Properties

Label 4864.2.a.bk.1.4
Level $4864$
Weight $2$
Character 4864.1
Self dual yes
Analytic conductor $38.839$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 4864 = 2^{8} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4864.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.8392355432\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{19})\)
Defining polynomial: \(x^{4} - 11 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1216)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(3.04547\) of defining polynomial
Character \(\chi\) \(=\) 4864.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.04547 q^{5} +0.418627 q^{7} -3.00000 q^{9} +O(q^{10})\) \(q+3.04547 q^{5} +0.418627 q^{7} -3.00000 q^{9} +5.27492 q^{11} +2.62685 q^{13} +3.27492 q^{17} -1.00000 q^{19} +3.46410 q^{23} +4.27492 q^{25} +2.62685 q^{29} -6.09095 q^{31} +1.27492 q^{35} +9.55505 q^{37} -4.54983 q^{41} +2.72508 q^{43} -9.13642 q^{45} +0.418627 q^{47} -6.82475 q^{49} +10.3923 q^{53} +16.0646 q^{55} -6.54983 q^{59} -3.04547 q^{61} -1.25588 q^{63} +8.00000 q^{65} +6.54983 q^{67} -11.3446 q^{71} +3.27492 q^{73} +2.20822 q^{77} -13.0192 q^{79} +9.00000 q^{81} -17.0997 q^{83} +9.97368 q^{85} +10.0000 q^{89} +1.09967 q^{91} -3.04547 q^{95} +16.5498 q^{97} -15.8248 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 12q^{9} + O(q^{10}) \) \( 4q - 12q^{9} + 6q^{11} - 2q^{17} - 4q^{19} + 2q^{25} - 10q^{35} + 12q^{41} + 26q^{43} + 18q^{49} + 4q^{59} + 32q^{65} - 4q^{67} - 2q^{73} + 36q^{81} - 8q^{83} + 40q^{89} - 56q^{91} + 36q^{97} - 18q^{99} + O(q^{100}) \)

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
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 0 0
\(5\) 3.04547 1.36198 0.680989 0.732294i \(-0.261550\pi\)
0.680989 + 0.732294i \(0.261550\pi\)
\(6\) 0 0
\(7\) 0.418627 0.158226 0.0791130 0.996866i \(-0.474791\pi\)
0.0791130 + 0.996866i \(0.474791\pi\)
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) 5.27492 1.59045 0.795224 0.606316i \(-0.207353\pi\)
0.795224 + 0.606316i \(0.207353\pi\)
\(12\) 0 0
\(13\) 2.62685 0.728557 0.364278 0.931290i \(-0.381316\pi\)
0.364278 + 0.931290i \(0.381316\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.27492 0.794284 0.397142 0.917757i \(-0.370002\pi\)
0.397142 + 0.917757i \(0.370002\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.46410 0.722315 0.361158 0.932505i \(-0.382382\pi\)
0.361158 + 0.932505i \(0.382382\pi\)
\(24\) 0 0
\(25\) 4.27492 0.854983
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.62685 0.487793 0.243897 0.969801i \(-0.421574\pi\)
0.243897 + 0.969801i \(0.421574\pi\)
\(30\) 0 0
\(31\) −6.09095 −1.09397 −0.546983 0.837143i \(-0.684224\pi\)
−0.546983 + 0.837143i \(0.684224\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.27492 0.215500
\(36\) 0 0
\(37\) 9.55505 1.57084 0.785420 0.618963i \(-0.212447\pi\)
0.785420 + 0.618963i \(0.212447\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4.54983 −0.710565 −0.355282 0.934759i \(-0.615615\pi\)
−0.355282 + 0.934759i \(0.615615\pi\)
\(42\) 0 0
\(43\) 2.72508 0.415571 0.207786 0.978174i \(-0.433374\pi\)
0.207786 + 0.978174i \(0.433374\pi\)
\(44\) 0 0
\(45\) −9.13642 −1.36198
\(46\) 0 0
\(47\) 0.418627 0.0610630 0.0305315 0.999534i \(-0.490280\pi\)
0.0305315 + 0.999534i \(0.490280\pi\)
\(48\) 0 0
\(49\) −6.82475 −0.974965
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 10.3923 1.42749 0.713746 0.700404i \(-0.246997\pi\)
0.713746 + 0.700404i \(0.246997\pi\)
\(54\) 0 0
\(55\) 16.0646 2.16615
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −6.54983 −0.852716 −0.426358 0.904555i \(-0.640204\pi\)
−0.426358 + 0.904555i \(0.640204\pi\)
\(60\) 0 0
\(61\) −3.04547 −0.389933 −0.194967 0.980810i \(-0.562460\pi\)
−0.194967 + 0.980810i \(0.562460\pi\)
\(62\) 0 0
\(63\) −1.25588 −0.158226
\(64\) 0 0
\(65\) 8.00000 0.992278
\(66\) 0 0
\(67\) 6.54983 0.800190 0.400095 0.916474i \(-0.368977\pi\)
0.400095 + 0.916474i \(0.368977\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −11.3446 −1.34636 −0.673181 0.739478i \(-0.735072\pi\)
−0.673181 + 0.739478i \(0.735072\pi\)
\(72\) 0 0
\(73\) 3.27492 0.383300 0.191650 0.981463i \(-0.438616\pi\)
0.191650 + 0.981463i \(0.438616\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.20822 0.251650
\(78\) 0 0
\(79\) −13.0192 −1.46477 −0.732385 0.680891i \(-0.761593\pi\)
−0.732385 + 0.680891i \(0.761593\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) −17.0997 −1.87693 −0.938466 0.345371i \(-0.887753\pi\)
−0.938466 + 0.345371i \(0.887753\pi\)
\(84\) 0 0
\(85\) 9.97368 1.08180
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 1.09967 0.115277
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −3.04547 −0.312459
\(96\) 0 0
\(97\) 16.5498 1.68038 0.840191 0.542291i \(-0.182443\pi\)
0.840191 + 0.542291i \(0.182443\pi\)
\(98\) 0 0
\(99\) −15.8248 −1.59045
\(100\) 0 0
\(101\) −13.8564 −1.37876 −0.689382 0.724398i \(-0.742118\pi\)
−0.689382 + 0.724398i \(0.742118\pi\)
\(102\) 0 0
\(103\) 6.09095 0.600159 0.300080 0.953914i \(-0.402987\pi\)
0.300080 + 0.953914i \(0.402987\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.45017 0.140193 0.0700964 0.997540i \(-0.477669\pi\)
0.0700964 + 0.997540i \(0.477669\pi\)
\(108\) 0 0
\(109\) 3.46410 0.331801 0.165900 0.986143i \(-0.446947\pi\)
0.165900 + 0.986143i \(0.446947\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 12.5498 1.18059 0.590295 0.807188i \(-0.299012\pi\)
0.590295 + 0.807188i \(0.299012\pi\)
\(114\) 0 0
\(115\) 10.5498 0.983777
\(116\) 0 0
\(117\) −7.88054 −0.728557
\(118\) 0 0
\(119\) 1.37097 0.125676
\(120\) 0 0
\(121\) 16.8248 1.52952
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −2.20822 −0.197509
\(126\) 0 0
\(127\) 17.4356 1.54716 0.773579 0.633699i \(-0.218464\pi\)
0.773579 + 0.633699i \(0.218464\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −5.27492 −0.460872 −0.230436 0.973088i \(-0.574015\pi\)
−0.230436 + 0.973088i \(0.574015\pi\)
\(132\) 0 0
\(133\) −0.418627 −0.0362995
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 12.7251 1.08718 0.543589 0.839352i \(-0.317065\pi\)
0.543589 + 0.839352i \(0.317065\pi\)
\(138\) 0 0
\(139\) 2.72508 0.231139 0.115569 0.993299i \(-0.463131\pi\)
0.115569 + 0.993299i \(0.463131\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 13.8564 1.15873
\(144\) 0 0
\(145\) 8.00000 0.664364
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −16.0646 −1.31607 −0.658033 0.752989i \(-0.728611\pi\)
−0.658033 + 0.752989i \(0.728611\pi\)
\(150\) 0 0
\(151\) −0.837253 −0.0681347 −0.0340674 0.999420i \(-0.510846\pi\)
−0.0340674 + 0.999420i \(0.510846\pi\)
\(152\) 0 0
\(153\) −9.82475 −0.794284
\(154\) 0 0
\(155\) −18.5498 −1.48996
\(156\) 0 0
\(157\) −12.1819 −0.972221 −0.486111 0.873897i \(-0.661585\pi\)
−0.486111 + 0.873897i \(0.661585\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1.45017 0.114289
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −19.1101 −1.47878 −0.739392 0.673275i \(-0.764887\pi\)
−0.739392 + 0.673275i \(0.764887\pi\)
\(168\) 0 0
\(169\) −6.09967 −0.469205
\(170\) 0 0
\(171\) 3.00000 0.229416
\(172\) 0 0
\(173\) 8.71780 0.662802 0.331401 0.943490i \(-0.392479\pi\)
0.331401 + 0.943490i \(0.392479\pi\)
\(174\) 0 0
\(175\) 1.78959 0.135281
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 21.0997 1.57706 0.788532 0.614994i \(-0.210842\pi\)
0.788532 + 0.614994i \(0.210842\pi\)
\(180\) 0 0
\(181\) 3.46410 0.257485 0.128742 0.991678i \(-0.458906\pi\)
0.128742 + 0.991678i \(0.458906\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 29.0997 2.13945
\(186\) 0 0
\(187\) 17.2749 1.26327
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −11.7633 −0.851161 −0.425580 0.904921i \(-0.639930\pi\)
−0.425580 + 0.904921i \(0.639930\pi\)
\(192\) 0 0
\(193\) 11.0997 0.798972 0.399486 0.916739i \(-0.369189\pi\)
0.399486 + 0.916739i \(0.369189\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 5.25370 0.374310 0.187155 0.982330i \(-0.440073\pi\)
0.187155 + 0.982330i \(0.440073\pi\)
\(198\) 0 0
\(199\) −14.2750 −1.01193 −0.505965 0.862554i \(-0.668864\pi\)
−0.505965 + 0.862554i \(0.668864\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.09967 0.0771816
\(204\) 0 0
\(205\) −13.8564 −0.967773
\(206\) 0 0
\(207\) −10.3923 −0.722315
\(208\) 0 0
\(209\) −5.27492 −0.364874
\(210\) 0 0
\(211\) 26.5498 1.82777 0.913883 0.405978i \(-0.133069\pi\)
0.913883 + 0.405978i \(0.133069\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 8.29917 0.565999
\(216\) 0 0
\(217\) −2.54983 −0.173094
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 8.60271 0.578681
\(222\) 0 0
\(223\) −17.4356 −1.16757 −0.583787 0.811907i \(-0.698430\pi\)
−0.583787 + 0.811907i \(0.698430\pi\)
\(224\) 0 0
\(225\) −12.8248 −0.854983
\(226\) 0 0
\(227\) −4.00000 −0.265489 −0.132745 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) 0 0
\(229\) −17.7391 −1.17224 −0.586118 0.810226i \(-0.699344\pi\)
−0.586118 + 0.810226i \(0.699344\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 20.3746 1.33478 0.667392 0.744707i \(-0.267411\pi\)
0.667392 + 0.744707i \(0.267411\pi\)
\(234\) 0 0
\(235\) 1.27492 0.0831664
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 25.6197 1.65720 0.828600 0.559842i \(-0.189138\pi\)
0.828600 + 0.559842i \(0.189138\pi\)
\(240\) 0 0
\(241\) −12.5498 −0.808406 −0.404203 0.914669i \(-0.632451\pi\)
−0.404203 + 0.914669i \(0.632451\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −20.7846 −1.32788
\(246\) 0 0
\(247\) −2.62685 −0.167142
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −10.7251 −0.676961 −0.338481 0.940973i \(-0.609913\pi\)
−0.338481 + 0.940973i \(0.609913\pi\)
\(252\) 0 0
\(253\) 18.2728 1.14880
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 7.45017 0.464729 0.232364 0.972629i \(-0.425354\pi\)
0.232364 + 0.972629i \(0.425354\pi\)
\(258\) 0 0
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) −7.88054 −0.487793
\(262\) 0 0
\(263\) 1.25588 0.0774409 0.0387204 0.999250i \(-0.487672\pi\)
0.0387204 + 0.999250i \(0.487672\pi\)
\(264\) 0 0
\(265\) 31.6495 1.94421
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 5.13861 0.313306 0.156653 0.987654i \(-0.449929\pi\)
0.156653 + 0.987654i \(0.449929\pi\)
\(270\) 0 0
\(271\) 27.8279 1.69042 0.845212 0.534431i \(-0.179474\pi\)
0.845212 + 0.534431i \(0.179474\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 22.5498 1.35981
\(276\) 0 0
\(277\) 16.0646 0.965230 0.482615 0.875833i \(-0.339687\pi\)
0.482615 + 0.875833i \(0.339687\pi\)
\(278\) 0 0
\(279\) 18.2728 1.09397
\(280\) 0 0
\(281\) −11.0997 −0.662151 −0.331075 0.943604i \(-0.607411\pi\)
−0.331075 + 0.943604i \(0.607411\pi\)
\(282\) 0 0
\(283\) −26.3746 −1.56781 −0.783903 0.620883i \(-0.786774\pi\)
−0.783903 + 0.620883i \(0.786774\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −1.90468 −0.112430
\(288\) 0 0
\(289\) −6.27492 −0.369113
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −30.3397 −1.77246 −0.886231 0.463244i \(-0.846685\pi\)
−0.886231 + 0.463244i \(0.846685\pi\)
\(294\) 0 0
\(295\) −19.9474 −1.16138
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 9.09967 0.526247
\(300\) 0 0
\(301\) 1.14079 0.0657542
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −9.27492 −0.531080
\(306\) 0 0
\(307\) 1.45017 0.0827653 0.0413827 0.999143i \(-0.486824\pi\)
0.0413827 + 0.999143i \(0.486824\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −24.7824 −1.40528 −0.702641 0.711544i \(-0.747996\pi\)
−0.702641 + 0.711544i \(0.747996\pi\)
\(312\) 0 0
\(313\) −15.0997 −0.853484 −0.426742 0.904373i \(-0.640339\pi\)
−0.426742 + 0.904373i \(0.640339\pi\)
\(314\) 0 0
\(315\) −3.82475 −0.215500
\(316\) 0 0
\(317\) −8.71780 −0.489640 −0.244820 0.969569i \(-0.578729\pi\)
−0.244820 + 0.969569i \(0.578729\pi\)
\(318\) 0 0
\(319\) 13.8564 0.775810
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −3.27492 −0.182221
\(324\) 0 0
\(325\) 11.2296 0.622904
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0.175248 0.00966175
\(330\) 0 0
\(331\) 9.45017 0.519428 0.259714 0.965686i \(-0.416372\pi\)
0.259714 + 0.965686i \(0.416372\pi\)
\(332\) 0 0
\(333\) −28.6652 −1.57084
\(334\) 0 0
\(335\) 19.9474 1.08984
\(336\) 0 0
\(337\) 33.6495 1.83301 0.916503 0.400029i \(-0.131000\pi\)
0.916503 + 0.400029i \(0.131000\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −32.1293 −1.73990
\(342\) 0 0
\(343\) −5.78741 −0.312491
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 7.82475 0.420055 0.210027 0.977696i \(-0.432645\pi\)
0.210027 + 0.977696i \(0.432645\pi\)
\(348\) 0 0
\(349\) 12.7156 0.680651 0.340326 0.940308i \(-0.389463\pi\)
0.340326 + 0.940308i \(0.389463\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −10.0000 −0.532246 −0.266123 0.963939i \(-0.585743\pi\)
−0.266123 + 0.963939i \(0.585743\pi\)
\(354\) 0 0
\(355\) −34.5498 −1.83371
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −12.6005 −0.665030 −0.332515 0.943098i \(-0.607897\pi\)
−0.332515 + 0.943098i \(0.607897\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 9.97368 0.522046
\(366\) 0 0
\(367\) −15.6460 −0.816715 −0.408357 0.912822i \(-0.633898\pi\)
−0.408357 + 0.912822i \(0.633898\pi\)
\(368\) 0 0
\(369\) 13.6495 0.710565
\(370\) 0 0
\(371\) 4.35050 0.225867
\(372\) 0 0
\(373\) −0.952341 −0.0493104 −0.0246552 0.999696i \(-0.507849\pi\)
−0.0246552 + 0.999696i \(0.507849\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6.90033 0.355385
\(378\) 0 0
\(379\) 23.6495 1.21479 0.607397 0.794399i \(-0.292214\pi\)
0.607397 + 0.794399i \(0.292214\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 32.1293 1.64173 0.820864 0.571124i \(-0.193492\pi\)
0.820864 + 0.571124i \(0.193492\pi\)
\(384\) 0 0
\(385\) 6.72508 0.342742
\(386\) 0 0
\(387\) −8.17525 −0.415571
\(388\) 0 0
\(389\) 4.71998 0.239313 0.119656 0.992815i \(-0.461821\pi\)
0.119656 + 0.992815i \(0.461821\pi\)
\(390\) 0 0
\(391\) 11.3446 0.573723
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −39.6495 −1.99498
\(396\) 0 0
\(397\) 32.6630 1.63931 0.819654 0.572859i \(-0.194166\pi\)
0.819654 + 0.572859i \(0.194166\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 35.0997 1.75279 0.876397 0.481590i \(-0.159940\pi\)
0.876397 + 0.481590i \(0.159940\pi\)
\(402\) 0 0
\(403\) −16.0000 −0.797017
\(404\) 0 0
\(405\) 27.4093 1.36198
\(406\) 0 0
\(407\) 50.4021 2.49834
\(408\) 0 0
\(409\) −18.0000 −0.890043 −0.445021 0.895520i \(-0.646804\pi\)
−0.445021 + 0.895520i \(0.646804\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −2.74194 −0.134922
\(414\) 0 0
\(415\) −52.0766 −2.55634
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 24.2487 1.18181 0.590905 0.806741i \(-0.298771\pi\)
0.590905 + 0.806741i \(0.298771\pi\)
\(422\) 0 0
\(423\) −1.25588 −0.0610630
\(424\) 0 0
\(425\) 14.0000 0.679100
\(426\) 0 0
\(427\) −1.27492 −0.0616976
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −1.67451 −0.0806582 −0.0403291 0.999186i \(-0.512841\pi\)
−0.0403291 + 0.999186i \(0.512841\pi\)
\(432\) 0 0
\(433\) −17.6495 −0.848181 −0.424091 0.905620i \(-0.639406\pi\)
−0.424091 + 0.905620i \(0.639406\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −3.46410 −0.165710
\(438\) 0 0
\(439\) −31.2920 −1.49349 −0.746743 0.665113i \(-0.768383\pi\)
−0.746743 + 0.665113i \(0.768383\pi\)
\(440\) 0 0
\(441\) 20.4743 0.974965
\(442\) 0 0
\(443\) −10.3746 −0.492911 −0.246456 0.969154i \(-0.579266\pi\)
−0.246456 + 0.969154i \(0.579266\pi\)
\(444\) 0 0
\(445\) 30.4547 1.44369
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −20.5498 −0.969807 −0.484903 0.874568i \(-0.661145\pi\)
−0.484903 + 0.874568i \(0.661145\pi\)
\(450\) 0 0
\(451\) −24.0000 −1.13012
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 3.34901 0.157004
\(456\) 0 0
\(457\) 15.2749 0.714530 0.357265 0.934003i \(-0.383709\pi\)
0.357265 + 0.934003i \(0.383709\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 12.7156 0.592225 0.296113 0.955153i \(-0.404310\pi\)
0.296113 + 0.955153i \(0.404310\pi\)
\(462\) 0 0
\(463\) 2.93039 0.136187 0.0680933 0.997679i \(-0.478308\pi\)
0.0680933 + 0.997679i \(0.478308\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 8.17525 0.378305 0.189153 0.981948i \(-0.439426\pi\)
0.189153 + 0.981948i \(0.439426\pi\)
\(468\) 0 0
\(469\) 2.74194 0.126611
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 14.3746 0.660944
\(474\) 0 0
\(475\) −4.27492 −0.196147
\(476\) 0 0
\(477\) −31.1769 −1.42749
\(478\) 0 0
\(479\) 17.3205 0.791394 0.395697 0.918381i \(-0.370503\pi\)
0.395697 + 0.918381i \(0.370503\pi\)
\(480\) 0 0
\(481\) 25.0997 1.14445
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 50.4021 2.28864
\(486\) 0 0
\(487\) 39.0575 1.76986 0.884931 0.465722i \(-0.154205\pi\)
0.884931 + 0.465722i \(0.154205\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) 8.60271 0.387447
\(494\) 0 0
\(495\) −48.1939 −2.16615
\(496\) 0 0
\(497\) −4.74917 −0.213029
\(498\) 0 0
\(499\) 26.7251 1.19638 0.598190 0.801355i \(-0.295887\pi\)
0.598190 + 0.801355i \(0.295887\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −41.6843 −1.85861 −0.929306 0.369311i \(-0.879594\pi\)
−0.929306 + 0.369311i \(0.879594\pi\)
\(504\) 0 0
\(505\) −42.1993 −1.87785
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −4.30136 −0.190654 −0.0953271 0.995446i \(-0.530390\pi\)
−0.0953271 + 0.995446i \(0.530390\pi\)
\(510\) 0 0
\(511\) 1.37097 0.0606480
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 18.5498 0.817403
\(516\) 0 0
\(517\) 2.20822 0.0971175
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 27.0997 1.18726 0.593629 0.804739i \(-0.297695\pi\)
0.593629 + 0.804739i \(0.297695\pi\)
\(522\) 0 0
\(523\) −5.45017 −0.238319 −0.119160 0.992875i \(-0.538020\pi\)
−0.119160 + 0.992875i \(0.538020\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −19.9474 −0.868920
\(528\) 0 0
\(529\) −11.0000 −0.478261
\(530\) 0 0
\(531\) 19.6495 0.852716
\(532\) 0 0
\(533\) −11.9517 −0.517687
\(534\) 0 0
\(535\) 4.41644 0.190939
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −36.0000 −1.55063
\(540\) 0 0
\(541\) −29.0838 −1.25041 −0.625205 0.780461i \(-0.714985\pi\)
−0.625205 + 0.780461i \(0.714985\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 10.5498 0.451905
\(546\) 0 0
\(547\) −32.0000 −1.36822 −0.684111 0.729378i \(-0.739809\pi\)
−0.684111 + 0.729378i \(0.739809\pi\)
\(548\) 0 0
\(549\) 9.13642 0.389933
\(550\) 0 0
\(551\) −2.62685 −0.111907
\(552\) 0 0
\(553\) −5.45017 −0.231765
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −29.9210 −1.26779 −0.633897 0.773417i \(-0.718546\pi\)
−0.633897 + 0.773417i \(0.718546\pi\)
\(558\) 0 0
\(559\) 7.15838 0.302767
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 0 0
\(565\) 38.2202 1.60794
\(566\) 0 0
\(567\) 3.76764 0.158226
\(568\) 0 0
\(569\) 19.4502 0.815393 0.407697 0.913117i \(-0.366332\pi\)
0.407697 + 0.913117i \(0.366332\pi\)
\(570\) 0 0
\(571\) 41.0997 1.71997 0.859984 0.510321i \(-0.170474\pi\)
0.859984 + 0.510321i \(0.170474\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 14.8087 0.617567
\(576\) 0 0
\(577\) 23.2749 0.968947 0.484474 0.874806i \(-0.339011\pi\)
0.484474 + 0.874806i \(0.339011\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −7.15838 −0.296980
\(582\) 0 0
\(583\) 54.8185 2.27035
\(584\) 0 0
\(585\) −24.0000 −0.992278
\(586\) 0 0
\(587\) 37.2749 1.53850 0.769250 0.638948i \(-0.220630\pi\)
0.769250 + 0.638948i \(0.220630\pi\)
\(588\) 0 0
\(589\) 6.09095 0.250973
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 19.0997 0.784329 0.392165 0.919895i \(-0.371726\pi\)
0.392165 + 0.919895i \(0.371726\pi\)
\(594\) 0 0
\(595\) 4.17525 0.171168
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −9.43996 −0.385706 −0.192853 0.981228i \(-0.561774\pi\)
−0.192853 + 0.981228i \(0.561774\pi\)
\(600\) 0 0
\(601\) −11.0997 −0.452765 −0.226382 0.974038i \(-0.572690\pi\)
−0.226382 + 0.974038i \(0.572690\pi\)
\(602\) 0 0
\(603\) −19.6495 −0.800190
\(604\) 0 0
\(605\) 51.2394 2.08318
\(606\) 0 0
\(607\) −29.3873 −1.19279 −0.596397 0.802689i \(-0.703402\pi\)
−0.596397 + 0.802689i \(0.703402\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.09967 0.0444878
\(612\) 0 0
\(613\) 29.9210 1.20850 0.604250 0.796795i \(-0.293473\pi\)
0.604250 + 0.796795i \(0.293473\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −12.7251 −0.512293 −0.256146 0.966638i \(-0.582453\pi\)
−0.256146 + 0.966638i \(0.582453\pi\)
\(618\) 0 0
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 4.18627 0.167719
\(624\) 0 0
\(625\) −28.0997 −1.12399
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 31.2920 1.24769
\(630\) 0 0
\(631\) 13.4378 0.534950 0.267475 0.963565i \(-0.413811\pi\)
0.267475 + 0.963565i \(0.413811\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 53.0997 2.10720
\(636\) 0 0
\(637\) −17.9276 −0.710317
\(638\) 0 0
\(639\) 34.0339 1.34636
\(640\) 0 0
\(641\) −12.5498 −0.495689 −0.247844 0.968800i \(-0.579722\pi\)
−0.247844 + 0.968800i \(0.579722\pi\)
\(642\) 0 0
\(643\) −44.9244 −1.77165 −0.885823 0.464023i \(-0.846405\pi\)
−0.885823 + 0.464023i \(0.846405\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 25.6197 1.00721 0.503607 0.863933i \(-0.332006\pi\)
0.503607 + 0.863933i \(0.332006\pi\)
\(648\) 0 0
\(649\) −34.5498 −1.35620
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −16.0646 −0.628657 −0.314329 0.949314i \(-0.601779\pi\)
−0.314329 + 0.949314i \(0.601779\pi\)
\(654\) 0 0
\(655\) −16.0646 −0.627697
\(656\) 0 0
\(657\) −9.82475 −0.383300
\(658\) 0 0
\(659\) −13.4502 −0.523944 −0.261972 0.965075i \(-0.584373\pi\)
−0.261972 + 0.965075i \(0.584373\pi\)
\(660\) 0 0
\(661\) 9.55505 0.371648 0.185824 0.982583i \(-0.440505\pi\)
0.185824 + 0.982583i \(0.440505\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −1.27492 −0.0494392
\(666\) 0 0
\(667\) 9.09967 0.352341
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −16.0646 −0.620168
\(672\) 0 0
\(673\) −47.0997 −1.81556 −0.907779 0.419448i \(-0.862224\pi\)
−0.907779 + 0.419448i \(0.862224\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0.722166 0.0277551 0.0138775 0.999904i \(-0.495582\pi\)
0.0138775 + 0.999904i \(0.495582\pi\)
\(678\) 0 0
\(679\) 6.92820 0.265880
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −22.1993 −0.849434 −0.424717 0.905326i \(-0.639626\pi\)
−0.424717 + 0.905326i \(0.639626\pi\)
\(684\) 0 0
\(685\) 38.7539 1.48071
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 27.2990 1.04001
\(690\) 0 0
\(691\) 23.4743 0.893003 0.446501 0.894783i \(-0.352670\pi\)
0.446501 + 0.894783i \(0.352670\pi\)
\(692\) 0 0
\(693\) −6.62466 −0.251650
\(694\) 0 0
\(695\) 8.29917 0.314806
\(696\) 0 0
\(697\) −14.9003 −0.564390
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −21.0148 −0.793717 −0.396859 0.917880i \(-0.629900\pi\)
−0.396859 + 0.917880i \(0.629900\pi\)
\(702\) 0 0
\(703\) −9.55505 −0.360376
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −5.80066 −0.218156
\(708\) 0 0
\(709\) −38.2202 −1.43539 −0.717695 0.696358i \(-0.754803\pi\)
−0.717695 + 0.696358i \(0.754803\pi\)
\(710\) 0 0
\(711\) 39.0575 1.46477
\(712\) 0 0
\(713\) −21.0997 −0.790189
\(714\) 0 0
\(715\) 42.1993 1.57817
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −51.6580 −1.92652 −0.963259 0.268575i \(-0.913447\pi\)
−0.963259 + 0.268575i \(0.913447\pi\)
\(720\) 0 0
\(721\) 2.54983 0.0949608
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 11.2296 0.417055
\(726\) 0 0
\(727\) −10.9260 −0.405224 −0.202612 0.979259i \(-0.564943\pi\)
−0.202612 + 0.979259i \(0.564943\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 8.92442 0.330082
\(732\) 0 0
\(733\) −19.1101 −0.705848 −0.352924 0.935652i \(-0.614813\pi\)
−0.352924 + 0.935652i \(0.614813\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 34.5498 1.27266
\(738\) 0 0
\(739\) 7.82475 0.287838 0.143919 0.989589i \(-0.454029\pi\)
0.143919 + 0.989589i \(0.454029\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −52.9139 −1.94122 −0.970611 0.240655i \(-0.922638\pi\)
−0.970611 + 0.240655i \(0.922638\pi\)
\(744\) 0 0
\(745\) −48.9244 −1.79245
\(746\) 0 0
\(747\) 51.2990 1.87693
\(748\) 0 0
\(749\) 0.607078 0.0221822
\(750\) 0 0
\(751\) 36.3155 1.32517 0.662586 0.748986i \(-0.269459\pi\)
0.662586 + 0.748986i \(0.269459\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −2.54983 −0.0927980
\(756\) 0 0
\(757\) −3.04547 −0.110690 −0.0553448 0.998467i \(-0.517626\pi\)
−0.0553448 + 0.998467i \(0.517626\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −3.27492 −0.118716 −0.0593578 0.998237i \(-0.518905\pi\)
−0.0593578 + 0.998237i \(0.518905\pi\)
\(762\) 0 0
\(763\) 1.45017 0.0524995
\(764\) 0 0
\(765\) −29.9210 −1.08180
\(766\) 0 0
\(767\) −17.2054 −0.621252
\(768\) 0 0
\(769\) −13.8248 −0.498533 −0.249267 0.968435i \(-0.580190\pi\)
−0.249267 + 0.968435i \(0.580190\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −34.7561 −1.25009 −0.625045 0.780589i \(-0.714919\pi\)
−0.625045 + 0.780589i \(0.714919\pi\)
\(774\) 0 0
\(775\) −26.0383 −0.935324
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 4.54983 0.163015
\(780\) 0 0
\(781\) −59.8421 −2.14132
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −37.0997 −1.32414
\(786\) 0 0
\(787\) −14.1993 −0.506152 −0.253076 0.967446i \(-0.581442\pi\)
−0.253076 + 0.967446i \(0.581442\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 5.25370 0.186800
\(792\) 0 0
\(793\) −8.00000 −0.284088
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 24.2487 0.858933 0.429467 0.903083i \(-0.358702\pi\)
0.429467 + 0.903083i \(0.358702\pi\)
\(798\) 0 0
\(799\) 1.37097 0.0485014
\(800\) 0 0
\(801\) −30.0000 −1.06000
\(802\) 0 0
\(803\) 17.2749 0.609619
\(804\) 0 0
\(805\) 4.41644 0.155659
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 5.82475 0.204787 0.102394 0.994744i \(-0.467350\pi\)
0.102394 + 0.994744i \(0.467350\pi\)
\(810\) 0 0
\(811\) −47.2990 −1.66089 −0.830446 0.557099i \(-0.811915\pi\)
−0.830446 + 0.557099i \(0.811915\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −12.1819 −0.426713
\(816\) 0 0
\(817\) −2.72508 −0.0953386
\(818\) 0 0
\(819\) −3.29901 −0.115277
\(820\) 0 0
\(821\) −26.5720 −0.927370 −0.463685 0.886000i \(-0.653473\pi\)
−0.463685 + 0.886000i \(0.653473\pi\)
\(822\) 0 0
\(823\) −21.4334 −0.747122 −0.373561 0.927606i \(-0.621863\pi\)
−0.373561 + 0.927606i \(0.621863\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −47.6495 −1.65694 −0.828468 0.560037i \(-0.810787\pi\)
−0.828468 + 0.560037i \(0.810787\pi\)
\(828\) 0 0
\(829\) −25.3161 −0.879266 −0.439633 0.898178i \(-0.644891\pi\)
−0.439633 + 0.898178i \(0.644891\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −22.3505 −0.774399
\(834\) 0 0
\(835\) −58.1993 −2.01407
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −47.6602 −1.64541 −0.822706 0.568467i \(-0.807537\pi\)
−0.822706 + 0.568467i \(0.807537\pi\)
\(840\) 0 0
\(841\) −22.0997 −0.762058
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −18.5764 −0.639047
\(846\) 0 0
\(847\) 7.04329 0.242010
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 33.0997 1.13464
\(852\) 0 0
\(853\) −32.9665 −1.12875 −0.564376 0.825518i \(-0.690883\pi\)
−0.564376 + 0.825518i \(0.690883\pi\)
\(854\) 0 0
\(855\) 9.13642 0.312459
\(856\) 0 0
\(857\) 16.1993 0.553359 0.276679 0.960962i \(-0.410766\pi\)
0.276679 + 0.960962i \(0.410766\pi\)
\(858\) 0 0
\(859\) −55.4743 −1.89276 −0.946379 0.323060i \(-0.895289\pi\)
−0.946379 + 0.323060i \(0.895289\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 41.7994 1.42287 0.711434 0.702753i \(-0.248046\pi\)
0.711434 + 0.702753i \(0.248046\pi\)
\(864\) 0 0
\(865\) 26.5498 0.902721
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −68.6750 −2.32964
\(870\) 0 0
\(871\) 17.2054 0.582983
\(872\) 0 0
\(873\) −49.6495 −1.68038
\(874\) 0 0
\(875\) −0.924421 −0.0312511
\(876\) 0 0
\(877\) −31.4071 −1.06054 −0.530271 0.847828i \(-0.677910\pi\)
−0.530271 + 0.847828i \(0.677910\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1.82475 0.0614774 0.0307387 0.999527i \(-0.490214\pi\)
0.0307387 + 0.999527i \(0.490214\pi\)
\(882\) 0 0
\(883\) 13.6254 0.458532 0.229266 0.973364i \(-0.426367\pi\)
0.229266 + 0.973364i \(0.426367\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0.837253 0.0281122 0.0140561 0.999901i \(-0.495526\pi\)
0.0140561 + 0.999901i \(0.495526\pi\)
\(888\) 0 0
\(889\) 7.29901 0.244801
\(890\) 0 0
\(891\) 47.4743 1.59045
\(892\) 0 0
\(893\) −0.418627 −0.0140088
\(894\) 0 0
\(895\) 64.2585 2.14793
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −16.0000 −0.533630
\(900\) 0 0
\(901\) 34.0339 1.13383
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 10.5498 0.350688
\(906\) 0 0
\(907\) −23.6495 −0.785269 −0.392634 0.919695i \(-0.628436\pi\)
−0.392634 + 0.919695i \(0.628436\pi\)
\(908\) 0 0
\(909\) 41.5692 1.37876
\(910\) 0 0
\(911\) −34.0339 −1.12759 −0.563797 0.825913i \(-0.690660\pi\)
−0.563797 + 0.825913i \(0.690660\pi\)
\(912\) 0 0
\(913\) −90.1993 −2.98516
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −2.20822 −0.0729219
\(918\) 0 0
\(919\) 0.115088 0.00379639 0.00189820 0.999998i \(-0.499396\pi\)
0.00189820 + 0.999998i \(0.499396\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −29.8007 −0.980901
\(924\) 0 0
\(925\) 40.8471 1.34304
\(926\) 0 0
\(927\) −18.2728 −0.600159
\(928\) 0 0
\(929\) −7.09967 −0.232933 −0.116466 0.993195i \(-0.537157\pi\)
−0.116466 + 0.993195i \(0.537157\pi\)
\(930\) 0 0
\(931\) 6.82475 0.223672
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 52.6103 1.72054
\(936\) 0 0
\(937\) 14.9244 0.487560 0.243780 0.969831i \(-0.421613\pi\)
0.243780 + 0.969831i \(0.421613\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −31.1769 −1.01634 −0.508169 0.861257i \(-0.669678\pi\)
−0.508169 + 0.861257i \(0.669678\pi\)
\(942\) 0 0
\(943\) −15.7611 −0.513252
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 30.1993 0.981347 0.490673 0.871344i \(-0.336751\pi\)
0.490673 + 0.871344i \(0.336751\pi\)
\(948\) 0 0
\(949\) 8.60271 0.279256
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 45.2990 1.46738 0.733689 0.679485i \(-0.237797\pi\)
0.733689 + 0.679485i \(0.237797\pi\)
\(954\) 0 0
\(955\) −35.8248 −1.15926
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 5.32706 0.172020
\(960\) 0 0
\(961\) 6.09967 0.196764
\(962\) 0 0
\(963\) −4.35050 −0.140193
\(964\) 0 0
\(965\) 33.8038 1.08818
\(966\) 0 0
\(967\) 55.5407 1.78607 0.893034 0.449988i \(-0.148572\pi\)
0.893034 + 0.449988i \(0.148572\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 44.0000 1.41203 0.706014 0.708198i \(-0.250492\pi\)
0.706014 + 0.708198i \(0.250492\pi\)
\(972\) 0 0
\(973\) 1.14079 0.0365721
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −28.1993 −0.902177 −0.451088 0.892479i \(-0.648964\pi\)
−0.451088 + 0.892479i \(0.648964\pi\)
\(978\) 0 0
\(979\) 52.7492 1.68587
\(980\) 0 0
\(981\) −10.3923 −0.331801
\(982\) 0 0
\(983\) −22.6893 −0.723676 −0.361838 0.932241i \(-0.617851\pi\)
−0.361838 + 0.932241i \(0.617851\pi\)
\(984\) 0 0
\(985\) 16.0000 0.509802
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 9.43996 0.300173
\(990\) 0 0
\(991\) −47.0531 −1.49469 −0.747345 0.664436i \(-0.768672\pi\)
−0.747345 + 0.664436i \(0.768672\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −43.4743 −1.37823
\(996\) 0 0
\(997\) 42.1029 1.33341 0.666707 0.745320i \(-0.267703\pi\)
0.666707 + 0.745320i \(0.267703\pi\)
\(998\) 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 4864.2.a.bk.1.4 4
4.3 odd 2 4864.2.a.bj.1.4 4
8.3 odd 2 inner 4864.2.a.bk.1.1 4
8.5 even 2 4864.2.a.bj.1.1 4
16.3 odd 4 1216.2.c.i.609.2 yes 8
16.5 even 4 1216.2.c.i.609.7 yes 8
16.11 odd 4 1216.2.c.i.609.8 yes 8
16.13 even 4 1216.2.c.i.609.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1216.2.c.i.609.1 8 16.13 even 4
1216.2.c.i.609.2 yes 8 16.3 odd 4
1216.2.c.i.609.7 yes 8 16.5 even 4
1216.2.c.i.609.8 yes 8 16.11 odd 4
4864.2.a.bj.1.1 4 8.5 even 2
4864.2.a.bj.1.4 4 4.3 odd 2
4864.2.a.bk.1.1 4 8.3 odd 2 inner
4864.2.a.bk.1.4 4 1.1 even 1 trivial