Properties

Label 1716.2.a.e.1.1
Level $1716$
Weight $2$
Character 1716.1
Self dual yes
Analytic conductor $13.702$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1716 = 2^{2} \cdot 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1716.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.7023289869\)
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: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 1716.1

$q$-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 \(-0.320491\pi\)
0.534522 + 0.845154i \(0.320491\pi\)
\(8\) 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 \(-0.667917\pi\)
−0.503398 + 0.864054i \(0.667917\pi\)
\(24\) 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 \(-0.661621\pi\)
−0.486211 + 0.873842i \(0.661621\pi\)
\(32\) 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 \(-0.505889\pi\)
−0.0185012 + 0.999829i \(0.505889\pi\)
\(44\) 0 0
\(45\) −3.41421 −0.508961
\(46\) 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
\(51\) −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 \(-0.441058\pi\)
0.184115 + 0.982905i \(0.441058\pi\)
\(60\) 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 \(-0.416554\pi\)
0.259161 + 0.965834i \(0.416554\pi\)
\(68\) 0 0
\(69\) −4.82843 −0.581274
\(70\) 0 0
\(71\) −10.8284 −1.28510 −0.642549 0.766245i \(-0.722123\pi\)
−0.642549 + 0.766245i \(0.722123\pi\)
\(72\) 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 \(-0.630219\pi\)
−0.397779 + 0.917481i \(0.630219\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −17.6569 −1.93809 −0.969046 0.246881i \(-0.920594\pi\)
−0.969046 + 0.246881i \(0.920594\pi\)
\(84\) 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 \(-0.544500\pi\)
−0.139347 + 0.990244i \(0.544500\pi\)
\(104\) 0 0
\(105\) −9.65685 −0.942412
\(106\) 0 0
\(107\) 1.17157 0.113260 0.0566301 0.998395i \(-0.481964\pi\)
0.0566301 + 0.998395i \(0.481964\pi\)
\(108\) 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 \(-0.601578\pi\)
−0.313728 + 0.949513i \(0.601578\pi\)
\(128\) 0 0
\(129\) −0.242641 −0.0213633
\(130\) 0 0
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) 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 \(-0.386329\pi\)
0.349566 + 0.936912i \(0.386329\pi\)
\(140\) 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 \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 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 \(-0.666102\pi\)
−0.498464 + 0.866910i \(0.666102\pi\)
\(164\) 0 0
\(165\) 3.41421 0.265796
\(166\) 0 0
\(167\) 11.3137 0.875481 0.437741 0.899101i \(-0.355779\pi\)
0.437741 + 0.899101i \(0.355779\pi\)
\(168\) 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 \(-0.442246\pi\)
0.180447 + 0.983585i \(0.442246\pi\)
\(180\) 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 \(-0.527016\pi\)
−0.0847720 + 0.996400i \(0.527016\pi\)
\(192\) 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 \(-0.550819\pi\)
−0.158977 + 0.987282i \(0.550819\pi\)
\(200\) 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 \(-0.381282\pi\)
0.364378 + 0.931251i \(0.381282\pi\)
\(212\) 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 \(-0.761997\pi\)
−0.733249 + 0.679960i \(0.761997\pi\)
\(224\) 0 0
\(225\) 6.65685 0.443790
\(226\) 0 0
\(227\) 12.1421 0.805902 0.402951 0.915222i \(-0.367985\pi\)
0.402951 + 0.915222i \(0.367985\pi\)
\(228\) 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 \(-0.274164\pi\)
0.651443 + 0.758697i \(0.274164\pi\)
\(240\) 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 \(-0.199617\pi\)
0.809723 + 0.586812i \(0.199617\pi\)
\(252\) 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 \(-0.415746\pi\)
0.261612 + 0.965173i \(0.415746\pi\)
\(264\) 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 \(-0.483975\pi\)
0.0503234 + 0.998733i \(0.483975\pi\)
\(272\) 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 \(-0.202378\pi\)
0.804604 + 0.593812i \(0.202378\pi\)
\(284\) 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 \(-0.205351\pi\)
0.799022 + 0.601302i \(0.205351\pi\)
\(308\) 0 0
\(309\) −2.82843 −0.160904
\(310\) 0 0
\(311\) −24.8284 −1.40789 −0.703945 0.710254i \(-0.748580\pi\)
−0.703945 + 0.710254i \(0.748580\pi\)
\(312\) 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 \(-0.165887\pi\)
0.867247 + 0.497878i \(0.165887\pi\)
\(332\) 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 \(-0.741803\pi\)
−0.688666 + 0.725079i \(0.741803\pi\)
\(348\) 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 \(-0.640069\pi\)
−0.425975 + 0.904735i \(0.640069\pi\)
\(360\) 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 \(-0.427203\pi\)
0.226710 + 0.973962i \(0.427203\pi\)
\(368\) 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 \(-0.223847\pi\)
0.762754 + 0.646688i \(0.223847\pi\)
\(380\) 0 0
\(381\) −7.07107 −0.362262
\(382\) 0 0
\(383\) −19.7990 −1.01168 −0.505841 0.862627i \(-0.668818\pi\)
−0.505841 + 0.862627i \(0.668818\pi\)
\(384\) 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 \(-0.251315\pi\)
0.704180 + 0.710021i \(0.251315\pi\)
\(420\) 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 \(-0.606683\pi\)
−0.328914 + 0.944360i \(0.606683\pi\)
\(432\) 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 \(-0.427874\pi\)
0.224658 + 0.974438i \(0.427874\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −23.3137 −1.10767 −0.553834 0.832627i \(-0.686836\pi\)
−0.553834 + 0.832627i \(0.686836\pi\)
\(444\) 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 \(-0.402870\pi\)
0.300429 + 0.953804i \(0.402870\pi\)
\(464\) 0 0
\(465\) 18.4853 0.857234
\(466\) 0 0
\(467\) −23.4558 −1.08541 −0.542704 0.839924i \(-0.682599\pi\)
−0.542704 + 0.839924i \(0.682599\pi\)
\(468\) 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 \(-0.314040\pi\)
0.551541 + 0.834148i \(0.314040\pi\)
\(480\) 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 \(-0.584327\pi\)
−0.261834 + 0.965113i \(0.584327\pi\)
\(488\) 0 0
\(489\) −12.7279 −0.575577
\(490\) 0 0
\(491\) 20.9706 0.946388 0.473194 0.880958i \(-0.343101\pi\)
0.473194 + 0.880958i \(0.343101\pi\)
\(492\) 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 \(-0.428878\pi\)
0.221581 + 0.975142i \(0.428878\pi\)
\(500\) 0 0
\(501\) 11.3137 0.505459
\(502\) 0 0
\(503\) −24.4853 −1.09174 −0.545872 0.837868i \(-0.683802\pi\)
−0.545872 + 0.837868i \(0.683802\pi\)
\(504\) 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 \(-0.514623\pi\)
−0.0459243 + 0.998945i \(0.514623\pi\)
\(524\) 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 \(-0.311573\pi\)
0.557989 + 0.829849i \(0.311573\pi\)
\(548\) 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 \(-0.511116\pi\)
−0.0349140 + 0.999390i \(0.511116\pi\)
\(564\) 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 \(-0.699594\pi\)
−0.586753 + 0.809766i \(0.699594\pi\)
\(572\) 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 \(-0.727855\pi\)
−0.656242 + 0.754550i \(0.727855\pi\)
\(588\) 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 \(-0.590004\pi\)
−0.279002 + 0.960291i \(0.590004\pi\)
\(600\) 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 \(-0.413586\pi\)
0.268154 + 0.963376i \(0.413586\pi\)
\(608\) 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 \(-0.439412\pi\)
0.189195 + 0.981940i \(0.439412\pi\)
\(620\) 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 \(-0.827870\pi\)
−0.857316 + 0.514790i \(0.827870\pi\)
\(632\) 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 \(-0.547604\pi\)
−0.148997 + 0.988838i \(0.547604\pi\)
\(644\) 0 0
\(645\) 0.828427 0.0326193
\(646\) 0 0
\(647\) 29.6569 1.16593 0.582966 0.812497i \(-0.301892\pi\)
0.582966 + 0.812497i \(0.301892\pi\)
\(648\) 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 \(-0.707223\pi\)
−0.605989 + 0.795473i \(0.707223\pi\)
\(660\) 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 \(-0.458296\pi\)
0.130641 + 0.991430i \(0.458296\pi\)
\(684\) 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 \(-0.551345\pi\)
−0.160606 + 0.987019i \(0.551345\pi\)
\(692\) 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 \(-0.728804\pi\)
−0.658490 + 0.752590i \(0.728804\pi\)
\(720\) 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 \(-0.603155\pi\)
−0.318429 + 0.947947i \(0.603155\pi\)
\(728\) 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 \(-0.446046\pi\)
0.168691 + 0.985669i \(0.446046\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 25.9411 0.951688 0.475844 0.879530i \(-0.342143\pi\)
0.475844 + 0.879530i \(0.342143\pi\)
\(744\) 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 \(-0.614629\pi\)
−0.352384 + 0.935856i \(0.614629\pi\)
\(752\) 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 \(-0.590861\pi\)
−0.281587 + 0.959536i \(0.590861\pi\)
\(788\) 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 \(-0.407704\pi\)
0.285909 + 0.958257i \(0.407704\pi\)
\(812\) 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 \(-0.491922\pi\)
0.0253738 + 0.999678i \(0.491922\pi\)
\(824\) 0 0
\(825\) −6.65685 −0.231762
\(826\) 0 0
\(827\) 42.7696 1.48724 0.743622 0.668601i \(-0.233106\pi\)
0.743622 + 0.668601i \(0.233106\pi\)
\(828\) 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 \(-0.755435\pi\)
−0.719077 + 0.694930i \(0.755435\pi\)
\(840\) 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 \(-0.627334\pi\)
−0.389448 + 0.921048i \(0.627334\pi\)
\(860\) 0 0
\(861\) 10.3431 0.352493
\(862\) 0 0
\(863\) 20.7696 0.707004 0.353502 0.935434i \(-0.384991\pi\)
0.353502 + 0.935434i \(0.384991\pi\)
\(864\) 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 \(-0.699312\pi\)
−0.586035 + 0.810286i \(0.699312\pi\)
\(884\) 0 0
\(885\) −9.65685 −0.324612
\(886\) 0 0
\(887\) 8.48528 0.284908 0.142454 0.989801i \(-0.454501\pi\)
0.142454 + 0.989801i \(0.454501\pi\)
\(888\) 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 \(-0.439854\pi\)
0.187833 + 0.982201i \(0.439854\pi\)
\(908\) 0 0
\(909\) −18.7279 −0.621166
\(910\) 0 0
\(911\) 44.2843 1.46720 0.733602 0.679580i \(-0.237838\pi\)
0.733602 + 0.679580i \(0.237838\pi\)
\(912\) 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 \(-0.739353\pi\)
−0.683064 + 0.730358i \(0.739353\pi\)
\(920\) 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 \(-0.555240\pi\)
−0.172672 + 0.984979i \(0.555240\pi\)
\(948\) 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 \(-0.231739\pi\)
0.746486 + 0.665401i \(0.231739\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −31.1716 −1.00034 −0.500172 0.865926i \(-0.666730\pi\)
−0.500172 + 0.865926i \(0.666730\pi\)
\(972\) 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 \(-0.354936\pi\)
0.440119 + 0.897939i \(0.354936\pi\)
\(984\) 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 \(-0.632490\pi\)
−0.404316 + 0.914619i \(0.632490\pi\)
\(992\) 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 1716.2.a.e.1.1 2
3.2 odd 2 5148.2.a.j.1.2 2
4.3 odd 2 6864.2.a.bb.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1716.2.a.e.1.1 2 1.1 even 1 trivial
5148.2.a.j.1.2 2 3.2 odd 2
6864.2.a.bb.1.1 2 4.3 odd 2