Properties

Label 6048.2.j.d.5615.7
Level 6048
Weight 2
Character 6048.5615
Analytic conductor 48.294
Analytic rank 0
Dimension 48
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6048.j (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(48\)
Coefficient ring index: multiple of None
Twist minimal: no (minimal twist has level 1512)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 5615.7
Character \(\chi\) = 6048.5615
Dual form 6048.2.j.d.5615.8

$q$-expansion

\(f(q)\) \(=\) \(q-2.90802 q^{5} +1.00000i q^{7} +O(q^{10})\) \(q-2.90802 q^{5} +1.00000i q^{7} -1.28828i q^{11} -1.86999i q^{13} +3.87222i q^{17} +2.04228 q^{19} -0.934839 q^{23} +3.45657 q^{25} -2.98295 q^{29} -1.85691i q^{31} -2.90802i q^{35} +3.85846i q^{37} -8.72247i q^{41} +1.19319 q^{43} -11.1537 q^{47} -1.00000 q^{49} +4.43885 q^{53} +3.74635i q^{55} +10.9714i q^{59} -4.70930i q^{61} +5.43797i q^{65} -4.41800 q^{67} -9.72888 q^{71} +9.40756 q^{73} +1.28828 q^{77} -16.4660i q^{79} +3.72150i q^{83} -11.2605i q^{85} +12.2297i q^{89} +1.86999 q^{91} -5.93900 q^{95} +15.8302 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48q + O(q^{10}) \) \( 48q - 16q^{19} + 48q^{25} - 64q^{43} - 48q^{49} - 32q^{67} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times\).

\(n\) \(2593\) \(3781\) \(3809\) \(4159\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −2.90802 −1.30051 −0.650253 0.759718i \(-0.725337\pi\)
−0.650253 + 0.759718i \(0.725337\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 1.28828i − 0.388432i −0.980959 0.194216i \(-0.937784\pi\)
0.980959 0.194216i \(-0.0622163\pi\)
\(12\) 0 0
\(13\) − 1.86999i − 0.518642i −0.965791 0.259321i \(-0.916501\pi\)
0.965791 0.259321i \(-0.0834988\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.87222i 0.939151i 0.882892 + 0.469576i \(0.155593\pi\)
−0.882892 + 0.469576i \(0.844407\pi\)
\(18\) 0 0
\(19\) 2.04228 0.468532 0.234266 0.972173i \(-0.424731\pi\)
0.234266 + 0.972173i \(0.424731\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −0.934839 −0.194927 −0.0974637 0.995239i \(-0.531073\pi\)
−0.0974637 + 0.995239i \(0.531073\pi\)
\(24\) 0 0
\(25\) 3.45657 0.691315
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.98295 −0.553920 −0.276960 0.960881i \(-0.589327\pi\)
−0.276960 + 0.960881i \(0.589327\pi\)
\(30\) 0 0
\(31\) − 1.85691i − 0.333511i −0.985998 0.166756i \(-0.946671\pi\)
0.985998 0.166756i \(-0.0533291\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 2.90802i − 0.491545i
\(36\) 0 0
\(37\) 3.85846i 0.634327i 0.948371 + 0.317163i \(0.102730\pi\)
−0.948371 + 0.317163i \(0.897270\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 8.72247i − 1.36222i −0.732181 0.681110i \(-0.761497\pi\)
0.732181 0.681110i \(-0.238503\pi\)
\(42\) 0 0
\(43\) 1.19319 0.181959 0.0909796 0.995853i \(-0.471000\pi\)
0.0909796 + 0.995853i \(0.471000\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −11.1537 −1.62694 −0.813468 0.581609i \(-0.802423\pi\)
−0.813468 + 0.581609i \(0.802423\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.43885 0.609723 0.304862 0.952397i \(-0.401390\pi\)
0.304862 + 0.952397i \(0.401390\pi\)
\(54\) 0 0
\(55\) 3.74635i 0.505158i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 10.9714i 1.42836i 0.699964 + 0.714178i \(0.253199\pi\)
−0.699964 + 0.714178i \(0.746801\pi\)
\(60\) 0 0
\(61\) − 4.70930i − 0.602964i −0.953472 0.301482i \(-0.902519\pi\)
0.953472 0.301482i \(-0.0974813\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 5.43797i 0.674497i
\(66\) 0 0
\(67\) −4.41800 −0.539745 −0.269872 0.962896i \(-0.586981\pi\)
−0.269872 + 0.962896i \(0.586981\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −9.72888 −1.15461 −0.577303 0.816530i \(-0.695895\pi\)
−0.577303 + 0.816530i \(0.695895\pi\)
\(72\) 0 0
\(73\) 9.40756 1.10107 0.550536 0.834812i \(-0.314423\pi\)
0.550536 + 0.834812i \(0.314423\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.28828 0.146813
\(78\) 0 0
\(79\) − 16.4660i − 1.85257i −0.376822 0.926286i \(-0.622983\pi\)
0.376822 0.926286i \(-0.377017\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.72150i 0.408488i 0.978920 + 0.204244i \(0.0654736\pi\)
−0.978920 + 0.204244i \(0.934526\pi\)
\(84\) 0 0
\(85\) − 11.2605i − 1.22137i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 12.2297i 1.29635i 0.761492 + 0.648174i \(0.224467\pi\)
−0.761492 + 0.648174i \(0.775533\pi\)
\(90\) 0 0
\(91\) 1.86999 0.196028
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −5.93900 −0.609329
\(96\) 0 0
\(97\) 15.8302 1.60731 0.803657 0.595093i \(-0.202885\pi\)
0.803657 + 0.595093i \(0.202885\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0.737825 0.0734163 0.0367081 0.999326i \(-0.488313\pi\)
0.0367081 + 0.999326i \(0.488313\pi\)
\(102\) 0 0
\(103\) 16.2518i 1.60134i 0.599109 + 0.800668i \(0.295522\pi\)
−0.599109 + 0.800668i \(0.704478\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 1.49407i − 0.144438i −0.997389 0.0722188i \(-0.976992\pi\)
0.997389 0.0722188i \(-0.0230080\pi\)
\(108\) 0 0
\(109\) 11.8362i 1.13370i 0.823821 + 0.566850i \(0.191838\pi\)
−0.823821 + 0.566850i \(0.808162\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 15.3287i − 1.44200i −0.692934 0.721001i \(-0.743682\pi\)
0.692934 0.721001i \(-0.256318\pi\)
\(114\) 0 0
\(115\) 2.71853 0.253504
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −3.87222 −0.354966
\(120\) 0 0
\(121\) 9.34033 0.849121
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 4.48831 0.401447
\(126\) 0 0
\(127\) 6.55275i 0.581462i 0.956805 + 0.290731i \(0.0938985\pi\)
−0.956805 + 0.290731i \(0.906101\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 11.6893i − 1.02130i −0.859790 0.510648i \(-0.829405\pi\)
0.859790 0.510648i \(-0.170595\pi\)
\(132\) 0 0
\(133\) 2.04228i 0.177088i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 7.35868i − 0.628694i −0.949308 0.314347i \(-0.898214\pi\)
0.949308 0.314347i \(-0.101786\pi\)
\(138\) 0 0
\(139\) 15.2683 1.29504 0.647520 0.762048i \(-0.275806\pi\)
0.647520 + 0.762048i \(0.275806\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −2.40908 −0.201457
\(144\) 0 0
\(145\) 8.67448 0.720376
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 16.6778 1.36630 0.683148 0.730280i \(-0.260611\pi\)
0.683148 + 0.730280i \(0.260611\pi\)
\(150\) 0 0
\(151\) − 16.4838i − 1.34143i −0.741714 0.670716i \(-0.765987\pi\)
0.741714 0.670716i \(-0.234013\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 5.39994i 0.433733i
\(156\) 0 0
\(157\) − 16.0196i − 1.27851i −0.768997 0.639253i \(-0.779244\pi\)
0.768997 0.639253i \(-0.220756\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 0.934839i − 0.0736756i
\(162\) 0 0
\(163\) −6.18368 −0.484343 −0.242172 0.970233i \(-0.577860\pi\)
−0.242172 + 0.970233i \(0.577860\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −12.7232 −0.984549 −0.492275 0.870440i \(-0.663834\pi\)
−0.492275 + 0.870440i \(0.663834\pi\)
\(168\) 0 0
\(169\) 9.50313 0.731010
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 11.2579 0.855925 0.427963 0.903796i \(-0.359231\pi\)
0.427963 + 0.903796i \(0.359231\pi\)
\(174\) 0 0
\(175\) 3.45657i 0.261292i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 8.68048i − 0.648809i −0.945918 0.324405i \(-0.894836\pi\)
0.945918 0.324405i \(-0.105164\pi\)
\(180\) 0 0
\(181\) 2.48587i 0.184773i 0.995723 + 0.0923866i \(0.0294496\pi\)
−0.995723 + 0.0923866i \(0.970550\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 11.2205i − 0.824945i
\(186\) 0 0
\(187\) 4.98851 0.364796
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3.40263 0.246206 0.123103 0.992394i \(-0.460715\pi\)
0.123103 + 0.992394i \(0.460715\pi\)
\(192\) 0 0
\(193\) 20.7031 1.49024 0.745121 0.666929i \(-0.232392\pi\)
0.745121 + 0.666929i \(0.232392\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 16.3546 1.16521 0.582607 0.812754i \(-0.302033\pi\)
0.582607 + 0.812754i \(0.302033\pi\)
\(198\) 0 0
\(199\) 11.3717i 0.806121i 0.915173 + 0.403061i \(0.132054\pi\)
−0.915173 + 0.403061i \(0.867946\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 2.98295i − 0.209362i
\(204\) 0 0
\(205\) 25.3651i 1.77158i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 2.63104i − 0.181993i
\(210\) 0 0
\(211\) 2.56405 0.176517 0.0882583 0.996098i \(-0.471870\pi\)
0.0882583 + 0.996098i \(0.471870\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −3.46981 −0.236639
\(216\) 0 0
\(217\) 1.85691 0.126055
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 7.24102 0.487083
\(222\) 0 0
\(223\) 1.37584i 0.0921329i 0.998938 + 0.0460665i \(0.0146686\pi\)
−0.998938 + 0.0460665i \(0.985331\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 27.3740i 1.81688i 0.418020 + 0.908438i \(0.362724\pi\)
−0.418020 + 0.908438i \(0.637276\pi\)
\(228\) 0 0
\(229\) − 12.3066i − 0.813243i −0.913597 0.406621i \(-0.866707\pi\)
0.913597 0.406621i \(-0.133293\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 24.0829i 1.57772i 0.614571 + 0.788862i \(0.289329\pi\)
−0.614571 + 0.788862i \(0.710671\pi\)
\(234\) 0 0
\(235\) 32.4352 2.11584
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 18.0426 1.16708 0.583540 0.812085i \(-0.301667\pi\)
0.583540 + 0.812085i \(0.301667\pi\)
\(240\) 0 0
\(241\) −10.8541 −0.699174 −0.349587 0.936904i \(-0.613678\pi\)
−0.349587 + 0.936904i \(0.613678\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.90802 0.185787
\(246\) 0 0
\(247\) − 3.81905i − 0.243001i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 6.63032i − 0.418502i −0.977862 0.209251i \(-0.932897\pi\)
0.977862 0.209251i \(-0.0671025\pi\)
\(252\) 0 0
\(253\) 1.20434i 0.0757160i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 12.2398i 0.763497i 0.924266 + 0.381748i \(0.124678\pi\)
−0.924266 + 0.381748i \(0.875322\pi\)
\(258\) 0 0
\(259\) −3.85846 −0.239753
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2.87474 0.177264 0.0886320 0.996064i \(-0.471750\pi\)
0.0886320 + 0.996064i \(0.471750\pi\)
\(264\) 0 0
\(265\) −12.9083 −0.792948
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 6.17565 0.376536 0.188268 0.982118i \(-0.439713\pi\)
0.188268 + 0.982118i \(0.439713\pi\)
\(270\) 0 0
\(271\) 19.5103i 1.18517i 0.805510 + 0.592583i \(0.201892\pi\)
−0.805510 + 0.592583i \(0.798108\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 4.45305i − 0.268529i
\(276\) 0 0
\(277\) − 2.82757i − 0.169892i −0.996386 0.0849460i \(-0.972928\pi\)
0.996386 0.0849460i \(-0.0270718\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 29.7569i − 1.77515i −0.460663 0.887575i \(-0.652388\pi\)
0.460663 0.887575i \(-0.347612\pi\)
\(282\) 0 0
\(283\) 30.7427 1.82746 0.913731 0.406319i \(-0.133188\pi\)
0.913731 + 0.406319i \(0.133188\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 8.72247 0.514871
\(288\) 0 0
\(289\) 2.00592 0.117995
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 5.70430 0.333249 0.166624 0.986020i \(-0.446713\pi\)
0.166624 + 0.986020i \(0.446713\pi\)
\(294\) 0 0
\(295\) − 31.9051i − 1.85758i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.74814i 0.101098i
\(300\) 0 0
\(301\) 1.19319i 0.0687741i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 13.6947i 0.784158i
\(306\) 0 0
\(307\) 6.47411 0.369497 0.184748 0.982786i \(-0.440853\pi\)
0.184748 + 0.982786i \(0.440853\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −2.44723 −0.138770 −0.0693848 0.997590i \(-0.522104\pi\)
−0.0693848 + 0.997590i \(0.522104\pi\)
\(312\) 0 0
\(313\) −8.03434 −0.454128 −0.227064 0.973880i \(-0.572913\pi\)
−0.227064 + 0.973880i \(0.572913\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 19.0098 1.06770 0.533850 0.845579i \(-0.320745\pi\)
0.533850 + 0.845579i \(0.320745\pi\)
\(318\) 0 0
\(319\) 3.84289i 0.215160i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 7.90817i 0.440022i
\(324\) 0 0
\(325\) − 6.46376i − 0.358545i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 11.1537i − 0.614924i
\(330\) 0 0
\(331\) 35.8597 1.97103 0.985515 0.169588i \(-0.0542438\pi\)
0.985515 + 0.169588i \(0.0542438\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 12.8476 0.701941
\(336\) 0 0
\(337\) −6.90274 −0.376016 −0.188008 0.982168i \(-0.560203\pi\)
−0.188008 + 0.982168i \(0.560203\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −2.39223 −0.129546
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 36.6932i − 1.96980i −0.173135 0.984898i \(-0.555390\pi\)
0.173135 0.984898i \(-0.444610\pi\)
\(348\) 0 0
\(349\) 28.2560i 1.51251i 0.654278 + 0.756254i \(0.272973\pi\)
−0.654278 + 0.756254i \(0.727027\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 24.3138i 1.29410i 0.762450 + 0.647048i \(0.223997\pi\)
−0.762450 + 0.647048i \(0.776003\pi\)
\(354\) 0 0
\(355\) 28.2918 1.50157
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1.95382 −0.103119 −0.0515594 0.998670i \(-0.516419\pi\)
−0.0515594 + 0.998670i \(0.516419\pi\)
\(360\) 0 0
\(361\) −14.8291 −0.780478
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −27.3574 −1.43195
\(366\) 0 0
\(367\) − 21.3819i − 1.11613i −0.829798 0.558064i \(-0.811544\pi\)
0.829798 0.558064i \(-0.188456\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 4.43885i 0.230454i
\(372\) 0 0
\(373\) 33.0682i 1.71221i 0.516804 + 0.856104i \(0.327122\pi\)
−0.516804 + 0.856104i \(0.672878\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 5.57809i 0.287286i
\(378\) 0 0
\(379\) 17.0331 0.874934 0.437467 0.899235i \(-0.355876\pi\)
0.437467 + 0.899235i \(0.355876\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 15.7372 0.804132 0.402066 0.915611i \(-0.368292\pi\)
0.402066 + 0.915611i \(0.368292\pi\)
\(384\) 0 0
\(385\) −3.74635 −0.190932
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −22.7854 −1.15527 −0.577634 0.816296i \(-0.696024\pi\)
−0.577634 + 0.816296i \(0.696024\pi\)
\(390\) 0 0
\(391\) − 3.61990i − 0.183066i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 47.8835i 2.40928i
\(396\) 0 0
\(397\) − 23.6829i − 1.18861i −0.804239 0.594306i \(-0.797427\pi\)
0.804239 0.594306i \(-0.202573\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 1.51029i − 0.0754204i −0.999289 0.0377102i \(-0.987994\pi\)
0.999289 0.0377102i \(-0.0120064\pi\)
\(402\) 0 0
\(403\) −3.47241 −0.172973
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 4.97079 0.246393
\(408\) 0 0
\(409\) 9.86160 0.487625 0.243812 0.969822i \(-0.421602\pi\)
0.243812 + 0.969822i \(0.421602\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −10.9714 −0.539868
\(414\) 0 0
\(415\) − 10.8222i − 0.531241i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 20.8183i − 1.01704i −0.861050 0.508520i \(-0.830193\pi\)
0.861050 0.508520i \(-0.169807\pi\)
\(420\) 0 0
\(421\) − 28.5459i − 1.39124i −0.718409 0.695621i \(-0.755129\pi\)
0.718409 0.695621i \(-0.244871\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 13.3846i 0.649249i
\(426\) 0 0
\(427\) 4.70930 0.227899
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 6.53111 0.314593 0.157296 0.987551i \(-0.449722\pi\)
0.157296 + 0.987551i \(0.449722\pi\)
\(432\) 0 0
\(433\) 14.2151 0.683136 0.341568 0.939857i \(-0.389042\pi\)
0.341568 + 0.939857i \(0.389042\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1.90921 −0.0913297
\(438\) 0 0
\(439\) 22.5811i 1.07773i 0.842391 + 0.538867i \(0.181148\pi\)
−0.842391 + 0.538867i \(0.818852\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 15.7932i 0.750359i 0.926952 + 0.375179i \(0.122419\pi\)
−0.926952 + 0.375179i \(0.877581\pi\)
\(444\) 0 0
\(445\) − 35.5643i − 1.68591i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 26.9861i − 1.27355i −0.771049 0.636776i \(-0.780267\pi\)
0.771049 0.636776i \(-0.219733\pi\)
\(450\) 0 0
\(451\) −11.2370 −0.529130
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −5.43797 −0.254936
\(456\) 0 0
\(457\) 37.9588 1.77564 0.887818 0.460195i \(-0.152220\pi\)
0.887818 + 0.460195i \(0.152220\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −8.35921 −0.389327 −0.194664 0.980870i \(-0.562362\pi\)
−0.194664 + 0.980870i \(0.562362\pi\)
\(462\) 0 0
\(463\) − 18.4723i − 0.858483i −0.903190 0.429241i \(-0.858781\pi\)
0.903190 0.429241i \(-0.141219\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 4.34051i 0.200855i 0.994944 + 0.100427i \(0.0320210\pi\)
−0.994944 + 0.100427i \(0.967979\pi\)
\(468\) 0 0
\(469\) − 4.41800i − 0.204004i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 1.53716i − 0.0706788i
\(474\) 0 0
\(475\) 7.05931 0.323903
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 33.7292 1.54113 0.770564 0.637363i \(-0.219975\pi\)
0.770564 + 0.637363i \(0.219975\pi\)
\(480\) 0 0
\(481\) 7.21528 0.328989
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −46.0345 −2.09032
\(486\) 0 0
\(487\) 6.00662i 0.272186i 0.990696 + 0.136093i \(0.0434546\pi\)
−0.990696 + 0.136093i \(0.956545\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 26.2731i − 1.18569i −0.805318 0.592844i \(-0.798005\pi\)
0.805318 0.592844i \(-0.201995\pi\)
\(492\) 0 0
\(493\) − 11.5506i − 0.520215i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 9.72888i − 0.436400i
\(498\) 0 0
\(499\) −35.9974 −1.61147 −0.805733 0.592279i \(-0.798228\pi\)
−0.805733 + 0.592279i \(0.798228\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 15.7775 0.703485 0.351743 0.936097i \(-0.385589\pi\)
0.351743 + 0.936097i \(0.385589\pi\)
\(504\) 0 0
\(505\) −2.14561 −0.0954783
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −6.09029 −0.269947 −0.134974 0.990849i \(-0.543095\pi\)
−0.134974 + 0.990849i \(0.543095\pi\)
\(510\) 0 0
\(511\) 9.40756i 0.416166i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 47.2605i − 2.08255i
\(516\) 0 0
\(517\) 14.3691i 0.631954i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 18.1215i 0.793916i 0.917837 + 0.396958i \(0.129934\pi\)
−0.917837 + 0.396958i \(0.870066\pi\)
\(522\) 0 0
\(523\) −4.00317 −0.175046 −0.0875232 0.996162i \(-0.527895\pi\)
−0.0875232 + 0.996162i \(0.527895\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 7.19037 0.313217
\(528\) 0 0
\(529\) −22.1261 −0.962003
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −16.3109 −0.706505
\(534\) 0 0
\(535\) 4.34480i 0.187842i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1.28828i 0.0554903i
\(540\) 0 0
\(541\) 4.52777i 0.194664i 0.995252 + 0.0973320i \(0.0310309\pi\)
−0.995252 + 0.0973320i \(0.968969\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 34.4198i − 1.47438i
\(546\) 0 0
\(547\) −1.99785 −0.0854218 −0.0427109 0.999087i \(-0.513599\pi\)
−0.0427109 + 0.999087i \(0.513599\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −6.09203 −0.259529
\(552\) 0 0
\(553\) 16.4660 0.700206
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 14.9242 0.632360 0.316180 0.948699i \(-0.397600\pi\)
0.316180 + 0.948699i \(0.397600\pi\)
\(558\) 0 0
\(559\) − 2.23125i − 0.0943717i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 0.157861i − 0.00665305i −0.999994 0.00332653i \(-0.998941\pi\)
0.999994 0.00332653i \(-0.00105887\pi\)
\(564\) 0 0
\(565\) 44.5761i 1.87533i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 11.8914i 0.498514i 0.968437 + 0.249257i \(0.0801864\pi\)
−0.968437 + 0.249257i \(0.919814\pi\)
\(570\) 0 0
\(571\) −14.0748 −0.589013 −0.294507 0.955649i \(-0.595155\pi\)
−0.294507 + 0.955649i \(0.595155\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −3.23134 −0.134756
\(576\) 0 0
\(577\) 25.5261 1.06267 0.531333 0.847163i \(-0.321691\pi\)
0.531333 + 0.847163i \(0.321691\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −3.72150 −0.154394
\(582\) 0 0
\(583\) − 5.71850i − 0.236836i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 9.47422i 0.391043i 0.980699 + 0.195521i \(0.0626399\pi\)
−0.980699 + 0.195521i \(0.937360\pi\)
\(588\) 0 0
\(589\) − 3.79234i − 0.156261i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 18.9405i 0.777792i 0.921282 + 0.388896i \(0.127143\pi\)
−0.921282 + 0.388896i \(0.872857\pi\)
\(594\) 0 0
\(595\) 11.2605 0.461635
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −34.9184 −1.42673 −0.713364 0.700794i \(-0.752829\pi\)
−0.713364 + 0.700794i \(0.752829\pi\)
\(600\) 0 0
\(601\) 25.8342 1.05380 0.526899 0.849928i \(-0.323355\pi\)
0.526899 + 0.849928i \(0.323355\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −27.1618 −1.10429
\(606\) 0 0
\(607\) 14.4361i 0.585943i 0.956121 + 0.292972i \(0.0946442\pi\)
−0.956121 + 0.292972i \(0.905356\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 20.8574i 0.843798i
\(612\) 0 0
\(613\) − 46.3647i − 1.87265i −0.351133 0.936325i \(-0.614204\pi\)
0.351133 0.936325i \(-0.385796\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 29.3648i − 1.18218i −0.806605 0.591091i \(-0.798697\pi\)
0.806605 0.591091i \(-0.201303\pi\)
\(618\) 0 0
\(619\) −13.7590 −0.553021 −0.276510 0.961011i \(-0.589178\pi\)
−0.276510 + 0.961011i \(0.589178\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −12.2297 −0.489974
\(624\) 0 0
\(625\) −30.3350 −1.21340
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −14.9408 −0.595729
\(630\) 0 0
\(631\) − 32.9228i − 1.31064i −0.755352 0.655319i \(-0.772534\pi\)
0.755352 0.655319i \(-0.227466\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 19.0555i − 0.756195i
\(636\) 0 0
\(637\) 1.86999i 0.0740918i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 39.5844i 1.56349i 0.623597 + 0.781746i \(0.285670\pi\)
−0.623597 + 0.781746i \(0.714330\pi\)
\(642\) 0 0
\(643\) −44.0571 −1.73744 −0.868720 0.495303i \(-0.835057\pi\)
−0.868720 + 0.495303i \(0.835057\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −39.4430 −1.55066 −0.775331 0.631555i \(-0.782417\pi\)
−0.775331 + 0.631555i \(0.782417\pi\)
\(648\) 0 0
\(649\) 14.1343 0.554819
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −43.4794 −1.70148 −0.850741 0.525585i \(-0.823846\pi\)
−0.850741 + 0.525585i \(0.823846\pi\)
\(654\) 0 0
\(655\) 33.9926i 1.32820i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 26.8653i 1.04652i 0.852172 + 0.523262i \(0.175285\pi\)
−0.852172 + 0.523262i \(0.824715\pi\)
\(660\) 0 0
\(661\) − 0.365746i − 0.0142259i −0.999975 0.00711293i \(-0.997736\pi\)
0.999975 0.00711293i \(-0.00226414\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 5.93900i − 0.230305i
\(666\) 0 0
\(667\) 2.78858 0.107974
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −6.06691 −0.234211
\(672\) 0 0
\(673\) 35.2522 1.35887 0.679435 0.733735i \(-0.262225\pi\)
0.679435 + 0.733735i \(0.262225\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 28.6841 1.10242 0.551211 0.834366i \(-0.314166\pi\)
0.551211 + 0.834366i \(0.314166\pi\)
\(678\) 0 0
\(679\) 15.8302i 0.607508i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 22.7609i − 0.870921i −0.900208 0.435460i \(-0.856586\pi\)
0.900208 0.435460i \(-0.143414\pi\)
\(684\) 0 0
\(685\) 21.3992i 0.817620i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 8.30061i − 0.316228i
\(690\) 0 0
\(691\) −36.1270 −1.37434 −0.687168 0.726499i \(-0.741146\pi\)
−0.687168 + 0.726499i \(0.741146\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −44.4005 −1.68421
\(696\) 0 0
\(697\) 33.7753 1.27933
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 24.7739 0.935699 0.467849 0.883808i \(-0.345029\pi\)
0.467849 + 0.883808i \(0.345029\pi\)
\(702\) 0 0
\(703\) 7.88007i 0.297202i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0.737825i 0.0277487i
\(708\) 0 0
\(709\) 23.4819i 0.881881i 0.897536 + 0.440941i \(0.145355\pi\)
−0.897536 + 0.440941i \(0.854645\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1.73591i 0.0650105i
\(714\) 0 0
\(715\) 7.00564 0.261996
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 38.0908 1.42055 0.710274 0.703926i \(-0.248571\pi\)
0.710274 + 0.703926i \(0.248571\pi\)
\(720\) 0 0
\(721\) −16.2518 −0.605248
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −10.3108 −0.382933
\(726\) 0 0
\(727\) − 6.53603i − 0.242408i −0.992628 0.121204i \(-0.961324\pi\)
0.992628 0.121204i \(-0.0386755\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 4.62028i 0.170887i
\(732\) 0 0
\(733\) 32.5549i 1.20244i 0.799082 + 0.601222i \(0.205319\pi\)
−0.799082 + 0.601222i \(0.794681\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 5.69163i 0.209654i
\(738\) 0 0
\(739\) 5.47095 0.201252 0.100626 0.994924i \(-0.467915\pi\)
0.100626 + 0.994924i \(0.467915\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 7.23548 0.265444 0.132722 0.991153i \(-0.457628\pi\)
0.132722 + 0.991153i \(0.457628\pi\)
\(744\) 0 0
\(745\) −48.4992 −1.77687
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1.49407 0.0545923
\(750\) 0 0
\(751\) 24.5924i 0.897391i 0.893685 + 0.448695i \(0.148111\pi\)
−0.893685 + 0.448695i \(0.851889\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 47.9352i 1.74454i
\(756\) 0 0
\(757\) − 32.5168i − 1.18184i −0.806728 0.590922i \(-0.798764\pi\)
0.806728 0.590922i \(-0.201236\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 21.5639i − 0.781689i −0.920457 0.390845i \(-0.872183\pi\)
0.920457 0.390845i \(-0.127817\pi\)
\(762\) 0 0
\(763\) −11.8362 −0.428499
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 20.5164 0.740806
\(768\) 0 0
\(769\) −1.56057 −0.0562756 −0.0281378 0.999604i \(-0.508958\pi\)
−0.0281378 + 0.999604i \(0.508958\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −24.0539 −0.865160 −0.432580 0.901596i \(-0.642397\pi\)
−0.432580 + 0.901596i \(0.642397\pi\)
\(774\) 0 0
\(775\) − 6.41855i − 0.230561i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 17.8138i − 0.638244i
\(780\) 0 0
\(781\) 12.5336i 0.448486i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 46.5854i 1.66270i
\(786\) 0 0
\(787\) −3.99996 −0.142583 −0.0712916 0.997456i \(-0.522712\pi\)
−0.0712916 + 0.997456i \(0.522712\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 15.3287 0.545025
\(792\) 0 0
\(793\) −8.80635 −0.312723
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 7.18205 0.254401 0.127201 0.991877i \(-0.459401\pi\)
0.127201 + 0.991877i \(0.459401\pi\)
\(798\) 0 0
\(799\) − 43.1896i − 1.52794i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 12.1196i − 0.427691i
\(804\) 0 0
\(805\) 2.71853i 0.0958155i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 48.2698i 1.69707i 0.529136 + 0.848537i \(0.322516\pi\)
−0.529136 + 0.848537i \(0.677484\pi\)
\(810\) 0 0
\(811\) −20.3182 −0.713469 −0.356735 0.934206i \(-0.616110\pi\)
−0.356735 + 0.934206i \(0.616110\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 17.9823 0.629891
\(816\) 0 0
\(817\) 2.43682 0.0852537
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 35.0745 1.22411 0.612055 0.790815i \(-0.290343\pi\)
0.612055 + 0.790815i \(0.290343\pi\)
\(822\) 0 0
\(823\) 15.4211i 0.537544i 0.963204 + 0.268772i \(0.0866179\pi\)
−0.963204 + 0.268772i \(0.913382\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 27.5561i 0.958218i 0.877755 + 0.479109i \(0.159040\pi\)
−0.877755 + 0.479109i \(0.840960\pi\)
\(828\) 0 0
\(829\) − 56.9495i − 1.97794i −0.148124 0.988969i \(-0.547323\pi\)
0.148124 0.988969i \(-0.452677\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 3.87222i − 0.134164i
\(834\) 0 0
\(835\) 36.9992 1.28041
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 2.54853 0.0879849 0.0439925 0.999032i \(-0.485992\pi\)
0.0439925 + 0.999032i \(0.485992\pi\)
\(840\) 0 0
\(841\) −20.1020 −0.693173
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −27.6353 −0.950683
\(846\) 0 0
\(847\) 9.34033i 0.320937i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 3.60704i − 0.123648i
\(852\) 0 0
\(853\) 23.1990i 0.794319i 0.917750 + 0.397159i \(0.130004\pi\)
−0.917750 + 0.397159i \(0.869996\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 19.5713i − 0.668544i −0.942477 0.334272i \(-0.891510\pi\)
0.942477 0.334272i \(-0.108490\pi\)
\(858\) 0 0
\(859\) 50.8322 1.73437 0.867186 0.497985i \(-0.165927\pi\)
0.867186 + 0.497985i \(0.165927\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 39.1106 1.33134 0.665671 0.746246i \(-0.268146\pi\)
0.665671 + 0.746246i \(0.268146\pi\)
\(864\) 0 0
\(865\) −32.7383 −1.11314
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −21.2129 −0.719598
\(870\) 0 0
\(871\) 8.26162i 0.279934i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 4.48831i 0.151733i
\(876\) 0 0
\(877\) − 26.1514i − 0.883070i −0.897244 0.441535i \(-0.854434\pi\)
0.897244 0.441535i \(-0.145566\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 49.2277i − 1.65852i −0.558860 0.829262i \(-0.688761\pi\)
0.558860 0.829262i \(-0.311239\pi\)
\(882\) 0 0
\(883\) −44.3014 −1.49086 −0.745430 0.666584i \(-0.767756\pi\)
−0.745430 + 0.666584i \(0.767756\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −6.83633 −0.229542 −0.114771 0.993392i \(-0.536613\pi\)
−0.114771 + 0.993392i \(0.536613\pi\)
\(888\) 0 0
\(889\) −6.55275 −0.219772
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −22.7791 −0.762272
\(894\) 0 0
\(895\) 25.2430i 0.843780i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 5.53908i 0.184739i
\(900\) 0 0
\(901\) 17.1882i 0.572622i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 7.22895i − 0.240298i
\(906\) 0 0
\(907\) −24.2295 −0.804528 −0.402264 0.915524i \(-0.631777\pi\)
−0.402264 + 0.915524i \(0.631777\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −6.75748 −0.223885 −0.111943 0.993715i \(-0.535707\pi\)
−0.111943 + 0.993715i \(0.535707\pi\)
\(912\) 0 0
\(913\) 4.79435 0.158670
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 11.6893 0.386014
\(918\) 0 0
\(919\) − 1.77205i − 0.0584546i −0.999573 0.0292273i \(-0.990695\pi\)
0.999573 0.0292273i \(-0.00930466\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 18.1929i 0.598827i
\(924\) 0 0
\(925\) 13.3370i 0.438519i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 15.3376i 0.503212i 0.967830 + 0.251606i \(0.0809587\pi\)
−0.967830 + 0.251606i \(0.919041\pi\)
\(930\) 0 0
\(931\) −2.04228 −0.0669332
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −14.5067 −0.474420
\(936\) 0 0
\(937\) −48.6373 −1.58891 −0.794456 0.607322i \(-0.792244\pi\)
−0.794456 + 0.607322i \(0.792244\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 34.4202 1.12207 0.561034 0.827793i \(-0.310404\pi\)
0.561034 + 0.827793i \(0.310404\pi\)
\(942\) 0 0
\(943\) 8.15410i 0.265534i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 27.8402i − 0.904684i −0.891844 0.452342i \(-0.850589\pi\)
0.891844 0.452342i \(-0.149411\pi\)
\(948\) 0 0
\(949\) − 17.5921i − 0.571062i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 22.7401i 0.736624i 0.929702 + 0.368312i \(0.120064\pi\)
−0.929702 + 0.368312i \(0.879936\pi\)
\(954\) 0 0
\(955\) −9.89491 −0.320192
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 7.35868 0.237624
\(960\) 0 0
\(961\) 27.5519 0.888770
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −60.2050 −1.93807
\(966\) 0 0
\(967\) − 28.0470i − 0.901931i −0.892541 0.450965i \(-0.851080\pi\)
0.892541 0.450965i \(-0.148920\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 10.0931i − 0.323904i −0.986799 0.161952i \(-0.948221\pi\)
0.986799 0.161952i \(-0.0517790\pi\)
\(972\) 0 0
\(973\) 15.2683i 0.489479i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 1.93536i − 0.0619176i −0.999521 0.0309588i \(-0.990144\pi\)
0.999521 0.0309588i \(-0.00985607\pi\)
\(978\) 0 0
\(979\) 15.7553 0.503543
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 45.8652 1.46287 0.731436 0.681910i \(-0.238850\pi\)
0.731436 + 0.681910i \(0.238850\pi\)
\(984\) 0 0
\(985\) −47.5594 −1.51537
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −1.11544 −0.0354688
\(990\) 0 0
\(991\) − 20.5946i − 0.654209i −0.944988 0.327105i \(-0.893927\pi\)
0.944988 0.327105i \(-0.106073\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 33.0692i − 1.04837i
\(996\) 0 0
\(997\) 11.0478i 0.349889i 0.984578 + 0.174944i \(0.0559745\pi\)
−0.984578 + 0.174944i \(0.944025\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 6048.2.j.d.5615.7 48
3.2 odd 2 inner 6048.2.j.d.5615.42 48
4.3 odd 2 1512.2.j.d.323.1 48
8.3 odd 2 inner 6048.2.j.d.5615.41 48
8.5 even 2 1512.2.j.d.323.47 yes 48
12.11 even 2 1512.2.j.d.323.48 yes 48
24.5 odd 2 1512.2.j.d.323.2 yes 48
24.11 even 2 inner 6048.2.j.d.5615.8 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.j.d.323.1 48 4.3 odd 2
1512.2.j.d.323.2 yes 48 24.5 odd 2
1512.2.j.d.323.47 yes 48 8.5 even 2
1512.2.j.d.323.48 yes 48 12.11 even 2
6048.2.j.d.5615.7 48 1.1 even 1 trivial
6048.2.j.d.5615.8 48 24.11 even 2 inner
6048.2.j.d.5615.41 48 8.3 odd 2 inner
6048.2.j.d.5615.42 48 3.2 odd 2 inner