Properties

Label 4851.2.a.be.1.2
Level $4851$
Weight $2$
Character 4851.1
Self dual yes
Analytic conductor $38.735$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4851 = 3^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4851.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.7354300205\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 4851.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.41421 q^{2} +3.82843 q^{4} +2.00000 q^{5} +4.41421 q^{8} +O(q^{10})\) \(q+2.41421 q^{2} +3.82843 q^{4} +2.00000 q^{5} +4.41421 q^{8} +4.82843 q^{10} -1.00000 q^{11} -0.828427 q^{13} +3.00000 q^{16} +4.41421 q^{17} +7.24264 q^{19} +7.65685 q^{20} -2.41421 q^{22} +7.00000 q^{23} -1.00000 q^{25} -2.00000 q^{26} -3.24264 q^{29} +5.65685 q^{31} -1.58579 q^{32} +10.6569 q^{34} -9.48528 q^{37} +17.4853 q^{38} +8.82843 q^{40} +1.17157 q^{41} -2.75736 q^{43} -3.82843 q^{44} +16.8995 q^{46} -9.82843 q^{47} -2.41421 q^{50} -3.17157 q^{52} +7.17157 q^{53} -2.00000 q^{55} -7.82843 q^{58} +8.65685 q^{59} -4.00000 q^{61} +13.6569 q^{62} -9.82843 q^{64} -1.65685 q^{65} -3.17157 q^{67} +16.8995 q^{68} +4.17157 q^{71} +0.343146 q^{73} -22.8995 q^{74} +27.7279 q^{76} +13.3137 q^{79} +6.00000 q^{80} +2.82843 q^{82} +2.82843 q^{83} +8.82843 q^{85} -6.65685 q^{86} -4.41421 q^{88} +14.1421 q^{89} +26.7990 q^{92} -23.7279 q^{94} +14.4853 q^{95} -11.4853 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} + 6 q^{8} + 4 q^{10} - 2 q^{11} + 4 q^{13} + 6 q^{16} + 6 q^{17} + 6 q^{19} + 4 q^{20} - 2 q^{22} + 14 q^{23} - 2 q^{25} - 4 q^{26} + 2 q^{29} - 6 q^{32} + 10 q^{34} - 2 q^{37} + 18 q^{38} + 12 q^{40} + 8 q^{41} - 14 q^{43} - 2 q^{44} + 14 q^{46} - 14 q^{47} - 2 q^{50} - 12 q^{52} + 20 q^{53} - 4 q^{55} - 10 q^{58} + 6 q^{59} - 8 q^{61} + 16 q^{62} - 14 q^{64} + 8 q^{65} - 12 q^{67} + 14 q^{68} + 14 q^{71} + 12 q^{73} - 26 q^{74} + 30 q^{76} + 4 q^{79} + 12 q^{80} + 12 q^{85} - 2 q^{86} - 6 q^{88} + 14 q^{92} - 22 q^{94} + 12 q^{95} - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.41421 1.70711 0.853553 0.521005i \(-0.174443\pi\)
0.853553 + 0.521005i \(0.174443\pi\)
\(3\) 0 0
\(4\) 3.82843 1.91421
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 4.41421 1.56066
\(9\) 0 0
\(10\) 4.82843 1.52688
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −0.828427 −0.229764 −0.114882 0.993379i \(-0.536649\pi\)
−0.114882 + 0.993379i \(0.536649\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) 4.41421 1.07060 0.535302 0.844661i \(-0.320198\pi\)
0.535302 + 0.844661i \(0.320198\pi\)
\(18\) 0 0
\(19\) 7.24264 1.66158 0.830788 0.556589i \(-0.187890\pi\)
0.830788 + 0.556589i \(0.187890\pi\)
\(20\) 7.65685 1.71212
\(21\) 0 0
\(22\) −2.41421 −0.514712
\(23\) 7.00000 1.45960 0.729800 0.683660i \(-0.239613\pi\)
0.729800 + 0.683660i \(0.239613\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) 0 0
\(29\) −3.24264 −0.602143 −0.301072 0.953602i \(-0.597344\pi\)
−0.301072 + 0.953602i \(0.597344\pi\)
\(30\) 0 0
\(31\) 5.65685 1.01600 0.508001 0.861357i \(-0.330385\pi\)
0.508001 + 0.861357i \(0.330385\pi\)
\(32\) −1.58579 −0.280330
\(33\) 0 0
\(34\) 10.6569 1.82764
\(35\) 0 0
\(36\) 0 0
\(37\) −9.48528 −1.55937 −0.779685 0.626172i \(-0.784621\pi\)
−0.779685 + 0.626172i \(0.784621\pi\)
\(38\) 17.4853 2.83649
\(39\) 0 0
\(40\) 8.82843 1.39590
\(41\) 1.17157 0.182969 0.0914845 0.995807i \(-0.470839\pi\)
0.0914845 + 0.995807i \(0.470839\pi\)
\(42\) 0 0
\(43\) −2.75736 −0.420493 −0.210247 0.977648i \(-0.567427\pi\)
−0.210247 + 0.977648i \(0.567427\pi\)
\(44\) −3.82843 −0.577157
\(45\) 0 0
\(46\) 16.8995 2.49169
\(47\) −9.82843 −1.43362 −0.716812 0.697267i \(-0.754399\pi\)
−0.716812 + 0.697267i \(0.754399\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −2.41421 −0.341421
\(51\) 0 0
\(52\) −3.17157 −0.439818
\(53\) 7.17157 0.985091 0.492546 0.870287i \(-0.336066\pi\)
0.492546 + 0.870287i \(0.336066\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) 0 0
\(58\) −7.82843 −1.02792
\(59\) 8.65685 1.12703 0.563513 0.826107i \(-0.309449\pi\)
0.563513 + 0.826107i \(0.309449\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) 13.6569 1.73442
\(63\) 0 0
\(64\) −9.82843 −1.22855
\(65\) −1.65685 −0.205507
\(66\) 0 0
\(67\) −3.17157 −0.387469 −0.193735 0.981054i \(-0.562060\pi\)
−0.193735 + 0.981054i \(0.562060\pi\)
\(68\) 16.8995 2.04936
\(69\) 0 0
\(70\) 0 0
\(71\) 4.17157 0.495075 0.247537 0.968878i \(-0.420379\pi\)
0.247537 + 0.968878i \(0.420379\pi\)
\(72\) 0 0
\(73\) 0.343146 0.0401622 0.0200811 0.999798i \(-0.493608\pi\)
0.0200811 + 0.999798i \(0.493608\pi\)
\(74\) −22.8995 −2.66201
\(75\) 0 0
\(76\) 27.7279 3.18061
\(77\) 0 0
\(78\) 0 0
\(79\) 13.3137 1.49791 0.748955 0.662621i \(-0.230556\pi\)
0.748955 + 0.662621i \(0.230556\pi\)
\(80\) 6.00000 0.670820
\(81\) 0 0
\(82\) 2.82843 0.312348
\(83\) 2.82843 0.310460 0.155230 0.987878i \(-0.450388\pi\)
0.155230 + 0.987878i \(0.450388\pi\)
\(84\) 0 0
\(85\) 8.82843 0.957577
\(86\) −6.65685 −0.717827
\(87\) 0 0
\(88\) −4.41421 −0.470557
\(89\) 14.1421 1.49906 0.749532 0.661968i \(-0.230279\pi\)
0.749532 + 0.661968i \(0.230279\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 26.7990 2.79399
\(93\) 0 0
\(94\) −23.7279 −2.44735
\(95\) 14.4853 1.48616
\(96\) 0 0
\(97\) −11.4853 −1.16615 −0.583077 0.812417i \(-0.698151\pi\)
−0.583077 + 0.812417i \(0.698151\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −3.82843 −0.382843
\(101\) −4.89949 −0.487518 −0.243759 0.969836i \(-0.578381\pi\)
−0.243759 + 0.969836i \(0.578381\pi\)
\(102\) 0 0
\(103\) 12.4853 1.23021 0.615106 0.788445i \(-0.289113\pi\)
0.615106 + 0.788445i \(0.289113\pi\)
\(104\) −3.65685 −0.358584
\(105\) 0 0
\(106\) 17.3137 1.68166
\(107\) −9.65685 −0.933563 −0.466782 0.884373i \(-0.654587\pi\)
−0.466782 + 0.884373i \(0.654587\pi\)
\(108\) 0 0
\(109\) −6.82843 −0.654045 −0.327022 0.945017i \(-0.606045\pi\)
−0.327022 + 0.945017i \(0.606045\pi\)
\(110\) −4.82843 −0.460372
\(111\) 0 0
\(112\) 0 0
\(113\) 7.65685 0.720296 0.360148 0.932895i \(-0.382726\pi\)
0.360148 + 0.932895i \(0.382726\pi\)
\(114\) 0 0
\(115\) 14.0000 1.30551
\(116\) −12.4142 −1.15263
\(117\) 0 0
\(118\) 20.8995 1.92395
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −9.65685 −0.874291
\(123\) 0 0
\(124\) 21.6569 1.94484
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) −16.0711 −1.42608 −0.713038 0.701125i \(-0.752681\pi\)
−0.713038 + 0.701125i \(0.752681\pi\)
\(128\) −20.5563 −1.81694
\(129\) 0 0
\(130\) −4.00000 −0.350823
\(131\) 5.17157 0.451842 0.225921 0.974146i \(-0.427461\pi\)
0.225921 + 0.974146i \(0.427461\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −7.65685 −0.661451
\(135\) 0 0
\(136\) 19.4853 1.67085
\(137\) −0.485281 −0.0414604 −0.0207302 0.999785i \(-0.506599\pi\)
−0.0207302 + 0.999785i \(0.506599\pi\)
\(138\) 0 0
\(139\) 8.41421 0.713684 0.356842 0.934165i \(-0.383853\pi\)
0.356842 + 0.934165i \(0.383853\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 10.0711 0.845145
\(143\) 0.828427 0.0692766
\(144\) 0 0
\(145\) −6.48528 −0.538573
\(146\) 0.828427 0.0685611
\(147\) 0 0
\(148\) −36.3137 −2.98497
\(149\) 22.2132 1.81978 0.909888 0.414853i \(-0.136167\pi\)
0.909888 + 0.414853i \(0.136167\pi\)
\(150\) 0 0
\(151\) −18.8995 −1.53802 −0.769010 0.639237i \(-0.779250\pi\)
−0.769010 + 0.639237i \(0.779250\pi\)
\(152\) 31.9706 2.59316
\(153\) 0 0
\(154\) 0 0
\(155\) 11.3137 0.908739
\(156\) 0 0
\(157\) 7.00000 0.558661 0.279330 0.960195i \(-0.409888\pi\)
0.279330 + 0.960195i \(0.409888\pi\)
\(158\) 32.1421 2.55709
\(159\) 0 0
\(160\) −3.17157 −0.250735
\(161\) 0 0
\(162\) 0 0
\(163\) −20.9706 −1.64254 −0.821271 0.570539i \(-0.806734\pi\)
−0.821271 + 0.570539i \(0.806734\pi\)
\(164\) 4.48528 0.350242
\(165\) 0 0
\(166\) 6.82843 0.529989
\(167\) −5.17157 −0.400188 −0.200094 0.979777i \(-0.564125\pi\)
−0.200094 + 0.979777i \(0.564125\pi\)
\(168\) 0 0
\(169\) −12.3137 −0.947208
\(170\) 21.3137 1.63469
\(171\) 0 0
\(172\) −10.5563 −0.804914
\(173\) 14.8284 1.12738 0.563692 0.825985i \(-0.309380\pi\)
0.563692 + 0.825985i \(0.309380\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −3.00000 −0.226134
\(177\) 0 0
\(178\) 34.1421 2.55906
\(179\) −17.8284 −1.33256 −0.666280 0.745702i \(-0.732114\pi\)
−0.666280 + 0.745702i \(0.732114\pi\)
\(180\) 0 0
\(181\) −11.6569 −0.866447 −0.433224 0.901286i \(-0.642624\pi\)
−0.433224 + 0.901286i \(0.642624\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 30.8995 2.27794
\(185\) −18.9706 −1.39474
\(186\) 0 0
\(187\) −4.41421 −0.322799
\(188\) −37.6274 −2.74426
\(189\) 0 0
\(190\) 34.9706 2.53703
\(191\) 21.6569 1.56703 0.783517 0.621370i \(-0.213423\pi\)
0.783517 + 0.621370i \(0.213423\pi\)
\(192\) 0 0
\(193\) −12.1421 −0.874010 −0.437005 0.899459i \(-0.643961\pi\)
−0.437005 + 0.899459i \(0.643961\pi\)
\(194\) −27.7279 −1.99075
\(195\) 0 0
\(196\) 0 0
\(197\) 16.4142 1.16946 0.584732 0.811226i \(-0.301200\pi\)
0.584732 + 0.811226i \(0.301200\pi\)
\(198\) 0 0
\(199\) −19.7990 −1.40351 −0.701757 0.712417i \(-0.747601\pi\)
−0.701757 + 0.712417i \(0.747601\pi\)
\(200\) −4.41421 −0.312132
\(201\) 0 0
\(202\) −11.8284 −0.832245
\(203\) 0 0
\(204\) 0 0
\(205\) 2.34315 0.163652
\(206\) 30.1421 2.10010
\(207\) 0 0
\(208\) −2.48528 −0.172323
\(209\) −7.24264 −0.500984
\(210\) 0 0
\(211\) 14.9706 1.03062 0.515308 0.857005i \(-0.327678\pi\)
0.515308 + 0.857005i \(0.327678\pi\)
\(212\) 27.4558 1.88568
\(213\) 0 0
\(214\) −23.3137 −1.59369
\(215\) −5.51472 −0.376101
\(216\) 0 0
\(217\) 0 0
\(218\) −16.4853 −1.11652
\(219\) 0 0
\(220\) −7.65685 −0.516225
\(221\) −3.65685 −0.245987
\(222\) 0 0
\(223\) −22.9706 −1.53822 −0.769111 0.639115i \(-0.779301\pi\)
−0.769111 + 0.639115i \(0.779301\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 18.4853 1.22962
\(227\) −14.9706 −0.993631 −0.496816 0.867856i \(-0.665497\pi\)
−0.496816 + 0.867856i \(0.665497\pi\)
\(228\) 0 0
\(229\) −15.6569 −1.03463 −0.517317 0.855794i \(-0.673069\pi\)
−0.517317 + 0.855794i \(0.673069\pi\)
\(230\) 33.7990 2.22864
\(231\) 0 0
\(232\) −14.3137 −0.939741
\(233\) −10.4142 −0.682258 −0.341129 0.940017i \(-0.610809\pi\)
−0.341129 + 0.940017i \(0.610809\pi\)
\(234\) 0 0
\(235\) −19.6569 −1.28227
\(236\) 33.1421 2.15737
\(237\) 0 0
\(238\) 0 0
\(239\) 6.48528 0.419498 0.209749 0.977755i \(-0.432735\pi\)
0.209749 + 0.977755i \(0.432735\pi\)
\(240\) 0 0
\(241\) 7.31371 0.471117 0.235559 0.971860i \(-0.424308\pi\)
0.235559 + 0.971860i \(0.424308\pi\)
\(242\) 2.41421 0.155192
\(243\) 0 0
\(244\) −15.3137 −0.980360
\(245\) 0 0
\(246\) 0 0
\(247\) −6.00000 −0.381771
\(248\) 24.9706 1.58563
\(249\) 0 0
\(250\) −28.9706 −1.83226
\(251\) 2.51472 0.158728 0.0793638 0.996846i \(-0.474711\pi\)
0.0793638 + 0.996846i \(0.474711\pi\)
\(252\) 0 0
\(253\) −7.00000 −0.440086
\(254\) −38.7990 −2.43447
\(255\) 0 0
\(256\) −29.9706 −1.87316
\(257\) −21.4558 −1.33838 −0.669189 0.743092i \(-0.733359\pi\)
−0.669189 + 0.743092i \(0.733359\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −6.34315 −0.393385
\(261\) 0 0
\(262\) 12.4853 0.771343
\(263\) −18.0000 −1.10993 −0.554964 0.831875i \(-0.687268\pi\)
−0.554964 + 0.831875i \(0.687268\pi\)
\(264\) 0 0
\(265\) 14.3431 0.881092
\(266\) 0 0
\(267\) 0 0
\(268\) −12.1421 −0.741699
\(269\) 27.7990 1.69493 0.847467 0.530848i \(-0.178126\pi\)
0.847467 + 0.530848i \(0.178126\pi\)
\(270\) 0 0
\(271\) −22.9706 −1.39536 −0.697681 0.716408i \(-0.745785\pi\)
−0.697681 + 0.716408i \(0.745785\pi\)
\(272\) 13.2426 0.802953
\(273\) 0 0
\(274\) −1.17157 −0.0707773
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) −25.6569 −1.54157 −0.770785 0.637095i \(-0.780136\pi\)
−0.770785 + 0.637095i \(0.780136\pi\)
\(278\) 20.3137 1.21834
\(279\) 0 0
\(280\) 0 0
\(281\) −14.4142 −0.859880 −0.429940 0.902857i \(-0.641465\pi\)
−0.429940 + 0.902857i \(0.641465\pi\)
\(282\) 0 0
\(283\) 20.3431 1.20927 0.604637 0.796501i \(-0.293318\pi\)
0.604637 + 0.796501i \(0.293318\pi\)
\(284\) 15.9706 0.947679
\(285\) 0 0
\(286\) 2.00000 0.118262
\(287\) 0 0
\(288\) 0 0
\(289\) 2.48528 0.146193
\(290\) −15.6569 −0.919402
\(291\) 0 0
\(292\) 1.31371 0.0768790
\(293\) −5.72792 −0.334629 −0.167314 0.985904i \(-0.553509\pi\)
−0.167314 + 0.985904i \(0.553509\pi\)
\(294\) 0 0
\(295\) 17.3137 1.00804
\(296\) −41.8701 −2.43365
\(297\) 0 0
\(298\) 53.6274 3.10655
\(299\) −5.79899 −0.335364
\(300\) 0 0
\(301\) 0 0
\(302\) −45.6274 −2.62556
\(303\) 0 0
\(304\) 21.7279 1.24618
\(305\) −8.00000 −0.458079
\(306\) 0 0
\(307\) −17.3137 −0.988146 −0.494073 0.869421i \(-0.664492\pi\)
−0.494073 + 0.869421i \(0.664492\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 27.3137 1.55131
\(311\) −29.6274 −1.68002 −0.840008 0.542573i \(-0.817450\pi\)
−0.840008 + 0.542573i \(0.817450\pi\)
\(312\) 0 0
\(313\) 4.79899 0.271255 0.135627 0.990760i \(-0.456695\pi\)
0.135627 + 0.990760i \(0.456695\pi\)
\(314\) 16.8995 0.953694
\(315\) 0 0
\(316\) 50.9706 2.86732
\(317\) 25.3137 1.42176 0.710880 0.703314i \(-0.248297\pi\)
0.710880 + 0.703314i \(0.248297\pi\)
\(318\) 0 0
\(319\) 3.24264 0.181553
\(320\) −19.6569 −1.09885
\(321\) 0 0
\(322\) 0 0
\(323\) 31.9706 1.77889
\(324\) 0 0
\(325\) 0.828427 0.0459529
\(326\) −50.6274 −2.80399
\(327\) 0 0
\(328\) 5.17157 0.285552
\(329\) 0 0
\(330\) 0 0
\(331\) 22.4853 1.23590 0.617951 0.786216i \(-0.287963\pi\)
0.617951 + 0.786216i \(0.287963\pi\)
\(332\) 10.8284 0.594287
\(333\) 0 0
\(334\) −12.4853 −0.683164
\(335\) −6.34315 −0.346563
\(336\) 0 0
\(337\) 3.85786 0.210151 0.105076 0.994464i \(-0.466492\pi\)
0.105076 + 0.994464i \(0.466492\pi\)
\(338\) −29.7279 −1.61699
\(339\) 0 0
\(340\) 33.7990 1.83301
\(341\) −5.65685 −0.306336
\(342\) 0 0
\(343\) 0 0
\(344\) −12.1716 −0.656247
\(345\) 0 0
\(346\) 35.7990 1.92457
\(347\) −14.9706 −0.803662 −0.401831 0.915714i \(-0.631626\pi\)
−0.401831 + 0.915714i \(0.631626\pi\)
\(348\) 0 0
\(349\) 10.9706 0.587241 0.293620 0.955922i \(-0.405140\pi\)
0.293620 + 0.955922i \(0.405140\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.58579 0.0845227
\(353\) −13.3137 −0.708617 −0.354309 0.935129i \(-0.615284\pi\)
−0.354309 + 0.935129i \(0.615284\pi\)
\(354\) 0 0
\(355\) 8.34315 0.442808
\(356\) 54.1421 2.86953
\(357\) 0 0
\(358\) −43.0416 −2.27482
\(359\) 8.00000 0.422224 0.211112 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) 0 0
\(361\) 33.4558 1.76083
\(362\) −28.1421 −1.47912
\(363\) 0 0
\(364\) 0 0
\(365\) 0.686292 0.0359221
\(366\) 0 0
\(367\) −4.14214 −0.216218 −0.108109 0.994139i \(-0.534480\pi\)
−0.108109 + 0.994139i \(0.534480\pi\)
\(368\) 21.0000 1.09470
\(369\) 0 0
\(370\) −45.7990 −2.38098
\(371\) 0 0
\(372\) 0 0
\(373\) −1.65685 −0.0857887 −0.0428943 0.999080i \(-0.513658\pi\)
−0.0428943 + 0.999080i \(0.513658\pi\)
\(374\) −10.6569 −0.551053
\(375\) 0 0
\(376\) −43.3848 −2.23740
\(377\) 2.68629 0.138351
\(378\) 0 0
\(379\) 18.8284 0.967151 0.483576 0.875303i \(-0.339338\pi\)
0.483576 + 0.875303i \(0.339338\pi\)
\(380\) 55.4558 2.84482
\(381\) 0 0
\(382\) 52.2843 2.67510
\(383\) −2.31371 −0.118225 −0.0591125 0.998251i \(-0.518827\pi\)
−0.0591125 + 0.998251i \(0.518827\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −29.3137 −1.49203
\(387\) 0 0
\(388\) −43.9706 −2.23227
\(389\) 11.7990 0.598233 0.299116 0.954217i \(-0.403308\pi\)
0.299116 + 0.954217i \(0.403308\pi\)
\(390\) 0 0
\(391\) 30.8995 1.56265
\(392\) 0 0
\(393\) 0 0
\(394\) 39.6274 1.99640
\(395\) 26.6274 1.33977
\(396\) 0 0
\(397\) 17.4853 0.877561 0.438781 0.898594i \(-0.355411\pi\)
0.438781 + 0.898594i \(0.355411\pi\)
\(398\) −47.7990 −2.39595
\(399\) 0 0
\(400\) −3.00000 −0.150000
\(401\) −7.51472 −0.375267 −0.187634 0.982239i \(-0.560082\pi\)
−0.187634 + 0.982239i \(0.560082\pi\)
\(402\) 0 0
\(403\) −4.68629 −0.233441
\(404\) −18.7574 −0.933214
\(405\) 0 0
\(406\) 0 0
\(407\) 9.48528 0.470168
\(408\) 0 0
\(409\) −7.79899 −0.385635 −0.192818 0.981235i \(-0.561763\pi\)
−0.192818 + 0.981235i \(0.561763\pi\)
\(410\) 5.65685 0.279372
\(411\) 0 0
\(412\) 47.7990 2.35489
\(413\) 0 0
\(414\) 0 0
\(415\) 5.65685 0.277684
\(416\) 1.31371 0.0644099
\(417\) 0 0
\(418\) −17.4853 −0.855233
\(419\) −14.7990 −0.722978 −0.361489 0.932376i \(-0.617731\pi\)
−0.361489 + 0.932376i \(0.617731\pi\)
\(420\) 0 0
\(421\) −7.00000 −0.341159 −0.170580 0.985344i \(-0.554564\pi\)
−0.170580 + 0.985344i \(0.554564\pi\)
\(422\) 36.1421 1.75937
\(423\) 0 0
\(424\) 31.6569 1.53739
\(425\) −4.41421 −0.214121
\(426\) 0 0
\(427\) 0 0
\(428\) −36.9706 −1.78704
\(429\) 0 0
\(430\) −13.3137 −0.642044
\(431\) −6.97056 −0.335760 −0.167880 0.985807i \(-0.553692\pi\)
−0.167880 + 0.985807i \(0.553692\pi\)
\(432\) 0 0
\(433\) 0.857864 0.0412263 0.0206132 0.999788i \(-0.493438\pi\)
0.0206132 + 0.999788i \(0.493438\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −26.1421 −1.25198
\(437\) 50.6985 2.42524
\(438\) 0 0
\(439\) 30.8995 1.47475 0.737376 0.675482i \(-0.236065\pi\)
0.737376 + 0.675482i \(0.236065\pi\)
\(440\) −8.82843 −0.420879
\(441\) 0 0
\(442\) −8.82843 −0.419925
\(443\) −33.6274 −1.59769 −0.798843 0.601539i \(-0.794554\pi\)
−0.798843 + 0.601539i \(0.794554\pi\)
\(444\) 0 0
\(445\) 28.2843 1.34080
\(446\) −55.4558 −2.62591
\(447\) 0 0
\(448\) 0 0
\(449\) 4.00000 0.188772 0.0943858 0.995536i \(-0.469911\pi\)
0.0943858 + 0.995536i \(0.469911\pi\)
\(450\) 0 0
\(451\) −1.17157 −0.0551672
\(452\) 29.3137 1.37880
\(453\) 0 0
\(454\) −36.1421 −1.69623
\(455\) 0 0
\(456\) 0 0
\(457\) 18.8284 0.880757 0.440378 0.897812i \(-0.354844\pi\)
0.440378 + 0.897812i \(0.354844\pi\)
\(458\) −37.7990 −1.76623
\(459\) 0 0
\(460\) 53.5980 2.49902
\(461\) −6.75736 −0.314722 −0.157361 0.987541i \(-0.550299\pi\)
−0.157361 + 0.987541i \(0.550299\pi\)
\(462\) 0 0
\(463\) 18.6274 0.865689 0.432845 0.901468i \(-0.357510\pi\)
0.432845 + 0.901468i \(0.357510\pi\)
\(464\) −9.72792 −0.451607
\(465\) 0 0
\(466\) −25.1421 −1.16469
\(467\) 8.31371 0.384713 0.192356 0.981325i \(-0.438387\pi\)
0.192356 + 0.981325i \(0.438387\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −47.4558 −2.18897
\(471\) 0 0
\(472\) 38.2132 1.75891
\(473\) 2.75736 0.126784
\(474\) 0 0
\(475\) −7.24264 −0.332315
\(476\) 0 0
\(477\) 0 0
\(478\) 15.6569 0.716128
\(479\) −14.0000 −0.639676 −0.319838 0.947472i \(-0.603629\pi\)
−0.319838 + 0.947472i \(0.603629\pi\)
\(480\) 0 0
\(481\) 7.85786 0.358288
\(482\) 17.6569 0.804248
\(483\) 0 0
\(484\) 3.82843 0.174019
\(485\) −22.9706 −1.04304
\(486\) 0 0
\(487\) −32.6274 −1.47849 −0.739245 0.673437i \(-0.764817\pi\)
−0.739245 + 0.673437i \(0.764817\pi\)
\(488\) −17.6569 −0.799288
\(489\) 0 0
\(490\) 0 0
\(491\) 1.51472 0.0683583 0.0341791 0.999416i \(-0.489118\pi\)
0.0341791 + 0.999416i \(0.489118\pi\)
\(492\) 0 0
\(493\) −14.3137 −0.644657
\(494\) −14.4853 −0.651724
\(495\) 0 0
\(496\) 16.9706 0.762001
\(497\) 0 0
\(498\) 0 0
\(499\) −31.7990 −1.42352 −0.711759 0.702424i \(-0.752101\pi\)
−0.711759 + 0.702424i \(0.752101\pi\)
\(500\) −45.9411 −2.05455
\(501\) 0 0
\(502\) 6.07107 0.270965
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) −9.79899 −0.436049
\(506\) −16.8995 −0.751274
\(507\) 0 0
\(508\) −61.5269 −2.72982
\(509\) 12.0000 0.531891 0.265945 0.963988i \(-0.414316\pi\)
0.265945 + 0.963988i \(0.414316\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −31.2426 −1.38074
\(513\) 0 0
\(514\) −51.7990 −2.28476
\(515\) 24.9706 1.10033
\(516\) 0 0
\(517\) 9.82843 0.432254
\(518\) 0 0
\(519\) 0 0
\(520\) −7.31371 −0.320727
\(521\) −31.1716 −1.36565 −0.682826 0.730581i \(-0.739249\pi\)
−0.682826 + 0.730581i \(0.739249\pi\)
\(522\) 0 0
\(523\) −22.2843 −0.974423 −0.487212 0.873284i \(-0.661986\pi\)
−0.487212 + 0.873284i \(0.661986\pi\)
\(524\) 19.7990 0.864923
\(525\) 0 0
\(526\) −43.4558 −1.89476
\(527\) 24.9706 1.08773
\(528\) 0 0
\(529\) 26.0000 1.13043
\(530\) 34.6274 1.50412
\(531\) 0 0
\(532\) 0 0
\(533\) −0.970563 −0.0420397
\(534\) 0 0
\(535\) −19.3137 −0.835004
\(536\) −14.0000 −0.604708
\(537\) 0 0
\(538\) 67.1127 2.89343
\(539\) 0 0
\(540\) 0 0
\(541\) 19.6569 0.845114 0.422557 0.906336i \(-0.361133\pi\)
0.422557 + 0.906336i \(0.361133\pi\)
\(542\) −55.4558 −2.38203
\(543\) 0 0
\(544\) −7.00000 −0.300123
\(545\) −13.6569 −0.584995
\(546\) 0 0
\(547\) −24.2132 −1.03528 −0.517641 0.855598i \(-0.673190\pi\)
−0.517641 + 0.855598i \(0.673190\pi\)
\(548\) −1.85786 −0.0793640
\(549\) 0 0
\(550\) 2.41421 0.102942
\(551\) −23.4853 −1.00051
\(552\) 0 0
\(553\) 0 0
\(554\) −61.9411 −2.63163
\(555\) 0 0
\(556\) 32.2132 1.36614
\(557\) 8.55635 0.362544 0.181272 0.983433i \(-0.441979\pi\)
0.181272 + 0.983433i \(0.441979\pi\)
\(558\) 0 0
\(559\) 2.28427 0.0966144
\(560\) 0 0
\(561\) 0 0
\(562\) −34.7990 −1.46791
\(563\) −24.8284 −1.04639 −0.523197 0.852212i \(-0.675261\pi\)
−0.523197 + 0.852212i \(0.675261\pi\)
\(564\) 0 0
\(565\) 15.3137 0.644253
\(566\) 49.1127 2.06436
\(567\) 0 0
\(568\) 18.4142 0.772643
\(569\) −1.24264 −0.0520942 −0.0260471 0.999661i \(-0.508292\pi\)
−0.0260471 + 0.999661i \(0.508292\pi\)
\(570\) 0 0
\(571\) −3.92893 −0.164421 −0.0822103 0.996615i \(-0.526198\pi\)
−0.0822103 + 0.996615i \(0.526198\pi\)
\(572\) 3.17157 0.132610
\(573\) 0 0
\(574\) 0 0
\(575\) −7.00000 −0.291920
\(576\) 0 0
\(577\) 7.65685 0.318759 0.159380 0.987217i \(-0.449051\pi\)
0.159380 + 0.987217i \(0.449051\pi\)
\(578\) 6.00000 0.249567
\(579\) 0 0
\(580\) −24.8284 −1.03094
\(581\) 0 0
\(582\) 0 0
\(583\) −7.17157 −0.297016
\(584\) 1.51472 0.0626795
\(585\) 0 0
\(586\) −13.8284 −0.571247
\(587\) −8.68629 −0.358522 −0.179261 0.983802i \(-0.557371\pi\)
−0.179261 + 0.983802i \(0.557371\pi\)
\(588\) 0 0
\(589\) 40.9706 1.68816
\(590\) 41.7990 1.72084
\(591\) 0 0
\(592\) −28.4558 −1.16953
\(593\) 22.2721 0.914605 0.457302 0.889311i \(-0.348816\pi\)
0.457302 + 0.889311i \(0.348816\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 85.0416 3.48344
\(597\) 0 0
\(598\) −14.0000 −0.572503
\(599\) 1.37258 0.0560822 0.0280411 0.999607i \(-0.491073\pi\)
0.0280411 + 0.999607i \(0.491073\pi\)
\(600\) 0 0
\(601\) −14.4853 −0.590867 −0.295433 0.955363i \(-0.595464\pi\)
−0.295433 + 0.955363i \(0.595464\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −72.3553 −2.94410
\(605\) 2.00000 0.0813116
\(606\) 0 0
\(607\) −18.6863 −0.758453 −0.379227 0.925304i \(-0.623810\pi\)
−0.379227 + 0.925304i \(0.623810\pi\)
\(608\) −11.4853 −0.465790
\(609\) 0 0
\(610\) −19.3137 −0.781989
\(611\) 8.14214 0.329396
\(612\) 0 0
\(613\) 25.1716 1.01667 0.508335 0.861159i \(-0.330261\pi\)
0.508335 + 0.861159i \(0.330261\pi\)
\(614\) −41.7990 −1.68687
\(615\) 0 0
\(616\) 0 0
\(617\) −13.1716 −0.530268 −0.265134 0.964212i \(-0.585416\pi\)
−0.265134 + 0.964212i \(0.585416\pi\)
\(618\) 0 0
\(619\) 34.6274 1.39179 0.695897 0.718142i \(-0.255007\pi\)
0.695897 + 0.718142i \(0.255007\pi\)
\(620\) 43.3137 1.73952
\(621\) 0 0
\(622\) −71.5269 −2.86797
\(623\) 0 0
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 11.5858 0.463061
\(627\) 0 0
\(628\) 26.7990 1.06940
\(629\) −41.8701 −1.66947
\(630\) 0 0
\(631\) 26.2843 1.04636 0.523180 0.852222i \(-0.324745\pi\)
0.523180 + 0.852222i \(0.324745\pi\)
\(632\) 58.7696 2.33773
\(633\) 0 0
\(634\) 61.1127 2.42710
\(635\) −32.1421 −1.27552
\(636\) 0 0
\(637\) 0 0
\(638\) 7.82843 0.309930
\(639\) 0 0
\(640\) −41.1127 −1.62512
\(641\) 12.4853 0.493139 0.246569 0.969125i \(-0.420697\pi\)
0.246569 + 0.969125i \(0.420697\pi\)
\(642\) 0 0
\(643\) 2.00000 0.0788723 0.0394362 0.999222i \(-0.487444\pi\)
0.0394362 + 0.999222i \(0.487444\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 77.1838 3.03675
\(647\) 19.3137 0.759300 0.379650 0.925130i \(-0.376044\pi\)
0.379650 + 0.925130i \(0.376044\pi\)
\(648\) 0 0
\(649\) −8.65685 −0.339811
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) −80.2843 −3.14417
\(653\) −31.1127 −1.21753 −0.608767 0.793349i \(-0.708336\pi\)
−0.608767 + 0.793349i \(0.708336\pi\)
\(654\) 0 0
\(655\) 10.3431 0.404140
\(656\) 3.51472 0.137227
\(657\) 0 0
\(658\) 0 0
\(659\) −34.4853 −1.34336 −0.671678 0.740843i \(-0.734426\pi\)
−0.671678 + 0.740843i \(0.734426\pi\)
\(660\) 0 0
\(661\) 11.9706 0.465601 0.232800 0.972525i \(-0.425211\pi\)
0.232800 + 0.972525i \(0.425211\pi\)
\(662\) 54.2843 2.10982
\(663\) 0 0
\(664\) 12.4853 0.484523
\(665\) 0 0
\(666\) 0 0
\(667\) −22.6985 −0.878889
\(668\) −19.7990 −0.766046
\(669\) 0 0
\(670\) −15.3137 −0.591620
\(671\) 4.00000 0.154418
\(672\) 0 0
\(673\) 27.1716 1.04739 0.523694 0.851907i \(-0.324554\pi\)
0.523694 + 0.851907i \(0.324554\pi\)
\(674\) 9.31371 0.358751
\(675\) 0 0
\(676\) −47.1421 −1.81316
\(677\) 32.0122 1.23033 0.615164 0.788399i \(-0.289090\pi\)
0.615164 + 0.788399i \(0.289090\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 38.9706 1.49445
\(681\) 0 0
\(682\) −13.6569 −0.522948
\(683\) −3.20101 −0.122483 −0.0612416 0.998123i \(-0.519506\pi\)
−0.0612416 + 0.998123i \(0.519506\pi\)
\(684\) 0 0
\(685\) −0.970563 −0.0370833
\(686\) 0 0
\(687\) 0 0
\(688\) −8.27208 −0.315370
\(689\) −5.94113 −0.226339
\(690\) 0 0
\(691\) −11.4558 −0.435801 −0.217900 0.975971i \(-0.569921\pi\)
−0.217900 + 0.975971i \(0.569921\pi\)
\(692\) 56.7696 2.15805
\(693\) 0 0
\(694\) −36.1421 −1.37194
\(695\) 16.8284 0.638339
\(696\) 0 0
\(697\) 5.17157 0.195887
\(698\) 26.4853 1.00248
\(699\) 0 0
\(700\) 0 0
\(701\) 2.89949 0.109512 0.0547562 0.998500i \(-0.482562\pi\)
0.0547562 + 0.998500i \(0.482562\pi\)
\(702\) 0 0
\(703\) −68.6985 −2.59101
\(704\) 9.82843 0.370423
\(705\) 0 0
\(706\) −32.1421 −1.20969
\(707\) 0 0
\(708\) 0 0
\(709\) 25.7696 0.967796 0.483898 0.875124i \(-0.339221\pi\)
0.483898 + 0.875124i \(0.339221\pi\)
\(710\) 20.1421 0.755921
\(711\) 0 0
\(712\) 62.4264 2.33953
\(713\) 39.5980 1.48296
\(714\) 0 0
\(715\) 1.65685 0.0619628
\(716\) −68.2548 −2.55080
\(717\) 0 0
\(718\) 19.3137 0.720781
\(719\) −11.2010 −0.417727 −0.208864 0.977945i \(-0.566976\pi\)
−0.208864 + 0.977945i \(0.566976\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 80.7696 3.00593
\(723\) 0 0
\(724\) −44.6274 −1.65856
\(725\) 3.24264 0.120429
\(726\) 0 0
\(727\) 46.0833 1.70913 0.854567 0.519342i \(-0.173823\pi\)
0.854567 + 0.519342i \(0.173823\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 1.65685 0.0613229
\(731\) −12.1716 −0.450182
\(732\) 0 0
\(733\) 9.65685 0.356684 0.178342 0.983969i \(-0.442927\pi\)
0.178342 + 0.983969i \(0.442927\pi\)
\(734\) −10.0000 −0.369107
\(735\) 0 0
\(736\) −11.1005 −0.409170
\(737\) 3.17157 0.116826
\(738\) 0 0
\(739\) −39.6569 −1.45880 −0.729400 0.684087i \(-0.760201\pi\)
−0.729400 + 0.684087i \(0.760201\pi\)
\(740\) −72.6274 −2.66984
\(741\) 0 0
\(742\) 0 0
\(743\) −33.7990 −1.23996 −0.619982 0.784616i \(-0.712860\pi\)
−0.619982 + 0.784616i \(0.712860\pi\)
\(744\) 0 0
\(745\) 44.4264 1.62766
\(746\) −4.00000 −0.146450
\(747\) 0 0
\(748\) −16.8995 −0.617907
\(749\) 0 0
\(750\) 0 0
\(751\) −29.3137 −1.06967 −0.534836 0.844956i \(-0.679627\pi\)
−0.534836 + 0.844956i \(0.679627\pi\)
\(752\) −29.4853 −1.07522
\(753\) 0 0
\(754\) 6.48528 0.236180
\(755\) −37.7990 −1.37565
\(756\) 0 0
\(757\) −7.68629 −0.279363 −0.139682 0.990196i \(-0.544608\pi\)
−0.139682 + 0.990196i \(0.544608\pi\)
\(758\) 45.4558 1.65103
\(759\) 0 0
\(760\) 63.9411 2.31939
\(761\) 0.201010 0.00728661 0.00364331 0.999993i \(-0.498840\pi\)
0.00364331 + 0.999993i \(0.498840\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 82.9117 2.99964
\(765\) 0 0
\(766\) −5.58579 −0.201823
\(767\) −7.17157 −0.258950
\(768\) 0 0
\(769\) 33.7990 1.21882 0.609411 0.792854i \(-0.291406\pi\)
0.609411 + 0.792854i \(0.291406\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −46.4853 −1.67304
\(773\) 51.6569 1.85797 0.928984 0.370120i \(-0.120683\pi\)
0.928984 + 0.370120i \(0.120683\pi\)
\(774\) 0 0
\(775\) −5.65685 −0.203200
\(776\) −50.6985 −1.81997
\(777\) 0 0
\(778\) 28.4853 1.02125
\(779\) 8.48528 0.304017
\(780\) 0 0
\(781\) −4.17157 −0.149271
\(782\) 74.5980 2.66762
\(783\) 0 0
\(784\) 0 0
\(785\) 14.0000 0.499681
\(786\) 0 0
\(787\) 29.5269 1.05252 0.526260 0.850323i \(-0.323594\pi\)
0.526260 + 0.850323i \(0.323594\pi\)
\(788\) 62.8406 2.23860
\(789\) 0 0
\(790\) 64.2843 2.28713
\(791\) 0 0
\(792\) 0 0
\(793\) 3.31371 0.117673
\(794\) 42.2132 1.49809
\(795\) 0 0
\(796\) −75.7990 −2.68662
\(797\) −11.1716 −0.395717 −0.197859 0.980231i \(-0.563399\pi\)
−0.197859 + 0.980231i \(0.563399\pi\)
\(798\) 0 0
\(799\) −43.3848 −1.53484
\(800\) 1.58579 0.0560660
\(801\) 0 0
\(802\) −18.1421 −0.640621
\(803\) −0.343146 −0.0121094
\(804\) 0 0
\(805\) 0 0
\(806\) −11.3137 −0.398508
\(807\) 0 0
\(808\) −21.6274 −0.760850
\(809\) −10.8284 −0.380707 −0.190354 0.981716i \(-0.560963\pi\)
−0.190354 + 0.981716i \(0.560963\pi\)
\(810\) 0 0
\(811\) 14.9706 0.525688 0.262844 0.964838i \(-0.415340\pi\)
0.262844 + 0.964838i \(0.415340\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 22.8995 0.802627
\(815\) −41.9411 −1.46913
\(816\) 0 0
\(817\) −19.9706 −0.698682
\(818\) −18.8284 −0.658321
\(819\) 0 0
\(820\) 8.97056 0.313266
\(821\) −38.8284 −1.35512 −0.677561 0.735467i \(-0.736963\pi\)
−0.677561 + 0.735467i \(0.736963\pi\)
\(822\) 0 0
\(823\) −33.1716 −1.15629 −0.578144 0.815935i \(-0.696223\pi\)
−0.578144 + 0.815935i \(0.696223\pi\)
\(824\) 55.1127 1.91994
\(825\) 0 0
\(826\) 0 0
\(827\) 37.3137 1.29752 0.648762 0.760991i \(-0.275287\pi\)
0.648762 + 0.760991i \(0.275287\pi\)
\(828\) 0 0
\(829\) −6.17157 −0.214348 −0.107174 0.994240i \(-0.534180\pi\)
−0.107174 + 0.994240i \(0.534180\pi\)
\(830\) 13.6569 0.474036
\(831\) 0 0
\(832\) 8.14214 0.282278
\(833\) 0 0
\(834\) 0 0
\(835\) −10.3431 −0.357939
\(836\) −27.7279 −0.958990
\(837\) 0 0
\(838\) −35.7279 −1.23420
\(839\) −8.97056 −0.309698 −0.154849 0.987938i \(-0.549489\pi\)
−0.154849 + 0.987938i \(0.549489\pi\)
\(840\) 0 0
\(841\) −18.4853 −0.637423
\(842\) −16.8995 −0.582395
\(843\) 0 0
\(844\) 57.3137 1.97282
\(845\) −24.6274 −0.847209
\(846\) 0 0
\(847\) 0 0
\(848\) 21.5147 0.738818
\(849\) 0 0
\(850\) −10.6569 −0.365527
\(851\) −66.3970 −2.27606
\(852\) 0 0
\(853\) −12.4853 −0.427488 −0.213744 0.976890i \(-0.568566\pi\)
−0.213744 + 0.976890i \(0.568566\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −42.6274 −1.45698
\(857\) 44.3553 1.51515 0.757575 0.652748i \(-0.226384\pi\)
0.757575 + 0.652748i \(0.226384\pi\)
\(858\) 0 0
\(859\) −24.4853 −0.835427 −0.417714 0.908579i \(-0.637168\pi\)
−0.417714 + 0.908579i \(0.637168\pi\)
\(860\) −21.1127 −0.719937
\(861\) 0 0
\(862\) −16.8284 −0.573179
\(863\) −2.62742 −0.0894383 −0.0447192 0.999000i \(-0.514239\pi\)
−0.0447192 + 0.999000i \(0.514239\pi\)
\(864\) 0 0
\(865\) 29.6569 1.00836
\(866\) 2.07107 0.0703777
\(867\) 0 0
\(868\) 0 0
\(869\) −13.3137 −0.451637
\(870\) 0 0
\(871\) 2.62742 0.0890266
\(872\) −30.1421 −1.02074
\(873\) 0 0
\(874\) 122.397 4.14014
\(875\) 0 0
\(876\) 0 0
\(877\) 2.48528 0.0839220 0.0419610 0.999119i \(-0.486639\pi\)
0.0419610 + 0.999119i \(0.486639\pi\)
\(878\) 74.5980 2.51756
\(879\) 0 0
\(880\) −6.00000 −0.202260
\(881\) −16.6863 −0.562175 −0.281088 0.959682i \(-0.590695\pi\)
−0.281088 + 0.959682i \(0.590695\pi\)
\(882\) 0 0
\(883\) 39.6569 1.33456 0.667280 0.744807i \(-0.267459\pi\)
0.667280 + 0.744807i \(0.267459\pi\)
\(884\) −14.0000 −0.470871
\(885\) 0 0
\(886\) −81.1838 −2.72742
\(887\) 13.3137 0.447031 0.223515 0.974700i \(-0.428247\pi\)
0.223515 + 0.974700i \(0.428247\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 68.2843 2.28889
\(891\) 0 0
\(892\) −87.9411 −2.94449
\(893\) −71.1838 −2.38207
\(894\) 0 0
\(895\) −35.6569 −1.19188
\(896\) 0 0
\(897\) 0 0
\(898\) 9.65685 0.322253
\(899\) −18.3431 −0.611778
\(900\) 0 0
\(901\) 31.6569 1.05464
\(902\) −2.82843 −0.0941763
\(903\) 0 0
\(904\) 33.7990 1.12414
\(905\) −23.3137 −0.774974
\(906\) 0 0
\(907\) −17.5147 −0.581567 −0.290783 0.956789i \(-0.593916\pi\)
−0.290783 + 0.956789i \(0.593916\pi\)
\(908\) −57.3137 −1.90202
\(909\) 0 0
\(910\) 0 0
\(911\) 4.51472 0.149579 0.0747897 0.997199i \(-0.476171\pi\)
0.0747897 + 0.997199i \(0.476171\pi\)
\(912\) 0 0
\(913\) −2.82843 −0.0936073
\(914\) 45.4558 1.50355
\(915\) 0 0
\(916\) −59.9411 −1.98051
\(917\) 0 0
\(918\) 0 0
\(919\) 56.3553 1.85899 0.929496 0.368833i \(-0.120243\pi\)
0.929496 + 0.368833i \(0.120243\pi\)
\(920\) 61.7990 2.03745
\(921\) 0 0
\(922\) −16.3137 −0.537263
\(923\) −3.45584 −0.113750
\(924\) 0 0
\(925\) 9.48528 0.311874
\(926\) 44.9706 1.47782
\(927\) 0 0
\(928\) 5.14214 0.168799
\(929\) 22.8284 0.748976 0.374488 0.927232i \(-0.377818\pi\)
0.374488 + 0.927232i \(0.377818\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −39.8701 −1.30599
\(933\) 0 0
\(934\) 20.0711 0.656745
\(935\) −8.82843 −0.288720
\(936\) 0 0
\(937\) 5.85786 0.191368 0.0956840 0.995412i \(-0.469496\pi\)
0.0956840 + 0.995412i \(0.469496\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −75.2548 −2.45454
\(941\) 12.0711 0.393506 0.196753 0.980453i \(-0.436960\pi\)
0.196753 + 0.980453i \(0.436960\pi\)
\(942\) 0 0
\(943\) 8.20101 0.267062
\(944\) 25.9706 0.845270
\(945\) 0 0
\(946\) 6.65685 0.216433
\(947\) −37.4853 −1.21811 −0.609054 0.793129i \(-0.708451\pi\)
−0.609054 + 0.793129i \(0.708451\pi\)
\(948\) 0 0
\(949\) −0.284271 −0.00922784
\(950\) −17.4853 −0.567297
\(951\) 0 0
\(952\) 0 0
\(953\) 14.1421 0.458109 0.229054 0.973414i \(-0.426437\pi\)
0.229054 + 0.973414i \(0.426437\pi\)
\(954\) 0 0
\(955\) 43.3137 1.40160
\(956\) 24.8284 0.803009
\(957\) 0 0
\(958\) −33.7990 −1.09200
\(959\) 0 0
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 18.9706 0.611635
\(963\) 0 0
\(964\) 28.0000 0.901819
\(965\) −24.2843 −0.781738
\(966\) 0 0
\(967\) −21.0416 −0.676653 −0.338327 0.941029i \(-0.609861\pi\)
−0.338327 + 0.941029i \(0.609861\pi\)
\(968\) 4.41421 0.141878
\(969\) 0 0
\(970\) −55.4558 −1.78058
\(971\) −4.97056 −0.159513 −0.0797565 0.996814i \(-0.525414\pi\)
−0.0797565 + 0.996814i \(0.525414\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −78.7696 −2.52394
\(975\) 0 0
\(976\) −12.0000 −0.384111
\(977\) 12.8284 0.410418 0.205209 0.978718i \(-0.434213\pi\)
0.205209 + 0.978718i \(0.434213\pi\)
\(978\) 0 0
\(979\) −14.1421 −0.451985
\(980\) 0 0
\(981\) 0 0
\(982\) 3.65685 0.116695
\(983\) 2.02944 0.0647290 0.0323645 0.999476i \(-0.489696\pi\)
0.0323645 + 0.999476i \(0.489696\pi\)
\(984\) 0 0
\(985\) 32.8284 1.04600
\(986\) −34.5563 −1.10050
\(987\) 0 0
\(988\) −22.9706 −0.730791
\(989\) −19.3015 −0.613752
\(990\) 0 0
\(991\) −24.6863 −0.784186 −0.392093 0.919926i \(-0.628249\pi\)
−0.392093 + 0.919926i \(0.628249\pi\)
\(992\) −8.97056 −0.284816
\(993\) 0 0
\(994\) 0 0
\(995\) −39.5980 −1.25534
\(996\) 0 0
\(997\) 9.85786 0.312202 0.156101 0.987741i \(-0.450108\pi\)
0.156101 + 0.987741i \(0.450108\pi\)
\(998\) −76.7696 −2.43010
\(999\) 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 4851.2.a.be.1.2 2
3.2 odd 2 1617.2.a.n.1.1 2
7.2 even 3 693.2.i.f.298.1 4
7.4 even 3 693.2.i.f.100.1 4
7.6 odd 2 4851.2.a.bd.1.2 2
21.2 odd 6 231.2.i.d.67.2 4
21.11 odd 6 231.2.i.d.100.2 yes 4
21.20 even 2 1617.2.a.m.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.i.d.67.2 4 21.2 odd 6
231.2.i.d.100.2 yes 4 21.11 odd 6
693.2.i.f.100.1 4 7.4 even 3
693.2.i.f.298.1 4 7.2 even 3
1617.2.a.m.1.1 2 21.20 even 2
1617.2.a.n.1.1 2 3.2 odd 2
4851.2.a.bd.1.2 2 7.6 odd 2
4851.2.a.be.1.2 2 1.1 even 1 trivial