Properties

Label 6048.2.h.g.2591.5
Level 6048
Weight 2
Character 6048.2591
Analytic conductor 48.294
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6048.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2591.5
Root \(-0.965926 + 0.258819i\)
Character \(\chi\) = 6048.2591
Dual form 6048.2.h.g.2591.4

$q$-expansion

\(f(q)\) \(=\) \(q+1.19980i q^{5} +1.00000i q^{7} +O(q^{10})\) \(q+1.19980i q^{5} +1.00000i q^{7} -1.62863 q^{11} -2.09638 q^{13} -0.585786i q^{17} +5.89898i q^{19} +7.09273 q^{23} +3.56048 q^{25} +7.08516i q^{29} -6.72741i q^{31} -1.19980 q^{35} +2.17157 q^{37} +0.164525i q^{41} -5.75323i q^{43} -3.41421 q^{47} -1.00000 q^{49} +3.29858i q^{53} -1.95403i q^{55} +2.08641 q^{59} +4.92820 q^{61} -2.51523i q^{65} +12.4887i q^{67} -11.2497 q^{71} +3.97418 q^{73} -1.62863i q^{77} +0.702827i q^{79} -16.8831 q^{83} +0.702827 q^{85} +6.33485i q^{89} -2.09638i q^{91} -7.07760 q^{95} -6.58506 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 8q^{11} - 8q^{13} + 8q^{23} - 8q^{25} - 8q^{35} + 40q^{37} - 16q^{47} - 8q^{49} + 64q^{59} - 16q^{61} - 72q^{71} + 24q^{73} - 32q^{83} + 8q^{85} - 56q^{95} + 48q^{97} + O(q^{100}) \)

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
\(5\) 1.19980i 0.536567i 0.963340 + 0.268284i \(0.0864564\pi\)
−0.963340 + 0.268284i \(0.913544\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.62863 −0.491049 −0.245525 0.969390i \(-0.578960\pi\)
−0.245525 + 0.969390i \(0.578960\pi\)
\(12\) 0 0
\(13\) −2.09638 −0.581430 −0.290715 0.956810i \(-0.593893\pi\)
−0.290715 + 0.956810i \(0.593893\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 0.585786i − 0.142074i −0.997474 0.0710370i \(-0.977369\pi\)
0.997474 0.0710370i \(-0.0226309\pi\)
\(18\) 0 0
\(19\) 5.89898i 1.35332i 0.736296 + 0.676659i \(0.236573\pi\)
−0.736296 + 0.676659i \(0.763427\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.09273 1.47894 0.739468 0.673192i \(-0.235077\pi\)
0.739468 + 0.673192i \(0.235077\pi\)
\(24\) 0 0
\(25\) 3.56048 0.712096
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7.08516i 1.31568i 0.753157 + 0.657841i \(0.228530\pi\)
−0.753157 + 0.657841i \(0.771470\pi\)
\(30\) 0 0
\(31\) − 6.72741i − 1.20828i −0.796879 0.604139i \(-0.793517\pi\)
0.796879 0.604139i \(-0.206483\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.19980 −0.202803
\(36\) 0 0
\(37\) 2.17157 0.357004 0.178502 0.983940i \(-0.442875\pi\)
0.178502 + 0.983940i \(0.442875\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.164525i 0.0256944i 0.999917 + 0.0128472i \(0.00408951\pi\)
−0.999917 + 0.0128472i \(0.995910\pi\)
\(42\) 0 0
\(43\) − 5.75323i − 0.877359i −0.898643 0.438680i \(-0.855446\pi\)
0.898643 0.438680i \(-0.144554\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.41421 −0.498014 −0.249007 0.968502i \(-0.580104\pi\)
−0.249007 + 0.968502i \(0.580104\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.29858i 0.453095i 0.974000 + 0.226547i \(0.0727438\pi\)
−0.974000 + 0.226547i \(0.927256\pi\)
\(54\) 0 0
\(55\) − 1.95403i − 0.263481i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.08641 0.271627 0.135814 0.990734i \(-0.456635\pi\)
0.135814 + 0.990734i \(0.456635\pi\)
\(60\) 0 0
\(61\) 4.92820 0.630992 0.315496 0.948927i \(-0.397829\pi\)
0.315496 + 0.948927i \(0.397829\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 2.51523i − 0.311976i
\(66\) 0 0
\(67\) 12.4887i 1.52574i 0.646555 + 0.762868i \(0.276209\pi\)
−0.646555 + 0.762868i \(0.723791\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −11.2497 −1.33509 −0.667546 0.744568i \(-0.732655\pi\)
−0.667546 + 0.744568i \(0.732655\pi\)
\(72\) 0 0
\(73\) 3.97418 0.465142 0.232571 0.972579i \(-0.425286\pi\)
0.232571 + 0.972579i \(0.425286\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 1.62863i − 0.185599i
\(78\) 0 0
\(79\) 0.702827i 0.0790742i 0.999218 + 0.0395371i \(0.0125883\pi\)
−0.999218 + 0.0395371i \(0.987412\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −16.8831 −1.85316 −0.926582 0.376093i \(-0.877267\pi\)
−0.926582 + 0.376093i \(0.877267\pi\)
\(84\) 0 0
\(85\) 0.702827 0.0762323
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.33485i 0.671493i 0.941952 + 0.335747i \(0.108989\pi\)
−0.941952 + 0.335747i \(0.891011\pi\)
\(90\) 0 0
\(91\) − 2.09638i − 0.219760i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −7.07760 −0.726146
\(96\) 0 0
\(97\) −6.58506 −0.668611 −0.334306 0.942465i \(-0.608502\pi\)
−0.334306 + 0.942465i \(0.608502\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 9.34366i − 0.929729i −0.885382 0.464865i \(-0.846103\pi\)
0.885382 0.464865i \(-0.153897\pi\)
\(102\) 0 0
\(103\) 0.586302i 0.0577700i 0.999583 + 0.0288850i \(0.00919567\pi\)
−0.999583 + 0.0288850i \(0.990804\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −3.14162 −0.303712 −0.151856 0.988403i \(-0.548525\pi\)
−0.151856 + 0.988403i \(0.548525\pi\)
\(108\) 0 0
\(109\) 11.6522 1.11608 0.558040 0.829814i \(-0.311554\pi\)
0.558040 + 0.829814i \(0.311554\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 19.5546i 1.83954i 0.392457 + 0.919771i \(0.371625\pi\)
−0.392457 + 0.919771i \(0.628375\pi\)
\(114\) 0 0
\(115\) 8.50986i 0.793549i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.585786 0.0536990
\(120\) 0 0
\(121\) −8.34758 −0.758871
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 10.2709i 0.918654i
\(126\) 0 0
\(127\) 4.16693i 0.369755i 0.982762 + 0.184878i \(0.0591889\pi\)
−0.982762 + 0.184878i \(0.940811\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −13.6410 −1.19182 −0.595910 0.803051i \(-0.703208\pi\)
−0.595910 + 0.803051i \(0.703208\pi\)
\(132\) 0 0
\(133\) −5.89898 −0.511506
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 8.15696i 0.696896i 0.937328 + 0.348448i \(0.113291\pi\)
−0.937328 + 0.348448i \(0.886709\pi\)
\(138\) 0 0
\(139\) 9.46410i 0.802735i 0.915917 + 0.401367i \(0.131465\pi\)
−0.915917 + 0.401367i \(0.868535\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 3.41421 0.285511
\(144\) 0 0
\(145\) −8.50079 −0.705952
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 20.6848i 1.69457i 0.531140 + 0.847284i \(0.321764\pi\)
−0.531140 + 0.847284i \(0.678236\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 8.07155 0.648322
\(156\) 0 0
\(157\) 0.246769 0.0196943 0.00984717 0.999952i \(-0.496865\pi\)
0.00984717 + 0.999952i \(0.496865\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 7.09273i 0.558985i
\(162\) 0 0
\(163\) − 7.89434i − 0.618332i −0.951008 0.309166i \(-0.899950\pi\)
0.951008 0.309166i \(-0.100050\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1.74150 −0.134762 −0.0673808 0.997727i \(-0.521464\pi\)
−0.0673808 + 0.997727i \(0.521464\pi\)
\(168\) 0 0
\(169\) −8.60521 −0.661939
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 10.2709i − 0.780880i −0.920628 0.390440i \(-0.872323\pi\)
0.920628 0.390440i \(-0.127677\pi\)
\(174\) 0 0
\(175\) 3.56048i 0.269147i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −2.81257 −0.210221 −0.105111 0.994461i \(-0.533520\pi\)
−0.105111 + 0.994461i \(0.533520\pi\)
\(180\) 0 0
\(181\) −8.14110 −0.605124 −0.302562 0.953130i \(-0.597842\pi\)
−0.302562 + 0.953130i \(0.597842\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 2.60545i 0.191557i
\(186\) 0 0
\(187\) 0.954027i 0.0697654i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −16.0482 −1.16120 −0.580602 0.814187i \(-0.697183\pi\)
−0.580602 + 0.814187i \(0.697183\pi\)
\(192\) 0 0
\(193\) −2.87904 −0.207238 −0.103619 0.994617i \(-0.533042\pi\)
−0.103619 + 0.994617i \(0.533042\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 8.09778i 0.576943i 0.957488 + 0.288472i \(0.0931471\pi\)
−0.957488 + 0.288472i \(0.906853\pi\)
\(198\) 0 0
\(199\) − 21.9024i − 1.55262i −0.630352 0.776309i \(-0.717090\pi\)
0.630352 0.776309i \(-0.282910\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −7.08516 −0.497281
\(204\) 0 0
\(205\) −0.197397 −0.0137868
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 9.60723i − 0.664546i
\(210\) 0 0
\(211\) 3.80725i 0.262102i 0.991376 + 0.131051i \(0.0418351\pi\)
−0.991376 + 0.131051i \(0.958165\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 6.90273 0.470762
\(216\) 0 0
\(217\) 6.72741 0.456686
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.22803i 0.0826062i
\(222\) 0 0
\(223\) − 27.4088i − 1.83543i −0.397237 0.917716i \(-0.630031\pi\)
0.397237 0.917716i \(-0.369969\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −4.79920 −0.318534 −0.159267 0.987236i \(-0.550913\pi\)
−0.159267 + 0.987236i \(0.550913\pi\)
\(228\) 0 0
\(229\) 11.9907 0.792369 0.396184 0.918171i \(-0.370334\pi\)
0.396184 + 0.918171i \(0.370334\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0.898979i 0.0588941i 0.999566 + 0.0294471i \(0.00937464\pi\)
−0.999566 + 0.0294471i \(0.990625\pi\)
\(234\) 0 0
\(235\) − 4.09638i − 0.267218i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 15.6702 1.01362 0.506811 0.862057i \(-0.330824\pi\)
0.506811 + 0.862057i \(0.330824\pi\)
\(240\) 0 0
\(241\) 27.1443 1.74852 0.874259 0.485460i \(-0.161348\pi\)
0.874259 + 0.485460i \(0.161348\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 1.19980i − 0.0766525i
\(246\) 0 0
\(247\) − 12.3665i − 0.786860i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −21.8977 −1.38217 −0.691086 0.722772i \(-0.742868\pi\)
−0.691086 + 0.722772i \(0.742868\pi\)
\(252\) 0 0
\(253\) −11.5514 −0.726230
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.89245i 0.180426i 0.995923 + 0.0902129i \(0.0287548\pi\)
−0.995923 + 0.0902129i \(0.971245\pi\)
\(258\) 0 0
\(259\) 2.17157i 0.134935i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −25.5901 −1.57795 −0.788977 0.614423i \(-0.789389\pi\)
−0.788977 + 0.614423i \(0.789389\pi\)
\(264\) 0 0
\(265\) −3.95764 −0.243116
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 25.6692i 1.56508i 0.622600 + 0.782540i \(0.286076\pi\)
−0.622600 + 0.782540i \(0.713924\pi\)
\(270\) 0 0
\(271\) − 6.05845i − 0.368024i −0.982924 0.184012i \(-0.941091\pi\)
0.982924 0.184012i \(-0.0589086\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −5.79869 −0.349674
\(276\) 0 0
\(277\) −31.8166 −1.91168 −0.955839 0.293892i \(-0.905049\pi\)
−0.955839 + 0.293892i \(0.905049\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 2.80295i − 0.167210i −0.996499 0.0836051i \(-0.973357\pi\)
0.996499 0.0836051i \(-0.0266434\pi\)
\(282\) 0 0
\(283\) 10.6367i 0.632287i 0.948711 + 0.316143i \(0.102388\pi\)
−0.948711 + 0.316143i \(0.897612\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −0.164525 −0.00971158
\(288\) 0 0
\(289\) 16.6569 0.979815
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 0.702148i − 0.0410199i −0.999790 0.0205100i \(-0.993471\pi\)
0.999790 0.0205100i \(-0.00652898\pi\)
\(294\) 0 0
\(295\) 2.50327i 0.145746i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −14.8690 −0.859898
\(300\) 0 0
\(301\) 5.75323 0.331611
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 5.91286i 0.338570i
\(306\) 0 0
\(307\) 21.4122i 1.22206i 0.791607 + 0.611031i \(0.209245\pi\)
−0.791607 + 0.611031i \(0.790755\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −14.3424 −0.813284 −0.406642 0.913588i \(-0.633300\pi\)
−0.406642 + 0.913588i \(0.633300\pi\)
\(312\) 0 0
\(313\) 17.8238 1.00746 0.503730 0.863861i \(-0.331961\pi\)
0.503730 + 0.863861i \(0.331961\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 26.7113i − 1.50026i −0.661291 0.750129i \(-0.729991\pi\)
0.661291 0.750129i \(-0.270009\pi\)
\(318\) 0 0
\(319\) − 11.5391i − 0.646065i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3.45554 0.192272
\(324\) 0 0
\(325\) −7.46410 −0.414034
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 3.41421i − 0.188232i
\(330\) 0 0
\(331\) 3.01086i 0.165492i 0.996571 + 0.0827459i \(0.0263690\pi\)
−0.996571 + 0.0827459i \(0.973631\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −14.9839 −0.818660
\(336\) 0 0
\(337\) −14.1210 −0.769217 −0.384609 0.923080i \(-0.625664\pi\)
−0.384609 + 0.923080i \(0.625664\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 10.9564i 0.593324i
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 11.5835 0.621837 0.310919 0.950437i \(-0.399363\pi\)
0.310919 + 0.950437i \(0.399363\pi\)
\(348\) 0 0
\(349\) 25.7887 1.38044 0.690218 0.723602i \(-0.257515\pi\)
0.690218 + 0.723602i \(0.257515\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 28.2326i 1.50267i 0.659922 + 0.751335i \(0.270590\pi\)
−0.659922 + 0.751335i \(0.729410\pi\)
\(354\) 0 0
\(355\) − 13.4974i − 0.716367i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −16.4976 −0.870711 −0.435355 0.900259i \(-0.643377\pi\)
−0.435355 + 0.900259i \(0.643377\pi\)
\(360\) 0 0
\(361\) −15.7980 −0.831472
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 4.76822i 0.249580i
\(366\) 0 0
\(367\) 7.06818i 0.368956i 0.982837 + 0.184478i \(0.0590595\pi\)
−0.982837 + 0.184478i \(0.940941\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −3.29858 −0.171254
\(372\) 0 0
\(373\) 2.88369 0.149312 0.0746559 0.997209i \(-0.476214\pi\)
0.0746559 + 0.997209i \(0.476214\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 14.8532i − 0.764977i
\(378\) 0 0
\(379\) 29.4076i 1.51057i 0.655398 + 0.755283i \(0.272501\pi\)
−0.655398 + 0.755283i \(0.727499\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −13.8977 −0.710141 −0.355071 0.934840i \(-0.615543\pi\)
−0.355071 + 0.934840i \(0.615543\pi\)
\(384\) 0 0
\(385\) 1.95403 0.0995864
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 22.4263i 1.13706i 0.822662 + 0.568530i \(0.192488\pi\)
−0.822662 + 0.568530i \(0.807512\pi\)
\(390\) 0 0
\(391\) − 4.15482i − 0.210118i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −0.843253 −0.0424286
\(396\) 0 0
\(397\) −14.3028 −0.717839 −0.358920 0.933368i \(-0.616855\pi\)
−0.358920 + 0.933368i \(0.616855\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 26.5964i 1.32816i 0.747660 + 0.664081i \(0.231177\pi\)
−0.747660 + 0.664081i \(0.768823\pi\)
\(402\) 0 0
\(403\) 14.1032i 0.702529i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −3.53668 −0.175307
\(408\) 0 0
\(409\) −16.2093 −0.801498 −0.400749 0.916188i \(-0.631250\pi\)
−0.400749 + 0.916188i \(0.631250\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 2.08641i 0.102665i
\(414\) 0 0
\(415\) − 20.2564i − 0.994347i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 24.5437 1.19904 0.599520 0.800360i \(-0.295358\pi\)
0.599520 + 0.800360i \(0.295358\pi\)
\(420\) 0 0
\(421\) −0.373614 −0.0182088 −0.00910441 0.999959i \(-0.502898\pi\)
−0.00910441 + 0.999959i \(0.502898\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 2.08568i − 0.101170i
\(426\) 0 0
\(427\) 4.92820i 0.238492i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 17.7801 0.856435 0.428217 0.903676i \(-0.359142\pi\)
0.428217 + 0.903676i \(0.359142\pi\)
\(432\) 0 0
\(433\) 10.7613 0.517154 0.258577 0.965991i \(-0.416746\pi\)
0.258577 + 0.965991i \(0.416746\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 41.8399i 2.00147i
\(438\) 0 0
\(439\) − 18.5758i − 0.886573i −0.896380 0.443287i \(-0.853812\pi\)
0.896380 0.443287i \(-0.146188\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −30.3699 −1.44292 −0.721459 0.692457i \(-0.756528\pi\)
−0.721459 + 0.692457i \(0.756528\pi\)
\(444\) 0 0
\(445\) −7.60056 −0.360301
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 31.4382i 1.48366i 0.670587 + 0.741831i \(0.266042\pi\)
−0.670587 + 0.741831i \(0.733958\pi\)
\(450\) 0 0
\(451\) − 0.267949i − 0.0126172i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 2.51523 0.117916
\(456\) 0 0
\(457\) −30.3629 −1.42032 −0.710158 0.704043i \(-0.751376\pi\)
−0.710158 + 0.704043i \(0.751376\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 2.28745i − 0.106537i −0.998580 0.0532687i \(-0.983036\pi\)
0.998580 0.0532687i \(-0.0169640\pi\)
\(462\) 0 0
\(463\) − 4.92140i − 0.228717i −0.993440 0.114359i \(-0.963519\pi\)
0.993440 0.114359i \(-0.0364813\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 17.1860 0.795272 0.397636 0.917543i \(-0.369831\pi\)
0.397636 + 0.917543i \(0.369831\pi\)
\(468\) 0 0
\(469\) −12.4887 −0.576674
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 9.36986i 0.430827i
\(474\) 0 0
\(475\) 21.0032i 0.963692i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −0.585057 −0.0267320 −0.0133660 0.999911i \(-0.504255\pi\)
−0.0133660 + 0.999911i \(0.504255\pi\)
\(480\) 0 0
\(481\) −4.55243 −0.207573
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 7.90076i − 0.358755i
\(486\) 0 0
\(487\) − 15.7318i − 0.712878i −0.934319 0.356439i \(-0.883991\pi\)
0.934319 0.356439i \(-0.116009\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 27.7267 1.25129 0.625645 0.780108i \(-0.284836\pi\)
0.625645 + 0.780108i \(0.284836\pi\)
\(492\) 0 0
\(493\) 4.15039 0.186924
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 11.2497i − 0.504618i
\(498\) 0 0
\(499\) 1.41688i 0.0634285i 0.999497 + 0.0317142i \(0.0100966\pi\)
−0.999497 + 0.0317142i \(0.989903\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −4.69870 −0.209505 −0.104752 0.994498i \(-0.533405\pi\)
−0.104752 + 0.994498i \(0.533405\pi\)
\(504\) 0 0
\(505\) 11.2105 0.498862
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 7.82645i − 0.346901i −0.984843 0.173451i \(-0.944508\pi\)
0.984843 0.173451i \(-0.0554917\pi\)
\(510\) 0 0
\(511\) 3.97418i 0.175807i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −0.703446 −0.0309975
\(516\) 0 0
\(517\) 5.56048 0.244550
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 4.83496i 0.211823i 0.994376 + 0.105912i \(0.0337761\pi\)
−0.994376 + 0.105912i \(0.966224\pi\)
\(522\) 0 0
\(523\) 6.46750i 0.282804i 0.989952 + 0.141402i \(0.0451610\pi\)
−0.989952 + 0.141402i \(0.954839\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −3.94082 −0.171665
\(528\) 0 0
\(529\) 27.3068 1.18725
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 0.344906i − 0.0149395i
\(534\) 0 0
\(535\) − 3.76932i − 0.162962i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1.62863 0.0701499
\(540\) 0 0
\(541\) 20.0864 0.863583 0.431791 0.901973i \(-0.357882\pi\)
0.431791 + 0.901973i \(0.357882\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 13.9803i 0.598852i
\(546\) 0 0
\(547\) 19.2689i 0.823877i 0.911212 + 0.411938i \(0.135148\pi\)
−0.911212 + 0.411938i \(0.864852\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −41.7952 −1.78054
\(552\) 0 0
\(553\) −0.702827 −0.0298872
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 2.90346i − 0.123024i −0.998106 0.0615118i \(-0.980408\pi\)
0.998106 0.0615118i \(-0.0195922\pi\)
\(558\) 0 0
\(559\) 12.0609i 0.510123i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −9.18743 −0.387204 −0.193602 0.981080i \(-0.562017\pi\)
−0.193602 + 0.981080i \(0.562017\pi\)
\(564\) 0 0
\(565\) −23.4616 −0.987037
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 2.31694i 0.0971314i 0.998820 + 0.0485657i \(0.0154650\pi\)
−0.998820 + 0.0485657i \(0.984535\pi\)
\(570\) 0 0
\(571\) 18.6770i 0.781608i 0.920474 + 0.390804i \(0.127803\pi\)
−0.920474 + 0.390804i \(0.872197\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 25.2535 1.05314
\(576\) 0 0
\(577\) 39.2011 1.63196 0.815982 0.578077i \(-0.196197\pi\)
0.815982 + 0.578077i \(0.196197\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 16.8831i − 0.700430i
\(582\) 0 0
\(583\) − 5.37216i − 0.222492i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −17.9839 −0.742275 −0.371138 0.928578i \(-0.621032\pi\)
−0.371138 + 0.928578i \(0.621032\pi\)
\(588\) 0 0
\(589\) 39.6848 1.63519
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 30.6480i − 1.25856i −0.777177 0.629282i \(-0.783349\pi\)
0.777177 0.629282i \(-0.216651\pi\)
\(594\) 0 0
\(595\) 0.702827i 0.0288131i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 23.8899 0.976116 0.488058 0.872811i \(-0.337705\pi\)
0.488058 + 0.872811i \(0.337705\pi\)
\(600\) 0 0
\(601\) −12.2161 −0.498305 −0.249152 0.968464i \(-0.580152\pi\)
−0.249152 + 0.968464i \(0.580152\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 10.0154i − 0.407185i
\(606\) 0 0
\(607\) 0.921404i 0.0373986i 0.999825 + 0.0186993i \(0.00595252\pi\)
−0.999825 + 0.0186993i \(0.994047\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 7.15748 0.289561
\(612\) 0 0
\(613\) −17.7716 −0.717788 −0.358894 0.933378i \(-0.616846\pi\)
−0.358894 + 0.933378i \(0.616846\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 18.5702i 0.747609i 0.927507 + 0.373805i \(0.121947\pi\)
−0.927507 + 0.373805i \(0.878053\pi\)
\(618\) 0 0
\(619\) − 28.9780i − 1.16473i −0.812929 0.582363i \(-0.802128\pi\)
0.812929 0.582363i \(-0.197872\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −6.33485 −0.253801
\(624\) 0 0
\(625\) 5.47939 0.219176
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 1.27208i − 0.0507211i
\(630\) 0 0
\(631\) − 32.3996i − 1.28981i −0.764265 0.644903i \(-0.776898\pi\)
0.764265 0.644903i \(-0.223102\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −4.99948 −0.198398
\(636\) 0 0
\(637\) 2.09638 0.0830615
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 41.1701i − 1.62612i −0.582179 0.813061i \(-0.697800\pi\)
0.582179 0.813061i \(-0.302200\pi\)
\(642\) 0 0
\(643\) 4.52433i 0.178422i 0.996013 + 0.0892112i \(0.0284346\pi\)
−0.996013 + 0.0892112i \(0.971565\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 24.4377 0.960746 0.480373 0.877064i \(-0.340501\pi\)
0.480373 + 0.877064i \(0.340501\pi\)
\(648\) 0 0
\(649\) −3.39798 −0.133382
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 2.82564i − 0.110576i −0.998470 0.0552878i \(-0.982392\pi\)
0.998470 0.0552878i \(-0.0176076\pi\)
\(654\) 0 0
\(655\) − 16.3665i − 0.639491i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −34.8305 −1.35680 −0.678401 0.734692i \(-0.737327\pi\)
−0.678401 + 0.734692i \(0.737327\pi\)
\(660\) 0 0
\(661\) 32.6230 1.26889 0.634444 0.772969i \(-0.281229\pi\)
0.634444 + 0.772969i \(0.281229\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 7.07760i − 0.274458i
\(666\) 0 0
\(667\) 50.2531i 1.94581i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −8.02620 −0.309848
\(672\) 0 0
\(673\) −10.0000 −0.385472 −0.192736 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 13.6947i − 0.526330i −0.964751 0.263165i \(-0.915234\pi\)
0.964751 0.263165i \(-0.0847664\pi\)
\(678\) 0 0
\(679\) − 6.58506i − 0.252711i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 31.7975 1.21670 0.608349 0.793670i \(-0.291832\pi\)
0.608349 + 0.793670i \(0.291832\pi\)
\(684\) 0 0
\(685\) −9.78673 −0.373932
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 6.91507i − 0.263443i
\(690\) 0 0
\(691\) 29.8496i 1.13553i 0.823190 + 0.567766i \(0.192192\pi\)
−0.823190 + 0.567766i \(0.807808\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −11.3550 −0.430721
\(696\) 0 0
\(697\) 0.0963763 0.00365051
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 29.0711i 1.09800i 0.835823 + 0.549000i \(0.184991\pi\)
−0.835823 + 0.549000i \(0.815009\pi\)
\(702\) 0 0
\(703\) 12.8101i 0.483141i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 9.34366 0.351405
\(708\) 0 0
\(709\) 10.4368 0.391963 0.195981 0.980608i \(-0.437211\pi\)
0.195981 + 0.980608i \(0.437211\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 47.7157i − 1.78697i
\(714\) 0 0
\(715\) 4.09638i 0.153196i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −43.2268 −1.61209 −0.806044 0.591856i \(-0.798395\pi\)
−0.806044 + 0.591856i \(0.798395\pi\)
\(720\) 0 0
\(721\) −0.586302 −0.0218350
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 25.2266i 0.936891i
\(726\) 0 0
\(727\) 5.31122i 0.196982i 0.995138 + 0.0984911i \(0.0314016\pi\)
−0.995138 + 0.0984911i \(0.968598\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −3.37016 −0.124650
\(732\) 0 0
\(733\) 16.1057 0.594876 0.297438 0.954741i \(-0.403868\pi\)
0.297438 + 0.954741i \(0.403868\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 20.3394i − 0.749211i
\(738\) 0 0
\(739\) 16.8673i 0.620472i 0.950660 + 0.310236i \(0.100408\pi\)
−0.950660 + 0.310236i \(0.899592\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 45.2019 1.65830 0.829148 0.559029i \(-0.188826\pi\)
0.829148 + 0.559029i \(0.188826\pi\)
\(744\) 0 0
\(745\) −24.8177 −0.909249
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 3.14162i − 0.114792i
\(750\) 0 0
\(751\) − 0.479065i − 0.0174813i −0.999962 0.00874067i \(-0.997218\pi\)
0.999962 0.00874067i \(-0.00278228\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −3.71530 −0.135035 −0.0675175 0.997718i \(-0.521508\pi\)
−0.0675175 + 0.997718i \(0.521508\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 22.2164i 0.805345i 0.915344 + 0.402673i \(0.131919\pi\)
−0.915344 + 0.402673i \(0.868081\pi\)
\(762\) 0 0
\(763\) 11.6522i 0.421838i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −4.37390 −0.157932
\(768\) 0 0
\(769\) −40.1576 −1.44812 −0.724061 0.689736i \(-0.757727\pi\)
−0.724061 + 0.689736i \(0.757727\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 4.25750i − 0.153132i −0.997065 0.0765658i \(-0.975604\pi\)
0.997065 0.0765658i \(-0.0243955\pi\)
\(774\) 0 0
\(775\) − 23.9528i − 0.860409i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −0.970528 −0.0347728
\(780\) 0 0
\(781\) 18.3215 0.655596
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0.296074i 0.0105673i
\(786\) 0 0
\(787\) − 5.70850i − 0.203486i −0.994811 0.101743i \(-0.967558\pi\)
0.994811 0.101743i \(-0.0324420\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −19.5546 −0.695281
\(792\) 0 0
\(793\) −10.3314 −0.366878
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 1.02499i − 0.0363071i −0.999835 0.0181535i \(-0.994221\pi\)
0.999835 0.0181535i \(-0.00577877\pi\)
\(798\) 0 0
\(799\) 2.00000i 0.0707549i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −6.47245 −0.228408
\(804\) 0 0
\(805\) −8.50986 −0.299933
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 15.4989i 0.544911i 0.962168 + 0.272455i \(0.0878358\pi\)
−0.962168 + 0.272455i \(0.912164\pi\)
\(810\) 0 0
\(811\) − 32.4913i − 1.14092i −0.821324 0.570462i \(-0.806764\pi\)
0.821324 0.570462i \(-0.193236\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 9.47163 0.331777
\(816\) 0 0
\(817\) 33.9382 1.18735
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 40.6559i − 1.41890i −0.704755 0.709451i \(-0.748943\pi\)
0.704755 0.709451i \(-0.251057\pi\)
\(822\) 0 0
\(823\) − 3.07180i − 0.107076i −0.998566 0.0535381i \(-0.982950\pi\)
0.998566 0.0535381i \(-0.0170498\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 27.7665 0.965536 0.482768 0.875748i \(-0.339631\pi\)
0.482768 + 0.875748i \(0.339631\pi\)
\(828\) 0 0
\(829\) −55.3673 −1.92299 −0.961493 0.274830i \(-0.911378\pi\)
−0.961493 + 0.274830i \(0.911378\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0.585786i 0.0202963i
\(834\) 0 0
\(835\) − 2.08946i − 0.0723086i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 5.38698 0.185979 0.0929896 0.995667i \(-0.470358\pi\)
0.0929896 + 0.995667i \(0.470358\pi\)
\(840\) 0 0
\(841\) −21.1996 −0.731019
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 10.3245i − 0.355175i
\(846\) 0 0
\(847\) − 8.34758i − 0.286826i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 15.4024 0.527987
\(852\) 0 0
\(853\) −15.9597 −0.546450 −0.273225 0.961950i \(-0.588090\pi\)
−0.273225 + 0.961950i \(0.588090\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 33.7050i − 1.15134i −0.817682 0.575671i \(-0.804741\pi\)
0.817682 0.575671i \(-0.195259\pi\)
\(858\) 0 0
\(859\) 20.3739i 0.695150i 0.937652 + 0.347575i \(0.112995\pi\)
−0.937652 + 0.347575i \(0.887005\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0.986419 0.0335781 0.0167890 0.999859i \(-0.494656\pi\)
0.0167890 + 0.999859i \(0.494656\pi\)
\(864\) 0 0
\(865\) 12.3230 0.418994
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 1.14464i − 0.0388293i
\(870\) 0 0
\(871\) − 26.1810i − 0.887109i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −10.2709 −0.347219
\(876\) 0 0
\(877\) 54.0958 1.82669 0.913343 0.407191i \(-0.133492\pi\)
0.913343 + 0.407191i \(0.133492\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 29.7649i 1.00281i 0.865214 + 0.501403i \(0.167183\pi\)
−0.865214 + 0.501403i \(0.832817\pi\)
\(882\) 0 0
\(883\) − 1.48582i − 0.0500019i −0.999687 0.0250010i \(-0.992041\pi\)
0.999687 0.0250010i \(-0.00795888\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 13.8854 0.466226 0.233113 0.972450i \(-0.425109\pi\)
0.233113 + 0.972450i \(0.425109\pi\)
\(888\) 0 0
\(889\) −4.16693 −0.139754
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 20.1404i − 0.673972i
\(894\) 0 0
\(895\) − 3.37452i − 0.112798i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 47.6648 1.58971
\(900\) 0 0
\(901\) 1.93226 0.0643731
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 9.76770i − 0.324689i
\(906\) 0 0
\(907\) − 4.42580i − 0.146956i −0.997297 0.0734782i \(-0.976590\pi\)
0.997297 0.0734782i \(-0.0234099\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 29.6657 0.982870 0.491435 0.870914i \(-0.336473\pi\)
0.491435 + 0.870914i \(0.336473\pi\)
\(912\) 0 0
\(913\) 27.4963 0.909995
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 13.6410i − 0.450465i
\(918\) 0 0
\(919\) 42.4838i 1.40141i 0.713450 + 0.700706i \(0.247132\pi\)
−0.713450 + 0.700706i \(0.752868\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 23.5836 0.776263
\(924\) 0 0
\(925\) 7.73184 0.254221
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 31.3523i − 1.02864i −0.857600 0.514318i \(-0.828045\pi\)
0.857600 0.514318i \(-0.171955\pi\)
\(930\) 0 0
\(931\) − 5.89898i − 0.193331i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −1.14464 −0.0374338
\(936\) 0 0
\(937\) 17.0557 0.557186 0.278593 0.960409i \(-0.410132\pi\)
0.278593 + 0.960409i \(0.410132\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 45.1949i 1.47331i 0.676268 + 0.736656i \(0.263596\pi\)
−0.676268 + 0.736656i \(0.736404\pi\)
\(942\) 0 0
\(943\) 1.16693i 0.0380004i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 31.6571 1.02872 0.514359 0.857575i \(-0.328030\pi\)
0.514359 + 0.857575i \(0.328030\pi\)
\(948\) 0 0
\(949\) −8.33137 −0.270448
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 39.1230i 1.26732i 0.773612 + 0.633660i \(0.218448\pi\)
−0.773612 + 0.633660i \(0.781552\pi\)
\(954\) 0 0
\(955\) − 19.2546i − 0.623064i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −8.15696 −0.263402
\(960\) 0 0
\(961\) −14.2580 −0.459935
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 3.45428i − 0.111197i
\(966\) 0 0
\(967\) − 50.8761i − 1.63607i −0.575171 0.818033i \(-0.695065\pi\)
0.575171 0.818033i \(-0.304935\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −45.9782 −1.47551 −0.737755 0.675068i \(-0.764114\pi\)
−0.737755 + 0.675068i \(0.764114\pi\)
\(972\) 0 0
\(973\) −9.46410 −0.303405
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 15.1822i − 0.485722i −0.970061 0.242861i \(-0.921914\pi\)
0.970061 0.242861i \(-0.0780859\pi\)
\(978\) 0 0
\(979\) − 10.3171i − 0.329736i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 42.6625 1.36072 0.680361 0.732877i \(-0.261823\pi\)
0.680361 + 0.732877i \(0.261823\pi\)
\(984\) 0 0
\(985\) −9.71573 −0.309569
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 40.8061i − 1.29756i
\(990\) 0 0
\(991\) 38.0072i 1.20734i 0.797234 + 0.603670i \(0.206296\pi\)
−0.797234 + 0.603670i \(0.793704\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 26.2785 0.833084
\(996\) 0 0
\(997\) −24.0560 −0.761860 −0.380930 0.924604i \(-0.624396\pi\)
−0.380930 + 0.924604i \(0.624396\pi\)
\(998\) 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.g.2591.5 yes 8
3.2 odd 2 6048.2.h.a.2591.4 8
4.3 odd 2 6048.2.h.a.2591.5 yes 8
12.11 even 2 inner 6048.2.h.g.2591.4 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.h.a.2591.4 8 3.2 odd 2
6048.2.h.a.2591.5 yes 8 4.3 odd 2
6048.2.h.g.2591.4 yes 8 12.11 even 2 inner
6048.2.h.g.2591.5 yes 8 1.1 even 1 trivial