Properties

Label 6048.2.j.d.5615.13
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 5615.13
Character \(\chi\) = 6048.5615
Dual form 6048.2.j.d.5615.14

$q$-expansion

\(f(q)\) \(=\) \(q-2.29653 q^{5} -1.00000i q^{7} +O(q^{10})\) \(q-2.29653 q^{5} -1.00000i q^{7} +4.02419i q^{11} +1.82461i q^{13} -0.430472i q^{17} -5.01046 q^{19} -3.49479 q^{23} +0.274031 q^{25} -2.16016 q^{29} -2.10635i q^{31} +2.29653i q^{35} +2.19775i q^{37} -4.35032i q^{41} +12.0354 q^{43} +1.72531 q^{47} -1.00000 q^{49} -10.0938 q^{53} -9.24166i q^{55} +10.6175i q^{59} -8.44575i q^{61} -4.19026i q^{65} +12.6358 q^{67} +8.95532 q^{71} -7.16051 q^{73} +4.02419 q^{77} -15.2405i q^{79} +13.4507i q^{83} +0.988589i q^{85} +7.44987i q^{89} +1.82461 q^{91} +11.5067 q^{95} +1.68878 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48q + O(q^{10}) \) \( 48q - 16q^{19} + 48q^{25} - 64q^{43} - 48q^{49} - 32q^{67} + O(q^{100}) \)

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
\(5\) −2.29653 −1.02704 −0.513519 0.858078i \(-0.671658\pi\)
−0.513519 + 0.858078i \(0.671658\pi\)
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.02419i 1.21334i 0.794954 + 0.606670i \(0.207495\pi\)
−0.794954 + 0.606670i \(0.792505\pi\)
\(12\) 0 0
\(13\) 1.82461i 0.506055i 0.967459 + 0.253027i \(0.0814263\pi\)
−0.967459 + 0.253027i \(0.918574\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 0.430472i − 0.104405i −0.998637 0.0522024i \(-0.983376\pi\)
0.998637 0.0522024i \(-0.0166241\pi\)
\(18\) 0 0
\(19\) −5.01046 −1.14948 −0.574739 0.818336i \(-0.694897\pi\)
−0.574739 + 0.818336i \(0.694897\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.49479 −0.728715 −0.364357 0.931259i \(-0.618711\pi\)
−0.364357 + 0.931259i \(0.618711\pi\)
\(24\) 0 0
\(25\) 0.274031 0.0548062
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.16016 −0.401131 −0.200565 0.979680i \(-0.564278\pi\)
−0.200565 + 0.979680i \(0.564278\pi\)
\(30\) 0 0
\(31\) − 2.10635i − 0.378311i −0.981947 0.189156i \(-0.939425\pi\)
0.981947 0.189156i \(-0.0605750\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.29653i 0.388184i
\(36\) 0 0
\(37\) 2.19775i 0.361309i 0.983547 + 0.180654i \(0.0578215\pi\)
−0.983547 + 0.180654i \(0.942178\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 4.35032i − 0.679405i −0.940533 0.339703i \(-0.889674\pi\)
0.940533 0.339703i \(-0.110326\pi\)
\(42\) 0 0
\(43\) 12.0354 1.83538 0.917690 0.397298i \(-0.130052\pi\)
0.917690 + 0.397298i \(0.130052\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.72531 0.251663 0.125831 0.992052i \(-0.459840\pi\)
0.125831 + 0.992052i \(0.459840\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −10.0938 −1.38649 −0.693246 0.720701i \(-0.743820\pi\)
−0.693246 + 0.720701i \(0.743820\pi\)
\(54\) 0 0
\(55\) − 9.24166i − 1.24615i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 10.6175i 1.38228i 0.722721 + 0.691140i \(0.242891\pi\)
−0.722721 + 0.691140i \(0.757109\pi\)
\(60\) 0 0
\(61\) − 8.44575i − 1.08137i −0.841226 0.540684i \(-0.818166\pi\)
0.841226 0.540684i \(-0.181834\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 4.19026i − 0.519737i
\(66\) 0 0
\(67\) 12.6358 1.54370 0.771852 0.635802i \(-0.219331\pi\)
0.771852 + 0.635802i \(0.219331\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.95532 1.06280 0.531401 0.847121i \(-0.321666\pi\)
0.531401 + 0.847121i \(0.321666\pi\)
\(72\) 0 0
\(73\) −7.16051 −0.838075 −0.419037 0.907969i \(-0.637632\pi\)
−0.419037 + 0.907969i \(0.637632\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.02419 0.458599
\(78\) 0 0
\(79\) − 15.2405i − 1.71468i −0.514747 0.857342i \(-0.672114\pi\)
0.514747 0.857342i \(-0.327886\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 13.4507i 1.47640i 0.674581 + 0.738201i \(0.264324\pi\)
−0.674581 + 0.738201i \(0.735676\pi\)
\(84\) 0 0
\(85\) 0.988589i 0.107228i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 7.44987i 0.789684i 0.918749 + 0.394842i \(0.129201\pi\)
−0.918749 + 0.394842i \(0.870799\pi\)
\(90\) 0 0
\(91\) 1.82461 0.191271
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 11.5067 1.18056
\(96\) 0 0
\(97\) 1.68878 0.171469 0.0857347 0.996318i \(-0.472676\pi\)
0.0857347 + 0.996318i \(0.472676\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −13.3668 −1.33005 −0.665024 0.746822i \(-0.731579\pi\)
−0.665024 + 0.746822i \(0.731579\pi\)
\(102\) 0 0
\(103\) 4.59723i 0.452978i 0.974014 + 0.226489i \(0.0727248\pi\)
−0.974014 + 0.226489i \(0.927275\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 15.7048i − 1.51824i −0.650951 0.759120i \(-0.725630\pi\)
0.650951 0.759120i \(-0.274370\pi\)
\(108\) 0 0
\(109\) − 16.0140i − 1.53386i −0.641728 0.766932i \(-0.721782\pi\)
0.641728 0.766932i \(-0.278218\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 1.46858i − 0.138152i −0.997611 0.0690760i \(-0.977995\pi\)
0.997611 0.0690760i \(-0.0220051\pi\)
\(114\) 0 0
\(115\) 8.02588 0.748417
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −0.430472 −0.0394613
\(120\) 0 0
\(121\) −5.19413 −0.472193
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 10.8533 0.970750
\(126\) 0 0
\(127\) 6.35551i 0.563960i 0.959420 + 0.281980i \(0.0909913\pi\)
−0.959420 + 0.281980i \(0.909009\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 7.75532i 0.677585i 0.940861 + 0.338793i \(0.110018\pi\)
−0.940861 + 0.338793i \(0.889982\pi\)
\(132\) 0 0
\(133\) 5.01046i 0.434462i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 3.09831i 0.264706i 0.991203 + 0.132353i \(0.0422533\pi\)
−0.991203 + 0.132353i \(0.957747\pi\)
\(138\) 0 0
\(139\) 3.76422 0.319277 0.159638 0.987176i \(-0.448967\pi\)
0.159638 + 0.987176i \(0.448967\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −7.34257 −0.614016
\(144\) 0 0
\(145\) 4.96085 0.411976
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 6.57894 0.538968 0.269484 0.963005i \(-0.413147\pi\)
0.269484 + 0.963005i \(0.413147\pi\)
\(150\) 0 0
\(151\) − 1.08336i − 0.0881627i −0.999028 0.0440814i \(-0.985964\pi\)
0.999028 0.0440814i \(-0.0140361\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 4.83728i 0.388540i
\(156\) 0 0
\(157\) − 16.6130i − 1.32586i −0.748680 0.662932i \(-0.769312\pi\)
0.748680 0.662932i \(-0.230688\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 3.49479i 0.275428i
\(162\) 0 0
\(163\) −11.4881 −0.899818 −0.449909 0.893074i \(-0.648544\pi\)
−0.449909 + 0.893074i \(0.648544\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 13.4761 1.04281 0.521405 0.853309i \(-0.325408\pi\)
0.521405 + 0.853309i \(0.325408\pi\)
\(168\) 0 0
\(169\) 9.67081 0.743909
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −4.98344 −0.378884 −0.189442 0.981892i \(-0.560668\pi\)
−0.189442 + 0.981892i \(0.560668\pi\)
\(174\) 0 0
\(175\) − 0.274031i − 0.0207148i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 3.45929i − 0.258559i −0.991608 0.129280i \(-0.958733\pi\)
0.991608 0.129280i \(-0.0412665\pi\)
\(180\) 0 0
\(181\) − 10.1409i − 0.753768i −0.926260 0.376884i \(-0.876996\pi\)
0.926260 0.376884i \(-0.123004\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 5.04720i − 0.371078i
\(186\) 0 0
\(187\) 1.73230 0.126678
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 10.9743 0.794074 0.397037 0.917803i \(-0.370038\pi\)
0.397037 + 0.917803i \(0.370038\pi\)
\(192\) 0 0
\(193\) 2.07226 0.149165 0.0745824 0.997215i \(-0.476238\pi\)
0.0745824 + 0.997215i \(0.476238\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −4.69547 −0.334539 −0.167269 0.985911i \(-0.553495\pi\)
−0.167269 + 0.985911i \(0.553495\pi\)
\(198\) 0 0
\(199\) − 16.7556i − 1.18777i −0.804549 0.593887i \(-0.797593\pi\)
0.804549 0.593887i \(-0.202407\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 2.16016i 0.151613i
\(204\) 0 0
\(205\) 9.99061i 0.697775i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 20.1631i − 1.39471i
\(210\) 0 0
\(211\) −15.2955 −1.05299 −0.526494 0.850179i \(-0.676494\pi\)
−0.526494 + 0.850179i \(0.676494\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −27.6396 −1.88500
\(216\) 0 0
\(217\) −2.10635 −0.142988
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0.785442 0.0528345
\(222\) 0 0
\(223\) − 23.5657i − 1.57808i −0.614343 0.789039i \(-0.710579\pi\)
0.614343 0.789039i \(-0.289421\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 16.6514i − 1.10519i −0.833450 0.552595i \(-0.813638\pi\)
0.833450 0.552595i \(-0.186362\pi\)
\(228\) 0 0
\(229\) − 24.4291i − 1.61432i −0.590332 0.807161i \(-0.701003\pi\)
0.590332 0.807161i \(-0.298997\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 18.3470i − 1.20195i −0.799268 0.600975i \(-0.794779\pi\)
0.799268 0.600975i \(-0.205221\pi\)
\(234\) 0 0
\(235\) −3.96222 −0.258467
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 30.1173 1.94812 0.974062 0.226280i \(-0.0726563\pi\)
0.974062 + 0.226280i \(0.0726563\pi\)
\(240\) 0 0
\(241\) −5.51878 −0.355496 −0.177748 0.984076i \(-0.556881\pi\)
−0.177748 + 0.984076i \(0.556881\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.29653 0.146720
\(246\) 0 0
\(247\) − 9.14212i − 0.581699i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 18.2010i − 1.14884i −0.818561 0.574420i \(-0.805228\pi\)
0.818561 0.574420i \(-0.194772\pi\)
\(252\) 0 0
\(253\) − 14.0637i − 0.884178i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 5.91116i − 0.368728i −0.982858 0.184364i \(-0.940977\pi\)
0.982858 0.184364i \(-0.0590226\pi\)
\(258\) 0 0
\(259\) 2.19775 0.136562
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 15.7132 0.968916 0.484458 0.874815i \(-0.339017\pi\)
0.484458 + 0.874815i \(0.339017\pi\)
\(264\) 0 0
\(265\) 23.1807 1.42398
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −26.9473 −1.64301 −0.821504 0.570202i \(-0.806865\pi\)
−0.821504 + 0.570202i \(0.806865\pi\)
\(270\) 0 0
\(271\) 5.68443i 0.345305i 0.984983 + 0.172652i \(0.0552337\pi\)
−0.984983 + 0.172652i \(0.944766\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.10275i 0.0664985i
\(276\) 0 0
\(277\) 7.86390i 0.472496i 0.971693 + 0.236248i \(0.0759178\pi\)
−0.971693 + 0.236248i \(0.924082\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0.186660i 0.0111352i 0.999985 + 0.00556760i \(0.00177223\pi\)
−0.999985 + 0.00556760i \(0.998228\pi\)
\(282\) 0 0
\(283\) 12.4545 0.740343 0.370171 0.928963i \(-0.379299\pi\)
0.370171 + 0.928963i \(0.379299\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −4.35032 −0.256791
\(288\) 0 0
\(289\) 16.8147 0.989100
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 8.52169 0.497843 0.248921 0.968524i \(-0.419924\pi\)
0.248921 + 0.968524i \(0.419924\pi\)
\(294\) 0 0
\(295\) − 24.3833i − 1.41965i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 6.37662i − 0.368770i
\(300\) 0 0
\(301\) − 12.0354i − 0.693708i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 19.3959i 1.11060i
\(306\) 0 0
\(307\) −23.4380 −1.33768 −0.668838 0.743408i \(-0.733208\pi\)
−0.668838 + 0.743408i \(0.733208\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −12.1365 −0.688199 −0.344099 0.938933i \(-0.611816\pi\)
−0.344099 + 0.938933i \(0.611816\pi\)
\(312\) 0 0
\(313\) −7.54204 −0.426301 −0.213151 0.977019i \(-0.568372\pi\)
−0.213151 + 0.977019i \(0.568372\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −29.7545 −1.67118 −0.835590 0.549354i \(-0.814874\pi\)
−0.835590 + 0.549354i \(0.814874\pi\)
\(318\) 0 0
\(319\) − 8.69288i − 0.486708i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2.15686i 0.120011i
\(324\) 0 0
\(325\) 0.499998i 0.0277349i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 1.72531i − 0.0951195i
\(330\) 0 0
\(331\) 16.2670 0.894117 0.447058 0.894505i \(-0.352472\pi\)
0.447058 + 0.894505i \(0.352472\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −29.0184 −1.58544
\(336\) 0 0
\(337\) 32.0056 1.74345 0.871727 0.489992i \(-0.163000\pi\)
0.871727 + 0.489992i \(0.163000\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 8.47634 0.459020
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 15.3562i − 0.824366i −0.911101 0.412183i \(-0.864766\pi\)
0.911101 0.412183i \(-0.135234\pi\)
\(348\) 0 0
\(349\) − 10.7044i − 0.572996i −0.958081 0.286498i \(-0.907509\pi\)
0.958081 0.286498i \(-0.0924912\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 17.0572i − 0.907866i −0.891036 0.453933i \(-0.850021\pi\)
0.891036 0.453933i \(-0.149979\pi\)
\(354\) 0 0
\(355\) −20.5661 −1.09154
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −17.2174 −0.908697 −0.454349 0.890824i \(-0.650128\pi\)
−0.454349 + 0.890824i \(0.650128\pi\)
\(360\) 0 0
\(361\) 6.10474 0.321302
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 16.4443 0.860734
\(366\) 0 0
\(367\) 25.6116i 1.33692i 0.743750 + 0.668458i \(0.233045\pi\)
−0.743750 + 0.668458i \(0.766955\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 10.0938i 0.524045i
\(372\) 0 0
\(373\) − 3.99889i − 0.207054i −0.994627 0.103527i \(-0.966987\pi\)
0.994627 0.103527i \(-0.0330129\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 3.94143i − 0.202994i
\(378\) 0 0
\(379\) 21.0603 1.08180 0.540898 0.841088i \(-0.318085\pi\)
0.540898 + 0.841088i \(0.318085\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 27.1156 1.38554 0.692771 0.721158i \(-0.256390\pi\)
0.692771 + 0.721158i \(0.256390\pi\)
\(384\) 0 0
\(385\) −9.24166 −0.470999
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 8.59148 0.435605 0.217803 0.975993i \(-0.430111\pi\)
0.217803 + 0.975993i \(0.430111\pi\)
\(390\) 0 0
\(391\) 1.50441i 0.0760813i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 35.0001i 1.76105i
\(396\) 0 0
\(397\) − 33.3145i − 1.67201i −0.548725 0.836003i \(-0.684887\pi\)
0.548725 0.836003i \(-0.315113\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 5.42643i − 0.270983i −0.990779 0.135491i \(-0.956739\pi\)
0.990779 0.135491i \(-0.0432613\pi\)
\(402\) 0 0
\(403\) 3.84325 0.191446
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −8.84419 −0.438390
\(408\) 0 0
\(409\) 37.1175 1.83534 0.917672 0.397339i \(-0.130066\pi\)
0.917672 + 0.397339i \(0.130066\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 10.6175 0.522452
\(414\) 0 0
\(415\) − 30.8898i − 1.51632i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 19.7739i 0.966016i 0.875616 + 0.483008i \(0.160456\pi\)
−0.875616 + 0.483008i \(0.839544\pi\)
\(420\) 0 0
\(421\) 19.4861i 0.949692i 0.880069 + 0.474846i \(0.157496\pi\)
−0.880069 + 0.474846i \(0.842504\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 0.117963i − 0.00572202i
\(426\) 0 0
\(427\) −8.44575 −0.408718
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 33.0237 1.59069 0.795347 0.606154i \(-0.207288\pi\)
0.795347 + 0.606154i \(0.207288\pi\)
\(432\) 0 0
\(433\) −33.2428 −1.59755 −0.798773 0.601632i \(-0.794517\pi\)
−0.798773 + 0.601632i \(0.794517\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 17.5105 0.837642
\(438\) 0 0
\(439\) 9.28199i 0.443005i 0.975160 + 0.221503i \(0.0710962\pi\)
−0.975160 + 0.221503i \(0.928904\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 2.61270i − 0.124133i −0.998072 0.0620667i \(-0.980231\pi\)
0.998072 0.0620667i \(-0.0197691\pi\)
\(444\) 0 0
\(445\) − 17.1088i − 0.811035i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 30.3318i − 1.43145i −0.698384 0.715723i \(-0.746097\pi\)
0.698384 0.715723i \(-0.253903\pi\)
\(450\) 0 0
\(451\) 17.5065 0.824349
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −4.19026 −0.196442
\(456\) 0 0
\(457\) 6.64077 0.310642 0.155321 0.987864i \(-0.450359\pi\)
0.155321 + 0.987864i \(0.450359\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 35.1189 1.63565 0.817826 0.575465i \(-0.195179\pi\)
0.817826 + 0.575465i \(0.195179\pi\)
\(462\) 0 0
\(463\) − 2.53401i − 0.117765i −0.998265 0.0588826i \(-0.981246\pi\)
0.998265 0.0588826i \(-0.0187538\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 16.0379i − 0.742145i −0.928604 0.371072i \(-0.878990\pi\)
0.928604 0.371072i \(-0.121010\pi\)
\(468\) 0 0
\(469\) − 12.6358i − 0.583465i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 48.4327i 2.22694i
\(474\) 0 0
\(475\) −1.37302 −0.0629985
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −29.2616 −1.33700 −0.668498 0.743714i \(-0.733062\pi\)
−0.668498 + 0.743714i \(0.733062\pi\)
\(480\) 0 0
\(481\) −4.01004 −0.182842
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −3.87832 −0.176106
\(486\) 0 0
\(487\) 2.81399i 0.127514i 0.997965 + 0.0637570i \(0.0203083\pi\)
−0.997965 + 0.0637570i \(0.979692\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 19.2668i 0.869497i 0.900552 + 0.434748i \(0.143163\pi\)
−0.900552 + 0.434748i \(0.856837\pi\)
\(492\) 0 0
\(493\) 0.929886i 0.0418799i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 8.95532i − 0.401701i
\(498\) 0 0
\(499\) −39.4436 −1.76574 −0.882869 0.469620i \(-0.844391\pi\)
−0.882869 + 0.469620i \(0.844391\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −33.3860 −1.48861 −0.744304 0.667841i \(-0.767219\pi\)
−0.744304 + 0.667841i \(0.767219\pi\)
\(504\) 0 0
\(505\) 30.6973 1.36601
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −9.16584 −0.406269 −0.203134 0.979151i \(-0.565113\pi\)
−0.203134 + 0.979151i \(0.565113\pi\)
\(510\) 0 0
\(511\) 7.16051i 0.316762i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 10.5577i − 0.465226i
\(516\) 0 0
\(517\) 6.94299i 0.305352i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.14191i 0.0500280i 0.999687 + 0.0250140i \(0.00796304\pi\)
−0.999687 + 0.0250140i \(0.992037\pi\)
\(522\) 0 0
\(523\) 19.5606 0.855323 0.427662 0.903939i \(-0.359337\pi\)
0.427662 + 0.903939i \(0.359337\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −0.906723 −0.0394975
\(528\) 0 0
\(529\) −10.7864 −0.468975
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 7.93762 0.343816
\(534\) 0 0
\(535\) 36.0665i 1.55929i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 4.02419i − 0.173334i
\(540\) 0 0
\(541\) 35.9815i 1.54696i 0.633818 + 0.773482i \(0.281487\pi\)
−0.633818 + 0.773482i \(0.718513\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 36.7766i 1.57534i
\(546\) 0 0
\(547\) 19.0634 0.815092 0.407546 0.913185i \(-0.366385\pi\)
0.407546 + 0.913185i \(0.366385\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 10.8234 0.461091
\(552\) 0 0
\(553\) −15.2405 −0.648090
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −18.4543 −0.781934 −0.390967 0.920405i \(-0.627859\pi\)
−0.390967 + 0.920405i \(0.627859\pi\)
\(558\) 0 0
\(559\) 21.9598i 0.928803i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 31.6916i − 1.33564i −0.744322 0.667821i \(-0.767227\pi\)
0.744322 0.667821i \(-0.232773\pi\)
\(564\) 0 0
\(565\) 3.37262i 0.141887i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 27.5168i 1.15357i 0.816897 + 0.576783i \(0.195692\pi\)
−0.816897 + 0.576783i \(0.804308\pi\)
\(570\) 0 0
\(571\) 27.7326 1.16057 0.580287 0.814412i \(-0.302940\pi\)
0.580287 + 0.814412i \(0.302940\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −0.957681 −0.0399381
\(576\) 0 0
\(577\) 38.3530 1.59666 0.798328 0.602223i \(-0.205718\pi\)
0.798328 + 0.602223i \(0.205718\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 13.4507 0.558027
\(582\) 0 0
\(583\) − 40.6194i − 1.68229i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 13.7550i 0.567731i 0.958864 + 0.283865i \(0.0916169\pi\)
−0.958864 + 0.283865i \(0.908383\pi\)
\(588\) 0 0
\(589\) 10.5538i 0.434861i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 31.5293i 1.29475i 0.762171 + 0.647376i \(0.224134\pi\)
−0.762171 + 0.647376i \(0.775866\pi\)
\(594\) 0 0
\(595\) 0.988589 0.0405282
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 14.6650 0.599196 0.299598 0.954066i \(-0.403147\pi\)
0.299598 + 0.954066i \(0.403147\pi\)
\(600\) 0 0
\(601\) −40.8677 −1.66703 −0.833515 0.552497i \(-0.813675\pi\)
−0.833515 + 0.552497i \(0.813675\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 11.9284 0.484960
\(606\) 0 0
\(607\) − 40.0141i − 1.62412i −0.583572 0.812061i \(-0.698345\pi\)
0.583572 0.812061i \(-0.301655\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 3.14802i 0.127355i
\(612\) 0 0
\(613\) 3.46053i 0.139770i 0.997555 + 0.0698848i \(0.0222632\pi\)
−0.997555 + 0.0698848i \(0.977737\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 36.6562i 1.47572i 0.674952 + 0.737861i \(0.264164\pi\)
−0.674952 + 0.737861i \(0.735836\pi\)
\(618\) 0 0
\(619\) 10.0770 0.405029 0.202514 0.979279i \(-0.435089\pi\)
0.202514 + 0.979279i \(0.435089\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 7.44987 0.298473
\(624\) 0 0
\(625\) −26.2951 −1.05180
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0.946071 0.0377223
\(630\) 0 0
\(631\) 28.1634i 1.12117i 0.828098 + 0.560583i \(0.189423\pi\)
−0.828098 + 0.560583i \(0.810577\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 14.5956i − 0.579209i
\(636\) 0 0
\(637\) − 1.82461i − 0.0722935i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 12.3588i 0.488142i 0.969757 + 0.244071i \(0.0784830\pi\)
−0.969757 + 0.244071i \(0.921517\pi\)
\(642\) 0 0
\(643\) −28.0875 −1.10766 −0.553831 0.832629i \(-0.686835\pi\)
−0.553831 + 0.832629i \(0.686835\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −43.3350 −1.70368 −0.851838 0.523806i \(-0.824512\pi\)
−0.851838 + 0.523806i \(0.824512\pi\)
\(648\) 0 0
\(649\) −42.7268 −1.67717
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 32.0509 1.25425 0.627125 0.778919i \(-0.284232\pi\)
0.627125 + 0.778919i \(0.284232\pi\)
\(654\) 0 0
\(655\) − 17.8103i − 0.695905i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 6.88401i − 0.268163i −0.990970 0.134081i \(-0.957192\pi\)
0.990970 0.134081i \(-0.0428084\pi\)
\(660\) 0 0
\(661\) − 47.2652i − 1.83840i −0.393788 0.919201i \(-0.628836\pi\)
0.393788 0.919201i \(-0.371164\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 11.5067i − 0.446209i
\(666\) 0 0
\(667\) 7.54929 0.292310
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 33.9873 1.31207
\(672\) 0 0
\(673\) 6.89911 0.265941 0.132971 0.991120i \(-0.457548\pi\)
0.132971 + 0.991120i \(0.457548\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −23.7479 −0.912707 −0.456353 0.889799i \(-0.650845\pi\)
−0.456353 + 0.889799i \(0.650845\pi\)
\(678\) 0 0
\(679\) − 1.68878i − 0.0648094i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 41.5546i − 1.59004i −0.606583 0.795020i \(-0.707460\pi\)
0.606583 0.795020i \(-0.292540\pi\)
\(684\) 0 0
\(685\) − 7.11534i − 0.271863i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 18.4172i − 0.701641i
\(690\) 0 0
\(691\) 17.5146 0.666285 0.333142 0.942876i \(-0.391891\pi\)
0.333142 + 0.942876i \(0.391891\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −8.64462 −0.327909
\(696\) 0 0
\(697\) −1.87269 −0.0709331
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1.39361 0.0526361 0.0263180 0.999654i \(-0.491622\pi\)
0.0263180 + 0.999654i \(0.491622\pi\)
\(702\) 0 0
\(703\) − 11.0118i − 0.415317i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 13.3668i 0.502711i
\(708\) 0 0
\(709\) − 42.1733i − 1.58385i −0.610616 0.791927i \(-0.709078\pi\)
0.610616 0.791927i \(-0.290922\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 7.36125i 0.275681i
\(714\) 0 0
\(715\) 16.8624 0.630618
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −11.8108 −0.440468 −0.220234 0.975447i \(-0.570682\pi\)
−0.220234 + 0.975447i \(0.570682\pi\)
\(720\) 0 0
\(721\) 4.59723 0.171210
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −0.591949 −0.0219844
\(726\) 0 0
\(727\) − 45.5504i − 1.68937i −0.535264 0.844685i \(-0.679788\pi\)
0.535264 0.844685i \(-0.320212\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 5.18089i − 0.191622i
\(732\) 0 0
\(733\) 38.4714i 1.42097i 0.703711 + 0.710487i \(0.251525\pi\)
−0.703711 + 0.710487i \(0.748475\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 50.8487i 1.87304i
\(738\) 0 0
\(739\) −12.1292 −0.446179 −0.223090 0.974798i \(-0.571614\pi\)
−0.223090 + 0.974798i \(0.571614\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −46.7546 −1.71526 −0.857630 0.514267i \(-0.828064\pi\)
−0.857630 + 0.514267i \(0.828064\pi\)
\(744\) 0 0
\(745\) −15.1087 −0.553540
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −15.7048 −0.573841
\(750\) 0 0
\(751\) 29.7196i 1.08448i 0.840222 + 0.542242i \(0.182424\pi\)
−0.840222 + 0.542242i \(0.817576\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 2.48797i 0.0905464i
\(756\) 0 0
\(757\) 16.0932i 0.584919i 0.956278 + 0.292459i \(0.0944737\pi\)
−0.956278 + 0.292459i \(0.905526\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 5.21482i 0.189037i 0.995523 + 0.0945186i \(0.0301312\pi\)
−0.995523 + 0.0945186i \(0.969869\pi\)
\(762\) 0 0
\(763\) −16.0140 −0.579746
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −19.3727 −0.699509
\(768\) 0 0
\(769\) 17.7856 0.641367 0.320683 0.947186i \(-0.396087\pi\)
0.320683 + 0.947186i \(0.396087\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −19.9688 −0.718228 −0.359114 0.933294i \(-0.616921\pi\)
−0.359114 + 0.933294i \(0.616921\pi\)
\(774\) 0 0
\(775\) − 0.577204i − 0.0207338i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 21.7971i 0.780962i
\(780\) 0 0
\(781\) 36.0379i 1.28954i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 38.1522i 1.36171i
\(786\) 0 0
\(787\) 32.1178 1.14487 0.572437 0.819948i \(-0.305998\pi\)
0.572437 + 0.819948i \(0.305998\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −1.46858 −0.0522166
\(792\) 0 0
\(793\) 15.4102 0.547231
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 10.4964 0.371800 0.185900 0.982569i \(-0.440480\pi\)
0.185900 + 0.982569i \(0.440480\pi\)
\(798\) 0 0
\(799\) − 0.742698i − 0.0262748i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 28.8153i − 1.01687i
\(804\) 0 0
\(805\) − 8.02588i − 0.282875i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 31.8333i − 1.11920i −0.828763 0.559599i \(-0.810955\pi\)
0.828763 0.559599i \(-0.189045\pi\)
\(810\) 0 0
\(811\) 1.39041 0.0488238 0.0244119 0.999702i \(-0.492229\pi\)
0.0244119 + 0.999702i \(0.492229\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 26.3827 0.924147
\(816\) 0 0
\(817\) −60.3029 −2.10973
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −18.4554 −0.644097 −0.322048 0.946723i \(-0.604371\pi\)
−0.322048 + 0.946723i \(0.604371\pi\)
\(822\) 0 0
\(823\) 41.1895i 1.43578i 0.696159 + 0.717888i \(0.254891\pi\)
−0.696159 + 0.717888i \(0.745109\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 6.08227i 0.211501i 0.994393 + 0.105751i \(0.0337245\pi\)
−0.994393 + 0.105751i \(0.966275\pi\)
\(828\) 0 0
\(829\) 44.2474i 1.53677i 0.639985 + 0.768387i \(0.278940\pi\)
−0.639985 + 0.768387i \(0.721060\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0.430472i 0.0149150i
\(834\) 0 0
\(835\) −30.9482 −1.07101
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −45.9488 −1.58633 −0.793164 0.609008i \(-0.791568\pi\)
−0.793164 + 0.609008i \(0.791568\pi\)
\(840\) 0 0
\(841\) −24.3337 −0.839094
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −22.2093 −0.764022
\(846\) 0 0
\(847\) 5.19413i 0.178472i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 7.68070i − 0.263291i
\(852\) 0 0
\(853\) 22.7862i 0.780185i 0.920776 + 0.390093i \(0.127557\pi\)
−0.920776 + 0.390093i \(0.872443\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 28.8660i − 0.986042i −0.870017 0.493021i \(-0.835892\pi\)
0.870017 0.493021i \(-0.164108\pi\)
\(858\) 0 0
\(859\) −53.1350 −1.81294 −0.906471 0.422268i \(-0.861234\pi\)
−0.906471 + 0.422268i \(0.861234\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −27.2397 −0.927250 −0.463625 0.886032i \(-0.653451\pi\)
−0.463625 + 0.886032i \(0.653451\pi\)
\(864\) 0 0
\(865\) 11.4446 0.389128
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 61.3305 2.08050
\(870\) 0 0
\(871\) 23.0553i 0.781199i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 10.8533i − 0.366909i
\(876\) 0 0
\(877\) 24.5161i 0.827850i 0.910311 + 0.413925i \(0.135842\pi\)
−0.910311 + 0.413925i \(0.864158\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 18.5560i − 0.625167i −0.949890 0.312584i \(-0.898806\pi\)
0.949890 0.312584i \(-0.101194\pi\)
\(882\) 0 0
\(883\) 48.1013 1.61874 0.809369 0.587300i \(-0.199809\pi\)
0.809369 + 0.587300i \(0.199809\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 5.36841 0.180254 0.0901268 0.995930i \(-0.471273\pi\)
0.0901268 + 0.995930i \(0.471273\pi\)
\(888\) 0 0
\(889\) 6.35551 0.213157
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −8.64461 −0.289281
\(894\) 0 0
\(895\) 7.94435i 0.265550i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 4.55004i 0.151752i
\(900\) 0 0
\(901\) 4.34510i 0.144756i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 23.2889i 0.774148i
\(906\) 0 0
\(907\) −4.80217 −0.159453 −0.0797267 0.996817i \(-0.525405\pi\)
−0.0797267 + 0.996817i \(0.525405\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −2.79257 −0.0925219 −0.0462609 0.998929i \(-0.514731\pi\)
−0.0462609 + 0.998929i \(0.514731\pi\)
\(912\) 0 0
\(913\) −54.1280 −1.79138
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 7.75532 0.256103
\(918\) 0 0
\(919\) 7.90433i 0.260740i 0.991465 + 0.130370i \(0.0416165\pi\)
−0.991465 + 0.130370i \(0.958384\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 16.3399i 0.537836i
\(924\) 0 0
\(925\) 0.602253i 0.0198019i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 4.24927i − 0.139414i −0.997568 0.0697070i \(-0.977794\pi\)
0.997568 0.0697070i \(-0.0222064\pi\)
\(930\) 0 0
\(931\) 5.01046 0.164211
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −3.97827 −0.130103
\(936\) 0 0
\(937\) 1.70393 0.0556649 0.0278324 0.999613i \(-0.491140\pi\)
0.0278324 + 0.999613i \(0.491140\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 53.7297 1.75154 0.875769 0.482730i \(-0.160355\pi\)
0.875769 + 0.482730i \(0.160355\pi\)
\(942\) 0 0
\(943\) 15.2035i 0.495093i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 48.4816i − 1.57544i −0.616033 0.787720i \(-0.711261\pi\)
0.616033 0.787720i \(-0.288739\pi\)
\(948\) 0 0
\(949\) − 13.0651i − 0.424112i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 51.7471i − 1.67625i −0.545476 0.838126i \(-0.683651\pi\)
0.545476 0.838126i \(-0.316349\pi\)
\(954\) 0 0
\(955\) −25.2028 −0.815544
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 3.09831 0.100050
\(960\) 0 0
\(961\) 26.5633 0.856881
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −4.75901 −0.153198
\(966\) 0 0
\(967\) − 7.89472i − 0.253877i −0.991911 0.126939i \(-0.959485\pi\)
0.991911 0.126939i \(-0.0405151\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 15.1077i − 0.484829i −0.970173 0.242414i \(-0.922061\pi\)
0.970173 0.242414i \(-0.0779393\pi\)
\(972\) 0 0
\(973\) − 3.76422i − 0.120675i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 11.3398i 0.362792i 0.983410 + 0.181396i \(0.0580616\pi\)
−0.983410 + 0.181396i \(0.941938\pi\)
\(978\) 0 0
\(979\) −29.9797 −0.958155
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −34.3365 −1.09517 −0.547583 0.836751i \(-0.684452\pi\)
−0.547583 + 0.836751i \(0.684452\pi\)
\(984\) 0 0
\(985\) 10.7833 0.343584
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −42.0612 −1.33747
\(990\) 0 0
\(991\) 20.9504i 0.665510i 0.943013 + 0.332755i \(0.107978\pi\)
−0.943013 + 0.332755i \(0.892022\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 38.4797i 1.21989i
\(996\) 0 0
\(997\) − 8.15182i − 0.258171i −0.991633 0.129085i \(-0.958796\pi\)
0.991633 0.129085i \(-0.0412041\pi\)
\(998\) 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.5615.13 48
3.2 odd 2 inner 6048.2.j.d.5615.36 48
4.3 odd 2 1512.2.j.d.323.19 48
8.3 odd 2 inner 6048.2.j.d.5615.35 48
8.5 even 2 1512.2.j.d.323.29 yes 48
12.11 even 2 1512.2.j.d.323.30 yes 48
24.5 odd 2 1512.2.j.d.323.20 yes 48
24.11 even 2 inner 6048.2.j.d.5615.14 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.j.d.323.19 48 4.3 odd 2
1512.2.j.d.323.20 yes 48 24.5 odd 2
1512.2.j.d.323.29 yes 48 8.5 even 2
1512.2.j.d.323.30 yes 48 12.11 even 2
6048.2.j.d.5615.13 48 1.1 even 1 trivial
6048.2.j.d.5615.14 48 24.11 even 2 inner
6048.2.j.d.5615.35 48 8.3 odd 2 inner
6048.2.j.d.5615.36 48 3.2 odd 2 inner