Properties

Label 5520.2.a.bx.1.3
Level $5520$
Weight $2$
Character 5520.1
Self dual yes
Analytic conductor $44.077$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5520.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.0774219157\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2760)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.470683\) of defining polynomial
Character \(\chi\) \(=\) 5520.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} +4.77846 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} +4.77846 q^{7} +1.00000 q^{9} -3.30777 q^{11} +3.19051 q^{13} +1.00000 q^{15} -2.41205 q^{17} -2.24914 q^{19} -4.77846 q^{21} -1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -7.96896 q^{29} +10.3354 q^{31} +3.30777 q^{33} -4.77846 q^{35} -11.2767 q^{37} -3.19051 q^{39} +1.02760 q^{41} -10.0552 q^{43} -1.00000 q^{45} -2.74742 q^{47} +15.8337 q^{49} +2.41205 q^{51} -5.58795 q^{53} +3.30777 q^{55} +2.24914 q^{57} +1.96896 q^{59} -8.24914 q^{61} +4.77846 q^{63} -3.19051 q^{65} +7.71982 q^{67} +1.00000 q^{69} -16.0242 q^{71} -7.92332 q^{73} -1.00000 q^{75} -15.8061 q^{77} +1.00000 q^{81} +12.5845 q^{83} +2.41205 q^{85} +7.96896 q^{87} -3.05863 q^{89} +15.2457 q^{91} -10.3354 q^{93} +2.24914 q^{95} -5.43965 q^{97} -3.30777 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} - 3 q^{5} + 6 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} - 3 q^{5} + 6 q^{7} + 3 q^{9} - 2 q^{11} + 3 q^{15} - 6 q^{17} + 2 q^{19} - 6 q^{21} - 3 q^{23} + 3 q^{25} - 3 q^{27} - 6 q^{29} + 6 q^{31} + 2 q^{33} - 6 q^{35} - 8 q^{37} - 14 q^{41} + 4 q^{43} - 3 q^{45} + 18 q^{47} + 5 q^{49} + 6 q^{51} - 18 q^{53} + 2 q^{55} - 2 q^{57} - 12 q^{59} - 16 q^{61} + 6 q^{63} + 14 q^{67} + 3 q^{69} + 4 q^{71} - 3 q^{75} - 22 q^{77} + 3 q^{81} + 4 q^{83} + 6 q^{85} + 6 q^{87} - 10 q^{89} + 2 q^{91} - 6 q^{93} - 2 q^{95} + 2 q^{97} - 2 q^{99}+O(q^{100}) \) 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
\(7\) 4.77846 1.80609 0.903044 0.429549i \(-0.141327\pi\)
0.903044 + 0.429549i \(0.141327\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.30777 −0.997331 −0.498666 0.866794i \(-0.666176\pi\)
−0.498666 + 0.866794i \(0.666176\pi\)
\(12\) 0 0
\(13\) 3.19051 0.884888 0.442444 0.896796i \(-0.354112\pi\)
0.442444 + 0.896796i \(0.354112\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −2.41205 −0.585008 −0.292504 0.956264i \(-0.594488\pi\)
−0.292504 + 0.956264i \(0.594488\pi\)
\(18\) 0 0
\(19\) −2.24914 −0.515988 −0.257994 0.966146i \(-0.583062\pi\)
−0.257994 + 0.966146i \(0.583062\pi\)
\(20\) 0 0
\(21\) −4.77846 −1.04274
\(22\) 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
\(29\) −7.96896 −1.47980 −0.739900 0.672717i \(-0.765127\pi\)
−0.739900 + 0.672717i \(0.765127\pi\)
\(30\) 0 0
\(31\) 10.3354 1.85629 0.928144 0.372222i \(-0.121404\pi\)
0.928144 + 0.372222i \(0.121404\pi\)
\(32\) 0 0
\(33\) 3.30777 0.575809
\(34\) 0 0
\(35\) −4.77846 −0.807707
\(36\) 0 0
\(37\) −11.2767 −1.85388 −0.926942 0.375204i \(-0.877573\pi\)
−0.926942 + 0.375204i \(0.877573\pi\)
\(38\) 0 0
\(39\) −3.19051 −0.510890
\(40\) 0 0
\(41\) 1.02760 0.160484 0.0802419 0.996775i \(-0.474431\pi\)
0.0802419 + 0.996775i \(0.474431\pi\)
\(42\) 0 0
\(43\) −10.0552 −1.53340 −0.766701 0.642004i \(-0.778103\pi\)
−0.766701 + 0.642004i \(0.778103\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −2.74742 −0.400753 −0.200376 0.979719i \(-0.564216\pi\)
−0.200376 + 0.979719i \(0.564216\pi\)
\(48\) 0 0
\(49\) 15.8337 2.26195
\(50\) 0 0
\(51\) 2.41205 0.337755
\(52\) 0 0
\(53\) −5.58795 −0.767564 −0.383782 0.923424i \(-0.625379\pi\)
−0.383782 + 0.923424i \(0.625379\pi\)
\(54\) 0 0
\(55\) 3.30777 0.446020
\(56\) 0 0
\(57\) 2.24914 0.297906
\(58\) 0 0
\(59\) 1.96896 0.256337 0.128169 0.991752i \(-0.459090\pi\)
0.128169 + 0.991752i \(0.459090\pi\)
\(60\) 0 0
\(61\) −8.24914 −1.05619 −0.528097 0.849184i \(-0.677094\pi\)
−0.528097 + 0.849184i \(0.677094\pi\)
\(62\) 0 0
\(63\) 4.77846 0.602029
\(64\) 0 0
\(65\) −3.19051 −0.395734
\(66\) 0 0
\(67\) 7.71982 0.943127 0.471563 0.881832i \(-0.343690\pi\)
0.471563 + 0.881832i \(0.343690\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −16.0242 −1.90172 −0.950859 0.309624i \(-0.899797\pi\)
−0.950859 + 0.309624i \(0.899797\pi\)
\(72\) 0 0
\(73\) −7.92332 −0.927355 −0.463677 0.886004i \(-0.653470\pi\)
−0.463677 + 0.886004i \(0.653470\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −15.8061 −1.80127
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 12.5845 1.38133 0.690665 0.723175i \(-0.257318\pi\)
0.690665 + 0.723175i \(0.257318\pi\)
\(84\) 0 0
\(85\) 2.41205 0.261624
\(86\) 0 0
\(87\) 7.96896 0.854363
\(88\) 0 0
\(89\) −3.05863 −0.324214 −0.162107 0.986773i \(-0.551829\pi\)
−0.162107 + 0.986773i \(0.551829\pi\)
\(90\) 0 0
\(91\) 15.2457 1.59818
\(92\) 0 0
\(93\) −10.3354 −1.07173
\(94\) 0 0
\(95\) 2.24914 0.230757
\(96\) 0 0
\(97\) −5.43965 −0.552313 −0.276156 0.961113i \(-0.589061\pi\)
−0.276156 + 0.961113i \(0.589061\pi\)
\(98\) 0 0
\(99\) −3.30777 −0.332444
\(100\) 0 0
\(101\) −8.52932 −0.848699 −0.424349 0.905499i \(-0.639497\pi\)
−0.424349 + 0.905499i \(0.639497\pi\)
\(102\) 0 0
\(103\) 9.32238 0.918562 0.459281 0.888291i \(-0.348107\pi\)
0.459281 + 0.888291i \(0.348107\pi\)
\(104\) 0 0
\(105\) 4.77846 0.466330
\(106\) 0 0
\(107\) 13.1449 1.27076 0.635381 0.772199i \(-0.280843\pi\)
0.635381 + 0.772199i \(0.280843\pi\)
\(108\) 0 0
\(109\) 10.0406 0.961714 0.480857 0.876799i \(-0.340326\pi\)
0.480857 + 0.876799i \(0.340326\pi\)
\(110\) 0 0
\(111\) 11.2767 1.07034
\(112\) 0 0
\(113\) −3.70522 −0.348557 −0.174279 0.984696i \(-0.555759\pi\)
−0.174279 + 0.984696i \(0.555759\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 0 0
\(117\) 3.19051 0.294963
\(118\) 0 0
\(119\) −11.5259 −1.05658
\(120\) 0 0
\(121\) −0.0586332 −0.00533029
\(122\) 0 0
\(123\) −1.02760 −0.0926554
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −1.68879 −0.149856 −0.0749279 0.997189i \(-0.523873\pi\)
−0.0749279 + 0.997189i \(0.523873\pi\)
\(128\) 0 0
\(129\) 10.0552 0.885311
\(130\) 0 0
\(131\) 8.99656 0.786033 0.393017 0.919531i \(-0.371431\pi\)
0.393017 + 0.919531i \(0.371431\pi\)
\(132\) 0 0
\(133\) −10.7474 −0.931920
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −19.5569 −1.67086 −0.835430 0.549597i \(-0.814781\pi\)
−0.835430 + 0.549597i \(0.814781\pi\)
\(138\) 0 0
\(139\) 15.3319 1.30044 0.650219 0.759747i \(-0.274677\pi\)
0.650219 + 0.759747i \(0.274677\pi\)
\(140\) 0 0
\(141\) 2.74742 0.231375
\(142\) 0 0
\(143\) −10.5535 −0.882526
\(144\) 0 0
\(145\) 7.96896 0.661786
\(146\) 0 0
\(147\) −15.8337 −1.30594
\(148\) 0 0
\(149\) 9.80605 0.803343 0.401672 0.915784i \(-0.368429\pi\)
0.401672 + 0.915784i \(0.368429\pi\)
\(150\) 0 0
\(151\) −14.0552 −1.14380 −0.571898 0.820325i \(-0.693793\pi\)
−0.571898 + 0.820325i \(0.693793\pi\)
\(152\) 0 0
\(153\) −2.41205 −0.195003
\(154\) 0 0
\(155\) −10.3354 −0.830157
\(156\) 0 0
\(157\) −2.28018 −0.181978 −0.0909889 0.995852i \(-0.529003\pi\)
−0.0909889 + 0.995852i \(0.529003\pi\)
\(158\) 0 0
\(159\) 5.58795 0.443153
\(160\) 0 0
\(161\) −4.77846 −0.376595
\(162\) 0 0
\(163\) −13.5569 −1.06186 −0.530930 0.847416i \(-0.678157\pi\)
−0.530930 + 0.847416i \(0.678157\pi\)
\(164\) 0 0
\(165\) −3.30777 −0.257510
\(166\) 0 0
\(167\) 12.8647 0.995499 0.497750 0.867321i \(-0.334160\pi\)
0.497750 + 0.867321i \(0.334160\pi\)
\(168\) 0 0
\(169\) −2.82066 −0.216974
\(170\) 0 0
\(171\) −2.24914 −0.171996
\(172\) 0 0
\(173\) −11.3224 −0.860825 −0.430412 0.902632i \(-0.641632\pi\)
−0.430412 + 0.902632i \(0.641632\pi\)
\(174\) 0 0
\(175\) 4.77846 0.361217
\(176\) 0 0
\(177\) −1.96896 −0.147996
\(178\) 0 0
\(179\) 20.9345 1.56472 0.782359 0.622828i \(-0.214016\pi\)
0.782359 + 0.622828i \(0.214016\pi\)
\(180\) 0 0
\(181\) 1.93793 0.144045 0.0720226 0.997403i \(-0.477055\pi\)
0.0720226 + 0.997403i \(0.477055\pi\)
\(182\) 0 0
\(183\) 8.24914 0.609794
\(184\) 0 0
\(185\) 11.2767 0.829082
\(186\) 0 0
\(187\) 7.97852 0.583447
\(188\) 0 0
\(189\) −4.77846 −0.347582
\(190\) 0 0
\(191\) 11.7440 0.849765 0.424882 0.905249i \(-0.360315\pi\)
0.424882 + 0.905249i \(0.360315\pi\)
\(192\) 0 0
\(193\) 3.29317 0.237047 0.118524 0.992951i \(-0.462184\pi\)
0.118524 + 0.992951i \(0.462184\pi\)
\(194\) 0 0
\(195\) 3.19051 0.228477
\(196\) 0 0
\(197\) −6.73281 −0.479693 −0.239847 0.970811i \(-0.577097\pi\)
−0.239847 + 0.970811i \(0.577097\pi\)
\(198\) 0 0
\(199\) −9.82066 −0.696168 −0.348084 0.937463i \(-0.613168\pi\)
−0.348084 + 0.937463i \(0.613168\pi\)
\(200\) 0 0
\(201\) −7.71982 −0.544514
\(202\) 0 0
\(203\) −38.0794 −2.67265
\(204\) 0 0
\(205\) −1.02760 −0.0717705
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) 7.43965 0.514611
\(210\) 0 0
\(211\) −27.0974 −1.86546 −0.932731 0.360573i \(-0.882581\pi\)
−0.932731 + 0.360573i \(0.882581\pi\)
\(212\) 0 0
\(213\) 16.0242 1.09796
\(214\) 0 0
\(215\) 10.0552 0.685759
\(216\) 0 0
\(217\) 49.3871 3.35262
\(218\) 0 0
\(219\) 7.92332 0.535408
\(220\) 0 0
\(221\) −7.69566 −0.517666
\(222\) 0 0
\(223\) −20.9053 −1.39992 −0.699960 0.714182i \(-0.746799\pi\)
−0.699960 + 0.714182i \(0.746799\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −4.00000 −0.265489 −0.132745 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) 0 0
\(229\) 29.0518 1.91979 0.959897 0.280353i \(-0.0904514\pi\)
0.959897 + 0.280353i \(0.0904514\pi\)
\(230\) 0 0
\(231\) 15.8061 1.03996
\(232\) 0 0
\(233\) −20.5535 −1.34650 −0.673252 0.739414i \(-0.735103\pi\)
−0.673252 + 0.739414i \(0.735103\pi\)
\(234\) 0 0
\(235\) 2.74742 0.179222
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −20.0862 −1.29927 −0.649635 0.760246i \(-0.725078\pi\)
−0.649635 + 0.760246i \(0.725078\pi\)
\(240\) 0 0
\(241\) −0.574960 −0.0370364 −0.0185182 0.999829i \(-0.505895\pi\)
−0.0185182 + 0.999829i \(0.505895\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −15.8337 −1.01157
\(246\) 0 0
\(247\) −7.17590 −0.456592
\(248\) 0 0
\(249\) −12.5845 −0.797511
\(250\) 0 0
\(251\) −5.32238 −0.335946 −0.167973 0.985792i \(-0.553722\pi\)
−0.167973 + 0.985792i \(0.553722\pi\)
\(252\) 0 0
\(253\) 3.30777 0.207958
\(254\) 0 0
\(255\) −2.41205 −0.151048
\(256\) 0 0
\(257\) 17.9164 1.11760 0.558799 0.829303i \(-0.311262\pi\)
0.558799 + 0.829303i \(0.311262\pi\)
\(258\) 0 0
\(259\) −53.8854 −3.34828
\(260\) 0 0
\(261\) −7.96896 −0.493267
\(262\) 0 0
\(263\) 18.9655 1.16946 0.584732 0.811226i \(-0.301200\pi\)
0.584732 + 0.811226i \(0.301200\pi\)
\(264\) 0 0
\(265\) 5.58795 0.343265
\(266\) 0 0
\(267\) 3.05863 0.187185
\(268\) 0 0
\(269\) −13.7604 −0.838987 −0.419494 0.907758i \(-0.637792\pi\)
−0.419494 + 0.907758i \(0.637792\pi\)
\(270\) 0 0
\(271\) −19.1595 −1.16386 −0.581928 0.813241i \(-0.697701\pi\)
−0.581928 + 0.813241i \(0.697701\pi\)
\(272\) 0 0
\(273\) −15.2457 −0.922712
\(274\) 0 0
\(275\) −3.30777 −0.199466
\(276\) 0 0
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 0 0
\(279\) 10.3354 0.618762
\(280\) 0 0
\(281\) −5.63359 −0.336072 −0.168036 0.985781i \(-0.553742\pi\)
−0.168036 + 0.985781i \(0.553742\pi\)
\(282\) 0 0
\(283\) −12.0456 −0.716039 −0.358020 0.933714i \(-0.616548\pi\)
−0.358020 + 0.933714i \(0.616548\pi\)
\(284\) 0 0
\(285\) −2.24914 −0.133228
\(286\) 0 0
\(287\) 4.91033 0.289848
\(288\) 0 0
\(289\) −11.1820 −0.657766
\(290\) 0 0
\(291\) 5.43965 0.318878
\(292\) 0 0
\(293\) −3.73443 −0.218168 −0.109084 0.994033i \(-0.534792\pi\)
−0.109084 + 0.994033i \(0.534792\pi\)
\(294\) 0 0
\(295\) −1.96896 −0.114638
\(296\) 0 0
\(297\) 3.30777 0.191936
\(298\) 0 0
\(299\) −3.19051 −0.184512
\(300\) 0 0
\(301\) −48.0483 −2.76946
\(302\) 0 0
\(303\) 8.52932 0.489996
\(304\) 0 0
\(305\) 8.24914 0.472344
\(306\) 0 0
\(307\) −20.6302 −1.17743 −0.588713 0.808342i \(-0.700365\pi\)
−0.588713 + 0.808342i \(0.700365\pi\)
\(308\) 0 0
\(309\) −9.32238 −0.530332
\(310\) 0 0
\(311\) −19.1138 −1.08385 −0.541923 0.840428i \(-0.682304\pi\)
−0.541923 + 0.840428i \(0.682304\pi\)
\(312\) 0 0
\(313\) −2.84053 −0.160556 −0.0802781 0.996773i \(-0.525581\pi\)
−0.0802781 + 0.996773i \(0.525581\pi\)
\(314\) 0 0
\(315\) −4.77846 −0.269236
\(316\) 0 0
\(317\) −18.9560 −1.06467 −0.532337 0.846533i \(-0.678686\pi\)
−0.532337 + 0.846533i \(0.678686\pi\)
\(318\) 0 0
\(319\) 26.3595 1.47585
\(320\) 0 0
\(321\) −13.1449 −0.733675
\(322\) 0 0
\(323\) 5.42504 0.301857
\(324\) 0 0
\(325\) 3.19051 0.176978
\(326\) 0 0
\(327\) −10.0406 −0.555246
\(328\) 0 0
\(329\) −13.1284 −0.723794
\(330\) 0 0
\(331\) −30.5078 −1.67686 −0.838431 0.545008i \(-0.816527\pi\)
−0.838431 + 0.545008i \(0.816527\pi\)
\(332\) 0 0
\(333\) −11.2767 −0.617961
\(334\) 0 0
\(335\) −7.71982 −0.421779
\(336\) 0 0
\(337\) −0.117266 −0.00638790 −0.00319395 0.999995i \(-0.501017\pi\)
−0.00319395 + 0.999995i \(0.501017\pi\)
\(338\) 0 0
\(339\) 3.70522 0.201240
\(340\) 0 0
\(341\) −34.1871 −1.85133
\(342\) 0 0
\(343\) 42.2112 2.27919
\(344\) 0 0
\(345\) −1.00000 −0.0538382
\(346\) 0 0
\(347\) −21.3845 −1.14798 −0.573989 0.818863i \(-0.694605\pi\)
−0.573989 + 0.818863i \(0.694605\pi\)
\(348\) 0 0
\(349\) 3.10428 0.166168 0.0830841 0.996543i \(-0.473523\pi\)
0.0830841 + 0.996543i \(0.473523\pi\)
\(350\) 0 0
\(351\) −3.19051 −0.170297
\(352\) 0 0
\(353\) 25.9164 1.37939 0.689697 0.724098i \(-0.257744\pi\)
0.689697 + 0.724098i \(0.257744\pi\)
\(354\) 0 0
\(355\) 16.0242 0.850474
\(356\) 0 0
\(357\) 11.5259 0.610014
\(358\) 0 0
\(359\) 2.01461 0.106327 0.0531635 0.998586i \(-0.483070\pi\)
0.0531635 + 0.998586i \(0.483070\pi\)
\(360\) 0 0
\(361\) −13.9414 −0.733756
\(362\) 0 0
\(363\) 0.0586332 0.00307744
\(364\) 0 0
\(365\) 7.92332 0.414726
\(366\) 0 0
\(367\) 12.7785 0.667030 0.333515 0.942745i \(-0.391765\pi\)
0.333515 + 0.942745i \(0.391765\pi\)
\(368\) 0 0
\(369\) 1.02760 0.0534946
\(370\) 0 0
\(371\) −26.7018 −1.38629
\(372\) 0 0
\(373\) 14.9345 0.773279 0.386639 0.922231i \(-0.373636\pi\)
0.386639 + 0.922231i \(0.373636\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −25.4250 −1.30946
\(378\) 0 0
\(379\) 13.9931 0.718779 0.359389 0.933188i \(-0.382985\pi\)
0.359389 + 0.933188i \(0.382985\pi\)
\(380\) 0 0
\(381\) 1.68879 0.0865193
\(382\) 0 0
\(383\) 18.1414 0.926984 0.463492 0.886101i \(-0.346596\pi\)
0.463492 + 0.886101i \(0.346596\pi\)
\(384\) 0 0
\(385\) 15.8061 0.805551
\(386\) 0 0
\(387\) −10.0552 −0.511134
\(388\) 0 0
\(389\) −0.850080 −0.0431008 −0.0215504 0.999768i \(-0.506860\pi\)
−0.0215504 + 0.999768i \(0.506860\pi\)
\(390\) 0 0
\(391\) 2.41205 0.121983
\(392\) 0 0
\(393\) −8.99656 −0.453817
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 4.67762 0.234763 0.117381 0.993087i \(-0.462550\pi\)
0.117381 + 0.993087i \(0.462550\pi\)
\(398\) 0 0
\(399\) 10.7474 0.538044
\(400\) 0 0
\(401\) 31.4948 1.57278 0.786389 0.617732i \(-0.211948\pi\)
0.786389 + 0.617732i \(0.211948\pi\)
\(402\) 0 0
\(403\) 32.9751 1.64261
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 37.3009 1.84894
\(408\) 0 0
\(409\) −32.6803 −1.61594 −0.807968 0.589226i \(-0.799433\pi\)
−0.807968 + 0.589226i \(0.799433\pi\)
\(410\) 0 0
\(411\) 19.5569 0.964671
\(412\) 0 0
\(413\) 9.40861 0.462968
\(414\) 0 0
\(415\) −12.5845 −0.617749
\(416\) 0 0
\(417\) −15.3319 −0.750808
\(418\) 0 0
\(419\) −0.926759 −0.0452751 −0.0226376 0.999744i \(-0.507206\pi\)
−0.0226376 + 0.999744i \(0.507206\pi\)
\(420\) 0 0
\(421\) −25.4182 −1.23881 −0.619403 0.785073i \(-0.712625\pi\)
−0.619403 + 0.785073i \(0.712625\pi\)
\(422\) 0 0
\(423\) −2.74742 −0.133584
\(424\) 0 0
\(425\) −2.41205 −0.117002
\(426\) 0 0
\(427\) −39.4182 −1.90758
\(428\) 0 0
\(429\) 10.5535 0.509527
\(430\) 0 0
\(431\) −31.6121 −1.52270 −0.761351 0.648340i \(-0.775464\pi\)
−0.761351 + 0.648340i \(0.775464\pi\)
\(432\) 0 0
\(433\) −27.1855 −1.30645 −0.653225 0.757164i \(-0.726584\pi\)
−0.653225 + 0.757164i \(0.726584\pi\)
\(434\) 0 0
\(435\) −7.96896 −0.382083
\(436\) 0 0
\(437\) 2.24914 0.107591
\(438\) 0 0
\(439\) −17.5569 −0.837946 −0.418973 0.907999i \(-0.637610\pi\)
−0.418973 + 0.907999i \(0.637610\pi\)
\(440\) 0 0
\(441\) 15.8337 0.753983
\(442\) 0 0
\(443\) 27.9785 1.32930 0.664650 0.747155i \(-0.268581\pi\)
0.664650 + 0.747155i \(0.268581\pi\)
\(444\) 0 0
\(445\) 3.05863 0.144993
\(446\) 0 0
\(447\) −9.80605 −0.463810
\(448\) 0 0
\(449\) −9.43459 −0.445246 −0.222623 0.974905i \(-0.571462\pi\)
−0.222623 + 0.974905i \(0.571462\pi\)
\(450\) 0 0
\(451\) −3.39906 −0.160056
\(452\) 0 0
\(453\) 14.0552 0.660371
\(454\) 0 0
\(455\) −15.2457 −0.714730
\(456\) 0 0
\(457\) 12.1629 0.568957 0.284478 0.958682i \(-0.408180\pi\)
0.284478 + 0.958682i \(0.408180\pi\)
\(458\) 0 0
\(459\) 2.41205 0.112585
\(460\) 0 0
\(461\) −14.7328 −0.686176 −0.343088 0.939303i \(-0.611473\pi\)
−0.343088 + 0.939303i \(0.611473\pi\)
\(462\) 0 0
\(463\) −20.2423 −0.940738 −0.470369 0.882470i \(-0.655879\pi\)
−0.470369 + 0.882470i \(0.655879\pi\)
\(464\) 0 0
\(465\) 10.3354 0.479291
\(466\) 0 0
\(467\) −34.5484 −1.59871 −0.799355 0.600859i \(-0.794825\pi\)
−0.799355 + 0.600859i \(0.794825\pi\)
\(468\) 0 0
\(469\) 36.8888 1.70337
\(470\) 0 0
\(471\) 2.28018 0.105065
\(472\) 0 0
\(473\) 33.2603 1.52931
\(474\) 0 0
\(475\) −2.24914 −0.103198
\(476\) 0 0
\(477\) −5.58795 −0.255855
\(478\) 0 0
\(479\) −0.0214836 −0.000981610 0 −0.000490805 1.00000i \(-0.500156\pi\)
−0.000490805 1.00000i \(0.500156\pi\)
\(480\) 0 0
\(481\) −35.9785 −1.64048
\(482\) 0 0
\(483\) 4.77846 0.217427
\(484\) 0 0
\(485\) 5.43965 0.247002
\(486\) 0 0
\(487\) 14.8387 0.672406 0.336203 0.941790i \(-0.390857\pi\)
0.336203 + 0.941790i \(0.390857\pi\)
\(488\) 0 0
\(489\) 13.5569 0.613065
\(490\) 0 0
\(491\) 4.81904 0.217480 0.108740 0.994070i \(-0.465318\pi\)
0.108740 + 0.994070i \(0.465318\pi\)
\(492\) 0 0
\(493\) 19.2215 0.865695
\(494\) 0 0
\(495\) 3.30777 0.148673
\(496\) 0 0
\(497\) −76.5708 −3.43467
\(498\) 0 0
\(499\) −40.0647 −1.79354 −0.896772 0.442493i \(-0.854094\pi\)
−0.896772 + 0.442493i \(0.854094\pi\)
\(500\) 0 0
\(501\) −12.8647 −0.574752
\(502\) 0 0
\(503\) 33.1449 1.47786 0.738928 0.673784i \(-0.235332\pi\)
0.738928 + 0.673784i \(0.235332\pi\)
\(504\) 0 0
\(505\) 8.52932 0.379550
\(506\) 0 0
\(507\) 2.82066 0.125270
\(508\) 0 0
\(509\) 34.9053 1.54715 0.773575 0.633705i \(-0.218467\pi\)
0.773575 + 0.633705i \(0.218467\pi\)
\(510\) 0 0
\(511\) −37.8613 −1.67488
\(512\) 0 0
\(513\) 2.24914 0.0993020
\(514\) 0 0
\(515\) −9.32238 −0.410793
\(516\) 0 0
\(517\) 9.08785 0.399683
\(518\) 0 0
\(519\) 11.3224 0.496997
\(520\) 0 0
\(521\) −15.0180 −0.657953 −0.328976 0.944338i \(-0.606704\pi\)
−0.328976 + 0.944338i \(0.606704\pi\)
\(522\) 0 0
\(523\) 3.00344 0.131331 0.0656656 0.997842i \(-0.479083\pi\)
0.0656656 + 0.997842i \(0.479083\pi\)
\(524\) 0 0
\(525\) −4.77846 −0.208549
\(526\) 0 0
\(527\) −24.9294 −1.08594
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 1.96896 0.0854458
\(532\) 0 0
\(533\) 3.27856 0.142010
\(534\) 0 0
\(535\) −13.1449 −0.568302
\(536\) 0 0
\(537\) −20.9345 −0.903390
\(538\) 0 0
\(539\) −52.3741 −2.25591
\(540\) 0 0
\(541\) 5.90871 0.254035 0.127018 0.991900i \(-0.459459\pi\)
0.127018 + 0.991900i \(0.459459\pi\)
\(542\) 0 0
\(543\) −1.93793 −0.0831645
\(544\) 0 0
\(545\) −10.0406 −0.430092
\(546\) 0 0
\(547\) −39.1138 −1.67239 −0.836193 0.548435i \(-0.815224\pi\)
−0.836193 + 0.548435i \(0.815224\pi\)
\(548\) 0 0
\(549\) −8.24914 −0.352065
\(550\) 0 0
\(551\) 17.9233 0.763559
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −11.2767 −0.478671
\(556\) 0 0
\(557\) −4.59139 −0.194543 −0.0972717 0.995258i \(-0.531012\pi\)
−0.0972717 + 0.995258i \(0.531012\pi\)
\(558\) 0 0
\(559\) −32.0812 −1.35689
\(560\) 0 0
\(561\) −7.97852 −0.336853
\(562\) 0 0
\(563\) 12.3208 0.519258 0.259629 0.965708i \(-0.416400\pi\)
0.259629 + 0.965708i \(0.416400\pi\)
\(564\) 0 0
\(565\) 3.70522 0.155880
\(566\) 0 0
\(567\) 4.77846 0.200676
\(568\) 0 0
\(569\) 25.1430 1.05405 0.527026 0.849849i \(-0.323307\pi\)
0.527026 + 0.849849i \(0.323307\pi\)
\(570\) 0 0
\(571\) 17.2197 0.720623 0.360311 0.932832i \(-0.382670\pi\)
0.360311 + 0.932832i \(0.382670\pi\)
\(572\) 0 0
\(573\) −11.7440 −0.490612
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) 0 0
\(577\) 32.7880 1.36498 0.682491 0.730894i \(-0.260896\pi\)
0.682491 + 0.730894i \(0.260896\pi\)
\(578\) 0 0
\(579\) −3.29317 −0.136859
\(580\) 0 0
\(581\) 60.1346 2.49480
\(582\) 0 0
\(583\) 18.4837 0.765516
\(584\) 0 0
\(585\) −3.19051 −0.131911
\(586\) 0 0
\(587\) 29.9018 1.23418 0.617090 0.786892i \(-0.288311\pi\)
0.617090 + 0.786892i \(0.288311\pi\)
\(588\) 0 0
\(589\) −23.2457 −0.957822
\(590\) 0 0
\(591\) 6.73281 0.276951
\(592\) 0 0
\(593\) 0.656135 0.0269442 0.0134721 0.999909i \(-0.495712\pi\)
0.0134721 + 0.999909i \(0.495712\pi\)
\(594\) 0 0
\(595\) 11.5259 0.472515
\(596\) 0 0
\(597\) 9.82066 0.401933
\(598\) 0 0
\(599\) −41.2603 −1.68585 −0.842925 0.538031i \(-0.819168\pi\)
−0.842925 + 0.538031i \(0.819168\pi\)
\(600\) 0 0
\(601\) −12.1008 −0.493604 −0.246802 0.969066i \(-0.579380\pi\)
−0.246802 + 0.969066i \(0.579380\pi\)
\(602\) 0 0
\(603\) 7.71982 0.314376
\(604\) 0 0
\(605\) 0.0586332 0.00238378
\(606\) 0 0
\(607\) 37.1475 1.50777 0.753886 0.657005i \(-0.228177\pi\)
0.753886 + 0.657005i \(0.228177\pi\)
\(608\) 0 0
\(609\) 38.0794 1.54305
\(610\) 0 0
\(611\) −8.76567 −0.354621
\(612\) 0 0
\(613\) −31.3415 −1.26587 −0.632935 0.774205i \(-0.718150\pi\)
−0.632935 + 0.774205i \(0.718150\pi\)
\(614\) 0 0
\(615\) 1.02760 0.0414367
\(616\) 0 0
\(617\) 1.26213 0.0508115 0.0254057 0.999677i \(-0.491912\pi\)
0.0254057 + 0.999677i \(0.491912\pi\)
\(618\) 0 0
\(619\) 31.1138 1.25057 0.625285 0.780396i \(-0.284983\pi\)
0.625285 + 0.780396i \(0.284983\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) −14.6155 −0.585560
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −7.43965 −0.297111
\(628\) 0 0
\(629\) 27.2001 1.08454
\(630\) 0 0
\(631\) 43.2648 1.72234 0.861172 0.508313i \(-0.169731\pi\)
0.861172 + 0.508313i \(0.169731\pi\)
\(632\) 0 0
\(633\) 27.0974 1.07702
\(634\) 0 0
\(635\) 1.68879 0.0670175
\(636\) 0 0
\(637\) 50.5174 2.00157
\(638\) 0 0
\(639\) −16.0242 −0.633906
\(640\) 0 0
\(641\) −8.01461 −0.316558 −0.158279 0.987394i \(-0.550595\pi\)
−0.158279 + 0.987394i \(0.550595\pi\)
\(642\) 0 0
\(643\) −27.0061 −1.06502 −0.532509 0.846425i \(-0.678751\pi\)
−0.532509 + 0.846425i \(0.678751\pi\)
\(644\) 0 0
\(645\) −10.0552 −0.395923
\(646\) 0 0
\(647\) −14.6854 −0.577341 −0.288670 0.957429i \(-0.593213\pi\)
−0.288670 + 0.957429i \(0.593213\pi\)
\(648\) 0 0
\(649\) −6.51289 −0.255653
\(650\) 0 0
\(651\) −49.3871 −1.93563
\(652\) 0 0
\(653\) 11.4871 0.449525 0.224763 0.974414i \(-0.427839\pi\)
0.224763 + 0.974414i \(0.427839\pi\)
\(654\) 0 0
\(655\) −8.99656 −0.351525
\(656\) 0 0
\(657\) −7.92332 −0.309118
\(658\) 0 0
\(659\) 18.9199 0.737014 0.368507 0.929625i \(-0.379869\pi\)
0.368507 + 0.929625i \(0.379869\pi\)
\(660\) 0 0
\(661\) −47.3707 −1.84251 −0.921253 0.388963i \(-0.872833\pi\)
−0.921253 + 0.388963i \(0.872833\pi\)
\(662\) 0 0
\(663\) 7.69566 0.298875
\(664\) 0 0
\(665\) 10.7474 0.416767
\(666\) 0 0
\(667\) 7.96896 0.308560
\(668\) 0 0
\(669\) 20.9053 0.808245
\(670\) 0 0
\(671\) 27.2863 1.05338
\(672\) 0 0
\(673\) 38.2130 1.47300 0.736502 0.676435i \(-0.236476\pi\)
0.736502 + 0.676435i \(0.236476\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 47.8156 1.83770 0.918852 0.394603i \(-0.129118\pi\)
0.918852 + 0.394603i \(0.129118\pi\)
\(678\) 0 0
\(679\) −25.9931 −0.997525
\(680\) 0 0
\(681\) 4.00000 0.153280
\(682\) 0 0
\(683\) 26.2491 1.00440 0.502198 0.864753i \(-0.332525\pi\)
0.502198 + 0.864753i \(0.332525\pi\)
\(684\) 0 0
\(685\) 19.5569 0.747231
\(686\) 0 0
\(687\) −29.0518 −1.10839
\(688\) 0 0
\(689\) −17.8284 −0.679208
\(690\) 0 0
\(691\) −18.0552 −0.686852 −0.343426 0.939180i \(-0.611587\pi\)
−0.343426 + 0.939180i \(0.611587\pi\)
\(692\) 0 0
\(693\) −15.8061 −0.600422
\(694\) 0 0
\(695\) −15.3319 −0.581573
\(696\) 0 0
\(697\) −2.47862 −0.0938843
\(698\) 0 0
\(699\) 20.5535 0.777404
\(700\) 0 0
\(701\) 0.0766789 0.00289612 0.00144806 0.999999i \(-0.499539\pi\)
0.00144806 + 0.999999i \(0.499539\pi\)
\(702\) 0 0
\(703\) 25.3630 0.956582
\(704\) 0 0
\(705\) −2.74742 −0.103474
\(706\) 0 0
\(707\) −40.7570 −1.53282
\(708\) 0 0
\(709\) −5.48024 −0.205815 −0.102907 0.994691i \(-0.532814\pi\)
−0.102907 + 0.994691i \(0.532814\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −10.3354 −0.387063
\(714\) 0 0
\(715\) 10.5535 0.394678
\(716\) 0 0
\(717\) 20.0862 0.750134
\(718\) 0 0
\(719\) 17.5811 0.655663 0.327832 0.944736i \(-0.393682\pi\)
0.327832 + 0.944736i \(0.393682\pi\)
\(720\) 0 0
\(721\) 44.5466 1.65900
\(722\) 0 0
\(723\) 0.574960 0.0213830
\(724\) 0 0
\(725\) −7.96896 −0.295960
\(726\) 0 0
\(727\) 28.2993 1.04956 0.524781 0.851237i \(-0.324147\pi\)
0.524781 + 0.851237i \(0.324147\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 24.2536 0.897053
\(732\) 0 0
\(733\) 13.9836 0.516495 0.258248 0.966079i \(-0.416855\pi\)
0.258248 + 0.966079i \(0.416855\pi\)
\(734\) 0 0
\(735\) 15.8337 0.584033
\(736\) 0 0
\(737\) −25.5354 −0.940610
\(738\) 0 0
\(739\) 12.9509 0.476407 0.238204 0.971215i \(-0.423441\pi\)
0.238204 + 0.971215i \(0.423441\pi\)
\(740\) 0 0
\(741\) 7.17590 0.263613
\(742\) 0 0
\(743\) −1.64820 −0.0604666 −0.0302333 0.999543i \(-0.509625\pi\)
−0.0302333 + 0.999543i \(0.509625\pi\)
\(744\) 0 0
\(745\) −9.80605 −0.359266
\(746\) 0 0
\(747\) 12.5845 0.460443
\(748\) 0 0
\(749\) 62.8122 2.29511
\(750\) 0 0
\(751\) 34.6233 1.26342 0.631711 0.775204i \(-0.282353\pi\)
0.631711 + 0.775204i \(0.282353\pi\)
\(752\) 0 0
\(753\) 5.32238 0.193958
\(754\) 0 0
\(755\) 14.0552 0.511521
\(756\) 0 0
\(757\) 15.9183 0.578559 0.289280 0.957245i \(-0.406584\pi\)
0.289280 + 0.957245i \(0.406584\pi\)
\(758\) 0 0
\(759\) −3.30777 −0.120065
\(760\) 0 0
\(761\) 1.58795 0.0575631 0.0287816 0.999586i \(-0.490837\pi\)
0.0287816 + 0.999586i \(0.490837\pi\)
\(762\) 0 0
\(763\) 47.9785 1.73694
\(764\) 0 0
\(765\) 2.41205 0.0872079
\(766\) 0 0
\(767\) 6.28200 0.226830
\(768\) 0 0
\(769\) 50.6302 1.82577 0.912885 0.408217i \(-0.133849\pi\)
0.912885 + 0.408217i \(0.133849\pi\)
\(770\) 0 0
\(771\) −17.9164 −0.645245
\(772\) 0 0
\(773\) 5.37758 0.193418 0.0967090 0.995313i \(-0.469168\pi\)
0.0967090 + 0.995313i \(0.469168\pi\)
\(774\) 0 0
\(775\) 10.3354 0.371257
\(776\) 0 0
\(777\) 53.8854 1.93313
\(778\) 0 0
\(779\) −2.31121 −0.0828077
\(780\) 0 0
\(781\) 53.0043 1.89664
\(782\) 0 0
\(783\) 7.96896 0.284788
\(784\) 0 0
\(785\) 2.28018 0.0813830
\(786\) 0 0
\(787\) −6.42666 −0.229086 −0.114543 0.993418i \(-0.536540\pi\)
−0.114543 + 0.993418i \(0.536540\pi\)
\(788\) 0 0
\(789\) −18.9655 −0.675191
\(790\) 0 0
\(791\) −17.7052 −0.629525
\(792\) 0 0
\(793\) −26.3189 −0.934613
\(794\) 0 0
\(795\) −5.58795 −0.198184
\(796\) 0 0
\(797\) −14.2295 −0.504034 −0.252017 0.967723i \(-0.581094\pi\)
−0.252017 + 0.967723i \(0.581094\pi\)
\(798\) 0 0
\(799\) 6.62692 0.234444
\(800\) 0 0
\(801\) −3.05863 −0.108071
\(802\) 0 0
\(803\) 26.2086 0.924880
\(804\) 0 0
\(805\) 4.77846 0.168418
\(806\) 0 0
\(807\) 13.7604 0.484389
\(808\) 0 0
\(809\) −45.5551 −1.60163 −0.800816 0.598911i \(-0.795600\pi\)
−0.800816 + 0.598911i \(0.795600\pi\)
\(810\) 0 0
\(811\) −17.1043 −0.600612 −0.300306 0.953843i \(-0.597089\pi\)
−0.300306 + 0.953843i \(0.597089\pi\)
\(812\) 0 0
\(813\) 19.1595 0.671952
\(814\) 0 0
\(815\) 13.5569 0.474878
\(816\) 0 0
\(817\) 22.6155 0.791218
\(818\) 0 0
\(819\) 15.2457 0.532728
\(820\) 0 0
\(821\) −47.9018 −1.67179 −0.835893 0.548893i \(-0.815050\pi\)
−0.835893 + 0.548893i \(0.815050\pi\)
\(822\) 0 0
\(823\) −19.5309 −0.680806 −0.340403 0.940280i \(-0.610563\pi\)
−0.340403 + 0.940280i \(0.610563\pi\)
\(824\) 0 0
\(825\) 3.30777 0.115162
\(826\) 0 0
\(827\) 19.5879 0.681140 0.340570 0.940219i \(-0.389380\pi\)
0.340570 + 0.940219i \(0.389380\pi\)
\(828\) 0 0
\(829\) −30.1560 −1.04736 −0.523681 0.851914i \(-0.675442\pi\)
−0.523681 + 0.851914i \(0.675442\pi\)
\(830\) 0 0
\(831\) −10.0000 −0.346896
\(832\) 0 0
\(833\) −38.1916 −1.32326
\(834\) 0 0
\(835\) −12.8647 −0.445201
\(836\) 0 0
\(837\) −10.3354 −0.357243
\(838\) 0 0
\(839\) −21.3224 −0.736130 −0.368065 0.929800i \(-0.619980\pi\)
−0.368065 + 0.929800i \(0.619980\pi\)
\(840\) 0 0
\(841\) 34.5044 1.18981
\(842\) 0 0
\(843\) 5.63359 0.194031
\(844\) 0 0
\(845\) 2.82066 0.0970337
\(846\) 0 0
\(847\) −0.280176 −0.00962696
\(848\) 0 0
\(849\) 12.0456 0.413405
\(850\) 0 0
\(851\) 11.2767 0.386562
\(852\) 0 0
\(853\) 46.3741 1.58782 0.793910 0.608035i \(-0.208042\pi\)
0.793910 + 0.608035i \(0.208042\pi\)
\(854\) 0 0
\(855\) 2.24914 0.0769190
\(856\) 0 0
\(857\) −50.9966 −1.74201 −0.871005 0.491275i \(-0.836531\pi\)
−0.871005 + 0.491275i \(0.836531\pi\)
\(858\) 0 0
\(859\) −12.6872 −0.432881 −0.216440 0.976296i \(-0.569445\pi\)
−0.216440 + 0.976296i \(0.569445\pi\)
\(860\) 0 0
\(861\) −4.91033 −0.167344
\(862\) 0 0
\(863\) 46.5827 1.58569 0.792847 0.609421i \(-0.208598\pi\)
0.792847 + 0.609421i \(0.208598\pi\)
\(864\) 0 0
\(865\) 11.3224 0.384973
\(866\) 0 0
\(867\) 11.1820 0.379761
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 24.6302 0.834561
\(872\) 0 0
\(873\) −5.43965 −0.184104
\(874\) 0 0
\(875\) −4.77846 −0.161541
\(876\) 0 0
\(877\) 0.270624 0.00913833 0.00456916 0.999990i \(-0.498546\pi\)
0.00456916 + 0.999990i \(0.498546\pi\)
\(878\) 0 0
\(879\) 3.73443 0.125959
\(880\) 0 0
\(881\) −22.5389 −0.759354 −0.379677 0.925119i \(-0.623965\pi\)
−0.379677 + 0.925119i \(0.623965\pi\)
\(882\) 0 0
\(883\) 48.3043 1.62557 0.812785 0.582564i \(-0.197950\pi\)
0.812785 + 0.582564i \(0.197950\pi\)
\(884\) 0 0
\(885\) 1.96896 0.0661860
\(886\) 0 0
\(887\) 11.9379 0.400836 0.200418 0.979710i \(-0.435770\pi\)
0.200418 + 0.979710i \(0.435770\pi\)
\(888\) 0 0
\(889\) −8.06980 −0.270653
\(890\) 0 0
\(891\) −3.30777 −0.110815
\(892\) 0 0
\(893\) 6.17934 0.206784
\(894\) 0 0
\(895\) −20.9345 −0.699763
\(896\) 0 0
\(897\) 3.19051 0.106528
\(898\) 0 0
\(899\) −82.3622 −2.74693
\(900\) 0 0
\(901\) 13.4784 0.449031
\(902\) 0 0
\(903\) 48.0483 1.59895
\(904\) 0 0
\(905\) −1.93793 −0.0644189
\(906\) 0 0
\(907\) 28.8268 0.957177 0.478589 0.878039i \(-0.341149\pi\)
0.478589 + 0.878039i \(0.341149\pi\)
\(908\) 0 0
\(909\) −8.52932 −0.282900
\(910\) 0 0
\(911\) −54.5726 −1.80807 −0.904035 0.427458i \(-0.859409\pi\)
−0.904035 + 0.427458i \(0.859409\pi\)
\(912\) 0 0
\(913\) −41.6267 −1.37764
\(914\) 0 0
\(915\) −8.24914 −0.272708
\(916\) 0 0
\(917\) 42.9897 1.41964
\(918\) 0 0
\(919\) −30.2277 −0.997118 −0.498559 0.866856i \(-0.666137\pi\)
−0.498559 + 0.866856i \(0.666137\pi\)
\(920\) 0 0
\(921\) 20.6302 0.679787
\(922\) 0 0
\(923\) −51.1252 −1.68281
\(924\) 0 0
\(925\) −11.2767 −0.370777
\(926\) 0 0
\(927\) 9.32238 0.306187
\(928\) 0 0
\(929\) 40.1706 1.31796 0.658978 0.752162i \(-0.270989\pi\)
0.658978 + 0.752162i \(0.270989\pi\)
\(930\) 0 0
\(931\) −35.6121 −1.16714
\(932\) 0 0
\(933\) 19.1138 0.625759
\(934\) 0 0
\(935\) −7.97852 −0.260925
\(936\) 0 0
\(937\) −10.3258 −0.337330 −0.168665 0.985673i \(-0.553946\pi\)
−0.168665 + 0.985673i \(0.553946\pi\)
\(938\) 0 0
\(939\) 2.84053 0.0926971
\(940\) 0 0
\(941\) −43.9096 −1.43141 −0.715706 0.698402i \(-0.753895\pi\)
−0.715706 + 0.698402i \(0.753895\pi\)
\(942\) 0 0
\(943\) −1.02760 −0.0334632
\(944\) 0 0
\(945\) 4.77846 0.155443
\(946\) 0 0
\(947\) 58.4914 1.90072 0.950358 0.311160i \(-0.100717\pi\)
0.950358 + 0.311160i \(0.100717\pi\)
\(948\) 0 0
\(949\) −25.2794 −0.820605
\(950\) 0 0
\(951\) 18.9560 0.614690
\(952\) 0 0
\(953\) 21.5309 0.697455 0.348728 0.937224i \(-0.386614\pi\)
0.348728 + 0.937224i \(0.386614\pi\)
\(954\) 0 0
\(955\) −11.7440 −0.380026
\(956\) 0 0
\(957\) −26.3595 −0.852083
\(958\) 0 0
\(959\) −93.4519 −3.01772
\(960\) 0 0
\(961\) 75.8199 2.44580
\(962\) 0 0
\(963\) 13.1449 0.423587
\(964\) 0 0
\(965\) −3.29317 −0.106011
\(966\) 0 0
\(967\) 14.9629 0.481173 0.240586 0.970628i \(-0.422660\pi\)
0.240586 + 0.970628i \(0.422660\pi\)
\(968\) 0 0
\(969\) −5.42504 −0.174277
\(970\) 0 0
\(971\) 41.0777 1.31825 0.659124 0.752035i \(-0.270927\pi\)
0.659124 + 0.752035i \(0.270927\pi\)
\(972\) 0 0
\(973\) 73.2630 2.34870
\(974\) 0 0
\(975\) −3.19051 −0.102178
\(976\) 0 0
\(977\) 5.76041 0.184292 0.0921459 0.995746i \(-0.470627\pi\)
0.0921459 + 0.995746i \(0.470627\pi\)
\(978\) 0 0
\(979\) 10.1173 0.323349
\(980\) 0 0
\(981\) 10.0406 0.320571
\(982\) 0 0
\(983\) 14.1223 0.450432 0.225216 0.974309i \(-0.427691\pi\)
0.225216 + 0.974309i \(0.427691\pi\)
\(984\) 0 0
\(985\) 6.73281 0.214525
\(986\) 0 0
\(987\) 13.1284 0.417883
\(988\) 0 0
\(989\) 10.0552 0.319737
\(990\) 0 0
\(991\) −30.4458 −0.967142 −0.483571 0.875305i \(-0.660660\pi\)
−0.483571 + 0.875305i \(0.660660\pi\)
\(992\) 0 0
\(993\) 30.5078 0.968137
\(994\) 0 0
\(995\) 9.82066 0.311336
\(996\) 0 0
\(997\) 1.17590 0.0372411 0.0186206 0.999827i \(-0.494073\pi\)
0.0186206 + 0.999827i \(0.494073\pi\)
\(998\) 0 0
\(999\) 11.2767 0.356780
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 5520.2.a.bx.1.3 3
4.3 odd 2 2760.2.a.r.1.1 3
12.11 even 2 8280.2.a.bm.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2760.2.a.r.1.1 3 4.3 odd 2
5520.2.a.bx.1.3 3 1.1 even 1 trivial
8280.2.a.bm.1.1 3 12.11 even 2