Properties

Label 6048.2.c.f.3025.7
Level 6048
Weight 2
Character 6048.3025
Analytic conductor 48.294
Analytic rank 0
Dimension 24
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.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(24\)
Coefficient ring index: multiple of None
Twist minimal: no (minimal twist has level 1512)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3025.7
Character \(\chi\) = 6048.3025
Dual form 6048.2.c.f.3025.18

$q$-expansion

\(f(q)\) \(=\) \(q-1.82986i q^{5} -1.00000 q^{7} +O(q^{10})\) \(q-1.82986i q^{5} -1.00000 q^{7} -3.75866i q^{11} -3.09942i q^{13} -3.61074 q^{17} +1.65297i q^{19} -7.47934 q^{23} +1.65160 q^{25} -2.38629i q^{29} -8.97301 q^{31} +1.82986i q^{35} +8.94392i q^{37} -3.04284 q^{41} +1.82597i q^{43} +6.21697 q^{47} +1.00000 q^{49} -3.21435i q^{53} -6.87784 q^{55} +12.4366i q^{59} -9.91962i q^{61} -5.67151 q^{65} +8.73156i q^{67} -12.4229 q^{71} +9.79655 q^{73} +3.75866i q^{77} +7.24388 q^{79} +8.99325i q^{83} +6.60717i q^{85} +1.54944 q^{89} +3.09942i q^{91} +3.02470 q^{95} -10.2583 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24q - 24q^{7} + O(q^{10}) \) \( 24q - 24q^{7} - 24q^{25} - 8q^{31} + 24q^{49} - 16q^{55} - 8q^{79} + O(q^{100}) \)

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
\(5\) − 1.82986i − 0.818340i −0.912458 0.409170i \(-0.865818\pi\)
0.912458 0.409170i \(-0.134182\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 3.75866i − 1.13328i −0.823966 0.566639i \(-0.808243\pi\)
0.823966 0.566639i \(-0.191757\pi\)
\(12\) 0 0
\(13\) − 3.09942i − 0.859623i −0.902919 0.429812i \(-0.858580\pi\)
0.902919 0.429812i \(-0.141420\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.61074 −0.875734 −0.437867 0.899040i \(-0.644266\pi\)
−0.437867 + 0.899040i \(0.644266\pi\)
\(18\) 0 0
\(19\) 1.65297i 0.379217i 0.981860 + 0.189608i \(0.0607218\pi\)
−0.981860 + 0.189608i \(0.939278\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −7.47934 −1.55955 −0.779775 0.626060i \(-0.784666\pi\)
−0.779775 + 0.626060i \(0.784666\pi\)
\(24\) 0 0
\(25\) 1.65160 0.330320
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 2.38629i − 0.443123i −0.975146 0.221561i \(-0.928885\pi\)
0.975146 0.221561i \(-0.0711153\pi\)
\(30\) 0 0
\(31\) −8.97301 −1.61160 −0.805800 0.592188i \(-0.798264\pi\)
−0.805800 + 0.592188i \(0.798264\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.82986i 0.309303i
\(36\) 0 0
\(37\) 8.94392i 1.47037i 0.677865 + 0.735186i \(0.262905\pi\)
−0.677865 + 0.735186i \(0.737095\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.04284 −0.475212 −0.237606 0.971362i \(-0.576363\pi\)
−0.237606 + 0.971362i \(0.576363\pi\)
\(42\) 0 0
\(43\) 1.82597i 0.278457i 0.990260 + 0.139229i \(0.0444623\pi\)
−0.990260 + 0.139229i \(0.955538\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.21697 0.906838 0.453419 0.891297i \(-0.350204\pi\)
0.453419 + 0.891297i \(0.350204\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 3.21435i − 0.441525i −0.975328 0.220762i \(-0.929145\pi\)
0.975328 0.220762i \(-0.0708546\pi\)
\(54\) 0 0
\(55\) −6.87784 −0.927407
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 12.4366i 1.61911i 0.587046 + 0.809554i \(0.300291\pi\)
−0.587046 + 0.809554i \(0.699709\pi\)
\(60\) 0 0
\(61\) − 9.91962i − 1.27008i −0.772480 0.635039i \(-0.780984\pi\)
0.772480 0.635039i \(-0.219016\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −5.67151 −0.703464
\(66\) 0 0
\(67\) 8.73156i 1.06673i 0.845885 + 0.533365i \(0.179073\pi\)
−0.845885 + 0.533365i \(0.820927\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −12.4229 −1.47432 −0.737161 0.675717i \(-0.763834\pi\)
−0.737161 + 0.675717i \(0.763834\pi\)
\(72\) 0 0
\(73\) 9.79655 1.14660 0.573300 0.819346i \(-0.305663\pi\)
0.573300 + 0.819346i \(0.305663\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.75866i 0.428339i
\(78\) 0 0
\(79\) 7.24388 0.815001 0.407500 0.913205i \(-0.366401\pi\)
0.407500 + 0.913205i \(0.366401\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 8.99325i 0.987138i 0.869707 + 0.493569i \(0.164308\pi\)
−0.869707 + 0.493569i \(0.835692\pi\)
\(84\) 0 0
\(85\) 6.60717i 0.716648i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.54944 0.164240 0.0821202 0.996622i \(-0.473831\pi\)
0.0821202 + 0.996622i \(0.473831\pi\)
\(90\) 0 0
\(91\) 3.09942i 0.324907i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.02470 0.310328
\(96\) 0 0
\(97\) −10.2583 −1.04158 −0.520788 0.853686i \(-0.674362\pi\)
−0.520788 + 0.853686i \(0.674362\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 5.97880i − 0.594913i −0.954735 0.297456i \(-0.903862\pi\)
0.954735 0.297456i \(-0.0961383\pi\)
\(102\) 0 0
\(103\) 16.9060 1.66579 0.832896 0.553429i \(-0.186681\pi\)
0.832896 + 0.553429i \(0.186681\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 2.14315i − 0.207186i −0.994620 0.103593i \(-0.966966\pi\)
0.994620 0.103593i \(-0.0330339\pi\)
\(108\) 0 0
\(109\) 8.34775i 0.799569i 0.916609 + 0.399785i \(0.130915\pi\)
−0.916609 + 0.399785i \(0.869085\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 17.5901 1.65474 0.827369 0.561659i \(-0.189837\pi\)
0.827369 + 0.561659i \(0.189837\pi\)
\(114\) 0 0
\(115\) 13.6862i 1.27624i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.61074 0.330996
\(120\) 0 0
\(121\) −3.12753 −0.284321
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 12.1715i − 1.08865i
\(126\) 0 0
\(127\) −9.73427 −0.863777 −0.431888 0.901927i \(-0.642153\pi\)
−0.431888 + 0.901927i \(0.642153\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 16.7317i − 1.46185i −0.682456 0.730926i \(-0.739088\pi\)
0.682456 0.730926i \(-0.260912\pi\)
\(132\) 0 0
\(133\) − 1.65297i − 0.143330i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 15.1897 1.29774 0.648872 0.760897i \(-0.275241\pi\)
0.648872 + 0.760897i \(0.275241\pi\)
\(138\) 0 0
\(139\) 9.32344i 0.790804i 0.918508 + 0.395402i \(0.129395\pi\)
−0.918508 + 0.395402i \(0.870605\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −11.6497 −0.974193
\(144\) 0 0
\(145\) −4.36658 −0.362625
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 3.32322i − 0.272248i −0.990692 0.136124i \(-0.956535\pi\)
0.990692 0.136124i \(-0.0434646\pi\)
\(150\) 0 0
\(151\) −12.3757 −1.00712 −0.503561 0.863960i \(-0.667977\pi\)
−0.503561 + 0.863960i \(0.667977\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 16.4194i 1.31884i
\(156\) 0 0
\(157\) 12.9811i 1.03601i 0.855378 + 0.518004i \(0.173325\pi\)
−0.855378 + 0.518004i \(0.826675\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 7.47934 0.589454
\(162\) 0 0
\(163\) 6.17108i 0.483356i 0.970356 + 0.241678i \(0.0776978\pi\)
−0.970356 + 0.241678i \(0.922302\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −5.75113 −0.445036 −0.222518 0.974929i \(-0.571428\pi\)
−0.222518 + 0.974929i \(0.571428\pi\)
\(168\) 0 0
\(169\) 3.39362 0.261048
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 8.09698i − 0.615602i −0.951451 0.307801i \(-0.900407\pi\)
0.951451 0.307801i \(-0.0995931\pi\)
\(174\) 0 0
\(175\) −1.65160 −0.124849
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 15.4128i 1.15201i 0.817447 + 0.576003i \(0.195389\pi\)
−0.817447 + 0.576003i \(0.804611\pi\)
\(180\) 0 0
\(181\) 6.06823i 0.451048i 0.974238 + 0.225524i \(0.0724095\pi\)
−0.974238 + 0.225524i \(0.927591\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 16.3662 1.20326
\(186\) 0 0
\(187\) 13.5716i 0.992450i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2.76632 −0.200164 −0.100082 0.994979i \(-0.531910\pi\)
−0.100082 + 0.994979i \(0.531910\pi\)
\(192\) 0 0
\(193\) −17.9782 −1.29410 −0.647048 0.762449i \(-0.723997\pi\)
−0.647048 + 0.762449i \(0.723997\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 10.5179i − 0.749369i −0.927152 0.374684i \(-0.877751\pi\)
0.927152 0.374684i \(-0.122249\pi\)
\(198\) 0 0
\(199\) −4.09973 −0.290622 −0.145311 0.989386i \(-0.546418\pi\)
−0.145311 + 0.989386i \(0.546418\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 2.38629i 0.167485i
\(204\) 0 0
\(205\) 5.56799i 0.388885i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 6.21294 0.429758
\(210\) 0 0
\(211\) 21.1789i 1.45802i 0.684504 + 0.729009i \(0.260019\pi\)
−0.684504 + 0.729009i \(0.739981\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 3.34127 0.227873
\(216\) 0 0
\(217\) 8.97301 0.609128
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 11.1912i 0.752801i
\(222\) 0 0
\(223\) 2.30718 0.154500 0.0772502 0.997012i \(-0.475386\pi\)
0.0772502 + 0.997012i \(0.475386\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2.69006i 0.178545i 0.996007 + 0.0892727i \(0.0284543\pi\)
−0.996007 + 0.0892727i \(0.971546\pi\)
\(228\) 0 0
\(229\) 15.3215i 1.01247i 0.862394 + 0.506237i \(0.168964\pi\)
−0.862394 + 0.506237i \(0.831036\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −21.3885 −1.40121 −0.700605 0.713549i \(-0.747086\pi\)
−0.700605 + 0.713549i \(0.747086\pi\)
\(234\) 0 0
\(235\) − 11.3762i − 0.742102i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −7.04548 −0.455734 −0.227867 0.973692i \(-0.573175\pi\)
−0.227867 + 0.973692i \(0.573175\pi\)
\(240\) 0 0
\(241\) −0.343612 −0.0221340 −0.0110670 0.999939i \(-0.503523\pi\)
−0.0110670 + 0.999939i \(0.503523\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 1.82986i − 0.116906i
\(246\) 0 0
\(247\) 5.12323 0.325983
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 7.65363i 0.483093i 0.970389 + 0.241546i \(0.0776546\pi\)
−0.970389 + 0.241546i \(0.922345\pi\)
\(252\) 0 0
\(253\) 28.1123i 1.76740i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −7.01134 −0.437356 −0.218678 0.975797i \(-0.570174\pi\)
−0.218678 + 0.975797i \(0.570174\pi\)
\(258\) 0 0
\(259\) − 8.94392i − 0.555748i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 24.0879 1.48532 0.742661 0.669667i \(-0.233563\pi\)
0.742661 + 0.669667i \(0.233563\pi\)
\(264\) 0 0
\(265\) −5.88182 −0.361317
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 14.4906i − 0.883507i −0.897136 0.441753i \(-0.854357\pi\)
0.897136 0.441753i \(-0.145643\pi\)
\(270\) 0 0
\(271\) −28.3057 −1.71945 −0.859726 0.510756i \(-0.829365\pi\)
−0.859726 + 0.510756i \(0.829365\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 6.20780i − 0.374344i
\(276\) 0 0
\(277\) 9.70296i 0.582994i 0.956572 + 0.291497i \(0.0941534\pi\)
−0.956572 + 0.291497i \(0.905847\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 16.0412 0.956937 0.478469 0.878105i \(-0.341192\pi\)
0.478469 + 0.878105i \(0.341192\pi\)
\(282\) 0 0
\(283\) 11.0326i 0.655818i 0.944709 + 0.327909i \(0.106344\pi\)
−0.944709 + 0.327909i \(0.893656\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3.04284 0.179613
\(288\) 0 0
\(289\) −3.96254 −0.233091
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 10.9744i 0.641131i 0.947226 + 0.320566i \(0.103873\pi\)
−0.947226 + 0.320566i \(0.896127\pi\)
\(294\) 0 0
\(295\) 22.7573 1.32498
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 23.1816i 1.34062i
\(300\) 0 0
\(301\) − 1.82597i − 0.105247i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −18.1516 −1.03936
\(306\) 0 0
\(307\) − 28.2184i − 1.61051i −0.592931 0.805254i \(-0.702029\pi\)
0.592931 0.805254i \(-0.297971\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −24.3156 −1.37881 −0.689407 0.724375i \(-0.742129\pi\)
−0.689407 + 0.724375i \(0.742129\pi\)
\(312\) 0 0
\(313\) −8.29548 −0.468889 −0.234444 0.972130i \(-0.575327\pi\)
−0.234444 + 0.972130i \(0.575327\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 0.906822i − 0.0509322i −0.999676 0.0254661i \(-0.991893\pi\)
0.999676 0.0254661i \(-0.00810699\pi\)
\(318\) 0 0
\(319\) −8.96925 −0.502182
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 5.96844i − 0.332093i
\(324\) 0 0
\(325\) − 5.11899i − 0.283950i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −6.21697 −0.342753
\(330\) 0 0
\(331\) − 31.7662i − 1.74603i −0.487697 0.873013i \(-0.662163\pi\)
0.487697 0.873013i \(-0.337837\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 15.9776 0.872948
\(336\) 0 0
\(337\) −34.4231 −1.87515 −0.937573 0.347788i \(-0.886933\pi\)
−0.937573 + 0.347788i \(0.886933\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 33.7265i 1.82639i
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 23.5611i − 1.26483i −0.774630 0.632414i \(-0.782064\pi\)
0.774630 0.632414i \(-0.217936\pi\)
\(348\) 0 0
\(349\) 28.6095i 1.53143i 0.643179 + 0.765716i \(0.277615\pi\)
−0.643179 + 0.765716i \(0.722385\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 7.03843 0.374618 0.187309 0.982301i \(-0.440024\pi\)
0.187309 + 0.982301i \(0.440024\pi\)
\(354\) 0 0
\(355\) 22.7321i 1.20650i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −13.2261 −0.698046 −0.349023 0.937114i \(-0.613487\pi\)
−0.349023 + 0.937114i \(0.613487\pi\)
\(360\) 0 0
\(361\) 16.2677 0.856195
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 17.9263i − 0.938308i
\(366\) 0 0
\(367\) −11.6530 −0.608281 −0.304140 0.952627i \(-0.598369\pi\)
−0.304140 + 0.952627i \(0.598369\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 3.21435i 0.166881i
\(372\) 0 0
\(373\) 9.85547i 0.510297i 0.966902 + 0.255148i \(0.0821243\pi\)
−0.966902 + 0.255148i \(0.917876\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −7.39610 −0.380919
\(378\) 0 0
\(379\) 34.5581i 1.77513i 0.460684 + 0.887564i \(0.347604\pi\)
−0.460684 + 0.887564i \(0.652396\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 11.4073 0.582884 0.291442 0.956589i \(-0.405865\pi\)
0.291442 + 0.956589i \(0.405865\pi\)
\(384\) 0 0
\(385\) 6.87784 0.350527
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 37.3093i 1.89166i 0.324667 + 0.945828i \(0.394748\pi\)
−0.324667 + 0.945828i \(0.605252\pi\)
\(390\) 0 0
\(391\) 27.0060 1.36575
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 13.2553i − 0.666947i
\(396\) 0 0
\(397\) − 10.8903i − 0.546567i −0.961934 0.273283i \(-0.911890\pi\)
0.961934 0.273283i \(-0.0881097\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −20.6784 −1.03263 −0.516316 0.856398i \(-0.672697\pi\)
−0.516316 + 0.856398i \(0.672697\pi\)
\(402\) 0 0
\(403\) 27.8111i 1.38537i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 33.6172 1.66634
\(408\) 0 0
\(409\) −16.1209 −0.797129 −0.398565 0.917140i \(-0.630492\pi\)
−0.398565 + 0.917140i \(0.630492\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 12.4366i − 0.611965i
\(414\) 0 0
\(415\) 16.4564 0.807814
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 26.5958i − 1.29929i −0.760237 0.649645i \(-0.774917\pi\)
0.760237 0.649645i \(-0.225083\pi\)
\(420\) 0 0
\(421\) 27.2198i 1.32661i 0.748348 + 0.663307i \(0.230847\pi\)
−0.748348 + 0.663307i \(0.769153\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −5.96350 −0.289272
\(426\) 0 0
\(427\) 9.91962i 0.480044i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 4.58135 0.220676 0.110338 0.993894i \(-0.464807\pi\)
0.110338 + 0.993894i \(0.464807\pi\)
\(432\) 0 0
\(433\) 4.43085 0.212933 0.106466 0.994316i \(-0.466046\pi\)
0.106466 + 0.994316i \(0.466046\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 12.3631i − 0.591407i
\(438\) 0 0
\(439\) 11.2867 0.538685 0.269343 0.963044i \(-0.413194\pi\)
0.269343 + 0.963044i \(0.413194\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 3.69129i − 0.175378i −0.996148 0.0876892i \(-0.972052\pi\)
0.996148 0.0876892i \(-0.0279482\pi\)
\(444\) 0 0
\(445\) − 2.83526i − 0.134404i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 18.0692 0.852737 0.426368 0.904550i \(-0.359793\pi\)
0.426368 + 0.904550i \(0.359793\pi\)
\(450\) 0 0
\(451\) 11.4370i 0.538548i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 5.67151 0.265884
\(456\) 0 0
\(457\) −21.4880 −1.00517 −0.502583 0.864529i \(-0.667617\pi\)
−0.502583 + 0.864529i \(0.667617\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 16.7622i − 0.780695i −0.920668 0.390348i \(-0.872355\pi\)
0.920668 0.390348i \(-0.127645\pi\)
\(462\) 0 0
\(463\) 9.74065 0.452686 0.226343 0.974048i \(-0.427323\pi\)
0.226343 + 0.974048i \(0.427323\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 33.8683i − 1.56724i −0.621241 0.783620i \(-0.713371\pi\)
0.621241 0.783620i \(-0.286629\pi\)
\(468\) 0 0
\(469\) − 8.73156i − 0.403186i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 6.86319 0.315570
\(474\) 0 0
\(475\) 2.73004i 0.125263i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −12.0611 −0.551084 −0.275542 0.961289i \(-0.588857\pi\)
−0.275542 + 0.961289i \(0.588857\pi\)
\(480\) 0 0
\(481\) 27.7209 1.26397
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 18.7714i 0.852364i
\(486\) 0 0
\(487\) 11.5752 0.524522 0.262261 0.964997i \(-0.415532\pi\)
0.262261 + 0.964997i \(0.415532\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 40.0965i 1.80953i 0.425911 + 0.904765i \(0.359954\pi\)
−0.425911 + 0.904765i \(0.640046\pi\)
\(492\) 0 0
\(493\) 8.61627i 0.388057i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 12.4229 0.557241
\(498\) 0 0
\(499\) − 33.2035i − 1.48639i −0.669073 0.743197i \(-0.733309\pi\)
0.669073 0.743197i \(-0.266691\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 4.46295 0.198993 0.0994967 0.995038i \(-0.468277\pi\)
0.0994967 + 0.995038i \(0.468277\pi\)
\(504\) 0 0
\(505\) −10.9404 −0.486841
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 28.2457i 1.25197i 0.779837 + 0.625983i \(0.215302\pi\)
−0.779837 + 0.625983i \(0.784698\pi\)
\(510\) 0 0
\(511\) −9.79655 −0.433374
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 30.9356i − 1.36318i
\(516\) 0 0
\(517\) − 23.3675i − 1.02770i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −30.4655 −1.33472 −0.667359 0.744737i \(-0.732575\pi\)
−0.667359 + 0.744737i \(0.732575\pi\)
\(522\) 0 0
\(523\) − 36.5909i − 1.60001i −0.599994 0.800004i \(-0.704831\pi\)
0.599994 0.800004i \(-0.295169\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 32.3992 1.41133
\(528\) 0 0
\(529\) 32.9405 1.43219
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 9.43104i 0.408504i
\(534\) 0 0
\(535\) −3.92167 −0.169548
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 3.75866i − 0.161897i
\(540\) 0 0
\(541\) 42.8045i 1.84031i 0.391556 + 0.920154i \(0.371937\pi\)
−0.391556 + 0.920154i \(0.628063\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 15.2752 0.654319
\(546\) 0 0
\(547\) − 6.57128i − 0.280968i −0.990083 0.140484i \(-0.955134\pi\)
0.990083 0.140484i \(-0.0448658\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 3.94446 0.168040
\(552\) 0 0
\(553\) −7.24388 −0.308041
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 31.2762i 1.32521i 0.748967 + 0.662607i \(0.230550\pi\)
−0.748967 + 0.662607i \(0.769450\pi\)
\(558\) 0 0
\(559\) 5.65943 0.239368
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 3.69436i 0.155699i 0.996965 + 0.0778493i \(0.0248053\pi\)
−0.996965 + 0.0778493i \(0.975195\pi\)
\(564\) 0 0
\(565\) − 32.1875i − 1.35414i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 11.6082 0.486641 0.243320 0.969946i \(-0.421763\pi\)
0.243320 + 0.969946i \(0.421763\pi\)
\(570\) 0 0
\(571\) 26.8739i 1.12464i 0.826920 + 0.562319i \(0.190091\pi\)
−0.826920 + 0.562319i \(0.809909\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −12.3529 −0.515150
\(576\) 0 0
\(577\) 10.2051 0.424843 0.212421 0.977178i \(-0.431865\pi\)
0.212421 + 0.977178i \(0.431865\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 8.99325i − 0.373103i
\(582\) 0 0
\(583\) −12.0817 −0.500371
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 25.0075i 1.03217i 0.856538 + 0.516085i \(0.172611\pi\)
−0.856538 + 0.516085i \(0.827389\pi\)
\(588\) 0 0
\(589\) − 14.8321i − 0.611146i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −43.4040 −1.78239 −0.891195 0.453620i \(-0.850132\pi\)
−0.891195 + 0.453620i \(0.850132\pi\)
\(594\) 0 0
\(595\) − 6.60717i − 0.270867i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −2.47057 −0.100945 −0.0504724 0.998725i \(-0.516073\pi\)
−0.0504724 + 0.998725i \(0.516073\pi\)
\(600\) 0 0
\(601\) −13.2142 −0.539018 −0.269509 0.962998i \(-0.586861\pi\)
−0.269509 + 0.962998i \(0.586861\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 5.72296i 0.232671i
\(606\) 0 0
\(607\) −29.4562 −1.19559 −0.597796 0.801649i \(-0.703957\pi\)
−0.597796 + 0.801649i \(0.703957\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 19.2690i − 0.779539i
\(612\) 0 0
\(613\) − 16.9457i − 0.684429i −0.939622 0.342215i \(-0.888823\pi\)
0.939622 0.342215i \(-0.111177\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −42.1917 −1.69858 −0.849288 0.527930i \(-0.822968\pi\)
−0.849288 + 0.527930i \(0.822968\pi\)
\(618\) 0 0
\(619\) 47.7508i 1.91927i 0.281256 + 0.959633i \(0.409249\pi\)
−0.281256 + 0.959633i \(0.590751\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −1.54944 −0.0620770
\(624\) 0 0
\(625\) −14.0142 −0.560569
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 32.2942i − 1.28765i
\(630\) 0 0
\(631\) −29.3539 −1.16856 −0.584279 0.811553i \(-0.698623\pi\)
−0.584279 + 0.811553i \(0.698623\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 17.8124i 0.706863i
\(636\) 0 0
\(637\) − 3.09942i − 0.122803i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −15.2796 −0.603509 −0.301754 0.953386i \(-0.597572\pi\)
−0.301754 + 0.953386i \(0.597572\pi\)
\(642\) 0 0
\(643\) − 26.8547i − 1.05905i −0.848295 0.529524i \(-0.822371\pi\)
0.848295 0.529524i \(-0.177629\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −13.6808 −0.537850 −0.268925 0.963161i \(-0.586668\pi\)
−0.268925 + 0.963161i \(0.586668\pi\)
\(648\) 0 0
\(649\) 46.7450 1.83490
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 5.65064i − 0.221127i −0.993869 0.110563i \(-0.964734\pi\)
0.993869 0.110563i \(-0.0352655\pi\)
\(654\) 0 0
\(655\) −30.6167 −1.19629
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 24.7127i − 0.962669i −0.876537 0.481334i \(-0.840152\pi\)
0.876537 0.481334i \(-0.159848\pi\)
\(660\) 0 0
\(661\) 20.1467i 0.783615i 0.920047 + 0.391807i \(0.128150\pi\)
−0.920047 + 0.391807i \(0.871850\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −3.02470 −0.117293
\(666\) 0 0
\(667\) 17.8479i 0.691072i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −37.2845 −1.43935
\(672\) 0 0
\(673\) −23.0926 −0.890155 −0.445077 0.895492i \(-0.646824\pi\)
−0.445077 + 0.895492i \(0.646824\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 5.41515i − 0.208121i −0.994571 0.104061i \(-0.966816\pi\)
0.994571 0.104061i \(-0.0331836\pi\)
\(678\) 0 0
\(679\) 10.2583 0.393679
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 16.5403i − 0.632898i −0.948609 0.316449i \(-0.897509\pi\)
0.948609 0.316449i \(-0.102491\pi\)
\(684\) 0 0
\(685\) − 27.7951i − 1.06200i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −9.96261 −0.379545
\(690\) 0 0
\(691\) − 17.8893i − 0.680542i −0.940327 0.340271i \(-0.889481\pi\)
0.940327 0.340271i \(-0.110519\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 17.0606 0.647147
\(696\) 0 0
\(697\) 10.9869 0.416160
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 28.6956i 1.08382i 0.840437 + 0.541909i \(0.182298\pi\)
−0.840437 + 0.541909i \(0.817702\pi\)
\(702\) 0 0
\(703\) −14.7840 −0.557590
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 5.97880i 0.224856i
\(708\) 0 0
\(709\) − 43.2083i − 1.62272i −0.584546 0.811360i \(-0.698728\pi\)
0.584546 0.811360i \(-0.301272\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 67.1122 2.51337
\(714\) 0 0
\(715\) 21.3173i 0.797221i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −40.7179 −1.51852 −0.759261 0.650786i \(-0.774440\pi\)
−0.759261 + 0.650786i \(0.774440\pi\)
\(720\) 0 0
\(721\) −16.9060 −0.629611
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 3.94119i − 0.146372i
\(726\) 0 0
\(727\) 1.96551 0.0728967 0.0364483 0.999336i \(-0.488396\pi\)
0.0364483 + 0.999336i \(0.488396\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 6.59310i − 0.243854i
\(732\) 0 0
\(733\) − 39.1211i − 1.44497i −0.691386 0.722486i \(-0.742999\pi\)
0.691386 0.722486i \(-0.257001\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 32.8190 1.20890
\(738\) 0 0
\(739\) 36.1851i 1.33109i 0.746357 + 0.665546i \(0.231801\pi\)
−0.746357 + 0.665546i \(0.768199\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −20.1222 −0.738210 −0.369105 0.929388i \(-0.620336\pi\)
−0.369105 + 0.929388i \(0.620336\pi\)
\(744\) 0 0
\(745\) −6.08103 −0.222792
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 2.14315i 0.0783089i
\(750\) 0 0
\(751\) 36.2627 1.32324 0.661622 0.749838i \(-0.269869\pi\)
0.661622 + 0.749838i \(0.269869\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 22.6459i 0.824167i
\(756\) 0 0
\(757\) 53.6865i 1.95127i 0.219401 + 0.975635i \(0.429590\pi\)
−0.219401 + 0.975635i \(0.570410\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −37.3332 −1.35333 −0.676664 0.736292i \(-0.736575\pi\)
−0.676664 + 0.736292i \(0.736575\pi\)
\(762\) 0 0
\(763\) − 8.34775i − 0.302209i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 38.5462 1.39182
\(768\) 0 0
\(769\) 15.5423 0.560469 0.280235 0.959932i \(-0.409588\pi\)
0.280235 + 0.959932i \(0.409588\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 39.0073i − 1.40299i −0.712672 0.701497i \(-0.752515\pi\)
0.712672 0.701497i \(-0.247485\pi\)
\(774\) 0 0
\(775\) −14.8198 −0.532343
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 5.02972i − 0.180209i
\(780\) 0 0
\(781\) 46.6933i 1.67082i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 23.7537 0.847807
\(786\) 0 0
\(787\) − 9.50805i − 0.338925i −0.985537 0.169463i \(-0.945797\pi\)
0.985537 0.169463i \(-0.0542032\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −17.5901 −0.625432
\(792\) 0 0
\(793\) −30.7450 −1.09179
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 40.4120i 1.43147i 0.698374 + 0.715733i \(0.253907\pi\)
−0.698374 + 0.715733i \(0.746093\pi\)
\(798\) 0 0
\(799\) −22.4479 −0.794149
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 36.8219i − 1.29942i
\(804\) 0 0
\(805\) − 13.6862i − 0.482374i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −32.4907 −1.14231 −0.571157 0.820841i \(-0.693505\pi\)
−0.571157 + 0.820841i \(0.693505\pi\)
\(810\) 0 0
\(811\) 8.34275i 0.292954i 0.989214 + 0.146477i \(0.0467934\pi\)
−0.989214 + 0.146477i \(0.953207\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 11.2922 0.395550
\(816\) 0 0
\(817\) −3.01826 −0.105596
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 30.6627i 1.07014i 0.844809 + 0.535068i \(0.179714\pi\)
−0.844809 + 0.535068i \(0.820286\pi\)
\(822\) 0 0
\(823\) 25.5627 0.891059 0.445530 0.895267i \(-0.353015\pi\)
0.445530 + 0.895267i \(0.353015\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 12.4138i − 0.431670i −0.976430 0.215835i \(-0.930753\pi\)
0.976430 0.215835i \(-0.0692474\pi\)
\(828\) 0 0
\(829\) − 21.6360i − 0.751449i −0.926731 0.375724i \(-0.877394\pi\)
0.926731 0.375724i \(-0.122606\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −3.61074 −0.125105
\(834\) 0 0
\(835\) 10.5238i 0.364191i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1.76288 0.0608612 0.0304306 0.999537i \(-0.490312\pi\)
0.0304306 + 0.999537i \(0.490312\pi\)
\(840\) 0 0
\(841\) 23.3056 0.803642
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 6.20987i − 0.213626i
\(846\) 0 0
\(847\) 3.12753 0.107463
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 66.8946i − 2.29312i
\(852\) 0 0
\(853\) − 0.627765i − 0.0214943i −0.999942 0.0107471i \(-0.996579\pi\)
0.999942 0.0107471i \(-0.00342098\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −35.3406 −1.20721 −0.603605 0.797283i \(-0.706270\pi\)
−0.603605 + 0.797283i \(0.706270\pi\)
\(858\) 0 0
\(859\) − 50.5205i − 1.72374i −0.507133 0.861868i \(-0.669295\pi\)
0.507133 0.861868i \(-0.330705\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −35.7081 −1.21552 −0.607758 0.794122i \(-0.707931\pi\)
−0.607758 + 0.794122i \(0.707931\pi\)
\(864\) 0 0
\(865\) −14.8164 −0.503771
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 27.2273i − 0.923623i
\(870\) 0 0
\(871\) 27.0627 0.916986
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 12.1715i 0.411472i
\(876\) 0 0
\(877\) − 43.7066i − 1.47587i −0.674873 0.737934i \(-0.735802\pi\)
0.674873 0.737934i \(-0.264198\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −39.3404 −1.32541 −0.662705 0.748880i \(-0.730592\pi\)
−0.662705 + 0.748880i \(0.730592\pi\)
\(882\) 0 0
\(883\) − 14.5600i − 0.489981i −0.969525 0.244991i \(-0.921215\pi\)
0.969525 0.244991i \(-0.0787849\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 24.2186 0.813182 0.406591 0.913610i \(-0.366717\pi\)
0.406591 + 0.913610i \(0.366717\pi\)
\(888\) 0 0
\(889\) 9.73427 0.326477
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 10.2764i 0.343888i
\(894\) 0 0
\(895\) 28.2033 0.942733
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 21.4122i 0.714137i
\(900\) 0 0
\(901\) 11.6062i 0.386658i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 11.1040 0.369111
\(906\) 0 0
\(907\) − 32.8766i − 1.09165i −0.837899 0.545825i \(-0.816216\pi\)
0.837899 0.545825i \(-0.183784\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 17.8694 0.592039 0.296020 0.955182i \(-0.404341\pi\)
0.296020 + 0.955182i \(0.404341\pi\)
\(912\) 0 0
\(913\) 33.8026 1.11870
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 16.7317i 0.552528i
\(918\) 0 0
\(919\) −25.3511 −0.836257 −0.418128 0.908388i \(-0.637314\pi\)
−0.418128 + 0.908388i \(0.637314\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 38.5036i 1.26736i
\(924\) 0 0
\(925\) 14.7718i 0.485693i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 39.3410 1.29074 0.645368 0.763872i \(-0.276704\pi\)
0.645368 + 0.763872i \(0.276704\pi\)
\(930\) 0 0
\(931\) 1.65297i 0.0541738i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 24.8341 0.812162
\(936\) 0 0
\(937\) 44.9769 1.46933 0.734665 0.678430i \(-0.237339\pi\)
0.734665 + 0.678430i \(0.237339\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0.488014i 0.0159088i 0.999968 + 0.00795440i \(0.00253199\pi\)
−0.999968 + 0.00795440i \(0.997468\pi\)
\(942\) 0 0
\(943\) 22.7585 0.741117
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 1.70377i − 0.0553652i −0.999617 0.0276826i \(-0.991187\pi\)
0.999617 0.0276826i \(-0.00881278\pi\)
\(948\) 0 0
\(949\) − 30.3636i − 0.985643i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 9.15979 0.296715 0.148357 0.988934i \(-0.452601\pi\)
0.148357 + 0.988934i \(0.452601\pi\)
\(954\) 0 0
\(955\) 5.06198i 0.163802i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −15.1897 −0.490501
\(960\) 0 0
\(961\) 49.5149 1.59726
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 32.8976i 1.05901i
\(966\) 0 0
\(967\) −36.1742 −1.16328 −0.581642 0.813445i \(-0.697589\pi\)
−0.581642 + 0.813445i \(0.697589\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 7.10607i − 0.228045i −0.993478 0.114022i \(-0.963626\pi\)
0.993478 0.114022i \(-0.0363735\pi\)
\(972\) 0 0
\(973\) − 9.32344i − 0.298896i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −56.6938 −1.81380 −0.906898 0.421349i \(-0.861557\pi\)
−0.906898 + 0.421349i \(0.861557\pi\)
\(978\) 0 0
\(979\) − 5.82382i − 0.186130i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −21.6213 −0.689614 −0.344807 0.938674i \(-0.612056\pi\)
−0.344807 + 0.938674i \(0.612056\pi\)
\(984\) 0 0
\(985\) −19.2463 −0.613238
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 13.6570i − 0.434268i
\(990\) 0 0
\(991\) −22.3945 −0.711384 −0.355692 0.934603i \(-0.615755\pi\)
−0.355692 + 0.934603i \(0.615755\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 7.50195i 0.237828i
\(996\) 0 0
\(997\) − 11.3181i − 0.358448i −0.983808 0.179224i \(-0.942641\pi\)
0.983808 0.179224i \(-0.0573587\pi\)
\(998\) 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.c.f.3025.7 24
3.2 odd 2 inner 6048.2.c.f.3025.17 24
4.3 odd 2 1512.2.c.g.757.4 yes 24
8.3 odd 2 1512.2.c.g.757.3 24
8.5 even 2 inner 6048.2.c.f.3025.18 24
12.11 even 2 1512.2.c.g.757.21 yes 24
24.5 odd 2 inner 6048.2.c.f.3025.8 24
24.11 even 2 1512.2.c.g.757.22 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.c.g.757.3 24 8.3 odd 2
1512.2.c.g.757.4 yes 24 4.3 odd 2
1512.2.c.g.757.21 yes 24 12.11 even 2
1512.2.c.g.757.22 yes 24 24.11 even 2
6048.2.c.f.3025.7 24 1.1 even 1 trivial
6048.2.c.f.3025.8 24 24.5 odd 2 inner
6048.2.c.f.3025.17 24 3.2 odd 2 inner
6048.2.c.f.3025.18 24 8.5 even 2 inner