Properties

Label 6048.2.a.bs.1.1
Level 6048
Weight 2
Character 6048.1
Self dual yes
Analytic conductor 48.294
Analytic rank 0
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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.39528.1
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.12265\)
Character \(\chi\) = 6048.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.73965 q^{5} -1.00000 q^{7} +O(q^{10})\) \(q-2.73965 q^{5} -1.00000 q^{7} +2.24531 q^{11} -0.458685 q^{13} -7.44364 q^{17} -4.24531 q^{19} +4.70399 q^{23} +2.50566 q^{25} -1.73965 q^{29} +1.73965 q^{31} +2.73965 q^{35} +2.49434 q^{37} -4.73965 q^{41} +1.96435 q^{43} -9.90233 q^{47} +1.00000 q^{49} +5.42303 q^{53} -6.15135 q^{55} -6.47929 q^{59} +5.23398 q^{61} +1.25664 q^{65} +2.77530 q^{67} -0.703994 q^{71} -7.96991 q^{73} -2.24531 q^{77} +6.66834 q^{79} -3.50566 q^{83} +20.3929 q^{85} -1.78662 q^{89} +0.458685 q^{91} +11.6306 q^{95} +13.1627 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{5} - 4q^{7} + O(q^{10}) \) \( 4q + 2q^{5} - 4q^{7} + 8q^{13} + 2q^{17} - 8q^{19} + 14q^{25} + 6q^{29} - 6q^{31} - 2q^{35} + 6q^{37} - 6q^{41} + 2q^{43} + 2q^{47} + 4q^{49} + 6q^{53} + 12q^{55} + 4q^{61} + 4q^{65} + 4q^{67} + 16q^{71} + 12q^{73} + 2q^{79} - 18q^{83} + 22q^{85} - 8q^{89} - 8q^{91} - 16q^{95} + 24q^{97} + O(q^{100}) \)

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
\(5\) −2.73965 −1.22521 −0.612604 0.790390i \(-0.709878\pi\)
−0.612604 + 0.790390i \(0.709878\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.24531 0.676986 0.338493 0.940969i \(-0.390083\pi\)
0.338493 + 0.940969i \(0.390083\pi\)
\(12\) 0 0
\(13\) −0.458685 −0.127216 −0.0636082 0.997975i \(-0.520261\pi\)
−0.0636082 + 0.997975i \(0.520261\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −7.44364 −1.80535 −0.902674 0.430325i \(-0.858399\pi\)
−0.902674 + 0.430325i \(0.858399\pi\)
\(18\) 0 0
\(19\) −4.24531 −0.973941 −0.486970 0.873418i \(-0.661898\pi\)
−0.486970 + 0.873418i \(0.661898\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.70399 0.980851 0.490425 0.871483i \(-0.336841\pi\)
0.490425 + 0.871483i \(0.336841\pi\)
\(24\) 0 0
\(25\) 2.50566 0.501133
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.73965 −0.323044 −0.161522 0.986869i \(-0.551640\pi\)
−0.161522 + 0.986869i \(0.551640\pi\)
\(30\) 0 0
\(31\) 1.73965 0.312450 0.156225 0.987722i \(-0.450068\pi\)
0.156225 + 0.987722i \(0.450068\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.73965 0.463085
\(36\) 0 0
\(37\) 2.49434 0.410066 0.205033 0.978755i \(-0.434270\pi\)
0.205033 + 0.978755i \(0.434270\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4.73965 −0.740208 −0.370104 0.928990i \(-0.620678\pi\)
−0.370104 + 0.928990i \(0.620678\pi\)
\(42\) 0 0
\(43\) 1.96435 0.299560 0.149780 0.988719i \(-0.452143\pi\)
0.149780 + 0.988719i \(0.452143\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −9.90233 −1.44440 −0.722201 0.691683i \(-0.756870\pi\)
−0.722201 + 0.691683i \(0.756870\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 5.42303 0.744911 0.372455 0.928050i \(-0.378516\pi\)
0.372455 + 0.928050i \(0.378516\pi\)
\(54\) 0 0
\(55\) −6.15135 −0.829448
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −6.47929 −0.843532 −0.421766 0.906705i \(-0.638590\pi\)
−0.421766 + 0.906705i \(0.638590\pi\)
\(60\) 0 0
\(61\) 5.23398 0.670143 0.335071 0.942193i \(-0.391240\pi\)
0.335071 + 0.942193i \(0.391240\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.25664 0.155866
\(66\) 0 0
\(67\) 2.77530 0.339057 0.169528 0.985525i \(-0.445776\pi\)
0.169528 + 0.985525i \(0.445776\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −0.703994 −0.0835488 −0.0417744 0.999127i \(-0.513301\pi\)
−0.0417744 + 0.999127i \(0.513301\pi\)
\(72\) 0 0
\(73\) −7.96991 −0.932808 −0.466404 0.884572i \(-0.654451\pi\)
−0.466404 + 0.884572i \(0.654451\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.24531 −0.255877
\(78\) 0 0
\(79\) 6.66834 0.750247 0.375124 0.926975i \(-0.377600\pi\)
0.375124 + 0.926975i \(0.377600\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.50566 −0.384796 −0.192398 0.981317i \(-0.561627\pi\)
−0.192398 + 0.981317i \(0.561627\pi\)
\(84\) 0 0
\(85\) 20.3929 2.21193
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.78662 −0.189382 −0.0946909 0.995507i \(-0.530186\pi\)
−0.0946909 + 0.995507i \(0.530186\pi\)
\(90\) 0 0
\(91\) 0.458685 0.0480833
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 11.6306 1.19328
\(96\) 0 0
\(97\) 13.1627 1.33647 0.668234 0.743951i \(-0.267051\pi\)
0.668234 + 0.743951i \(0.267051\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −2.49062 −0.247826 −0.123913 0.992293i \(-0.539544\pi\)
−0.123913 + 0.992293i \(0.539544\pi\)
\(102\) 0 0
\(103\) 10.3816 1.02293 0.511466 0.859304i \(-0.329103\pi\)
0.511466 + 0.859304i \(0.329103\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 0 0
\(109\) 8.49434 0.813610 0.406805 0.913515i \(-0.366643\pi\)
0.406805 + 0.913515i \(0.366643\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1.75469 −0.165067 −0.0825337 0.996588i \(-0.526301\pi\)
−0.0825337 + 0.996588i \(0.526301\pi\)
\(114\) 0 0
\(115\) −12.8873 −1.20175
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 7.44364 0.682357
\(120\) 0 0
\(121\) −5.95859 −0.541690
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 6.83360 0.611216
\(126\) 0 0
\(127\) 2.24903 0.199569 0.0997845 0.995009i \(-0.468185\pi\)
0.0997845 + 0.995009i \(0.468185\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −17.9549 −1.56872 −0.784362 0.620303i \(-0.787010\pi\)
−0.784362 + 0.620303i \(0.787010\pi\)
\(132\) 0 0
\(133\) 4.24531 0.368115
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −8.24531 −0.704444 −0.352222 0.935916i \(-0.614574\pi\)
−0.352222 + 0.935916i \(0.614574\pi\)
\(138\) 0 0
\(139\) 12.2865 1.04213 0.521065 0.853517i \(-0.325535\pi\)
0.521065 + 0.853517i \(0.325535\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −1.02989 −0.0861237
\(144\) 0 0
\(145\) 4.76602 0.395796
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 8.23027 0.674250 0.337125 0.941460i \(-0.390546\pi\)
0.337125 + 0.941460i \(0.390546\pi\)
\(150\) 0 0
\(151\) −13.6269 −1.10894 −0.554472 0.832202i \(-0.687080\pi\)
−0.554472 + 0.832202i \(0.687080\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −4.76602 −0.382816
\(156\) 0 0
\(157\) 19.4173 1.54967 0.774833 0.632165i \(-0.217834\pi\)
0.774833 + 0.632165i \(0.217834\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −4.70399 −0.370727
\(162\) 0 0
\(163\) −12.0056 −0.940348 −0.470174 0.882574i \(-0.655809\pi\)
−0.470174 + 0.882574i \(0.655809\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.06759 0.469524 0.234762 0.972053i \(-0.424569\pi\)
0.234762 + 0.972053i \(0.424569\pi\)
\(168\) 0 0
\(169\) −12.7896 −0.983816
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 8.98867 0.683396 0.341698 0.939810i \(-0.388998\pi\)
0.341698 + 0.939810i \(0.388998\pi\)
\(174\) 0 0
\(175\) −2.50566 −0.189410
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 21.2039 1.58485 0.792427 0.609967i \(-0.208817\pi\)
0.792427 + 0.609967i \(0.208817\pi\)
\(180\) 0 0
\(181\) 18.1833 1.35155 0.675777 0.737107i \(-0.263808\pi\)
0.675777 + 0.737107i \(0.263808\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −6.83360 −0.502416
\(186\) 0 0
\(187\) −16.7133 −1.22220
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −25.9699 −1.87912 −0.939558 0.342389i \(-0.888764\pi\)
−0.939558 + 0.342389i \(0.888764\pi\)
\(192\) 0 0
\(193\) 18.9699 1.36548 0.682742 0.730659i \(-0.260787\pi\)
0.682742 + 0.730659i \(0.260787\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.31661 0.165052 0.0825259 0.996589i \(-0.473701\pi\)
0.0825259 + 0.996589i \(0.473701\pi\)
\(198\) 0 0
\(199\) 21.3929 1.51651 0.758253 0.651961i \(-0.226053\pi\)
0.758253 + 0.651961i \(0.226053\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.73965 0.122099
\(204\) 0 0
\(205\) 12.9850 0.906909
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −9.53203 −0.659344
\(210\) 0 0
\(211\) 25.1007 1.72800 0.864000 0.503491i \(-0.167951\pi\)
0.864000 + 0.503491i \(0.167951\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −5.38162 −0.367023
\(216\) 0 0
\(217\) −1.73965 −0.118095
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 3.41429 0.229670
\(222\) 0 0
\(223\) −25.1025 −1.68099 −0.840494 0.541821i \(-0.817735\pi\)
−0.840494 + 0.541821i \(0.817735\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −20.4529 −1.35751 −0.678754 0.734366i \(-0.737480\pi\)
−0.678754 + 0.734366i \(0.737480\pi\)
\(228\) 0 0
\(229\) −2.88728 −0.190797 −0.0953985 0.995439i \(-0.530413\pi\)
−0.0953985 + 0.995439i \(0.530413\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 16.6119 1.08828 0.544140 0.838994i \(-0.316856\pi\)
0.544140 + 0.838994i \(0.316856\pi\)
\(234\) 0 0
\(235\) 27.1289 1.76969
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 15.7658 1.01981 0.509903 0.860232i \(-0.329681\pi\)
0.509903 + 0.860232i \(0.329681\pi\)
\(240\) 0 0
\(241\) 1.82599 0.117623 0.0588113 0.998269i \(-0.481269\pi\)
0.0588113 + 0.998269i \(0.481269\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −2.73965 −0.175030
\(246\) 0 0
\(247\) 1.94726 0.123901
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 19.4529 1.22786 0.613929 0.789361i \(-0.289588\pi\)
0.613929 + 0.789361i \(0.289588\pi\)
\(252\) 0 0
\(253\) 10.5619 0.664022
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −7.43808 −0.463975 −0.231987 0.972719i \(-0.574523\pi\)
−0.231987 + 0.972719i \(0.574523\pi\)
\(258\) 0 0
\(259\) −2.49434 −0.154991
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 7.90789 0.487621 0.243811 0.969823i \(-0.421602\pi\)
0.243811 + 0.969823i \(0.421602\pi\)
\(264\) 0 0
\(265\) −14.8572 −0.912670
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 28.7096 1.75045 0.875226 0.483714i \(-0.160712\pi\)
0.875226 + 0.483714i \(0.160712\pi\)
\(270\) 0 0
\(271\) −21.3929 −1.29953 −0.649764 0.760136i \(-0.725132\pi\)
−0.649764 + 0.760136i \(0.725132\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 5.62599 0.339260
\(276\) 0 0
\(277\) −1.47557 −0.0886587 −0.0443293 0.999017i \(-0.514115\pi\)
−0.0443293 + 0.999017i \(0.514115\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 7.96991 0.475445 0.237723 0.971333i \(-0.423599\pi\)
0.237723 + 0.971333i \(0.423599\pi\)
\(282\) 0 0
\(283\) 12.3166 0.732147 0.366073 0.930586i \(-0.380702\pi\)
0.366073 + 0.930586i \(0.380702\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4.73965 0.279772
\(288\) 0 0
\(289\) 38.4078 2.25928
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 33.0349 1.92992 0.964960 0.262396i \(-0.0845127\pi\)
0.964960 + 0.262396i \(0.0845127\pi\)
\(294\) 0 0
\(295\) 17.7510 1.03350
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −2.15765 −0.124780
\(300\) 0 0
\(301\) −1.96435 −0.113223
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −14.3393 −0.821064
\(306\) 0 0
\(307\) 10.9285 0.623722 0.311861 0.950128i \(-0.399048\pi\)
0.311861 + 0.950128i \(0.399048\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −20.4342 −1.15871 −0.579357 0.815074i \(-0.696696\pi\)
−0.579357 + 0.815074i \(0.696696\pi\)
\(312\) 0 0
\(313\) 33.3066 1.88260 0.941300 0.337571i \(-0.109605\pi\)
0.941300 + 0.337571i \(0.109605\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 23.9699 1.34628 0.673142 0.739513i \(-0.264944\pi\)
0.673142 + 0.739513i \(0.264944\pi\)
\(318\) 0 0
\(319\) −3.90604 −0.218697
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 31.6006 1.75830
\(324\) 0 0
\(325\) −1.14931 −0.0637523
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 9.90233 0.545933
\(330\) 0 0
\(331\) −18.9530 −1.04175 −0.520876 0.853632i \(-0.674395\pi\)
−0.520876 + 0.853632i \(0.674395\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −7.60334 −0.415415
\(336\) 0 0
\(337\) 23.7933 1.29611 0.648053 0.761596i \(-0.275584\pi\)
0.648053 + 0.761596i \(0.275584\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 3.90604 0.211524
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −5.58943 −0.300056 −0.150028 0.988682i \(-0.547936\pi\)
−0.150028 + 0.988682i \(0.547936\pi\)
\(348\) 0 0
\(349\) 12.2772 0.657186 0.328593 0.944472i \(-0.393426\pi\)
0.328593 + 0.944472i \(0.393426\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 8.92293 0.474920 0.237460 0.971397i \(-0.423685\pi\)
0.237460 + 0.971397i \(0.423685\pi\)
\(354\) 0 0
\(355\) 1.92870 0.102365
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −6.81651 −0.359762 −0.179881 0.983688i \(-0.557571\pi\)
−0.179881 + 0.983688i \(0.557571\pi\)
\(360\) 0 0
\(361\) −0.977349 −0.0514394
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 21.8347 1.14288
\(366\) 0 0
\(367\) 17.4230 0.909475 0.454737 0.890626i \(-0.349733\pi\)
0.454737 + 0.890626i \(0.349733\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −5.42303 −0.281550
\(372\) 0 0
\(373\) −27.3214 −1.41465 −0.707325 0.706888i \(-0.750098\pi\)
−0.707325 + 0.706888i \(0.750098\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0.797950 0.0410965
\(378\) 0 0
\(379\) −13.5968 −0.698423 −0.349211 0.937044i \(-0.613550\pi\)
−0.349211 + 0.937044i \(0.613550\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 14.5582 0.743889 0.371945 0.928255i \(-0.378691\pi\)
0.371945 + 0.928255i \(0.378691\pi\)
\(384\) 0 0
\(385\) 6.15135 0.313502
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 7.79591 0.395268 0.197634 0.980276i \(-0.436674\pi\)
0.197634 + 0.980276i \(0.436674\pi\)
\(390\) 0 0
\(391\) −35.0148 −1.77078
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −18.2689 −0.919208
\(396\) 0 0
\(397\) −6.53183 −0.327823 −0.163912 0.986475i \(-0.552411\pi\)
−0.163912 + 0.986475i \(0.552411\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −28.3254 −1.41450 −0.707250 0.706963i \(-0.750065\pi\)
−0.707250 + 0.706963i \(0.750065\pi\)
\(402\) 0 0
\(403\) −0.797950 −0.0397487
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 5.60056 0.277609
\(408\) 0 0
\(409\) 36.8572 1.82247 0.911235 0.411886i \(-0.135130\pi\)
0.911235 + 0.411886i \(0.135130\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 6.47929 0.318825
\(414\) 0 0
\(415\) 9.60428 0.471455
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 21.3254 1.04181 0.520906 0.853614i \(-0.325594\pi\)
0.520906 + 0.853614i \(0.325594\pi\)
\(420\) 0 0
\(421\) −20.0111 −0.975283 −0.487641 0.873044i \(-0.662143\pi\)
−0.487641 + 0.873044i \(0.662143\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −18.6513 −0.904719
\(426\) 0 0
\(427\) −5.23398 −0.253290
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 7.50938 0.361714 0.180857 0.983509i \(-0.442113\pi\)
0.180857 + 0.983509i \(0.442113\pi\)
\(432\) 0 0
\(433\) 11.2752 0.541852 0.270926 0.962600i \(-0.412670\pi\)
0.270926 + 0.962600i \(0.412670\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −19.9699 −0.955290
\(438\) 0 0
\(439\) 24.8421 1.18565 0.592826 0.805331i \(-0.298012\pi\)
0.592826 + 0.805331i \(0.298012\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −32.1851 −1.52916 −0.764581 0.644528i \(-0.777054\pi\)
−0.764581 + 0.644528i \(0.777054\pi\)
\(444\) 0 0
\(445\) 4.89472 0.232032
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −19.2340 −0.907708 −0.453854 0.891076i \(-0.649951\pi\)
−0.453854 + 0.891076i \(0.649951\pi\)
\(450\) 0 0
\(451\) −10.6420 −0.501111
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −1.25664 −0.0589120
\(456\) 0 0
\(457\) 17.8197 0.833570 0.416785 0.909005i \(-0.363157\pi\)
0.416785 + 0.909005i \(0.363157\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 4.69843 0.218828 0.109414 0.993996i \(-0.465103\pi\)
0.109414 + 0.993996i \(0.465103\pi\)
\(462\) 0 0
\(463\) 12.2902 0.571176 0.285588 0.958352i \(-0.407811\pi\)
0.285588 + 0.958352i \(0.407811\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 29.2839 1.35510 0.677550 0.735477i \(-0.263042\pi\)
0.677550 + 0.735477i \(0.263042\pi\)
\(468\) 0 0
\(469\) −2.77530 −0.128151
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 4.41057 0.202798
\(474\) 0 0
\(475\) −10.6373 −0.488073
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −37.3289 −1.70560 −0.852800 0.522238i \(-0.825097\pi\)
−0.852800 + 0.522238i \(0.825097\pi\)
\(480\) 0 0
\(481\) −1.14412 −0.0521672
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −36.0611 −1.63745
\(486\) 0 0
\(487\) −18.9285 −0.857732 −0.428866 0.903368i \(-0.641087\pi\)
−0.428866 + 0.903368i \(0.641087\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 32.0273 1.44537 0.722686 0.691176i \(-0.242907\pi\)
0.722686 + 0.691176i \(0.242907\pi\)
\(492\) 0 0
\(493\) 12.9493 0.583207
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0.703994 0.0315785
\(498\) 0 0
\(499\) −36.5554 −1.63645 −0.818223 0.574901i \(-0.805040\pi\)
−0.818223 + 0.574901i \(0.805040\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 9.57697 0.427016 0.213508 0.976941i \(-0.431511\pi\)
0.213508 + 0.976941i \(0.431511\pi\)
\(504\) 0 0
\(505\) 6.82341 0.303638
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −4.54430 −0.201422 −0.100711 0.994916i \(-0.532112\pi\)
−0.100711 + 0.994916i \(0.532112\pi\)
\(510\) 0 0
\(511\) 7.96991 0.352568
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −28.4420 −1.25330
\(516\) 0 0
\(517\) −22.2338 −0.977841
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −32.2370 −1.41233 −0.706164 0.708049i \(-0.749576\pi\)
−0.706164 + 0.708049i \(0.749576\pi\)
\(522\) 0 0
\(523\) −24.5707 −1.07440 −0.537200 0.843455i \(-0.680518\pi\)
−0.537200 + 0.843455i \(0.680518\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −12.9493 −0.564081
\(528\) 0 0
\(529\) −0.872436 −0.0379320
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 2.17401 0.0941666
\(534\) 0 0
\(535\) −10.9586 −0.473781
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 2.24531 0.0967123
\(540\) 0 0
\(541\) 38.8007 1.66817 0.834087 0.551633i \(-0.185995\pi\)
0.834087 + 0.551633i \(0.185995\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −23.2715 −0.996841
\(546\) 0 0
\(547\) 24.8035 1.06052 0.530261 0.847835i \(-0.322094\pi\)
0.530261 + 0.847835i \(0.322094\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 7.38534 0.314626
\(552\) 0 0
\(553\) −6.66834 −0.283567
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 2.29919 0.0974197 0.0487099 0.998813i \(-0.484489\pi\)
0.0487099 + 0.998813i \(0.484489\pi\)
\(558\) 0 0
\(559\) −0.901017 −0.0381090
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −35.9021 −1.51309 −0.756547 0.653939i \(-0.773115\pi\)
−0.756547 + 0.653939i \(0.773115\pi\)
\(564\) 0 0
\(565\) 4.80723 0.202242
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −26.3591 −1.10503 −0.552516 0.833502i \(-0.686332\pi\)
−0.552516 + 0.833502i \(0.686332\pi\)
\(570\) 0 0
\(571\) 16.8591 0.705530 0.352765 0.935712i \(-0.385242\pi\)
0.352765 + 0.935712i \(0.385242\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 11.7866 0.491536
\(576\) 0 0
\(577\) 3.43808 0.143129 0.0715645 0.997436i \(-0.477201\pi\)
0.0715645 + 0.997436i \(0.477201\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 3.50566 0.145439
\(582\) 0 0
\(583\) 12.1764 0.504294
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −23.3291 −0.962894 −0.481447 0.876475i \(-0.659889\pi\)
−0.481447 + 0.876475i \(0.659889\pi\)
\(588\) 0 0
\(589\) −7.38534 −0.304308
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −15.2604 −0.626668 −0.313334 0.949643i \(-0.601446\pi\)
−0.313334 + 0.949643i \(0.601446\pi\)
\(594\) 0 0
\(595\) −20.3929 −0.836029
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −11.8366 −0.483630 −0.241815 0.970322i \(-0.577743\pi\)
−0.241815 + 0.970322i \(0.577743\pi\)
\(600\) 0 0
\(601\) −13.7473 −0.560762 −0.280381 0.959889i \(-0.590461\pi\)
−0.280381 + 0.959889i \(0.590461\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 16.3244 0.663682
\(606\) 0 0
\(607\) −44.5330 −1.80754 −0.903769 0.428021i \(-0.859211\pi\)
−0.903769 + 0.428021i \(0.859211\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 4.54205 0.183752
\(612\) 0 0
\(613\) −41.2951 −1.66789 −0.833946 0.551846i \(-0.813924\pi\)
−0.833946 + 0.551846i \(0.813924\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −8.46053 −0.340608 −0.170304 0.985392i \(-0.554475\pi\)
−0.170304 + 0.985392i \(0.554475\pi\)
\(618\) 0 0
\(619\) 33.9259 1.36360 0.681799 0.731540i \(-0.261198\pi\)
0.681799 + 0.731540i \(0.261198\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 1.78662 0.0715796
\(624\) 0 0
\(625\) −31.2500 −1.25000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −18.5670 −0.740313
\(630\) 0 0
\(631\) 45.6982 1.81922 0.909609 0.415465i \(-0.136381\pi\)
0.909609 + 0.415465i \(0.136381\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −6.16154 −0.244513
\(636\) 0 0
\(637\) −0.458685 −0.0181738
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −48.0685 −1.89859 −0.949296 0.314383i \(-0.898202\pi\)
−0.949296 + 0.314383i \(0.898202\pi\)
\(642\) 0 0
\(643\) 24.2013 0.954407 0.477203 0.878793i \(-0.341651\pi\)
0.477203 + 0.878793i \(0.341651\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 12.5207 0.492240 0.246120 0.969239i \(-0.420844\pi\)
0.246120 + 0.969239i \(0.420844\pi\)
\(648\) 0 0
\(649\) −14.5480 −0.571060
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 18.0951 0.708116 0.354058 0.935224i \(-0.384802\pi\)
0.354058 + 0.935224i \(0.384802\pi\)
\(654\) 0 0
\(655\) 49.1900 1.92201
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −7.86483 −0.306370 −0.153185 0.988198i \(-0.548953\pi\)
−0.153185 + 0.988198i \(0.548953\pi\)
\(660\) 0 0
\(661\) 17.1738 0.667984 0.333992 0.942576i \(-0.391604\pi\)
0.333992 + 0.942576i \(0.391604\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −11.6306 −0.451017
\(666\) 0 0
\(667\) −8.18329 −0.316858
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 11.7519 0.453678
\(672\) 0 0
\(673\) 2.64083 0.101797 0.0508983 0.998704i \(-0.483792\pi\)
0.0508983 + 0.998704i \(0.483792\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 14.5545 0.559374 0.279687 0.960091i \(-0.409769\pi\)
0.279687 + 0.960091i \(0.409769\pi\)
\(678\) 0 0
\(679\) −13.1627 −0.505137
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 7.23398 0.276801 0.138400 0.990376i \(-0.455804\pi\)
0.138400 + 0.990376i \(0.455804\pi\)
\(684\) 0 0
\(685\) 22.5892 0.863090
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −2.48746 −0.0947649
\(690\) 0 0
\(691\) −6.79610 −0.258536 −0.129268 0.991610i \(-0.541263\pi\)
−0.129268 + 0.991610i \(0.541263\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −33.6607 −1.27682
\(696\) 0 0
\(697\) 35.2802 1.33633
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 8.83732 0.333781 0.166890 0.985975i \(-0.446627\pi\)
0.166890 + 0.985975i \(0.446627\pi\)
\(702\) 0 0
\(703\) −10.5892 −0.399380
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 2.49062 0.0936694
\(708\) 0 0
\(709\) −25.8195 −0.969672 −0.484836 0.874605i \(-0.661121\pi\)
−0.484836 + 0.874605i \(0.661121\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 8.18329 0.306467
\(714\) 0 0
\(715\) 2.82153 0.105519
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 46.3216 1.72751 0.863753 0.503916i \(-0.168108\pi\)
0.863753 + 0.503916i \(0.168108\pi\)
\(720\) 0 0
\(721\) −10.3816 −0.386632
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −4.35897 −0.161888
\(726\) 0 0
\(727\) −31.3240 −1.16174 −0.580872 0.813995i \(-0.697288\pi\)
−0.580872 + 0.813995i \(0.697288\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −14.6219 −0.540811
\(732\) 0 0
\(733\) 35.0405 1.29425 0.647125 0.762384i \(-0.275971\pi\)
0.647125 + 0.762384i \(0.275971\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 6.23140 0.229537
\(738\) 0 0
\(739\) 16.0824 0.591602 0.295801 0.955250i \(-0.404414\pi\)
0.295801 + 0.955250i \(0.404414\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 3.39871 0.124686 0.0623432 0.998055i \(-0.480143\pi\)
0.0623432 + 0.998055i \(0.480143\pi\)
\(744\) 0 0
\(745\) −22.5480 −0.826095
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −4.00000 −0.146157
\(750\) 0 0
\(751\) −14.4041 −0.525613 −0.262807 0.964849i \(-0.584648\pi\)
−0.262807 + 0.964849i \(0.584648\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 37.3330 1.35869
\(756\) 0 0
\(757\) −15.2691 −0.554965 −0.277482 0.960731i \(-0.589500\pi\)
−0.277482 + 0.960731i \(0.589500\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 32.7276 1.18637 0.593187 0.805065i \(-0.297869\pi\)
0.593187 + 0.805065i \(0.297869\pi\)
\(762\) 0 0
\(763\) −8.49434 −0.307516
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 2.97196 0.107311
\(768\) 0 0
\(769\) 7.40799 0.267139 0.133569 0.991039i \(-0.457356\pi\)
0.133569 + 0.991039i \(0.457356\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 13.6982 0.492691 0.246346 0.969182i \(-0.420770\pi\)
0.246346 + 0.969182i \(0.420770\pi\)
\(774\) 0 0
\(775\) 4.35897 0.156579
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 20.1213 0.720919
\(780\) 0 0
\(781\) −1.58069 −0.0565614
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −53.1965 −1.89866
\(786\) 0 0
\(787\) 35.3367 1.25962 0.629808 0.776751i \(-0.283133\pi\)
0.629808 + 0.776751i \(0.283133\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1.75469 0.0623896
\(792\) 0 0
\(793\) −2.40075 −0.0852531
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −5.40055 −0.191297 −0.0956487 0.995415i \(-0.530493\pi\)
−0.0956487 + 0.995415i \(0.530493\pi\)
\(798\) 0 0
\(799\) 73.7094 2.60765
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −17.8949 −0.631498
\(804\) 0 0
\(805\) 12.8873 0.454217
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 35.7445 1.25671 0.628354 0.777927i \(-0.283729\pi\)
0.628354 + 0.777927i \(0.283729\pi\)
\(810\) 0 0
\(811\) −8.34670 −0.293092 −0.146546 0.989204i \(-0.546816\pi\)
−0.146546 + 0.989204i \(0.546816\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 32.8910 1.15212
\(816\) 0 0
\(817\) −8.33926 −0.291754
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 16.8797 0.589105 0.294552 0.955635i \(-0.404829\pi\)
0.294552 + 0.955635i \(0.404829\pi\)
\(822\) 0 0
\(823\) −16.3315 −0.569279 −0.284639 0.958635i \(-0.591874\pi\)
−0.284639 + 0.958635i \(0.591874\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 35.6533 1.23979 0.619893 0.784686i \(-0.287176\pi\)
0.619893 + 0.784686i \(0.287176\pi\)
\(828\) 0 0
\(829\) −47.0011 −1.63242 −0.816208 0.577758i \(-0.803928\pi\)
−0.816208 + 0.577758i \(0.803928\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −7.44364 −0.257907
\(834\) 0 0
\(835\) −16.6230 −0.575264
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −16.8724 −0.582501 −0.291251 0.956647i \(-0.594071\pi\)
−0.291251 + 0.956647i \(0.594071\pi\)
\(840\) 0 0
\(841\) −25.9736 −0.895642
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 35.0390 1.20538
\(846\) 0 0
\(847\) 5.95859 0.204739
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 11.7333 0.402214
\(852\) 0 0
\(853\) −7.22470 −0.247369 −0.123685 0.992322i \(-0.539471\pi\)
−0.123685 + 0.992322i \(0.539471\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 44.3796 1.51598 0.757989 0.652268i \(-0.226182\pi\)
0.757989 + 0.652268i \(0.226182\pi\)
\(858\) 0 0
\(859\) −53.2464 −1.81674 −0.908372 0.418163i \(-0.862674\pi\)
−0.908372 + 0.418163i \(0.862674\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −17.0594 −0.580710 −0.290355 0.956919i \(-0.593773\pi\)
−0.290355 + 0.956919i \(0.593773\pi\)
\(864\) 0 0
\(865\) −24.6258 −0.837302
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 14.9725 0.507907
\(870\) 0 0
\(871\) −1.27299 −0.0431336
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −6.83360 −0.231018
\(876\) 0 0
\(877\) −7.59553 −0.256483 −0.128241 0.991743i \(-0.540933\pi\)
−0.128241 + 0.991743i \(0.540933\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −22.7040 −0.764917 −0.382459 0.923973i \(-0.624923\pi\)
−0.382459 + 0.923973i \(0.624923\pi\)
\(882\) 0 0
\(883\) 31.8704 1.07252 0.536262 0.844052i \(-0.319836\pi\)
0.536262 + 0.844052i \(0.319836\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 35.3029 1.18535 0.592677 0.805440i \(-0.298071\pi\)
0.592677 + 0.805440i \(0.298071\pi\)
\(888\) 0 0
\(889\) −2.24903 −0.0754300
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 42.0384 1.40676
\(894\) 0 0
\(895\) −58.0912 −1.94177
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −3.02637 −0.100935
\(900\) 0 0
\(901\) −40.3671 −1.34482
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −49.8158 −1.65593
\(906\) 0 0
\(907\) −15.1890 −0.504344 −0.252172 0.967682i \(-0.581145\pi\)
−0.252172 + 0.967682i \(0.581145\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −44.1138 −1.46156 −0.730778 0.682615i \(-0.760843\pi\)
−0.730778 + 0.682615i \(0.760843\pi\)
\(912\) 0 0
\(913\) −7.87130 −0.260502
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 17.9549 0.592922
\(918\) 0 0
\(919\) −11.1590 −0.368100 −0.184050 0.982917i \(-0.558921\pi\)
−0.184050 + 0.982917i \(0.558921\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0.322912 0.0106288
\(924\) 0 0
\(925\) 6.24997 0.205498
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0.0536799 0.00176118 0.000880591 1.00000i \(-0.499720\pi\)
0.000880591 1.00000i \(0.499720\pi\)
\(930\) 0 0
\(931\) −4.24531 −0.139134
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 45.7885 1.49744
\(936\) 0 0
\(937\) −0.275398 −0.00899685 −0.00449843 0.999990i \(-0.501432\pi\)
−0.00449843 + 0.999990i \(0.501432\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −0.301768 −0.00983734 −0.00491867 0.999988i \(-0.501566\pi\)
−0.00491867 + 0.999988i \(0.501566\pi\)
\(942\) 0 0
\(943\) −22.2953 −0.726034
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 38.7859 1.26037 0.630186 0.776444i \(-0.282979\pi\)
0.630186 + 0.776444i \(0.282979\pi\)
\(948\) 0 0
\(949\) 3.65568 0.118668
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 8.09396 0.262189 0.131094 0.991370i \(-0.458151\pi\)
0.131094 + 0.991370i \(0.458151\pi\)
\(954\) 0 0
\(955\) 71.1484 2.30231
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 8.24531 0.266255
\(960\) 0 0
\(961\) −27.9736 −0.902375
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −51.9709 −1.67300
\(966\) 0 0
\(967\) 12.9475 0.416362 0.208181 0.978090i \(-0.433246\pi\)
0.208181 + 0.978090i \(0.433246\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −44.0886 −1.41487 −0.707435 0.706778i \(-0.750148\pi\)
−0.707435 + 0.706778i \(0.750148\pi\)
\(972\) 0 0
\(973\) −12.2865 −0.393888
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 30.3365 0.970550 0.485275 0.874362i \(-0.338719\pi\)
0.485275 + 0.874362i \(0.338719\pi\)
\(978\) 0 0
\(979\) −4.01152 −0.128209
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −3.95507 −0.126147 −0.0630735 0.998009i \(-0.520090\pi\)
−0.0630735 + 0.998009i \(0.520090\pi\)
\(984\) 0 0
\(985\) −6.34670 −0.202223
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 9.24028 0.293824
\(990\) 0 0
\(991\) 11.0650 0.351491 0.175746 0.984436i \(-0.443766\pi\)
0.175746 + 0.984436i \(0.443766\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −58.6091 −1.85803
\(996\) 0 0
\(997\) −11.1146 −0.352002 −0.176001 0.984390i \(-0.556316\pi\)
−0.176001 + 0.984390i \(0.556316\pi\)
\(998\) 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.bs.1.1 yes 4
3.2 odd 2 6048.2.a.bk.1.4 4
4.3 odd 2 6048.2.a.bt.1.1 yes 4
12.11 even 2 6048.2.a.bp.1.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.a.bk.1.4 4 3.2 odd 2
6048.2.a.bp.1.4 yes 4 12.11 even 2
6048.2.a.bs.1.1 yes 4 1.1 even 1 trivial
6048.2.a.bt.1.1 yes 4 4.3 odd 2