Properties

Label 6048.2.j.d.5615.5
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 5615.5
Character \(\chi\) = 6048.5615
Dual form 6048.2.j.d.5615.6

$q$-expansion

\(f(q)\) \(=\) \(q-2.99711 q^{5} -1.00000i q^{7} +O(q^{10})\) \(q-2.99711 q^{5} -1.00000i q^{7} -5.36291i q^{11} +4.55519i q^{13} +0.958700i q^{17} -0.532227 q^{19} +2.78680 q^{23} +3.98265 q^{25} -5.15616 q^{29} +4.66569i q^{31} +2.99711i q^{35} -6.10293i q^{37} -12.3255i q^{41} -4.45583 q^{43} +1.40285 q^{47} -1.00000 q^{49} -9.57019 q^{53} +16.0732i q^{55} -0.0356973i q^{59} -5.13661i q^{61} -13.6524i q^{65} -8.59546 q^{67} +13.3269 q^{71} +6.50808 q^{73} -5.36291 q^{77} -1.90581i q^{79} -3.13371i q^{83} -2.87332i q^{85} +5.91323i q^{89} +4.55519 q^{91} +1.59514 q^{95} -16.0057 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48q + O(q^{10}) \) \( 48q - 16q^{19} + 48q^{25} - 64q^{43} - 48q^{49} - 32q^{67} + O(q^{100}) \)

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
\(5\) −2.99711 −1.34035 −0.670173 0.742205i \(-0.733780\pi\)
−0.670173 + 0.742205i \(0.733780\pi\)
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 5.36291i − 1.61698i −0.588512 0.808488i \(-0.700286\pi\)
0.588512 0.808488i \(-0.299714\pi\)
\(12\) 0 0
\(13\) 4.55519i 1.26338i 0.775221 + 0.631691i \(0.217639\pi\)
−0.775221 + 0.631691i \(0.782361\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.958700i 0.232519i 0.993219 + 0.116259i \(0.0370904\pi\)
−0.993219 + 0.116259i \(0.962910\pi\)
\(18\) 0 0
\(19\) −0.532227 −0.122101 −0.0610506 0.998135i \(-0.519445\pi\)
−0.0610506 + 0.998135i \(0.519445\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.78680 0.581088 0.290544 0.956862i \(-0.406164\pi\)
0.290544 + 0.956862i \(0.406164\pi\)
\(24\) 0 0
\(25\) 3.98265 0.796529
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −5.15616 −0.957475 −0.478737 0.877958i \(-0.658905\pi\)
−0.478737 + 0.877958i \(0.658905\pi\)
\(30\) 0 0
\(31\) 4.66569i 0.837983i 0.907990 + 0.418992i \(0.137616\pi\)
−0.907990 + 0.418992i \(0.862384\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.99711i 0.506603i
\(36\) 0 0
\(37\) − 6.10293i − 1.00332i −0.865066 0.501658i \(-0.832724\pi\)
0.865066 0.501658i \(-0.167276\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 12.3255i − 1.92492i −0.271423 0.962460i \(-0.587494\pi\)
0.271423 0.962460i \(-0.412506\pi\)
\(42\) 0 0
\(43\) −4.45583 −0.679508 −0.339754 0.940514i \(-0.610344\pi\)
−0.339754 + 0.940514i \(0.610344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.40285 0.204627 0.102314 0.994752i \(-0.467375\pi\)
0.102314 + 0.994752i \(0.467375\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −9.57019 −1.31457 −0.657283 0.753644i \(-0.728295\pi\)
−0.657283 + 0.753644i \(0.728295\pi\)
\(54\) 0 0
\(55\) 16.0732i 2.16731i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 0.0356973i − 0.00464739i −0.999997 0.00232369i \(-0.999260\pi\)
0.999997 0.00232369i \(-0.000739655\pi\)
\(60\) 0 0
\(61\) − 5.13661i − 0.657675i −0.944386 0.328838i \(-0.893343\pi\)
0.944386 0.328838i \(-0.106657\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 13.6524i − 1.69337i
\(66\) 0 0
\(67\) −8.59546 −1.05010 −0.525051 0.851071i \(-0.675954\pi\)
−0.525051 + 0.851071i \(0.675954\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 13.3269 1.58161 0.790804 0.612070i \(-0.209663\pi\)
0.790804 + 0.612070i \(0.209663\pi\)
\(72\) 0 0
\(73\) 6.50808 0.761713 0.380857 0.924634i \(-0.375629\pi\)
0.380857 + 0.924634i \(0.375629\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −5.36291 −0.611160
\(78\) 0 0
\(79\) − 1.90581i − 0.214420i −0.994236 0.107210i \(-0.965808\pi\)
0.994236 0.107210i \(-0.0341917\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 3.13371i − 0.343970i −0.985100 0.171985i \(-0.944982\pi\)
0.985100 0.171985i \(-0.0550180\pi\)
\(84\) 0 0
\(85\) − 2.87332i − 0.311656i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 5.91323i 0.626801i 0.949621 + 0.313401i \(0.101468\pi\)
−0.949621 + 0.313401i \(0.898532\pi\)
\(90\) 0 0
\(91\) 4.55519 0.477513
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.59514 0.163658
\(96\) 0 0
\(97\) −16.0057 −1.62514 −0.812568 0.582867i \(-0.801931\pi\)
−0.812568 + 0.582867i \(0.801931\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 16.1028 1.60229 0.801143 0.598473i \(-0.204226\pi\)
0.801143 + 0.598473i \(0.204226\pi\)
\(102\) 0 0
\(103\) 18.0349i 1.77703i 0.458850 + 0.888514i \(0.348261\pi\)
−0.458850 + 0.888514i \(0.651739\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 9.08345i − 0.878131i −0.898455 0.439065i \(-0.855310\pi\)
0.898455 0.439065i \(-0.144690\pi\)
\(108\) 0 0
\(109\) 17.2591i 1.65312i 0.562848 + 0.826560i \(0.309706\pi\)
−0.562848 + 0.826560i \(0.690294\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 1.27590i − 0.120026i −0.998198 0.0600131i \(-0.980886\pi\)
0.998198 0.0600131i \(-0.0191142\pi\)
\(114\) 0 0
\(115\) −8.35233 −0.778859
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.958700 0.0878838
\(120\) 0 0
\(121\) −17.7608 −1.61461
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 3.04912 0.272721
\(126\) 0 0
\(127\) 12.7265i 1.12929i 0.825333 + 0.564646i \(0.190987\pi\)
−0.825333 + 0.564646i \(0.809013\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 13.3457i 1.16602i 0.812465 + 0.583011i \(0.198125\pi\)
−0.812465 + 0.583011i \(0.801875\pi\)
\(132\) 0 0
\(133\) 0.532227i 0.0461499i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 17.0901i 1.46011i 0.683390 + 0.730053i \(0.260505\pi\)
−0.683390 + 0.730053i \(0.739495\pi\)
\(138\) 0 0
\(139\) 16.0609 1.36227 0.681134 0.732158i \(-0.261487\pi\)
0.681134 + 0.732158i \(0.261487\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 24.4290 2.04286
\(144\) 0 0
\(145\) 15.4536 1.28335
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1.37457 −0.112609 −0.0563047 0.998414i \(-0.517932\pi\)
−0.0563047 + 0.998414i \(0.517932\pi\)
\(150\) 0 0
\(151\) − 5.50504i − 0.447994i −0.974590 0.223997i \(-0.928089\pi\)
0.974590 0.223997i \(-0.0719105\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 13.9836i − 1.12319i
\(156\) 0 0
\(157\) 18.7801i 1.49881i 0.662110 + 0.749407i \(0.269661\pi\)
−0.662110 + 0.749407i \(0.730339\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 2.78680i − 0.219630i
\(162\) 0 0
\(163\) 17.7671 1.39163 0.695813 0.718223i \(-0.255044\pi\)
0.695813 + 0.718223i \(0.255044\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −18.1777 −1.40663 −0.703315 0.710878i \(-0.748298\pi\)
−0.703315 + 0.710878i \(0.748298\pi\)
\(168\) 0 0
\(169\) −7.74971 −0.596132
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −2.38643 −0.181437 −0.0907186 0.995877i \(-0.528916\pi\)
−0.0907186 + 0.995877i \(0.528916\pi\)
\(174\) 0 0
\(175\) − 3.98265i − 0.301060i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 6.73152i 0.503137i 0.967839 + 0.251569i \(0.0809464\pi\)
−0.967839 + 0.251569i \(0.919054\pi\)
\(180\) 0 0
\(181\) − 1.20606i − 0.0896460i −0.998995 0.0448230i \(-0.985728\pi\)
0.998995 0.0448230i \(-0.0142724\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 18.2911i 1.34479i
\(186\) 0 0
\(187\) 5.14142 0.375978
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 15.5066 1.12202 0.561008 0.827810i \(-0.310414\pi\)
0.561008 + 0.827810i \(0.310414\pi\)
\(192\) 0 0
\(193\) 7.67734 0.552627 0.276314 0.961068i \(-0.410887\pi\)
0.276314 + 0.961068i \(0.410887\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −23.1208 −1.64729 −0.823644 0.567107i \(-0.808062\pi\)
−0.823644 + 0.567107i \(0.808062\pi\)
\(198\) 0 0
\(199\) − 1.14987i − 0.0815122i −0.999169 0.0407561i \(-0.987023\pi\)
0.999169 0.0407561i \(-0.0129767\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 5.15616i 0.361891i
\(204\) 0 0
\(205\) 36.9408i 2.58006i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 2.85428i 0.197435i
\(210\) 0 0
\(211\) −15.6643 −1.07837 −0.539186 0.842187i \(-0.681268\pi\)
−0.539186 + 0.842187i \(0.681268\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 13.3546 0.910776
\(216\) 0 0
\(217\) 4.66569 0.316728
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −4.36705 −0.293760
\(222\) 0 0
\(223\) 13.9655i 0.935198i 0.883941 + 0.467599i \(0.154881\pi\)
−0.883941 + 0.467599i \(0.845119\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 26.1607i 1.73635i 0.496262 + 0.868173i \(0.334705\pi\)
−0.496262 + 0.868173i \(0.665295\pi\)
\(228\) 0 0
\(229\) − 6.01986i − 0.397803i −0.980019 0.198902i \(-0.936263\pi\)
0.980019 0.198902i \(-0.0637374\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 13.4333i − 0.880046i −0.897986 0.440023i \(-0.854970\pi\)
0.897986 0.440023i \(-0.145030\pi\)
\(234\) 0 0
\(235\) −4.20450 −0.274271
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −26.3499 −1.70443 −0.852215 0.523191i \(-0.824741\pi\)
−0.852215 + 0.523191i \(0.824741\pi\)
\(240\) 0 0
\(241\) −12.9481 −0.834063 −0.417031 0.908892i \(-0.636929\pi\)
−0.417031 + 0.908892i \(0.636929\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.99711 0.191478
\(246\) 0 0
\(247\) − 2.42439i − 0.154260i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 20.5607i 1.29778i 0.760881 + 0.648891i \(0.224767\pi\)
−0.760881 + 0.648891i \(0.775233\pi\)
\(252\) 0 0
\(253\) − 14.9453i − 0.939605i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.25071i 0.0780174i 0.999239 + 0.0390087i \(0.0124200\pi\)
−0.999239 + 0.0390087i \(0.987580\pi\)
\(258\) 0 0
\(259\) −6.10293 −0.379218
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −25.0372 −1.54386 −0.771931 0.635706i \(-0.780709\pi\)
−0.771931 + 0.635706i \(0.780709\pi\)
\(264\) 0 0
\(265\) 28.6829 1.76197
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −0.520900 −0.0317598 −0.0158799 0.999874i \(-0.505055\pi\)
−0.0158799 + 0.999874i \(0.505055\pi\)
\(270\) 0 0
\(271\) 7.99100i 0.485419i 0.970099 + 0.242709i \(0.0780361\pi\)
−0.970099 + 0.242709i \(0.921964\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 21.3586i − 1.28797i
\(276\) 0 0
\(277\) 32.8049i 1.97106i 0.169509 + 0.985529i \(0.445782\pi\)
−0.169509 + 0.985529i \(0.554218\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 7.13786i 0.425809i 0.977073 + 0.212905i \(0.0682924\pi\)
−0.977073 + 0.212905i \(0.931708\pi\)
\(282\) 0 0
\(283\) 4.47861 0.266226 0.133113 0.991101i \(-0.457503\pi\)
0.133113 + 0.991101i \(0.457503\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −12.3255 −0.727551
\(288\) 0 0
\(289\) 16.0809 0.945935
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −23.4454 −1.36969 −0.684846 0.728688i \(-0.740131\pi\)
−0.684846 + 0.728688i \(0.740131\pi\)
\(294\) 0 0
\(295\) 0.106988i 0.00622911i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 12.6944i 0.734135i
\(300\) 0 0
\(301\) 4.45583i 0.256830i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 15.3950i 0.881513i
\(306\) 0 0
\(307\) 5.07164 0.289454 0.144727 0.989472i \(-0.453770\pi\)
0.144727 + 0.989472i \(0.453770\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 5.43039 0.307929 0.153964 0.988076i \(-0.450796\pi\)
0.153964 + 0.988076i \(0.450796\pi\)
\(312\) 0 0
\(313\) 12.5124 0.707245 0.353622 0.935388i \(-0.384950\pi\)
0.353622 + 0.935388i \(0.384950\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9.29939 0.522306 0.261153 0.965297i \(-0.415897\pi\)
0.261153 + 0.965297i \(0.415897\pi\)
\(318\) 0 0
\(319\) 27.6520i 1.54821i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 0.510246i − 0.0283908i
\(324\) 0 0
\(325\) 18.1417i 1.00632i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 1.40285i − 0.0773418i
\(330\) 0 0
\(331\) 21.4879 1.18108 0.590541 0.807008i \(-0.298914\pi\)
0.590541 + 0.807008i \(0.298914\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 25.7615 1.40750
\(336\) 0 0
\(337\) −28.1982 −1.53605 −0.768026 0.640419i \(-0.778761\pi\)
−0.768026 + 0.640419i \(0.778761\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 25.0217 1.35500
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 21.2134i − 1.13880i −0.822062 0.569398i \(-0.807176\pi\)
0.822062 0.569398i \(-0.192824\pi\)
\(348\) 0 0
\(349\) 31.7541i 1.69976i 0.526979 + 0.849878i \(0.323325\pi\)
−0.526979 + 0.849878i \(0.676675\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 7.37308i − 0.392429i −0.980561 0.196215i \(-0.937135\pi\)
0.980561 0.196215i \(-0.0628650\pi\)
\(354\) 0 0
\(355\) −39.9420 −2.11990
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 14.8384 0.783142 0.391571 0.920148i \(-0.371932\pi\)
0.391571 + 0.920148i \(0.371932\pi\)
\(360\) 0 0
\(361\) −18.7167 −0.985091
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −19.5054 −1.02096
\(366\) 0 0
\(367\) − 13.3215i − 0.695379i −0.937610 0.347690i \(-0.886966\pi\)
0.937610 0.347690i \(-0.113034\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 9.57019i 0.496859i
\(372\) 0 0
\(373\) − 24.7297i − 1.28046i −0.768185 0.640228i \(-0.778840\pi\)
0.768185 0.640228i \(-0.221160\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 23.4873i − 1.20966i
\(378\) 0 0
\(379\) 20.7764 1.06721 0.533606 0.845733i \(-0.320837\pi\)
0.533606 + 0.845733i \(0.320837\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −4.92022 −0.251411 −0.125706 0.992068i \(-0.540119\pi\)
−0.125706 + 0.992068i \(0.540119\pi\)
\(384\) 0 0
\(385\) 16.0732 0.819166
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −34.4298 −1.74566 −0.872830 0.488025i \(-0.837717\pi\)
−0.872830 + 0.488025i \(0.837717\pi\)
\(390\) 0 0
\(391\) 2.67170i 0.135114i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 5.71191i 0.287397i
\(396\) 0 0
\(397\) 21.8751i 1.09788i 0.835861 + 0.548941i \(0.184969\pi\)
−0.835861 + 0.548941i \(0.815031\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 29.0637i 1.45137i 0.688027 + 0.725685i \(0.258477\pi\)
−0.688027 + 0.725685i \(0.741523\pi\)
\(402\) 0 0
\(403\) −21.2531 −1.05869
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −32.7295 −1.62234
\(408\) 0 0
\(409\) −4.15951 −0.205674 −0.102837 0.994698i \(-0.532792\pi\)
−0.102837 + 0.994698i \(0.532792\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −0.0356973 −0.00175655
\(414\) 0 0
\(415\) 9.39207i 0.461039i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 22.5241i 1.10038i 0.835041 + 0.550188i \(0.185444\pi\)
−0.835041 + 0.550188i \(0.814556\pi\)
\(420\) 0 0
\(421\) 32.7375i 1.59553i 0.602970 + 0.797764i \(0.293984\pi\)
−0.602970 + 0.797764i \(0.706016\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 3.81816i 0.185208i
\(426\) 0 0
\(427\) −5.13661 −0.248578
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 28.1991 1.35830 0.679152 0.733998i \(-0.262348\pi\)
0.679152 + 0.733998i \(0.262348\pi\)
\(432\) 0 0
\(433\) −22.1414 −1.06405 −0.532024 0.846729i \(-0.678569\pi\)
−0.532024 + 0.846729i \(0.678569\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1.48321 −0.0709515
\(438\) 0 0
\(439\) − 24.6966i − 1.17870i −0.807876 0.589352i \(-0.799383\pi\)
0.807876 0.589352i \(-0.200617\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 24.6550i 1.17139i 0.810531 + 0.585696i \(0.199179\pi\)
−0.810531 + 0.585696i \(0.800821\pi\)
\(444\) 0 0
\(445\) − 17.7226i − 0.840131i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 4.80960i 0.226979i 0.993539 + 0.113489i \(0.0362028\pi\)
−0.993539 + 0.113489i \(0.963797\pi\)
\(450\) 0 0
\(451\) −66.1005 −3.11255
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −13.6524 −0.640033
\(456\) 0 0
\(457\) −20.5555 −0.961548 −0.480774 0.876845i \(-0.659644\pi\)
−0.480774 + 0.876845i \(0.659644\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 6.96146 0.324228 0.162114 0.986772i \(-0.448169\pi\)
0.162114 + 0.986772i \(0.448169\pi\)
\(462\) 0 0
\(463\) − 0.214819i − 0.00998347i −0.999988 0.00499174i \(-0.998411\pi\)
0.999988 0.00499174i \(-0.00158893\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 30.4215i − 1.40774i −0.710328 0.703870i \(-0.751454\pi\)
0.710328 0.703870i \(-0.248546\pi\)
\(468\) 0 0
\(469\) 8.59546i 0.396901i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 23.8962i 1.09875i
\(474\) 0 0
\(475\) −2.11967 −0.0972572
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 36.9888 1.69006 0.845030 0.534719i \(-0.179582\pi\)
0.845030 + 0.534719i \(0.179582\pi\)
\(480\) 0 0
\(481\) 27.8000 1.26757
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 47.9709 2.17824
\(486\) 0 0
\(487\) − 10.2932i − 0.466431i −0.972425 0.233215i \(-0.925075\pi\)
0.972425 0.233215i \(-0.0749248\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 1.77521i − 0.0801142i −0.999197 0.0400571i \(-0.987246\pi\)
0.999197 0.0400571i \(-0.0127540\pi\)
\(492\) 0 0
\(493\) − 4.94321i − 0.222631i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 13.3269i − 0.597791i
\(498\) 0 0
\(499\) 27.4254 1.22773 0.613864 0.789412i \(-0.289614\pi\)
0.613864 + 0.789412i \(0.289614\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −31.1546 −1.38911 −0.694557 0.719437i \(-0.744400\pi\)
−0.694557 + 0.719437i \(0.744400\pi\)
\(504\) 0 0
\(505\) −48.2617 −2.14762
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 3.20979 0.142271 0.0711356 0.997467i \(-0.477338\pi\)
0.0711356 + 0.997467i \(0.477338\pi\)
\(510\) 0 0
\(511\) − 6.50808i − 0.287901i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 54.0524i − 2.38183i
\(516\) 0 0
\(517\) − 7.52337i − 0.330877i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 34.3057i − 1.50296i −0.659756 0.751480i \(-0.729340\pi\)
0.659756 0.751480i \(-0.270660\pi\)
\(522\) 0 0
\(523\) 37.4283 1.63662 0.818312 0.574774i \(-0.194910\pi\)
0.818312 + 0.574774i \(0.194910\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −4.47300 −0.194847
\(528\) 0 0
\(529\) −15.2338 −0.662337
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 56.1449 2.43191
\(534\) 0 0
\(535\) 27.2241i 1.17700i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 5.36291i 0.230997i
\(540\) 0 0
\(541\) 1.04111i 0.0447610i 0.999750 + 0.0223805i \(0.00712453\pi\)
−0.999750 + 0.0223805i \(0.992875\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 51.7273i − 2.21576i
\(546\) 0 0
\(547\) −3.89746 −0.166643 −0.0833216 0.996523i \(-0.526553\pi\)
−0.0833216 + 0.996523i \(0.526553\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 2.74425 0.116909
\(552\) 0 0
\(553\) −1.90581 −0.0810432
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 32.1046 1.36031 0.680157 0.733067i \(-0.261912\pi\)
0.680157 + 0.733067i \(0.261912\pi\)
\(558\) 0 0
\(559\) − 20.2971i − 0.858477i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 9.40478i − 0.396364i −0.980165 0.198182i \(-0.936496\pi\)
0.980165 0.198182i \(-0.0635037\pi\)
\(564\) 0 0
\(565\) 3.82399i 0.160877i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 5.32036i 0.223041i 0.993762 + 0.111521i \(0.0355721\pi\)
−0.993762 + 0.111521i \(0.964428\pi\)
\(570\) 0 0
\(571\) 47.3101 1.97986 0.989932 0.141542i \(-0.0452059\pi\)
0.989932 + 0.141542i \(0.0452059\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 11.0988 0.462853
\(576\) 0 0
\(577\) 21.0280 0.875407 0.437704 0.899119i \(-0.355792\pi\)
0.437704 + 0.899119i \(0.355792\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −3.13371 −0.130008
\(582\) 0 0
\(583\) 51.3240i 2.12562i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 18.8579i − 0.778348i −0.921164 0.389174i \(-0.872761\pi\)
0.921164 0.389174i \(-0.127239\pi\)
\(588\) 0 0
\(589\) − 2.48321i − 0.102319i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 15.7984i 0.648764i 0.945926 + 0.324382i \(0.105156\pi\)
−0.945926 + 0.324382i \(0.894844\pi\)
\(594\) 0 0
\(595\) −2.87332 −0.117795
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −24.1841 −0.988136 −0.494068 0.869423i \(-0.664491\pi\)
−0.494068 + 0.869423i \(0.664491\pi\)
\(600\) 0 0
\(601\) 31.5679 1.28768 0.643842 0.765159i \(-0.277340\pi\)
0.643842 + 0.765159i \(0.277340\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 53.2309 2.16414
\(606\) 0 0
\(607\) 23.4349i 0.951194i 0.879663 + 0.475597i \(0.157768\pi\)
−0.879663 + 0.475597i \(0.842232\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 6.39025i 0.258522i
\(612\) 0 0
\(613\) − 21.4865i − 0.867832i −0.900953 0.433916i \(-0.857132\pi\)
0.900953 0.433916i \(-0.142868\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 12.4334i − 0.500549i −0.968175 0.250275i \(-0.919479\pi\)
0.968175 0.250275i \(-0.0805209\pi\)
\(618\) 0 0
\(619\) 20.4500 0.821956 0.410978 0.911645i \(-0.365187\pi\)
0.410978 + 0.911645i \(0.365187\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 5.91323 0.236909
\(624\) 0 0
\(625\) −29.0518 −1.16207
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 5.85088 0.233290
\(630\) 0 0
\(631\) 14.3682i 0.571990i 0.958231 + 0.285995i \(0.0923240\pi\)
−0.958231 + 0.285995i \(0.907676\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 38.1426i − 1.51364i
\(636\) 0 0
\(637\) − 4.55519i − 0.180483i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 13.7983i 0.544998i 0.962156 + 0.272499i \(0.0878502\pi\)
−0.962156 + 0.272499i \(0.912150\pi\)
\(642\) 0 0
\(643\) 36.7203 1.44811 0.724053 0.689745i \(-0.242277\pi\)
0.724053 + 0.689745i \(0.242277\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −1.59099 −0.0625483 −0.0312742 0.999511i \(-0.509956\pi\)
−0.0312742 + 0.999511i \(0.509956\pi\)
\(648\) 0 0
\(649\) −0.191441 −0.00751472
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −22.5953 −0.884223 −0.442111 0.896960i \(-0.645770\pi\)
−0.442111 + 0.896960i \(0.645770\pi\)
\(654\) 0 0
\(655\) − 39.9986i − 1.56287i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 24.8372i 0.967520i 0.875201 + 0.483760i \(0.160729\pi\)
−0.875201 + 0.483760i \(0.839271\pi\)
\(660\) 0 0
\(661\) − 28.3986i − 1.10458i −0.833653 0.552288i \(-0.813755\pi\)
0.833653 0.552288i \(-0.186245\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 1.59514i − 0.0618569i
\(666\) 0 0
\(667\) −14.3692 −0.556377
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −27.5471 −1.06345
\(672\) 0 0
\(673\) 3.94402 0.152031 0.0760153 0.997107i \(-0.475780\pi\)
0.0760153 + 0.997107i \(0.475780\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −8.72694 −0.335403 −0.167702 0.985838i \(-0.553635\pi\)
−0.167702 + 0.985838i \(0.553635\pi\)
\(678\) 0 0
\(679\) 16.0057i 0.614243i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 23.0246i − 0.881014i −0.897749 0.440507i \(-0.854799\pi\)
0.897749 0.440507i \(-0.145201\pi\)
\(684\) 0 0
\(685\) − 51.2209i − 1.95705i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 43.5940i − 1.66080i
\(690\) 0 0
\(691\) 19.5528 0.743823 0.371912 0.928268i \(-0.378702\pi\)
0.371912 + 0.928268i \(0.378702\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −48.1363 −1.82591
\(696\) 0 0
\(697\) 11.8165 0.447580
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −22.1070 −0.834971 −0.417486 0.908684i \(-0.637089\pi\)
−0.417486 + 0.908684i \(0.637089\pi\)
\(702\) 0 0
\(703\) 3.24814i 0.122506i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 16.1028i − 0.605607i
\(708\) 0 0
\(709\) 15.3427i 0.576207i 0.957599 + 0.288104i \(0.0930247\pi\)
−0.957599 + 0.288104i \(0.906975\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 13.0024i 0.486942i
\(714\) 0 0
\(715\) −73.2164 −2.73814
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −15.1069 −0.563392 −0.281696 0.959504i \(-0.590897\pi\)
−0.281696 + 0.959504i \(0.590897\pi\)
\(720\) 0 0
\(721\) 18.0349 0.671653
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −20.5352 −0.762657
\(726\) 0 0
\(727\) − 20.4051i − 0.756784i −0.925645 0.378392i \(-0.876477\pi\)
0.925645 0.378392i \(-0.123523\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 4.27180i − 0.157998i
\(732\) 0 0
\(733\) 39.8824i 1.47309i 0.676389 + 0.736544i \(0.263544\pi\)
−0.676389 + 0.736544i \(0.736456\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 46.0966i 1.69799i
\(738\) 0 0
\(739\) −31.8675 −1.17226 −0.586132 0.810216i \(-0.699350\pi\)
−0.586132 + 0.810216i \(0.699350\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −36.1243 −1.32527 −0.662636 0.748942i \(-0.730562\pi\)
−0.662636 + 0.748942i \(0.730562\pi\)
\(744\) 0 0
\(745\) 4.11974 0.150936
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −9.08345 −0.331902
\(750\) 0 0
\(751\) 7.07701i 0.258244i 0.991629 + 0.129122i \(0.0412159\pi\)
−0.991629 + 0.129122i \(0.958784\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 16.4992i 0.600467i
\(756\) 0 0
\(757\) 25.8712i 0.940304i 0.882585 + 0.470152i \(0.155801\pi\)
−0.882585 + 0.470152i \(0.844199\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 3.07176i 0.111351i 0.998449 + 0.0556757i \(0.0177313\pi\)
−0.998449 + 0.0556757i \(0.982269\pi\)
\(762\) 0 0
\(763\) 17.2591 0.624821
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0.162608 0.00587142
\(768\) 0 0
\(769\) 39.3610 1.41940 0.709698 0.704506i \(-0.248831\pi\)
0.709698 + 0.704506i \(0.248831\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 48.9129 1.75928 0.879638 0.475644i \(-0.157785\pi\)
0.879638 + 0.475644i \(0.157785\pi\)
\(774\) 0 0
\(775\) 18.5818i 0.667478i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 6.55996i 0.235035i
\(780\) 0 0
\(781\) − 71.4707i − 2.55742i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 56.2859i − 2.00893i
\(786\) 0 0
\(787\) −16.5486 −0.589893 −0.294946 0.955514i \(-0.595302\pi\)
−0.294946 + 0.955514i \(0.595302\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −1.27590 −0.0453656
\(792\) 0 0
\(793\) 23.3982 0.830895
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −6.15737 −0.218105 −0.109053 0.994036i \(-0.534782\pi\)
−0.109053 + 0.994036i \(0.534782\pi\)
\(798\) 0 0
\(799\) 1.34491i 0.0475797i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 34.9022i − 1.23167i
\(804\) 0 0
\(805\) 8.35233i 0.294381i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 10.7152i 0.376728i 0.982099 + 0.188364i \(0.0603185\pi\)
−0.982099 + 0.188364i \(0.939682\pi\)
\(810\) 0 0
\(811\) −1.60272 −0.0562791 −0.0281395 0.999604i \(-0.508958\pi\)
−0.0281395 + 0.999604i \(0.508958\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −53.2498 −1.86526
\(816\) 0 0
\(817\) 2.37151 0.0829687
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 5.62139 0.196188 0.0980940 0.995177i \(-0.468725\pi\)
0.0980940 + 0.995177i \(0.468725\pi\)
\(822\) 0 0
\(823\) 10.5605i 0.368114i 0.982916 + 0.184057i \(0.0589232\pi\)
−0.982916 + 0.184057i \(0.941077\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 37.6285i 1.30847i 0.756290 + 0.654236i \(0.227010\pi\)
−0.756290 + 0.654236i \(0.772990\pi\)
\(828\) 0 0
\(829\) − 2.85388i − 0.0991193i −0.998771 0.0495596i \(-0.984218\pi\)
0.998771 0.0495596i \(-0.0157818\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 0.958700i − 0.0332170i
\(834\) 0 0
\(835\) 54.4804 1.88537
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 9.63628 0.332681 0.166341 0.986068i \(-0.446805\pi\)
0.166341 + 0.986068i \(0.446805\pi\)
\(840\) 0 0
\(841\) −2.41401 −0.0832418
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 23.2267 0.799023
\(846\) 0 0
\(847\) 17.7608i 0.610267i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 17.0076i − 0.583015i
\(852\) 0 0
\(853\) − 4.13385i − 0.141540i −0.997493 0.0707701i \(-0.977454\pi\)
0.997493 0.0707701i \(-0.0225457\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 32.0416i 1.09452i 0.836963 + 0.547260i \(0.184329\pi\)
−0.836963 + 0.547260i \(0.815671\pi\)
\(858\) 0 0
\(859\) 17.8213 0.608055 0.304027 0.952663i \(-0.401669\pi\)
0.304027 + 0.952663i \(0.401669\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 2.53777 0.0863868 0.0431934 0.999067i \(-0.486247\pi\)
0.0431934 + 0.999067i \(0.486247\pi\)
\(864\) 0 0
\(865\) 7.15239 0.243189
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −10.2207 −0.346713
\(870\) 0 0
\(871\) − 39.1539i − 1.32668i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 3.04912i − 0.103079i
\(876\) 0 0
\(877\) − 46.9723i − 1.58614i −0.609129 0.793072i \(-0.708481\pi\)
0.609129 0.793072i \(-0.291519\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 45.8003i 1.54305i 0.636198 + 0.771525i \(0.280506\pi\)
−0.636198 + 0.771525i \(0.719494\pi\)
\(882\) 0 0
\(883\) −35.1176 −1.18180 −0.590901 0.806744i \(-0.701228\pi\)
−0.590901 + 0.806744i \(0.701228\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −10.1586 −0.341091 −0.170546 0.985350i \(-0.554553\pi\)
−0.170546 + 0.985350i \(0.554553\pi\)
\(888\) 0 0
\(889\) 12.7265 0.426833
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −0.746636 −0.0249852
\(894\) 0 0
\(895\) − 20.1751i − 0.674378i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 24.0571i − 0.802348i
\(900\) 0 0
\(901\) − 9.17494i − 0.305661i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 3.61470i 0.120157i
\(906\) 0 0
\(907\) −35.4243 −1.17625 −0.588123 0.808772i \(-0.700133\pi\)
−0.588123 + 0.808772i \(0.700133\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 37.6389 1.24703 0.623517 0.781810i \(-0.285703\pi\)
0.623517 + 0.781810i \(0.285703\pi\)
\(912\) 0 0
\(913\) −16.8058 −0.556191
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 13.3457 0.440715
\(918\) 0 0
\(919\) − 39.1964i − 1.29297i −0.762927 0.646485i \(-0.776238\pi\)
0.762927 0.646485i \(-0.223762\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 60.7063i 1.99817i
\(924\) 0 0
\(925\) − 24.3058i − 0.799171i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 9.19387i − 0.301641i −0.988561 0.150821i \(-0.951808\pi\)
0.988561 0.150821i \(-0.0481916\pi\)
\(930\) 0 0
\(931\) 0.532227 0.0174430
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −15.4094 −0.503940
\(936\) 0 0
\(937\) −48.7873 −1.59381 −0.796905 0.604104i \(-0.793531\pi\)
−0.796905 + 0.604104i \(0.793531\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −0.0818594 −0.00266854 −0.00133427 0.999999i \(-0.500425\pi\)
−0.00133427 + 0.999999i \(0.500425\pi\)
\(942\) 0 0
\(943\) − 34.3487i − 1.11855i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 21.5632i 0.700711i 0.936617 + 0.350356i \(0.113939\pi\)
−0.936617 + 0.350356i \(0.886061\pi\)
\(948\) 0 0
\(949\) 29.6455i 0.962334i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 13.2536i 0.429325i 0.976688 + 0.214662i \(0.0688651\pi\)
−0.976688 + 0.214662i \(0.931135\pi\)
\(954\) 0 0
\(955\) −46.4748 −1.50389
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 17.0901 0.551868
\(960\) 0 0
\(961\) 9.23129 0.297784
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −23.0098 −0.740712
\(966\) 0 0
\(967\) − 6.74105i − 0.216778i −0.994109 0.108389i \(-0.965431\pi\)
0.994109 0.108389i \(-0.0345691\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 23.5075i − 0.754391i −0.926134 0.377196i \(-0.876888\pi\)
0.926134 0.377196i \(-0.123112\pi\)
\(972\) 0 0
\(973\) − 16.0609i − 0.514889i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 13.3757i − 0.427927i −0.976842 0.213964i \(-0.931363\pi\)
0.976842 0.213964i \(-0.0686374\pi\)
\(978\) 0 0
\(979\) 31.7121 1.01352
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −18.3093 −0.583975 −0.291987 0.956422i \(-0.594317\pi\)
−0.291987 + 0.956422i \(0.594317\pi\)
\(984\) 0 0
\(985\) 69.2955 2.20794
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −12.4175 −0.394853
\(990\) 0 0
\(991\) 58.0580i 1.84427i 0.386865 + 0.922136i \(0.373558\pi\)
−0.386865 + 0.922136i \(0.626442\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 3.44629i 0.109255i
\(996\) 0 0
\(997\) − 53.3202i − 1.68867i −0.535818 0.844334i \(-0.679997\pi\)
0.535818 0.844334i \(-0.320003\pi\)
\(998\) 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.5615.5 48
3.2 odd 2 inner 6048.2.j.d.5615.44 48
4.3 odd 2 1512.2.j.d.323.28 yes 48
8.3 odd 2 inner 6048.2.j.d.5615.43 48
8.5 even 2 1512.2.j.d.323.22 yes 48
12.11 even 2 1512.2.j.d.323.21 48
24.5 odd 2 1512.2.j.d.323.27 yes 48
24.11 even 2 inner 6048.2.j.d.5615.6 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.j.d.323.21 48 12.11 even 2
1512.2.j.d.323.22 yes 48 8.5 even 2
1512.2.j.d.323.27 yes 48 24.5 odd 2
1512.2.j.d.323.28 yes 48 4.3 odd 2
6048.2.j.d.5615.5 48 1.1 even 1 trivial
6048.2.j.d.5615.6 48 24.11 even 2 inner
6048.2.j.d.5615.43 48 8.3 odd 2 inner
6048.2.j.d.5615.44 48 3.2 odd 2 inner