Properties

Label 6048.2.a.bh.1.1
Level 6048
Weight 2
Character 6048.1
Self dual yes
Analytic conductor 48.294
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6048.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.2935231425\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\)
Character \(\chi\) = 6048.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.82843 q^{5} -1.00000 q^{7} +O(q^{10})\) \(q-1.82843 q^{5} -1.00000 q^{7} +4.82843 q^{11} +2.82843 q^{13} -0.171573 q^{17} -6.82843 q^{19} -4.00000 q^{23} -1.65685 q^{25} -2.82843 q^{29} +6.82843 q^{31} +1.82843 q^{35} +2.65685 q^{37} -3.82843 q^{41} -7.82843 q^{43} +8.65685 q^{47} +1.00000 q^{49} +2.00000 q^{53} -8.82843 q^{55} +0.656854 q^{59} +3.17157 q^{61} -5.17157 q^{65} -4.00000 q^{67} +1.65685 q^{71} -5.65685 q^{73} -4.82843 q^{77} +1.82843 q^{79} -5.34315 q^{83} +0.313708 q^{85} +6.00000 q^{89} -2.82843 q^{91} +12.4853 q^{95} -18.1421 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{5} - 2q^{7} + O(q^{10}) \) \( 2q + 2q^{5} - 2q^{7} + 4q^{11} - 6q^{17} - 8q^{19} - 8q^{23} + 8q^{25} + 8q^{31} - 2q^{35} - 6q^{37} - 2q^{41} - 10q^{43} + 6q^{47} + 2q^{49} + 4q^{53} - 12q^{55} - 10q^{59} + 12q^{61} - 16q^{65} - 8q^{67} - 8q^{71} - 4q^{77} - 2q^{79} - 22q^{83} - 22q^{85} + 12q^{89} + 8q^{95} - 8q^{97} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −1.82843 −0.817697 −0.408849 0.912602i \(-0.634070\pi\)
−0.408849 + 0.912602i \(0.634070\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.82843 1.45583 0.727913 0.685670i \(-0.240491\pi\)
0.727913 + 0.685670i \(0.240491\pi\)
\(12\) 0 0
\(13\) 2.82843 0.784465 0.392232 0.919866i \(-0.371703\pi\)
0.392232 + 0.919866i \(0.371703\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.171573 −0.0416125 −0.0208063 0.999784i \(-0.506623\pi\)
−0.0208063 + 0.999784i \(0.506623\pi\)
\(18\) 0 0
\(19\) −6.82843 −1.56655 −0.783274 0.621676i \(-0.786452\pi\)
−0.783274 + 0.621676i \(0.786452\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −1.65685 −0.331371
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.82843 −0.525226 −0.262613 0.964901i \(-0.584584\pi\)
−0.262613 + 0.964901i \(0.584584\pi\)
\(30\) 0 0
\(31\) 6.82843 1.22642 0.613211 0.789919i \(-0.289878\pi\)
0.613211 + 0.789919i \(0.289878\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.82843 0.309061
\(36\) 0 0
\(37\) 2.65685 0.436784 0.218392 0.975861i \(-0.429919\pi\)
0.218392 + 0.975861i \(0.429919\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.82843 −0.597900 −0.298950 0.954269i \(-0.596636\pi\)
−0.298950 + 0.954269i \(0.596636\pi\)
\(42\) 0 0
\(43\) −7.82843 −1.19382 −0.596912 0.802307i \(-0.703606\pi\)
−0.596912 + 0.802307i \(0.703606\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.65685 1.26273 0.631366 0.775485i \(-0.282495\pi\)
0.631366 + 0.775485i \(0.282495\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) −8.82843 −1.19042
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0.656854 0.0855151 0.0427576 0.999085i \(-0.486386\pi\)
0.0427576 + 0.999085i \(0.486386\pi\)
\(60\) 0 0
\(61\) 3.17157 0.406078 0.203039 0.979171i \(-0.434918\pi\)
0.203039 + 0.979171i \(0.434918\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −5.17157 −0.641455
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1.65685 0.196632 0.0983162 0.995155i \(-0.468654\pi\)
0.0983162 + 0.995155i \(0.468654\pi\)
\(72\) 0 0
\(73\) −5.65685 −0.662085 −0.331042 0.943616i \(-0.607400\pi\)
−0.331042 + 0.943616i \(0.607400\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.82843 −0.550250
\(78\) 0 0
\(79\) 1.82843 0.205714 0.102857 0.994696i \(-0.467202\pi\)
0.102857 + 0.994696i \(0.467202\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −5.34315 −0.586486 −0.293243 0.956038i \(-0.594735\pi\)
−0.293243 + 0.956038i \(0.594735\pi\)
\(84\) 0 0
\(85\) 0.313708 0.0340265
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −2.82843 −0.296500
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 12.4853 1.28096
\(96\) 0 0
\(97\) −18.1421 −1.84205 −0.921027 0.389498i \(-0.872649\pi\)
−0.921027 + 0.389498i \(0.872649\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 18.9706 1.88764 0.943821 0.330458i \(-0.107203\pi\)
0.943821 + 0.330458i \(0.107203\pi\)
\(102\) 0 0
\(103\) −2.34315 −0.230877 −0.115439 0.993315i \(-0.536827\pi\)
−0.115439 + 0.993315i \(0.536827\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −5.65685 −0.546869 −0.273434 0.961891i \(-0.588160\pi\)
−0.273434 + 0.961891i \(0.588160\pi\)
\(108\) 0 0
\(109\) −10.6569 −1.02074 −0.510371 0.859954i \(-0.670492\pi\)
−0.510371 + 0.859954i \(0.670492\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 7.17157 0.674645 0.337322 0.941389i \(-0.390479\pi\)
0.337322 + 0.941389i \(0.390479\pi\)
\(114\) 0 0
\(115\) 7.31371 0.682007
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.171573 0.0157281
\(120\) 0 0
\(121\) 12.3137 1.11943
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 12.1716 1.08866
\(126\) 0 0
\(127\) −12.1716 −1.08005 −0.540026 0.841648i \(-0.681586\pi\)
−0.540026 + 0.841648i \(0.681586\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) 6.82843 0.592100
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −14.8284 −1.26688 −0.633439 0.773793i \(-0.718357\pi\)
−0.633439 + 0.773793i \(0.718357\pi\)
\(138\) 0 0
\(139\) −11.1716 −0.947560 −0.473780 0.880643i \(-0.657111\pi\)
−0.473780 + 0.880643i \(0.657111\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 13.6569 1.14204
\(144\) 0 0
\(145\) 5.17157 0.429476
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 20.1421 1.65011 0.825054 0.565054i \(-0.191145\pi\)
0.825054 + 0.565054i \(0.191145\pi\)
\(150\) 0 0
\(151\) −17.8284 −1.45086 −0.725428 0.688298i \(-0.758358\pi\)
−0.725428 + 0.688298i \(0.758358\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −12.4853 −1.00284
\(156\) 0 0
\(157\) 2.82843 0.225733 0.112867 0.993610i \(-0.463997\pi\)
0.112867 + 0.993610i \(0.463997\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 4.00000 0.315244
\(162\) 0 0
\(163\) 1.48528 0.116336 0.0581681 0.998307i \(-0.481474\pi\)
0.0581681 + 0.998307i \(0.481474\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −17.9706 −1.39060 −0.695302 0.718718i \(-0.744729\pi\)
−0.695302 + 0.718718i \(0.744729\pi\)
\(168\) 0 0
\(169\) −5.00000 −0.384615
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2.00000 0.152057 0.0760286 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) 0 0
\(175\) 1.65685 0.125246
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −4.14214 −0.309598 −0.154799 0.987946i \(-0.549473\pi\)
−0.154799 + 0.987946i \(0.549473\pi\)
\(180\) 0 0
\(181\) 19.6569 1.46108 0.730541 0.682869i \(-0.239268\pi\)
0.730541 + 0.682869i \(0.239268\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −4.85786 −0.357157
\(186\) 0 0
\(187\) −0.828427 −0.0605806
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −24.6274 −1.78198 −0.890989 0.454026i \(-0.849987\pi\)
−0.890989 + 0.454026i \(0.849987\pi\)
\(192\) 0 0
\(193\) 10.3137 0.742397 0.371198 0.928554i \(-0.378947\pi\)
0.371198 + 0.928554i \(0.378947\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −11.7990 −0.840643 −0.420322 0.907375i \(-0.638083\pi\)
−0.420322 + 0.907375i \(0.638083\pi\)
\(198\) 0 0
\(199\) −18.9706 −1.34479 −0.672394 0.740194i \(-0.734734\pi\)
−0.672394 + 0.740194i \(0.734734\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 2.82843 0.198517
\(204\) 0 0
\(205\) 7.00000 0.488901
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −32.9706 −2.28062
\(210\) 0 0
\(211\) −16.9706 −1.16830 −0.584151 0.811645i \(-0.698572\pi\)
−0.584151 + 0.811645i \(0.698572\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 14.3137 0.976187
\(216\) 0 0
\(217\) −6.82843 −0.463544
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −0.485281 −0.0326436
\(222\) 0 0
\(223\) −18.1421 −1.21489 −0.607444 0.794363i \(-0.707805\pi\)
−0.607444 + 0.794363i \(0.707805\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 17.6569 1.17193 0.585963 0.810338i \(-0.300716\pi\)
0.585963 + 0.810338i \(0.300716\pi\)
\(228\) 0 0
\(229\) 1.31371 0.0868123 0.0434062 0.999058i \(-0.486179\pi\)
0.0434062 + 0.999058i \(0.486179\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 10.1421 0.664433 0.332217 0.943203i \(-0.392203\pi\)
0.332217 + 0.943203i \(0.392203\pi\)
\(234\) 0 0
\(235\) −15.8284 −1.03253
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −12.1421 −0.785409 −0.392705 0.919665i \(-0.628461\pi\)
−0.392705 + 0.919665i \(0.628461\pi\)
\(240\) 0 0
\(241\) −18.1421 −1.16864 −0.584319 0.811524i \(-0.698638\pi\)
−0.584319 + 0.811524i \(0.698638\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.82843 −0.116814
\(246\) 0 0
\(247\) −19.3137 −1.22890
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −3.97056 −0.250620 −0.125310 0.992118i \(-0.539992\pi\)
−0.125310 + 0.992118i \(0.539992\pi\)
\(252\) 0 0
\(253\) −19.3137 −1.21424
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 0 0
\(259\) −2.65685 −0.165089
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −10.8284 −0.667709 −0.333855 0.942625i \(-0.608349\pi\)
−0.333855 + 0.942625i \(0.608349\pi\)
\(264\) 0 0
\(265\) −3.65685 −0.224639
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 15.1421 0.923232 0.461616 0.887080i \(-0.347270\pi\)
0.461616 + 0.887080i \(0.347270\pi\)
\(270\) 0 0
\(271\) 6.00000 0.364474 0.182237 0.983255i \(-0.441666\pi\)
0.182237 + 0.983255i \(0.441666\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −8.00000 −0.482418
\(276\) 0 0
\(277\) −24.6569 −1.48149 −0.740743 0.671788i \(-0.765527\pi\)
−0.740743 + 0.671788i \(0.765527\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −32.9706 −1.96686 −0.983429 0.181291i \(-0.941972\pi\)
−0.983429 + 0.181291i \(0.941972\pi\)
\(282\) 0 0
\(283\) −23.1716 −1.37741 −0.688704 0.725043i \(-0.741820\pi\)
−0.688704 + 0.725043i \(0.741820\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3.82843 0.225985
\(288\) 0 0
\(289\) −16.9706 −0.998268
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 12.1716 0.711071 0.355535 0.934663i \(-0.384299\pi\)
0.355535 + 0.934663i \(0.384299\pi\)
\(294\) 0 0
\(295\) −1.20101 −0.0699255
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −11.3137 −0.654289
\(300\) 0 0
\(301\) 7.82843 0.451223
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −5.79899 −0.332049
\(306\) 0 0
\(307\) 24.2843 1.38598 0.692988 0.720949i \(-0.256294\pi\)
0.692988 + 0.720949i \(0.256294\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −9.97056 −0.565379 −0.282689 0.959212i \(-0.591227\pi\)
−0.282689 + 0.959212i \(0.591227\pi\)
\(312\) 0 0
\(313\) 24.6274 1.39202 0.696012 0.718030i \(-0.254956\pi\)
0.696012 + 0.718030i \(0.254956\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −29.6569 −1.66569 −0.832847 0.553503i \(-0.813291\pi\)
−0.832847 + 0.553503i \(0.813291\pi\)
\(318\) 0 0
\(319\) −13.6569 −0.764637
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1.17157 0.0651881
\(324\) 0 0
\(325\) −4.68629 −0.259949
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −8.65685 −0.477268
\(330\) 0 0
\(331\) −28.7990 −1.58294 −0.791468 0.611211i \(-0.790683\pi\)
−0.791468 + 0.611211i \(0.790683\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 7.31371 0.399591
\(336\) 0 0
\(337\) −20.3137 −1.10656 −0.553279 0.832996i \(-0.686624\pi\)
−0.553279 + 0.832996i \(0.686624\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 32.9706 1.78546
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6.14214 0.329727 0.164864 0.986316i \(-0.447282\pi\)
0.164864 + 0.986316i \(0.447282\pi\)
\(348\) 0 0
\(349\) −20.2843 −1.08579 −0.542896 0.839800i \(-0.682672\pi\)
−0.542896 + 0.839800i \(0.682672\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −4.85786 −0.258558 −0.129279 0.991608i \(-0.541266\pi\)
−0.129279 + 0.991608i \(0.541266\pi\)
\(354\) 0 0
\(355\) −3.02944 −0.160786
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −16.3431 −0.862558 −0.431279 0.902219i \(-0.641938\pi\)
−0.431279 + 0.902219i \(0.641938\pi\)
\(360\) 0 0
\(361\) 27.6274 1.45407
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 10.3431 0.541385
\(366\) 0 0
\(367\) −22.0000 −1.14839 −0.574195 0.818718i \(-0.694685\pi\)
−0.574195 + 0.818718i \(0.694685\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −2.00000 −0.103835
\(372\) 0 0
\(373\) 3.68629 0.190869 0.0954345 0.995436i \(-0.469576\pi\)
0.0954345 + 0.995436i \(0.469576\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −8.00000 −0.412021
\(378\) 0 0
\(379\) 17.4853 0.898159 0.449079 0.893492i \(-0.351752\pi\)
0.449079 + 0.893492i \(0.351752\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1.00000 −0.0510976 −0.0255488 0.999674i \(-0.508133\pi\)
−0.0255488 + 0.999674i \(0.508133\pi\)
\(384\) 0 0
\(385\) 8.82843 0.449938
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −1.79899 −0.0912124 −0.0456062 0.998959i \(-0.514522\pi\)
−0.0456062 + 0.998959i \(0.514522\pi\)
\(390\) 0 0
\(391\) 0.686292 0.0347073
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −3.34315 −0.168212
\(396\) 0 0
\(397\) 14.3431 0.719862 0.359931 0.932979i \(-0.382800\pi\)
0.359931 + 0.932979i \(0.382800\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 25.3137 1.26411 0.632053 0.774925i \(-0.282212\pi\)
0.632053 + 0.774925i \(0.282212\pi\)
\(402\) 0 0
\(403\) 19.3137 0.962084
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 12.8284 0.635882
\(408\) 0 0
\(409\) −3.02944 −0.149796 −0.0748980 0.997191i \(-0.523863\pi\)
−0.0748980 + 0.997191i \(0.523863\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −0.656854 −0.0323217
\(414\) 0 0
\(415\) 9.76955 0.479568
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 18.6569 0.911447 0.455723 0.890121i \(-0.349381\pi\)
0.455723 + 0.890121i \(0.349381\pi\)
\(420\) 0 0
\(421\) 14.9706 0.729621 0.364810 0.931082i \(-0.381134\pi\)
0.364810 + 0.931082i \(0.381134\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0.284271 0.0137892
\(426\) 0 0
\(427\) −3.17157 −0.153483
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −24.9706 −1.20279 −0.601395 0.798952i \(-0.705388\pi\)
−0.601395 + 0.798952i \(0.705388\pi\)
\(432\) 0 0
\(433\) 34.4853 1.65726 0.828628 0.559799i \(-0.189122\pi\)
0.828628 + 0.559799i \(0.189122\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 27.3137 1.30659
\(438\) 0 0
\(439\) −22.0000 −1.05000 −0.525001 0.851101i \(-0.675935\pi\)
−0.525001 + 0.851101i \(0.675935\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 34.8284 1.65475 0.827374 0.561651i \(-0.189834\pi\)
0.827374 + 0.561651i \(0.189834\pi\)
\(444\) 0 0
\(445\) −10.9706 −0.520055
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.17157 0.0552899 0.0276450 0.999618i \(-0.491199\pi\)
0.0276450 + 0.999618i \(0.491199\pi\)
\(450\) 0 0
\(451\) −18.4853 −0.870438
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 5.17157 0.242447
\(456\) 0 0
\(457\) 41.3137 1.93257 0.966287 0.257468i \(-0.0828881\pi\)
0.966287 + 0.257468i \(0.0828881\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −0.514719 −0.0239728 −0.0119864 0.999928i \(-0.503815\pi\)
−0.0119864 + 0.999928i \(0.503815\pi\)
\(462\) 0 0
\(463\) 18.1716 0.844505 0.422252 0.906478i \(-0.361240\pi\)
0.422252 + 0.906478i \(0.361240\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −40.2843 −1.86413 −0.932067 0.362286i \(-0.881996\pi\)
−0.932067 + 0.362286i \(0.881996\pi\)
\(468\) 0 0
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −37.7990 −1.73800
\(474\) 0 0
\(475\) 11.3137 0.519109
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 1.00000 0.0456912 0.0228456 0.999739i \(-0.492727\pi\)
0.0228456 + 0.999739i \(0.492727\pi\)
\(480\) 0 0
\(481\) 7.51472 0.342642
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 33.1716 1.50624
\(486\) 0 0
\(487\) 16.0000 0.725029 0.362515 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 30.1421 1.36030 0.680148 0.733075i \(-0.261916\pi\)
0.680148 + 0.733075i \(0.261916\pi\)
\(492\) 0 0
\(493\) 0.485281 0.0218560
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1.65685 −0.0743201
\(498\) 0 0
\(499\) −43.7696 −1.95939 −0.979697 0.200483i \(-0.935749\pi\)
−0.979697 + 0.200483i \(0.935749\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 9.62742 0.429265 0.214633 0.976695i \(-0.431145\pi\)
0.214633 + 0.976695i \(0.431145\pi\)
\(504\) 0 0
\(505\) −34.6863 −1.54352
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 5.48528 0.243131 0.121565 0.992583i \(-0.461209\pi\)
0.121565 + 0.992583i \(0.461209\pi\)
\(510\) 0 0
\(511\) 5.65685 0.250244
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 4.28427 0.188788
\(516\) 0 0
\(517\) 41.7990 1.83832
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 38.4558 1.68478 0.842391 0.538867i \(-0.181148\pi\)
0.842391 + 0.538867i \(0.181148\pi\)
\(522\) 0 0
\(523\) −40.4264 −1.76772 −0.883862 0.467748i \(-0.845065\pi\)
−0.883862 + 0.467748i \(0.845065\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −1.17157 −0.0510345
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −10.8284 −0.469031
\(534\) 0 0
\(535\) 10.3431 0.447173
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 4.82843 0.207975
\(540\) 0 0
\(541\) −27.2843 −1.17304 −0.586521 0.809934i \(-0.699503\pi\)
−0.586521 + 0.809934i \(0.699503\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 19.4853 0.834658
\(546\) 0 0
\(547\) −5.48528 −0.234534 −0.117267 0.993100i \(-0.537413\pi\)
−0.117267 + 0.993100i \(0.537413\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 19.3137 0.822792
\(552\) 0 0
\(553\) −1.82843 −0.0777526
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 35.3137 1.49629 0.748145 0.663535i \(-0.230945\pi\)
0.748145 + 0.663535i \(0.230945\pi\)
\(558\) 0 0
\(559\) −22.1421 −0.936513
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −39.3137 −1.65688 −0.828438 0.560081i \(-0.810770\pi\)
−0.828438 + 0.560081i \(0.810770\pi\)
\(564\) 0 0
\(565\) −13.1127 −0.551655
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −26.3431 −1.10436 −0.552181 0.833724i \(-0.686204\pi\)
−0.552181 + 0.833724i \(0.686204\pi\)
\(570\) 0 0
\(571\) 31.7696 1.32951 0.664757 0.747059i \(-0.268535\pi\)
0.664757 + 0.747059i \(0.268535\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 6.62742 0.276382
\(576\) 0 0
\(577\) 10.9706 0.456711 0.228355 0.973578i \(-0.426665\pi\)
0.228355 + 0.973578i \(0.426665\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 5.34315 0.221671
\(582\) 0 0
\(583\) 9.65685 0.399946
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −4.97056 −0.205157 −0.102579 0.994725i \(-0.532709\pi\)
−0.102579 + 0.994725i \(0.532709\pi\)
\(588\) 0 0
\(589\) −46.6274 −1.92125
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −11.4853 −0.471644 −0.235822 0.971796i \(-0.575778\pi\)
−0.235822 + 0.971796i \(0.575778\pi\)
\(594\) 0 0
\(595\) −0.313708 −0.0128608
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 27.1716 1.11020 0.555100 0.831783i \(-0.312680\pi\)
0.555100 + 0.831783i \(0.312680\pi\)
\(600\) 0 0
\(601\) 19.1127 0.779623 0.389812 0.920895i \(-0.372540\pi\)
0.389812 + 0.920895i \(0.372540\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −22.5147 −0.915353
\(606\) 0 0
\(607\) 4.14214 0.168124 0.0840620 0.996461i \(-0.473211\pi\)
0.0840620 + 0.996461i \(0.473211\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 24.4853 0.990568
\(612\) 0 0
\(613\) −14.0000 −0.565455 −0.282727 0.959200i \(-0.591239\pi\)
−0.282727 + 0.959200i \(0.591239\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −35.3137 −1.42168 −0.710838 0.703356i \(-0.751684\pi\)
−0.710838 + 0.703356i \(0.751684\pi\)
\(618\) 0 0
\(619\) 1.17157 0.0470895 0.0235447 0.999723i \(-0.492505\pi\)
0.0235447 + 0.999723i \(0.492505\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) −13.9706 −0.558823
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −0.455844 −0.0181757
\(630\) 0 0
\(631\) 28.1127 1.11915 0.559574 0.828780i \(-0.310965\pi\)
0.559574 + 0.828780i \(0.310965\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 22.2548 0.883156
\(636\) 0 0
\(637\) 2.82843 0.112066
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 11.4558 0.452479 0.226239 0.974072i \(-0.427357\pi\)
0.226239 + 0.974072i \(0.427357\pi\)
\(642\) 0 0
\(643\) −26.0000 −1.02534 −0.512670 0.858586i \(-0.671344\pi\)
−0.512670 + 0.858586i \(0.671344\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 21.6569 0.851419 0.425709 0.904860i \(-0.360025\pi\)
0.425709 + 0.904860i \(0.360025\pi\)
\(648\) 0 0
\(649\) 3.17157 0.124495
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 6.48528 0.253789 0.126894 0.991916i \(-0.459499\pi\)
0.126894 + 0.991916i \(0.459499\pi\)
\(654\) 0 0
\(655\) −7.31371 −0.285770
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −12.3431 −0.480821 −0.240410 0.970671i \(-0.577282\pi\)
−0.240410 + 0.970671i \(0.577282\pi\)
\(660\) 0 0
\(661\) −17.1127 −0.665607 −0.332803 0.942996i \(-0.607995\pi\)
−0.332803 + 0.942996i \(0.607995\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −12.4853 −0.484158
\(666\) 0 0
\(667\) 11.3137 0.438069
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 15.3137 0.591179
\(672\) 0 0
\(673\) −35.6569 −1.37447 −0.687235 0.726435i \(-0.741176\pi\)
−0.687235 + 0.726435i \(0.741176\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −25.3137 −0.972885 −0.486442 0.873713i \(-0.661706\pi\)
−0.486442 + 0.873713i \(0.661706\pi\)
\(678\) 0 0
\(679\) 18.1421 0.696231
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −21.1716 −0.810108 −0.405054 0.914293i \(-0.632747\pi\)
−0.405054 + 0.914293i \(0.632747\pi\)
\(684\) 0 0
\(685\) 27.1127 1.03592
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 5.65685 0.215509
\(690\) 0 0
\(691\) 20.8284 0.792351 0.396175 0.918175i \(-0.370337\pi\)
0.396175 + 0.918175i \(0.370337\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 20.4264 0.774818
\(696\) 0 0
\(697\) 0.656854 0.0248801
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 46.4853 1.75572 0.877862 0.478913i \(-0.158969\pi\)
0.877862 + 0.478913i \(0.158969\pi\)
\(702\) 0 0
\(703\) −18.1421 −0.684244
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −18.9706 −0.713461
\(708\) 0 0
\(709\) 31.2843 1.17491 0.587453 0.809258i \(-0.300131\pi\)
0.587453 + 0.809258i \(0.300131\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −27.3137 −1.02291
\(714\) 0 0
\(715\) −24.9706 −0.933846
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1.34315 0.0500909 0.0250454 0.999686i \(-0.492027\pi\)
0.0250454 + 0.999686i \(0.492027\pi\)
\(720\) 0 0
\(721\) 2.34315 0.0872633
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 4.68629 0.174044
\(726\) 0 0
\(727\) −10.1421 −0.376151 −0.188075 0.982155i \(-0.560225\pi\)
−0.188075 + 0.982155i \(0.560225\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1.34315 0.0496780
\(732\) 0 0
\(733\) 2.68629 0.0992204 0.0496102 0.998769i \(-0.484202\pi\)
0.0496102 + 0.998769i \(0.484202\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −19.3137 −0.711430
\(738\) 0 0
\(739\) −37.9411 −1.39569 −0.697843 0.716250i \(-0.745857\pi\)
−0.697843 + 0.716250i \(0.745857\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1.17157 −0.0429808 −0.0214904 0.999769i \(-0.506841\pi\)
−0.0214904 + 0.999769i \(0.506841\pi\)
\(744\) 0 0
\(745\) −36.8284 −1.34929
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 5.65685 0.206697
\(750\) 0 0
\(751\) 24.0000 0.875772 0.437886 0.899030i \(-0.355727\pi\)
0.437886 + 0.899030i \(0.355727\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 32.5980 1.18636
\(756\) 0 0
\(757\) −35.6274 −1.29490 −0.647450 0.762108i \(-0.724165\pi\)
−0.647450 + 0.762108i \(0.724165\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −2.17157 −0.0787195 −0.0393597 0.999225i \(-0.512532\pi\)
−0.0393597 + 0.999225i \(0.512532\pi\)
\(762\) 0 0
\(763\) 10.6569 0.385804
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.85786 0.0670836
\(768\) 0 0
\(769\) 4.34315 0.156618 0.0783089 0.996929i \(-0.475048\pi\)
0.0783089 + 0.996929i \(0.475048\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 37.8284 1.36059 0.680297 0.732937i \(-0.261851\pi\)
0.680297 + 0.732937i \(0.261851\pi\)
\(774\) 0 0
\(775\) −11.3137 −0.406400
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 26.1421 0.936639
\(780\) 0 0
\(781\) 8.00000 0.286263
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −5.17157 −0.184581
\(786\) 0 0
\(787\) 25.9411 0.924701 0.462351 0.886697i \(-0.347006\pi\)
0.462351 + 0.886697i \(0.347006\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −7.17157 −0.254992
\(792\) 0 0
\(793\) 8.97056 0.318554
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −13.3137 −0.471596 −0.235798 0.971802i \(-0.575770\pi\)
−0.235798 + 0.971802i \(0.575770\pi\)
\(798\) 0 0
\(799\) −1.48528 −0.0525455
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −27.3137 −0.963880
\(804\) 0 0
\(805\) −7.31371 −0.257774
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −31.9411 −1.12299 −0.561495 0.827480i \(-0.689774\pi\)
−0.561495 + 0.827480i \(0.689774\pi\)
\(810\) 0 0
\(811\) 42.7696 1.50184 0.750921 0.660392i \(-0.229610\pi\)
0.750921 + 0.660392i \(0.229610\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −2.71573 −0.0951278
\(816\) 0 0
\(817\) 53.4558 1.87018
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 21.6569 0.755829 0.377915 0.925840i \(-0.376641\pi\)
0.377915 + 0.925840i \(0.376641\pi\)
\(822\) 0 0
\(823\) 26.1716 0.912284 0.456142 0.889907i \(-0.349231\pi\)
0.456142 + 0.889907i \(0.349231\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 3.17157 0.110286 0.0551432 0.998478i \(-0.482438\pi\)
0.0551432 + 0.998478i \(0.482438\pi\)
\(828\) 0 0
\(829\) 9.17157 0.318542 0.159271 0.987235i \(-0.449086\pi\)
0.159271 + 0.987235i \(0.449086\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −0.171573 −0.00594465
\(834\) 0 0
\(835\) 32.8579 1.13709
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −24.3137 −0.839402 −0.419701 0.907662i \(-0.637865\pi\)
−0.419701 + 0.907662i \(0.637865\pi\)
\(840\) 0 0
\(841\) −21.0000 −0.724138
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 9.14214 0.314499
\(846\) 0 0
\(847\) −12.3137 −0.423104
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −10.6274 −0.364303
\(852\) 0 0
\(853\) 14.9706 0.512582 0.256291 0.966600i \(-0.417499\pi\)
0.256291 + 0.966600i \(0.417499\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −43.1421 −1.47371 −0.736854 0.676052i \(-0.763689\pi\)
−0.736854 + 0.676052i \(0.763689\pi\)
\(858\) 0 0
\(859\) 11.6569 0.397727 0.198863 0.980027i \(-0.436275\pi\)
0.198863 + 0.980027i \(0.436275\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 18.6863 0.636089 0.318044 0.948076i \(-0.396974\pi\)
0.318044 + 0.948076i \(0.396974\pi\)
\(864\) 0 0
\(865\) −3.65685 −0.124337
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 8.82843 0.299484
\(870\) 0 0
\(871\) −11.3137 −0.383350
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −12.1716 −0.411474
\(876\) 0 0
\(877\) −31.9706 −1.07957 −0.539785 0.841803i \(-0.681494\pi\)
−0.539785 + 0.841803i \(0.681494\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 26.9706 0.908661 0.454331 0.890833i \(-0.349878\pi\)
0.454331 + 0.890833i \(0.349878\pi\)
\(882\) 0 0
\(883\) −30.7990 −1.03647 −0.518234 0.855239i \(-0.673410\pi\)
−0.518234 + 0.855239i \(0.673410\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −52.5980 −1.76607 −0.883034 0.469310i \(-0.844503\pi\)
−0.883034 + 0.469310i \(0.844503\pi\)
\(888\) 0 0
\(889\) 12.1716 0.408221
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −59.1127 −1.97813
\(894\) 0 0
\(895\) 7.57359 0.253157
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −19.3137 −0.644148
\(900\) 0 0
\(901\) −0.343146 −0.0114318
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −35.9411 −1.19472
\(906\) 0 0
\(907\) 29.7696 0.988482 0.494241 0.869325i \(-0.335446\pi\)
0.494241 + 0.869325i \(0.335446\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −30.4853 −1.01002 −0.505011 0.863113i \(-0.668512\pi\)
−0.505011 + 0.863113i \(0.668512\pi\)
\(912\) 0 0
\(913\) −25.7990 −0.853822
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −4.00000 −0.132092
\(918\) 0 0
\(919\) 45.0833 1.48716 0.743580 0.668647i \(-0.233126\pi\)
0.743580 + 0.668647i \(0.233126\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 4.68629 0.154251
\(924\) 0 0
\(925\) −4.40202 −0.144738
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −15.8284 −0.519314 −0.259657 0.965701i \(-0.583609\pi\)
−0.259657 + 0.965701i \(0.583609\pi\)
\(930\) 0 0
\(931\) −6.82843 −0.223793
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1.51472 0.0495366
\(936\) 0 0
\(937\) 33.1716 1.08367 0.541834 0.840486i \(-0.317730\pi\)
0.541834 + 0.840486i \(0.317730\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −42.7990 −1.39521 −0.697604 0.716484i \(-0.745750\pi\)
−0.697604 + 0.716484i \(0.745750\pi\)
\(942\) 0 0
\(943\) 15.3137 0.498683
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 57.9411 1.88283 0.941417 0.337245i \(-0.109495\pi\)
0.941417 + 0.337245i \(0.109495\pi\)
\(948\) 0 0
\(949\) −16.0000 −0.519382
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 3.31371 0.107342 0.0536708 0.998559i \(-0.482908\pi\)
0.0536708 + 0.998559i \(0.482908\pi\)
\(954\) 0 0
\(955\) 45.0294 1.45712
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 14.8284 0.478835
\(960\) 0 0
\(961\) 15.6274 0.504110
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −18.8579 −0.607056
\(966\) 0 0
\(967\) 5.65685 0.181912 0.0909561 0.995855i \(-0.471008\pi\)
0.0909561 + 0.995855i \(0.471008\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 10.6569 0.341995 0.170997 0.985271i \(-0.445301\pi\)
0.170997 + 0.985271i \(0.445301\pi\)
\(972\) 0 0
\(973\) 11.1716 0.358144
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 42.9706 1.37475 0.687375 0.726303i \(-0.258763\pi\)
0.687375 + 0.726303i \(0.258763\pi\)
\(978\) 0 0
\(979\) 28.9706 0.925903
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 2.71573 0.0866183 0.0433091 0.999062i \(-0.486210\pi\)
0.0433091 + 0.999062i \(0.486210\pi\)
\(984\) 0 0
\(985\) 21.5736 0.687392
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 31.3137 0.995718
\(990\) 0 0
\(991\) −27.4264 −0.871229 −0.435614 0.900133i \(-0.643469\pi\)
−0.435614 + 0.900133i \(0.643469\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 34.6863 1.09963
\(996\) 0 0
\(997\) −17.8579 −0.565564 −0.282782 0.959184i \(-0.591257\pi\)
−0.282782 + 0.959184i \(0.591257\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 6048.2.a.bh.1.1 yes 2
3.2 odd 2 6048.2.a.y.1.2 2
4.3 odd 2 6048.2.a.bi.1.1 yes 2
12.11 even 2 6048.2.a.bb.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.a.y.1.2 2 3.2 odd 2
6048.2.a.bb.1.2 yes 2 12.11 even 2
6048.2.a.bh.1.1 yes 2 1.1 even 1 trivial
6048.2.a.bi.1.1 yes 2 4.3 odd 2