Properties

Label 6048.2.j.b.5615.4
Level 6048
Weight 2
Character 6048.5615
Analytic conductor 48.294
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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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: \(8\)
Coefficient field: 8.0.56070144.2
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 1512)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 5615.4
Root \(0.500000 - 1.56488i\)
Character \(\chi\) = 6048.5615
Dual form 6048.2.j.b.5615.3

$q$-expansion

\(f(q)\) \(=\) \(q-1.12976 q^{5} +1.00000i q^{7} +O(q^{10})\) \(q-1.12976 q^{5} +1.00000i q^{7} -3.86182i q^{11} -5.08658i q^{13} +2.00000i q^{17} -7.99158 q^{19} -7.68044 q^{23} -3.72363 q^{25} +4.73205 q^{29} -4.08658i q^{31} -1.12976i q^{35} +8.28273i q^{37} +8.41249i q^{41} +10.0148 q^{43} -1.93662 q^{47} -1.00000 q^{49} +7.72363 q^{53} +4.36294i q^{55} +10.9916i q^{59} -11.4641i q^{61} +5.74663i q^{65} -5.08658 q^{67} +13.6804 q^{71} -2.63706 q^{73} +3.86182 q^{77} +7.17295i q^{79} +14.9052i q^{83} -2.25953i q^{85} +12.6889i q^{89} +5.08658 q^{91} +9.02861 q^{95} +4.17315 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{5} + O(q^{10}) \) \( 8q + 8q^{5} - 16q^{19} + 16q^{23} + 32q^{25} + 24q^{29} - 8q^{43} - 8q^{47} - 8q^{49} - 8q^{67} + 32q^{71} + 8q^{73} + 8q^{91} + 112q^{95} - 32q^{97} + O(q^{100}) \)

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
\(5\) −1.12976 −0.505246 −0.252623 0.967565i \(-0.581293\pi\)
−0.252623 + 0.967565i \(0.581293\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 3.86182i − 1.16438i −0.813052 0.582191i \(-0.802196\pi\)
0.813052 0.582191i \(-0.197804\pi\)
\(12\) 0 0
\(13\) − 5.08658i − 1.41076i −0.708828 0.705381i \(-0.750776\pi\)
0.708828 0.705381i \(-0.249224\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 0 0
\(19\) −7.99158 −1.83339 −0.916697 0.399583i \(-0.869155\pi\)
−0.916697 + 0.399583i \(0.869155\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −7.68044 −1.60148 −0.800741 0.599010i \(-0.795561\pi\)
−0.800741 + 0.599010i \(0.795561\pi\)
\(24\) 0 0
\(25\) −3.72363 −0.744726
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.73205 0.878720 0.439360 0.898311i \(-0.355205\pi\)
0.439360 + 0.898311i \(0.355205\pi\)
\(30\) 0 0
\(31\) − 4.08658i − 0.733971i −0.930227 0.366985i \(-0.880390\pi\)
0.930227 0.366985i \(-0.119610\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 1.12976i − 0.190965i
\(36\) 0 0
\(37\) 8.28273i 1.36167i 0.732436 + 0.680836i \(0.238383\pi\)
−0.732436 + 0.680836i \(0.761617\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 8.41249i 1.31381i 0.753973 + 0.656905i \(0.228135\pi\)
−0.753973 + 0.656905i \(0.771865\pi\)
\(42\) 0 0
\(43\) 10.0148 1.52724 0.763620 0.645666i \(-0.223420\pi\)
0.763620 + 0.645666i \(0.223420\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.93662 −0.282485 −0.141243 0.989975i \(-0.545110\pi\)
−0.141243 + 0.989975i \(0.545110\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 7.72363 1.06092 0.530461 0.847709i \(-0.322019\pi\)
0.530461 + 0.847709i \(0.322019\pi\)
\(54\) 0 0
\(55\) 4.36294i 0.588299i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 10.9916i 1.43098i 0.698622 + 0.715491i \(0.253797\pi\)
−0.698622 + 0.715491i \(0.746203\pi\)
\(60\) 0 0
\(61\) − 11.4641i − 1.46783i −0.679243 0.733914i \(-0.737692\pi\)
0.679243 0.733914i \(-0.262308\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 5.74663i 0.712782i
\(66\) 0 0
\(67\) −5.08658 −0.621424 −0.310712 0.950504i \(-0.600567\pi\)
−0.310712 + 0.950504i \(0.600567\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 13.6804 1.62357 0.811785 0.583957i \(-0.198496\pi\)
0.811785 + 0.583957i \(0.198496\pi\)
\(72\) 0 0
\(73\) −2.63706 −0.308644 −0.154322 0.988021i \(-0.549319\pi\)
−0.154322 + 0.988021i \(0.549319\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.86182 0.440095
\(78\) 0 0
\(79\) 7.17295i 0.807020i 0.914975 + 0.403510i \(0.132210\pi\)
−0.914975 + 0.403510i \(0.867790\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 14.9052i 1.63606i 0.575177 + 0.818029i \(0.304933\pi\)
−0.575177 + 0.818029i \(0.695067\pi\)
\(84\) 0 0
\(85\) − 2.25953i − 0.245080i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 12.6889i 1.34502i 0.740090 + 0.672508i \(0.234783\pi\)
−0.740090 + 0.672508i \(0.765217\pi\)
\(90\) 0 0
\(91\) 5.08658 0.533218
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 9.02861 0.926316
\(96\) 0 0
\(97\) 4.17315 0.423719 0.211860 0.977300i \(-0.432048\pi\)
0.211860 + 0.977300i \(0.432048\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 0 0
\(103\) 5.37753i 0.529863i 0.964267 + 0.264932i \(0.0853494\pi\)
−0.964267 + 0.264932i \(0.914651\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 12.8418i − 1.24147i −0.784022 0.620733i \(-0.786835\pi\)
0.784022 0.620733i \(-0.213165\pi\)
\(108\) 0 0
\(109\) − 13.0782i − 1.25266i −0.779558 0.626330i \(-0.784556\pi\)
0.779558 0.626330i \(-0.215444\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 7.05496i − 0.663675i −0.943337 0.331837i \(-0.892331\pi\)
0.943337 0.331837i \(-0.107669\pi\)
\(114\) 0 0
\(115\) 8.67709 0.809143
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) −3.91362 −0.355784
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 9.85565 0.881516
\(126\) 0 0
\(127\) − 4.63706i − 0.411472i −0.978608 0.205736i \(-0.934041\pi\)
0.978608 0.205736i \(-0.0659589\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 8.36930i − 0.731229i −0.930766 0.365615i \(-0.880859\pi\)
0.930766 0.365615i \(-0.119141\pi\)
\(132\) 0 0
\(133\) − 7.99158i − 0.692958i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 13.9366i 1.19069i 0.803472 + 0.595343i \(0.202984\pi\)
−0.803472 + 0.595343i \(0.797016\pi\)
\(138\) 0 0
\(139\) 13.9832 1.18604 0.593018 0.805189i \(-0.297936\pi\)
0.593018 + 0.805189i \(0.297936\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −19.6434 −1.64266
\(144\) 0 0
\(145\) −5.34610 −0.443970
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −21.7700 −1.78347 −0.891735 0.452558i \(-0.850512\pi\)
−0.891735 + 0.452558i \(0.850512\pi\)
\(150\) 0 0
\(151\) 16.0464i 1.30584i 0.757428 + 0.652919i \(0.226456\pi\)
−0.757428 + 0.652919i \(0.773544\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 4.61687i 0.370836i
\(156\) 0 0
\(157\) − 5.49572i − 0.438606i −0.975657 0.219303i \(-0.929622\pi\)
0.975657 0.219303i \(-0.0703783\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 7.68044i − 0.605304i
\(162\) 0 0
\(163\) −1.62247 −0.127082 −0.0635410 0.997979i \(-0.520239\pi\)
−0.0635410 + 0.997979i \(0.520239\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.70905 0.673926 0.336963 0.941518i \(-0.390600\pi\)
0.336963 + 0.941518i \(0.390600\pi\)
\(168\) 0 0
\(169\) −12.8732 −0.990250
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.53891 0.117001 0.0585005 0.998287i \(-0.481368\pi\)
0.0585005 + 0.998287i \(0.481368\pi\)
\(174\) 0 0
\(175\) − 3.72363i − 0.281480i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 26.7386i 1.99854i 0.0382400 + 0.999269i \(0.487825\pi\)
−0.0382400 + 0.999269i \(0.512175\pi\)
\(180\) 0 0
\(181\) 3.10135i 0.230522i 0.993335 + 0.115261i \(0.0367704\pi\)
−0.993335 + 0.115261i \(0.963230\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 9.35753i − 0.687980i
\(186\) 0 0
\(187\) 7.72363 0.564808
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 13.8132 0.999489 0.499745 0.866173i \(-0.333427\pi\)
0.499745 + 0.866173i \(0.333427\pi\)
\(192\) 0 0
\(193\) 7.05496 0.507827 0.253913 0.967227i \(-0.418282\pi\)
0.253913 + 0.967227i \(0.418282\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0.362748 0.0258448 0.0129224 0.999917i \(-0.495887\pi\)
0.0129224 + 0.999917i \(0.495887\pi\)
\(198\) 0 0
\(199\) − 7.70029i − 0.545859i −0.962034 0.272930i \(-0.912007\pi\)
0.962034 0.272930i \(-0.0879926\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 4.73205i 0.332125i
\(204\) 0 0
\(205\) − 9.50414i − 0.663798i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 30.8620i 2.13477i
\(210\) 0 0
\(211\) 14.5191 0.999533 0.499767 0.866160i \(-0.333419\pi\)
0.499767 + 0.866160i \(0.333419\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −11.3143 −0.771632
\(216\) 0 0
\(217\) 4.08658 0.277415
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 10.1732 0.684320
\(222\) 0 0
\(223\) 19.2743i 1.29070i 0.763886 + 0.645352i \(0.223289\pi\)
−0.763886 + 0.645352i \(0.776711\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 23.2341i 1.54210i 0.636773 + 0.771052i \(0.280269\pi\)
−0.636773 + 0.771052i \(0.719731\pi\)
\(228\) 0 0
\(229\) − 13.4641i − 0.889733i −0.895597 0.444866i \(-0.853251\pi\)
0.895597 0.444866i \(-0.146749\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 17.4641i 1.14411i 0.820215 + 0.572056i \(0.193854\pi\)
−0.820215 + 0.572056i \(0.806146\pi\)
\(234\) 0 0
\(235\) 2.18793 0.142725
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1.82724 −0.118194 −0.0590972 0.998252i \(-0.518822\pi\)
−0.0590972 + 0.998252i \(0.518822\pi\)
\(240\) 0 0
\(241\) −21.5193 −1.38618 −0.693089 0.720852i \(-0.743751\pi\)
−0.693089 + 0.720852i \(0.743751\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.12976 0.0721780
\(246\) 0 0
\(247\) 40.6498i 2.58648i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1.15818i 0.0731035i 0.999332 + 0.0365517i \(0.0116374\pi\)
−0.999332 + 0.0365517i \(0.988363\pi\)
\(252\) 0 0
\(253\) 29.6604i 1.86474i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 11.0812i − 0.691224i −0.938377 0.345612i \(-0.887671\pi\)
0.938377 0.345612i \(-0.112329\pi\)
\(258\) 0 0
\(259\) −8.28273 −0.514664
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −10.7354 −0.661973 −0.330987 0.943635i \(-0.607381\pi\)
−0.330987 + 0.943635i \(0.607381\pi\)
\(264\) 0 0
\(265\) −8.72589 −0.536027
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 5.75204 0.350708 0.175354 0.984505i \(-0.443893\pi\)
0.175354 + 0.984505i \(0.443893\pi\)
\(270\) 0 0
\(271\) − 1.33735i − 0.0812380i −0.999175 0.0406190i \(-0.987067\pi\)
0.999175 0.0406190i \(-0.0129330\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 14.3800i 0.867145i
\(276\) 0 0
\(277\) 3.13537i 0.188386i 0.995554 + 0.0941930i \(0.0300271\pi\)
−0.995554 + 0.0941930i \(0.969973\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 13.3839i 0.798416i 0.916860 + 0.399208i \(0.130715\pi\)
−0.916860 + 0.399208i \(0.869285\pi\)
\(282\) 0 0
\(283\) −21.8104 −1.29649 −0.648247 0.761430i \(-0.724498\pi\)
−0.648247 + 0.761430i \(0.724498\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −8.41249 −0.496574
\(288\) 0 0
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −30.6982 −1.79341 −0.896705 0.442629i \(-0.854046\pi\)
−0.896705 + 0.442629i \(0.854046\pi\)
\(294\) 0 0
\(295\) − 12.4179i − 0.722998i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 39.0671i 2.25931i
\(300\) 0 0
\(301\) 10.0148i 0.577242i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 12.9517i 0.741614i
\(306\) 0 0
\(307\) −9.10977 −0.519922 −0.259961 0.965619i \(-0.583710\pi\)
−0.259961 + 0.965619i \(0.583710\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 27.5275 1.56094 0.780470 0.625193i \(-0.214980\pi\)
0.780470 + 0.625193i \(0.214980\pi\)
\(312\) 0 0
\(313\) 5.36294 0.303131 0.151566 0.988447i \(-0.451568\pi\)
0.151566 + 0.988447i \(0.451568\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 7.30832 0.410476 0.205238 0.978712i \(-0.434203\pi\)
0.205238 + 0.978712i \(0.434203\pi\)
\(318\) 0 0
\(319\) − 18.2743i − 1.02316i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 15.9832i − 0.889327i
\(324\) 0 0
\(325\) 18.9405i 1.05063i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 1.93662i − 0.106769i
\(330\) 0 0
\(331\) −10.5823 −0.581655 −0.290828 0.956775i \(-0.593931\pi\)
−0.290828 + 0.956775i \(0.593931\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 5.74663 0.313972
\(336\) 0 0
\(337\) −4.29095 −0.233743 −0.116872 0.993147i \(-0.537287\pi\)
−0.116872 + 0.993147i \(0.537287\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −15.7816 −0.854622
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 10.3405i 0.555107i 0.960710 + 0.277554i \(0.0895236\pi\)
−0.960710 + 0.277554i \(0.910476\pi\)
\(348\) 0 0
\(349\) 11.1182i 0.595143i 0.954700 + 0.297572i \(0.0961767\pi\)
−0.954700 + 0.297572i \(0.903823\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 23.3407i 1.24230i 0.783692 + 0.621150i \(0.213334\pi\)
−0.783692 + 0.621150i \(0.786666\pi\)
\(354\) 0 0
\(355\) −15.4557 −0.820302
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 22.3923 1.18182 0.590910 0.806737i \(-0.298769\pi\)
0.590910 + 0.806737i \(0.298769\pi\)
\(360\) 0 0
\(361\) 44.8654 2.36133
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 2.97925 0.155941
\(366\) 0 0
\(367\) 1.93696i 0.101109i 0.998721 + 0.0505543i \(0.0160988\pi\)
−0.998721 + 0.0505543i \(0.983901\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 7.72363i 0.400991i
\(372\) 0 0
\(373\) − 15.1246i − 0.783120i −0.920153 0.391560i \(-0.871936\pi\)
0.920153 0.391560i \(-0.128064\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 24.0699i − 1.23966i
\(378\) 0 0
\(379\) −6.15837 −0.316334 −0.158167 0.987412i \(-0.550558\pi\)
−0.158167 + 0.987412i \(0.550558\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 14.8822 0.760445 0.380222 0.924895i \(-0.375847\pi\)
0.380222 + 0.924895i \(0.375847\pi\)
\(384\) 0 0
\(385\) −4.36294 −0.222356
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 1.82685 0.0926250 0.0463125 0.998927i \(-0.485253\pi\)
0.0463125 + 0.998927i \(0.485253\pi\)
\(390\) 0 0
\(391\) − 15.3609i − 0.776833i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 8.10375i − 0.407744i
\(396\) 0 0
\(397\) − 2.18999i − 0.109912i −0.998489 0.0549562i \(-0.982498\pi\)
0.998489 0.0549562i \(-0.0175019\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 3.59071i 0.179312i 0.995973 + 0.0896558i \(0.0285767\pi\)
−0.995973 + 0.0896558i \(0.971423\pi\)
\(402\) 0 0
\(403\) −20.7867 −1.03546
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 31.9864 1.58551
\(408\) 0 0
\(409\) −35.3588 −1.74838 −0.874191 0.485583i \(-0.838607\pi\)
−0.874191 + 0.485583i \(0.838607\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −10.9916 −0.540860
\(414\) 0 0
\(415\) − 16.8394i − 0.826612i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 2.44952i 0.119667i 0.998208 + 0.0598334i \(0.0190569\pi\)
−0.998208 + 0.0598334i \(0.980943\pi\)
\(420\) 0 0
\(421\) − 6.26589i − 0.305381i −0.988274 0.152690i \(-0.951206\pi\)
0.988274 0.152690i \(-0.0487937\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 7.44726i − 0.361245i
\(426\) 0 0
\(427\) 11.4641 0.554787
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 3.01177 0.145072 0.0725359 0.997366i \(-0.476891\pi\)
0.0725359 + 0.997366i \(0.476891\pi\)
\(432\) 0 0
\(433\) −7.00020 −0.336408 −0.168204 0.985752i \(-0.553797\pi\)
−0.168204 + 0.985752i \(0.553797\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 61.3789 2.93615
\(438\) 0 0
\(439\) 1.33735i 0.0638281i 0.999491 + 0.0319140i \(0.0101603\pi\)
−0.999491 + 0.0319140i \(0.989840\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 1.90821i − 0.0906618i −0.998972 0.0453309i \(-0.985566\pi\)
0.998972 0.0453309i \(-0.0144342\pi\)
\(444\) 0 0
\(445\) − 14.3354i − 0.679565i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 10.0695i 0.475211i 0.971362 + 0.237606i \(0.0763625\pi\)
−0.971362 + 0.237606i \(0.923637\pi\)
\(450\) 0 0
\(451\) 32.4875 1.52978
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −5.74663 −0.269406
\(456\) 0 0
\(457\) 10.3205 0.482773 0.241387 0.970429i \(-0.422398\pi\)
0.241387 + 0.970429i \(0.422398\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −17.9420 −0.835644 −0.417822 0.908529i \(-0.637206\pi\)
−0.417822 + 0.908529i \(0.637206\pi\)
\(462\) 0 0
\(463\) − 21.6664i − 1.00692i −0.864017 0.503462i \(-0.832059\pi\)
0.864017 0.503462i \(-0.167941\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 18.3525i 0.849251i 0.905369 + 0.424625i \(0.139594\pi\)
−0.905369 + 0.424625i \(0.860406\pi\)
\(468\) 0 0
\(469\) − 5.08658i − 0.234876i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 38.6752i − 1.77829i
\(474\) 0 0
\(475\) 29.7577 1.36538
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 14.6518 0.669459 0.334730 0.942314i \(-0.391355\pi\)
0.334730 + 0.942314i \(0.391355\pi\)
\(480\) 0 0
\(481\) 42.1307 1.92100
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −4.71468 −0.214083
\(486\) 0 0
\(487\) − 6.60789i − 0.299432i −0.988729 0.149716i \(-0.952164\pi\)
0.988729 0.149716i \(-0.0478360\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 7.08323i − 0.319662i −0.987144 0.159831i \(-0.948905\pi\)
0.987144 0.159831i \(-0.0510949\pi\)
\(492\) 0 0
\(493\) 9.46410i 0.426242i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 13.6804i 0.613652i
\(498\) 0 0
\(499\) 7.13297 0.319316 0.159658 0.987172i \(-0.448961\pi\)
0.159658 + 0.987172i \(0.448961\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −14.7321 −0.656870 −0.328435 0.944527i \(-0.606521\pi\)
−0.328435 + 0.944527i \(0.606521\pi\)
\(504\) 0 0
\(505\) −2.25953 −0.100548
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −6.21955 −0.275677 −0.137838 0.990455i \(-0.544015\pi\)
−0.137838 + 0.990455i \(0.544015\pi\)
\(510\) 0 0
\(511\) − 2.63706i − 0.116657i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 6.07534i − 0.267711i
\(516\) 0 0
\(517\) 7.47888i 0.328921i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 20.0897i − 0.880147i −0.897962 0.440073i \(-0.854952\pi\)
0.897962 0.440073i \(-0.145048\pi\)
\(522\) 0 0
\(523\) −25.5885 −1.11891 −0.559453 0.828862i \(-0.688989\pi\)
−0.559453 + 0.828862i \(0.688989\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8.17315 0.356028
\(528\) 0 0
\(529\) 35.9892 1.56475
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 42.7908 1.85347
\(534\) 0 0
\(535\) 14.5082i 0.627246i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 3.86182i 0.166340i
\(540\) 0 0
\(541\) 25.7932i 1.10894i 0.832205 + 0.554469i \(0.187078\pi\)
−0.832205 + 0.554469i \(0.812922\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 14.7752i 0.632902i
\(546\) 0 0
\(547\) 1.41771 0.0606167 0.0303084 0.999541i \(-0.490351\pi\)
0.0303084 + 0.999541i \(0.490351\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −37.8166 −1.61104
\(552\) 0 0
\(553\) −7.17295 −0.305025
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 4.36275 0.184856 0.0924278 0.995719i \(-0.470537\pi\)
0.0924278 + 0.995719i \(0.470537\pi\)
\(558\) 0 0
\(559\) − 50.9409i − 2.15457i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 25.4007i − 1.07051i −0.844690 0.535256i \(-0.820215\pi\)
0.844690 0.535256i \(-0.179785\pi\)
\(564\) 0 0
\(565\) 7.97044i 0.335319i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 24.9746i 1.04699i 0.852029 + 0.523495i \(0.175372\pi\)
−0.852029 + 0.523495i \(0.824628\pi\)
\(570\) 0 0
\(571\) 8.02956 0.336026 0.168013 0.985785i \(-0.446265\pi\)
0.168013 + 0.985785i \(0.446265\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 28.5991 1.19267
\(576\) 0 0
\(577\) −21.7808 −0.906748 −0.453374 0.891320i \(-0.649780\pi\)
−0.453374 + 0.891320i \(0.649780\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −14.9052 −0.618372
\(582\) 0 0
\(583\) − 29.8272i − 1.23532i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 13.2913i − 0.548592i −0.961645 0.274296i \(-0.911555\pi\)
0.961645 0.274296i \(-0.0884449\pi\)
\(588\) 0 0
\(589\) 32.6582i 1.34566i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0.746075i 0.0306376i 0.999883 + 0.0153188i \(0.00487632\pi\)
−0.999883 + 0.0153188i \(0.995124\pi\)
\(594\) 0 0
\(595\) 2.25953 0.0926317
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −42.1595 −1.72259 −0.861296 0.508104i \(-0.830346\pi\)
−0.861296 + 0.508104i \(0.830346\pi\)
\(600\) 0 0
\(601\) −12.6371 −0.515476 −0.257738 0.966215i \(-0.582977\pi\)
−0.257738 + 0.966215i \(0.582977\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 4.42147 0.179758
\(606\) 0 0
\(607\) 44.5950i 1.81006i 0.425352 + 0.905028i \(0.360150\pi\)
−0.425352 + 0.905028i \(0.639850\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 9.85078i 0.398520i
\(612\) 0 0
\(613\) 16.6859i 0.673935i 0.941516 + 0.336968i \(0.109401\pi\)
−0.941516 + 0.336968i \(0.890599\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 30.1225i 1.21269i 0.795203 + 0.606343i \(0.207364\pi\)
−0.795203 + 0.606343i \(0.792636\pi\)
\(618\) 0 0
\(619\) −32.5002 −1.30629 −0.653147 0.757231i \(-0.726552\pi\)
−0.653147 + 0.757231i \(0.726552\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −12.6889 −0.508368
\(624\) 0 0
\(625\) 7.48359 0.299343
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −16.5655 −0.660508
\(630\) 0 0
\(631\) 18.8734i 0.751340i 0.926754 + 0.375670i \(0.122587\pi\)
−0.926754 + 0.375670i \(0.877413\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 5.23878i 0.207895i
\(636\) 0 0
\(637\) 5.08658i 0.201537i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 13.1417i − 0.519067i −0.965734 0.259534i \(-0.916431\pi\)
0.965734 0.259534i \(-0.0835688\pi\)
\(642\) 0 0
\(643\) −11.3121 −0.446105 −0.223053 0.974806i \(-0.571602\pi\)
−0.223053 + 0.974806i \(0.571602\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 10.1038 0.397219 0.198610 0.980079i \(-0.436357\pi\)
0.198610 + 0.980079i \(0.436357\pi\)
\(648\) 0 0
\(649\) 42.4475 1.66621
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −18.2661 −0.714807 −0.357404 0.933950i \(-0.616338\pi\)
−0.357404 + 0.933950i \(0.616338\pi\)
\(654\) 0 0
\(655\) 9.45534i 0.369451i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 37.9781i 1.47942i 0.672927 + 0.739709i \(0.265037\pi\)
−0.672927 + 0.739709i \(0.734963\pi\)
\(660\) 0 0
\(661\) − 8.98899i − 0.349631i −0.984601 0.174816i \(-0.944067\pi\)
0.984601 0.174816i \(-0.0559329\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 9.02861i 0.350114i
\(666\) 0 0
\(667\) −36.3442 −1.40725
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −44.2722 −1.70911
\(672\) 0 0
\(673\) 21.4937 0.828520 0.414260 0.910159i \(-0.364040\pi\)
0.414260 + 0.910159i \(0.364040\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 13.5920 0.522383 0.261192 0.965287i \(-0.415885\pi\)
0.261192 + 0.965287i \(0.415885\pi\)
\(678\) 0 0
\(679\) 4.17315i 0.160151i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 13.0200i 0.498196i 0.968478 + 0.249098i \(0.0801342\pi\)
−0.968478 + 0.249098i \(0.919866\pi\)
\(684\) 0 0
\(685\) − 15.7451i − 0.601590i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 39.2868i − 1.49671i
\(690\) 0 0
\(691\) 8.07556 0.307209 0.153604 0.988132i \(-0.450912\pi\)
0.153604 + 0.988132i \(0.450912\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −15.7977 −0.599240
\(696\) 0 0
\(697\) −16.8250 −0.637292
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 9.91978 0.374665 0.187333 0.982297i \(-0.440016\pi\)
0.187333 + 0.982297i \(0.440016\pi\)
\(702\) 0 0
\(703\) − 66.1921i − 2.49648i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 2.00000i 0.0752177i
\(708\) 0 0
\(709\) − 33.4709i − 1.25702i −0.777800 0.628512i \(-0.783664\pi\)
0.777800 0.628512i \(-0.216336\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 31.3867i 1.17544i
\(714\) 0 0
\(715\) 22.1924 0.829950
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −29.1075 −1.08553 −0.542764 0.839886i \(-0.682622\pi\)
−0.542764 + 0.839886i \(0.682622\pi\)
\(720\) 0 0
\(721\) −5.37753 −0.200270
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −17.6204 −0.654406
\(726\) 0 0
\(727\) 2.39230i 0.0887257i 0.999015 + 0.0443628i \(0.0141258\pi\)
−0.999015 + 0.0443628i \(0.985874\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 20.0296i 0.740820i
\(732\) 0 0
\(733\) 20.3315i 0.750962i 0.926830 + 0.375481i \(0.122522\pi\)
−0.926830 + 0.375481i \(0.877478\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 19.6434i 0.723574i
\(738\) 0 0
\(739\) −36.5823 −1.34570 −0.672851 0.739778i \(-0.734930\pi\)
−0.672851 + 0.739778i \(0.734930\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −13.9864 −0.513110 −0.256555 0.966530i \(-0.582587\pi\)
−0.256555 + 0.966530i \(0.582587\pi\)
\(744\) 0 0
\(745\) 24.5950 0.901091
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 12.8418 0.469230
\(750\) 0 0
\(751\) − 47.5657i − 1.73570i −0.496830 0.867848i \(-0.665503\pi\)
0.496830 0.867848i \(-0.334497\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 18.1287i − 0.659769i
\(756\) 0 0
\(757\) − 41.3844i − 1.50414i −0.659082 0.752071i \(-0.729055\pi\)
0.659082 0.752071i \(-0.270945\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 19.8335i 0.718966i 0.933152 + 0.359483i \(0.117047\pi\)
−0.933152 + 0.359483i \(0.882953\pi\)
\(762\) 0 0
\(763\) 13.0782 0.473461
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 55.9095 2.01878
\(768\) 0 0
\(769\) −20.2739 −0.731096 −0.365548 0.930792i \(-0.619118\pi\)
−0.365548 + 0.930792i \(0.619118\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −0.271102 −0.00975088 −0.00487544 0.999988i \(-0.501552\pi\)
−0.00487544 + 0.999988i \(0.501552\pi\)
\(774\) 0 0
\(775\) 15.2169i 0.546607i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 67.2291i − 2.40873i
\(780\) 0 0
\(781\) − 52.8313i − 1.89045i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 6.20887i 0.221604i
\(786\) 0 0
\(787\) 50.0760 1.78501 0.892507 0.451033i \(-0.148944\pi\)
0.892507 + 0.451033i \(0.148944\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 7.05496 0.250845
\(792\) 0 0
\(793\) −58.3130 −2.07076
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −27.1762 −0.962629 −0.481314 0.876548i \(-0.659840\pi\)
−0.481314 + 0.876548i \(0.659840\pi\)
\(798\) 0 0
\(799\) − 3.87325i − 0.137026i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 10.1838i 0.359379i
\(804\) 0 0
\(805\) 8.67709i 0.305827i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 33.5566i − 1.17979i −0.807480 0.589894i \(-0.799169\pi\)
0.807480 0.589894i \(-0.200831\pi\)
\(810\) 0 0
\(811\) 3.54206 0.124379 0.0621893 0.998064i \(-0.480192\pi\)
0.0621893 + 0.998064i \(0.480192\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1.83301 0.0642077
\(816\) 0 0
\(817\) −80.0339 −2.80003
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −31.0847 −1.08486 −0.542431 0.840100i \(-0.682496\pi\)
−0.542431 + 0.840100i \(0.682496\pi\)
\(822\) 0 0
\(823\) 36.2323i 1.26298i 0.775385 + 0.631489i \(0.217556\pi\)
−0.775385 + 0.631489i \(0.782444\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 53.4991i − 1.86034i −0.367123 0.930172i \(-0.619657\pi\)
0.367123 0.930172i \(-0.380343\pi\)
\(828\) 0 0
\(829\) 46.2003i 1.60460i 0.596920 + 0.802301i \(0.296391\pi\)
−0.596920 + 0.802301i \(0.703609\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 2.00000i − 0.0692959i
\(834\) 0 0
\(835\) −9.83918 −0.340499
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 29.0211 1.00192 0.500960 0.865470i \(-0.332980\pi\)
0.500960 + 0.865470i \(0.332980\pi\)
\(840\) 0 0
\(841\) −6.60770 −0.227852
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 14.5437 0.500320
\(846\) 0 0
\(847\) − 3.91362i − 0.134474i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 63.6150i − 2.18069i
\(852\) 0 0
\(853\) 50.8777i 1.74202i 0.491266 + 0.871009i \(0.336534\pi\)
−0.491266 + 0.871009i \(0.663466\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 22.3553i 0.763642i 0.924236 + 0.381821i \(0.124703\pi\)
−0.924236 + 0.381821i \(0.875297\pi\)
\(858\) 0 0
\(859\) −44.4134 −1.51537 −0.757684 0.652622i \(-0.773669\pi\)
−0.757684 + 0.652622i \(0.773669\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −8.80145 −0.299605 −0.149802 0.988716i \(-0.547864\pi\)
−0.149802 + 0.988716i \(0.547864\pi\)
\(864\) 0 0
\(865\) −1.73860 −0.0591143
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 27.7006 0.939679
\(870\) 0 0
\(871\) 25.8732i 0.876681i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 9.85565i 0.333182i
\(876\) 0 0
\(877\) 25.5280i 0.862020i 0.902347 + 0.431010i \(0.141843\pi\)
−0.902347 + 0.431010i \(0.858157\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 41.0812i − 1.38406i −0.721869 0.692030i \(-0.756717\pi\)
0.721869 0.692030i \(-0.243283\pi\)
\(882\) 0 0
\(883\) −2.80145 −0.0942763 −0.0471381 0.998888i \(-0.515010\pi\)
−0.0471381 + 0.998888i \(0.515010\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −45.4366 −1.52561 −0.762806 0.646628i \(-0.776179\pi\)
−0.762806 + 0.646628i \(0.776179\pi\)
\(888\) 0 0
\(889\) 4.63706 0.155522
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 15.4767 0.517907
\(894\) 0 0
\(895\) − 30.2083i − 1.00975i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 19.3379i − 0.644954i
\(900\) 0 0
\(901\) 15.4473i 0.514623i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 3.50380i − 0.116470i
\(906\) 0 0
\(907\) −46.5318 −1.54506 −0.772531 0.634977i \(-0.781010\pi\)
−0.772531 + 0.634977i \(0.781010\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 24.9174 0.825550 0.412775 0.910833i \(-0.364560\pi\)
0.412775 + 0.910833i \(0.364560\pi\)
\(912\) 0 0
\(913\) 57.5611 1.90500
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 8.36930 0.276379
\(918\) 0 0
\(919\) 8.77228i 0.289371i 0.989478 + 0.144685i \(0.0462170\pi\)
−0.989478 + 0.144685i \(0.953783\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 69.5866i − 2.29047i
\(924\) 0 0
\(925\) − 30.8418i − 1.01407i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 3.96002i − 0.129924i −0.997888 0.0649620i \(-0.979307\pi\)
0.997888 0.0649620i \(-0.0206926\pi\)
\(930\) 0 0
\(931\) 7.99158 0.261913
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −8.72589 −0.285367
\(936\) 0 0
\(937\) −11.6077 −0.379207 −0.189603 0.981861i \(-0.560720\pi\)
−0.189603 + 0.981861i \(0.560720\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 2.05758 0.0670751 0.0335376 0.999437i \(-0.489323\pi\)
0.0335376 + 0.999437i \(0.489323\pi\)
\(942\) 0 0
\(943\) − 64.6117i − 2.10404i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 53.3723i 1.73437i 0.497989 + 0.867184i \(0.334072\pi\)
−0.497989 + 0.867184i \(0.665928\pi\)
\(948\) 0 0
\(949\) 13.4136i 0.435423i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 45.9433i − 1.48825i −0.668040 0.744125i \(-0.732867\pi\)
0.668040 0.744125i \(-0.267133\pi\)
\(954\) 0 0
\(955\) −15.6057 −0.504988
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −13.9366 −0.450037
\(960\) 0 0
\(961\) 14.2999 0.461287
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −7.97044 −0.256578
\(966\) 0 0
\(967\) 1.43455i 0.0461319i 0.999734 + 0.0230659i \(0.00734276\pi\)
−0.999734 + 0.0230659i \(0.992657\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 13.6030i − 0.436542i −0.975888 0.218271i \(-0.929958\pi\)
0.975888 0.218271i \(-0.0700417\pi\)
\(972\) 0 0
\(973\) 13.9832i 0.448280i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 10.3860i 0.332278i 0.986102 + 0.166139i \(0.0531300\pi\)
−0.986102 + 0.166139i \(0.946870\pi\)
\(978\) 0 0
\(979\) 49.0020 1.56611
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −20.0062 −0.638098 −0.319049 0.947738i \(-0.603363\pi\)
−0.319049 + 0.947738i \(0.603363\pi\)
\(984\) 0 0
\(985\) −0.409820 −0.0130580
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −76.9179 −2.44585
\(990\) 0 0
\(991\) 45.7220i 1.45241i 0.687480 + 0.726203i \(0.258717\pi\)
−0.687480 + 0.726203i \(0.741283\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 8.69952i 0.275793i
\(996\) 0 0
\(997\) 41.6964i 1.32054i 0.751030 + 0.660269i \(0.229558\pi\)
−0.751030 + 0.660269i \(0.770442\pi\)
\(998\) 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.b.5615.4 8
3.2 odd 2 6048.2.j.a.5615.6 8
4.3 odd 2 1512.2.j.b.323.7 yes 8
8.3 odd 2 6048.2.j.a.5615.5 8
8.5 even 2 1512.2.j.a.323.4 yes 8
12.11 even 2 1512.2.j.a.323.2 8
24.5 odd 2 1512.2.j.b.323.5 yes 8
24.11 even 2 inner 6048.2.j.b.5615.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.j.a.323.2 8 12.11 even 2
1512.2.j.a.323.4 yes 8 8.5 even 2
1512.2.j.b.323.5 yes 8 24.5 odd 2
1512.2.j.b.323.7 yes 8 4.3 odd 2
6048.2.j.a.5615.5 8 8.3 odd 2
6048.2.j.a.5615.6 8 3.2 odd 2
6048.2.j.b.5615.3 8 24.11 even 2 inner
6048.2.j.b.5615.4 8 1.1 even 1 trivial