Properties

Label 1716.2.a.e.1.2
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 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 1716.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -0.585786 q^{5} -2.82843 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -0.585786 q^{5} -2.82843 q^{7} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{13} -0.585786 q^{15} +2.24264 q^{17} -2.82843 q^{21} +0.828427 q^{23} -4.65685 q^{25} +1.00000 q^{27} -3.41421 q^{29} -2.58579 q^{31} -1.00000 q^{33} +1.65685 q^{35} -4.82843 q^{37} -1.00000 q^{39} -7.65685 q^{41} +8.24264 q^{43} -0.585786 q^{45} -8.00000 q^{47} +1.00000 q^{49} +2.24264 q^{51} -13.3137 q^{53} +0.585786 q^{55} -2.82843 q^{59} -8.82843 q^{61} -2.82843 q^{63} +0.585786 q^{65} -4.24264 q^{67} +0.828427 q^{69} -5.17157 q^{71} +5.65685 q^{73} -4.65685 q^{75} +2.82843 q^{77} +7.07107 q^{79} +1.00000 q^{81} -6.34315 q^{83} -1.31371 q^{85} -3.41421 q^{87} +7.89949 q^{89} +2.82843 q^{91} -2.58579 q^{93} -7.17157 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 4q^{5} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 4q^{5} + 2q^{9} - 2q^{11} - 2q^{13} - 4q^{15} - 4q^{17} - 4q^{23} + 2q^{25} + 2q^{27} - 4q^{29} - 8q^{31} - 2q^{33} - 8q^{35} - 4q^{37} - 2q^{39} - 4q^{41} + 8q^{43} - 4q^{45} - 16q^{47} + 2q^{49} - 4q^{51} - 4q^{53} + 4q^{55} - 12q^{61} + 4q^{65} - 4q^{69} - 16q^{71} + 2q^{75} + 2q^{81} - 24q^{83} + 20q^{85} - 4q^{87} - 4q^{89} - 8q^{93} - 20q^{97} - 2q^{99} + O(q^{100}) \)

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
\(5\) −0.585786 −0.261972 −0.130986 0.991384i \(-0.541814\pi\)
−0.130986 + 0.991384i \(0.541814\pi\)
\(6\) 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
\(15\) −0.585786 −0.151249
\(16\) 0 0
\(17\) 2.24264 0.543920 0.271960 0.962309i \(-0.412328\pi\)
0.271960 + 0.962309i \(0.412328\pi\)
\(18\) 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
\(23\) 0.828427 0.172739 0.0863695 0.996263i \(-0.472473\pi\)
0.0863695 + 0.996263i \(0.472473\pi\)
\(24\) 0 0
\(25\) −4.65685 −0.931371
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −3.41421 −0.634004 −0.317002 0.948425i \(-0.602676\pi\)
−0.317002 + 0.948425i \(0.602676\pi\)
\(30\) 0 0
\(31\) −2.58579 −0.464421 −0.232210 0.972666i \(-0.574596\pi\)
−0.232210 + 0.972666i \(0.574596\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) 1.65685 0.280059
\(36\) 0 0
\(37\) −4.82843 −0.793789 −0.396894 0.917864i \(-0.629912\pi\)
−0.396894 + 0.917864i \(0.629912\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −7.65685 −1.19580 −0.597900 0.801571i \(-0.703998\pi\)
−0.597900 + 0.801571i \(0.703998\pi\)
\(42\) 0 0
\(43\) 8.24264 1.25699 0.628495 0.777813i \(-0.283671\pi\)
0.628495 + 0.777813i \(0.283671\pi\)
\(44\) 0 0
\(45\) −0.585786 −0.0873239
\(46\) 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
\(51\) 2.24264 0.314033
\(52\) 0 0
\(53\) −13.3137 −1.82878 −0.914389 0.404836i \(-0.867329\pi\)
−0.914389 + 0.404836i \(0.867329\pi\)
\(54\) 0 0
\(55\) 0.585786 0.0789874
\(56\) 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
\(61\) −8.82843 −1.13036 −0.565182 0.824966i \(-0.691194\pi\)
−0.565182 + 0.824966i \(0.691194\pi\)
\(62\) 0 0
\(63\) −2.82843 −0.356348
\(64\) 0 0
\(65\) 0.585786 0.0726579
\(66\) 0 0
\(67\) −4.24264 −0.518321 −0.259161 0.965834i \(-0.583446\pi\)
−0.259161 + 0.965834i \(0.583446\pi\)
\(68\) 0 0
\(69\) 0.828427 0.0997309
\(70\) 0 0
\(71\) −5.17157 −0.613753 −0.306876 0.951749i \(-0.599284\pi\)
−0.306876 + 0.951749i \(0.599284\pi\)
\(72\) 0 0
\(73\) 5.65685 0.662085 0.331042 0.943616i \(-0.392600\pi\)
0.331042 + 0.943616i \(0.392600\pi\)
\(74\) 0 0
\(75\) −4.65685 −0.537727
\(76\) 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
\(83\) −6.34315 −0.696251 −0.348125 0.937448i \(-0.613182\pi\)
−0.348125 + 0.937448i \(0.613182\pi\)
\(84\) 0 0
\(85\) −1.31371 −0.142492
\(86\) 0 0
\(87\) −3.41421 −0.366042
\(88\) 0 0
\(89\) 7.89949 0.837345 0.418672 0.908137i \(-0.362496\pi\)
0.418672 + 0.908137i \(0.362496\pi\)
\(90\) 0 0
\(91\) 2.82843 0.296500
\(92\) 0 0
\(93\) −2.58579 −0.268134
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −7.17157 −0.728163 −0.364081 0.931367i \(-0.618617\pi\)
−0.364081 + 0.931367i \(0.618617\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) 6.72792 0.669453 0.334727 0.942315i \(-0.391356\pi\)
0.334727 + 0.942315i \(0.391356\pi\)
\(102\) 0 0
\(103\) 2.82843 0.278693 0.139347 0.990244i \(-0.455500\pi\)
0.139347 + 0.990244i \(0.455500\pi\)
\(104\) 0 0
\(105\) 1.65685 0.161692
\(106\) 0 0
\(107\) 6.82843 0.660129 0.330064 0.943958i \(-0.392930\pi\)
0.330064 + 0.943958i \(0.392930\pi\)
\(108\) 0 0
\(109\) 7.31371 0.700526 0.350263 0.936651i \(-0.386092\pi\)
0.350263 + 0.936651i \(0.386092\pi\)
\(110\) 0 0
\(111\) −4.82843 −0.458294
\(112\) 0 0
\(113\) −14.9706 −1.40831 −0.704156 0.710045i \(-0.748674\pi\)
−0.704156 + 0.710045i \(0.748674\pi\)
\(114\) 0 0
\(115\) −0.485281 −0.0452527
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) −6.34315 −0.581475
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −7.65685 −0.690395
\(124\) 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
\(129\) 8.24264 0.725724
\(130\) 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
\(135\) −0.585786 −0.0504165
\(136\) 0 0
\(137\) −1.07107 −0.0915075 −0.0457537 0.998953i \(-0.514569\pi\)
−0.0457537 + 0.998953i \(0.514569\pi\)
\(138\) 0 0
\(139\) −0.242641 −0.0205805 −0.0102903 0.999947i \(-0.503276\pi\)
−0.0102903 + 0.999947i \(0.503276\pi\)
\(140\) 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
\(149\) −14.4853 −1.18668 −0.593340 0.804952i \(-0.702191\pi\)
−0.593340 + 0.804952i \(0.702191\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 0 0
\(153\) 2.24264 0.181307
\(154\) 0 0
\(155\) 1.51472 0.121665
\(156\) 0 0
\(157\) 17.6569 1.40917 0.704585 0.709619i \(-0.251133\pi\)
0.704585 + 0.709619i \(0.251133\pi\)
\(158\) 0 0
\(159\) −13.3137 −1.05585
\(160\) 0 0
\(161\) −2.34315 −0.184666
\(162\) 0 0
\(163\) 12.7279 0.996928 0.498464 0.866910i \(-0.333898\pi\)
0.498464 + 0.866910i \(0.333898\pi\)
\(164\) 0 0
\(165\) 0.585786 0.0456034
\(166\) 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
\(173\) 0.100505 0.00764126 0.00382063 0.999993i \(-0.498784\pi\)
0.00382063 + 0.999993i \(0.498784\pi\)
\(174\) 0 0
\(175\) 13.1716 0.995677
\(176\) 0 0
\(177\) −2.82843 −0.212598
\(178\) 0 0
\(179\) −0.828427 −0.0619196 −0.0309598 0.999521i \(-0.509856\pi\)
−0.0309598 + 0.999521i \(0.509856\pi\)
\(180\) 0 0
\(181\) −8.97056 −0.666777 −0.333388 0.942790i \(-0.608192\pi\)
−0.333388 + 0.942790i \(0.608192\pi\)
\(182\) 0 0
\(183\) −8.82843 −0.652616
\(184\) 0 0
\(185\) 2.82843 0.207950
\(186\) 0 0
\(187\) −2.24264 −0.163998
\(188\) 0 0
\(189\) −2.82843 −0.205738
\(190\) 0 0
\(191\) −13.6569 −0.988175 −0.494088 0.869412i \(-0.664498\pi\)
−0.494088 + 0.869412i \(0.664498\pi\)
\(192\) 0 0
\(193\) 5.65685 0.407189 0.203595 0.979055i \(-0.434738\pi\)
0.203595 + 0.979055i \(0.434738\pi\)
\(194\) 0 0
\(195\) 0.585786 0.0419490
\(196\) 0 0
\(197\) 16.1421 1.15008 0.575040 0.818125i \(-0.304987\pi\)
0.575040 + 0.818125i \(0.304987\pi\)
\(198\) 0 0
\(199\) 12.4853 0.885058 0.442529 0.896754i \(-0.354081\pi\)
0.442529 + 0.896754i \(0.354081\pi\)
\(200\) 0 0
\(201\) −4.24264 −0.299253
\(202\) 0 0
\(203\) 9.65685 0.677778
\(204\) 0 0
\(205\) 4.48528 0.313266
\(206\) 0 0
\(207\) 0.828427 0.0575797
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 13.4142 0.923473 0.461736 0.887017i \(-0.347227\pi\)
0.461736 + 0.887017i \(0.347227\pi\)
\(212\) 0 0
\(213\) −5.17157 −0.354350
\(214\) 0 0
\(215\) −4.82843 −0.329296
\(216\) 0 0
\(217\) 7.31371 0.496487
\(218\) 0 0
\(219\) 5.65685 0.382255
\(220\) 0 0
\(221\) −2.24264 −0.150856
\(222\) 0 0
\(223\) −2.10051 −0.140660 −0.0703301 0.997524i \(-0.522405\pi\)
−0.0703301 + 0.997524i \(0.522405\pi\)
\(224\) 0 0
\(225\) −4.65685 −0.310457
\(226\) 0 0
\(227\) −16.1421 −1.07139 −0.535696 0.844411i \(-0.679951\pi\)
−0.535696 + 0.844411i \(0.679951\pi\)
\(228\) 0 0
\(229\) −21.3137 −1.40845 −0.704225 0.709977i \(-0.748705\pi\)
−0.704225 + 0.709977i \(0.748705\pi\)
\(230\) 0 0
\(231\) 2.82843 0.186097
\(232\) 0 0
\(233\) 11.4142 0.747770 0.373885 0.927475i \(-0.378025\pi\)
0.373885 + 0.927475i \(0.378025\pi\)
\(234\) 0 0
\(235\) 4.68629 0.305700
\(236\) 0 0
\(237\) 7.07107 0.459315
\(238\) 0 0
\(239\) −8.14214 −0.526671 −0.263335 0.964704i \(-0.584823\pi\)
−0.263335 + 0.964704i \(0.584823\pi\)
\(240\) 0 0
\(241\) −11.6569 −0.750884 −0.375442 0.926846i \(-0.622509\pi\)
−0.375442 + 0.926846i \(0.622509\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −0.585786 −0.0374245
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −6.34315 −0.401981
\(250\) 0 0
\(251\) 14.3431 0.905331 0.452666 0.891680i \(-0.350473\pi\)
0.452666 + 0.891680i \(0.350473\pi\)
\(252\) 0 0
\(253\) −0.828427 −0.0520828
\(254\) 0 0
\(255\) −1.31371 −0.0822676
\(256\) 0 0
\(257\) 4.82843 0.301189 0.150595 0.988596i \(-0.451881\pi\)
0.150595 + 0.988596i \(0.451881\pi\)
\(258\) 0 0
\(259\) 13.6569 0.848596
\(260\) 0 0
\(261\) −3.41421 −0.211335
\(262\) 0 0
\(263\) −8.48528 −0.523225 −0.261612 0.965173i \(-0.584254\pi\)
−0.261612 + 0.965173i \(0.584254\pi\)
\(264\) 0 0
\(265\) 7.79899 0.479088
\(266\) 0 0
\(267\) 7.89949 0.483441
\(268\) 0 0
\(269\) −3.17157 −0.193374 −0.0966871 0.995315i \(-0.530825\pi\)
−0.0966871 + 0.995315i \(0.530825\pi\)
\(270\) 0 0
\(271\) −9.65685 −0.586612 −0.293306 0.956019i \(-0.594755\pi\)
−0.293306 + 0.956019i \(0.594755\pi\)
\(272\) 0 0
\(273\) 2.82843 0.171184
\(274\) 0 0
\(275\) 4.65685 0.280819
\(276\) 0 0
\(277\) −18.9706 −1.13983 −0.569915 0.821703i \(-0.693024\pi\)
−0.569915 + 0.821703i \(0.693024\pi\)
\(278\) 0 0
\(279\) −2.58579 −0.154807
\(280\) 0 0
\(281\) −12.3431 −0.736330 −0.368165 0.929760i \(-0.620014\pi\)
−0.368165 + 0.929760i \(0.620014\pi\)
\(282\) 0 0
\(283\) 12.9289 0.768545 0.384273 0.923220i \(-0.374452\pi\)
0.384273 + 0.923220i \(0.374452\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 21.6569 1.27836
\(288\) 0 0
\(289\) −11.9706 −0.704151
\(290\) 0 0
\(291\) −7.17157 −0.420405
\(292\) 0 0
\(293\) −18.4853 −1.07992 −0.539961 0.841690i \(-0.681561\pi\)
−0.539961 + 0.841690i \(0.681561\pi\)
\(294\) 0 0
\(295\) 1.65685 0.0964658
\(296\) 0 0
\(297\) −1.00000 −0.0580259
\(298\) 0 0
\(299\) −0.828427 −0.0479092
\(300\) 0 0
\(301\) −23.3137 −1.34378
\(302\) 0 0
\(303\) 6.72792 0.386509
\(304\) 0 0
\(305\) 5.17157 0.296123
\(306\) 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
\(311\) −19.1716 −1.08712 −0.543560 0.839370i \(-0.682924\pi\)
−0.543560 + 0.839370i \(0.682924\pi\)
\(312\) 0 0
\(313\) 22.6274 1.27898 0.639489 0.768801i \(-0.279146\pi\)
0.639489 + 0.768801i \(0.279146\pi\)
\(314\) 0 0
\(315\) 1.65685 0.0933532
\(316\) 0 0
\(317\) −21.0711 −1.18347 −0.591735 0.806133i \(-0.701557\pi\)
−0.591735 + 0.806133i \(0.701557\pi\)
\(318\) 0 0
\(319\) 3.41421 0.191159
\(320\) 0 0
\(321\) 6.82843 0.381126
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 4.65685 0.258316
\(326\) 0 0
\(327\) 7.31371 0.404449
\(328\) 0 0
\(329\) 22.6274 1.24749
\(330\) 0 0
\(331\) 0.443651 0.0243853 0.0121926 0.999926i \(-0.496119\pi\)
0.0121926 + 0.999926i \(0.496119\pi\)
\(332\) 0 0
\(333\) −4.82843 −0.264596
\(334\) 0 0
\(335\) 2.48528 0.135785
\(336\) 0 0
\(337\) 26.4853 1.44275 0.721373 0.692547i \(-0.243512\pi\)
0.721373 + 0.692547i \(0.243512\pi\)
\(338\) 0 0
\(339\) −14.9706 −0.813089
\(340\) 0 0
\(341\) 2.58579 0.140028
\(342\) 0 0
\(343\) 16.9706 0.916324
\(344\) 0 0
\(345\) −0.485281 −0.0261267
\(346\) 0 0
\(347\) −14.3431 −0.769980 −0.384990 0.922921i \(-0.625795\pi\)
−0.384990 + 0.922921i \(0.625795\pi\)
\(348\) 0 0
\(349\) 31.9411 1.70977 0.854885 0.518818i \(-0.173628\pi\)
0.854885 + 0.518818i \(0.173628\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) −10.7279 −0.570990 −0.285495 0.958380i \(-0.592158\pi\)
−0.285495 + 0.958380i \(0.592158\pi\)
\(354\) 0 0
\(355\) 3.02944 0.160786
\(356\) 0 0
\(357\) −6.34315 −0.335715
\(358\) 0 0
\(359\) 12.1421 0.640837 0.320419 0.947276i \(-0.396176\pi\)
0.320419 + 0.947276i \(0.396176\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) −3.31371 −0.173447
\(366\) 0 0
\(367\) 31.3137 1.63456 0.817281 0.576239i \(-0.195480\pi\)
0.817281 + 0.576239i \(0.195480\pi\)
\(368\) 0 0
\(369\) −7.65685 −0.398600
\(370\) 0 0
\(371\) 37.6569 1.95505
\(372\) 0 0
\(373\) −31.4558 −1.62872 −0.814361 0.580359i \(-0.802912\pi\)
−0.814361 + 0.580359i \(0.802912\pi\)
\(374\) 0 0
\(375\) 5.65685 0.292119
\(376\) 0 0
\(377\) 3.41421 0.175841
\(378\) 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
\(385\) −1.65685 −0.0844411
\(386\) 0 0
\(387\) 8.24264 0.418997
\(388\) 0 0
\(389\) 21.7990 1.10525 0.552626 0.833429i \(-0.313626\pi\)
0.552626 + 0.833429i \(0.313626\pi\)
\(390\) 0 0
\(391\) 1.85786 0.0939562
\(392\) 0 0
\(393\) 8.00000 0.403547
\(394\) 0 0
\(395\) −4.14214 −0.208413
\(396\) 0 0
\(397\) 1.79899 0.0902887 0.0451444 0.998980i \(-0.485625\pi\)
0.0451444 + 0.998980i \(0.485625\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 23.2132 1.15921 0.579606 0.814897i \(-0.303206\pi\)
0.579606 + 0.814897i \(0.303206\pi\)
\(402\) 0 0
\(403\) 2.58579 0.128807
\(404\) 0 0
\(405\) −0.585786 −0.0291080
\(406\) 0 0
\(407\) 4.82843 0.239336
\(408\) 0 0
\(409\) 13.6569 0.675288 0.337644 0.941274i \(-0.390370\pi\)
0.337644 + 0.941274i \(0.390370\pi\)
\(410\) 0 0
\(411\) −1.07107 −0.0528319
\(412\) 0 0
\(413\) 8.00000 0.393654
\(414\) 0 0
\(415\) 3.71573 0.182398
\(416\) 0 0
\(417\) −0.242641 −0.0118822
\(418\) 0 0
\(419\) 23.1716 1.13201 0.566003 0.824403i \(-0.308489\pi\)
0.566003 + 0.824403i \(0.308489\pi\)
\(420\) 0 0
\(421\) −14.6863 −0.715766 −0.357883 0.933766i \(-0.616501\pi\)
−0.357883 + 0.933766i \(0.616501\pi\)
\(422\) 0 0
\(423\) −8.00000 −0.388973
\(424\) 0 0
\(425\) −10.4437 −0.506591
\(426\) 0 0
\(427\) 24.9706 1.20841
\(428\) 0 0
\(429\) 1.00000 0.0482805
\(430\) 0 0
\(431\) −2.34315 −0.112865 −0.0564327 0.998406i \(-0.517973\pi\)
−0.0564327 + 0.998406i \(0.517973\pi\)
\(432\) 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
\(439\) 6.58579 0.314322 0.157161 0.987573i \(-0.449766\pi\)
0.157161 + 0.987573i \(0.449766\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −0.686292 −0.0326067 −0.0163033 0.999867i \(-0.505190\pi\)
−0.0163033 + 0.999867i \(0.505190\pi\)
\(444\) 0 0
\(445\) −4.62742 −0.219361
\(446\) 0 0
\(447\) −14.4853 −0.685130
\(448\) 0 0
\(449\) 11.4142 0.538670 0.269335 0.963047i \(-0.413196\pi\)
0.269335 + 0.963047i \(0.413196\pi\)
\(450\) 0 0
\(451\) 7.65685 0.360547
\(452\) 0 0
\(453\) 12.0000 0.563809
\(454\) 0 0
\(455\) −1.65685 −0.0776745
\(456\) 0 0
\(457\) 2.68629 0.125659 0.0628297 0.998024i \(-0.479988\pi\)
0.0628297 + 0.998024i \(0.479988\pi\)
\(458\) 0 0
\(459\) 2.24264 0.104678
\(460\) 0 0
\(461\) −22.9706 −1.06985 −0.534923 0.844901i \(-0.679659\pi\)
−0.534923 + 0.844901i \(0.679659\pi\)
\(462\) 0 0
\(463\) 27.0711 1.25810 0.629050 0.777365i \(-0.283444\pi\)
0.629050 + 0.777365i \(0.283444\pi\)
\(464\) 0 0
\(465\) 1.51472 0.0702434
\(466\) 0 0
\(467\) 27.4558 1.27050 0.635252 0.772305i \(-0.280896\pi\)
0.635252 + 0.772305i \(0.280896\pi\)
\(468\) 0 0
\(469\) 12.0000 0.554109
\(470\) 0 0
\(471\) 17.6569 0.813585
\(472\) 0 0
\(473\) −8.24264 −0.378997
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −13.3137 −0.609593
\(478\) 0 0
\(479\) −4.14214 −0.189259 −0.0946295 0.995513i \(-0.530167\pi\)
−0.0946295 + 0.995513i \(0.530167\pi\)
\(480\) 0 0
\(481\) 4.82843 0.220157
\(482\) 0 0
\(483\) −2.34315 −0.106617
\(484\) 0 0
\(485\) 4.20101 0.190758
\(486\) 0 0
\(487\) 19.5563 0.886183 0.443091 0.896476i \(-0.353882\pi\)
0.443091 + 0.896476i \(0.353882\pi\)
\(488\) 0 0
\(489\) 12.7279 0.575577
\(490\) 0 0
\(491\) −12.9706 −0.585353 −0.292677 0.956211i \(-0.594546\pi\)
−0.292677 + 0.956211i \(0.594546\pi\)
\(492\) 0 0
\(493\) −7.65685 −0.344847
\(494\) 0 0
\(495\) 0.585786 0.0263291
\(496\) 0 0
\(497\) 14.6274 0.656129
\(498\) 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
\(503\) −7.51472 −0.335065 −0.167532 0.985867i \(-0.553580\pi\)
−0.167532 + 0.985867i \(0.553580\pi\)
\(504\) 0 0
\(505\) −3.94113 −0.175378
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 0 0
\(509\) −21.5563 −0.955468 −0.477734 0.878504i \(-0.658542\pi\)
−0.477734 + 0.878504i \(0.658542\pi\)
\(510\) 0 0
\(511\) −16.0000 −0.707798
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1.65685 −0.0730097
\(516\) 0 0
\(517\) 8.00000 0.351840
\(518\) 0 0
\(519\) 0.100505 0.00441168
\(520\) 0 0
\(521\) −18.9706 −0.831115 −0.415558 0.909567i \(-0.636414\pi\)
−0.415558 + 0.909567i \(0.636414\pi\)
\(522\) 0 0
\(523\) −21.8995 −0.957598 −0.478799 0.877925i \(-0.658928\pi\)
−0.478799 + 0.877925i \(0.658928\pi\)
\(524\) 0 0
\(525\) 13.1716 0.574855
\(526\) 0 0
\(527\) −5.79899 −0.252608
\(528\) 0 0
\(529\) −22.3137 −0.970161
\(530\) 0 0
\(531\) −2.82843 −0.122743
\(532\) 0 0
\(533\) 7.65685 0.331655
\(534\) 0 0
\(535\) −4.00000 −0.172935
\(536\) 0 0
\(537\) −0.828427 −0.0357493
\(538\) 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
\(543\) −8.97056 −0.384964
\(544\) 0 0
\(545\) −4.28427 −0.183518
\(546\) 0 0
\(547\) 45.8995 1.96252 0.981260 0.192687i \(-0.0617201\pi\)
0.981260 + 0.192687i \(0.0617201\pi\)
\(548\) 0 0
\(549\) −8.82843 −0.376788
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −20.0000 −0.850487
\(554\) 0 0
\(555\) 2.82843 0.120060
\(556\) 0 0
\(557\) 9.51472 0.403152 0.201576 0.979473i \(-0.435394\pi\)
0.201576 + 0.979473i \(0.435394\pi\)
\(558\) 0 0
\(559\) −8.24264 −0.348627
\(560\) 0 0
\(561\) −2.24264 −0.0946844
\(562\) 0 0
\(563\) 9.65685 0.406988 0.203494 0.979076i \(-0.434770\pi\)
0.203494 + 0.979076i \(0.434770\pi\)
\(564\) 0 0
\(565\) 8.76955 0.368938
\(566\) 0 0
\(567\) −2.82843 −0.118783
\(568\) 0 0
\(569\) −17.5563 −0.736000 −0.368000 0.929826i \(-0.619957\pi\)
−0.368000 + 0.929826i \(0.619957\pi\)
\(570\) 0 0
\(571\) 20.0416 0.838716 0.419358 0.907821i \(-0.362255\pi\)
0.419358 + 0.907821i \(0.362255\pi\)
\(572\) 0 0
\(573\) −13.6569 −0.570523
\(574\) 0 0
\(575\) −3.85786 −0.160884
\(576\) 0 0
\(577\) −6.97056 −0.290188 −0.145094 0.989418i \(-0.546349\pi\)
−0.145094 + 0.989418i \(0.546349\pi\)
\(578\) 0 0
\(579\) 5.65685 0.235091
\(580\) 0 0
\(581\) 17.9411 0.744323
\(582\) 0 0
\(583\) 13.3137 0.551397
\(584\) 0 0
\(585\) 0.585786 0.0242193
\(586\) 0 0
\(587\) 7.79899 0.321899 0.160949 0.986963i \(-0.448544\pi\)
0.160949 + 0.986963i \(0.448544\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 16.1421 0.663999
\(592\) 0 0
\(593\) −24.8284 −1.01958 −0.509791 0.860298i \(-0.670277\pi\)
−0.509791 + 0.860298i \(0.670277\pi\)
\(594\) 0 0
\(595\) 3.71573 0.152330
\(596\) 0 0
\(597\) 12.4853 0.510989
\(598\) 0 0
\(599\) −2.34315 −0.0957383 −0.0478692 0.998854i \(-0.515243\pi\)
−0.0478692 + 0.998854i \(0.515243\pi\)
\(600\) 0 0
\(601\) −47.2548 −1.92756 −0.963782 0.266690i \(-0.914070\pi\)
−0.963782 + 0.266690i \(0.914070\pi\)
\(602\) 0 0
\(603\) −4.24264 −0.172774
\(604\) 0 0
\(605\) −0.585786 −0.0238156
\(606\) 0 0
\(607\) −29.2132 −1.18573 −0.592864 0.805303i \(-0.702003\pi\)
−0.592864 + 0.805303i \(0.702003\pi\)
\(608\) 0 0
\(609\) 9.65685 0.391315
\(610\) 0 0
\(611\) 8.00000 0.323645
\(612\) 0 0
\(613\) 29.9411 1.20931 0.604655 0.796487i \(-0.293311\pi\)
0.604655 + 0.796487i \(0.293311\pi\)
\(614\) 0 0
\(615\) 4.48528 0.180864
\(616\) 0 0
\(617\) 41.0711 1.65346 0.826729 0.562600i \(-0.190199\pi\)
0.826729 + 0.562600i \(0.190199\pi\)
\(618\) 0 0
\(619\) 6.58579 0.264705 0.132353 0.991203i \(-0.457747\pi\)
0.132353 + 0.991203i \(0.457747\pi\)
\(620\) 0 0
\(621\) 0.828427 0.0332436
\(622\) 0 0
\(623\) −22.3431 −0.895159
\(624\) 0 0
\(625\) 19.9706 0.798823
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −10.8284 −0.431758
\(630\) 0 0
\(631\) −28.9289 −1.15164 −0.575821 0.817576i \(-0.695318\pi\)
−0.575821 + 0.817576i \(0.695318\pi\)
\(632\) 0 0
\(633\) 13.4142 0.533167
\(634\) 0 0
\(635\) −4.14214 −0.164376
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 0 0
\(639\) −5.17157 −0.204584
\(640\) 0 0
\(641\) −41.5980 −1.64302 −0.821511 0.570193i \(-0.806868\pi\)
−0.821511 + 0.570193i \(0.806868\pi\)
\(642\) 0 0
\(643\) 23.5563 0.928972 0.464486 0.885581i \(-0.346239\pi\)
0.464486 + 0.885581i \(0.346239\pi\)
\(644\) 0 0
\(645\) −4.82843 −0.190119
\(646\) 0 0
\(647\) 18.3431 0.721143 0.360572 0.932731i \(-0.382582\pi\)
0.360572 + 0.932731i \(0.382582\pi\)
\(648\) 0 0
\(649\) 2.82843 0.111025
\(650\) 0 0
\(651\) 7.31371 0.286647
\(652\) 0 0
\(653\) −23.1716 −0.906774 −0.453387 0.891314i \(-0.649784\pi\)
−0.453387 + 0.891314i \(0.649784\pi\)
\(654\) 0 0
\(655\) −4.68629 −0.183109
\(656\) 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
\(661\) 3.17157 0.123360 0.0616799 0.998096i \(-0.480354\pi\)
0.0616799 + 0.998096i \(0.480354\pi\)
\(662\) 0 0
\(663\) −2.24264 −0.0870969
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −2.82843 −0.109517
\(668\) 0 0
\(669\) −2.10051 −0.0812102
\(670\) 0 0
\(671\) 8.82843 0.340818
\(672\) 0 0
\(673\) −0.343146 −0.0132273 −0.00661365 0.999978i \(-0.502105\pi\)
−0.00661365 + 0.999978i \(0.502105\pi\)
\(674\) 0 0
\(675\) −4.65685 −0.179242
\(676\) 0 0
\(677\) −22.2426 −0.854854 −0.427427 0.904050i \(-0.640580\pi\)
−0.427427 + 0.904050i \(0.640580\pi\)
\(678\) 0 0
\(679\) 20.2843 0.778439
\(680\) 0 0
\(681\) −16.1421 −0.618568
\(682\) 0 0
\(683\) 1.17157 0.0448290 0.0224145 0.999749i \(-0.492865\pi\)
0.0224145 + 0.999749i \(0.492865\pi\)
\(684\) 0 0
\(685\) 0.627417 0.0239724
\(686\) 0 0
\(687\) −21.3137 −0.813169
\(688\) 0 0
\(689\) 13.3137 0.507212
\(690\) 0 0
\(691\) −39.5563 −1.50479 −0.752397 0.658710i \(-0.771103\pi\)
−0.752397 + 0.658710i \(0.771103\pi\)
\(692\) 0 0
\(693\) 2.82843 0.107443
\(694\) 0 0
\(695\) 0.142136 0.00539151
\(696\) 0 0
\(697\) −17.1716 −0.650420
\(698\) 0 0
\(699\) 11.4142 0.431725
\(700\) 0 0
\(701\) 21.7574 0.821764 0.410882 0.911689i \(-0.365221\pi\)
0.410882 + 0.911689i \(0.365221\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 4.68629 0.176496
\(706\) 0 0
\(707\) −19.0294 −0.715676
\(708\) 0 0
\(709\) −3.37258 −0.126660 −0.0633300 0.997993i \(-0.520172\pi\)
−0.0633300 + 0.997993i \(0.520172\pi\)
\(710\) 0 0
\(711\) 7.07107 0.265186
\(712\) 0 0
\(713\) −2.14214 −0.0802236
\(714\) 0 0
\(715\) −0.585786 −0.0219072
\(716\) 0 0
\(717\) −8.14214 −0.304074
\(718\) 0 0
\(719\) −12.6863 −0.473119 −0.236559 0.971617i \(-0.576020\pi\)
−0.236559 + 0.971617i \(0.576020\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 0 0
\(723\) −11.6569 −0.433523
\(724\) 0 0
\(725\) 15.8995 0.590492
\(726\) 0 0
\(727\) −22.8284 −0.846659 −0.423330 0.905976i \(-0.639139\pi\)
−0.423330 + 0.905976i \(0.639139\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 18.4853 0.683703
\(732\) 0 0
\(733\) −30.6274 −1.13125 −0.565625 0.824663i \(-0.691365\pi\)
−0.565625 + 0.824663i \(0.691365\pi\)
\(734\) 0 0
\(735\) −0.585786 −0.0216071
\(736\) 0 0
\(737\) 4.24264 0.156280
\(738\) 0 0
\(739\) 14.8284 0.545473 0.272736 0.962089i \(-0.412071\pi\)
0.272736 + 0.962089i \(0.412071\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −41.9411 −1.53867 −0.769335 0.638845i \(-0.779412\pi\)
−0.769335 + 0.638845i \(0.779412\pi\)
\(744\) 0 0
\(745\) 8.48528 0.310877
\(746\) 0 0
\(747\) −6.34315 −0.232084
\(748\) 0 0
\(749\) −19.3137 −0.705708
\(750\) 0 0
\(751\) 3.31371 0.120919 0.0604595 0.998171i \(-0.480743\pi\)
0.0604595 + 0.998171i \(0.480743\pi\)
\(752\) 0 0
\(753\) 14.3431 0.522693
\(754\) 0 0
\(755\) −7.02944 −0.255827
\(756\) 0 0
\(757\) 32.2843 1.17339 0.586696 0.809807i \(-0.300428\pi\)
0.586696 + 0.809807i \(0.300428\pi\)
\(758\) 0 0
\(759\) −0.828427 −0.0300700
\(760\) 0 0
\(761\) −24.8284 −0.900030 −0.450015 0.893021i \(-0.648581\pi\)
−0.450015 + 0.893021i \(0.648581\pi\)
\(762\) 0 0
\(763\) −20.6863 −0.748894
\(764\) 0 0
\(765\) −1.31371 −0.0474972
\(766\) 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
\(771\) 4.82843 0.173892
\(772\) 0 0
\(773\) −7.41421 −0.266671 −0.133335 0.991071i \(-0.542569\pi\)
−0.133335 + 0.991071i \(0.542569\pi\)
\(774\) 0 0
\(775\) 12.0416 0.432548
\(776\) 0 0
\(777\) 13.6569 0.489937
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 5.17157 0.185053
\(782\) 0 0
\(783\) −3.41421 −0.122014
\(784\) 0 0
\(785\) −10.3431 −0.369163
\(786\) 0 0
\(787\) 23.7990 0.848342 0.424171 0.905582i \(-0.360565\pi\)
0.424171 + 0.905582i \(0.360565\pi\)
\(788\) 0 0
\(789\) −8.48528 −0.302084
\(790\) 0 0
\(791\) 42.3431 1.50555
\(792\) 0 0
\(793\) 8.82843 0.313507
\(794\) 0 0
\(795\) 7.79899 0.276602
\(796\) 0 0
\(797\) 22.4853 0.796470 0.398235 0.917284i \(-0.369623\pi\)
0.398235 + 0.917284i \(0.369623\pi\)
\(798\) 0 0
\(799\) −17.9411 −0.634711
\(800\) 0 0
\(801\) 7.89949 0.279115
\(802\) 0 0
\(803\) −5.65685 −0.199626
\(804\) 0 0
\(805\) 1.37258 0.0483772
\(806\) 0 0
\(807\) −3.17157 −0.111645
\(808\) 0 0
\(809\) 16.3848 0.576058 0.288029 0.957622i \(-0.407000\pi\)
0.288029 + 0.957622i \(0.407000\pi\)
\(810\) 0 0
\(811\) −40.2843 −1.41457 −0.707286 0.706927i \(-0.750081\pi\)
−0.707286 + 0.706927i \(0.750081\pi\)
\(812\) 0 0
\(813\) −9.65685 −0.338681
\(814\) 0 0
\(815\) −7.45584 −0.261167
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 2.82843 0.0988332
\(820\) 0 0
\(821\) 4.62742 0.161498 0.0807490 0.996734i \(-0.474269\pi\)
0.0807490 + 0.996734i \(0.474269\pi\)
\(822\) 0 0
\(823\) −49.4558 −1.72392 −0.861961 0.506974i \(-0.830764\pi\)
−0.861961 + 0.506974i \(0.830764\pi\)
\(824\) 0 0
\(825\) 4.65685 0.162131
\(826\) 0 0
\(827\) −30.7696 −1.06996 −0.534981 0.844864i \(-0.679681\pi\)
−0.534981 + 0.844864i \(0.679681\pi\)
\(828\) 0 0
\(829\) −18.6863 −0.649002 −0.324501 0.945885i \(-0.605196\pi\)
−0.324501 + 0.945885i \(0.605196\pi\)
\(830\) 0 0
\(831\) −18.9706 −0.658082
\(832\) 0 0
\(833\) 2.24264 0.0777029
\(834\) 0 0
\(835\) 6.62742 0.229351
\(836\) 0 0
\(837\) −2.58579 −0.0893779
\(838\) 0 0
\(839\) −30.3431 −1.04756 −0.523781 0.851853i \(-0.675479\pi\)
−0.523781 + 0.851853i \(0.675479\pi\)
\(840\) 0 0
\(841\) −17.3431 −0.598040
\(842\) 0 0
\(843\) −12.3431 −0.425121
\(844\) 0 0
\(845\) −0.585786 −0.0201517
\(846\) 0 0
\(847\) −2.82843 −0.0971859
\(848\) 0 0
\(849\) 12.9289 0.443720
\(850\) 0 0
\(851\) −4.00000 −0.137118
\(852\) 0 0
\(853\) 5.31371 0.181938 0.0909690 0.995854i \(-0.471004\pi\)
0.0909690 + 0.995854i \(0.471004\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 10.9289 0.373325 0.186663 0.982424i \(-0.440233\pi\)
0.186663 + 0.982424i \(0.440233\pi\)
\(858\) 0 0
\(859\) −17.1716 −0.585887 −0.292943 0.956130i \(-0.594635\pi\)
−0.292943 + 0.956130i \(0.594635\pi\)
\(860\) 0 0
\(861\) 21.6569 0.738064
\(862\) 0 0
\(863\) −52.7696 −1.79630 −0.898148 0.439693i \(-0.855087\pi\)
−0.898148 + 0.439693i \(0.855087\pi\)
\(864\) 0 0
\(865\) −0.0588745 −0.00200179
\(866\) 0 0
\(867\) −11.9706 −0.406542
\(868\) 0 0
\(869\) −7.07107 −0.239870
\(870\) 0 0
\(871\) 4.24264 0.143756
\(872\) 0 0
\(873\) −7.17157 −0.242721
\(874\) 0 0
\(875\) −16.0000 −0.540899
\(876\) 0 0
\(877\) 16.3431 0.551869 0.275934 0.961176i \(-0.411013\pi\)
0.275934 + 0.961176i \(0.411013\pi\)
\(878\) 0 0
\(879\) −18.4853 −0.623493
\(880\) 0 0
\(881\) 46.9706 1.58248 0.791239 0.611507i \(-0.209436\pi\)
0.791239 + 0.611507i \(0.209436\pi\)
\(882\) 0 0
\(883\) −29.1716 −0.981702 −0.490851 0.871244i \(-0.663314\pi\)
−0.490851 + 0.871244i \(0.663314\pi\)
\(884\) 0 0
\(885\) 1.65685 0.0556945
\(886\) 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
\(895\) 0.485281 0.0162212
\(896\) 0 0
\(897\) −0.828427 −0.0276604
\(898\) 0 0
\(899\) 8.82843 0.294445
\(900\) 0 0
\(901\) −29.8579 −0.994710
\(902\) 0 0
\(903\) −23.3137 −0.775832
\(904\) 0 0
\(905\) 5.25483 0.174677
\(906\) 0 0
\(907\) −11.3137 −0.375666 −0.187833 0.982201i \(-0.560146\pi\)
−0.187833 + 0.982201i \(0.560146\pi\)
\(908\) 0 0
\(909\) 6.72792 0.223151
\(910\) 0 0
\(911\) −12.2843 −0.406996 −0.203498 0.979075i \(-0.565231\pi\)
−0.203498 + 0.979075i \(0.565231\pi\)
\(912\) 0 0
\(913\) 6.34315 0.209927
\(914\) 0 0
\(915\) 5.17157 0.170967
\(916\) 0 0
\(917\) −22.6274 −0.747223
\(918\) 0 0
\(919\) −38.5858 −1.27283 −0.636414 0.771348i \(-0.719583\pi\)
−0.636414 + 0.771348i \(0.719583\pi\)
\(920\) 0 0
\(921\) 28.0000 0.922631
\(922\) 0 0
\(923\) 5.17157 0.170224
\(924\) 0 0
\(925\) 22.4853 0.739311
\(926\) 0 0
\(927\) 2.82843 0.0928977
\(928\) 0 0
\(929\) −33.0711 −1.08503 −0.542513 0.840047i \(-0.682527\pi\)
−0.542513 + 0.840047i \(0.682527\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −19.1716 −0.627649
\(934\) 0 0
\(935\) 1.31371 0.0429629
\(936\) 0 0
\(937\) −34.2843 −1.12002 −0.560009 0.828486i \(-0.689202\pi\)
−0.560009 + 0.828486i \(0.689202\pi\)
\(938\) 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
\(943\) −6.34315 −0.206561
\(944\) 0 0
\(945\) 1.65685 0.0538975
\(946\) 0 0
\(947\) 34.6274 1.12524 0.562620 0.826716i \(-0.309793\pi\)
0.562620 + 0.826716i \(0.309793\pi\)
\(948\) 0 0
\(949\) −5.65685 −0.183629
\(950\) 0 0
\(951\) −21.0711 −0.683276
\(952\) 0 0
\(953\) 32.5858 1.05556 0.527779 0.849382i \(-0.323025\pi\)
0.527779 + 0.849382i \(0.323025\pi\)
\(954\) 0 0
\(955\) 8.00000 0.258874
\(956\) 0 0
\(957\) 3.41421 0.110366
\(958\) 0 0
\(959\) 3.02944 0.0978256
\(960\) 0 0
\(961\) −24.3137 −0.784313
\(962\) 0 0
\(963\) 6.82843 0.220043
\(964\) 0 0
\(965\) −3.31371 −0.106672
\(966\) 0 0
\(967\) −38.4264 −1.23571 −0.617855 0.786292i \(-0.711998\pi\)
−0.617855 + 0.786292i \(0.711998\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −36.8284 −1.18188 −0.590940 0.806715i \(-0.701243\pi\)
−0.590940 + 0.806715i \(0.701243\pi\)
\(972\) 0 0
\(973\) 0.686292 0.0220015
\(974\) 0 0
\(975\) 4.65685 0.149139
\(976\) 0 0
\(977\) −17.5563 −0.561677 −0.280839 0.959755i \(-0.590613\pi\)
−0.280839 + 0.959755i \(0.590613\pi\)
\(978\) 0 0
\(979\) −7.89949 −0.252469
\(980\) 0 0
\(981\) 7.31371 0.233509
\(982\) 0 0
\(983\) −51.5980 −1.64572 −0.822860 0.568244i \(-0.807623\pi\)
−0.822860 + 0.568244i \(0.807623\pi\)
\(984\) 0 0
\(985\) −9.45584 −0.301288
\(986\) 0 0
\(987\) 22.6274 0.720239
\(988\) 0 0
\(989\) 6.82843 0.217131
\(990\) 0 0
\(991\) 25.4558 0.808632 0.404316 0.914619i \(-0.367510\pi\)
0.404316 + 0.914619i \(0.367510\pi\)
\(992\) 0 0
\(993\) 0.443651 0.0140788
\(994\) 0 0
\(995\) −7.31371 −0.231860
\(996\) 0 0
\(997\) −32.1421 −1.01795 −0.508976 0.860781i \(-0.669976\pi\)
−0.508976 + 0.860781i \(0.669976\pi\)
\(998\) 0 0
\(999\) −4.82843 −0.152765
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.1.2 2
3.2 odd 2 5148.2.a.j.1.1 2
4.3 odd 2 6864.2.a.bb.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1716.2.a.e.1.2 2 1.1 even 1 trivial
5148.2.a.j.1.1 2 3.2 odd 2
6864.2.a.bb.1.2 2 4.3 odd 2