Properties

Label 6048.2.a.bt.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 \(-0.609917\pi\)
−0.338493 + 0.940969i \(0.609917\pi\)
\(12\) 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 \(-0.338102\pi\)
0.486970 + 0.873418i \(0.338102\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.70399 −0.980851 −0.490425 0.871483i \(-0.663159\pi\)
−0.490425 + 0.871483i \(0.663159\pi\)
\(24\) 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 \(-0.549932\pi\)
−0.156225 + 0.987722i \(0.549932\pi\)
\(32\) 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 \(-0.547857\pi\)
−0.149780 + 0.988719i \(0.547857\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.90233 1.44440 0.722201 0.691683i \(-0.243130\pi\)
0.722201 + 0.691683i \(0.243130\pi\)
\(48\) 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 \(-0.361410\pi\)
0.421766 + 0.906705i \(0.361410\pi\)
\(60\) 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 \(-0.554224\pi\)
−0.169528 + 0.985525i \(0.554224\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0.703994 0.0835488 0.0417744 0.999127i \(-0.486699\pi\)
0.0417744 + 0.999127i \(0.486699\pi\)
\(72\) 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 \(-0.622400\pi\)
−0.375124 + 0.926975i \(0.622400\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.50566 0.384796 0.192398 0.981317i \(-0.438373\pi\)
0.192398 + 0.981317i \(0.438373\pi\)
\(84\) 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 \(-0.670897\pi\)
−0.511466 + 0.859304i \(0.670897\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 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 \(-0.531815\pi\)
−0.0997845 + 0.995009i \(0.531815\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 17.9549 1.56872 0.784362 0.620303i \(-0.212990\pi\)
0.784362 + 0.620303i \(0.212990\pi\)
\(132\) 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 \(-0.674465\pi\)
−0.521065 + 0.853517i \(0.674465\pi\)
\(140\) 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 \(-0.312920\pi\)
0.554472 + 0.832202i \(0.312920\pi\)
\(152\) 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 \(-0.344191\pi\)
0.470174 + 0.882574i \(0.344191\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −6.06759 −0.469524 −0.234762 0.972053i \(-0.575431\pi\)
−0.234762 + 0.972053i \(0.575431\pi\)
\(168\) 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 \(-0.791183\pi\)
−0.792427 + 0.609967i \(0.791183\pi\)
\(180\) 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 \(-0.111236\pi\)
0.939558 + 0.342389i \(0.111236\pi\)
\(192\) 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 \(-0.773947\pi\)
−0.758253 + 0.651961i \(0.773947\pi\)
\(200\) 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 \(-0.832049\pi\)
−0.864000 + 0.503491i \(0.832049\pi\)
\(212\) 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 \(-0.182265\pi\)
0.840494 + 0.541821i \(0.182265\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 20.4529 1.35751 0.678754 0.734366i \(-0.262520\pi\)
0.678754 + 0.734366i \(0.262520\pi\)
\(228\) 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 \(-0.670319\pi\)
−0.509903 + 0.860232i \(0.670319\pi\)
\(240\) 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 \(-0.710412\pi\)
−0.613929 + 0.789361i \(0.710412\pi\)
\(252\) 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 \(-0.578398\pi\)
−0.243811 + 0.969823i \(0.578398\pi\)
\(264\) 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 \(-0.274868\pi\)
0.649764 + 0.760136i \(0.274868\pi\)
\(272\) 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 \(-0.619298\pi\)
−0.366073 + 0.930586i \(0.619298\pi\)
\(284\) 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 \(-0.600952\pi\)
−0.311861 + 0.950128i \(0.600952\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 20.4342 1.15871 0.579357 0.815074i \(-0.303304\pi\)
0.579357 + 0.815074i \(0.303304\pi\)
\(312\) 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 \(-0.325605\pi\)
0.520876 + 0.853632i \(0.325605\pi\)
\(332\) 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 \(-0.452064\pi\)
0.150028 + 0.988682i \(0.452064\pi\)
\(348\) 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 \(-0.442429\pi\)
0.179881 + 0.983688i \(0.442429\pi\)
\(360\) 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 \(-0.650267\pi\)
−0.454737 + 0.890626i \(0.650267\pi\)
\(368\) 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 \(-0.386450\pi\)
0.349211 + 0.937044i \(0.386450\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −14.5582 −0.743889 −0.371945 0.928255i \(-0.621309\pi\)
−0.371945 + 0.928255i \(0.621309\pi\)
\(384\) 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 \(-0.674406\pi\)
−0.520906 + 0.853614i \(0.674406\pi\)
\(420\) 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 \(-0.557887\pi\)
−0.180857 + 0.983509i \(0.557887\pi\)
\(432\) 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 \(-0.701988\pi\)
−0.592826 + 0.805331i \(0.701988\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 32.1851 1.52916 0.764581 0.644528i \(-0.222946\pi\)
0.764581 + 0.644528i \(0.222946\pi\)
\(444\) 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 \(-0.592189\pi\)
−0.285588 + 0.958352i \(0.592189\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −29.2839 −1.35510 −0.677550 0.735477i \(-0.736958\pi\)
−0.677550 + 0.735477i \(0.736958\pi\)
\(468\) 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 \(-0.174903\pi\)
0.852800 + 0.522238i \(0.174903\pi\)
\(480\) 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 \(-0.358913\pi\)
0.428866 + 0.903368i \(0.358913\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −32.0273 −1.44537 −0.722686 0.691176i \(-0.757093\pi\)
−0.722686 + 0.691176i \(0.757093\pi\)
\(492\) 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 \(-0.194960\pi\)
0.818223 + 0.574901i \(0.194960\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −9.57697 −0.427016 −0.213508 0.976941i \(-0.568489\pi\)
−0.213508 + 0.976941i \(0.568489\pi\)
\(504\) 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 \(-0.319482\pi\)
0.537200 + 0.843455i \(0.319482\pi\)
\(524\) 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 \(-0.677906\pi\)
−0.530261 + 0.847835i \(0.677906\pi\)
\(548\) 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 \(-0.226885\pi\)
0.756547 + 0.653939i \(0.226885\pi\)
\(564\) 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 \(-0.614758\pi\)
−0.352765 + 0.935712i \(0.614758\pi\)
\(572\) 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 \(-0.340111\pi\)
0.481447 + 0.876475i \(0.340111\pi\)
\(588\) 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 \(-0.422257\pi\)
0.241815 + 0.970322i \(0.422257\pi\)
\(600\) 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 \(-0.140789\pi\)
0.903769 + 0.428021i \(0.140789\pi\)
\(608\) 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 \(-0.738802\pi\)
−0.681799 + 0.731540i \(0.738802\pi\)
\(620\) 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 \(-0.863619\pi\)
−0.909609 + 0.415465i \(0.863619\pi\)
\(632\) 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 \(-0.658349\pi\)
−0.477203 + 0.878793i \(0.658349\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −12.5207 −0.492240 −0.246120 0.969239i \(-0.579156\pi\)
−0.246120 + 0.969239i \(0.579156\pi\)
\(648\) 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 \(-0.451047\pi\)
0.153185 + 0.988198i \(0.451047\pi\)
\(660\) 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 \(-0.544196\pi\)
−0.138400 + 0.990376i \(0.544196\pi\)
\(684\) 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 \(-0.458737\pi\)
0.129268 + 0.991610i \(0.458737\pi\)
\(692\) 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 \(-0.831892\pi\)
−0.863753 + 0.503916i \(0.831892\pi\)
\(720\) 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 \(-0.302712\pi\)
0.580872 + 0.813995i \(0.302712\pi\)
\(728\) 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 \(-0.595586\pi\)
−0.295801 + 0.955250i \(0.595586\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −3.39871 −0.124686 −0.0623432 0.998055i \(-0.519857\pi\)
−0.0623432 + 0.998055i \(0.519857\pi\)
\(744\) 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 \(-0.415352\pi\)
0.262807 + 0.964849i \(0.415352\pi\)
\(752\) 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 \(-0.716867\pi\)
−0.629808 + 0.776751i \(0.716867\pi\)
\(788\) 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 \(-0.453184\pi\)
0.146546 + 0.989204i \(0.453184\pi\)
\(812\) 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 \(-0.408126\pi\)
0.284639 + 0.958635i \(0.408126\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −35.6533 −1.23979 −0.619893 0.784686i \(-0.712824\pi\)
−0.619893 + 0.784686i \(0.712824\pi\)
\(828\) 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 \(-0.405929\pi\)
0.291251 + 0.956647i \(0.405929\pi\)
\(840\) 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 \(-0.137326\pi\)
0.908372 + 0.418163i \(0.137326\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 17.0594 0.580710 0.290355 0.956919i \(-0.406227\pi\)
0.290355 + 0.956919i \(0.406227\pi\)
\(864\) 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 \(-0.680164\pi\)
−0.536262 + 0.844052i \(0.680164\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −35.3029 −1.18535 −0.592677 0.805440i \(-0.701929\pi\)
−0.592677 + 0.805440i \(0.701929\pi\)
\(888\) 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 \(-0.418855\pi\)
0.252172 + 0.967682i \(0.418855\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 44.1138 1.46156 0.730778 0.682615i \(-0.239157\pi\)
0.730778 + 0.682615i \(0.239157\pi\)
\(912\) 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 \(-0.441079\pi\)
0.184050 + 0.982917i \(0.441079\pi\)
\(920\) 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 \(-0.717021\pi\)
−0.630186 + 0.776444i \(0.717021\pi\)
\(948\) 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 \(-0.566754\pi\)
−0.208181 + 0.978090i \(0.566754\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 44.0886 1.41487 0.707435 0.706778i \(-0.249852\pi\)
0.707435 + 0.706778i \(0.249852\pi\)
\(972\) 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 \(-0.479910\pi\)
0.0630735 + 0.998009i \(0.479910\pi\)
\(984\) 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 \(-0.556234\pi\)
−0.175746 + 0.984436i \(0.556234\pi\)
\(992\) 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.bt.1.1 yes 4
3.2 odd 2 6048.2.a.bp.1.4 yes 4
4.3 odd 2 6048.2.a.bs.1.1 yes 4
12.11 even 2 6048.2.a.bk.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.a.bk.1.4 4 12.11 even 2
6048.2.a.bp.1.4 yes 4 3.2 odd 2
6048.2.a.bs.1.1 yes 4 4.3 odd 2
6048.2.a.bt.1.1 yes 4 1.1 even 1 trivial