Properties

Label 2736.2.a.bf.1.1
Level $2736$
Weight $2$
Character 2736.1
Self dual yes
Analytic conductor $21.847$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.13068.1
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 171)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.04374\) of defining polynomial
Character \(\chi\) \(=\) 2736.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.22060 q^{5} +2.37228 q^{7} +O(q^{10})\) \(q-3.22060 q^{5} +2.37228 q^{7} +2.20979 q^{11} +2.00000 q^{13} -3.22060 q^{17} -1.00000 q^{19} -1.01082 q^{23} +5.37228 q^{25} +1.01082 q^{29} -4.74456 q^{31} -7.64018 q^{35} +10.7446 q^{37} -5.43039 q^{41} +11.1168 q^{43} +4.23142 q^{47} -1.37228 q^{49} +9.84996 q^{53} -7.11684 q^{55} -10.8608 q^{59} -5.11684 q^{61} -6.44121 q^{65} +4.00000 q^{67} -2.02163 q^{71} -5.11684 q^{73} +5.24224 q^{77} +4.00000 q^{79} +11.8716 q^{83} +10.3723 q^{85} +9.84996 q^{89} +4.74456 q^{91} +3.22060 q^{95} +7.48913 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{7} + 8 q^{13} - 4 q^{19} + 10 q^{25} + 4 q^{31} + 20 q^{37} + 10 q^{43} + 6 q^{49} + 6 q^{55} + 14 q^{61} + 16 q^{67} + 14 q^{73} + 16 q^{79} + 30 q^{85} - 4 q^{91} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) −3.22060 −1.44030 −0.720149 0.693820i \(-0.755926\pi\)
−0.720149 + 0.693820i \(0.755926\pi\)
\(6\) 0 0
\(7\) 2.37228 0.896638 0.448319 0.893874i \(-0.352023\pi\)
0.448319 + 0.893874i \(0.352023\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.20979 0.666276 0.333138 0.942878i \(-0.391893\pi\)
0.333138 + 0.942878i \(0.391893\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.22060 −0.781111 −0.390555 0.920579i \(-0.627717\pi\)
−0.390555 + 0.920579i \(0.627717\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.01082 −0.210770 −0.105385 0.994432i \(-0.533607\pi\)
−0.105385 + 0.994432i \(0.533607\pi\)
\(24\) 0 0
\(25\) 5.37228 1.07446
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.01082 0.187704 0.0938519 0.995586i \(-0.470082\pi\)
0.0938519 + 0.995586i \(0.470082\pi\)
\(30\) 0 0
\(31\) −4.74456 −0.852149 −0.426074 0.904688i \(-0.640104\pi\)
−0.426074 + 0.904688i \(0.640104\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −7.64018 −1.29143
\(36\) 0 0
\(37\) 10.7446 1.76640 0.883198 0.469001i \(-0.155386\pi\)
0.883198 + 0.469001i \(0.155386\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −5.43039 −0.848084 −0.424042 0.905642i \(-0.639389\pi\)
−0.424042 + 0.905642i \(0.639389\pi\)
\(42\) 0 0
\(43\) 11.1168 1.69530 0.847651 0.530554i \(-0.178016\pi\)
0.847651 + 0.530554i \(0.178016\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.23142 0.617216 0.308608 0.951189i \(-0.400137\pi\)
0.308608 + 0.951189i \(0.400137\pi\)
\(48\) 0 0
\(49\) −1.37228 −0.196040
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 9.84996 1.35300 0.676498 0.736444i \(-0.263497\pi\)
0.676498 + 0.736444i \(0.263497\pi\)
\(54\) 0 0
\(55\) −7.11684 −0.959635
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −10.8608 −1.41395 −0.706976 0.707237i \(-0.749941\pi\)
−0.706976 + 0.707237i \(0.749941\pi\)
\(60\) 0 0
\(61\) −5.11684 −0.655145 −0.327572 0.944826i \(-0.606231\pi\)
−0.327572 + 0.944826i \(0.606231\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −6.44121 −0.798933
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −2.02163 −0.239924 −0.119962 0.992779i \(-0.538277\pi\)
−0.119962 + 0.992779i \(0.538277\pi\)
\(72\) 0 0
\(73\) −5.11684 −0.598881 −0.299441 0.954115i \(-0.596800\pi\)
−0.299441 + 0.954115i \(0.596800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.24224 0.597408
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 11.8716 1.30308 0.651538 0.758616i \(-0.274124\pi\)
0.651538 + 0.758616i \(0.274124\pi\)
\(84\) 0 0
\(85\) 10.3723 1.12503
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 9.84996 1.04409 0.522047 0.852917i \(-0.325169\pi\)
0.522047 + 0.852917i \(0.325169\pi\)
\(90\) 0 0
\(91\) 4.74456 0.497365
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.22060 0.330427
\(96\) 0 0
\(97\) 7.48913 0.760405 0.380203 0.924903i \(-0.375854\pi\)
0.380203 + 0.924903i \(0.375854\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 12.8824 1.28185 0.640924 0.767604i \(-0.278551\pi\)
0.640924 + 0.767604i \(0.278551\pi\)
\(102\) 0 0
\(103\) 7.25544 0.714899 0.357450 0.933932i \(-0.383646\pi\)
0.357450 + 0.933932i \(0.383646\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.41957 −0.427256 −0.213628 0.976915i \(-0.568528\pi\)
−0.213628 + 0.976915i \(0.568528\pi\)
\(108\) 0 0
\(109\) 5.25544 0.503380 0.251690 0.967808i \(-0.419014\pi\)
0.251690 + 0.967808i \(0.419014\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −11.8716 −1.11679 −0.558393 0.829577i \(-0.688582\pi\)
−0.558393 + 0.829577i \(0.688582\pi\)
\(114\) 0 0
\(115\) 3.25544 0.303571
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −7.64018 −0.700374
\(120\) 0 0
\(121\) −6.11684 −0.556077
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.19897 −0.107239
\(126\) 0 0
\(127\) 12.7446 1.13090 0.565449 0.824784i \(-0.308703\pi\)
0.565449 + 0.824784i \(0.308703\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 15.0922 1.31861 0.659306 0.751875i \(-0.270850\pi\)
0.659306 + 0.751875i \(0.270850\pi\)
\(132\) 0 0
\(133\) −2.37228 −0.205703
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 12.0597 1.03033 0.515167 0.857090i \(-0.327730\pi\)
0.515167 + 0.857090i \(0.327730\pi\)
\(138\) 0 0
\(139\) −3.11684 −0.264367 −0.132184 0.991225i \(-0.542199\pi\)
−0.132184 + 0.991225i \(0.542199\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.41957 0.369583
\(144\) 0 0
\(145\) −3.25544 −0.270349
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 20.5226 1.68128 0.840638 0.541598i \(-0.182180\pi\)
0.840638 + 0.541598i \(0.182180\pi\)
\(150\) 0 0
\(151\) 4.00000 0.325515 0.162758 0.986666i \(-0.447961\pi\)
0.162758 + 0.986666i \(0.447961\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 15.2804 1.22735
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −2.39794 −0.188984
\(162\) 0 0
\(163\) 21.4891 1.68316 0.841579 0.540134i \(-0.181626\pi\)
0.841579 + 0.540134i \(0.181626\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.44121 0.498435 0.249218 0.968447i \(-0.419827\pi\)
0.249218 + 0.968447i \(0.419827\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1.01082 −0.0768509 −0.0384255 0.999261i \(-0.512234\pi\)
−0.0384255 + 0.999261i \(0.512234\pi\)
\(174\) 0 0
\(175\) 12.7446 0.963398
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 4.41957 0.330334 0.165167 0.986266i \(-0.447184\pi\)
0.165167 + 0.986266i \(0.447184\pi\)
\(180\) 0 0
\(181\) 19.4891 1.44862 0.724308 0.689477i \(-0.242160\pi\)
0.724308 + 0.689477i \(0.242160\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −34.6040 −2.54413
\(186\) 0 0
\(187\) −7.11684 −0.520435
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 21.5334 1.55810 0.779051 0.626960i \(-0.215701\pi\)
0.779051 + 0.626960i \(0.215701\pi\)
\(192\) 0 0
\(193\) 16.2337 1.16853 0.584263 0.811564i \(-0.301384\pi\)
0.584263 + 0.811564i \(0.301384\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −23.7432 −1.69163 −0.845816 0.533475i \(-0.820886\pi\)
−0.845816 + 0.533475i \(0.820886\pi\)
\(198\) 0 0
\(199\) −0.883156 −0.0626053 −0.0313026 0.999510i \(-0.509966\pi\)
−0.0313026 + 0.999510i \(0.509966\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 2.39794 0.168302
\(204\) 0 0
\(205\) 17.4891 1.22149
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −2.20979 −0.152854
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −35.8029 −2.44174
\(216\) 0 0
\(217\) −11.2554 −0.764069
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −6.44121 −0.433282
\(222\) 0 0
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −19.3236 −1.28255 −0.641277 0.767310i \(-0.721595\pi\)
−0.641277 + 0.767310i \(0.721595\pi\)
\(228\) 0 0
\(229\) 15.6277 1.03271 0.516354 0.856375i \(-0.327289\pi\)
0.516354 + 0.856375i \(0.327289\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −7.64018 −0.500525 −0.250262 0.968178i \(-0.580517\pi\)
−0.250262 + 0.968178i \(0.580517\pi\)
\(234\) 0 0
\(235\) −13.6277 −0.888974
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 12.6943 0.821123 0.410562 0.911833i \(-0.365333\pi\)
0.410562 + 0.911833i \(0.365333\pi\)
\(240\) 0 0
\(241\) −24.2337 −1.56103 −0.780515 0.625138i \(-0.785043\pi\)
−0.780515 + 0.625138i \(0.785043\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 4.41957 0.282356
\(246\) 0 0
\(247\) −2.00000 −0.127257
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −23.9313 −1.51053 −0.755266 0.655418i \(-0.772493\pi\)
−0.755266 + 0.655418i \(0.772493\pi\)
\(252\) 0 0
\(253\) −2.23369 −0.140431
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 5.43039 0.338738 0.169369 0.985553i \(-0.445827\pi\)
0.169369 + 0.985553i \(0.445827\pi\)
\(258\) 0 0
\(259\) 25.4891 1.58382
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −13.0706 −0.805966 −0.402983 0.915208i \(-0.632027\pi\)
−0.402983 + 0.915208i \(0.632027\pi\)
\(264\) 0 0
\(265\) −31.7228 −1.94872
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 7.82833 0.477302 0.238651 0.971105i \(-0.423295\pi\)
0.238651 + 0.971105i \(0.423295\pi\)
\(270\) 0 0
\(271\) −25.4891 −1.54835 −0.774177 0.632969i \(-0.781836\pi\)
−0.774177 + 0.632969i \(0.781836\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 11.8716 0.715884
\(276\) 0 0
\(277\) 9.11684 0.547778 0.273889 0.961761i \(-0.411690\pi\)
0.273889 + 0.961761i \(0.411690\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 24.7540 1.47670 0.738350 0.674418i \(-0.235605\pi\)
0.738350 + 0.674418i \(0.235605\pi\)
\(282\) 0 0
\(283\) −6.37228 −0.378793 −0.189396 0.981901i \(-0.560653\pi\)
−0.189396 + 0.981901i \(0.560653\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −12.8824 −0.760425
\(288\) 0 0
\(289\) −6.62772 −0.389866
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −25.1303 −1.46813 −0.734064 0.679080i \(-0.762379\pi\)
−0.734064 + 0.679080i \(0.762379\pi\)
\(294\) 0 0
\(295\) 34.9783 2.03651
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −2.02163 −0.116914
\(300\) 0 0
\(301\) 26.3723 1.52007
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 16.4793 0.943603
\(306\) 0 0
\(307\) −25.4891 −1.45474 −0.727371 0.686245i \(-0.759258\pi\)
−0.727371 + 0.686245i \(0.759258\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −12.6943 −0.719825 −0.359913 0.932986i \(-0.617194\pi\)
−0.359913 + 0.932986i \(0.617194\pi\)
\(312\) 0 0
\(313\) 7.48913 0.423310 0.211655 0.977344i \(-0.432115\pi\)
0.211655 + 0.977344i \(0.432115\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −12.2479 −0.687911 −0.343955 0.938986i \(-0.611767\pi\)
−0.343955 + 0.938986i \(0.611767\pi\)
\(318\) 0 0
\(319\) 2.23369 0.125063
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3.22060 0.179199
\(324\) 0 0
\(325\) 10.7446 0.596001
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 10.0381 0.553419
\(330\) 0 0
\(331\) −18.9783 −1.04314 −0.521569 0.853209i \(-0.674653\pi\)
−0.521569 + 0.853209i \(0.674653\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −12.8824 −0.703841
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −10.4845 −0.567766
\(342\) 0 0
\(343\) −19.8614 −1.07242
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −32.3942 −1.73901 −0.869505 0.493924i \(-0.835562\pi\)
−0.869505 + 0.493924i \(0.835562\pi\)
\(348\) 0 0
\(349\) 17.8614 0.956099 0.478050 0.878333i \(-0.341344\pi\)
0.478050 + 0.878333i \(0.341344\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 14.9040 0.793262 0.396631 0.917978i \(-0.370179\pi\)
0.396631 + 0.917978i \(0.370179\pi\)
\(354\) 0 0
\(355\) 6.51087 0.345561
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 4.60773 0.243187 0.121593 0.992580i \(-0.461200\pi\)
0.121593 + 0.992580i \(0.461200\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 16.4793 0.862567
\(366\) 0 0
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 23.3669 1.21315
\(372\) 0 0
\(373\) −24.2337 −1.25477 −0.627386 0.778708i \(-0.715875\pi\)
−0.627386 + 0.778708i \(0.715875\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.02163 0.104119
\(378\) 0 0
\(379\) −28.7446 −1.47651 −0.738255 0.674522i \(-0.764350\pi\)
−0.738255 + 0.674522i \(0.764350\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 17.3020 0.884090 0.442045 0.896993i \(-0.354253\pi\)
0.442045 + 0.896993i \(0.354253\pi\)
\(384\) 0 0
\(385\) −16.8832 −0.860445
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 12.0597 0.611454 0.305727 0.952119i \(-0.401101\pi\)
0.305727 + 0.952119i \(0.401101\pi\)
\(390\) 0 0
\(391\) 3.25544 0.164635
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −12.8824 −0.648184
\(396\) 0 0
\(397\) −17.1168 −0.859070 −0.429535 0.903050i \(-0.641322\pi\)
−0.429535 + 0.903050i \(0.641322\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 3.40876 0.170225 0.0851126 0.996371i \(-0.472875\pi\)
0.0851126 + 0.996371i \(0.472875\pi\)
\(402\) 0 0
\(403\) −9.48913 −0.472687
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 23.7432 1.17691
\(408\) 0 0
\(409\) 7.48913 0.370313 0.185157 0.982709i \(-0.440721\pi\)
0.185157 + 0.982709i \(0.440721\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −25.7648 −1.26780
\(414\) 0 0
\(415\) −38.2337 −1.87682
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −11.8716 −0.579965 −0.289983 0.957032i \(-0.593650\pi\)
−0.289983 + 0.957032i \(0.593650\pi\)
\(420\) 0 0
\(421\) −8.97825 −0.437573 −0.218787 0.975773i \(-0.570210\pi\)
−0.218787 + 0.975773i \(0.570210\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −17.3020 −0.839269
\(426\) 0 0
\(427\) −12.1386 −0.587428
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 30.1844 1.45393 0.726966 0.686674i \(-0.240930\pi\)
0.726966 + 0.686674i \(0.240930\pi\)
\(432\) 0 0
\(433\) −22.0000 −1.05725 −0.528626 0.848855i \(-0.677293\pi\)
−0.528626 + 0.848855i \(0.677293\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.01082 0.0483539
\(438\) 0 0
\(439\) 18.2337 0.870246 0.435123 0.900371i \(-0.356705\pi\)
0.435123 + 0.900371i \(0.356705\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 27.9746 1.32911 0.664557 0.747238i \(-0.268620\pi\)
0.664557 + 0.747238i \(0.268620\pi\)
\(444\) 0 0
\(445\) −31.7228 −1.50381
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 22.3561 1.05505 0.527524 0.849540i \(-0.323120\pi\)
0.527524 + 0.849540i \(0.323120\pi\)
\(450\) 0 0
\(451\) −12.0000 −0.565058
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −15.2804 −0.716354
\(456\) 0 0
\(457\) −8.37228 −0.391639 −0.195819 0.980640i \(-0.562737\pi\)
−0.195819 + 0.980640i \(0.562737\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −26.9638 −1.25583 −0.627914 0.778282i \(-0.716091\pi\)
−0.627914 + 0.778282i \(0.716091\pi\)
\(462\) 0 0
\(463\) 2.37228 0.110249 0.0551246 0.998479i \(-0.482444\pi\)
0.0551246 + 0.998479i \(0.482444\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 36.4374 1.68612 0.843062 0.537816i \(-0.180751\pi\)
0.843062 + 0.537816i \(0.180751\pi\)
\(468\) 0 0
\(469\) 9.48913 0.438167
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 24.5659 1.12954
\(474\) 0 0
\(475\) −5.37228 −0.246497
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −33.5932 −1.53491 −0.767455 0.641103i \(-0.778477\pi\)
−0.767455 + 0.641103i \(0.778477\pi\)
\(480\) 0 0
\(481\) 21.4891 0.979820
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −24.1195 −1.09521
\(486\) 0 0
\(487\) −8.00000 −0.362515 −0.181257 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 1.01082 0.0456175 0.0228087 0.999740i \(-0.492739\pi\)
0.0228087 + 0.999740i \(0.492739\pi\)
\(492\) 0 0
\(493\) −3.25544 −0.146618
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −4.79588 −0.215125
\(498\) 0 0
\(499\) 11.1168 0.497658 0.248829 0.968547i \(-0.419954\pi\)
0.248829 + 0.968547i \(0.419954\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 31.5715 1.40770 0.703852 0.710346i \(-0.251462\pi\)
0.703852 + 0.710346i \(0.251462\pi\)
\(504\) 0 0
\(505\) −41.4891 −1.84624
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 24.7540 1.09720 0.548601 0.836084i \(-0.315161\pi\)
0.548601 + 0.836084i \(0.315161\pi\)
\(510\) 0 0
\(511\) −12.1386 −0.536980
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −23.3669 −1.02967
\(516\) 0 0
\(517\) 9.35053 0.411236
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 16.2912 0.713729 0.356864 0.934156i \(-0.383846\pi\)
0.356864 + 0.934156i \(0.383846\pi\)
\(522\) 0 0
\(523\) 18.2337 0.797304 0.398652 0.917102i \(-0.369478\pi\)
0.398652 + 0.917102i \(0.369478\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 15.2804 0.665623
\(528\) 0 0
\(529\) −21.9783 −0.955576
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −10.8608 −0.470433
\(534\) 0 0
\(535\) 14.2337 0.615376
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −3.03245 −0.130617
\(540\) 0 0
\(541\) 21.1168 0.907884 0.453942 0.891031i \(-0.350017\pi\)
0.453942 + 0.891031i \(0.350017\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −16.9257 −0.725016
\(546\) 0 0
\(547\) −22.2337 −0.950644 −0.475322 0.879812i \(-0.657668\pi\)
−0.475322 + 0.879812i \(0.657668\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −1.01082 −0.0430622
\(552\) 0 0
\(553\) 9.48913 0.403519
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −20.5226 −0.869570 −0.434785 0.900534i \(-0.643176\pi\)
−0.434785 + 0.900534i \(0.643176\pi\)
\(558\) 0 0
\(559\) 22.2337 0.940385
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 38.6472 1.62879 0.814393 0.580313i \(-0.197070\pi\)
0.814393 + 0.580313i \(0.197070\pi\)
\(564\) 0 0
\(565\) 38.2337 1.60850
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 3.03245 0.127127 0.0635634 0.997978i \(-0.479753\pi\)
0.0635634 + 0.997978i \(0.479753\pi\)
\(570\) 0 0
\(571\) −2.51087 −0.105077 −0.0525384 0.998619i \(-0.516731\pi\)
−0.0525384 + 0.998619i \(0.516731\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −5.43039 −0.226463
\(576\) 0 0
\(577\) −41.1168 −1.71172 −0.855858 0.517210i \(-0.826970\pi\)
−0.855858 + 0.517210i \(0.826970\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 28.1628 1.16839
\(582\) 0 0
\(583\) 21.7663 0.901469
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.83348 0.0756758 0.0378379 0.999284i \(-0.487953\pi\)
0.0378379 + 0.999284i \(0.487953\pi\)
\(588\) 0 0
\(589\) 4.74456 0.195496
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 4.04326 0.166037 0.0830185 0.996548i \(-0.473544\pi\)
0.0830185 + 0.996548i \(0.473544\pi\)
\(594\) 0 0
\(595\) 24.6060 1.00875
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −12.8824 −0.526361 −0.263181 0.964747i \(-0.584771\pi\)
−0.263181 + 0.964747i \(0.584771\pi\)
\(600\) 0 0
\(601\) −25.2554 −1.03019 −0.515095 0.857133i \(-0.672244\pi\)
−0.515095 + 0.857133i \(0.672244\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 19.6999 0.800916
\(606\) 0 0
\(607\) −28.7446 −1.16671 −0.583353 0.812219i \(-0.698260\pi\)
−0.583353 + 0.812219i \(0.698260\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 8.46284 0.342370
\(612\) 0 0
\(613\) 2.60597 0.105254 0.0526271 0.998614i \(-0.483241\pi\)
0.0526271 + 0.998614i \(0.483241\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 3.22060 0.129657 0.0648283 0.997896i \(-0.479350\pi\)
0.0648283 + 0.997896i \(0.479350\pi\)
\(618\) 0 0
\(619\) −6.97825 −0.280480 −0.140240 0.990118i \(-0.544787\pi\)
−0.140240 + 0.990118i \(0.544787\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 23.3669 0.936174
\(624\) 0 0
\(625\) −23.0000 −0.920000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −34.6040 −1.37975
\(630\) 0 0
\(631\) −15.1168 −0.601792 −0.300896 0.953657i \(-0.597286\pi\)
−0.300896 + 0.953657i \(0.597286\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −41.0452 −1.62883
\(636\) 0 0
\(637\) −2.74456 −0.108744
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −24.7540 −0.977724 −0.488862 0.872361i \(-0.662588\pi\)
−0.488862 + 0.872361i \(0.662588\pi\)
\(642\) 0 0
\(643\) 35.1168 1.38487 0.692437 0.721479i \(-0.256537\pi\)
0.692437 + 0.721479i \(0.256537\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 17.1138 0.672814 0.336407 0.941717i \(-0.390788\pi\)
0.336407 + 0.941717i \(0.390788\pi\)
\(648\) 0 0
\(649\) −24.0000 −0.942082
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 14.0814 0.551047 0.275524 0.961294i \(-0.411149\pi\)
0.275524 + 0.961294i \(0.411149\pi\)
\(654\) 0 0
\(655\) −48.6060 −1.89919
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −11.2371 −0.437735 −0.218867 0.975755i \(-0.570236\pi\)
−0.218867 + 0.975755i \(0.570236\pi\)
\(660\) 0 0
\(661\) −6.74456 −0.262333 −0.131167 0.991360i \(-0.541872\pi\)
−0.131167 + 0.991360i \(0.541872\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 7.64018 0.296273
\(666\) 0 0
\(667\) −1.02175 −0.0395623
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −11.3071 −0.436507
\(672\) 0 0
\(673\) −25.2554 −0.973526 −0.486763 0.873534i \(-0.661822\pi\)
−0.486763 + 0.873534i \(0.661822\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 44.4539 1.70850 0.854252 0.519860i \(-0.174016\pi\)
0.854252 + 0.519860i \(0.174016\pi\)
\(678\) 0 0
\(679\) 17.7663 0.681808
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −34.2277 −1.30968 −0.654842 0.755765i \(-0.727265\pi\)
−0.654842 + 0.755765i \(0.727265\pi\)
\(684\) 0 0
\(685\) −38.8397 −1.48399
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 19.6999 0.750507
\(690\) 0 0
\(691\) −27.1168 −1.03157 −0.515787 0.856717i \(-0.672500\pi\)
−0.515787 + 0.856717i \(0.672500\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 10.0381 0.380767
\(696\) 0 0
\(697\) 17.4891 0.662448
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −30.5607 −1.15426 −0.577131 0.816652i \(-0.695828\pi\)
−0.577131 + 0.816652i \(0.695828\pi\)
\(702\) 0 0
\(703\) −10.7446 −0.405239
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 30.5607 1.14935
\(708\) 0 0
\(709\) 38.0000 1.42712 0.713560 0.700594i \(-0.247082\pi\)
0.713560 + 0.700594i \(0.247082\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 4.79588 0.179607
\(714\) 0 0
\(715\) −14.2337 −0.532310
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 38.8354 1.44832 0.724158 0.689634i \(-0.242229\pi\)
0.724158 + 0.689634i \(0.242229\pi\)
\(720\) 0 0
\(721\) 17.2119 0.641006
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 5.43039 0.201680
\(726\) 0 0
\(727\) 37.3505 1.38525 0.692627 0.721296i \(-0.256453\pi\)
0.692627 + 0.721296i \(0.256453\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −35.8029 −1.32422
\(732\) 0 0
\(733\) 18.4674 0.682108 0.341054 0.940044i \(-0.389216\pi\)
0.341054 + 0.940044i \(0.389216\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8.83915 0.325594
\(738\) 0 0
\(739\) −27.1168 −0.997509 −0.498755 0.866743i \(-0.666209\pi\)
−0.498755 + 0.866743i \(0.666209\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −28.5391 −1.04700 −0.523498 0.852027i \(-0.675373\pi\)
−0.523498 + 0.852027i \(0.675373\pi\)
\(744\) 0 0
\(745\) −66.0951 −2.42154
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −10.4845 −0.383094
\(750\) 0 0
\(751\) −24.4674 −0.892827 −0.446414 0.894827i \(-0.647299\pi\)
−0.446414 + 0.894827i \(0.647299\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −12.8824 −0.468839
\(756\) 0 0
\(757\) −5.11684 −0.185975 −0.0929874 0.995667i \(-0.529642\pi\)
−0.0929874 + 0.995667i \(0.529642\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0.822662 0.0298215 0.0149107 0.999889i \(-0.495254\pi\)
0.0149107 + 0.999889i \(0.495254\pi\)
\(762\) 0 0
\(763\) 12.4674 0.451349
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −21.7216 −0.784320
\(768\) 0 0
\(769\) 6.88316 0.248213 0.124106 0.992269i \(-0.460394\pi\)
0.124106 + 0.992269i \(0.460394\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 5.43039 0.195318 0.0976588 0.995220i \(-0.468865\pi\)
0.0976588 + 0.995220i \(0.468865\pi\)
\(774\) 0 0
\(775\) −25.4891 −0.915596
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 5.43039 0.194564
\(780\) 0 0
\(781\) −4.46738 −0.159855
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −6.44121 −0.229896
\(786\) 0 0
\(787\) 49.9565 1.78076 0.890378 0.455221i \(-0.150440\pi\)
0.890378 + 0.455221i \(0.150440\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −28.1628 −1.00135
\(792\) 0 0
\(793\) −10.2337 −0.363409
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −7.45202 −0.263964 −0.131982 0.991252i \(-0.542134\pi\)
−0.131982 + 0.991252i \(0.542134\pi\)
\(798\) 0 0
\(799\) −13.6277 −0.482114
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −11.3071 −0.399020
\(804\) 0 0
\(805\) 7.72281 0.272193
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −3.59691 −0.126461 −0.0632303 0.997999i \(-0.520140\pi\)
−0.0632303 + 0.997999i \(0.520140\pi\)
\(810\) 0 0
\(811\) 36.7446 1.29028 0.645138 0.764066i \(-0.276800\pi\)
0.645138 + 0.764066i \(0.276800\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −69.2079 −2.42425
\(816\) 0 0
\(817\) −11.1168 −0.388929
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −29.3617 −1.02473 −0.512366 0.858767i \(-0.671231\pi\)
−0.512366 + 0.858767i \(0.671231\pi\)
\(822\) 0 0
\(823\) −8.60597 −0.299985 −0.149993 0.988687i \(-0.547925\pi\)
−0.149993 + 0.988687i \(0.547925\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 28.5391 0.992401 0.496200 0.868208i \(-0.334728\pi\)
0.496200 + 0.868208i \(0.334728\pi\)
\(828\) 0 0
\(829\) −1.25544 −0.0436031 −0.0218016 0.999762i \(-0.506940\pi\)
−0.0218016 + 0.999762i \(0.506940\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 4.41957 0.153129
\(834\) 0 0
\(835\) −20.7446 −0.717895
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −30.1844 −1.04208 −0.521041 0.853532i \(-0.674456\pi\)
−0.521041 + 0.853532i \(0.674456\pi\)
\(840\) 0 0
\(841\) −27.9783 −0.964767
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 28.9854 0.997129
\(846\) 0 0
\(847\) −14.5109 −0.498600
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −10.8608 −0.372303
\(852\) 0 0
\(853\) −38.4674 −1.31710 −0.658549 0.752538i \(-0.728829\pi\)
−0.658549 + 0.752538i \(0.728829\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −35.2385 −1.20372 −0.601862 0.798600i \(-0.705574\pi\)
−0.601862 + 0.798600i \(0.705574\pi\)
\(858\) 0 0
\(859\) −3.11684 −0.106345 −0.0531727 0.998585i \(-0.516933\pi\)
−0.0531727 + 0.998585i \(0.516933\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 14.9040 0.507340 0.253670 0.967291i \(-0.418362\pi\)
0.253670 + 0.967291i \(0.418362\pi\)
\(864\) 0 0
\(865\) 3.25544 0.110688
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 8.83915 0.299847
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −2.84429 −0.0961547
\(876\) 0 0
\(877\) 14.0000 0.472746 0.236373 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −0.822662 −0.0277162 −0.0138581 0.999904i \(-0.504411\pi\)
−0.0138581 + 0.999904i \(0.504411\pi\)
\(882\) 0 0
\(883\) −3.11684 −0.104890 −0.0524451 0.998624i \(-0.516701\pi\)
−0.0524451 + 0.998624i \(0.516701\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −21.7216 −0.729338 −0.364669 0.931137i \(-0.618818\pi\)
−0.364669 + 0.931137i \(0.618818\pi\)
\(888\) 0 0
\(889\) 30.2337 1.01401
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −4.23142 −0.141599
\(894\) 0 0
\(895\) −14.2337 −0.475780
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −4.79588 −0.159952
\(900\) 0 0
\(901\) −31.7228 −1.05684
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −62.7667 −2.08644
\(906\) 0 0
\(907\) −23.2554 −0.772184 −0.386092 0.922460i \(-0.626175\pi\)
−0.386092 + 0.922460i \(0.626175\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 37.0019 1.22593 0.612964 0.790111i \(-0.289977\pi\)
0.612964 + 0.790111i \(0.289977\pi\)
\(912\) 0 0
\(913\) 26.2337 0.868208
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 35.8029 1.18232
\(918\) 0 0
\(919\) −18.9783 −0.626035 −0.313017 0.949747i \(-0.601340\pi\)
−0.313017 + 0.949747i \(0.601340\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −4.04326 −0.133086
\(924\) 0 0
\(925\) 57.7228 1.89791
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 4.79588 0.157348 0.0786739 0.996900i \(-0.474931\pi\)
0.0786739 + 0.996900i \(0.474931\pi\)
\(930\) 0 0
\(931\) 1.37228 0.0449747
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 22.9205 0.749581
\(936\) 0 0
\(937\) −5.11684 −0.167160 −0.0835800 0.996501i \(-0.526635\pi\)
−0.0835800 + 0.996501i \(0.526635\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 24.7540 0.806957 0.403479 0.914989i \(-0.367801\pi\)
0.403479 + 0.914989i \(0.367801\pi\)
\(942\) 0 0
\(943\) 5.48913 0.178751
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 9.84996 0.320081 0.160040 0.987110i \(-0.448838\pi\)
0.160040 + 0.987110i \(0.448838\pi\)
\(948\) 0 0
\(949\) −10.2337 −0.332200
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 9.47365 0.306882 0.153441 0.988158i \(-0.450965\pi\)
0.153441 + 0.988158i \(0.450965\pi\)
\(954\) 0 0
\(955\) −69.3505 −2.24413
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 28.6091 0.923837
\(960\) 0 0
\(961\) −8.48913 −0.273843
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −52.2823 −1.68303
\(966\) 0 0
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 51.5296 1.65366 0.826832 0.562448i \(-0.190140\pi\)
0.826832 + 0.562448i \(0.190140\pi\)
\(972\) 0 0
\(973\) −7.39403 −0.237042
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −20.7107 −0.662595 −0.331298 0.943526i \(-0.607486\pi\)
−0.331298 + 0.943526i \(0.607486\pi\)
\(978\) 0 0
\(979\) 21.7663 0.695654
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −52.2823 −1.66755 −0.833773 0.552108i \(-0.813824\pi\)
−0.833773 + 0.552108i \(0.813824\pi\)
\(984\) 0 0
\(985\) 76.4674 2.43645
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −11.2371 −0.357319
\(990\) 0 0
\(991\) −32.0000 −1.01651 −0.508257 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 2.84429 0.0901702
\(996\) 0 0
\(997\) −19.3505 −0.612837 −0.306419 0.951897i \(-0.599131\pi\)
−0.306419 + 0.951897i \(0.599131\pi\)
\(998\) 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 2736.2.a.bf.1.1 4
3.2 odd 2 inner 2736.2.a.bf.1.4 4
4.3 odd 2 171.2.a.e.1.4 yes 4
12.11 even 2 171.2.a.e.1.1 4
20.19 odd 2 4275.2.a.bp.1.1 4
28.27 even 2 8379.2.a.bw.1.4 4
60.59 even 2 4275.2.a.bp.1.4 4
76.75 even 2 3249.2.a.bf.1.1 4
84.83 odd 2 8379.2.a.bw.1.1 4
228.227 odd 2 3249.2.a.bf.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
171.2.a.e.1.1 4 12.11 even 2
171.2.a.e.1.4 yes 4 4.3 odd 2
2736.2.a.bf.1.1 4 1.1 even 1 trivial
2736.2.a.bf.1.4 4 3.2 odd 2 inner
3249.2.a.bf.1.1 4 76.75 even 2
3249.2.a.bf.1.4 4 228.227 odd 2
4275.2.a.bp.1.1 4 20.19 odd 2
4275.2.a.bp.1.4 4 60.59 even 2
8379.2.a.bw.1.1 4 84.83 odd 2
8379.2.a.bw.1.4 4 28.27 even 2