Properties

Label 4560.2.a.bg.1.2
Level $4560$
Weight $2$
Character 4560.1
Self dual yes
Analytic conductor $36.412$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4560.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.4117833217\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2280)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 4560.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} +3.41421 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} +3.41421 q^{7} +1.00000 q^{9} +1.41421 q^{11} +4.24264 q^{13} +1.00000 q^{15} +2.82843 q^{17} +1.00000 q^{19} -3.41421 q^{21} +4.82843 q^{23} +1.00000 q^{25} -1.00000 q^{27} +2.24264 q^{29} +8.82843 q^{31} -1.41421 q^{33} -3.41421 q^{35} -7.07107 q^{37} -4.24264 q^{39} +2.24264 q^{41} +1.75736 q^{43} -1.00000 q^{45} -4.82843 q^{47} +4.65685 q^{49} -2.82843 q^{51} -12.4853 q^{53} -1.41421 q^{55} -1.00000 q^{57} +2.82843 q^{59} -8.00000 q^{61} +3.41421 q^{63} -4.24264 q^{65} +11.3137 q^{67} -4.82843 q^{69} +5.17157 q^{71} -3.65685 q^{73} -1.00000 q^{75} +4.82843 q^{77} +2.34315 q^{79} +1.00000 q^{81} +6.00000 q^{83} -2.82843 q^{85} -2.24264 q^{87} -13.5563 q^{89} +14.4853 q^{91} -8.82843 q^{93} -1.00000 q^{95} +9.89949 q^{97} +1.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{5} + 4 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{3} - 2 q^{5} + 4 q^{7} + 2 q^{9} + 2 q^{15} + 2 q^{19} - 4 q^{21} + 4 q^{23} + 2 q^{25} - 2 q^{27} - 4 q^{29} + 12 q^{31} - 4 q^{35} - 4 q^{41} + 12 q^{43} - 2 q^{45} - 4 q^{47} - 2 q^{49} - 8 q^{53} - 2 q^{57} - 16 q^{61} + 4 q^{63} - 4 q^{69} + 16 q^{71} + 4 q^{73} - 2 q^{75} + 4 q^{77} + 16 q^{79} + 2 q^{81} + 12 q^{83} + 4 q^{87} + 4 q^{89} + 12 q^{91} - 12 q^{93} - 2 q^{95} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 3.41421 1.29045 0.645226 0.763992i \(-0.276763\pi\)
0.645226 + 0.763992i \(0.276763\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.41421 0.426401 0.213201 0.977008i \(-0.431611\pi\)
0.213201 + 0.977008i \(0.431611\pi\)
\(12\) 0 0
\(13\) 4.24264 1.17670 0.588348 0.808608i \(-0.299778\pi\)
0.588348 + 0.808608i \(0.299778\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 2.82843 0.685994 0.342997 0.939336i \(-0.388558\pi\)
0.342997 + 0.939336i \(0.388558\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −3.41421 −0.745042
\(22\) 0 0
\(23\) 4.82843 1.00680 0.503398 0.864054i \(-0.332083\pi\)
0.503398 + 0.864054i \(0.332083\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 2.24264 0.416448 0.208224 0.978081i \(-0.433232\pi\)
0.208224 + 0.978081i \(0.433232\pi\)
\(30\) 0 0
\(31\) 8.82843 1.58563 0.792816 0.609461i \(-0.208614\pi\)
0.792816 + 0.609461i \(0.208614\pi\)
\(32\) 0 0
\(33\) −1.41421 −0.246183
\(34\) 0 0
\(35\) −3.41421 −0.577107
\(36\) 0 0
\(37\) −7.07107 −1.16248 −0.581238 0.813733i \(-0.697432\pi\)
−0.581238 + 0.813733i \(0.697432\pi\)
\(38\) 0 0
\(39\) −4.24264 −0.679366
\(40\) 0 0
\(41\) 2.24264 0.350242 0.175121 0.984547i \(-0.443968\pi\)
0.175121 + 0.984547i \(0.443968\pi\)
\(42\) 0 0
\(43\) 1.75736 0.267995 0.133997 0.990982i \(-0.457219\pi\)
0.133997 + 0.990982i \(0.457219\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −4.82843 −0.704298 −0.352149 0.935944i \(-0.614549\pi\)
−0.352149 + 0.935944i \(0.614549\pi\)
\(48\) 0 0
\(49\) 4.65685 0.665265
\(50\) 0 0
\(51\) −2.82843 −0.396059
\(52\) 0 0
\(53\) −12.4853 −1.71499 −0.857493 0.514496i \(-0.827979\pi\)
−0.857493 + 0.514496i \(0.827979\pi\)
\(54\) 0 0
\(55\) −1.41421 −0.190693
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) 2.82843 0.368230 0.184115 0.982905i \(-0.441058\pi\)
0.184115 + 0.982905i \(0.441058\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 0 0
\(63\) 3.41421 0.430150
\(64\) 0 0
\(65\) −4.24264 −0.526235
\(66\) 0 0
\(67\) 11.3137 1.38219 0.691095 0.722764i \(-0.257129\pi\)
0.691095 + 0.722764i \(0.257129\pi\)
\(68\) 0 0
\(69\) −4.82843 −0.581274
\(70\) 0 0
\(71\) 5.17157 0.613753 0.306876 0.951749i \(-0.400716\pi\)
0.306876 + 0.951749i \(0.400716\pi\)
\(72\) 0 0
\(73\) −3.65685 −0.428002 −0.214001 0.976833i \(-0.568650\pi\)
−0.214001 + 0.976833i \(0.568650\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 4.82843 0.550250
\(78\) 0 0
\(79\) 2.34315 0.263624 0.131812 0.991275i \(-0.457920\pi\)
0.131812 + 0.991275i \(0.457920\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) −2.82843 −0.306786
\(86\) 0 0
\(87\) −2.24264 −0.240436
\(88\) 0 0
\(89\) −13.5563 −1.43697 −0.718485 0.695542i \(-0.755164\pi\)
−0.718485 + 0.695542i \(0.755164\pi\)
\(90\) 0 0
\(91\) 14.4853 1.51847
\(92\) 0 0
\(93\) −8.82843 −0.915465
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) 9.89949 1.00514 0.502571 0.864536i \(-0.332388\pi\)
0.502571 + 0.864536i \(0.332388\pi\)
\(98\) 0 0
\(99\) 1.41421 0.142134
\(100\) 0 0
\(101\) −12.8284 −1.27648 −0.638238 0.769839i \(-0.720336\pi\)
−0.638238 + 0.769839i \(0.720336\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) 3.41421 0.333193
\(106\) 0 0
\(107\) −13.6569 −1.32026 −0.660129 0.751152i \(-0.729498\pi\)
−0.660129 + 0.751152i \(0.729498\pi\)
\(108\) 0 0
\(109\) 12.1421 1.16301 0.581503 0.813544i \(-0.302465\pi\)
0.581503 + 0.813544i \(0.302465\pi\)
\(110\) 0 0
\(111\) 7.07107 0.671156
\(112\) 0 0
\(113\) 9.65685 0.908440 0.454220 0.890889i \(-0.349918\pi\)
0.454220 + 0.890889i \(0.349918\pi\)
\(114\) 0 0
\(115\) −4.82843 −0.450253
\(116\) 0 0
\(117\) 4.24264 0.392232
\(118\) 0 0
\(119\) 9.65685 0.885242
\(120\) 0 0
\(121\) −9.00000 −0.818182
\(122\) 0 0
\(123\) −2.24264 −0.202212
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −16.9706 −1.50589 −0.752947 0.658081i \(-0.771368\pi\)
−0.752947 + 0.658081i \(0.771368\pi\)
\(128\) 0 0
\(129\) −1.75736 −0.154727
\(130\) 0 0
\(131\) −4.92893 −0.430643 −0.215321 0.976543i \(-0.569080\pi\)
−0.215321 + 0.976543i \(0.569080\pi\)
\(132\) 0 0
\(133\) 3.41421 0.296050
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −14.9706 −1.27902 −0.639511 0.768782i \(-0.720863\pi\)
−0.639511 + 0.768782i \(0.720863\pi\)
\(138\) 0 0
\(139\) 12.4853 1.05899 0.529494 0.848314i \(-0.322382\pi\)
0.529494 + 0.848314i \(0.322382\pi\)
\(140\) 0 0
\(141\) 4.82843 0.406627
\(142\) 0 0
\(143\) 6.00000 0.501745
\(144\) 0 0
\(145\) −2.24264 −0.186241
\(146\) 0 0
\(147\) −4.65685 −0.384091
\(148\) 0 0
\(149\) 11.6569 0.954967 0.477483 0.878641i \(-0.341549\pi\)
0.477483 + 0.878641i \(0.341549\pi\)
\(150\) 0 0
\(151\) 11.1716 0.909130 0.454565 0.890714i \(-0.349795\pi\)
0.454565 + 0.890714i \(0.349795\pi\)
\(152\) 0 0
\(153\) 2.82843 0.228665
\(154\) 0 0
\(155\) −8.82843 −0.709116
\(156\) 0 0
\(157\) −8.14214 −0.649813 −0.324907 0.945746i \(-0.605333\pi\)
−0.324907 + 0.945746i \(0.605333\pi\)
\(158\) 0 0
\(159\) 12.4853 0.990147
\(160\) 0 0
\(161\) 16.4853 1.29922
\(162\) 0 0
\(163\) 0.100505 0.00787216 0.00393608 0.999992i \(-0.498747\pi\)
0.00393608 + 0.999992i \(0.498747\pi\)
\(164\) 0 0
\(165\) 1.41421 0.110096
\(166\) 0 0
\(167\) −15.6569 −1.21156 −0.605782 0.795631i \(-0.707140\pi\)
−0.605782 + 0.795631i \(0.707140\pi\)
\(168\) 0 0
\(169\) 5.00000 0.384615
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 0 0
\(173\) 13.6569 1.03831 0.519156 0.854680i \(-0.326246\pi\)
0.519156 + 0.854680i \(0.326246\pi\)
\(174\) 0 0
\(175\) 3.41421 0.258090
\(176\) 0 0
\(177\) −2.82843 −0.212598
\(178\) 0 0
\(179\) −23.7990 −1.77882 −0.889410 0.457110i \(-0.848884\pi\)
−0.889410 + 0.457110i \(0.848884\pi\)
\(180\) 0 0
\(181\) 4.82843 0.358894 0.179447 0.983768i \(-0.442569\pi\)
0.179447 + 0.983768i \(0.442569\pi\)
\(182\) 0 0
\(183\) 8.00000 0.591377
\(184\) 0 0
\(185\) 7.07107 0.519875
\(186\) 0 0
\(187\) 4.00000 0.292509
\(188\) 0 0
\(189\) −3.41421 −0.248347
\(190\) 0 0
\(191\) −13.4142 −0.970618 −0.485309 0.874343i \(-0.661293\pi\)
−0.485309 + 0.874343i \(0.661293\pi\)
\(192\) 0 0
\(193\) 23.0711 1.66069 0.830346 0.557248i \(-0.188143\pi\)
0.830346 + 0.557248i \(0.188143\pi\)
\(194\) 0 0
\(195\) 4.24264 0.303822
\(196\) 0 0
\(197\) 8.48528 0.604551 0.302276 0.953221i \(-0.402254\pi\)
0.302276 + 0.953221i \(0.402254\pi\)
\(198\) 0 0
\(199\) 14.1421 1.00251 0.501255 0.865300i \(-0.332872\pi\)
0.501255 + 0.865300i \(0.332872\pi\)
\(200\) 0 0
\(201\) −11.3137 −0.798007
\(202\) 0 0
\(203\) 7.65685 0.537406
\(204\) 0 0
\(205\) −2.24264 −0.156633
\(206\) 0 0
\(207\) 4.82843 0.335599
\(208\) 0 0
\(209\) 1.41421 0.0978232
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 0 0
\(213\) −5.17157 −0.354350
\(214\) 0 0
\(215\) −1.75736 −0.119851
\(216\) 0 0
\(217\) 30.1421 2.04618
\(218\) 0 0
\(219\) 3.65685 0.247107
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) 27.3137 1.82906 0.914531 0.404517i \(-0.132560\pi\)
0.914531 + 0.404517i \(0.132560\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 5.31371 0.352683 0.176342 0.984329i \(-0.443574\pi\)
0.176342 + 0.984329i \(0.443574\pi\)
\(228\) 0 0
\(229\) −18.6274 −1.23093 −0.615467 0.788163i \(-0.711033\pi\)
−0.615467 + 0.788163i \(0.711033\pi\)
\(230\) 0 0
\(231\) −4.82843 −0.317687
\(232\) 0 0
\(233\) 7.65685 0.501617 0.250809 0.968037i \(-0.419304\pi\)
0.250809 + 0.968037i \(0.419304\pi\)
\(234\) 0 0
\(235\) 4.82843 0.314972
\(236\) 0 0
\(237\) −2.34315 −0.152204
\(238\) 0 0
\(239\) 12.2426 0.791911 0.395955 0.918270i \(-0.370414\pi\)
0.395955 + 0.918270i \(0.370414\pi\)
\(240\) 0 0
\(241\) −14.9706 −0.964339 −0.482169 0.876078i \(-0.660151\pi\)
−0.482169 + 0.876078i \(0.660151\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −4.65685 −0.297516
\(246\) 0 0
\(247\) 4.24264 0.269953
\(248\) 0 0
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) −24.7279 −1.56081 −0.780406 0.625273i \(-0.784988\pi\)
−0.780406 + 0.625273i \(0.784988\pi\)
\(252\) 0 0
\(253\) 6.82843 0.429300
\(254\) 0 0
\(255\) 2.82843 0.177123
\(256\) 0 0
\(257\) 17.6569 1.10140 0.550702 0.834702i \(-0.314360\pi\)
0.550702 + 0.834702i \(0.314360\pi\)
\(258\) 0 0
\(259\) −24.1421 −1.50012
\(260\) 0 0
\(261\) 2.24264 0.138816
\(262\) 0 0
\(263\) 19.6569 1.21209 0.606047 0.795429i \(-0.292754\pi\)
0.606047 + 0.795429i \(0.292754\pi\)
\(264\) 0 0
\(265\) 12.4853 0.766965
\(266\) 0 0
\(267\) 13.5563 0.829635
\(268\) 0 0
\(269\) 4.10051 0.250012 0.125006 0.992156i \(-0.460105\pi\)
0.125006 + 0.992156i \(0.460105\pi\)
\(270\) 0 0
\(271\) −2.14214 −0.130125 −0.0650627 0.997881i \(-0.520725\pi\)
−0.0650627 + 0.997881i \(0.520725\pi\)
\(272\) 0 0
\(273\) −14.4853 −0.876689
\(274\) 0 0
\(275\) 1.41421 0.0852803
\(276\) 0 0
\(277\) −5.31371 −0.319270 −0.159635 0.987176i \(-0.551032\pi\)
−0.159635 + 0.987176i \(0.551032\pi\)
\(278\) 0 0
\(279\) 8.82843 0.528544
\(280\) 0 0
\(281\) 14.2426 0.849645 0.424822 0.905277i \(-0.360337\pi\)
0.424822 + 0.905277i \(0.360337\pi\)
\(282\) 0 0
\(283\) 4.10051 0.243750 0.121875 0.992545i \(-0.461109\pi\)
0.121875 + 0.992545i \(0.461109\pi\)
\(284\) 0 0
\(285\) 1.00000 0.0592349
\(286\) 0 0
\(287\) 7.65685 0.451970
\(288\) 0 0
\(289\) −9.00000 −0.529412
\(290\) 0 0
\(291\) −9.89949 −0.580319
\(292\) 0 0
\(293\) 4.00000 0.233682 0.116841 0.993151i \(-0.462723\pi\)
0.116841 + 0.993151i \(0.462723\pi\)
\(294\) 0 0
\(295\) −2.82843 −0.164677
\(296\) 0 0
\(297\) −1.41421 −0.0820610
\(298\) 0 0
\(299\) 20.4853 1.18469
\(300\) 0 0
\(301\) 6.00000 0.345834
\(302\) 0 0
\(303\) 12.8284 0.736974
\(304\) 0 0
\(305\) 8.00000 0.458079
\(306\) 0 0
\(307\) −1.17157 −0.0668652 −0.0334326 0.999441i \(-0.510644\pi\)
−0.0334326 + 0.999441i \(0.510644\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) 9.41421 0.533831 0.266916 0.963720i \(-0.413996\pi\)
0.266916 + 0.963720i \(0.413996\pi\)
\(312\) 0 0
\(313\) 29.7990 1.68434 0.842169 0.539213i \(-0.181278\pi\)
0.842169 + 0.539213i \(0.181278\pi\)
\(314\) 0 0
\(315\) −3.41421 −0.192369
\(316\) 0 0
\(317\) 8.48528 0.476581 0.238290 0.971194i \(-0.423413\pi\)
0.238290 + 0.971194i \(0.423413\pi\)
\(318\) 0 0
\(319\) 3.17157 0.177574
\(320\) 0 0
\(321\) 13.6569 0.762251
\(322\) 0 0
\(323\) 2.82843 0.157378
\(324\) 0 0
\(325\) 4.24264 0.235339
\(326\) 0 0
\(327\) −12.1421 −0.671462
\(328\) 0 0
\(329\) −16.4853 −0.908863
\(330\) 0 0
\(331\) 12.8284 0.705114 0.352557 0.935790i \(-0.385312\pi\)
0.352557 + 0.935790i \(0.385312\pi\)
\(332\) 0 0
\(333\) −7.07107 −0.387492
\(334\) 0 0
\(335\) −11.3137 −0.618134
\(336\) 0 0
\(337\) −1.89949 −0.103472 −0.0517360 0.998661i \(-0.516475\pi\)
−0.0517360 + 0.998661i \(0.516475\pi\)
\(338\) 0 0
\(339\) −9.65685 −0.524488
\(340\) 0 0
\(341\) 12.4853 0.676116
\(342\) 0 0
\(343\) −8.00000 −0.431959
\(344\) 0 0
\(345\) 4.82843 0.259954
\(346\) 0 0
\(347\) −0.343146 −0.0184210 −0.00921051 0.999958i \(-0.502932\pi\)
−0.00921051 + 0.999958i \(0.502932\pi\)
\(348\) 0 0
\(349\) −22.0000 −1.17763 −0.588817 0.808267i \(-0.700406\pi\)
−0.588817 + 0.808267i \(0.700406\pi\)
\(350\) 0 0
\(351\) −4.24264 −0.226455
\(352\) 0 0
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 0 0
\(355\) −5.17157 −0.274479
\(356\) 0 0
\(357\) −9.65685 −0.511095
\(358\) 0 0
\(359\) −20.0416 −1.05776 −0.528878 0.848698i \(-0.677387\pi\)
−0.528878 + 0.848698i \(0.677387\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 9.00000 0.472377
\(364\) 0 0
\(365\) 3.65685 0.191408
\(366\) 0 0
\(367\) 5.07107 0.264708 0.132354 0.991203i \(-0.457746\pi\)
0.132354 + 0.991203i \(0.457746\pi\)
\(368\) 0 0
\(369\) 2.24264 0.116747
\(370\) 0 0
\(371\) −42.6274 −2.21311
\(372\) 0 0
\(373\) −28.0416 −1.45194 −0.725970 0.687726i \(-0.758609\pi\)
−0.725970 + 0.687726i \(0.758609\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 9.51472 0.490033
\(378\) 0 0
\(379\) 10.4853 0.538593 0.269296 0.963057i \(-0.413209\pi\)
0.269296 + 0.963057i \(0.413209\pi\)
\(380\) 0 0
\(381\) 16.9706 0.869428
\(382\) 0 0
\(383\) 23.3137 1.19127 0.595637 0.803253i \(-0.296900\pi\)
0.595637 + 0.803253i \(0.296900\pi\)
\(384\) 0 0
\(385\) −4.82843 −0.246079
\(386\) 0 0
\(387\) 1.75736 0.0893316
\(388\) 0 0
\(389\) −23.6569 −1.19945 −0.599725 0.800206i \(-0.704723\pi\)
−0.599725 + 0.800206i \(0.704723\pi\)
\(390\) 0 0
\(391\) 13.6569 0.690657
\(392\) 0 0
\(393\) 4.92893 0.248632
\(394\) 0 0
\(395\) −2.34315 −0.117896
\(396\) 0 0
\(397\) 26.0000 1.30490 0.652451 0.757831i \(-0.273741\pi\)
0.652451 + 0.757831i \(0.273741\pi\)
\(398\) 0 0
\(399\) −3.41421 −0.170924
\(400\) 0 0
\(401\) −11.2132 −0.559961 −0.279980 0.960006i \(-0.590328\pi\)
−0.279980 + 0.960006i \(0.590328\pi\)
\(402\) 0 0
\(403\) 37.4558 1.86581
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −10.0000 −0.495682
\(408\) 0 0
\(409\) −36.1421 −1.78711 −0.893557 0.448950i \(-0.851798\pi\)
−0.893557 + 0.448950i \(0.851798\pi\)
\(410\) 0 0
\(411\) 14.9706 0.738443
\(412\) 0 0
\(413\) 9.65685 0.475183
\(414\) 0 0
\(415\) −6.00000 −0.294528
\(416\) 0 0
\(417\) −12.4853 −0.611407
\(418\) 0 0
\(419\) −28.2426 −1.37974 −0.689872 0.723932i \(-0.742333\pi\)
−0.689872 + 0.723932i \(0.742333\pi\)
\(420\) 0 0
\(421\) 25.3137 1.23371 0.616857 0.787075i \(-0.288406\pi\)
0.616857 + 0.787075i \(0.288406\pi\)
\(422\) 0 0
\(423\) −4.82843 −0.234766
\(424\) 0 0
\(425\) 2.82843 0.137199
\(426\) 0 0
\(427\) −27.3137 −1.32180
\(428\) 0 0
\(429\) −6.00000 −0.289683
\(430\) 0 0
\(431\) 34.1421 1.64457 0.822284 0.569077i \(-0.192699\pi\)
0.822284 + 0.569077i \(0.192699\pi\)
\(432\) 0 0
\(433\) 11.7574 0.565023 0.282511 0.959264i \(-0.408833\pi\)
0.282511 + 0.959264i \(0.408833\pi\)
\(434\) 0 0
\(435\) 2.24264 0.107526
\(436\) 0 0
\(437\) 4.82843 0.230975
\(438\) 0 0
\(439\) 4.68629 0.223664 0.111832 0.993727i \(-0.464328\pi\)
0.111832 + 0.993727i \(0.464328\pi\)
\(440\) 0 0
\(441\) 4.65685 0.221755
\(442\) 0 0
\(443\) −4.14214 −0.196799 −0.0983994 0.995147i \(-0.531372\pi\)
−0.0983994 + 0.995147i \(0.531372\pi\)
\(444\) 0 0
\(445\) 13.5563 0.642633
\(446\) 0 0
\(447\) −11.6569 −0.551350
\(448\) 0 0
\(449\) 26.5269 1.25188 0.625941 0.779870i \(-0.284715\pi\)
0.625941 + 0.779870i \(0.284715\pi\)
\(450\) 0 0
\(451\) 3.17157 0.149344
\(452\) 0 0
\(453\) −11.1716 −0.524886
\(454\) 0 0
\(455\) −14.4853 −0.679080
\(456\) 0 0
\(457\) 20.1421 0.942209 0.471105 0.882077i \(-0.343855\pi\)
0.471105 + 0.882077i \(0.343855\pi\)
\(458\) 0 0
\(459\) −2.82843 −0.132020
\(460\) 0 0
\(461\) −20.6274 −0.960715 −0.480357 0.877073i \(-0.659493\pi\)
−0.480357 + 0.877073i \(0.659493\pi\)
\(462\) 0 0
\(463\) 25.7574 1.19705 0.598523 0.801106i \(-0.295755\pi\)
0.598523 + 0.801106i \(0.295755\pi\)
\(464\) 0 0
\(465\) 8.82843 0.409409
\(466\) 0 0
\(467\) 20.1421 0.932067 0.466033 0.884767i \(-0.345683\pi\)
0.466033 + 0.884767i \(0.345683\pi\)
\(468\) 0 0
\(469\) 38.6274 1.78365
\(470\) 0 0
\(471\) 8.14214 0.375170
\(472\) 0 0
\(473\) 2.48528 0.114273
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) −12.4853 −0.571662
\(478\) 0 0
\(479\) −1.61522 −0.0738015 −0.0369007 0.999319i \(-0.511749\pi\)
−0.0369007 + 0.999319i \(0.511749\pi\)
\(480\) 0 0
\(481\) −30.0000 −1.36788
\(482\) 0 0
\(483\) −16.4853 −0.750106
\(484\) 0 0
\(485\) −9.89949 −0.449513
\(486\) 0 0
\(487\) −18.8284 −0.853197 −0.426599 0.904441i \(-0.640288\pi\)
−0.426599 + 0.904441i \(0.640288\pi\)
\(488\) 0 0
\(489\) −0.100505 −0.00454500
\(490\) 0 0
\(491\) −31.7574 −1.43319 −0.716595 0.697490i \(-0.754300\pi\)
−0.716595 + 0.697490i \(0.754300\pi\)
\(492\) 0 0
\(493\) 6.34315 0.285681
\(494\) 0 0
\(495\) −1.41421 −0.0635642
\(496\) 0 0
\(497\) 17.6569 0.792018
\(498\) 0 0
\(499\) −14.8284 −0.663812 −0.331906 0.943313i \(-0.607692\pi\)
−0.331906 + 0.943313i \(0.607692\pi\)
\(500\) 0 0
\(501\) 15.6569 0.699497
\(502\) 0 0
\(503\) 26.2843 1.17196 0.585979 0.810326i \(-0.300710\pi\)
0.585979 + 0.810326i \(0.300710\pi\)
\(504\) 0 0
\(505\) 12.8284 0.570858
\(506\) 0 0
\(507\) −5.00000 −0.222058
\(508\) 0 0
\(509\) 13.0711 0.579365 0.289682 0.957123i \(-0.406450\pi\)
0.289682 + 0.957123i \(0.406450\pi\)
\(510\) 0 0
\(511\) −12.4853 −0.552316
\(512\) 0 0
\(513\) −1.00000 −0.0441511
\(514\) 0 0
\(515\) −8.00000 −0.352522
\(516\) 0 0
\(517\) −6.82843 −0.300314
\(518\) 0 0
\(519\) −13.6569 −0.599469
\(520\) 0 0
\(521\) −20.3848 −0.893073 −0.446537 0.894765i \(-0.647343\pi\)
−0.446537 + 0.894765i \(0.647343\pi\)
\(522\) 0 0
\(523\) −38.1421 −1.66784 −0.833920 0.551886i \(-0.813908\pi\)
−0.833920 + 0.551886i \(0.813908\pi\)
\(524\) 0 0
\(525\) −3.41421 −0.149008
\(526\) 0 0
\(527\) 24.9706 1.08773
\(528\) 0 0
\(529\) 0.313708 0.0136395
\(530\) 0 0
\(531\) 2.82843 0.122743
\(532\) 0 0
\(533\) 9.51472 0.412128
\(534\) 0 0
\(535\) 13.6569 0.590437
\(536\) 0 0
\(537\) 23.7990 1.02700
\(538\) 0 0
\(539\) 6.58579 0.283670
\(540\) 0 0
\(541\) 18.6863 0.803386 0.401693 0.915774i \(-0.368422\pi\)
0.401693 + 0.915774i \(0.368422\pi\)
\(542\) 0 0
\(543\) −4.82843 −0.207208
\(544\) 0 0
\(545\) −12.1421 −0.520112
\(546\) 0 0
\(547\) −28.4853 −1.21794 −0.608971 0.793192i \(-0.708418\pi\)
−0.608971 + 0.793192i \(0.708418\pi\)
\(548\) 0 0
\(549\) −8.00000 −0.341432
\(550\) 0 0
\(551\) 2.24264 0.0955397
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) 0 0
\(555\) −7.07107 −0.300150
\(556\) 0 0
\(557\) 10.9706 0.464838 0.232419 0.972616i \(-0.425336\pi\)
0.232419 + 0.972616i \(0.425336\pi\)
\(558\) 0 0
\(559\) 7.45584 0.315349
\(560\) 0 0
\(561\) −4.00000 −0.168880
\(562\) 0 0
\(563\) −33.3137 −1.40401 −0.702003 0.712174i \(-0.747711\pi\)
−0.702003 + 0.712174i \(0.747711\pi\)
\(564\) 0 0
\(565\) −9.65685 −0.406267
\(566\) 0 0
\(567\) 3.41421 0.143383
\(568\) 0 0
\(569\) −4.10051 −0.171902 −0.0859511 0.996299i \(-0.527393\pi\)
−0.0859511 + 0.996299i \(0.527393\pi\)
\(570\) 0 0
\(571\) −0.485281 −0.0203084 −0.0101542 0.999948i \(-0.503232\pi\)
−0.0101542 + 0.999948i \(0.503232\pi\)
\(572\) 0 0
\(573\) 13.4142 0.560387
\(574\) 0 0
\(575\) 4.82843 0.201359
\(576\) 0 0
\(577\) 1.51472 0.0630586 0.0315293 0.999503i \(-0.489962\pi\)
0.0315293 + 0.999503i \(0.489962\pi\)
\(578\) 0 0
\(579\) −23.0711 −0.958801
\(580\) 0 0
\(581\) 20.4853 0.849873
\(582\) 0 0
\(583\) −17.6569 −0.731272
\(584\) 0 0
\(585\) −4.24264 −0.175412
\(586\) 0 0
\(587\) −7.17157 −0.296002 −0.148001 0.988987i \(-0.547284\pi\)
−0.148001 + 0.988987i \(0.547284\pi\)
\(588\) 0 0
\(589\) 8.82843 0.363769
\(590\) 0 0
\(591\) −8.48528 −0.349038
\(592\) 0 0
\(593\) −21.3137 −0.875249 −0.437625 0.899158i \(-0.644180\pi\)
−0.437625 + 0.899158i \(0.644180\pi\)
\(594\) 0 0
\(595\) −9.65685 −0.395892
\(596\) 0 0
\(597\) −14.1421 −0.578799
\(598\) 0 0
\(599\) 16.9706 0.693398 0.346699 0.937976i \(-0.387302\pi\)
0.346699 + 0.937976i \(0.387302\pi\)
\(600\) 0 0
\(601\) −40.1421 −1.63743 −0.818716 0.574199i \(-0.805314\pi\)
−0.818716 + 0.574199i \(0.805314\pi\)
\(602\) 0 0
\(603\) 11.3137 0.460730
\(604\) 0 0
\(605\) 9.00000 0.365902
\(606\) 0 0
\(607\) −34.1421 −1.38579 −0.692893 0.721040i \(-0.743664\pi\)
−0.692893 + 0.721040i \(0.743664\pi\)
\(608\) 0 0
\(609\) −7.65685 −0.310271
\(610\) 0 0
\(611\) −20.4853 −0.828746
\(612\) 0 0
\(613\) 0.142136 0.00574080 0.00287040 0.999996i \(-0.499086\pi\)
0.00287040 + 0.999996i \(0.499086\pi\)
\(614\) 0 0
\(615\) 2.24264 0.0904320
\(616\) 0 0
\(617\) −31.1127 −1.25255 −0.626275 0.779602i \(-0.715421\pi\)
−0.626275 + 0.779602i \(0.715421\pi\)
\(618\) 0 0
\(619\) 5.17157 0.207863 0.103932 0.994584i \(-0.466858\pi\)
0.103932 + 0.994584i \(0.466858\pi\)
\(620\) 0 0
\(621\) −4.82843 −0.193758
\(622\) 0 0
\(623\) −46.2843 −1.85434
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −1.41421 −0.0564782
\(628\) 0 0
\(629\) −20.0000 −0.797452
\(630\) 0 0
\(631\) −8.97056 −0.357112 −0.178556 0.983930i \(-0.557143\pi\)
−0.178556 + 0.983930i \(0.557143\pi\)
\(632\) 0 0
\(633\) −12.0000 −0.476957
\(634\) 0 0
\(635\) 16.9706 0.673456
\(636\) 0 0
\(637\) 19.7574 0.782815
\(638\) 0 0
\(639\) 5.17157 0.204584
\(640\) 0 0
\(641\) −34.2426 −1.35250 −0.676251 0.736671i \(-0.736397\pi\)
−0.676251 + 0.736671i \(0.736397\pi\)
\(642\) 0 0
\(643\) −1.75736 −0.0693035 −0.0346517 0.999399i \(-0.511032\pi\)
−0.0346517 + 0.999399i \(0.511032\pi\)
\(644\) 0 0
\(645\) 1.75736 0.0691960
\(646\) 0 0
\(647\) −20.3431 −0.799772 −0.399886 0.916565i \(-0.630950\pi\)
−0.399886 + 0.916565i \(0.630950\pi\)
\(648\) 0 0
\(649\) 4.00000 0.157014
\(650\) 0 0
\(651\) −30.1421 −1.18136
\(652\) 0 0
\(653\) −12.4853 −0.488587 −0.244293 0.969701i \(-0.578556\pi\)
−0.244293 + 0.969701i \(0.578556\pi\)
\(654\) 0 0
\(655\) 4.92893 0.192589
\(656\) 0 0
\(657\) −3.65685 −0.142667
\(658\) 0 0
\(659\) 6.34315 0.247094 0.123547 0.992339i \(-0.460573\pi\)
0.123547 + 0.992339i \(0.460573\pi\)
\(660\) 0 0
\(661\) 25.1127 0.976771 0.488385 0.872628i \(-0.337586\pi\)
0.488385 + 0.872628i \(0.337586\pi\)
\(662\) 0 0
\(663\) −12.0000 −0.466041
\(664\) 0 0
\(665\) −3.41421 −0.132398
\(666\) 0 0
\(667\) 10.8284 0.419278
\(668\) 0 0
\(669\) −27.3137 −1.05601
\(670\) 0 0
\(671\) −11.3137 −0.436761
\(672\) 0 0
\(673\) −29.8995 −1.15254 −0.576270 0.817259i \(-0.695492\pi\)
−0.576270 + 0.817259i \(0.695492\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 39.7990 1.52960 0.764800 0.644268i \(-0.222838\pi\)
0.764800 + 0.644268i \(0.222838\pi\)
\(678\) 0 0
\(679\) 33.7990 1.29709
\(680\) 0 0
\(681\) −5.31371 −0.203622
\(682\) 0 0
\(683\) 20.6863 0.791539 0.395769 0.918350i \(-0.370478\pi\)
0.395769 + 0.918350i \(0.370478\pi\)
\(684\) 0 0
\(685\) 14.9706 0.571996
\(686\) 0 0
\(687\) 18.6274 0.710680
\(688\) 0 0
\(689\) −52.9706 −2.01802
\(690\) 0 0
\(691\) 31.1127 1.18358 0.591791 0.806091i \(-0.298421\pi\)
0.591791 + 0.806091i \(0.298421\pi\)
\(692\) 0 0
\(693\) 4.82843 0.183417
\(694\) 0 0
\(695\) −12.4853 −0.473594
\(696\) 0 0
\(697\) 6.34315 0.240264
\(698\) 0 0
\(699\) −7.65685 −0.289609
\(700\) 0 0
\(701\) 0.343146 0.0129604 0.00648022 0.999979i \(-0.497937\pi\)
0.00648022 + 0.999979i \(0.497937\pi\)
\(702\) 0 0
\(703\) −7.07107 −0.266690
\(704\) 0 0
\(705\) −4.82843 −0.181849
\(706\) 0 0
\(707\) −43.7990 −1.64723
\(708\) 0 0
\(709\) −18.3431 −0.688891 −0.344446 0.938806i \(-0.611933\pi\)
−0.344446 + 0.938806i \(0.611933\pi\)
\(710\) 0 0
\(711\) 2.34315 0.0878748
\(712\) 0 0
\(713\) 42.6274 1.59641
\(714\) 0 0
\(715\) −6.00000 −0.224387
\(716\) 0 0
\(717\) −12.2426 −0.457210
\(718\) 0 0
\(719\) −46.3848 −1.72986 −0.864930 0.501892i \(-0.832637\pi\)
−0.864930 + 0.501892i \(0.832637\pi\)
\(720\) 0 0
\(721\) 27.3137 1.01722
\(722\) 0 0
\(723\) 14.9706 0.556761
\(724\) 0 0
\(725\) 2.24264 0.0832896
\(726\) 0 0
\(727\) −8.38478 −0.310974 −0.155487 0.987838i \(-0.549695\pi\)
−0.155487 + 0.987838i \(0.549695\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 4.97056 0.183843
\(732\) 0 0
\(733\) 26.0000 0.960332 0.480166 0.877178i \(-0.340576\pi\)
0.480166 + 0.877178i \(0.340576\pi\)
\(734\) 0 0
\(735\) 4.65685 0.171771
\(736\) 0 0
\(737\) 16.0000 0.589368
\(738\) 0 0
\(739\) −8.68629 −0.319530 −0.159765 0.987155i \(-0.551074\pi\)
−0.159765 + 0.987155i \(0.551074\pi\)
\(740\) 0 0
\(741\) −4.24264 −0.155857
\(742\) 0 0
\(743\) −45.9411 −1.68542 −0.842708 0.538371i \(-0.819040\pi\)
−0.842708 + 0.538371i \(0.819040\pi\)
\(744\) 0 0
\(745\) −11.6569 −0.427074
\(746\) 0 0
\(747\) 6.00000 0.219529
\(748\) 0 0
\(749\) −46.6274 −1.70373
\(750\) 0 0
\(751\) 0.142136 0.00518660 0.00259330 0.999997i \(-0.499175\pi\)
0.00259330 + 0.999997i \(0.499175\pi\)
\(752\) 0 0
\(753\) 24.7279 0.901136
\(754\) 0 0
\(755\) −11.1716 −0.406575
\(756\) 0 0
\(757\) 46.7696 1.69987 0.849934 0.526889i \(-0.176642\pi\)
0.849934 + 0.526889i \(0.176642\pi\)
\(758\) 0 0
\(759\) −6.82843 −0.247856
\(760\) 0 0
\(761\) 14.9706 0.542682 0.271341 0.962483i \(-0.412533\pi\)
0.271341 + 0.962483i \(0.412533\pi\)
\(762\) 0 0
\(763\) 41.4558 1.50080
\(764\) 0 0
\(765\) −2.82843 −0.102262
\(766\) 0 0
\(767\) 12.0000 0.433295
\(768\) 0 0
\(769\) −19.6569 −0.708844 −0.354422 0.935086i \(-0.615322\pi\)
−0.354422 + 0.935086i \(0.615322\pi\)
\(770\) 0 0
\(771\) −17.6569 −0.635896
\(772\) 0 0
\(773\) 24.4853 0.880674 0.440337 0.897832i \(-0.354859\pi\)
0.440337 + 0.897832i \(0.354859\pi\)
\(774\) 0 0
\(775\) 8.82843 0.317126
\(776\) 0 0
\(777\) 24.1421 0.866094
\(778\) 0 0
\(779\) 2.24264 0.0803509
\(780\) 0 0
\(781\) 7.31371 0.261705
\(782\) 0 0
\(783\) −2.24264 −0.0801454
\(784\) 0 0
\(785\) 8.14214 0.290605
\(786\) 0 0
\(787\) 8.48528 0.302468 0.151234 0.988498i \(-0.451675\pi\)
0.151234 + 0.988498i \(0.451675\pi\)
\(788\) 0 0
\(789\) −19.6569 −0.699803
\(790\) 0 0
\(791\) 32.9706 1.17230
\(792\) 0 0
\(793\) −33.9411 −1.20528
\(794\) 0 0
\(795\) −12.4853 −0.442807
\(796\) 0 0
\(797\) −40.9706 −1.45125 −0.725626 0.688089i \(-0.758450\pi\)
−0.725626 + 0.688089i \(0.758450\pi\)
\(798\) 0 0
\(799\) −13.6569 −0.483145
\(800\) 0 0
\(801\) −13.5563 −0.478990
\(802\) 0 0
\(803\) −5.17157 −0.182501
\(804\) 0 0
\(805\) −16.4853 −0.581030
\(806\) 0 0
\(807\) −4.10051 −0.144345
\(808\) 0 0
\(809\) 2.68629 0.0944450 0.0472225 0.998884i \(-0.484963\pi\)
0.0472225 + 0.998884i \(0.484963\pi\)
\(810\) 0 0
\(811\) 48.2843 1.69549 0.847745 0.530404i \(-0.177960\pi\)
0.847745 + 0.530404i \(0.177960\pi\)
\(812\) 0 0
\(813\) 2.14214 0.0751280
\(814\) 0 0
\(815\) −0.100505 −0.00352054
\(816\) 0 0
\(817\) 1.75736 0.0614822
\(818\) 0 0
\(819\) 14.4853 0.506157
\(820\) 0 0
\(821\) −22.4853 −0.784742 −0.392371 0.919807i \(-0.628345\pi\)
−0.392371 + 0.919807i \(0.628345\pi\)
\(822\) 0 0
\(823\) −19.2132 −0.669730 −0.334865 0.942266i \(-0.608691\pi\)
−0.334865 + 0.942266i \(0.608691\pi\)
\(824\) 0 0
\(825\) −1.41421 −0.0492366
\(826\) 0 0
\(827\) −26.0000 −0.904109 −0.452054 0.891990i \(-0.649309\pi\)
−0.452054 + 0.891990i \(0.649309\pi\)
\(828\) 0 0
\(829\) −1.79899 −0.0624815 −0.0312408 0.999512i \(-0.509946\pi\)
−0.0312408 + 0.999512i \(0.509946\pi\)
\(830\) 0 0
\(831\) 5.31371 0.184331
\(832\) 0 0
\(833\) 13.1716 0.456368
\(834\) 0 0
\(835\) 15.6569 0.541828
\(836\) 0 0
\(837\) −8.82843 −0.305155
\(838\) 0 0
\(839\) 23.5147 0.811818 0.405909 0.913913i \(-0.366955\pi\)
0.405909 + 0.913913i \(0.366955\pi\)
\(840\) 0 0
\(841\) −23.9706 −0.826571
\(842\) 0 0
\(843\) −14.2426 −0.490543
\(844\) 0 0
\(845\) −5.00000 −0.172005
\(846\) 0 0
\(847\) −30.7279 −1.05582
\(848\) 0 0
\(849\) −4.10051 −0.140729
\(850\) 0 0
\(851\) −34.1421 −1.17038
\(852\) 0 0
\(853\) −47.4558 −1.62486 −0.812429 0.583061i \(-0.801855\pi\)
−0.812429 + 0.583061i \(0.801855\pi\)
\(854\) 0 0
\(855\) −1.00000 −0.0341993
\(856\) 0 0
\(857\) 44.0833 1.50586 0.752928 0.658103i \(-0.228641\pi\)
0.752928 + 0.658103i \(0.228641\pi\)
\(858\) 0 0
\(859\) −49.9411 −1.70397 −0.851985 0.523567i \(-0.824601\pi\)
−0.851985 + 0.523567i \(0.824601\pi\)
\(860\) 0 0
\(861\) −7.65685 −0.260945
\(862\) 0 0
\(863\) 36.0000 1.22545 0.612727 0.790295i \(-0.290072\pi\)
0.612727 + 0.790295i \(0.290072\pi\)
\(864\) 0 0
\(865\) −13.6569 −0.464347
\(866\) 0 0
\(867\) 9.00000 0.305656
\(868\) 0 0
\(869\) 3.31371 0.112410
\(870\) 0 0
\(871\) 48.0000 1.62642
\(872\) 0 0
\(873\) 9.89949 0.335047
\(874\) 0 0
\(875\) −3.41421 −0.115421
\(876\) 0 0
\(877\) 37.4142 1.26339 0.631694 0.775218i \(-0.282360\pi\)
0.631694 + 0.775218i \(0.282360\pi\)
\(878\) 0 0
\(879\) −4.00000 −0.134917
\(880\) 0 0
\(881\) 33.1127 1.11560 0.557798 0.829977i \(-0.311647\pi\)
0.557798 + 0.829977i \(0.311647\pi\)
\(882\) 0 0
\(883\) −9.27208 −0.312030 −0.156015 0.987755i \(-0.549865\pi\)
−0.156015 + 0.987755i \(0.549865\pi\)
\(884\) 0 0
\(885\) 2.82843 0.0950765
\(886\) 0 0
\(887\) −16.9706 −0.569816 −0.284908 0.958555i \(-0.591963\pi\)
−0.284908 + 0.958555i \(0.591963\pi\)
\(888\) 0 0
\(889\) −57.9411 −1.94328
\(890\) 0 0
\(891\) 1.41421 0.0473779
\(892\) 0 0
\(893\) −4.82843 −0.161577
\(894\) 0 0
\(895\) 23.7990 0.795512
\(896\) 0 0
\(897\) −20.4853 −0.683984
\(898\) 0 0
\(899\) 19.7990 0.660333
\(900\) 0 0
\(901\) −35.3137 −1.17647
\(902\) 0 0
\(903\) −6.00000 −0.199667
\(904\) 0 0
\(905\) −4.82843 −0.160502
\(906\) 0 0
\(907\) 13.8579 0.460143 0.230071 0.973174i \(-0.426104\pi\)
0.230071 + 0.973174i \(0.426104\pi\)
\(908\) 0 0
\(909\) −12.8284 −0.425492
\(910\) 0 0
\(911\) 20.0000 0.662630 0.331315 0.943520i \(-0.392508\pi\)
0.331315 + 0.943520i \(0.392508\pi\)
\(912\) 0 0
\(913\) 8.48528 0.280822
\(914\) 0 0
\(915\) −8.00000 −0.264472
\(916\) 0 0
\(917\) −16.8284 −0.555724
\(918\) 0 0
\(919\) 17.6569 0.582446 0.291223 0.956655i \(-0.405938\pi\)
0.291223 + 0.956655i \(0.405938\pi\)
\(920\) 0 0
\(921\) 1.17157 0.0386046
\(922\) 0 0
\(923\) 21.9411 0.722201
\(924\) 0 0
\(925\) −7.07107 −0.232495
\(926\) 0 0
\(927\) 8.00000 0.262754
\(928\) 0 0
\(929\) −24.8284 −0.814594 −0.407297 0.913296i \(-0.633529\pi\)
−0.407297 + 0.913296i \(0.633529\pi\)
\(930\) 0 0
\(931\) 4.65685 0.152622
\(932\) 0 0
\(933\) −9.41421 −0.308208
\(934\) 0 0
\(935\) −4.00000 −0.130814
\(936\) 0 0
\(937\) −13.1127 −0.428373 −0.214187 0.976793i \(-0.568710\pi\)
−0.214187 + 0.976793i \(0.568710\pi\)
\(938\) 0 0
\(939\) −29.7990 −0.972453
\(940\) 0 0
\(941\) 22.0416 0.718537 0.359268 0.933234i \(-0.383026\pi\)
0.359268 + 0.933234i \(0.383026\pi\)
\(942\) 0 0
\(943\) 10.8284 0.352622
\(944\) 0 0
\(945\) 3.41421 0.111064
\(946\) 0 0
\(947\) −25.3137 −0.822585 −0.411292 0.911503i \(-0.634923\pi\)
−0.411292 + 0.911503i \(0.634923\pi\)
\(948\) 0 0
\(949\) −15.5147 −0.503629
\(950\) 0 0
\(951\) −8.48528 −0.275154
\(952\) 0 0
\(953\) −21.9411 −0.710743 −0.355371 0.934725i \(-0.615646\pi\)
−0.355371 + 0.934725i \(0.615646\pi\)
\(954\) 0 0
\(955\) 13.4142 0.434074
\(956\) 0 0
\(957\) −3.17157 −0.102522
\(958\) 0 0
\(959\) −51.1127 −1.65052
\(960\) 0 0
\(961\) 46.9411 1.51423
\(962\) 0 0
\(963\) −13.6569 −0.440086
\(964\) 0 0
\(965\) −23.0711 −0.742684
\(966\) 0 0
\(967\) 7.89949 0.254031 0.127015 0.991901i \(-0.459460\pi\)
0.127015 + 0.991901i \(0.459460\pi\)
\(968\) 0 0
\(969\) −2.82843 −0.0908622
\(970\) 0 0
\(971\) −0.686292 −0.0220241 −0.0110121 0.999939i \(-0.503505\pi\)
−0.0110121 + 0.999939i \(0.503505\pi\)
\(972\) 0 0
\(973\) 42.6274 1.36657
\(974\) 0 0
\(975\) −4.24264 −0.135873
\(976\) 0 0
\(977\) −20.2010 −0.646288 −0.323144 0.946350i \(-0.604740\pi\)
−0.323144 + 0.946350i \(0.604740\pi\)
\(978\) 0 0
\(979\) −19.1716 −0.612726
\(980\) 0 0
\(981\) 12.1421 0.387669
\(982\) 0 0
\(983\) −21.5980 −0.688869 −0.344434 0.938810i \(-0.611929\pi\)
−0.344434 + 0.938810i \(0.611929\pi\)
\(984\) 0 0
\(985\) −8.48528 −0.270364
\(986\) 0 0
\(987\) 16.4853 0.524732
\(988\) 0 0
\(989\) 8.48528 0.269816
\(990\) 0 0
\(991\) 15.0294 0.477426 0.238713 0.971090i \(-0.423275\pi\)
0.238713 + 0.971090i \(0.423275\pi\)
\(992\) 0 0
\(993\) −12.8284 −0.407098
\(994\) 0 0
\(995\) −14.1421 −0.448336
\(996\) 0 0
\(997\) 49.7990 1.57715 0.788575 0.614939i \(-0.210819\pi\)
0.788575 + 0.614939i \(0.210819\pi\)
\(998\) 0 0
\(999\) 7.07107 0.223719
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 4560.2.a.bg.1.2 2
4.3 odd 2 2280.2.a.o.1.1 2
12.11 even 2 6840.2.a.x.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2280.2.a.o.1.1 2 4.3 odd 2
4560.2.a.bg.1.2 2 1.1 even 1 trivial
6840.2.a.x.1.1 2 12.11 even 2