Properties

Label 5520.2.a.bo.1.1
Level $5520$
Weight $2$
Character 5520.1
Self dual yes
Analytic conductor $44.077$
Analytic rank $0$
Dimension $2$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{11}) \)
Defining polynomial: \( x^{2} - 11 \) 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 1.1
Root \(-3.31662\) of defining polynomial
Character \(\chi\) \(=\) 5520.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} +3.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000 q^{5} +3.00000 q^{7} +1.00000 q^{9} -2.31662 q^{11} +4.31662 q^{13} -1.00000 q^{15} +3.31662 q^{17} +8.31662 q^{19} +3.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -7.31662 q^{29} -3.63325 q^{31} -2.31662 q^{33} -3.00000 q^{35} -1.63325 q^{37} +4.31662 q^{39} -3.31662 q^{41} +10.6332 q^{43} -1.00000 q^{45} +8.94987 q^{47} +2.00000 q^{49} +3.31662 q^{51} +9.94987 q^{53} +2.31662 q^{55} +8.31662 q^{57} -7.94987 q^{59} +10.3166 q^{61} +3.00000 q^{63} -4.31662 q^{65} -15.6332 q^{67} -1.00000 q^{69} -11.9499 q^{71} +8.31662 q^{73} +1.00000 q^{75} -6.94987 q^{77} +1.00000 q^{81} +1.31662 q^{83} -3.31662 q^{85} -7.31662 q^{87} -4.63325 q^{89} +12.9499 q^{91} -3.63325 q^{93} -8.31662 q^{95} +12.0000 q^{97} -2.31662 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} + 6 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{5} + 6 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{13} - 2 q^{15} + 10 q^{19} + 6 q^{21} - 2 q^{23} + 2 q^{25} + 2 q^{27} - 8 q^{29} + 6 q^{31} + 2 q^{33} - 6 q^{35} + 10 q^{37} + 2 q^{39} + 8 q^{43} - 2 q^{45} - 2 q^{47} + 4 q^{49} - 2 q^{55} + 10 q^{57} + 4 q^{59} + 14 q^{61} + 6 q^{63} - 2 q^{65} - 18 q^{67} - 2 q^{69} - 4 q^{71} + 10 q^{73} + 2 q^{75} + 6 q^{77} + 2 q^{81} - 4 q^{83} - 8 q^{87} + 4 q^{89} + 6 q^{91} + 6 q^{93} - 10 q^{95} + 24 q^{97} + 2 q^{99}+O(q^{100}) \) 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
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.31662 −0.698489 −0.349244 0.937032i \(-0.613562\pi\)
−0.349244 + 0.937032i \(0.613562\pi\)
\(12\) 0 0
\(13\) 4.31662 1.19722 0.598608 0.801042i \(-0.295721\pi\)
0.598608 + 0.801042i \(0.295721\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 3.31662 0.804400 0.402200 0.915552i \(-0.368246\pi\)
0.402200 + 0.915552i \(0.368246\pi\)
\(18\) 0 0
\(19\) 8.31662 1.90796 0.953982 0.299863i \(-0.0969410\pi\)
0.953982 + 0.299863i \(0.0969410\pi\)
\(20\) 0 0
\(21\) 3.00000 0.654654
\(22\) 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
\(29\) −7.31662 −1.35866 −0.679332 0.733831i \(-0.737730\pi\)
−0.679332 + 0.733831i \(0.737730\pi\)
\(30\) 0 0
\(31\) −3.63325 −0.652551 −0.326275 0.945275i \(-0.605794\pi\)
−0.326275 + 0.945275i \(0.605794\pi\)
\(32\) 0 0
\(33\) −2.31662 −0.403273
\(34\) 0 0
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) −1.63325 −0.268505 −0.134252 0.990947i \(-0.542863\pi\)
−0.134252 + 0.990947i \(0.542863\pi\)
\(38\) 0 0
\(39\) 4.31662 0.691213
\(40\) 0 0
\(41\) −3.31662 −0.517970 −0.258985 0.965881i \(-0.583388\pi\)
−0.258985 + 0.965881i \(0.583388\pi\)
\(42\) 0 0
\(43\) 10.6332 1.62156 0.810778 0.585354i \(-0.199045\pi\)
0.810778 + 0.585354i \(0.199045\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 8.94987 1.30547 0.652737 0.757585i \(-0.273621\pi\)
0.652737 + 0.757585i \(0.273621\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) 3.31662 0.464420
\(52\) 0 0
\(53\) 9.94987 1.36672 0.683360 0.730081i \(-0.260518\pi\)
0.683360 + 0.730081i \(0.260518\pi\)
\(54\) 0 0
\(55\) 2.31662 0.312374
\(56\) 0 0
\(57\) 8.31662 1.10156
\(58\) 0 0
\(59\) −7.94987 −1.03499 −0.517493 0.855688i \(-0.673135\pi\)
−0.517493 + 0.855688i \(0.673135\pi\)
\(60\) 0 0
\(61\) 10.3166 1.32091 0.660454 0.750866i \(-0.270364\pi\)
0.660454 + 0.750866i \(0.270364\pi\)
\(62\) 0 0
\(63\) 3.00000 0.377964
\(64\) 0 0
\(65\) −4.31662 −0.535411
\(66\) 0 0
\(67\) −15.6332 −1.90991 −0.954953 0.296758i \(-0.904095\pi\)
−0.954953 + 0.296758i \(0.904095\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −11.9499 −1.41819 −0.709095 0.705113i \(-0.750896\pi\)
−0.709095 + 0.705113i \(0.750896\pi\)
\(72\) 0 0
\(73\) 8.31662 0.973387 0.486694 0.873573i \(-0.338203\pi\)
0.486694 + 0.873573i \(0.338203\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −6.94987 −0.792012
\(78\) 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
\(83\) 1.31662 0.144518 0.0722592 0.997386i \(-0.476979\pi\)
0.0722592 + 0.997386i \(0.476979\pi\)
\(84\) 0 0
\(85\) −3.31662 −0.359738
\(86\) 0 0
\(87\) −7.31662 −0.784425
\(88\) 0 0
\(89\) −4.63325 −0.491123 −0.245562 0.969381i \(-0.578972\pi\)
−0.245562 + 0.969381i \(0.578972\pi\)
\(90\) 0 0
\(91\) 12.9499 1.35752
\(92\) 0 0
\(93\) −3.63325 −0.376750
\(94\) 0 0
\(95\) −8.31662 −0.853268
\(96\) 0 0
\(97\) 12.0000 1.21842 0.609208 0.793011i \(-0.291488\pi\)
0.609208 + 0.793011i \(0.291488\pi\)
\(98\) 0 0
\(99\) −2.31662 −0.232830
\(100\) 0 0
\(101\) −1.31662 −0.131009 −0.0655045 0.997852i \(-0.520866\pi\)
−0.0655045 + 0.997852i \(0.520866\pi\)
\(102\) 0 0
\(103\) −18.0000 −1.77359 −0.886796 0.462160i \(-0.847074\pi\)
−0.886796 + 0.462160i \(0.847074\pi\)
\(104\) 0 0
\(105\) −3.00000 −0.292770
\(106\) 0 0
\(107\) −5.94987 −0.575196 −0.287598 0.957751i \(-0.592857\pi\)
−0.287598 + 0.957751i \(0.592857\pi\)
\(108\) 0 0
\(109\) 8.94987 0.857242 0.428621 0.903484i \(-0.358999\pi\)
0.428621 + 0.903484i \(0.358999\pi\)
\(110\) 0 0
\(111\) −1.63325 −0.155021
\(112\) 0 0
\(113\) 8.68338 0.816863 0.408432 0.912789i \(-0.366076\pi\)
0.408432 + 0.912789i \(0.366076\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 0 0
\(117\) 4.31662 0.399072
\(118\) 0 0
\(119\) 9.94987 0.912103
\(120\) 0 0
\(121\) −5.63325 −0.512114
\(122\) 0 0
\(123\) −3.31662 −0.299050
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −14.9499 −1.32659 −0.663293 0.748359i \(-0.730842\pi\)
−0.663293 + 0.748359i \(0.730842\pi\)
\(128\) 0 0
\(129\) 10.6332 0.936205
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 24.9499 2.16343
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 13.2665 1.13343 0.566717 0.823913i \(-0.308213\pi\)
0.566717 + 0.823913i \(0.308213\pi\)
\(138\) 0 0
\(139\) −11.6332 −0.986719 −0.493360 0.869825i \(-0.664231\pi\)
−0.493360 + 0.869825i \(0.664231\pi\)
\(140\) 0 0
\(141\) 8.94987 0.753715
\(142\) 0 0
\(143\) −10.0000 −0.836242
\(144\) 0 0
\(145\) 7.31662 0.607613
\(146\) 0 0
\(147\) 2.00000 0.164957
\(148\) 0 0
\(149\) −4.31662 −0.353632 −0.176816 0.984244i \(-0.556580\pi\)
−0.176816 + 0.984244i \(0.556580\pi\)
\(150\) 0 0
\(151\) 14.6332 1.19084 0.595418 0.803416i \(-0.296986\pi\)
0.595418 + 0.803416i \(0.296986\pi\)
\(152\) 0 0
\(153\) 3.31662 0.268133
\(154\) 0 0
\(155\) 3.63325 0.291830
\(156\) 0 0
\(157\) 19.6332 1.56690 0.783452 0.621452i \(-0.213457\pi\)
0.783452 + 0.621452i \(0.213457\pi\)
\(158\) 0 0
\(159\) 9.94987 0.789076
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) 0 0
\(163\) 23.2665 1.82237 0.911186 0.411994i \(-0.135168\pi\)
0.911186 + 0.411994i \(0.135168\pi\)
\(164\) 0 0
\(165\) 2.31662 0.180349
\(166\) 0 0
\(167\) 13.5831 1.05109 0.525547 0.850765i \(-0.323861\pi\)
0.525547 + 0.850765i \(0.323861\pi\)
\(168\) 0 0
\(169\) 5.63325 0.433327
\(170\) 0 0
\(171\) 8.31662 0.635988
\(172\) 0 0
\(173\) 4.00000 0.304114 0.152057 0.988372i \(-0.451410\pi\)
0.152057 + 0.988372i \(0.451410\pi\)
\(174\) 0 0
\(175\) 3.00000 0.226779
\(176\) 0 0
\(177\) −7.94987 −0.597549
\(178\) 0 0
\(179\) 22.6332 1.69169 0.845844 0.533430i \(-0.179097\pi\)
0.845844 + 0.533430i \(0.179097\pi\)
\(180\) 0 0
\(181\) 7.36675 0.547566 0.273783 0.961791i \(-0.411725\pi\)
0.273783 + 0.961791i \(0.411725\pi\)
\(182\) 0 0
\(183\) 10.3166 0.762627
\(184\) 0 0
\(185\) 1.63325 0.120079
\(186\) 0 0
\(187\) −7.68338 −0.561864
\(188\) 0 0
\(189\) 3.00000 0.218218
\(190\) 0 0
\(191\) 13.5831 0.982840 0.491420 0.870923i \(-0.336478\pi\)
0.491420 + 0.870923i \(0.336478\pi\)
\(192\) 0 0
\(193\) −0.633250 −0.0455823 −0.0227912 0.999740i \(-0.507255\pi\)
−0.0227912 + 0.999740i \(0.507255\pi\)
\(194\) 0 0
\(195\) −4.31662 −0.309120
\(196\) 0 0
\(197\) 11.8997 0.847822 0.423911 0.905704i \(-0.360657\pi\)
0.423911 + 0.905704i \(0.360657\pi\)
\(198\) 0 0
\(199\) −14.6332 −1.03732 −0.518662 0.854980i \(-0.673570\pi\)
−0.518662 + 0.854980i \(0.673570\pi\)
\(200\) 0 0
\(201\) −15.6332 −1.10268
\(202\) 0 0
\(203\) −21.9499 −1.54058
\(204\) 0 0
\(205\) 3.31662 0.231643
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) −19.2665 −1.33269
\(210\) 0 0
\(211\) 3.63325 0.250123 0.125062 0.992149i \(-0.460087\pi\)
0.125062 + 0.992149i \(0.460087\pi\)
\(212\) 0 0
\(213\) −11.9499 −0.818792
\(214\) 0 0
\(215\) −10.6332 −0.725182
\(216\) 0 0
\(217\) −10.8997 −0.739923
\(218\) 0 0
\(219\) 8.31662 0.561985
\(220\) 0 0
\(221\) 14.3166 0.963040
\(222\) 0 0
\(223\) −23.2665 −1.55804 −0.779020 0.626999i \(-0.784283\pi\)
−0.779020 + 0.626999i \(0.784283\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 28.0000 1.85843 0.929213 0.369546i \(-0.120487\pi\)
0.929213 + 0.369546i \(0.120487\pi\)
\(228\) 0 0
\(229\) 3.36675 0.222481 0.111241 0.993794i \(-0.464518\pi\)
0.111241 + 0.993794i \(0.464518\pi\)
\(230\) 0 0
\(231\) −6.94987 −0.457268
\(232\) 0 0
\(233\) 8.00000 0.524097 0.262049 0.965055i \(-0.415602\pi\)
0.262049 + 0.965055i \(0.415602\pi\)
\(234\) 0 0
\(235\) −8.94987 −0.583825
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 15.9499 1.03171 0.515856 0.856675i \(-0.327474\pi\)
0.515856 + 0.856675i \(0.327474\pi\)
\(240\) 0 0
\(241\) −26.2164 −1.68875 −0.844373 0.535756i \(-0.820027\pi\)
−0.844373 + 0.535756i \(0.820027\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −2.00000 −0.127775
\(246\) 0 0
\(247\) 35.8997 2.28425
\(248\) 0 0
\(249\) 1.31662 0.0834377
\(250\) 0 0
\(251\) −23.2665 −1.46857 −0.734284 0.678842i \(-0.762482\pi\)
−0.734284 + 0.678842i \(0.762482\pi\)
\(252\) 0 0
\(253\) 2.31662 0.145645
\(254\) 0 0
\(255\) −3.31662 −0.207695
\(256\) 0 0
\(257\) 0.949874 0.0592515 0.0296258 0.999561i \(-0.490568\pi\)
0.0296258 + 0.999561i \(0.490568\pi\)
\(258\) 0 0
\(259\) −4.89975 −0.304456
\(260\) 0 0
\(261\) −7.31662 −0.452888
\(262\) 0 0
\(263\) −28.5831 −1.76251 −0.881255 0.472640i \(-0.843301\pi\)
−0.881255 + 0.472640i \(0.843301\pi\)
\(264\) 0 0
\(265\) −9.94987 −0.611216
\(266\) 0 0
\(267\) −4.63325 −0.283550
\(268\) 0 0
\(269\) −22.5831 −1.37692 −0.688459 0.725276i \(-0.741712\pi\)
−0.688459 + 0.725276i \(0.741712\pi\)
\(270\) 0 0
\(271\) −24.2665 −1.47408 −0.737042 0.675846i \(-0.763778\pi\)
−0.737042 + 0.675846i \(0.763778\pi\)
\(272\) 0 0
\(273\) 12.9499 0.783762
\(274\) 0 0
\(275\) −2.31662 −0.139698
\(276\) 0 0
\(277\) 18.0000 1.08152 0.540758 0.841178i \(-0.318138\pi\)
0.540758 + 0.841178i \(0.318138\pi\)
\(278\) 0 0
\(279\) −3.63325 −0.217517
\(280\) 0 0
\(281\) 2.94987 0.175975 0.0879874 0.996122i \(-0.471956\pi\)
0.0879874 + 0.996122i \(0.471956\pi\)
\(282\) 0 0
\(283\) 5.63325 0.334862 0.167431 0.985884i \(-0.446453\pi\)
0.167431 + 0.985884i \(0.446453\pi\)
\(284\) 0 0
\(285\) −8.31662 −0.492634
\(286\) 0 0
\(287\) −9.94987 −0.587323
\(288\) 0 0
\(289\) −6.00000 −0.352941
\(290\) 0 0
\(291\) 12.0000 0.703452
\(292\) 0 0
\(293\) −7.94987 −0.464437 −0.232218 0.972664i \(-0.574598\pi\)
−0.232218 + 0.972664i \(0.574598\pi\)
\(294\) 0 0
\(295\) 7.94987 0.462860
\(296\) 0 0
\(297\) −2.31662 −0.134424
\(298\) 0 0
\(299\) −4.31662 −0.249637
\(300\) 0 0
\(301\) 31.8997 1.83867
\(302\) 0 0
\(303\) −1.31662 −0.0756381
\(304\) 0 0
\(305\) −10.3166 −0.590728
\(306\) 0 0
\(307\) 24.9499 1.42396 0.711982 0.702197i \(-0.247798\pi\)
0.711982 + 0.702197i \(0.247798\pi\)
\(308\) 0 0
\(309\) −18.0000 −1.02398
\(310\) 0 0
\(311\) −2.73350 −0.155003 −0.0775013 0.996992i \(-0.524694\pi\)
−0.0775013 + 0.996992i \(0.524694\pi\)
\(312\) 0 0
\(313\) 33.6332 1.90106 0.950532 0.310627i \(-0.100539\pi\)
0.950532 + 0.310627i \(0.100539\pi\)
\(314\) 0 0
\(315\) −3.00000 −0.169031
\(316\) 0 0
\(317\) −16.3166 −0.916433 −0.458216 0.888841i \(-0.651512\pi\)
−0.458216 + 0.888841i \(0.651512\pi\)
\(318\) 0 0
\(319\) 16.9499 0.949011
\(320\) 0 0
\(321\) −5.94987 −0.332090
\(322\) 0 0
\(323\) 27.5831 1.53477
\(324\) 0 0
\(325\) 4.31662 0.239443
\(326\) 0 0
\(327\) 8.94987 0.494929
\(328\) 0 0
\(329\) 26.8496 1.48027
\(330\) 0 0
\(331\) −21.5330 −1.18356 −0.591780 0.806099i \(-0.701575\pi\)
−0.591780 + 0.806099i \(0.701575\pi\)
\(332\) 0 0
\(333\) −1.63325 −0.0895015
\(334\) 0 0
\(335\) 15.6332 0.854136
\(336\) 0 0
\(337\) 4.73350 0.257850 0.128925 0.991654i \(-0.458847\pi\)
0.128925 + 0.991654i \(0.458847\pi\)
\(338\) 0 0
\(339\) 8.68338 0.471616
\(340\) 0 0
\(341\) 8.41688 0.455799
\(342\) 0 0
\(343\) −15.0000 −0.809924
\(344\) 0 0
\(345\) 1.00000 0.0538382
\(346\) 0 0
\(347\) 24.6332 1.32238 0.661191 0.750218i \(-0.270051\pi\)
0.661191 + 0.750218i \(0.270051\pi\)
\(348\) 0 0
\(349\) −8.36675 −0.447862 −0.223931 0.974605i \(-0.571889\pi\)
−0.223931 + 0.974605i \(0.571889\pi\)
\(350\) 0 0
\(351\) 4.31662 0.230404
\(352\) 0 0
\(353\) 26.2164 1.39536 0.697678 0.716411i \(-0.254216\pi\)
0.697678 + 0.716411i \(0.254216\pi\)
\(354\) 0 0
\(355\) 11.9499 0.634233
\(356\) 0 0
\(357\) 9.94987 0.526603
\(358\) 0 0
\(359\) −30.8496 −1.62818 −0.814090 0.580738i \(-0.802764\pi\)
−0.814090 + 0.580738i \(0.802764\pi\)
\(360\) 0 0
\(361\) 50.1662 2.64033
\(362\) 0 0
\(363\) −5.63325 −0.295669
\(364\) 0 0
\(365\) −8.31662 −0.435312
\(366\) 0 0
\(367\) −35.5330 −1.85481 −0.927404 0.374062i \(-0.877965\pi\)
−0.927404 + 0.374062i \(0.877965\pi\)
\(368\) 0 0
\(369\) −3.31662 −0.172657
\(370\) 0 0
\(371\) 29.8496 1.54972
\(372\) 0 0
\(373\) −9.89975 −0.512590 −0.256295 0.966599i \(-0.582502\pi\)
−0.256295 + 0.966599i \(0.582502\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −31.5831 −1.62661
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) −14.9499 −0.765905
\(382\) 0 0
\(383\) −12.6834 −0.648090 −0.324045 0.946042i \(-0.605043\pi\)
−0.324045 + 0.946042i \(0.605043\pi\)
\(384\) 0 0
\(385\) 6.94987 0.354198
\(386\) 0 0
\(387\) 10.6332 0.540518
\(388\) 0 0
\(389\) 15.8997 0.806149 0.403075 0.915167i \(-0.367942\pi\)
0.403075 + 0.915167i \(0.367942\pi\)
\(390\) 0 0
\(391\) −3.31662 −0.167729
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 12.0000 0.602263 0.301131 0.953583i \(-0.402636\pi\)
0.301131 + 0.953583i \(0.402636\pi\)
\(398\) 0 0
\(399\) 24.9499 1.24906
\(400\) 0 0
\(401\) 19.8997 0.993746 0.496873 0.867823i \(-0.334481\pi\)
0.496873 + 0.867823i \(0.334481\pi\)
\(402\) 0 0
\(403\) −15.6834 −0.781245
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 3.78363 0.187547
\(408\) 0 0
\(409\) 13.6332 0.674121 0.337060 0.941483i \(-0.390567\pi\)
0.337060 + 0.941483i \(0.390567\pi\)
\(410\) 0 0
\(411\) 13.2665 0.654388
\(412\) 0 0
\(413\) −23.8496 −1.17356
\(414\) 0 0
\(415\) −1.31662 −0.0646306
\(416\) 0 0
\(417\) −11.6332 −0.569683
\(418\) 0 0
\(419\) 18.3166 0.894826 0.447413 0.894328i \(-0.352345\pi\)
0.447413 + 0.894328i \(0.352345\pi\)
\(420\) 0 0
\(421\) −9.58312 −0.467053 −0.233526 0.972350i \(-0.575027\pi\)
−0.233526 + 0.972350i \(0.575027\pi\)
\(422\) 0 0
\(423\) 8.94987 0.435158
\(424\) 0 0
\(425\) 3.31662 0.160880
\(426\) 0 0
\(427\) 30.9499 1.49777
\(428\) 0 0
\(429\) −10.0000 −0.482805
\(430\) 0 0
\(431\) 3.36675 0.162171 0.0810853 0.996707i \(-0.474161\pi\)
0.0810853 + 0.996707i \(0.474161\pi\)
\(432\) 0 0
\(433\) −23.6332 −1.13574 −0.567871 0.823118i \(-0.692233\pi\)
−0.567871 + 0.823118i \(0.692233\pi\)
\(434\) 0 0
\(435\) 7.31662 0.350805
\(436\) 0 0
\(437\) −8.31662 −0.397838
\(438\) 0 0
\(439\) 22.0000 1.05000 0.525001 0.851101i \(-0.324065\pi\)
0.525001 + 0.851101i \(0.324065\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 0 0
\(443\) 28.3166 1.34536 0.672682 0.739932i \(-0.265142\pi\)
0.672682 + 0.739932i \(0.265142\pi\)
\(444\) 0 0
\(445\) 4.63325 0.219637
\(446\) 0 0
\(447\) −4.31662 −0.204169
\(448\) 0 0
\(449\) −7.31662 −0.345293 −0.172646 0.984984i \(-0.555232\pi\)
−0.172646 + 0.984984i \(0.555232\pi\)
\(450\) 0 0
\(451\) 7.68338 0.361796
\(452\) 0 0
\(453\) 14.6332 0.687530
\(454\) 0 0
\(455\) −12.9499 −0.607099
\(456\) 0 0
\(457\) 25.6332 1.19907 0.599536 0.800347i \(-0.295352\pi\)
0.599536 + 0.800347i \(0.295352\pi\)
\(458\) 0 0
\(459\) 3.31662 0.154807
\(460\) 0 0
\(461\) 2.63325 0.122643 0.0613213 0.998118i \(-0.480469\pi\)
0.0613213 + 0.998118i \(0.480469\pi\)
\(462\) 0 0
\(463\) −38.2164 −1.77607 −0.888033 0.459780i \(-0.847928\pi\)
−0.888033 + 0.459780i \(0.847928\pi\)
\(464\) 0 0
\(465\) 3.63325 0.168488
\(466\) 0 0
\(467\) 10.0501 0.465064 0.232532 0.972589i \(-0.425299\pi\)
0.232532 + 0.972589i \(0.425299\pi\)
\(468\) 0 0
\(469\) −46.8997 −2.16563
\(470\) 0 0
\(471\) 19.6332 0.904652
\(472\) 0 0
\(473\) −24.6332 −1.13264
\(474\) 0 0
\(475\) 8.31662 0.381593
\(476\) 0 0
\(477\) 9.94987 0.455573
\(478\) 0 0
\(479\) −11.6834 −0.533827 −0.266914 0.963720i \(-0.586004\pi\)
−0.266914 + 0.963720i \(0.586004\pi\)
\(480\) 0 0
\(481\) −7.05013 −0.321458
\(482\) 0 0
\(483\) −3.00000 −0.136505
\(484\) 0 0
\(485\) −12.0000 −0.544892
\(486\) 0 0
\(487\) 19.5831 0.887396 0.443698 0.896176i \(-0.353666\pi\)
0.443698 + 0.896176i \(0.353666\pi\)
\(488\) 0 0
\(489\) 23.2665 1.05215
\(490\) 0 0
\(491\) 9.94987 0.449032 0.224516 0.974470i \(-0.427920\pi\)
0.224516 + 0.974470i \(0.427920\pi\)
\(492\) 0 0
\(493\) −24.2665 −1.09291
\(494\) 0 0
\(495\) 2.31662 0.104125
\(496\) 0 0
\(497\) −35.8496 −1.60808
\(498\) 0 0
\(499\) 8.26650 0.370059 0.185030 0.982733i \(-0.440762\pi\)
0.185030 + 0.982733i \(0.440762\pi\)
\(500\) 0 0
\(501\) 13.5831 0.606849
\(502\) 0 0
\(503\) 21.8496 0.974227 0.487113 0.873339i \(-0.338050\pi\)
0.487113 + 0.873339i \(0.338050\pi\)
\(504\) 0 0
\(505\) 1.31662 0.0585890
\(506\) 0 0
\(507\) 5.63325 0.250181
\(508\) 0 0
\(509\) 8.00000 0.354594 0.177297 0.984157i \(-0.443265\pi\)
0.177297 + 0.984157i \(0.443265\pi\)
\(510\) 0 0
\(511\) 24.9499 1.10372
\(512\) 0 0
\(513\) 8.31662 0.367188
\(514\) 0 0
\(515\) 18.0000 0.793175
\(516\) 0 0
\(517\) −20.7335 −0.911858
\(518\) 0 0
\(519\) 4.00000 0.175581
\(520\) 0 0
\(521\) 6.31662 0.276736 0.138368 0.990381i \(-0.455814\pi\)
0.138368 + 0.990381i \(0.455814\pi\)
\(522\) 0 0
\(523\) −30.5330 −1.33511 −0.667557 0.744558i \(-0.732660\pi\)
−0.667557 + 0.744558i \(0.732660\pi\)
\(524\) 0 0
\(525\) 3.00000 0.130931
\(526\) 0 0
\(527\) −12.0501 −0.524912
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −7.94987 −0.344995
\(532\) 0 0
\(533\) −14.3166 −0.620122
\(534\) 0 0
\(535\) 5.94987 0.257236
\(536\) 0 0
\(537\) 22.6332 0.976697
\(538\) 0 0
\(539\) −4.63325 −0.199568
\(540\) 0 0
\(541\) −5.26650 −0.226424 −0.113212 0.993571i \(-0.536114\pi\)
−0.113212 + 0.993571i \(0.536114\pi\)
\(542\) 0 0
\(543\) 7.36675 0.316138
\(544\) 0 0
\(545\) −8.94987 −0.383370
\(546\) 0 0
\(547\) −1.26650 −0.0541516 −0.0270758 0.999633i \(-0.508620\pi\)
−0.0270758 + 0.999633i \(0.508620\pi\)
\(548\) 0 0
\(549\) 10.3166 0.440303
\(550\) 0 0
\(551\) −60.8496 −2.59228
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 1.63325 0.0693276
\(556\) 0 0
\(557\) −6.05013 −0.256352 −0.128176 0.991751i \(-0.540912\pi\)
−0.128176 + 0.991751i \(0.540912\pi\)
\(558\) 0 0
\(559\) 45.8997 1.94135
\(560\) 0 0
\(561\) −7.68338 −0.324392
\(562\) 0 0
\(563\) −43.3166 −1.82558 −0.912789 0.408431i \(-0.866076\pi\)
−0.912789 + 0.408431i \(0.866076\pi\)
\(564\) 0 0
\(565\) −8.68338 −0.365312
\(566\) 0 0
\(567\) 3.00000 0.125988
\(568\) 0 0
\(569\) −35.8997 −1.50500 −0.752498 0.658595i \(-0.771151\pi\)
−0.752498 + 0.658595i \(0.771151\pi\)
\(570\) 0 0
\(571\) −5.05013 −0.211341 −0.105671 0.994401i \(-0.533699\pi\)
−0.105671 + 0.994401i \(0.533699\pi\)
\(572\) 0 0
\(573\) 13.5831 0.567443
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) 0 0
\(577\) 17.2665 0.718814 0.359407 0.933181i \(-0.382979\pi\)
0.359407 + 0.933181i \(0.382979\pi\)
\(578\) 0 0
\(579\) −0.633250 −0.0263170
\(580\) 0 0
\(581\) 3.94987 0.163868
\(582\) 0 0
\(583\) −23.0501 −0.954639
\(584\) 0 0
\(585\) −4.31662 −0.178470
\(586\) 0 0
\(587\) −2.00000 −0.0825488 −0.0412744 0.999148i \(-0.513142\pi\)
−0.0412744 + 0.999148i \(0.513142\pi\)
\(588\) 0 0
\(589\) −30.2164 −1.24504
\(590\) 0 0
\(591\) 11.8997 0.489490
\(592\) 0 0
\(593\) −15.6834 −0.644039 −0.322020 0.946733i \(-0.604362\pi\)
−0.322020 + 0.946733i \(0.604362\pi\)
\(594\) 0 0
\(595\) −9.94987 −0.407905
\(596\) 0 0
\(597\) −14.6332 −0.598899
\(598\) 0 0
\(599\) −15.3668 −0.627868 −0.313934 0.949445i \(-0.601647\pi\)
−0.313934 + 0.949445i \(0.601647\pi\)
\(600\) 0 0
\(601\) −3.63325 −0.148203 −0.0741017 0.997251i \(-0.523609\pi\)
−0.0741017 + 0.997251i \(0.523609\pi\)
\(602\) 0 0
\(603\) −15.6332 −0.636635
\(604\) 0 0
\(605\) 5.63325 0.229024
\(606\) 0 0
\(607\) −3.58312 −0.145435 −0.0727173 0.997353i \(-0.523167\pi\)
−0.0727173 + 0.997353i \(0.523167\pi\)
\(608\) 0 0
\(609\) −21.9499 −0.889454
\(610\) 0 0
\(611\) 38.6332 1.56293
\(612\) 0 0
\(613\) −34.0000 −1.37325 −0.686624 0.727013i \(-0.740908\pi\)
−0.686624 + 0.727013i \(0.740908\pi\)
\(614\) 0 0
\(615\) 3.31662 0.133739
\(616\) 0 0
\(617\) 12.0501 0.485120 0.242560 0.970136i \(-0.422013\pi\)
0.242560 + 0.970136i \(0.422013\pi\)
\(618\) 0 0
\(619\) 35.7995 1.43890 0.719452 0.694543i \(-0.244393\pi\)
0.719452 + 0.694543i \(0.244393\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) −13.8997 −0.556882
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −19.2665 −0.769430
\(628\) 0 0
\(629\) −5.41688 −0.215985
\(630\) 0 0
\(631\) 2.94987 0.117433 0.0587163 0.998275i \(-0.481299\pi\)
0.0587163 + 0.998275i \(0.481299\pi\)
\(632\) 0 0
\(633\) 3.63325 0.144409
\(634\) 0 0
\(635\) 14.9499 0.593268
\(636\) 0 0
\(637\) 8.63325 0.342062
\(638\) 0 0
\(639\) −11.9499 −0.472730
\(640\) 0 0
\(641\) 23.5831 0.931477 0.465739 0.884922i \(-0.345789\pi\)
0.465739 + 0.884922i \(0.345789\pi\)
\(642\) 0 0
\(643\) −8.26650 −0.325999 −0.162999 0.986626i \(-0.552117\pi\)
−0.162999 + 0.986626i \(0.552117\pi\)
\(644\) 0 0
\(645\) −10.6332 −0.418684
\(646\) 0 0
\(647\) 15.5831 0.612636 0.306318 0.951929i \(-0.400903\pi\)
0.306318 + 0.951929i \(0.400903\pi\)
\(648\) 0 0
\(649\) 18.4169 0.722926
\(650\) 0 0
\(651\) −10.8997 −0.427195
\(652\) 0 0
\(653\) −3.78363 −0.148065 −0.0740324 0.997256i \(-0.523587\pi\)
−0.0740324 + 0.997256i \(0.523587\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 8.31662 0.324462
\(658\) 0 0
\(659\) 4.41688 0.172057 0.0860285 0.996293i \(-0.472582\pi\)
0.0860285 + 0.996293i \(0.472582\pi\)
\(660\) 0 0
\(661\) −21.3668 −0.831070 −0.415535 0.909577i \(-0.636406\pi\)
−0.415535 + 0.909577i \(0.636406\pi\)
\(662\) 0 0
\(663\) 14.3166 0.556012
\(664\) 0 0
\(665\) −24.9499 −0.967515
\(666\) 0 0
\(667\) 7.31662 0.283301
\(668\) 0 0
\(669\) −23.2665 −0.899535
\(670\) 0 0
\(671\) −23.8997 −0.922640
\(672\) 0 0
\(673\) −22.9499 −0.884653 −0.442326 0.896854i \(-0.645847\pi\)
−0.442326 + 0.896854i \(0.645847\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −45.9499 −1.76600 −0.882999 0.469376i \(-0.844479\pi\)
−0.882999 + 0.469376i \(0.844479\pi\)
\(678\) 0 0
\(679\) 36.0000 1.38155
\(680\) 0 0
\(681\) 28.0000 1.07296
\(682\) 0 0
\(683\) −20.3166 −0.777394 −0.388697 0.921366i \(-0.627075\pi\)
−0.388697 + 0.921366i \(0.627075\pi\)
\(684\) 0 0
\(685\) −13.2665 −0.506887
\(686\) 0 0
\(687\) 3.36675 0.128450
\(688\) 0 0
\(689\) 42.9499 1.63626
\(690\) 0 0
\(691\) 2.63325 0.100174 0.0500868 0.998745i \(-0.484050\pi\)
0.0500868 + 0.998745i \(0.484050\pi\)
\(692\) 0 0
\(693\) −6.94987 −0.264004
\(694\) 0 0
\(695\) 11.6332 0.441274
\(696\) 0 0
\(697\) −11.0000 −0.416655
\(698\) 0 0
\(699\) 8.00000 0.302588
\(700\) 0 0
\(701\) −3.68338 −0.139119 −0.0695596 0.997578i \(-0.522159\pi\)
−0.0695596 + 0.997578i \(0.522159\pi\)
\(702\) 0 0
\(703\) −13.5831 −0.512297
\(704\) 0 0
\(705\) −8.94987 −0.337072
\(706\) 0 0
\(707\) −3.94987 −0.148550
\(708\) 0 0
\(709\) 12.8496 0.482578 0.241289 0.970453i \(-0.422430\pi\)
0.241289 + 0.970453i \(0.422430\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 3.63325 0.136066
\(714\) 0 0
\(715\) 10.0000 0.373979
\(716\) 0 0
\(717\) 15.9499 0.595659
\(718\) 0 0
\(719\) −21.8496 −0.814853 −0.407427 0.913238i \(-0.633574\pi\)
−0.407427 + 0.913238i \(0.633574\pi\)
\(720\) 0 0
\(721\) −54.0000 −2.01107
\(722\) 0 0
\(723\) −26.2164 −0.974997
\(724\) 0 0
\(725\) −7.31662 −0.271733
\(726\) 0 0
\(727\) −23.6332 −0.876509 −0.438254 0.898851i \(-0.644403\pi\)
−0.438254 + 0.898851i \(0.644403\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 35.2665 1.30438
\(732\) 0 0
\(733\) −15.7335 −0.581130 −0.290565 0.956855i \(-0.593843\pi\)
−0.290565 + 0.956855i \(0.593843\pi\)
\(734\) 0 0
\(735\) −2.00000 −0.0737711
\(736\) 0 0
\(737\) 36.2164 1.33405
\(738\) 0 0
\(739\) 37.0000 1.36107 0.680534 0.732717i \(-0.261748\pi\)
0.680534 + 0.732717i \(0.261748\pi\)
\(740\) 0 0
\(741\) 35.8997 1.31881
\(742\) 0 0
\(743\) 31.7995 1.16661 0.583305 0.812253i \(-0.301759\pi\)
0.583305 + 0.812253i \(0.301759\pi\)
\(744\) 0 0
\(745\) 4.31662 0.158149
\(746\) 0 0
\(747\) 1.31662 0.0481728
\(748\) 0 0
\(749\) −17.8496 −0.652211
\(750\) 0 0
\(751\) 10.8496 0.395908 0.197954 0.980211i \(-0.436570\pi\)
0.197954 + 0.980211i \(0.436570\pi\)
\(752\) 0 0
\(753\) −23.2665 −0.847878
\(754\) 0 0
\(755\) −14.6332 −0.532558
\(756\) 0 0
\(757\) −23.5330 −0.855321 −0.427661 0.903939i \(-0.640662\pi\)
−0.427661 + 0.903939i \(0.640662\pi\)
\(758\) 0 0
\(759\) 2.31662 0.0840882
\(760\) 0 0
\(761\) 5.41688 0.196362 0.0981808 0.995169i \(-0.468698\pi\)
0.0981808 + 0.995169i \(0.468698\pi\)
\(762\) 0 0
\(763\) 26.8496 0.972022
\(764\) 0 0
\(765\) −3.31662 −0.119913
\(766\) 0 0
\(767\) −34.3166 −1.23910
\(768\) 0 0
\(769\) 11.5831 0.417698 0.208849 0.977948i \(-0.433028\pi\)
0.208849 + 0.977948i \(0.433028\pi\)
\(770\) 0 0
\(771\) 0.949874 0.0342089
\(772\) 0 0
\(773\) −45.1662 −1.62452 −0.812259 0.583298i \(-0.801762\pi\)
−0.812259 + 0.583298i \(0.801762\pi\)
\(774\) 0 0
\(775\) −3.63325 −0.130510
\(776\) 0 0
\(777\) −4.89975 −0.175778
\(778\) 0 0
\(779\) −27.5831 −0.988268
\(780\) 0 0
\(781\) 27.6834 0.990589
\(782\) 0 0
\(783\) −7.31662 −0.261475
\(784\) 0 0
\(785\) −19.6332 −0.700741
\(786\) 0 0
\(787\) −0.467002 −0.0166468 −0.00832341 0.999965i \(-0.502649\pi\)
−0.00832341 + 0.999965i \(0.502649\pi\)
\(788\) 0 0
\(789\) −28.5831 −1.01759
\(790\) 0 0
\(791\) 26.0501 0.926236
\(792\) 0 0
\(793\) 44.5330 1.58141
\(794\) 0 0
\(795\) −9.94987 −0.352886
\(796\) 0 0
\(797\) 17.8496 0.632266 0.316133 0.948715i \(-0.397615\pi\)
0.316133 + 0.948715i \(0.397615\pi\)
\(798\) 0 0
\(799\) 29.6834 1.05012
\(800\) 0 0
\(801\) −4.63325 −0.163708
\(802\) 0 0
\(803\) −19.2665 −0.679900
\(804\) 0 0
\(805\) 3.00000 0.105736
\(806\) 0 0
\(807\) −22.5831 −0.794964
\(808\) 0 0
\(809\) −51.2164 −1.80067 −0.900336 0.435196i \(-0.856679\pi\)
−0.900336 + 0.435196i \(0.856679\pi\)
\(810\) 0 0
\(811\) −24.1662 −0.848592 −0.424296 0.905524i \(-0.639478\pi\)
−0.424296 + 0.905524i \(0.639478\pi\)
\(812\) 0 0
\(813\) −24.2665 −0.851063
\(814\) 0 0
\(815\) −23.2665 −0.814990
\(816\) 0 0
\(817\) 88.4327 3.09387
\(818\) 0 0
\(819\) 12.9499 0.452505
\(820\) 0 0
\(821\) 2.73350 0.0953998 0.0476999 0.998862i \(-0.484811\pi\)
0.0476999 + 0.998862i \(0.484811\pi\)
\(822\) 0 0
\(823\) −26.7335 −0.931871 −0.465936 0.884819i \(-0.654282\pi\)
−0.465936 + 0.884819i \(0.654282\pi\)
\(824\) 0 0
\(825\) −2.31662 −0.0806545
\(826\) 0 0
\(827\) 14.5831 0.507105 0.253552 0.967322i \(-0.418401\pi\)
0.253552 + 0.967322i \(0.418401\pi\)
\(828\) 0 0
\(829\) 19.0000 0.659897 0.329949 0.943999i \(-0.392969\pi\)
0.329949 + 0.943999i \(0.392969\pi\)
\(830\) 0 0
\(831\) 18.0000 0.624413
\(832\) 0 0
\(833\) 6.63325 0.229828
\(834\) 0 0
\(835\) −13.5831 −0.470063
\(836\) 0 0
\(837\) −3.63325 −0.125583
\(838\) 0 0
\(839\) 19.2665 0.665153 0.332577 0.943076i \(-0.392082\pi\)
0.332577 + 0.943076i \(0.392082\pi\)
\(840\) 0 0
\(841\) 24.5330 0.845965
\(842\) 0 0
\(843\) 2.94987 0.101599
\(844\) 0 0
\(845\) −5.63325 −0.193790
\(846\) 0 0
\(847\) −16.8997 −0.580682
\(848\) 0 0
\(849\) 5.63325 0.193333
\(850\) 0 0
\(851\) 1.63325 0.0559871
\(852\) 0 0
\(853\) 13.3668 0.457669 0.228834 0.973465i \(-0.426509\pi\)
0.228834 + 0.973465i \(0.426509\pi\)
\(854\) 0 0
\(855\) −8.31662 −0.284423
\(856\) 0 0
\(857\) −28.5330 −0.974669 −0.487334 0.873215i \(-0.662031\pi\)
−0.487334 + 0.873215i \(0.662031\pi\)
\(858\) 0 0
\(859\) −33.6332 −1.14755 −0.573776 0.819012i \(-0.694522\pi\)
−0.573776 + 0.819012i \(0.694522\pi\)
\(860\) 0 0
\(861\) −9.94987 −0.339091
\(862\) 0 0
\(863\) 14.6332 0.498122 0.249061 0.968488i \(-0.419878\pi\)
0.249061 + 0.968488i \(0.419878\pi\)
\(864\) 0 0
\(865\) −4.00000 −0.136004
\(866\) 0 0
\(867\) −6.00000 −0.203771
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −67.4829 −2.28657
\(872\) 0 0
\(873\) 12.0000 0.406138
\(874\) 0 0
\(875\) −3.00000 −0.101419
\(876\) 0 0
\(877\) 11.8997 0.401826 0.200913 0.979609i \(-0.435609\pi\)
0.200913 + 0.979609i \(0.435609\pi\)
\(878\) 0 0
\(879\) −7.94987 −0.268143
\(880\) 0 0
\(881\) −41.5831 −1.40097 −0.700486 0.713667i \(-0.747033\pi\)
−0.700486 + 0.713667i \(0.747033\pi\)
\(882\) 0 0
\(883\) 18.3166 0.616404 0.308202 0.951321i \(-0.400273\pi\)
0.308202 + 0.951321i \(0.400273\pi\)
\(884\) 0 0
\(885\) 7.94987 0.267232
\(886\) 0 0
\(887\) −46.6332 −1.56579 −0.782896 0.622153i \(-0.786258\pi\)
−0.782896 + 0.622153i \(0.786258\pi\)
\(888\) 0 0
\(889\) −44.8496 −1.50421
\(890\) 0 0
\(891\) −2.31662 −0.0776099
\(892\) 0 0
\(893\) 74.4327 2.49080
\(894\) 0 0
\(895\) −22.6332 −0.756546
\(896\) 0 0
\(897\) −4.31662 −0.144128
\(898\) 0 0
\(899\) 26.5831 0.886597
\(900\) 0 0
\(901\) 33.0000 1.09939
\(902\) 0 0
\(903\) 31.8997 1.06156
\(904\) 0 0
\(905\) −7.36675 −0.244879
\(906\) 0 0
\(907\) 5.63325 0.187049 0.0935245 0.995617i \(-0.470187\pi\)
0.0935245 + 0.995617i \(0.470187\pi\)
\(908\) 0 0
\(909\) −1.31662 −0.0436697
\(910\) 0 0
\(911\) 31.7995 1.05356 0.526782 0.850000i \(-0.323398\pi\)
0.526782 + 0.850000i \(0.323398\pi\)
\(912\) 0 0
\(913\) −3.05013 −0.100944
\(914\) 0 0
\(915\) −10.3166 −0.341057
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −9.26650 −0.305674 −0.152837 0.988251i \(-0.548841\pi\)
−0.152837 + 0.988251i \(0.548841\pi\)
\(920\) 0 0
\(921\) 24.9499 0.822126
\(922\) 0 0
\(923\) −51.5831 −1.69788
\(924\) 0 0
\(925\) −1.63325 −0.0537009
\(926\) 0 0
\(927\) −18.0000 −0.591198
\(928\) 0 0
\(929\) −10.6834 −0.350510 −0.175255 0.984523i \(-0.556075\pi\)
−0.175255 + 0.984523i \(0.556075\pi\)
\(930\) 0 0
\(931\) 16.6332 0.545133
\(932\) 0 0
\(933\) −2.73350 −0.0894908
\(934\) 0 0
\(935\) 7.68338 0.251273
\(936\) 0 0
\(937\) 40.0000 1.30674 0.653372 0.757037i \(-0.273354\pi\)
0.653372 + 0.757037i \(0.273354\pi\)
\(938\) 0 0
\(939\) 33.6332 1.09758
\(940\) 0 0
\(941\) 37.5831 1.22517 0.612587 0.790403i \(-0.290129\pi\)
0.612587 + 0.790403i \(0.290129\pi\)
\(942\) 0 0
\(943\) 3.31662 0.108004
\(944\) 0 0
\(945\) −3.00000 −0.0975900
\(946\) 0 0
\(947\) 12.6332 0.410525 0.205263 0.978707i \(-0.434195\pi\)
0.205263 + 0.978707i \(0.434195\pi\)
\(948\) 0 0
\(949\) 35.8997 1.16536
\(950\) 0 0
\(951\) −16.3166 −0.529103
\(952\) 0 0
\(953\) −52.5330 −1.70171 −0.850855 0.525400i \(-0.823916\pi\)
−0.850855 + 0.525400i \(0.823916\pi\)
\(954\) 0 0
\(955\) −13.5831 −0.439540
\(956\) 0 0
\(957\) 16.9499 0.547912
\(958\) 0 0
\(959\) 39.7995 1.28519
\(960\) 0 0
\(961\) −17.7995 −0.574177
\(962\) 0 0
\(963\) −5.94987 −0.191732
\(964\) 0 0
\(965\) 0.633250 0.0203850
\(966\) 0 0
\(967\) −18.9499 −0.609387 −0.304693 0.952450i \(-0.598554\pi\)
−0.304693 + 0.952450i \(0.598554\pi\)
\(968\) 0 0
\(969\) 27.5831 0.886098
\(970\) 0 0
\(971\) 4.63325 0.148688 0.0743440 0.997233i \(-0.476314\pi\)
0.0743440 + 0.997233i \(0.476314\pi\)
\(972\) 0 0
\(973\) −34.8997 −1.11883
\(974\) 0 0
\(975\) 4.31662 0.138243
\(976\) 0 0
\(977\) −45.8496 −1.46686 −0.733430 0.679765i \(-0.762082\pi\)
−0.733430 + 0.679765i \(0.762082\pi\)
\(978\) 0 0
\(979\) 10.7335 0.343044
\(980\) 0 0
\(981\) 8.94987 0.285747
\(982\) 0 0
\(983\) −54.4829 −1.73773 −0.868867 0.495046i \(-0.835151\pi\)
−0.868867 + 0.495046i \(0.835151\pi\)
\(984\) 0 0
\(985\) −11.8997 −0.379158
\(986\) 0 0
\(987\) 26.8496 0.854633
\(988\) 0 0
\(989\) −10.6332 −0.338118
\(990\) 0 0
\(991\) 14.1662 0.450006 0.225003 0.974358i \(-0.427761\pi\)
0.225003 + 0.974358i \(0.427761\pi\)
\(992\) 0 0
\(993\) −21.5330 −0.683329
\(994\) 0 0
\(995\) 14.6332 0.463905
\(996\) 0 0
\(997\) 11.3668 0.359989 0.179994 0.983668i \(-0.442392\pi\)
0.179994 + 0.983668i \(0.442392\pi\)
\(998\) 0 0
\(999\) −1.63325 −0.0516737
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.bo.1.1 2
4.3 odd 2 2760.2.a.l.1.2 2
12.11 even 2 8280.2.a.bc.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2760.2.a.l.1.2 2 4.3 odd 2
5520.2.a.bo.1.1 2 1.1 even 1 trivial
8280.2.a.bc.1.1 2 12.11 even 2