Properties

Label 5733.2.a.bu.1.2
Level $5733$
Weight $2$
Character 5733.1
Self dual yes
Analytic conductor $45.778$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5733 = 3^{2} \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5733.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(45.7782354788\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.4507648.1
Defining polynomial: \( x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 637)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.146243\) of defining polynomial
Character \(\chi\) \(=\) 5733.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.83237 q^{2} +1.35758 q^{4} +2.62555 q^{5} +1.17715 q^{8} +O(q^{10})\) \(q-1.83237 q^{2} +1.35758 q^{4} +2.62555 q^{5} +1.17715 q^{8} -4.81098 q^{10} -3.26469 q^{11} -1.00000 q^{13} -4.87214 q^{16} +4.53021 q^{17} -4.06615 q^{19} +3.56440 q^{20} +5.98212 q^{22} +4.53266 q^{23} +1.89352 q^{25} +1.83237 q^{26} +1.42268 q^{29} +2.80328 q^{31} +6.57326 q^{32} -8.30102 q^{34} -10.0503 q^{37} +7.45070 q^{38} +3.09067 q^{40} -2.84271 q^{41} +9.72632 q^{43} -4.43208 q^{44} -8.30551 q^{46} -9.44956 q^{47} -3.46963 q^{50} -1.35758 q^{52} -5.26439 q^{53} -8.57161 q^{55} -2.60687 q^{58} -2.56791 q^{59} +11.1830 q^{61} -5.13664 q^{62} -2.30037 q^{64} -2.62555 q^{65} -1.98172 q^{67} +6.15013 q^{68} +11.7544 q^{71} -12.1391 q^{73} +18.4159 q^{74} -5.52013 q^{76} +11.9089 q^{79} -12.7920 q^{80} +5.20889 q^{82} +13.2233 q^{83} +11.8943 q^{85} -17.8222 q^{86} -3.84303 q^{88} +10.6666 q^{89} +6.15345 q^{92} +17.3151 q^{94} -10.6759 q^{95} +13.7422 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{4} + 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 4 q^{4} + 6 q^{5} - 4 q^{10} - 4 q^{11} - 6 q^{13} + 16 q^{17} - 2 q^{19} + 16 q^{20} - 12 q^{22} + 6 q^{23} - 4 q^{25} + 6 q^{29} - 6 q^{31} + 20 q^{32} + 8 q^{38} - 4 q^{40} - 8 q^{41} + 2 q^{43} + 4 q^{44} + 8 q^{46} + 30 q^{47} - 8 q^{50} - 4 q^{52} + 14 q^{53} + 8 q^{55} - 8 q^{58} + 24 q^{59} + 28 q^{62} - 20 q^{64} - 6 q^{65} + 16 q^{67} + 28 q^{68} - 8 q^{71} + 6 q^{73} + 12 q^{74} + 16 q^{76} - 22 q^{79} - 28 q^{80} + 40 q^{82} + 50 q^{83} - 8 q^{85} + 16 q^{86} - 44 q^{88} + 26 q^{89} - 20 q^{92} + 32 q^{94} + 6 q^{95} + 14 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.83237 −1.29568 −0.647841 0.761776i \(-0.724328\pi\)
−0.647841 + 0.761776i \(0.724328\pi\)
\(3\) 0 0
\(4\) 1.35758 0.678791
\(5\) 2.62555 1.17418 0.587091 0.809521i \(-0.300273\pi\)
0.587091 + 0.809521i \(0.300273\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 1.17715 0.416185
\(9\) 0 0
\(10\) −4.81098 −1.52137
\(11\) −3.26469 −0.984341 −0.492170 0.870499i \(-0.663796\pi\)
−0.492170 + 0.870499i \(0.663796\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) −4.87214 −1.21803
\(17\) 4.53021 1.09874 0.549369 0.835580i \(-0.314868\pi\)
0.549369 + 0.835580i \(0.314868\pi\)
\(18\) 0 0
\(19\) −4.06615 −0.932839 −0.466420 0.884564i \(-0.654456\pi\)
−0.466420 + 0.884564i \(0.654456\pi\)
\(20\) 3.56440 0.797024
\(21\) 0 0
\(22\) 5.98212 1.27539
\(23\) 4.53266 0.945125 0.472562 0.881297i \(-0.343329\pi\)
0.472562 + 0.881297i \(0.343329\pi\)
\(24\) 0 0
\(25\) 1.89352 0.378705
\(26\) 1.83237 0.359357
\(27\) 0 0
\(28\) 0 0
\(29\) 1.42268 0.264184 0.132092 0.991237i \(-0.457831\pi\)
0.132092 + 0.991237i \(0.457831\pi\)
\(30\) 0 0
\(31\) 2.80328 0.503484 0.251742 0.967794i \(-0.418997\pi\)
0.251742 + 0.967794i \(0.418997\pi\)
\(32\) 6.57326 1.16200
\(33\) 0 0
\(34\) −8.30102 −1.42361
\(35\) 0 0
\(36\) 0 0
\(37\) −10.0503 −1.65227 −0.826133 0.563475i \(-0.809464\pi\)
−0.826133 + 0.563475i \(0.809464\pi\)
\(38\) 7.45070 1.20866
\(39\) 0 0
\(40\) 3.09067 0.488677
\(41\) −2.84271 −0.443956 −0.221978 0.975052i \(-0.571251\pi\)
−0.221978 + 0.975052i \(0.571251\pi\)
\(42\) 0 0
\(43\) 9.72632 1.48325 0.741625 0.670815i \(-0.234056\pi\)
0.741625 + 0.670815i \(0.234056\pi\)
\(44\) −4.43208 −0.668161
\(45\) 0 0
\(46\) −8.30551 −1.22458
\(47\) −9.44956 −1.37836 −0.689180 0.724590i \(-0.742029\pi\)
−0.689180 + 0.724590i \(0.742029\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −3.46963 −0.490680
\(51\) 0 0
\(52\) −1.35758 −0.188263
\(53\) −5.26439 −0.723119 −0.361559 0.932349i \(-0.617756\pi\)
−0.361559 + 0.932349i \(0.617756\pi\)
\(54\) 0 0
\(55\) −8.57161 −1.15580
\(56\) 0 0
\(57\) 0 0
\(58\) −2.60687 −0.342299
\(59\) −2.56791 −0.334313 −0.167156 0.985930i \(-0.553458\pi\)
−0.167156 + 0.985930i \(0.553458\pi\)
\(60\) 0 0
\(61\) 11.1830 1.43183 0.715917 0.698185i \(-0.246009\pi\)
0.715917 + 0.698185i \(0.246009\pi\)
\(62\) −5.13664 −0.652354
\(63\) 0 0
\(64\) −2.30037 −0.287546
\(65\) −2.62555 −0.325660
\(66\) 0 0
\(67\) −1.98172 −0.242106 −0.121053 0.992646i \(-0.538627\pi\)
−0.121053 + 0.992646i \(0.538627\pi\)
\(68\) 6.15013 0.745813
\(69\) 0 0
\(70\) 0 0
\(71\) 11.7544 1.39499 0.697495 0.716590i \(-0.254298\pi\)
0.697495 + 0.716590i \(0.254298\pi\)
\(72\) 0 0
\(73\) −12.1391 −1.42078 −0.710388 0.703810i \(-0.751481\pi\)
−0.710388 + 0.703810i \(0.751481\pi\)
\(74\) 18.4159 2.14081
\(75\) 0 0
\(76\) −5.52013 −0.633202
\(77\) 0 0
\(78\) 0 0
\(79\) 11.9089 1.33986 0.669928 0.742426i \(-0.266325\pi\)
0.669928 + 0.742426i \(0.266325\pi\)
\(80\) −12.7920 −1.43019
\(81\) 0 0
\(82\) 5.20889 0.575226
\(83\) 13.2233 1.45145 0.725723 0.687987i \(-0.241505\pi\)
0.725723 + 0.687987i \(0.241505\pi\)
\(84\) 0 0
\(85\) 11.8943 1.29012
\(86\) −17.8222 −1.92182
\(87\) 0 0
\(88\) −3.84303 −0.409668
\(89\) 10.6666 1.13065 0.565326 0.824867i \(-0.308750\pi\)
0.565326 + 0.824867i \(0.308750\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 6.15345 0.641542
\(93\) 0 0
\(94\) 17.3151 1.78592
\(95\) −10.6759 −1.09532
\(96\) 0 0
\(97\) 13.7422 1.39531 0.697655 0.716433i \(-0.254227\pi\)
0.697655 + 0.716433i \(0.254227\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 2.57061 0.257061
\(101\) 5.89458 0.586533 0.293266 0.956031i \(-0.405258\pi\)
0.293266 + 0.956031i \(0.405258\pi\)
\(102\) 0 0
\(103\) −2.78737 −0.274647 −0.137324 0.990526i \(-0.543850\pi\)
−0.137324 + 0.990526i \(0.543850\pi\)
\(104\) −1.17715 −0.115429
\(105\) 0 0
\(106\) 9.64630 0.936932
\(107\) 17.6647 1.70771 0.853856 0.520509i \(-0.174258\pi\)
0.853856 + 0.520509i \(0.174258\pi\)
\(108\) 0 0
\(109\) −9.56450 −0.916113 −0.458057 0.888923i \(-0.651454\pi\)
−0.458057 + 0.888923i \(0.651454\pi\)
\(110\) 15.7064 1.49754
\(111\) 0 0
\(112\) 0 0
\(113\) 17.6017 1.65583 0.827916 0.560852i \(-0.189526\pi\)
0.827916 + 0.560852i \(0.189526\pi\)
\(114\) 0 0
\(115\) 11.9007 1.10975
\(116\) 1.93140 0.179326
\(117\) 0 0
\(118\) 4.70535 0.433163
\(119\) 0 0
\(120\) 0 0
\(121\) −0.341808 −0.0310735
\(122\) −20.4914 −1.85520
\(123\) 0 0
\(124\) 3.80568 0.341760
\(125\) −8.15622 −0.729514
\(126\) 0 0
\(127\) −1.59482 −0.141517 −0.0707586 0.997493i \(-0.522542\pi\)
−0.0707586 + 0.997493i \(0.522542\pi\)
\(128\) −8.93138 −0.789430
\(129\) 0 0
\(130\) 4.81098 0.421951
\(131\) 4.30279 0.375937 0.187968 0.982175i \(-0.439810\pi\)
0.187968 + 0.982175i \(0.439810\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 3.63125 0.313692
\(135\) 0 0
\(136\) 5.33273 0.457278
\(137\) −9.82234 −0.839179 −0.419590 0.907714i \(-0.637826\pi\)
−0.419590 + 0.907714i \(0.637826\pi\)
\(138\) 0 0
\(139\) −10.0811 −0.855070 −0.427535 0.903999i \(-0.640618\pi\)
−0.427535 + 0.903999i \(0.640618\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −21.5384 −1.80746
\(143\) 3.26469 0.273007
\(144\) 0 0
\(145\) 3.73531 0.310201
\(146\) 22.2434 1.84087
\(147\) 0 0
\(148\) −13.6442 −1.12154
\(149\) 13.8124 1.13155 0.565777 0.824558i \(-0.308576\pi\)
0.565777 + 0.824558i \(0.308576\pi\)
\(150\) 0 0
\(151\) 21.2123 1.72623 0.863117 0.505004i \(-0.168509\pi\)
0.863117 + 0.505004i \(0.168509\pi\)
\(152\) −4.78647 −0.388234
\(153\) 0 0
\(154\) 0 0
\(155\) 7.36015 0.591182
\(156\) 0 0
\(157\) −23.5155 −1.87674 −0.938372 0.345627i \(-0.887666\pi\)
−0.938372 + 0.345627i \(0.887666\pi\)
\(158\) −21.8215 −1.73603
\(159\) 0 0
\(160\) 17.2584 1.36440
\(161\) 0 0
\(162\) 0 0
\(163\) 7.83062 0.613341 0.306671 0.951816i \(-0.400785\pi\)
0.306671 + 0.951816i \(0.400785\pi\)
\(164\) −3.85920 −0.301353
\(165\) 0 0
\(166\) −24.2300 −1.88061
\(167\) 12.7116 0.983654 0.491827 0.870693i \(-0.336329\pi\)
0.491827 + 0.870693i \(0.336329\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −21.7948 −1.67158
\(171\) 0 0
\(172\) 13.2043 1.00682
\(173\) 5.24907 0.399079 0.199540 0.979890i \(-0.436055\pi\)
0.199540 + 0.979890i \(0.436055\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 15.9060 1.19896
\(177\) 0 0
\(178\) −19.5451 −1.46497
\(179\) 10.5255 0.786714 0.393357 0.919386i \(-0.371314\pi\)
0.393357 + 0.919386i \(0.371314\pi\)
\(180\) 0 0
\(181\) 18.8177 1.39871 0.699356 0.714774i \(-0.253470\pi\)
0.699356 + 0.714774i \(0.253470\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 5.33562 0.393347
\(185\) −26.3877 −1.94006
\(186\) 0 0
\(187\) −14.7897 −1.08153
\(188\) −12.8285 −0.935618
\(189\) 0 0
\(190\) 19.5622 1.41919
\(191\) −14.9144 −1.07917 −0.539585 0.841931i \(-0.681419\pi\)
−0.539585 + 0.841931i \(0.681419\pi\)
\(192\) 0 0
\(193\) −0.0531356 −0.00382479 −0.00191239 0.999998i \(-0.500609\pi\)
−0.00191239 + 0.999998i \(0.500609\pi\)
\(194\) −25.1808 −1.80788
\(195\) 0 0
\(196\) 0 0
\(197\) −10.0478 −0.715875 −0.357938 0.933745i \(-0.616520\pi\)
−0.357938 + 0.933745i \(0.616520\pi\)
\(198\) 0 0
\(199\) −23.2914 −1.65109 −0.825543 0.564340i \(-0.809131\pi\)
−0.825543 + 0.564340i \(0.809131\pi\)
\(200\) 2.22896 0.157611
\(201\) 0 0
\(202\) −10.8011 −0.759959
\(203\) 0 0
\(204\) 0 0
\(205\) −7.46367 −0.521285
\(206\) 5.10749 0.355855
\(207\) 0 0
\(208\) 4.87214 0.337822
\(209\) 13.2747 0.918232
\(210\) 0 0
\(211\) −2.47457 −0.170356 −0.0851780 0.996366i \(-0.527146\pi\)
−0.0851780 + 0.996366i \(0.527146\pi\)
\(212\) −7.14683 −0.490846
\(213\) 0 0
\(214\) −32.3683 −2.21265
\(215\) 25.5369 1.74161
\(216\) 0 0
\(217\) 0 0
\(218\) 17.5257 1.18699
\(219\) 0 0
\(220\) −11.6367 −0.784543
\(221\) −4.53021 −0.304735
\(222\) 0 0
\(223\) −6.17027 −0.413192 −0.206596 0.978426i \(-0.566239\pi\)
−0.206596 + 0.978426i \(0.566239\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −32.2529 −2.14543
\(227\) 11.5939 0.769513 0.384757 0.923018i \(-0.374285\pi\)
0.384757 + 0.923018i \(0.374285\pi\)
\(228\) 0 0
\(229\) 10.2382 0.676558 0.338279 0.941046i \(-0.390155\pi\)
0.338279 + 0.941046i \(0.390155\pi\)
\(230\) −21.8065 −1.43788
\(231\) 0 0
\(232\) 1.67470 0.109950
\(233\) −21.4822 −1.40735 −0.703673 0.710524i \(-0.748458\pi\)
−0.703673 + 0.710524i \(0.748458\pi\)
\(234\) 0 0
\(235\) −24.8103 −1.61845
\(236\) −3.48614 −0.226928
\(237\) 0 0
\(238\) 0 0
\(239\) −1.08591 −0.0702414 −0.0351207 0.999383i \(-0.511182\pi\)
−0.0351207 + 0.999383i \(0.511182\pi\)
\(240\) 0 0
\(241\) 19.9830 1.28722 0.643608 0.765355i \(-0.277437\pi\)
0.643608 + 0.765355i \(0.277437\pi\)
\(242\) 0.626319 0.0402613
\(243\) 0 0
\(244\) 15.1818 0.971915
\(245\) 0 0
\(246\) 0 0
\(247\) 4.06615 0.258723
\(248\) 3.29988 0.209542
\(249\) 0 0
\(250\) 14.9452 0.945218
\(251\) 20.1497 1.27184 0.635920 0.771755i \(-0.280621\pi\)
0.635920 + 0.771755i \(0.280621\pi\)
\(252\) 0 0
\(253\) −14.7977 −0.930325
\(254\) 2.92230 0.183361
\(255\) 0 0
\(256\) 20.9663 1.31040
\(257\) −15.0174 −0.936759 −0.468379 0.883527i \(-0.655162\pi\)
−0.468379 + 0.883527i \(0.655162\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −3.56440 −0.221055
\(261\) 0 0
\(262\) −7.88431 −0.487094
\(263\) 21.1662 1.30516 0.652582 0.757718i \(-0.273686\pi\)
0.652582 + 0.757718i \(0.273686\pi\)
\(264\) 0 0
\(265\) −13.8219 −0.849074
\(266\) 0 0
\(267\) 0 0
\(268\) −2.69035 −0.164339
\(269\) −4.10480 −0.250274 −0.125137 0.992139i \(-0.539937\pi\)
−0.125137 + 0.992139i \(0.539937\pi\)
\(270\) 0 0
\(271\) −3.43244 −0.208506 −0.104253 0.994551i \(-0.533245\pi\)
−0.104253 + 0.994551i \(0.533245\pi\)
\(272\) −22.0718 −1.33830
\(273\) 0 0
\(274\) 17.9982 1.08731
\(275\) −6.18176 −0.372774
\(276\) 0 0
\(277\) −0.361012 −0.0216911 −0.0108455 0.999941i \(-0.503452\pi\)
−0.0108455 + 0.999941i \(0.503452\pi\)
\(278\) 18.4724 1.10790
\(279\) 0 0
\(280\) 0 0
\(281\) −18.5213 −1.10489 −0.552445 0.833550i \(-0.686305\pi\)
−0.552445 + 0.833550i \(0.686305\pi\)
\(282\) 0 0
\(283\) 1.61158 0.0957983 0.0478991 0.998852i \(-0.484747\pi\)
0.0478991 + 0.998852i \(0.484747\pi\)
\(284\) 15.9575 0.946906
\(285\) 0 0
\(286\) −5.98212 −0.353730
\(287\) 0 0
\(288\) 0 0
\(289\) 3.52281 0.207224
\(290\) −6.84447 −0.401921
\(291\) 0 0
\(292\) −16.4798 −0.964410
\(293\) −4.41671 −0.258027 −0.129013 0.991643i \(-0.541181\pi\)
−0.129013 + 0.991643i \(0.541181\pi\)
\(294\) 0 0
\(295\) −6.74217 −0.392544
\(296\) −11.8308 −0.687649
\(297\) 0 0
\(298\) −25.3094 −1.46613
\(299\) −4.53266 −0.262130
\(300\) 0 0
\(301\) 0 0
\(302\) −38.8688 −2.23665
\(303\) 0 0
\(304\) 19.8108 1.13623
\(305\) 29.3615 1.68123
\(306\) 0 0
\(307\) 5.78353 0.330083 0.165042 0.986287i \(-0.447224\pi\)
0.165042 + 0.986287i \(0.447224\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −13.4865 −0.765983
\(311\) 14.2895 0.810284 0.405142 0.914254i \(-0.367222\pi\)
0.405142 + 0.914254i \(0.367222\pi\)
\(312\) 0 0
\(313\) −2.73725 −0.154718 −0.0773592 0.997003i \(-0.524649\pi\)
−0.0773592 + 0.997003i \(0.524649\pi\)
\(314\) 43.0892 2.43166
\(315\) 0 0
\(316\) 16.1673 0.909481
\(317\) −10.6512 −0.598231 −0.299116 0.954217i \(-0.596692\pi\)
−0.299116 + 0.954217i \(0.596692\pi\)
\(318\) 0 0
\(319\) −4.64460 −0.260047
\(320\) −6.03975 −0.337632
\(321\) 0 0
\(322\) 0 0
\(323\) −18.4205 −1.02495
\(324\) 0 0
\(325\) −1.89352 −0.105034
\(326\) −14.3486 −0.794695
\(327\) 0 0
\(328\) −3.34629 −0.184768
\(329\) 0 0
\(330\) 0 0
\(331\) −3.41626 −0.187774 −0.0938872 0.995583i \(-0.529929\pi\)
−0.0938872 + 0.995583i \(0.529929\pi\)
\(332\) 17.9517 0.985228
\(333\) 0 0
\(334\) −23.2924 −1.27450
\(335\) −5.20312 −0.284277
\(336\) 0 0
\(337\) 24.9606 1.35969 0.679844 0.733357i \(-0.262047\pi\)
0.679844 + 0.733357i \(0.262047\pi\)
\(338\) −1.83237 −0.0996678
\(339\) 0 0
\(340\) 16.1475 0.875720
\(341\) −9.15183 −0.495599
\(342\) 0 0
\(343\) 0 0
\(344\) 11.4493 0.617306
\(345\) 0 0
\(346\) −9.61824 −0.517080
\(347\) 19.1833 1.02981 0.514907 0.857246i \(-0.327827\pi\)
0.514907 + 0.857246i \(0.327827\pi\)
\(348\) 0 0
\(349\) 3.94421 0.211129 0.105564 0.994412i \(-0.466335\pi\)
0.105564 + 0.994412i \(0.466335\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −21.4596 −1.14380
\(353\) 28.0232 1.49152 0.745762 0.666212i \(-0.232086\pi\)
0.745762 + 0.666212i \(0.232086\pi\)
\(354\) 0 0
\(355\) 30.8618 1.63797
\(356\) 14.4807 0.767476
\(357\) 0 0
\(358\) −19.2866 −1.01933
\(359\) −33.2232 −1.75345 −0.876726 0.480989i \(-0.840278\pi\)
−0.876726 + 0.480989i \(0.840278\pi\)
\(360\) 0 0
\(361\) −2.46641 −0.129811
\(362\) −34.4811 −1.81228
\(363\) 0 0
\(364\) 0 0
\(365\) −31.8719 −1.66825
\(366\) 0 0
\(367\) 28.3091 1.47772 0.738862 0.673857i \(-0.235363\pi\)
0.738862 + 0.673857i \(0.235363\pi\)
\(368\) −22.0837 −1.15119
\(369\) 0 0
\(370\) 48.3520 2.51370
\(371\) 0 0
\(372\) 0 0
\(373\) −12.9515 −0.670602 −0.335301 0.942111i \(-0.608838\pi\)
−0.335301 + 0.942111i \(0.608838\pi\)
\(374\) 27.1003 1.40132
\(375\) 0 0
\(376\) −11.1235 −0.573653
\(377\) −1.42268 −0.0732716
\(378\) 0 0
\(379\) 0.168981 0.00867995 0.00433997 0.999991i \(-0.498619\pi\)
0.00433997 + 0.999991i \(0.498619\pi\)
\(380\) −14.4934 −0.743495
\(381\) 0 0
\(382\) 27.3288 1.39826
\(383\) 10.1933 0.520854 0.260427 0.965494i \(-0.416137\pi\)
0.260427 + 0.965494i \(0.416137\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0.0973641 0.00495570
\(387\) 0 0
\(388\) 18.6562 0.947124
\(389\) 28.6665 1.45345 0.726723 0.686930i \(-0.241042\pi\)
0.726723 + 0.686930i \(0.241042\pi\)
\(390\) 0 0
\(391\) 20.5339 1.03844
\(392\) 0 0
\(393\) 0 0
\(394\) 18.4113 0.927547
\(395\) 31.2674 1.57323
\(396\) 0 0
\(397\) 28.6411 1.43746 0.718728 0.695292i \(-0.244725\pi\)
0.718728 + 0.695292i \(0.244725\pi\)
\(398\) 42.6785 2.13928
\(399\) 0 0
\(400\) −9.22550 −0.461275
\(401\) 14.2963 0.713922 0.356961 0.934119i \(-0.383813\pi\)
0.356961 + 0.934119i \(0.383813\pi\)
\(402\) 0 0
\(403\) −2.80328 −0.139641
\(404\) 8.00237 0.398133
\(405\) 0 0
\(406\) 0 0
\(407\) 32.8112 1.62639
\(408\) 0 0
\(409\) −25.6988 −1.27072 −0.635362 0.772215i \(-0.719149\pi\)
−0.635362 + 0.772215i \(0.719149\pi\)
\(410\) 13.6762 0.675420
\(411\) 0 0
\(412\) −3.78408 −0.186428
\(413\) 0 0
\(414\) 0 0
\(415\) 34.7185 1.70426
\(416\) −6.57326 −0.322281
\(417\) 0 0
\(418\) −24.3242 −1.18974
\(419\) 18.7999 0.918433 0.459216 0.888324i \(-0.348130\pi\)
0.459216 + 0.888324i \(0.348130\pi\)
\(420\) 0 0
\(421\) −18.0283 −0.878645 −0.439322 0.898329i \(-0.644781\pi\)
−0.439322 + 0.898329i \(0.644781\pi\)
\(422\) 4.53432 0.220727
\(423\) 0 0
\(424\) −6.19697 −0.300951
\(425\) 8.57806 0.416097
\(426\) 0 0
\(427\) 0 0
\(428\) 23.9813 1.15918
\(429\) 0 0
\(430\) −46.7931 −2.25657
\(431\) −20.5583 −0.990256 −0.495128 0.868820i \(-0.664879\pi\)
−0.495128 + 0.868820i \(0.664879\pi\)
\(432\) 0 0
\(433\) −18.9235 −0.909404 −0.454702 0.890644i \(-0.650254\pi\)
−0.454702 + 0.890644i \(0.650254\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −12.9846 −0.621849
\(437\) −18.4305 −0.881650
\(438\) 0 0
\(439\) 14.6550 0.699445 0.349723 0.936853i \(-0.386276\pi\)
0.349723 + 0.936853i \(0.386276\pi\)
\(440\) −10.0901 −0.481025
\(441\) 0 0
\(442\) 8.30102 0.394839
\(443\) 11.9592 0.568201 0.284100 0.958795i \(-0.408305\pi\)
0.284100 + 0.958795i \(0.408305\pi\)
\(444\) 0 0
\(445\) 28.0056 1.32759
\(446\) 11.3062 0.535365
\(447\) 0 0
\(448\) 0 0
\(449\) 2.32245 0.109603 0.0548015 0.998497i \(-0.482547\pi\)
0.0548015 + 0.998497i \(0.482547\pi\)
\(450\) 0 0
\(451\) 9.28055 0.437004
\(452\) 23.8958 1.12396
\(453\) 0 0
\(454\) −21.2443 −0.997044
\(455\) 0 0
\(456\) 0 0
\(457\) −18.5805 −0.869157 −0.434579 0.900634i \(-0.643103\pi\)
−0.434579 + 0.900634i \(0.643103\pi\)
\(458\) −18.7601 −0.876604
\(459\) 0 0
\(460\) 16.1562 0.753287
\(461\) 27.8926 1.29909 0.649543 0.760325i \(-0.274960\pi\)
0.649543 + 0.760325i \(0.274960\pi\)
\(462\) 0 0
\(463\) −3.66462 −0.170309 −0.0851547 0.996368i \(-0.527138\pi\)
−0.0851547 + 0.996368i \(0.527138\pi\)
\(464\) −6.93147 −0.321786
\(465\) 0 0
\(466\) 39.3634 1.82347
\(467\) 38.7532 1.79328 0.896641 0.442757i \(-0.146000\pi\)
0.896641 + 0.442757i \(0.146000\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 45.4617 2.09699
\(471\) 0 0
\(472\) −3.02281 −0.139136
\(473\) −31.7534 −1.46002
\(474\) 0 0
\(475\) −7.69935 −0.353270
\(476\) 0 0
\(477\) 0 0
\(478\) 1.98978 0.0910105
\(479\) −6.03430 −0.275714 −0.137857 0.990452i \(-0.544021\pi\)
−0.137857 + 0.990452i \(0.544021\pi\)
\(480\) 0 0
\(481\) 10.0503 0.458256
\(482\) −36.6162 −1.66782
\(483\) 0 0
\(484\) −0.464032 −0.0210924
\(485\) 36.0809 1.63835
\(486\) 0 0
\(487\) −3.80249 −0.172307 −0.0861537 0.996282i \(-0.527458\pi\)
−0.0861537 + 0.996282i \(0.527458\pi\)
\(488\) 13.1640 0.595908
\(489\) 0 0
\(490\) 0 0
\(491\) 0.381464 0.0172152 0.00860761 0.999963i \(-0.497260\pi\)
0.00860761 + 0.999963i \(0.497260\pi\)
\(492\) 0 0
\(493\) 6.44502 0.290269
\(494\) −7.45070 −0.335223
\(495\) 0 0
\(496\) −13.6580 −0.613260
\(497\) 0 0
\(498\) 0 0
\(499\) 34.5739 1.54774 0.773871 0.633343i \(-0.218318\pi\)
0.773871 + 0.633343i \(0.218318\pi\)
\(500\) −11.0727 −0.495187
\(501\) 0 0
\(502\) −36.9218 −1.64790
\(503\) −2.41090 −0.107497 −0.0537485 0.998555i \(-0.517117\pi\)
−0.0537485 + 0.998555i \(0.517117\pi\)
\(504\) 0 0
\(505\) 15.4765 0.688696
\(506\) 27.1149 1.20540
\(507\) 0 0
\(508\) −2.16509 −0.0960605
\(509\) −21.9153 −0.971380 −0.485690 0.874131i \(-0.661432\pi\)
−0.485690 + 0.874131i \(0.661432\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −20.5553 −0.908426
\(513\) 0 0
\(514\) 27.5174 1.21374
\(515\) −7.31837 −0.322486
\(516\) 0 0
\(517\) 30.8499 1.35678
\(518\) 0 0
\(519\) 0 0
\(520\) −3.09067 −0.135535
\(521\) −30.1450 −1.32068 −0.660338 0.750968i \(-0.729587\pi\)
−0.660338 + 0.750968i \(0.729587\pi\)
\(522\) 0 0
\(523\) 18.0993 0.791428 0.395714 0.918374i \(-0.370497\pi\)
0.395714 + 0.918374i \(0.370497\pi\)
\(524\) 5.84139 0.255182
\(525\) 0 0
\(526\) −38.7843 −1.69108
\(527\) 12.6994 0.553196
\(528\) 0 0
\(529\) −2.45500 −0.106739
\(530\) 25.3269 1.10013
\(531\) 0 0
\(532\) 0 0
\(533\) 2.84271 0.123131
\(534\) 0 0
\(535\) 46.3796 2.00517
\(536\) −2.33278 −0.100761
\(537\) 0 0
\(538\) 7.52151 0.324275
\(539\) 0 0
\(540\) 0 0
\(541\) 4.08890 0.175795 0.0878977 0.996130i \(-0.471985\pi\)
0.0878977 + 0.996130i \(0.471985\pi\)
\(542\) 6.28950 0.270157
\(543\) 0 0
\(544\) 29.7782 1.27673
\(545\) −25.1121 −1.07568
\(546\) 0 0
\(547\) −40.4264 −1.72851 −0.864255 0.503055i \(-0.832209\pi\)
−0.864255 + 0.503055i \(0.832209\pi\)
\(548\) −13.3346 −0.569627
\(549\) 0 0
\(550\) 11.3273 0.482997
\(551\) −5.78482 −0.246442
\(552\) 0 0
\(553\) 0 0
\(554\) 0.661507 0.0281047
\(555\) 0 0
\(556\) −13.6859 −0.580413
\(557\) −19.7690 −0.837640 −0.418820 0.908069i \(-0.637556\pi\)
−0.418820 + 0.908069i \(0.637556\pi\)
\(558\) 0 0
\(559\) −9.72632 −0.411379
\(560\) 0 0
\(561\) 0 0
\(562\) 33.9379 1.43158
\(563\) 8.22392 0.346597 0.173298 0.984869i \(-0.444557\pi\)
0.173298 + 0.984869i \(0.444557\pi\)
\(564\) 0 0
\(565\) 46.2143 1.94425
\(566\) −2.95300 −0.124124
\(567\) 0 0
\(568\) 13.8367 0.580574
\(569\) −14.5770 −0.611099 −0.305550 0.952176i \(-0.598840\pi\)
−0.305550 + 0.952176i \(0.598840\pi\)
\(570\) 0 0
\(571\) −2.58822 −0.108314 −0.0541568 0.998532i \(-0.517247\pi\)
−0.0541568 + 0.998532i \(0.517247\pi\)
\(572\) 4.43208 0.185315
\(573\) 0 0
\(574\) 0 0
\(575\) 8.58269 0.357923
\(576\) 0 0
\(577\) 23.8937 0.994707 0.497353 0.867548i \(-0.334305\pi\)
0.497353 + 0.867548i \(0.334305\pi\)
\(578\) −6.45509 −0.268496
\(579\) 0 0
\(580\) 5.07099 0.210561
\(581\) 0 0
\(582\) 0 0
\(583\) 17.1866 0.711795
\(584\) −14.2896 −0.591306
\(585\) 0 0
\(586\) 8.09304 0.334320
\(587\) 28.5759 1.17945 0.589726 0.807604i \(-0.299236\pi\)
0.589726 + 0.807604i \(0.299236\pi\)
\(588\) 0 0
\(589\) −11.3986 −0.469669
\(590\) 12.3541 0.508612
\(591\) 0 0
\(592\) 48.9666 2.01252
\(593\) −19.9575 −0.819556 −0.409778 0.912185i \(-0.634394\pi\)
−0.409778 + 0.912185i \(0.634394\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 18.7514 0.768088
\(597\) 0 0
\(598\) 8.30551 0.339638
\(599\) −1.17716 −0.0480973 −0.0240486 0.999711i \(-0.507656\pi\)
−0.0240486 + 0.999711i \(0.507656\pi\)
\(600\) 0 0
\(601\) 30.9250 1.26146 0.630729 0.776003i \(-0.282756\pi\)
0.630729 + 0.776003i \(0.282756\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 28.7974 1.17175
\(605\) −0.897435 −0.0364859
\(606\) 0 0
\(607\) −2.40708 −0.0977005 −0.0488503 0.998806i \(-0.515556\pi\)
−0.0488503 + 0.998806i \(0.515556\pi\)
\(608\) −26.7279 −1.08396
\(609\) 0 0
\(610\) −53.8011 −2.17834
\(611\) 9.44956 0.382288
\(612\) 0 0
\(613\) −27.7742 −1.12179 −0.560895 0.827887i \(-0.689543\pi\)
−0.560895 + 0.827887i \(0.689543\pi\)
\(614\) −10.5976 −0.427683
\(615\) 0 0
\(616\) 0 0
\(617\) −26.2125 −1.05527 −0.527637 0.849470i \(-0.676922\pi\)
−0.527637 + 0.849470i \(0.676922\pi\)
\(618\) 0 0
\(619\) 17.2357 0.692763 0.346381 0.938094i \(-0.387410\pi\)
0.346381 + 0.938094i \(0.387410\pi\)
\(620\) 9.99200 0.401288
\(621\) 0 0
\(622\) −26.1837 −1.04987
\(623\) 0 0
\(624\) 0 0
\(625\) −30.8822 −1.23529
\(626\) 5.01565 0.200466
\(627\) 0 0
\(628\) −31.9242 −1.27392
\(629\) −45.5302 −1.81541
\(630\) 0 0
\(631\) 39.2125 1.56103 0.780513 0.625140i \(-0.214958\pi\)
0.780513 + 0.625140i \(0.214958\pi\)
\(632\) 14.0185 0.557628
\(633\) 0 0
\(634\) 19.5170 0.775117
\(635\) −4.18728 −0.166167
\(636\) 0 0
\(637\) 0 0
\(638\) 8.51062 0.336939
\(639\) 0 0
\(640\) −23.4498 −0.926935
\(641\) 22.8735 0.903451 0.451725 0.892157i \(-0.350809\pi\)
0.451725 + 0.892157i \(0.350809\pi\)
\(642\) 0 0
\(643\) 46.7072 1.84195 0.920976 0.389620i \(-0.127394\pi\)
0.920976 + 0.389620i \(0.127394\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 33.7532 1.32800
\(647\) 5.93075 0.233162 0.116581 0.993181i \(-0.462807\pi\)
0.116581 + 0.993181i \(0.462807\pi\)
\(648\) 0 0
\(649\) 8.38341 0.329078
\(650\) 3.46963 0.136090
\(651\) 0 0
\(652\) 10.6307 0.416330
\(653\) 18.8046 0.735881 0.367940 0.929849i \(-0.380063\pi\)
0.367940 + 0.929849i \(0.380063\pi\)
\(654\) 0 0
\(655\) 11.2972 0.441419
\(656\) 13.8500 0.540754
\(657\) 0 0
\(658\) 0 0
\(659\) −23.1086 −0.900184 −0.450092 0.892982i \(-0.648609\pi\)
−0.450092 + 0.892982i \(0.648609\pi\)
\(660\) 0 0
\(661\) 33.9087 1.31889 0.659447 0.751751i \(-0.270790\pi\)
0.659447 + 0.751751i \(0.270790\pi\)
\(662\) 6.25985 0.243296
\(663\) 0 0
\(664\) 15.5658 0.604071
\(665\) 0 0
\(666\) 0 0
\(667\) 6.44851 0.249687
\(668\) 17.2570 0.667695
\(669\) 0 0
\(670\) 9.53404 0.368332
\(671\) −36.5090 −1.40941
\(672\) 0 0
\(673\) 6.00430 0.231449 0.115724 0.993281i \(-0.463081\pi\)
0.115724 + 0.993281i \(0.463081\pi\)
\(674\) −45.7370 −1.76172
\(675\) 0 0
\(676\) 1.35758 0.0522147
\(677\) 20.2075 0.776637 0.388318 0.921525i \(-0.373056\pi\)
0.388318 + 0.921525i \(0.373056\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 14.0014 0.536928
\(681\) 0 0
\(682\) 16.7695 0.642139
\(683\) 31.6508 1.21108 0.605541 0.795814i \(-0.292957\pi\)
0.605541 + 0.795814i \(0.292957\pi\)
\(684\) 0 0
\(685\) −25.7891 −0.985350
\(686\) 0 0
\(687\) 0 0
\(688\) −47.3879 −1.80665
\(689\) 5.26439 0.200557
\(690\) 0 0
\(691\) −9.69199 −0.368701 −0.184350 0.982861i \(-0.559018\pi\)
−0.184350 + 0.982861i \(0.559018\pi\)
\(692\) 7.12604 0.270891
\(693\) 0 0
\(694\) −35.1509 −1.33431
\(695\) −26.4685 −1.00401
\(696\) 0 0
\(697\) −12.8781 −0.487791
\(698\) −7.22725 −0.273556
\(699\) 0 0
\(700\) 0 0
\(701\) 5.10365 0.192762 0.0963811 0.995345i \(-0.469273\pi\)
0.0963811 + 0.995345i \(0.469273\pi\)
\(702\) 0 0
\(703\) 40.8662 1.54130
\(704\) 7.51000 0.283044
\(705\) 0 0
\(706\) −51.3489 −1.93254
\(707\) 0 0
\(708\) 0 0
\(709\) 26.1445 0.981878 0.490939 0.871194i \(-0.336654\pi\)
0.490939 + 0.871194i \(0.336654\pi\)
\(710\) −56.5502 −2.12229
\(711\) 0 0
\(712\) 12.5561 0.470561
\(713\) 12.7063 0.475855
\(714\) 0 0
\(715\) 8.57161 0.320560
\(716\) 14.2892 0.534014
\(717\) 0 0
\(718\) 60.8772 2.27192
\(719\) −8.72884 −0.325531 −0.162765 0.986665i \(-0.552041\pi\)
−0.162765 + 0.986665i \(0.552041\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 4.51937 0.168193
\(723\) 0 0
\(724\) 25.5466 0.949432
\(725\) 2.69387 0.100048
\(726\) 0 0
\(727\) 21.8712 0.811158 0.405579 0.914060i \(-0.367070\pi\)
0.405579 + 0.914060i \(0.367070\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 58.4011 2.16152
\(731\) 44.0623 1.62970
\(732\) 0 0
\(733\) 35.5640 1.31359 0.656793 0.754071i \(-0.271912\pi\)
0.656793 + 0.754071i \(0.271912\pi\)
\(734\) −51.8728 −1.91466
\(735\) 0 0
\(736\) 29.7943 1.09823
\(737\) 6.46971 0.238315
\(738\) 0 0
\(739\) −36.4894 −1.34228 −0.671142 0.741329i \(-0.734196\pi\)
−0.671142 + 0.741329i \(0.734196\pi\)
\(740\) −35.8234 −1.31690
\(741\) 0 0
\(742\) 0 0
\(743\) −12.4588 −0.457071 −0.228535 0.973536i \(-0.573394\pi\)
−0.228535 + 0.973536i \(0.573394\pi\)
\(744\) 0 0
\(745\) 36.2651 1.32865
\(746\) 23.7319 0.868886
\(747\) 0 0
\(748\) −20.0783 −0.734134
\(749\) 0 0
\(750\) 0 0
\(751\) −18.9424 −0.691217 −0.345608 0.938379i \(-0.612327\pi\)
−0.345608 + 0.938379i \(0.612327\pi\)
\(752\) 46.0395 1.67889
\(753\) 0 0
\(754\) 2.60687 0.0949366
\(755\) 55.6940 2.02691
\(756\) 0 0
\(757\) −25.0956 −0.912114 −0.456057 0.889951i \(-0.650739\pi\)
−0.456057 + 0.889951i \(0.650739\pi\)
\(758\) −0.309635 −0.0112464
\(759\) 0 0
\(760\) −12.5671 −0.455857
\(761\) 28.0617 1.01723 0.508617 0.860993i \(-0.330157\pi\)
0.508617 + 0.860993i \(0.330157\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −20.2476 −0.732531
\(765\) 0 0
\(766\) −18.6779 −0.674861
\(767\) 2.56791 0.0927217
\(768\) 0 0
\(769\) 23.2636 0.838907 0.419454 0.907777i \(-0.362222\pi\)
0.419454 + 0.907777i \(0.362222\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −0.0721359 −0.00259623
\(773\) 45.8118 1.64774 0.823869 0.566781i \(-0.191811\pi\)
0.823869 + 0.566781i \(0.191811\pi\)
\(774\) 0 0
\(775\) 5.30807 0.190672
\(776\) 16.1766 0.580708
\(777\) 0 0
\(778\) −52.5276 −1.88320
\(779\) 11.5589 0.414140
\(780\) 0 0
\(781\) −38.3744 −1.37315
\(782\) −37.6257 −1.34549
\(783\) 0 0
\(784\) 0 0
\(785\) −61.7413 −2.20364
\(786\) 0 0
\(787\) −31.2715 −1.11471 −0.557355 0.830274i \(-0.688184\pi\)
−0.557355 + 0.830274i \(0.688184\pi\)
\(788\) −13.6407 −0.485929
\(789\) 0 0
\(790\) −57.2935 −2.03841
\(791\) 0 0
\(792\) 0 0
\(793\) −11.1830 −0.397119
\(794\) −52.4811 −1.86248
\(795\) 0 0
\(796\) −31.6200 −1.12074
\(797\) −11.6121 −0.411323 −0.205661 0.978623i \(-0.565935\pi\)
−0.205661 + 0.978623i \(0.565935\pi\)
\(798\) 0 0
\(799\) −42.8085 −1.51446
\(800\) 12.4466 0.440054
\(801\) 0 0
\(802\) −26.1961 −0.925016
\(803\) 39.6304 1.39853
\(804\) 0 0
\(805\) 0 0
\(806\) 5.13664 0.180931
\(807\) 0 0
\(808\) 6.93880 0.244106
\(809\) −7.30665 −0.256888 −0.128444 0.991717i \(-0.540998\pi\)
−0.128444 + 0.991717i \(0.540998\pi\)
\(810\) 0 0
\(811\) 15.9662 0.560651 0.280325 0.959905i \(-0.409558\pi\)
0.280325 + 0.959905i \(0.409558\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −60.1223 −2.10729
\(815\) 20.5597 0.720175
\(816\) 0 0
\(817\) −39.5487 −1.38363
\(818\) 47.0897 1.64645
\(819\) 0 0
\(820\) −10.1325 −0.353844
\(821\) −1.12788 −0.0393634 −0.0196817 0.999806i \(-0.506265\pi\)
−0.0196817 + 0.999806i \(0.506265\pi\)
\(822\) 0 0
\(823\) 22.4044 0.780968 0.390484 0.920610i \(-0.372308\pi\)
0.390484 + 0.920610i \(0.372308\pi\)
\(824\) −3.28115 −0.114304
\(825\) 0 0
\(826\) 0 0
\(827\) 6.32296 0.219871 0.109935 0.993939i \(-0.464936\pi\)
0.109935 + 0.993939i \(0.464936\pi\)
\(828\) 0 0
\(829\) 44.3704 1.54105 0.770523 0.637412i \(-0.219995\pi\)
0.770523 + 0.637412i \(0.219995\pi\)
\(830\) −63.6171 −2.20818
\(831\) 0 0
\(832\) 2.30037 0.0797510
\(833\) 0 0
\(834\) 0 0
\(835\) 33.3750 1.15499
\(836\) 18.0215 0.623287
\(837\) 0 0
\(838\) −34.4483 −1.19000
\(839\) −9.89476 −0.341605 −0.170803 0.985305i \(-0.554636\pi\)
−0.170803 + 0.985305i \(0.554636\pi\)
\(840\) 0 0
\(841\) −26.9760 −0.930207
\(842\) 33.0345 1.13844
\(843\) 0 0
\(844\) −3.35942 −0.115636
\(845\) 2.62555 0.0903217
\(846\) 0 0
\(847\) 0 0
\(848\) 25.6488 0.880783
\(849\) 0 0
\(850\) −15.7182 −0.539129
\(851\) −45.5548 −1.56160
\(852\) 0 0
\(853\) −25.9407 −0.888193 −0.444097 0.895979i \(-0.646475\pi\)
−0.444097 + 0.895979i \(0.646475\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 20.7940 0.710725
\(857\) 10.0244 0.342426 0.171213 0.985234i \(-0.445231\pi\)
0.171213 + 0.985234i \(0.445231\pi\)
\(858\) 0 0
\(859\) −37.4378 −1.27736 −0.638680 0.769472i \(-0.720519\pi\)
−0.638680 + 0.769472i \(0.720519\pi\)
\(860\) 34.6685 1.18219
\(861\) 0 0
\(862\) 37.6703 1.28306
\(863\) −21.0026 −0.714938 −0.357469 0.933925i \(-0.616360\pi\)
−0.357469 + 0.933925i \(0.616360\pi\)
\(864\) 0 0
\(865\) 13.7817 0.468592
\(866\) 34.6748 1.17830
\(867\) 0 0
\(868\) 0 0
\(869\) −38.8788 −1.31887
\(870\) 0 0
\(871\) 1.98172 0.0671481
\(872\) −11.2588 −0.381273
\(873\) 0 0
\(874\) 33.7715 1.14234
\(875\) 0 0
\(876\) 0 0
\(877\) 7.56302 0.255385 0.127692 0.991814i \(-0.459243\pi\)
0.127692 + 0.991814i \(0.459243\pi\)
\(878\) −26.8534 −0.906258
\(879\) 0 0
\(880\) 41.7620 1.40780
\(881\) −34.3550 −1.15745 −0.578725 0.815522i \(-0.696450\pi\)
−0.578725 + 0.815522i \(0.696450\pi\)
\(882\) 0 0
\(883\) 49.6697 1.67152 0.835759 0.549096i \(-0.185028\pi\)
0.835759 + 0.549096i \(0.185028\pi\)
\(884\) −6.15013 −0.206851
\(885\) 0 0
\(886\) −21.9138 −0.736207
\(887\) −48.0325 −1.61277 −0.806386 0.591389i \(-0.798580\pi\)
−0.806386 + 0.591389i \(0.798580\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −51.3166 −1.72014
\(891\) 0 0
\(892\) −8.37665 −0.280471
\(893\) 38.4234 1.28579
\(894\) 0 0
\(895\) 27.6353 0.923745
\(896\) 0 0
\(897\) 0 0
\(898\) −4.25558 −0.142011
\(899\) 3.98816 0.133013
\(900\) 0 0
\(901\) −23.8488 −0.794518
\(902\) −17.0054 −0.566218
\(903\) 0 0
\(904\) 20.7199 0.689133
\(905\) 49.4069 1.64234
\(906\) 0 0
\(907\) 2.40195 0.0797556 0.0398778 0.999205i \(-0.487303\pi\)
0.0398778 + 0.999205i \(0.487303\pi\)
\(908\) 15.7396 0.522338
\(909\) 0 0
\(910\) 0 0
\(911\) 33.9555 1.12500 0.562499 0.826798i \(-0.309840\pi\)
0.562499 + 0.826798i \(0.309840\pi\)
\(912\) 0 0
\(913\) −43.1700 −1.42872
\(914\) 34.0463 1.12615
\(915\) 0 0
\(916\) 13.8992 0.459241
\(917\) 0 0
\(918\) 0 0
\(919\) −9.37134 −0.309132 −0.154566 0.987982i \(-0.549398\pi\)
−0.154566 + 0.987982i \(0.549398\pi\)
\(920\) 14.0089 0.461861
\(921\) 0 0
\(922\) −51.1095 −1.68320
\(923\) −11.7544 −0.386901
\(924\) 0 0
\(925\) −19.0306 −0.625721
\(926\) 6.71494 0.220667
\(927\) 0 0
\(928\) 9.35162 0.306982
\(929\) −43.4273 −1.42480 −0.712402 0.701771i \(-0.752393\pi\)
−0.712402 + 0.701771i \(0.752393\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −29.1639 −0.955294
\(933\) 0 0
\(934\) −71.0102 −2.32352
\(935\) −38.8312 −1.26992
\(936\) 0 0
\(937\) −2.59416 −0.0847476 −0.0423738 0.999102i \(-0.513492\pi\)
−0.0423738 + 0.999102i \(0.513492\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −33.6820 −1.09859
\(941\) 43.4060 1.41499 0.707497 0.706716i \(-0.249824\pi\)
0.707497 + 0.706716i \(0.249824\pi\)
\(942\) 0 0
\(943\) −12.8850 −0.419594
\(944\) 12.5112 0.407204
\(945\) 0 0
\(946\) 58.1840 1.89172
\(947\) 7.50058 0.243736 0.121868 0.992546i \(-0.461112\pi\)
0.121868 + 0.992546i \(0.461112\pi\)
\(948\) 0 0
\(949\) 12.1391 0.394052
\(950\) 14.1081 0.457726
\(951\) 0 0
\(952\) 0 0
\(953\) 0.708439 0.0229486 0.0114743 0.999934i \(-0.496348\pi\)
0.0114743 + 0.999934i \(0.496348\pi\)
\(954\) 0 0
\(955\) −39.1586 −1.26714
\(956\) −1.47421 −0.0476792
\(957\) 0 0
\(958\) 11.0571 0.357238
\(959\) 0 0
\(960\) 0 0
\(961\) −23.1416 −0.746504
\(962\) −18.4159 −0.593754
\(963\) 0 0
\(964\) 27.1285 0.873750
\(965\) −0.139510 −0.00449100
\(966\) 0 0
\(967\) −16.9761 −0.545915 −0.272957 0.962026i \(-0.588002\pi\)
−0.272957 + 0.962026i \(0.588002\pi\)
\(968\) −0.402359 −0.0129323
\(969\) 0 0
\(970\) −66.1136 −2.12278
\(971\) −6.86359 −0.220263 −0.110132 0.993917i \(-0.535127\pi\)
−0.110132 + 0.993917i \(0.535127\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 6.96757 0.223255
\(975\) 0 0
\(976\) −54.4850 −1.74402
\(977\) −0.581913 −0.0186171 −0.00930853 0.999957i \(-0.502963\pi\)
−0.00930853 + 0.999957i \(0.502963\pi\)
\(978\) 0 0
\(979\) −34.8230 −1.11295
\(980\) 0 0
\(981\) 0 0
\(982\) −0.698983 −0.0223054
\(983\) 34.7195 1.10738 0.553689 0.832723i \(-0.313220\pi\)
0.553689 + 0.832723i \(0.313220\pi\)
\(984\) 0 0
\(985\) −26.3810 −0.840568
\(986\) −11.8097 −0.376097
\(987\) 0 0
\(988\) 5.52013 0.175619
\(989\) 44.0861 1.40186
\(990\) 0 0
\(991\) 24.1688 0.767748 0.383874 0.923385i \(-0.374590\pi\)
0.383874 + 0.923385i \(0.374590\pi\)
\(992\) 18.4267 0.585047
\(993\) 0 0
\(994\) 0 0
\(995\) −61.1528 −1.93868
\(996\) 0 0
\(997\) 35.1852 1.11433 0.557163 0.830403i \(-0.311890\pi\)
0.557163 + 0.830403i \(0.311890\pi\)
\(998\) −63.3523 −2.00538
\(999\) 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 5733.2.a.bu.1.2 6
3.2 odd 2 637.2.a.m.1.5 6
7.6 odd 2 5733.2.a.br.1.2 6
21.2 odd 6 637.2.e.o.508.2 12
21.5 even 6 637.2.e.n.508.2 12
21.11 odd 6 637.2.e.o.79.2 12
21.17 even 6 637.2.e.n.79.2 12
21.20 even 2 637.2.a.n.1.5 yes 6
39.38 odd 2 8281.2.a.cc.1.2 6
273.272 even 2 8281.2.a.cd.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
637.2.a.m.1.5 6 3.2 odd 2
637.2.a.n.1.5 yes 6 21.20 even 2
637.2.e.n.79.2 12 21.17 even 6
637.2.e.n.508.2 12 21.5 even 6
637.2.e.o.79.2 12 21.11 odd 6
637.2.e.o.508.2 12 21.2 odd 6
5733.2.a.br.1.2 6 7.6 odd 2
5733.2.a.bu.1.2 6 1.1 even 1 trivial
8281.2.a.cc.1.2 6 39.38 odd 2
8281.2.a.cd.1.2 6 273.272 even 2