Properties

Label 177.2.a.b.1.1
Level $177$
Weight $2$
Character 177.1
Self dual yes
Analytic conductor $1.413$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

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 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} +2.23607 q^{5} +1.61803 q^{6} -4.61803 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} +2.23607 q^{5} +1.61803 q^{6} -4.61803 q^{7} +2.23607 q^{8} +1.00000 q^{9} -3.61803 q^{10} -2.23607 q^{11} -0.618034 q^{12} -1.76393 q^{13} +7.47214 q^{14} -2.23607 q^{15} -4.85410 q^{16} -4.85410 q^{17} -1.61803 q^{18} -8.09017 q^{19} +1.38197 q^{20} +4.61803 q^{21} +3.61803 q^{22} -2.38197 q^{23} -2.23607 q^{24} +2.85410 q^{26} -1.00000 q^{27} -2.85410 q^{28} +8.61803 q^{29} +3.61803 q^{30} +9.56231 q^{31} +3.38197 q^{32} +2.23607 q^{33} +7.85410 q^{34} -10.3262 q^{35} +0.618034 q^{36} -6.85410 q^{37} +13.0902 q^{38} +1.76393 q^{39} +5.00000 q^{40} -3.09017 q^{41} -7.47214 q^{42} +4.70820 q^{43} -1.38197 q^{44} +2.23607 q^{45} +3.85410 q^{46} -4.14590 q^{47} +4.85410 q^{48} +14.3262 q^{49} +4.85410 q^{51} -1.09017 q^{52} +1.76393 q^{53} +1.61803 q^{54} -5.00000 q^{55} -10.3262 q^{56} +8.09017 q^{57} -13.9443 q^{58} -1.00000 q^{59} -1.38197 q^{60} -9.85410 q^{61} -15.4721 q^{62} -4.61803 q^{63} +4.23607 q^{64} -3.94427 q^{65} -3.61803 q^{66} -2.70820 q^{67} -3.00000 q^{68} +2.38197 q^{69} +16.7082 q^{70} +9.94427 q^{71} +2.23607 q^{72} -5.85410 q^{73} +11.0902 q^{74} -5.00000 q^{76} +10.3262 q^{77} -2.85410 q^{78} -3.00000 q^{79} -10.8541 q^{80} +1.00000 q^{81} +5.00000 q^{82} +0.618034 q^{83} +2.85410 q^{84} -10.8541 q^{85} -7.61803 q^{86} -8.61803 q^{87} -5.00000 q^{88} +10.7984 q^{89} -3.61803 q^{90} +8.14590 q^{91} -1.47214 q^{92} -9.56231 q^{93} +6.70820 q^{94} -18.0902 q^{95} -3.38197 q^{96} +3.00000 q^{97} -23.1803 q^{98} -2.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - 2q^{3} - q^{4} + q^{6} - 7q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - q^{2} - 2q^{3} - q^{4} + q^{6} - 7q^{7} + 2q^{9} - 5q^{10} + q^{12} - 8q^{13} + 6q^{14} - 3q^{16} - 3q^{17} - q^{18} - 5q^{19} + 5q^{20} + 7q^{21} + 5q^{22} - 7q^{23} - q^{26} - 2q^{27} + q^{28} + 15q^{29} + 5q^{30} - q^{31} + 9q^{32} + 9q^{34} - 5q^{35} - q^{36} - 7q^{37} + 15q^{38} + 8q^{39} + 10q^{40} + 5q^{41} - 6q^{42} - 4q^{43} - 5q^{44} + q^{46} - 15q^{47} + 3q^{48} + 13q^{49} + 3q^{51} + 9q^{52} + 8q^{53} + q^{54} - 10q^{55} - 5q^{56} + 5q^{57} - 10q^{58} - 2q^{59} - 5q^{60} - 13q^{61} - 22q^{62} - 7q^{63} + 4q^{64} + 10q^{65} - 5q^{66} + 8q^{67} - 6q^{68} + 7q^{69} + 20q^{70} + 2q^{71} - 5q^{73} + 11q^{74} - 10q^{76} + 5q^{77} + q^{78} - 6q^{79} - 15q^{80} + 2q^{81} + 10q^{82} - q^{83} - q^{84} - 15q^{85} - 13q^{86} - 15q^{87} - 10q^{88} - 3q^{89} - 5q^{90} + 23q^{91} + 6q^{92} + q^{93} - 25q^{95} - 9q^{96} + 6q^{97} - 24q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.61803 −1.14412 −0.572061 0.820211i \(-0.693856\pi\)
−0.572061 + 0.820211i \(0.693856\pi\)
\(3\) −1.00000 −0.577350
\(4\) 0.618034 0.309017
\(5\) 2.23607 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(6\) 1.61803 0.660560
\(7\) −4.61803 −1.74545 −0.872726 0.488210i \(-0.837650\pi\)
−0.872726 + 0.488210i \(0.837650\pi\)
\(8\) 2.23607 0.790569
\(9\) 1.00000 0.333333
\(10\) −3.61803 −1.14412
\(11\) −2.23607 −0.674200 −0.337100 0.941469i \(-0.609446\pi\)
−0.337100 + 0.941469i \(0.609446\pi\)
\(12\) −0.618034 −0.178411
\(13\) −1.76393 −0.489227 −0.244613 0.969621i \(-0.578661\pi\)
−0.244613 + 0.969621i \(0.578661\pi\)
\(14\) 7.47214 1.99701
\(15\) −2.23607 −0.577350
\(16\) −4.85410 −1.21353
\(17\) −4.85410 −1.17729 −0.588646 0.808391i \(-0.700339\pi\)
−0.588646 + 0.808391i \(0.700339\pi\)
\(18\) −1.61803 −0.381374
\(19\) −8.09017 −1.85601 −0.928006 0.372565i \(-0.878478\pi\)
−0.928006 + 0.372565i \(0.878478\pi\)
\(20\) 1.38197 0.309017
\(21\) 4.61803 1.00774
\(22\) 3.61803 0.771367
\(23\) −2.38197 −0.496674 −0.248337 0.968674i \(-0.579884\pi\)
−0.248337 + 0.968674i \(0.579884\pi\)
\(24\) −2.23607 −0.456435
\(25\) 0 0
\(26\) 2.85410 0.559735
\(27\) −1.00000 −0.192450
\(28\) −2.85410 −0.539375
\(29\) 8.61803 1.60033 0.800164 0.599781i \(-0.204746\pi\)
0.800164 + 0.599781i \(0.204746\pi\)
\(30\) 3.61803 0.660560
\(31\) 9.56231 1.71744 0.858720 0.512444i \(-0.171260\pi\)
0.858720 + 0.512444i \(0.171260\pi\)
\(32\) 3.38197 0.597853
\(33\) 2.23607 0.389249
\(34\) 7.85410 1.34697
\(35\) −10.3262 −1.74545
\(36\) 0.618034 0.103006
\(37\) −6.85410 −1.12681 −0.563404 0.826182i \(-0.690508\pi\)
−0.563404 + 0.826182i \(0.690508\pi\)
\(38\) 13.0902 2.12351
\(39\) 1.76393 0.282455
\(40\) 5.00000 0.790569
\(41\) −3.09017 −0.482603 −0.241302 0.970450i \(-0.577574\pi\)
−0.241302 + 0.970450i \(0.577574\pi\)
\(42\) −7.47214 −1.15298
\(43\) 4.70820 0.717994 0.358997 0.933339i \(-0.383119\pi\)
0.358997 + 0.933339i \(0.383119\pi\)
\(44\) −1.38197 −0.208339
\(45\) 2.23607 0.333333
\(46\) 3.85410 0.568256
\(47\) −4.14590 −0.604741 −0.302371 0.953190i \(-0.597778\pi\)
−0.302371 + 0.953190i \(0.597778\pi\)
\(48\) 4.85410 0.700629
\(49\) 14.3262 2.04661
\(50\) 0 0
\(51\) 4.85410 0.679710
\(52\) −1.09017 −0.151179
\(53\) 1.76393 0.242295 0.121147 0.992635i \(-0.461343\pi\)
0.121147 + 0.992635i \(0.461343\pi\)
\(54\) 1.61803 0.220187
\(55\) −5.00000 −0.674200
\(56\) −10.3262 −1.37990
\(57\) 8.09017 1.07157
\(58\) −13.9443 −1.83097
\(59\) −1.00000 −0.130189
\(60\) −1.38197 −0.178411
\(61\) −9.85410 −1.26169 −0.630844 0.775909i \(-0.717291\pi\)
−0.630844 + 0.775909i \(0.717291\pi\)
\(62\) −15.4721 −1.96496
\(63\) −4.61803 −0.581818
\(64\) 4.23607 0.529508
\(65\) −3.94427 −0.489227
\(66\) −3.61803 −0.445349
\(67\) −2.70820 −0.330860 −0.165430 0.986222i \(-0.552901\pi\)
−0.165430 + 0.986222i \(0.552901\pi\)
\(68\) −3.00000 −0.363803
\(69\) 2.38197 0.286755
\(70\) 16.7082 1.99701
\(71\) 9.94427 1.18017 0.590084 0.807342i \(-0.299095\pi\)
0.590084 + 0.807342i \(0.299095\pi\)
\(72\) 2.23607 0.263523
\(73\) −5.85410 −0.685171 −0.342585 0.939487i \(-0.611303\pi\)
−0.342585 + 0.939487i \(0.611303\pi\)
\(74\) 11.0902 1.28921
\(75\) 0 0
\(76\) −5.00000 −0.573539
\(77\) 10.3262 1.17678
\(78\) −2.85410 −0.323163
\(79\) −3.00000 −0.337526 −0.168763 0.985657i \(-0.553977\pi\)
−0.168763 + 0.985657i \(0.553977\pi\)
\(80\) −10.8541 −1.21353
\(81\) 1.00000 0.111111
\(82\) 5.00000 0.552158
\(83\) 0.618034 0.0678380 0.0339190 0.999425i \(-0.489201\pi\)
0.0339190 + 0.999425i \(0.489201\pi\)
\(84\) 2.85410 0.311408
\(85\) −10.8541 −1.17729
\(86\) −7.61803 −0.821474
\(87\) −8.61803 −0.923950
\(88\) −5.00000 −0.533002
\(89\) 10.7984 1.14463 0.572313 0.820035i \(-0.306046\pi\)
0.572313 + 0.820035i \(0.306046\pi\)
\(90\) −3.61803 −0.381374
\(91\) 8.14590 0.853922
\(92\) −1.47214 −0.153481
\(93\) −9.56231 −0.991565
\(94\) 6.70820 0.691898
\(95\) −18.0902 −1.85601
\(96\) −3.38197 −0.345170
\(97\) 3.00000 0.304604 0.152302 0.988334i \(-0.451331\pi\)
0.152302 + 0.988334i \(0.451331\pi\)
\(98\) −23.1803 −2.34157
\(99\) −2.23607 −0.224733
\(100\) 0 0
\(101\) −9.70820 −0.966002 −0.483001 0.875620i \(-0.660453\pi\)
−0.483001 + 0.875620i \(0.660453\pi\)
\(102\) −7.85410 −0.777672
\(103\) 1.23607 0.121793 0.0608967 0.998144i \(-0.480604\pi\)
0.0608967 + 0.998144i \(0.480604\pi\)
\(104\) −3.94427 −0.386768
\(105\) 10.3262 1.00774
\(106\) −2.85410 −0.277215
\(107\) 12.0902 1.16880 0.584400 0.811465i \(-0.301330\pi\)
0.584400 + 0.811465i \(0.301330\pi\)
\(108\) −0.618034 −0.0594703
\(109\) −10.8541 −1.03963 −0.519817 0.854278i \(-0.674000\pi\)
−0.519817 + 0.854278i \(0.674000\pi\)
\(110\) 8.09017 0.771367
\(111\) 6.85410 0.650563
\(112\) 22.4164 2.11815
\(113\) −9.00000 −0.846649 −0.423324 0.905978i \(-0.639137\pi\)
−0.423324 + 0.905978i \(0.639137\pi\)
\(114\) −13.0902 −1.22601
\(115\) −5.32624 −0.496674
\(116\) 5.32624 0.494529
\(117\) −1.76393 −0.163076
\(118\) 1.61803 0.148952
\(119\) 22.4164 2.05491
\(120\) −5.00000 −0.456435
\(121\) −6.00000 −0.545455
\(122\) 15.9443 1.44353
\(123\) 3.09017 0.278631
\(124\) 5.90983 0.530718
\(125\) −11.1803 −1.00000
\(126\) 7.47214 0.665671
\(127\) −1.94427 −0.172526 −0.0862631 0.996272i \(-0.527493\pi\)
−0.0862631 + 0.996272i \(0.527493\pi\)
\(128\) −13.6180 −1.20368
\(129\) −4.70820 −0.414534
\(130\) 6.38197 0.559735
\(131\) 10.6525 0.930711 0.465356 0.885124i \(-0.345926\pi\)
0.465356 + 0.885124i \(0.345926\pi\)
\(132\) 1.38197 0.120285
\(133\) 37.3607 3.23958
\(134\) 4.38197 0.378544
\(135\) −2.23607 −0.192450
\(136\) −10.8541 −0.930732
\(137\) 7.76393 0.663317 0.331659 0.943399i \(-0.392392\pi\)
0.331659 + 0.943399i \(0.392392\pi\)
\(138\) −3.85410 −0.328083
\(139\) 16.2361 1.37713 0.688563 0.725177i \(-0.258242\pi\)
0.688563 + 0.725177i \(0.258242\pi\)
\(140\) −6.38197 −0.539375
\(141\) 4.14590 0.349148
\(142\) −16.0902 −1.35026
\(143\) 3.94427 0.329837
\(144\) −4.85410 −0.404508
\(145\) 19.2705 1.60033
\(146\) 9.47214 0.783920
\(147\) −14.3262 −1.18161
\(148\) −4.23607 −0.348203
\(149\) 7.90983 0.647999 0.323999 0.946057i \(-0.394972\pi\)
0.323999 + 0.946057i \(0.394972\pi\)
\(150\) 0 0
\(151\) −17.5623 −1.42920 −0.714600 0.699533i \(-0.753391\pi\)
−0.714600 + 0.699533i \(0.753391\pi\)
\(152\) −18.0902 −1.46731
\(153\) −4.85410 −0.392431
\(154\) −16.7082 −1.34639
\(155\) 21.3820 1.71744
\(156\) 1.09017 0.0872835
\(157\) 9.00000 0.718278 0.359139 0.933284i \(-0.383070\pi\)
0.359139 + 0.933284i \(0.383070\pi\)
\(158\) 4.85410 0.386172
\(159\) −1.76393 −0.139889
\(160\) 7.56231 0.597853
\(161\) 11.0000 0.866921
\(162\) −1.61803 −0.127125
\(163\) 1.56231 0.122369 0.0611846 0.998126i \(-0.480512\pi\)
0.0611846 + 0.998126i \(0.480512\pi\)
\(164\) −1.90983 −0.149133
\(165\) 5.00000 0.389249
\(166\) −1.00000 −0.0776151
\(167\) −22.0344 −1.70508 −0.852538 0.522665i \(-0.824938\pi\)
−0.852538 + 0.522665i \(0.824938\pi\)
\(168\) 10.3262 0.796687
\(169\) −9.88854 −0.760657
\(170\) 17.5623 1.34697
\(171\) −8.09017 −0.618671
\(172\) 2.90983 0.221872
\(173\) 14.6180 1.11139 0.555694 0.831387i \(-0.312453\pi\)
0.555694 + 0.831387i \(0.312453\pi\)
\(174\) 13.9443 1.05711
\(175\) 0 0
\(176\) 10.8541 0.818159
\(177\) 1.00000 0.0751646
\(178\) −17.4721 −1.30959
\(179\) −9.47214 −0.707981 −0.353990 0.935249i \(-0.615175\pi\)
−0.353990 + 0.935249i \(0.615175\pi\)
\(180\) 1.38197 0.103006
\(181\) −11.2705 −0.837730 −0.418865 0.908048i \(-0.637572\pi\)
−0.418865 + 0.908048i \(0.637572\pi\)
\(182\) −13.1803 −0.976992
\(183\) 9.85410 0.728436
\(184\) −5.32624 −0.392655
\(185\) −15.3262 −1.12681
\(186\) 15.4721 1.13447
\(187\) 10.8541 0.793731
\(188\) −2.56231 −0.186875
\(189\) 4.61803 0.335913
\(190\) 29.2705 2.12351
\(191\) −10.4164 −0.753705 −0.376852 0.926273i \(-0.622994\pi\)
−0.376852 + 0.926273i \(0.622994\pi\)
\(192\) −4.23607 −0.305712
\(193\) 8.00000 0.575853 0.287926 0.957653i \(-0.407034\pi\)
0.287926 + 0.957653i \(0.407034\pi\)
\(194\) −4.85410 −0.348504
\(195\) 3.94427 0.282455
\(196\) 8.85410 0.632436
\(197\) 10.6525 0.758957 0.379479 0.925200i \(-0.376103\pi\)
0.379479 + 0.925200i \(0.376103\pi\)
\(198\) 3.61803 0.257122
\(199\) 3.56231 0.252525 0.126263 0.991997i \(-0.459702\pi\)
0.126263 + 0.991997i \(0.459702\pi\)
\(200\) 0 0
\(201\) 2.70820 0.191022
\(202\) 15.7082 1.10523
\(203\) −39.7984 −2.79330
\(204\) 3.00000 0.210042
\(205\) −6.90983 −0.482603
\(206\) −2.00000 −0.139347
\(207\) −2.38197 −0.165558
\(208\) 8.56231 0.593689
\(209\) 18.0902 1.25132
\(210\) −16.7082 −1.15298
\(211\) −8.85410 −0.609542 −0.304771 0.952426i \(-0.598580\pi\)
−0.304771 + 0.952426i \(0.598580\pi\)
\(212\) 1.09017 0.0748732
\(213\) −9.94427 −0.681370
\(214\) −19.5623 −1.33725
\(215\) 10.5279 0.717994
\(216\) −2.23607 −0.152145
\(217\) −44.1591 −2.99771
\(218\) 17.5623 1.18947
\(219\) 5.85410 0.395584
\(220\) −3.09017 −0.208339
\(221\) 8.56231 0.575963
\(222\) −11.0902 −0.744323
\(223\) −18.4721 −1.23699 −0.618493 0.785790i \(-0.712256\pi\)
−0.618493 + 0.785790i \(0.712256\pi\)
\(224\) −15.6180 −1.04352
\(225\) 0 0
\(226\) 14.5623 0.968670
\(227\) −28.8541 −1.91511 −0.957557 0.288244i \(-0.906929\pi\)
−0.957557 + 0.288244i \(0.906929\pi\)
\(228\) 5.00000 0.331133
\(229\) −15.8541 −1.04767 −0.523834 0.851820i \(-0.675499\pi\)
−0.523834 + 0.851820i \(0.675499\pi\)
\(230\) 8.61803 0.568256
\(231\) −10.3262 −0.679417
\(232\) 19.2705 1.26517
\(233\) −21.7082 −1.42215 −0.711076 0.703115i \(-0.751792\pi\)
−0.711076 + 0.703115i \(0.751792\pi\)
\(234\) 2.85410 0.186578
\(235\) −9.27051 −0.604741
\(236\) −0.618034 −0.0402306
\(237\) 3.00000 0.194871
\(238\) −36.2705 −2.35107
\(239\) 7.47214 0.483332 0.241666 0.970359i \(-0.422306\pi\)
0.241666 + 0.970359i \(0.422306\pi\)
\(240\) 10.8541 0.700629
\(241\) 3.41641 0.220070 0.110035 0.993928i \(-0.464904\pi\)
0.110035 + 0.993928i \(0.464904\pi\)
\(242\) 9.70820 0.624067
\(243\) −1.00000 −0.0641500
\(244\) −6.09017 −0.389883
\(245\) 32.0344 2.04661
\(246\) −5.00000 −0.318788
\(247\) 14.2705 0.908011
\(248\) 21.3820 1.35776
\(249\) −0.618034 −0.0391663
\(250\) 18.0902 1.14412
\(251\) −7.18034 −0.453219 −0.226610 0.973986i \(-0.572764\pi\)
−0.226610 + 0.973986i \(0.572764\pi\)
\(252\) −2.85410 −0.179792
\(253\) 5.32624 0.334858
\(254\) 3.14590 0.197391
\(255\) 10.8541 0.679710
\(256\) 13.5623 0.847644
\(257\) 7.41641 0.462623 0.231311 0.972880i \(-0.425698\pi\)
0.231311 + 0.972880i \(0.425698\pi\)
\(258\) 7.61803 0.474278
\(259\) 31.6525 1.96679
\(260\) −2.43769 −0.151179
\(261\) 8.61803 0.533443
\(262\) −17.2361 −1.06485
\(263\) −0.618034 −0.0381096 −0.0190548 0.999818i \(-0.506066\pi\)
−0.0190548 + 0.999818i \(0.506066\pi\)
\(264\) 5.00000 0.307729
\(265\) 3.94427 0.242295
\(266\) −60.4508 −3.70648
\(267\) −10.7984 −0.660850
\(268\) −1.67376 −0.102241
\(269\) −2.52786 −0.154127 −0.0770633 0.997026i \(-0.524554\pi\)
−0.0770633 + 0.997026i \(0.524554\pi\)
\(270\) 3.61803 0.220187
\(271\) −12.2361 −0.743288 −0.371644 0.928375i \(-0.621206\pi\)
−0.371644 + 0.928375i \(0.621206\pi\)
\(272\) 23.5623 1.42867
\(273\) −8.14590 −0.493012
\(274\) −12.5623 −0.758917
\(275\) 0 0
\(276\) 1.47214 0.0886122
\(277\) 3.47214 0.208620 0.104310 0.994545i \(-0.466737\pi\)
0.104310 + 0.994545i \(0.466737\pi\)
\(278\) −26.2705 −1.57560
\(279\) 9.56231 0.572480
\(280\) −23.0902 −1.37990
\(281\) −3.70820 −0.221213 −0.110606 0.993864i \(-0.535279\pi\)
−0.110606 + 0.993864i \(0.535279\pi\)
\(282\) −6.70820 −0.399468
\(283\) 10.2705 0.610518 0.305259 0.952269i \(-0.401257\pi\)
0.305259 + 0.952269i \(0.401257\pi\)
\(284\) 6.14590 0.364692
\(285\) 18.0902 1.07157
\(286\) −6.38197 −0.377374
\(287\) 14.2705 0.842362
\(288\) 3.38197 0.199284
\(289\) 6.56231 0.386018
\(290\) −31.1803 −1.83097
\(291\) −3.00000 −0.175863
\(292\) −3.61803 −0.211729
\(293\) 21.6180 1.26294 0.631470 0.775401i \(-0.282452\pi\)
0.631470 + 0.775401i \(0.282452\pi\)
\(294\) 23.1803 1.35190
\(295\) −2.23607 −0.130189
\(296\) −15.3262 −0.890819
\(297\) 2.23607 0.129750
\(298\) −12.7984 −0.741390
\(299\) 4.20163 0.242986
\(300\) 0 0
\(301\) −21.7426 −1.25323
\(302\) 28.4164 1.63518
\(303\) 9.70820 0.557722
\(304\) 39.2705 2.25232
\(305\) −22.0344 −1.26169
\(306\) 7.85410 0.448989
\(307\) 9.88854 0.564369 0.282185 0.959360i \(-0.408941\pi\)
0.282185 + 0.959360i \(0.408941\pi\)
\(308\) 6.38197 0.363646
\(309\) −1.23607 −0.0703175
\(310\) −34.5967 −1.96496
\(311\) −28.4508 −1.61330 −0.806650 0.591030i \(-0.798722\pi\)
−0.806650 + 0.591030i \(0.798722\pi\)
\(312\) 3.94427 0.223300
\(313\) 3.79837 0.214697 0.107348 0.994221i \(-0.465764\pi\)
0.107348 + 0.994221i \(0.465764\pi\)
\(314\) −14.5623 −0.821798
\(315\) −10.3262 −0.581818
\(316\) −1.85410 −0.104301
\(317\) −6.81966 −0.383030 −0.191515 0.981490i \(-0.561340\pi\)
−0.191515 + 0.981490i \(0.561340\pi\)
\(318\) 2.85410 0.160050
\(319\) −19.2705 −1.07894
\(320\) 9.47214 0.529508
\(321\) −12.0902 −0.674807
\(322\) −17.7984 −0.991865
\(323\) 39.2705 2.18507
\(324\) 0.618034 0.0343352
\(325\) 0 0
\(326\) −2.52786 −0.140005
\(327\) 10.8541 0.600233
\(328\) −6.90983 −0.381532
\(329\) 19.1459 1.05555
\(330\) −8.09017 −0.445349
\(331\) 29.1246 1.60083 0.800417 0.599444i \(-0.204612\pi\)
0.800417 + 0.599444i \(0.204612\pi\)
\(332\) 0.381966 0.0209631
\(333\) −6.85410 −0.375602
\(334\) 35.6525 1.95082
\(335\) −6.05573 −0.330860
\(336\) −22.4164 −1.22292
\(337\) −26.0902 −1.42122 −0.710611 0.703585i \(-0.751581\pi\)
−0.710611 + 0.703585i \(0.751581\pi\)
\(338\) 16.0000 0.870285
\(339\) 9.00000 0.488813
\(340\) −6.70820 −0.363803
\(341\) −21.3820 −1.15790
\(342\) 13.0902 0.707835
\(343\) −33.8328 −1.82680
\(344\) 10.5279 0.567624
\(345\) 5.32624 0.286755
\(346\) −23.6525 −1.27156
\(347\) 1.96556 0.105517 0.0527583 0.998607i \(-0.483199\pi\)
0.0527583 + 0.998607i \(0.483199\pi\)
\(348\) −5.32624 −0.285516
\(349\) −27.0000 −1.44528 −0.722638 0.691226i \(-0.757071\pi\)
−0.722638 + 0.691226i \(0.757071\pi\)
\(350\) 0 0
\(351\) 1.76393 0.0941517
\(352\) −7.56231 −0.403072
\(353\) 13.0344 0.693753 0.346877 0.937911i \(-0.387242\pi\)
0.346877 + 0.937911i \(0.387242\pi\)
\(354\) −1.61803 −0.0859975
\(355\) 22.2361 1.18017
\(356\) 6.67376 0.353709
\(357\) −22.4164 −1.18640
\(358\) 15.3262 0.810017
\(359\) 13.8885 0.733009 0.366505 0.930416i \(-0.380554\pi\)
0.366505 + 0.930416i \(0.380554\pi\)
\(360\) 5.00000 0.263523
\(361\) 46.4508 2.44478
\(362\) 18.2361 0.958466
\(363\) 6.00000 0.314918
\(364\) 5.03444 0.263876
\(365\) −13.0902 −0.685171
\(366\) −15.9443 −0.833420
\(367\) −23.8328 −1.24406 −0.622031 0.782992i \(-0.713692\pi\)
−0.622031 + 0.782992i \(0.713692\pi\)
\(368\) 11.5623 0.602727
\(369\) −3.09017 −0.160868
\(370\) 24.7984 1.28921
\(371\) −8.14590 −0.422914
\(372\) −5.90983 −0.306410
\(373\) 18.6738 0.966891 0.483445 0.875375i \(-0.339385\pi\)
0.483445 + 0.875375i \(0.339385\pi\)
\(374\) −17.5623 −0.908125
\(375\) 11.1803 0.577350
\(376\) −9.27051 −0.478090
\(377\) −15.2016 −0.782924
\(378\) −7.47214 −0.384325
\(379\) −4.41641 −0.226856 −0.113428 0.993546i \(-0.536183\pi\)
−0.113428 + 0.993546i \(0.536183\pi\)
\(380\) −11.1803 −0.573539
\(381\) 1.94427 0.0996081
\(382\) 16.8541 0.862331
\(383\) −13.7639 −0.703304 −0.351652 0.936131i \(-0.614380\pi\)
−0.351652 + 0.936131i \(0.614380\pi\)
\(384\) 13.6180 0.694942
\(385\) 23.0902 1.17678
\(386\) −12.9443 −0.658846
\(387\) 4.70820 0.239331
\(388\) 1.85410 0.0941278
\(389\) 23.5279 1.19291 0.596455 0.802646i \(-0.296575\pi\)
0.596455 + 0.802646i \(0.296575\pi\)
\(390\) −6.38197 −0.323163
\(391\) 11.5623 0.584731
\(392\) 32.0344 1.61798
\(393\) −10.6525 −0.537346
\(394\) −17.2361 −0.868341
\(395\) −6.70820 −0.337526
\(396\) −1.38197 −0.0694464
\(397\) 3.00000 0.150566 0.0752828 0.997162i \(-0.476014\pi\)
0.0752828 + 0.997162i \(0.476014\pi\)
\(398\) −5.76393 −0.288920
\(399\) −37.3607 −1.87037
\(400\) 0 0
\(401\) −5.90983 −0.295123 −0.147561 0.989053i \(-0.547142\pi\)
−0.147561 + 0.989053i \(0.547142\pi\)
\(402\) −4.38197 −0.218553
\(403\) −16.8673 −0.840218
\(404\) −6.00000 −0.298511
\(405\) 2.23607 0.111111
\(406\) 64.3951 3.19588
\(407\) 15.3262 0.759693
\(408\) 10.8541 0.537358
\(409\) −37.4164 −1.85012 −0.925061 0.379818i \(-0.875987\pi\)
−0.925061 + 0.379818i \(0.875987\pi\)
\(410\) 11.1803 0.552158
\(411\) −7.76393 −0.382967
\(412\) 0.763932 0.0376362
\(413\) 4.61803 0.227239
\(414\) 3.85410 0.189419
\(415\) 1.38197 0.0678380
\(416\) −5.96556 −0.292486
\(417\) −16.2361 −0.795084
\(418\) −29.2705 −1.43167
\(419\) 31.3050 1.52935 0.764673 0.644418i \(-0.222900\pi\)
0.764673 + 0.644418i \(0.222900\pi\)
\(420\) 6.38197 0.311408
\(421\) 1.00000 0.0487370 0.0243685 0.999703i \(-0.492242\pi\)
0.0243685 + 0.999703i \(0.492242\pi\)
\(422\) 14.3262 0.697390
\(423\) −4.14590 −0.201580
\(424\) 3.94427 0.191551
\(425\) 0 0
\(426\) 16.0902 0.779571
\(427\) 45.5066 2.20222
\(428\) 7.47214 0.361179
\(429\) −3.94427 −0.190431
\(430\) −17.0344 −0.821474
\(431\) 14.6180 0.704126 0.352063 0.935976i \(-0.385480\pi\)
0.352063 + 0.935976i \(0.385480\pi\)
\(432\) 4.85410 0.233543
\(433\) 19.3262 0.928760 0.464380 0.885636i \(-0.346277\pi\)
0.464380 + 0.885636i \(0.346277\pi\)
\(434\) 71.4508 3.42975
\(435\) −19.2705 −0.923950
\(436\) −6.70820 −0.321265
\(437\) 19.2705 0.921833
\(438\) −9.47214 −0.452596
\(439\) −32.3820 −1.54551 −0.772753 0.634706i \(-0.781121\pi\)
−0.772753 + 0.634706i \(0.781121\pi\)
\(440\) −11.1803 −0.533002
\(441\) 14.3262 0.682202
\(442\) −13.8541 −0.658972
\(443\) 5.61803 0.266921 0.133460 0.991054i \(-0.457391\pi\)
0.133460 + 0.991054i \(0.457391\pi\)
\(444\) 4.23607 0.201035
\(445\) 24.1459 1.14463
\(446\) 29.8885 1.41526
\(447\) −7.90983 −0.374122
\(448\) −19.5623 −0.924232
\(449\) −28.8885 −1.36333 −0.681667 0.731662i \(-0.738745\pi\)
−0.681667 + 0.731662i \(0.738745\pi\)
\(450\) 0 0
\(451\) 6.90983 0.325371
\(452\) −5.56231 −0.261629
\(453\) 17.5623 0.825149
\(454\) 46.6869 2.19113
\(455\) 18.2148 0.853922
\(456\) 18.0902 0.847150
\(457\) −26.6869 −1.24836 −0.624181 0.781280i \(-0.714567\pi\)
−0.624181 + 0.781280i \(0.714567\pi\)
\(458\) 25.6525 1.19866
\(459\) 4.85410 0.226570
\(460\) −3.29180 −0.153481
\(461\) −18.7426 −0.872932 −0.436466 0.899721i \(-0.643770\pi\)
−0.436466 + 0.899721i \(0.643770\pi\)
\(462\) 16.7082 0.777336
\(463\) 10.8541 0.504433 0.252216 0.967671i \(-0.418841\pi\)
0.252216 + 0.967671i \(0.418841\pi\)
\(464\) −41.8328 −1.94204
\(465\) −21.3820 −0.991565
\(466\) 35.1246 1.62712
\(467\) −38.8328 −1.79697 −0.898484 0.439006i \(-0.855331\pi\)
−0.898484 + 0.439006i \(0.855331\pi\)
\(468\) −1.09017 −0.0503931
\(469\) 12.5066 0.577500
\(470\) 15.0000 0.691898
\(471\) −9.00000 −0.414698
\(472\) −2.23607 −0.102923
\(473\) −10.5279 −0.484072
\(474\) −4.85410 −0.222956
\(475\) 0 0
\(476\) 13.8541 0.635002
\(477\) 1.76393 0.0807649
\(478\) −12.0902 −0.552992
\(479\) −26.0902 −1.19209 −0.596045 0.802951i \(-0.703262\pi\)
−0.596045 + 0.802951i \(0.703262\pi\)
\(480\) −7.56231 −0.345170
\(481\) 12.0902 0.551264
\(482\) −5.52786 −0.251787
\(483\) −11.0000 −0.500517
\(484\) −3.70820 −0.168555
\(485\) 6.70820 0.304604
\(486\) 1.61803 0.0733955
\(487\) 36.7426 1.66497 0.832484 0.554049i \(-0.186918\pi\)
0.832484 + 0.554049i \(0.186918\pi\)
\(488\) −22.0344 −0.997452
\(489\) −1.56231 −0.0706499
\(490\) −51.8328 −2.34157
\(491\) 26.5066 1.19623 0.598113 0.801412i \(-0.295918\pi\)
0.598113 + 0.801412i \(0.295918\pi\)
\(492\) 1.90983 0.0861018
\(493\) −41.8328 −1.88406
\(494\) −23.0902 −1.03888
\(495\) −5.00000 −0.224733
\(496\) −46.4164 −2.08416
\(497\) −45.9230 −2.05993
\(498\) 1.00000 0.0448111
\(499\) 12.4164 0.555835 0.277917 0.960605i \(-0.410356\pi\)
0.277917 + 0.960605i \(0.410356\pi\)
\(500\) −6.90983 −0.309017
\(501\) 22.0344 0.984426
\(502\) 11.6180 0.518538
\(503\) −20.7984 −0.927354 −0.463677 0.886004i \(-0.653470\pi\)
−0.463677 + 0.886004i \(0.653470\pi\)
\(504\) −10.3262 −0.459967
\(505\) −21.7082 −0.966002
\(506\) −8.61803 −0.383118
\(507\) 9.88854 0.439166
\(508\) −1.20163 −0.0533135
\(509\) −29.9230 −1.32631 −0.663157 0.748481i \(-0.730784\pi\)
−0.663157 + 0.748481i \(0.730784\pi\)
\(510\) −17.5623 −0.777672
\(511\) 27.0344 1.19593
\(512\) 5.29180 0.233867
\(513\) 8.09017 0.357190
\(514\) −12.0000 −0.529297
\(515\) 2.76393 0.121793
\(516\) −2.90983 −0.128098
\(517\) 9.27051 0.407717
\(518\) −51.2148 −2.25025
\(519\) −14.6180 −0.641660
\(520\) −8.81966 −0.386768
\(521\) −14.5066 −0.635545 −0.317772 0.948167i \(-0.602935\pi\)
−0.317772 + 0.948167i \(0.602935\pi\)
\(522\) −13.9443 −0.610324
\(523\) 31.9443 1.39683 0.698413 0.715695i \(-0.253890\pi\)
0.698413 + 0.715695i \(0.253890\pi\)
\(524\) 6.58359 0.287606
\(525\) 0 0
\(526\) 1.00000 0.0436021
\(527\) −46.4164 −2.02193
\(528\) −10.8541 −0.472364
\(529\) −17.3262 −0.753315
\(530\) −6.38197 −0.277215
\(531\) −1.00000 −0.0433963
\(532\) 23.0902 1.00109
\(533\) 5.45085 0.236103
\(534\) 17.4721 0.756093
\(535\) 27.0344 1.16880
\(536\) −6.05573 −0.261568
\(537\) 9.47214 0.408753
\(538\) 4.09017 0.176340
\(539\) −32.0344 −1.37982
\(540\) −1.38197 −0.0594703
\(541\) 14.1246 0.607264 0.303632 0.952789i \(-0.401801\pi\)
0.303632 + 0.952789i \(0.401801\pi\)
\(542\) 19.7984 0.850413
\(543\) 11.2705 0.483664
\(544\) −16.4164 −0.703848
\(545\) −24.2705 −1.03963
\(546\) 13.1803 0.564066
\(547\) 32.5623 1.39226 0.696132 0.717914i \(-0.254903\pi\)
0.696132 + 0.717914i \(0.254903\pi\)
\(548\) 4.79837 0.204976
\(549\) −9.85410 −0.420563
\(550\) 0 0
\(551\) −69.7214 −2.97023
\(552\) 5.32624 0.226700
\(553\) 13.8541 0.589136
\(554\) −5.61803 −0.238687
\(555\) 15.3262 0.650563
\(556\) 10.0344 0.425555
\(557\) 1.41641 0.0600151 0.0300076 0.999550i \(-0.490447\pi\)
0.0300076 + 0.999550i \(0.490447\pi\)
\(558\) −15.4721 −0.654988
\(559\) −8.30495 −0.351262
\(560\) 50.1246 2.11815
\(561\) −10.8541 −0.458261
\(562\) 6.00000 0.253095
\(563\) 28.5967 1.20521 0.602605 0.798040i \(-0.294130\pi\)
0.602605 + 0.798040i \(0.294130\pi\)
\(564\) 2.56231 0.107893
\(565\) −20.1246 −0.846649
\(566\) −16.6180 −0.698508
\(567\) −4.61803 −0.193939
\(568\) 22.2361 0.933005
\(569\) 26.5066 1.11121 0.555607 0.831445i \(-0.312486\pi\)
0.555607 + 0.831445i \(0.312486\pi\)
\(570\) −29.2705 −1.22601
\(571\) 28.4164 1.18919 0.594595 0.804025i \(-0.297312\pi\)
0.594595 + 0.804025i \(0.297312\pi\)
\(572\) 2.43769 0.101925
\(573\) 10.4164 0.435152
\(574\) −23.0902 −0.963765
\(575\) 0 0
\(576\) 4.23607 0.176503
\(577\) 21.4721 0.893897 0.446948 0.894560i \(-0.352511\pi\)
0.446948 + 0.894560i \(0.352511\pi\)
\(578\) −10.6180 −0.441652
\(579\) −8.00000 −0.332469
\(580\) 11.9098 0.494529
\(581\) −2.85410 −0.118408
\(582\) 4.85410 0.201209
\(583\) −3.94427 −0.163355
\(584\) −13.0902 −0.541675
\(585\) −3.94427 −0.163076
\(586\) −34.9787 −1.44496
\(587\) 29.3607 1.21184 0.605922 0.795524i \(-0.292804\pi\)
0.605922 + 0.795524i \(0.292804\pi\)
\(588\) −8.85410 −0.365137
\(589\) −77.3607 −3.18759
\(590\) 3.61803 0.148952
\(591\) −10.6525 −0.438184
\(592\) 33.2705 1.36741
\(593\) 4.90983 0.201623 0.100811 0.994906i \(-0.467856\pi\)
0.100811 + 0.994906i \(0.467856\pi\)
\(594\) −3.61803 −0.148450
\(595\) 50.1246 2.05491
\(596\) 4.88854 0.200243
\(597\) −3.56231 −0.145795
\(598\) −6.79837 −0.278006
\(599\) −28.6525 −1.17071 −0.585354 0.810778i \(-0.699045\pi\)
−0.585354 + 0.810778i \(0.699045\pi\)
\(600\) 0 0
\(601\) 25.6180 1.04498 0.522491 0.852645i \(-0.325003\pi\)
0.522491 + 0.852645i \(0.325003\pi\)
\(602\) 35.1803 1.43384
\(603\) −2.70820 −0.110287
\(604\) −10.8541 −0.441647
\(605\) −13.4164 −0.545455
\(606\) −15.7082 −0.638102
\(607\) 17.6525 0.716492 0.358246 0.933627i \(-0.383375\pi\)
0.358246 + 0.933627i \(0.383375\pi\)
\(608\) −27.3607 −1.10962
\(609\) 39.7984 1.61271
\(610\) 35.6525 1.44353
\(611\) 7.31308 0.295856
\(612\) −3.00000 −0.121268
\(613\) −7.34752 −0.296764 −0.148382 0.988930i \(-0.547406\pi\)
−0.148382 + 0.988930i \(0.547406\pi\)
\(614\) −16.0000 −0.645707
\(615\) 6.90983 0.278631
\(616\) 23.0902 0.930329
\(617\) 28.8541 1.16162 0.580811 0.814038i \(-0.302735\pi\)
0.580811 + 0.814038i \(0.302735\pi\)
\(618\) 2.00000 0.0804518
\(619\) −28.1246 −1.13042 −0.565212 0.824946i \(-0.691206\pi\)
−0.565212 + 0.824946i \(0.691206\pi\)
\(620\) 13.2148 0.530718
\(621\) 2.38197 0.0955850
\(622\) 46.0344 1.84581
\(623\) −49.8673 −1.99789
\(624\) −8.56231 −0.342767
\(625\) −25.0000 −1.00000
\(626\) −6.14590 −0.245639
\(627\) −18.0902 −0.722452
\(628\) 5.56231 0.221960
\(629\) 33.2705 1.32658
\(630\) 16.7082 0.665671
\(631\) −13.5836 −0.540754 −0.270377 0.962754i \(-0.587148\pi\)
−0.270377 + 0.962754i \(0.587148\pi\)
\(632\) −6.70820 −0.266838
\(633\) 8.85410 0.351919
\(634\) 11.0344 0.438234
\(635\) −4.34752 −0.172526
\(636\) −1.09017 −0.0432281
\(637\) −25.2705 −1.00125
\(638\) 31.1803 1.23444
\(639\) 9.94427 0.393389
\(640\) −30.4508 −1.20368
\(641\) −10.0557 −0.397177 −0.198589 0.980083i \(-0.563636\pi\)
−0.198589 + 0.980083i \(0.563636\pi\)
\(642\) 19.5623 0.772063
\(643\) −13.1459 −0.518424 −0.259212 0.965821i \(-0.583463\pi\)
−0.259212 + 0.965821i \(0.583463\pi\)
\(644\) 6.79837 0.267893
\(645\) −10.5279 −0.414534
\(646\) −63.5410 −2.49999
\(647\) 27.0557 1.06367 0.531835 0.846848i \(-0.321503\pi\)
0.531835 + 0.846848i \(0.321503\pi\)
\(648\) 2.23607 0.0878410
\(649\) 2.23607 0.0877733
\(650\) 0 0
\(651\) 44.1591 1.73073
\(652\) 0.965558 0.0378142
\(653\) 28.9098 1.13133 0.565665 0.824635i \(-0.308620\pi\)
0.565665 + 0.824635i \(0.308620\pi\)
\(654\) −17.5623 −0.686741
\(655\) 23.8197 0.930711
\(656\) 15.0000 0.585652
\(657\) −5.85410 −0.228390
\(658\) −30.9787 −1.20768
\(659\) −4.85410 −0.189089 −0.0945445 0.995521i \(-0.530139\pi\)
−0.0945445 + 0.995521i \(0.530139\pi\)
\(660\) 3.09017 0.120285
\(661\) −11.7426 −0.456736 −0.228368 0.973575i \(-0.573339\pi\)
−0.228368 + 0.973575i \(0.573339\pi\)
\(662\) −47.1246 −1.83155
\(663\) −8.56231 −0.332532
\(664\) 1.38197 0.0536307
\(665\) 83.5410 3.23958
\(666\) 11.0902 0.429735
\(667\) −20.5279 −0.794842
\(668\) −13.6180 −0.526898
\(669\) 18.4721 0.714174
\(670\) 9.79837 0.378544
\(671\) 22.0344 0.850630
\(672\) 15.6180 0.602479
\(673\) 0.472136 0.0181995 0.00909975 0.999959i \(-0.497103\pi\)
0.00909975 + 0.999959i \(0.497103\pi\)
\(674\) 42.2148 1.62605
\(675\) 0 0
\(676\) −6.11146 −0.235056
\(677\) −50.1803 −1.92859 −0.964294 0.264836i \(-0.914682\pi\)
−0.964294 + 0.264836i \(0.914682\pi\)
\(678\) −14.5623 −0.559262
\(679\) −13.8541 −0.531672
\(680\) −24.2705 −0.930732
\(681\) 28.8541 1.10569
\(682\) 34.5967 1.32478
\(683\) 39.0000 1.49229 0.746147 0.665782i \(-0.231902\pi\)
0.746147 + 0.665782i \(0.231902\pi\)
\(684\) −5.00000 −0.191180
\(685\) 17.3607 0.663317
\(686\) 54.7426 2.09008
\(687\) 15.8541 0.604872
\(688\) −22.8541 −0.871304
\(689\) −3.11146 −0.118537
\(690\) −8.61803 −0.328083
\(691\) −5.12461 −0.194949 −0.0974747 0.995238i \(-0.531077\pi\)
−0.0974747 + 0.995238i \(0.531077\pi\)
\(692\) 9.03444 0.343438
\(693\) 10.3262 0.392261
\(694\) −3.18034 −0.120724
\(695\) 36.3050 1.37713
\(696\) −19.2705 −0.730447
\(697\) 15.0000 0.568166
\(698\) 43.6869 1.65357
\(699\) 21.7082 0.821080
\(700\) 0 0
\(701\) 42.7984 1.61647 0.808236 0.588859i \(-0.200422\pi\)
0.808236 + 0.588859i \(0.200422\pi\)
\(702\) −2.85410 −0.107721
\(703\) 55.4508 2.09137
\(704\) −9.47214 −0.356995
\(705\) 9.27051 0.349148
\(706\) −21.0902 −0.793739
\(707\) 44.8328 1.68611
\(708\) 0.618034 0.0232271
\(709\) −14.2918 −0.536740 −0.268370 0.963316i \(-0.586485\pi\)
−0.268370 + 0.963316i \(0.586485\pi\)
\(710\) −35.9787 −1.35026
\(711\) −3.00000 −0.112509
\(712\) 24.1459 0.904906
\(713\) −22.7771 −0.853009
\(714\) 36.2705 1.35739
\(715\) 8.81966 0.329837
\(716\) −5.85410 −0.218778
\(717\) −7.47214 −0.279052
\(718\) −22.4721 −0.838653
\(719\) 12.3820 0.461769 0.230885 0.972981i \(-0.425838\pi\)
0.230885 + 0.972981i \(0.425838\pi\)
\(720\) −10.8541 −0.404508
\(721\) −5.70820 −0.212585
\(722\) −75.1591 −2.79713
\(723\) −3.41641 −0.127058
\(724\) −6.96556 −0.258873
\(725\) 0 0
\(726\) −9.70820 −0.360305
\(727\) 26.0000 0.964287 0.482143 0.876092i \(-0.339858\pi\)
0.482143 + 0.876092i \(0.339858\pi\)
\(728\) 18.2148 0.675085
\(729\) 1.00000 0.0370370
\(730\) 21.1803 0.783920
\(731\) −22.8541 −0.845289
\(732\) 6.09017 0.225099
\(733\) 29.4721 1.08858 0.544289 0.838898i \(-0.316799\pi\)
0.544289 + 0.838898i \(0.316799\pi\)
\(734\) 38.5623 1.42336
\(735\) −32.0344 −1.18161
\(736\) −8.05573 −0.296938
\(737\) 6.05573 0.223066
\(738\) 5.00000 0.184053
\(739\) −39.1033 −1.43844 −0.719220 0.694783i \(-0.755500\pi\)
−0.719220 + 0.694783i \(0.755500\pi\)
\(740\) −9.47214 −0.348203
\(741\) −14.2705 −0.524240
\(742\) 13.1803 0.483865
\(743\) −22.3820 −0.821115 −0.410557 0.911835i \(-0.634666\pi\)
−0.410557 + 0.911835i \(0.634666\pi\)
\(744\) −21.3820 −0.783901
\(745\) 17.6869 0.647999
\(746\) −30.2148 −1.10624
\(747\) 0.618034 0.0226127
\(748\) 6.70820 0.245276
\(749\) −55.8328 −2.04009
\(750\) −18.0902 −0.660560
\(751\) −45.1246 −1.64662 −0.823310 0.567592i \(-0.807875\pi\)
−0.823310 + 0.567592i \(0.807875\pi\)
\(752\) 20.1246 0.733869
\(753\) 7.18034 0.261666
\(754\) 24.5967 0.895761
\(755\) −39.2705 −1.42920
\(756\) 2.85410 0.103803
\(757\) −43.6180 −1.58532 −0.792662 0.609661i \(-0.791306\pi\)
−0.792662 + 0.609661i \(0.791306\pi\)
\(758\) 7.14590 0.259551
\(759\) −5.32624 −0.193330
\(760\) −40.4508 −1.46731
\(761\) 17.2918 0.626827 0.313414 0.949617i \(-0.398527\pi\)
0.313414 + 0.949617i \(0.398527\pi\)
\(762\) −3.14590 −0.113964
\(763\) 50.1246 1.81463
\(764\) −6.43769 −0.232908
\(765\) −10.8541 −0.392431
\(766\) 22.2705 0.804666
\(767\) 1.76393 0.0636919
\(768\) −13.5623 −0.489388
\(769\) −35.9787 −1.29743 −0.648713 0.761033i \(-0.724692\pi\)
−0.648713 + 0.761033i \(0.724692\pi\)
\(770\) −37.3607 −1.34639
\(771\) −7.41641 −0.267095
\(772\) 4.94427 0.177948
\(773\) −2.65248 −0.0954029 −0.0477015 0.998862i \(-0.515190\pi\)
−0.0477015 + 0.998862i \(0.515190\pi\)
\(774\) −7.61803 −0.273825
\(775\) 0 0
\(776\) 6.70820 0.240810
\(777\) −31.6525 −1.13553
\(778\) −38.0689 −1.36484
\(779\) 25.0000 0.895718
\(780\) 2.43769 0.0872835
\(781\) −22.2361 −0.795669
\(782\) −18.7082 −0.669004
\(783\) −8.61803 −0.307983
\(784\) −69.5410 −2.48361
\(785\) 20.1246 0.718278
\(786\) 17.2361 0.614790
\(787\) −17.7082 −0.631229 −0.315615 0.948887i \(-0.602211\pi\)
−0.315615 + 0.948887i \(0.602211\pi\)
\(788\) 6.58359 0.234531
\(789\) 0.618034 0.0220026
\(790\) 10.8541 0.386172
\(791\) 41.5623 1.47779
\(792\) −5.00000 −0.177667
\(793\) 17.3820 0.617252
\(794\) −4.85410 −0.172266
\(795\) −3.94427 −0.139889
\(796\) 2.20163 0.0780346
\(797\) 3.23607 0.114627 0.0573137 0.998356i \(-0.481746\pi\)
0.0573137 + 0.998356i \(0.481746\pi\)
\(798\) 60.4508 2.13994
\(799\) 20.1246 0.711958
\(800\) 0 0
\(801\) 10.7984 0.381542
\(802\) 9.56231 0.337657
\(803\) 13.0902 0.461942
\(804\) 1.67376 0.0590290
\(805\) 24.5967 0.866921
\(806\) 27.2918 0.961313
\(807\) 2.52786 0.0889850
\(808\) −21.7082 −0.763692
\(809\) 23.3262 0.820107 0.410053 0.912062i \(-0.365510\pi\)
0.410053 + 0.912062i \(0.365510\pi\)
\(810\) −3.61803 −0.127125
\(811\) −16.6525 −0.584748 −0.292374 0.956304i \(-0.594445\pi\)
−0.292374 + 0.956304i \(0.594445\pi\)
\(812\) −24.5967 −0.863177
\(813\) 12.2361 0.429138
\(814\) −24.7984 −0.869183
\(815\) 3.49342 0.122369
\(816\) −23.5623 −0.824846
\(817\) −38.0902 −1.33261
\(818\) 60.5410 2.11677
\(819\) 8.14590 0.284641
\(820\) −4.27051 −0.149133
\(821\) 23.9230 0.834918 0.417459 0.908696i \(-0.362921\pi\)
0.417459 + 0.908696i \(0.362921\pi\)
\(822\) 12.5623 0.438161
\(823\) 24.7082 0.861274 0.430637 0.902525i \(-0.358289\pi\)
0.430637 + 0.902525i \(0.358289\pi\)
\(824\) 2.76393 0.0962861
\(825\) 0 0
\(826\) −7.47214 −0.259989
\(827\) −25.1803 −0.875606 −0.437803 0.899071i \(-0.644243\pi\)
−0.437803 + 0.899071i \(0.644243\pi\)
\(828\) −1.47214 −0.0511603
\(829\) 14.6869 0.510098 0.255049 0.966928i \(-0.417908\pi\)
0.255049 + 0.966928i \(0.417908\pi\)
\(830\) −2.23607 −0.0776151
\(831\) −3.47214 −0.120447
\(832\) −7.47214 −0.259050
\(833\) −69.5410 −2.40945
\(834\) 26.2705 0.909673
\(835\) −49.2705 −1.70508
\(836\) 11.1803 0.386680
\(837\) −9.56231 −0.330522
\(838\) −50.6525 −1.74976
\(839\) −47.1803 −1.62885 −0.814423 0.580271i \(-0.802946\pi\)
−0.814423 + 0.580271i \(0.802946\pi\)
\(840\) 23.0902 0.796687
\(841\) 45.2705 1.56105
\(842\) −1.61803 −0.0557611
\(843\) 3.70820 0.127717
\(844\) −5.47214 −0.188359
\(845\) −22.1115 −0.760657
\(846\) 6.70820 0.230633
\(847\) 27.7082 0.952065
\(848\) −8.56231 −0.294031
\(849\) −10.2705 −0.352483
\(850\) 0 0
\(851\) 16.3262 0.559656
\(852\) −6.14590 −0.210555
\(853\) −33.0344 −1.13108 −0.565539 0.824722i \(-0.691332\pi\)
−0.565539 + 0.824722i \(0.691332\pi\)
\(854\) −73.6312 −2.51961
\(855\) −18.0902 −0.618671
\(856\) 27.0344 0.924018
\(857\) 42.7771 1.46124 0.730619 0.682786i \(-0.239232\pi\)
0.730619 + 0.682786i \(0.239232\pi\)
\(858\) 6.38197 0.217877
\(859\) 24.4721 0.834979 0.417489 0.908682i \(-0.362910\pi\)
0.417489 + 0.908682i \(0.362910\pi\)
\(860\) 6.50658 0.221872
\(861\) −14.2705 −0.486338
\(862\) −23.6525 −0.805607
\(863\) 43.7984 1.49091 0.745457 0.666554i \(-0.232231\pi\)
0.745457 + 0.666554i \(0.232231\pi\)
\(864\) −3.38197 −0.115057
\(865\) 32.6869 1.11139
\(866\) −31.2705 −1.06262
\(867\) −6.56231 −0.222868
\(868\) −27.2918 −0.926344
\(869\) 6.70820 0.227560
\(870\) 31.1803 1.05711
\(871\) 4.77709 0.161865
\(872\) −24.2705 −0.821903
\(873\) 3.00000 0.101535
\(874\) −31.1803 −1.05469
\(875\) 51.6312 1.74545
\(876\) 3.61803 0.122242
\(877\) −51.8885 −1.75215 −0.876076 0.482173i \(-0.839848\pi\)
−0.876076 + 0.482173i \(0.839848\pi\)
\(878\) 52.3951 1.76825
\(879\) −21.6180 −0.729158
\(880\) 24.2705 0.818159
\(881\) −2.29180 −0.0772126 −0.0386063 0.999254i \(-0.512292\pi\)
−0.0386063 + 0.999254i \(0.512292\pi\)
\(882\) −23.1803 −0.780523
\(883\) 3.41641 0.114971 0.0574856 0.998346i \(-0.481692\pi\)
0.0574856 + 0.998346i \(0.481692\pi\)
\(884\) 5.29180 0.177982
\(885\) 2.23607 0.0751646
\(886\) −9.09017 −0.305390
\(887\) −31.4721 −1.05673 −0.528365 0.849017i \(-0.677195\pi\)
−0.528365 + 0.849017i \(0.677195\pi\)
\(888\) 15.3262 0.514315
\(889\) 8.97871 0.301136
\(890\) −39.0689 −1.30959
\(891\) −2.23607 −0.0749111
\(892\) −11.4164 −0.382250
\(893\) 33.5410 1.12241
\(894\) 12.7984 0.428042
\(895\) −21.1803 −0.707981
\(896\) 62.8885 2.10096
\(897\) −4.20163 −0.140288
\(898\) 46.7426 1.55982
\(899\) 82.4083 2.74847
\(900\) 0 0
\(901\) −8.56231 −0.285252
\(902\) −11.1803 −0.372265
\(903\) 21.7426 0.723550
\(904\) −20.1246 −0.669335
\(905\) −25.2016 −0.837730
\(906\) −28.4164 −0.944072
\(907\) −2.05573 −0.0682593 −0.0341297 0.999417i \(-0.510866\pi\)
−0.0341297 + 0.999417i \(0.510866\pi\)
\(908\) −17.8328 −0.591803
\(909\) −9.70820 −0.322001
\(910\) −29.4721 −0.976992
\(911\) 42.1033 1.39495 0.697473 0.716611i \(-0.254308\pi\)
0.697473 + 0.716611i \(0.254308\pi\)
\(912\) −39.2705 −1.30038
\(913\) −1.38197 −0.0457364
\(914\) 43.1803 1.42828
\(915\) 22.0344 0.728436
\(916\) −9.79837 −0.323747
\(917\) −49.1935 −1.62451
\(918\) −7.85410 −0.259224
\(919\) 35.5967 1.17423 0.587114 0.809504i \(-0.300264\pi\)
0.587114 + 0.809504i \(0.300264\pi\)
\(920\) −11.9098 −0.392655
\(921\) −9.88854 −0.325839
\(922\) 30.3262 0.998741
\(923\) −17.5410 −0.577370
\(924\) −6.38197 −0.209951
\(925\) 0 0
\(926\) −17.5623 −0.577133
\(927\) 1.23607 0.0405978
\(928\) 29.1459 0.956761
\(929\) −35.2918 −1.15789 −0.578943 0.815368i \(-0.696535\pi\)
−0.578943 + 0.815368i \(0.696535\pi\)
\(930\) 34.5967 1.13447
\(931\) −115.902 −3.79852
\(932\) −13.4164 −0.439469
\(933\) 28.4508 0.931439
\(934\) 62.8328 2.05595
\(935\) 24.2705 0.793731
\(936\) −3.94427 −0.128923
\(937\) −37.7214 −1.23230 −0.616152 0.787628i \(-0.711309\pi\)
−0.616152 + 0.787628i \(0.711309\pi\)
\(938\) −20.2361 −0.660731
\(939\) −3.79837 −0.123955
\(940\) −5.72949 −0.186875
\(941\) −22.3050 −0.727121 −0.363560 0.931571i \(-0.618439\pi\)
−0.363560 + 0.931571i \(0.618439\pi\)
\(942\) 14.5623 0.474466
\(943\) 7.36068 0.239697
\(944\) 4.85410 0.157988
\(945\) 10.3262 0.335913
\(946\) 17.0344 0.553837
\(947\) −4.03444 −0.131102 −0.0655509 0.997849i \(-0.520880\pi\)
−0.0655509 + 0.997849i \(0.520880\pi\)
\(948\) 1.85410 0.0602184
\(949\) 10.3262 0.335204
\(950\) 0 0
\(951\) 6.81966 0.221143
\(952\) 50.1246 1.62455
\(953\) −12.9443 −0.419306 −0.209653 0.977776i \(-0.567233\pi\)
−0.209653 + 0.977776i \(0.567233\pi\)
\(954\) −2.85410 −0.0924050
\(955\) −23.2918 −0.753705
\(956\) 4.61803 0.149358
\(957\) 19.2705 0.622927
\(958\) 42.2148 1.36390
\(959\) −35.8541 −1.15779
\(960\) −9.47214 −0.305712
\(961\) 60.4377 1.94960
\(962\) −19.5623 −0.630714
\(963\) 12.0902 0.389600
\(964\) 2.11146 0.0680054
\(965\) 17.8885 0.575853
\(966\) 17.7984 0.572653
\(967\) −18.2705 −0.587540 −0.293770 0.955876i \(-0.594910\pi\)
−0.293770 + 0.955876i \(0.594910\pi\)
\(968\) −13.4164 −0.431220
\(969\) −39.2705 −1.26155
\(970\) −10.8541 −0.348504
\(971\) 22.2016 0.712484 0.356242 0.934394i \(-0.384058\pi\)
0.356242 + 0.934394i \(0.384058\pi\)
\(972\) −0.618034 −0.0198234
\(973\) −74.9787 −2.40371
\(974\) −59.4508 −1.90493
\(975\) 0 0
\(976\) 47.8328 1.53109
\(977\) 14.8885 0.476327 0.238163 0.971225i \(-0.423455\pi\)
0.238163 + 0.971225i \(0.423455\pi\)
\(978\) 2.52786 0.0808322
\(979\) −24.1459 −0.771706
\(980\) 19.7984 0.632436
\(981\) −10.8541 −0.346545
\(982\) −42.8885 −1.36863
\(983\) −58.5967 −1.86895 −0.934473 0.356034i \(-0.884129\pi\)
−0.934473 + 0.356034i \(0.884129\pi\)
\(984\) 6.90983 0.220277
\(985\) 23.8197 0.758957
\(986\) 67.6869 2.15559
\(987\) −19.1459 −0.609421
\(988\) 8.81966 0.280591
\(989\) −11.2148 −0.356609
\(990\) 8.09017 0.257122
\(991\) −18.7426 −0.595380 −0.297690 0.954663i \(-0.596216\pi\)
−0.297690 + 0.954663i \(0.596216\pi\)
\(992\) 32.3394 1.02678
\(993\) −29.1246 −0.924242
\(994\) 74.3050 2.35681
\(995\) 7.96556 0.252525
\(996\) −0.381966 −0.0121031
\(997\) 1.78522 0.0565384 0.0282692 0.999600i \(-0.491000\pi\)
0.0282692 + 0.999600i \(0.491000\pi\)
\(998\) −20.0902 −0.635943
\(999\) 6.85410 0.216854
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.1.1 2
3.2 odd 2 531.2.a.b.1.2 2
4.3 odd 2 2832.2.a.o.1.2 2
5.4 even 2 4425.2.a.t.1.2 2
7.6 odd 2 8673.2.a.k.1.1 2
12.11 even 2 8496.2.a.bb.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.2.a.b.1.1 2 1.1 even 1 trivial
531.2.a.b.1.2 2 3.2 odd 2
2832.2.a.o.1.2 2 4.3 odd 2
4425.2.a.t.1.2 2 5.4 even 2
8496.2.a.bb.1.1 2 12.11 even 2
8673.2.a.k.1.1 2 7.6 odd 2