Properties

Label 177.2.a.b.1.2
Level $177$
Weight $2$
Character 177.1
Self dual yes
Analytic conductor $1.413$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.41335211578\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} -2.23607 q^{5} -0.618034 q^{6} -2.38197 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} -2.23607 q^{5} -0.618034 q^{6} -2.38197 q^{7} -2.23607 q^{8} +1.00000 q^{9} -1.38197 q^{10} +2.23607 q^{11} +1.61803 q^{12} -6.23607 q^{13} -1.47214 q^{14} +2.23607 q^{15} +1.85410 q^{16} +1.85410 q^{17} +0.618034 q^{18} +3.09017 q^{19} +3.61803 q^{20} +2.38197 q^{21} +1.38197 q^{22} -4.61803 q^{23} +2.23607 q^{24} -3.85410 q^{26} -1.00000 q^{27} +3.85410 q^{28} +6.38197 q^{29} +1.38197 q^{30} -10.5623 q^{31} +5.61803 q^{32} -2.23607 q^{33} +1.14590 q^{34} +5.32624 q^{35} -1.61803 q^{36} -0.145898 q^{37} +1.90983 q^{38} +6.23607 q^{39} +5.00000 q^{40} +8.09017 q^{41} +1.47214 q^{42} -8.70820 q^{43} -3.61803 q^{44} -2.23607 q^{45} -2.85410 q^{46} -10.8541 q^{47} -1.85410 q^{48} -1.32624 q^{49} -1.85410 q^{51} +10.0902 q^{52} +6.23607 q^{53} -0.618034 q^{54} -5.00000 q^{55} +5.32624 q^{56} -3.09017 q^{57} +3.94427 q^{58} -1.00000 q^{59} -3.61803 q^{60} -3.14590 q^{61} -6.52786 q^{62} -2.38197 q^{63} -0.236068 q^{64} +13.9443 q^{65} -1.38197 q^{66} +10.7082 q^{67} -3.00000 q^{68} +4.61803 q^{69} +3.29180 q^{70} -7.94427 q^{71} -2.23607 q^{72} +0.854102 q^{73} -0.0901699 q^{74} -5.00000 q^{76} -5.32624 q^{77} +3.85410 q^{78} -3.00000 q^{79} -4.14590 q^{80} +1.00000 q^{81} +5.00000 q^{82} -1.61803 q^{83} -3.85410 q^{84} -4.14590 q^{85} -5.38197 q^{86} -6.38197 q^{87} -5.00000 q^{88} -13.7984 q^{89} -1.38197 q^{90} +14.8541 q^{91} +7.47214 q^{92} +10.5623 q^{93} -6.70820 q^{94} -6.90983 q^{95} -5.61803 q^{96} +3.00000 q^{97} -0.819660 q^{98} +2.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - 2q^{3} - q^{4} + q^{6} - 7q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - q^{2} - 2q^{3} - q^{4} + q^{6} - 7q^{7} + 2q^{9} - 5q^{10} + q^{12} - 8q^{13} + 6q^{14} - 3q^{16} - 3q^{17} - q^{18} - 5q^{19} + 5q^{20} + 7q^{21} + 5q^{22} - 7q^{23} - q^{26} - 2q^{27} + q^{28} + 15q^{29} + 5q^{30} - q^{31} + 9q^{32} + 9q^{34} - 5q^{35} - q^{36} - 7q^{37} + 15q^{38} + 8q^{39} + 10q^{40} + 5q^{41} - 6q^{42} - 4q^{43} - 5q^{44} + q^{46} - 15q^{47} + 3q^{48} + 13q^{49} + 3q^{51} + 9q^{52} + 8q^{53} + q^{54} - 10q^{55} - 5q^{56} + 5q^{57} - 10q^{58} - 2q^{59} - 5q^{60} - 13q^{61} - 22q^{62} - 7q^{63} + 4q^{64} + 10q^{65} - 5q^{66} + 8q^{67} - 6q^{68} + 7q^{69} + 20q^{70} + 2q^{71} - 5q^{73} + 11q^{74} - 10q^{76} + 5q^{77} + q^{78} - 6q^{79} - 15q^{80} + 2q^{81} + 10q^{82} - q^{83} - q^{84} - 15q^{85} - 13q^{86} - 15q^{87} - 10q^{88} - 3q^{89} - 5q^{90} + 23q^{91} + 6q^{92} + q^{93} - 25q^{95} - 9q^{96} + 6q^{97} - 24q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.618034 0.437016 0.218508 0.975835i \(-0.429881\pi\)
0.218508 + 0.975835i \(0.429881\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.61803 −0.809017
\(5\) −2.23607 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(6\) −0.618034 −0.252311
\(7\) −2.38197 −0.900299 −0.450149 0.892953i \(-0.648629\pi\)
−0.450149 + 0.892953i \(0.648629\pi\)
\(8\) −2.23607 −0.790569
\(9\) 1.00000 0.333333
\(10\) −1.38197 −0.437016
\(11\) 2.23607 0.674200 0.337100 0.941469i \(-0.390554\pi\)
0.337100 + 0.941469i \(0.390554\pi\)
\(12\) 1.61803 0.467086
\(13\) −6.23607 −1.72957 −0.864787 0.502139i \(-0.832547\pi\)
−0.864787 + 0.502139i \(0.832547\pi\)
\(14\) −1.47214 −0.393445
\(15\) 2.23607 0.577350
\(16\) 1.85410 0.463525
\(17\) 1.85410 0.449686 0.224843 0.974395i \(-0.427813\pi\)
0.224843 + 0.974395i \(0.427813\pi\)
\(18\) 0.618034 0.145672
\(19\) 3.09017 0.708934 0.354467 0.935069i \(-0.384662\pi\)
0.354467 + 0.935069i \(0.384662\pi\)
\(20\) 3.61803 0.809017
\(21\) 2.38197 0.519788
\(22\) 1.38197 0.294636
\(23\) −4.61803 −0.962927 −0.481463 0.876466i \(-0.659895\pi\)
−0.481463 + 0.876466i \(0.659895\pi\)
\(24\) 2.23607 0.456435
\(25\) 0 0
\(26\) −3.85410 −0.755852
\(27\) −1.00000 −0.192450
\(28\) 3.85410 0.728357
\(29\) 6.38197 1.18510 0.592551 0.805533i \(-0.298121\pi\)
0.592551 + 0.805533i \(0.298121\pi\)
\(30\) 1.38197 0.252311
\(31\) −10.5623 −1.89705 −0.948523 0.316708i \(-0.897422\pi\)
−0.948523 + 0.316708i \(0.897422\pi\)
\(32\) 5.61803 0.993137
\(33\) −2.23607 −0.389249
\(34\) 1.14590 0.196520
\(35\) 5.32624 0.900299
\(36\) −1.61803 −0.269672
\(37\) −0.145898 −0.0239855 −0.0119927 0.999928i \(-0.503818\pi\)
−0.0119927 + 0.999928i \(0.503818\pi\)
\(38\) 1.90983 0.309815
\(39\) 6.23607 0.998570
\(40\) 5.00000 0.790569
\(41\) 8.09017 1.26347 0.631736 0.775183i \(-0.282343\pi\)
0.631736 + 0.775183i \(0.282343\pi\)
\(42\) 1.47214 0.227156
\(43\) −8.70820 −1.32799 −0.663994 0.747738i \(-0.731140\pi\)
−0.663994 + 0.747738i \(0.731140\pi\)
\(44\) −3.61803 −0.545439
\(45\) −2.23607 −0.333333
\(46\) −2.85410 −0.420814
\(47\) −10.8541 −1.58323 −0.791617 0.611018i \(-0.790760\pi\)
−0.791617 + 0.611018i \(0.790760\pi\)
\(48\) −1.85410 −0.267617
\(49\) −1.32624 −0.189463
\(50\) 0 0
\(51\) −1.85410 −0.259626
\(52\) 10.0902 1.39925
\(53\) 6.23607 0.856590 0.428295 0.903639i \(-0.359114\pi\)
0.428295 + 0.903639i \(0.359114\pi\)
\(54\) −0.618034 −0.0841038
\(55\) −5.00000 −0.674200
\(56\) 5.32624 0.711748
\(57\) −3.09017 −0.409303
\(58\) 3.94427 0.517908
\(59\) −1.00000 −0.130189
\(60\) −3.61803 −0.467086
\(61\) −3.14590 −0.402791 −0.201395 0.979510i \(-0.564548\pi\)
−0.201395 + 0.979510i \(0.564548\pi\)
\(62\) −6.52786 −0.829040
\(63\) −2.38197 −0.300100
\(64\) −0.236068 −0.0295085
\(65\) 13.9443 1.72957
\(66\) −1.38197 −0.170108
\(67\) 10.7082 1.30822 0.654108 0.756401i \(-0.273044\pi\)
0.654108 + 0.756401i \(0.273044\pi\)
\(68\) −3.00000 −0.363803
\(69\) 4.61803 0.555946
\(70\) 3.29180 0.393445
\(71\) −7.94427 −0.942812 −0.471406 0.881916i \(-0.656253\pi\)
−0.471406 + 0.881916i \(0.656253\pi\)
\(72\) −2.23607 −0.263523
\(73\) 0.854102 0.0999651 0.0499825 0.998750i \(-0.484083\pi\)
0.0499825 + 0.998750i \(0.484083\pi\)
\(74\) −0.0901699 −0.0104820
\(75\) 0 0
\(76\) −5.00000 −0.573539
\(77\) −5.32624 −0.606981
\(78\) 3.85410 0.436391
\(79\) −3.00000 −0.337526 −0.168763 0.985657i \(-0.553977\pi\)
−0.168763 + 0.985657i \(0.553977\pi\)
\(80\) −4.14590 −0.463525
\(81\) 1.00000 0.111111
\(82\) 5.00000 0.552158
\(83\) −1.61803 −0.177602 −0.0888012 0.996049i \(-0.528304\pi\)
−0.0888012 + 0.996049i \(0.528304\pi\)
\(84\) −3.85410 −0.420517
\(85\) −4.14590 −0.449686
\(86\) −5.38197 −0.580352
\(87\) −6.38197 −0.684219
\(88\) −5.00000 −0.533002
\(89\) −13.7984 −1.46262 −0.731312 0.682043i \(-0.761092\pi\)
−0.731312 + 0.682043i \(0.761092\pi\)
\(90\) −1.38197 −0.145672
\(91\) 14.8541 1.55713
\(92\) 7.47214 0.779024
\(93\) 10.5623 1.09526
\(94\) −6.70820 −0.691898
\(95\) −6.90983 −0.708934
\(96\) −5.61803 −0.573388
\(97\) 3.00000 0.304604 0.152302 0.988334i \(-0.451331\pi\)
0.152302 + 0.988334i \(0.451331\pi\)
\(98\) −0.819660 −0.0827982
\(99\) 2.23607 0.224733
\(100\) 0 0
\(101\) 3.70820 0.368980 0.184490 0.982834i \(-0.440937\pi\)
0.184490 + 0.982834i \(0.440937\pi\)
\(102\) −1.14590 −0.113461
\(103\) −3.23607 −0.318859 −0.159430 0.987209i \(-0.550966\pi\)
−0.159430 + 0.987209i \(0.550966\pi\)
\(104\) 13.9443 1.36735
\(105\) −5.32624 −0.519788
\(106\) 3.85410 0.374343
\(107\) 0.909830 0.0879566 0.0439783 0.999032i \(-0.485997\pi\)
0.0439783 + 0.999032i \(0.485997\pi\)
\(108\) 1.61803 0.155695
\(109\) −4.14590 −0.397105 −0.198553 0.980090i \(-0.563624\pi\)
−0.198553 + 0.980090i \(0.563624\pi\)
\(110\) −3.09017 −0.294636
\(111\) 0.145898 0.0138480
\(112\) −4.41641 −0.417311
\(113\) −9.00000 −0.846649 −0.423324 0.905978i \(-0.639137\pi\)
−0.423324 + 0.905978i \(0.639137\pi\)
\(114\) −1.90983 −0.178872
\(115\) 10.3262 0.962927
\(116\) −10.3262 −0.958767
\(117\) −6.23607 −0.576525
\(118\) −0.618034 −0.0568946
\(119\) −4.41641 −0.404851
\(120\) −5.00000 −0.456435
\(121\) −6.00000 −0.545455
\(122\) −1.94427 −0.176026
\(123\) −8.09017 −0.729466
\(124\) 17.0902 1.53474
\(125\) 11.1803 1.00000
\(126\) −1.47214 −0.131148
\(127\) 15.9443 1.41483 0.707413 0.706801i \(-0.249862\pi\)
0.707413 + 0.706801i \(0.249862\pi\)
\(128\) −11.3820 −1.00603
\(129\) 8.70820 0.766715
\(130\) 8.61803 0.755852
\(131\) −20.6525 −1.80442 −0.902208 0.431302i \(-0.858054\pi\)
−0.902208 + 0.431302i \(0.858054\pi\)
\(132\) 3.61803 0.314909
\(133\) −7.36068 −0.638252
\(134\) 6.61803 0.571711
\(135\) 2.23607 0.192450
\(136\) −4.14590 −0.355508
\(137\) 12.2361 1.04540 0.522699 0.852517i \(-0.324925\pi\)
0.522699 + 0.852517i \(0.324925\pi\)
\(138\) 2.85410 0.242957
\(139\) 11.7639 0.997804 0.498902 0.866658i \(-0.333737\pi\)
0.498902 + 0.866658i \(0.333737\pi\)
\(140\) −8.61803 −0.728357
\(141\) 10.8541 0.914080
\(142\) −4.90983 −0.412024
\(143\) −13.9443 −1.16608
\(144\) 1.85410 0.154508
\(145\) −14.2705 −1.18510
\(146\) 0.527864 0.0436863
\(147\) 1.32624 0.109386
\(148\) 0.236068 0.0194047
\(149\) 19.0902 1.56393 0.781964 0.623324i \(-0.214218\pi\)
0.781964 + 0.623324i \(0.214218\pi\)
\(150\) 0 0
\(151\) 2.56231 0.208517 0.104259 0.994550i \(-0.466753\pi\)
0.104259 + 0.994550i \(0.466753\pi\)
\(152\) −6.90983 −0.560461
\(153\) 1.85410 0.149895
\(154\) −3.29180 −0.265260
\(155\) 23.6180 1.89705
\(156\) −10.0902 −0.807860
\(157\) 9.00000 0.718278 0.359139 0.933284i \(-0.383070\pi\)
0.359139 + 0.933284i \(0.383070\pi\)
\(158\) −1.85410 −0.147504
\(159\) −6.23607 −0.494552
\(160\) −12.5623 −0.993137
\(161\) 11.0000 0.866921
\(162\) 0.618034 0.0485573
\(163\) −18.5623 −1.45391 −0.726956 0.686684i \(-0.759066\pi\)
−0.726956 + 0.686684i \(0.759066\pi\)
\(164\) −13.0902 −1.02217
\(165\) 5.00000 0.389249
\(166\) −1.00000 −0.0776151
\(167\) 7.03444 0.544341 0.272171 0.962249i \(-0.412258\pi\)
0.272171 + 0.962249i \(0.412258\pi\)
\(168\) −5.32624 −0.410928
\(169\) 25.8885 1.99143
\(170\) −2.56231 −0.196520
\(171\) 3.09017 0.236311
\(172\) 14.0902 1.07437
\(173\) 12.3820 0.941383 0.470692 0.882298i \(-0.344004\pi\)
0.470692 + 0.882298i \(0.344004\pi\)
\(174\) −3.94427 −0.299014
\(175\) 0 0
\(176\) 4.14590 0.312509
\(177\) 1.00000 0.0751646
\(178\) −8.52786 −0.639190
\(179\) −0.527864 −0.0394544 −0.0197272 0.999805i \(-0.506280\pi\)
−0.0197272 + 0.999805i \(0.506280\pi\)
\(180\) 3.61803 0.269672
\(181\) 22.2705 1.65535 0.827677 0.561205i \(-0.189662\pi\)
0.827677 + 0.561205i \(0.189662\pi\)
\(182\) 9.18034 0.680492
\(183\) 3.14590 0.232551
\(184\) 10.3262 0.761260
\(185\) 0.326238 0.0239855
\(186\) 6.52786 0.478646
\(187\) 4.14590 0.303178
\(188\) 17.5623 1.28086
\(189\) 2.38197 0.173263
\(190\) −4.27051 −0.309815
\(191\) 16.4164 1.18785 0.593925 0.804521i \(-0.297578\pi\)
0.593925 + 0.804521i \(0.297578\pi\)
\(192\) 0.236068 0.0170367
\(193\) 8.00000 0.575853 0.287926 0.957653i \(-0.407034\pi\)
0.287926 + 0.957653i \(0.407034\pi\)
\(194\) 1.85410 0.133117
\(195\) −13.9443 −0.998570
\(196\) 2.14590 0.153278
\(197\) −20.6525 −1.47143 −0.735714 0.677292i \(-0.763153\pi\)
−0.735714 + 0.677292i \(0.763153\pi\)
\(198\) 1.38197 0.0982120
\(199\) −16.5623 −1.17407 −0.587035 0.809561i \(-0.699705\pi\)
−0.587035 + 0.809561i \(0.699705\pi\)
\(200\) 0 0
\(201\) −10.7082 −0.755298
\(202\) 2.29180 0.161250
\(203\) −15.2016 −1.06694
\(204\) 3.00000 0.210042
\(205\) −18.0902 −1.26347
\(206\) −2.00000 −0.139347
\(207\) −4.61803 −0.320976
\(208\) −11.5623 −0.801702
\(209\) 6.90983 0.477963
\(210\) −3.29180 −0.227156
\(211\) −2.14590 −0.147730 −0.0738649 0.997268i \(-0.523533\pi\)
−0.0738649 + 0.997268i \(0.523533\pi\)
\(212\) −10.0902 −0.692996
\(213\) 7.94427 0.544333
\(214\) 0.562306 0.0384384
\(215\) 19.4721 1.32799
\(216\) 2.23607 0.152145
\(217\) 25.1591 1.70791
\(218\) −2.56231 −0.173541
\(219\) −0.854102 −0.0577149
\(220\) 8.09017 0.545439
\(221\) −11.5623 −0.777765
\(222\) 0.0901699 0.00605181
\(223\) −9.52786 −0.638033 −0.319016 0.947749i \(-0.603353\pi\)
−0.319016 + 0.947749i \(0.603353\pi\)
\(224\) −13.3820 −0.894120
\(225\) 0 0
\(226\) −5.56231 −0.369999
\(227\) −22.1459 −1.46987 −0.734937 0.678135i \(-0.762789\pi\)
−0.734937 + 0.678135i \(0.762789\pi\)
\(228\) 5.00000 0.331133
\(229\) −9.14590 −0.604378 −0.302189 0.953248i \(-0.597717\pi\)
−0.302189 + 0.953248i \(0.597717\pi\)
\(230\) 6.38197 0.420814
\(231\) 5.32624 0.350441
\(232\) −14.2705 −0.936905
\(233\) −8.29180 −0.543214 −0.271607 0.962408i \(-0.587555\pi\)
−0.271607 + 0.962408i \(0.587555\pi\)
\(234\) −3.85410 −0.251951
\(235\) 24.2705 1.58323
\(236\) 1.61803 0.105325
\(237\) 3.00000 0.194871
\(238\) −2.72949 −0.176927
\(239\) −1.47214 −0.0952246 −0.0476123 0.998866i \(-0.515161\pi\)
−0.0476123 + 0.998866i \(0.515161\pi\)
\(240\) 4.14590 0.267617
\(241\) −23.4164 −1.50838 −0.754192 0.656654i \(-0.771971\pi\)
−0.754192 + 0.656654i \(0.771971\pi\)
\(242\) −3.70820 −0.238372
\(243\) −1.00000 −0.0641500
\(244\) 5.09017 0.325865
\(245\) 2.96556 0.189463
\(246\) −5.00000 −0.318788
\(247\) −19.2705 −1.22615
\(248\) 23.6180 1.49975
\(249\) 1.61803 0.102539
\(250\) 6.90983 0.437016
\(251\) 15.1803 0.958175 0.479087 0.877767i \(-0.340968\pi\)
0.479087 + 0.877767i \(0.340968\pi\)
\(252\) 3.85410 0.242786
\(253\) −10.3262 −0.649205
\(254\) 9.85410 0.618301
\(255\) 4.14590 0.259626
\(256\) −6.56231 −0.410144
\(257\) −19.4164 −1.21116 −0.605581 0.795784i \(-0.707059\pi\)
−0.605581 + 0.795784i \(0.707059\pi\)
\(258\) 5.38197 0.335067
\(259\) 0.347524 0.0215941
\(260\) −22.5623 −1.39925
\(261\) 6.38197 0.395034
\(262\) −12.7639 −0.788558
\(263\) 1.61803 0.0997722 0.0498861 0.998755i \(-0.484114\pi\)
0.0498861 + 0.998755i \(0.484114\pi\)
\(264\) 5.00000 0.307729
\(265\) −13.9443 −0.856590
\(266\) −4.54915 −0.278926
\(267\) 13.7984 0.844447
\(268\) −17.3262 −1.05837
\(269\) −11.4721 −0.699468 −0.349734 0.936849i \(-0.613728\pi\)
−0.349734 + 0.936849i \(0.613728\pi\)
\(270\) 1.38197 0.0841038
\(271\) −7.76393 −0.471625 −0.235813 0.971799i \(-0.575775\pi\)
−0.235813 + 0.971799i \(0.575775\pi\)
\(272\) 3.43769 0.208441
\(273\) −14.8541 −0.899011
\(274\) 7.56231 0.456856
\(275\) 0 0
\(276\) −7.47214 −0.449770
\(277\) −5.47214 −0.328789 −0.164394 0.986395i \(-0.552567\pi\)
−0.164394 + 0.986395i \(0.552567\pi\)
\(278\) 7.27051 0.436056
\(279\) −10.5623 −0.632349
\(280\) −11.9098 −0.711748
\(281\) 9.70820 0.579143 0.289571 0.957156i \(-0.406487\pi\)
0.289571 + 0.957156i \(0.406487\pi\)
\(282\) 6.70820 0.399468
\(283\) −23.2705 −1.38329 −0.691644 0.722238i \(-0.743113\pi\)
−0.691644 + 0.722238i \(0.743113\pi\)
\(284\) 12.8541 0.762751
\(285\) 6.90983 0.409303
\(286\) −8.61803 −0.509595
\(287\) −19.2705 −1.13750
\(288\) 5.61803 0.331046
\(289\) −13.5623 −0.797783
\(290\) −8.81966 −0.517908
\(291\) −3.00000 −0.175863
\(292\) −1.38197 −0.0808734
\(293\) 19.3820 1.13231 0.566153 0.824300i \(-0.308431\pi\)
0.566153 + 0.824300i \(0.308431\pi\)
\(294\) 0.819660 0.0478035
\(295\) 2.23607 0.130189
\(296\) 0.326238 0.0189622
\(297\) −2.23607 −0.129750
\(298\) 11.7984 0.683461
\(299\) 28.7984 1.66545
\(300\) 0 0
\(301\) 20.7426 1.19559
\(302\) 1.58359 0.0911255
\(303\) −3.70820 −0.213031
\(304\) 5.72949 0.328609
\(305\) 7.03444 0.402791
\(306\) 1.14590 0.0655066
\(307\) −25.8885 −1.47754 −0.738769 0.673959i \(-0.764592\pi\)
−0.738769 + 0.673959i \(0.764592\pi\)
\(308\) 8.61803 0.491058
\(309\) 3.23607 0.184093
\(310\) 14.5967 0.829040
\(311\) 27.4508 1.55659 0.778297 0.627896i \(-0.216084\pi\)
0.778297 + 0.627896i \(0.216084\pi\)
\(312\) −13.9443 −0.789439
\(313\) −20.7984 −1.17559 −0.587797 0.809009i \(-0.700005\pi\)
−0.587797 + 0.809009i \(0.700005\pi\)
\(314\) 5.56231 0.313899
\(315\) 5.32624 0.300100
\(316\) 4.85410 0.273065
\(317\) −29.1803 −1.63893 −0.819466 0.573128i \(-0.805730\pi\)
−0.819466 + 0.573128i \(0.805730\pi\)
\(318\) −3.85410 −0.216127
\(319\) 14.2705 0.798995
\(320\) 0.527864 0.0295085
\(321\) −0.909830 −0.0507818
\(322\) 6.79837 0.378859
\(323\) 5.72949 0.318797
\(324\) −1.61803 −0.0898908
\(325\) 0 0
\(326\) −11.4721 −0.635383
\(327\) 4.14590 0.229269
\(328\) −18.0902 −0.998863
\(329\) 25.8541 1.42538
\(330\) 3.09017 0.170108
\(331\) −11.1246 −0.611464 −0.305732 0.952118i \(-0.598901\pi\)
−0.305732 + 0.952118i \(0.598901\pi\)
\(332\) 2.61803 0.143683
\(333\) −0.145898 −0.00799516
\(334\) 4.34752 0.237886
\(335\) −23.9443 −1.30822
\(336\) 4.41641 0.240935
\(337\) −14.9098 −0.812190 −0.406095 0.913831i \(-0.633110\pi\)
−0.406095 + 0.913831i \(0.633110\pi\)
\(338\) 16.0000 0.870285
\(339\) 9.00000 0.488813
\(340\) 6.70820 0.363803
\(341\) −23.6180 −1.27899
\(342\) 1.90983 0.103272
\(343\) 19.8328 1.07087
\(344\) 19.4721 1.04987
\(345\) −10.3262 −0.555946
\(346\) 7.65248 0.411400
\(347\) 31.0344 1.66602 0.833008 0.553261i \(-0.186617\pi\)
0.833008 + 0.553261i \(0.186617\pi\)
\(348\) 10.3262 0.553544
\(349\) −27.0000 −1.44528 −0.722638 0.691226i \(-0.757071\pi\)
−0.722638 + 0.691226i \(0.757071\pi\)
\(350\) 0 0
\(351\) 6.23607 0.332857
\(352\) 12.5623 0.669573
\(353\) −16.0344 −0.853427 −0.426714 0.904387i \(-0.640329\pi\)
−0.426714 + 0.904387i \(0.640329\pi\)
\(354\) 0.618034 0.0328481
\(355\) 17.7639 0.942812
\(356\) 22.3262 1.18329
\(357\) 4.41641 0.233741
\(358\) −0.326238 −0.0172422
\(359\) −21.8885 −1.15523 −0.577617 0.816308i \(-0.696017\pi\)
−0.577617 + 0.816308i \(0.696017\pi\)
\(360\) 5.00000 0.263523
\(361\) −9.45085 −0.497413
\(362\) 13.7639 0.723416
\(363\) 6.00000 0.314918
\(364\) −24.0344 −1.25975
\(365\) −1.90983 −0.0999651
\(366\) 1.94427 0.101629
\(367\) 29.8328 1.55726 0.778630 0.627483i \(-0.215915\pi\)
0.778630 + 0.627483i \(0.215915\pi\)
\(368\) −8.56231 −0.446341
\(369\) 8.09017 0.421157
\(370\) 0.201626 0.0104820
\(371\) −14.8541 −0.771187
\(372\) −17.0902 −0.886084
\(373\) 34.3262 1.77735 0.888673 0.458542i \(-0.151628\pi\)
0.888673 + 0.458542i \(0.151628\pi\)
\(374\) 2.56231 0.132494
\(375\) −11.1803 −0.577350
\(376\) 24.2705 1.25166
\(377\) −39.7984 −2.04972
\(378\) 1.47214 0.0757185
\(379\) 22.4164 1.15145 0.575727 0.817642i \(-0.304719\pi\)
0.575727 + 0.817642i \(0.304719\pi\)
\(380\) 11.1803 0.573539
\(381\) −15.9443 −0.816850
\(382\) 10.1459 0.519109
\(383\) −18.2361 −0.931820 −0.465910 0.884832i \(-0.654273\pi\)
−0.465910 + 0.884832i \(0.654273\pi\)
\(384\) 11.3820 0.580834
\(385\) 11.9098 0.606981
\(386\) 4.94427 0.251657
\(387\) −8.70820 −0.442663
\(388\) −4.85410 −0.246430
\(389\) 32.4721 1.64640 0.823201 0.567750i \(-0.192186\pi\)
0.823201 + 0.567750i \(0.192186\pi\)
\(390\) −8.61803 −0.436391
\(391\) −8.56231 −0.433014
\(392\) 2.96556 0.149783
\(393\) 20.6525 1.04178
\(394\) −12.7639 −0.643038
\(395\) 6.70820 0.337526
\(396\) −3.61803 −0.181813
\(397\) 3.00000 0.150566 0.0752828 0.997162i \(-0.476014\pi\)
0.0752828 + 0.997162i \(0.476014\pi\)
\(398\) −10.2361 −0.513088
\(399\) 7.36068 0.368495
\(400\) 0 0
\(401\) −17.0902 −0.853442 −0.426721 0.904383i \(-0.640331\pi\)
−0.426721 + 0.904383i \(0.640331\pi\)
\(402\) −6.61803 −0.330078
\(403\) 65.8673 3.28108
\(404\) −6.00000 −0.298511
\(405\) −2.23607 −0.111111
\(406\) −9.39512 −0.466272
\(407\) −0.326238 −0.0161710
\(408\) 4.14590 0.205253
\(409\) −10.5836 −0.523325 −0.261662 0.965159i \(-0.584271\pi\)
−0.261662 + 0.965159i \(0.584271\pi\)
\(410\) −11.1803 −0.552158
\(411\) −12.2361 −0.603561
\(412\) 5.23607 0.257963
\(413\) 2.38197 0.117209
\(414\) −2.85410 −0.140271
\(415\) 3.61803 0.177602
\(416\) −35.0344 −1.71770
\(417\) −11.7639 −0.576082
\(418\) 4.27051 0.208877
\(419\) −31.3050 −1.52935 −0.764673 0.644418i \(-0.777100\pi\)
−0.764673 + 0.644418i \(0.777100\pi\)
\(420\) 8.61803 0.420517
\(421\) 1.00000 0.0487370 0.0243685 0.999703i \(-0.492242\pi\)
0.0243685 + 0.999703i \(0.492242\pi\)
\(422\) −1.32624 −0.0645603
\(423\) −10.8541 −0.527744
\(424\) −13.9443 −0.677194
\(425\) 0 0
\(426\) 4.90983 0.237882
\(427\) 7.49342 0.362632
\(428\) −1.47214 −0.0711584
\(429\) 13.9443 0.673236
\(430\) 12.0344 0.580352
\(431\) 12.3820 0.596418 0.298209 0.954501i \(-0.403611\pi\)
0.298209 + 0.954501i \(0.403611\pi\)
\(432\) −1.85410 −0.0892055
\(433\) 3.67376 0.176550 0.0882749 0.996096i \(-0.471865\pi\)
0.0882749 + 0.996096i \(0.471865\pi\)
\(434\) 15.5492 0.746383
\(435\) 14.2705 0.684219
\(436\) 6.70820 0.321265
\(437\) −14.2705 −0.682651
\(438\) −0.527864 −0.0252223
\(439\) −34.6180 −1.65223 −0.826114 0.563503i \(-0.809453\pi\)
−0.826114 + 0.563503i \(0.809453\pi\)
\(440\) 11.1803 0.533002
\(441\) −1.32624 −0.0631542
\(442\) −7.14590 −0.339896
\(443\) 3.38197 0.160682 0.0803410 0.996767i \(-0.474399\pi\)
0.0803410 + 0.996767i \(0.474399\pi\)
\(444\) −0.236068 −0.0112033
\(445\) 30.8541 1.46262
\(446\) −5.88854 −0.278831
\(447\) −19.0902 −0.902934
\(448\) 0.562306 0.0265665
\(449\) 6.88854 0.325090 0.162545 0.986701i \(-0.448030\pi\)
0.162545 + 0.986701i \(0.448030\pi\)
\(450\) 0 0
\(451\) 18.0902 0.851833
\(452\) 14.5623 0.684953
\(453\) −2.56231 −0.120388
\(454\) −13.6869 −0.642359
\(455\) −33.2148 −1.55713
\(456\) 6.90983 0.323582
\(457\) 33.6869 1.57581 0.787904 0.615798i \(-0.211166\pi\)
0.787904 + 0.615798i \(0.211166\pi\)
\(458\) −5.65248 −0.264123
\(459\) −1.85410 −0.0865421
\(460\) −16.7082 −0.779024
\(461\) 23.7426 1.10581 0.552903 0.833246i \(-0.313520\pi\)
0.552903 + 0.833246i \(0.313520\pi\)
\(462\) 3.29180 0.153148
\(463\) 4.14590 0.192676 0.0963381 0.995349i \(-0.469287\pi\)
0.0963381 + 0.995349i \(0.469287\pi\)
\(464\) 11.8328 0.549325
\(465\) −23.6180 −1.09526
\(466\) −5.12461 −0.237393
\(467\) 14.8328 0.686381 0.343190 0.939266i \(-0.388492\pi\)
0.343190 + 0.939266i \(0.388492\pi\)
\(468\) 10.0902 0.466418
\(469\) −25.5066 −1.17778
\(470\) 15.0000 0.691898
\(471\) −9.00000 −0.414698
\(472\) 2.23607 0.102923
\(473\) −19.4721 −0.895330
\(474\) 1.85410 0.0851617
\(475\) 0 0
\(476\) 7.14590 0.327532
\(477\) 6.23607 0.285530
\(478\) −0.909830 −0.0416147
\(479\) −14.9098 −0.681248 −0.340624 0.940200i \(-0.610638\pi\)
−0.340624 + 0.940200i \(0.610638\pi\)
\(480\) 12.5623 0.573388
\(481\) 0.909830 0.0414847
\(482\) −14.4721 −0.659188
\(483\) −11.0000 −0.500517
\(484\) 9.70820 0.441282
\(485\) −6.70820 −0.304604
\(486\) −0.618034 −0.0280346
\(487\) −5.74265 −0.260224 −0.130112 0.991499i \(-0.541534\pi\)
−0.130112 + 0.991499i \(0.541534\pi\)
\(488\) 7.03444 0.318434
\(489\) 18.5623 0.839416
\(490\) 1.83282 0.0827982
\(491\) −11.5066 −0.519285 −0.259642 0.965705i \(-0.583605\pi\)
−0.259642 + 0.965705i \(0.583605\pi\)
\(492\) 13.0902 0.590150
\(493\) 11.8328 0.532923
\(494\) −11.9098 −0.535849
\(495\) −5.00000 −0.224733
\(496\) −19.5836 −0.879329
\(497\) 18.9230 0.848812
\(498\) 1.00000 0.0448111
\(499\) −14.4164 −0.645367 −0.322684 0.946507i \(-0.604585\pi\)
−0.322684 + 0.946507i \(0.604585\pi\)
\(500\) −18.0902 −0.809017
\(501\) −7.03444 −0.314276
\(502\) 9.38197 0.418738
\(503\) 3.79837 0.169361 0.0846806 0.996408i \(-0.473013\pi\)
0.0846806 + 0.996408i \(0.473013\pi\)
\(504\) 5.32624 0.237249
\(505\) −8.29180 −0.368980
\(506\) −6.38197 −0.283713
\(507\) −25.8885 −1.14975
\(508\) −25.7984 −1.14462
\(509\) 34.9230 1.54793 0.773967 0.633226i \(-0.218270\pi\)
0.773967 + 0.633226i \(0.218270\pi\)
\(510\) 2.56231 0.113461
\(511\) −2.03444 −0.0899984
\(512\) 18.7082 0.826794
\(513\) −3.09017 −0.136434
\(514\) −12.0000 −0.529297
\(515\) 7.23607 0.318859
\(516\) −14.0902 −0.620285
\(517\) −24.2705 −1.06742
\(518\) 0.214782 0.00943697
\(519\) −12.3820 −0.543508
\(520\) −31.1803 −1.36735
\(521\) 23.5066 1.02984 0.514921 0.857238i \(-0.327821\pi\)
0.514921 + 0.857238i \(0.327821\pi\)
\(522\) 3.94427 0.172636
\(523\) 14.0557 0.614614 0.307307 0.951610i \(-0.400572\pi\)
0.307307 + 0.951610i \(0.400572\pi\)
\(524\) 33.4164 1.45980
\(525\) 0 0
\(526\) 1.00000 0.0436021
\(527\) −19.5836 −0.853075
\(528\) −4.14590 −0.180427
\(529\) −1.67376 −0.0727723
\(530\) −8.61803 −0.374343
\(531\) −1.00000 −0.0433963
\(532\) 11.9098 0.516357
\(533\) −50.4508 −2.18527
\(534\) 8.52786 0.369037
\(535\) −2.03444 −0.0879566
\(536\) −23.9443 −1.03424
\(537\) 0.527864 0.0227790
\(538\) −7.09017 −0.305679
\(539\) −2.96556 −0.127736
\(540\) −3.61803 −0.155695
\(541\) −26.1246 −1.12318 −0.561592 0.827414i \(-0.689811\pi\)
−0.561592 + 0.827414i \(0.689811\pi\)
\(542\) −4.79837 −0.206108
\(543\) −22.2705 −0.955719
\(544\) 10.4164 0.446600
\(545\) 9.27051 0.397105
\(546\) −9.18034 −0.392882
\(547\) 12.4377 0.531797 0.265899 0.964001i \(-0.414331\pi\)
0.265899 + 0.964001i \(0.414331\pi\)
\(548\) −19.7984 −0.845745
\(549\) −3.14590 −0.134264
\(550\) 0 0
\(551\) 19.7214 0.840158
\(552\) −10.3262 −0.439514
\(553\) 7.14590 0.303874
\(554\) −3.38197 −0.143686
\(555\) −0.326238 −0.0138480
\(556\) −19.0344 −0.807240
\(557\) −25.4164 −1.07693 −0.538464 0.842649i \(-0.680995\pi\)
−0.538464 + 0.842649i \(0.680995\pi\)
\(558\) −6.52786 −0.276347
\(559\) 54.3050 2.29685
\(560\) 9.87539 0.417311
\(561\) −4.14590 −0.175040
\(562\) 6.00000 0.253095
\(563\) −20.5967 −0.868049 −0.434025 0.900901i \(-0.642907\pi\)
−0.434025 + 0.900901i \(0.642907\pi\)
\(564\) −17.5623 −0.739506
\(565\) 20.1246 0.846649
\(566\) −14.3820 −0.604519
\(567\) −2.38197 −0.100033
\(568\) 17.7639 0.745358
\(569\) −11.5066 −0.482381 −0.241190 0.970478i \(-0.577538\pi\)
−0.241190 + 0.970478i \(0.577538\pi\)
\(570\) 4.27051 0.178872
\(571\) 1.58359 0.0662713 0.0331356 0.999451i \(-0.489451\pi\)
0.0331356 + 0.999451i \(0.489451\pi\)
\(572\) 22.5623 0.943377
\(573\) −16.4164 −0.685805
\(574\) −11.9098 −0.497107
\(575\) 0 0
\(576\) −0.236068 −0.00983617
\(577\) 12.5279 0.521542 0.260771 0.965401i \(-0.416023\pi\)
0.260771 + 0.965401i \(0.416023\pi\)
\(578\) −8.38197 −0.348644
\(579\) −8.00000 −0.332469
\(580\) 23.0902 0.958767
\(581\) 3.85410 0.159895
\(582\) −1.85410 −0.0768550
\(583\) 13.9443 0.577513
\(584\) −1.90983 −0.0790293
\(585\) 13.9443 0.576525
\(586\) 11.9787 0.494836
\(587\) −15.3607 −0.634003 −0.317002 0.948425i \(-0.602676\pi\)
−0.317002 + 0.948425i \(0.602676\pi\)
\(588\) −2.14590 −0.0884953
\(589\) −32.6393 −1.34488
\(590\) 1.38197 0.0568946
\(591\) 20.6525 0.849529
\(592\) −0.270510 −0.0111179
\(593\) 16.0902 0.660744 0.330372 0.943851i \(-0.392826\pi\)
0.330372 + 0.943851i \(0.392826\pi\)
\(594\) −1.38197 −0.0567028
\(595\) 9.87539 0.404851
\(596\) −30.8885 −1.26524
\(597\) 16.5623 0.677850
\(598\) 17.7984 0.727830
\(599\) 2.65248 0.108377 0.0541886 0.998531i \(-0.482743\pi\)
0.0541886 + 0.998531i \(0.482743\pi\)
\(600\) 0 0
\(601\) 23.3820 0.953770 0.476885 0.878966i \(-0.341766\pi\)
0.476885 + 0.878966i \(0.341766\pi\)
\(602\) 12.8197 0.522490
\(603\) 10.7082 0.436072
\(604\) −4.14590 −0.168694
\(605\) 13.4164 0.545455
\(606\) −2.29180 −0.0930979
\(607\) −13.6525 −0.554137 −0.277068 0.960850i \(-0.589363\pi\)
−0.277068 + 0.960850i \(0.589363\pi\)
\(608\) 17.3607 0.704069
\(609\) 15.2016 0.616001
\(610\) 4.34752 0.176026
\(611\) 67.6869 2.73832
\(612\) −3.00000 −0.121268
\(613\) −38.6525 −1.56116 −0.780579 0.625057i \(-0.785076\pi\)
−0.780579 + 0.625057i \(0.785076\pi\)
\(614\) −16.0000 −0.645707
\(615\) 18.0902 0.729466
\(616\) 11.9098 0.479861
\(617\) 22.1459 0.891560 0.445780 0.895142i \(-0.352926\pi\)
0.445780 + 0.895142i \(0.352926\pi\)
\(618\) 2.00000 0.0804518
\(619\) 12.1246 0.487329 0.243665 0.969860i \(-0.421650\pi\)
0.243665 + 0.969860i \(0.421650\pi\)
\(620\) −38.2148 −1.53474
\(621\) 4.61803 0.185315
\(622\) 16.9656 0.680257
\(623\) 32.8673 1.31680
\(624\) 11.5623 0.462863
\(625\) −25.0000 −1.00000
\(626\) −12.8541 −0.513753
\(627\) −6.90983 −0.275952
\(628\) −14.5623 −0.581099
\(629\) −0.270510 −0.0107859
\(630\) 3.29180 0.131148
\(631\) −40.4164 −1.60895 −0.804476 0.593985i \(-0.797554\pi\)
−0.804476 + 0.593985i \(0.797554\pi\)
\(632\) 6.70820 0.266838
\(633\) 2.14590 0.0852918
\(634\) −18.0344 −0.716239
\(635\) −35.6525 −1.41483
\(636\) 10.0902 0.400101
\(637\) 8.27051 0.327690
\(638\) 8.81966 0.349174
\(639\) −7.94427 −0.314271
\(640\) 25.4508 1.00603
\(641\) −27.9443 −1.10373 −0.551866 0.833933i \(-0.686084\pi\)
−0.551866 + 0.833933i \(0.686084\pi\)
\(642\) −0.562306 −0.0221924
\(643\) −19.8541 −0.782969 −0.391485 0.920185i \(-0.628038\pi\)
−0.391485 + 0.920185i \(0.628038\pi\)
\(644\) −17.7984 −0.701354
\(645\) −19.4721 −0.766715
\(646\) 3.54102 0.139320
\(647\) 44.9443 1.76694 0.883471 0.468486i \(-0.155200\pi\)
0.883471 + 0.468486i \(0.155200\pi\)
\(648\) −2.23607 −0.0878410
\(649\) −2.23607 −0.0877733
\(650\) 0 0
\(651\) −25.1591 −0.986061
\(652\) 30.0344 1.17624
\(653\) 40.0902 1.56885 0.784425 0.620224i \(-0.212958\pi\)
0.784425 + 0.620224i \(0.212958\pi\)
\(654\) 2.56231 0.100194
\(655\) 46.1803 1.80442
\(656\) 15.0000 0.585652
\(657\) 0.854102 0.0333217
\(658\) 15.9787 0.622915
\(659\) 1.85410 0.0722256 0.0361128 0.999348i \(-0.488502\pi\)
0.0361128 + 0.999348i \(0.488502\pi\)
\(660\) −8.09017 −0.314909
\(661\) 30.7426 1.19575 0.597875 0.801589i \(-0.296012\pi\)
0.597875 + 0.801589i \(0.296012\pi\)
\(662\) −6.87539 −0.267220
\(663\) 11.5623 0.449043
\(664\) 3.61803 0.140407
\(665\) 16.4590 0.638252
\(666\) −0.0901699 −0.00349401
\(667\) −29.4721 −1.14117
\(668\) −11.3820 −0.440381
\(669\) 9.52786 0.368369
\(670\) −14.7984 −0.571711
\(671\) −7.03444 −0.271562
\(672\) 13.3820 0.516221
\(673\) −8.47214 −0.326577 −0.163288 0.986578i \(-0.552210\pi\)
−0.163288 + 0.986578i \(0.552210\pi\)
\(674\) −9.21478 −0.354940
\(675\) 0 0
\(676\) −41.8885 −1.61110
\(677\) −27.8197 −1.06920 −0.534598 0.845106i \(-0.679537\pi\)
−0.534598 + 0.845106i \(0.679537\pi\)
\(678\) 5.56231 0.213619
\(679\) −7.14590 −0.274234
\(680\) 9.27051 0.355508
\(681\) 22.1459 0.848633
\(682\) −14.5967 −0.558938
\(683\) 39.0000 1.49229 0.746147 0.665782i \(-0.231902\pi\)
0.746147 + 0.665782i \(0.231902\pi\)
\(684\) −5.00000 −0.191180
\(685\) −27.3607 −1.04540
\(686\) 12.2574 0.467988
\(687\) 9.14590 0.348938
\(688\) −16.1459 −0.615557
\(689\) −38.8885 −1.48154
\(690\) −6.38197 −0.242957
\(691\) 35.1246 1.33620 0.668102 0.744070i \(-0.267107\pi\)
0.668102 + 0.744070i \(0.267107\pi\)
\(692\) −20.0344 −0.761595
\(693\) −5.32624 −0.202327
\(694\) 19.1803 0.728076
\(695\) −26.3050 −0.997804
\(696\) 14.2705 0.540922
\(697\) 15.0000 0.568166
\(698\) −16.6869 −0.631609
\(699\) 8.29180 0.313625
\(700\) 0 0
\(701\) 18.2016 0.687466 0.343733 0.939067i \(-0.388308\pi\)
0.343733 + 0.939067i \(0.388308\pi\)
\(702\) 3.85410 0.145464
\(703\) −0.450850 −0.0170041
\(704\) −0.527864 −0.0198946
\(705\) −24.2705 −0.914080
\(706\) −9.90983 −0.372961
\(707\) −8.83282 −0.332192
\(708\) −1.61803 −0.0608094
\(709\) −27.7082 −1.04060 −0.520302 0.853983i \(-0.674181\pi\)
−0.520302 + 0.853983i \(0.674181\pi\)
\(710\) 10.9787 0.412024
\(711\) −3.00000 −0.112509
\(712\) 30.8541 1.15631
\(713\) 48.7771 1.82672
\(714\) 2.72949 0.102149
\(715\) 31.1803 1.16608
\(716\) 0.854102 0.0319193
\(717\) 1.47214 0.0549779
\(718\) −13.5279 −0.504855
\(719\) 14.6180 0.545161 0.272580 0.962133i \(-0.412123\pi\)
0.272580 + 0.962133i \(0.412123\pi\)
\(720\) −4.14590 −0.154508
\(721\) 7.70820 0.287069
\(722\) −5.84095 −0.217378
\(723\) 23.4164 0.870866
\(724\) −36.0344 −1.33921
\(725\) 0 0
\(726\) 3.70820 0.137624
\(727\) 26.0000 0.964287 0.482143 0.876092i \(-0.339858\pi\)
0.482143 + 0.876092i \(0.339858\pi\)
\(728\) −33.2148 −1.23102
\(729\) 1.00000 0.0370370
\(730\) −1.18034 −0.0436863
\(731\) −16.1459 −0.597178
\(732\) −5.09017 −0.188138
\(733\) 20.5279 0.758214 0.379107 0.925353i \(-0.376231\pi\)
0.379107 + 0.925353i \(0.376231\pi\)
\(734\) 18.4377 0.680548
\(735\) −2.96556 −0.109386
\(736\) −25.9443 −0.956319
\(737\) 23.9443 0.881999
\(738\) 5.00000 0.184053
\(739\) 48.1033 1.76951 0.884755 0.466057i \(-0.154326\pi\)
0.884755 + 0.466057i \(0.154326\pi\)
\(740\) −0.527864 −0.0194047
\(741\) 19.2705 0.707920
\(742\) −9.18034 −0.337021
\(743\) −24.6180 −0.903148 −0.451574 0.892234i \(-0.649137\pi\)
−0.451574 + 0.892234i \(0.649137\pi\)
\(744\) −23.6180 −0.865879
\(745\) −42.6869 −1.56393
\(746\) 21.2148 0.776728
\(747\) −1.61803 −0.0592008
\(748\) −6.70820 −0.245276
\(749\) −2.16718 −0.0791872
\(750\) −6.90983 −0.252311
\(751\) −4.87539 −0.177905 −0.0889527 0.996036i \(-0.528352\pi\)
−0.0889527 + 0.996036i \(0.528352\pi\)
\(752\) −20.1246 −0.733869
\(753\) −15.1803 −0.553202
\(754\) −24.5967 −0.895761
\(755\) −5.72949 −0.208517
\(756\) −3.85410 −0.140172
\(757\) −41.3820 −1.50405 −0.752027 0.659133i \(-0.770924\pi\)
−0.752027 + 0.659133i \(0.770924\pi\)
\(758\) 13.8541 0.503204
\(759\) 10.3262 0.374819
\(760\) 15.4508 0.560461
\(761\) 30.7082 1.11317 0.556586 0.830790i \(-0.312111\pi\)
0.556586 + 0.830790i \(0.312111\pi\)
\(762\) −9.85410 −0.356976
\(763\) 9.87539 0.357513
\(764\) −26.5623 −0.960991
\(765\) −4.14590 −0.149895
\(766\) −11.2705 −0.407220
\(767\) 6.23607 0.225171
\(768\) 6.56231 0.236797
\(769\) 10.9787 0.395903 0.197951 0.980212i \(-0.436571\pi\)
0.197951 + 0.980212i \(0.436571\pi\)
\(770\) 7.36068 0.265260
\(771\) 19.4164 0.699265
\(772\) −12.9443 −0.465875
\(773\) 28.6525 1.03056 0.515279 0.857023i \(-0.327688\pi\)
0.515279 + 0.857023i \(0.327688\pi\)
\(774\) −5.38197 −0.193451
\(775\) 0 0
\(776\) −6.70820 −0.240810
\(777\) −0.347524 −0.0124674
\(778\) 20.0689 0.719504
\(779\) 25.0000 0.895718
\(780\) 22.5623 0.807860
\(781\) −17.7639 −0.635643
\(782\) −5.29180 −0.189234
\(783\) −6.38197 −0.228073
\(784\) −2.45898 −0.0878207
\(785\) −20.1246 −0.718278
\(786\) 12.7639 0.455274
\(787\) −4.29180 −0.152986 −0.0764930 0.997070i \(-0.524372\pi\)
−0.0764930 + 0.997070i \(0.524372\pi\)
\(788\) 33.4164 1.19041
\(789\) −1.61803 −0.0576035
\(790\) 4.14590 0.147504
\(791\) 21.4377 0.762237
\(792\) −5.00000 −0.177667
\(793\) 19.6180 0.696657
\(794\) 1.85410 0.0657996
\(795\) 13.9443 0.494552
\(796\) 26.7984 0.949843
\(797\) −1.23607 −0.0437838 −0.0218919 0.999760i \(-0.506969\pi\)
−0.0218919 + 0.999760i \(0.506969\pi\)
\(798\) 4.54915 0.161038
\(799\) −20.1246 −0.711958
\(800\) 0 0
\(801\) −13.7984 −0.487542
\(802\) −10.5623 −0.372968
\(803\) 1.90983 0.0673964
\(804\) 17.3262 0.611049
\(805\) −24.5967 −0.866921
\(806\) 40.7082 1.43389
\(807\) 11.4721 0.403838
\(808\) −8.29180 −0.291704
\(809\) 7.67376 0.269795 0.134898 0.990860i \(-0.456929\pi\)
0.134898 + 0.990860i \(0.456929\pi\)
\(810\) −1.38197 −0.0485573
\(811\) 14.6525 0.514518 0.257259 0.966342i \(-0.417181\pi\)
0.257259 + 0.966342i \(0.417181\pi\)
\(812\) 24.5967 0.863177
\(813\) 7.76393 0.272293
\(814\) −0.201626 −0.00706699
\(815\) 41.5066 1.45391
\(816\) −3.43769 −0.120343
\(817\) −26.9098 −0.941456
\(818\) −6.54102 −0.228701
\(819\) 14.8541 0.519044
\(820\) 29.2705 1.02217
\(821\) −40.9230 −1.42822 −0.714111 0.700032i \(-0.753169\pi\)
−0.714111 + 0.700032i \(0.753169\pi\)
\(822\) −7.56231 −0.263766
\(823\) 11.2918 0.393607 0.196804 0.980443i \(-0.436944\pi\)
0.196804 + 0.980443i \(0.436944\pi\)
\(824\) 7.23607 0.252080
\(825\) 0 0
\(826\) 1.47214 0.0512222
\(827\) −2.81966 −0.0980492 −0.0490246 0.998798i \(-0.515611\pi\)
−0.0490246 + 0.998798i \(0.515611\pi\)
\(828\) 7.47214 0.259675
\(829\) −45.6869 −1.58677 −0.793386 0.608719i \(-0.791684\pi\)
−0.793386 + 0.608719i \(0.791684\pi\)
\(830\) 2.23607 0.0776151
\(831\) 5.47214 0.189826
\(832\) 1.47214 0.0510371
\(833\) −2.45898 −0.0851986
\(834\) −7.27051 −0.251757
\(835\) −15.7295 −0.544341
\(836\) −11.1803 −0.386680
\(837\) 10.5623 0.365087
\(838\) −19.3475 −0.668349
\(839\) −24.8197 −0.856870 −0.428435 0.903573i \(-0.640935\pi\)
−0.428435 + 0.903573i \(0.640935\pi\)
\(840\) 11.9098 0.410928
\(841\) 11.7295 0.404465
\(842\) 0.618034 0.0212989
\(843\) −9.70820 −0.334368
\(844\) 3.47214 0.119516
\(845\) −57.8885 −1.99143
\(846\) −6.70820 −0.230633
\(847\) 14.2918 0.491072
\(848\) 11.5623 0.397051
\(849\) 23.2705 0.798642
\(850\) 0 0
\(851\) 0.673762 0.0230963
\(852\) −12.8541 −0.440374
\(853\) −3.96556 −0.135778 −0.0678891 0.997693i \(-0.521626\pi\)
−0.0678891 + 0.997693i \(0.521626\pi\)
\(854\) 4.63119 0.158476
\(855\) −6.90983 −0.236311
\(856\) −2.03444 −0.0695358
\(857\) −28.7771 −0.983007 −0.491503 0.870876i \(-0.663552\pi\)
−0.491503 + 0.870876i \(0.663552\pi\)
\(858\) 8.61803 0.294215
\(859\) 15.5279 0.529804 0.264902 0.964275i \(-0.414660\pi\)
0.264902 + 0.964275i \(0.414660\pi\)
\(860\) −31.5066 −1.07437
\(861\) 19.2705 0.656737
\(862\) 7.65248 0.260644
\(863\) 19.2016 0.653631 0.326815 0.945088i \(-0.394024\pi\)
0.326815 + 0.945088i \(0.394024\pi\)
\(864\) −5.61803 −0.191129
\(865\) −27.6869 −0.941383
\(866\) 2.27051 0.0771551
\(867\) 13.5623 0.460600
\(868\) −40.7082 −1.38173
\(869\) −6.70820 −0.227560
\(870\) 8.81966 0.299014
\(871\) −66.7771 −2.26266
\(872\) 9.27051 0.313939
\(873\) 3.00000 0.101535
\(874\) −8.81966 −0.298329
\(875\) −26.6312 −0.900299
\(876\) 1.38197 0.0466923
\(877\) −16.1115 −0.544045 −0.272023 0.962291i \(-0.587693\pi\)
−0.272023 + 0.962291i \(0.587693\pi\)
\(878\) −21.3951 −0.722050
\(879\) −19.3820 −0.653737
\(880\) −9.27051 −0.312509
\(881\) −15.7082 −0.529223 −0.264611 0.964355i \(-0.585244\pi\)
−0.264611 + 0.964355i \(0.585244\pi\)
\(882\) −0.819660 −0.0275994
\(883\) −23.4164 −0.788025 −0.394012 0.919105i \(-0.628913\pi\)
−0.394012 + 0.919105i \(0.628913\pi\)
\(884\) 18.7082 0.629225
\(885\) −2.23607 −0.0751646
\(886\) 2.09017 0.0702206
\(887\) −22.5279 −0.756412 −0.378206 0.925722i \(-0.623459\pi\)
−0.378206 + 0.925722i \(0.623459\pi\)
\(888\) −0.326238 −0.0109478
\(889\) −37.9787 −1.27377
\(890\) 19.0689 0.639190
\(891\) 2.23607 0.0749111
\(892\) 15.4164 0.516180
\(893\) −33.5410 −1.12241
\(894\) −11.7984 −0.394597
\(895\) 1.18034 0.0394544
\(896\) 27.1115 0.905730
\(897\) −28.7984 −0.961550
\(898\) 4.25735 0.142070
\(899\) −67.4083 −2.24819
\(900\) 0 0
\(901\) 11.5623 0.385196
\(902\) 11.1803 0.372265
\(903\) −20.7426 −0.690272
\(904\) 20.1246 0.669335
\(905\) −49.7984 −1.65535
\(906\) −1.58359 −0.0526113
\(907\) −19.9443 −0.662239 −0.331119 0.943589i \(-0.607426\pi\)
−0.331119 + 0.943589i \(0.607426\pi\)
\(908\) 35.8328 1.18915
\(909\) 3.70820 0.122993
\(910\) −20.5279 −0.680492
\(911\) −45.1033 −1.49434 −0.747170 0.664633i \(-0.768588\pi\)
−0.747170 + 0.664633i \(0.768588\pi\)
\(912\) −5.72949 −0.189722
\(913\) −3.61803 −0.119739
\(914\) 20.8197 0.688653
\(915\) −7.03444 −0.232551
\(916\) 14.7984 0.488952
\(917\) 49.1935 1.62451
\(918\) −1.14590 −0.0378203
\(919\) −13.5967 −0.448515 −0.224258 0.974530i \(-0.571996\pi\)
−0.224258 + 0.974530i \(0.571996\pi\)
\(920\) −23.0902 −0.761260
\(921\) 25.8885 0.853057
\(922\) 14.6738 0.483255
\(923\) 49.5410 1.63066
\(924\) −8.61803 −0.283513
\(925\) 0 0
\(926\) 2.56231 0.0842026
\(927\) −3.23607 −0.106286
\(928\) 35.8541 1.17697
\(929\) −48.7082 −1.59806 −0.799032 0.601288i \(-0.794654\pi\)
−0.799032 + 0.601288i \(0.794654\pi\)
\(930\) −14.5967 −0.478646
\(931\) −4.09830 −0.134316
\(932\) 13.4164 0.439469
\(933\) −27.4508 −0.898700
\(934\) 9.16718 0.299959
\(935\) −9.27051 −0.303178
\(936\) 13.9443 0.455783
\(937\) 51.7214 1.68966 0.844832 0.535032i \(-0.179701\pi\)
0.844832 + 0.535032i \(0.179701\pi\)
\(938\) −15.7639 −0.514711
\(939\) 20.7984 0.678729
\(940\) −39.2705 −1.28086
\(941\) 40.3050 1.31390 0.656952 0.753932i \(-0.271845\pi\)
0.656952 + 0.753932i \(0.271845\pi\)
\(942\) −5.56231 −0.181230
\(943\) −37.3607 −1.21663
\(944\) −1.85410 −0.0603459
\(945\) −5.32624 −0.173263
\(946\) −12.0344 −0.391273
\(947\) 25.0344 0.813510 0.406755 0.913537i \(-0.366660\pi\)
0.406755 + 0.913537i \(0.366660\pi\)
\(948\) −4.85410 −0.157654
\(949\) −5.32624 −0.172897
\(950\) 0 0
\(951\) 29.1803 0.946237
\(952\) 9.87539 0.320063
\(953\) 4.94427 0.160161 0.0800803 0.996788i \(-0.474482\pi\)
0.0800803 + 0.996788i \(0.474482\pi\)
\(954\) 3.85410 0.124781
\(955\) −36.7082 −1.18785
\(956\) 2.38197 0.0770383
\(957\) −14.2705 −0.461300
\(958\) −9.21478 −0.297716
\(959\) −29.1459 −0.941170
\(960\) −0.527864 −0.0170367
\(961\) 80.5623 2.59878
\(962\) 0.562306 0.0181295
\(963\) 0.909830 0.0293189
\(964\) 37.8885 1.22031
\(965\) −17.8885 −0.575853
\(966\) −6.79837 −0.218734
\(967\) 15.2705 0.491066 0.245533 0.969388i \(-0.421037\pi\)
0.245533 + 0.969388i \(0.421037\pi\)
\(968\) 13.4164 0.431220
\(969\) −5.72949 −0.184058
\(970\) −4.14590 −0.133117
\(971\) 46.7984 1.50183 0.750916 0.660398i \(-0.229612\pi\)
0.750916 + 0.660398i \(0.229612\pi\)
\(972\) 1.61803 0.0518985
\(973\) −28.0213 −0.898321
\(974\) −3.54915 −0.113722
\(975\) 0 0
\(976\) −5.83282 −0.186704
\(977\) −20.8885 −0.668284 −0.334142 0.942523i \(-0.608446\pi\)
−0.334142 + 0.942523i \(0.608446\pi\)
\(978\) 11.4721 0.366838
\(979\) −30.8541 −0.986101
\(980\) −4.79837 −0.153278
\(981\) −4.14590 −0.132368
\(982\) −7.11146 −0.226936
\(983\) −9.40325 −0.299917 −0.149959 0.988692i \(-0.547914\pi\)
−0.149959 + 0.988692i \(0.547914\pi\)
\(984\) 18.0902 0.576694
\(985\) 46.1803 1.47143
\(986\) 7.31308 0.232896
\(987\) −25.8541 −0.822945
\(988\) 31.1803 0.991979
\(989\) 40.2148 1.27876
\(990\) −3.09017 −0.0982120
\(991\) 23.7426 0.754210 0.377105 0.926171i \(-0.376920\pi\)
0.377105 + 0.926171i \(0.376920\pi\)
\(992\) −59.3394 −1.88403
\(993\) 11.1246 0.353029
\(994\) 11.6950 0.370944
\(995\) 37.0344 1.17407
\(996\) −2.61803 −0.0829556
\(997\) 53.2148 1.68533 0.842665 0.538439i \(-0.180986\pi\)
0.842665 + 0.538439i \(0.180986\pi\)
\(998\) −8.90983 −0.282036
\(999\) 0.145898 0.00461601
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 177.2.a.b.1.2 2
3.2 odd 2 531.2.a.b.1.1 2
4.3 odd 2 2832.2.a.o.1.1 2
5.4 even 2 4425.2.a.t.1.1 2
7.6 odd 2 8673.2.a.k.1.2 2
12.11 even 2 8496.2.a.bb.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.2.a.b.1.2 2 1.1 even 1 trivial
531.2.a.b.1.1 2 3.2 odd 2
2832.2.a.o.1.1 2 4.3 odd 2
4425.2.a.t.1.1 2 5.4 even 2
8496.2.a.bb.1.2 2 12.11 even 2
8673.2.a.k.1.2 2 7.6 odd 2