Properties

Label 1792.2.a.o.1.2
Level $1792$
Weight $2$
Character 1792.1
Self dual yes
Analytic conductor $14.309$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1792.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.3091920422\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 896)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 1792.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.82843 q^{5} +1.00000 q^{7} -3.00000 q^{9} +O(q^{10})\) \(q+2.82843 q^{5} +1.00000 q^{7} -3.00000 q^{9} -2.82843 q^{11} -2.82843 q^{13} -2.00000 q^{17} -5.65685 q^{19} -8.00000 q^{23} +3.00000 q^{25} -5.65685 q^{29} -8.00000 q^{31} +2.82843 q^{35} +5.65685 q^{37} +6.00000 q^{41} -2.82843 q^{43} -8.48528 q^{45} -8.00000 q^{47} +1.00000 q^{49} +11.3137 q^{53} -8.00000 q^{55} +11.3137 q^{59} +2.82843 q^{61} -3.00000 q^{63} -8.00000 q^{65} +8.48528 q^{67} -8.00000 q^{71} -6.00000 q^{73} -2.82843 q^{77} +8.00000 q^{79} +9.00000 q^{81} +5.65685 q^{83} -5.65685 q^{85} -6.00000 q^{89} -2.82843 q^{91} -16.0000 q^{95} +14.0000 q^{97} +8.48528 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{7} - 6q^{9} + O(q^{10}) \) \( 2q + 2q^{7} - 6q^{9} - 4q^{17} - 16q^{23} + 6q^{25} - 16q^{31} + 12q^{41} - 16q^{47} + 2q^{49} - 16q^{55} - 6q^{63} - 16q^{65} - 16q^{71} - 12q^{73} + 16q^{79} + 18q^{81} - 12q^{89} - 32q^{95} + 28q^{97} + O(q^{100}) \)

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
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 0 0
\(5\) 2.82843 1.26491 0.632456 0.774597i \(-0.282047\pi\)
0.632456 + 0.774597i \(0.282047\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) −2.82843 −0.852803 −0.426401 0.904534i \(-0.640219\pi\)
−0.426401 + 0.904534i \(0.640219\pi\)
\(12\) 0 0
\(13\) −2.82843 −0.784465 −0.392232 0.919866i \(-0.628297\pi\)
−0.392232 + 0.919866i \(0.628297\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) −5.65685 −1.29777 −0.648886 0.760886i \(-0.724765\pi\)
−0.648886 + 0.760886i \(0.724765\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 0 0
\(25\) 3.00000 0.600000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −5.65685 −1.05045 −0.525226 0.850963i \(-0.676019\pi\)
−0.525226 + 0.850963i \(0.676019\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.82843 0.478091
\(36\) 0 0
\(37\) 5.65685 0.929981 0.464991 0.885316i \(-0.346058\pi\)
0.464991 + 0.885316i \(0.346058\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −2.82843 −0.431331 −0.215666 0.976467i \(-0.569192\pi\)
−0.215666 + 0.976467i \(0.569192\pi\)
\(44\) 0 0
\(45\) −8.48528 −1.26491
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 11.3137 1.55406 0.777029 0.629465i \(-0.216726\pi\)
0.777029 + 0.629465i \(0.216726\pi\)
\(54\) 0 0
\(55\) −8.00000 −1.07872
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 11.3137 1.47292 0.736460 0.676481i \(-0.236496\pi\)
0.736460 + 0.676481i \(0.236496\pi\)
\(60\) 0 0
\(61\) 2.82843 0.362143 0.181071 0.983470i \(-0.442043\pi\)
0.181071 + 0.983470i \(0.442043\pi\)
\(62\) 0 0
\(63\) −3.00000 −0.377964
\(64\) 0 0
\(65\) −8.00000 −0.992278
\(66\) 0 0
\(67\) 8.48528 1.03664 0.518321 0.855186i \(-0.326557\pi\)
0.518321 + 0.855186i \(0.326557\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.82843 −0.322329
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 5.65685 0.620920 0.310460 0.950586i \(-0.399517\pi\)
0.310460 + 0.950586i \(0.399517\pi\)
\(84\) 0 0
\(85\) −5.65685 −0.613572
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −2.82843 −0.296500
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −16.0000 −1.64157
\(96\) 0 0
\(97\) 14.0000 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 0 0
\(99\) 8.48528 0.852803
\(100\) 0 0
\(101\) −8.48528 −0.844317 −0.422159 0.906522i \(-0.638727\pi\)
−0.422159 + 0.906522i \(0.638727\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 14.1421 1.36717 0.683586 0.729870i \(-0.260419\pi\)
0.683586 + 0.729870i \(0.260419\pi\)
\(108\) 0 0
\(109\) 16.9706 1.62549 0.812743 0.582623i \(-0.197974\pi\)
0.812743 + 0.582623i \(0.197974\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) −22.6274 −2.11002
\(116\) 0 0
\(117\) 8.48528 0.784465
\(118\) 0 0
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) −3.00000 −0.272727
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −5.65685 −0.505964
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 5.65685 0.494242 0.247121 0.968985i \(-0.420516\pi\)
0.247121 + 0.968985i \(0.420516\pi\)
\(132\) 0 0
\(133\) −5.65685 −0.490511
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) −16.9706 −1.43942 −0.719712 0.694273i \(-0.755726\pi\)
−0.719712 + 0.694273i \(0.755726\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 8.00000 0.668994
\(144\) 0 0
\(145\) −16.0000 −1.32873
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −11.3137 −0.926855 −0.463428 0.886135i \(-0.653381\pi\)
−0.463428 + 0.886135i \(0.653381\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) −22.6274 −1.81748
\(156\) 0 0
\(157\) 8.48528 0.677199 0.338600 0.940931i \(-0.390047\pi\)
0.338600 + 0.940931i \(0.390047\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −8.00000 −0.630488
\(162\) 0 0
\(163\) −19.7990 −1.55078 −0.775388 0.631485i \(-0.782446\pi\)
−0.775388 + 0.631485i \(0.782446\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −5.00000 −0.384615
\(170\) 0 0
\(171\) 16.9706 1.29777
\(172\) 0 0
\(173\) 8.48528 0.645124 0.322562 0.946548i \(-0.395456\pi\)
0.322562 + 0.946548i \(0.395456\pi\)
\(174\) 0 0
\(175\) 3.00000 0.226779
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 8.48528 0.634220 0.317110 0.948389i \(-0.397288\pi\)
0.317110 + 0.948389i \(0.397288\pi\)
\(180\) 0 0
\(181\) −25.4558 −1.89212 −0.946059 0.323994i \(-0.894974\pi\)
−0.946059 + 0.323994i \(0.894974\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 16.0000 1.17634
\(186\) 0 0
\(187\) 5.65685 0.413670
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −24.0000 −1.73658 −0.868290 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −5.65685 −0.397033
\(204\) 0 0
\(205\) 16.9706 1.18528
\(206\) 0 0
\(207\) 24.0000 1.66812
\(208\) 0 0
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) 19.7990 1.36302 0.681509 0.731809i \(-0.261324\pi\)
0.681509 + 0.731809i \(0.261324\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −8.00000 −0.545595
\(216\) 0 0
\(217\) −8.00000 −0.543075
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 5.65685 0.380521
\(222\) 0 0
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 0 0
\(225\) −9.00000 −0.600000
\(226\) 0 0
\(227\) −28.2843 −1.87729 −0.938647 0.344881i \(-0.887919\pi\)
−0.938647 + 0.344881i \(0.887919\pi\)
\(228\) 0 0
\(229\) −25.4558 −1.68217 −0.841085 0.540903i \(-0.818082\pi\)
−0.841085 + 0.540903i \(0.818082\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −14.0000 −0.917170 −0.458585 0.888650i \(-0.651644\pi\)
−0.458585 + 0.888650i \(0.651644\pi\)
\(234\) 0 0
\(235\) −22.6274 −1.47605
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.82843 0.180702
\(246\) 0 0
\(247\) 16.0000 1.01806
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 5.65685 0.357057 0.178529 0.983935i \(-0.442866\pi\)
0.178529 + 0.983935i \(0.442866\pi\)
\(252\) 0 0
\(253\) 22.6274 1.42257
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) 5.65685 0.351500
\(260\) 0 0
\(261\) 16.9706 1.05045
\(262\) 0 0
\(263\) 8.00000 0.493301 0.246651 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(264\) 0 0
\(265\) 32.0000 1.96574
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −2.82843 −0.172452 −0.0862261 0.996276i \(-0.527481\pi\)
−0.0862261 + 0.996276i \(0.527481\pi\)
\(270\) 0 0
\(271\) 32.0000 1.94386 0.971931 0.235267i \(-0.0755965\pi\)
0.971931 + 0.235267i \(0.0755965\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −8.48528 −0.511682
\(276\) 0 0
\(277\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(278\) 0 0
\(279\) 24.0000 1.43684
\(280\) 0 0
\(281\) −30.0000 −1.78965 −0.894825 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 0 0
\(283\) 22.6274 1.34506 0.672530 0.740070i \(-0.265208\pi\)
0.672530 + 0.740070i \(0.265208\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6.00000 0.354169
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 14.1421 0.826192 0.413096 0.910687i \(-0.364447\pi\)
0.413096 + 0.910687i \(0.364447\pi\)
\(294\) 0 0
\(295\) 32.0000 1.86311
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 22.6274 1.30858
\(300\) 0 0
\(301\) −2.82843 −0.163028
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 8.00000 0.458079
\(306\) 0 0
\(307\) 28.2843 1.61427 0.807134 0.590368i \(-0.201017\pi\)
0.807134 + 0.590368i \(0.201017\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 0 0
\(315\) −8.48528 −0.478091
\(316\) 0 0
\(317\) −22.6274 −1.27088 −0.635441 0.772149i \(-0.719182\pi\)
−0.635441 + 0.772149i \(0.719182\pi\)
\(318\) 0 0
\(319\) 16.0000 0.895828
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 11.3137 0.629512
\(324\) 0 0
\(325\) −8.48528 −0.470679
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) 8.48528 0.466393 0.233197 0.972430i \(-0.425081\pi\)
0.233197 + 0.972430i \(0.425081\pi\)
\(332\) 0 0
\(333\) −16.9706 −0.929981
\(334\) 0 0
\(335\) 24.0000 1.31126
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 22.6274 1.22534
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −25.4558 −1.36654 −0.683271 0.730165i \(-0.739443\pi\)
−0.683271 + 0.730165i \(0.739443\pi\)
\(348\) 0 0
\(349\) 8.48528 0.454207 0.227103 0.973871i \(-0.427074\pi\)
0.227103 + 0.973871i \(0.427074\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 0 0
\(355\) −22.6274 −1.20094
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) 13.0000 0.684211
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −16.9706 −0.888280
\(366\) 0 0
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) 0 0
\(369\) −18.0000 −0.937043
\(370\) 0 0
\(371\) 11.3137 0.587378
\(372\) 0 0
\(373\) −33.9411 −1.75740 −0.878702 0.477370i \(-0.841590\pi\)
−0.878702 + 0.477370i \(0.841590\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 16.0000 0.824042
\(378\) 0 0
\(379\) 8.48528 0.435860 0.217930 0.975964i \(-0.430070\pi\)
0.217930 + 0.975964i \(0.430070\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 0 0
\(385\) −8.00000 −0.407718
\(386\) 0 0
\(387\) 8.48528 0.431331
\(388\) 0 0
\(389\) −16.9706 −0.860442 −0.430221 0.902724i \(-0.641564\pi\)
−0.430221 + 0.902724i \(0.641564\pi\)
\(390\) 0 0
\(391\) 16.0000 0.809155
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 22.6274 1.13851
\(396\) 0 0
\(397\) −25.4558 −1.27759 −0.638796 0.769376i \(-0.720567\pi\)
−0.638796 + 0.769376i \(0.720567\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) 22.6274 1.12715
\(404\) 0 0
\(405\) 25.4558 1.26491
\(406\) 0 0
\(407\) −16.0000 −0.793091
\(408\) 0 0
\(409\) 38.0000 1.87898 0.939490 0.342578i \(-0.111300\pi\)
0.939490 + 0.342578i \(0.111300\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 11.3137 0.556711
\(414\) 0 0
\(415\) 16.0000 0.785409
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 22.6274 1.10279 0.551396 0.834243i \(-0.314095\pi\)
0.551396 + 0.834243i \(0.314095\pi\)
\(422\) 0 0
\(423\) 24.0000 1.16692
\(424\) 0 0
\(425\) −6.00000 −0.291043
\(426\) 0 0
\(427\) 2.82843 0.136877
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 0 0
\(433\) 30.0000 1.44171 0.720854 0.693087i \(-0.243750\pi\)
0.720854 + 0.693087i \(0.243750\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 45.2548 2.16483
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) 2.82843 0.134383 0.0671913 0.997740i \(-0.478596\pi\)
0.0671913 + 0.997740i \(0.478596\pi\)
\(444\) 0 0
\(445\) −16.9706 −0.804482
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −10.0000 −0.471929 −0.235965 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) 0 0
\(451\) −16.9706 −0.799113
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −8.00000 −0.375046
\(456\) 0 0
\(457\) −34.0000 −1.59045 −0.795226 0.606313i \(-0.792648\pi\)
−0.795226 + 0.606313i \(0.792648\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 8.48528 0.395199 0.197599 0.980283i \(-0.436685\pi\)
0.197599 + 0.980283i \(0.436685\pi\)
\(462\) 0 0
\(463\) 24.0000 1.11537 0.557687 0.830051i \(-0.311689\pi\)
0.557687 + 0.830051i \(0.311689\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −5.65685 −0.261768 −0.130884 0.991398i \(-0.541782\pi\)
−0.130884 + 0.991398i \(0.541782\pi\)
\(468\) 0 0
\(469\) 8.48528 0.391814
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 8.00000 0.367840
\(474\) 0 0
\(475\) −16.9706 −0.778663
\(476\) 0 0
\(477\) −33.9411 −1.55406
\(478\) 0 0
\(479\) 8.00000 0.365529 0.182765 0.983157i \(-0.441495\pi\)
0.182765 + 0.983157i \(0.441495\pi\)
\(480\) 0 0
\(481\) −16.0000 −0.729537
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 39.5980 1.79805
\(486\) 0 0
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −31.1127 −1.40410 −0.702048 0.712129i \(-0.747731\pi\)
−0.702048 + 0.712129i \(0.747731\pi\)
\(492\) 0 0
\(493\) 11.3137 0.509544
\(494\) 0 0
\(495\) 24.0000 1.07872
\(496\) 0 0
\(497\) −8.00000 −0.358849
\(498\) 0 0
\(499\) −25.4558 −1.13956 −0.569780 0.821797i \(-0.692972\pi\)
−0.569780 + 0.821797i \(0.692972\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −40.0000 −1.78351 −0.891756 0.452517i \(-0.850526\pi\)
−0.891756 + 0.452517i \(0.850526\pi\)
\(504\) 0 0
\(505\) −24.0000 −1.06799
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −2.82843 −0.125368 −0.0626839 0.998033i \(-0.519966\pi\)
−0.0626839 + 0.998033i \(0.519966\pi\)
\(510\) 0 0
\(511\) −6.00000 −0.265424
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 22.6274 0.995153
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −26.0000 −1.13908 −0.569540 0.821963i \(-0.692879\pi\)
−0.569540 + 0.821963i \(0.692879\pi\)
\(522\) 0 0
\(523\) −11.3137 −0.494714 −0.247357 0.968924i \(-0.579562\pi\)
−0.247357 + 0.968924i \(0.579562\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 16.0000 0.696971
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) −33.9411 −1.47292
\(532\) 0 0
\(533\) −16.9706 −0.735077
\(534\) 0 0
\(535\) 40.0000 1.72935
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −2.82843 −0.121829
\(540\) 0 0
\(541\) −11.3137 −0.486414 −0.243207 0.969974i \(-0.578199\pi\)
−0.243207 + 0.969974i \(0.578199\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 48.0000 2.05609
\(546\) 0 0
\(547\) 2.82843 0.120935 0.0604674 0.998170i \(-0.480741\pi\)
0.0604674 + 0.998170i \(0.480741\pi\)
\(548\) 0 0
\(549\) −8.48528 −0.362143
\(550\) 0 0
\(551\) 32.0000 1.36325
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −45.2548 −1.91751 −0.958754 0.284236i \(-0.908260\pi\)
−0.958754 + 0.284236i \(0.908260\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −11.3137 −0.476816 −0.238408 0.971165i \(-0.576626\pi\)
−0.238408 + 0.971165i \(0.576626\pi\)
\(564\) 0 0
\(565\) −16.9706 −0.713957
\(566\) 0 0
\(567\) 9.00000 0.377964
\(568\) 0 0
\(569\) −2.00000 −0.0838444 −0.0419222 0.999121i \(-0.513348\pi\)
−0.0419222 + 0.999121i \(0.513348\pi\)
\(570\) 0 0
\(571\) 19.7990 0.828562 0.414281 0.910149i \(-0.364033\pi\)
0.414281 + 0.910149i \(0.364033\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −24.0000 −1.00087
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 5.65685 0.234686
\(582\) 0 0
\(583\) −32.0000 −1.32530
\(584\) 0 0
\(585\) 24.0000 0.992278
\(586\) 0 0
\(587\) −5.65685 −0.233483 −0.116742 0.993162i \(-0.537245\pi\)
−0.116742 + 0.993162i \(0.537245\pi\)
\(588\) 0 0
\(589\) 45.2548 1.86469
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −14.0000 −0.574911 −0.287456 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) 0 0
\(595\) −5.65685 −0.231908
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) −25.4558 −1.03664
\(604\) 0 0
\(605\) −8.48528 −0.344976
\(606\) 0 0
\(607\) −16.0000 −0.649420 −0.324710 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 22.6274 0.915407
\(612\) 0 0
\(613\) −28.2843 −1.14239 −0.571195 0.820814i \(-0.693520\pi\)
−0.571195 + 0.820814i \(0.693520\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 26.0000 1.04672 0.523360 0.852111i \(-0.324678\pi\)
0.523360 + 0.852111i \(0.324678\pi\)
\(618\) 0 0
\(619\) −11.3137 −0.454736 −0.227368 0.973809i \(-0.573012\pi\)
−0.227368 + 0.973809i \(0.573012\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) −31.0000 −1.24000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −11.3137 −0.451107
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −22.6274 −0.897942
\(636\) 0 0
\(637\) −2.82843 −0.112066
\(638\) 0 0
\(639\) 24.0000 0.949425
\(640\) 0 0
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) 0 0
\(643\) 33.9411 1.33851 0.669254 0.743034i \(-0.266614\pi\)
0.669254 + 0.743034i \(0.266614\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −16.0000 −0.629025 −0.314512 0.949253i \(-0.601841\pi\)
−0.314512 + 0.949253i \(0.601841\pi\)
\(648\) 0 0
\(649\) −32.0000 −1.25611
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 5.65685 0.221370 0.110685 0.993856i \(-0.464696\pi\)
0.110685 + 0.993856i \(0.464696\pi\)
\(654\) 0 0
\(655\) 16.0000 0.625172
\(656\) 0 0
\(657\) 18.0000 0.702247
\(658\) 0 0
\(659\) −8.48528 −0.330540 −0.165270 0.986248i \(-0.552849\pi\)
−0.165270 + 0.986248i \(0.552849\pi\)
\(660\) 0 0
\(661\) 2.82843 0.110013 0.0550065 0.998486i \(-0.482482\pi\)
0.0550065 + 0.998486i \(0.482482\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −16.0000 −0.620453
\(666\) 0 0
\(667\) 45.2548 1.75227
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −8.00000 −0.308837
\(672\) 0 0
\(673\) −26.0000 −1.00223 −0.501113 0.865382i \(-0.667076\pi\)
−0.501113 + 0.865382i \(0.667076\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −2.82843 −0.108705 −0.0543526 0.998522i \(-0.517310\pi\)
−0.0543526 + 0.998522i \(0.517310\pi\)
\(678\) 0 0
\(679\) 14.0000 0.537271
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 36.7696 1.40695 0.703474 0.710721i \(-0.251631\pi\)
0.703474 + 0.710721i \(0.251631\pi\)
\(684\) 0 0
\(685\) 16.9706 0.648412
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −32.0000 −1.21910
\(690\) 0 0
\(691\) 16.9706 0.645591 0.322795 0.946469i \(-0.395377\pi\)
0.322795 + 0.946469i \(0.395377\pi\)
\(692\) 0 0
\(693\) 8.48528 0.322329
\(694\) 0 0
\(695\) −48.0000 −1.82074
\(696\) 0 0
\(697\) −12.0000 −0.454532
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 5.65685 0.213656 0.106828 0.994277i \(-0.465931\pi\)
0.106828 + 0.994277i \(0.465931\pi\)
\(702\) 0 0
\(703\) −32.0000 −1.20690
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −8.48528 −0.319122
\(708\) 0 0
\(709\) −16.9706 −0.637343 −0.318671 0.947865i \(-0.603237\pi\)
−0.318671 + 0.947865i \(0.603237\pi\)
\(710\) 0 0
\(711\) −24.0000 −0.900070
\(712\) 0 0
\(713\) 64.0000 2.39682
\(714\) 0 0
\(715\) 22.6274 0.846217
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −16.9706 −0.630271
\(726\) 0 0
\(727\) 16.0000 0.593407 0.296704 0.954970i \(-0.404113\pi\)
0.296704 + 0.954970i \(0.404113\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 5.65685 0.209226
\(732\) 0 0
\(733\) 14.1421 0.522352 0.261176 0.965291i \(-0.415890\pi\)
0.261176 + 0.965291i \(0.415890\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −24.0000 −0.884051
\(738\) 0 0
\(739\) 25.4558 0.936408 0.468204 0.883620i \(-0.344901\pi\)
0.468204 + 0.883620i \(0.344901\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) −32.0000 −1.17239
\(746\) 0 0
\(747\) −16.9706 −0.620920
\(748\) 0 0
\(749\) 14.1421 0.516742
\(750\) 0 0
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −22.6274 −0.823496
\(756\) 0 0
\(757\) 39.5980 1.43921 0.719607 0.694382i \(-0.244322\pi\)
0.719607 + 0.694382i \(0.244322\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 22.0000 0.797499 0.398750 0.917060i \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) 0 0
\(763\) 16.9706 0.614376
\(764\) 0 0
\(765\) 16.9706 0.613572
\(766\) 0 0
\(767\) −32.0000 −1.15545
\(768\) 0 0
\(769\) −2.00000 −0.0721218 −0.0360609 0.999350i \(-0.511481\pi\)
−0.0360609 + 0.999350i \(0.511481\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −19.7990 −0.712120 −0.356060 0.934463i \(-0.615880\pi\)
−0.356060 + 0.934463i \(0.615880\pi\)
\(774\) 0 0
\(775\) −24.0000 −0.862105
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −33.9411 −1.21607
\(780\) 0 0
\(781\) 22.6274 0.809673
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 24.0000 0.856597
\(786\) 0 0
\(787\) 28.2843 1.00823 0.504113 0.863638i \(-0.331820\pi\)
0.504113 + 0.863638i \(0.331820\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) 0 0
\(793\) −8.00000 −0.284088
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −2.82843 −0.100188 −0.0500940 0.998745i \(-0.515952\pi\)
−0.0500940 + 0.998745i \(0.515952\pi\)
\(798\) 0 0
\(799\) 16.0000 0.566039
\(800\) 0 0
\(801\) 18.0000 0.635999
\(802\) 0 0
\(803\) 16.9706 0.598878
\(804\) 0 0
\(805\) −22.6274 −0.797512
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −56.0000 −1.96159
\(816\) 0 0
\(817\) 16.0000 0.559769
\(818\) 0 0
\(819\) 8.48528 0.296500
\(820\) 0 0
\(821\) −45.2548 −1.57940 −0.789702 0.613490i \(-0.789765\pi\)
−0.789702 + 0.613490i \(0.789765\pi\)
\(822\) 0 0
\(823\) −24.0000 −0.836587 −0.418294 0.908312i \(-0.637372\pi\)
−0.418294 + 0.908312i \(0.637372\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 36.7696 1.27860 0.639301 0.768956i \(-0.279224\pi\)
0.639301 + 0.768956i \(0.279224\pi\)
\(828\) 0 0
\(829\) 48.0833 1.67000 0.835000 0.550249i \(-0.185467\pi\)
0.835000 + 0.550249i \(0.185467\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −2.00000 −0.0692959
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −32.0000 −1.10476 −0.552381 0.833592i \(-0.686281\pi\)
−0.552381 + 0.833592i \(0.686281\pi\)
\(840\) 0 0
\(841\) 3.00000 0.103448
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −14.1421 −0.486504
\(846\) 0 0
\(847\) −3.00000 −0.103081
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −45.2548 −1.55132
\(852\) 0 0
\(853\) −25.4558 −0.871592 −0.435796 0.900046i \(-0.643533\pi\)
−0.435796 + 0.900046i \(0.643533\pi\)
\(854\) 0 0
\(855\) 48.0000 1.64157
\(856\) 0 0
\(857\) 22.0000 0.751506 0.375753 0.926720i \(-0.377384\pi\)
0.375753 + 0.926720i \(0.377384\pi\)
\(858\) 0 0
\(859\) 39.5980 1.35107 0.675533 0.737330i \(-0.263914\pi\)
0.675533 + 0.737330i \(0.263914\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 0 0
\(865\) 24.0000 0.816024
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −22.6274 −0.767583
\(870\) 0 0
\(871\) −24.0000 −0.813209
\(872\) 0 0
\(873\) −42.0000 −1.42148
\(874\) 0 0
\(875\) −5.65685 −0.191237
\(876\) 0 0
\(877\) −16.9706 −0.573055 −0.286528 0.958072i \(-0.592501\pi\)
−0.286528 + 0.958072i \(0.592501\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 0 0
\(883\) −25.4558 −0.856657 −0.428329 0.903623i \(-0.640897\pi\)
−0.428329 + 0.903623i \(0.640897\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −32.0000 −1.07445 −0.537227 0.843437i \(-0.680528\pi\)
−0.537227 + 0.843437i \(0.680528\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) −25.4558 −0.852803
\(892\) 0 0
\(893\) 45.2548 1.51440
\(894\) 0 0
\(895\) 24.0000 0.802232
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 45.2548 1.50933
\(900\) 0 0
\(901\) −22.6274 −0.753829
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −72.0000 −2.39336
\(906\) 0 0
\(907\) −19.7990 −0.657415 −0.328707 0.944432i \(-0.606613\pi\)
−0.328707 + 0.944432i \(0.606613\pi\)
\(908\) 0 0
\(909\) 25.4558 0.844317
\(910\) 0 0
\(911\) −8.00000 −0.265052 −0.132526 0.991180i \(-0.542309\pi\)
−0.132526 + 0.991180i \(0.542309\pi\)
\(912\) 0 0
\(913\) −16.0000 −0.529523
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 5.65685 0.186806
\(918\) 0 0
\(919\) 24.0000 0.791687 0.395843 0.918318i \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 22.6274 0.744791
\(924\) 0 0
\(925\) 16.9706 0.557989
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −34.0000 −1.11550 −0.557752 0.830008i \(-0.688336\pi\)
−0.557752 + 0.830008i \(0.688336\pi\)
\(930\) 0 0
\(931\) −5.65685 −0.185396
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 16.0000 0.523256
\(936\) 0 0
\(937\) 10.0000 0.326686 0.163343 0.986569i \(-0.447772\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 2.82843 0.0922041 0.0461020 0.998937i \(-0.485320\pi\)
0.0461020 + 0.998937i \(0.485320\pi\)
\(942\) 0 0
\(943\) −48.0000 −1.56310
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 25.4558 0.827204 0.413602 0.910458i \(-0.364271\pi\)
0.413602 + 0.910458i \(0.364271\pi\)
\(948\) 0 0
\(949\) 16.9706 0.550888
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −10.0000 −0.323932 −0.161966 0.986796i \(-0.551783\pi\)
−0.161966 + 0.986796i \(0.551783\pi\)
\(954\) 0 0
\(955\) −67.8823 −2.19662
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 6.00000 0.193750
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) −42.4264 −1.36717
\(964\) 0 0
\(965\) −5.65685 −0.182101
\(966\) 0 0
\(967\) −40.0000 −1.28631 −0.643157 0.765735i \(-0.722376\pi\)
−0.643157 + 0.765735i \(0.722376\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −11.3137 −0.363074 −0.181537 0.983384i \(-0.558107\pi\)
−0.181537 + 0.983384i \(0.558107\pi\)
\(972\) 0 0
\(973\) −16.9706 −0.544051
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −10.0000 −0.319928 −0.159964 0.987123i \(-0.551138\pi\)
−0.159964 + 0.987123i \(0.551138\pi\)
\(978\) 0 0
\(979\) 16.9706 0.542382
\(980\) 0 0
\(981\) −50.9117 −1.62549
\(982\) 0 0
\(983\) −48.0000 −1.53096 −0.765481 0.643458i \(-0.777499\pi\)
−0.765481 + 0.643458i \(0.777499\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 22.6274 0.719510
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 25.4558 0.806195 0.403097 0.915157i \(-0.367934\pi\)
0.403097 + 0.915157i \(0.367934\pi\)
\(998\) 0 0
\(999\) 0 0
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 1792.2.a.o.1.2 2
4.3 odd 2 1792.2.a.m.1.2 2
8.3 odd 2 1792.2.a.m.1.1 2
8.5 even 2 inner 1792.2.a.o.1.1 2
16.3 odd 4 896.2.b.d.449.1 yes 2
16.5 even 4 896.2.b.b.449.2 yes 2
16.11 odd 4 896.2.b.d.449.2 yes 2
16.13 even 4 896.2.b.b.449.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
896.2.b.b.449.1 2 16.13 even 4
896.2.b.b.449.2 yes 2 16.5 even 4
896.2.b.d.449.1 yes 2 16.3 odd 4
896.2.b.d.449.2 yes 2 16.11 odd 4
1792.2.a.m.1.1 2 8.3 odd 2
1792.2.a.m.1.2 2 4.3 odd 2
1792.2.a.o.1.1 2 8.5 even 2 inner
1792.2.a.o.1.2 2 1.1 even 1 trivial