Properties

Label 6864.2.a.bb.1.1
Level $6864$
Weight $2$
Character 6864.1
Self dual yes
Analytic conductor $54.809$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6864.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(54.8093159474\)
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: \( 1 \)
Twist minimal: no (minimal twist has level 1716)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 6864.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -3.41421 q^{5} -2.82843 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -3.41421 q^{5} -2.82843 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} +3.41421 q^{15} -6.24264 q^{17} +2.82843 q^{21} +4.82843 q^{23} +6.65685 q^{25} -1.00000 q^{27} -0.585786 q^{29} +5.41421 q^{31} -1.00000 q^{33} +9.65685 q^{35} +0.828427 q^{37} +1.00000 q^{39} +3.65685 q^{41} +0.242641 q^{43} -3.41421 q^{45} +8.00000 q^{47} +1.00000 q^{49} +6.24264 q^{51} +9.31371 q^{53} -3.41421 q^{55} -2.82843 q^{59} -3.17157 q^{61} -2.82843 q^{63} +3.41421 q^{65} -4.24264 q^{67} -4.82843 q^{69} +10.8284 q^{71} -5.65685 q^{73} -6.65685 q^{75} -2.82843 q^{77} +7.07107 q^{79} +1.00000 q^{81} +17.6569 q^{83} +21.3137 q^{85} +0.585786 q^{87} -11.8995 q^{89} +2.82843 q^{91} -5.41421 q^{93} -12.8284 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 4q^{5} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 4q^{5} + 2q^{9} + 2q^{11} - 2q^{13} + 4q^{15} - 4q^{17} + 4q^{23} + 2q^{25} - 2q^{27} - 4q^{29} + 8q^{31} - 2q^{33} + 8q^{35} - 4q^{37} + 2q^{39} - 4q^{41} - 8q^{43} - 4q^{45} + 16q^{47} + 2q^{49} + 4q^{51} - 4q^{53} - 4q^{55} - 12q^{61} + 4q^{65} - 4q^{69} + 16q^{71} - 2q^{75} + 2q^{81} + 24q^{83} + 20q^{85} + 4q^{87} - 4q^{89} - 8q^{93} - 20q^{97} + 2q^{99} + O(q^{100}) \)

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
\(5\) −3.41421 −1.52688 −0.763441 0.645877i \(-0.776492\pi\)
−0.763441 + 0.645877i \(0.776492\pi\)
\(6\) 0 0
\(7\) −2.82843 −1.06904 −0.534522 0.845154i \(-0.679509\pi\)
−0.534522 + 0.845154i \(0.679509\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 3.41421 0.881546
\(16\) 0 0
\(17\) −6.24264 −1.51406 −0.757031 0.653379i \(-0.773351\pi\)
−0.757031 + 0.653379i \(0.773351\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 2.82843 0.617213
\(22\) 0 0
\(23\) 4.82843 1.00680 0.503398 0.864054i \(-0.332083\pi\)
0.503398 + 0.864054i \(0.332083\pi\)
\(24\) 0 0
\(25\) 6.65685 1.33137
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −0.585786 −0.108778 −0.0543889 0.998520i \(-0.517321\pi\)
−0.0543889 + 0.998520i \(0.517321\pi\)
\(30\) 0 0
\(31\) 5.41421 0.972421 0.486211 0.873842i \(-0.338379\pi\)
0.486211 + 0.873842i \(0.338379\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) 9.65685 1.63231
\(36\) 0 0
\(37\) 0.828427 0.136193 0.0680963 0.997679i \(-0.478307\pi\)
0.0680963 + 0.997679i \(0.478307\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 3.65685 0.571105 0.285552 0.958363i \(-0.407823\pi\)
0.285552 + 0.958363i \(0.407823\pi\)
\(42\) 0 0
\(43\) 0.242641 0.0370024 0.0185012 0.999829i \(-0.494111\pi\)
0.0185012 + 0.999829i \(0.494111\pi\)
\(44\) 0 0
\(45\) −3.41421 −0.508961
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 6.24264 0.874145
\(52\) 0 0
\(53\) 9.31371 1.27934 0.639668 0.768651i \(-0.279072\pi\)
0.639668 + 0.768651i \(0.279072\pi\)
\(54\) 0 0
\(55\) −3.41421 −0.460372
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.82843 −0.368230 −0.184115 0.982905i \(-0.558942\pi\)
−0.184115 + 0.982905i \(0.558942\pi\)
\(60\) 0 0
\(61\) −3.17157 −0.406078 −0.203039 0.979171i \(-0.565082\pi\)
−0.203039 + 0.979171i \(0.565082\pi\)
\(62\) 0 0
\(63\) −2.82843 −0.356348
\(64\) 0 0
\(65\) 3.41421 0.423481
\(66\) 0 0
\(67\) −4.24264 −0.518321 −0.259161 0.965834i \(-0.583446\pi\)
−0.259161 + 0.965834i \(0.583446\pi\)
\(68\) 0 0
\(69\) −4.82843 −0.581274
\(70\) 0 0
\(71\) 10.8284 1.28510 0.642549 0.766245i \(-0.277877\pi\)
0.642549 + 0.766245i \(0.277877\pi\)
\(72\) 0 0
\(73\) −5.65685 −0.662085 −0.331042 0.943616i \(-0.607400\pi\)
−0.331042 + 0.943616i \(0.607400\pi\)
\(74\) 0 0
\(75\) −6.65685 −0.768667
\(76\) 0 0
\(77\) −2.82843 −0.322329
\(78\) 0 0
\(79\) 7.07107 0.795557 0.397779 0.917481i \(-0.369781\pi\)
0.397779 + 0.917481i \(0.369781\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 17.6569 1.93809 0.969046 0.246881i \(-0.0794057\pi\)
0.969046 + 0.246881i \(0.0794057\pi\)
\(84\) 0 0
\(85\) 21.3137 2.31180
\(86\) 0 0
\(87\) 0.585786 0.0628029
\(88\) 0 0
\(89\) −11.8995 −1.26134 −0.630672 0.776050i \(-0.717221\pi\)
−0.630672 + 0.776050i \(0.717221\pi\)
\(90\) 0 0
\(91\) 2.82843 0.296500
\(92\) 0 0
\(93\) −5.41421 −0.561428
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −12.8284 −1.30253 −0.651265 0.758851i \(-0.725761\pi\)
−0.651265 + 0.758851i \(0.725761\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) −18.7279 −1.86350 −0.931749 0.363103i \(-0.881717\pi\)
−0.931749 + 0.363103i \(0.881717\pi\)
\(102\) 0 0
\(103\) 2.82843 0.278693 0.139347 0.990244i \(-0.455500\pi\)
0.139347 + 0.990244i \(0.455500\pi\)
\(104\) 0 0
\(105\) −9.65685 −0.942412
\(106\) 0 0
\(107\) −1.17157 −0.113260 −0.0566301 0.998395i \(-0.518036\pi\)
−0.0566301 + 0.998395i \(0.518036\pi\)
\(108\) 0 0
\(109\) −15.3137 −1.46679 −0.733394 0.679804i \(-0.762065\pi\)
−0.733394 + 0.679804i \(0.762065\pi\)
\(110\) 0 0
\(111\) −0.828427 −0.0786308
\(112\) 0 0
\(113\) 18.9706 1.78460 0.892300 0.451442i \(-0.149090\pi\)
0.892300 + 0.451442i \(0.149090\pi\)
\(114\) 0 0
\(115\) −16.4853 −1.53726
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) 17.6569 1.61860
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −3.65685 −0.329727
\(124\) 0 0
\(125\) −5.65685 −0.505964
\(126\) 0 0
\(127\) 7.07107 0.627456 0.313728 0.949513i \(-0.398422\pi\)
0.313728 + 0.949513i \(0.398422\pi\)
\(128\) 0 0
\(129\) −0.242641 −0.0213633
\(130\) 0 0
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 3.41421 0.293849
\(136\) 0 0
\(137\) 13.0711 1.11674 0.558368 0.829593i \(-0.311428\pi\)
0.558368 + 0.829593i \(0.311428\pi\)
\(138\) 0 0
\(139\) −8.24264 −0.699132 −0.349566 0.936912i \(-0.613671\pi\)
−0.349566 + 0.936912i \(0.613671\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) −1.00000 −0.0836242
\(144\) 0 0
\(145\) 2.00000 0.166091
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) 2.48528 0.203602 0.101801 0.994805i \(-0.467539\pi\)
0.101801 + 0.994805i \(0.467539\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 0 0
\(153\) −6.24264 −0.504688
\(154\) 0 0
\(155\) −18.4853 −1.48477
\(156\) 0 0
\(157\) 6.34315 0.506238 0.253119 0.967435i \(-0.418544\pi\)
0.253119 + 0.967435i \(0.418544\pi\)
\(158\) 0 0
\(159\) −9.31371 −0.738625
\(160\) 0 0
\(161\) −13.6569 −1.07631
\(162\) 0 0
\(163\) 12.7279 0.996928 0.498464 0.866910i \(-0.333898\pi\)
0.498464 + 0.866910i \(0.333898\pi\)
\(164\) 0 0
\(165\) 3.41421 0.265796
\(166\) 0 0
\(167\) −11.3137 −0.875481 −0.437741 0.899101i \(-0.644221\pi\)
−0.437741 + 0.899101i \(0.644221\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 19.8995 1.51293 0.756465 0.654034i \(-0.226925\pi\)
0.756465 + 0.654034i \(0.226925\pi\)
\(174\) 0 0
\(175\) −18.8284 −1.42330
\(176\) 0 0
\(177\) 2.82843 0.212598
\(178\) 0 0
\(179\) −4.82843 −0.360894 −0.180447 0.983585i \(-0.557754\pi\)
−0.180447 + 0.983585i \(0.557754\pi\)
\(180\) 0 0
\(181\) 24.9706 1.85605 0.928024 0.372521i \(-0.121507\pi\)
0.928024 + 0.372521i \(0.121507\pi\)
\(182\) 0 0
\(183\) 3.17157 0.234449
\(184\) 0 0
\(185\) −2.82843 −0.207950
\(186\) 0 0
\(187\) −6.24264 −0.456507
\(188\) 0 0
\(189\) 2.82843 0.205738
\(190\) 0 0
\(191\) 2.34315 0.169544 0.0847720 0.996400i \(-0.472984\pi\)
0.0847720 + 0.996400i \(0.472984\pi\)
\(192\) 0 0
\(193\) −5.65685 −0.407189 −0.203595 0.979055i \(-0.565262\pi\)
−0.203595 + 0.979055i \(0.565262\pi\)
\(194\) 0 0
\(195\) −3.41421 −0.244497
\(196\) 0 0
\(197\) −12.1421 −0.865091 −0.432546 0.901612i \(-0.642385\pi\)
−0.432546 + 0.901612i \(0.642385\pi\)
\(198\) 0 0
\(199\) 4.48528 0.317953 0.158977 0.987282i \(-0.449181\pi\)
0.158977 + 0.987282i \(0.449181\pi\)
\(200\) 0 0
\(201\) 4.24264 0.299253
\(202\) 0 0
\(203\) 1.65685 0.116288
\(204\) 0 0
\(205\) −12.4853 −0.872010
\(206\) 0 0
\(207\) 4.82843 0.335599
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −10.5858 −0.728756 −0.364378 0.931251i \(-0.618718\pi\)
−0.364378 + 0.931251i \(0.618718\pi\)
\(212\) 0 0
\(213\) −10.8284 −0.741952
\(214\) 0 0
\(215\) −0.828427 −0.0564983
\(216\) 0 0
\(217\) −15.3137 −1.03956
\(218\) 0 0
\(219\) 5.65685 0.382255
\(220\) 0 0
\(221\) 6.24264 0.419925
\(222\) 0 0
\(223\) 21.8995 1.46650 0.733249 0.679960i \(-0.238003\pi\)
0.733249 + 0.679960i \(0.238003\pi\)
\(224\) 0 0
\(225\) 6.65685 0.443790
\(226\) 0 0
\(227\) −12.1421 −0.805902 −0.402951 0.915222i \(-0.632015\pi\)
−0.402951 + 0.915222i \(0.632015\pi\)
\(228\) 0 0
\(229\) 1.31371 0.0868123 0.0434062 0.999058i \(-0.486179\pi\)
0.0434062 + 0.999058i \(0.486179\pi\)
\(230\) 0 0
\(231\) 2.82843 0.186097
\(232\) 0 0
\(233\) 8.58579 0.562474 0.281237 0.959638i \(-0.409255\pi\)
0.281237 + 0.959638i \(0.409255\pi\)
\(234\) 0 0
\(235\) −27.3137 −1.78175
\(236\) 0 0
\(237\) −7.07107 −0.459315
\(238\) 0 0
\(239\) −20.1421 −1.30289 −0.651443 0.758697i \(-0.725836\pi\)
−0.651443 + 0.758697i \(0.725836\pi\)
\(240\) 0 0
\(241\) −0.343146 −0.0221040 −0.0110520 0.999939i \(-0.503518\pi\)
−0.0110520 + 0.999939i \(0.503518\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −3.41421 −0.218126
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −17.6569 −1.11896
\(250\) 0 0
\(251\) −25.6569 −1.61945 −0.809723 0.586812i \(-0.800383\pi\)
−0.809723 + 0.586812i \(0.800383\pi\)
\(252\) 0 0
\(253\) 4.82843 0.303561
\(254\) 0 0
\(255\) −21.3137 −1.33472
\(256\) 0 0
\(257\) −0.828427 −0.0516759 −0.0258379 0.999666i \(-0.508225\pi\)
−0.0258379 + 0.999666i \(0.508225\pi\)
\(258\) 0 0
\(259\) −2.34315 −0.145596
\(260\) 0 0
\(261\) −0.585786 −0.0362593
\(262\) 0 0
\(263\) −8.48528 −0.523225 −0.261612 0.965173i \(-0.584254\pi\)
−0.261612 + 0.965173i \(0.584254\pi\)
\(264\) 0 0
\(265\) −31.7990 −1.95340
\(266\) 0 0
\(267\) 11.8995 0.728237
\(268\) 0 0
\(269\) −8.82843 −0.538279 −0.269139 0.963101i \(-0.586739\pi\)
−0.269139 + 0.963101i \(0.586739\pi\)
\(270\) 0 0
\(271\) −1.65685 −0.100647 −0.0503234 0.998733i \(-0.516025\pi\)
−0.0503234 + 0.998733i \(0.516025\pi\)
\(272\) 0 0
\(273\) −2.82843 −0.171184
\(274\) 0 0
\(275\) 6.65685 0.401423
\(276\) 0 0
\(277\) 14.9706 0.899494 0.449747 0.893156i \(-0.351514\pi\)
0.449747 + 0.893156i \(0.351514\pi\)
\(278\) 0 0
\(279\) 5.41421 0.324140
\(280\) 0 0
\(281\) −23.6569 −1.41125 −0.705625 0.708586i \(-0.749334\pi\)
−0.705625 + 0.708586i \(0.749334\pi\)
\(282\) 0 0
\(283\) −27.0711 −1.60921 −0.804604 0.593812i \(-0.797622\pi\)
−0.804604 + 0.593812i \(0.797622\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −10.3431 −0.610537
\(288\) 0 0
\(289\) 21.9706 1.29239
\(290\) 0 0
\(291\) 12.8284 0.752016
\(292\) 0 0
\(293\) −1.51472 −0.0884908 −0.0442454 0.999021i \(-0.514088\pi\)
−0.0442454 + 0.999021i \(0.514088\pi\)
\(294\) 0 0
\(295\) 9.65685 0.562244
\(296\) 0 0
\(297\) −1.00000 −0.0580259
\(298\) 0 0
\(299\) −4.82843 −0.279235
\(300\) 0 0
\(301\) −0.686292 −0.0395572
\(302\) 0 0
\(303\) 18.7279 1.07589
\(304\) 0 0
\(305\) 10.8284 0.620034
\(306\) 0 0
\(307\) −28.0000 −1.59804 −0.799022 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) 0 0
\(309\) −2.82843 −0.160904
\(310\) 0 0
\(311\) 24.8284 1.40789 0.703945 0.710254i \(-0.251420\pi\)
0.703945 + 0.710254i \(0.251420\pi\)
\(312\) 0 0
\(313\) −22.6274 −1.27898 −0.639489 0.768801i \(-0.720854\pi\)
−0.639489 + 0.768801i \(0.720854\pi\)
\(314\) 0 0
\(315\) 9.65685 0.544102
\(316\) 0 0
\(317\) −6.92893 −0.389168 −0.194584 0.980886i \(-0.562336\pi\)
−0.194584 + 0.980886i \(0.562336\pi\)
\(318\) 0 0
\(319\) −0.585786 −0.0327977
\(320\) 0 0
\(321\) 1.17157 0.0653908
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −6.65685 −0.369256
\(326\) 0 0
\(327\) 15.3137 0.846850
\(328\) 0 0
\(329\) −22.6274 −1.24749
\(330\) 0 0
\(331\) −31.5563 −1.73449 −0.867247 0.497878i \(-0.834113\pi\)
−0.867247 + 0.497878i \(0.834113\pi\)
\(332\) 0 0
\(333\) 0.828427 0.0453975
\(334\) 0 0
\(335\) 14.4853 0.791415
\(336\) 0 0
\(337\) 9.51472 0.518300 0.259150 0.965837i \(-0.416558\pi\)
0.259150 + 0.965837i \(0.416558\pi\)
\(338\) 0 0
\(339\) −18.9706 −1.03034
\(340\) 0 0
\(341\) 5.41421 0.293196
\(342\) 0 0
\(343\) 16.9706 0.916324
\(344\) 0 0
\(345\) 16.4853 0.887538
\(346\) 0 0
\(347\) 25.6569 1.37733 0.688666 0.725079i \(-0.258197\pi\)
0.688666 + 0.725079i \(0.258197\pi\)
\(348\) 0 0
\(349\) −35.9411 −1.92388 −0.961942 0.273253i \(-0.911900\pi\)
−0.961942 + 0.273253i \(0.911900\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) 14.7279 0.783888 0.391944 0.919989i \(-0.371803\pi\)
0.391944 + 0.919989i \(0.371803\pi\)
\(354\) 0 0
\(355\) −36.9706 −1.96219
\(356\) 0 0
\(357\) −17.6569 −0.934500
\(358\) 0 0
\(359\) 16.1421 0.851949 0.425975 0.904735i \(-0.359931\pi\)
0.425975 + 0.904735i \(0.359931\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) 19.3137 1.01093
\(366\) 0 0
\(367\) −8.68629 −0.453421 −0.226710 0.973962i \(-0.572797\pi\)
−0.226710 + 0.973962i \(0.572797\pi\)
\(368\) 0 0
\(369\) 3.65685 0.190368
\(370\) 0 0
\(371\) −26.3431 −1.36767
\(372\) 0 0
\(373\) 19.4558 1.00739 0.503693 0.863883i \(-0.331974\pi\)
0.503693 + 0.863883i \(0.331974\pi\)
\(374\) 0 0
\(375\) 5.65685 0.292119
\(376\) 0 0
\(377\) 0.585786 0.0301695
\(378\) 0 0
\(379\) −29.6985 −1.52551 −0.762754 0.646688i \(-0.776153\pi\)
−0.762754 + 0.646688i \(0.776153\pi\)
\(380\) 0 0
\(381\) −7.07107 −0.362262
\(382\) 0 0
\(383\) 19.7990 1.01168 0.505841 0.862627i \(-0.331182\pi\)
0.505841 + 0.862627i \(0.331182\pi\)
\(384\) 0 0
\(385\) 9.65685 0.492159
\(386\) 0 0
\(387\) 0.242641 0.0123341
\(388\) 0 0
\(389\) −17.7990 −0.902445 −0.451222 0.892412i \(-0.649012\pi\)
−0.451222 + 0.892412i \(0.649012\pi\)
\(390\) 0 0
\(391\) −30.1421 −1.52435
\(392\) 0 0
\(393\) 8.00000 0.403547
\(394\) 0 0
\(395\) −24.1421 −1.21472
\(396\) 0 0
\(397\) −37.7990 −1.89708 −0.948538 0.316662i \(-0.897438\pi\)
−0.948538 + 0.316662i \(0.897438\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −19.2132 −0.959462 −0.479731 0.877416i \(-0.659266\pi\)
−0.479731 + 0.877416i \(0.659266\pi\)
\(402\) 0 0
\(403\) −5.41421 −0.269701
\(404\) 0 0
\(405\) −3.41421 −0.169654
\(406\) 0 0
\(407\) 0.828427 0.0410636
\(408\) 0 0
\(409\) 2.34315 0.115861 0.0579306 0.998321i \(-0.481550\pi\)
0.0579306 + 0.998321i \(0.481550\pi\)
\(410\) 0 0
\(411\) −13.0711 −0.644748
\(412\) 0 0
\(413\) 8.00000 0.393654
\(414\) 0 0
\(415\) −60.2843 −2.95924
\(416\) 0 0
\(417\) 8.24264 0.403644
\(418\) 0 0
\(419\) −28.8284 −1.40836 −0.704180 0.710021i \(-0.748685\pi\)
−0.704180 + 0.710021i \(0.748685\pi\)
\(420\) 0 0
\(421\) −37.3137 −1.81856 −0.909279 0.416186i \(-0.863366\pi\)
−0.909279 + 0.416186i \(0.863366\pi\)
\(422\) 0 0
\(423\) 8.00000 0.388973
\(424\) 0 0
\(425\) −41.5563 −2.01578
\(426\) 0 0
\(427\) 8.97056 0.434116
\(428\) 0 0
\(429\) 1.00000 0.0482805
\(430\) 0 0
\(431\) 13.6569 0.657828 0.328914 0.944360i \(-0.393317\pi\)
0.328914 + 0.944360i \(0.393317\pi\)
\(432\) 0 0
\(433\) −26.0000 −1.24948 −0.624740 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) 0 0
\(435\) −2.00000 −0.0958927
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −9.41421 −0.449316 −0.224658 0.974438i \(-0.572126\pi\)
−0.224658 + 0.974438i \(0.572126\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 23.3137 1.10767 0.553834 0.832627i \(-0.313164\pi\)
0.553834 + 0.832627i \(0.313164\pi\)
\(444\) 0 0
\(445\) 40.6274 1.92592
\(446\) 0 0
\(447\) −2.48528 −0.117550
\(448\) 0 0
\(449\) 8.58579 0.405188 0.202594 0.979263i \(-0.435063\pi\)
0.202594 + 0.979263i \(0.435063\pi\)
\(450\) 0 0
\(451\) 3.65685 0.172195
\(452\) 0 0
\(453\) 12.0000 0.563809
\(454\) 0 0
\(455\) −9.65685 −0.452720
\(456\) 0 0
\(457\) 25.3137 1.18413 0.592063 0.805892i \(-0.298314\pi\)
0.592063 + 0.805892i \(0.298314\pi\)
\(458\) 0 0
\(459\) 6.24264 0.291382
\(460\) 0 0
\(461\) 10.9706 0.510950 0.255475 0.966816i \(-0.417768\pi\)
0.255475 + 0.966816i \(0.417768\pi\)
\(462\) 0 0
\(463\) −12.9289 −0.600858 −0.300429 0.953804i \(-0.597130\pi\)
−0.300429 + 0.953804i \(0.597130\pi\)
\(464\) 0 0
\(465\) 18.4853 0.857234
\(466\) 0 0
\(467\) 23.4558 1.08541 0.542704 0.839924i \(-0.317401\pi\)
0.542704 + 0.839924i \(0.317401\pi\)
\(468\) 0 0
\(469\) 12.0000 0.554109
\(470\) 0 0
\(471\) −6.34315 −0.292277
\(472\) 0 0
\(473\) 0.242641 0.0111566
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 9.31371 0.426445
\(478\) 0 0
\(479\) −24.1421 −1.10308 −0.551541 0.834148i \(-0.685960\pi\)
−0.551541 + 0.834148i \(0.685960\pi\)
\(480\) 0 0
\(481\) −0.828427 −0.0377730
\(482\) 0 0
\(483\) 13.6569 0.621408
\(484\) 0 0
\(485\) 43.7990 1.98881
\(486\) 0 0
\(487\) 11.5563 0.523668 0.261834 0.965113i \(-0.415673\pi\)
0.261834 + 0.965113i \(0.415673\pi\)
\(488\) 0 0
\(489\) −12.7279 −0.575577
\(490\) 0 0
\(491\) −20.9706 −0.946388 −0.473194 0.880958i \(-0.656899\pi\)
−0.473194 + 0.880958i \(0.656899\pi\)
\(492\) 0 0
\(493\) 3.65685 0.164696
\(494\) 0 0
\(495\) −3.41421 −0.153457
\(496\) 0 0
\(497\) −30.6274 −1.37383
\(498\) 0 0
\(499\) −9.89949 −0.443162 −0.221581 0.975142i \(-0.571122\pi\)
−0.221581 + 0.975142i \(0.571122\pi\)
\(500\) 0 0
\(501\) 11.3137 0.505459
\(502\) 0 0
\(503\) 24.4853 1.09174 0.545872 0.837868i \(-0.316198\pi\)
0.545872 + 0.837868i \(0.316198\pi\)
\(504\) 0 0
\(505\) 63.9411 2.84534
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) 9.55635 0.423578 0.211789 0.977315i \(-0.432071\pi\)
0.211789 + 0.977315i \(0.432071\pi\)
\(510\) 0 0
\(511\) 16.0000 0.707798
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −9.65685 −0.425532
\(516\) 0 0
\(517\) 8.00000 0.351840
\(518\) 0 0
\(519\) −19.8995 −0.873491
\(520\) 0 0
\(521\) 14.9706 0.655872 0.327936 0.944700i \(-0.393647\pi\)
0.327936 + 0.944700i \(0.393647\pi\)
\(522\) 0 0
\(523\) 2.10051 0.0918487 0.0459243 0.998945i \(-0.485377\pi\)
0.0459243 + 0.998945i \(0.485377\pi\)
\(524\) 0 0
\(525\) 18.8284 0.821740
\(526\) 0 0
\(527\) −33.7990 −1.47231
\(528\) 0 0
\(529\) 0.313708 0.0136395
\(530\) 0 0
\(531\) −2.82843 −0.122743
\(532\) 0 0
\(533\) −3.65685 −0.158396
\(534\) 0 0
\(535\) 4.00000 0.172935
\(536\) 0 0
\(537\) 4.82843 0.208362
\(538\) 0 0
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 18.0000 0.773880 0.386940 0.922105i \(-0.373532\pi\)
0.386940 + 0.922105i \(0.373532\pi\)
\(542\) 0 0
\(543\) −24.9706 −1.07159
\(544\) 0 0
\(545\) 52.2843 2.23961
\(546\) 0 0
\(547\) −26.1005 −1.11598 −0.557989 0.829849i \(-0.688427\pi\)
−0.557989 + 0.829849i \(0.688427\pi\)
\(548\) 0 0
\(549\) −3.17157 −0.135359
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −20.0000 −0.850487
\(554\) 0 0
\(555\) 2.82843 0.120060
\(556\) 0 0
\(557\) 26.4853 1.12222 0.561109 0.827742i \(-0.310375\pi\)
0.561109 + 0.827742i \(0.310375\pi\)
\(558\) 0 0
\(559\) −0.242641 −0.0102626
\(560\) 0 0
\(561\) 6.24264 0.263564
\(562\) 0 0
\(563\) 1.65685 0.0698281 0.0349140 0.999390i \(-0.488884\pi\)
0.0349140 + 0.999390i \(0.488884\pi\)
\(564\) 0 0
\(565\) −64.7696 −2.72488
\(566\) 0 0
\(567\) −2.82843 −0.118783
\(568\) 0 0
\(569\) 13.5563 0.568312 0.284156 0.958778i \(-0.408287\pi\)
0.284156 + 0.958778i \(0.408287\pi\)
\(570\) 0 0
\(571\) 28.0416 1.17351 0.586753 0.809766i \(-0.300406\pi\)
0.586753 + 0.809766i \(0.300406\pi\)
\(572\) 0 0
\(573\) −2.34315 −0.0978863
\(574\) 0 0
\(575\) 32.1421 1.34042
\(576\) 0 0
\(577\) 26.9706 1.12280 0.561400 0.827545i \(-0.310263\pi\)
0.561400 + 0.827545i \(0.310263\pi\)
\(578\) 0 0
\(579\) 5.65685 0.235091
\(580\) 0 0
\(581\) −49.9411 −2.07191
\(582\) 0 0
\(583\) 9.31371 0.385734
\(584\) 0 0
\(585\) 3.41421 0.141160
\(586\) 0 0
\(587\) 31.7990 1.31248 0.656242 0.754550i \(-0.272145\pi\)
0.656242 + 0.754550i \(0.272145\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 12.1421 0.499461
\(592\) 0 0
\(593\) −19.1716 −0.787282 −0.393641 0.919264i \(-0.628785\pi\)
−0.393641 + 0.919264i \(0.628785\pi\)
\(594\) 0 0
\(595\) −60.2843 −2.47141
\(596\) 0 0
\(597\) −4.48528 −0.183570
\(598\) 0 0
\(599\) 13.6569 0.558004 0.279002 0.960291i \(-0.409996\pi\)
0.279002 + 0.960291i \(0.409996\pi\)
\(600\) 0 0
\(601\) 43.2548 1.76440 0.882201 0.470874i \(-0.156061\pi\)
0.882201 + 0.470874i \(0.156061\pi\)
\(602\) 0 0
\(603\) −4.24264 −0.172774
\(604\) 0 0
\(605\) −3.41421 −0.138808
\(606\) 0 0
\(607\) −13.2132 −0.536307 −0.268154 0.963376i \(-0.586414\pi\)
−0.268154 + 0.963376i \(0.586414\pi\)
\(608\) 0 0
\(609\) −1.65685 −0.0671391
\(610\) 0 0
\(611\) −8.00000 −0.323645
\(612\) 0 0
\(613\) −37.9411 −1.53243 −0.766214 0.642586i \(-0.777862\pi\)
−0.766214 + 0.642586i \(0.777862\pi\)
\(614\) 0 0
\(615\) 12.4853 0.503455
\(616\) 0 0
\(617\) 26.9289 1.08412 0.542059 0.840340i \(-0.317645\pi\)
0.542059 + 0.840340i \(0.317645\pi\)
\(618\) 0 0
\(619\) −9.41421 −0.378389 −0.189195 0.981940i \(-0.560588\pi\)
−0.189195 + 0.981940i \(0.560588\pi\)
\(620\) 0 0
\(621\) −4.82843 −0.193758
\(622\) 0 0
\(623\) 33.6569 1.34843
\(624\) 0 0
\(625\) −13.9706 −0.558823
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −5.17157 −0.206204
\(630\) 0 0
\(631\) 43.0711 1.71463 0.857316 0.514790i \(-0.172130\pi\)
0.857316 + 0.514790i \(0.172130\pi\)
\(632\) 0 0
\(633\) 10.5858 0.420747
\(634\) 0 0
\(635\) −24.1421 −0.958051
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 0 0
\(639\) 10.8284 0.428366
\(640\) 0 0
\(641\) 37.5980 1.48503 0.742515 0.669829i \(-0.233633\pi\)
0.742515 + 0.669829i \(0.233633\pi\)
\(642\) 0 0
\(643\) 7.55635 0.297993 0.148997 0.988838i \(-0.452396\pi\)
0.148997 + 0.988838i \(0.452396\pi\)
\(644\) 0 0
\(645\) 0.828427 0.0326193
\(646\) 0 0
\(647\) −29.6569 −1.16593 −0.582966 0.812497i \(-0.698108\pi\)
−0.582966 + 0.812497i \(0.698108\pi\)
\(648\) 0 0
\(649\) −2.82843 −0.111025
\(650\) 0 0
\(651\) 15.3137 0.600192
\(652\) 0 0
\(653\) −28.8284 −1.12814 −0.564072 0.825726i \(-0.690766\pi\)
−0.564072 + 0.825726i \(0.690766\pi\)
\(654\) 0 0
\(655\) 27.3137 1.06723
\(656\) 0 0
\(657\) −5.65685 −0.220695
\(658\) 0 0
\(659\) 31.1127 1.21198 0.605989 0.795473i \(-0.292777\pi\)
0.605989 + 0.795473i \(0.292777\pi\)
\(660\) 0 0
\(661\) 8.82843 0.343386 0.171693 0.985151i \(-0.445076\pi\)
0.171693 + 0.985151i \(0.445076\pi\)
\(662\) 0 0
\(663\) −6.24264 −0.242444
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −2.82843 −0.109517
\(668\) 0 0
\(669\) −21.8995 −0.846683
\(670\) 0 0
\(671\) −3.17157 −0.122437
\(672\) 0 0
\(673\) −11.6569 −0.449339 −0.224669 0.974435i \(-0.572130\pi\)
−0.224669 + 0.974435i \(0.572130\pi\)
\(674\) 0 0
\(675\) −6.65685 −0.256222
\(676\) 0 0
\(677\) −13.7574 −0.528738 −0.264369 0.964422i \(-0.585164\pi\)
−0.264369 + 0.964422i \(0.585164\pi\)
\(678\) 0 0
\(679\) 36.2843 1.39246
\(680\) 0 0
\(681\) 12.1421 0.465288
\(682\) 0 0
\(683\) −6.82843 −0.261283 −0.130641 0.991430i \(-0.541704\pi\)
−0.130641 + 0.991430i \(0.541704\pi\)
\(684\) 0 0
\(685\) −44.6274 −1.70513
\(686\) 0 0
\(687\) −1.31371 −0.0501211
\(688\) 0 0
\(689\) −9.31371 −0.354824
\(690\) 0 0
\(691\) 8.44365 0.321212 0.160606 0.987019i \(-0.448655\pi\)
0.160606 + 0.987019i \(0.448655\pi\)
\(692\) 0 0
\(693\) −2.82843 −0.107443
\(694\) 0 0
\(695\) 28.1421 1.06749
\(696\) 0 0
\(697\) −22.8284 −0.864688
\(698\) 0 0
\(699\) −8.58579 −0.324744
\(700\) 0 0
\(701\) 30.2426 1.14225 0.571124 0.820864i \(-0.306507\pi\)
0.571124 + 0.820864i \(0.306507\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 27.3137 1.02869
\(706\) 0 0
\(707\) 52.9706 1.99216
\(708\) 0 0
\(709\) −48.6274 −1.82624 −0.913120 0.407690i \(-0.866334\pi\)
−0.913120 + 0.407690i \(0.866334\pi\)
\(710\) 0 0
\(711\) 7.07107 0.265186
\(712\) 0 0
\(713\) 26.1421 0.979031
\(714\) 0 0
\(715\) 3.41421 0.127684
\(716\) 0 0
\(717\) 20.1421 0.752222
\(718\) 0 0
\(719\) 35.3137 1.31698 0.658490 0.752590i \(-0.271196\pi\)
0.658490 + 0.752590i \(0.271196\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 0 0
\(723\) 0.343146 0.0127617
\(724\) 0 0
\(725\) −3.89949 −0.144824
\(726\) 0 0
\(727\) 17.1716 0.636858 0.318429 0.947947i \(-0.396845\pi\)
0.318429 + 0.947947i \(0.396845\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −1.51472 −0.0560239
\(732\) 0 0
\(733\) 14.6274 0.540276 0.270138 0.962822i \(-0.412931\pi\)
0.270138 + 0.962822i \(0.412931\pi\)
\(734\) 0 0
\(735\) 3.41421 0.125935
\(736\) 0 0
\(737\) −4.24264 −0.156280
\(738\) 0 0
\(739\) −9.17157 −0.337382 −0.168691 0.985669i \(-0.553954\pi\)
−0.168691 + 0.985669i \(0.553954\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −25.9411 −0.951688 −0.475844 0.879530i \(-0.657857\pi\)
−0.475844 + 0.879530i \(0.657857\pi\)
\(744\) 0 0
\(745\) −8.48528 −0.310877
\(746\) 0 0
\(747\) 17.6569 0.646031
\(748\) 0 0
\(749\) 3.31371 0.121080
\(750\) 0 0
\(751\) 19.3137 0.704767 0.352384 0.935856i \(-0.385371\pi\)
0.352384 + 0.935856i \(0.385371\pi\)
\(752\) 0 0
\(753\) 25.6569 0.934988
\(754\) 0 0
\(755\) 40.9706 1.49107
\(756\) 0 0
\(757\) −24.2843 −0.882627 −0.441313 0.897353i \(-0.645487\pi\)
−0.441313 + 0.897353i \(0.645487\pi\)
\(758\) 0 0
\(759\) −4.82843 −0.175261
\(760\) 0 0
\(761\) −19.1716 −0.694969 −0.347484 0.937686i \(-0.612964\pi\)
−0.347484 + 0.937686i \(0.612964\pi\)
\(762\) 0 0
\(763\) 43.3137 1.56806
\(764\) 0 0
\(765\) 21.3137 0.770599
\(766\) 0 0
\(767\) 2.82843 0.102129
\(768\) 0 0
\(769\) −18.0000 −0.649097 −0.324548 0.945869i \(-0.605212\pi\)
−0.324548 + 0.945869i \(0.605212\pi\)
\(770\) 0 0
\(771\) 0.828427 0.0298351
\(772\) 0 0
\(773\) −4.58579 −0.164939 −0.0824696 0.996594i \(-0.526281\pi\)
−0.0824696 + 0.996594i \(0.526281\pi\)
\(774\) 0 0
\(775\) 36.0416 1.29465
\(776\) 0 0
\(777\) 2.34315 0.0840599
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 10.8284 0.387472
\(782\) 0 0
\(783\) 0.585786 0.0209343
\(784\) 0 0
\(785\) −21.6569 −0.772966
\(786\) 0 0
\(787\) 15.7990 0.563173 0.281587 0.959536i \(-0.409139\pi\)
0.281587 + 0.959536i \(0.409139\pi\)
\(788\) 0 0
\(789\) 8.48528 0.302084
\(790\) 0 0
\(791\) −53.6569 −1.90782
\(792\) 0 0
\(793\) 3.17157 0.112626
\(794\) 0 0
\(795\) 31.7990 1.12779
\(796\) 0 0
\(797\) 5.51472 0.195341 0.0976707 0.995219i \(-0.468861\pi\)
0.0976707 + 0.995219i \(0.468861\pi\)
\(798\) 0 0
\(799\) −49.9411 −1.76679
\(800\) 0 0
\(801\) −11.8995 −0.420448
\(802\) 0 0
\(803\) −5.65685 −0.199626
\(804\) 0 0
\(805\) 46.6274 1.64340
\(806\) 0 0
\(807\) 8.82843 0.310775
\(808\) 0 0
\(809\) −20.3848 −0.716691 −0.358345 0.933589i \(-0.616659\pi\)
−0.358345 + 0.933589i \(0.616659\pi\)
\(810\) 0 0
\(811\) −16.2843 −0.571818 −0.285909 0.958257i \(-0.592296\pi\)
−0.285909 + 0.958257i \(0.592296\pi\)
\(812\) 0 0
\(813\) 1.65685 0.0581084
\(814\) 0 0
\(815\) −43.4558 −1.52219
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 2.82843 0.0988332
\(820\) 0 0
\(821\) −40.6274 −1.41791 −0.708953 0.705255i \(-0.750832\pi\)
−0.708953 + 0.705255i \(0.750832\pi\)
\(822\) 0 0
\(823\) −1.45584 −0.0507475 −0.0253738 0.999678i \(-0.508078\pi\)
−0.0253738 + 0.999678i \(0.508078\pi\)
\(824\) 0 0
\(825\) −6.65685 −0.231762
\(826\) 0 0
\(827\) −42.7696 −1.48724 −0.743622 0.668601i \(-0.766894\pi\)
−0.743622 + 0.668601i \(0.766894\pi\)
\(828\) 0 0
\(829\) −41.3137 −1.43488 −0.717442 0.696618i \(-0.754687\pi\)
−0.717442 + 0.696618i \(0.754687\pi\)
\(830\) 0 0
\(831\) −14.9706 −0.519323
\(832\) 0 0
\(833\) −6.24264 −0.216295
\(834\) 0 0
\(835\) 38.6274 1.33676
\(836\) 0 0
\(837\) −5.41421 −0.187143
\(838\) 0 0
\(839\) 41.6569 1.43815 0.719077 0.694930i \(-0.244565\pi\)
0.719077 + 0.694930i \(0.244565\pi\)
\(840\) 0 0
\(841\) −28.6569 −0.988167
\(842\) 0 0
\(843\) 23.6569 0.814785
\(844\) 0 0
\(845\) −3.41421 −0.117453
\(846\) 0 0
\(847\) −2.82843 −0.0971859
\(848\) 0 0
\(849\) 27.0711 0.929077
\(850\) 0 0
\(851\) 4.00000 0.137118
\(852\) 0 0
\(853\) −17.3137 −0.592810 −0.296405 0.955062i \(-0.595788\pi\)
−0.296405 + 0.955062i \(0.595788\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 25.0711 0.856411 0.428206 0.903681i \(-0.359146\pi\)
0.428206 + 0.903681i \(0.359146\pi\)
\(858\) 0 0
\(859\) 22.8284 0.778896 0.389448 0.921048i \(-0.372666\pi\)
0.389448 + 0.921048i \(0.372666\pi\)
\(860\) 0 0
\(861\) 10.3431 0.352493
\(862\) 0 0
\(863\) −20.7696 −0.707004 −0.353502 0.935434i \(-0.615009\pi\)
−0.353502 + 0.935434i \(0.615009\pi\)
\(864\) 0 0
\(865\) −67.9411 −2.31007
\(866\) 0 0
\(867\) −21.9706 −0.746159
\(868\) 0 0
\(869\) 7.07107 0.239870
\(870\) 0 0
\(871\) 4.24264 0.143756
\(872\) 0 0
\(873\) −12.8284 −0.434176
\(874\) 0 0
\(875\) 16.0000 0.540899
\(876\) 0 0
\(877\) 27.6569 0.933906 0.466953 0.884282i \(-0.345352\pi\)
0.466953 + 0.884282i \(0.345352\pi\)
\(878\) 0 0
\(879\) 1.51472 0.0510902
\(880\) 0 0
\(881\) 13.0294 0.438973 0.219486 0.975616i \(-0.429562\pi\)
0.219486 + 0.975616i \(0.429562\pi\)
\(882\) 0 0
\(883\) 34.8284 1.17207 0.586035 0.810286i \(-0.300688\pi\)
0.586035 + 0.810286i \(0.300688\pi\)
\(884\) 0 0
\(885\) −9.65685 −0.324612
\(886\) 0 0
\(887\) −8.48528 −0.284908 −0.142454 0.989801i \(-0.545499\pi\)
−0.142454 + 0.989801i \(0.545499\pi\)
\(888\) 0 0
\(889\) −20.0000 −0.670778
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 16.4853 0.551042
\(896\) 0 0
\(897\) 4.82843 0.161216
\(898\) 0 0
\(899\) −3.17157 −0.105778
\(900\) 0 0
\(901\) −58.1421 −1.93700
\(902\) 0 0
\(903\) 0.686292 0.0228384
\(904\) 0 0
\(905\) −85.2548 −2.83397
\(906\) 0 0
\(907\) −11.3137 −0.375666 −0.187833 0.982201i \(-0.560146\pi\)
−0.187833 + 0.982201i \(0.560146\pi\)
\(908\) 0 0
\(909\) −18.7279 −0.621166
\(910\) 0 0
\(911\) −44.2843 −1.46720 −0.733602 0.679580i \(-0.762162\pi\)
−0.733602 + 0.679580i \(0.762162\pi\)
\(912\) 0 0
\(913\) 17.6569 0.584357
\(914\) 0 0
\(915\) −10.8284 −0.357977
\(916\) 0 0
\(917\) 22.6274 0.747223
\(918\) 0 0
\(919\) 41.4142 1.36613 0.683064 0.730358i \(-0.260647\pi\)
0.683064 + 0.730358i \(0.260647\pi\)
\(920\) 0 0
\(921\) 28.0000 0.922631
\(922\) 0 0
\(923\) −10.8284 −0.356422
\(924\) 0 0
\(925\) 5.51472 0.181323
\(926\) 0 0
\(927\) 2.82843 0.0928977
\(928\) 0 0
\(929\) −18.9289 −0.621038 −0.310519 0.950567i \(-0.600503\pi\)
−0.310519 + 0.950567i \(0.600503\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −24.8284 −0.812846
\(934\) 0 0
\(935\) 21.3137 0.697033
\(936\) 0 0
\(937\) 22.2843 0.727995 0.363998 0.931400i \(-0.381412\pi\)
0.363998 + 0.931400i \(0.381412\pi\)
\(938\) 0 0
\(939\) 22.6274 0.738418
\(940\) 0 0
\(941\) 30.0000 0.977972 0.488986 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) 0 0
\(943\) 17.6569 0.574986
\(944\) 0 0
\(945\) −9.65685 −0.314137
\(946\) 0 0
\(947\) 10.6274 0.345345 0.172672 0.984979i \(-0.444760\pi\)
0.172672 + 0.984979i \(0.444760\pi\)
\(948\) 0 0
\(949\) 5.65685 0.183629
\(950\) 0 0
\(951\) 6.92893 0.224686
\(952\) 0 0
\(953\) 35.4142 1.14718 0.573589 0.819143i \(-0.305550\pi\)
0.573589 + 0.819143i \(0.305550\pi\)
\(954\) 0 0
\(955\) −8.00000 −0.258874
\(956\) 0 0
\(957\) 0.585786 0.0189358
\(958\) 0 0
\(959\) −36.9706 −1.19384
\(960\) 0 0
\(961\) −1.68629 −0.0543965
\(962\) 0 0
\(963\) −1.17157 −0.0377534
\(964\) 0 0
\(965\) 19.3137 0.621730
\(966\) 0 0
\(967\) −46.4264 −1.49297 −0.746486 0.665401i \(-0.768261\pi\)
−0.746486 + 0.665401i \(0.768261\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 31.1716 1.00034 0.500172 0.865926i \(-0.333270\pi\)
0.500172 + 0.865926i \(0.333270\pi\)
\(972\) 0 0
\(973\) 23.3137 0.747403
\(974\) 0 0
\(975\) 6.65685 0.213190
\(976\) 0 0
\(977\) 13.5563 0.433706 0.216853 0.976204i \(-0.430421\pi\)
0.216853 + 0.976204i \(0.430421\pi\)
\(978\) 0 0
\(979\) −11.8995 −0.380310
\(980\) 0 0
\(981\) −15.3137 −0.488929
\(982\) 0 0
\(983\) −27.5980 −0.880239 −0.440119 0.897939i \(-0.645064\pi\)
−0.440119 + 0.897939i \(0.645064\pi\)
\(984\) 0 0
\(985\) 41.4558 1.32089
\(986\) 0 0
\(987\) 22.6274 0.720239
\(988\) 0 0
\(989\) 1.17157 0.0372539
\(990\) 0 0
\(991\) 25.4558 0.808632 0.404316 0.914619i \(-0.367510\pi\)
0.404316 + 0.914619i \(0.367510\pi\)
\(992\) 0 0
\(993\) 31.5563 1.00141
\(994\) 0 0
\(995\) −15.3137 −0.485477
\(996\) 0 0
\(997\) −3.85786 −0.122180 −0.0610899 0.998132i \(-0.519458\pi\)
−0.0610899 + 0.998132i \(0.519458\pi\)
\(998\) 0 0
\(999\) −0.828427 −0.0262103
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 6864.2.a.bb.1.1 2
4.3 odd 2 1716.2.a.e.1.1 2
12.11 even 2 5148.2.a.j.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1716.2.a.e.1.1 2 4.3 odd 2
5148.2.a.j.1.2 2 12.11 even 2
6864.2.a.bb.1.1 2 1.1 even 1 trivial