Properties

Label 6048.2.a.bq.1.4
Level 6048
Weight 2
Character 6048.1
Self dual yes
Analytic conductor 48.294
Analytic rank 1
Dimension 4
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: \(4\)
Coefficient field: 4.4.25808.1
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.55466\)
Character \(\chi\) = 6048.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.95805 q^{5} -1.00000 q^{7} +O(q^{10})\) \(q+3.95805 q^{5} -1.00000 q^{7} -0.848723 q^{11} -5.85936 q^{13} +2.75004 q^{17} -7.66613 q^{19} +2.70808 q^{23} +10.6661 q^{25} -5.85936 q^{29} -5.80677 q^{31} -3.95805 q^{35} -3.35929 q^{37} +1.90131 q^{41} -2.35929 q^{43} +3.55681 q^{47} +1.00000 q^{49} +10.2186 q^{53} -3.35929 q^{55} +5.77545 q^{59} -5.50008 q^{61} -23.1916 q^{65} -11.4729 q^{67} -9.45812 q^{71} -7.97282 q^{73} +0.848723 q^{77} -10.0780 q^{79} -9.85936 q^{83} +10.8848 q^{85} -1.81741 q^{89} +5.85936 q^{91} -30.3429 q^{95} +15.8322 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{5} - 4q^{7} + O(q^{10}) \) \( 4q + 2q^{5} - 4q^{7} - 2q^{11} - 4q^{13} + 4q^{17} + 4q^{19} - 10q^{23} + 8q^{25} - 4q^{29} - 8q^{31} - 2q^{35} - 8q^{37} + 2q^{41} - 4q^{43} - 8q^{47} + 4q^{49} + 16q^{53} - 8q^{55} - 24q^{59} - 8q^{61} - 4q^{65} + 4q^{67} - 10q^{71} + 4q^{73} + 2q^{77} + 4q^{79} - 20q^{83} - 16q^{85} + 26q^{89} + 4q^{91} - 34q^{95} + 8q^{97} + O(q^{100}) \)

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
\(5\) 3.95805 1.77009 0.885046 0.465503i \(-0.154127\pi\)
0.885046 + 0.465503i \(0.154127\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.848723 −0.255900 −0.127950 0.991781i \(-0.540840\pi\)
−0.127950 + 0.991781i \(0.540840\pi\)
\(12\) 0 0
\(13\) −5.85936 −1.62509 −0.812547 0.582895i \(-0.801920\pi\)
−0.812547 + 0.582895i \(0.801920\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.75004 0.666982 0.333491 0.942753i \(-0.391773\pi\)
0.333491 + 0.942753i \(0.391773\pi\)
\(18\) 0 0
\(19\) −7.66613 −1.75873 −0.879365 0.476147i \(-0.842033\pi\)
−0.879365 + 0.476147i \(0.842033\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.70808 0.564675 0.282337 0.959315i \(-0.408890\pi\)
0.282337 + 0.959315i \(0.408890\pi\)
\(24\) 0 0
\(25\) 10.6661 2.13323
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −5.85936 −1.08806 −0.544028 0.839067i \(-0.683101\pi\)
−0.544028 + 0.839067i \(0.683101\pi\)
\(30\) 0 0
\(31\) −5.80677 −1.04293 −0.521463 0.853274i \(-0.674614\pi\)
−0.521463 + 0.853274i \(0.674614\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.95805 −0.669032
\(36\) 0 0
\(37\) −3.35929 −0.552263 −0.276132 0.961120i \(-0.589053\pi\)
−0.276132 + 0.961120i \(0.589053\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.90131 0.296935 0.148468 0.988917i \(-0.452566\pi\)
0.148468 + 0.988917i \(0.452566\pi\)
\(42\) 0 0
\(43\) −2.35929 −0.359788 −0.179894 0.983686i \(-0.557575\pi\)
−0.179894 + 0.983686i \(0.557575\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.55681 0.518814 0.259407 0.965768i \(-0.416473\pi\)
0.259407 + 0.965768i \(0.416473\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 10.2186 1.40364 0.701820 0.712355i \(-0.252371\pi\)
0.701820 + 0.712355i \(0.252371\pi\)
\(54\) 0 0
\(55\) −3.35929 −0.452966
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 5.77545 0.751900 0.375950 0.926640i \(-0.377316\pi\)
0.375950 + 0.926640i \(0.377316\pi\)
\(60\) 0 0
\(61\) −5.50008 −0.704212 −0.352106 0.935960i \(-0.614534\pi\)
−0.352106 + 0.935960i \(0.614534\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −23.1916 −2.87657
\(66\) 0 0
\(67\) −11.4729 −1.40164 −0.700819 0.713339i \(-0.747182\pi\)
−0.700819 + 0.713339i \(0.747182\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −9.45812 −1.12247 −0.561236 0.827656i \(-0.689674\pi\)
−0.561236 + 0.827656i \(0.689674\pi\)
\(72\) 0 0
\(73\) −7.97282 −0.933148 −0.466574 0.884482i \(-0.654512\pi\)
−0.466574 + 0.884482i \(0.654512\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.848723 0.0967210
\(78\) 0 0
\(79\) −10.0780 −1.13386 −0.566932 0.823764i \(-0.691870\pi\)
−0.566932 + 0.823764i \(0.691870\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −9.85936 −1.08221 −0.541103 0.840957i \(-0.681993\pi\)
−0.541103 + 0.840957i \(0.681993\pi\)
\(84\) 0 0
\(85\) 10.8848 1.18062
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.81741 −0.192645 −0.0963224 0.995350i \(-0.530708\pi\)
−0.0963224 + 0.995350i \(0.530708\pi\)
\(90\) 0 0
\(91\) 5.85936 0.614228
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −30.3429 −3.11312
\(96\) 0 0
\(97\) 15.8322 1.60751 0.803757 0.594957i \(-0.202831\pi\)
0.803757 + 0.594957i \(0.202831\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 14.5822 1.45099 0.725493 0.688230i \(-0.241612\pi\)
0.725493 + 0.688230i \(0.241612\pi\)
\(102\) 0 0
\(103\) 5.91195 0.582522 0.291261 0.956644i \(-0.405925\pi\)
0.291261 + 0.956644i \(0.405925\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −18.6661 −1.80452 −0.902261 0.431189i \(-0.858094\pi\)
−0.902261 + 0.431189i \(0.858094\pi\)
\(108\) 0 0
\(109\) −5.80677 −0.556188 −0.278094 0.960554i \(-0.589703\pi\)
−0.278094 + 0.960554i \(0.589703\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −19.9161 −1.87355 −0.936774 0.349934i \(-0.886204\pi\)
−0.936774 + 0.349934i \(0.886204\pi\)
\(114\) 0 0
\(115\) 10.7187 0.999526
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −2.75004 −0.252096
\(120\) 0 0
\(121\) −10.2797 −0.934515
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 22.4268 2.00591
\(126\) 0 0
\(127\) 0.359285 0.0318814 0.0159407 0.999873i \(-0.494926\pi\)
0.0159407 + 0.999873i \(0.494926\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −5.94327 −0.519266 −0.259633 0.965707i \(-0.583602\pi\)
−0.259633 + 0.965707i \(0.583602\pi\)
\(132\) 0 0
\(133\) 7.66613 0.664738
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 12.9730 1.10836 0.554178 0.832398i \(-0.313033\pi\)
0.554178 + 0.832398i \(0.313033\pi\)
\(138\) 0 0
\(139\) 5.61354 0.476134 0.238067 0.971249i \(-0.423486\pi\)
0.238067 + 0.971249i \(0.423486\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.97298 0.415861
\(144\) 0 0
\(145\) −23.1916 −1.92596
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −7.80263 −0.639216 −0.319608 0.947550i \(-0.603551\pi\)
−0.319608 + 0.947550i \(0.603551\pi\)
\(150\) 0 0
\(151\) 0.499925 0.0406833 0.0203416 0.999793i \(-0.493525\pi\)
0.0203416 + 0.999793i \(0.493525\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −22.9835 −1.84608
\(156\) 0 0
\(157\) −20.1915 −1.61145 −0.805727 0.592287i \(-0.798225\pi\)
−0.805727 + 0.592287i \(0.798225\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −2.70808 −0.213427
\(162\) 0 0
\(163\) 9.25425 0.724849 0.362425 0.932013i \(-0.381949\pi\)
0.362425 + 0.932013i \(0.381949\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −25.5296 −1.97554 −0.987771 0.155911i \(-0.950169\pi\)
−0.987771 + 0.155911i \(0.950169\pi\)
\(168\) 0 0
\(169\) 21.3321 1.64093
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2.70808 0.205892 0.102946 0.994687i \(-0.467173\pi\)
0.102946 + 0.994687i \(0.467173\pi\)
\(174\) 0 0
\(175\) −10.6661 −0.806284
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 8.16621 0.610371 0.305185 0.952293i \(-0.401282\pi\)
0.305185 + 0.952293i \(0.401282\pi\)
\(180\) 0 0
\(181\) −5.39489 −0.400999 −0.200500 0.979694i \(-0.564257\pi\)
−0.200500 + 0.979694i \(0.564257\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −13.2962 −0.977557
\(186\) 0 0
\(187\) −2.33402 −0.170680
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 16.8428 1.21870 0.609352 0.792900i \(-0.291430\pi\)
0.609352 + 0.792900i \(0.291430\pi\)
\(192\) 0 0
\(193\) 15.6135 1.12389 0.561944 0.827176i \(-0.310054\pi\)
0.561944 + 0.827176i \(0.310054\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −3.80263 −0.270926 −0.135463 0.990782i \(-0.543252\pi\)
−0.135463 + 0.990782i \(0.543252\pi\)
\(198\) 0 0
\(199\) −16.3594 −1.15969 −0.579845 0.814727i \(-0.696887\pi\)
−0.579845 + 0.814727i \(0.696887\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 5.85936 0.411247
\(204\) 0 0
\(205\) 7.52549 0.525603
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 6.50642 0.450059
\(210\) 0 0
\(211\) 2.11346 0.145497 0.0727484 0.997350i \(-0.476823\pi\)
0.0727484 + 0.997350i \(0.476823\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −9.33816 −0.636857
\(216\) 0 0
\(217\) 5.80677 0.394189
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −16.1135 −1.08391
\(222\) 0 0
\(223\) 5.35929 0.358884 0.179442 0.983769i \(-0.442571\pi\)
0.179442 + 0.983769i \(0.442571\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −7.80263 −0.517879 −0.258939 0.965894i \(-0.583373\pi\)
−0.258939 + 0.965894i \(0.583373\pi\)
\(228\) 0 0
\(229\) −10.4999 −0.693855 −0.346927 0.937892i \(-0.612775\pi\)
−0.346927 + 0.937892i \(0.612775\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 27.0297 1.77077 0.885387 0.464854i \(-0.153893\pi\)
0.885387 + 0.464854i \(0.153893\pi\)
\(234\) 0 0
\(235\) 14.0780 0.918348
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 10.3849 0.671740 0.335870 0.941908i \(-0.390970\pi\)
0.335870 + 0.941908i \(0.390970\pi\)
\(240\) 0 0
\(241\) −18.1915 −1.17182 −0.585908 0.810378i \(-0.699262\pi\)
−0.585908 + 0.810378i \(0.699262\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 3.95805 0.252870
\(246\) 0 0
\(247\) 44.9186 2.85810
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 29.1644 1.84084 0.920422 0.390927i \(-0.127845\pi\)
0.920422 + 0.390927i \(0.127845\pi\)
\(252\) 0 0
\(253\) −2.29841 −0.144500
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −10.8700 −0.678052 −0.339026 0.940777i \(-0.610097\pi\)
−0.339026 + 0.940777i \(0.610097\pi\)
\(258\) 0 0
\(259\) 3.35929 0.208736
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 12.6455 0.779752 0.389876 0.920867i \(-0.372518\pi\)
0.389876 + 0.920867i \(0.372518\pi\)
\(264\) 0 0
\(265\) 40.4459 2.48457
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −22.2590 −1.35715 −0.678577 0.734529i \(-0.737403\pi\)
−0.678577 + 0.734529i \(0.737403\pi\)
\(270\) 0 0
\(271\) 25.0510 1.52174 0.760869 0.648905i \(-0.224773\pi\)
0.760869 + 0.648905i \(0.224773\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −9.05259 −0.545892
\(276\) 0 0
\(277\) 20.3051 1.22001 0.610007 0.792396i \(-0.291167\pi\)
0.610007 + 0.792396i \(0.291167\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −9.94327 −0.593166 −0.296583 0.955007i \(-0.595847\pi\)
−0.296583 + 0.955007i \(0.595847\pi\)
\(282\) 0 0
\(283\) 15.7187 0.934381 0.467191 0.884157i \(-0.345266\pi\)
0.467191 + 0.884157i \(0.345266\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −1.90131 −0.112231
\(288\) 0 0
\(289\) −9.43729 −0.555135
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 24.1958 1.41353 0.706766 0.707448i \(-0.250153\pi\)
0.706766 + 0.707448i \(0.250153\pi\)
\(294\) 0 0
\(295\) 22.8595 1.33093
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −15.8676 −0.917649
\(300\) 0 0
\(301\) 2.35929 0.135987
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −21.7696 −1.24652
\(306\) 0 0
\(307\) 17.6661 1.00826 0.504130 0.863628i \(-0.331813\pi\)
0.504130 + 0.863628i \(0.331813\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −25.8050 −1.46327 −0.731634 0.681698i \(-0.761242\pi\)
−0.731634 + 0.681698i \(0.761242\pi\)
\(312\) 0 0
\(313\) 25.8594 1.46166 0.730829 0.682561i \(-0.239134\pi\)
0.730829 + 0.682561i \(0.239134\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0.329730 0.0185195 0.00925973 0.999957i \(-0.497052\pi\)
0.00925973 + 0.999957i \(0.497052\pi\)
\(318\) 0 0
\(319\) 4.97298 0.278433
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −21.0821 −1.17304
\(324\) 0 0
\(325\) −62.4967 −3.46669
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −3.55681 −0.196093
\(330\) 0 0
\(331\) −25.6135 −1.40785 −0.703924 0.710276i \(-0.748570\pi\)
−0.703924 + 0.710276i \(0.748570\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −45.4103 −2.48103
\(336\) 0 0
\(337\) −8.71872 −0.474939 −0.237470 0.971395i \(-0.576318\pi\)
−0.237470 + 0.971395i \(0.576318\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 4.92834 0.266885
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −17.5066 −0.939802 −0.469901 0.882719i \(-0.655710\pi\)
−0.469901 + 0.882719i \(0.655710\pi\)
\(348\) 0 0
\(349\) −9.71872 −0.520231 −0.260116 0.965577i \(-0.583761\pi\)
−0.260116 + 0.965577i \(0.583761\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −0.682517 −0.0363267 −0.0181634 0.999835i \(-0.505782\pi\)
−0.0181634 + 0.999835i \(0.505782\pi\)
\(354\) 0 0
\(355\) −37.4357 −1.98688
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −7.55252 −0.398607 −0.199303 0.979938i \(-0.563868\pi\)
−0.199303 + 0.979938i \(0.563868\pi\)
\(360\) 0 0
\(361\) 39.7696 2.09313
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −31.5568 −1.65176
\(366\) 0 0
\(367\) 24.6915 1.28889 0.644444 0.764651i \(-0.277089\pi\)
0.644444 + 0.764651i \(0.277089\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −10.2186 −0.530526
\(372\) 0 0
\(373\) −35.1390 −1.81943 −0.909715 0.415233i \(-0.863700\pi\)
−0.909715 + 0.415233i \(0.863700\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 34.3321 1.76819
\(378\) 0 0
\(379\) −5.35944 −0.275296 −0.137648 0.990481i \(-0.543954\pi\)
−0.137648 + 0.990481i \(0.543954\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −23.6348 −1.20768 −0.603841 0.797105i \(-0.706364\pi\)
−0.603841 + 0.797105i \(0.706364\pi\)
\(384\) 0 0
\(385\) 3.35929 0.171205
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −12.8950 −0.653801 −0.326900 0.945059i \(-0.606004\pi\)
−0.326900 + 0.945059i \(0.606004\pi\)
\(390\) 0 0
\(391\) 7.44733 0.376628
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −39.8892 −2.00704
\(396\) 0 0
\(397\) 26.3324 1.32159 0.660793 0.750568i \(-0.270220\pi\)
0.660793 + 0.750568i \(0.270220\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 28.6917 1.43279 0.716397 0.697692i \(-0.245790\pi\)
0.716397 + 0.697692i \(0.245790\pi\)
\(402\) 0 0
\(403\) 34.0240 1.69485
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2.85110 0.141324
\(408\) 0 0
\(409\) −22.5239 −1.11373 −0.556867 0.830602i \(-0.687997\pi\)
−0.556867 + 0.830602i \(0.687997\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −5.77545 −0.284191
\(414\) 0 0
\(415\) −39.0238 −1.91560
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 37.6856 1.84106 0.920532 0.390667i \(-0.127756\pi\)
0.920532 + 0.390667i \(0.127756\pi\)
\(420\) 0 0
\(421\) 2.69155 0.131178 0.0655890 0.997847i \(-0.479107\pi\)
0.0655890 + 0.997847i \(0.479107\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 29.3323 1.42282
\(426\) 0 0
\(427\) 5.50008 0.266167
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −2.70808 −0.130444 −0.0652219 0.997871i \(-0.520776\pi\)
−0.0652219 + 0.997871i \(0.520776\pi\)
\(432\) 0 0
\(433\) 0.964393 0.0463458 0.0231729 0.999731i \(-0.492623\pi\)
0.0231729 + 0.999731i \(0.492623\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −20.7605 −0.993111
\(438\) 0 0
\(439\) 20.4373 0.975419 0.487709 0.873006i \(-0.337833\pi\)
0.487709 + 0.873006i \(0.337833\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −30.8156 −1.46410 −0.732048 0.681253i \(-0.761435\pi\)
−0.732048 + 0.681253i \(0.761435\pi\)
\(444\) 0 0
\(445\) −7.19338 −0.340999
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 23.3535 1.10212 0.551061 0.834465i \(-0.314223\pi\)
0.551061 + 0.834465i \(0.314223\pi\)
\(450\) 0 0
\(451\) −1.61369 −0.0759857
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 23.1916 1.08724
\(456\) 0 0
\(457\) −18.6644 −0.873082 −0.436541 0.899684i \(-0.643797\pi\)
−0.436541 + 0.899684i \(0.643797\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −26.7802 −1.24728 −0.623639 0.781713i \(-0.714346\pi\)
−0.623639 + 0.781713i \(0.714346\pi\)
\(462\) 0 0
\(463\) −5.71872 −0.265772 −0.132886 0.991131i \(-0.542424\pi\)
−0.132886 + 0.991131i \(0.542424\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −39.7128 −1.83769 −0.918845 0.394618i \(-0.870877\pi\)
−0.918845 + 0.394618i \(0.870877\pi\)
\(468\) 0 0
\(469\) 11.4729 0.529769
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 2.00238 0.0920695
\(474\) 0 0
\(475\) −81.7679 −3.75177
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −10.8323 −0.494942 −0.247471 0.968895i \(-0.579600\pi\)
−0.247471 + 0.968895i \(0.579600\pi\)
\(480\) 0 0
\(481\) 19.6833 0.897480
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 62.6645 2.84545
\(486\) 0 0
\(487\) −31.5781 −1.43094 −0.715470 0.698644i \(-0.753787\pi\)
−0.715470 + 0.698644i \(0.753787\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 21.5361 0.971912 0.485956 0.873983i \(-0.338472\pi\)
0.485956 + 0.873983i \(0.338472\pi\)
\(492\) 0 0
\(493\) −16.1135 −0.725714
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 9.45812 0.424255
\(498\) 0 0
\(499\) 35.2425 1.57767 0.788834 0.614606i \(-0.210685\pi\)
0.788834 + 0.614606i \(0.210685\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −16.1915 −0.721942 −0.360971 0.932577i \(-0.617555\pi\)
−0.360971 + 0.932577i \(0.617555\pi\)
\(504\) 0 0
\(505\) 57.7171 2.56838
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −18.6957 −0.828672 −0.414336 0.910124i \(-0.635986\pi\)
−0.414336 + 0.910124i \(0.635986\pi\)
\(510\) 0 0
\(511\) 7.97282 0.352697
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 23.3998 1.03112
\(516\) 0 0
\(517\) −3.01874 −0.132764
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 10.3698 0.454308 0.227154 0.973859i \(-0.427058\pi\)
0.227154 + 0.973859i \(0.427058\pi\)
\(522\) 0 0
\(523\) 23.6137 1.03255 0.516277 0.856421i \(-0.327317\pi\)
0.516277 + 0.856421i \(0.327317\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −15.9688 −0.695613
\(528\) 0 0
\(529\) −15.6663 −0.681143
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −11.1405 −0.482548
\(534\) 0 0
\(535\) −73.8814 −3.19417
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −0.848723 −0.0365571
\(540\) 0 0
\(541\) −23.9730 −1.03068 −0.515339 0.856986i \(-0.672334\pi\)
−0.515339 + 0.856986i \(0.672334\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −22.9835 −0.984503
\(546\) 0 0
\(547\) −38.9458 −1.66520 −0.832601 0.553873i \(-0.813149\pi\)
−0.832601 + 0.553873i \(0.813149\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 44.9186 1.91360
\(552\) 0 0
\(553\) 10.0780 0.428560
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 15.9161 0.674386 0.337193 0.941435i \(-0.390522\pi\)
0.337193 + 0.941435i \(0.390522\pi\)
\(558\) 0 0
\(559\) 13.8239 0.584689
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 29.4103 1.23949 0.619747 0.784801i \(-0.287235\pi\)
0.619747 + 0.784801i \(0.287235\pi\)
\(564\) 0 0
\(565\) −78.8288 −3.31635
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 14.5295 0.609108 0.304554 0.952495i \(-0.401493\pi\)
0.304554 + 0.952495i \(0.401493\pi\)
\(570\) 0 0
\(571\) −9.55091 −0.399693 −0.199847 0.979827i \(-0.564044\pi\)
−0.199847 + 0.979827i \(0.564044\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 28.8848 1.20458
\(576\) 0 0
\(577\) 31.0510 1.29267 0.646335 0.763054i \(-0.276301\pi\)
0.646335 + 0.763054i \(0.276301\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 9.85936 0.409035
\(582\) 0 0
\(583\) −8.67280 −0.359191
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 11.7483 0.484903 0.242452 0.970163i \(-0.422048\pi\)
0.242452 + 0.970163i \(0.422048\pi\)
\(588\) 0 0
\(589\) 44.5155 1.83423
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 36.6514 1.50509 0.752545 0.658540i \(-0.228826\pi\)
0.752545 + 0.658540i \(0.228826\pi\)
\(594\) 0 0
\(595\) −10.8848 −0.446232
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −17.6453 −0.720967 −0.360484 0.932766i \(-0.617388\pi\)
−0.360484 + 0.932766i \(0.617388\pi\)
\(600\) 0 0
\(601\) 11.6407 0.474835 0.237417 0.971408i \(-0.423699\pi\)
0.237417 + 0.971408i \(0.423699\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −40.6874 −1.65418
\(606\) 0 0
\(607\) −42.0508 −1.70679 −0.853395 0.521264i \(-0.825461\pi\)
−0.853395 + 0.521264i \(0.825461\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −20.8406 −0.843121
\(612\) 0 0
\(613\) −21.1934 −0.855993 −0.427996 0.903780i \(-0.640780\pi\)
−0.427996 + 0.903780i \(0.640780\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −14.2245 −0.572659 −0.286329 0.958131i \(-0.592435\pi\)
−0.286329 + 0.958131i \(0.592435\pi\)
\(618\) 0 0
\(619\) −8.82390 −0.354663 −0.177331 0.984151i \(-0.556746\pi\)
−0.177331 + 0.984151i \(0.556746\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 1.81741 0.0728129
\(624\) 0 0
\(625\) 35.4357 1.41743
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −9.23816 −0.368350
\(630\) 0 0
\(631\) −12.1406 −0.483311 −0.241656 0.970362i \(-0.577690\pi\)
−0.241656 + 0.970362i \(0.577690\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1.42207 0.0564331
\(636\) 0 0
\(637\) −5.85936 −0.232156
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −5.30270 −0.209444 −0.104722 0.994502i \(-0.533395\pi\)
−0.104722 + 0.994502i \(0.533395\pi\)
\(642\) 0 0
\(643\) 37.5409 1.48047 0.740234 0.672350i \(-0.234715\pi\)
0.740234 + 0.672350i \(0.234715\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −45.6076 −1.79302 −0.896511 0.443022i \(-0.853906\pi\)
−0.896511 + 0.443022i \(0.853906\pi\)
\(648\) 0 0
\(649\) −4.90176 −0.192411
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −28.5652 −1.11784 −0.558922 0.829220i \(-0.688785\pi\)
−0.558922 + 0.829220i \(0.688785\pi\)
\(654\) 0 0
\(655\) −23.5237 −0.919148
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −12.2436 −0.476944 −0.238472 0.971149i \(-0.576646\pi\)
−0.238472 + 0.971149i \(0.576646\pi\)
\(660\) 0 0
\(661\) −29.9728 −1.16581 −0.582904 0.812541i \(-0.698084\pi\)
−0.582904 + 0.812541i \(0.698084\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 30.3429 1.17665
\(666\) 0 0
\(667\) −15.8676 −0.614398
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 4.66804 0.180208
\(672\) 0 0
\(673\) 3.38661 0.130544 0.0652722 0.997867i \(-0.479208\pi\)
0.0652722 + 0.997867i \(0.479208\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 18.9792 0.729429 0.364714 0.931119i \(-0.381167\pi\)
0.364714 + 0.931119i \(0.381167\pi\)
\(678\) 0 0
\(679\) −15.8322 −0.607583
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 49.5555 1.89619 0.948094 0.317990i \(-0.103008\pi\)
0.948094 + 0.317990i \(0.103008\pi\)
\(684\) 0 0
\(685\) 51.3476 1.96189
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −59.8747 −2.28105
\(690\) 0 0
\(691\) 28.5593 1.08645 0.543224 0.839588i \(-0.317203\pi\)
0.543224 + 0.839588i \(0.317203\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 22.2186 0.842801
\(696\) 0 0
\(697\) 5.22869 0.198051
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −17.1913 −0.649307 −0.324654 0.945833i \(-0.605248\pi\)
−0.324654 + 0.945833i \(0.605248\pi\)
\(702\) 0 0
\(703\) 25.7527 0.971282
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −14.5822 −0.548421
\(708\) 0 0
\(709\) −46.8578 −1.75978 −0.879890 0.475178i \(-0.842384\pi\)
−0.879890 + 0.475178i \(0.842384\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −15.7252 −0.588914
\(714\) 0 0
\(715\) 19.6833 0.736112
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 48.3265 1.80227 0.901137 0.433534i \(-0.142733\pi\)
0.901137 + 0.433534i \(0.142733\pi\)
\(720\) 0 0
\(721\) −5.91195 −0.220173
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −62.4967 −2.32107
\(726\) 0 0
\(727\) 15.2814 0.566757 0.283378 0.959008i \(-0.408545\pi\)
0.283378 + 0.959008i \(0.408545\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −6.48812 −0.239972
\(732\) 0 0
\(733\) −9.07816 −0.335309 −0.167655 0.985846i \(-0.553619\pi\)
−0.167655 + 0.985846i \(0.553619\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 9.73732 0.358679
\(738\) 0 0
\(739\) 19.7814 0.727669 0.363834 0.931464i \(-0.381467\pi\)
0.363834 + 0.931464i \(0.381467\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 16.1554 0.592685 0.296342 0.955082i \(-0.404233\pi\)
0.296342 + 0.955082i \(0.404233\pi\)
\(744\) 0 0
\(745\) −30.8832 −1.13147
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 18.6661 0.682046
\(750\) 0 0
\(751\) 21.4100 0.781261 0.390630 0.920548i \(-0.372257\pi\)
0.390630 + 0.920548i \(0.372257\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1.97872 0.0720132
\(756\) 0 0
\(757\) −6.49164 −0.235943 −0.117971 0.993017i \(-0.537639\pi\)
−0.117971 + 0.993017i \(0.537639\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 50.5704 1.83318 0.916588 0.399833i \(-0.130932\pi\)
0.916588 + 0.399833i \(0.130932\pi\)
\(762\) 0 0
\(763\) 5.80677 0.210219
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −33.8405 −1.22191
\(768\) 0 0
\(769\) −21.1290 −0.761931 −0.380965 0.924589i \(-0.624408\pi\)
−0.380965 + 0.924589i \(0.624408\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 27.3116 0.982329 0.491165 0.871067i \(-0.336571\pi\)
0.491165 + 0.871067i \(0.336571\pi\)
\(774\) 0 0
\(775\) −61.9358 −2.22480
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −14.5757 −0.522230
\(780\) 0 0
\(781\) 8.02733 0.287240
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −79.9188 −2.85242
\(786\) 0 0
\(787\) −22.7694 −0.811642 −0.405821 0.913953i \(-0.633014\pi\)
−0.405821 + 0.913953i \(0.633014\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 19.9161 0.708135
\(792\) 0 0
\(793\) 32.2269 1.14441
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −48.9041 −1.73227 −0.866137 0.499807i \(-0.833404\pi\)
−0.866137 + 0.499807i \(0.833404\pi\)
\(798\) 0 0
\(799\) 9.78135 0.346039
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 6.76672 0.238792
\(804\) 0 0
\(805\) −10.7187 −0.377785
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 7.24597 0.254755 0.127377 0.991854i \(-0.459344\pi\)
0.127377 + 0.991854i \(0.459344\pi\)
\(810\) 0 0
\(811\) −53.3832 −1.87454 −0.937270 0.348605i \(-0.886655\pi\)
−0.937270 + 0.348605i \(0.886655\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 36.6288 1.28305
\(816\) 0 0
\(817\) 18.0866 0.632770
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −8.99395 −0.313891 −0.156945 0.987607i \(-0.550165\pi\)
−0.156945 + 0.987607i \(0.550165\pi\)
\(822\) 0 0
\(823\) −48.6645 −1.69634 −0.848169 0.529725i \(-0.822295\pi\)
−0.848169 + 0.529725i \(0.822295\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 26.3046 0.914702 0.457351 0.889286i \(-0.348798\pi\)
0.457351 + 0.889286i \(0.348798\pi\)
\(828\) 0 0
\(829\) 26.8052 0.930982 0.465491 0.885053i \(-0.345878\pi\)
0.465491 + 0.885053i \(0.345878\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 2.75004 0.0952832
\(834\) 0 0
\(835\) −101.047 −3.49689
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −1.85936 −0.0641923 −0.0320961 0.999485i \(-0.510218\pi\)
−0.0320961 + 0.999485i \(0.510218\pi\)
\(840\) 0 0
\(841\) 5.33211 0.183866
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 84.4335 2.90460
\(846\) 0 0
\(847\) 10.2797 0.353214
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −9.09723 −0.311849
\(852\) 0 0
\(853\) 10.8323 0.370892 0.185446 0.982654i \(-0.440627\pi\)
0.185446 + 0.982654i \(0.440627\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −0.598610 −0.0204481 −0.0102241 0.999948i \(-0.503254\pi\)
−0.0102241 + 0.999948i \(0.503254\pi\)
\(858\) 0 0
\(859\) 18.4391 0.629132 0.314566 0.949236i \(-0.398141\pi\)
0.314566 + 0.949236i \(0.398141\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 50.2761 1.71142 0.855710 0.517456i \(-0.173121\pi\)
0.855710 + 0.517456i \(0.173121\pi\)
\(864\) 0 0
\(865\) 10.7187 0.364447
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 8.55344 0.290156
\(870\) 0 0
\(871\) 67.2239 2.27779
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −22.4268 −0.758164
\(876\) 0 0
\(877\) 3.66789 0.123856 0.0619279 0.998081i \(-0.480275\pi\)
0.0619279 + 0.998081i \(0.480275\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 23.7122 0.798885 0.399443 0.916758i \(-0.369204\pi\)
0.399443 + 0.916758i \(0.369204\pi\)
\(882\) 0 0
\(883\) −13.9374 −0.469030 −0.234515 0.972113i \(-0.575350\pi\)
−0.234515 + 0.972113i \(0.575350\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −6.64087 −0.222978 −0.111489 0.993766i \(-0.535562\pi\)
−0.111489 + 0.993766i \(0.535562\pi\)
\(888\) 0 0
\(889\) −0.359285 −0.0120500
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −27.2669 −0.912454
\(894\) 0 0
\(895\) 32.3222 1.08041
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 34.0240 1.13476
\(900\) 0 0
\(901\) 28.1017 0.936202
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −21.3532 −0.709806
\(906\) 0 0
\(907\) 32.2778 1.07177 0.535883 0.844292i \(-0.319979\pi\)
0.535883 + 0.844292i \(0.319979\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −17.7168 −0.586984 −0.293492 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(912\) 0 0
\(913\) 8.36787 0.276936
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 5.94327 0.196264
\(918\) 0 0
\(919\) 54.7071 1.80462 0.902310 0.431088i \(-0.141870\pi\)
0.902310 + 0.431088i \(0.141870\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 55.4185 1.82412
\(924\) 0 0
\(925\) −35.8306 −1.17810
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 13.4264 0.440505 0.220252 0.975443i \(-0.429312\pi\)
0.220252 + 0.975443i \(0.429312\pi\)
\(930\) 0 0
\(931\) −7.66613 −0.251247
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −9.23816 −0.302120
\(936\) 0 0
\(937\) −43.1647 −1.41013 −0.705065 0.709142i \(-0.749082\pi\)
−0.705065 + 0.709142i \(0.749082\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 50.1735 1.63561 0.817804 0.575496i \(-0.195191\pi\)
0.817804 + 0.575496i \(0.195191\pi\)
\(942\) 0 0
\(943\) 5.14892 0.167672
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 49.7421 1.61640 0.808200 0.588908i \(-0.200442\pi\)
0.808200 + 0.588908i \(0.200442\pi\)
\(948\) 0 0
\(949\) 46.7157 1.51645
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −20.8595 −0.675706 −0.337853 0.941199i \(-0.609701\pi\)
−0.337853 + 0.941199i \(0.609701\pi\)
\(954\) 0 0
\(955\) 66.6647 2.15722
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −12.9730 −0.418919
\(960\) 0 0
\(961\) 2.71857 0.0876958
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 61.7991 1.98938
\(966\) 0 0
\(967\) 44.2068 1.42160 0.710798 0.703396i \(-0.248334\pi\)
0.710798 + 0.703396i \(0.248334\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 35.3680 1.13501 0.567507 0.823369i \(-0.307908\pi\)
0.567507 + 0.823369i \(0.307908\pi\)
\(972\) 0 0
\(973\) −5.61354 −0.179962
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 6.94580 0.222216 0.111108 0.993808i \(-0.464560\pi\)
0.111108 + 0.993808i \(0.464560\pi\)
\(978\) 0 0
\(979\) 1.54248 0.0492977
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 27.4433 0.875307 0.437653 0.899144i \(-0.355810\pi\)
0.437653 + 0.899144i \(0.355810\pi\)
\(984\) 0 0
\(985\) −15.0510 −0.479564
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −6.38914 −0.203163
\(990\) 0 0
\(991\) 19.9102 0.632468 0.316234 0.948681i \(-0.397582\pi\)
0.316234 + 0.948681i \(0.397582\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −64.7514 −2.05276
\(996\) 0 0
\(997\) −28.6645 −0.907814 −0.453907 0.891049i \(-0.649970\pi\)
−0.453907 + 0.891049i \(0.649970\pi\)
\(998\) 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.bq.1.4 yes 4
3.2 odd 2 6048.2.a.bl.1.1 4
4.3 odd 2 6048.2.a.bu.1.4 yes 4
12.11 even 2 6048.2.a.bn.1.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.a.bl.1.1 4 3.2 odd 2
6048.2.a.bn.1.1 yes 4 12.11 even 2
6048.2.a.bq.1.4 yes 4 1.1 even 1 trivial
6048.2.a.bu.1.4 yes 4 4.3 odd 2