Properties

Label 6048.2.c.d.3025.15
Level 6048
Weight 2
Character 6048.3025
Analytic conductor 48.294
Analytic rank 0
Dimension 16
CM no
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{40})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{8} \)
Twist minimal: no (minimal twist has level 1512)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3025.15
Root \(-0.156434 + 0.987688i\)
Character \(\chi\) = 6048.3025
Dual form 6048.2.c.d.3025.2

$q$-expansion

\(f(q)\) \(=\) \(q+4.29757i q^{5} +1.00000 q^{7} +O(q^{10})\) \(q+4.29757i q^{5} +1.00000 q^{7} -1.60758i q^{11} -6.02967i q^{13} -3.16228 q^{17} +3.07768i q^{19} +5.95080 q^{23} -13.4691 q^{25} -3.53972i q^{29} +2.08831 q^{31} +4.29757i q^{35} -4.48276i q^{37} +6.69568 q^{41} -12.2811i q^{43} +3.82734 q^{47} +1.00000 q^{49} -8.63711i q^{53} +6.90868 q^{55} +1.55265i q^{59} -3.64886i q^{61} +25.9129 q^{65} +0.772361i q^{67} +7.36501 q^{71} +1.32739 q^{73} -1.60758i q^{77} -12.7401 q^{79} -7.08446i q^{83} -13.5901i q^{85} -9.73307 q^{89} -6.02967i q^{91} -13.2266 q^{95} -10.1803 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 16q^{7} + O(q^{10}) \) \( 16q + 16q^{7} - 32q^{25} - 40q^{31} + 16q^{49} + 72q^{55} + 24q^{73} - 24q^{79} + 16q^{97} + O(q^{100}) \)

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
\(5\) 4.29757i 1.92193i 0.276666 + 0.960966i \(0.410770\pi\)
−0.276666 + 0.960966i \(0.589230\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 1.60758i − 0.484703i −0.970189 0.242351i \(-0.922081\pi\)
0.970189 0.242351i \(-0.0779187\pi\)
\(12\) 0 0
\(13\) − 6.02967i − 1.67233i −0.548478 0.836165i \(-0.684792\pi\)
0.548478 0.836165i \(-0.315208\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.16228 −0.766965 −0.383482 0.923548i \(-0.625275\pi\)
−0.383482 + 0.923548i \(0.625275\pi\)
\(18\) 0 0
\(19\) 3.07768i 0.706069i 0.935610 + 0.353035i \(0.114850\pi\)
−0.935610 + 0.353035i \(0.885150\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.95080 1.24083 0.620413 0.784275i \(-0.286965\pi\)
0.620413 + 0.784275i \(0.286965\pi\)
\(24\) 0 0
\(25\) −13.4691 −2.69382
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 3.53972i − 0.657310i −0.944450 0.328655i \(-0.893405\pi\)
0.944450 0.328655i \(-0.106595\pi\)
\(30\) 0 0
\(31\) 2.08831 0.375071 0.187535 0.982258i \(-0.439950\pi\)
0.187535 + 0.982258i \(0.439950\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.29757i 0.726422i
\(36\) 0 0
\(37\) − 4.48276i − 0.736961i −0.929636 0.368480i \(-0.879878\pi\)
0.929636 0.368480i \(-0.120122\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.69568 1.04569 0.522845 0.852428i \(-0.324871\pi\)
0.522845 + 0.852428i \(0.324871\pi\)
\(42\) 0 0
\(43\) − 12.2811i − 1.87284i −0.350875 0.936422i \(-0.614116\pi\)
0.350875 0.936422i \(-0.385884\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.82734 0.558275 0.279138 0.960251i \(-0.409951\pi\)
0.279138 + 0.960251i \(0.409951\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 8.63711i − 1.18640i −0.805056 0.593199i \(-0.797865\pi\)
0.805056 0.593199i \(-0.202135\pi\)
\(54\) 0 0
\(55\) 6.90868 0.931566
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.55265i 0.202137i 0.994879 + 0.101069i \(0.0322262\pi\)
−0.994879 + 0.101069i \(0.967774\pi\)
\(60\) 0 0
\(61\) − 3.64886i − 0.467189i −0.972334 0.233594i \(-0.924951\pi\)
0.972334 0.233594i \(-0.0750487\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 25.9129 3.21411
\(66\) 0 0
\(67\) 0.772361i 0.0943590i 0.998886 + 0.0471795i \(0.0150233\pi\)
−0.998886 + 0.0471795i \(0.984977\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 7.36501 0.874066 0.437033 0.899446i \(-0.356029\pi\)
0.437033 + 0.899446i \(0.356029\pi\)
\(72\) 0 0
\(73\) 1.32739 0.155359 0.0776797 0.996978i \(-0.475249\pi\)
0.0776797 + 0.996978i \(0.475249\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 1.60758i − 0.183200i
\(78\) 0 0
\(79\) −12.7401 −1.43337 −0.716685 0.697397i \(-0.754341\pi\)
−0.716685 + 0.697397i \(0.754341\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 7.08446i − 0.777620i −0.921318 0.388810i \(-0.872886\pi\)
0.921318 0.388810i \(-0.127114\pi\)
\(84\) 0 0
\(85\) − 13.5901i − 1.47405i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −9.73307 −1.03170 −0.515852 0.856678i \(-0.672525\pi\)
−0.515852 + 0.856678i \(0.672525\pi\)
\(90\) 0 0
\(91\) − 6.02967i − 0.632081i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −13.2266 −1.35702
\(96\) 0 0
\(97\) −10.1803 −1.03366 −0.516828 0.856089i \(-0.672887\pi\)
−0.516828 + 0.856089i \(0.672887\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1.02749i 0.102239i 0.998693 + 0.0511194i \(0.0162789\pi\)
−0.998693 + 0.0511194i \(0.983721\pi\)
\(102\) 0 0
\(103\) 0.965118 0.0950959 0.0475479 0.998869i \(-0.484859\pi\)
0.0475479 + 0.998869i \(0.484859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 16.0949i 1.55595i 0.628294 + 0.777976i \(0.283754\pi\)
−0.628294 + 0.777976i \(0.716246\pi\)
\(108\) 0 0
\(109\) − 20.2686i − 1.94138i −0.240327 0.970692i \(-0.577255\pi\)
0.240327 0.970692i \(-0.422745\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0.951214 0.0894826 0.0447413 0.998999i \(-0.485754\pi\)
0.0447413 + 0.998999i \(0.485754\pi\)
\(114\) 0 0
\(115\) 25.5740i 2.38478i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −3.16228 −0.289886
\(120\) 0 0
\(121\) 8.41570 0.765063
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 36.3966i − 3.25541i
\(126\) 0 0
\(127\) 11.6132 1.03050 0.515250 0.857040i \(-0.327699\pi\)
0.515250 + 0.857040i \(0.327699\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 10.0120i − 0.874752i −0.899279 0.437376i \(-0.855908\pi\)
0.899279 0.437376i \(-0.144092\pi\)
\(132\) 0 0
\(133\) 3.07768i 0.266869i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 8.40383 0.717988 0.358994 0.933340i \(-0.383120\pi\)
0.358994 + 0.933340i \(0.383120\pi\)
\(138\) 0 0
\(139\) − 6.61437i − 0.561024i −0.959851 0.280512i \(-0.909496\pi\)
0.959851 0.280512i \(-0.0905042\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −9.69316 −0.810583
\(144\) 0 0
\(145\) 15.2122 1.26330
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 16.9866i − 1.39160i −0.718238 0.695798i \(-0.755051\pi\)
0.718238 0.695798i \(-0.244949\pi\)
\(150\) 0 0
\(151\) 11.5256 0.937937 0.468968 0.883215i \(-0.344626\pi\)
0.468968 + 0.883215i \(0.344626\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 8.97464i 0.720860i
\(156\) 0 0
\(157\) 10.3889i 0.829127i 0.910020 + 0.414563i \(0.136066\pi\)
−0.910020 + 0.414563i \(0.863934\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 5.95080 0.468988
\(162\) 0 0
\(163\) 11.8743i 0.930067i 0.885293 + 0.465034i \(0.153958\pi\)
−0.885293 + 0.465034i \(0.846042\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 15.0596 1.16535 0.582673 0.812706i \(-0.302007\pi\)
0.582673 + 0.812706i \(0.302007\pi\)
\(168\) 0 0
\(169\) −23.3570 −1.79669
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 23.7854i 1.80837i 0.427142 + 0.904185i \(0.359521\pi\)
−0.427142 + 0.904185i \(0.640479\pi\)
\(174\) 0 0
\(175\) −13.4691 −1.01817
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 22.1919i − 1.65870i −0.558728 0.829351i \(-0.688710\pi\)
0.558728 0.829351i \(-0.311290\pi\)
\(180\) 0 0
\(181\) − 8.60253i − 0.639421i −0.947515 0.319711i \(-0.896414\pi\)
0.947515 0.319711i \(-0.103586\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 19.2650 1.41639
\(186\) 0 0
\(187\) 5.08361i 0.371750i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −8.72989 −0.631673 −0.315836 0.948814i \(-0.602285\pi\)
−0.315836 + 0.948814i \(0.602285\pi\)
\(192\) 0 0
\(193\) −12.8708 −0.926459 −0.463229 0.886238i \(-0.653309\pi\)
−0.463229 + 0.886238i \(0.653309\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 22.4345i 1.59839i 0.601072 + 0.799195i \(0.294741\pi\)
−0.601072 + 0.799195i \(0.705259\pi\)
\(198\) 0 0
\(199\) 14.9799 1.06189 0.530947 0.847405i \(-0.321836\pi\)
0.530947 + 0.847405i \(0.321836\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 3.53972i − 0.248440i
\(204\) 0 0
\(205\) 28.7752i 2.00974i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 4.94761 0.342234
\(210\) 0 0
\(211\) 8.50651i 0.585612i 0.956172 + 0.292806i \(0.0945890\pi\)
−0.956172 + 0.292806i \(0.905411\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 52.7787 3.59948
\(216\) 0 0
\(217\) 2.08831 0.141763
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 19.0675i 1.28262i
\(222\) 0 0
\(223\) −16.3214 −1.09296 −0.546479 0.837473i \(-0.684032\pi\)
−0.546479 + 0.837473i \(0.684032\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 10.8279i 0.718671i 0.933208 + 0.359336i \(0.116997\pi\)
−0.933208 + 0.359336i \(0.883003\pi\)
\(228\) 0 0
\(229\) 24.9251i 1.64710i 0.567245 + 0.823549i \(0.308009\pi\)
−0.567245 + 0.823549i \(0.691991\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 13.5950 0.890641 0.445321 0.895371i \(-0.353090\pi\)
0.445321 + 0.895371i \(0.353090\pi\)
\(234\) 0 0
\(235\) 16.4483i 1.07297i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 3.82998 0.247741 0.123870 0.992298i \(-0.460469\pi\)
0.123870 + 0.992298i \(0.460469\pi\)
\(240\) 0 0
\(241\) 5.32509 0.343019 0.171509 0.985182i \(-0.445136\pi\)
0.171509 + 0.985182i \(0.445136\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 4.29757i 0.274562i
\(246\) 0 0
\(247\) 18.5574 1.18078
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 19.2193i − 1.21311i −0.795040 0.606556i \(-0.792550\pi\)
0.795040 0.606556i \(-0.207450\pi\)
\(252\) 0 0
\(253\) − 9.56636i − 0.601432i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −0.655642 −0.0408979 −0.0204489 0.999791i \(-0.506510\pi\)
−0.0204489 + 0.999791i \(0.506510\pi\)
\(258\) 0 0
\(259\) − 4.48276i − 0.278545i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 25.8211 1.59220 0.796098 0.605168i \(-0.206894\pi\)
0.796098 + 0.605168i \(0.206894\pi\)
\(264\) 0 0
\(265\) 37.1186 2.28018
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 14.2078i − 0.866266i −0.901330 0.433133i \(-0.857408\pi\)
0.901330 0.433133i \(-0.142592\pi\)
\(270\) 0 0
\(271\) −3.52786 −0.214302 −0.107151 0.994243i \(-0.534173\pi\)
−0.107151 + 0.994243i \(0.534173\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 21.6526i 1.30570i
\(276\) 0 0
\(277\) − 13.9396i − 0.837548i −0.908091 0.418774i \(-0.862460\pi\)
0.908091 0.418774i \(-0.137540\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −20.4105 −1.21759 −0.608793 0.793329i \(-0.708346\pi\)
−0.608793 + 0.793329i \(0.708346\pi\)
\(282\) 0 0
\(283\) 29.6867i 1.76469i 0.470600 + 0.882347i \(0.344037\pi\)
−0.470600 + 0.882347i \(0.655963\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6.69568 0.395233
\(288\) 0 0
\(289\) −7.00000 −0.411765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 16.4504i 0.961044i 0.876983 + 0.480522i \(0.159553\pi\)
−0.876983 + 0.480522i \(0.840447\pi\)
\(294\) 0 0
\(295\) −6.67261 −0.388494
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 35.8813i − 2.07507i
\(300\) 0 0
\(301\) − 12.2811i − 0.707869i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 15.6812 0.897905
\(306\) 0 0
\(307\) − 19.8393i − 1.13229i −0.824306 0.566145i \(-0.808434\pi\)
0.824306 0.566145i \(-0.191566\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 32.3919 1.83677 0.918387 0.395683i \(-0.129492\pi\)
0.918387 + 0.395683i \(0.129492\pi\)
\(312\) 0 0
\(313\) −6.02156 −0.340359 −0.170179 0.985413i \(-0.554435\pi\)
−0.170179 + 0.985413i \(0.554435\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 5.23318i 0.293925i 0.989142 + 0.146962i \(0.0469496\pi\)
−0.989142 + 0.146962i \(0.953050\pi\)
\(318\) 0 0
\(319\) −5.69038 −0.318600
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 9.73249i − 0.541530i
\(324\) 0 0
\(325\) 81.2144i 4.50496i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 3.82734 0.211008
\(330\) 0 0
\(331\) 16.5740i 0.910988i 0.890239 + 0.455494i \(0.150537\pi\)
−0.890239 + 0.455494i \(0.849463\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −3.31928 −0.181352
\(336\) 0 0
\(337\) 1.70020 0.0926159 0.0463080 0.998927i \(-0.485254\pi\)
0.0463080 + 0.998927i \(0.485254\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 3.35711i − 0.181798i
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 24.0032i 1.28856i 0.764790 + 0.644279i \(0.222843\pi\)
−0.764790 + 0.644279i \(0.777157\pi\)
\(348\) 0 0
\(349\) 21.3679i 1.14380i 0.820324 + 0.571898i \(0.193793\pi\)
−0.820324 + 0.571898i \(0.806207\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −25.1560 −1.33892 −0.669460 0.742849i \(-0.733474\pi\)
−0.669460 + 0.742849i \(0.733474\pi\)
\(354\) 0 0
\(355\) 31.6517i 1.67990i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −18.1318 −0.956957 −0.478479 0.878099i \(-0.658812\pi\)
−0.478479 + 0.878099i \(0.658812\pi\)
\(360\) 0 0
\(361\) 9.52786 0.501467
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 5.70456i 0.298590i
\(366\) 0 0
\(367\) 29.6249 1.54641 0.773203 0.634158i \(-0.218653\pi\)
0.773203 + 0.634158i \(0.218653\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 8.63711i − 0.448416i
\(372\) 0 0
\(373\) − 29.3696i − 1.52070i −0.649514 0.760349i \(-0.725028\pi\)
0.649514 0.760349i \(-0.274972\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −21.3434 −1.09924
\(378\) 0 0
\(379\) − 16.1375i − 0.828930i −0.910065 0.414465i \(-0.863969\pi\)
0.910065 0.414465i \(-0.136031\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 21.9227 1.12020 0.560099 0.828426i \(-0.310763\pi\)
0.560099 + 0.828426i \(0.310763\pi\)
\(384\) 0 0
\(385\) 6.90868 0.352099
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 27.7725i − 1.40812i −0.710140 0.704061i \(-0.751368\pi\)
0.710140 0.704061i \(-0.248632\pi\)
\(390\) 0 0
\(391\) −18.8181 −0.951671
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 54.7514i − 2.75484i
\(396\) 0 0
\(397\) − 27.6193i − 1.38617i −0.720855 0.693086i \(-0.756251\pi\)
0.720855 0.693086i \(-0.243749\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −10.4439 −0.521546 −0.260773 0.965400i \(-0.583977\pi\)
−0.260773 + 0.965400i \(0.583977\pi\)
\(402\) 0 0
\(403\) − 12.5918i − 0.627242i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −7.20638 −0.357207
\(408\) 0 0
\(409\) 15.1424 0.748745 0.374373 0.927278i \(-0.377858\pi\)
0.374373 + 0.927278i \(0.377858\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 1.55265i 0.0764007i
\(414\) 0 0
\(415\) 30.4460 1.49453
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 19.9623i 0.975222i 0.873061 + 0.487611i \(0.162131\pi\)
−0.873061 + 0.487611i \(0.837869\pi\)
\(420\) 0 0
\(421\) − 12.8339i − 0.625486i −0.949838 0.312743i \(-0.898752\pi\)
0.949838 0.312743i \(-0.101248\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 42.5931 2.06607
\(426\) 0 0
\(427\) − 3.64886i − 0.176581i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −2.66364 −0.128303 −0.0641514 0.997940i \(-0.520434\pi\)
−0.0641514 + 0.997940i \(0.520434\pi\)
\(432\) 0 0
\(433\) −19.2122 −0.923280 −0.461640 0.887067i \(-0.652739\pi\)
−0.461640 + 0.887067i \(0.652739\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 18.3147i 0.876109i
\(438\) 0 0
\(439\) 35.1209 1.67623 0.838114 0.545495i \(-0.183658\pi\)
0.838114 + 0.545495i \(0.183658\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 8.40943i − 0.399544i −0.979842 0.199772i \(-0.935980\pi\)
0.979842 0.199772i \(-0.0640202\pi\)
\(444\) 0 0
\(445\) − 41.8286i − 1.98286i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 4.81067 0.227030 0.113515 0.993536i \(-0.463789\pi\)
0.113515 + 0.993536i \(0.463789\pi\)
\(450\) 0 0
\(451\) − 10.7638i − 0.506848i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 25.9129 1.21482
\(456\) 0 0
\(457\) −37.8291 −1.76957 −0.884785 0.465999i \(-0.845695\pi\)
−0.884785 + 0.465999i \(0.845695\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 10.5113i 0.489558i 0.969579 + 0.244779i \(0.0787154\pi\)
−0.969579 + 0.244779i \(0.921285\pi\)
\(462\) 0 0
\(463\) −4.76163 −0.221292 −0.110646 0.993860i \(-0.535292\pi\)
−0.110646 + 0.993860i \(0.535292\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 19.7216i − 0.912609i −0.889824 0.456305i \(-0.849173\pi\)
0.889824 0.456305i \(-0.150827\pi\)
\(468\) 0 0
\(469\) 0.772361i 0.0356643i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −19.7428 −0.907773
\(474\) 0 0
\(475\) − 41.4537i − 1.90203i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 2.49558 0.114026 0.0570131 0.998373i \(-0.481842\pi\)
0.0570131 + 0.998373i \(0.481842\pi\)
\(480\) 0 0
\(481\) −27.0296 −1.23244
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 43.7507i − 1.98662i
\(486\) 0 0
\(487\) 10.0178 0.453951 0.226976 0.973900i \(-0.427116\pi\)
0.226976 + 0.973900i \(0.427116\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 4.23469i 0.191109i 0.995424 + 0.0955544i \(0.0304624\pi\)
−0.995424 + 0.0955544i \(0.969538\pi\)
\(492\) 0 0
\(493\) 11.1936i 0.504134i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 7.36501 0.330366
\(498\) 0 0
\(499\) 10.1176i 0.452925i 0.974020 + 0.226463i \(0.0727161\pi\)
−0.974020 + 0.226463i \(0.927284\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 21.6693 0.966186 0.483093 0.875569i \(-0.339513\pi\)
0.483093 + 0.875569i \(0.339513\pi\)
\(504\) 0 0
\(505\) −4.41570 −0.196496
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 13.8362i − 0.613280i −0.951826 0.306640i \(-0.900795\pi\)
0.951826 0.306640i \(-0.0992048\pi\)
\(510\) 0 0
\(511\) 1.32739 0.0587203
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 4.14766i 0.182768i
\(516\) 0 0
\(517\) − 6.15275i − 0.270597i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 19.5067 0.854603 0.427302 0.904109i \(-0.359464\pi\)
0.427302 + 0.904109i \(0.359464\pi\)
\(522\) 0 0
\(523\) − 0.616566i − 0.0269606i −0.999909 0.0134803i \(-0.995709\pi\)
0.999909 0.0134803i \(-0.00429104\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −6.60380 −0.287666
\(528\) 0 0
\(529\) 12.4120 0.539651
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 40.3727i − 1.74874i
\(534\) 0 0
\(535\) −69.1690 −2.99044
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 1.60758i − 0.0692433i
\(540\) 0 0
\(541\) 33.0638i 1.42152i 0.703432 + 0.710762i \(0.251650\pi\)
−0.703432 + 0.710762i \(0.748350\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 87.1059 3.73121
\(546\) 0 0
\(547\) − 45.6463i − 1.95170i −0.218450 0.975848i \(-0.570100\pi\)
0.218450 0.975848i \(-0.429900\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 10.8941 0.464106
\(552\) 0 0
\(553\) −12.7401 −0.541763
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 0.0518694i − 0.00219778i −0.999999 0.00109889i \(-0.999650\pi\)
0.999999 0.00109889i \(-0.000349787\pi\)
\(558\) 0 0
\(559\) −74.0508 −3.13201
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 10.9105i − 0.459822i −0.973212 0.229911i \(-0.926156\pi\)
0.973212 0.229911i \(-0.0738436\pi\)
\(564\) 0 0
\(565\) 4.08791i 0.171980i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 26.2746 1.10149 0.550745 0.834673i \(-0.314344\pi\)
0.550745 + 0.834673i \(0.314344\pi\)
\(570\) 0 0
\(571\) − 24.4590i − 1.02358i −0.859111 0.511789i \(-0.828983\pi\)
0.859111 0.511789i \(-0.171017\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −80.1520 −3.34257
\(576\) 0 0
\(577\) −10.2538 −0.426873 −0.213436 0.976957i \(-0.568466\pi\)
−0.213436 + 0.976957i \(0.568466\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 7.08446i − 0.293913i
\(582\) 0 0
\(583\) −13.8848 −0.575050
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 25.3720i 1.04722i 0.851959 + 0.523608i \(0.175414\pi\)
−0.851959 + 0.523608i \(0.824586\pi\)
\(588\) 0 0
\(589\) 6.42714i 0.264826i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −7.14310 −0.293332 −0.146666 0.989186i \(-0.546854\pi\)
−0.146666 + 0.989186i \(0.546854\pi\)
\(594\) 0 0
\(595\) − 13.5901i − 0.557140i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 3.00175 0.122648 0.0613242 0.998118i \(-0.480468\pi\)
0.0613242 + 0.998118i \(0.480468\pi\)
\(600\) 0 0
\(601\) 21.2516 0.866871 0.433435 0.901185i \(-0.357301\pi\)
0.433435 + 0.901185i \(0.357301\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 36.1671i 1.47040i
\(606\) 0 0
\(607\) −24.0614 −0.976623 −0.488312 0.872669i \(-0.662387\pi\)
−0.488312 + 0.872669i \(0.662387\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 23.0776i − 0.933620i
\(612\) 0 0
\(613\) − 20.9197i − 0.844939i −0.906377 0.422469i \(-0.861163\pi\)
0.906377 0.422469i \(-0.138837\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −3.95122 −0.159070 −0.0795350 0.996832i \(-0.525344\pi\)
−0.0795350 + 0.996832i \(0.525344\pi\)
\(618\) 0 0
\(619\) − 23.8651i − 0.959220i −0.877482 0.479610i \(-0.840778\pi\)
0.877482 0.479610i \(-0.159222\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −9.73307 −0.389947
\(624\) 0 0
\(625\) 89.0716 3.56286
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 14.1757i 0.565223i
\(630\) 0 0
\(631\) −36.3725 −1.44797 −0.723983 0.689818i \(-0.757690\pi\)
−0.723983 + 0.689818i \(0.757690\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 49.9083i 1.98055i
\(636\) 0 0
\(637\) − 6.02967i − 0.238904i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −0.876669 −0.0346264 −0.0173132 0.999850i \(-0.505511\pi\)
−0.0173132 + 0.999850i \(0.505511\pi\)
\(642\) 0 0
\(643\) − 15.2486i − 0.601347i −0.953727 0.300674i \(-0.902789\pi\)
0.953727 0.300674i \(-0.0972114\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −9.53027 −0.374673 −0.187337 0.982296i \(-0.559986\pi\)
−0.187337 + 0.982296i \(0.559986\pi\)
\(648\) 0 0
\(649\) 2.49600 0.0979765
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 9.29496i − 0.363740i −0.983323 0.181870i \(-0.941785\pi\)
0.983323 0.181870i \(-0.0582150\pi\)
\(654\) 0 0
\(655\) 43.0273 1.68121
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 36.1879i 1.40968i 0.709366 + 0.704841i \(0.248982\pi\)
−0.709366 + 0.704841i \(0.751018\pi\)
\(660\) 0 0
\(661\) 19.8091i 0.770483i 0.922816 + 0.385242i \(0.125882\pi\)
−0.922816 + 0.385242i \(0.874118\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −13.2266 −0.512904
\(666\) 0 0
\(667\) − 21.0642i − 0.815607i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −5.86582 −0.226448
\(672\) 0 0
\(673\) −0.186375 −0.00718421 −0.00359211 0.999994i \(-0.501143\pi\)
−0.00359211 + 0.999994i \(0.501143\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 3.31885i − 0.127554i −0.997964 0.0637770i \(-0.979685\pi\)
0.997964 0.0637770i \(-0.0203146\pi\)
\(678\) 0 0
\(679\) −10.1803 −0.390686
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 17.9992i 0.688720i 0.938838 + 0.344360i \(0.111904\pi\)
−0.938838 + 0.344360i \(0.888096\pi\)
\(684\) 0 0
\(685\) 36.1161i 1.37992i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −52.0789 −1.98405
\(690\) 0 0
\(691\) − 3.67153i − 0.139671i −0.997559 0.0698357i \(-0.977752\pi\)
0.997559 0.0698357i \(-0.0222475\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 28.4257 1.07825
\(696\) 0 0
\(697\) −21.1736 −0.802007
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 7.06843i − 0.266971i −0.991051 0.133485i \(-0.957383\pi\)
0.991051 0.133485i \(-0.0426169\pi\)
\(702\) 0 0
\(703\) 13.7965 0.520345
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1.02749i 0.0386426i
\(708\) 0 0
\(709\) − 20.4607i − 0.768417i −0.923246 0.384209i \(-0.874474\pi\)
0.923246 0.384209i \(-0.125526\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 12.4271 0.465398
\(714\) 0 0
\(715\) − 41.6571i − 1.55789i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −40.5191 −1.51111 −0.755554 0.655087i \(-0.772632\pi\)
−0.755554 + 0.655087i \(0.772632\pi\)
\(720\) 0 0
\(721\) 0.965118 0.0359429
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 47.6769i 1.77068i
\(726\) 0 0
\(727\) −6.87078 −0.254823 −0.127412 0.991850i \(-0.540667\pi\)
−0.127412 + 0.991850i \(0.540667\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 38.8361i 1.43641i
\(732\) 0 0
\(733\) − 18.5847i − 0.686442i −0.939255 0.343221i \(-0.888482\pi\)
0.939255 0.343221i \(-0.111518\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.24163 0.0457360
\(738\) 0 0
\(739\) 4.69519i 0.172715i 0.996264 + 0.0863576i \(0.0275228\pi\)
−0.996264 + 0.0863576i \(0.972477\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −15.5867 −0.571821 −0.285911 0.958256i \(-0.592296\pi\)
−0.285911 + 0.958256i \(0.592296\pi\)
\(744\) 0 0
\(745\) 73.0011 2.67455
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 16.0949i 0.588095i
\(750\) 0 0
\(751\) 18.7401 0.683835 0.341917 0.939730i \(-0.388924\pi\)
0.341917 + 0.939730i \(0.388924\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 49.5319i 1.80265i
\(756\) 0 0
\(757\) − 24.9095i − 0.905352i −0.891675 0.452676i \(-0.850469\pi\)
0.891675 0.452676i \(-0.149531\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −4.70991 −0.170734 −0.0853670 0.996350i \(-0.527206\pi\)
−0.0853670 + 0.996350i \(0.527206\pi\)
\(762\) 0 0
\(763\) − 20.2686i − 0.733774i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 9.36195 0.338040
\(768\) 0 0
\(769\) 13.8989 0.501208 0.250604 0.968090i \(-0.419371\pi\)
0.250604 + 0.968090i \(0.419371\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 14.8995i 0.535899i 0.963433 + 0.267949i \(0.0863460\pi\)
−0.963433 + 0.267949i \(0.913654\pi\)
\(774\) 0 0
\(775\) −28.1276 −1.01037
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 20.6072i 0.738329i
\(780\) 0 0
\(781\) − 11.8398i − 0.423662i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −44.6472 −1.59353
\(786\) 0 0
\(787\) 10.1839i 0.363018i 0.983389 + 0.181509i \(0.0580981\pi\)
−0.983389 + 0.181509i \(0.941902\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0.951214 0.0338213
\(792\) 0 0
\(793\) −22.0014 −0.781293
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 14.0210i 0.496648i 0.968677 + 0.248324i \(0.0798798\pi\)
−0.968677 + 0.248324i \(0.920120\pi\)
\(798\) 0 0
\(799\) −12.1031 −0.428177
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 2.13388i − 0.0753031i
\(804\) 0 0
\(805\) 25.5740i 0.901364i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −28.0399 −0.985831 −0.492916 0.870077i \(-0.664069\pi\)
−0.492916 + 0.870077i \(0.664069\pi\)
\(810\) 0 0
\(811\) − 13.2362i − 0.464785i −0.972622 0.232393i \(-0.925345\pi\)
0.972622 0.232393i \(-0.0746554\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −51.0307 −1.78753
\(816\) 0 0
\(817\) 37.7972 1.32236
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 13.8343i − 0.482821i −0.970423 0.241411i \(-0.922390\pi\)
0.970423 0.241411i \(-0.0776100\pi\)
\(822\) 0 0
\(823\) −29.4862 −1.02782 −0.513912 0.857843i \(-0.671804\pi\)
−0.513912 + 0.857843i \(0.671804\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 28.9797i − 1.00772i −0.863784 0.503861i \(-0.831912\pi\)
0.863784 0.503861i \(-0.168088\pi\)
\(828\) 0 0
\(829\) − 32.5800i − 1.13155i −0.824560 0.565774i \(-0.808577\pi\)
0.824560 0.565774i \(-0.191423\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −3.16228 −0.109566
\(834\) 0 0
\(835\) 64.7197i 2.23972i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 19.8215 0.684313 0.342157 0.939643i \(-0.388843\pi\)
0.342157 + 0.939643i \(0.388843\pi\)
\(840\) 0 0
\(841\) 16.4704 0.567944
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 100.378i − 3.45311i
\(846\) 0 0
\(847\) 8.41570 0.289167
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 26.6760i − 0.914441i
\(852\) 0 0
\(853\) 17.4764i 0.598381i 0.954193 + 0.299190i \(0.0967166\pi\)
−0.954193 + 0.299190i \(0.903283\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −26.0767 −0.890764 −0.445382 0.895341i \(-0.646932\pi\)
−0.445382 + 0.895341i \(0.646932\pi\)
\(858\) 0 0
\(859\) − 40.6339i − 1.38641i −0.720740 0.693205i \(-0.756198\pi\)
0.720740 0.693205i \(-0.243802\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 41.4216 1.41001 0.705004 0.709204i \(-0.250945\pi\)
0.705004 + 0.709204i \(0.250945\pi\)
\(864\) 0 0
\(865\) −102.219 −3.47556
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 20.4806i 0.694758i
\(870\) 0 0
\(871\) 4.65709 0.157799
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 36.3966i − 1.23043i
\(876\) 0 0
\(877\) 33.0093i 1.11464i 0.830296 + 0.557322i \(0.188171\pi\)
−0.830296 + 0.557322i \(0.811829\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 18.0587 0.608414 0.304207 0.952606i \(-0.401609\pi\)
0.304207 + 0.952606i \(0.401609\pi\)
\(882\) 0 0
\(883\) 46.6771i 1.57081i 0.618983 + 0.785404i \(0.287545\pi\)
−0.618983 + 0.785404i \(0.712455\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −29.2001 −0.980444 −0.490222 0.871598i \(-0.663084\pi\)
−0.490222 + 0.871598i \(0.663084\pi\)
\(888\) 0 0
\(889\) 11.6132 0.389493
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 11.7793i 0.394181i
\(894\) 0 0
\(895\) 95.3714 3.18791
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 7.39202i − 0.246538i
\(900\) 0 0
\(901\) 27.3129i 0.909926i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 36.9700 1.22892
\(906\) 0 0
\(907\) − 9.06592i − 0.301029i −0.988608 0.150514i \(-0.951907\pi\)
0.988608 0.150514i \(-0.0480930\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −14.3571 −0.475673 −0.237837 0.971305i \(-0.576438\pi\)
−0.237837 + 0.971305i \(0.576438\pi\)
\(912\) 0 0
\(913\) −11.3888 −0.376915
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 10.0120i − 0.330625i
\(918\) 0 0
\(919\) −49.8788 −1.64535 −0.822675 0.568513i \(-0.807519\pi\)
−0.822675 + 0.568513i \(0.807519\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 44.4086i − 1.46173i
\(924\) 0 0
\(925\) 60.3788i 1.98524i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 39.3069 1.28962 0.644809 0.764343i \(-0.276937\pi\)
0.644809 + 0.764343i \(0.276937\pi\)
\(930\) 0 0
\(931\) 3.07768i 0.100867i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −21.8472 −0.714478
\(936\) 0 0
\(937\) 37.3504 1.22018 0.610092 0.792331i \(-0.291133\pi\)
0.610092 + 0.792331i \(0.291133\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 38.7781i 1.26413i 0.774915 + 0.632065i \(0.217792\pi\)
−0.774915 + 0.632065i \(0.782208\pi\)
\(942\) 0 0
\(943\) 39.8446 1.29752
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 3.75328i − 0.121965i −0.998139 0.0609826i \(-0.980577\pi\)
0.998139 0.0609826i \(-0.0194234\pi\)
\(948\) 0 0
\(949\) − 8.00373i − 0.259812i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −46.5443 −1.50772 −0.753859 0.657036i \(-0.771810\pi\)
−0.753859 + 0.657036i \(0.771810\pi\)
\(954\) 0 0
\(955\) − 37.5173i − 1.21403i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 8.40383 0.271374
\(960\) 0 0
\(961\) −26.6390 −0.859322
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 55.3131i − 1.78059i
\(966\) 0 0
\(967\) 16.6487 0.535388 0.267694 0.963504i \(-0.413738\pi\)
0.267694 + 0.963504i \(0.413738\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 5.74652i − 0.184415i −0.995740 0.0922073i \(-0.970608\pi\)
0.995740 0.0922073i \(-0.0293922\pi\)
\(972\) 0 0
\(973\) − 6.61437i − 0.212047i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −27.6236 −0.883758 −0.441879 0.897075i \(-0.645688\pi\)
−0.441879 + 0.897075i \(0.645688\pi\)
\(978\) 0 0
\(979\) 15.6467i 0.500070i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 2.19812 0.0701091 0.0350545 0.999385i \(-0.488840\pi\)
0.0350545 + 0.999385i \(0.488840\pi\)
\(984\) 0 0
\(985\) −96.4138 −3.07200
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 73.0821i − 2.32388i
\(990\) 0 0
\(991\) −2.62346 −0.0833369 −0.0416684 0.999131i \(-0.513267\pi\)
−0.0416684 + 0.999131i \(0.513267\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 64.3770i 2.04089i
\(996\) 0 0
\(997\) − 29.1361i − 0.922749i −0.887205 0.461375i \(-0.847356\pi\)
0.887205 0.461375i \(-0.152644\pi\)
\(998\) 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.d.3025.15 16
3.2 odd 2 inner 6048.2.c.d.3025.1 16
4.3 odd 2 1512.2.c.d.757.8 yes 16
8.3 odd 2 1512.2.c.d.757.5 16
8.5 even 2 inner 6048.2.c.d.3025.2 16
12.11 even 2 1512.2.c.d.757.9 yes 16
24.5 odd 2 inner 6048.2.c.d.3025.16 16
24.11 even 2 1512.2.c.d.757.12 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.c.d.757.5 16 8.3 odd 2
1512.2.c.d.757.8 yes 16 4.3 odd 2
1512.2.c.d.757.9 yes 16 12.11 even 2
1512.2.c.d.757.12 yes 16 24.11 even 2
6048.2.c.d.3025.1 16 3.2 odd 2 inner
6048.2.c.d.3025.2 16 8.5 even 2 inner
6048.2.c.d.3025.15 16 1.1 even 1 trivial
6048.2.c.d.3025.16 16 24.5 odd 2 inner