Properties

Label 8004.2.a.e.1.7
Level 8004
Weight 2
Character 8004.1
Self dual yes
Analytic conductor 63.912
Analytic rank 1
Dimension 9
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9122617778\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - 3 x^{8} - 13 x^{7} + 32 x^{6} + 40 x^{5} - 79 x^{4} - 39 x^{3} + 58 x^{2} + 9 x - 11\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(1.39711\) of defining polynomial
Character \(\chi\) \(=\) 8004.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.20115 q^{5} -1.61263 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.20115 q^{5} -1.61263 q^{7} +1.00000 q^{9} +0.338239 q^{11} +3.11150 q^{13} -1.20115 q^{15} -0.639136 q^{17} +0.596248 q^{19} +1.61263 q^{21} -1.00000 q^{23} -3.55724 q^{25} -1.00000 q^{27} +1.00000 q^{29} +6.81971 q^{31} -0.338239 q^{33} -1.93701 q^{35} -9.39475 q^{37} -3.11150 q^{39} -7.25774 q^{41} -10.1340 q^{43} +1.20115 q^{45} +8.35438 q^{47} -4.39942 q^{49} +0.639136 q^{51} -3.58270 q^{53} +0.406277 q^{55} -0.596248 q^{57} -7.07731 q^{59} +2.15381 q^{61} -1.61263 q^{63} +3.73739 q^{65} +14.3862 q^{67} +1.00000 q^{69} +2.94972 q^{71} -2.80159 q^{73} +3.55724 q^{75} -0.545455 q^{77} -0.0498286 q^{79} +1.00000 q^{81} -1.56713 q^{83} -0.767698 q^{85} -1.00000 q^{87} +1.00920 q^{89} -5.01771 q^{91} -6.81971 q^{93} +0.716184 q^{95} +6.80229 q^{97} +0.338239 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q - 9q^{3} - 3q^{5} + 7q^{7} + 9q^{9} + O(q^{10}) \) \( 9q - 9q^{3} - 3q^{5} + 7q^{7} + 9q^{9} - 2q^{11} - 7q^{13} + 3q^{15} - q^{19} - 7q^{21} - 9q^{23} + 2q^{25} - 9q^{27} + 9q^{29} + 8q^{31} + 2q^{33} - 5q^{35} - 8q^{37} + 7q^{39} - 19q^{41} - 3q^{43} - 3q^{45} - 3q^{47} - 18q^{49} - 17q^{53} + 9q^{55} + q^{57} - 10q^{59} + q^{61} + 7q^{63} - 16q^{65} + 12q^{67} + 9q^{69} - 7q^{71} + 13q^{73} - 2q^{75} - 15q^{77} - 10q^{79} + 9q^{81} + 9q^{83} - 6q^{85} - 9q^{87} - 5q^{89} - 18q^{91} - 8q^{93} + 31q^{95} - 7q^{97} - 2q^{99} + O(q^{100}) \)

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
\(5\) 1.20115 0.537171 0.268585 0.963256i \(-0.413444\pi\)
0.268585 + 0.963256i \(0.413444\pi\)
\(6\) 0 0
\(7\) −1.61263 −0.609517 −0.304759 0.952430i \(-0.598576\pi\)
−0.304759 + 0.952430i \(0.598576\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0.338239 0.101983 0.0509915 0.998699i \(-0.483762\pi\)
0.0509915 + 0.998699i \(0.483762\pi\)
\(12\) 0 0
\(13\) 3.11150 0.862976 0.431488 0.902119i \(-0.357989\pi\)
0.431488 + 0.902119i \(0.357989\pi\)
\(14\) 0 0
\(15\) −1.20115 −0.310136
\(16\) 0 0
\(17\) −0.639136 −0.155013 −0.0775066 0.996992i \(-0.524696\pi\)
−0.0775066 + 0.996992i \(0.524696\pi\)
\(18\) 0 0
\(19\) 0.596248 0.136789 0.0683944 0.997658i \(-0.478212\pi\)
0.0683944 + 0.997658i \(0.478212\pi\)
\(20\) 0 0
\(21\) 1.61263 0.351905
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −3.55724 −0.711447
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 6.81971 1.22486 0.612428 0.790527i \(-0.290193\pi\)
0.612428 + 0.790527i \(0.290193\pi\)
\(32\) 0 0
\(33\) −0.338239 −0.0588799
\(34\) 0 0
\(35\) −1.93701 −0.327415
\(36\) 0 0
\(37\) −9.39475 −1.54449 −0.772243 0.635327i \(-0.780865\pi\)
−0.772243 + 0.635327i \(0.780865\pi\)
\(38\) 0 0
\(39\) −3.11150 −0.498239
\(40\) 0 0
\(41\) −7.25774 −1.13347 −0.566734 0.823901i \(-0.691793\pi\)
−0.566734 + 0.823901i \(0.691793\pi\)
\(42\) 0 0
\(43\) −10.1340 −1.54543 −0.772713 0.634756i \(-0.781101\pi\)
−0.772713 + 0.634756i \(0.781101\pi\)
\(44\) 0 0
\(45\) 1.20115 0.179057
\(46\) 0 0
\(47\) 8.35438 1.21861 0.609306 0.792935i \(-0.291448\pi\)
0.609306 + 0.792935i \(0.291448\pi\)
\(48\) 0 0
\(49\) −4.39942 −0.628489
\(50\) 0 0
\(51\) 0.639136 0.0894969
\(52\) 0 0
\(53\) −3.58270 −0.492122 −0.246061 0.969254i \(-0.579136\pi\)
−0.246061 + 0.969254i \(0.579136\pi\)
\(54\) 0 0
\(55\) 0.406277 0.0547823
\(56\) 0 0
\(57\) −0.596248 −0.0789750
\(58\) 0 0
\(59\) −7.07731 −0.921388 −0.460694 0.887559i \(-0.652399\pi\)
−0.460694 + 0.887559i \(0.652399\pi\)
\(60\) 0 0
\(61\) 2.15381 0.275767 0.137883 0.990448i \(-0.455970\pi\)
0.137883 + 0.990448i \(0.455970\pi\)
\(62\) 0 0
\(63\) −1.61263 −0.203172
\(64\) 0 0
\(65\) 3.73739 0.463566
\(66\) 0 0
\(67\) 14.3862 1.75756 0.878780 0.477227i \(-0.158358\pi\)
0.878780 + 0.477227i \(0.158358\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 2.94972 0.350067 0.175034 0.984562i \(-0.443997\pi\)
0.175034 + 0.984562i \(0.443997\pi\)
\(72\) 0 0
\(73\) −2.80159 −0.327901 −0.163951 0.986469i \(-0.552424\pi\)
−0.163951 + 0.986469i \(0.552424\pi\)
\(74\) 0 0
\(75\) 3.55724 0.410754
\(76\) 0 0
\(77\) −0.545455 −0.0621604
\(78\) 0 0
\(79\) −0.0498286 −0.00560616 −0.00280308 0.999996i \(-0.500892\pi\)
−0.00280308 + 0.999996i \(0.500892\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −1.56713 −0.172015 −0.0860074 0.996294i \(-0.527411\pi\)
−0.0860074 + 0.996294i \(0.527411\pi\)
\(84\) 0 0
\(85\) −0.767698 −0.0832685
\(86\) 0 0
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) 1.00920 0.106975 0.0534876 0.998569i \(-0.482966\pi\)
0.0534876 + 0.998569i \(0.482966\pi\)
\(90\) 0 0
\(91\) −5.01771 −0.525999
\(92\) 0 0
\(93\) −6.81971 −0.707171
\(94\) 0 0
\(95\) 0.716184 0.0734789
\(96\) 0 0
\(97\) 6.80229 0.690667 0.345334 0.938480i \(-0.387766\pi\)
0.345334 + 0.938480i \(0.387766\pi\)
\(98\) 0 0
\(99\) 0.338239 0.0339943
\(100\) 0 0
\(101\) −17.8893 −1.78005 −0.890024 0.455913i \(-0.849313\pi\)
−0.890024 + 0.455913i \(0.849313\pi\)
\(102\) 0 0
\(103\) 0.803404 0.0791618 0.0395809 0.999216i \(-0.487398\pi\)
0.0395809 + 0.999216i \(0.487398\pi\)
\(104\) 0 0
\(105\) 1.93701 0.189033
\(106\) 0 0
\(107\) 6.26733 0.605886 0.302943 0.953009i \(-0.402031\pi\)
0.302943 + 0.953009i \(0.402031\pi\)
\(108\) 0 0
\(109\) 5.86541 0.561804 0.280902 0.959736i \(-0.409366\pi\)
0.280902 + 0.959736i \(0.409366\pi\)
\(110\) 0 0
\(111\) 9.39475 0.891710
\(112\) 0 0
\(113\) 18.1926 1.71141 0.855707 0.517461i \(-0.173123\pi\)
0.855707 + 0.517461i \(0.173123\pi\)
\(114\) 0 0
\(115\) −1.20115 −0.112008
\(116\) 0 0
\(117\) 3.11150 0.287659
\(118\) 0 0
\(119\) 1.03069 0.0944832
\(120\) 0 0
\(121\) −10.8856 −0.989599
\(122\) 0 0
\(123\) 7.25774 0.654408
\(124\) 0 0
\(125\) −10.2785 −0.919340
\(126\) 0 0
\(127\) 4.35561 0.386498 0.193249 0.981150i \(-0.438097\pi\)
0.193249 + 0.981150i \(0.438097\pi\)
\(128\) 0 0
\(129\) 10.1340 0.892252
\(130\) 0 0
\(131\) 8.07738 0.705724 0.352862 0.935675i \(-0.385209\pi\)
0.352862 + 0.935675i \(0.385209\pi\)
\(132\) 0 0
\(133\) −0.961528 −0.0833751
\(134\) 0 0
\(135\) −1.20115 −0.103379
\(136\) 0 0
\(137\) −21.7999 −1.86249 −0.931245 0.364392i \(-0.881277\pi\)
−0.931245 + 0.364392i \(0.881277\pi\)
\(138\) 0 0
\(139\) 8.45828 0.717422 0.358711 0.933449i \(-0.383216\pi\)
0.358711 + 0.933449i \(0.383216\pi\)
\(140\) 0 0
\(141\) −8.35438 −0.703566
\(142\) 0 0
\(143\) 1.05243 0.0880089
\(144\) 0 0
\(145\) 1.20115 0.0997501
\(146\) 0 0
\(147\) 4.39942 0.362858
\(148\) 0 0
\(149\) −2.71469 −0.222396 −0.111198 0.993798i \(-0.535469\pi\)
−0.111198 + 0.993798i \(0.535469\pi\)
\(150\) 0 0
\(151\) −15.0192 −1.22225 −0.611123 0.791536i \(-0.709282\pi\)
−0.611123 + 0.791536i \(0.709282\pi\)
\(152\) 0 0
\(153\) −0.639136 −0.0516710
\(154\) 0 0
\(155\) 8.19150 0.657957
\(156\) 0 0
\(157\) −14.5542 −1.16155 −0.580775 0.814064i \(-0.697251\pi\)
−0.580775 + 0.814064i \(0.697251\pi\)
\(158\) 0 0
\(159\) 3.58270 0.284127
\(160\) 0 0
\(161\) 1.61263 0.127093
\(162\) 0 0
\(163\) 0.681185 0.0533545 0.0266773 0.999644i \(-0.491507\pi\)
0.0266773 + 0.999644i \(0.491507\pi\)
\(164\) 0 0
\(165\) −0.406277 −0.0316286
\(166\) 0 0
\(167\) −11.6805 −0.903863 −0.451931 0.892053i \(-0.649265\pi\)
−0.451931 + 0.892053i \(0.649265\pi\)
\(168\) 0 0
\(169\) −3.31854 −0.255272
\(170\) 0 0
\(171\) 0.596248 0.0455962
\(172\) 0 0
\(173\) −9.91936 −0.754155 −0.377077 0.926182i \(-0.623071\pi\)
−0.377077 + 0.926182i \(0.623071\pi\)
\(174\) 0 0
\(175\) 5.73651 0.433639
\(176\) 0 0
\(177\) 7.07731 0.531963
\(178\) 0 0
\(179\) 0.993143 0.0742310 0.0371155 0.999311i \(-0.488183\pi\)
0.0371155 + 0.999311i \(0.488183\pi\)
\(180\) 0 0
\(181\) −11.2303 −0.834738 −0.417369 0.908737i \(-0.637048\pi\)
−0.417369 + 0.908737i \(0.637048\pi\)
\(182\) 0 0
\(183\) −2.15381 −0.159214
\(184\) 0 0
\(185\) −11.2845 −0.829653
\(186\) 0 0
\(187\) −0.216181 −0.0158087
\(188\) 0 0
\(189\) 1.61263 0.117302
\(190\) 0 0
\(191\) −6.98963 −0.505752 −0.252876 0.967499i \(-0.581376\pi\)
−0.252876 + 0.967499i \(0.581376\pi\)
\(192\) 0 0
\(193\) 23.6537 1.70263 0.851316 0.524653i \(-0.175805\pi\)
0.851316 + 0.524653i \(0.175805\pi\)
\(194\) 0 0
\(195\) −3.73739 −0.267640
\(196\) 0 0
\(197\) −12.5658 −0.895279 −0.447640 0.894214i \(-0.647735\pi\)
−0.447640 + 0.894214i \(0.647735\pi\)
\(198\) 0 0
\(199\) −13.2327 −0.938043 −0.469022 0.883187i \(-0.655393\pi\)
−0.469022 + 0.883187i \(0.655393\pi\)
\(200\) 0 0
\(201\) −14.3862 −1.01473
\(202\) 0 0
\(203\) −1.61263 −0.113184
\(204\) 0 0
\(205\) −8.71764 −0.608866
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) 0.201675 0.0139501
\(210\) 0 0
\(211\) 17.0732 1.17537 0.587684 0.809091i \(-0.300040\pi\)
0.587684 + 0.809091i \(0.300040\pi\)
\(212\) 0 0
\(213\) −2.94972 −0.202111
\(214\) 0 0
\(215\) −12.1725 −0.830158
\(216\) 0 0
\(217\) −10.9977 −0.746570
\(218\) 0 0
\(219\) 2.80159 0.189314
\(220\) 0 0
\(221\) −1.98867 −0.133773
\(222\) 0 0
\(223\) −2.73917 −0.183428 −0.0917141 0.995785i \(-0.529235\pi\)
−0.0917141 + 0.995785i \(0.529235\pi\)
\(224\) 0 0
\(225\) −3.55724 −0.237149
\(226\) 0 0
\(227\) 15.3610 1.01954 0.509772 0.860309i \(-0.329730\pi\)
0.509772 + 0.860309i \(0.329730\pi\)
\(228\) 0 0
\(229\) −13.5282 −0.893966 −0.446983 0.894543i \(-0.647502\pi\)
−0.446983 + 0.894543i \(0.647502\pi\)
\(230\) 0 0
\(231\) 0.545455 0.0358883
\(232\) 0 0
\(233\) 2.16344 0.141732 0.0708660 0.997486i \(-0.477424\pi\)
0.0708660 + 0.997486i \(0.477424\pi\)
\(234\) 0 0
\(235\) 10.0349 0.654603
\(236\) 0 0
\(237\) 0.0498286 0.00323672
\(238\) 0 0
\(239\) −0.642665 −0.0415705 −0.0207853 0.999784i \(-0.506617\pi\)
−0.0207853 + 0.999784i \(0.506617\pi\)
\(240\) 0 0
\(241\) −19.7133 −1.26985 −0.634924 0.772575i \(-0.718969\pi\)
−0.634924 + 0.772575i \(0.718969\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −5.28437 −0.337606
\(246\) 0 0
\(247\) 1.85523 0.118045
\(248\) 0 0
\(249\) 1.56713 0.0993128
\(250\) 0 0
\(251\) 22.4803 1.41895 0.709473 0.704732i \(-0.248933\pi\)
0.709473 + 0.704732i \(0.248933\pi\)
\(252\) 0 0
\(253\) −0.338239 −0.0212649
\(254\) 0 0
\(255\) 0.767698 0.0480751
\(256\) 0 0
\(257\) 3.21026 0.200250 0.100125 0.994975i \(-0.468076\pi\)
0.100125 + 0.994975i \(0.468076\pi\)
\(258\) 0 0
\(259\) 15.1503 0.941391
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 0 0
\(263\) 11.0263 0.679911 0.339955 0.940442i \(-0.389588\pi\)
0.339955 + 0.940442i \(0.389588\pi\)
\(264\) 0 0
\(265\) −4.30337 −0.264354
\(266\) 0 0
\(267\) −1.00920 −0.0617621
\(268\) 0 0
\(269\) 29.7003 1.81086 0.905429 0.424498i \(-0.139549\pi\)
0.905429 + 0.424498i \(0.139549\pi\)
\(270\) 0 0
\(271\) −5.89141 −0.357877 −0.178939 0.983860i \(-0.557266\pi\)
−0.178939 + 0.983860i \(0.557266\pi\)
\(272\) 0 0
\(273\) 5.01771 0.303685
\(274\) 0 0
\(275\) −1.20320 −0.0725556
\(276\) 0 0
\(277\) 2.94396 0.176886 0.0884428 0.996081i \(-0.471811\pi\)
0.0884428 + 0.996081i \(0.471811\pi\)
\(278\) 0 0
\(279\) 6.81971 0.408285
\(280\) 0 0
\(281\) −4.01684 −0.239625 −0.119812 0.992797i \(-0.538229\pi\)
−0.119812 + 0.992797i \(0.538229\pi\)
\(282\) 0 0
\(283\) −4.45516 −0.264831 −0.132416 0.991194i \(-0.542273\pi\)
−0.132416 + 0.991194i \(0.542273\pi\)
\(284\) 0 0
\(285\) −0.716184 −0.0424231
\(286\) 0 0
\(287\) 11.7041 0.690868
\(288\) 0 0
\(289\) −16.5915 −0.975971
\(290\) 0 0
\(291\) −6.80229 −0.398757
\(292\) 0 0
\(293\) −21.7830 −1.27258 −0.636288 0.771452i \(-0.719531\pi\)
−0.636288 + 0.771452i \(0.719531\pi\)
\(294\) 0 0
\(295\) −8.50092 −0.494943
\(296\) 0 0
\(297\) −0.338239 −0.0196266
\(298\) 0 0
\(299\) −3.11150 −0.179943
\(300\) 0 0
\(301\) 16.3425 0.941963
\(302\) 0 0
\(303\) 17.8893 1.02771
\(304\) 0 0
\(305\) 2.58705 0.148134
\(306\) 0 0
\(307\) −9.41091 −0.537109 −0.268554 0.963265i \(-0.586546\pi\)
−0.268554 + 0.963265i \(0.586546\pi\)
\(308\) 0 0
\(309\) −0.803404 −0.0457041
\(310\) 0 0
\(311\) −27.3265 −1.54954 −0.774770 0.632243i \(-0.782135\pi\)
−0.774770 + 0.632243i \(0.782135\pi\)
\(312\) 0 0
\(313\) −4.68271 −0.264682 −0.132341 0.991204i \(-0.542249\pi\)
−0.132341 + 0.991204i \(0.542249\pi\)
\(314\) 0 0
\(315\) −1.93701 −0.109138
\(316\) 0 0
\(317\) 10.0704 0.565608 0.282804 0.959178i \(-0.408735\pi\)
0.282804 + 0.959178i \(0.408735\pi\)
\(318\) 0 0
\(319\) 0.338239 0.0189378
\(320\) 0 0
\(321\) −6.26733 −0.349808
\(322\) 0 0
\(323\) −0.381083 −0.0212041
\(324\) 0 0
\(325\) −11.0684 −0.613962
\(326\) 0 0
\(327\) −5.86541 −0.324358
\(328\) 0 0
\(329\) −13.4725 −0.742765
\(330\) 0 0
\(331\) −11.1249 −0.611481 −0.305741 0.952115i \(-0.598904\pi\)
−0.305741 + 0.952115i \(0.598904\pi\)
\(332\) 0 0
\(333\) −9.39475 −0.514829
\(334\) 0 0
\(335\) 17.2800 0.944110
\(336\) 0 0
\(337\) −6.15480 −0.335273 −0.167637 0.985849i \(-0.553614\pi\)
−0.167637 + 0.985849i \(0.553614\pi\)
\(338\) 0 0
\(339\) −18.1926 −0.988085
\(340\) 0 0
\(341\) 2.30669 0.124914
\(342\) 0 0
\(343\) 18.3831 0.992592
\(344\) 0 0
\(345\) 1.20115 0.0646678
\(346\) 0 0
\(347\) −21.9824 −1.18008 −0.590039 0.807375i \(-0.700888\pi\)
−0.590039 + 0.807375i \(0.700888\pi\)
\(348\) 0 0
\(349\) −3.02475 −0.161911 −0.0809557 0.996718i \(-0.525797\pi\)
−0.0809557 + 0.996718i \(0.525797\pi\)
\(350\) 0 0
\(351\) −3.11150 −0.166080
\(352\) 0 0
\(353\) −7.90540 −0.420762 −0.210381 0.977619i \(-0.567470\pi\)
−0.210381 + 0.977619i \(0.567470\pi\)
\(354\) 0 0
\(355\) 3.54306 0.188046
\(356\) 0 0
\(357\) −1.03069 −0.0545499
\(358\) 0 0
\(359\) −31.1058 −1.64170 −0.820850 0.571143i \(-0.806500\pi\)
−0.820850 + 0.571143i \(0.806500\pi\)
\(360\) 0 0
\(361\) −18.6445 −0.981289
\(362\) 0 0
\(363\) 10.8856 0.571346
\(364\) 0 0
\(365\) −3.36513 −0.176139
\(366\) 0 0
\(367\) −27.8879 −1.45574 −0.727868 0.685717i \(-0.759489\pi\)
−0.727868 + 0.685717i \(0.759489\pi\)
\(368\) 0 0
\(369\) −7.25774 −0.377823
\(370\) 0 0
\(371\) 5.77758 0.299957
\(372\) 0 0
\(373\) 8.19547 0.424345 0.212173 0.977232i \(-0.431946\pi\)
0.212173 + 0.977232i \(0.431946\pi\)
\(374\) 0 0
\(375\) 10.2785 0.530781
\(376\) 0 0
\(377\) 3.11150 0.160251
\(378\) 0 0
\(379\) −0.785868 −0.0403673 −0.0201837 0.999796i \(-0.506425\pi\)
−0.0201837 + 0.999796i \(0.506425\pi\)
\(380\) 0 0
\(381\) −4.35561 −0.223145
\(382\) 0 0
\(383\) 1.00046 0.0511212 0.0255606 0.999673i \(-0.491863\pi\)
0.0255606 + 0.999673i \(0.491863\pi\)
\(384\) 0 0
\(385\) −0.655174 −0.0333908
\(386\) 0 0
\(387\) −10.1340 −0.515142
\(388\) 0 0
\(389\) 1.73378 0.0879061 0.0439531 0.999034i \(-0.486005\pi\)
0.0439531 + 0.999034i \(0.486005\pi\)
\(390\) 0 0
\(391\) 0.639136 0.0323225
\(392\) 0 0
\(393\) −8.07738 −0.407450
\(394\) 0 0
\(395\) −0.0598517 −0.00301147
\(396\) 0 0
\(397\) 4.10969 0.206259 0.103130 0.994668i \(-0.467114\pi\)
0.103130 + 0.994668i \(0.467114\pi\)
\(398\) 0 0
\(399\) 0.961528 0.0481366
\(400\) 0 0
\(401\) −25.7878 −1.28778 −0.643890 0.765118i \(-0.722680\pi\)
−0.643890 + 0.765118i \(0.722680\pi\)
\(402\) 0 0
\(403\) 21.2195 1.05702
\(404\) 0 0
\(405\) 1.20115 0.0596857
\(406\) 0 0
\(407\) −3.17767 −0.157511
\(408\) 0 0
\(409\) 16.9038 0.835837 0.417919 0.908484i \(-0.362760\pi\)
0.417919 + 0.908484i \(0.362760\pi\)
\(410\) 0 0
\(411\) 21.7999 1.07531
\(412\) 0 0
\(413\) 11.4131 0.561602
\(414\) 0 0
\(415\) −1.88236 −0.0924013
\(416\) 0 0
\(417\) −8.45828 −0.414204
\(418\) 0 0
\(419\) −23.4626 −1.14622 −0.573111 0.819478i \(-0.694264\pi\)
−0.573111 + 0.819478i \(0.694264\pi\)
\(420\) 0 0
\(421\) −7.82889 −0.381557 −0.190778 0.981633i \(-0.561101\pi\)
−0.190778 + 0.981633i \(0.561101\pi\)
\(422\) 0 0
\(423\) 8.35438 0.406204
\(424\) 0 0
\(425\) 2.27356 0.110284
\(426\) 0 0
\(427\) −3.47329 −0.168085
\(428\) 0 0
\(429\) −1.05243 −0.0508120
\(430\) 0 0
\(431\) −3.45370 −0.166359 −0.0831795 0.996535i \(-0.526507\pi\)
−0.0831795 + 0.996535i \(0.526507\pi\)
\(432\) 0 0
\(433\) −7.84877 −0.377188 −0.188594 0.982055i \(-0.560393\pi\)
−0.188594 + 0.982055i \(0.560393\pi\)
\(434\) 0 0
\(435\) −1.20115 −0.0575908
\(436\) 0 0
\(437\) −0.596248 −0.0285224
\(438\) 0 0
\(439\) −40.5186 −1.93385 −0.966925 0.255062i \(-0.917904\pi\)
−0.966925 + 0.255062i \(0.917904\pi\)
\(440\) 0 0
\(441\) −4.39942 −0.209496
\(442\) 0 0
\(443\) −29.7763 −1.41472 −0.707358 0.706856i \(-0.750113\pi\)
−0.707358 + 0.706856i \(0.750113\pi\)
\(444\) 0 0
\(445\) 1.21220 0.0574639
\(446\) 0 0
\(447\) 2.71469 0.128401
\(448\) 0 0
\(449\) 33.0566 1.56004 0.780018 0.625757i \(-0.215210\pi\)
0.780018 + 0.625757i \(0.215210\pi\)
\(450\) 0 0
\(451\) −2.45485 −0.115595
\(452\) 0 0
\(453\) 15.0192 0.705664
\(454\) 0 0
\(455\) −6.02702 −0.282551
\(456\) 0 0
\(457\) 15.8554 0.741682 0.370841 0.928696i \(-0.379069\pi\)
0.370841 + 0.928696i \(0.379069\pi\)
\(458\) 0 0
\(459\) 0.639136 0.0298323
\(460\) 0 0
\(461\) −1.78319 −0.0830516 −0.0415258 0.999137i \(-0.513222\pi\)
−0.0415258 + 0.999137i \(0.513222\pi\)
\(462\) 0 0
\(463\) 34.3951 1.59848 0.799238 0.601015i \(-0.205237\pi\)
0.799238 + 0.601015i \(0.205237\pi\)
\(464\) 0 0
\(465\) −8.19150 −0.379872
\(466\) 0 0
\(467\) −17.8508 −0.826038 −0.413019 0.910722i \(-0.635526\pi\)
−0.413019 + 0.910722i \(0.635526\pi\)
\(468\) 0 0
\(469\) −23.1997 −1.07126
\(470\) 0 0
\(471\) 14.5542 0.670622
\(472\) 0 0
\(473\) −3.42773 −0.157607
\(474\) 0 0
\(475\) −2.12100 −0.0973180
\(476\) 0 0
\(477\) −3.58270 −0.164041
\(478\) 0 0
\(479\) −14.8933 −0.680492 −0.340246 0.940337i \(-0.610510\pi\)
−0.340246 + 0.940337i \(0.610510\pi\)
\(480\) 0 0
\(481\) −29.2318 −1.33285
\(482\) 0 0
\(483\) −1.61263 −0.0733772
\(484\) 0 0
\(485\) 8.17057 0.371006
\(486\) 0 0
\(487\) 31.3135 1.41895 0.709476 0.704730i \(-0.248932\pi\)
0.709476 + 0.704730i \(0.248932\pi\)
\(488\) 0 0
\(489\) −0.681185 −0.0308042
\(490\) 0 0
\(491\) 0.662539 0.0299000 0.0149500 0.999888i \(-0.495241\pi\)
0.0149500 + 0.999888i \(0.495241\pi\)
\(492\) 0 0
\(493\) −0.639136 −0.0287852
\(494\) 0 0
\(495\) 0.406277 0.0182608
\(496\) 0 0
\(497\) −4.75681 −0.213372
\(498\) 0 0
\(499\) 5.89940 0.264093 0.132047 0.991244i \(-0.457845\pi\)
0.132047 + 0.991244i \(0.457845\pi\)
\(500\) 0 0
\(501\) 11.6805 0.521845
\(502\) 0 0
\(503\) −5.24014 −0.233646 −0.116823 0.993153i \(-0.537271\pi\)
−0.116823 + 0.993153i \(0.537271\pi\)
\(504\) 0 0
\(505\) −21.4877 −0.956190
\(506\) 0 0
\(507\) 3.31854 0.147382
\(508\) 0 0
\(509\) 4.61806 0.204692 0.102346 0.994749i \(-0.467365\pi\)
0.102346 + 0.994749i \(0.467365\pi\)
\(510\) 0 0
\(511\) 4.51793 0.199862
\(512\) 0 0
\(513\) −0.596248 −0.0263250
\(514\) 0 0
\(515\) 0.965010 0.0425234
\(516\) 0 0
\(517\) 2.82578 0.124278
\(518\) 0 0
\(519\) 9.91936 0.435412
\(520\) 0 0
\(521\) −42.4262 −1.85873 −0.929363 0.369168i \(-0.879643\pi\)
−0.929363 + 0.369168i \(0.879643\pi\)
\(522\) 0 0
\(523\) 8.74382 0.382341 0.191170 0.981557i \(-0.438772\pi\)
0.191170 + 0.981557i \(0.438772\pi\)
\(524\) 0 0
\(525\) −5.73651 −0.250362
\(526\) 0 0
\(527\) −4.35872 −0.189869
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −7.07731 −0.307129
\(532\) 0 0
\(533\) −22.5825 −0.978156
\(534\) 0 0
\(535\) 7.52801 0.325464
\(536\) 0 0
\(537\) −0.993143 −0.0428573
\(538\) 0 0
\(539\) −1.48806 −0.0640952
\(540\) 0 0
\(541\) −23.6159 −1.01533 −0.507663 0.861556i \(-0.669490\pi\)
−0.507663 + 0.861556i \(0.669490\pi\)
\(542\) 0 0
\(543\) 11.2303 0.481936
\(544\) 0 0
\(545\) 7.04524 0.301785
\(546\) 0 0
\(547\) 6.05500 0.258893 0.129447 0.991586i \(-0.458680\pi\)
0.129447 + 0.991586i \(0.458680\pi\)
\(548\) 0 0
\(549\) 2.15381 0.0919222
\(550\) 0 0
\(551\) 0.596248 0.0254010
\(552\) 0 0
\(553\) 0.0803552 0.00341705
\(554\) 0 0
\(555\) 11.2845 0.479001
\(556\) 0 0
\(557\) −39.7879 −1.68587 −0.842935 0.538016i \(-0.819174\pi\)
−0.842935 + 0.538016i \(0.819174\pi\)
\(558\) 0 0
\(559\) −31.5321 −1.33366
\(560\) 0 0
\(561\) 0.216181 0.00912716
\(562\) 0 0
\(563\) 16.8818 0.711483 0.355741 0.934584i \(-0.384228\pi\)
0.355741 + 0.934584i \(0.384228\pi\)
\(564\) 0 0
\(565\) 21.8520 0.919321
\(566\) 0 0
\(567\) −1.61263 −0.0677241
\(568\) 0 0
\(569\) −5.00577 −0.209853 −0.104926 0.994480i \(-0.533461\pi\)
−0.104926 + 0.994480i \(0.533461\pi\)
\(570\) 0 0
\(571\) 42.7880 1.79062 0.895310 0.445444i \(-0.146954\pi\)
0.895310 + 0.445444i \(0.146954\pi\)
\(572\) 0 0
\(573\) 6.98963 0.291996
\(574\) 0 0
\(575\) 3.55724 0.148347
\(576\) 0 0
\(577\) 17.5686 0.731391 0.365696 0.930734i \(-0.380831\pi\)
0.365696 + 0.930734i \(0.380831\pi\)
\(578\) 0 0
\(579\) −23.6537 −0.983015
\(580\) 0 0
\(581\) 2.52720 0.104846
\(582\) 0 0
\(583\) −1.21181 −0.0501881
\(584\) 0 0
\(585\) 3.73739 0.154522
\(586\) 0 0
\(587\) −21.9948 −0.907821 −0.453910 0.891047i \(-0.649971\pi\)
−0.453910 + 0.891047i \(0.649971\pi\)
\(588\) 0 0
\(589\) 4.06624 0.167546
\(590\) 0 0
\(591\) 12.5658 0.516890
\(592\) 0 0
\(593\) −20.4911 −0.841468 −0.420734 0.907184i \(-0.638227\pi\)
−0.420734 + 0.907184i \(0.638227\pi\)
\(594\) 0 0
\(595\) 1.23801 0.0507536
\(596\) 0 0
\(597\) 13.2327 0.541580
\(598\) 0 0
\(599\) −30.4985 −1.24613 −0.623067 0.782169i \(-0.714113\pi\)
−0.623067 + 0.782169i \(0.714113\pi\)
\(600\) 0 0
\(601\) 32.9470 1.34394 0.671969 0.740580i \(-0.265449\pi\)
0.671969 + 0.740580i \(0.265449\pi\)
\(602\) 0 0
\(603\) 14.3862 0.585853
\(604\) 0 0
\(605\) −13.0752 −0.531584
\(606\) 0 0
\(607\) 15.9288 0.646531 0.323266 0.946308i \(-0.395219\pi\)
0.323266 + 0.946308i \(0.395219\pi\)
\(608\) 0 0
\(609\) 1.61263 0.0653471
\(610\) 0 0
\(611\) 25.9947 1.05163
\(612\) 0 0
\(613\) 41.2801 1.66729 0.833643 0.552303i \(-0.186251\pi\)
0.833643 + 0.552303i \(0.186251\pi\)
\(614\) 0 0
\(615\) 8.71764 0.351529
\(616\) 0 0
\(617\) 24.0064 0.966460 0.483230 0.875493i \(-0.339464\pi\)
0.483230 + 0.875493i \(0.339464\pi\)
\(618\) 0 0
\(619\) −26.4600 −1.06352 −0.531759 0.846896i \(-0.678469\pi\)
−0.531759 + 0.846896i \(0.678469\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) −1.62747 −0.0652032
\(624\) 0 0
\(625\) 5.44012 0.217605
\(626\) 0 0
\(627\) −0.201675 −0.00805411
\(628\) 0 0
\(629\) 6.00452 0.239416
\(630\) 0 0
\(631\) −3.25857 −0.129722 −0.0648609 0.997894i \(-0.520660\pi\)
−0.0648609 + 0.997894i \(0.520660\pi\)
\(632\) 0 0
\(633\) −17.0732 −0.678599
\(634\) 0 0
\(635\) 5.23175 0.207615
\(636\) 0 0
\(637\) −13.6888 −0.542371
\(638\) 0 0
\(639\) 2.94972 0.116689
\(640\) 0 0
\(641\) 0.165723 0.00654567 0.00327283 0.999995i \(-0.498958\pi\)
0.00327283 + 0.999995i \(0.498958\pi\)
\(642\) 0 0
\(643\) −30.6352 −1.20813 −0.604067 0.796934i \(-0.706454\pi\)
−0.604067 + 0.796934i \(0.706454\pi\)
\(644\) 0 0
\(645\) 12.1725 0.479292
\(646\) 0 0
\(647\) −5.14154 −0.202135 −0.101067 0.994880i \(-0.532226\pi\)
−0.101067 + 0.994880i \(0.532226\pi\)
\(648\) 0 0
\(649\) −2.39383 −0.0939659
\(650\) 0 0
\(651\) 10.9977 0.431033
\(652\) 0 0
\(653\) 18.3580 0.718402 0.359201 0.933260i \(-0.383049\pi\)
0.359201 + 0.933260i \(0.383049\pi\)
\(654\) 0 0
\(655\) 9.70215 0.379094
\(656\) 0 0
\(657\) −2.80159 −0.109300
\(658\) 0 0
\(659\) −41.1689 −1.60371 −0.801857 0.597516i \(-0.796154\pi\)
−0.801857 + 0.597516i \(0.796154\pi\)
\(660\) 0 0
\(661\) 33.4208 1.29992 0.649958 0.759970i \(-0.274786\pi\)
0.649958 + 0.759970i \(0.274786\pi\)
\(662\) 0 0
\(663\) 1.98867 0.0772337
\(664\) 0 0
\(665\) −1.15494 −0.0447867
\(666\) 0 0
\(667\) −1.00000 −0.0387202
\(668\) 0 0
\(669\) 2.73917 0.105902
\(670\) 0 0
\(671\) 0.728502 0.0281235
\(672\) 0 0
\(673\) 35.2889 1.36029 0.680143 0.733080i \(-0.261918\pi\)
0.680143 + 0.733080i \(0.261918\pi\)
\(674\) 0 0
\(675\) 3.55724 0.136918
\(676\) 0 0
\(677\) −16.2575 −0.624827 −0.312414 0.949946i \(-0.601137\pi\)
−0.312414 + 0.949946i \(0.601137\pi\)
\(678\) 0 0
\(679\) −10.9696 −0.420974
\(680\) 0 0
\(681\) −15.3610 −0.588634
\(682\) 0 0
\(683\) −38.3035 −1.46564 −0.732821 0.680421i \(-0.761797\pi\)
−0.732821 + 0.680421i \(0.761797\pi\)
\(684\) 0 0
\(685\) −26.1850 −1.00048
\(686\) 0 0
\(687\) 13.5282 0.516131
\(688\) 0 0
\(689\) −11.1476 −0.424690
\(690\) 0 0
\(691\) 5.64282 0.214663 0.107331 0.994223i \(-0.465769\pi\)
0.107331 + 0.994223i \(0.465769\pi\)
\(692\) 0 0
\(693\) −0.545455 −0.0207201
\(694\) 0 0
\(695\) 10.1597 0.385378
\(696\) 0 0
\(697\) 4.63868 0.175702
\(698\) 0 0
\(699\) −2.16344 −0.0818290
\(700\) 0 0
\(701\) 14.2253 0.537283 0.268641 0.963240i \(-0.413425\pi\)
0.268641 + 0.963240i \(0.413425\pi\)
\(702\) 0 0
\(703\) −5.60160 −0.211268
\(704\) 0 0
\(705\) −10.0349 −0.377935
\(706\) 0 0
\(707\) 28.8488 1.08497
\(708\) 0 0
\(709\) −19.2154 −0.721650 −0.360825 0.932633i \(-0.617505\pi\)
−0.360825 + 0.932633i \(0.617505\pi\)
\(710\) 0 0
\(711\) −0.0498286 −0.00186872
\(712\) 0 0
\(713\) −6.81971 −0.255400
\(714\) 0 0
\(715\) 1.26413 0.0472758
\(716\) 0 0
\(717\) 0.642665 0.0240008
\(718\) 0 0
\(719\) −3.53093 −0.131682 −0.0658408 0.997830i \(-0.520973\pi\)
−0.0658408 + 0.997830i \(0.520973\pi\)
\(720\) 0 0
\(721\) −1.29559 −0.0482505
\(722\) 0 0
\(723\) 19.7133 0.733147
\(724\) 0 0
\(725\) −3.55724 −0.132112
\(726\) 0 0
\(727\) 4.63590 0.171936 0.0859679 0.996298i \(-0.472602\pi\)
0.0859679 + 0.996298i \(0.472602\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 6.47702 0.239561
\(732\) 0 0
\(733\) −45.6038 −1.68442 −0.842208 0.539153i \(-0.818745\pi\)
−0.842208 + 0.539153i \(0.818745\pi\)
\(734\) 0 0
\(735\) 5.28437 0.194917
\(736\) 0 0
\(737\) 4.86600 0.179241
\(738\) 0 0
\(739\) 4.44161 0.163387 0.0816936 0.996657i \(-0.473967\pi\)
0.0816936 + 0.996657i \(0.473967\pi\)
\(740\) 0 0
\(741\) −1.85523 −0.0681535
\(742\) 0 0
\(743\) −1.66431 −0.0610578 −0.0305289 0.999534i \(-0.509719\pi\)
−0.0305289 + 0.999534i \(0.509719\pi\)
\(744\) 0 0
\(745\) −3.26076 −0.119465
\(746\) 0 0
\(747\) −1.56713 −0.0573383
\(748\) 0 0
\(749\) −10.1069 −0.369298
\(750\) 0 0
\(751\) 28.9356 1.05588 0.527938 0.849283i \(-0.322965\pi\)
0.527938 + 0.849283i \(0.322965\pi\)
\(752\) 0 0
\(753\) −22.4803 −0.819229
\(754\) 0 0
\(755\) −18.0403 −0.656555
\(756\) 0 0
\(757\) −10.0491 −0.365240 −0.182620 0.983184i \(-0.558458\pi\)
−0.182620 + 0.983184i \(0.558458\pi\)
\(758\) 0 0
\(759\) 0.338239 0.0122773
\(760\) 0 0
\(761\) −4.78753 −0.173548 −0.0867740 0.996228i \(-0.527656\pi\)
−0.0867740 + 0.996228i \(0.527656\pi\)
\(762\) 0 0
\(763\) −9.45873 −0.342429
\(764\) 0 0
\(765\) −0.767698 −0.0277562
\(766\) 0 0
\(767\) −22.0211 −0.795135
\(768\) 0 0
\(769\) −27.8260 −1.00343 −0.501715 0.865033i \(-0.667297\pi\)
−0.501715 + 0.865033i \(0.667297\pi\)
\(770\) 0 0
\(771\) −3.21026 −0.115615
\(772\) 0 0
\(773\) 39.4542 1.41907 0.709534 0.704672i \(-0.248906\pi\)
0.709534 + 0.704672i \(0.248906\pi\)
\(774\) 0 0
\(775\) −24.2593 −0.871420
\(776\) 0 0
\(777\) −15.1503 −0.543512
\(778\) 0 0
\(779\) −4.32741 −0.155046
\(780\) 0 0
\(781\) 0.997711 0.0357009
\(782\) 0 0
\(783\) −1.00000 −0.0357371
\(784\) 0 0
\(785\) −17.4818 −0.623951
\(786\) 0 0
\(787\) 31.7369 1.13130 0.565649 0.824646i \(-0.308626\pi\)
0.565649 + 0.824646i \(0.308626\pi\)
\(788\) 0 0
\(789\) −11.0263 −0.392547
\(790\) 0 0
\(791\) −29.3379 −1.04314
\(792\) 0 0
\(793\) 6.70158 0.237980
\(794\) 0 0
\(795\) 4.30337 0.152625
\(796\) 0 0
\(797\) 2.81985 0.0998843 0.0499422 0.998752i \(-0.484096\pi\)
0.0499422 + 0.998752i \(0.484096\pi\)
\(798\) 0 0
\(799\) −5.33958 −0.188901
\(800\) 0 0
\(801\) 1.00920 0.0356584
\(802\) 0 0
\(803\) −0.947609 −0.0334404
\(804\) 0 0
\(805\) 1.93701 0.0682707
\(806\) 0 0
\(807\) −29.7003 −1.04550
\(808\) 0 0
\(809\) −21.1839 −0.744788 −0.372394 0.928075i \(-0.621463\pi\)
−0.372394 + 0.928075i \(0.621463\pi\)
\(810\) 0 0
\(811\) 13.9950 0.491432 0.245716 0.969342i \(-0.420977\pi\)
0.245716 + 0.969342i \(0.420977\pi\)
\(812\) 0 0
\(813\) 5.89141 0.206621
\(814\) 0 0
\(815\) 0.818206 0.0286605
\(816\) 0 0
\(817\) −6.04240 −0.211397
\(818\) 0 0
\(819\) −5.01771 −0.175333
\(820\) 0 0
\(821\) 0.808941 0.0282322 0.0141161 0.999900i \(-0.495507\pi\)
0.0141161 + 0.999900i \(0.495507\pi\)
\(822\) 0 0
\(823\) 16.6231 0.579444 0.289722 0.957111i \(-0.406437\pi\)
0.289722 + 0.957111i \(0.406437\pi\)
\(824\) 0 0
\(825\) 1.20320 0.0418900
\(826\) 0 0
\(827\) −46.0968 −1.60294 −0.801472 0.598033i \(-0.795949\pi\)
−0.801472 + 0.598033i \(0.795949\pi\)
\(828\) 0 0
\(829\) −33.7905 −1.17359 −0.586796 0.809735i \(-0.699611\pi\)
−0.586796 + 0.809735i \(0.699611\pi\)
\(830\) 0 0
\(831\) −2.94396 −0.102125
\(832\) 0 0
\(833\) 2.81183 0.0974240
\(834\) 0 0
\(835\) −14.0300 −0.485529
\(836\) 0 0
\(837\) −6.81971 −0.235724
\(838\) 0 0
\(839\) 1.78694 0.0616919 0.0308460 0.999524i \(-0.490180\pi\)
0.0308460 + 0.999524i \(0.490180\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) 4.01684 0.138347
\(844\) 0 0
\(845\) −3.98607 −0.137125
\(846\) 0 0
\(847\) 17.5544 0.603178
\(848\) 0 0
\(849\) 4.45516 0.152901
\(850\) 0 0
\(851\) 9.39475 0.322048
\(852\) 0 0
\(853\) 56.6350 1.93915 0.969573 0.244802i \(-0.0787231\pi\)
0.969573 + 0.244802i \(0.0787231\pi\)
\(854\) 0 0
\(855\) 0.716184 0.0244930
\(856\) 0 0
\(857\) −23.1805 −0.791830 −0.395915 0.918287i \(-0.629573\pi\)
−0.395915 + 0.918287i \(0.629573\pi\)
\(858\) 0 0
\(859\) −10.7068 −0.365312 −0.182656 0.983177i \(-0.558469\pi\)
−0.182656 + 0.983177i \(0.558469\pi\)
\(860\) 0 0
\(861\) −11.7041 −0.398873
\(862\) 0 0
\(863\) −12.3716 −0.421133 −0.210567 0.977580i \(-0.567531\pi\)
−0.210567 + 0.977580i \(0.567531\pi\)
\(864\) 0 0
\(865\) −11.9146 −0.405110
\(866\) 0 0
\(867\) 16.5915 0.563477
\(868\) 0 0
\(869\) −0.0168540 −0.000571733 0
\(870\) 0 0
\(871\) 44.7629 1.51673
\(872\) 0 0
\(873\) 6.80229 0.230222
\(874\) 0 0
\(875\) 16.5755 0.560353
\(876\) 0 0
\(877\) −41.1193 −1.38850 −0.694250 0.719734i \(-0.744264\pi\)
−0.694250 + 0.719734i \(0.744264\pi\)
\(878\) 0 0
\(879\) 21.7830 0.734722
\(880\) 0 0
\(881\) −33.6496 −1.13368 −0.566842 0.823826i \(-0.691835\pi\)
−0.566842 + 0.823826i \(0.691835\pi\)
\(882\) 0 0
\(883\) −36.7557 −1.23693 −0.618463 0.785814i \(-0.712244\pi\)
−0.618463 + 0.785814i \(0.712244\pi\)
\(884\) 0 0
\(885\) 8.50092 0.285755
\(886\) 0 0
\(887\) 39.9939 1.34287 0.671433 0.741066i \(-0.265679\pi\)
0.671433 + 0.741066i \(0.265679\pi\)
\(888\) 0 0
\(889\) −7.02399 −0.235577
\(890\) 0 0
\(891\) 0.338239 0.0113314
\(892\) 0 0
\(893\) 4.98129 0.166692
\(894\) 0 0
\(895\) 1.19291 0.0398747
\(896\) 0 0
\(897\) 3.11150 0.103890
\(898\) 0 0
\(899\) 6.81971 0.227450
\(900\) 0 0
\(901\) 2.28983 0.0762854
\(902\) 0 0
\(903\) −16.3425 −0.543843
\(904\) 0 0
\(905\) −13.4892 −0.448397
\(906\) 0 0
\(907\) 29.5832 0.982294 0.491147 0.871077i \(-0.336578\pi\)
0.491147 + 0.871077i \(0.336578\pi\)
\(908\) 0 0
\(909\) −17.8893 −0.593350
\(910\) 0 0
\(911\) −6.07788 −0.201369 −0.100685 0.994918i \(-0.532103\pi\)
−0.100685 + 0.994918i \(0.532103\pi\)
\(912\) 0 0
\(913\) −0.530065 −0.0175426
\(914\) 0 0
\(915\) −2.58705 −0.0855251
\(916\) 0 0
\(917\) −13.0258 −0.430151
\(918\) 0 0
\(919\) 28.9627 0.955392 0.477696 0.878525i \(-0.341472\pi\)
0.477696 + 0.878525i \(0.341472\pi\)
\(920\) 0 0
\(921\) 9.41091 0.310100
\(922\) 0 0
\(923\) 9.17806 0.302100
\(924\) 0 0
\(925\) 33.4193 1.09882
\(926\) 0 0
\(927\) 0.803404 0.0263873
\(928\) 0 0
\(929\) −8.76667 −0.287625 −0.143813 0.989605i \(-0.545936\pi\)
−0.143813 + 0.989605i \(0.545936\pi\)
\(930\) 0 0
\(931\) −2.62315 −0.0859702
\(932\) 0 0
\(933\) 27.3265 0.894628
\(934\) 0 0
\(935\) −0.259666 −0.00849198
\(936\) 0 0
\(937\) −2.90935 −0.0950442 −0.0475221 0.998870i \(-0.515132\pi\)
−0.0475221 + 0.998870i \(0.515132\pi\)
\(938\) 0 0
\(939\) 4.68271 0.152814
\(940\) 0 0
\(941\) 10.4834 0.341748 0.170874 0.985293i \(-0.445341\pi\)
0.170874 + 0.985293i \(0.445341\pi\)
\(942\) 0 0
\(943\) 7.25774 0.236344
\(944\) 0 0
\(945\) 1.93701 0.0630110
\(946\) 0 0
\(947\) −31.4324 −1.02142 −0.510708 0.859754i \(-0.670617\pi\)
−0.510708 + 0.859754i \(0.670617\pi\)
\(948\) 0 0
\(949\) −8.71716 −0.282971
\(950\) 0 0
\(951\) −10.0704 −0.326554
\(952\) 0 0
\(953\) −29.1720 −0.944973 −0.472487 0.881338i \(-0.656643\pi\)
−0.472487 + 0.881338i \(0.656643\pi\)
\(954\) 0 0
\(955\) −8.39560 −0.271675
\(956\) 0 0
\(957\) −0.338239 −0.0109337
\(958\) 0 0
\(959\) 35.1552 1.13522
\(960\) 0 0
\(961\) 15.5084 0.500271
\(962\) 0 0
\(963\) 6.26733 0.201962
\(964\) 0 0
\(965\) 28.4117 0.914605
\(966\) 0 0
\(967\) 45.4493 1.46155 0.730775 0.682618i \(-0.239159\pi\)
0.730775 + 0.682618i \(0.239159\pi\)
\(968\) 0 0
\(969\) 0.381083 0.0122422
\(970\) 0 0
\(971\) −3.45579 −0.110902 −0.0554508 0.998461i \(-0.517660\pi\)
−0.0554508 + 0.998461i \(0.517660\pi\)
\(972\) 0 0
\(973\) −13.6401 −0.437281
\(974\) 0 0
\(975\) 11.0684 0.354471
\(976\) 0 0
\(977\) −30.5352 −0.976907 −0.488453 0.872590i \(-0.662439\pi\)
−0.488453 + 0.872590i \(0.662439\pi\)
\(978\) 0 0
\(979\) 0.341352 0.0109097
\(980\) 0 0
\(981\) 5.86541 0.187268
\(982\) 0 0
\(983\) −42.9381 −1.36951 −0.684756 0.728773i \(-0.740091\pi\)
−0.684756 + 0.728773i \(0.740091\pi\)
\(984\) 0 0
\(985\) −15.0935 −0.480918
\(986\) 0 0
\(987\) 13.4725 0.428836
\(988\) 0 0
\(989\) 10.1340 0.322243
\(990\) 0 0
\(991\) −0.0808830 −0.00256933 −0.00128467 0.999999i \(-0.500409\pi\)
−0.00128467 + 0.999999i \(0.500409\pi\)
\(992\) 0 0
\(993\) 11.1249 0.353039
\(994\) 0 0
\(995\) −15.8945 −0.503890
\(996\) 0 0
\(997\) 41.0203 1.29912 0.649562 0.760308i \(-0.274952\pi\)
0.649562 + 0.760308i \(0.274952\pi\)
\(998\) 0 0
\(999\) 9.39475 0.297237
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 8004.2.a.e.1.7 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.e.1.7 9 1.1 even 1 trivial