Properties

Label 3864.2.a.o.1.2
Level $3864$
Weight $2$
Character 3864.1
Self dual yes
Analytic conductor $30.854$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3864.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.8541953410\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.733.1
Defining polynomial: \(x^{3} - x^{2} - 7 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.51820\) of defining polynomial
Character \(\chi\) \(=\) 3864.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -2.34132 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -2.34132 q^{5} -1.00000 q^{7} +1.00000 q^{9} +2.85952 q^{11} -0.518199 q^{13} -2.34132 q^{15} -2.17687 q^{17} -1.14048 q^{19} -1.00000 q^{21} +1.00000 q^{23} +0.481801 q^{25} +1.00000 q^{27} -2.17687 q^{29} +5.89592 q^{31} +2.85952 q^{33} +2.34132 q^{35} -9.89592 q^{37} -0.518199 q^{39} +6.85952 q^{41} +2.23725 q^{43} -2.34132 q^{45} -5.03640 q^{47} +1.00000 q^{49} -2.17687 q^{51} -1.30493 q^{53} -6.69507 q^{55} -1.14048 q^{57} -8.41412 q^{59} -6.16445 q^{61} -1.00000 q^{63} +1.21327 q^{65} -14.4141 q^{67} +1.00000 q^{69} +2.51820 q^{71} +7.21327 q^{73} +0.481801 q^{75} -2.85952 q^{77} -8.85952 q^{79} +1.00000 q^{81} +12.5786 q^{83} +5.09677 q^{85} -2.17687 q^{87} -14.5910 q^{89} +0.518199 q^{91} +5.89592 q^{93} +2.67023 q^{95} +1.82313 q^{97} +2.85952 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9} + O(q^{10}) \) \( 3 q + 3 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9} - 2 q^{11} + 5 q^{13} - 3 q^{15} - 4 q^{17} - 14 q^{19} - 3 q^{21} + 3 q^{23} + 8 q^{25} + 3 q^{27} - 4 q^{29} - 6 q^{31} - 2 q^{33} + 3 q^{35} - 6 q^{37} + 5 q^{39} + 10 q^{41} - 21 q^{43} - 3 q^{45} - 2 q^{47} + 3 q^{49} - 4 q^{51} - 13 q^{53} - 11 q^{55} - 14 q^{57} + 5 q^{59} - 17 q^{61} - 3 q^{63} - 12 q^{65} - 13 q^{67} + 3 q^{69} + q^{71} + 6 q^{73} + 8 q^{75} + 2 q^{77} - 16 q^{79} + 3 q^{81} + 6 q^{83} - 23 q^{85} - 4 q^{87} - 11 q^{89} - 5 q^{91} - 6 q^{93} + q^{95} + 8 q^{97} - 2 q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −2.34132 −1.04707 −0.523536 0.852003i \(-0.675387\pi\)
−0.523536 + 0.852003i \(0.675387\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.85952 0.862179 0.431089 0.902309i \(-0.358129\pi\)
0.431089 + 0.902309i \(0.358129\pi\)
\(12\) 0 0
\(13\) −0.518199 −0.143722 −0.0718612 0.997415i \(-0.522894\pi\)
−0.0718612 + 0.997415i \(0.522894\pi\)
\(14\) 0 0
\(15\) −2.34132 −0.604527
\(16\) 0 0
\(17\) −2.17687 −0.527969 −0.263985 0.964527i \(-0.585037\pi\)
−0.263985 + 0.964527i \(0.585037\pi\)
\(18\) 0 0
\(19\) −1.14048 −0.261643 −0.130822 0.991406i \(-0.541762\pi\)
−0.130822 + 0.991406i \(0.541762\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 0.481801 0.0963603
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −2.17687 −0.404235 −0.202118 0.979361i \(-0.564782\pi\)
−0.202118 + 0.979361i \(0.564782\pi\)
\(30\) 0 0
\(31\) 5.89592 1.05894 0.529469 0.848329i \(-0.322391\pi\)
0.529469 + 0.848329i \(0.322391\pi\)
\(32\) 0 0
\(33\) 2.85952 0.497779
\(34\) 0 0
\(35\) 2.34132 0.395756
\(36\) 0 0
\(37\) −9.89592 −1.62688 −0.813440 0.581649i \(-0.802408\pi\)
−0.813440 + 0.581649i \(0.802408\pi\)
\(38\) 0 0
\(39\) −0.518199 −0.0829782
\(40\) 0 0
\(41\) 6.85952 1.07128 0.535639 0.844447i \(-0.320071\pi\)
0.535639 + 0.844447i \(0.320071\pi\)
\(42\) 0 0
\(43\) 2.23725 0.341177 0.170588 0.985342i \(-0.445433\pi\)
0.170588 + 0.985342i \(0.445433\pi\)
\(44\) 0 0
\(45\) −2.34132 −0.349024
\(46\) 0 0
\(47\) −5.03640 −0.734634 −0.367317 0.930096i \(-0.619724\pi\)
−0.367317 + 0.930096i \(0.619724\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −2.17687 −0.304823
\(52\) 0 0
\(53\) −1.30493 −0.179246 −0.0896228 0.995976i \(-0.528566\pi\)
−0.0896228 + 0.995976i \(0.528566\pi\)
\(54\) 0 0
\(55\) −6.69507 −0.902763
\(56\) 0 0
\(57\) −1.14048 −0.151060
\(58\) 0 0
\(59\) −8.41412 −1.09543 −0.547713 0.836667i \(-0.684501\pi\)
−0.547713 + 0.836667i \(0.684501\pi\)
\(60\) 0 0
\(61\) −6.16445 −0.789277 −0.394639 0.918836i \(-0.629130\pi\)
−0.394639 + 0.918836i \(0.629130\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 1.21327 0.150488
\(66\) 0 0
\(67\) −14.4141 −1.76096 −0.880482 0.474079i \(-0.842781\pi\)
−0.880482 + 0.474079i \(0.842781\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 2.51820 0.298855 0.149428 0.988773i \(-0.452257\pi\)
0.149428 + 0.988773i \(0.452257\pi\)
\(72\) 0 0
\(73\) 7.21327 0.844250 0.422125 0.906538i \(-0.361284\pi\)
0.422125 + 0.906538i \(0.361284\pi\)
\(74\) 0 0
\(75\) 0.481801 0.0556336
\(76\) 0 0
\(77\) −2.85952 −0.325873
\(78\) 0 0
\(79\) −8.85952 −0.996774 −0.498387 0.866955i \(-0.666074\pi\)
−0.498387 + 0.866955i \(0.666074\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 12.5786 1.38068 0.690339 0.723486i \(-0.257462\pi\)
0.690339 + 0.723486i \(0.257462\pi\)
\(84\) 0 0
\(85\) 5.09677 0.552822
\(86\) 0 0
\(87\) −2.17687 −0.233385
\(88\) 0 0
\(89\) −14.5910 −1.54664 −0.773321 0.634015i \(-0.781406\pi\)
−0.773321 + 0.634015i \(0.781406\pi\)
\(90\) 0 0
\(91\) 0.518199 0.0543220
\(92\) 0 0
\(93\) 5.89592 0.611379
\(94\) 0 0
\(95\) 2.67023 0.273959
\(96\) 0 0
\(97\) 1.82313 0.185110 0.0925552 0.995708i \(-0.470497\pi\)
0.0925552 + 0.995708i \(0.470497\pi\)
\(98\) 0 0
\(99\) 2.85952 0.287393
\(100\) 0 0
\(101\) −11.0968 −1.10417 −0.552085 0.833788i \(-0.686168\pi\)
−0.552085 + 0.833788i \(0.686168\pi\)
\(102\) 0 0
\(103\) 4.70750 0.463843 0.231922 0.972734i \(-0.425499\pi\)
0.231922 + 0.972734i \(0.425499\pi\)
\(104\) 0 0
\(105\) 2.34132 0.228490
\(106\) 0 0
\(107\) 11.9563 1.15586 0.577929 0.816087i \(-0.303861\pi\)
0.577929 + 0.816087i \(0.303861\pi\)
\(108\) 0 0
\(109\) −7.70662 −0.738161 −0.369080 0.929398i \(-0.620327\pi\)
−0.369080 + 0.929398i \(0.620327\pi\)
\(110\) 0 0
\(111\) −9.89592 −0.939279
\(112\) 0 0
\(113\) −10.0604 −0.946400 −0.473200 0.880955i \(-0.656901\pi\)
−0.473200 + 0.880955i \(0.656901\pi\)
\(114\) 0 0
\(115\) −2.34132 −0.218330
\(116\) 0 0
\(117\) −0.518199 −0.0479075
\(118\) 0 0
\(119\) 2.17687 0.199554
\(120\) 0 0
\(121\) −2.82313 −0.256648
\(122\) 0 0
\(123\) 6.85952 0.618502
\(124\) 0 0
\(125\) 10.5786 0.946176
\(126\) 0 0
\(127\) −16.5661 −1.47001 −0.735004 0.678063i \(-0.762820\pi\)
−0.735004 + 0.678063i \(0.762820\pi\)
\(128\) 0 0
\(129\) 2.23725 0.196978
\(130\) 0 0
\(131\) 9.56702 0.835874 0.417937 0.908476i \(-0.362753\pi\)
0.417937 + 0.908476i \(0.362753\pi\)
\(132\) 0 0
\(133\) 1.14048 0.0988919
\(134\) 0 0
\(135\) −2.34132 −0.201509
\(136\) 0 0
\(137\) −8.57857 −0.732917 −0.366458 0.930434i \(-0.619430\pi\)
−0.366458 + 0.930434i \(0.619430\pi\)
\(138\) 0 0
\(139\) −4.97603 −0.422061 −0.211031 0.977479i \(-0.567682\pi\)
−0.211031 + 0.977479i \(0.567682\pi\)
\(140\) 0 0
\(141\) −5.03640 −0.424141
\(142\) 0 0
\(143\) −1.48180 −0.123914
\(144\) 0 0
\(145\) 5.09677 0.423264
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) −22.4017 −1.83522 −0.917609 0.397484i \(-0.869884\pi\)
−0.917609 + 0.397484i \(0.869884\pi\)
\(150\) 0 0
\(151\) −16.4745 −1.34068 −0.670338 0.742056i \(-0.733851\pi\)
−0.670338 + 0.742056i \(0.733851\pi\)
\(152\) 0 0
\(153\) −2.17687 −0.175990
\(154\) 0 0
\(155\) −13.8043 −1.10879
\(156\) 0 0
\(157\) −4.68265 −0.373716 −0.186858 0.982387i \(-0.559830\pi\)
−0.186858 + 0.982387i \(0.559830\pi\)
\(158\) 0 0
\(159\) −1.30493 −0.103487
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) −16.4141 −1.28565 −0.642827 0.766012i \(-0.722238\pi\)
−0.642827 + 0.766012i \(0.722238\pi\)
\(164\) 0 0
\(165\) −6.69507 −0.521211
\(166\) 0 0
\(167\) 10.5786 0.818594 0.409297 0.912401i \(-0.365774\pi\)
0.409297 + 0.912401i \(0.365774\pi\)
\(168\) 0 0
\(169\) −12.7315 −0.979344
\(170\) 0 0
\(171\) −1.14048 −0.0872144
\(172\) 0 0
\(173\) 7.21327 0.548415 0.274207 0.961671i \(-0.411585\pi\)
0.274207 + 0.961671i \(0.411585\pi\)
\(174\) 0 0
\(175\) −0.481801 −0.0364208
\(176\) 0 0
\(177\) −8.41412 −0.632444
\(178\) 0 0
\(179\) 6.69507 0.500413 0.250207 0.968192i \(-0.419501\pi\)
0.250207 + 0.968192i \(0.419501\pi\)
\(180\) 0 0
\(181\) 23.0531 1.71352 0.856760 0.515715i \(-0.172474\pi\)
0.856760 + 0.515715i \(0.172474\pi\)
\(182\) 0 0
\(183\) −6.16445 −0.455689
\(184\) 0 0
\(185\) 23.1696 1.70346
\(186\) 0 0
\(187\) −6.22482 −0.455204
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −18.4017 −1.33150 −0.665750 0.746175i \(-0.731888\pi\)
−0.665750 + 0.746175i \(0.731888\pi\)
\(192\) 0 0
\(193\) 15.0364 1.08234 0.541172 0.840912i \(-0.317981\pi\)
0.541172 + 0.840912i \(0.317981\pi\)
\(194\) 0 0
\(195\) 1.21327 0.0868842
\(196\) 0 0
\(197\) 11.6274 0.828417 0.414209 0.910182i \(-0.364058\pi\)
0.414209 + 0.910182i \(0.364058\pi\)
\(198\) 0 0
\(199\) −24.7679 −1.75575 −0.877874 0.478892i \(-0.841039\pi\)
−0.877874 + 0.478892i \(0.841039\pi\)
\(200\) 0 0
\(201\) −14.4141 −1.01669
\(202\) 0 0
\(203\) 2.17687 0.152787
\(204\) 0 0
\(205\) −16.0604 −1.12170
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −3.26122 −0.225583
\(210\) 0 0
\(211\) −18.0167 −1.24032 −0.620159 0.784476i \(-0.712932\pi\)
−0.620159 + 0.784476i \(0.712932\pi\)
\(212\) 0 0
\(213\) 2.51820 0.172544
\(214\) 0 0
\(215\) −5.23812 −0.357237
\(216\) 0 0
\(217\) −5.89592 −0.400241
\(218\) 0 0
\(219\) 7.21327 0.487428
\(220\) 0 0
\(221\) 1.12805 0.0758811
\(222\) 0 0
\(223\) −12.9760 −0.868939 −0.434469 0.900687i \(-0.643064\pi\)
−0.434469 + 0.900687i \(0.643064\pi\)
\(224\) 0 0
\(225\) 0.481801 0.0321201
\(226\) 0 0
\(227\) 1.73147 0.114922 0.0574609 0.998348i \(-0.481700\pi\)
0.0574609 + 0.998348i \(0.481700\pi\)
\(228\) 0 0
\(229\) −15.4985 −1.02417 −0.512084 0.858936i \(-0.671126\pi\)
−0.512084 + 0.858936i \(0.671126\pi\)
\(230\) 0 0
\(231\) −2.85952 −0.188143
\(232\) 0 0
\(233\) −8.01242 −0.524911 −0.262456 0.964944i \(-0.584532\pi\)
−0.262456 + 0.964944i \(0.584532\pi\)
\(234\) 0 0
\(235\) 11.7918 0.769215
\(236\) 0 0
\(237\) −8.85952 −0.575488
\(238\) 0 0
\(239\) 2.36617 0.153055 0.0765275 0.997067i \(-0.475617\pi\)
0.0765275 + 0.997067i \(0.475617\pi\)
\(240\) 0 0
\(241\) 3.54217 0.228171 0.114086 0.993471i \(-0.463606\pi\)
0.114086 + 0.993471i \(0.463606\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −2.34132 −0.149582
\(246\) 0 0
\(247\) 0.590993 0.0376040
\(248\) 0 0
\(249\) 12.5786 0.797134
\(250\) 0 0
\(251\) −24.5473 −1.54941 −0.774705 0.632322i \(-0.782102\pi\)
−0.774705 + 0.632322i \(0.782102\pi\)
\(252\) 0 0
\(253\) 2.85952 0.179777
\(254\) 0 0
\(255\) 5.09677 0.319172
\(256\) 0 0
\(257\) 21.4629 1.33882 0.669411 0.742893i \(-0.266547\pi\)
0.669411 + 0.742893i \(0.266547\pi\)
\(258\) 0 0
\(259\) 9.89592 0.614903
\(260\) 0 0
\(261\) −2.17687 −0.134745
\(262\) 0 0
\(263\) 14.9075 0.919234 0.459617 0.888117i \(-0.347987\pi\)
0.459617 + 0.888117i \(0.347987\pi\)
\(264\) 0 0
\(265\) 3.05526 0.187683
\(266\) 0 0
\(267\) −14.5910 −0.892954
\(268\) 0 0
\(269\) 9.52975 0.581039 0.290520 0.956869i \(-0.406172\pi\)
0.290520 + 0.956869i \(0.406172\pi\)
\(270\) 0 0
\(271\) −8.50578 −0.516689 −0.258345 0.966053i \(-0.583177\pi\)
−0.258345 + 0.966053i \(0.583177\pi\)
\(272\) 0 0
\(273\) 0.518199 0.0313628
\(274\) 0 0
\(275\) 1.37772 0.0830798
\(276\) 0 0
\(277\) −8.51820 −0.511809 −0.255904 0.966702i \(-0.582373\pi\)
−0.255904 + 0.966702i \(0.582373\pi\)
\(278\) 0 0
\(279\) 5.89592 0.352980
\(280\) 0 0
\(281\) 18.3289 1.09341 0.546705 0.837325i \(-0.315882\pi\)
0.546705 + 0.837325i \(0.315882\pi\)
\(282\) 0 0
\(283\) −4.69507 −0.279093 −0.139547 0.990216i \(-0.544564\pi\)
−0.139547 + 0.990216i \(0.544564\pi\)
\(284\) 0 0
\(285\) 2.67023 0.158171
\(286\) 0 0
\(287\) −6.85952 −0.404905
\(288\) 0 0
\(289\) −12.2612 −0.721248
\(290\) 0 0
\(291\) 1.82313 0.106874
\(292\) 0 0
\(293\) 2.63470 0.153921 0.0769604 0.997034i \(-0.475478\pi\)
0.0769604 + 0.997034i \(0.475478\pi\)
\(294\) 0 0
\(295\) 19.7002 1.14699
\(296\) 0 0
\(297\) 2.85952 0.165926
\(298\) 0 0
\(299\) −0.518199 −0.0299682
\(300\) 0 0
\(301\) −2.23725 −0.128953
\(302\) 0 0
\(303\) −11.0968 −0.637493
\(304\) 0 0
\(305\) 14.4330 0.826430
\(306\) 0 0
\(307\) −6.45783 −0.368568 −0.184284 0.982873i \(-0.558997\pi\)
−0.184284 + 0.982873i \(0.558997\pi\)
\(308\) 0 0
\(309\) 4.70750 0.267800
\(310\) 0 0
\(311\) −15.6753 −0.888867 −0.444433 0.895812i \(-0.646595\pi\)
−0.444433 + 0.895812i \(0.646595\pi\)
\(312\) 0 0
\(313\) 34.4496 1.94721 0.973605 0.228242i \(-0.0732976\pi\)
0.973605 + 0.228242i \(0.0732976\pi\)
\(314\) 0 0
\(315\) 2.34132 0.131919
\(316\) 0 0
\(317\) −3.98758 −0.223965 −0.111982 0.993710i \(-0.535720\pi\)
−0.111982 + 0.993710i \(0.535720\pi\)
\(318\) 0 0
\(319\) −6.22482 −0.348523
\(320\) 0 0
\(321\) 11.9563 0.667335
\(322\) 0 0
\(323\) 2.48267 0.138140
\(324\) 0 0
\(325\) −0.249669 −0.0138491
\(326\) 0 0
\(327\) −7.70662 −0.426177
\(328\) 0 0
\(329\) 5.03640 0.277666
\(330\) 0 0
\(331\) 13.7190 0.754067 0.377034 0.926200i \(-0.376944\pi\)
0.377034 + 0.926200i \(0.376944\pi\)
\(332\) 0 0
\(333\) −9.89592 −0.542293
\(334\) 0 0
\(335\) 33.7481 1.84386
\(336\) 0 0
\(337\) 22.3580 1.21792 0.608959 0.793202i \(-0.291588\pi\)
0.608959 + 0.793202i \(0.291588\pi\)
\(338\) 0 0
\(339\) −10.0604 −0.546404
\(340\) 0 0
\(341\) 16.8595 0.912994
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −2.34132 −0.126053
\(346\) 0 0
\(347\) 31.7357 1.70366 0.851831 0.523817i \(-0.175492\pi\)
0.851831 + 0.523817i \(0.175492\pi\)
\(348\) 0 0
\(349\) 0.670226 0.0358764 0.0179382 0.999839i \(-0.494290\pi\)
0.0179382 + 0.999839i \(0.494290\pi\)
\(350\) 0 0
\(351\) −0.518199 −0.0276594
\(352\) 0 0
\(353\) 28.9323 1.53991 0.769956 0.638097i \(-0.220278\pi\)
0.769956 + 0.638097i \(0.220278\pi\)
\(354\) 0 0
\(355\) −5.89592 −0.312923
\(356\) 0 0
\(357\) 2.17687 0.115212
\(358\) 0 0
\(359\) 16.0355 0.846323 0.423161 0.906054i \(-0.360920\pi\)
0.423161 + 0.906054i \(0.360920\pi\)
\(360\) 0 0
\(361\) −17.6993 −0.931543
\(362\) 0 0
\(363\) −2.82313 −0.148176
\(364\) 0 0
\(365\) −16.8886 −0.883990
\(366\) 0 0
\(367\) 0.566147 0.0295526 0.0147763 0.999891i \(-0.495296\pi\)
0.0147763 + 0.999891i \(0.495296\pi\)
\(368\) 0 0
\(369\) 6.85952 0.357093
\(370\) 0 0
\(371\) 1.30493 0.0677484
\(372\) 0 0
\(373\) −27.9687 −1.44817 −0.724083 0.689713i \(-0.757737\pi\)
−0.724083 + 0.689713i \(0.757737\pi\)
\(374\) 0 0
\(375\) 10.5786 0.546275
\(376\) 0 0
\(377\) 1.12805 0.0580977
\(378\) 0 0
\(379\) −0.426543 −0.0219100 −0.0109550 0.999940i \(-0.503487\pi\)
−0.0109550 + 0.999940i \(0.503487\pi\)
\(380\) 0 0
\(381\) −16.5661 −0.848709
\(382\) 0 0
\(383\) −26.9323 −1.37618 −0.688089 0.725627i \(-0.741550\pi\)
−0.688089 + 0.725627i \(0.741550\pi\)
\(384\) 0 0
\(385\) 6.69507 0.341213
\(386\) 0 0
\(387\) 2.23725 0.113726
\(388\) 0 0
\(389\) 13.6878 0.693997 0.346999 0.937866i \(-0.387201\pi\)
0.346999 + 0.937866i \(0.387201\pi\)
\(390\) 0 0
\(391\) −2.17687 −0.110089
\(392\) 0 0
\(393\) 9.56702 0.482592
\(394\) 0 0
\(395\) 20.7430 1.04369
\(396\) 0 0
\(397\) 17.4132 0.873946 0.436973 0.899475i \(-0.356051\pi\)
0.436973 + 0.899475i \(0.356051\pi\)
\(398\) 0 0
\(399\) 1.14048 0.0570952
\(400\) 0 0
\(401\) −9.82313 −0.490544 −0.245272 0.969454i \(-0.578877\pi\)
−0.245272 + 0.969454i \(0.578877\pi\)
\(402\) 0 0
\(403\) −3.05526 −0.152193
\(404\) 0 0
\(405\) −2.34132 −0.116341
\(406\) 0 0
\(407\) −28.2976 −1.40266
\(408\) 0 0
\(409\) 28.6993 1.41909 0.709545 0.704660i \(-0.248901\pi\)
0.709545 + 0.704660i \(0.248901\pi\)
\(410\) 0 0
\(411\) −8.57857 −0.423150
\(412\) 0 0
\(413\) 8.41412 0.414032
\(414\) 0 0
\(415\) −29.4505 −1.44567
\(416\) 0 0
\(417\) −4.97603 −0.243677
\(418\) 0 0
\(419\) −10.3893 −0.507549 −0.253775 0.967263i \(-0.581672\pi\)
−0.253775 + 0.967263i \(0.581672\pi\)
\(420\) 0 0
\(421\) −3.55460 −0.173240 −0.0866202 0.996241i \(-0.527607\pi\)
−0.0866202 + 0.996241i \(0.527607\pi\)
\(422\) 0 0
\(423\) −5.03640 −0.244878
\(424\) 0 0
\(425\) −1.04882 −0.0508753
\(426\) 0 0
\(427\) 6.16445 0.298319
\(428\) 0 0
\(429\) −1.48180 −0.0715420
\(430\) 0 0
\(431\) 30.6389 1.47583 0.737913 0.674896i \(-0.235811\pi\)
0.737913 + 0.674896i \(0.235811\pi\)
\(432\) 0 0
\(433\) 26.3289 1.26529 0.632643 0.774443i \(-0.281970\pi\)
0.632643 + 0.774443i \(0.281970\pi\)
\(434\) 0 0
\(435\) 5.09677 0.244371
\(436\) 0 0
\(437\) −1.14048 −0.0545564
\(438\) 0 0
\(439\) −12.1769 −0.581170 −0.290585 0.956849i \(-0.593850\pi\)
−0.290585 + 0.956849i \(0.593850\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −12.8034 −0.608308 −0.304154 0.952623i \(-0.598374\pi\)
−0.304154 + 0.952623i \(0.598374\pi\)
\(444\) 0 0
\(445\) 34.1623 1.61945
\(446\) 0 0
\(447\) −22.4017 −1.05956
\(448\) 0 0
\(449\) 16.6702 0.786717 0.393358 0.919385i \(-0.371313\pi\)
0.393358 + 0.919385i \(0.371313\pi\)
\(450\) 0 0
\(451\) 19.6150 0.923633
\(452\) 0 0
\(453\) −16.4745 −0.774039
\(454\) 0 0
\(455\) −1.21327 −0.0568790
\(456\) 0 0
\(457\) 15.4754 0.723907 0.361953 0.932196i \(-0.382110\pi\)
0.361953 + 0.932196i \(0.382110\pi\)
\(458\) 0 0
\(459\) −2.17687 −0.101608
\(460\) 0 0
\(461\) −25.1696 −1.17226 −0.586132 0.810216i \(-0.699350\pi\)
−0.586132 + 0.810216i \(0.699350\pi\)
\(462\) 0 0
\(463\) 6.55372 0.304577 0.152289 0.988336i \(-0.451336\pi\)
0.152289 + 0.988336i \(0.451336\pi\)
\(464\) 0 0
\(465\) −13.8043 −0.640157
\(466\) 0 0
\(467\) 3.56702 0.165062 0.0825310 0.996588i \(-0.473700\pi\)
0.0825310 + 0.996588i \(0.473700\pi\)
\(468\) 0 0
\(469\) 14.4141 0.665582
\(470\) 0 0
\(471\) −4.68265 −0.215765
\(472\) 0 0
\(473\) 6.39746 0.294155
\(474\) 0 0
\(475\) −0.549483 −0.0252120
\(476\) 0 0
\(477\) −1.30493 −0.0597485
\(478\) 0 0
\(479\) −14.6034 −0.667247 −0.333624 0.942706i \(-0.608271\pi\)
−0.333624 + 0.942706i \(0.608271\pi\)
\(480\) 0 0
\(481\) 5.12805 0.233819
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) −4.26853 −0.193824
\(486\) 0 0
\(487\) 27.7606 1.25795 0.628975 0.777425i \(-0.283475\pi\)
0.628975 + 0.777425i \(0.283475\pi\)
\(488\) 0 0
\(489\) −16.4141 −0.742272
\(490\) 0 0
\(491\) 16.3828 0.739347 0.369673 0.929162i \(-0.379470\pi\)
0.369673 + 0.929162i \(0.379470\pi\)
\(492\) 0 0
\(493\) 4.73878 0.213424
\(494\) 0 0
\(495\) −6.69507 −0.300921
\(496\) 0 0
\(497\) −2.51820 −0.112957
\(498\) 0 0
\(499\) −30.6951 −1.37410 −0.687050 0.726610i \(-0.741095\pi\)
−0.687050 + 0.726610i \(0.741095\pi\)
\(500\) 0 0
\(501\) 10.5786 0.472616
\(502\) 0 0
\(503\) 8.74302 0.389832 0.194916 0.980820i \(-0.437557\pi\)
0.194916 + 0.980820i \(0.437557\pi\)
\(504\) 0 0
\(505\) 25.9811 1.15615
\(506\) 0 0
\(507\) −12.7315 −0.565424
\(508\) 0 0
\(509\) −3.24456 −0.143812 −0.0719062 0.997411i \(-0.522908\pi\)
−0.0719062 + 0.997411i \(0.522908\pi\)
\(510\) 0 0
\(511\) −7.21327 −0.319096
\(512\) 0 0
\(513\) −1.14048 −0.0503533
\(514\) 0 0
\(515\) −11.0218 −0.485678
\(516\) 0 0
\(517\) −14.4017 −0.633386
\(518\) 0 0
\(519\) 7.21327 0.316627
\(520\) 0 0
\(521\) 4.48911 0.196672 0.0983358 0.995153i \(-0.468648\pi\)
0.0983358 + 0.995153i \(0.468648\pi\)
\(522\) 0 0
\(523\) −11.4150 −0.499143 −0.249571 0.968356i \(-0.580290\pi\)
−0.249571 + 0.968356i \(0.580290\pi\)
\(524\) 0 0
\(525\) −0.481801 −0.0210275
\(526\) 0 0
\(527\) −12.8347 −0.559087
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −8.41412 −0.365142
\(532\) 0 0
\(533\) −3.55460 −0.153967
\(534\) 0 0
\(535\) −27.9936 −1.21027
\(536\) 0 0
\(537\) 6.69507 0.288914
\(538\) 0 0
\(539\) 2.85952 0.123168
\(540\) 0 0
\(541\) 12.1935 0.524241 0.262121 0.965035i \(-0.415578\pi\)
0.262121 + 0.965035i \(0.415578\pi\)
\(542\) 0 0
\(543\) 23.0531 0.989302
\(544\) 0 0
\(545\) 18.0437 0.772908
\(546\) 0 0
\(547\) −28.7679 −1.23003 −0.615013 0.788517i \(-0.710849\pi\)
−0.615013 + 0.788517i \(0.710849\pi\)
\(548\) 0 0
\(549\) −6.16445 −0.263092
\(550\) 0 0
\(551\) 2.48267 0.105765
\(552\) 0 0
\(553\) 8.85952 0.376745
\(554\) 0 0
\(555\) 23.1696 0.983493
\(556\) 0 0
\(557\) 20.8282 0.882521 0.441260 0.897379i \(-0.354532\pi\)
0.441260 + 0.897379i \(0.354532\pi\)
\(558\) 0 0
\(559\) −1.15934 −0.0490348
\(560\) 0 0
\(561\) −6.22482 −0.262812
\(562\) 0 0
\(563\) −18.6869 −0.787559 −0.393779 0.919205i \(-0.628833\pi\)
−0.393779 + 0.919205i \(0.628833\pi\)
\(564\) 0 0
\(565\) 23.5546 0.990949
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −21.6463 −0.907458 −0.453729 0.891140i \(-0.649907\pi\)
−0.453729 + 0.891140i \(0.649907\pi\)
\(570\) 0 0
\(571\) −25.6398 −1.07299 −0.536496 0.843903i \(-0.680253\pi\)
−0.536496 + 0.843903i \(0.680253\pi\)
\(572\) 0 0
\(573\) −18.4017 −0.768742
\(574\) 0 0
\(575\) 0.481801 0.0200925
\(576\) 0 0
\(577\) −26.9490 −1.12190 −0.560950 0.827849i \(-0.689564\pi\)
−0.560950 + 0.827849i \(0.689564\pi\)
\(578\) 0 0
\(579\) 15.0364 0.624891
\(580\) 0 0
\(581\) −12.5786 −0.521847
\(582\) 0 0
\(583\) −3.73147 −0.154542
\(584\) 0 0
\(585\) 1.21327 0.0501626
\(586\) 0 0
\(587\) −35.4672 −1.46389 −0.731944 0.681365i \(-0.761387\pi\)
−0.731944 + 0.681365i \(0.761387\pi\)
\(588\) 0 0
\(589\) −6.72416 −0.277064
\(590\) 0 0
\(591\) 11.6274 0.478287
\(592\) 0 0
\(593\) 8.96360 0.368091 0.184046 0.982918i \(-0.441081\pi\)
0.184046 + 0.982918i \(0.441081\pi\)
\(594\) 0 0
\(595\) −5.09677 −0.208947
\(596\) 0 0
\(597\) −24.7679 −1.01368
\(598\) 0 0
\(599\) 0.799152 0.0326525 0.0163262 0.999867i \(-0.494803\pi\)
0.0163262 + 0.999867i \(0.494803\pi\)
\(600\) 0 0
\(601\) 40.4869 1.65150 0.825748 0.564039i \(-0.190753\pi\)
0.825748 + 0.564039i \(0.190753\pi\)
\(602\) 0 0
\(603\) −14.4141 −0.586988
\(604\) 0 0
\(605\) 6.60985 0.268729
\(606\) 0 0
\(607\) 21.2985 0.864479 0.432240 0.901759i \(-0.357724\pi\)
0.432240 + 0.901759i \(0.357724\pi\)
\(608\) 0 0
\(609\) 2.17687 0.0882114
\(610\) 0 0
\(611\) 2.60985 0.105583
\(612\) 0 0
\(613\) −18.5786 −0.750381 −0.375191 0.926948i \(-0.622423\pi\)
−0.375191 + 0.926948i \(0.622423\pi\)
\(614\) 0 0
\(615\) −16.0604 −0.647617
\(616\) 0 0
\(617\) 19.4818 0.784308 0.392154 0.919900i \(-0.371730\pi\)
0.392154 + 0.919900i \(0.371730\pi\)
\(618\) 0 0
\(619\) 20.3349 0.817328 0.408664 0.912685i \(-0.365995\pi\)
0.408664 + 0.912685i \(0.365995\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 14.5910 0.584576
\(624\) 0 0
\(625\) −27.1769 −1.08707
\(626\) 0 0
\(627\) −3.26122 −0.130241
\(628\) 0 0
\(629\) 21.5422 0.858943
\(630\) 0 0
\(631\) 21.3340 0.849294 0.424647 0.905359i \(-0.360398\pi\)
0.424647 + 0.905359i \(0.360398\pi\)
\(632\) 0 0
\(633\) −18.0167 −0.716098
\(634\) 0 0
\(635\) 38.7867 1.53920
\(636\) 0 0
\(637\) −0.518199 −0.0205318
\(638\) 0 0
\(639\) 2.51820 0.0996184
\(640\) 0 0
\(641\) 38.7117 1.52902 0.764511 0.644611i \(-0.222981\pi\)
0.764511 + 0.644611i \(0.222981\pi\)
\(642\) 0 0
\(643\) −38.0291 −1.49972 −0.749860 0.661596i \(-0.769879\pi\)
−0.749860 + 0.661596i \(0.769879\pi\)
\(644\) 0 0
\(645\) −5.23812 −0.206251
\(646\) 0 0
\(647\) 40.1811 1.57968 0.789841 0.613311i \(-0.210163\pi\)
0.789841 + 0.613311i \(0.210163\pi\)
\(648\) 0 0
\(649\) −24.0604 −0.944452
\(650\) 0 0
\(651\) −5.89592 −0.231079
\(652\) 0 0
\(653\) 18.2060 0.712454 0.356227 0.934399i \(-0.384063\pi\)
0.356227 + 0.934399i \(0.384063\pi\)
\(654\) 0 0
\(655\) −22.3995 −0.875221
\(656\) 0 0
\(657\) 7.21327 0.281417
\(658\) 0 0
\(659\) −48.7242 −1.89802 −0.949012 0.315240i \(-0.897915\pi\)
−0.949012 + 0.315240i \(0.897915\pi\)
\(660\) 0 0
\(661\) −32.0646 −1.24717 −0.623584 0.781756i \(-0.714324\pi\)
−0.623584 + 0.781756i \(0.714324\pi\)
\(662\) 0 0
\(663\) 1.12805 0.0438100
\(664\) 0 0
\(665\) −2.67023 −0.103547
\(666\) 0 0
\(667\) −2.17687 −0.0842889
\(668\) 0 0
\(669\) −12.9760 −0.501682
\(670\) 0 0
\(671\) −17.6274 −0.680498
\(672\) 0 0
\(673\) 13.2612 0.511182 0.255591 0.966785i \(-0.417730\pi\)
0.255591 + 0.966785i \(0.417730\pi\)
\(674\) 0 0
\(675\) 0.481801 0.0185445
\(676\) 0 0
\(677\) −13.6274 −0.523743 −0.261872 0.965103i \(-0.584340\pi\)
−0.261872 + 0.965103i \(0.584340\pi\)
\(678\) 0 0
\(679\) −1.82313 −0.0699652
\(680\) 0 0
\(681\) 1.73147 0.0663501
\(682\) 0 0
\(683\) 9.77518 0.374037 0.187018 0.982356i \(-0.440118\pi\)
0.187018 + 0.982356i \(0.440118\pi\)
\(684\) 0 0
\(685\) 20.0852 0.767417
\(686\) 0 0
\(687\) −15.4985 −0.591303
\(688\) 0 0
\(689\) 0.676212 0.0257616
\(690\) 0 0
\(691\) 19.7002 0.749430 0.374715 0.927140i \(-0.377740\pi\)
0.374715 + 0.927140i \(0.377740\pi\)
\(692\) 0 0
\(693\) −2.85952 −0.108624
\(694\) 0 0
\(695\) 11.6505 0.441928
\(696\) 0 0
\(697\) −14.9323 −0.565602
\(698\) 0 0
\(699\) −8.01242 −0.303058
\(700\) 0 0
\(701\) −16.6223 −0.627815 −0.313907 0.949454i \(-0.601638\pi\)
−0.313907 + 0.949454i \(0.601638\pi\)
\(702\) 0 0
\(703\) 11.2861 0.425662
\(704\) 0 0
\(705\) 11.7918 0.444106
\(706\) 0 0
\(707\) 11.0968 0.417337
\(708\) 0 0
\(709\) 38.6389 1.45112 0.725558 0.688161i \(-0.241582\pi\)
0.725558 + 0.688161i \(0.241582\pi\)
\(710\) 0 0
\(711\) −8.85952 −0.332258
\(712\) 0 0
\(713\) 5.89592 0.220804
\(714\) 0 0
\(715\) 3.46938 0.129747
\(716\) 0 0
\(717\) 2.36617 0.0883663
\(718\) 0 0
\(719\) 25.6878 0.957992 0.478996 0.877817i \(-0.341001\pi\)
0.478996 + 0.877817i \(0.341001\pi\)
\(720\) 0 0
\(721\) −4.70750 −0.175316
\(722\) 0 0
\(723\) 3.54217 0.131735
\(724\) 0 0
\(725\) −1.04882 −0.0389522
\(726\) 0 0
\(727\) 18.2248 0.675921 0.337961 0.941160i \(-0.390263\pi\)
0.337961 + 0.941160i \(0.390263\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −4.87020 −0.180131
\(732\) 0 0
\(733\) 7.46294 0.275650 0.137825 0.990457i \(-0.455989\pi\)
0.137825 + 0.990457i \(0.455989\pi\)
\(734\) 0 0
\(735\) −2.34132 −0.0863611
\(736\) 0 0
\(737\) −41.2175 −1.51827
\(738\) 0 0
\(739\) 22.3704 0.822909 0.411454 0.911430i \(-0.365021\pi\)
0.411454 + 0.911430i \(0.365021\pi\)
\(740\) 0 0
\(741\) 0.590993 0.0217107
\(742\) 0 0
\(743\) −23.3529 −0.856734 −0.428367 0.903605i \(-0.640911\pi\)
−0.428367 + 0.903605i \(0.640911\pi\)
\(744\) 0 0
\(745\) 52.4496 1.92161
\(746\) 0 0
\(747\) 12.5786 0.460226
\(748\) 0 0
\(749\) −11.9563 −0.436873
\(750\) 0 0
\(751\) −22.3165 −0.814340 −0.407170 0.913352i \(-0.633484\pi\)
−0.407170 + 0.913352i \(0.633484\pi\)
\(752\) 0 0
\(753\) −24.5473 −0.894553
\(754\) 0 0
\(755\) 38.5721 1.40378
\(756\) 0 0
\(757\) 5.23812 0.190383 0.0951913 0.995459i \(-0.469654\pi\)
0.0951913 + 0.995459i \(0.469654\pi\)
\(758\) 0 0
\(759\) 2.85952 0.103794
\(760\) 0 0
\(761\) 11.7190 0.424815 0.212408 0.977181i \(-0.431870\pi\)
0.212408 + 0.977181i \(0.431870\pi\)
\(762\) 0 0
\(763\) 7.70662 0.278999
\(764\) 0 0
\(765\) 5.09677 0.184274
\(766\) 0 0
\(767\) 4.36019 0.157437
\(768\) 0 0
\(769\) −15.7439 −0.567739 −0.283870 0.958863i \(-0.591618\pi\)
−0.283870 + 0.958863i \(0.591618\pi\)
\(770\) 0 0
\(771\) 21.4629 0.772969
\(772\) 0 0
\(773\) 20.6514 0.742778 0.371389 0.928477i \(-0.378882\pi\)
0.371389 + 0.928477i \(0.378882\pi\)
\(774\) 0 0
\(775\) 2.84066 0.102040
\(776\) 0 0
\(777\) 9.89592 0.355014
\(778\) 0 0
\(779\) −7.82313 −0.280293
\(780\) 0 0
\(781\) 7.20085 0.257667
\(782\) 0 0
\(783\) −2.17687 −0.0777951
\(784\) 0 0
\(785\) 10.9636 0.391308
\(786\) 0 0
\(787\) −6.83044 −0.243479 −0.121739 0.992562i \(-0.538847\pi\)
−0.121739 + 0.992562i \(0.538847\pi\)
\(788\) 0 0
\(789\) 14.9075 0.530720
\(790\) 0 0
\(791\) 10.0604 0.357706
\(792\) 0 0
\(793\) 3.19441 0.113437
\(794\) 0 0
\(795\) 3.05526 0.108359
\(796\) 0 0
\(797\) −3.67110 −0.130037 −0.0650185 0.997884i \(-0.520711\pi\)
−0.0650185 + 0.997884i \(0.520711\pi\)
\(798\) 0 0
\(799\) 10.9636 0.387864
\(800\) 0 0
\(801\) −14.5910 −0.515547
\(802\) 0 0
\(803\) 20.6265 0.727894
\(804\) 0 0
\(805\) 2.34132 0.0825209
\(806\) 0 0
\(807\) 9.52975 0.335463
\(808\) 0 0
\(809\) −13.9083 −0.488991 −0.244496 0.969650i \(-0.578622\pi\)
−0.244496 + 0.969650i \(0.578622\pi\)
\(810\) 0 0
\(811\) −19.6398 −0.689647 −0.344824 0.938668i \(-0.612061\pi\)
−0.344824 + 0.938668i \(0.612061\pi\)
\(812\) 0 0
\(813\) −8.50578 −0.298311
\(814\) 0 0
\(815\) 38.4308 1.34617
\(816\) 0 0
\(817\) −2.55153 −0.0892666
\(818\) 0 0
\(819\) 0.518199 0.0181073
\(820\) 0 0
\(821\) 17.9752 0.627337 0.313669 0.949533i \(-0.398442\pi\)
0.313669 + 0.949533i \(0.398442\pi\)
\(822\) 0 0
\(823\) 3.96448 0.138193 0.0690965 0.997610i \(-0.477988\pi\)
0.0690965 + 0.997610i \(0.477988\pi\)
\(824\) 0 0
\(825\) 1.37772 0.0479661
\(826\) 0 0
\(827\) −21.2570 −0.739178 −0.369589 0.929195i \(-0.620501\pi\)
−0.369589 + 0.929195i \(0.620501\pi\)
\(828\) 0 0
\(829\) −5.82313 −0.202246 −0.101123 0.994874i \(-0.532243\pi\)
−0.101123 + 0.994874i \(0.532243\pi\)
\(830\) 0 0
\(831\) −8.51820 −0.295493
\(832\) 0 0
\(833\) −2.17687 −0.0754242
\(834\) 0 0
\(835\) −24.7679 −0.857127
\(836\) 0 0
\(837\) 5.89592 0.203793
\(838\) 0 0
\(839\) −19.1760 −0.662029 −0.331015 0.943626i \(-0.607391\pi\)
−0.331015 + 0.943626i \(0.607391\pi\)
\(840\) 0 0
\(841\) −24.2612 −0.836594
\(842\) 0 0
\(843\) 18.3289 0.631281
\(844\) 0 0
\(845\) 29.8085 1.02544
\(846\) 0 0
\(847\) 2.82313 0.0970038
\(848\) 0 0
\(849\) −4.69507 −0.161134
\(850\) 0 0
\(851\) −9.89592 −0.339228
\(852\) 0 0
\(853\) 6.32890 0.216697 0.108349 0.994113i \(-0.465444\pi\)
0.108349 + 0.994113i \(0.465444\pi\)
\(854\) 0 0
\(855\) 2.67023 0.0913198
\(856\) 0 0
\(857\) 24.4017 0.833546 0.416773 0.909011i \(-0.363161\pi\)
0.416773 + 0.909011i \(0.363161\pi\)
\(858\) 0 0
\(859\) 10.9884 0.374921 0.187461 0.982272i \(-0.439974\pi\)
0.187461 + 0.982272i \(0.439974\pi\)
\(860\) 0 0
\(861\) −6.85952 −0.233772
\(862\) 0 0
\(863\) 56.8282 1.93446 0.967228 0.253910i \(-0.0817167\pi\)
0.967228 + 0.253910i \(0.0817167\pi\)
\(864\) 0 0
\(865\) −16.8886 −0.574230
\(866\) 0 0
\(867\) −12.2612 −0.416413
\(868\) 0 0
\(869\) −25.3340 −0.859398
\(870\) 0 0
\(871\) 7.46938 0.253090
\(872\) 0 0
\(873\) 1.82313 0.0617035
\(874\) 0 0
\(875\) −10.5786 −0.357621
\(876\) 0 0
\(877\) −7.71905 −0.260654 −0.130327 0.991471i \(-0.541603\pi\)
−0.130327 + 0.991471i \(0.541603\pi\)
\(878\) 0 0
\(879\) 2.63470 0.0888663
\(880\) 0 0
\(881\) 31.8895 1.07438 0.537192 0.843460i \(-0.319485\pi\)
0.537192 + 0.843460i \(0.319485\pi\)
\(882\) 0 0
\(883\) 18.1396 0.610446 0.305223 0.952281i \(-0.401269\pi\)
0.305223 + 0.952281i \(0.401269\pi\)
\(884\) 0 0
\(885\) 19.7002 0.662214
\(886\) 0 0
\(887\) −25.0903 −0.842451 −0.421225 0.906956i \(-0.638400\pi\)
−0.421225 + 0.906956i \(0.638400\pi\)
\(888\) 0 0
\(889\) 16.5661 0.555611
\(890\) 0 0
\(891\) 2.85952 0.0957976
\(892\) 0 0
\(893\) 5.74389 0.192212
\(894\) 0 0
\(895\) −15.6753 −0.523969
\(896\) 0 0
\(897\) −0.518199 −0.0173022
\(898\) 0 0
\(899\) −12.8347 −0.428060
\(900\) 0 0
\(901\) 2.84066 0.0946362
\(902\) 0 0
\(903\) −2.23725 −0.0744509
\(904\) 0 0
\(905\) −53.9747 −1.79418
\(906\) 0 0
\(907\) 12.0373 0.399691 0.199845 0.979827i \(-0.435956\pi\)
0.199845 + 0.979827i \(0.435956\pi\)
\(908\) 0 0
\(909\) −11.0968 −0.368057
\(910\) 0 0
\(911\) 14.9636 0.495766 0.247883 0.968790i \(-0.420265\pi\)
0.247883 + 0.968790i \(0.420265\pi\)
\(912\) 0 0
\(913\) 35.9687 1.19039
\(914\) 0 0
\(915\) 14.4330 0.477140
\(916\) 0 0
\(917\) −9.56702 −0.315931
\(918\) 0 0
\(919\) 1.04284 0.0344000 0.0172000 0.999852i \(-0.494525\pi\)
0.0172000 + 0.999852i \(0.494525\pi\)
\(920\) 0 0
\(921\) −6.45783 −0.212793
\(922\) 0 0
\(923\) −1.30493 −0.0429522
\(924\) 0 0
\(925\) −4.76787 −0.156767
\(926\) 0 0
\(927\) 4.70750 0.154614
\(928\) 0 0
\(929\) −5.91478 −0.194058 −0.0970288 0.995282i \(-0.530934\pi\)
−0.0970288 + 0.995282i \(0.530934\pi\)
\(930\) 0 0
\(931\) −1.14048 −0.0373776
\(932\) 0 0
\(933\) −15.6753 −0.513188
\(934\) 0 0
\(935\) 14.5743 0.476632
\(936\) 0 0
\(937\) −8.88262 −0.290183 −0.145091 0.989418i \(-0.546348\pi\)
−0.145091 + 0.989418i \(0.546348\pi\)
\(938\) 0 0
\(939\) 34.4496 1.12422
\(940\) 0 0
\(941\) −2.87619 −0.0937610 −0.0468805 0.998901i \(-0.514928\pi\)
−0.0468805 + 0.998901i \(0.514928\pi\)
\(942\) 0 0
\(943\) 6.85952 0.223377
\(944\) 0 0
\(945\) 2.34132 0.0761633
\(946\) 0 0
\(947\) −40.3537 −1.31132 −0.655660 0.755056i \(-0.727610\pi\)
−0.655660 + 0.755056i \(0.727610\pi\)
\(948\) 0 0
\(949\) −3.73791 −0.121338
\(950\) 0 0
\(951\) −3.98758 −0.129306
\(952\) 0 0
\(953\) 10.8702 0.352120 0.176060 0.984379i \(-0.443665\pi\)
0.176060 + 0.984379i \(0.443665\pi\)
\(954\) 0 0
\(955\) 43.0843 1.39418
\(956\) 0 0
\(957\) −6.22482 −0.201220
\(958\) 0 0
\(959\) 8.57857 0.277016
\(960\) 0 0
\(961\) 3.76188 0.121351
\(962\) 0 0
\(963\) 11.9563 0.385286
\(964\) 0 0
\(965\) −35.2051 −1.13329
\(966\) 0 0
\(967\) 9.01155 0.289792 0.144896 0.989447i \(-0.453715\pi\)
0.144896 + 0.989447i \(0.453715\pi\)
\(968\) 0 0
\(969\) 2.48267 0.0797550
\(970\) 0 0
\(971\) −15.0655 −0.483474 −0.241737 0.970342i \(-0.577717\pi\)
−0.241737 + 0.970342i \(0.577717\pi\)
\(972\) 0 0
\(973\) 4.97603 0.159524
\(974\) 0 0
\(975\) −0.249669 −0.00799580
\(976\) 0 0
\(977\) 17.9083 0.572939 0.286469 0.958089i \(-0.407518\pi\)
0.286469 + 0.958089i \(0.407518\pi\)
\(978\) 0 0
\(979\) −41.7233 −1.33348
\(980\) 0 0
\(981\) −7.70662 −0.246054
\(982\) 0 0
\(983\) 9.77518 0.311780 0.155890 0.987774i \(-0.450176\pi\)
0.155890 + 0.987774i \(0.450176\pi\)
\(984\) 0 0
\(985\) −27.2235 −0.867413
\(986\) 0 0
\(987\) 5.03640 0.160310
\(988\) 0 0
\(989\) 2.23725 0.0711403
\(990\) 0 0
\(991\) 21.1281 0.671155 0.335577 0.942013i \(-0.391069\pi\)
0.335577 + 0.942013i \(0.391069\pi\)
\(992\) 0 0
\(993\) 13.7190 0.435361
\(994\) 0 0
\(995\) 57.9896 1.83839
\(996\) 0 0
\(997\) 34.1769 1.08239 0.541196 0.840896i \(-0.317972\pi\)
0.541196 + 0.840896i \(0.317972\pi\)
\(998\) 0 0
\(999\) −9.89592 −0.313093
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 3864.2.a.o.1.2 3
4.3 odd 2 7728.2.a.bq.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3864.2.a.o.1.2 3 1.1 even 1 trivial
7728.2.a.bq.1.2 3 4.3 odd 2