Properties

Label 8281.2.a.cd.1.2
Level $8281$
Weight $2$
Character 8281.1
Self dual yes
Analytic conductor $66.124$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8281 = 7^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8281.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(66.1241179138\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.4507648.1
Defining polynomial: \(x^{6} - 2 x^{5} - 5 x^{4} + 8 x^{3} + 7 x^{2} - 6 x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\chi\) \(=\) 8281.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.83237 q^{2} +0.853757 q^{3} +1.35758 q^{4} -2.62555 q^{5} -1.56440 q^{6} +1.17715 q^{8} -2.27110 q^{9} +O(q^{10})\) \(q-1.83237 q^{2} +0.853757 q^{3} +1.35758 q^{4} -2.62555 q^{5} -1.56440 q^{6} +1.17715 q^{8} -2.27110 q^{9} +4.81098 q^{10} -3.26469 q^{11} +1.15904 q^{12} -2.24158 q^{15} -4.87214 q^{16} +4.53021 q^{17} +4.16149 q^{18} -4.06615 q^{19} -3.56440 q^{20} +5.98212 q^{22} -4.53266 q^{23} +1.00500 q^{24} +1.89352 q^{25} -4.50024 q^{27} -1.42268 q^{29} +4.10741 q^{30} +2.80328 q^{31} +6.57326 q^{32} -2.78725 q^{33} -8.30102 q^{34} -3.08320 q^{36} +10.0503 q^{37} +7.45070 q^{38} -3.09067 q^{40} +2.84271 q^{41} +9.72632 q^{43} -4.43208 q^{44} +5.96289 q^{45} +8.30551 q^{46} +9.44956 q^{47} -4.15962 q^{48} -3.46963 q^{50} +3.86770 q^{51} +5.26439 q^{53} +8.24610 q^{54} +8.57161 q^{55} -3.47151 q^{57} +2.60687 q^{58} +2.56791 q^{59} -3.04313 q^{60} -11.1830 q^{61} -5.13664 q^{62} -2.30037 q^{64} +5.10728 q^{66} +1.98172 q^{67} +6.15013 q^{68} -3.86979 q^{69} +11.7544 q^{71} -2.67342 q^{72} -12.1391 q^{73} -18.4159 q^{74} +1.61661 q^{75} -5.52013 q^{76} +11.9089 q^{79} +12.7920 q^{80} +2.97118 q^{81} -5.20889 q^{82} -13.2233 q^{83} -11.8943 q^{85} -17.8222 q^{86} -1.21462 q^{87} -3.84303 q^{88} -10.6666 q^{89} -10.9262 q^{90} -6.15345 q^{92} +2.39332 q^{93} -17.3151 q^{94} +10.6759 q^{95} +5.61197 q^{96} +13.7422 q^{97} +7.41443 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 8 q^{3} + 4 q^{4} - 6 q^{5} - 4 q^{6} + 6 q^{9} + O(q^{10}) \) \( 6 q + 8 q^{3} + 4 q^{4} - 6 q^{5} - 4 q^{6} + 6 q^{9} + 4 q^{10} - 4 q^{11} - 4 q^{12} - 12 q^{15} + 16 q^{17} + 4 q^{18} - 2 q^{19} - 16 q^{20} - 12 q^{22} - 6 q^{23} - 12 q^{24} - 4 q^{25} + 20 q^{27} - 6 q^{29} - 6 q^{31} + 20 q^{32} - 4 q^{33} - 24 q^{36} + 8 q^{38} + 4 q^{40} + 8 q^{41} + 2 q^{43} + 4 q^{44} - 14 q^{45} - 8 q^{46} - 30 q^{47} - 8 q^{48} - 8 q^{50} - 4 q^{51} - 14 q^{53} + 48 q^{54} - 8 q^{55} - 4 q^{57} + 8 q^{58} - 24 q^{59} - 12 q^{60} + 28 q^{62} - 20 q^{64} - 4 q^{66} - 16 q^{67} + 28 q^{68} - 20 q^{69} - 8 q^{71} - 28 q^{72} + 6 q^{73} - 12 q^{74} + 12 q^{75} + 16 q^{76} - 22 q^{79} + 28 q^{80} + 46 q^{81} - 40 q^{82} - 50 q^{83} + 8 q^{85} + 16 q^{86} - 16 q^{87} - 44 q^{88} - 26 q^{89} - 40 q^{90} + 20 q^{92} - 16 q^{93} - 32 q^{94} - 6 q^{95} + 20 q^{96} + 14 q^{97} - 12 q^{99} + O(q^{100}) \)

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 \(0.724328\pi\)
\(3\) 0.853757 0.492917 0.246458 0.969153i \(-0.420733\pi\)
0.246458 + 0.969153i \(0.420733\pi\)
\(4\) 1.35758 0.678791
\(5\) −2.62555 −1.17418 −0.587091 0.809521i \(-0.699727\pi\)
−0.587091 + 0.809521i \(0.699727\pi\)
\(6\) −1.56440 −0.638663
\(7\) 0 0
\(8\) 1.17715 0.416185
\(9\) −2.27110 −0.757033
\(10\) 4.81098 1.52137
\(11\) −3.26469 −0.984341 −0.492170 0.870499i \(-0.663796\pi\)
−0.492170 + 0.870499i \(0.663796\pi\)
\(12\) 1.15904 0.334587
\(13\) 0 0
\(14\) 0 0
\(15\) −2.24158 −0.578775
\(16\) −4.87214 −1.21803
\(17\) 4.53021 1.09874 0.549369 0.835580i \(-0.314868\pi\)
0.549369 + 0.835580i \(0.314868\pi\)
\(18\) 4.16149 0.980873
\(19\) −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 \(-0.656671\pi\)
−0.472562 + 0.881297i \(0.656671\pi\)
\(24\) 1.00500 0.205145
\(25\) 1.89352 0.378705
\(26\) 0 0
\(27\) −4.50024 −0.866071
\(28\) 0 0
\(29\) −1.42268 −0.264184 −0.132092 0.991237i \(-0.542169\pi\)
−0.132092 + 0.991237i \(0.542169\pi\)
\(30\) 4.10741 0.749907
\(31\) 2.80328 0.503484 0.251742 0.967794i \(-0.418997\pi\)
0.251742 + 0.967794i \(0.418997\pi\)
\(32\) 6.57326 1.16200
\(33\) −2.78725 −0.485198
\(34\) −8.30102 −1.42361
\(35\) 0 0
\(36\) −3.08320 −0.513867
\(37\) 10.0503 1.65227 0.826133 0.563475i \(-0.190536\pi\)
0.826133 + 0.563475i \(0.190536\pi\)
\(38\) 7.45070 1.20866
\(39\) 0 0
\(40\) −3.09067 −0.488677
\(41\) 2.84271 0.443956 0.221978 0.975052i \(-0.428749\pi\)
0.221978 + 0.975052i \(0.428749\pi\)
\(42\) 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
\(45\) 5.96289 0.888895
\(46\) 8.30551 1.22458
\(47\) 9.44956 1.37836 0.689180 0.724590i \(-0.257971\pi\)
0.689180 + 0.724590i \(0.257971\pi\)
\(48\) −4.15962 −0.600390
\(49\) 0 0
\(50\) −3.46963 −0.490680
\(51\) 3.86770 0.541586
\(52\) 0 0
\(53\) 5.26439 0.723119 0.361559 0.932349i \(-0.382244\pi\)
0.361559 + 0.932349i \(0.382244\pi\)
\(54\) 8.24610 1.12215
\(55\) 8.57161 1.15580
\(56\) 0 0
\(57\) −3.47151 −0.459812
\(58\) 2.60687 0.342299
\(59\) 2.56791 0.334313 0.167156 0.985930i \(-0.446542\pi\)
0.167156 + 0.985930i \(0.446542\pi\)
\(60\) −3.04313 −0.392867
\(61\) −11.1830 −1.43183 −0.715917 0.698185i \(-0.753991\pi\)
−0.715917 + 0.698185i \(0.753991\pi\)
\(62\) −5.13664 −0.652354
\(63\) 0 0
\(64\) −2.30037 −0.287546
\(65\) 0 0
\(66\) 5.10728 0.628662
\(67\) 1.98172 0.242106 0.121053 0.992646i \(-0.461373\pi\)
0.121053 + 0.992646i \(0.461373\pi\)
\(68\) 6.15013 0.745813
\(69\) −3.86979 −0.465868
\(70\) 0 0
\(71\) 11.7544 1.39499 0.697495 0.716590i \(-0.254298\pi\)
0.697495 + 0.716590i \(0.254298\pi\)
\(72\) −2.67342 −0.315066
\(73\) −12.1391 −1.42078 −0.710388 0.703810i \(-0.751481\pi\)
−0.710388 + 0.703810i \(0.751481\pi\)
\(74\) −18.4159 −2.14081
\(75\) 1.61661 0.186670
\(76\) −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
\(81\) 2.97118 0.330132
\(82\) −5.20889 −0.575226
\(83\) −13.2233 −1.45145 −0.725723 0.687987i \(-0.758495\pi\)
−0.725723 + 0.687987i \(0.758495\pi\)
\(84\) 0 0
\(85\) −11.8943 −1.29012
\(86\) −17.8222 −1.92182
\(87\) −1.21462 −0.130221
\(88\) −3.84303 −0.409668
\(89\) −10.6666 −1.13065 −0.565326 0.824867i \(-0.691250\pi\)
−0.565326 + 0.824867i \(0.691250\pi\)
\(90\) −10.9262 −1.15172
\(91\) 0 0
\(92\) −6.15345 −0.641542
\(93\) 2.39332 0.248176
\(94\) −17.3151 −1.78592
\(95\) 10.6759 1.09532
\(96\) 5.61197 0.572769
\(97\) 13.7422 1.39531 0.697655 0.716433i \(-0.254227\pi\)
0.697655 + 0.716433i \(0.254227\pi\)
\(98\) 0 0
\(99\) 7.41443 0.745178
\(100\) 2.57061 0.257061
\(101\) 5.89458 0.586533 0.293266 0.956031i \(-0.405258\pi\)
0.293266 + 0.956031i \(0.405258\pi\)
\(102\) −7.08706 −0.701723
\(103\) 2.78737 0.274647 0.137324 0.990526i \(-0.456150\pi\)
0.137324 + 0.990526i \(0.456150\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −9.64630 −0.936932
\(107\) −17.6647 −1.70771 −0.853856 0.520509i \(-0.825742\pi\)
−0.853856 + 0.520509i \(0.825742\pi\)
\(108\) −6.10944 −0.587881
\(109\) 9.56450 0.916113 0.458057 0.888923i \(-0.348546\pi\)
0.458057 + 0.888923i \(0.348546\pi\)
\(110\) −15.7064 −1.49754
\(111\) 8.58055 0.814430
\(112\) 0 0
\(113\) −17.6017 −1.65583 −0.827916 0.560852i \(-0.810474\pi\)
−0.827916 + 0.560852i \(0.810474\pi\)
\(114\) 6.36109 0.595770
\(115\) 11.9007 1.10975
\(116\) −1.93140 −0.179326
\(117\) 0 0
\(118\) −4.70535 −0.433163
\(119\) 0 0
\(120\) −2.63868 −0.240877
\(121\) −0.341808 −0.0310735
\(122\) 20.4914 1.85520
\(123\) 2.42698 0.218833
\(124\) 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
\(129\) 8.30391 0.731119
\(130\) 0 0
\(131\) 4.30279 0.375937 0.187968 0.982175i \(-0.439810\pi\)
0.187968 + 0.982175i \(0.439810\pi\)
\(132\) −3.78392 −0.329348
\(133\) 0 0
\(134\) −3.63125 −0.313692
\(135\) 11.8156 1.01693
\(136\) 5.33273 0.457278
\(137\) −9.82234 −0.839179 −0.419590 0.907714i \(-0.637826\pi\)
−0.419590 + 0.907714i \(0.637826\pi\)
\(138\) 7.09089 0.603617
\(139\) 10.0811 0.855070 0.427535 0.903999i \(-0.359382\pi\)
0.427535 + 0.903999i \(0.359382\pi\)
\(140\) 0 0
\(141\) 8.06763 0.679417
\(142\) −21.5384 −1.80746
\(143\) 0 0
\(144\) 11.0651 0.922092
\(145\) 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 \(0.308576\pi\)
\(150\) −2.96223 −0.241865
\(151\) −21.2123 −1.72623 −0.863117 0.505004i \(-0.831491\pi\)
−0.863117 + 0.505004i \(0.831491\pi\)
\(152\) −4.78647 −0.388234
\(153\) −10.2886 −0.831780
\(154\) 0 0
\(155\) −7.36015 −0.591182
\(156\) 0 0
\(157\) 23.5155 1.87674 0.938372 0.345627i \(-0.112334\pi\)
0.938372 + 0.345627i \(0.112334\pi\)
\(158\) −21.8215 −1.73603
\(159\) 4.49451 0.356438
\(160\) −17.2584 −1.36440
\(161\) 0 0
\(162\) −5.44431 −0.427745
\(163\) −7.83062 −0.613341 −0.306671 0.951816i \(-0.599215\pi\)
−0.306671 + 0.951816i \(0.599215\pi\)
\(164\) 3.85920 0.301353
\(165\) 7.31807 0.569711
\(166\) 24.2300 1.88061
\(167\) −12.7116 −0.983654 −0.491827 0.870693i \(-0.663671\pi\)
−0.491827 + 0.870693i \(0.663671\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 21.7948 1.67158
\(171\) 9.23463 0.706190
\(172\) 13.2043 1.00682
\(173\) 5.24907 0.399079 0.199540 0.979890i \(-0.436055\pi\)
0.199540 + 0.979890i \(0.436055\pi\)
\(174\) 2.22563 0.168725
\(175\) 0 0
\(176\) 15.9060 1.19896
\(177\) 2.19237 0.164788
\(178\) 19.5451 1.46497
\(179\) −10.5255 −0.786714 −0.393357 0.919386i \(-0.628686\pi\)
−0.393357 + 0.919386i \(0.628686\pi\)
\(180\) 8.09510 0.603373
\(181\) −18.8177 −1.39871 −0.699356 0.714774i \(-0.746530\pi\)
−0.699356 + 0.714774i \(0.746530\pi\)
\(182\) 0 0
\(183\) −9.54755 −0.705775
\(184\) −5.33562 −0.393347
\(185\) −26.3877 −1.94006
\(186\) −4.38545 −0.321557
\(187\) −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 \(-0.318581\pi\)
0.539585 + 0.841931i \(0.318581\pi\)
\(192\) −1.96396 −0.141737
\(193\) 0.0531356 0.00382479 0.00191239 0.999998i \(-0.499391\pi\)
0.00191239 + 0.999998i \(0.499391\pi\)
\(194\) −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 \(0.616520\pi\)
\(198\) −13.5860 −0.965514
\(199\) 23.2914 1.65109 0.825543 0.564340i \(-0.190869\pi\)
0.825543 + 0.564340i \(0.190869\pi\)
\(200\) 2.22896 0.157611
\(201\) 1.69191 0.119338
\(202\) −10.8011 −0.759959
\(203\) 0 0
\(204\) 5.25072 0.367624
\(205\) −7.46367 −0.521285
\(206\) −5.10749 −0.355855
\(207\) 10.2941 0.715491
\(208\) 0 0
\(209\) 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
\(213\) 10.0354 0.687614
\(214\) 32.3683 2.21265
\(215\) −25.5369 −1.74161
\(216\) −5.29745 −0.360446
\(217\) 0 0
\(218\) −17.5257 −1.18699
\(219\) −10.3639 −0.700325
\(220\) 11.6367 0.784543
\(221\) 0 0
\(222\) −15.7227 −1.05524
\(223\) −6.17027 −0.413192 −0.206596 0.978426i \(-0.566239\pi\)
−0.206596 + 0.978426i \(0.566239\pi\)
\(224\) 0 0
\(225\) −4.30038 −0.286692
\(226\) 32.2529 2.14543
\(227\) −11.5939 −0.769513 −0.384757 0.923018i \(-0.625715\pi\)
−0.384757 + 0.923018i \(0.625715\pi\)
\(228\) −4.71285 −0.312116
\(229\) 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 \(-0.251542\pi\)
0.703673 + 0.710524i \(0.251542\pi\)
\(234\) 0 0
\(235\) −24.8103 −1.61845
\(236\) 3.48614 0.226928
\(237\) 10.1673 0.660437
\(238\) 0 0
\(239\) −1.08591 −0.0702414 −0.0351207 0.999383i \(-0.511182\pi\)
−0.0351207 + 0.999383i \(0.511182\pi\)
\(240\) 10.9213 0.704967
\(241\) 19.9830 1.28722 0.643608 0.765355i \(-0.277437\pi\)
0.643608 + 0.765355i \(0.277437\pi\)
\(242\) 0.626319 0.0402613
\(243\) 16.0374 1.02880
\(244\) −15.1818 −0.971915
\(245\) 0 0
\(246\) −4.44713 −0.283538
\(247\) 0 0
\(248\) 3.29988 0.209542
\(249\) −11.2895 −0.715443
\(250\) −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
\(255\) −10.1548 −0.635921
\(256\) 20.9663 1.31040
\(257\) −15.0174 −0.936759 −0.468379 0.883527i \(-0.655162\pi\)
−0.468379 + 0.883527i \(0.655162\pi\)
\(258\) −15.2158 −0.947297
\(259\) 0 0
\(260\) 0 0
\(261\) 3.23104 0.199996
\(262\) −7.88431 −0.487094
\(263\) −21.1662 −1.30516 −0.652582 0.757718i \(-0.726314\pi\)
−0.652582 + 0.757718i \(0.726314\pi\)
\(264\) −3.28101 −0.201932
\(265\) −13.8219 −0.849074
\(266\) 0 0
\(267\) −9.10665 −0.557318
\(268\) 2.69035 0.164339
\(269\) −4.10480 −0.250274 −0.125137 0.992139i \(-0.539937\pi\)
−0.125137 + 0.992139i \(0.539937\pi\)
\(270\) −21.6506 −1.31761
\(271\) −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
\(276\) −5.25355 −0.316227
\(277\) −0.361012 −0.0216911 −0.0108455 0.999941i \(-0.503452\pi\)
−0.0108455 + 0.999941i \(0.503452\pi\)
\(278\) −18.4724 −1.10790
\(279\) −6.36652 −0.381154
\(280\) 0 0
\(281\) −18.5213 −1.10489 −0.552445 0.833550i \(-0.686305\pi\)
−0.552445 + 0.833550i \(0.686305\pi\)
\(282\) −14.7829 −0.880308
\(283\) −1.61158 −0.0957983 −0.0478991 0.998852i \(-0.515253\pi\)
−0.0478991 + 0.998852i \(0.515253\pi\)
\(284\) 15.9575 0.946906
\(285\) 9.11462 0.539904
\(286\) 0 0
\(287\) 0 0
\(288\) −14.9285 −0.879671
\(289\) 3.52281 0.207224
\(290\) −6.84447 −0.401921
\(291\) 11.7325 0.687772
\(292\) −16.4798 −0.964410
\(293\) 4.41671 0.258027 0.129013 0.991643i \(-0.458819\pi\)
0.129013 + 0.991643i \(0.458819\pi\)
\(294\) 0 0
\(295\) −6.74217 −0.392544
\(296\) 11.8308 0.687649
\(297\) 14.6919 0.852509
\(298\) −25.3094 −1.46613
\(299\) 0 0
\(300\) 2.19468 0.126710
\(301\) 0 0
\(302\) 38.8688 2.23665
\(303\) 5.03254 0.289112
\(304\) 19.8108 1.13623
\(305\) 29.3615 1.68123
\(306\) 18.8524 1.07772
\(307\) 5.78353 0.330083 0.165042 0.986287i \(-0.447224\pi\)
0.165042 + 0.986287i \(0.447224\pi\)
\(308\) 0 0
\(309\) 2.37973 0.135378
\(310\) 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 \(-0.475351\pi\)
0.0773592 + 0.997003i \(0.475351\pi\)
\(314\) −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 \(0.596692\pi\)
\(318\) −8.23560 −0.461830
\(319\) 4.64460 0.260047
\(320\) 6.03975 0.337632
\(321\) −15.0814 −0.841761
\(322\) 0 0
\(323\) −18.4205 −1.02495
\(324\) 4.03362 0.224090
\(325\) 0 0
\(326\) 14.3486 0.794695
\(327\) 8.16576 0.451568
\(328\) 3.34629 0.184768
\(329\) 0 0
\(330\) −13.4094 −0.738164
\(331\) 3.41626 0.187774 0.0938872 0.995583i \(-0.470071\pi\)
0.0938872 + 0.995583i \(0.470071\pi\)
\(332\) −17.9517 −0.985228
\(333\) −22.8253 −1.25082
\(334\) 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 \(0.262047\pi\)
\(338\) 0 0
\(339\) −15.0276 −0.816188
\(340\) −16.1475 −0.875720
\(341\) −9.15183 −0.495599
\(342\) −16.9213 −0.914997
\(343\) 0 0
\(344\) 11.4493 0.617306
\(345\) 10.1603 0.547014
\(346\) −9.61824 −0.517080
\(347\) −19.1833 −1.02981 −0.514907 0.857246i \(-0.672173\pi\)
−0.514907 + 0.857246i \(0.672173\pi\)
\(348\) −1.64895 −0.0883928
\(349\) 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 \(-0.767914\pi\)
−0.745762 + 0.666212i \(0.767914\pi\)
\(354\) −4.01723 −0.213513
\(355\) −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 \(0.840278\pi\)
\(360\) 7.01921 0.369945
\(361\) −2.46641 −0.129811
\(362\) 34.4811 1.81228
\(363\) −0.291821 −0.0153166
\(364\) 0 0
\(365\) 31.8719 1.66825
\(366\) 17.4947 0.914460
\(367\) −28.3091 −1.47772 −0.738862 0.673857i \(-0.764637\pi\)
−0.738862 + 0.673857i \(0.764637\pi\)
\(368\) 22.0837 1.15119
\(369\) −6.45606 −0.336089
\(370\) 48.3520 2.51370
\(371\) 0 0
\(372\) 3.24912 0.168459
\(373\) −12.9515 −0.670602 −0.335301 0.942111i \(-0.608838\pi\)
−0.335301 + 0.942111i \(0.608838\pi\)
\(374\) 27.1003 1.40132
\(375\) 6.96343 0.359590
\(376\) 11.1235 0.573653
\(377\) 0 0
\(378\) 0 0
\(379\) −0.168981 −0.00867995 −0.00433997 0.999991i \(-0.501381\pi\)
−0.00433997 + 0.999991i \(0.501381\pi\)
\(380\) 14.4934 0.743495
\(381\) −1.36159 −0.0697562
\(382\) −27.3288 −1.39826
\(383\) −10.1933 −0.520854 −0.260427 0.965494i \(-0.583863\pi\)
−0.260427 + 0.965494i \(0.583863\pi\)
\(384\) −7.62523 −0.389124
\(385\) 0 0
\(386\) −0.0973641 −0.00495570
\(387\) −22.0894 −1.12287
\(388\) 18.6562 0.947124
\(389\) −28.6665 −1.45345 −0.726723 0.686930i \(-0.758958\pi\)
−0.726723 + 0.686930i \(0.758958\pi\)
\(390\) 0 0
\(391\) −20.5339 −1.03844
\(392\) 0 0
\(393\) 3.67354 0.185306
\(394\) 18.4113 0.927547
\(395\) −31.2674 −1.57323
\(396\) 10.0657 0.505820
\(397\) 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 \(0.383813\pi\)
\(402\) −3.10021 −0.154624
\(403\) 0 0
\(404\) 8.00237 0.398133
\(405\) −7.80100 −0.387635
\(406\) 0 0
\(407\) −32.8112 −1.62639
\(408\) 4.55286 0.225400
\(409\) −25.6988 −1.27072 −0.635362 0.772215i \(-0.719149\pi\)
−0.635362 + 0.772215i \(0.719149\pi\)
\(410\) 13.6762 0.675420
\(411\) −8.38590 −0.413646
\(412\) 3.78408 0.186428
\(413\) 0 0
\(414\) −18.8626 −0.927048
\(415\) 34.7185 1.70426
\(416\) 0 0
\(417\) 8.60684 0.421479
\(418\) −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 \(-0.355219\pi\)
0.439322 + 0.898329i \(0.355219\pi\)
\(422\) 4.53432 0.220727
\(423\) −21.4609 −1.04346
\(424\) 6.19697 0.300951
\(425\) 8.57806 0.416097
\(426\) −18.3886 −0.890929
\(427\) 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 \(0.664879\pi\)
\(432\) 21.9258 1.05490
\(433\) 18.9235 0.909404 0.454702 0.890644i \(-0.349746\pi\)
0.454702 + 0.890644i \(0.349746\pi\)
\(434\) 0 0
\(435\) 3.18905 0.152903
\(436\) 12.9846 0.621849
\(437\) 18.4305 0.881650
\(438\) 18.9904 0.907398
\(439\) −14.6550 −0.699445 −0.349723 0.936853i \(-0.613724\pi\)
−0.349723 + 0.936853i \(0.613724\pi\)
\(440\) 10.0901 0.481025
\(441\) 0 0
\(442\) 0 0
\(443\) −11.9592 −0.568201 −0.284100 0.958795i \(-0.591695\pi\)
−0.284100 + 0.958795i \(0.591695\pi\)
\(444\) 11.6488 0.552827
\(445\) 28.0056 1.32759
\(446\) 11.3062 0.535365
\(447\) 11.7924 0.557762
\(448\) 0 0
\(449\) 2.32245 0.109603 0.0548015 0.998497i \(-0.482547\pi\)
0.0548015 + 0.998497i \(0.482547\pi\)
\(450\) 7.87988 0.371461
\(451\) −9.28055 −0.437004
\(452\) −23.8958 −1.12396
\(453\) −18.1102 −0.850890
\(454\) 21.2443 0.997044
\(455\) 0 0
\(456\) −4.08648 −0.191367
\(457\) 18.5805 0.869157 0.434579 0.900634i \(-0.356897\pi\)
0.434579 + 0.900634i \(0.356897\pi\)
\(458\) −18.7601 −0.876604
\(459\) −20.3870 −0.951585
\(460\) 16.1562 0.753287
\(461\) −27.8926 −1.29909 −0.649543 0.760325i \(-0.725040\pi\)
−0.649543 + 0.760325i \(0.725040\pi\)
\(462\) 0 0
\(463\) 3.66462 0.170309 0.0851547 0.996368i \(-0.472862\pi\)
0.0851547 + 0.996368i \(0.472862\pi\)
\(464\) 6.93147 0.321786
\(465\) −6.28378 −0.291403
\(466\) −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
\(471\) 20.0766 0.925079
\(472\) 3.02281 0.139136
\(473\) −31.7534 −1.46002
\(474\) −18.6303 −0.855716
\(475\) −7.69935 −0.353270
\(476\) 0 0
\(477\) −11.9559 −0.547425
\(478\) 1.98978 0.0910105
\(479\) 6.03430 0.275714 0.137857 0.990452i \(-0.455979\pi\)
0.137857 + 0.990452i \(0.455979\pi\)
\(480\) −14.7345 −0.672535
\(481\) 0 0
\(482\) −36.6162 −1.66782
\(483\) 0 0
\(484\) −0.464032 −0.0210924
\(485\) −36.0809 −1.63835
\(486\) −29.3864 −1.33300
\(487\) 3.80249 0.172307 0.0861537 0.996282i \(-0.472542\pi\)
0.0861537 + 0.996282i \(0.472542\pi\)
\(488\) −13.1640 −0.595908
\(489\) −6.68545 −0.302326
\(490\) 0 0
\(491\) −0.381464 −0.0172152 −0.00860761 0.999963i \(-0.502740\pi\)
−0.00860761 + 0.999963i \(0.502740\pi\)
\(492\) 3.29482 0.148542
\(493\) −6.44502 −0.290269
\(494\) 0 0
\(495\) −19.4670 −0.874975
\(496\) −13.6580 −0.613260
\(497\) 0 0
\(498\) 20.6865 0.926986
\(499\) −34.5739 −1.54774 −0.773871 0.633343i \(-0.781682\pi\)
−0.773871 + 0.633343i \(0.781682\pi\)
\(500\) 11.0727 0.495187
\(501\) −10.8526 −0.484860
\(502\) −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 \(-0.338568\pi\)
0.485690 + 0.874131i \(0.338568\pi\)
\(510\) 18.6074 0.823951
\(511\) 0 0
\(512\) −20.5553 −0.908426
\(513\) 18.2987 0.807905
\(514\) 27.5174 1.21374
\(515\) −7.31837 −0.322486
\(516\) 11.2732 0.496276
\(517\) −30.8499 −1.35678
\(518\) 0 0
\(519\) 4.48143 0.196713
\(520\) 0 0
\(521\) −30.1450 −1.32068 −0.660338 0.750968i \(-0.729587\pi\)
−0.660338 + 0.750968i \(0.729587\pi\)
\(522\) −5.92046 −0.259131
\(523\) −18.0993 −0.791428 −0.395714 0.918374i \(-0.629503\pi\)
−0.395714 + 0.918374i \(0.629503\pi\)
\(524\) 5.84139 0.255182
\(525\) 0 0
\(526\) 38.7843 1.69108
\(527\) 12.6994 0.553196
\(528\) 13.5799 0.590988
\(529\) −2.45500 −0.106739
\(530\) 25.3269 1.10013
\(531\) −5.83197 −0.253086
\(532\) 0 0
\(533\) 0 0
\(534\) 16.6868 0.722106
\(535\) 46.3796 2.00517
\(536\) 2.33278 0.100761
\(537\) −8.98623 −0.387784
\(538\) 7.52151 0.324275
\(539\) 0 0
\(540\) 16.0406 0.690280
\(541\) −4.08890 −0.175795 −0.0878977 0.996130i \(-0.528015\pi\)
−0.0878977 + 0.996130i \(0.528015\pi\)
\(542\) 6.28950 0.270157
\(543\) −16.0658 −0.689449
\(544\) 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
\(549\) 25.3977 1.08395
\(550\) 11.3273 0.482997
\(551\) 5.78482 0.246442
\(552\) −4.55532 −0.193887
\(553\) 0 0
\(554\) 0.661507 0.0281047
\(555\) −22.5287 −0.956289
\(556\) 13.6859 0.580413
\(557\) −19.7690 −0.837640 −0.418820 0.908069i \(-0.637556\pi\)
−0.418820 + 0.908069i \(0.637556\pi\)
\(558\) 11.6658 0.493854
\(559\) 0 0
\(560\) 0 0
\(561\) −12.6268 −0.533105
\(562\) 33.9379 1.43158
\(563\) 8.22392 0.346597 0.173298 0.984869i \(-0.444557\pi\)
0.173298 + 0.984869i \(0.444557\pi\)
\(564\) 10.9525 0.461182
\(565\) 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 \(-0.401160\pi\)
0.305550 + 0.952176i \(0.401160\pi\)
\(570\) −16.7014 −0.699543
\(571\) −2.58822 −0.108314 −0.0541568 0.998532i \(-0.517247\pi\)
−0.0541568 + 0.998532i \(0.517247\pi\)
\(572\) 0 0
\(573\) 12.7333 0.531942
\(574\) 0 0
\(575\) −8.58269 −0.357923
\(576\) 5.22437 0.217682
\(577\) 23.8937 0.994707 0.497353 0.867548i \(-0.334305\pi\)
0.497353 + 0.867548i \(0.334305\pi\)
\(578\) −6.45509 −0.268496
\(579\) 0.0453649 0.00188530
\(580\) 5.07099 0.210561
\(581\) 0 0
\(582\) −21.4983 −0.891134
\(583\) −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 \(-0.700764\pi\)
−0.589726 + 0.807604i \(0.700764\pi\)
\(588\) 0 0
\(589\) −11.3986 −0.469669
\(590\) 12.3541 0.508612
\(591\) −8.57837 −0.352867
\(592\) −48.9666 −2.01252
\(593\) 19.9575 0.819556 0.409778 0.912185i \(-0.365606\pi\)
0.409778 + 0.912185i \(0.365606\pi\)
\(594\) −26.9210 −1.10458
\(595\) 0 0
\(596\) 18.7514 0.768088
\(597\) 19.8852 0.813848
\(598\) 0 0
\(599\) 1.17716 0.0480973 0.0240486 0.999711i \(-0.492344\pi\)
0.0240486 + 0.999711i \(0.492344\pi\)
\(600\) 1.90299 0.0776892
\(601\) −30.9250 −1.26146 −0.630729 0.776003i \(-0.717244\pi\)
−0.630729 + 0.776003i \(0.717244\pi\)
\(602\) 0 0
\(603\) −4.50069 −0.183282
\(604\) −28.7974 −1.17175
\(605\) 0.897435 0.0364859
\(606\) −9.22148 −0.374597
\(607\) 2.40708 0.0977005 0.0488503 0.998806i \(-0.484444\pi\)
0.0488503 + 0.998806i \(0.484444\pi\)
\(608\) −26.7279 −1.08396
\(609\) 0 0
\(610\) −53.8011 −2.17834
\(611\) 0 0
\(612\) −13.9675 −0.564605
\(613\) 27.7742 1.12179 0.560895 0.827887i \(-0.310457\pi\)
0.560895 + 0.827887i \(0.310457\pi\)
\(614\) −10.5976 −0.427683
\(615\) −6.37216 −0.256950
\(616\) 0 0
\(617\) −26.2125 −1.05527 −0.527637 0.849470i \(-0.676922\pi\)
−0.527637 + 0.849470i \(0.676922\pi\)
\(618\) −4.36055 −0.175407
\(619\) 17.2357 0.692763 0.346381 0.938094i \(-0.387410\pi\)
0.346381 + 0.938094i \(0.387410\pi\)
\(620\) −9.99200 −0.401288
\(621\) 20.3980 0.818546
\(622\) −26.1837 −1.04987
\(623\) 0 0
\(624\) 0 0
\(625\) −30.8822 −1.23529
\(626\) −5.01565 −0.200466
\(627\) 11.3334 0.452612
\(628\) 31.9242 1.27392
\(629\) 45.5302 1.81541
\(630\) 0 0
\(631\) −39.2125 −1.56103 −0.780513 0.625140i \(-0.785042\pi\)
−0.780513 + 0.625140i \(0.785042\pi\)
\(632\) 14.0185 0.557628
\(633\) −2.11268 −0.0839714
\(634\) 19.5170 0.775117
\(635\) 4.18728 0.166167
\(636\) 6.10166 0.241946
\(637\) 0 0
\(638\) −8.51062 −0.336939
\(639\) −26.6954 −1.05605
\(640\) 23.4498 0.926935
\(641\) −22.8735 −0.903451 −0.451725 0.892157i \(-0.649191\pi\)
−0.451725 + 0.892157i \(0.649191\pi\)
\(642\) 27.6347 1.09065
\(643\) 46.7072 1.84195 0.920976 0.389620i \(-0.127394\pi\)
0.920976 + 0.389620i \(0.127394\pi\)
\(644\) 0 0
\(645\) −21.8024 −0.858467
\(646\) 33.7532 1.32800
\(647\) 5.93075 0.233162 0.116581 0.993181i \(-0.462807\pi\)
0.116581 + 0.993181i \(0.462807\pi\)
\(648\) 3.49753 0.137396
\(649\) −8.38341 −0.329078
\(650\) 0 0
\(651\) 0 0
\(652\) −10.6307 −0.416330
\(653\) −18.8046 −0.735881 −0.367940 0.929849i \(-0.619937\pi\)
−0.367940 + 0.929849i \(0.619937\pi\)
\(654\) −14.9627 −0.585088
\(655\) −11.2972 −0.441419
\(656\) −13.8500 −0.540754
\(657\) 27.5691 1.07557
\(658\) 0 0
\(659\) 23.1086 0.900184 0.450092 0.892982i \(-0.351391\pi\)
0.450092 + 0.892982i \(0.351391\pi\)
\(660\) 9.93488 0.386715
\(661\) 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
\(666\) 41.8244 1.62066
\(667\) 6.44851 0.249687
\(668\) −17.2570 −0.667695
\(669\) −5.26792 −0.203669
\(670\) 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
\(675\) −8.52130 −0.327985
\(676\) 0 0
\(677\) 20.2075 0.776637 0.388318 0.921525i \(-0.373056\pi\)
0.388318 + 0.921525i \(0.373056\pi\)
\(678\) 27.5361 1.05752
\(679\) 0 0
\(680\) −14.0014 −0.536928
\(681\) −9.89837 −0.379306
\(682\) 16.7695 0.642139
\(683\) 31.6508 1.21108 0.605541 0.795814i \(-0.292957\pi\)
0.605541 + 0.795814i \(0.292957\pi\)
\(684\) 12.5368 0.479355
\(685\) 25.7891 0.985350
\(686\) 0 0
\(687\) 8.74092 0.333487
\(688\) −47.3879 −1.80665
\(689\) 0 0
\(690\) −18.6175 −0.708756
\(691\) −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
\(696\) −1.42979 −0.0541960
\(697\) 12.8781 0.487791
\(698\) −7.22725 −0.273556
\(699\) 18.3406 0.693705
\(700\) 0 0
\(701\) −5.10365 −0.192762 −0.0963811 0.995345i \(-0.530727\pi\)
−0.0963811 + 0.995345i \(0.530727\pi\)
\(702\) 0 0
\(703\) −40.8662 −1.54130
\(704\) 7.51000 0.283044
\(705\) −21.1820 −0.797760
\(706\) 51.3489 1.93254
\(707\) 0 0
\(708\) 2.97632 0.111857
\(709\) −26.1445 −0.981878 −0.490939 0.871194i \(-0.663346\pi\)
−0.490939 + 0.871194i \(0.663346\pi\)
\(710\) 56.5502 2.12229
\(711\) −27.0463 −1.01431
\(712\) −12.5561 −0.470561
\(713\) −12.7063 −0.475855
\(714\) 0 0
\(715\) 0 0
\(716\) −14.2892 −0.534014
\(717\) −0.927100 −0.0346232
\(718\) 60.8772 2.27192
\(719\) −8.72884 −0.325531 −0.162765 0.986665i \(-0.552041\pi\)
−0.162765 + 0.986665i \(0.552041\pi\)
\(720\) −29.0520 −1.08270
\(721\) 0 0
\(722\) 4.51937 0.168193
\(723\) 17.0606 0.634491
\(724\) −25.5466 −0.949432
\(725\) −2.69387 −0.100048
\(726\) 0.534725 0.0198455
\(727\) −21.8712 −0.811158 −0.405579 0.914060i \(-0.632930\pi\)
−0.405579 + 0.914060i \(0.632930\pi\)
\(728\) 0 0
\(729\) 4.77848 0.176981
\(730\) −58.4011 −2.16152
\(731\) 44.0623 1.62970
\(732\) −12.9616 −0.479074
\(733\) 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
\(738\) 11.8299 0.435465
\(739\) 36.4894 1.34228 0.671142 0.741329i \(-0.265804\pi\)
0.671142 + 0.741329i \(0.265804\pi\)
\(740\) −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 \(0.573394\pi\)
\(744\) 2.81729 0.103287
\(745\) −36.2651 −1.32865
\(746\) 23.7319 0.868886
\(747\) 30.0314 1.09879
\(748\) −20.0783 −0.734134
\(749\) 0 0
\(750\) −12.7596 −0.465914
\(751\) −18.9424 −0.691217 −0.345608 0.938379i \(-0.612327\pi\)
−0.345608 + 0.938379i \(0.612327\pi\)
\(752\) −46.0395 −1.67889
\(753\) 17.2030 0.626912
\(754\) 0 0
\(755\) 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
\(759\) 12.6337 0.458573
\(760\) 12.5671 0.455857
\(761\) −28.0617 −1.01723 −0.508617 0.860993i \(-0.669843\pi\)
−0.508617 + 0.860993i \(0.669843\pi\)
\(762\) 2.49493 0.0903818
\(763\) 0 0
\(764\) 20.2476 0.732531
\(765\) 27.0131 0.976662
\(766\) 18.6779 0.674861
\(767\) 0 0
\(768\) 17.9002 0.645917
\(769\) 23.2636 0.838907 0.419454 0.907777i \(-0.362222\pi\)
0.419454 + 0.907777i \(0.362222\pi\)
\(770\) 0 0
\(771\) −12.8212 −0.461744
\(772\) 0.0721359 0.00259623
\(773\) −45.8118 −1.64774 −0.823869 0.566781i \(-0.808189\pi\)
−0.823869 + 0.566781i \(0.808189\pi\)
\(774\) 40.4760 1.45488
\(775\) 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
\(783\) 6.40238 0.228803
\(784\) 0 0
\(785\) −61.7413 −2.20364
\(786\) −6.73129 −0.240097
\(787\) −31.2715 −1.11471 −0.557355 0.830274i \(-0.688184\pi\)
−0.557355 + 0.830274i \(0.688184\pi\)
\(788\) −13.6407 −0.485929
\(789\) −18.0708 −0.643338
\(790\) 57.2935 2.03841
\(791\) 0 0
\(792\) 8.72789 0.310132
\(793\) 0 0
\(794\) −52.4811 −1.86248
\(795\) −11.8006 −0.418523
\(796\) 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
\(801\) 24.2248 0.855941
\(802\) −26.1961 −0.925016
\(803\) 39.6304 1.39853
\(804\) 2.29691 0.0810056
\(805\) 0 0
\(806\) 0 0
\(807\) −3.50450 −0.123364
\(808\) 6.93880 0.244106
\(809\) 7.30665 0.256888 0.128444 0.991717i \(-0.459002\pi\)
0.128444 + 0.991717i \(0.459002\pi\)
\(810\) 14.2943 0.502251
\(811\) 15.9662 0.560651 0.280325 0.959905i \(-0.409558\pi\)
0.280325 + 0.959905i \(0.409558\pi\)
\(812\) 0 0
\(813\) −2.93047 −0.102776
\(814\) 60.1223 2.10729
\(815\) 20.5597 0.720175
\(816\) −18.8440 −0.659671
\(817\) −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 \(0.506265\pi\)
\(822\) 15.3661 0.535953
\(823\) 22.4044 0.780968 0.390484 0.920610i \(-0.372308\pi\)
0.390484 + 0.920610i \(0.372308\pi\)
\(824\) 3.28115 0.114304
\(825\) −5.27772 −0.183747
\(826\) 0 0
\(827\) 6.32296 0.219871 0.109935 0.993939i \(-0.464936\pi\)
0.109935 + 0.993939i \(0.464936\pi\)
\(828\) 13.9751 0.485668
\(829\) −44.3704 −1.54105 −0.770523 0.637412i \(-0.780005\pi\)
−0.770523 + 0.637412i \(0.780005\pi\)
\(830\) −63.6171 −2.20818
\(831\) −0.308216 −0.0106919
\(832\) 0 0
\(833\) 0 0
\(834\) −15.7709 −0.546102
\(835\) 33.3750 1.15499
\(836\) 18.0215 0.623287
\(837\) −12.6154 −0.436053
\(838\) −34.4483 −1.19000
\(839\) 9.89476 0.341605 0.170803 0.985305i \(-0.445364\pi\)
0.170803 + 0.985305i \(0.445364\pi\)
\(840\) 0 0
\(841\) −26.9760 −0.930207
\(842\) −33.0345 −1.13844
\(843\) −15.8127 −0.544619
\(844\) −3.35942 −0.115636
\(845\) 0 0
\(846\) 39.3243 1.35200
\(847\) 0 0
\(848\) −25.6488 −0.880783
\(849\) −1.37589 −0.0472206
\(850\) −15.7182 −0.539129
\(851\) −45.5548 −1.56160
\(852\) 13.6239 0.466746
\(853\) −25.9407 −0.888193 −0.444097 0.895979i \(-0.646475\pi\)
−0.444097 + 0.895979i \(0.646475\pi\)
\(854\) 0 0
\(855\) −24.2460 −0.829196
\(856\) −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 \(-0.279481\pi\)
0.638680 + 0.769472i \(0.279481\pi\)
\(860\) −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 \(0.616360\pi\)
\(864\) −29.5812 −1.00637
\(865\) −13.7817 −0.468592
\(866\) −34.6748 −1.17830
\(867\) 3.00762 0.102144
\(868\) 0 0
\(869\) −38.8788 −1.31887
\(870\) −5.84352 −0.198114
\(871\) 0 0
\(872\) 11.2588 0.381273
\(873\) −31.2099 −1.05630
\(874\) −33.7715 −1.14234
\(875\) 0 0
\(876\) −14.0698 −0.475374
\(877\) −7.56302 −0.255385 −0.127692 0.991814i \(-0.540757\pi\)
−0.127692 + 0.991814i \(0.540757\pi\)
\(878\) 26.8534 0.906258
\(879\) 3.77080 0.127186
\(880\) −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 \(0.185028\pi\)
\(884\) 0 0
\(885\) −5.75617 −0.193492
\(886\) 21.9138 0.736207
\(887\) −48.0325 −1.61277 −0.806386 0.591389i \(-0.798580\pi\)
−0.806386 + 0.591389i \(0.798580\pi\)
\(888\) 10.1006 0.338954
\(889\) 0 0
\(890\) −51.3166 −1.72014
\(891\) −9.69999 −0.324962
\(892\) −8.37665 −0.280471
\(893\) −38.4234 −1.28579
\(894\) −21.6081 −0.722682
\(895\) 27.6353 0.923745
\(896\) 0 0
\(897\) 0 0
\(898\) −4.25558 −0.142011
\(899\) −3.98816 −0.133013
\(900\) −5.83811 −0.194604
\(901\) 23.8488 0.794518
\(902\) 17.0054 0.566218
\(903\) 0 0
\(904\) −20.7199 −0.689133
\(905\) 49.4069 1.64234
\(906\) 33.1845 1.10248
\(907\) 2.40195 0.0797556 0.0398778 0.999205i \(-0.487303\pi\)
0.0398778 + 0.999205i \(0.487303\pi\)
\(908\) −15.7396 −0.522338
\(909\) −13.3872 −0.444024
\(910\) 0 0
\(911\) −33.9555 −1.12500 −0.562499 0.826798i \(-0.690160\pi\)
−0.562499 + 0.826798i \(0.690160\pi\)
\(912\) 16.9137 0.560067
\(913\) 43.1700 1.42872
\(914\) −34.0463 −1.12615
\(915\) 25.0676 0.828709
\(916\) 13.8992 0.459241
\(917\) 0 0
\(918\) 37.3566 1.23295
\(919\) −9.37134 −0.309132 −0.154566 0.987982i \(-0.549398\pi\)
−0.154566 + 0.987982i \(0.549398\pi\)
\(920\) 14.0089 0.461861
\(921\) 4.93773 0.162704
\(922\) 51.1095 1.68320
\(923\) 0 0
\(924\) 0 0
\(925\) 19.0306 0.625721
\(926\) −6.71494 −0.220667
\(927\) −6.33038 −0.207917
\(928\) −9.35162 −0.306982
\(929\) 43.4273 1.42480 0.712402 0.701771i \(-0.247607\pi\)
0.712402 + 0.701771i \(0.247607\pi\)
\(930\) 11.5142 0.377566
\(931\) 0 0
\(932\) 29.1639 0.955294
\(933\) 12.1998 0.399403
\(934\) −71.0102 −2.32352
\(935\) 38.8312 1.26992
\(936\) 0 0
\(937\) 2.59416 0.0847476 0.0423738 0.999102i \(-0.486508\pi\)
0.0423738 + 0.999102i \(0.486508\pi\)
\(938\) 0 0
\(939\) 2.33695 0.0762633
\(940\) −33.6820 −1.09859
\(941\) −43.4060 −1.41499 −0.707497 0.706716i \(-0.750176\pi\)
−0.707497 + 0.706716i \(0.750176\pi\)
\(942\) −36.7877 −1.19861
\(943\) −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 \(0.461112\pi\)
\(948\) 13.8029 0.448299
\(949\) 0 0
\(950\) 14.1081 0.457726
\(951\) −9.09355 −0.294878
\(952\) 0 0
\(953\) −0.708439 −0.0229486 −0.0114743 0.999934i \(-0.503652\pi\)
−0.0114743 + 0.999934i \(0.503652\pi\)
\(954\) 21.9077 0.709288
\(955\) −39.1586 −1.26714
\(956\) −1.47421 −0.0476792
\(957\) 3.96536 0.128182
\(958\) −11.0571 −0.357238
\(959\) 0 0
\(960\) 5.15648 0.166425
\(961\) −23.1416 −0.746504
\(962\) 0 0
\(963\) 40.1183 1.29279
\(964\) 27.1285 0.873750
\(965\) −0.139510 −0.00449100
\(966\) 0 0
\(967\) 16.9761 0.545915 0.272957 0.962026i \(-0.411998\pi\)
0.272957 + 0.962026i \(0.411998\pi\)
\(968\) −0.402359 −0.0129323
\(969\) −15.7267 −0.505213
\(970\) 66.1136 2.12278
\(971\) −6.86359 −0.220263 −0.110132 0.993917i \(-0.535127\pi\)
−0.110132 + 0.993917i \(0.535127\pi\)
\(972\) 21.7721 0.698339
\(973\) 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 \(0.502963\pi\)
\(978\) 12.2502 0.391719
\(979\) 34.8230 1.11295
\(980\) 0 0
\(981\) −21.7219 −0.693528
\(982\) 0.698983 0.0223054
\(983\) −34.7195 −1.10738 −0.553689 0.832723i \(-0.686780\pi\)
−0.553689 + 0.832723i \(0.686780\pi\)
\(984\) 2.85692 0.0910752
\(985\) 26.3810 0.840568
\(986\) 11.8097 0.376097
\(987\) 0 0
\(988\) 0 0
\(989\) −44.0861 −1.40186
\(990\) 35.6707 1.13369
\(991\) 24.1688 0.767748 0.383874 0.923385i \(-0.374590\pi\)
0.383874 + 0.923385i \(0.374590\pi\)
\(992\) 18.4267 0.585047
\(993\) 2.91665 0.0925572
\(994\) 0 0
\(995\) −61.1528 −1.93868
\(996\) −15.3264 −0.485636
\(997\) −35.1852 −1.11433 −0.557163 0.830403i \(-0.688110\pi\)
−0.557163 + 0.830403i \(0.688110\pi\)
\(998\) 63.3523 2.00538
\(999\) −45.2289 −1.43098
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 8281.2.a.cd.1.2 6
7.6 odd 2 8281.2.a.cc.1.2 6
13.12 even 2 637.2.a.n.1.5 yes 6
39.38 odd 2 5733.2.a.br.1.2 6
91.12 odd 6 637.2.e.o.508.2 12
91.25 even 6 637.2.e.n.79.2 12
91.38 odd 6 637.2.e.o.79.2 12
91.51 even 6 637.2.e.n.508.2 12
91.90 odd 2 637.2.a.m.1.5 6
273.272 even 2 5733.2.a.bu.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
637.2.a.m.1.5 6 91.90 odd 2
637.2.a.n.1.5 yes 6 13.12 even 2
637.2.e.n.79.2 12 91.25 even 6
637.2.e.n.508.2 12 91.51 even 6
637.2.e.o.79.2 12 91.38 odd 6
637.2.e.o.508.2 12 91.12 odd 6
5733.2.a.br.1.2 6 39.38 odd 2
5733.2.a.bu.1.2 6 273.272 even 2
8281.2.a.cc.1.2 6 7.6 odd 2
8281.2.a.cd.1.2 6 1.1 even 1 trivial