i \(0.342003\pi\)
\(572\) 0 0
\(573\) −1.78465 −0.0745549
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) 40.2310 1.67484 0.837419 0.546561i \(-0.184063\pi\)
0.837419 + 0.546561i \(0.184063\pi\)
\(578\) 0 0
\(579\) 11.4524 0.475945
\(580\) 0 0
\(581\) −6.34945 −0.263420
\(582\) 0 0
\(583\) 66.0858 2.73699
\(584\) 0 0
\(585\) −5.23704 −0.216525
\(586\) 0 0
\(587\) −42.7278 −1.76357 −0.881783 0.471655i \(-0.843657\pi\)
−0.881783 + 0.471655i \(0.843657\pi\)
\(588\) 0 0
\(589\) 10.1637 0.418786
\(590\) 0 0
\(591\) −2.23874 −0.0920897
\(592\) 0 0
\(593\) 22.1852 0.911036 0.455518 0.890227i \(-0.349454\pi\)
0.455518 + 0.890227i \(0.349454\pi\)
\(594\) 0 0
\(595\) 3.99629 0.163832
\(596\) 0 0
\(597\) −18.5530 −0.759322
\(598\) 0 0
\(599\) −9.85632 −0.402718 −0.201359 0.979517i \(-0.564536\pi\)
−0.201359 + 0.979517i \(0.564536\pi\)
\(600\) 0 0
\(601\) 12.1157 0.494208 0.247104 0.968989i \(-0.420521\pi\)
0.247104 + 0.968989i \(0.420521\pi\)
\(602\) 0 0
\(603\) −7.85849 −0.320022
\(604\) 0 0
\(605\) −9.78294 −0.397733
\(606\) 0 0
\(607\) 14.4984 0.588471 0.294235 0.955733i \(-0.404935\pi\)
0.294235 + 0.955733i \(0.404935\pi\)
\(608\) 0 0
\(609\) 5.97786 0.242235
\(610\) 0 0
\(611\) 29.0441 1.17500
\(612\) 0 0
\(613\) −36.0312 −1.45529 −0.727643 0.685956i \(-0.759384\pi\)
−0.727643 + 0.685956i \(0.759384\pi\)
\(614\) 0 0
\(615\) 5.71327 0.230381
\(616\) 0 0
\(617\) −44.0792 −1.77456 −0.887280 0.461230i \(-0.847408\pi\)
−0.887280 + 0.461230i \(0.847408\pi\)
\(618\) 0 0
\(619\) 32.9998 1.32638 0.663188 0.748453i \(-0.269203\pi\)
0.663188 + 0.748453i \(0.269203\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) −2.52576 −0.101193
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −25.2829 −1.00970
\(628\) 0 0
\(629\) −46.9660 −1.87266
\(630\) 0 0
\(631\) −12.8200 −0.510356 −0.255178 0.966894i \(-0.582134\pi\)
−0.255178 + 0.966894i \(0.582134\pi\)
\(632\) 0 0
\(633\) 25.0204 0.994473
\(634\) 0 0
\(635\) 13.2629 0.526321
\(636\) 0 0
\(637\) −32.9149 −1.30414
\(638\) 0 0
\(639\) 3.71327 0.146895
\(640\) 0 0
\(641\) 38.8126 1.53301 0.766503 0.642241i \(-0.221995\pi\)
0.766503 + 0.642241i \(0.221995\pi\)
\(642\) 0 0
\(643\) −33.1339 −1.30667 −0.653337 0.757067i \(-0.726632\pi\)
−0.653337 + 0.757067i \(0.726632\pi\)
\(644\) 0 0
\(645\) 4.78294 0.188328
\(646\) 0 0
\(647\) 24.3547 0.957482 0.478741 0.877956i \(-0.341093\pi\)
0.478741 + 0.877956i \(0.341093\pi\)
\(648\) 0 0
\(649\) −23.1118 −0.907217
\(650\) 0 0
\(651\) −1.54961 −0.0607341
\(652\) 0 0
\(653\) 12.5943 0.492855 0.246427 0.969161i \(-0.420743\pi\)
0.246427 + 0.969161i \(0.420743\pi\)
\(654\) 0 0
\(655\) 7.38225 0.288449
\(656\) 0 0
\(657\) 1.23704 0.0482613
\(658\) 0 0
\(659\) −41.9259 −1.63320 −0.816601 0.577202i \(-0.804144\pi\)
−0.816601 + 0.577202i \(0.804144\pi\)
\(660\) 0 0
\(661\) 45.1664 1.75677 0.878384 0.477955i \(-0.158622\pi\)
0.878384 + 0.477955i \(0.158622\pi\)
\(662\) 0 0
\(663\) −24.7512 −0.961258
\(664\) 0 0
\(665\) −4.68942 −0.181848
\(666\) 0 0
\(667\) 7.06968 0.273739
\(668\) 0 0
\(669\) 7.71697 0.298355
\(670\) 0 0
\(671\) 28.3747 1.09539
\(672\) 0 0
\(673\) −13.4022 −0.516618 −0.258309 0.966062i \(-0.583165\pi\)
−0.258309 + 0.966062i \(0.583165\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −41.5662 −1.59752 −0.798759 0.601651i \(-0.794510\pi\)
−0.798759 + 0.601651i \(0.794510\pi\)
\(678\) 0 0
\(679\) 1.97415 0.0757611
\(680\) 0 0
\(681\) 14.4395 0.553321
\(682\) 0 0
\(683\) −24.1778 −0.925136 −0.462568 0.886584i \(-0.653072\pi\)
−0.462568 + 0.886584i \(0.653072\pi\)
\(684\) 0 0
\(685\) −10.1306 −0.387070
\(686\) 0 0
\(687\) −15.8660 −0.605325
\(688\) 0 0
\(689\) −75.9172 −2.89221
\(690\) 0 0
\(691\) 4.99242 0.189921 0.0949603 0.995481i \(-0.469728\pi\)
0.0949603 + 0.995481i \(0.469728\pi\)
\(692\) 0 0
\(693\) 3.85478 0.146431
\(694\) 0 0
\(695\) −11.7332 −0.445067
\(696\) 0 0
\(697\) 27.0020 1.02277
\(698\) 0 0
\(699\) 8.36057 0.316226
\(700\) 0 0
\(701\) 28.0384 1.05900 0.529498 0.848311i \(-0.322380\pi\)
0.529498 + 0.848311i \(0.322380\pi\)
\(702\) 0 0
\(703\) 55.1118 2.07858
\(704\) 0 0
\(705\) 5.54591 0.208871
\(706\) 0 0
\(707\) 10.3052 0.387566
\(708\) 0 0
\(709\) 40.9594 1.53826 0.769130 0.639092i \(-0.220690\pi\)
0.769130 + 0.639092i \(0.220690\pi\)
\(710\) 0 0
\(711\) −13.7700 −0.516416
\(712\) 0 0
\(713\) −1.83264 −0.0686329
\(714\) 0 0
\(715\) 23.8748 0.892865
\(716\) 0 0
\(717\) −11.6173 −0.433856
\(718\) 0 0
\(719\) 26.8568 1.00159 0.500794 0.865566i \(-0.333041\pi\)
0.500794 + 0.865566i \(0.333041\pi\)
\(720\) 0 0
\(721\) 15.3087 0.570126
\(722\) 0 0
\(723\) 25.0675 0.932271
\(724\) 0 0
\(725\) 7.06968 0.262561
\(726\) 0 0
\(727\) −34.9849 −1.29752 −0.648759 0.760994i \(-0.724712\pi\)
−0.648759 + 0.760994i \(0.724712\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 22.6051 0.836080
\(732\) 0 0
\(733\) 28.2170 1.04222 0.521109 0.853490i \(-0.325518\pi\)
0.521109 + 0.853490i \(0.325518\pi\)
\(734\) 0 0
\(735\) −6.28502 −0.231827
\(736\) 0 0
\(737\) 35.8255 1.31965
\(738\) 0 0
\(739\) 30.8251 1.13392 0.566959 0.823746i \(-0.308120\pi\)
0.566959 + 0.823746i \(0.308120\pi\)
\(740\) 0 0
\(741\) 29.0441 1.06696
\(742\) 0 0
\(743\) 53.2093 1.95206 0.976030 0.217635i \(-0.0698343\pi\)
0.976030 + 0.217635i \(0.0698343\pi\)
\(744\) 0 0
\(745\) 10.5934 0.388114
\(746\) 0 0
\(747\) 7.50913 0.274745
\(748\) 0 0
\(749\) 5.40439 0.197472
\(750\) 0 0
\(751\) 25.0513 0.914136 0.457068 0.889432i \(-0.348900\pi\)
0.457068 + 0.889432i \(0.348900\pi\)
\(752\) 0 0
\(753\) 13.4870 0.491493
\(754\) 0 0
\(755\) −8.16520 −0.297162
\(756\) 0 0
\(757\) 2.72032 0.0988718 0.0494359 0.998777i \(-0.484258\pi\)
0.0494359 + 0.998777i \(0.484258\pi\)
\(758\) 0 0
\(759\) 4.55883 0.165475
\(760\) 0 0
\(761\) −11.9779 −0.434197 −0.217099 0.976150i \(-0.569659\pi\)
−0.217099 + 0.976150i \(0.569659\pi\)
\(762\) 0 0
\(763\) −5.54591 −0.200775
\(764\) 0 0
\(765\) −4.72619 −0.170876
\(766\) 0 0
\(767\) 26.5501 0.958667
\(768\) 0 0
\(769\) 14.2412 0.513551 0.256775 0.966471i \(-0.417340\pi\)
0.256775 + 0.966471i \(0.417340\pi\)
\(770\) 0 0
\(771\) 1.85478 0.0667983
\(772\) 0 0
\(773\) 15.0660 0.541885 0.270943 0.962595i \(-0.412665\pi\)
0.270943 + 0.962595i \(0.412665\pi\)
\(774\) 0 0
\(775\) −1.83264 −0.0658304
\(776\) 0 0
\(777\) −8.40269 −0.301445
\(778\) 0 0
\(779\) −31.6853 −1.13524
\(780\) 0 0
\(781\) −16.9282 −0.605737
\(782\) 0 0
\(783\) −7.06968 −0.252650
\(784\) 0 0
\(785\) 10.6068 0.378574
\(786\) 0 0
\(787\) 4.79922 0.171074 0.0855368 0.996335i \(-0.472739\pi\)
0.0855368 + 0.996335i \(0.472739\pi\)
\(788\) 0 0
\(789\) −16.2132 −0.577205
\(790\) 0 0
\(791\) 11.9077 0.423390
\(792\) 0 0
\(793\) −32.5959 −1.15751
\(794\) 0 0
\(795\) −14.4962 −0.514128
\(796\) 0 0
\(797\) 51.6538 1.82967 0.914836 0.403825i \(-0.132320\pi\)
0.914836 + 0.403825i \(0.132320\pi\)
\(798\) 0 0
\(799\) 26.2110 0.927279
\(800\) 0 0
\(801\) 2.98708 0.105543
\(802\) 0 0
\(803\) −5.63943 −0.199011
\(804\) 0 0
\(805\) 0.845563 0.0298022
\(806\) 0 0
\(807\) 8.80508 0.309954
\(808\) 0 0
\(809\) −9.30500 −0.327146 −0.163573 0.986531i \(-0.552302\pi\)
−0.163573 + 0.986531i \(0.552302\pi\)
\(810\) 0 0
\(811\) −39.2041 −1.37664 −0.688320 0.725407i \(-0.741652\pi\)
−0.688320 + 0.725407i \(0.741652\pi\)
\(812\) 0 0
\(813\) −3.23333 −0.113398
\(814\) 0 0
\(815\) 15.6653 0.548731
\(816\) 0 0
\(817\) −26.5258 −0.928019
\(818\) 0 0
\(819\) −4.42825 −0.154735
\(820\) 0 0
\(821\) −9.23948 −0.322460 −0.161230 0.986917i \(-0.551546\pi\)
−0.161230 + 0.986917i \(0.551546\pi\)
\(822\) 0 0
\(823\) −26.3931 −0.920006 −0.460003 0.887917i \(-0.652152\pi\)
−0.460003 + 0.887917i \(0.652152\pi\)
\(824\) 0 0
\(825\) 4.55883 0.158718
\(826\) 0 0
\(827\) −16.7520 −0.582525 −0.291263 0.956643i \(-0.594075\pi\)
−0.291263 + 0.956643i \(0.594075\pi\)
\(828\) 0 0
\(829\) −32.1381 −1.11620 −0.558101 0.829773i \(-0.688470\pi\)
−0.558101 + 0.829773i \(0.688470\pi\)
\(830\) 0 0
\(831\) −2.61775 −0.0908086
\(832\) 0 0
\(833\) −29.7042 −1.02919
\(834\) 0 0
\(835\) 2.16365 0.0748763
\(836\) 0 0
\(837\) 1.83264 0.0633453
\(838\) 0 0
\(839\) 15.8748 0.548058 0.274029 0.961721i \(-0.411644\pi\)
0.274029 + 0.961721i \(0.411644\pi\)
\(840\) 0 0
\(841\) 20.9803 0.723459
\(842\) 0 0
\(843\) 14.6031 0.502958
\(844\) 0 0
\(845\) −14.4265 −0.496288
\(846\) 0 0
\(847\) −8.27210 −0.284233
\(848\) 0 0
\(849\) 22.7808 0.781834
\(850\) 0 0
\(851\) −9.93738 −0.340649
\(852\) 0 0
\(853\) −27.4481 −0.939804 −0.469902 0.882719i \(-0.655711\pi\)
−0.469902 + 0.882719i \(0.655711\pi\)
\(854\) 0 0
\(855\) 5.54591 0.189666
\(856\) 0 0
\(857\) −49.3929 −1.68723 −0.843615 0.536948i \(-0.819577\pi\)
−0.843615 + 0.536948i \(0.819577\pi\)
\(858\) 0 0
\(859\) −18.5863 −0.634157 −0.317078 0.948399i \(-0.602702\pi\)
−0.317078 + 0.948399i \(0.602702\pi\)
\(860\) 0 0
\(861\) 4.83093 0.164638
\(862\) 0 0
\(863\) −33.6619 −1.14586 −0.572932 0.819603i \(-0.694194\pi\)
−0.572932 + 0.819603i \(0.694194\pi\)
\(864\) 0 0
\(865\) 8.30887 0.282510
\(866\) 0 0
\(867\) −5.33688 −0.181250
\(868\) 0 0
\(869\) 62.7752 2.12950
\(870\) 0 0
\(871\) −41.1552 −1.39449
\(872\) 0 0
\(873\) −2.33472 −0.0790183
\(874\) 0 0
\(875\) 0.845563 0.0285853
\(876\) 0 0
\(877\) 29.2321 0.987097 0.493548 0.869718i \(-0.335700\pi\)
0.493548 + 0.869718i \(0.335700\pi\)
\(878\) 0 0
\(879\) 25.5880 0.863063
\(880\) 0 0
\(881\) 1.20242 0.0405107 0.0202553 0.999795i \(-0.493552\pi\)
0.0202553 + 0.999795i \(0.493552\pi\)
\(882\) 0 0
\(883\) −2.82877 −0.0951956 −0.0475978 0.998867i \(-0.515157\pi\)
−0.0475978 + 0.998867i \(0.515157\pi\)
\(884\) 0 0
\(885\) 5.06968 0.170415
\(886\) 0 0
\(887\) −6.86048 −0.230352 −0.115176 0.993345i \(-0.536743\pi\)
−0.115176 + 0.993345i \(0.536743\pi\)
\(888\) 0 0
\(889\) 11.2146 0.376126
\(890\) 0 0
\(891\) −4.55883 −0.152727
\(892\) 0 0
\(893\) −30.7571 −1.02925
\(894\) 0 0
\(895\) 10.7571 0.359570
\(896\) 0 0
\(897\) −5.23704 −0.174860
\(898\) 0 0
\(899\) −12.9562 −0.432113
\(900\) 0 0
\(901\) −68.5119 −2.28246
\(902\) 0 0
\(903\) 4.04428 0.134585
\(904\) 0 0
\(905\) −19.2482 −0.639833
\(906\) 0 0
\(907\) −15.1198 −0.502046 −0.251023 0.967981i \(-0.580767\pi\)
−0.251023 + 0.967981i \(0.580767\pi\)
\(908\) 0 0
\(909\) −12.1873 −0.404228
\(910\) 0 0
\(911\) −50.3931 −1.66960 −0.834799 0.550555i \(-0.814416\pi\)
−0.834799 + 0.550555i \(0.814416\pi\)
\(912\) 0 0
\(913\) −34.2329 −1.13294
\(914\) 0 0
\(915\) −6.22411 −0.205763
\(916\) 0 0
\(917\) 6.24216 0.206134
\(918\) 0 0
\(919\) −40.0053 −1.31965 −0.659827 0.751417i \(-0.729371\pi\)
−0.659827 + 0.751417i \(0.729371\pi\)
\(920\) 0 0
\(921\) −12.4507 −0.410264
\(922\) 0 0
\(923\) 19.4465 0.640090
\(924\) 0 0
\(925\) −9.93738 −0.326739
\(926\) 0 0
\(927\) −18.1047 −0.594638
\(928\) 0 0
\(929\) 39.0921 1.28257 0.641285 0.767303i \(-0.278402\pi\)
0.641285 + 0.767303i \(0.278402\pi\)
\(930\) 0 0
\(931\) 34.8562 1.14236
\(932\) 0 0
\(933\) −10.9048 −0.357006
\(934\) 0 0
\(935\) 21.5459 0.704627
\(936\) 0 0
\(937\) 48.2310 1.57564 0.787819 0.615907i \(-0.211210\pi\)
0.787819 + 0.615907i \(0.211210\pi\)
\(938\) 0 0
\(939\) 30.7549 1.00365
\(940\) 0 0
\(941\) 10.7337 0.349909 0.174954 0.984577i \(-0.444022\pi\)
0.174954 + 0.984577i \(0.444022\pi\)
\(942\) 0 0
\(943\) 5.71327 0.186050
\(944\) 0 0
\(945\) −0.845563 −0.0275062
\(946\) 0 0
\(947\) −12.9257 −0.420029 −0.210015 0.977698i \(-0.567351\pi\)
−0.210015 + 0.977698i \(0.567351\pi\)
\(948\) 0 0
\(949\) 6.47840 0.210298
\(950\) 0 0
\(951\) −24.9983 −0.810625
\(952\) 0 0
\(953\) −53.4051 −1.72996 −0.864981 0.501805i \(-0.832670\pi\)
−0.864981 + 0.501805i \(0.832670\pi\)
\(954\) 0 0
\(955\) −1.78465 −0.0577500
\(956\) 0 0
\(957\) 32.2295 1.04183
\(958\) 0 0
\(959\) −8.56605 −0.276612
\(960\) 0 0
\(961\) −27.6414 −0.891659
\(962\) 0 0
\(963\) −6.39147 −0.205962
\(964\) 0 0
\(965\) 11.4524 0.368665
\(966\) 0 0
\(967\) −28.1852 −0.906374 −0.453187 0.891415i \(-0.649713\pi\)
−0.453187 + 0.891415i \(0.649713\pi\)
\(968\) 0 0
\(969\) 26.2110 0.842019
\(970\) 0 0
\(971\) 33.1922 1.06519 0.532595 0.846370i \(-0.321217\pi\)
0.532595 + 0.846370i \(0.321217\pi\)
\(972\) 0 0
\(973\) −9.92120 −0.318059
\(974\) 0 0
\(975\) −5.23704 −0.167719
\(976\) 0 0
\(977\) 11.0750 0.354321 0.177161 0.984182i \(-0.443309\pi\)
0.177161 + 0.984182i \(0.443309\pi\)
\(978\) 0 0
\(979\) −13.6176 −0.435220
\(980\) 0 0
\(981\) 6.55883 0.209407
\(982\) 0 0
\(983\) −11.8785 −0.378864 −0.189432 0.981894i \(-0.560665\pi\)
−0.189432 + 0.981894i \(0.560665\pi\)
\(984\) 0 0
\(985\) −2.23874 −0.0713323
\(986\) 0 0
\(987\) 4.68942 0.149266
\(988\) 0 0
\(989\) 4.78294 0.152089
\(990\) 0 0
\(991\) −33.8800 −1.07623 −0.538117 0.842870i \(-0.680864\pi\)
−0.538117 + 0.842870i \(0.680864\pi\)
\(992\) 0 0
\(993\) 9.59390 0.304453
\(994\) 0 0
\(995\) −18.5530 −0.588168
\(996\) 0 0
\(997\) −9.33797 −0.295737 −0.147868 0.989007i \(-0.547241\pi\)
−0.147868 + 0.989007i \(0.547241\pi\)
\(998\) 0 0
\(999\) 9.93738 0.314405
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 2760.2.a.v.1.2 4
3.2 odd 2 8280.2.a.bq.1.2 4
4.3 odd 2 5520.2.a.cb.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2760.2.a.v.1.2 4 1.1 even 1 trivial
5520.2.a.cb.1.3 4 4.3 odd 2
8280.2.a.bq.1.2 4 3.2 odd 2