Properties

Label 6048.2.h.h.2591.1
Level 6048
Weight 2
Character 6048.2591
Analytic conductor 48.294
Analytic rank 0
Dimension 16
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.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2591.1
Root \(-0.826679 - 0.826679i\)
Character \(\chi\) = 6048.2591
Dual form 6048.2.h.h.2591.15

$q$-expansion

\(f(q)\) \(=\) \(q-1.93185i q^{5} -1.00000i q^{7} +O(q^{10})\) \(q-1.93185i q^{5} -1.00000i q^{7} -1.13572 q^{11} -3.19404 q^{13} +5.93127i q^{17} -1.33820i q^{19} +2.96713 q^{23} +1.26795 q^{25} +1.65336i q^{29} +10.3902i q^{31} -1.93185 q^{35} +0.538007 q^{37} -9.51648i q^{41} -6.65815i q^{43} +11.1739 q^{47} -1.00000 q^{49} -1.68863i q^{53} +2.19404i q^{55} -1.82842 q^{59} +14.1904 q^{61} +6.17042i q^{65} +9.53225i q^{67} +10.9690 q^{71} +0.319943 q^{73} +1.13572i q^{77} +12.9465i q^{79} +3.24562 q^{83} +11.4583 q^{85} -13.7621i q^{89} +3.19404i q^{91} -2.58521 q^{95} +2.33820 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q + 8q^{13} + 48q^{25} + 40q^{37} - 16q^{49} - 56q^{73} - 16q^{85} - 16q^{97} + O(q^{100}) \)

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
\(5\) − 1.93185i − 0.863950i −0.901886 0.431975i \(-0.857817\pi\)
0.901886 0.431975i \(-0.142183\pi\)
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.13572 −0.342433 −0.171216 0.985233i \(-0.554770\pi\)
−0.171216 + 0.985233i \(0.554770\pi\)
\(12\) 0 0
\(13\) −3.19404 −0.885868 −0.442934 0.896554i \(-0.646062\pi\)
−0.442934 + 0.896554i \(0.646062\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.93127i 1.43855i 0.694728 + 0.719273i \(0.255525\pi\)
−0.694728 + 0.719273i \(0.744475\pi\)
\(18\) 0 0
\(19\) − 1.33820i − 0.307005i −0.988148 0.153502i \(-0.950945\pi\)
0.988148 0.153502i \(-0.0490553\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.96713 0.618689 0.309344 0.950950i \(-0.399890\pi\)
0.309344 + 0.950950i \(0.399890\pi\)
\(24\) 0 0
\(25\) 1.26795 0.253590
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.65336i 0.307021i 0.988147 + 0.153510i \(0.0490579\pi\)
−0.988147 + 0.153510i \(0.950942\pi\)
\(30\) 0 0
\(31\) 10.3902i 1.86613i 0.359702 + 0.933067i \(0.382878\pi\)
−0.359702 + 0.933067i \(0.617122\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.93185 −0.326543
\(36\) 0 0
\(37\) 0.538007 0.0884479 0.0442239 0.999022i \(-0.485918\pi\)
0.0442239 + 0.999022i \(0.485918\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 9.51648i − 1.48623i −0.669166 0.743113i \(-0.733349\pi\)
0.669166 0.743113i \(-0.266651\pi\)
\(42\) 0 0
\(43\) − 6.65815i − 1.01536i −0.861546 0.507679i \(-0.830504\pi\)
0.861546 0.507679i \(-0.169496\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 11.1739 1.62988 0.814942 0.579543i \(-0.196769\pi\)
0.814942 + 0.579543i \(0.196769\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 1.68863i − 0.231952i −0.993252 0.115976i \(-0.963001\pi\)
0.993252 0.115976i \(-0.0369995\pi\)
\(54\) 0 0
\(55\) 2.19404i 0.295845i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.82842 −0.238041 −0.119020 0.992892i \(-0.537975\pi\)
−0.119020 + 0.992892i \(0.537975\pi\)
\(60\) 0 0
\(61\) 14.1904 1.81689 0.908447 0.418001i \(-0.137269\pi\)
0.908447 + 0.418001i \(0.137269\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.17042i 0.765346i
\(66\) 0 0
\(67\) 9.53225i 1.16455i 0.812992 + 0.582275i \(0.197837\pi\)
−0.812992 + 0.582275i \(0.802163\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.9690 1.30178 0.650888 0.759174i \(-0.274397\pi\)
0.650888 + 0.759174i \(0.274397\pi\)
\(72\) 0 0
\(73\) 0.319943 0.0374465 0.0187232 0.999825i \(-0.494040\pi\)
0.0187232 + 0.999825i \(0.494040\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.13572i 0.129427i
\(78\) 0 0
\(79\) 12.9465i 1.45659i 0.685263 + 0.728295i \(0.259687\pi\)
−0.685263 + 0.728295i \(0.740313\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.24562 0.356253 0.178127 0.984008i \(-0.442996\pi\)
0.178127 + 0.984008i \(0.442996\pi\)
\(84\) 0 0
\(85\) 11.4583 1.24283
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 13.7621i − 1.45878i −0.684098 0.729390i \(-0.739804\pi\)
0.684098 0.729390i \(-0.260196\pi\)
\(90\) 0 0
\(91\) 3.19404i 0.334827i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.58521 −0.265237
\(96\) 0 0
\(97\) 2.33820 0.237408 0.118704 0.992930i \(-0.462126\pi\)
0.118704 + 0.992930i \(0.462126\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 16.2075i − 1.61271i −0.591433 0.806354i \(-0.701437\pi\)
0.591433 0.806354i \(-0.298563\pi\)
\(102\) 0 0
\(103\) − 1.80019i − 0.177378i −0.996059 0.0886892i \(-0.971732\pi\)
0.996059 0.0886892i \(-0.0282678\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0.0993527 0.00960479 0.00480239 0.999988i \(-0.498471\pi\)
0.00480239 + 0.999988i \(0.498471\pi\)
\(108\) 0 0
\(109\) −3.40635 −0.326269 −0.163134 0.986604i \(-0.552160\pi\)
−0.163134 + 0.986604i \(0.552160\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 11.2092i − 1.05447i −0.849719 0.527236i \(-0.823228\pi\)
0.849719 0.527236i \(-0.176772\pi\)
\(114\) 0 0
\(115\) − 5.73205i − 0.534516i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 5.93127 0.543719
\(120\) 0 0
\(121\) −9.71014 −0.882740
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 12.1087i − 1.08304i
\(126\) 0 0
\(127\) − 12.1180i − 1.07530i −0.843168 0.537651i \(-0.819312\pi\)
0.843168 0.537651i \(-0.180688\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1.54884 0.135323 0.0676614 0.997708i \(-0.478446\pi\)
0.0676614 + 0.997708i \(0.478446\pi\)
\(132\) 0 0
\(133\) −1.33820 −0.116037
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 0.895484i − 0.0765063i −0.999268 0.0382532i \(-0.987821\pi\)
0.999268 0.0382532i \(-0.0121793\pi\)
\(138\) 0 0
\(139\) − 13.9021i − 1.17916i −0.807711 0.589579i \(-0.799294\pi\)
0.807711 0.589579i \(-0.200706\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 3.62754 0.303350
\(144\) 0 0
\(145\) 3.19404 0.265251
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 4.72391i − 0.386998i −0.981100 0.193499i \(-0.938016\pi\)
0.981100 0.193499i \(-0.0619836\pi\)
\(150\) 0 0
\(151\) 3.51399i 0.285964i 0.989725 + 0.142982i \(0.0456691\pi\)
−0.989725 + 0.142982i \(0.954331\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 20.0723 1.61225
\(156\) 0 0
\(157\) −21.8747 −1.74579 −0.872894 0.487910i \(-0.837760\pi\)
−0.872894 + 0.487910i \(0.837760\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 2.96713i − 0.233842i
\(162\) 0 0
\(163\) − 11.5863i − 0.907513i −0.891126 0.453756i \(-0.850084\pi\)
0.891126 0.453756i \(-0.149916\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2.74973 −0.212781 −0.106390 0.994324i \(-0.533929\pi\)
−0.106390 + 0.994324i \(0.533929\pi\)
\(168\) 0 0
\(169\) −2.79809 −0.215237
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0.893593i 0.0679386i 0.999423 + 0.0339693i \(0.0108148\pi\)
−0.999423 + 0.0339693i \(0.989185\pi\)
\(174\) 0 0
\(175\) − 1.26795i − 0.0958479i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 14.7645 1.10355 0.551775 0.833993i \(-0.313951\pi\)
0.551775 + 0.833993i \(0.313951\pi\)
\(180\) 0 0
\(181\) 16.1904 1.20342 0.601711 0.798714i \(-0.294486\pi\)
0.601711 + 0.798714i \(0.294486\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 1.03935i − 0.0764146i
\(186\) 0 0
\(187\) − 6.73627i − 0.492605i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 9.20679 0.666180 0.333090 0.942895i \(-0.391909\pi\)
0.333090 + 0.942895i \(0.391909\pi\)
\(192\) 0 0
\(193\) 8.04988 0.579443 0.289722 0.957111i \(-0.406437\pi\)
0.289722 + 0.957111i \(0.406437\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 23.9319i 1.70508i 0.522662 + 0.852540i \(0.324939\pi\)
−0.522662 + 0.852540i \(0.675061\pi\)
\(198\) 0 0
\(199\) 11.6581i 0.826424i 0.910635 + 0.413212i \(0.135593\pi\)
−0.910635 + 0.413212i \(0.864407\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.65336 0.116043
\(204\) 0 0
\(205\) −18.3844 −1.28402
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1.51982i 0.105128i
\(210\) 0 0
\(211\) 6.97387i 0.480101i 0.970760 + 0.240051i \(0.0771640\pi\)
−0.970760 + 0.240051i \(0.922836\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −12.8625 −0.877219
\(216\) 0 0
\(217\) 10.3902 0.705332
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 18.9447i − 1.27436i
\(222\) 0 0
\(223\) − 9.74455i − 0.652543i −0.945276 0.326272i \(-0.894208\pi\)
0.945276 0.326272i \(-0.105792\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 12.7309 0.844980 0.422490 0.906368i \(-0.361156\pi\)
0.422490 + 0.906368i \(0.361156\pi\)
\(228\) 0 0
\(229\) −15.1259 −0.999548 −0.499774 0.866156i \(-0.666584\pi\)
−0.499774 + 0.866156i \(0.666584\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 22.1211i − 1.44920i −0.689168 0.724601i \(-0.742024\pi\)
0.689168 0.724601i \(-0.257976\pi\)
\(234\) 0 0
\(235\) − 21.5863i − 1.40814i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 11.1077 0.718494 0.359247 0.933242i \(-0.383033\pi\)
0.359247 + 0.933242i \(0.383033\pi\)
\(240\) 0 0
\(241\) −15.4458 −0.994954 −0.497477 0.867477i \(-0.665740\pi\)
−0.497477 + 0.867477i \(0.665740\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.93185i 0.123421i
\(246\) 0 0
\(247\) 4.27428i 0.271966i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −28.6241 −1.80674 −0.903368 0.428867i \(-0.858913\pi\)
−0.903368 + 0.428867i \(0.858913\pi\)
\(252\) 0 0
\(253\) −3.36983 −0.211859
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 7.74617i 0.483193i 0.970377 + 0.241596i \(0.0776710\pi\)
−0.970377 + 0.241596i \(0.922329\pi\)
\(258\) 0 0
\(259\) − 0.538007i − 0.0334301i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 14.4877 0.893349 0.446675 0.894696i \(-0.352608\pi\)
0.446675 + 0.894696i \(0.352608\pi\)
\(264\) 0 0
\(265\) −3.26219 −0.200395
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 10.7174i − 0.653450i −0.945119 0.326725i \(-0.894055\pi\)
0.945119 0.326725i \(-0.105945\pi\)
\(270\) 0 0
\(271\) 19.7086i 1.19721i 0.801044 + 0.598606i \(0.204278\pi\)
−0.801044 + 0.598606i \(0.795722\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.44004 −0.0868374
\(276\) 0 0
\(277\) 9.26430 0.556638 0.278319 0.960489i \(-0.410223\pi\)
0.278319 + 0.960489i \(0.410223\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 4.37727i − 0.261126i −0.991440 0.130563i \(-0.958322\pi\)
0.991440 0.130563i \(-0.0416785\pi\)
\(282\) 0 0
\(283\) 5.12168i 0.304452i 0.988346 + 0.152226i \(0.0486442\pi\)
−0.988346 + 0.152226i \(0.951356\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −9.51648 −0.561740
\(288\) 0 0
\(289\) −18.1800 −1.06941
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 23.0042i − 1.34392i −0.740588 0.671959i \(-0.765453\pi\)
0.740588 0.671959i \(-0.234547\pi\)
\(294\) 0 0
\(295\) 3.53225i 0.205655i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −9.47714 −0.548077
\(300\) 0 0
\(301\) −6.65815 −0.383769
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 27.4137i − 1.56971i
\(306\) 0 0
\(307\) − 9.84180i − 0.561701i −0.959751 0.280851i \(-0.909383\pi\)
0.959751 0.280851i \(-0.0906165\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 15.7593 0.893630 0.446815 0.894626i \(-0.352558\pi\)
0.446815 + 0.894626i \(0.352558\pi\)
\(312\) 0 0
\(313\) −19.0828 −1.07862 −0.539310 0.842107i \(-0.681315\pi\)
−0.539310 + 0.842107i \(0.681315\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 8.04008i − 0.451576i −0.974176 0.225788i \(-0.927504\pi\)
0.974176 0.225788i \(-0.0724957\pi\)
\(318\) 0 0
\(319\) − 1.87775i − 0.105134i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 7.93724 0.441640
\(324\) 0 0
\(325\) −4.04988 −0.224647
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 11.1739i − 0.616038i
\(330\) 0 0
\(331\) 9.60039i 0.527685i 0.964566 + 0.263843i \(0.0849899\pi\)
−0.964566 + 0.263843i \(0.915010\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 18.4149 1.00611
\(336\) 0 0
\(337\) 16.7981 0.915050 0.457525 0.889197i \(-0.348736\pi\)
0.457525 + 0.889197i \(0.348736\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 11.8004i − 0.639025i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 11.5900 0.622185 0.311092 0.950380i \(-0.399305\pi\)
0.311092 + 0.950380i \(0.399305\pi\)
\(348\) 0 0
\(349\) 13.2123 0.707238 0.353619 0.935389i \(-0.384951\pi\)
0.353619 + 0.935389i \(0.384951\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 25.9360i 1.38044i 0.723602 + 0.690218i \(0.242485\pi\)
−0.723602 + 0.690218i \(0.757515\pi\)
\(354\) 0 0
\(355\) − 21.1904i − 1.12467i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 17.2368 0.909725 0.454863 0.890562i \(-0.349688\pi\)
0.454863 + 0.890562i \(0.349688\pi\)
\(360\) 0 0
\(361\) 17.2092 0.905748
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 0.618082i − 0.0323519i
\(366\) 0 0
\(367\) 25.1763i 1.31419i 0.753806 + 0.657097i \(0.228216\pi\)
−0.753806 + 0.657097i \(0.771784\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −1.68863 −0.0876694
\(372\) 0 0
\(373\) 9.46467 0.490062 0.245031 0.969515i \(-0.421202\pi\)
0.245031 + 0.969515i \(0.421202\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 5.28090i − 0.271980i
\(378\) 0 0
\(379\) − 18.1265i − 0.931094i −0.885023 0.465547i \(-0.845858\pi\)
0.885023 0.465547i \(-0.154142\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −10.2526 −0.523884 −0.261942 0.965084i \(-0.584363\pi\)
−0.261942 + 0.965084i \(0.584363\pi\)
\(384\) 0 0
\(385\) 2.19404 0.111819
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 27.6933i 1.40411i 0.712124 + 0.702053i \(0.247733\pi\)
−0.712124 + 0.702053i \(0.752267\pi\)
\(390\) 0 0
\(391\) 17.5988i 0.890012i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 25.0106 1.25842
\(396\) 0 0
\(397\) 8.91977 0.447670 0.223835 0.974627i \(-0.428142\pi\)
0.223835 + 0.974627i \(0.428142\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 16.3761i − 0.817784i −0.912583 0.408892i \(-0.865915\pi\)
0.912583 0.408892i \(-0.134085\pi\)
\(402\) 0 0
\(403\) − 33.1867i − 1.65315i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −0.611026 −0.0302874
\(408\) 0 0
\(409\) 30.3625 1.50133 0.750665 0.660683i \(-0.229733\pi\)
0.750665 + 0.660683i \(0.229733\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 1.82842i 0.0899709i
\(414\) 0 0
\(415\) − 6.27006i − 0.307785i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 8.02467 0.392031 0.196015 0.980601i \(-0.437200\pi\)
0.196015 + 0.980601i \(0.437200\pi\)
\(420\) 0 0
\(421\) 4.45160 0.216958 0.108479 0.994099i \(-0.465402\pi\)
0.108479 + 0.994099i \(0.465402\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 7.52055i 0.364800i
\(426\) 0 0
\(427\) − 14.1904i − 0.686721i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −0.0363656 −0.00175167 −0.000875835 1.00000i \(-0.500279\pi\)
−0.000875835 1.00000i \(0.500279\pi\)
\(432\) 0 0
\(433\) −26.4033 −1.26886 −0.634430 0.772980i \(-0.718765\pi\)
−0.634430 + 0.772980i \(0.718765\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 3.97062i − 0.189940i
\(438\) 0 0
\(439\) 9.21652i 0.439881i 0.975513 + 0.219940i \(0.0705862\pi\)
−0.975513 + 0.219940i \(0.929414\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 38.3496 1.82204 0.911022 0.412358i \(-0.135295\pi\)
0.911022 + 0.412358i \(0.135295\pi\)
\(444\) 0 0
\(445\) −26.5863 −1.26031
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 8.83323i 0.416866i 0.978037 + 0.208433i \(0.0668363\pi\)
−0.978037 + 0.208433i \(0.933164\pi\)
\(450\) 0 0
\(451\) 10.8081i 0.508932i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 6.17042 0.289274
\(456\) 0 0
\(457\) 35.6825 1.66916 0.834578 0.550890i \(-0.185712\pi\)
0.834578 + 0.550890i \(0.185712\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 15.6670i − 0.729687i −0.931069 0.364843i \(-0.881122\pi\)
0.931069 0.364843i \(-0.118878\pi\)
\(462\) 0 0
\(463\) 26.2445i 1.21969i 0.792523 + 0.609843i \(0.208767\pi\)
−0.792523 + 0.609843i \(0.791233\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 24.7956 1.14741 0.573703 0.819064i \(-0.305506\pi\)
0.573703 + 0.819064i \(0.305506\pi\)
\(468\) 0 0
\(469\) 9.53225 0.440158
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 7.56179i 0.347692i
\(474\) 0 0
\(475\) − 1.69677i − 0.0778533i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 32.5665 1.48800 0.744000 0.668180i \(-0.232926\pi\)
0.744000 + 0.668180i \(0.232926\pi\)
\(480\) 0 0
\(481\) −1.71842 −0.0783531
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 4.51706i − 0.205109i
\(486\) 0 0
\(487\) 16.0724i 0.728308i 0.931339 + 0.364154i \(0.118642\pi\)
−0.931339 + 0.364154i \(0.881358\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 6.39378 0.288547 0.144274 0.989538i \(-0.453915\pi\)
0.144274 + 0.989538i \(0.453915\pi\)
\(492\) 0 0
\(493\) −9.80652 −0.441664
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 10.9690i − 0.492025i
\(498\) 0 0
\(499\) − 9.52494i − 0.426395i −0.977009 0.213197i \(-0.931612\pi\)
0.977009 0.213197i \(-0.0683878\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −2.31486 −0.103215 −0.0516073 0.998667i \(-0.516434\pi\)
−0.0516073 + 0.998667i \(0.516434\pi\)
\(504\) 0 0
\(505\) −31.3105 −1.39330
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 34.8925i − 1.54658i −0.634050 0.773292i \(-0.718609\pi\)
0.634050 0.773292i \(-0.281391\pi\)
\(510\) 0 0
\(511\) − 0.319943i − 0.0141534i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −3.47771 −0.153246
\(516\) 0 0
\(517\) −12.6904 −0.558125
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 14.9776i − 0.656181i −0.944646 0.328091i \(-0.893595\pi\)
0.944646 0.328091i \(-0.106405\pi\)
\(522\) 0 0
\(523\) 31.1347i 1.36143i 0.732550 + 0.680714i \(0.238330\pi\)
−0.732550 + 0.680714i \(0.761670\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −61.6271 −2.68452
\(528\) 0 0
\(529\) −14.1962 −0.617224
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 30.3961i 1.31660i
\(534\) 0 0
\(535\) − 0.191935i − 0.00829806i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1.13572 0.0489189
\(540\) 0 0
\(541\) 9.89996 0.425633 0.212816 0.977092i \(-0.431736\pi\)
0.212816 + 0.977092i \(0.431736\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 6.58056i 0.281880i
\(546\) 0 0
\(547\) − 42.7324i − 1.82710i −0.406723 0.913552i \(-0.633328\pi\)
0.406723 0.913552i \(-0.366672\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 2.21253 0.0942569
\(552\) 0 0
\(553\) 12.9465 0.550540
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 3.02932i − 0.128356i −0.997938 0.0641781i \(-0.979557\pi\)
0.997938 0.0641781i \(-0.0204426\pi\)
\(558\) 0 0
\(559\) 21.2664i 0.899473i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −29.4593 −1.24156 −0.620782 0.783984i \(-0.713185\pi\)
−0.620782 + 0.783984i \(0.713185\pi\)
\(564\) 0 0
\(565\) −21.6545 −0.911011
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 9.31152i − 0.390359i −0.980768 0.195180i \(-0.937471\pi\)
0.980768 0.195180i \(-0.0625290\pi\)
\(570\) 0 0
\(571\) − 30.5869i − 1.28002i −0.768366 0.640011i \(-0.778930\pi\)
0.768366 0.640011i \(-0.221070\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 3.76217 0.156893
\(576\) 0 0
\(577\) −6.38387 −0.265764 −0.132882 0.991132i \(-0.542423\pi\)
−0.132882 + 0.991132i \(0.542423\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 3.24562i − 0.134651i
\(582\) 0 0
\(583\) 1.91781i 0.0794278i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 20.0962 0.829461 0.414730 0.909944i \(-0.363876\pi\)
0.414730 + 0.909944i \(0.363876\pi\)
\(588\) 0 0
\(589\) 13.9042 0.572912
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 45.6194i − 1.87336i −0.350183 0.936681i \(-0.613881\pi\)
0.350183 0.936681i \(-0.386119\pi\)
\(594\) 0 0
\(595\) − 11.4583i − 0.469746i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −2.66390 −0.108844 −0.0544221 0.998518i \(-0.517332\pi\)
−0.0544221 + 0.998518i \(0.517332\pi\)
\(600\) 0 0
\(601\) 26.8851 1.09666 0.548332 0.836261i \(-0.315263\pi\)
0.548332 + 0.836261i \(0.315263\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 18.7585i 0.762643i
\(606\) 0 0
\(607\) 17.4027i 0.706354i 0.935557 + 0.353177i \(0.114899\pi\)
−0.935557 + 0.353177i \(0.885101\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −35.6900 −1.44386
\(612\) 0 0
\(613\) 24.6435 0.995343 0.497672 0.867366i \(-0.334188\pi\)
0.497672 + 0.867366i \(0.334188\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 16.0367i 0.645615i 0.946465 + 0.322807i \(0.104627\pi\)
−0.946465 + 0.322807i \(0.895373\pi\)
\(618\) 0 0
\(619\) − 8.90053i − 0.357742i −0.983872 0.178871i \(-0.942755\pi\)
0.983872 0.178871i \(-0.0572446\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −13.7621 −0.551367
\(624\) 0 0
\(625\) −17.0526 −0.682102
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 3.19107i 0.127236i
\(630\) 0 0
\(631\) 2.32416i 0.0925234i 0.998929 + 0.0462617i \(0.0147308\pi\)
−0.998929 + 0.0462617i \(0.985269\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −23.4102 −0.929007
\(636\) 0 0
\(637\) 3.19404 0.126553
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 46.6557i 1.84279i 0.388627 + 0.921395i \(0.372950\pi\)
−0.388627 + 0.921395i \(0.627050\pi\)
\(642\) 0 0
\(643\) 6.59617i 0.260128i 0.991506 + 0.130064i \(0.0415182\pi\)
−0.991506 + 0.130064i \(0.958482\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −23.9672 −0.942249 −0.471124 0.882067i \(-0.656152\pi\)
−0.471124 + 0.882067i \(0.656152\pi\)
\(648\) 0 0
\(649\) 2.07658 0.0815129
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 0.280888i − 0.0109920i −0.999985 0.00549600i \(-0.998251\pi\)
0.999985 0.00549600i \(-0.00174944\pi\)
\(654\) 0 0
\(655\) − 2.99213i − 0.116912i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 47.5965 1.85410 0.927049 0.374941i \(-0.122337\pi\)
0.927049 + 0.374941i \(0.122337\pi\)
\(660\) 0 0
\(661\) −6.56752 −0.255447 −0.127724 0.991810i \(-0.540767\pi\)
−0.127724 + 0.991810i \(0.540767\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2.58521i 0.100250i
\(666\) 0 0
\(667\) 4.90573i 0.189950i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −16.1163 −0.622164
\(672\) 0 0
\(673\) 3.50977 0.135292 0.0676458 0.997709i \(-0.478451\pi\)
0.0676458 + 0.997709i \(0.478451\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1.30650i 0.0502129i 0.999685 + 0.0251065i \(0.00799247\pi\)
−0.999685 + 0.0251065i \(0.992008\pi\)
\(678\) 0 0
\(679\) − 2.33820i − 0.0897320i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 17.8633 0.683519 0.341760 0.939787i \(-0.388977\pi\)
0.341760 + 0.939787i \(0.388977\pi\)
\(684\) 0 0
\(685\) −1.72994 −0.0660977
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 5.39357i 0.205478i
\(690\) 0 0
\(691\) 24.3267i 0.925430i 0.886507 + 0.462715i \(0.153125\pi\)
−0.886507 + 0.462715i \(0.846875\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −26.8567 −1.01873
\(696\) 0 0
\(697\) 56.4449 2.13800
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 2.57815i − 0.0973754i −0.998814 0.0486877i \(-0.984496\pi\)
0.998814 0.0486877i \(-0.0155039\pi\)
\(702\) 0 0
\(703\) − 0.719962i − 0.0271539i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −16.2075 −0.609547
\(708\) 0 0
\(709\) 51.7590 1.94385 0.971926 0.235287i \(-0.0756031\pi\)
0.971926 + 0.235287i \(0.0756031\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 30.8290i 1.15456i
\(714\) 0 0
\(715\) − 7.00787i − 0.262079i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −9.17988 −0.342352 −0.171176 0.985240i \(-0.554757\pi\)
−0.171176 + 0.985240i \(0.554757\pi\)
\(720\) 0 0
\(721\) −1.80019 −0.0670428
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 2.09637i 0.0778574i
\(726\) 0 0
\(727\) 27.9032i 1.03487i 0.855722 + 0.517436i \(0.173114\pi\)
−0.855722 + 0.517436i \(0.826886\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 39.4913 1.46064
\(732\) 0 0
\(733\) −27.7402 −1.02461 −0.512304 0.858804i \(-0.671208\pi\)
−0.512304 + 0.858804i \(0.671208\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 10.8260i − 0.398780i
\(738\) 0 0
\(739\) 24.0426i 0.884421i 0.896911 + 0.442210i \(0.145806\pi\)
−0.896911 + 0.442210i \(0.854194\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 38.7022 1.41985 0.709923 0.704279i \(-0.248730\pi\)
0.709923 + 0.704279i \(0.248730\pi\)
\(744\) 0 0
\(745\) −9.12590 −0.334347
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 0.0993527i − 0.00363027i
\(750\) 0 0
\(751\) 17.9337i 0.654410i 0.944953 + 0.327205i \(0.106107\pi\)
−0.944953 + 0.327205i \(0.893893\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 6.78850 0.247059
\(756\) 0 0
\(757\) 32.6326 1.18605 0.593026 0.805184i \(-0.297933\pi\)
0.593026 + 0.805184i \(0.297933\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 52.2699i − 1.89478i −0.320078 0.947391i \(-0.603709\pi\)
0.320078 0.947391i \(-0.396291\pi\)
\(762\) 0 0
\(763\) 3.40635i 0.123318i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 5.84007 0.210873
\(768\) 0 0
\(769\) 14.8516 0.535563 0.267782 0.963480i \(-0.413709\pi\)
0.267782 + 0.963480i \(0.413709\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 22.6513i 0.814709i 0.913270 + 0.407355i \(0.133549\pi\)
−0.913270 + 0.407355i \(0.866451\pi\)
\(774\) 0 0
\(775\) 13.1742i 0.473233i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −12.7350 −0.456278
\(780\) 0 0
\(781\) −12.4577 −0.445770
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 42.2586i 1.50827i
\(786\) 0 0
\(787\) 18.0749i 0.644300i 0.946689 + 0.322150i \(0.104406\pi\)
−0.946689 + 0.322150i \(0.895594\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −11.2092 −0.398553
\(792\) 0 0
\(793\) −45.3247 −1.60953
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 47.1587i − 1.67045i −0.549911 0.835223i \(-0.685338\pi\)
0.549911 0.835223i \(-0.314662\pi\)
\(798\) 0 0
\(799\) 66.2755i 2.34466i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −0.363366 −0.0128229
\(804\) 0 0
\(805\) −5.73205 −0.202028
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 48.6123i 1.70912i 0.519354 + 0.854559i \(0.326172\pi\)
−0.519354 + 0.854559i \(0.673828\pi\)
\(810\) 0 0
\(811\) − 43.2254i − 1.51785i −0.651179 0.758924i \(-0.725725\pi\)
0.651179 0.758924i \(-0.274275\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −22.3831 −0.784046
\(816\) 0 0
\(817\) −8.90994 −0.311719
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 35.5347i 1.24017i 0.784535 + 0.620085i \(0.212902\pi\)
−0.784535 + 0.620085i \(0.787098\pi\)
\(822\) 0 0
\(823\) 1.90023i 0.0662379i 0.999451 + 0.0331189i \(0.0105440\pi\)
−0.999451 + 0.0331189i \(0.989456\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −41.1105 −1.42955 −0.714775 0.699354i \(-0.753471\pi\)
−0.714775 + 0.699354i \(0.753471\pi\)
\(828\) 0 0
\(829\) −10.1265 −0.351707 −0.175853 0.984416i \(-0.556268\pi\)
−0.175853 + 0.984416i \(0.556268\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 5.93127i − 0.205506i
\(834\) 0 0
\(835\) 5.31207i 0.183832i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −46.5726 −1.60786 −0.803932 0.594721i \(-0.797263\pi\)
−0.803932 + 0.594721i \(0.797263\pi\)
\(840\) 0 0
\(841\) 26.2664 0.905738
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 5.40549i 0.185954i
\(846\) 0 0
\(847\) 9.71014i 0.333644i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1.59634 0.0547217
\(852\) 0 0
\(853\) 23.1228 0.791710 0.395855 0.918313i \(-0.370448\pi\)
0.395855 + 0.918313i \(0.370448\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 16.9208i 0.578004i 0.957329 + 0.289002i \(0.0933234\pi\)
−0.957329 + 0.289002i \(0.906677\pi\)
\(858\) 0 0
\(859\) − 37.0925i − 1.26558i −0.774324 0.632789i \(-0.781910\pi\)
0.774324 0.632789i \(-0.218090\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 24.0782 0.819632 0.409816 0.912168i \(-0.365593\pi\)
0.409816 + 0.912168i \(0.365593\pi\)
\(864\) 0 0
\(865\) 1.72629 0.0586956
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 14.7036i − 0.498784i
\(870\) 0 0
\(871\) − 30.4464i − 1.03164i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −12.1087 −0.409350
\(876\) 0 0
\(877\) 14.9708 0.505527 0.252764 0.967528i \(-0.418660\pi\)
0.252764 + 0.967528i \(0.418660\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 36.4601i 1.22837i 0.789162 + 0.614186i \(0.210515\pi\)
−0.789162 + 0.614186i \(0.789485\pi\)
\(882\) 0 0
\(883\) 9.37770i 0.315585i 0.987472 + 0.157792i \(0.0504377\pi\)
−0.987472 + 0.157792i \(0.949562\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −51.8028 −1.73937 −0.869684 0.493609i \(-0.835677\pi\)
−0.869684 + 0.493609i \(0.835677\pi\)
\(888\) 0 0
\(889\) −12.1180 −0.406426
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 14.9530i − 0.500382i
\(894\) 0 0
\(895\) − 28.5228i − 0.953413i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −17.1787 −0.572942
\(900\) 0 0
\(901\) 10.0157 0.333673
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 31.2774i − 1.03970i
\(906\) 0 0
\(907\) 15.0718i 0.500451i 0.968188 + 0.250225i \(0.0805047\pi\)
−0.968188 + 0.250225i \(0.919495\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 57.0842 1.89128 0.945642 0.325209i \(-0.105435\pi\)
0.945642 + 0.325209i \(0.105435\pi\)
\(912\) 0 0
\(913\) −3.68612 −0.121993
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 1.54884i − 0.0511472i
\(918\) 0 0
\(919\) − 2.23843i − 0.0738391i −0.999318 0.0369195i \(-0.988245\pi\)
0.999318 0.0369195i \(-0.0117545\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −35.0353 −1.15320
\(924\) 0 0
\(925\) 0.682166 0.0224295
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 25.4145i − 0.833824i −0.908947 0.416912i \(-0.863112\pi\)
0.908947 0.416912i \(-0.136888\pi\)
\(930\) 0 0
\(931\) 1.33820i 0.0438578i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −13.0135 −0.425586
\(936\) 0 0
\(937\) −38.0906 −1.24437 −0.622183 0.782872i \(-0.713754\pi\)
−0.622183 + 0.782872i \(0.713754\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 58.5266i 1.90791i 0.299943 + 0.953957i \(0.403032\pi\)
−0.299943 + 0.953957i \(0.596968\pi\)
\(942\) 0 0
\(943\) − 28.2366i − 0.919511i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −34.7722 −1.12995 −0.564973 0.825109i \(-0.691113\pi\)
−0.564973 + 0.825109i \(0.691113\pi\)
\(948\) 0 0
\(949\) −1.02191 −0.0331727
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 16.7143i − 0.541430i −0.962660 0.270715i \(-0.912740\pi\)
0.962660 0.270715i \(-0.0872601\pi\)
\(954\) 0 0
\(955\) − 17.7862i − 0.575546i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −0.895484 −0.0289167
\(960\) 0 0
\(961\) −76.9562 −2.48246
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 15.5512i − 0.500610i
\(966\) 0 0
\(967\) − 52.2257i − 1.67946i −0.543001 0.839732i \(-0.682712\pi\)
0.543001 0.839732i \(-0.317288\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −25.7723 −0.827073 −0.413536 0.910488i \(-0.635707\pi\)
−0.413536 + 0.910488i \(0.635707\pi\)
\(972\) 0 0
\(973\) −13.9021 −0.445680
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 12.2747i − 0.392701i −0.980534 0.196351i \(-0.937091\pi\)
0.980534 0.196351i \(-0.0629090\pi\)
\(978\) 0 0
\(979\) 15.6299i 0.499534i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 40.5111 1.29210 0.646052 0.763293i \(-0.276419\pi\)
0.646052 + 0.763293i \(0.276419\pi\)
\(984\) 0 0
\(985\) 46.2330 1.47310
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 19.7556i − 0.628191i
\(990\) 0 0
\(991\) 15.4002i 0.489203i 0.969624 + 0.244601i \(0.0786571\pi\)
−0.969624 + 0.244601i \(0.921343\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 22.5218 0.713989
\(996\) 0 0
\(997\) −26.6284 −0.843329 −0.421664 0.906752i \(-0.638554\pi\)
−0.421664 + 0.906752i \(0.638554\pi\)
\(998\) 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.h.h.2591.1 16
3.2 odd 2 inner 6048.2.h.h.2591.14 yes 16
4.3 odd 2 inner 6048.2.h.h.2591.4 yes 16
12.11 even 2 inner 6048.2.h.h.2591.15 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.h.h.2591.1 16 1.1 even 1 trivial
6048.2.h.h.2591.4 yes 16 4.3 odd 2 inner
6048.2.h.h.2591.14 yes 16 3.2 odd 2 inner
6048.2.h.h.2591.15 yes 16 12.11 even 2 inner