Properties

Label 6048.2.j.d.5615.12
Level 6048
Weight 2
Character 6048.5615
Analytic conductor 48.294
Analytic rank 0
Dimension 48
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6048.j (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(48\)
Coefficient ring index: multiple of None
Twist minimal: no (minimal twist has level 1512)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 5615.12
Character \(\chi\) = 6048.5615
Dual form 6048.2.j.d.5615.11

$q$-expansion

\(f(q)\) \(=\) \(q-2.35025 q^{5} -1.00000i q^{7} +O(q^{10})\) \(q-2.35025 q^{5} -1.00000i q^{7} -1.09095i q^{11} -4.10613i q^{13} +4.74991i q^{17} +7.10196 q^{19} -3.99240 q^{23} +0.523653 q^{25} -4.09983 q^{29} +10.8933i q^{31} +2.35025i q^{35} +2.27336i q^{37} +1.29246i q^{41} +8.59200 q^{43} +6.79438 q^{47} -1.00000 q^{49} +0.688525 q^{53} +2.56400i q^{55} -7.72675i q^{59} -9.70563i q^{61} +9.65042i q^{65} -8.23313 q^{67} +6.46122 q^{71} +8.44644 q^{73} -1.09095 q^{77} -5.70061i q^{79} +10.0681i q^{83} -11.1634i q^{85} -2.54950i q^{89} -4.10613 q^{91} -16.6913 q^{95} -15.6447 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48q + O(q^{10}) \) \( 48q - 16q^{19} + 48q^{25} - 64q^{43} - 48q^{49} - 32q^{67} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times\).

\(n\) \(2593\) \(3781\) \(3809\) \(4159\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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
\(5\) −2.35025 −1.05106 −0.525531 0.850775i \(-0.676133\pi\)
−0.525531 + 0.850775i \(0.676133\pi\)
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 1.09095i − 0.328934i −0.986383 0.164467i \(-0.947410\pi\)
0.986383 0.164467i \(-0.0525904\pi\)
\(12\) 0 0
\(13\) − 4.10613i − 1.13884i −0.822048 0.569418i \(-0.807168\pi\)
0.822048 0.569418i \(-0.192832\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.74991i 1.15202i 0.817442 + 0.576011i \(0.195391\pi\)
−0.817442 + 0.576011i \(0.804609\pi\)
\(18\) 0 0
\(19\) 7.10196 1.62930 0.814650 0.579953i \(-0.196929\pi\)
0.814650 + 0.579953i \(0.196929\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.99240 −0.832474 −0.416237 0.909256i \(-0.636651\pi\)
−0.416237 + 0.909256i \(0.636651\pi\)
\(24\) 0 0
\(25\) 0.523653 0.104731
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.09983 −0.761319 −0.380660 0.924715i \(-0.624303\pi\)
−0.380660 + 0.924715i \(0.624303\pi\)
\(30\) 0 0
\(31\) 10.8933i 1.95649i 0.207454 + 0.978245i \(0.433482\pi\)
−0.207454 + 0.978245i \(0.566518\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.35025i 0.397264i
\(36\) 0 0
\(37\) 2.27336i 0.373737i 0.982385 + 0.186869i \(0.0598339\pi\)
−0.982385 + 0.186869i \(0.940166\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.29246i 0.201848i 0.994894 + 0.100924i \(0.0321799\pi\)
−0.994894 + 0.100924i \(0.967820\pi\)
\(42\) 0 0
\(43\) 8.59200 1.31027 0.655134 0.755513i \(-0.272612\pi\)
0.655134 + 0.755513i \(0.272612\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.79438 0.991062 0.495531 0.868590i \(-0.334974\pi\)
0.495531 + 0.868590i \(0.334974\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.688525 0.0945762 0.0472881 0.998881i \(-0.484942\pi\)
0.0472881 + 0.998881i \(0.484942\pi\)
\(54\) 0 0
\(55\) 2.56400i 0.345730i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 7.72675i − 1.00594i −0.864305 0.502968i \(-0.832241\pi\)
0.864305 0.502968i \(-0.167759\pi\)
\(60\) 0 0
\(61\) − 9.70563i − 1.24268i −0.783542 0.621339i \(-0.786589\pi\)
0.783542 0.621339i \(-0.213411\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 9.65042i 1.19699i
\(66\) 0 0
\(67\) −8.23313 −1.00584 −0.502918 0.864334i \(-0.667740\pi\)
−0.502918 + 0.864334i \(0.667740\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6.46122 0.766806 0.383403 0.923581i \(-0.374752\pi\)
0.383403 + 0.923581i \(0.374752\pi\)
\(72\) 0 0
\(73\) 8.44644 0.988581 0.494290 0.869297i \(-0.335428\pi\)
0.494290 + 0.869297i \(0.335428\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.09095 −0.124325
\(78\) 0 0
\(79\) − 5.70061i − 0.641369i −0.947186 0.320684i \(-0.896087\pi\)
0.947186 0.320684i \(-0.103913\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 10.0681i 1.10512i 0.833474 + 0.552558i \(0.186348\pi\)
−0.833474 + 0.552558i \(0.813652\pi\)
\(84\) 0 0
\(85\) − 11.1634i − 1.21085i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 2.54950i − 0.270246i −0.990829 0.135123i \(-0.956857\pi\)
0.990829 0.135123i \(-0.0431430\pi\)
\(90\) 0 0
\(91\) −4.10613 −0.430440
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −16.6913 −1.71250
\(96\) 0 0
\(97\) −15.6447 −1.58847 −0.794237 0.607607i \(-0.792129\pi\)
−0.794237 + 0.607607i \(0.792129\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −12.3962 −1.23347 −0.616736 0.787170i \(-0.711546\pi\)
−0.616736 + 0.787170i \(0.711546\pi\)
\(102\) 0 0
\(103\) − 13.2234i − 1.30294i −0.758673 0.651472i \(-0.774152\pi\)
0.758673 0.651472i \(-0.225848\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 5.44530i − 0.526417i −0.964739 0.263209i \(-0.915219\pi\)
0.964739 0.263209i \(-0.0847808\pi\)
\(108\) 0 0
\(109\) 0.947776i 0.0907805i 0.998969 + 0.0453902i \(0.0144531\pi\)
−0.998969 + 0.0453902i \(0.985547\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 16.8819i 1.58812i 0.607840 + 0.794059i \(0.292036\pi\)
−0.607840 + 0.794059i \(0.707964\pi\)
\(114\) 0 0
\(115\) 9.38313 0.874981
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 4.74991 0.435423
\(120\) 0 0
\(121\) 9.80982 0.891802
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 10.5205 0.940983
\(126\) 0 0
\(127\) − 10.2310i − 0.907852i −0.891039 0.453926i \(-0.850023\pi\)
0.891039 0.453926i \(-0.149977\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 15.9013i − 1.38930i −0.719346 0.694651i \(-0.755559\pi\)
0.719346 0.694651i \(-0.244441\pi\)
\(132\) 0 0
\(133\) − 7.10196i − 0.615818i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 10.0619i 0.859647i 0.902913 + 0.429823i \(0.141424\pi\)
−0.902913 + 0.429823i \(0.858576\pi\)
\(138\) 0 0
\(139\) 4.44750 0.377232 0.188616 0.982051i \(-0.439600\pi\)
0.188616 + 0.982051i \(0.439600\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −4.47959 −0.374602
\(144\) 0 0
\(145\) 9.63561 0.800193
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −12.3264 −1.00981 −0.504907 0.863174i \(-0.668473\pi\)
−0.504907 + 0.863174i \(0.668473\pi\)
\(150\) 0 0
\(151\) − 9.25492i − 0.753154i −0.926385 0.376577i \(-0.877101\pi\)
0.926385 0.376577i \(-0.122899\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 25.6019i − 2.05639i
\(156\) 0 0
\(157\) − 4.19774i − 0.335016i −0.985871 0.167508i \(-0.946428\pi\)
0.985871 0.167508i \(-0.0535720\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 3.99240i 0.314645i
\(162\) 0 0
\(163\) 23.6952 1.85595 0.927977 0.372639i \(-0.121547\pi\)
0.927977 + 0.372639i \(0.121547\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 4.79250 0.370855 0.185427 0.982658i \(-0.440633\pi\)
0.185427 + 0.982658i \(0.440633\pi\)
\(168\) 0 0
\(169\) −3.86033 −0.296949
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 18.2897 1.39054 0.695271 0.718747i \(-0.255284\pi\)
0.695271 + 0.718747i \(0.255284\pi\)
\(174\) 0 0
\(175\) − 0.523653i − 0.0395844i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 6.64806i 0.496899i 0.968645 + 0.248450i \(0.0799210\pi\)
−0.968645 + 0.248450i \(0.920079\pi\)
\(180\) 0 0
\(181\) 2.46631i 0.183320i 0.995790 + 0.0916598i \(0.0292172\pi\)
−0.995790 + 0.0916598i \(0.970783\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 5.34294i − 0.392821i
\(186\) 0 0
\(187\) 5.18192 0.378939
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −12.9048 −0.933760 −0.466880 0.884321i \(-0.654622\pi\)
−0.466880 + 0.884321i \(0.654622\pi\)
\(192\) 0 0
\(193\) 18.3689 1.32222 0.661112 0.750287i \(-0.270085\pi\)
0.661112 + 0.750287i \(0.270085\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 15.0957 1.07552 0.537761 0.843098i \(-0.319270\pi\)
0.537761 + 0.843098i \(0.319270\pi\)
\(198\) 0 0
\(199\) − 3.51621i − 0.249258i −0.992203 0.124629i \(-0.960226\pi\)
0.992203 0.124629i \(-0.0397740\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 4.09983i 0.287752i
\(204\) 0 0
\(205\) − 3.03759i − 0.212155i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 7.74789i − 0.535933i
\(210\) 0 0
\(211\) 20.2502 1.39408 0.697041 0.717031i \(-0.254500\pi\)
0.697041 + 0.717031i \(0.254500\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −20.1933 −1.37717
\(216\) 0 0
\(217\) 10.8933 0.739484
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 19.5038 1.31196
\(222\) 0 0
\(223\) − 12.9963i − 0.870295i −0.900359 0.435148i \(-0.856696\pi\)
0.900359 0.435148i \(-0.143304\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 28.8974i − 1.91799i −0.283430 0.958993i \(-0.591473\pi\)
0.283430 0.958993i \(-0.408527\pi\)
\(228\) 0 0
\(229\) 3.65438i 0.241488i 0.992684 + 0.120744i \(0.0385281\pi\)
−0.992684 + 0.120744i \(0.961472\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 13.7096i − 0.898143i −0.893496 0.449071i \(-0.851755\pi\)
0.893496 0.449071i \(-0.148245\pi\)
\(234\) 0 0
\(235\) −15.9685 −1.04167
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 20.5360 1.32836 0.664181 0.747572i \(-0.268780\pi\)
0.664181 + 0.747572i \(0.268780\pi\)
\(240\) 0 0
\(241\) 22.1694 1.42806 0.714030 0.700116i \(-0.246868\pi\)
0.714030 + 0.700116i \(0.246868\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.35025 0.150152
\(246\) 0 0
\(247\) − 29.1616i − 1.85551i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 2.82901i − 0.178565i −0.996006 0.0892827i \(-0.971543\pi\)
0.996006 0.0892827i \(-0.0284575\pi\)
\(252\) 0 0
\(253\) 4.35552i 0.273829i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 19.2158i − 1.19865i −0.800505 0.599325i \(-0.795436\pi\)
0.800505 0.599325i \(-0.204564\pi\)
\(258\) 0 0
\(259\) 2.27336 0.141259
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 26.5713 1.63846 0.819230 0.573465i \(-0.194401\pi\)
0.819230 + 0.573465i \(0.194401\pi\)
\(264\) 0 0
\(265\) −1.61820 −0.0994054
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −7.88420 −0.480708 −0.240354 0.970685i \(-0.577264\pi\)
−0.240354 + 0.970685i \(0.577264\pi\)
\(270\) 0 0
\(271\) 15.1206i 0.918513i 0.888304 + 0.459256i \(0.151884\pi\)
−0.888304 + 0.459256i \(0.848116\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 0.571280i − 0.0344495i
\(276\) 0 0
\(277\) − 10.1215i − 0.608141i −0.952650 0.304071i \(-0.901654\pi\)
0.952650 0.304071i \(-0.0983459\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 23.6645i 1.41170i 0.708359 + 0.705852i \(0.249436\pi\)
−0.708359 + 0.705852i \(0.750564\pi\)
\(282\) 0 0
\(283\) −18.1838 −1.08091 −0.540457 0.841371i \(-0.681749\pi\)
−0.540457 + 0.841371i \(0.681749\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.29246 0.0762914
\(288\) 0 0
\(289\) −5.56163 −0.327155
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −20.8327 −1.21706 −0.608529 0.793532i \(-0.708240\pi\)
−0.608529 + 0.793532i \(0.708240\pi\)
\(294\) 0 0
\(295\) 18.1597i 1.05730i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 16.3933i 0.948052i
\(300\) 0 0
\(301\) − 8.59200i − 0.495235i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 22.8106i 1.30613i
\(306\) 0 0
\(307\) 4.57939 0.261360 0.130680 0.991425i \(-0.458284\pi\)
0.130680 + 0.991425i \(0.458284\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 6.20035 0.351589 0.175795 0.984427i \(-0.443751\pi\)
0.175795 + 0.984427i \(0.443751\pi\)
\(312\) 0 0
\(313\) −22.0230 −1.24481 −0.622406 0.782695i \(-0.713845\pi\)
−0.622406 + 0.782695i \(0.713845\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −2.35768 −0.132420 −0.0662102 0.997806i \(-0.521091\pi\)
−0.0662102 + 0.997806i \(0.521091\pi\)
\(318\) 0 0
\(319\) 4.47272i 0.250424i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 33.7336i 1.87699i
\(324\) 0 0
\(325\) − 2.15019i − 0.119271i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 6.79438i − 0.374586i
\(330\) 0 0
\(331\) −2.15331 −0.118357 −0.0591784 0.998247i \(-0.518848\pi\)
−0.0591784 + 0.998247i \(0.518848\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 19.3499 1.05720
\(336\) 0 0
\(337\) 23.2469 1.26634 0.633169 0.774014i \(-0.281754\pi\)
0.633169 + 0.774014i \(0.281754\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 11.8840 0.643556
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 26.1260i − 1.40252i −0.712906 0.701260i \(-0.752621\pi\)
0.712906 0.701260i \(-0.247379\pi\)
\(348\) 0 0
\(349\) 3.68463i 0.197234i 0.995125 + 0.0986169i \(0.0314418\pi\)
−0.995125 + 0.0986169i \(0.968558\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 15.5433i − 0.827285i −0.910439 0.413642i \(-0.864256\pi\)
0.910439 0.413642i \(-0.135744\pi\)
\(354\) 0 0
\(355\) −15.1855 −0.805960
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 23.0367 1.21583 0.607914 0.794003i \(-0.292006\pi\)
0.607914 + 0.794003i \(0.292006\pi\)
\(360\) 0 0
\(361\) 31.4378 1.65462
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −19.8512 −1.03906
\(366\) 0 0
\(367\) − 16.1135i − 0.841116i −0.907266 0.420558i \(-0.861834\pi\)
0.907266 0.420558i \(-0.138166\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 0.688525i − 0.0357464i
\(372\) 0 0
\(373\) − 8.37018i − 0.433392i −0.976239 0.216696i \(-0.930472\pi\)
0.976239 0.216696i \(-0.0695280\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 16.8344i 0.867018i
\(378\) 0 0
\(379\) 4.66739 0.239748 0.119874 0.992789i \(-0.461751\pi\)
0.119874 + 0.992789i \(0.461751\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −11.1113 −0.567761 −0.283880 0.958860i \(-0.591622\pi\)
−0.283880 + 0.958860i \(0.591622\pi\)
\(384\) 0 0
\(385\) 2.56400 0.130674
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 33.8483 1.71618 0.858090 0.513500i \(-0.171651\pi\)
0.858090 + 0.513500i \(0.171651\pi\)
\(390\) 0 0
\(391\) − 18.9636i − 0.959028i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 13.3978i 0.674118i
\(396\) 0 0
\(397\) − 30.2554i − 1.51848i −0.650813 0.759238i \(-0.725572\pi\)
0.650813 0.759238i \(-0.274428\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 35.7603i − 1.78578i −0.450272 0.892891i \(-0.648673\pi\)
0.450272 0.892891i \(-0.351327\pi\)
\(402\) 0 0
\(403\) 44.7292 2.22812
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2.48012 0.122935
\(408\) 0 0
\(409\) 25.9283 1.28207 0.641036 0.767511i \(-0.278505\pi\)
0.641036 + 0.767511i \(0.278505\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −7.72675 −0.380208
\(414\) 0 0
\(415\) − 23.6625i − 1.16155i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 6.32586i − 0.309038i −0.987990 0.154519i \(-0.950617\pi\)
0.987990 0.154519i \(-0.0493829\pi\)
\(420\) 0 0
\(421\) 27.3586i 1.33338i 0.745336 + 0.666689i \(0.232289\pi\)
−0.745336 + 0.666689i \(0.767711\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 2.48730i 0.120652i
\(426\) 0 0
\(427\) −9.70563 −0.469688
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −27.5526 −1.32716 −0.663582 0.748104i \(-0.730965\pi\)
−0.663582 + 0.748104i \(0.730965\pi\)
\(432\) 0 0
\(433\) 13.0479 0.627040 0.313520 0.949582i \(-0.398492\pi\)
0.313520 + 0.949582i \(0.398492\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −28.3539 −1.35635
\(438\) 0 0
\(439\) − 6.42338i − 0.306571i −0.988182 0.153286i \(-0.951015\pi\)
0.988182 0.153286i \(-0.0489854\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 22.1824i 1.05392i 0.849891 + 0.526959i \(0.176668\pi\)
−0.849891 + 0.526959i \(0.823332\pi\)
\(444\) 0 0
\(445\) 5.99195i 0.284046i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.86593i 0.0880586i 0.999030 + 0.0440293i \(0.0140195\pi\)
−0.999030 + 0.0440293i \(0.985981\pi\)
\(450\) 0 0
\(451\) 1.41001 0.0663947
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 9.65042 0.452419
\(456\) 0 0
\(457\) −28.3835 −1.32773 −0.663863 0.747854i \(-0.731084\pi\)
−0.663863 + 0.747854i \(0.731084\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −33.2364 −1.54797 −0.773986 0.633202i \(-0.781740\pi\)
−0.773986 + 0.633202i \(0.781740\pi\)
\(462\) 0 0
\(463\) − 2.11779i − 0.0984219i −0.998788 0.0492110i \(-0.984329\pi\)
0.998788 0.0492110i \(-0.0156707\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 13.7416i 0.635884i 0.948110 + 0.317942i \(0.102992\pi\)
−0.948110 + 0.317942i \(0.897008\pi\)
\(468\) 0 0
\(469\) 8.23313i 0.380171i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 9.37345i − 0.430992i
\(474\) 0 0
\(475\) 3.71896 0.170638
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −8.78287 −0.401299 −0.200650 0.979663i \(-0.564305\pi\)
−0.200650 + 0.979663i \(0.564305\pi\)
\(480\) 0 0
\(481\) 9.33470 0.425626
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 36.7688 1.66959
\(486\) 0 0
\(487\) 4.42586i 0.200555i 0.994960 + 0.100277i \(0.0319730\pi\)
−0.994960 + 0.100277i \(0.968027\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 18.5511i − 0.837199i −0.908171 0.418599i \(-0.862521\pi\)
0.908171 0.418599i \(-0.137479\pi\)
\(492\) 0 0
\(493\) − 19.4738i − 0.877056i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 6.46122i − 0.289825i
\(498\) 0 0
\(499\) −5.23350 −0.234284 −0.117142 0.993115i \(-0.537373\pi\)
−0.117142 + 0.993115i \(0.537373\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 27.2800 1.21635 0.608177 0.793802i \(-0.291901\pi\)
0.608177 + 0.793802i \(0.291901\pi\)
\(504\) 0 0
\(505\) 29.1342 1.29646
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −34.0131 −1.50761 −0.753803 0.657101i \(-0.771783\pi\)
−0.753803 + 0.657101i \(0.771783\pi\)
\(510\) 0 0
\(511\) − 8.44644i − 0.373648i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 31.0783i 1.36947i
\(516\) 0 0
\(517\) − 7.41234i − 0.325994i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 34.1052i − 1.49417i −0.664726 0.747087i \(-0.731452\pi\)
0.664726 0.747087i \(-0.268548\pi\)
\(522\) 0 0
\(523\) −24.7371 −1.08168 −0.540840 0.841126i \(-0.681894\pi\)
−0.540840 + 0.841126i \(0.681894\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −51.7420 −2.25392
\(528\) 0 0
\(529\) −7.06071 −0.306987
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 5.30700 0.229872
\(534\) 0 0
\(535\) 12.7978i 0.553297i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1.09095i 0.0469906i
\(540\) 0 0
\(541\) − 1.11043i − 0.0477412i −0.999715 0.0238706i \(-0.992401\pi\)
0.999715 0.0238706i \(-0.00759897\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 2.22751i − 0.0954159i
\(546\) 0 0
\(547\) 43.6930 1.86818 0.934089 0.357040i \(-0.116214\pi\)
0.934089 + 0.357040i \(0.116214\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −29.1168 −1.24042
\(552\) 0 0
\(553\) −5.70061 −0.242415
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 33.0536 1.40053 0.700263 0.713884i \(-0.253066\pi\)
0.700263 + 0.713884i \(0.253066\pi\)
\(558\) 0 0
\(559\) − 35.2799i − 1.49218i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 17.6097i − 0.742159i −0.928601 0.371080i \(-0.878988\pi\)
0.928601 0.371080i \(-0.121012\pi\)
\(564\) 0 0
\(565\) − 39.6767i − 1.66921i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 12.0355i 0.504555i 0.967655 + 0.252277i \(0.0811795\pi\)
−0.967655 + 0.252277i \(0.918820\pi\)
\(570\) 0 0
\(571\) 6.04104 0.252810 0.126405 0.991979i \(-0.459656\pi\)
0.126405 + 0.991979i \(0.459656\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −2.09063 −0.0871854
\(576\) 0 0
\(577\) −31.2605 −1.30139 −0.650696 0.759339i \(-0.725523\pi\)
−0.650696 + 0.759339i \(0.725523\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 10.0681 0.417695
\(582\) 0 0
\(583\) − 0.751147i − 0.0311093i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 29.7168i 1.22654i 0.789872 + 0.613272i \(0.210147\pi\)
−0.789872 + 0.613272i \(0.789853\pi\)
\(588\) 0 0
\(589\) 77.3635i 3.18771i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 19.9618i 0.819732i 0.912146 + 0.409866i \(0.134425\pi\)
−0.912146 + 0.409866i \(0.865575\pi\)
\(594\) 0 0
\(595\) −11.1634 −0.457657
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −26.5569 −1.08509 −0.542543 0.840028i \(-0.682539\pi\)
−0.542543 + 0.840028i \(0.682539\pi\)
\(600\) 0 0
\(601\) −19.6439 −0.801289 −0.400645 0.916233i \(-0.631214\pi\)
−0.400645 + 0.916233i \(0.631214\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −23.0555 −0.937339
\(606\) 0 0
\(607\) − 29.0877i − 1.18063i −0.807172 0.590316i \(-0.799003\pi\)
0.807172 0.590316i \(-0.200997\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 27.8986i − 1.12866i
\(612\) 0 0
\(613\) − 9.44223i − 0.381368i −0.981651 0.190684i \(-0.938929\pi\)
0.981651 0.190684i \(-0.0610706\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 29.4177i − 1.18431i −0.805823 0.592157i \(-0.798277\pi\)
0.805823 0.592157i \(-0.201723\pi\)
\(618\) 0 0
\(619\) −4.45956 −0.179245 −0.0896225 0.995976i \(-0.528566\pi\)
−0.0896225 + 0.995976i \(0.528566\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −2.54950 −0.102144
\(624\) 0 0
\(625\) −27.3441 −1.09376
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −10.7982 −0.430554
\(630\) 0 0
\(631\) 23.6013i 0.939553i 0.882786 + 0.469776i \(0.155665\pi\)
−0.882786 + 0.469776i \(0.844335\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 24.0453i 0.954208i
\(636\) 0 0
\(637\) 4.10613i 0.162691i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 6.81316i 0.269104i 0.990907 + 0.134552i \(0.0429595\pi\)
−0.990907 + 0.134552i \(0.957041\pi\)
\(642\) 0 0
\(643\) 15.8069 0.623361 0.311681 0.950187i \(-0.399108\pi\)
0.311681 + 0.950187i \(0.399108\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 42.5205 1.67165 0.835827 0.548992i \(-0.184988\pi\)
0.835827 + 0.548992i \(0.184988\pi\)
\(648\) 0 0
\(649\) −8.42950 −0.330887
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −2.40359 −0.0940597 −0.0470298 0.998893i \(-0.514976\pi\)
−0.0470298 + 0.998893i \(0.514976\pi\)
\(654\) 0 0
\(655\) 37.3720i 1.46024i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 26.3292i 1.02564i 0.858497 + 0.512819i \(0.171399\pi\)
−0.858497 + 0.512819i \(0.828601\pi\)
\(660\) 0 0
\(661\) 8.09353i 0.314802i 0.987535 + 0.157401i \(0.0503115\pi\)
−0.987535 + 0.157401i \(0.949689\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 16.6913i 0.647262i
\(666\) 0 0
\(667\) 16.3682 0.633778
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −10.5884 −0.408759
\(672\) 0 0
\(673\) 3.41009 0.131449 0.0657246 0.997838i \(-0.479064\pi\)
0.0657246 + 0.997838i \(0.479064\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 12.4265 0.477589 0.238794 0.971070i \(-0.423248\pi\)
0.238794 + 0.971070i \(0.423248\pi\)
\(678\) 0 0
\(679\) 15.6447i 0.600387i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 19.4879i 0.745682i 0.927895 + 0.372841i \(0.121616\pi\)
−0.927895 + 0.372841i \(0.878384\pi\)
\(684\) 0 0
\(685\) − 23.6480i − 0.903542i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 2.82717i − 0.107707i
\(690\) 0 0
\(691\) −2.44853 −0.0931464 −0.0465732 0.998915i \(-0.514830\pi\)
−0.0465732 + 0.998915i \(0.514830\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −10.4527 −0.396494
\(696\) 0 0
\(697\) −6.13906 −0.232533
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −21.8472 −0.825157 −0.412578 0.910922i \(-0.635372\pi\)
−0.412578 + 0.910922i \(0.635372\pi\)
\(702\) 0 0
\(703\) 16.1453i 0.608930i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 12.3962i 0.466209i
\(708\) 0 0
\(709\) 50.4998i 1.89656i 0.317437 + 0.948279i \(0.397178\pi\)
−0.317437 + 0.948279i \(0.602822\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 43.4903i − 1.62873i
\(714\) 0 0
\(715\) 10.5281 0.393730
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 13.7799 0.513904 0.256952 0.966424i \(-0.417282\pi\)
0.256952 + 0.966424i \(0.417282\pi\)
\(720\) 0 0
\(721\) −13.2234 −0.492467
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −2.14689 −0.0797334
\(726\) 0 0
\(727\) 12.5617i 0.465888i 0.972490 + 0.232944i \(0.0748358\pi\)
−0.972490 + 0.232944i \(0.925164\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 40.8112i 1.50946i
\(732\) 0 0
\(733\) − 1.04203i − 0.0384883i −0.999815 0.0192442i \(-0.993874\pi\)
0.999815 0.0192442i \(-0.00612599\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8.98195i 0.330854i
\(738\) 0 0
\(739\) −3.73631 −0.137442 −0.0687212 0.997636i \(-0.521892\pi\)
−0.0687212 + 0.997636i \(0.521892\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −4.44956 −0.163239 −0.0816193 0.996664i \(-0.526009\pi\)
−0.0816193 + 0.996664i \(0.526009\pi\)
\(744\) 0 0
\(745\) 28.9700 1.06138
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −5.44530 −0.198967
\(750\) 0 0
\(751\) 23.5520i 0.859426i 0.902965 + 0.429713i \(0.141385\pi\)
−0.902965 + 0.429713i \(0.858615\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 21.7513i 0.791612i
\(756\) 0 0
\(757\) − 16.5240i − 0.600576i −0.953849 0.300288i \(-0.902917\pi\)
0.953849 0.300288i \(-0.0970829\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 9.27488i − 0.336214i −0.985769 0.168107i \(-0.946235\pi\)
0.985769 0.168107i \(-0.0537655\pi\)
\(762\) 0 0
\(763\) 0.947776 0.0343118
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −31.7270 −1.14560
\(768\) 0 0
\(769\) 18.2106 0.656690 0.328345 0.944558i \(-0.393509\pi\)
0.328345 + 0.944558i \(0.393509\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1.51105 0.0543486 0.0271743 0.999631i \(-0.491349\pi\)
0.0271743 + 0.999631i \(0.491349\pi\)
\(774\) 0 0
\(775\) 5.70429i 0.204904i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 9.17898i 0.328871i
\(780\) 0 0
\(781\) − 7.04888i − 0.252229i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 9.86571i 0.352122i
\(786\) 0 0
\(787\) 3.25985 0.116201 0.0581005 0.998311i \(-0.481496\pi\)
0.0581005 + 0.998311i \(0.481496\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 16.8819 0.600252
\(792\) 0 0
\(793\) −39.8526 −1.41521
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 50.5236 1.78964 0.894819 0.446428i \(-0.147304\pi\)
0.894819 + 0.446428i \(0.147304\pi\)
\(798\) 0 0
\(799\) 32.2727i 1.14172i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 9.21465i − 0.325178i
\(804\) 0 0
\(805\) − 9.38313i − 0.330712i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 17.9908i − 0.632523i −0.948672 0.316261i \(-0.897572\pi\)
0.948672 0.316261i \(-0.102428\pi\)
\(810\) 0 0
\(811\) −22.2314 −0.780650 −0.390325 0.920677i \(-0.627637\pi\)
−0.390325 + 0.920677i \(0.627637\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −55.6896 −1.95072
\(816\) 0 0
\(817\) 61.0200 2.13482
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 28.7163 1.00221 0.501103 0.865388i \(-0.332928\pi\)
0.501103 + 0.865388i \(0.332928\pi\)
\(822\) 0 0
\(823\) 46.1721i 1.60946i 0.593641 + 0.804730i \(0.297690\pi\)
−0.593641 + 0.804730i \(0.702310\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 55.5900i 1.93305i 0.256568 + 0.966526i \(0.417408\pi\)
−0.256568 + 0.966526i \(0.582592\pi\)
\(828\) 0 0
\(829\) − 7.13019i − 0.247642i −0.992305 0.123821i \(-0.960485\pi\)
0.992305 0.123821i \(-0.0395148\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 4.74991i − 0.164575i
\(834\) 0 0
\(835\) −11.2636 −0.389791
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 33.3725 1.15215 0.576073 0.817399i \(-0.304585\pi\)
0.576073 + 0.817399i \(0.304585\pi\)
\(840\) 0 0
\(841\) −12.1914 −0.420393
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 9.07273 0.312111
\(846\) 0 0
\(847\) − 9.80982i − 0.337070i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 9.07615i − 0.311127i
\(852\) 0 0
\(853\) − 34.4031i − 1.17794i −0.808155 0.588969i \(-0.799534\pi\)
0.808155 0.588969i \(-0.200466\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 24.5369i 0.838164i 0.907948 + 0.419082i \(0.137648\pi\)
−0.907948 + 0.419082i \(0.862352\pi\)
\(858\) 0 0
\(859\) −49.8496 −1.70085 −0.850424 0.526098i \(-0.823655\pi\)
−0.850424 + 0.526098i \(0.823655\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1.18405 0.0403055 0.0201528 0.999797i \(-0.493585\pi\)
0.0201528 + 0.999797i \(0.493585\pi\)
\(864\) 0 0
\(865\) −42.9854 −1.46155
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −6.21909 −0.210968
\(870\) 0 0
\(871\) 33.8063i 1.14548i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 10.5205i − 0.355658i
\(876\) 0 0
\(877\) − 52.6756i − 1.77873i −0.457200 0.889364i \(-0.651147\pi\)
0.457200 0.889364i \(-0.348853\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 48.5193i 1.63466i 0.576172 + 0.817329i \(0.304546\pi\)
−0.576172 + 0.817329i \(0.695454\pi\)
\(882\) 0 0
\(883\) −36.3496 −1.22326 −0.611630 0.791144i \(-0.709486\pi\)
−0.611630 + 0.791144i \(0.709486\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −14.5979 −0.490149 −0.245075 0.969504i \(-0.578812\pi\)
−0.245075 + 0.969504i \(0.578812\pi\)
\(888\) 0 0
\(889\) −10.2310 −0.343136
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 48.2534 1.61474
\(894\) 0 0
\(895\) − 15.6246i − 0.522272i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 44.6606i − 1.48951i
\(900\) 0 0
\(901\) 3.27043i 0.108954i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 5.79644i − 0.192680i
\(906\) 0 0
\(907\) −3.77723 −0.125421 −0.0627104 0.998032i \(-0.519974\pi\)
−0.0627104 + 0.998032i \(0.519974\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 20.5255 0.680040 0.340020 0.940418i \(-0.389566\pi\)
0.340020 + 0.940418i \(0.389566\pi\)
\(912\) 0 0
\(913\) 10.9838 0.363511
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −15.9013 −0.525107
\(918\) 0 0
\(919\) − 19.8082i − 0.653413i −0.945126 0.326706i \(-0.894061\pi\)
0.945126 0.326706i \(-0.105939\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 26.5306i − 0.873267i
\(924\) 0 0
\(925\) 1.19045i 0.0391417i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 2.90178i 0.0952044i 0.998866 + 0.0476022i \(0.0151580\pi\)
−0.998866 + 0.0476022i \(0.984842\pi\)
\(930\) 0 0
\(931\) −7.10196 −0.232757
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −12.1788 −0.398289
\(936\) 0 0
\(937\) −9.02197 −0.294735 −0.147368 0.989082i \(-0.547080\pi\)
−0.147368 + 0.989082i \(0.547080\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −26.0744 −0.850000 −0.425000 0.905193i \(-0.639726\pi\)
−0.425000 + 0.905193i \(0.639726\pi\)
\(942\) 0 0
\(943\) − 5.16001i − 0.168033i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 46.0960i − 1.49792i −0.662616 0.748960i \(-0.730554\pi\)
0.662616 0.748960i \(-0.269446\pi\)
\(948\) 0 0
\(949\) − 34.6822i − 1.12583i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 43.0803i − 1.39551i −0.716338 0.697754i \(-0.754183\pi\)
0.716338 0.697754i \(-0.245817\pi\)
\(954\) 0 0
\(955\) 30.3295 0.981439
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 10.0619 0.324916
\(960\) 0 0
\(961\) −87.6634 −2.82785
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −43.1715 −1.38974
\(966\) 0 0
\(967\) − 0.389484i − 0.0125249i −0.999980 0.00626247i \(-0.998007\pi\)
0.999980 0.00626247i \(-0.00199342\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 41.1601i − 1.32089i −0.750875 0.660445i \(-0.770368\pi\)
0.750875 0.660445i \(-0.229632\pi\)
\(972\) 0 0
\(973\) − 4.44750i − 0.142580i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 46.9292i − 1.50140i −0.660644 0.750699i \(-0.729717\pi\)
0.660644 0.750699i \(-0.270283\pi\)
\(978\) 0 0
\(979\) −2.78138 −0.0888933
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 39.0802 1.24646 0.623232 0.782037i \(-0.285819\pi\)
0.623232 + 0.782037i \(0.285819\pi\)
\(984\) 0 0
\(985\) −35.4785 −1.13044
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −34.3027 −1.09076
\(990\) 0 0
\(991\) − 23.1043i − 0.733932i −0.930234 0.366966i \(-0.880397\pi\)
0.930234 0.366966i \(-0.119603\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 8.26396i 0.261985i
\(996\) 0 0
\(997\) 47.9056i 1.51719i 0.651565 + 0.758593i \(0.274113\pi\)
−0.651565 + 0.758593i \(0.725887\pi\)
\(998\) 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.j.d.5615.12 48
3.2 odd 2 inner 6048.2.j.d.5615.37 48
4.3 odd 2 1512.2.j.d.323.38 yes 48
8.3 odd 2 inner 6048.2.j.d.5615.38 48
8.5 even 2 1512.2.j.d.323.12 yes 48
12.11 even 2 1512.2.j.d.323.11 48
24.5 odd 2 1512.2.j.d.323.37 yes 48
24.11 even 2 inner 6048.2.j.d.5615.11 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.j.d.323.11 48 12.11 even 2
1512.2.j.d.323.12 yes 48 8.5 even 2
1512.2.j.d.323.37 yes 48 24.5 odd 2
1512.2.j.d.323.38 yes 48 4.3 odd 2
6048.2.j.d.5615.11 48 24.11 even 2 inner
6048.2.j.d.5615.12 48 1.1 even 1 trivial
6048.2.j.d.5615.37 48 3.2 odd 2 inner
6048.2.j.d.5615.38 48 8.3 odd 2 inner