Properties

Label 637.2.a.m.1.5
Level $637$
Weight $2$
Character 637.1
Self dual yes
Analytic conductor $5.086$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.08647060876\)
Analytic rank: \(1\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-0.146243\) of defining polynomial
Character \(\chi\) \(=\) 637.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.83237 q^{2} -0.853757 q^{3} +1.35758 q^{4} -2.62555 q^{5} -1.56440 q^{6} -1.17715 q^{8} -2.27110 q^{9} +O(q^{10})\) \(q+1.83237 q^{2} -0.853757 q^{3} +1.35758 q^{4} -2.62555 q^{5} -1.56440 q^{6} -1.17715 q^{8} -2.27110 q^{9} -4.81098 q^{10} +3.26469 q^{11} -1.15904 q^{12} -1.00000 q^{13} +2.24158 q^{15} -4.87214 q^{16} -4.53021 q^{17} -4.16149 q^{18} -4.06615 q^{19} -3.56440 q^{20} +5.98212 q^{22} -4.53266 q^{23} +1.00500 q^{24} +1.89352 q^{25} -1.83237 q^{26} +4.50024 q^{27} -1.42268 q^{29} +4.10741 q^{30} +2.80328 q^{31} -6.57326 q^{32} -2.78725 q^{33} -8.30102 q^{34} -3.08320 q^{36} -10.0503 q^{37} -7.45070 q^{38} +0.853757 q^{39} +3.09067 q^{40} +2.84271 q^{41} +9.72632 q^{43} +4.43208 q^{44} +5.96289 q^{45} -8.30551 q^{46} +9.44956 q^{47} +4.15962 q^{48} +3.46963 q^{50} +3.86770 q^{51} -1.35758 q^{52} +5.26439 q^{53} +8.24610 q^{54} -8.57161 q^{55} +3.47151 q^{57} -2.60687 q^{58} +2.56791 q^{59} +3.04313 q^{60} +11.1830 q^{61} +5.13664 q^{62} -2.30037 q^{64} +2.62555 q^{65} -5.10728 q^{66} -1.98172 q^{67} -6.15013 q^{68} +3.86979 q^{69} -11.7544 q^{71} +2.67342 q^{72} -12.1391 q^{73} -18.4159 q^{74} -1.61661 q^{75} -5.52013 q^{76} +1.56440 q^{78} +11.9089 q^{79} +12.7920 q^{80} +2.97118 q^{81} +5.20889 q^{82} -13.2233 q^{83} +11.8943 q^{85} +17.8222 q^{86} +1.21462 q^{87} -3.84303 q^{88} -10.6666 q^{89} +10.9262 q^{90} -6.15345 q^{92} -2.39332 q^{93} +17.3151 q^{94} +10.6759 q^{95} +5.61197 q^{96} +13.7422 q^{97} -7.41443 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 8 q^{3} + 4 q^{4} - 6 q^{5} - 4 q^{6} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 8 q^{3} + 4 q^{4} - 6 q^{5} - 4 q^{6} + 6 q^{9} - 4 q^{10} + 4 q^{11} + 4 q^{12} - 6 q^{13} + 12 q^{15} - 16 q^{17} - 4 q^{18} - 2 q^{19} - 16 q^{20} - 12 q^{22} - 6 q^{23} - 12 q^{24} - 4 q^{25} - 20 q^{27} - 6 q^{29} - 6 q^{31} - 20 q^{32} - 4 q^{33} - 24 q^{36} - 8 q^{38} + 8 q^{39} - 4 q^{40} + 8 q^{41} + 2 q^{43} - 4 q^{44} - 14 q^{45} + 8 q^{46} - 30 q^{47} + 8 q^{48} + 8 q^{50} - 4 q^{51} - 4 q^{52} - 14 q^{53} + 48 q^{54} + 8 q^{55} + 4 q^{57} - 8 q^{58} - 24 q^{59} + 12 q^{60} - 28 q^{62} - 20 q^{64} + 6 q^{65} + 4 q^{66} + 16 q^{67} - 28 q^{68} + 20 q^{69} + 8 q^{71} + 28 q^{72} + 6 q^{73} - 12 q^{74} - 12 q^{75} + 16 q^{76} + 4 q^{78} - 22 q^{79} + 28 q^{80} + 46 q^{81} + 40 q^{82} - 50 q^{83} - 8 q^{85} - 16 q^{86} + 16 q^{87} - 44 q^{88} - 26 q^{89} + 40 q^{90} + 20 q^{92} + 16 q^{93} + 32 q^{94} - 6 q^{95} + 20 q^{96} + 14 q^{97} + 12 q^{99}+O(q^{100}) \) 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 \(-0.275672\pi\)
0.647841 + 0.761776i \(0.275672\pi\)
\(3\) −0.853757 −0.492917 −0.246458 0.969153i \(-0.579267\pi\)
−0.246458 + 0.969153i \(0.579267\pi\)
\(4\) 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 \(-0.336204\pi\)
0.492170 + 0.870499i \(0.336204\pi\)
\(12\) −1.15904 −0.334587
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 2.24158 0.578775
\(16\) −4.87214 −1.21803
\(17\) −4.53021 −1.09874 −0.549369 0.835580i \(-0.685132\pi\)
−0.549369 + 0.835580i \(0.685132\pi\)
\(18\) −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
\(26\) −1.83237 −0.359357
\(27\) 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 \(-0.809464\pi\)
−0.826133 + 0.563475i \(0.809464\pi\)
\(38\) −7.45070 −1.20866
\(39\) 0.853757 0.136711
\(40\) 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
\(52\) −1.35758 −0.188263
\(53\) 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 \(-0.246009\pi\)
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
\(66\) −5.10728 −0.628662
\(67\) −1.98172 −0.242106 −0.121053 0.992646i \(-0.538627\pi\)
−0.121053 + 0.992646i \(0.538627\pi\)
\(68\) −6.15013 −0.745813
\(69\) 3.86979 0.465868
\(70\) 0 0
\(71\) −11.7544 −1.39499 −0.697495 0.716590i \(-0.745702\pi\)
−0.697495 + 0.716590i \(0.745702\pi\)
\(72\) 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
\(78\) 1.56440 0.177133
\(79\) 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 \(-0.594742\pi\)
−0.293266 + 0.956031i \(0.594742\pi\)
\(102\) 7.08706 0.701723
\(103\) −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 \(-0.825742\pi\)
−0.853856 + 0.520509i \(0.825742\pi\)
\(108\) 6.10944 0.587881
\(109\) −9.56450 −0.916113 −0.458057 0.888923i \(-0.651454\pi\)
−0.458057 + 0.888923i \(0.651454\pi\)
\(110\) −15.7064 −1.49754
\(111\) 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
\(117\) 2.27110 0.209963
\(118\) 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
\(130\) 4.81098 0.421951
\(131\) −4.30279 −0.375937 −0.187968 0.982175i \(-0.560190\pi\)
−0.187968 + 0.982175i \(0.560190\pi\)
\(132\) −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 \(-0.362174\pi\)
0.419590 + 0.907714i \(0.362174\pi\)
\(138\) 7.09089 0.603617
\(139\) −10.0811 −0.855070 −0.427535 0.903999i \(-0.640618\pi\)
−0.427535 + 0.903999i \(0.640618\pi\)
\(140\) 0 0
\(141\) −8.06763 −0.679417
\(142\) −21.5384 −1.80746
\(143\) −3.26469 −0.273007
\(144\) 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 \(-0.691424\pi\)
−0.565777 + 0.824558i \(0.691424\pi\)
\(150\) −2.96223 −0.241865
\(151\) 21.2123 1.72623 0.863117 0.505004i \(-0.168509\pi\)
0.863117 + 0.505004i \(0.168509\pi\)
\(152\) 4.78647 0.388234
\(153\) 10.2886 0.831780
\(154\) 0 0
\(155\) −7.36015 −0.591182
\(156\) 1.15904 0.0927978
\(157\) −23.5155 −1.87674 −0.938372 0.345627i \(-0.887666\pi\)
−0.938372 + 0.345627i \(0.887666\pi\)
\(158\) 21.8215 1.73603
\(159\) −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 \(-0.400785\pi\)
0.306671 + 0.951816i \(0.400785\pi\)
\(164\) 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
\(169\) 1.00000 0.0769231
\(170\) 21.7948 1.67158
\(171\) 9.23463 0.706190
\(172\) 13.2043 1.00682
\(173\) −5.24907 −0.399079 −0.199540 0.979890i \(-0.563945\pi\)
−0.199540 + 0.979890i \(0.563945\pi\)
\(174\) 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 \(-0.253470\pi\)
0.699356 + 0.714774i \(0.253470\pi\)
\(182\) 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 \(-0.500609\pi\)
−0.00191239 + 0.999998i \(0.500609\pi\)
\(194\) 25.1808 1.80788
\(195\) −2.24158 −0.160523
\(196\) 0 0
\(197\) 10.0478 0.715875 0.357938 0.933745i \(-0.383480\pi\)
0.357938 + 0.933745i \(0.383480\pi\)
\(198\) −13.5860 −0.965514
\(199\) −23.2914 −1.65109 −0.825543 0.564340i \(-0.809131\pi\)
−0.825543 + 0.564340i \(0.809131\pi\)
\(200\) −2.22896 −0.157611
\(201\) 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
\(208\) 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
\(213\) 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
\(221\) 4.53021 0.304735
\(222\) 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 \(0.251542\pi\)
\(234\) 4.16149 0.272045
\(235\) −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 \(-0.488818\pi\)
0.0351207 + 0.999383i \(0.488818\pi\)
\(240\) −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
\(247\) 4.06615 0.258723
\(248\) −3.29988 −0.209542
\(249\) 11.2895 0.715443
\(250\) 14.9452 0.945218
\(251\) −20.1497 −1.27184 −0.635920 0.771755i \(-0.719379\pi\)
−0.635920 + 0.771755i \(0.719379\pi\)
\(252\) 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 \(-0.344838\pi\)
0.468379 + 0.883527i \(0.344838\pi\)
\(258\) −15.2158 −0.947297
\(259\) 0 0
\(260\) 3.56440 0.221055
\(261\) 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 \(-0.460063\pi\)
0.125137 + 0.992139i \(0.460063\pi\)
\(270\) −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 \(-0.313695\pi\)
0.552445 + 0.833550i \(0.313695\pi\)
\(282\) −14.7829 −0.880308
\(283\) 1.61158 0.0957983 0.0478991 0.998852i \(-0.484747\pi\)
0.0478991 + 0.998852i \(0.484747\pi\)
\(284\) −15.9575 −0.946906
\(285\) −9.11462 −0.539904
\(286\) −5.98212 −0.353730
\(287\) 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
\(299\) 4.53266 0.262130
\(300\) −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 \(-0.632778\pi\)
−0.405142 + 0.914254i \(0.632778\pi\)
\(312\) −1.00500 −0.0568969
\(313\) −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 \(-0.403308\pi\)
0.299116 + 0.954217i \(0.403308\pi\)
\(318\) −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
\(325\) −1.89352 −0.105034
\(326\) 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 \(-0.529929\pi\)
−0.0938872 + 0.995583i \(0.529929\pi\)
\(332\) −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 \(0.262047\pi\)
\(338\) 1.83237 0.0996678
\(339\) 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
\(351\) −4.50024 −0.240205
\(352\) −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 \(-0.159722\pi\)
0.876726 + 0.480989i \(0.159722\pi\)
\(360\) −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 \(-0.235363\pi\)
0.738862 + 0.673857i \(0.235363\pi\)
\(368\) 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
\(377\) 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
\(381\) 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 \(0.758958\pi\)
\(390\) −4.10741 −0.207987
\(391\) 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 \(-0.616187\pi\)
−0.356961 + 0.934119i \(0.616187\pi\)
\(402\) 3.10021 0.154624
\(403\) −2.80328 −0.139641
\(404\) −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
\(416\) 6.57326 0.322281
\(417\) 8.60684 0.421479
\(418\) −24.3242 −1.18974
\(419\) −18.7999 −0.918433 −0.459216 0.888324i \(-0.651870\pi\)
−0.459216 + 0.888324i \(0.651870\pi\)
\(420\) 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
\(423\) −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
\(429\) 2.78725 0.134570
\(430\) −46.7931 −2.25657
\(431\) 20.5583 0.990256 0.495128 0.868820i \(-0.335121\pi\)
0.495128 + 0.868820i \(0.335121\pi\)
\(432\) −21.9258 −1.05490
\(433\) −18.9235 −0.909404 −0.454702 0.890644i \(-0.650254\pi\)
−0.454702 + 0.890644i \(0.650254\pi\)
\(434\) 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 \(-0.386276\pi\)
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 \(-0.591695\pi\)
−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 \(-0.517453\pi\)
−0.0548015 + 0.998497i \(0.517453\pi\)
\(450\) −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 \(-0.643103\pi\)
−0.434579 + 0.900634i \(0.643103\pi\)
\(458\) 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 \(-0.527138\pi\)
−0.0851547 + 0.996368i \(0.527138\pi\)
\(464\) 6.93147 0.321786
\(465\) 6.28378 0.291403
\(466\) 39.3634 1.82347
\(467\) −38.7532 −1.79328 −0.896641 0.442757i \(-0.854000\pi\)
−0.896641 + 0.442757i \(0.854000\pi\)
\(468\) 3.08320 0.142521
\(469\) 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
\(481\) 10.0503 0.458256
\(482\) 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 \(-0.527458\pi\)
−0.0861537 + 0.996282i \(0.527458\pi\)
\(488\) −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
\(494\) 7.45070 0.335223
\(495\) 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 \(-0.218318\pi\)
0.773871 + 0.633343i \(0.218318\pi\)
\(500\) 11.0727 0.495187
\(501\) 10.8526 0.484860
\(502\) −36.9218 −1.64790
\(503\) 2.41090 0.107497 0.0537485 0.998555i \(-0.482883\pi\)
0.0537485 + 0.998555i \(0.482883\pi\)
\(504\) 0 0
\(505\) 15.4765 0.688696
\(506\) −27.1149 −1.20540
\(507\) −0.853757 −0.0379167
\(508\) −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
\(520\) −3.09067 −0.135535
\(521\) 30.1450 1.32068 0.660338 0.750968i \(-0.270413\pi\)
0.660338 + 0.750968i \(0.270413\pi\)
\(522\) 5.92046 0.259131
\(523\) 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
\(528\) 13.5799 0.590988
\(529\) −2.45500 −0.106739
\(530\) −25.3269 −1.10013
\(531\) −5.83197 −0.253086
\(532\) 0 0
\(533\) −2.84271 −0.123131
\(534\) 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 \(-0.471985\pi\)
0.0878977 + 0.996130i \(0.471985\pi\)
\(542\) −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 \(-0.362444\pi\)
0.418820 + 0.908069i \(0.362444\pi\)
\(558\) −11.6658 −0.493854
\(559\) −9.72632 −0.411379
\(560\) 0 0
\(561\) 12.6268 0.533105
\(562\) 33.9379 1.43158
\(563\) −8.22392 −0.346597 −0.173298 0.984869i \(-0.555443\pi\)
−0.173298 + 0.984869i \(0.555443\pi\)
\(564\) −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 \(0.517247\pi\)
\(572\) −4.43208 −0.185315
\(573\) −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
\(585\) −5.96289 −0.246535
\(586\) 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
\(598\) 8.30551 0.339638
\(599\) 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 \(-0.282756\pi\)
0.630729 + 0.776003i \(0.282756\pi\)
\(602\) 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 \(-0.515556\pi\)
−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
\(612\) 13.9675 0.564605
\(613\) −27.7742 −1.12179 −0.560895 0.827887i \(-0.689543\pi\)
−0.560895 + 0.827887i \(0.689543\pi\)
\(614\) 10.5976 0.427683
\(615\) 6.37216 0.256950
\(616\) 0 0
\(617\) 26.2125 1.05527 0.527637 0.849470i \(-0.323078\pi\)
0.527637 + 0.849470i \(0.323078\pi\)
\(618\) 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
\(624\) −4.15962 −0.166518
\(625\) −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 \(-0.214958\pi\)
0.780513 + 0.625140i \(0.214958\pi\)
\(632\) −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 \(-0.537193\pi\)
−0.116581 + 0.993181i \(0.537193\pi\)
\(648\) −3.49753 −0.137396
\(649\) 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 \(-0.619937\pi\)
−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
\(663\) −3.86770 −0.150209
\(664\) 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
\(676\) 1.35758 0.0522147
\(677\) −20.2075 −0.776637 −0.388318 0.921525i \(-0.626944\pi\)
−0.388318 + 0.921525i \(0.626944\pi\)
\(678\) 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 \(-0.707043\pi\)
−0.605541 + 0.795814i \(0.707043\pi\)
\(684\) 12.5368 0.479355
\(685\) −25.7891 −0.985350
\(686\) 0 0
\(687\) −8.74092 −0.333487
\(688\) −47.3879 −1.80665
\(689\) −5.26439 −0.200557
\(690\) −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 \(0.530727\pi\)
\(702\) −8.24610 −0.311229
\(703\) 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 \(-0.336654\pi\)
0.490939 + 0.871194i \(0.336654\pi\)
\(710\) 56.5502 2.12229
\(711\) −27.0463 −1.01431
\(712\) 12.5561 0.470561
\(713\) −12.7063 −0.475855
\(714\) 0 0
\(715\) 8.57161 0.320560
\(716\) −14.2892 −0.534014
\(717\) −0.927100 −0.0346232
\(718\) 60.8772 2.27192
\(719\) 8.72884 0.325531 0.162765 0.986665i \(-0.447959\pi\)
0.162765 + 0.986665i \(0.447959\pi\)
\(720\) −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 \(-0.367070\pi\)
0.405579 + 0.914060i \(0.367070\pi\)
\(728\) 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 \(-0.734196\pi\)
−0.671142 + 0.741329i \(0.734196\pi\)
\(740\) 35.8234 1.31690
\(741\) −3.47151 −0.127529
\(742\) 0 0
\(743\) 12.4588 0.457071 0.228535 0.973536i \(-0.426606\pi\)
0.228535 + 0.973536i \(0.426606\pi\)
\(744\) 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
\(754\) 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
\(759\) 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
\(767\) −2.56791 −0.0927217
\(768\) −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
\(780\) −3.04313 −0.108962
\(781\) −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
\(793\) −11.1830 −0.397119
\(794\) 52.4811 1.86248
\(795\) 11.8006 0.418523
\(796\) −31.6200 −1.12074
\(797\) 11.6121 0.411323 0.205661 0.978623i \(-0.434065\pi\)
0.205661 + 0.978623i \(0.434065\pi\)
\(798\) 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
\(806\) −5.13664 −0.180931
\(807\) −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 \(-0.493735\pi\)
0.0196817 + 0.999806i \(0.493735\pi\)
\(822\) −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 \(-0.535064\pi\)
−0.109935 + 0.993939i \(0.535064\pi\)
\(828\) 13.9751 0.485668
\(829\) 44.3704 1.54105 0.770523 0.637412i \(-0.219995\pi\)
0.770523 + 0.637412i \(0.219995\pi\)
\(830\) 63.6171 2.20818
\(831\) 0.308216 0.0106919
\(832\) 2.30037 0.0797510
\(833\) 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
\(845\) −2.62555 −0.0903217
\(846\) −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 \(-0.554769\pi\)
−0.171213 + 0.985234i \(0.554769\pi\)
\(858\) 5.10728 0.174360
\(859\) −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 \(-0.383640\pi\)
0.357469 + 0.933925i \(0.383640\pi\)
\(864\) −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
\(871\) 1.98172 0.0671481
\(872\) 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 \(-0.459243\pi\)
0.127692 + 0.991814i \(0.459243\pi\)
\(878\) 26.8534 0.906258
\(879\) −3.77080 −0.127186
\(880\) 41.7620 1.40780
\(881\) 34.3550 1.15745 0.578725 0.815522i \(-0.303550\pi\)
0.578725 + 0.815522i \(0.303550\pi\)
\(882\) 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
\(885\) 5.75617 0.193492
\(886\) −21.9138 −0.736207
\(887\) 48.0325 1.61277 0.806386 0.591389i \(-0.201420\pi\)
0.806386 + 0.591389i \(0.201420\pi\)
\(888\) −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
\(897\) −3.86979 −0.129209
\(898\) −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
\(923\) 11.7544 0.386901
\(924\) 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
\(936\) −2.67342 −0.0873835
\(937\) −2.59416 −0.0847476 −0.0423738 0.999102i \(-0.513492\pi\)
−0.0423738 + 0.999102i \(0.513492\pi\)
\(938\) 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 \(-0.538888\pi\)
−0.121868 + 0.992546i \(0.538888\pi\)
\(948\) −13.8029 −0.448299
\(949\) 12.1391 0.394052
\(950\) −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
\(962\) 18.4159 0.593754
\(963\) 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 \(-0.588002\pi\)
−0.272957 + 0.962026i \(0.588002\pi\)
\(968\) 0.402359 0.0129323
\(969\) −15.7267 −0.505213
\(970\) −66.1136 −2.12278
\(971\) 6.86359 0.220263 0.110132 0.993917i \(-0.464873\pi\)
0.110132 + 0.993917i \(0.464873\pi\)
\(972\) −21.7721 −0.698339
\(973\) 0 0
\(974\) −6.96757 −0.223255
\(975\) 1.61661 0.0517729
\(976\) −54.4850 −1.74402
\(977\) 0.581913 0.0186171 0.00930853 0.999957i \(-0.497037\pi\)
0.00930853 + 0.999957i \(0.497037\pi\)
\(978\) −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
\(988\) 5.52013 0.175619
\(989\) −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 \(-0.311890\pi\)
0.557163 + 0.830403i \(0.311890\pi\)
\(998\) 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 637.2.a.m.1.5 6
3.2 odd 2 5733.2.a.bu.1.2 6
7.2 even 3 637.2.e.o.508.2 12
7.3 odd 6 637.2.e.n.79.2 12
7.4 even 3 637.2.e.o.79.2 12
7.5 odd 6 637.2.e.n.508.2 12
7.6 odd 2 637.2.a.n.1.5 yes 6
13.12 even 2 8281.2.a.cc.1.2 6
21.20 even 2 5733.2.a.br.1.2 6
91.90 odd 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 1.1 even 1 trivial
637.2.a.n.1.5 yes 6 7.6 odd 2
637.2.e.n.79.2 12 7.3 odd 6
637.2.e.n.508.2 12 7.5 odd 6
637.2.e.o.79.2 12 7.4 even 3
637.2.e.o.508.2 12 7.2 even 3
5733.2.a.br.1.2 6 21.20 even 2
5733.2.a.bu.1.2 6 3.2 odd 2
8281.2.a.cc.1.2 6 13.12 even 2
8281.2.a.cd.1.2 6 91.90 odd 2