Properties

Label 6048.2.h.c.2591.6
Level 6048
Weight 2
Character 6048.2591
Analytic conductor 48.294
Analytic rank 0
Dimension 8
CM no
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2591.6
Root \(0.258819 - 0.965926i\)
Character \(\chi\) = 6048.2591
Dual form 6048.2.h.c.2591.3

$q$-expansion

\(f(q)\) \(=\) \(q+0.317837i q^{5} +1.00000i q^{7} +O(q^{10})\) \(q+0.317837i q^{5} +1.00000i q^{7} -1.41421 q^{11} -4.44949 q^{13} +5.97469i q^{17} +2.44949i q^{19} +5.65685 q^{23} +4.89898 q^{25} -5.51399i q^{29} -6.44949i q^{31} -0.317837 q^{35} +9.00000 q^{37} -7.24604i q^{41} +6.34847i q^{43} +10.8530 q^{47} -1.00000 q^{49} +11.9494i q^{53} -0.449490i q^{55} -5.83183 q^{59} -10.4495 q^{61} -1.41421i q^{65} +5.79796i q^{67} -5.65685 q^{71} -9.79796 q^{73} -1.41421i q^{77} +14.3485i q^{79} -8.02458 q^{83} -1.89898 q^{85} -15.4135i q^{89} -4.44949i q^{91} -0.778539 q^{95} -14.2474 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 16q^{13} + 72q^{37} - 8q^{49} - 64q^{61} + 24q^{85} - 16q^{97} + O(q^{100}) \)

Character values

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

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

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 0.317837i 0.142141i 0.997471 + 0.0710706i \(0.0226416\pi\)
−0.997471 + 0.0710706i \(0.977358\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.41421 −0.426401 −0.213201 0.977008i \(-0.568389\pi\)
−0.213201 + 0.977008i \(0.568389\pi\)
\(12\) 0 0
\(13\) −4.44949 −1.23407 −0.617033 0.786937i \(-0.711666\pi\)
−0.617033 + 0.786937i \(0.711666\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.97469i 1.44908i 0.689235 + 0.724538i \(0.257947\pi\)
−0.689235 + 0.724538i \(0.742053\pi\)
\(18\) 0 0
\(19\) 2.44949i 0.561951i 0.959715 + 0.280976i \(0.0906580\pi\)
−0.959715 + 0.280976i \(0.909342\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.65685 1.17954 0.589768 0.807573i \(-0.299219\pi\)
0.589768 + 0.807573i \(0.299219\pi\)
\(24\) 0 0
\(25\) 4.89898 0.979796
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 5.51399i − 1.02392i −0.859009 0.511961i \(-0.828919\pi\)
0.859009 0.511961i \(-0.171081\pi\)
\(30\) 0 0
\(31\) − 6.44949i − 1.15836i −0.815199 0.579181i \(-0.803372\pi\)
0.815199 0.579181i \(-0.196628\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.317837 −0.0537243
\(36\) 0 0
\(37\) 9.00000 1.47959 0.739795 0.672832i \(-0.234922\pi\)
0.739795 + 0.672832i \(0.234922\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 7.24604i − 1.13164i −0.824528 0.565821i \(-0.808559\pi\)
0.824528 0.565821i \(-0.191441\pi\)
\(42\) 0 0
\(43\) 6.34847i 0.968132i 0.875031 + 0.484066i \(0.160841\pi\)
−0.875031 + 0.484066i \(0.839159\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 10.8530 1.58307 0.791537 0.611121i \(-0.209281\pi\)
0.791537 + 0.611121i \(0.209281\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 11.9494i 1.64137i 0.571378 + 0.820687i \(0.306409\pi\)
−0.571378 + 0.820687i \(0.693591\pi\)
\(54\) 0 0
\(55\) − 0.449490i − 0.0606092i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −5.83183 −0.759239 −0.379620 0.925143i \(-0.623945\pi\)
−0.379620 + 0.925143i \(0.623945\pi\)
\(60\) 0 0
\(61\) −10.4495 −1.33792 −0.668960 0.743298i \(-0.733260\pi\)
−0.668960 + 0.743298i \(0.733260\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 1.41421i − 0.175412i
\(66\) 0 0
\(67\) 5.79796i 0.708333i 0.935182 + 0.354167i \(0.115235\pi\)
−0.935182 + 0.354167i \(0.884765\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −5.65685 −0.671345 −0.335673 0.941979i \(-0.608964\pi\)
−0.335673 + 0.941979i \(0.608964\pi\)
\(72\) 0 0
\(73\) −9.79796 −1.14676 −0.573382 0.819288i \(-0.694369\pi\)
−0.573382 + 0.819288i \(0.694369\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 1.41421i − 0.161165i
\(78\) 0 0
\(79\) 14.3485i 1.61433i 0.590327 + 0.807164i \(0.298999\pi\)
−0.590327 + 0.807164i \(0.701001\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −8.02458 −0.880812 −0.440406 0.897799i \(-0.645165\pi\)
−0.440406 + 0.897799i \(0.645165\pi\)
\(84\) 0 0
\(85\) −1.89898 −0.205973
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 15.4135i − 1.63383i −0.576761 0.816913i \(-0.695684\pi\)
0.576761 0.816913i \(-0.304316\pi\)
\(90\) 0 0
\(91\) − 4.44949i − 0.466433i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −0.778539 −0.0798764
\(96\) 0 0
\(97\) −14.2474 −1.44661 −0.723305 0.690529i \(-0.757378\pi\)
−0.723305 + 0.690529i \(0.757378\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 5.65685i − 0.562878i −0.959579 0.281439i \(-0.909188\pi\)
0.959579 0.281439i \(-0.0908117\pi\)
\(102\) 0 0
\(103\) 13.7980i 1.35955i 0.733419 + 0.679777i \(0.237923\pi\)
−0.733419 + 0.679777i \(0.762077\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −13.8564 −1.33955 −0.669775 0.742564i \(-0.733609\pi\)
−0.669775 + 0.742564i \(0.733609\pi\)
\(108\) 0 0
\(109\) −18.7980 −1.80052 −0.900259 0.435355i \(-0.856623\pi\)
−0.900259 + 0.435355i \(0.856623\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 3.60697i 0.339315i 0.985503 + 0.169657i \(0.0542661\pi\)
−0.985503 + 0.169657i \(0.945734\pi\)
\(114\) 0 0
\(115\) 1.79796i 0.167661i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −5.97469 −0.547699
\(120\) 0 0
\(121\) −9.00000 −0.818182
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 3.14626i 0.281410i
\(126\) 0 0
\(127\) 17.4495i 1.54839i 0.632946 + 0.774196i \(0.281845\pi\)
−0.632946 + 0.774196i \(0.718155\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 14.7778 1.29114 0.645572 0.763700i \(-0.276619\pi\)
0.645572 + 0.763700i \(0.276619\pi\)
\(132\) 0 0
\(133\) −2.44949 −0.212398
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 15.2706i 1.30466i 0.757936 + 0.652329i \(0.226208\pi\)
−0.757936 + 0.652329i \(0.773792\pi\)
\(138\) 0 0
\(139\) − 4.44949i − 0.377401i −0.982035 0.188700i \(-0.939572\pi\)
0.982035 0.188700i \(-0.0604275\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 6.29253 0.526208
\(144\) 0 0
\(145\) 1.75255 0.145541
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3.32124i 0.272086i 0.990703 + 0.136043i \(0.0434386\pi\)
−0.990703 + 0.136043i \(0.956561\pi\)
\(150\) 0 0
\(151\) − 6.34847i − 0.516631i −0.966061 0.258316i \(-0.916833\pi\)
0.966061 0.258316i \(-0.0831674\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2.04989 0.164651
\(156\) 0 0
\(157\) 14.2474 1.13707 0.568535 0.822659i \(-0.307510\pi\)
0.568535 + 0.822659i \(0.307510\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 5.65685i 0.445823i
\(162\) 0 0
\(163\) 5.44949i 0.426837i 0.976961 + 0.213418i \(0.0684598\pi\)
−0.976961 + 0.213418i \(0.931540\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 13.0458 1.00951 0.504756 0.863262i \(-0.331583\pi\)
0.504756 + 0.863262i \(0.331583\pi\)
\(168\) 0 0
\(169\) 6.79796 0.522920
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 14.7778i 1.12354i 0.827295 + 0.561768i \(0.189879\pi\)
−0.827295 + 0.561768i \(0.810121\pi\)
\(174\) 0 0
\(175\) 4.89898i 0.370328i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 2.68556 0.200728 0.100364 0.994951i \(-0.467999\pi\)
0.100364 + 0.994951i \(0.467999\pi\)
\(180\) 0 0
\(181\) −15.7980 −1.17425 −0.587127 0.809495i \(-0.699741\pi\)
−0.587127 + 0.809495i \(0.699741\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 2.86054i 0.210311i
\(186\) 0 0
\(187\) − 8.44949i − 0.617888i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −14.7778 −1.06928 −0.534642 0.845078i \(-0.679554\pi\)
−0.534642 + 0.845078i \(0.679554\pi\)
\(192\) 0 0
\(193\) −18.5959 −1.33856 −0.669282 0.743009i \(-0.733398\pi\)
−0.669282 + 0.743009i \(0.733398\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 9.89949i 0.705310i 0.935753 + 0.352655i \(0.114721\pi\)
−0.935753 + 0.352655i \(0.885279\pi\)
\(198\) 0 0
\(199\) 16.6969i 1.18361i 0.806079 + 0.591807i \(0.201585\pi\)
−0.806079 + 0.591807i \(0.798415\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 5.51399 0.387006
\(204\) 0 0
\(205\) 2.30306 0.160853
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 3.46410i − 0.239617i
\(210\) 0 0
\(211\) 21.5959i 1.48672i 0.668889 + 0.743362i \(0.266770\pi\)
−0.668889 + 0.743362i \(0.733230\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −2.01778 −0.137611
\(216\) 0 0
\(217\) 6.44949 0.437820
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 26.5843i − 1.78826i
\(222\) 0 0
\(223\) − 3.75255i − 0.251289i −0.992075 0.125645i \(-0.959900\pi\)
0.992075 0.125645i \(-0.0400999\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −11.0280 −0.731953 −0.365976 0.930624i \(-0.619265\pi\)
−0.365976 + 0.930624i \(0.619265\pi\)
\(228\) 0 0
\(229\) 4.20204 0.277679 0.138839 0.990315i \(-0.455663\pi\)
0.138839 + 0.990315i \(0.455663\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 7.07107i 0.463241i 0.972806 + 0.231621i \(0.0744028\pi\)
−0.972806 + 0.231621i \(0.925597\pi\)
\(234\) 0 0
\(235\) 3.44949i 0.225020i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 22.1988 1.43592 0.717961 0.696083i \(-0.245075\pi\)
0.717961 + 0.696083i \(0.245075\pi\)
\(240\) 0 0
\(241\) −5.34847 −0.344525 −0.172263 0.985051i \(-0.555108\pi\)
−0.172263 + 0.985051i \(0.555108\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 0.317837i − 0.0203059i
\(246\) 0 0
\(247\) − 10.8990i − 0.693485i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −3.28913 −0.207608 −0.103804 0.994598i \(-0.533101\pi\)
−0.103804 + 0.994598i \(0.533101\pi\)
\(252\) 0 0
\(253\) −8.00000 −0.502956
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 14.1421i − 0.882162i −0.897467 0.441081i \(-0.854595\pi\)
0.897467 0.441081i \(-0.145405\pi\)
\(258\) 0 0
\(259\) 9.00000i 0.559233i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 15.9063 0.980824 0.490412 0.871491i \(-0.336846\pi\)
0.490412 + 0.871491i \(0.336846\pi\)
\(264\) 0 0
\(265\) −3.79796 −0.233307
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 19.4812i − 1.18779i −0.804544 0.593893i \(-0.797590\pi\)
0.804544 0.593893i \(-0.202410\pi\)
\(270\) 0 0
\(271\) − 20.6969i − 1.25725i −0.777709 0.628625i \(-0.783618\pi\)
0.777709 0.628625i \(-0.216382\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −6.92820 −0.417786
\(276\) 0 0
\(277\) −1.00000 −0.0600842 −0.0300421 0.999549i \(-0.509564\pi\)
−0.0300421 + 0.999549i \(0.509564\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 18.2419i − 1.08822i −0.839013 0.544111i \(-0.816867\pi\)
0.839013 0.544111i \(-0.183133\pi\)
\(282\) 0 0
\(283\) 17.3485i 1.03126i 0.856812 + 0.515630i \(0.172442\pi\)
−0.856812 + 0.515630i \(0.827558\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 7.24604 0.427720
\(288\) 0 0
\(289\) −18.6969 −1.09982
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 8.16744i − 0.477147i −0.971124 0.238574i \(-0.923320\pi\)
0.971124 0.238574i \(-0.0766798\pi\)
\(294\) 0 0
\(295\) − 1.85357i − 0.107919i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −25.1701 −1.45563
\(300\) 0 0
\(301\) −6.34847 −0.365920
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 3.32124i − 0.190173i
\(306\) 0 0
\(307\) − 21.7980i − 1.24408i −0.782987 0.622038i \(-0.786305\pi\)
0.782987 0.622038i \(-0.213695\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 24.3595 1.38130 0.690649 0.723190i \(-0.257325\pi\)
0.690649 + 0.723190i \(0.257325\pi\)
\(312\) 0 0
\(313\) −22.0000 −1.24351 −0.621757 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 1.27135i − 0.0714061i −0.999362 0.0357030i \(-0.988633\pi\)
0.999362 0.0357030i \(-0.0113670\pi\)
\(318\) 0 0
\(319\) 7.79796i 0.436602i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −14.6349 −0.814310
\(324\) 0 0
\(325\) −21.7980 −1.20913
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 10.8530i 0.598346i
\(330\) 0 0
\(331\) 6.14643i 0.337838i 0.985630 + 0.168919i \(0.0540277\pi\)
−0.985630 + 0.168919i \(0.945972\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −1.84281 −0.100683
\(336\) 0 0
\(337\) 18.5959 1.01298 0.506492 0.862245i \(-0.330942\pi\)
0.506492 + 0.862245i \(0.330942\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 9.12096i 0.493927i
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −29.7627 −1.59775 −0.798873 0.601500i \(-0.794570\pi\)
−0.798873 + 0.601500i \(0.794570\pi\)
\(348\) 0 0
\(349\) −16.0000 −0.856460 −0.428230 0.903670i \(-0.640863\pi\)
−0.428230 + 0.903670i \(0.640863\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 8.51739i 0.453335i 0.973972 + 0.226667i \(0.0727831\pi\)
−0.973972 + 0.226667i \(0.927217\pi\)
\(354\) 0 0
\(355\) − 1.79796i − 0.0954258i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 10.3923 0.548485 0.274242 0.961661i \(-0.411573\pi\)
0.274242 + 0.961661i \(0.411573\pi\)
\(360\) 0 0
\(361\) 13.0000 0.684211
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 3.11416i − 0.163002i
\(366\) 0 0
\(367\) 26.8990i 1.40412i 0.712120 + 0.702058i \(0.247735\pi\)
−0.712120 + 0.702058i \(0.752265\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −11.9494 −0.620381
\(372\) 0 0
\(373\) −21.8990 −1.13389 −0.566943 0.823757i \(-0.691874\pi\)
−0.566943 + 0.823757i \(0.691874\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 24.5344i 1.26359i
\(378\) 0 0
\(379\) 33.0454i 1.69743i 0.528852 + 0.848714i \(0.322623\pi\)
−0.528852 + 0.848714i \(0.677377\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −2.36773 −0.120985 −0.0604926 0.998169i \(-0.519267\pi\)
−0.0604926 + 0.998169i \(0.519267\pi\)
\(384\) 0 0
\(385\) 0.449490 0.0229081
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 5.16404i 0.261827i 0.991394 + 0.130914i \(0.0417911\pi\)
−0.991394 + 0.130914i \(0.958209\pi\)
\(390\) 0 0
\(391\) 33.7980i 1.70924i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −4.56048 −0.229463
\(396\) 0 0
\(397\) 2.20204 0.110517 0.0552586 0.998472i \(-0.482402\pi\)
0.0552586 + 0.998472i \(0.482402\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 37.1195i − 1.85366i −0.375482 0.926830i \(-0.622523\pi\)
0.375482 0.926830i \(-0.377477\pi\)
\(402\) 0 0
\(403\) 28.6969i 1.42950i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −12.7279 −0.630900
\(408\) 0 0
\(409\) −13.7980 −0.682265 −0.341133 0.940015i \(-0.610811\pi\)
−0.341133 + 0.940015i \(0.610811\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 5.83183i − 0.286965i
\(414\) 0 0
\(415\) − 2.55051i − 0.125200i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −8.02458 −0.392026 −0.196013 0.980601i \(-0.562800\pi\)
−0.196013 + 0.980601i \(0.562800\pi\)
\(420\) 0 0
\(421\) 23.1010 1.12587 0.562937 0.826500i \(-0.309671\pi\)
0.562937 + 0.826500i \(0.309671\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 29.2699i 1.41980i
\(426\) 0 0
\(427\) − 10.4495i − 0.505686i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −6.92820 −0.333720 −0.166860 0.985981i \(-0.553363\pi\)
−0.166860 + 0.985981i \(0.553363\pi\)
\(432\) 0 0
\(433\) 21.5505 1.03565 0.517826 0.855486i \(-0.326742\pi\)
0.517826 + 0.855486i \(0.326742\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 13.8564i 0.662842i
\(438\) 0 0
\(439\) 14.8990i 0.711089i 0.934659 + 0.355545i \(0.115705\pi\)
−0.934659 + 0.355545i \(0.884295\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 38.8194 1.84437 0.922184 0.386752i \(-0.126403\pi\)
0.922184 + 0.386752i \(0.126403\pi\)
\(444\) 0 0
\(445\) 4.89898 0.232234
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 1.12848i − 0.0532565i −0.999645 0.0266282i \(-0.991523\pi\)
0.999645 0.0266282i \(-0.00847703\pi\)
\(450\) 0 0
\(451\) 10.2474i 0.482534i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1.41421 0.0662994
\(456\) 0 0
\(457\) −27.5959 −1.29088 −0.645441 0.763810i \(-0.723327\pi\)
−0.645441 + 0.763810i \(0.723327\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 6.26042i 0.291577i 0.989316 + 0.145788i \(0.0465719\pi\)
−0.989316 + 0.145788i \(0.953428\pi\)
\(462\) 0 0
\(463\) − 6.75255i − 0.313818i −0.987613 0.156909i \(-0.949847\pi\)
0.987613 0.156909i \(-0.0501529\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −36.1981 −1.67505 −0.837524 0.546401i \(-0.815998\pi\)
−0.837524 + 0.546401i \(0.815998\pi\)
\(468\) 0 0
\(469\) −5.79796 −0.267725
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 8.97809i − 0.412813i
\(474\) 0 0
\(475\) 12.0000i 0.550598i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 24.9951 1.14206 0.571029 0.820930i \(-0.306545\pi\)
0.571029 + 0.820930i \(0.306545\pi\)
\(480\) 0 0
\(481\) −40.0454 −1.82591
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 4.52837i − 0.205623i
\(486\) 0 0
\(487\) 19.5959i 0.887976i 0.896033 + 0.443988i \(0.146437\pi\)
−0.896033 + 0.443988i \(0.853563\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −3.32124 −0.149885 −0.0749427 0.997188i \(-0.523877\pi\)
−0.0749427 + 0.997188i \(0.523877\pi\)
\(492\) 0 0
\(493\) 32.9444 1.48374
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 5.65685i − 0.253745i
\(498\) 0 0
\(499\) 0.752551i 0.0336888i 0.999858 + 0.0168444i \(0.00536200\pi\)
−0.999858 + 0.0168444i \(0.994638\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 12.7600 0.568942 0.284471 0.958685i \(-0.408182\pi\)
0.284471 + 0.958685i \(0.408182\pi\)
\(504\) 0 0
\(505\) 1.79796 0.0800081
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 12.2672i − 0.543735i −0.962335 0.271867i \(-0.912359\pi\)
0.962335 0.271867i \(-0.0876412\pi\)
\(510\) 0 0
\(511\) − 9.79796i − 0.433436i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −4.38551 −0.193248
\(516\) 0 0
\(517\) −15.3485 −0.675025
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 25.4880i 1.11665i 0.829623 + 0.558324i \(0.188555\pi\)
−0.829623 + 0.558324i \(0.811445\pi\)
\(522\) 0 0
\(523\) 32.0454i 1.40125i 0.713531 + 0.700624i \(0.247095\pi\)
−0.713531 + 0.700624i \(0.752905\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 38.5337 1.67855
\(528\) 0 0
\(529\) 9.00000 0.391304
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 32.2412i 1.39652i
\(534\) 0 0
\(535\) − 4.40408i − 0.190405i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1.41421 0.0609145
\(540\) 0 0
\(541\) −29.0000 −1.24681 −0.623404 0.781900i \(-0.714251\pi\)
−0.623404 + 0.781900i \(0.714251\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 5.97469i − 0.255928i
\(546\) 0 0
\(547\) − 34.1464i − 1.46000i −0.683449 0.729998i \(-0.739521\pi\)
0.683449 0.729998i \(-0.260479\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 13.5065 0.575395
\(552\) 0 0
\(553\) −14.3485 −0.610159
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 27.7128i − 1.17423i −0.809504 0.587115i \(-0.800264\pi\)
0.809504 0.587115i \(-0.199736\pi\)
\(558\) 0 0
\(559\) − 28.2474i − 1.19474i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 46.5262 1.96084 0.980422 0.196906i \(-0.0630893\pi\)
0.980422 + 0.196906i \(0.0630893\pi\)
\(564\) 0 0
\(565\) −1.14643 −0.0482306
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 18.2419i 0.764741i 0.924009 + 0.382370i \(0.124892\pi\)
−0.924009 + 0.382370i \(0.875108\pi\)
\(570\) 0 0
\(571\) 0.348469i 0.0145830i 0.999973 + 0.00729149i \(0.00232097\pi\)
−0.999973 + 0.00729149i \(0.997679\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 27.7128 1.15570
\(576\) 0 0
\(577\) 37.5959 1.56514 0.782569 0.622564i \(-0.213909\pi\)
0.782569 + 0.622564i \(0.213909\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 8.02458i − 0.332916i
\(582\) 0 0
\(583\) − 16.8990i − 0.699884i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −30.5412 −1.26057 −0.630286 0.776363i \(-0.717062\pi\)
−0.630286 + 0.776363i \(0.717062\pi\)
\(588\) 0 0
\(589\) 15.7980 0.650944
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 24.8523i 1.02056i 0.860008 + 0.510280i \(0.170458\pi\)
−0.860008 + 0.510280i \(0.829542\pi\)
\(594\) 0 0
\(595\) − 1.89898i − 0.0778506i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 26.2986 1.07453 0.537266 0.843413i \(-0.319457\pi\)
0.537266 + 0.843413i \(0.319457\pi\)
\(600\) 0 0
\(601\) −26.7423 −1.09084 −0.545422 0.838162i \(-0.683630\pi\)
−0.545422 + 0.838162i \(0.683630\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 2.86054i − 0.116297i
\(606\) 0 0
\(607\) − 26.2474i − 1.06535i −0.846320 0.532676i \(-0.821187\pi\)
0.846320 0.532676i \(-0.178813\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −48.2903 −1.95362
\(612\) 0 0
\(613\) 17.3939 0.702532 0.351266 0.936276i \(-0.385751\pi\)
0.351266 + 0.936276i \(0.385751\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 18.2419i − 0.734392i −0.930144 0.367196i \(-0.880318\pi\)
0.930144 0.367196i \(-0.119682\pi\)
\(618\) 0 0
\(619\) − 15.8434i − 0.636799i −0.947957 0.318399i \(-0.896855\pi\)
0.947957 0.318399i \(-0.103145\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 15.4135 0.617528
\(624\) 0 0
\(625\) 23.4949 0.939796
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 53.7722i 2.14404i
\(630\) 0 0
\(631\) 39.7423i 1.58212i 0.611740 + 0.791059i \(0.290470\pi\)
−0.611740 + 0.791059i \(0.709530\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −5.54610 −0.220090
\(636\) 0 0
\(637\) 4.44949 0.176295
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 5.16404i 0.203967i 0.994786 + 0.101984i \(0.0325190\pi\)
−0.994786 + 0.101984i \(0.967481\pi\)
\(642\) 0 0
\(643\) − 10.4949i − 0.413878i −0.978354 0.206939i \(-0.933650\pi\)
0.978354 0.206939i \(-0.0663502\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −49.3546 −1.94033 −0.970165 0.242446i \(-0.922050\pi\)
−0.970165 + 0.242446i \(0.922050\pi\)
\(648\) 0 0
\(649\) 8.24745 0.323741
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 13.3636i − 0.522958i −0.965209 0.261479i \(-0.915790\pi\)
0.965209 0.261479i \(-0.0842102\pi\)
\(654\) 0 0
\(655\) 4.69694i 0.183525i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −33.0197 −1.28627 −0.643133 0.765755i \(-0.722366\pi\)
−0.643133 + 0.765755i \(0.722366\pi\)
\(660\) 0 0
\(661\) −2.04541 −0.0795571 −0.0397786 0.999209i \(-0.512665\pi\)
−0.0397786 + 0.999209i \(0.512665\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 0.778539i − 0.0301905i
\(666\) 0 0
\(667\) − 31.1918i − 1.20775i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 14.7778 0.570491
\(672\) 0 0
\(673\) −3.10102 −0.119536 −0.0597678 0.998212i \(-0.519036\pi\)
−0.0597678 + 0.998212i \(0.519036\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 27.9985i − 1.07607i −0.842922 0.538036i \(-0.819167\pi\)
0.842922 0.538036i \(-0.180833\pi\)
\(678\) 0 0
\(679\) − 14.2474i − 0.546767i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −26.2344 −1.00383 −0.501915 0.864917i \(-0.667371\pi\)
−0.501915 + 0.864917i \(0.667371\pi\)
\(684\) 0 0
\(685\) −4.85357 −0.185445
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 53.1687i − 2.02556i
\(690\) 0 0
\(691\) − 12.9444i − 0.492428i −0.969216 0.246214i \(-0.920813\pi\)
0.969216 0.246214i \(-0.0791866\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1.41421 0.0536442
\(696\) 0 0
\(697\) 43.2929 1.63983
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 50.1331i 1.89350i 0.321965 + 0.946751i \(0.395657\pi\)
−0.321965 + 0.946751i \(0.604343\pi\)
\(702\) 0 0
\(703\) 22.0454i 0.831458i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 5.65685 0.212748
\(708\) 0 0
\(709\) −6.39388 −0.240127 −0.120064 0.992766i \(-0.538310\pi\)
−0.120064 + 0.992766i \(0.538310\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 36.4838i − 1.36633i
\(714\) 0 0
\(715\) 2.00000i 0.0747958i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1.38211 0.0515438 0.0257719 0.999668i \(-0.491796\pi\)
0.0257719 + 0.999668i \(0.491796\pi\)
\(720\) 0 0
\(721\) −13.7980 −0.513863
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 27.0129i − 1.00323i
\(726\) 0 0
\(727\) − 18.0454i − 0.669267i −0.942348 0.334634i \(-0.891387\pi\)
0.942348 0.334634i \(-0.108613\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −37.9301 −1.40290
\(732\) 0 0
\(733\) −21.5959 −0.797663 −0.398832 0.917024i \(-0.630584\pi\)
−0.398832 + 0.917024i \(0.630584\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 8.19955i − 0.302034i
\(738\) 0 0
\(739\) − 20.0000i − 0.735712i −0.929883 0.367856i \(-0.880092\pi\)
0.929883 0.367856i \(-0.119908\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1.76416 −0.0647207 −0.0323604 0.999476i \(-0.510302\pi\)
−0.0323604 + 0.999476i \(0.510302\pi\)
\(744\) 0 0
\(745\) −1.05561 −0.0386747
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 13.8564i − 0.506302i
\(750\) 0 0
\(751\) 6.20204i 0.226316i 0.993577 + 0.113158i \(0.0360966\pi\)
−0.993577 + 0.113158i \(0.963903\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 2.01778 0.0734345
\(756\) 0 0
\(757\) −4.30306 −0.156397 −0.0781987 0.996938i \(-0.524917\pi\)
−0.0781987 + 0.996938i \(0.524917\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 5.05329i 0.183182i 0.995797 + 0.0915908i \(0.0291952\pi\)
−0.995797 + 0.0915908i \(0.970805\pi\)
\(762\) 0 0
\(763\) − 18.7980i − 0.680532i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 25.9487 0.936952
\(768\) 0 0
\(769\) 16.2020 0.584261 0.292130 0.956379i \(-0.405636\pi\)
0.292130 + 0.956379i \(0.405636\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 29.8735i − 1.07447i −0.843431 0.537237i \(-0.819468\pi\)
0.843431 0.537237i \(-0.180532\pi\)
\(774\) 0 0
\(775\) − 31.5959i − 1.13496i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 17.7491 0.635928
\(780\) 0 0
\(781\) 8.00000 0.286263
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 4.52837i 0.161624i
\(786\) 0 0
\(787\) 21.3939i 0.762609i 0.924449 + 0.381305i \(0.124525\pi\)
−0.924449 + 0.381305i \(0.875475\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −3.60697 −0.128249
\(792\) 0 0
\(793\) 46.4949 1.65108
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0.985620i 0.0349124i 0.999848 + 0.0174562i \(0.00555677\pi\)
−0.999848 + 0.0174562i \(0.994443\pi\)
\(798\) 0 0
\(799\) 64.8434i 2.29399i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 13.8564 0.488982
\(804\) 0 0
\(805\) −1.79796 −0.0633697
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 42.7764i 1.50394i 0.659199 + 0.751968i \(0.270895\pi\)
−0.659199 + 0.751968i \(0.729105\pi\)
\(810\) 0 0
\(811\) 28.0454i 0.984807i 0.870367 + 0.492404i \(0.163882\pi\)
−0.870367 + 0.492404i \(0.836118\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −1.73205 −0.0606711
\(816\) 0 0
\(817\) −15.5505 −0.544043
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 47.2261i 1.64820i 0.566443 + 0.824101i \(0.308319\pi\)
−0.566443 + 0.824101i \(0.691681\pi\)
\(822\) 0 0
\(823\) − 20.8434i − 0.726554i −0.931681 0.363277i \(-0.881658\pi\)
0.931681 0.363277i \(-0.118342\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −2.68556 −0.0933862 −0.0466931 0.998909i \(-0.514868\pi\)
−0.0466931 + 0.998909i \(0.514868\pi\)
\(828\) 0 0
\(829\) −7.95459 −0.276274 −0.138137 0.990413i \(-0.544111\pi\)
−0.138137 + 0.990413i \(0.544111\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 5.97469i − 0.207011i
\(834\) 0 0
\(835\) 4.14643i 0.143493i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −1.09638 −0.0378511 −0.0189256 0.999821i \(-0.506025\pi\)
−0.0189256 + 0.999821i \(0.506025\pi\)
\(840\) 0 0
\(841\) −1.40408 −0.0484166
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 2.16064i 0.0743284i
\(846\) 0 0
\(847\) − 9.00000i − 0.309244i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 50.9117 1.74523
\(852\) 0 0
\(853\) 43.7980 1.49961 0.749807 0.661657i \(-0.230146\pi\)
0.749807 + 0.661657i \(0.230146\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 28.6021i 0.977029i 0.872556 + 0.488515i \(0.162461\pi\)
−0.872556 + 0.488515i \(0.837539\pi\)
\(858\) 0 0
\(859\) − 38.8990i − 1.32722i −0.748080 0.663608i \(-0.769024\pi\)
0.748080 0.663608i \(-0.230976\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −13.5065 −0.459765 −0.229883 0.973218i \(-0.573834\pi\)
−0.229883 + 0.973218i \(0.573834\pi\)
\(864\) 0 0
\(865\) −4.69694 −0.159701
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 20.2918i − 0.688352i
\(870\) 0 0
\(871\) − 25.7980i − 0.874130i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −3.14626 −0.106363
\(876\) 0 0
\(877\) −0.595918 −0.0201227 −0.0100614 0.999949i \(-0.503203\pi\)
−0.0100614 + 0.999949i \(0.503203\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 48.4974i − 1.63392i −0.576695 0.816960i \(-0.695658\pi\)
0.576695 0.816960i \(-0.304342\pi\)
\(882\) 0 0
\(883\) − 22.3485i − 0.752086i −0.926602 0.376043i \(-0.877285\pi\)
0.926602 0.376043i \(-0.122715\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 11.2030 0.376158 0.188079 0.982154i \(-0.439774\pi\)
0.188079 + 0.982154i \(0.439774\pi\)
\(888\) 0 0
\(889\) −17.4495 −0.585237
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 26.5843i 0.889611i
\(894\) 0 0
\(895\) 0.853572i 0.0285318i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −35.5624 −1.18607
\(900\) 0 0
\(901\) −71.3939 −2.37847
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 5.02118i − 0.166910i
\(906\) 0 0
\(907\) − 1.24745i − 0.0414209i −0.999786 0.0207104i \(-0.993407\pi\)
0.999786 0.0207104i \(-0.00659281\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −21.4989 −0.712291 −0.356146 0.934430i \(-0.615909\pi\)
−0.356146 + 0.934430i \(0.615909\pi\)
\(912\) 0 0
\(913\) 11.3485 0.375580
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 14.7778i 0.488006i
\(918\) 0 0
\(919\) − 34.8434i − 1.14938i −0.818372 0.574688i \(-0.805123\pi\)
0.818372 0.574688i \(-0.194877\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 25.1701 0.828484
\(924\) 0 0
\(925\) 44.0908 1.44970
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 0.317837i − 0.0104279i −0.999986 0.00521395i \(-0.998340\pi\)
0.999986 0.00521395i \(-0.00165966\pi\)
\(930\) 0 0
\(931\) − 2.44949i − 0.0802788i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 2.68556 0.0878273
\(936\) 0 0
\(937\) −18.7423 −0.612286 −0.306143 0.951986i \(-0.599039\pi\)
−0.306143 + 0.951986i \(0.599039\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 27.7449i − 0.904459i −0.891902 0.452229i \(-0.850629\pi\)
0.891902 0.452229i \(-0.149371\pi\)
\(942\) 0 0
\(943\) − 40.9898i − 1.33481i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1.84281 −0.0598832 −0.0299416 0.999552i \(-0.509532\pi\)
−0.0299416 + 0.999552i \(0.509532\pi\)
\(948\) 0 0
\(949\) 43.5959 1.41518
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 37.7552i 1.22301i 0.791241 + 0.611505i \(0.209436\pi\)
−0.791241 + 0.611505i \(0.790564\pi\)
\(954\) 0 0
\(955\) − 4.69694i − 0.151989i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −15.2706 −0.493114
\(960\) 0 0
\(961\) −10.5959 −0.341804
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 5.91048i − 0.190265i
\(966\) 0 0
\(967\) 3.10102i 0.0997221i 0.998756 + 0.0498610i \(0.0158778\pi\)
−0.998756 + 0.0498610i \(0.984122\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 33.5446 1.07650 0.538249 0.842786i \(-0.319086\pi\)
0.538249 + 0.842786i \(0.319086\pi\)
\(972\) 0 0
\(973\) 4.44949 0.142644
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 3.17837i 0.101685i 0.998707 + 0.0508426i \(0.0161907\pi\)
−0.998707 + 0.0508426i \(0.983809\pi\)
\(978\) 0 0
\(979\) 21.7980i 0.696666i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 13.0458 0.416095 0.208048 0.978119i \(-0.433289\pi\)
0.208048 + 0.978119i \(0.433289\pi\)
\(984\) 0 0
\(985\) −3.14643 −0.100254
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 35.9124i 1.14195i
\(990\) 0 0
\(991\) 22.3485i 0.709923i 0.934881 + 0.354961i \(0.115506\pi\)
−0.934881 + 0.354961i \(0.884494\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −5.30691 −0.168240
\(996\) 0 0
\(997\) 21.8434 0.691786 0.345893 0.938274i \(-0.387576\pi\)
0.345893 + 0.938274i \(0.387576\pi\)
\(998\) 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.c.2591.6 yes 8
3.2 odd 2 inner 6048.2.h.c.2591.4 yes 8
4.3 odd 2 inner 6048.2.h.c.2591.5 yes 8
12.11 even 2 inner 6048.2.h.c.2591.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.h.c.2591.3 8 12.11 even 2 inner
6048.2.h.c.2591.4 yes 8 3.2 odd 2 inner
6048.2.h.c.2591.5 yes 8 4.3 odd 2 inner
6048.2.h.c.2591.6 yes 8 1.1 even 1 trivial