Properties

Label 6048.2.h.h.2591.6
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 2591.6
Root \(1.85032 + 1.85032i\)
Character \(\chi\) = 6048.2591
Dual form 6048.2.h.h.2591.12

$q$-expansion

\(f(q)\) \(=\) \(q-0.517638i q^{5} -1.00000i q^{7} +O(q^{10})\) \(q-0.517638i q^{5} -1.00000i q^{7} +5.63249 q^{11} +1.91559 q^{13} +1.29484i q^{17} -4.23349i q^{19} +4.38134 q^{23} +4.73205 q^{25} -3.70064i q^{29} -5.11174i q^{31} -0.517638 q^{35} +2.18354 q^{37} +1.88816i q^{41} +5.37969i q^{43} -8.32221 q^{47} -1.00000 q^{49} -5.53748i q^{53} -2.91559i q^{55} -2.54599 q^{59} -0.0617920 q^{61} -0.991583i q^{65} +7.31790i q^{67} +13.4036 q^{71} -14.6132 q^{73} -5.63249i q^{77} +16.9185i q^{79} +8.59962 q^{83} +0.670259 q^{85} -1.33704i q^{89} -1.91559i q^{91} -2.19142 q^{95} +5.23349 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q + 8q^{13} + 48q^{25} + 40q^{37} - 16q^{49} - 56q^{73} - 16q^{85} - 16q^{97} + O(q^{100}) \)

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
\(5\) − 0.517638i − 0.231495i −0.993279 0.115747i \(-0.963074\pi\)
0.993279 0.115747i \(-0.0369263\pi\)
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.63249 1.69826 0.849130 0.528185i \(-0.177127\pi\)
0.849130 + 0.528185i \(0.177127\pi\)
\(12\) 0 0
\(13\) 1.91559 0.531289 0.265645 0.964071i \(-0.414415\pi\)
0.265645 + 0.964071i \(0.414415\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.29484i 0.314045i 0.987595 + 0.157022i \(0.0501895\pi\)
−0.987595 + 0.157022i \(0.949810\pi\)
\(18\) 0 0
\(19\) − 4.23349i − 0.971229i −0.874173 0.485615i \(-0.838596\pi\)
0.874173 0.485615i \(-0.161404\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.38134 0.913573 0.456786 0.889576i \(-0.349000\pi\)
0.456786 + 0.889576i \(0.349000\pi\)
\(24\) 0 0
\(25\) 4.73205 0.946410
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 3.70064i − 0.687191i −0.939118 0.343595i \(-0.888355\pi\)
0.939118 0.343595i \(-0.111645\pi\)
\(30\) 0 0
\(31\) − 5.11174i − 0.918096i −0.888411 0.459048i \(-0.848191\pi\)
0.888411 0.459048i \(-0.151809\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.517638 −0.0874968
\(36\) 0 0
\(37\) 2.18354 0.358972 0.179486 0.983761i \(-0.442557\pi\)
0.179486 + 0.983761i \(0.442557\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.88816i 0.294881i 0.989071 + 0.147440i \(0.0471035\pi\)
−0.989071 + 0.147440i \(0.952897\pi\)
\(42\) 0 0
\(43\) 5.37969i 0.820395i 0.911997 + 0.410198i \(0.134540\pi\)
−0.911997 + 0.410198i \(0.865460\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.32221 −1.21392 −0.606960 0.794732i \(-0.707611\pi\)
−0.606960 + 0.794732i \(0.707611\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 5.53748i − 0.760632i −0.924857 0.380316i \(-0.875815\pi\)
0.924857 0.380316i \(-0.124185\pi\)
\(54\) 0 0
\(55\) − 2.91559i − 0.393138i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.54599 −0.331459 −0.165730 0.986171i \(-0.552998\pi\)
−0.165730 + 0.986171i \(0.552998\pi\)
\(60\) 0 0
\(61\) −0.0617920 −0.00791166 −0.00395583 0.999992i \(-0.501259\pi\)
−0.00395583 + 0.999992i \(0.501259\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 0.991583i − 0.122991i
\(66\) 0 0
\(67\) 7.31790i 0.894024i 0.894528 + 0.447012i \(0.147512\pi\)
−0.894528 + 0.447012i \(0.852488\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 13.4036 1.59071 0.795357 0.606142i \(-0.207284\pi\)
0.795357 + 0.606142i \(0.207284\pi\)
\(72\) 0 0
\(73\) −14.6132 −1.71034 −0.855172 0.518345i \(-0.826548\pi\)
−0.855172 + 0.518345i \(0.826548\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 5.63249i − 0.641882i
\(78\) 0 0
\(79\) 16.9185i 1.90348i 0.306911 + 0.951738i \(0.400705\pi\)
−0.306911 + 0.951738i \(0.599295\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 8.59962 0.943931 0.471965 0.881617i \(-0.343545\pi\)
0.471965 + 0.881617i \(0.343545\pi\)
\(84\) 0 0
\(85\) 0.670259 0.0726998
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 1.33704i − 0.141726i −0.997486 0.0708632i \(-0.977425\pi\)
0.997486 0.0708632i \(-0.0225754\pi\)
\(90\) 0 0
\(91\) − 1.91559i − 0.200808i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.19142 −0.224835
\(96\) 0 0
\(97\) 5.23349 0.531380 0.265690 0.964058i \(-0.414400\pi\)
0.265690 + 0.964058i \(0.414400\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 6.51995i − 0.648760i −0.945927 0.324380i \(-0.894844\pi\)
0.945927 0.324380i \(-0.105156\pi\)
\(102\) 0 0
\(103\) − 3.04995i − 0.300521i −0.988646 0.150260i \(-0.951989\pi\)
0.988646 0.150260i \(-0.0480112\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0.705070 0.0681617 0.0340809 0.999419i \(-0.489150\pi\)
0.0340809 + 0.999419i \(0.489150\pi\)
\(108\) 0 0
\(109\) −11.0155 −1.05509 −0.527546 0.849526i \(-0.676888\pi\)
−0.527546 + 0.849526i \(0.676888\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 0.915903i − 0.0861609i −0.999072 0.0430805i \(-0.986283\pi\)
0.999072 0.0430805i \(-0.0137172\pi\)
\(114\) 0 0
\(115\) − 2.26795i − 0.211487i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1.29484 0.118698
\(120\) 0 0
\(121\) 20.7249 1.88408
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 5.03768i − 0.450584i
\(126\) 0 0
\(127\) − 3.71733i − 0.329860i −0.986305 0.164930i \(-0.947260\pi\)
0.986305 0.164930i \(-0.0527398\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 8.52897 0.745180 0.372590 0.927996i \(-0.378470\pi\)
0.372590 + 0.927996i \(0.378470\pi\)
\(132\) 0 0
\(133\) −4.23349 −0.367090
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 6.85520i − 0.585679i −0.956162 0.292840i \(-0.905400\pi\)
0.956162 0.292840i \(-0.0946002\pi\)
\(138\) 0 0
\(139\) 16.3600i 1.38763i 0.720152 + 0.693817i \(0.244072\pi\)
−0.720152 + 0.693817i \(0.755928\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 10.7895 0.902267
\(144\) 0 0
\(145\) −1.91559 −0.159081
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1.34764i 0.110403i 0.998475 + 0.0552017i \(0.0175802\pi\)
−0.998475 + 0.0552017i \(0.982420\pi\)
\(150\) 0 0
\(151\) − 16.5288i − 1.34509i −0.740055 0.672546i \(-0.765201\pi\)
0.740055 0.672546i \(-0.234799\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2.64603 −0.212534
\(156\) 0 0
\(157\) −11.9903 −0.956927 −0.478464 0.878107i \(-0.658806\pi\)
−0.478464 + 0.878107i \(0.658806\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 4.38134i − 0.345298i
\(162\) 0 0
\(163\) 14.3079i 1.12068i 0.828262 + 0.560340i \(0.189330\pi\)
−0.828262 + 0.560340i \(0.810670\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 15.8378 1.22557 0.612784 0.790251i \(-0.290050\pi\)
0.612784 + 0.790251i \(0.290050\pi\)
\(168\) 0 0
\(169\) −9.33051 −0.717732
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 10.8139i − 0.822166i −0.911598 0.411083i \(-0.865151\pi\)
0.911598 0.411083i \(-0.134849\pi\)
\(174\) 0 0
\(175\) − 4.73205i − 0.357709i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 25.7053 1.92131 0.960653 0.277750i \(-0.0895887\pi\)
0.960653 + 0.277750i \(0.0895887\pi\)
\(180\) 0 0
\(181\) 1.93821 0.144066 0.0720329 0.997402i \(-0.477051\pi\)
0.0720329 + 0.997402i \(0.477051\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 1.13028i − 0.0831001i
\(186\) 0 0
\(187\) 7.29317i 0.533330i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −18.0780 −1.30808 −0.654038 0.756462i \(-0.726926\pi\)
−0.654038 + 0.756462i \(0.726926\pi\)
\(192\) 0 0
\(193\) −5.06467 −0.364563 −0.182282 0.983246i \(-0.558348\pi\)
−0.182282 + 0.983246i \(0.558348\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.12266i 0.0799864i 0.999200 + 0.0399932i \(0.0127336\pi\)
−0.999200 + 0.0399932i \(0.987266\pi\)
\(198\) 0 0
\(199\) − 0.379692i − 0.0269157i −0.999909 0.0134578i \(-0.995716\pi\)
0.999909 0.0134578i \(-0.00428389\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −3.70064 −0.259734
\(204\) 0 0
\(205\) 0.977383 0.0682634
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 23.8451i − 1.64940i
\(210\) 0 0
\(211\) − 9.43175i − 0.649309i −0.945833 0.324654i \(-0.894752\pi\)
0.945833 0.324654i \(-0.105248\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 2.78473 0.189917
\(216\) 0 0
\(217\) −5.11174 −0.347008
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2.48038i 0.166849i
\(222\) 0 0
\(223\) − 20.2490i − 1.35597i −0.735075 0.677986i \(-0.762853\pi\)
0.735075 0.677986i \(-0.237147\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −5.26008 −0.349124 −0.174562 0.984646i \(-0.555851\pi\)
−0.174562 + 0.984646i \(0.555851\pi\)
\(228\) 0 0
\(229\) −5.30241 −0.350393 −0.175196 0.984533i \(-0.556056\pi\)
−0.175196 + 0.984533i \(0.556056\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 27.5870i − 1.80728i −0.428291 0.903641i \(-0.640884\pi\)
0.428291 0.903641i \(-0.359116\pi\)
\(234\) 0 0
\(235\) 4.30790i 0.281016i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 20.6134 1.33337 0.666684 0.745341i \(-0.267713\pi\)
0.666684 + 0.745341i \(0.267713\pi\)
\(240\) 0 0
\(241\) 9.31078 0.599760 0.299880 0.953977i \(-0.403053\pi\)
0.299880 + 0.953977i \(0.403053\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0.517638i 0.0330707i
\(246\) 0 0
\(247\) − 8.10963i − 0.516004i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −10.0783 −0.636139 −0.318070 0.948067i \(-0.603035\pi\)
−0.318070 + 0.948067i \(0.603035\pi\)
\(252\) 0 0
\(253\) 24.6779 1.55148
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 27.9029i − 1.74053i −0.492581 0.870267i \(-0.663946\pi\)
0.492581 0.870267i \(-0.336054\pi\)
\(258\) 0 0
\(259\) − 2.18354i − 0.135679i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −10.9891 −0.677615 −0.338807 0.940856i \(-0.610024\pi\)
−0.338807 + 0.940856i \(0.610024\pi\)
\(264\) 0 0
\(265\) −2.86641 −0.176082
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 26.2550i 1.60079i 0.599471 + 0.800396i \(0.295378\pi\)
−0.599471 + 0.800396i \(0.704622\pi\)
\(270\) 0 0
\(271\) − 25.1517i − 1.52786i −0.645301 0.763928i \(-0.723268\pi\)
0.645301 0.763928i \(-0.276732\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 26.6532 1.60725
\(276\) 0 0
\(277\) 3.58585 0.215453 0.107726 0.994181i \(-0.465643\pi\)
0.107726 + 0.994181i \(0.465643\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 5.70055i − 0.340066i −0.985438 0.170033i \(-0.945613\pi\)
0.985438 0.170033i \(-0.0543875\pi\)
\(282\) 0 0
\(283\) 5.86353i 0.348551i 0.984697 + 0.174275i \(0.0557583\pi\)
−0.984697 + 0.174275i \(0.944242\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.88816 0.111454
\(288\) 0 0
\(289\) 15.3234 0.901376
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 7.50281i 0.438319i 0.975689 + 0.219159i \(0.0703315\pi\)
−0.975689 + 0.219159i \(0.929669\pi\)
\(294\) 0 0
\(295\) 1.31790i 0.0767311i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 8.39286 0.485371
\(300\) 0 0
\(301\) 5.37969 0.310080
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0.0319859i 0.00183151i
\(306\) 0 0
\(307\) 26.5569i 1.51568i 0.652440 + 0.757841i \(0.273746\pi\)
−0.652440 + 0.757841i \(0.726254\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 33.2656 1.88632 0.943160 0.332340i \(-0.107838\pi\)
0.943160 + 0.332340i \(0.107838\pi\)
\(312\) 0 0
\(313\) −32.4825 −1.83602 −0.918009 0.396560i \(-0.870204\pi\)
−0.918009 + 0.396560i \(0.870204\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 25.7282i − 1.44504i −0.691350 0.722520i \(-0.742984\pi\)
0.691350 0.722520i \(-0.257016\pi\)
\(318\) 0 0
\(319\) − 20.8438i − 1.16703i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 5.48170 0.305010
\(324\) 0 0
\(325\) 9.06467 0.502818
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 8.32221i 0.458819i
\(330\) 0 0
\(331\) 12.0999i 0.665071i 0.943091 + 0.332535i \(0.107904\pi\)
−0.943091 + 0.332535i \(0.892096\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 3.78802 0.206962
\(336\) 0 0
\(337\) 23.3305 1.27089 0.635447 0.772144i \(-0.280816\pi\)
0.635447 + 0.772144i \(0.280816\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 28.7918i − 1.55917i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 13.3071 0.714362 0.357181 0.934035i \(-0.383738\pi\)
0.357181 + 0.934035i \(0.383738\pi\)
\(348\) 0 0
\(349\) 25.9311 1.38806 0.694030 0.719947i \(-0.255834\pi\)
0.694030 + 0.719947i \(0.255834\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 12.3596i − 0.657834i −0.944359 0.328917i \(-0.893316\pi\)
0.944359 0.328917i \(-0.106684\pi\)
\(354\) 0 0
\(355\) − 6.93821i − 0.368242i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −4.55203 −0.240247 −0.120123 0.992759i \(-0.538329\pi\)
−0.120123 + 0.992759i \(0.538329\pi\)
\(360\) 0 0
\(361\) 1.07756 0.0567135
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 7.56434i 0.395936i
\(366\) 0 0
\(367\) − 17.4696i − 0.911905i −0.890004 0.455953i \(-0.849299\pi\)
0.890004 0.455953i \(-0.150701\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −5.53748 −0.287492
\(372\) 0 0
\(373\) −17.1714 −0.889103 −0.444551 0.895753i \(-0.646637\pi\)
−0.444551 + 0.895753i \(0.646637\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 7.08890i − 0.365097i
\(378\) 0 0
\(379\) 11.4049i 0.585831i 0.956138 + 0.292916i \(0.0946255\pi\)
−0.956138 + 0.292916i \(0.905374\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −10.0616 −0.514124 −0.257062 0.966395i \(-0.582754\pi\)
−0.257062 + 0.966395i \(0.582754\pi\)
\(384\) 0 0
\(385\) −2.91559 −0.148592
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 6.01497i − 0.304971i −0.988306 0.152486i \(-0.951272\pi\)
0.988306 0.152486i \(-0.0487278\pi\)
\(390\) 0 0
\(391\) 5.67314i 0.286903i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 8.75764 0.440645
\(396\) 0 0
\(397\) 16.1940 0.812756 0.406378 0.913705i \(-0.366792\pi\)
0.406378 + 0.913705i \(0.366792\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 14.2427i 0.711247i 0.934629 + 0.355623i \(0.115731\pi\)
−0.934629 + 0.355623i \(0.884269\pi\)
\(402\) 0 0
\(403\) − 9.79201i − 0.487775i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 12.2988 0.609627
\(408\) 0 0
\(409\) −15.9703 −0.789678 −0.394839 0.918750i \(-0.629200\pi\)
−0.394839 + 0.918750i \(0.629200\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 2.54599i 0.125280i
\(414\) 0 0
\(415\) − 4.45149i − 0.218515i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −23.6846 −1.15707 −0.578534 0.815658i \(-0.696375\pi\)
−0.578534 + 0.815658i \(0.696375\pi\)
\(420\) 0 0
\(421\) −16.4451 −0.801487 −0.400743 0.916190i \(-0.631248\pi\)
−0.400743 + 0.916190i \(0.631248\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 6.12725i 0.297215i
\(426\) 0 0
\(427\) 0.0617920i 0.00299032i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0.963144 0.0463930 0.0231965 0.999731i \(-0.492616\pi\)
0.0231965 + 0.999731i \(0.492616\pi\)
\(432\) 0 0
\(433\) −5.16197 −0.248068 −0.124034 0.992278i \(-0.539583\pi\)
−0.124034 + 0.992278i \(0.539583\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 18.5484i − 0.887289i
\(438\) 0 0
\(439\) 11.3700i 0.542659i 0.962487 + 0.271329i \(0.0874632\pi\)
−0.962487 + 0.271329i \(0.912537\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −16.2690 −0.772965 −0.386483 0.922297i \(-0.626310\pi\)
−0.386483 + 0.922297i \(0.626310\pi\)
\(444\) 0 0
\(445\) −0.692105 −0.0328089
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 24.4105i 1.15200i 0.817449 + 0.576001i \(0.195388\pi\)
−0.817449 + 0.576001i \(0.804612\pi\)
\(450\) 0 0
\(451\) 10.6350i 0.500784i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −0.991583 −0.0464861
\(456\) 0 0
\(457\) −25.5834 −1.19674 −0.598371 0.801219i \(-0.704185\pi\)
−0.598371 + 0.801219i \(0.704185\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 37.2950i − 1.73700i −0.495690 0.868500i \(-0.665085\pi\)
0.495690 0.868500i \(-0.334915\pi\)
\(462\) 0 0
\(463\) − 11.6876i − 0.543168i −0.962415 0.271584i \(-0.912452\pi\)
0.962415 0.271584i \(-0.0875475\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 18.2812 0.845952 0.422976 0.906141i \(-0.360985\pi\)
0.422976 + 0.906141i \(0.360985\pi\)
\(468\) 0 0
\(469\) 7.31790 0.337909
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 30.3011i 1.39324i
\(474\) 0 0
\(475\) − 20.0331i − 0.919181i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 24.1230 1.10221 0.551104 0.834436i \(-0.314207\pi\)
0.551104 + 0.834436i \(0.314207\pi\)
\(480\) 0 0
\(481\) 4.18277 0.190718
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 2.70905i − 0.123012i
\(486\) 0 0
\(487\) 10.2209i 0.463152i 0.972817 + 0.231576i \(0.0743882\pi\)
−0.972817 + 0.231576i \(0.925612\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 34.1633 1.54177 0.770883 0.636976i \(-0.219815\pi\)
0.770883 + 0.636976i \(0.219815\pi\)
\(492\) 0 0
\(493\) 4.79173 0.215809
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 13.4036i − 0.601233i
\(498\) 0 0
\(499\) 10.9745i 0.491286i 0.969360 + 0.245643i \(0.0789991\pi\)
−0.969360 + 0.245643i \(0.921001\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 7.49370 0.334128 0.167064 0.985946i \(-0.446571\pi\)
0.167064 + 0.985946i \(0.446571\pi\)
\(504\) 0 0
\(505\) −3.37498 −0.150184
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 30.1521i 1.33647i 0.743950 + 0.668235i \(0.232950\pi\)
−0.743950 + 0.668235i \(0.767050\pi\)
\(510\) 0 0
\(511\) 14.6132i 0.646449i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1.57877 −0.0695690
\(516\) 0 0
\(517\) −46.8748 −2.06155
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.48731i 0.0651603i 0.999469 + 0.0325801i \(0.0103724\pi\)
−0.999469 + 0.0325801i \(0.989628\pi\)
\(522\) 0 0
\(523\) 26.1372i 1.14290i 0.820636 + 0.571451i \(0.193619\pi\)
−0.820636 + 0.571451i \(0.806381\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6.61889 0.288323
\(528\) 0 0
\(529\) −3.80385 −0.165385
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 3.61694i 0.156667i
\(534\) 0 0
\(535\) − 0.364971i − 0.0157791i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −5.63249 −0.242608
\(540\) 0 0
\(541\) −15.0794 −0.648314 −0.324157 0.946003i \(-0.605081\pi\)
−0.324157 + 0.946003i \(0.605081\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 5.70204i 0.244248i
\(546\) 0 0
\(547\) 31.6481i 1.35318i 0.736362 + 0.676588i \(0.236542\pi\)
−0.736362 + 0.676588i \(0.763458\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −15.6666 −0.667420
\(552\) 0 0
\(553\) 16.9185 0.719447
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 21.8208i 0.924577i 0.886729 + 0.462289i \(0.152972\pi\)
−0.886729 + 0.462289i \(0.847028\pi\)
\(558\) 0 0
\(559\) 10.3053i 0.435867i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 27.4121 1.15528 0.577640 0.816291i \(-0.303974\pi\)
0.577640 + 0.816291i \(0.303974\pi\)
\(564\) 0 0
\(565\) −0.474106 −0.0199458
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 19.8376i − 0.831637i −0.909448 0.415819i \(-0.863495\pi\)
0.909448 0.415819i \(-0.136505\pi\)
\(570\) 0 0
\(571\) 15.0152i 0.628367i 0.949362 + 0.314184i \(0.101731\pi\)
−0.949362 + 0.314184i \(0.898269\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 20.7327 0.864615
\(576\) 0 0
\(577\) −6.72994 −0.280171 −0.140086 0.990139i \(-0.544738\pi\)
−0.140086 + 0.990139i \(0.544738\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 8.59962i − 0.356772i
\(582\) 0 0
\(583\) − 31.1898i − 1.29175i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −45.5541 −1.88022 −0.940110 0.340870i \(-0.889278\pi\)
−0.940110 + 0.340870i \(0.889278\pi\)
\(588\) 0 0
\(589\) −21.6405 −0.891682
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 21.9300i 0.900556i 0.892888 + 0.450278i \(0.148675\pi\)
−0.892888 + 0.450278i \(0.851325\pi\)
\(594\) 0 0
\(595\) − 0.670259i − 0.0274779i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −15.2008 −0.621088 −0.310544 0.950559i \(-0.600511\pi\)
−0.310544 + 0.950559i \(0.600511\pi\)
\(600\) 0 0
\(601\) 36.2519 1.47874 0.739372 0.673297i \(-0.235122\pi\)
0.739372 + 0.673297i \(0.235122\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 10.7280i − 0.436156i
\(606\) 0 0
\(607\) 15.8693i 0.644115i 0.946720 + 0.322057i \(0.104374\pi\)
−0.946720 + 0.322057i \(0.895626\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −15.9420 −0.644943
\(612\) 0 0
\(613\) 3.91984 0.158321 0.0791603 0.996862i \(-0.474776\pi\)
0.0791603 + 0.996862i \(0.474776\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 47.6851i 1.91973i 0.280460 + 0.959866i \(0.409513\pi\)
−0.280460 + 0.959866i \(0.590487\pi\)
\(618\) 0 0
\(619\) 35.7867i 1.43839i 0.694808 + 0.719195i \(0.255489\pi\)
−0.694808 + 0.719195i \(0.744511\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −1.33704 −0.0535675
\(624\) 0 0
\(625\) 21.0526 0.842102
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 2.82734i 0.112733i
\(630\) 0 0
\(631\) − 23.1743i − 0.922555i −0.887256 0.461277i \(-0.847391\pi\)
0.887256 0.461277i \(-0.152609\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1.92423 −0.0763608
\(636\) 0 0
\(637\) −1.91559 −0.0758985
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 28.2675i − 1.11650i −0.829673 0.558250i \(-0.811473\pi\)
0.829673 0.558250i \(-0.188527\pi\)
\(642\) 0 0
\(643\) 19.6610i 0.775355i 0.921795 + 0.387678i \(0.126723\pi\)
−0.921795 + 0.387678i \(0.873277\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −10.3608 −0.407324 −0.203662 0.979041i \(-0.565284\pi\)
−0.203662 + 0.979041i \(0.565284\pi\)
\(648\) 0 0
\(649\) −14.3402 −0.562904
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 33.9610i − 1.32900i −0.747290 0.664498i \(-0.768645\pi\)
0.747290 0.664498i \(-0.231355\pi\)
\(654\) 0 0
\(655\) − 4.41492i − 0.172505i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0.0280540 0.00109283 0.000546414 1.00000i \(-0.499826\pi\)
0.000546414 1.00000i \(0.499826\pi\)
\(660\) 0 0
\(661\) 17.4472 0.678619 0.339310 0.940675i \(-0.389807\pi\)
0.339310 + 0.940675i \(0.389807\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2.19142i 0.0849795i
\(666\) 0 0
\(667\) − 16.2138i − 0.627799i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −0.348043 −0.0134360
\(672\) 0 0
\(673\) −5.96765 −0.230036 −0.115018 0.993363i \(-0.536693\pi\)
−0.115018 + 0.993363i \(0.536693\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 46.7977i − 1.79858i −0.437352 0.899290i \(-0.644084\pi\)
0.437352 0.899290i \(-0.355916\pi\)
\(678\) 0 0
\(679\) − 5.23349i − 0.200843i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 32.5620 1.24595 0.622976 0.782241i \(-0.285924\pi\)
0.622976 + 0.782241i \(0.285924\pi\)
\(684\) 0 0
\(685\) −3.54851 −0.135582
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 10.6075i − 0.404115i
\(690\) 0 0
\(691\) 19.5022i 0.741899i 0.928653 + 0.370950i \(0.120968\pi\)
−0.928653 + 0.370950i \(0.879032\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 8.46854 0.321230
\(696\) 0 0
\(697\) −2.44486 −0.0926058
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 2.54301i 0.0960480i 0.998846 + 0.0480240i \(0.0152924\pi\)
−0.998846 + 0.0480240i \(0.984708\pi\)
\(702\) 0 0
\(703\) − 9.24400i − 0.348644i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −6.51995 −0.245208
\(708\) 0 0
\(709\) −25.9237 −0.973584 −0.486792 0.873518i \(-0.661833\pi\)
−0.486792 + 0.873518i \(0.661833\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 22.3963i − 0.838748i
\(714\) 0 0
\(715\) − 5.58508i − 0.208870i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −17.3623 −0.647504 −0.323752 0.946142i \(-0.604944\pi\)
−0.323752 + 0.946142i \(0.604944\pi\)
\(720\) 0 0
\(721\) −3.04995 −0.113586
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 17.5116i − 0.650365i
\(726\) 0 0
\(727\) − 41.7746i − 1.54933i −0.632369 0.774667i \(-0.717917\pi\)
0.632369 0.774667i \(-0.282083\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −6.96584 −0.257641
\(732\) 0 0
\(733\) 48.0630 1.77525 0.887624 0.460568i \(-0.152354\pi\)
0.887624 + 0.460568i \(0.152354\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 41.2180i 1.51828i
\(738\) 0 0
\(739\) − 7.35707i − 0.270634i −0.990802 0.135317i \(-0.956795\pi\)
0.990802 0.135317i \(-0.0432053\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −8.38155 −0.307489 −0.153745 0.988111i \(-0.549133\pi\)
−0.153745 + 0.988111i \(0.549133\pi\)
\(744\) 0 0
\(745\) 0.697592 0.0255578
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 0.705070i − 0.0257627i
\(750\) 0 0
\(751\) − 43.2713i − 1.57899i −0.613756 0.789496i \(-0.710342\pi\)
0.613756 0.789496i \(-0.289658\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −8.55592 −0.311382
\(756\) 0 0
\(757\) −15.5188 −0.564039 −0.282020 0.959409i \(-0.591004\pi\)
−0.282020 + 0.959409i \(0.591004\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 38.9572i − 1.41220i −0.708114 0.706098i \(-0.750454\pi\)
0.708114 0.706098i \(-0.249546\pi\)
\(762\) 0 0
\(763\) 11.0155i 0.398788i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −4.87707 −0.176101
\(768\) 0 0
\(769\) 17.4120 0.627894 0.313947 0.949440i \(-0.398349\pi\)
0.313947 + 0.949440i \(0.398349\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 22.6813i 0.815791i 0.913029 + 0.407895i \(0.133737\pi\)
−0.913029 + 0.407895i \(0.866263\pi\)
\(774\) 0 0
\(775\) − 24.1890i − 0.868896i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 7.99350 0.286397
\(780\) 0 0
\(781\) 75.4956 2.70144
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 6.20662i 0.221524i
\(786\) 0 0
\(787\) 32.8974i 1.17267i 0.810070 + 0.586333i \(0.199429\pi\)
−0.810070 + 0.586333i \(0.800571\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −0.915903 −0.0325658
\(792\) 0 0
\(793\) −0.118368 −0.00420338
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 0.904336i − 0.0320332i −0.999872 0.0160166i \(-0.994902\pi\)
0.999872 0.0160166i \(-0.00509847\pi\)
\(798\) 0 0
\(799\) − 10.7759i − 0.381225i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −82.3086 −2.90461
\(804\) 0 0
\(805\) −2.26795 −0.0799347
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 42.7876i − 1.50433i −0.658973 0.752167i \(-0.729009\pi\)
0.658973 0.752167i \(-0.270991\pi\)
\(810\) 0 0
\(811\) − 50.2048i − 1.76293i −0.472251 0.881464i \(-0.656558\pi\)
0.472251 0.881464i \(-0.343442\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 7.40631 0.259432
\(816\) 0 0
\(817\) 22.7749 0.796792
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 18.3031i 0.638782i 0.947623 + 0.319391i \(0.103478\pi\)
−0.947623 + 0.319391i \(0.896522\pi\)
\(822\) 0 0
\(823\) 28.1293i 0.980527i 0.871574 + 0.490264i \(0.163100\pi\)
−0.871574 + 0.490264i \(0.836900\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 30.1059 1.04688 0.523442 0.852062i \(-0.324648\pi\)
0.523442 + 0.852062i \(0.324648\pi\)
\(828\) 0 0
\(829\) 19.4049 0.673961 0.336980 0.941512i \(-0.390594\pi\)
0.336980 + 0.941512i \(0.390594\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 1.29484i − 0.0448636i
\(834\) 0 0
\(835\) − 8.19826i − 0.283713i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −48.3106 −1.66787 −0.833933 0.551865i \(-0.813916\pi\)
−0.833933 + 0.551865i \(0.813916\pi\)
\(840\) 0 0
\(841\) 15.3053 0.527769
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 4.82983i 0.166151i
\(846\) 0 0
\(847\) − 20.7249i − 0.712117i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 9.56683 0.327947
\(852\) 0 0
\(853\) −15.5511 −0.532460 −0.266230 0.963910i \(-0.585778\pi\)
−0.266230 + 0.963910i \(0.585778\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 51.1766i 1.74816i 0.485784 + 0.874079i \(0.338534\pi\)
−0.485784 + 0.874079i \(0.661466\pi\)
\(858\) 0 0
\(859\) 7.42175i 0.253227i 0.991952 + 0.126613i \(0.0404107\pi\)
−0.991952 + 0.126613i \(0.959589\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 25.1405 0.855791 0.427895 0.903828i \(-0.359255\pi\)
0.427895 + 0.903828i \(0.359255\pi\)
\(864\) 0 0
\(865\) −5.59769 −0.190327
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 95.2931i 3.23260i
\(870\) 0 0
\(871\) 14.0181i 0.474985i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −5.03768 −0.170305
\(876\) 0 0
\(877\) −30.2853 −1.02266 −0.511331 0.859384i \(-0.670847\pi\)
−0.511331 + 0.859384i \(0.670847\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 2.81414i − 0.0948107i −0.998876 0.0474054i \(-0.984905\pi\)
0.998876 0.0474054i \(-0.0150952\pi\)
\(882\) 0 0
\(883\) − 20.0928i − 0.676176i −0.941115 0.338088i \(-0.890220\pi\)
0.941115 0.338088i \(-0.109780\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 3.25144 0.109173 0.0545864 0.998509i \(-0.482616\pi\)
0.0545864 + 0.998509i \(0.482616\pi\)
\(888\) 0 0
\(889\) −3.71733 −0.124675
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 35.2320i 1.17899i
\(894\) 0 0
\(895\) − 13.3061i − 0.444773i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −18.9167 −0.630907
\(900\) 0 0
\(901\) 7.17016 0.238873
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 1.00329i − 0.0333505i
\(906\) 0 0
\(907\) 28.9282i 0.960545i 0.877119 + 0.480273i \(0.159462\pi\)
−0.877119 + 0.480273i \(0.840538\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 15.9111 0.527157 0.263578 0.964638i \(-0.415097\pi\)
0.263578 + 0.964638i \(0.415097\pi\)
\(912\) 0 0
\(913\) 48.4372 1.60304
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 8.52897i − 0.281652i
\(918\) 0 0
\(919\) − 31.3628i − 1.03456i −0.855815 0.517282i \(-0.826944\pi\)
0.855815 0.517282i \(-0.173056\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 25.6758 0.845129
\(924\) 0 0
\(925\) 10.3326 0.339734
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 38.5872i − 1.26601i −0.774150 0.633003i \(-0.781822\pi\)
0.774150 0.633003i \(-0.218178\pi\)
\(930\) 0 0
\(931\) 4.23349i 0.138747i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 3.77522 0.123463
\(936\) 0 0
\(937\) −50.0676 −1.63564 −0.817818 0.575477i \(-0.804816\pi\)
−0.817818 + 0.575477i \(0.804816\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 19.0005i 0.619400i 0.950834 + 0.309700i \(0.100229\pi\)
−0.950834 + 0.309700i \(0.899771\pi\)
\(942\) 0 0
\(943\) 8.27267i 0.269395i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −19.5187 −0.634274 −0.317137 0.948380i \(-0.602722\pi\)
−0.317137 + 0.948380i \(0.602722\pi\)
\(948\) 0 0
\(949\) −27.9929 −0.908687
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 31.6208i 1.02430i 0.858896 + 0.512150i \(0.171151\pi\)
−0.858896 + 0.512150i \(0.828849\pi\)
\(954\) 0 0
\(955\) 9.35785i 0.302813i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −6.85520 −0.221366
\(960\) 0 0
\(961\) 4.87008 0.157099
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 2.62167i 0.0843944i
\(966\) 0 0
\(967\) 11.7113i 0.376609i 0.982111 + 0.188305i \(0.0602992\pi\)
−0.982111 + 0.188305i \(0.939701\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −33.5400 −1.07635 −0.538175 0.842833i \(-0.680886\pi\)
−0.538175 + 0.842833i \(0.680886\pi\)
\(972\) 0 0
\(973\) 16.3600 0.524476
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 61.2645i 1.96002i 0.198940 + 0.980012i \(0.436250\pi\)
−0.198940 + 0.980012i \(0.563750\pi\)
\(978\) 0 0
\(979\) − 7.53088i − 0.240688i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −52.5150 −1.67497 −0.837483 0.546463i \(-0.815974\pi\)
−0.837483 + 0.546463i \(0.815974\pi\)
\(984\) 0 0
\(985\) 0.581133 0.0185164
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 23.5703i 0.749491i
\(990\) 0 0
\(991\) − 6.80723i − 0.216239i −0.994138 0.108119i \(-0.965517\pi\)
0.994138 0.108119i \(-0.0344829\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −0.196543 −0.00623084
\(996\) 0 0
\(997\) 10.9576 0.347032 0.173516 0.984831i \(-0.444487\pi\)
0.173516 + 0.984831i \(0.444487\pi\)
\(998\) 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.2591.6 16
3.2 odd 2 inner 6048.2.h.h.2591.9 yes 16
4.3 odd 2 inner 6048.2.h.h.2591.7 yes 16
12.11 even 2 inner 6048.2.h.h.2591.12 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.h.h.2591.6 16 1.1 even 1 trivial
6048.2.h.h.2591.7 yes 16 4.3 odd 2 inner
6048.2.h.h.2591.9 yes 16 3.2 odd 2 inner
6048.2.h.h.2591.12 yes 16 12.11 even 2 inner