Properties

Label 6720.2.a.cy.1.2
Level $6720$
Weight $2$
Character 6720.1
Self dual yes
Analytic conductor $53.659$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6720 = 2^{6} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6720.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(53.6594701583\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 3360)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 6720.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +2.82843 q^{11} +4.82843 q^{13} +1.00000 q^{15} -3.65685 q^{17} +2.82843 q^{19} +1.00000 q^{21} +4.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -3.65685 q^{29} -2.82843 q^{31} +2.82843 q^{33} +1.00000 q^{35} +0.343146 q^{37} +4.82843 q^{39} +3.65685 q^{41} -9.65685 q^{43} +1.00000 q^{45} +11.3137 q^{47} +1.00000 q^{49} -3.65685 q^{51} +6.48528 q^{53} +2.82843 q^{55} +2.82843 q^{57} +13.3137 q^{61} +1.00000 q^{63} +4.82843 q^{65} -4.00000 q^{67} +4.00000 q^{69} +5.17157 q^{71} +0.828427 q^{73} +1.00000 q^{75} +2.82843 q^{77} -7.31371 q^{79} +1.00000 q^{81} -6.34315 q^{83} -3.65685 q^{85} -3.65685 q^{87} +0.343146 q^{89} +4.82843 q^{91} -2.82843 q^{93} +2.82843 q^{95} -12.8284 q^{97} +2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + 2q^{5} + 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} + 2q^{5} + 2q^{7} + 2q^{9} + 4q^{13} + 2q^{15} + 4q^{17} + 2q^{21} + 8q^{23} + 2q^{25} + 2q^{27} + 4q^{29} + 2q^{35} + 12q^{37} + 4q^{39} - 4q^{41} - 8q^{43} + 2q^{45} + 2q^{49} + 4q^{51} - 4q^{53} + 4q^{61} + 2q^{63} + 4q^{65} - 8q^{67} + 8q^{69} + 16q^{71} - 4q^{73} + 2q^{75} + 8q^{79} + 2q^{81} - 24q^{83} + 4q^{85} + 4q^{87} + 12q^{89} + 4q^{91} - 20q^{97} + O(q^{100}) \)

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
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.82843 0.852803 0.426401 0.904534i \(-0.359781\pi\)
0.426401 + 0.904534i \(0.359781\pi\)
\(12\) 0 0
\(13\) 4.82843 1.33916 0.669582 0.742738i \(-0.266473\pi\)
0.669582 + 0.742738i \(0.266473\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −3.65685 −0.886917 −0.443459 0.896295i \(-0.646249\pi\)
−0.443459 + 0.896295i \(0.646249\pi\)
\(18\) 0 0
\(19\) 2.82843 0.648886 0.324443 0.945905i \(-0.394823\pi\)
0.324443 + 0.945905i \(0.394823\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −3.65685 −0.679061 −0.339530 0.940595i \(-0.610268\pi\)
−0.339530 + 0.940595i \(0.610268\pi\)
\(30\) 0 0
\(31\) −2.82843 −0.508001 −0.254000 0.967204i \(-0.581746\pi\)
−0.254000 + 0.967204i \(0.581746\pi\)
\(32\) 0 0
\(33\) 2.82843 0.492366
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) 0.343146 0.0564128 0.0282064 0.999602i \(-0.491020\pi\)
0.0282064 + 0.999602i \(0.491020\pi\)
\(38\) 0 0
\(39\) 4.82843 0.773167
\(40\) 0 0
\(41\) 3.65685 0.571105 0.285552 0.958363i \(-0.407823\pi\)
0.285552 + 0.958363i \(0.407823\pi\)
\(42\) 0 0
\(43\) −9.65685 −1.47266 −0.736328 0.676625i \(-0.763442\pi\)
−0.736328 + 0.676625i \(0.763442\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 11.3137 1.65027 0.825137 0.564933i \(-0.191098\pi\)
0.825137 + 0.564933i \(0.191098\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −3.65685 −0.512062
\(52\) 0 0
\(53\) 6.48528 0.890822 0.445411 0.895326i \(-0.353058\pi\)
0.445411 + 0.895326i \(0.353058\pi\)
\(54\) 0 0
\(55\) 2.82843 0.381385
\(56\) 0 0
\(57\) 2.82843 0.374634
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 13.3137 1.70465 0.852323 0.523016i \(-0.175193\pi\)
0.852323 + 0.523016i \(0.175193\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 4.82843 0.598893
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 5.17157 0.613753 0.306876 0.951749i \(-0.400716\pi\)
0.306876 + 0.951749i \(0.400716\pi\)
\(72\) 0 0
\(73\) 0.828427 0.0969601 0.0484800 0.998824i \(-0.484562\pi\)
0.0484800 + 0.998824i \(0.484562\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 2.82843 0.322329
\(78\) 0 0
\(79\) −7.31371 −0.822856 −0.411428 0.911442i \(-0.634970\pi\)
−0.411428 + 0.911442i \(0.634970\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −6.34315 −0.696251 −0.348125 0.937448i \(-0.613182\pi\)
−0.348125 + 0.937448i \(0.613182\pi\)
\(84\) 0 0
\(85\) −3.65685 −0.396642
\(86\) 0 0
\(87\) −3.65685 −0.392056
\(88\) 0 0
\(89\) 0.343146 0.0363734 0.0181867 0.999835i \(-0.494211\pi\)
0.0181867 + 0.999835i \(0.494211\pi\)
\(90\) 0 0
\(91\) 4.82843 0.506157
\(92\) 0 0
\(93\) −2.82843 −0.293294
\(94\) 0 0
\(95\) 2.82843 0.290191
\(96\) 0 0
\(97\) −12.8284 −1.30253 −0.651265 0.758851i \(-0.725761\pi\)
−0.651265 + 0.758851i \(0.725761\pi\)
\(98\) 0 0
\(99\) 2.82843 0.284268
\(100\) 0 0
\(101\) −5.31371 −0.528734 −0.264367 0.964422i \(-0.585163\pi\)
−0.264367 + 0.964422i \(0.585163\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) 1.00000 0.0975900
\(106\) 0 0
\(107\) −8.00000 −0.773389 −0.386695 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) 0 0
\(109\) −9.31371 −0.892091 −0.446046 0.895010i \(-0.647168\pi\)
−0.446046 + 0.895010i \(0.647168\pi\)
\(110\) 0 0
\(111\) 0.343146 0.0325700
\(112\) 0 0
\(113\) 15.1716 1.42722 0.713611 0.700542i \(-0.247059\pi\)
0.713611 + 0.700542i \(0.247059\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) 0 0
\(117\) 4.82843 0.446388
\(118\) 0 0
\(119\) −3.65685 −0.335223
\(120\) 0 0
\(121\) −3.00000 −0.272727
\(122\) 0 0
\(123\) 3.65685 0.329727
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −5.65685 −0.501965 −0.250982 0.967992i \(-0.580754\pi\)
−0.250982 + 0.967992i \(0.580754\pi\)
\(128\) 0 0
\(129\) −9.65685 −0.850239
\(130\) 0 0
\(131\) −5.65685 −0.494242 −0.247121 0.968985i \(-0.579484\pi\)
−0.247121 + 0.968985i \(0.579484\pi\)
\(132\) 0 0
\(133\) 2.82843 0.245256
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 4.82843 0.412520 0.206260 0.978497i \(-0.433871\pi\)
0.206260 + 0.978497i \(0.433871\pi\)
\(138\) 0 0
\(139\) −19.7990 −1.67933 −0.839664 0.543106i \(-0.817248\pi\)
−0.839664 + 0.543106i \(0.817248\pi\)
\(140\) 0 0
\(141\) 11.3137 0.952786
\(142\) 0 0
\(143\) 13.6569 1.14204
\(144\) 0 0
\(145\) −3.65685 −0.303685
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 5.31371 0.435316 0.217658 0.976025i \(-0.430158\pi\)
0.217658 + 0.976025i \(0.430158\pi\)
\(150\) 0 0
\(151\) 9.65685 0.785864 0.392932 0.919568i \(-0.371461\pi\)
0.392932 + 0.919568i \(0.371461\pi\)
\(152\) 0 0
\(153\) −3.65685 −0.295639
\(154\) 0 0
\(155\) −2.82843 −0.227185
\(156\) 0 0
\(157\) 7.17157 0.572354 0.286177 0.958177i \(-0.407615\pi\)
0.286177 + 0.958177i \(0.407615\pi\)
\(158\) 0 0
\(159\) 6.48528 0.514316
\(160\) 0 0
\(161\) 4.00000 0.315244
\(162\) 0 0
\(163\) −6.34315 −0.496834 −0.248417 0.968653i \(-0.579910\pi\)
−0.248417 + 0.968653i \(0.579910\pi\)
\(164\) 0 0
\(165\) 2.82843 0.220193
\(166\) 0 0
\(167\) −8.97056 −0.694163 −0.347081 0.937835i \(-0.612827\pi\)
−0.347081 + 0.937835i \(0.612827\pi\)
\(168\) 0 0
\(169\) 10.3137 0.793362
\(170\) 0 0
\(171\) 2.82843 0.216295
\(172\) 0 0
\(173\) −18.9706 −1.44231 −0.721153 0.692776i \(-0.756387\pi\)
−0.721153 + 0.692776i \(0.756387\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 10.8284 0.809355 0.404677 0.914460i \(-0.367384\pi\)
0.404677 + 0.914460i \(0.367384\pi\)
\(180\) 0 0
\(181\) 23.6569 1.75840 0.879200 0.476453i \(-0.158078\pi\)
0.879200 + 0.476453i \(0.158078\pi\)
\(182\) 0 0
\(183\) 13.3137 0.984178
\(184\) 0 0
\(185\) 0.343146 0.0252286
\(186\) 0 0
\(187\) −10.3431 −0.756366
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 18.8284 1.36238 0.681189 0.732108i \(-0.261463\pi\)
0.681189 + 0.732108i \(0.261463\pi\)
\(192\) 0 0
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) 0 0
\(195\) 4.82843 0.345771
\(196\) 0 0
\(197\) −7.17157 −0.510953 −0.255477 0.966815i \(-0.582232\pi\)
−0.255477 + 0.966815i \(0.582232\pi\)
\(198\) 0 0
\(199\) 8.48528 0.601506 0.300753 0.953702i \(-0.402762\pi\)
0.300753 + 0.953702i \(0.402762\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 0 0
\(203\) −3.65685 −0.256661
\(204\) 0 0
\(205\) 3.65685 0.255406
\(206\) 0 0
\(207\) 4.00000 0.278019
\(208\) 0 0
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 0 0
\(213\) 5.17157 0.354350
\(214\) 0 0
\(215\) −9.65685 −0.658592
\(216\) 0 0
\(217\) −2.82843 −0.192006
\(218\) 0 0
\(219\) 0.828427 0.0559799
\(220\) 0 0
\(221\) −17.6569 −1.18773
\(222\) 0 0
\(223\) 16.9706 1.13643 0.568216 0.822879i \(-0.307634\pi\)
0.568216 + 0.822879i \(0.307634\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −17.6569 −1.17193 −0.585963 0.810338i \(-0.699284\pi\)
−0.585963 + 0.810338i \(0.699284\pi\)
\(228\) 0 0
\(229\) −25.3137 −1.67278 −0.836388 0.548137i \(-0.815337\pi\)
−0.836388 + 0.548137i \(0.815337\pi\)
\(230\) 0 0
\(231\) 2.82843 0.186097
\(232\) 0 0
\(233\) 23.1716 1.51802 0.759010 0.651079i \(-0.225683\pi\)
0.759010 + 0.651079i \(0.225683\pi\)
\(234\) 0 0
\(235\) 11.3137 0.738025
\(236\) 0 0
\(237\) −7.31371 −0.475076
\(238\) 0 0
\(239\) −18.8284 −1.21791 −0.608955 0.793205i \(-0.708411\pi\)
−0.608955 + 0.793205i \(0.708411\pi\)
\(240\) 0 0
\(241\) 7.65685 0.493221 0.246611 0.969115i \(-0.420683\pi\)
0.246611 + 0.969115i \(0.420683\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 13.6569 0.868965
\(248\) 0 0
\(249\) −6.34315 −0.401981
\(250\) 0 0
\(251\) −11.3137 −0.714115 −0.357057 0.934082i \(-0.616220\pi\)
−0.357057 + 0.934082i \(0.616220\pi\)
\(252\) 0 0
\(253\) 11.3137 0.711287
\(254\) 0 0
\(255\) −3.65685 −0.229001
\(256\) 0 0
\(257\) 29.3137 1.82854 0.914269 0.405107i \(-0.132766\pi\)
0.914269 + 0.405107i \(0.132766\pi\)
\(258\) 0 0
\(259\) 0.343146 0.0213220
\(260\) 0 0
\(261\) −3.65685 −0.226354
\(262\) 0 0
\(263\) −12.9706 −0.799799 −0.399900 0.916559i \(-0.630955\pi\)
−0.399900 + 0.916559i \(0.630955\pi\)
\(264\) 0 0
\(265\) 6.48528 0.398388
\(266\) 0 0
\(267\) 0.343146 0.0210002
\(268\) 0 0
\(269\) 17.3137 1.05564 0.527818 0.849358i \(-0.323010\pi\)
0.527818 + 0.849358i \(0.323010\pi\)
\(270\) 0 0
\(271\) −5.17157 −0.314151 −0.157075 0.987587i \(-0.550207\pi\)
−0.157075 + 0.987587i \(0.550207\pi\)
\(272\) 0 0
\(273\) 4.82843 0.292230
\(274\) 0 0
\(275\) 2.82843 0.170561
\(276\) 0 0
\(277\) 30.9706 1.86084 0.930420 0.366494i \(-0.119442\pi\)
0.930420 + 0.366494i \(0.119442\pi\)
\(278\) 0 0
\(279\) −2.82843 −0.169334
\(280\) 0 0
\(281\) −14.9706 −0.893069 −0.446534 0.894766i \(-0.647342\pi\)
−0.446534 + 0.894766i \(0.647342\pi\)
\(282\) 0 0
\(283\) 7.31371 0.434755 0.217377 0.976088i \(-0.430250\pi\)
0.217377 + 0.976088i \(0.430250\pi\)
\(284\) 0 0
\(285\) 2.82843 0.167542
\(286\) 0 0
\(287\) 3.65685 0.215857
\(288\) 0 0
\(289\) −3.62742 −0.213377
\(290\) 0 0
\(291\) −12.8284 −0.752016
\(292\) 0 0
\(293\) 8.34315 0.487412 0.243706 0.969849i \(-0.421637\pi\)
0.243706 + 0.969849i \(0.421637\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 2.82843 0.164122
\(298\) 0 0
\(299\) 19.3137 1.11694
\(300\) 0 0
\(301\) −9.65685 −0.556612
\(302\) 0 0
\(303\) −5.31371 −0.305265
\(304\) 0 0
\(305\) 13.3137 0.762341
\(306\) 0 0
\(307\) −6.34315 −0.362022 −0.181011 0.983481i \(-0.557937\pi\)
−0.181011 + 0.983481i \(0.557937\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 0 0
\(313\) −4.82843 −0.272919 −0.136459 0.990646i \(-0.543572\pi\)
−0.136459 + 0.990646i \(0.543572\pi\)
\(314\) 0 0
\(315\) 1.00000 0.0563436
\(316\) 0 0
\(317\) −12.8284 −0.720516 −0.360258 0.932853i \(-0.617311\pi\)
−0.360258 + 0.932853i \(0.617311\pi\)
\(318\) 0 0
\(319\) −10.3431 −0.579105
\(320\) 0 0
\(321\) −8.00000 −0.446516
\(322\) 0 0
\(323\) −10.3431 −0.575508
\(324\) 0 0
\(325\) 4.82843 0.267833
\(326\) 0 0
\(327\) −9.31371 −0.515049
\(328\) 0 0
\(329\) 11.3137 0.623745
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 0 0
\(333\) 0.343146 0.0188043
\(334\) 0 0
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) 21.3137 1.16103 0.580516 0.814249i \(-0.302851\pi\)
0.580516 + 0.814249i \(0.302851\pi\)
\(338\) 0 0
\(339\) 15.1716 0.824007
\(340\) 0 0
\(341\) −8.00000 −0.433224
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 4.00000 0.215353
\(346\) 0 0
\(347\) 18.3431 0.984712 0.492356 0.870394i \(-0.336136\pi\)
0.492356 + 0.870394i \(0.336136\pi\)
\(348\) 0 0
\(349\) 29.3137 1.56913 0.784563 0.620049i \(-0.212887\pi\)
0.784563 + 0.620049i \(0.212887\pi\)
\(350\) 0 0
\(351\) 4.82843 0.257722
\(352\) 0 0
\(353\) 26.9706 1.43550 0.717749 0.696302i \(-0.245172\pi\)
0.717749 + 0.696302i \(0.245172\pi\)
\(354\) 0 0
\(355\) 5.17157 0.274479
\(356\) 0 0
\(357\) −3.65685 −0.193541
\(358\) 0 0
\(359\) 17.4558 0.921284 0.460642 0.887586i \(-0.347619\pi\)
0.460642 + 0.887586i \(0.347619\pi\)
\(360\) 0 0
\(361\) −11.0000 −0.578947
\(362\) 0 0
\(363\) −3.00000 −0.157459
\(364\) 0 0
\(365\) 0.828427 0.0433619
\(366\) 0 0
\(367\) −18.3431 −0.957504 −0.478752 0.877950i \(-0.658911\pi\)
−0.478752 + 0.877950i \(0.658911\pi\)
\(368\) 0 0
\(369\) 3.65685 0.190368
\(370\) 0 0
\(371\) 6.48528 0.336699
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −17.6569 −0.909374
\(378\) 0 0
\(379\) 9.65685 0.496039 0.248020 0.968755i \(-0.420220\pi\)
0.248020 + 0.968755i \(0.420220\pi\)
\(380\) 0 0
\(381\) −5.65685 −0.289809
\(382\) 0 0
\(383\) −28.2843 −1.44526 −0.722629 0.691236i \(-0.757067\pi\)
−0.722629 + 0.691236i \(0.757067\pi\)
\(384\) 0 0
\(385\) 2.82843 0.144150
\(386\) 0 0
\(387\) −9.65685 −0.490885
\(388\) 0 0
\(389\) 13.3137 0.675032 0.337516 0.941320i \(-0.390413\pi\)
0.337516 + 0.941320i \(0.390413\pi\)
\(390\) 0 0
\(391\) −14.6274 −0.739740
\(392\) 0 0
\(393\) −5.65685 −0.285351
\(394\) 0 0
\(395\) −7.31371 −0.367993
\(396\) 0 0
\(397\) −14.4853 −0.726995 −0.363498 0.931595i \(-0.618418\pi\)
−0.363498 + 0.931595i \(0.618418\pi\)
\(398\) 0 0
\(399\) 2.82843 0.141598
\(400\) 0 0
\(401\) 21.3137 1.06436 0.532178 0.846633i \(-0.321374\pi\)
0.532178 + 0.846633i \(0.321374\pi\)
\(402\) 0 0
\(403\) −13.6569 −0.680296
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 0.970563 0.0481090
\(408\) 0 0
\(409\) −9.31371 −0.460533 −0.230267 0.973128i \(-0.573960\pi\)
−0.230267 + 0.973128i \(0.573960\pi\)
\(410\) 0 0
\(411\) 4.82843 0.238169
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −6.34315 −0.311373
\(416\) 0 0
\(417\) −19.7990 −0.969561
\(418\) 0 0
\(419\) 16.9706 0.829066 0.414533 0.910034i \(-0.363945\pi\)
0.414533 + 0.910034i \(0.363945\pi\)
\(420\) 0 0
\(421\) −1.31371 −0.0640262 −0.0320131 0.999487i \(-0.510192\pi\)
−0.0320131 + 0.999487i \(0.510192\pi\)
\(422\) 0 0
\(423\) 11.3137 0.550091
\(424\) 0 0
\(425\) −3.65685 −0.177383
\(426\) 0 0
\(427\) 13.3137 0.644296
\(428\) 0 0
\(429\) 13.6569 0.659359
\(430\) 0 0
\(431\) 16.4853 0.794068 0.397034 0.917804i \(-0.370039\pi\)
0.397034 + 0.917804i \(0.370039\pi\)
\(432\) 0 0
\(433\) 30.4853 1.46503 0.732515 0.680751i \(-0.238347\pi\)
0.732515 + 0.680751i \(0.238347\pi\)
\(434\) 0 0
\(435\) −3.65685 −0.175333
\(436\) 0 0
\(437\) 11.3137 0.541208
\(438\) 0 0
\(439\) −18.8284 −0.898632 −0.449316 0.893373i \(-0.648332\pi\)
−0.449316 + 0.893373i \(0.648332\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −8.97056 −0.426204 −0.213102 0.977030i \(-0.568357\pi\)
−0.213102 + 0.977030i \(0.568357\pi\)
\(444\) 0 0
\(445\) 0.343146 0.0162667
\(446\) 0 0
\(447\) 5.31371 0.251330
\(448\) 0 0
\(449\) 2.00000 0.0943858 0.0471929 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) 0 0
\(451\) 10.3431 0.487040
\(452\) 0 0
\(453\) 9.65685 0.453719
\(454\) 0 0
\(455\) 4.82843 0.226360
\(456\) 0 0
\(457\) −19.6569 −0.919509 −0.459754 0.888046i \(-0.652063\pi\)
−0.459754 + 0.888046i \(0.652063\pi\)
\(458\) 0 0
\(459\) −3.65685 −0.170687
\(460\) 0 0
\(461\) 28.6274 1.33331 0.666656 0.745366i \(-0.267725\pi\)
0.666656 + 0.745366i \(0.267725\pi\)
\(462\) 0 0
\(463\) −36.2843 −1.68627 −0.843137 0.537700i \(-0.819293\pi\)
−0.843137 + 0.537700i \(0.819293\pi\)
\(464\) 0 0
\(465\) −2.82843 −0.131165
\(466\) 0 0
\(467\) −4.97056 −0.230010 −0.115005 0.993365i \(-0.536688\pi\)
−0.115005 + 0.993365i \(0.536688\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) 7.17157 0.330449
\(472\) 0 0
\(473\) −27.3137 −1.25589
\(474\) 0 0
\(475\) 2.82843 0.129777
\(476\) 0 0
\(477\) 6.48528 0.296941
\(478\) 0 0
\(479\) −19.3137 −0.882466 −0.441233 0.897393i \(-0.645459\pi\)
−0.441233 + 0.897393i \(0.645459\pi\)
\(480\) 0 0
\(481\) 1.65685 0.0755461
\(482\) 0 0
\(483\) 4.00000 0.182006
\(484\) 0 0
\(485\) −12.8284 −0.582509
\(486\) 0 0
\(487\) 5.65685 0.256337 0.128168 0.991752i \(-0.459090\pi\)
0.128168 + 0.991752i \(0.459090\pi\)
\(488\) 0 0
\(489\) −6.34315 −0.286847
\(490\) 0 0
\(491\) 17.4558 0.787771 0.393886 0.919159i \(-0.371131\pi\)
0.393886 + 0.919159i \(0.371131\pi\)
\(492\) 0 0
\(493\) 13.3726 0.602271
\(494\) 0 0
\(495\) 2.82843 0.127128
\(496\) 0 0
\(497\) 5.17157 0.231977
\(498\) 0 0
\(499\) 17.6569 0.790429 0.395215 0.918589i \(-0.370670\pi\)
0.395215 + 0.918589i \(0.370670\pi\)
\(500\) 0 0
\(501\) −8.97056 −0.400775
\(502\) 0 0
\(503\) 39.5980 1.76559 0.882793 0.469762i \(-0.155660\pi\)
0.882793 + 0.469762i \(0.155660\pi\)
\(504\) 0 0
\(505\) −5.31371 −0.236457
\(506\) 0 0
\(507\) 10.3137 0.458048
\(508\) 0 0
\(509\) 1.31371 0.0582291 0.0291146 0.999576i \(-0.490731\pi\)
0.0291146 + 0.999576i \(0.490731\pi\)
\(510\) 0 0
\(511\) 0.828427 0.0366475
\(512\) 0 0
\(513\) 2.82843 0.124878
\(514\) 0 0
\(515\) 8.00000 0.352522
\(516\) 0 0
\(517\) 32.0000 1.40736
\(518\) 0 0
\(519\) −18.9706 −0.832715
\(520\) 0 0
\(521\) −18.9706 −0.831115 −0.415558 0.909567i \(-0.636414\pi\)
−0.415558 + 0.909567i \(0.636414\pi\)
\(522\) 0 0
\(523\) −33.6569 −1.47171 −0.735856 0.677138i \(-0.763220\pi\)
−0.735856 + 0.677138i \(0.763220\pi\)
\(524\) 0 0
\(525\) 1.00000 0.0436436
\(526\) 0 0
\(527\) 10.3431 0.450555
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 17.6569 0.764803
\(534\) 0 0
\(535\) −8.00000 −0.345870
\(536\) 0 0
\(537\) 10.8284 0.467281
\(538\) 0 0
\(539\) 2.82843 0.121829
\(540\) 0 0
\(541\) −44.6274 −1.91868 −0.959341 0.282249i \(-0.908920\pi\)
−0.959341 + 0.282249i \(0.908920\pi\)
\(542\) 0 0
\(543\) 23.6569 1.01521
\(544\) 0 0
\(545\) −9.31371 −0.398955
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 0 0
\(549\) 13.3137 0.568215
\(550\) 0 0
\(551\) −10.3431 −0.440633
\(552\) 0 0
\(553\) −7.31371 −0.311011
\(554\) 0 0
\(555\) 0.343146 0.0145657
\(556\) 0 0
\(557\) −20.8284 −0.882529 −0.441264 0.897377i \(-0.645470\pi\)
−0.441264 + 0.897377i \(0.645470\pi\)
\(558\) 0 0
\(559\) −46.6274 −1.97213
\(560\) 0 0
\(561\) −10.3431 −0.436688
\(562\) 0 0
\(563\) −23.3137 −0.982556 −0.491278 0.871003i \(-0.663470\pi\)
−0.491278 + 0.871003i \(0.663470\pi\)
\(564\) 0 0
\(565\) 15.1716 0.638273
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 24.6274 1.03244 0.516218 0.856458i \(-0.327340\pi\)
0.516218 + 0.856458i \(0.327340\pi\)
\(570\) 0 0
\(571\) −43.5980 −1.82452 −0.912259 0.409613i \(-0.865664\pi\)
−0.912259 + 0.409613i \(0.865664\pi\)
\(572\) 0 0
\(573\) 18.8284 0.786569
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) −4.82843 −0.201010 −0.100505 0.994937i \(-0.532046\pi\)
−0.100505 + 0.994937i \(0.532046\pi\)
\(578\) 0 0
\(579\) 10.0000 0.415586
\(580\) 0 0
\(581\) −6.34315 −0.263158
\(582\) 0 0
\(583\) 18.3431 0.759695
\(584\) 0 0
\(585\) 4.82843 0.199631
\(586\) 0 0
\(587\) 7.31371 0.301869 0.150935 0.988544i \(-0.451772\pi\)
0.150935 + 0.988544i \(0.451772\pi\)
\(588\) 0 0
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) −7.17157 −0.294999
\(592\) 0 0
\(593\) 28.3431 1.16391 0.581957 0.813220i \(-0.302287\pi\)
0.581957 + 0.813220i \(0.302287\pi\)
\(594\) 0 0
\(595\) −3.65685 −0.149916
\(596\) 0 0
\(597\) 8.48528 0.347279
\(598\) 0 0
\(599\) −32.4853 −1.32731 −0.663656 0.748038i \(-0.730996\pi\)
−0.663656 + 0.748038i \(0.730996\pi\)
\(600\) 0 0
\(601\) −35.6569 −1.45447 −0.727237 0.686387i \(-0.759196\pi\)
−0.727237 + 0.686387i \(0.759196\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) 0 0
\(605\) −3.00000 −0.121967
\(606\) 0 0
\(607\) −29.6569 −1.20373 −0.601867 0.798596i \(-0.705576\pi\)
−0.601867 + 0.798596i \(0.705576\pi\)
\(608\) 0 0
\(609\) −3.65685 −0.148183
\(610\) 0 0
\(611\) 54.6274 2.20999
\(612\) 0 0
\(613\) 16.3431 0.660093 0.330047 0.943965i \(-0.392935\pi\)
0.330047 + 0.943965i \(0.392935\pi\)
\(614\) 0 0
\(615\) 3.65685 0.147459
\(616\) 0 0
\(617\) 34.4853 1.38833 0.694163 0.719818i \(-0.255775\pi\)
0.694163 + 0.719818i \(0.255775\pi\)
\(618\) 0 0
\(619\) 41.4558 1.66625 0.833126 0.553084i \(-0.186549\pi\)
0.833126 + 0.553084i \(0.186549\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 0 0
\(623\) 0.343146 0.0137478
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 8.00000 0.319489
\(628\) 0 0
\(629\) −1.25483 −0.0500335
\(630\) 0 0
\(631\) 18.6274 0.741546 0.370773 0.928724i \(-0.379093\pi\)
0.370773 + 0.928724i \(0.379093\pi\)
\(632\) 0 0
\(633\) −20.0000 −0.794929
\(634\) 0 0
\(635\) −5.65685 −0.224485
\(636\) 0 0
\(637\) 4.82843 0.191309
\(638\) 0 0
\(639\) 5.17157 0.204584
\(640\) 0 0
\(641\) −11.6569 −0.460418 −0.230209 0.973141i \(-0.573941\pi\)
−0.230209 + 0.973141i \(0.573941\pi\)
\(642\) 0 0
\(643\) −22.3431 −0.881128 −0.440564 0.897721i \(-0.645221\pi\)
−0.440564 + 0.897721i \(0.645221\pi\)
\(644\) 0 0
\(645\) −9.65685 −0.380238
\(646\) 0 0
\(647\) −8.97056 −0.352669 −0.176335 0.984330i \(-0.556424\pi\)
−0.176335 + 0.984330i \(0.556424\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −2.82843 −0.110855
\(652\) 0 0
\(653\) −26.4853 −1.03645 −0.518225 0.855245i \(-0.673407\pi\)
−0.518225 + 0.855245i \(0.673407\pi\)
\(654\) 0 0
\(655\) −5.65685 −0.221032
\(656\) 0 0
\(657\) 0.828427 0.0323200
\(658\) 0 0
\(659\) −36.7696 −1.43234 −0.716169 0.697927i \(-0.754106\pi\)
−0.716169 + 0.697927i \(0.754106\pi\)
\(660\) 0 0
\(661\) −6.00000 −0.233373 −0.116686 0.993169i \(-0.537227\pi\)
−0.116686 + 0.993169i \(0.537227\pi\)
\(662\) 0 0
\(663\) −17.6569 −0.685735
\(664\) 0 0
\(665\) 2.82843 0.109682
\(666\) 0 0
\(667\) −14.6274 −0.566376
\(668\) 0 0
\(669\) 16.9706 0.656120
\(670\) 0 0
\(671\) 37.6569 1.45373
\(672\) 0 0
\(673\) 39.6569 1.52866 0.764330 0.644826i \(-0.223070\pi\)
0.764330 + 0.644826i \(0.223070\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −26.9706 −1.03656 −0.518281 0.855210i \(-0.673428\pi\)
−0.518281 + 0.855210i \(0.673428\pi\)
\(678\) 0 0
\(679\) −12.8284 −0.492310
\(680\) 0 0
\(681\) −17.6569 −0.676612
\(682\) 0 0
\(683\) 21.6569 0.828676 0.414338 0.910123i \(-0.364013\pi\)
0.414338 + 0.910123i \(0.364013\pi\)
\(684\) 0 0
\(685\) 4.82843 0.184485
\(686\) 0 0
\(687\) −25.3137 −0.965778
\(688\) 0 0
\(689\) 31.3137 1.19296
\(690\) 0 0
\(691\) −12.2010 −0.464148 −0.232074 0.972698i \(-0.574551\pi\)
−0.232074 + 0.972698i \(0.574551\pi\)
\(692\) 0 0
\(693\) 2.82843 0.107443
\(694\) 0 0
\(695\) −19.7990 −0.751018
\(696\) 0 0
\(697\) −13.3726 −0.506523
\(698\) 0 0
\(699\) 23.1716 0.876429
\(700\) 0 0
\(701\) −46.9706 −1.77405 −0.887027 0.461718i \(-0.847233\pi\)
−0.887027 + 0.461718i \(0.847233\pi\)
\(702\) 0 0
\(703\) 0.970563 0.0366055
\(704\) 0 0
\(705\) 11.3137 0.426099
\(706\) 0 0
\(707\) −5.31371 −0.199843
\(708\) 0 0
\(709\) −17.3137 −0.650230 −0.325115 0.945674i \(-0.605403\pi\)
−0.325115 + 0.945674i \(0.605403\pi\)
\(710\) 0 0
\(711\) −7.31371 −0.274285
\(712\) 0 0
\(713\) −11.3137 −0.423702
\(714\) 0 0
\(715\) 13.6569 0.510737
\(716\) 0 0
\(717\) −18.8284 −0.703160
\(718\) 0 0
\(719\) −26.3431 −0.982434 −0.491217 0.871037i \(-0.663448\pi\)
−0.491217 + 0.871037i \(0.663448\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) 0 0
\(723\) 7.65685 0.284761
\(724\) 0 0
\(725\) −3.65685 −0.135812
\(726\) 0 0
\(727\) −26.3431 −0.977013 −0.488507 0.872560i \(-0.662458\pi\)
−0.488507 + 0.872560i \(0.662458\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 35.3137 1.30612
\(732\) 0 0
\(733\) 30.7696 1.13650 0.568250 0.822856i \(-0.307621\pi\)
0.568250 + 0.822856i \(0.307621\pi\)
\(734\) 0 0
\(735\) 1.00000 0.0368856
\(736\) 0 0
\(737\) −11.3137 −0.416746
\(738\) 0 0
\(739\) 14.3431 0.527621 0.263811 0.964575i \(-0.415021\pi\)
0.263811 + 0.964575i \(0.415021\pi\)
\(740\) 0 0
\(741\) 13.6569 0.501697
\(742\) 0 0
\(743\) 25.6569 0.941259 0.470629 0.882331i \(-0.344027\pi\)
0.470629 + 0.882331i \(0.344027\pi\)
\(744\) 0 0
\(745\) 5.31371 0.194679
\(746\) 0 0
\(747\) −6.34315 −0.232084
\(748\) 0 0
\(749\) −8.00000 −0.292314
\(750\) 0 0
\(751\) −20.0000 −0.729810 −0.364905 0.931045i \(-0.618899\pi\)
−0.364905 + 0.931045i \(0.618899\pi\)
\(752\) 0 0
\(753\) −11.3137 −0.412294
\(754\) 0 0
\(755\) 9.65685 0.351449
\(756\) 0 0
\(757\) 22.0000 0.799604 0.399802 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) 0 0
\(759\) 11.3137 0.410662
\(760\) 0 0
\(761\) −7.65685 −0.277561 −0.138780 0.990323i \(-0.544318\pi\)
−0.138780 + 0.990323i \(0.544318\pi\)
\(762\) 0 0
\(763\) −9.31371 −0.337179
\(764\) 0 0
\(765\) −3.65685 −0.132214
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 10.9706 0.395609 0.197804 0.980242i \(-0.436619\pi\)
0.197804 + 0.980242i \(0.436619\pi\)
\(770\) 0 0
\(771\) 29.3137 1.05571
\(772\) 0 0
\(773\) 30.9706 1.11393 0.556967 0.830535i \(-0.311965\pi\)
0.556967 + 0.830535i \(0.311965\pi\)
\(774\) 0 0
\(775\) −2.82843 −0.101600
\(776\) 0 0
\(777\) 0.343146 0.0123103
\(778\) 0 0
\(779\) 10.3431 0.370582
\(780\) 0 0
\(781\) 14.6274 0.523410
\(782\) 0 0
\(783\) −3.65685 −0.130685
\(784\) 0 0
\(785\) 7.17157 0.255964
\(786\) 0 0
\(787\) −41.6569 −1.48491 −0.742453 0.669898i \(-0.766338\pi\)
−0.742453 + 0.669898i \(0.766338\pi\)
\(788\) 0 0
\(789\) −12.9706 −0.461764
\(790\) 0 0
\(791\) 15.1716 0.539439
\(792\) 0 0
\(793\) 64.2843 2.28280
\(794\) 0 0
\(795\) 6.48528 0.230009
\(796\) 0 0
\(797\) −34.9706 −1.23872 −0.619360 0.785107i \(-0.712608\pi\)
−0.619360 + 0.785107i \(0.712608\pi\)
\(798\) 0 0
\(799\) −41.3726 −1.46366
\(800\) 0 0
\(801\) 0.343146 0.0121245
\(802\) 0 0
\(803\) 2.34315 0.0826878
\(804\) 0 0
\(805\) 4.00000 0.140981
\(806\) 0 0
\(807\) 17.3137 0.609471
\(808\) 0 0
\(809\) −5.02944 −0.176826 −0.0884128 0.996084i \(-0.528179\pi\)
−0.0884128 + 0.996084i \(0.528179\pi\)
\(810\) 0 0
\(811\) 49.4558 1.73663 0.868315 0.496014i \(-0.165203\pi\)
0.868315 + 0.496014i \(0.165203\pi\)
\(812\) 0 0
\(813\) −5.17157 −0.181375
\(814\) 0 0
\(815\) −6.34315 −0.222191
\(816\) 0 0
\(817\) −27.3137 −0.955586
\(818\) 0 0
\(819\) 4.82843 0.168719
\(820\) 0 0
\(821\) 46.2843 1.61533 0.807666 0.589640i \(-0.200730\pi\)
0.807666 + 0.589640i \(0.200730\pi\)
\(822\) 0 0
\(823\) −36.2843 −1.26479 −0.632395 0.774646i \(-0.717928\pi\)
−0.632395 + 0.774646i \(0.717928\pi\)
\(824\) 0 0
\(825\) 2.82843 0.0984732
\(826\) 0 0
\(827\) 45.2548 1.57366 0.786832 0.617167i \(-0.211720\pi\)
0.786832 + 0.617167i \(0.211720\pi\)
\(828\) 0 0
\(829\) 22.2843 0.773965 0.386982 0.922087i \(-0.373517\pi\)
0.386982 + 0.922087i \(0.373517\pi\)
\(830\) 0 0
\(831\) 30.9706 1.07436
\(832\) 0 0
\(833\) −3.65685 −0.126702
\(834\) 0 0
\(835\) −8.97056 −0.310439
\(836\) 0 0
\(837\) −2.82843 −0.0977647
\(838\) 0 0
\(839\) −15.0294 −0.518874 −0.259437 0.965760i \(-0.583537\pi\)
−0.259437 + 0.965760i \(0.583537\pi\)
\(840\) 0 0
\(841\) −15.6274 −0.538876
\(842\) 0 0
\(843\) −14.9706 −0.515614
\(844\) 0 0
\(845\) 10.3137 0.354802
\(846\) 0 0
\(847\) −3.00000 −0.103081
\(848\) 0 0
\(849\) 7.31371 0.251006
\(850\) 0 0
\(851\) 1.37258 0.0470515
\(852\) 0 0
\(853\) 36.8284 1.26098 0.630491 0.776197i \(-0.282854\pi\)
0.630491 + 0.776197i \(0.282854\pi\)
\(854\) 0 0
\(855\) 2.82843 0.0967302
\(856\) 0 0
\(857\) 43.9411 1.50100 0.750500 0.660870i \(-0.229813\pi\)
0.750500 + 0.660870i \(0.229813\pi\)
\(858\) 0 0
\(859\) −33.4558 −1.14150 −0.570749 0.821124i \(-0.693347\pi\)
−0.570749 + 0.821124i \(0.693347\pi\)
\(860\) 0 0
\(861\) 3.65685 0.124625
\(862\) 0 0
\(863\) 36.0000 1.22545 0.612727 0.790295i \(-0.290072\pi\)
0.612727 + 0.790295i \(0.290072\pi\)
\(864\) 0 0
\(865\) −18.9706 −0.645018
\(866\) 0 0
\(867\) −3.62742 −0.123194
\(868\) 0 0
\(869\) −20.6863 −0.701734
\(870\) 0 0
\(871\) −19.3137 −0.654420
\(872\) 0 0
\(873\) −12.8284 −0.434176
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) 2.28427 0.0771344 0.0385672 0.999256i \(-0.487721\pi\)
0.0385672 + 0.999256i \(0.487721\pi\)
\(878\) 0 0
\(879\) 8.34315 0.281407
\(880\) 0 0
\(881\) −9.02944 −0.304209 −0.152105 0.988364i \(-0.548605\pi\)
−0.152105 + 0.988364i \(0.548605\pi\)
\(882\) 0 0
\(883\) −26.6274 −0.896084 −0.448042 0.894013i \(-0.647878\pi\)
−0.448042 + 0.894013i \(0.647878\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −35.3137 −1.18572 −0.592859 0.805306i \(-0.702001\pi\)
−0.592859 + 0.805306i \(0.702001\pi\)
\(888\) 0 0
\(889\) −5.65685 −0.189725
\(890\) 0 0
\(891\) 2.82843 0.0947559
\(892\) 0 0
\(893\) 32.0000 1.07084
\(894\) 0 0
\(895\) 10.8284 0.361954
\(896\) 0 0
\(897\) 19.3137 0.644866
\(898\) 0 0
\(899\) 10.3431 0.344963
\(900\) 0 0
\(901\) −23.7157 −0.790085
\(902\) 0 0
\(903\) −9.65685 −0.321360
\(904\) 0 0
\(905\) 23.6569 0.786380
\(906\) 0 0
\(907\) 34.6274 1.14978 0.574892 0.818229i \(-0.305044\pi\)
0.574892 + 0.818229i \(0.305044\pi\)
\(908\) 0 0
\(909\) −5.31371 −0.176245
\(910\) 0 0
\(911\) −0.485281 −0.0160781 −0.00803904 0.999968i \(-0.502559\pi\)
−0.00803904 + 0.999968i \(0.502559\pi\)
\(912\) 0 0
\(913\) −17.9411 −0.593765
\(914\) 0 0
\(915\) 13.3137 0.440138
\(916\) 0 0
\(917\) −5.65685 −0.186806
\(918\) 0 0
\(919\) −0.686292 −0.0226387 −0.0113193 0.999936i \(-0.503603\pi\)
−0.0113193 + 0.999936i \(0.503603\pi\)
\(920\) 0 0
\(921\) −6.34315 −0.209014
\(922\) 0 0
\(923\) 24.9706 0.821916
\(924\) 0 0
\(925\) 0.343146 0.0112826
\(926\) 0 0
\(927\) 8.00000 0.262754
\(928\) 0 0
\(929\) −22.2843 −0.731123 −0.365562 0.930787i \(-0.619123\pi\)
−0.365562 + 0.930787i \(0.619123\pi\)
\(930\) 0 0
\(931\) 2.82843 0.0926980
\(932\) 0 0
\(933\) 8.00000 0.261908
\(934\) 0 0
\(935\) −10.3431 −0.338257
\(936\) 0 0
\(937\) −58.4853 −1.91063 −0.955315 0.295588i \(-0.904484\pi\)
−0.955315 + 0.295588i \(0.904484\pi\)
\(938\) 0 0
\(939\) −4.82843 −0.157570
\(940\) 0 0
\(941\) −34.0000 −1.10837 −0.554184 0.832394i \(-0.686970\pi\)
−0.554184 + 0.832394i \(0.686970\pi\)
\(942\) 0 0
\(943\) 14.6274 0.476334
\(944\) 0 0
\(945\) 1.00000 0.0325300
\(946\) 0 0
\(947\) 50.9117 1.65441 0.827204 0.561902i \(-0.189930\pi\)
0.827204 + 0.561902i \(0.189930\pi\)
\(948\) 0 0
\(949\) 4.00000 0.129845
\(950\) 0 0
\(951\) −12.8284 −0.415990
\(952\) 0 0
\(953\) 28.4264 0.920822 0.460411 0.887706i \(-0.347702\pi\)
0.460411 + 0.887706i \(0.347702\pi\)
\(954\) 0 0
\(955\) 18.8284 0.609274
\(956\) 0 0
\(957\) −10.3431 −0.334346
\(958\) 0 0
\(959\) 4.82843 0.155918
\(960\) 0 0
\(961\) −23.0000 −0.741935
\(962\) 0 0
\(963\) −8.00000 −0.257796
\(964\) 0 0
\(965\) 10.0000 0.321911
\(966\) 0 0
\(967\) 25.9411 0.834210 0.417105 0.908858i \(-0.363045\pi\)
0.417105 + 0.908858i \(0.363045\pi\)
\(968\) 0 0
\(969\) −10.3431 −0.332270
\(970\) 0 0
\(971\) −8.00000 −0.256732 −0.128366 0.991727i \(-0.540973\pi\)
−0.128366 + 0.991727i \(0.540973\pi\)
\(972\) 0 0
\(973\) −19.7990 −0.634726
\(974\) 0 0
\(975\) 4.82843 0.154633
\(976\) 0 0
\(977\) −36.1421 −1.15629 −0.578145 0.815934i \(-0.696223\pi\)
−0.578145 + 0.815934i \(0.696223\pi\)
\(978\) 0 0
\(979\) 0.970563 0.0310193
\(980\) 0 0
\(981\) −9.31371 −0.297364
\(982\) 0 0
\(983\) 28.6863 0.914951 0.457475 0.889222i \(-0.348754\pi\)
0.457475 + 0.889222i \(0.348754\pi\)
\(984\) 0 0
\(985\) −7.17157 −0.228505
\(986\) 0 0
\(987\) 11.3137 0.360119
\(988\) 0 0
\(989\) −38.6274 −1.22828
\(990\) 0 0
\(991\) −13.9411 −0.442854 −0.221427 0.975177i \(-0.571072\pi\)
−0.221427 + 0.975177i \(0.571072\pi\)
\(992\) 0 0
\(993\) −4.00000 −0.126936
\(994\) 0 0
\(995\) 8.48528 0.269002
\(996\) 0 0
\(997\) 24.1421 0.764589 0.382295 0.924041i \(-0.375134\pi\)
0.382295 + 0.924041i \(0.375134\pi\)
\(998\) 0 0
\(999\) 0.343146 0.0108567
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 6720.2.a.cy.1.2 2
4.3 odd 2 6720.2.a.cr.1.1 2
8.3 odd 2 3360.2.a.be.1.2 yes 2
8.5 even 2 3360.2.a.bc.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3360.2.a.bc.1.1 2 8.5 even 2
3360.2.a.be.1.2 yes 2 8.3 odd 2
6720.2.a.cr.1.1 2 4.3 odd 2
6720.2.a.cy.1.2 2 1.1 even 1 trivial