Properties

Label 567.2.a.d.1.1
Level $567$
Weight $2$
Character 567.1
Self dual yes
Analytic conductor $4.528$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.52751779461\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.321.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.46050\) of defining polynomial
Character \(\chi\) \(=\) 567.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.46050 q^{2} +4.05408 q^{4} -2.59358 q^{5} -1.00000 q^{7} -5.05408 q^{8} +O(q^{10})\) \(q-2.46050 q^{2} +4.05408 q^{4} -2.59358 q^{5} -1.00000 q^{7} -5.05408 q^{8} +6.38151 q^{10} +4.51459 q^{11} +1.00000 q^{13} +2.46050 q^{14} +4.32743 q^{16} -0.945916 q^{17} -4.05408 q^{19} -10.5146 q^{20} -11.1082 q^{22} -0.273346 q^{23} +1.72665 q^{25} -2.46050 q^{26} -4.05408 q^{28} +2.46050 q^{29} +2.32743 q^{31} -0.539495 q^{32} +2.32743 q^{34} +2.59358 q^{35} +1.78074 q^{37} +9.97509 q^{38} +13.1082 q^{40} -6.40642 q^{41} -10.4356 q^{43} +18.3025 q^{44} +0.672570 q^{46} -12.1623 q^{47} +1.00000 q^{49} -4.24844 q^{50} +4.05408 q^{52} -6.27335 q^{53} -11.7089 q^{55} +5.05408 q^{56} -6.05408 q^{58} -2.72665 q^{59} -2.27335 q^{61} -5.72665 q^{62} -7.32743 q^{64} -2.59358 q^{65} -15.8171 q^{67} -3.83482 q^{68} -6.38151 q^{70} +3.27335 q^{71} -1.50739 q^{73} -4.38151 q^{74} -16.4356 q^{76} -4.51459 q^{77} +14.7089 q^{79} -11.2235 q^{80} +15.7630 q^{82} -0.945916 q^{83} +2.45331 q^{85} +25.6768 q^{86} -22.8171 q^{88} -14.3566 q^{89} -1.00000 q^{91} -1.10817 q^{92} +29.9253 q^{94} +10.5146 q^{95} -11.4897 q^{97} -2.46050 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - q^{2} + 3q^{4} - 5q^{5} - 3q^{7} - 6q^{8} + O(q^{10}) \) \( 3q - q^{2} + 3q^{4} - 5q^{5} - 3q^{7} - 6q^{8} - 2q^{11} + 3q^{13} + q^{14} + 3q^{16} - 12q^{17} - 3q^{19} - 16q^{20} - 15q^{22} + 6q^{25} - q^{26} - 3q^{28} + q^{29} - 3q^{31} - 8q^{32} - 3q^{34} + 5q^{35} - 3q^{37} + 8q^{38} + 21q^{40} - 22q^{41} - 3q^{43} + 23q^{44} + 12q^{46} - 9q^{47} + 3q^{49} + 10q^{50} + 3q^{52} - 18q^{53} - 6q^{55} + 6q^{56} - 9q^{58} - 9q^{59} - 6q^{61} - 18q^{62} - 12q^{64} - 5q^{65} + 6q^{68} + 9q^{71} + 3q^{73} + 6q^{74} - 21q^{76} + 2q^{77} + 15q^{79} + 11q^{80} + 9q^{82} - 12q^{83} + 9q^{85} + 34q^{86} - 21q^{88} - 2q^{89} - 3q^{91} + 15q^{92} + 24q^{94} + 16q^{95} + 3q^{97} - q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.46050 −1.73984 −0.869920 0.493193i \(-0.835830\pi\)
−0.869920 + 0.493193i \(0.835830\pi\)
\(3\) 0 0
\(4\) 4.05408 2.02704
\(5\) −2.59358 −1.15988 −0.579942 0.814658i \(-0.696925\pi\)
−0.579942 + 0.814658i \(0.696925\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −5.05408 −1.78689
\(9\) 0 0
\(10\) 6.38151 2.01801
\(11\) 4.51459 1.36120 0.680600 0.732655i \(-0.261719\pi\)
0.680600 + 0.732655i \(0.261719\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 2.46050 0.657598
\(15\) 0 0
\(16\) 4.32743 1.08186
\(17\) −0.945916 −0.229418 −0.114709 0.993399i \(-0.536594\pi\)
−0.114709 + 0.993399i \(0.536594\pi\)
\(18\) 0 0
\(19\) −4.05408 −0.930071 −0.465035 0.885292i \(-0.653958\pi\)
−0.465035 + 0.885292i \(0.653958\pi\)
\(20\) −10.5146 −2.35113
\(21\) 0 0
\(22\) −11.1082 −2.36827
\(23\) −0.273346 −0.0569966 −0.0284983 0.999594i \(-0.509073\pi\)
−0.0284983 + 0.999594i \(0.509073\pi\)
\(24\) 0 0
\(25\) 1.72665 0.345331
\(26\) −2.46050 −0.482545
\(27\) 0 0
\(28\) −4.05408 −0.766150
\(29\) 2.46050 0.456904 0.228452 0.973555i \(-0.426634\pi\)
0.228452 + 0.973555i \(0.426634\pi\)
\(30\) 0 0
\(31\) 2.32743 0.418019 0.209009 0.977914i \(-0.432976\pi\)
0.209009 + 0.977914i \(0.432976\pi\)
\(32\) −0.539495 −0.0953702
\(33\) 0 0
\(34\) 2.32743 0.399151
\(35\) 2.59358 0.438395
\(36\) 0 0
\(37\) 1.78074 0.292752 0.146376 0.989229i \(-0.453239\pi\)
0.146376 + 0.989229i \(0.453239\pi\)
\(38\) 9.97509 1.61817
\(39\) 0 0
\(40\) 13.1082 2.07258
\(41\) −6.40642 −1.00051 −0.500257 0.865877i \(-0.666761\pi\)
−0.500257 + 0.865877i \(0.666761\pi\)
\(42\) 0 0
\(43\) −10.4356 −1.59141 −0.795707 0.605682i \(-0.792900\pi\)
−0.795707 + 0.605682i \(0.792900\pi\)
\(44\) 18.3025 2.75921
\(45\) 0 0
\(46\) 0.672570 0.0991650
\(47\) −12.1623 −1.77405 −0.887023 0.461724i \(-0.847231\pi\)
−0.887023 + 0.461724i \(0.847231\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −4.24844 −0.600820
\(51\) 0 0
\(52\) 4.05408 0.562200
\(53\) −6.27335 −0.861710 −0.430855 0.902421i \(-0.641788\pi\)
−0.430855 + 0.902421i \(0.641788\pi\)
\(54\) 0 0
\(55\) −11.7089 −1.57883
\(56\) 5.05408 0.675380
\(57\) 0 0
\(58\) −6.05408 −0.794940
\(59\) −2.72665 −0.354980 −0.177490 0.984123i \(-0.556798\pi\)
−0.177490 + 0.984123i \(0.556798\pi\)
\(60\) 0 0
\(61\) −2.27335 −0.291072 −0.145536 0.989353i \(-0.546491\pi\)
−0.145536 + 0.989353i \(0.546491\pi\)
\(62\) −5.72665 −0.727286
\(63\) 0 0
\(64\) −7.32743 −0.915929
\(65\) −2.59358 −0.321694
\(66\) 0 0
\(67\) −15.8171 −1.93237 −0.966184 0.257854i \(-0.916985\pi\)
−0.966184 + 0.257854i \(0.916985\pi\)
\(68\) −3.83482 −0.465041
\(69\) 0 0
\(70\) −6.38151 −0.762737
\(71\) 3.27335 0.388475 0.194237 0.980955i \(-0.437777\pi\)
0.194237 + 0.980955i \(0.437777\pi\)
\(72\) 0 0
\(73\) −1.50739 −0.176427 −0.0882134 0.996102i \(-0.528116\pi\)
−0.0882134 + 0.996102i \(0.528116\pi\)
\(74\) −4.38151 −0.509341
\(75\) 0 0
\(76\) −16.4356 −1.88529
\(77\) −4.51459 −0.514485
\(78\) 0 0
\(79\) 14.7089 1.65489 0.827443 0.561550i \(-0.189795\pi\)
0.827443 + 0.561550i \(0.189795\pi\)
\(80\) −11.2235 −1.25483
\(81\) 0 0
\(82\) 15.7630 1.74074
\(83\) −0.945916 −0.103828 −0.0519139 0.998652i \(-0.516532\pi\)
−0.0519139 + 0.998652i \(0.516532\pi\)
\(84\) 0 0
\(85\) 2.45331 0.266099
\(86\) 25.6768 2.76881
\(87\) 0 0
\(88\) −22.8171 −2.43231
\(89\) −14.3566 −1.52180 −0.760899 0.648871i \(-0.775242\pi\)
−0.760899 + 0.648871i \(0.775242\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) −1.10817 −0.115535
\(93\) 0 0
\(94\) 29.9253 3.08656
\(95\) 10.5146 1.07877
\(96\) 0 0
\(97\) −11.4897 −1.16660 −0.583300 0.812257i \(-0.698239\pi\)
−0.583300 + 0.812257i \(0.698239\pi\)
\(98\) −2.46050 −0.248549
\(99\) 0 0
\(100\) 7.00000 0.700000
\(101\) −3.67977 −0.366150 −0.183075 0.983099i \(-0.558605\pi\)
−0.183075 + 0.983099i \(0.558605\pi\)
\(102\) 0 0
\(103\) −9.72665 −0.958396 −0.479198 0.877707i \(-0.659072\pi\)
−0.479198 + 0.877707i \(0.659072\pi\)
\(104\) −5.05408 −0.495594
\(105\) 0 0
\(106\) 15.4356 1.49924
\(107\) −1.37432 −0.132860 −0.0664301 0.997791i \(-0.521161\pi\)
−0.0664301 + 0.997791i \(0.521161\pi\)
\(108\) 0 0
\(109\) −3.39922 −0.325587 −0.162793 0.986660i \(-0.552050\pi\)
−0.162793 + 0.986660i \(0.552050\pi\)
\(110\) 28.8099 2.74692
\(111\) 0 0
\(112\) −4.32743 −0.408904
\(113\) 10.3887 0.977288 0.488644 0.872483i \(-0.337492\pi\)
0.488644 + 0.872483i \(0.337492\pi\)
\(114\) 0 0
\(115\) 0.708945 0.0661095
\(116\) 9.97509 0.926164
\(117\) 0 0
\(118\) 6.70895 0.617608
\(119\) 0.945916 0.0867120
\(120\) 0 0
\(121\) 9.38151 0.852865
\(122\) 5.59358 0.506419
\(123\) 0 0
\(124\) 9.43560 0.847342
\(125\) 8.48968 0.759340
\(126\) 0 0
\(127\) 0.672570 0.0596809 0.0298405 0.999555i \(-0.490500\pi\)
0.0298405 + 0.999555i \(0.490500\pi\)
\(128\) 19.1082 1.68894
\(129\) 0 0
\(130\) 6.38151 0.559696
\(131\) 7.91381 0.691433 0.345717 0.938339i \(-0.387636\pi\)
0.345717 + 0.938339i \(0.387636\pi\)
\(132\) 0 0
\(133\) 4.05408 0.351534
\(134\) 38.9181 3.36201
\(135\) 0 0
\(136\) 4.78074 0.409945
\(137\) −3.67257 −0.313769 −0.156884 0.987617i \(-0.550145\pi\)
−0.156884 + 0.987617i \(0.550145\pi\)
\(138\) 0 0
\(139\) −2.05408 −0.174225 −0.0871126 0.996198i \(-0.527764\pi\)
−0.0871126 + 0.996198i \(0.527764\pi\)
\(140\) 10.5146 0.888645
\(141\) 0 0
\(142\) −8.05408 −0.675884
\(143\) 4.51459 0.377529
\(144\) 0 0
\(145\) −6.38151 −0.529956
\(146\) 3.70895 0.306954
\(147\) 0 0
\(148\) 7.21926 0.593420
\(149\) −13.5438 −1.10955 −0.554774 0.832001i \(-0.687195\pi\)
−0.554774 + 0.832001i \(0.687195\pi\)
\(150\) 0 0
\(151\) 9.92821 0.807946 0.403973 0.914771i \(-0.367629\pi\)
0.403973 + 0.914771i \(0.367629\pi\)
\(152\) 20.4897 1.66193
\(153\) 0 0
\(154\) 11.1082 0.895122
\(155\) −6.03638 −0.484853
\(156\) 0 0
\(157\) 6.05408 0.483169 0.241584 0.970380i \(-0.422333\pi\)
0.241584 + 0.970380i \(0.422333\pi\)
\(158\) −36.1914 −2.87924
\(159\) 0 0
\(160\) 1.39922 0.110618
\(161\) 0.273346 0.0215427
\(162\) 0 0
\(163\) 17.8171 1.39554 0.697772 0.716320i \(-0.254175\pi\)
0.697772 + 0.716320i \(0.254175\pi\)
\(164\) −25.9722 −2.02809
\(165\) 0 0
\(166\) 2.32743 0.180644
\(167\) −8.46770 −0.655250 −0.327625 0.944808i \(-0.606248\pi\)
−0.327625 + 0.944808i \(0.606248\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) −6.03638 −0.462969
\(171\) 0 0
\(172\) −42.3068 −3.22586
\(173\) 17.3566 1.31960 0.659799 0.751442i \(-0.270641\pi\)
0.659799 + 0.751442i \(0.270641\pi\)
\(174\) 0 0
\(175\) −1.72665 −0.130523
\(176\) 19.5366 1.47262
\(177\) 0 0
\(178\) 35.3245 2.64768
\(179\) −11.3494 −0.848295 −0.424147 0.905593i \(-0.639426\pi\)
−0.424147 + 0.905593i \(0.639426\pi\)
\(180\) 0 0
\(181\) 21.8889 1.62699 0.813495 0.581572i \(-0.197562\pi\)
0.813495 + 0.581572i \(0.197562\pi\)
\(182\) 2.46050 0.182385
\(183\) 0 0
\(184\) 1.38151 0.101847
\(185\) −4.61849 −0.339558
\(186\) 0 0
\(187\) −4.27042 −0.312284
\(188\) −49.3068 −3.59607
\(189\) 0 0
\(190\) −25.8712 −1.87689
\(191\) −0.701748 −0.0507767 −0.0253883 0.999678i \(-0.508082\pi\)
−0.0253883 + 0.999678i \(0.508082\pi\)
\(192\) 0 0
\(193\) 12.1445 0.874183 0.437092 0.899417i \(-0.356009\pi\)
0.437092 + 0.899417i \(0.356009\pi\)
\(194\) 28.2704 2.02970
\(195\) 0 0
\(196\) 4.05408 0.289577
\(197\) −16.4107 −1.16921 −0.584607 0.811317i \(-0.698751\pi\)
−0.584607 + 0.811317i \(0.698751\pi\)
\(198\) 0 0
\(199\) 22.7060 1.60959 0.804794 0.593555i \(-0.202276\pi\)
0.804794 + 0.593555i \(0.202276\pi\)
\(200\) −8.72665 −0.617068
\(201\) 0 0
\(202\) 9.05408 0.637043
\(203\) −2.46050 −0.172694
\(204\) 0 0
\(205\) 16.6156 1.16048
\(206\) 23.9325 1.66745
\(207\) 0 0
\(208\) 4.32743 0.300053
\(209\) −18.3025 −1.26601
\(210\) 0 0
\(211\) 4.56148 0.314025 0.157012 0.987597i \(-0.449814\pi\)
0.157012 + 0.987597i \(0.449814\pi\)
\(212\) −25.4327 −1.74672
\(213\) 0 0
\(214\) 3.38151 0.231156
\(215\) 27.0656 1.84586
\(216\) 0 0
\(217\) −2.32743 −0.157996
\(218\) 8.36381 0.566468
\(219\) 0 0
\(220\) −47.4690 −3.20036
\(221\) −0.945916 −0.0636292
\(222\) 0 0
\(223\) 13.3245 0.892275 0.446137 0.894964i \(-0.352799\pi\)
0.446137 + 0.894964i \(0.352799\pi\)
\(224\) 0.539495 0.0360465
\(225\) 0 0
\(226\) −25.5615 −1.70032
\(227\) 1.38151 0.0916943 0.0458472 0.998948i \(-0.485401\pi\)
0.0458472 + 0.998948i \(0.485401\pi\)
\(228\) 0 0
\(229\) −17.9794 −1.18811 −0.594055 0.804424i \(-0.702474\pi\)
−0.594055 + 0.804424i \(0.702474\pi\)
\(230\) −1.74436 −0.115020
\(231\) 0 0
\(232\) −12.4356 −0.816437
\(233\) −18.9823 −1.24357 −0.621786 0.783187i \(-0.713592\pi\)
−0.621786 + 0.783187i \(0.713592\pi\)
\(234\) 0 0
\(235\) 31.5438 2.05769
\(236\) −11.0541 −0.719560
\(237\) 0 0
\(238\) −2.32743 −0.150865
\(239\) 4.89183 0.316426 0.158213 0.987405i \(-0.449427\pi\)
0.158213 + 0.987405i \(0.449427\pi\)
\(240\) 0 0
\(241\) −26.1593 −1.68507 −0.842535 0.538641i \(-0.818938\pi\)
−0.842535 + 0.538641i \(0.818938\pi\)
\(242\) −23.0833 −1.48385
\(243\) 0 0
\(244\) −9.21634 −0.590016
\(245\) −2.59358 −0.165698
\(246\) 0 0
\(247\) −4.05408 −0.257955
\(248\) −11.7630 −0.746953
\(249\) 0 0
\(250\) −20.8889 −1.32113
\(251\) 18.4576 1.16503 0.582516 0.812819i \(-0.302068\pi\)
0.582516 + 0.812819i \(0.302068\pi\)
\(252\) 0 0
\(253\) −1.23405 −0.0775838
\(254\) −1.65486 −0.103835
\(255\) 0 0
\(256\) −32.3609 −2.02256
\(257\) −11.7339 −0.731938 −0.365969 0.930627i \(-0.619262\pi\)
−0.365969 + 0.930627i \(0.619262\pi\)
\(258\) 0 0
\(259\) −1.78074 −0.110650
\(260\) −10.5146 −0.652087
\(261\) 0 0
\(262\) −19.4720 −1.20298
\(263\) −7.52179 −0.463813 −0.231907 0.972738i \(-0.574496\pi\)
−0.231907 + 0.972738i \(0.574496\pi\)
\(264\) 0 0
\(265\) 16.2704 0.999484
\(266\) −9.97509 −0.611612
\(267\) 0 0
\(268\) −64.1239 −3.91699
\(269\) −18.8348 −1.14838 −0.574190 0.818722i \(-0.694683\pi\)
−0.574190 + 0.818722i \(0.694683\pi\)
\(270\) 0 0
\(271\) −23.9823 −1.45682 −0.728410 0.685141i \(-0.759740\pi\)
−0.728410 + 0.685141i \(0.759740\pi\)
\(272\) −4.09338 −0.248198
\(273\) 0 0
\(274\) 9.03638 0.545907
\(275\) 7.79513 0.470064
\(276\) 0 0
\(277\) 7.16225 0.430338 0.215169 0.976577i \(-0.430970\pi\)
0.215169 + 0.976577i \(0.430970\pi\)
\(278\) 5.05408 0.303124
\(279\) 0 0
\(280\) −13.1082 −0.783363
\(281\) 14.8817 0.887768 0.443884 0.896084i \(-0.353600\pi\)
0.443884 + 0.896084i \(0.353600\pi\)
\(282\) 0 0
\(283\) 19.9971 1.18870 0.594351 0.804205i \(-0.297409\pi\)
0.594351 + 0.804205i \(0.297409\pi\)
\(284\) 13.2704 0.787455
\(285\) 0 0
\(286\) −11.1082 −0.656840
\(287\) 6.40642 0.378159
\(288\) 0 0
\(289\) −16.1052 −0.947367
\(290\) 15.7017 0.922038
\(291\) 0 0
\(292\) −6.11109 −0.357625
\(293\) 15.0656 0.880139 0.440070 0.897964i \(-0.354954\pi\)
0.440070 + 0.897964i \(0.354954\pi\)
\(294\) 0 0
\(295\) 7.07179 0.411736
\(296\) −9.00000 −0.523114
\(297\) 0 0
\(298\) 33.3245 1.93044
\(299\) −0.273346 −0.0158080
\(300\) 0 0
\(301\) 10.4356 0.601498
\(302\) −24.4284 −1.40570
\(303\) 0 0
\(304\) −17.5438 −1.00620
\(305\) 5.89610 0.337610
\(306\) 0 0
\(307\) −27.2704 −1.55641 −0.778203 0.628013i \(-0.783868\pi\)
−0.778203 + 0.628013i \(0.783868\pi\)
\(308\) −18.3025 −1.04288
\(309\) 0 0
\(310\) 14.8525 0.843567
\(311\) −15.9823 −0.906273 −0.453136 0.891441i \(-0.649695\pi\)
−0.453136 + 0.891441i \(0.649695\pi\)
\(312\) 0 0
\(313\) 11.5979 0.655549 0.327775 0.944756i \(-0.393701\pi\)
0.327775 + 0.944756i \(0.393701\pi\)
\(314\) −14.8961 −0.840636
\(315\) 0 0
\(316\) 59.6313 3.35452
\(317\) −2.01771 −0.113326 −0.0566629 0.998393i \(-0.518046\pi\)
−0.0566629 + 0.998393i \(0.518046\pi\)
\(318\) 0 0
\(319\) 11.1082 0.621938
\(320\) 19.0043 1.06237
\(321\) 0 0
\(322\) −0.672570 −0.0374808
\(323\) 3.83482 0.213375
\(324\) 0 0
\(325\) 1.72665 0.0957775
\(326\) −43.8391 −2.42802
\(327\) 0 0
\(328\) 32.3786 1.78781
\(329\) 12.1623 0.670527
\(330\) 0 0
\(331\) −19.7089 −1.08330 −0.541651 0.840604i \(-0.682200\pi\)
−0.541651 + 0.840604i \(0.682200\pi\)
\(332\) −3.83482 −0.210463
\(333\) 0 0
\(334\) 20.8348 1.14003
\(335\) 41.0229 2.24132
\(336\) 0 0
\(337\) −29.0512 −1.58252 −0.791259 0.611481i \(-0.790574\pi\)
−0.791259 + 0.611481i \(0.790574\pi\)
\(338\) 29.5261 1.60601
\(339\) 0 0
\(340\) 9.94592 0.539393
\(341\) 10.5074 0.569007
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 52.7424 2.84368
\(345\) 0 0
\(346\) −42.7060 −2.29589
\(347\) 29.0833 1.56127 0.780636 0.624986i \(-0.214895\pi\)
0.780636 + 0.624986i \(0.214895\pi\)
\(348\) 0 0
\(349\) 24.7630 1.32553 0.662767 0.748825i \(-0.269382\pi\)
0.662767 + 0.748825i \(0.269382\pi\)
\(350\) 4.24844 0.227089
\(351\) 0 0
\(352\) −2.43560 −0.129818
\(353\) −33.3025 −1.77251 −0.886257 0.463193i \(-0.846704\pi\)
−0.886257 + 0.463193i \(0.846704\pi\)
\(354\) 0 0
\(355\) −8.48968 −0.450586
\(356\) −58.2029 −3.08475
\(357\) 0 0
\(358\) 27.9253 1.47590
\(359\) 25.5366 1.34777 0.673884 0.738837i \(-0.264625\pi\)
0.673884 + 0.738837i \(0.264625\pi\)
\(360\) 0 0
\(361\) −2.56440 −0.134968
\(362\) −53.8578 −2.83070
\(363\) 0 0
\(364\) −4.05408 −0.212492
\(365\) 3.90954 0.204635
\(366\) 0 0
\(367\) 27.4504 1.43290 0.716449 0.697639i \(-0.245766\pi\)
0.716449 + 0.697639i \(0.245766\pi\)
\(368\) −1.18289 −0.0616622
\(369\) 0 0
\(370\) 11.3638 0.590776
\(371\) 6.27335 0.325696
\(372\) 0 0
\(373\) 16.3274 0.845402 0.422701 0.906269i \(-0.361082\pi\)
0.422701 + 0.906269i \(0.361082\pi\)
\(374\) 10.5074 0.543324
\(375\) 0 0
\(376\) 61.4690 3.17002
\(377\) 2.46050 0.126722
\(378\) 0 0
\(379\) 12.0364 0.618267 0.309134 0.951019i \(-0.399961\pi\)
0.309134 + 0.951019i \(0.399961\pi\)
\(380\) 42.6270 2.18672
\(381\) 0 0
\(382\) 1.72665 0.0883433
\(383\) −12.4356 −0.635429 −0.317715 0.948186i \(-0.602915\pi\)
−0.317715 + 0.948186i \(0.602915\pi\)
\(384\) 0 0
\(385\) 11.7089 0.596743
\(386\) −29.8817 −1.52094
\(387\) 0 0
\(388\) −46.5801 −2.36475
\(389\) 20.6008 1.04450 0.522250 0.852792i \(-0.325093\pi\)
0.522250 + 0.852792i \(0.325093\pi\)
\(390\) 0 0
\(391\) 0.258562 0.0130761
\(392\) −5.05408 −0.255270
\(393\) 0 0
\(394\) 40.3786 2.03424
\(395\) −38.1488 −1.91948
\(396\) 0 0
\(397\) −23.6372 −1.18631 −0.593157 0.805087i \(-0.702119\pi\)
−0.593157 + 0.805087i \(0.702119\pi\)
\(398\) −55.8683 −2.80042
\(399\) 0 0
\(400\) 7.47197 0.373599
\(401\) −2.56440 −0.128060 −0.0640300 0.997948i \(-0.520395\pi\)
−0.0640300 + 0.997948i \(0.520395\pi\)
\(402\) 0 0
\(403\) 2.32743 0.115938
\(404\) −14.9181 −0.742202
\(405\) 0 0
\(406\) 6.05408 0.300459
\(407\) 8.03930 0.398493
\(408\) 0 0
\(409\) −34.3245 −1.69724 −0.848619 0.529005i \(-0.822565\pi\)
−0.848619 + 0.529005i \(0.822565\pi\)
\(410\) −40.8827 −2.01905
\(411\) 0 0
\(412\) −39.4327 −1.94271
\(413\) 2.72665 0.134170
\(414\) 0 0
\(415\) 2.45331 0.120428
\(416\) −0.539495 −0.0264509
\(417\) 0 0
\(418\) 45.0335 2.20266
\(419\) 4.05701 0.198198 0.0990989 0.995078i \(-0.468404\pi\)
0.0990989 + 0.995078i \(0.468404\pi\)
\(420\) 0 0
\(421\) −21.0689 −1.02683 −0.513417 0.858139i \(-0.671621\pi\)
−0.513417 + 0.858139i \(0.671621\pi\)
\(422\) −11.2235 −0.546353
\(423\) 0 0
\(424\) 31.7060 1.53978
\(425\) −1.63327 −0.0792252
\(426\) 0 0
\(427\) 2.27335 0.110015
\(428\) −5.57160 −0.269313
\(429\) 0 0
\(430\) −66.5949 −3.21149
\(431\) 22.6185 1.08949 0.544747 0.838600i \(-0.316626\pi\)
0.544747 + 0.838600i \(0.316626\pi\)
\(432\) 0 0
\(433\) 2.41789 0.116196 0.0580982 0.998311i \(-0.481496\pi\)
0.0580982 + 0.998311i \(0.481496\pi\)
\(434\) 5.72665 0.274888
\(435\) 0 0
\(436\) −13.7807 −0.659978
\(437\) 1.10817 0.0530109
\(438\) 0 0
\(439\) −23.4897 −1.12110 −0.560551 0.828120i \(-0.689411\pi\)
−0.560551 + 0.828120i \(0.689411\pi\)
\(440\) 59.1780 2.82120
\(441\) 0 0
\(442\) 2.32743 0.110705
\(443\) −13.4179 −0.637503 −0.318752 0.947838i \(-0.603264\pi\)
−0.318752 + 0.947838i \(0.603264\pi\)
\(444\) 0 0
\(445\) 37.2350 1.76511
\(446\) −32.7850 −1.55242
\(447\) 0 0
\(448\) 7.32743 0.346189
\(449\) −9.16225 −0.432393 −0.216197 0.976350i \(-0.569365\pi\)
−0.216197 + 0.976350i \(0.569365\pi\)
\(450\) 0 0
\(451\) −28.9224 −1.36190
\(452\) 42.1167 1.98100
\(453\) 0 0
\(454\) −3.39922 −0.159533
\(455\) 2.59358 0.121589
\(456\) 0 0
\(457\) 8.81711 0.412447 0.206224 0.978505i \(-0.433883\pi\)
0.206224 + 0.978505i \(0.433883\pi\)
\(458\) 44.2383 2.06712
\(459\) 0 0
\(460\) 2.87412 0.134007
\(461\) −5.65913 −0.263572 −0.131786 0.991278i \(-0.542071\pi\)
−0.131786 + 0.991278i \(0.542071\pi\)
\(462\) 0 0
\(463\) 15.7267 0.730880 0.365440 0.930835i \(-0.380919\pi\)
0.365440 + 0.930835i \(0.380919\pi\)
\(464\) 10.6477 0.494305
\(465\) 0 0
\(466\) 46.7060 2.16361
\(467\) −21.9971 −1.01790 −0.508952 0.860795i \(-0.669967\pi\)
−0.508952 + 0.860795i \(0.669967\pi\)
\(468\) 0 0
\(469\) 15.8171 0.730366
\(470\) −77.6136 −3.58005
\(471\) 0 0
\(472\) 13.7807 0.634310
\(473\) −47.1124 −2.16623
\(474\) 0 0
\(475\) −7.00000 −0.321182
\(476\) 3.83482 0.175769
\(477\) 0 0
\(478\) −12.0364 −0.550531
\(479\) 24.9751 1.14114 0.570571 0.821249i \(-0.306722\pi\)
0.570571 + 0.821249i \(0.306722\pi\)
\(480\) 0 0
\(481\) 1.78074 0.0811947
\(482\) 64.3652 2.93175
\(483\) 0 0
\(484\) 38.0335 1.72879
\(485\) 29.7994 1.35312
\(486\) 0 0
\(487\) −17.5979 −0.797435 −0.398717 0.917074i \(-0.630545\pi\)
−0.398717 + 0.917074i \(0.630545\pi\)
\(488\) 11.4897 0.520114
\(489\) 0 0
\(490\) 6.38151 0.288287
\(491\) 13.7951 0.622566 0.311283 0.950317i \(-0.399241\pi\)
0.311283 + 0.950317i \(0.399241\pi\)
\(492\) 0 0
\(493\) −2.32743 −0.104822
\(494\) 9.97509 0.448801
\(495\) 0 0
\(496\) 10.0718 0.452237
\(497\) −3.27335 −0.146830
\(498\) 0 0
\(499\) 13.0875 0.585879 0.292939 0.956131i \(-0.405367\pi\)
0.292939 + 0.956131i \(0.405367\pi\)
\(500\) 34.4179 1.53921
\(501\) 0 0
\(502\) −45.4150 −2.02697
\(503\) −22.3068 −0.994611 −0.497305 0.867576i \(-0.665677\pi\)
−0.497305 + 0.867576i \(0.665677\pi\)
\(504\) 0 0
\(505\) 9.54377 0.424692
\(506\) 3.03638 0.134983
\(507\) 0 0
\(508\) 2.72665 0.120976
\(509\) −15.8932 −0.704453 −0.352226 0.935915i \(-0.614575\pi\)
−0.352226 + 0.935915i \(0.614575\pi\)
\(510\) 0 0
\(511\) 1.50739 0.0666831
\(512\) 41.4078 1.82998
\(513\) 0 0
\(514\) 28.8712 1.27345
\(515\) 25.2268 1.11163
\(516\) 0 0
\(517\) −54.9076 −2.41483
\(518\) 4.38151 0.192513
\(519\) 0 0
\(520\) 13.1082 0.574831
\(521\) 4.41789 0.193551 0.0967756 0.995306i \(-0.469147\pi\)
0.0967756 + 0.995306i \(0.469147\pi\)
\(522\) 0 0
\(523\) 25.2733 1.10513 0.552563 0.833471i \(-0.313650\pi\)
0.552563 + 0.833471i \(0.313650\pi\)
\(524\) 32.0833 1.40156
\(525\) 0 0
\(526\) 18.5074 0.806961
\(527\) −2.20155 −0.0959012
\(528\) 0 0
\(529\) −22.9253 −0.996751
\(530\) −40.0335 −1.73894
\(531\) 0 0
\(532\) 16.4356 0.712574
\(533\) −6.40642 −0.277493
\(534\) 0 0
\(535\) 3.56440 0.154103
\(536\) 79.9410 3.45293
\(537\) 0 0
\(538\) 46.3432 1.99800
\(539\) 4.51459 0.194457
\(540\) 0 0
\(541\) −3.43852 −0.147834 −0.0739168 0.997264i \(-0.523550\pi\)
−0.0739168 + 0.997264i \(0.523550\pi\)
\(542\) 59.0085 2.53463
\(543\) 0 0
\(544\) 0.510317 0.0218797
\(545\) 8.81616 0.377643
\(546\) 0 0
\(547\) −6.92821 −0.296229 −0.148114 0.988970i \(-0.547320\pi\)
−0.148114 + 0.988970i \(0.547320\pi\)
\(548\) −14.8889 −0.636023
\(549\) 0 0
\(550\) −19.1800 −0.817836
\(551\) −9.97509 −0.424953
\(552\) 0 0
\(553\) −14.7089 −0.625488
\(554\) −17.6228 −0.748719
\(555\) 0 0
\(556\) −8.32743 −0.353162
\(557\) 33.5835 1.42298 0.711488 0.702698i \(-0.248021\pi\)
0.711488 + 0.702698i \(0.248021\pi\)
\(558\) 0 0
\(559\) −10.4356 −0.441379
\(560\) 11.2235 0.474281
\(561\) 0 0
\(562\) −36.6165 −1.54457
\(563\) 42.4792 1.79028 0.895142 0.445781i \(-0.147074\pi\)
0.895142 + 0.445781i \(0.147074\pi\)
\(564\) 0 0
\(565\) −26.9439 −1.13354
\(566\) −49.2029 −2.06815
\(567\) 0 0
\(568\) −16.5438 −0.694161
\(569\) 10.4035 0.436137 0.218069 0.975933i \(-0.430024\pi\)
0.218069 + 0.975933i \(0.430024\pi\)
\(570\) 0 0
\(571\) 17.8496 0.746983 0.373491 0.927634i \(-0.378161\pi\)
0.373491 + 0.927634i \(0.378161\pi\)
\(572\) 18.3025 0.765267
\(573\) 0 0
\(574\) −15.7630 −0.657936
\(575\) −0.471974 −0.0196827
\(576\) 0 0
\(577\) 11.9430 0.497193 0.248597 0.968607i \(-0.420031\pi\)
0.248597 + 0.968607i \(0.420031\pi\)
\(578\) 39.6270 1.64827
\(579\) 0 0
\(580\) −25.8712 −1.07424
\(581\) 0.945916 0.0392432
\(582\) 0 0
\(583\) −28.3216 −1.17296
\(584\) 7.61849 0.315255
\(585\) 0 0
\(586\) −37.0689 −1.53130
\(587\) 23.8597 0.984796 0.492398 0.870370i \(-0.336120\pi\)
0.492398 + 0.870370i \(0.336120\pi\)
\(588\) 0 0
\(589\) −9.43560 −0.388787
\(590\) −17.4002 −0.716354
\(591\) 0 0
\(592\) 7.70602 0.316715
\(593\) 19.5801 0.804060 0.402030 0.915626i \(-0.368305\pi\)
0.402030 + 0.915626i \(0.368305\pi\)
\(594\) 0 0
\(595\) −2.45331 −0.100576
\(596\) −54.9076 −2.24910
\(597\) 0 0
\(598\) 0.672570 0.0275034
\(599\) 18.5467 0.757797 0.378899 0.925438i \(-0.376303\pi\)
0.378899 + 0.925438i \(0.376303\pi\)
\(600\) 0 0
\(601\) −18.1986 −0.742338 −0.371169 0.928565i \(-0.621043\pi\)
−0.371169 + 0.928565i \(0.621043\pi\)
\(602\) −25.6768 −1.04651
\(603\) 0 0
\(604\) 40.2498 1.63774
\(605\) −24.3317 −0.989224
\(606\) 0 0
\(607\) −22.3097 −0.905524 −0.452762 0.891631i \(-0.649561\pi\)
−0.452762 + 0.891631i \(0.649561\pi\)
\(608\) 2.18716 0.0887010
\(609\) 0 0
\(610\) −14.5074 −0.587387
\(611\) −12.1623 −0.492032
\(612\) 0 0
\(613\) 10.2370 0.413467 0.206734 0.978397i \(-0.433717\pi\)
0.206734 + 0.978397i \(0.433717\pi\)
\(614\) 67.0990 2.70790
\(615\) 0 0
\(616\) 22.8171 0.919328
\(617\) −11.3274 −0.456025 −0.228013 0.973658i \(-0.573223\pi\)
−0.228013 + 0.973658i \(0.573223\pi\)
\(618\) 0 0
\(619\) 8.63327 0.347000 0.173500 0.984834i \(-0.444492\pi\)
0.173500 + 0.984834i \(0.444492\pi\)
\(620\) −24.4720 −0.982818
\(621\) 0 0
\(622\) 39.3245 1.57677
\(623\) 14.3566 0.575185
\(624\) 0 0
\(625\) −30.6519 −1.22608
\(626\) −28.5366 −1.14055
\(627\) 0 0
\(628\) 24.5438 0.979403
\(629\) −1.68443 −0.0671626
\(630\) 0 0
\(631\) −14.8535 −0.591308 −0.295654 0.955295i \(-0.595538\pi\)
−0.295654 + 0.955295i \(0.595538\pi\)
\(632\) −74.3402 −2.95710
\(633\) 0 0
\(634\) 4.96458 0.197169
\(635\) −1.74436 −0.0692229
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) −27.3317 −1.08207
\(639\) 0 0
\(640\) −49.5586 −1.95897
\(641\) −34.1593 −1.34921 −0.674606 0.738178i \(-0.735687\pi\)
−0.674606 + 0.738178i \(0.735687\pi\)
\(642\) 0 0
\(643\) −10.8348 −0.427284 −0.213642 0.976912i \(-0.568533\pi\)
−0.213642 + 0.976912i \(0.568533\pi\)
\(644\) 1.10817 0.0436680
\(645\) 0 0
\(646\) −9.43560 −0.371239
\(647\) 32.9692 1.29615 0.648077 0.761575i \(-0.275573\pi\)
0.648077 + 0.761575i \(0.275573\pi\)
\(648\) 0 0
\(649\) −12.3097 −0.483199
\(650\) −4.24844 −0.166638
\(651\) 0 0
\(652\) 72.2321 2.82883
\(653\) −3.93113 −0.153837 −0.0769185 0.997037i \(-0.524508\pi\)
−0.0769185 + 0.997037i \(0.524508\pi\)
\(654\) 0 0
\(655\) −20.5251 −0.801982
\(656\) −27.7233 −1.08241
\(657\) 0 0
\(658\) −29.9253 −1.16661
\(659\) 16.8171 0.655102 0.327551 0.944834i \(-0.393777\pi\)
0.327551 + 0.944834i \(0.393777\pi\)
\(660\) 0 0
\(661\) −17.0216 −0.662063 −0.331032 0.943620i \(-0.607397\pi\)
−0.331032 + 0.943620i \(0.607397\pi\)
\(662\) 48.4940 1.88477
\(663\) 0 0
\(664\) 4.78074 0.185529
\(665\) −10.5146 −0.407738
\(666\) 0 0
\(667\) −0.672570 −0.0260420
\(668\) −34.3288 −1.32822
\(669\) 0 0
\(670\) −100.937 −3.89954
\(671\) −10.2632 −0.396207
\(672\) 0 0
\(673\) 28.7453 1.10805 0.554025 0.832500i \(-0.313091\pi\)
0.554025 + 0.832500i \(0.313091\pi\)
\(674\) 71.4805 2.75333
\(675\) 0 0
\(676\) −48.6490 −1.87112
\(677\) −6.03638 −0.231997 −0.115998 0.993249i \(-0.537007\pi\)
−0.115998 + 0.993249i \(0.537007\pi\)
\(678\) 0 0
\(679\) 11.4897 0.440934
\(680\) −12.3992 −0.475489
\(681\) 0 0
\(682\) −25.8535 −0.989981
\(683\) 20.5113 0.784842 0.392421 0.919786i \(-0.371638\pi\)
0.392421 + 0.919786i \(0.371638\pi\)
\(684\) 0 0
\(685\) 9.52510 0.363935
\(686\) 2.46050 0.0939425
\(687\) 0 0
\(688\) −45.1593 −1.72168
\(689\) −6.27335 −0.238995
\(690\) 0 0
\(691\) −15.0029 −0.570738 −0.285369 0.958418i \(-0.592116\pi\)
−0.285369 + 0.958418i \(0.592116\pi\)
\(692\) 70.3652 2.67488
\(693\) 0 0
\(694\) −71.5595 −2.71636
\(695\) 5.32743 0.202081
\(696\) 0 0
\(697\) 6.05993 0.229536
\(698\) −60.9296 −2.30622
\(699\) 0 0
\(700\) −7.00000 −0.264575
\(701\) 38.5113 1.45455 0.727275 0.686346i \(-0.240786\pi\)
0.727275 + 0.686346i \(0.240786\pi\)
\(702\) 0 0
\(703\) −7.21926 −0.272280
\(704\) −33.0803 −1.24676
\(705\) 0 0
\(706\) 81.9410 3.08389
\(707\) 3.67977 0.138392
\(708\) 0 0
\(709\) 7.64008 0.286929 0.143465 0.989655i \(-0.454176\pi\)
0.143465 + 0.989655i \(0.454176\pi\)
\(710\) 20.8889 0.783947
\(711\) 0 0
\(712\) 72.5595 2.71928
\(713\) −0.636194 −0.0238257
\(714\) 0 0
\(715\) −11.7089 −0.437890
\(716\) −46.0115 −1.71953
\(717\) 0 0
\(718\) −62.8329 −2.34490
\(719\) −30.0364 −1.12017 −0.560084 0.828436i \(-0.689231\pi\)
−0.560084 + 0.828436i \(0.689231\pi\)
\(720\) 0 0
\(721\) 9.72665 0.362240
\(722\) 6.30972 0.234824
\(723\) 0 0
\(724\) 88.7395 3.29798
\(725\) 4.24844 0.157783
\(726\) 0 0
\(727\) 3.45623 0.128185 0.0640923 0.997944i \(-0.479585\pi\)
0.0640923 + 0.997944i \(0.479585\pi\)
\(728\) 5.05408 0.187317
\(729\) 0 0
\(730\) −9.61944 −0.356032
\(731\) 9.87120 0.365099
\(732\) 0 0
\(733\) 38.5261 1.42299 0.711496 0.702690i \(-0.248018\pi\)
0.711496 + 0.702690i \(0.248018\pi\)
\(734\) −67.5418 −2.49301
\(735\) 0 0
\(736\) 0.147469 0.00543578
\(737\) −71.4078 −2.63034
\(738\) 0 0
\(739\) 45.1239 1.65991 0.829955 0.557830i \(-0.188366\pi\)
0.829955 + 0.557830i \(0.188366\pi\)
\(740\) −18.7237 −0.688298
\(741\) 0 0
\(742\) −15.4356 −0.566659
\(743\) 9.48676 0.348035 0.174018 0.984743i \(-0.444325\pi\)
0.174018 + 0.984743i \(0.444325\pi\)
\(744\) 0 0
\(745\) 35.1268 1.28695
\(746\) −40.1737 −1.47086
\(747\) 0 0
\(748\) −17.3126 −0.633013
\(749\) 1.37432 0.0502165
\(750\) 0 0
\(751\) −9.83190 −0.358771 −0.179386 0.983779i \(-0.557411\pi\)
−0.179386 + 0.983779i \(0.557411\pi\)
\(752\) −52.6313 −1.91927
\(753\) 0 0
\(754\) −6.05408 −0.220477
\(755\) −25.7496 −0.937124
\(756\) 0 0
\(757\) −41.8171 −1.51987 −0.759934 0.650000i \(-0.774769\pi\)
−0.759934 + 0.650000i \(0.774769\pi\)
\(758\) −29.6156 −1.07569
\(759\) 0 0
\(760\) −53.1416 −1.92765
\(761\) 22.9794 0.833001 0.416501 0.909135i \(-0.363256\pi\)
0.416501 + 0.909135i \(0.363256\pi\)
\(762\) 0 0
\(763\) 3.39922 0.123060
\(764\) −2.84494 −0.102926
\(765\) 0 0
\(766\) 30.5979 1.10555
\(767\) −2.72665 −0.0984538
\(768\) 0 0
\(769\) 6.08658 0.219488 0.109744 0.993960i \(-0.464997\pi\)
0.109744 + 0.993960i \(0.464997\pi\)
\(770\) −28.8099 −1.03824
\(771\) 0 0
\(772\) 49.2350 1.77201
\(773\) −41.8214 −1.50421 −0.752105 0.659043i \(-0.770962\pi\)
−0.752105 + 0.659043i \(0.770962\pi\)
\(774\) 0 0
\(775\) 4.01867 0.144355
\(776\) 58.0698 2.08459
\(777\) 0 0
\(778\) −50.6883 −1.81726
\(779\) 25.9722 0.930550
\(780\) 0 0
\(781\) 14.7778 0.528792
\(782\) −0.636194 −0.0227503
\(783\) 0 0
\(784\) 4.32743 0.154551
\(785\) −15.7017 −0.560419
\(786\) 0 0
\(787\) 32.2920 1.15109 0.575543 0.817772i \(-0.304791\pi\)
0.575543 + 0.817772i \(0.304791\pi\)
\(788\) −66.5303 −2.37004
\(789\) 0 0
\(790\) 93.8653 3.33958
\(791\) −10.3887 −0.369380
\(792\) 0 0
\(793\) −2.27335 −0.0807289
\(794\) 58.1593 2.06400
\(795\) 0 0
\(796\) 92.0521 3.26270
\(797\) 46.5657 1.64944 0.824722 0.565539i \(-0.191332\pi\)
0.824722 + 0.565539i \(0.191332\pi\)
\(798\) 0 0
\(799\) 11.5045 0.406999
\(800\) −0.931521 −0.0329343
\(801\) 0 0
\(802\) 6.30972 0.222804
\(803\) −6.80525 −0.240152
\(804\) 0 0
\(805\) −0.708945 −0.0249870
\(806\) −5.72665 −0.201713
\(807\) 0 0
\(808\) 18.5979 0.654270
\(809\) −10.8023 −0.379790 −0.189895 0.981804i \(-0.560815\pi\)
−0.189895 + 0.981804i \(0.560815\pi\)
\(810\) 0 0
\(811\) 5.58307 0.196048 0.0980240 0.995184i \(-0.468748\pi\)
0.0980240 + 0.995184i \(0.468748\pi\)
\(812\) −9.97509 −0.350057
\(813\) 0 0
\(814\) −19.7807 −0.693315
\(815\) −46.2101 −1.61867
\(816\) 0 0
\(817\) 42.3068 1.48013
\(818\) 84.4556 2.95292
\(819\) 0 0
\(820\) 67.3609 2.35234
\(821\) −31.7879 −1.10941 −0.554703 0.832048i \(-0.687168\pi\)
−0.554703 + 0.832048i \(0.687168\pi\)
\(822\) 0 0
\(823\) −36.0000 −1.25488 −0.627441 0.778664i \(-0.715897\pi\)
−0.627441 + 0.778664i \(0.715897\pi\)
\(824\) 49.1593 1.71255
\(825\) 0 0
\(826\) −6.70895 −0.233434
\(827\) 15.9224 0.553675 0.276837 0.960917i \(-0.410714\pi\)
0.276837 + 0.960917i \(0.410714\pi\)
\(828\) 0 0
\(829\) 35.4720 1.23199 0.615996 0.787749i \(-0.288754\pi\)
0.615996 + 0.787749i \(0.288754\pi\)
\(830\) −6.03638 −0.209526
\(831\) 0 0
\(832\) −7.32743 −0.254033
\(833\) −0.945916 −0.0327740
\(834\) 0 0
\(835\) 21.9617 0.760014
\(836\) −74.2000 −2.56626
\(837\) 0 0
\(838\) −9.98229 −0.344833
\(839\) −54.6782 −1.88770 −0.943850 0.330373i \(-0.892825\pi\)
−0.943850 + 0.330373i \(0.892825\pi\)
\(840\) 0 0
\(841\) −22.9459 −0.791238
\(842\) 51.8401 1.78653
\(843\) 0 0
\(844\) 18.4926 0.636542
\(845\) 31.1230 1.07066
\(846\) 0 0
\(847\) −9.38151 −0.322353
\(848\) −27.1475 −0.932248
\(849\) 0 0
\(850\) 4.01867 0.137839
\(851\) −0.486758 −0.0166858
\(852\) 0 0
\(853\) −2.19767 −0.0752468 −0.0376234 0.999292i \(-0.511979\pi\)
−0.0376234 + 0.999292i \(0.511979\pi\)
\(854\) −5.59358 −0.191408
\(855\) 0 0
\(856\) 6.94592 0.237407
\(857\) −15.7765 −0.538914 −0.269457 0.963012i \(-0.586844\pi\)
−0.269457 + 0.963012i \(0.586844\pi\)
\(858\) 0 0
\(859\) 5.57626 0.190260 0.0951298 0.995465i \(-0.469673\pi\)
0.0951298 + 0.995465i \(0.469673\pi\)
\(860\) 109.726 3.74163
\(861\) 0 0
\(862\) −55.6529 −1.89555
\(863\) 23.1268 0.787247 0.393623 0.919272i \(-0.371221\pi\)
0.393623 + 0.919272i \(0.371221\pi\)
\(864\) 0 0
\(865\) −45.0157 −1.53058
\(866\) −5.94923 −0.202163
\(867\) 0 0
\(868\) −9.43560 −0.320265
\(869\) 66.4048 2.25263
\(870\) 0 0
\(871\) −15.8171 −0.535942
\(872\) 17.1800 0.581787
\(873\) 0 0
\(874\) −2.72665 −0.0922304
\(875\) −8.48968 −0.287004
\(876\) 0 0
\(877\) 3.92528 0.132547 0.0662737 0.997801i \(-0.478889\pi\)
0.0662737 + 0.997801i \(0.478889\pi\)
\(878\) 57.7965 1.95054
\(879\) 0 0
\(880\) −50.6696 −1.70807
\(881\) 27.1986 0.916345 0.458173 0.888863i \(-0.348504\pi\)
0.458173 + 0.888863i \(0.348504\pi\)
\(882\) 0 0
\(883\) 8.21341 0.276403 0.138202 0.990404i \(-0.455868\pi\)
0.138202 + 0.990404i \(0.455868\pi\)
\(884\) −3.83482 −0.128979
\(885\) 0 0
\(886\) 33.0148 1.10915
\(887\) −6.48114 −0.217615 −0.108808 0.994063i \(-0.534703\pi\)
−0.108808 + 0.994063i \(0.534703\pi\)
\(888\) 0 0
\(889\) −0.672570 −0.0225573
\(890\) −91.6169 −3.07101
\(891\) 0 0
\(892\) 54.0187 1.80868
\(893\) 49.3068 1.64999
\(894\) 0 0
\(895\) 29.4356 0.983924
\(896\) −19.1082 −0.638359
\(897\) 0 0
\(898\) 22.5438 0.752295
\(899\) 5.72665 0.190995
\(900\) 0 0
\(901\) 5.93406 0.197692
\(902\) 71.1636 2.36949
\(903\) 0 0
\(904\) −52.5054 −1.74630
\(905\) −56.7706 −1.88712
\(906\) 0 0
\(907\) 10.1288 0.336321 0.168161 0.985760i \(-0.446217\pi\)
0.168161 + 0.985760i \(0.446217\pi\)
\(908\) 5.60078 0.185868
\(909\) 0 0
\(910\) −6.38151 −0.211545
\(911\) 45.9224 1.52148 0.760738 0.649059i \(-0.224837\pi\)
0.760738 + 0.649059i \(0.224837\pi\)
\(912\) 0 0
\(913\) −4.27042 −0.141330
\(914\) −21.6946 −0.717592
\(915\) 0 0
\(916\) −72.8899 −2.40835
\(917\) −7.91381 −0.261337
\(918\) 0 0
\(919\) −4.92432 −0.162438 −0.0812192 0.996696i \(-0.525881\pi\)
−0.0812192 + 0.996696i \(0.525881\pi\)
\(920\) −3.58307 −0.118130
\(921\) 0 0
\(922\) 13.9243 0.458573
\(923\) 3.27335 0.107744
\(924\) 0 0
\(925\) 3.07472 0.101096
\(926\) −38.6955 −1.27161
\(927\) 0 0
\(928\) −1.32743 −0.0435750
\(929\) 0.00758649 0.000248905 0 0.000124452 1.00000i \(-0.499960\pi\)
0.000124452 1.00000i \(0.499960\pi\)
\(930\) 0 0
\(931\) −4.05408 −0.132867
\(932\) −76.9558 −2.52077
\(933\) 0 0
\(934\) 54.1239 1.77099
\(935\) 11.0757 0.362213
\(936\) 0 0
\(937\) 21.1623 0.691341 0.345670 0.938356i \(-0.387652\pi\)
0.345670 + 0.938356i \(0.387652\pi\)
\(938\) −38.9181 −1.27072
\(939\) 0 0
\(940\) 127.881 4.17102
\(941\) 4.55816 0.148592 0.0742959 0.997236i \(-0.476329\pi\)
0.0742959 + 0.997236i \(0.476329\pi\)
\(942\) 0 0
\(943\) 1.75117 0.0570260
\(944\) −11.7994 −0.384038
\(945\) 0 0
\(946\) 115.920 3.76890
\(947\) 13.7352 0.446334 0.223167 0.974780i \(-0.428360\pi\)
0.223167 + 0.974780i \(0.428360\pi\)
\(948\) 0 0
\(949\) −1.50739 −0.0489320
\(950\) 17.2235 0.558805
\(951\) 0 0
\(952\) −4.78074 −0.154945
\(953\) 8.80699 0.285286 0.142643 0.989774i \(-0.454440\pi\)
0.142643 + 0.989774i \(0.454440\pi\)
\(954\) 0 0
\(955\) 1.82004 0.0588951
\(956\) 19.8319 0.641409
\(957\) 0 0
\(958\) −61.4513 −1.98540
\(959\) 3.67257 0.118593
\(960\) 0 0
\(961\) −25.5831 −0.825260
\(962\) −4.38151 −0.141266
\(963\) 0 0
\(964\) −106.052 −3.41571
\(965\) −31.4978 −1.01395
\(966\) 0 0
\(967\) 38.3284 1.23256 0.616279 0.787528i \(-0.288639\pi\)
0.616279 + 0.787528i \(0.288639\pi\)
\(968\) −47.4150 −1.52397
\(969\) 0 0
\(970\) −73.3216 −2.35421
\(971\) −31.0187 −0.995436 −0.497718 0.867339i \(-0.665829\pi\)
−0.497718 + 0.867339i \(0.665829\pi\)
\(972\) 0 0
\(973\) 2.05408 0.0658509
\(974\) 43.2996 1.38741
\(975\) 0 0
\(976\) −9.83775 −0.314899
\(977\) −52.7424 −1.68738 −0.843689 0.536832i \(-0.819621\pi\)
−0.843689 + 0.536832i \(0.819621\pi\)
\(978\) 0 0
\(979\) −64.8142 −2.07147
\(980\) −10.5146 −0.335876
\(981\) 0 0
\(982\) −33.9430 −1.08316
\(983\) −18.3029 −0.583772 −0.291886 0.956453i \(-0.594283\pi\)
−0.291886 + 0.956453i \(0.594283\pi\)
\(984\) 0 0
\(985\) 42.5624 1.35615
\(986\) 5.72665 0.182374
\(987\) 0 0
\(988\) −16.4356 −0.522886
\(989\) 2.85253 0.0907052
\(990\) 0 0
\(991\) −12.6008 −0.400277 −0.200138 0.979768i \(-0.564139\pi\)
−0.200138 + 0.979768i \(0.564139\pi\)
\(992\) −1.25564 −0.0398665
\(993\) 0 0
\(994\) 8.05408 0.255460
\(995\) −58.8899 −1.86693
\(996\) 0 0
\(997\) 11.7424 0.371885 0.185943 0.982561i \(-0.440466\pi\)
0.185943 + 0.982561i \(0.440466\pi\)
\(998\) −32.2019 −1.01933
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 567.2.a.d.1.1 3
3.2 odd 2 567.2.a.g.1.3 3
4.3 odd 2 9072.2.a.bq.1.2 3
7.6 odd 2 3969.2.a.m.1.1 3
9.2 odd 6 189.2.f.a.64.1 6
9.4 even 3 63.2.f.b.43.3 yes 6
9.5 odd 6 189.2.f.a.127.1 6
9.7 even 3 63.2.f.b.22.3 6
12.11 even 2 9072.2.a.cd.1.2 3
21.20 even 2 3969.2.a.p.1.3 3
36.7 odd 6 1008.2.r.k.337.3 6
36.11 even 6 3024.2.r.g.1009.2 6
36.23 even 6 3024.2.r.g.2017.2 6
36.31 odd 6 1008.2.r.k.673.3 6
63.2 odd 6 1323.2.g.c.361.1 6
63.4 even 3 441.2.g.e.79.3 6
63.5 even 6 1323.2.h.e.802.3 6
63.11 odd 6 1323.2.h.d.226.3 6
63.13 odd 6 441.2.f.d.295.3 6
63.16 even 3 441.2.g.e.67.3 6
63.20 even 6 1323.2.f.c.442.1 6
63.23 odd 6 1323.2.h.d.802.3 6
63.25 even 3 441.2.h.c.373.1 6
63.31 odd 6 441.2.g.d.79.3 6
63.32 odd 6 1323.2.g.c.667.1 6
63.34 odd 6 441.2.f.d.148.3 6
63.38 even 6 1323.2.h.e.226.3 6
63.40 odd 6 441.2.h.b.214.1 6
63.41 even 6 1323.2.f.c.883.1 6
63.47 even 6 1323.2.g.b.361.1 6
63.52 odd 6 441.2.h.b.373.1 6
63.58 even 3 441.2.h.c.214.1 6
63.59 even 6 1323.2.g.b.667.1 6
63.61 odd 6 441.2.g.d.67.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.b.22.3 6 9.7 even 3
63.2.f.b.43.3 yes 6 9.4 even 3
189.2.f.a.64.1 6 9.2 odd 6
189.2.f.a.127.1 6 9.5 odd 6
441.2.f.d.148.3 6 63.34 odd 6
441.2.f.d.295.3 6 63.13 odd 6
441.2.g.d.67.3 6 63.61 odd 6
441.2.g.d.79.3 6 63.31 odd 6
441.2.g.e.67.3 6 63.16 even 3
441.2.g.e.79.3 6 63.4 even 3
441.2.h.b.214.1 6 63.40 odd 6
441.2.h.b.373.1 6 63.52 odd 6
441.2.h.c.214.1 6 63.58 even 3
441.2.h.c.373.1 6 63.25 even 3
567.2.a.d.1.1 3 1.1 even 1 trivial
567.2.a.g.1.3 3 3.2 odd 2
1008.2.r.k.337.3 6 36.7 odd 6
1008.2.r.k.673.3 6 36.31 odd 6
1323.2.f.c.442.1 6 63.20 even 6
1323.2.f.c.883.1 6 63.41 even 6
1323.2.g.b.361.1 6 63.47 even 6
1323.2.g.b.667.1 6 63.59 even 6
1323.2.g.c.361.1 6 63.2 odd 6
1323.2.g.c.667.1 6 63.32 odd 6
1323.2.h.d.226.3 6 63.11 odd 6
1323.2.h.d.802.3 6 63.23 odd 6
1323.2.h.e.226.3 6 63.38 even 6
1323.2.h.e.802.3 6 63.5 even 6
3024.2.r.g.1009.2 6 36.11 even 6
3024.2.r.g.2017.2 6 36.23 even 6
3969.2.a.m.1.1 3 7.6 odd 2
3969.2.a.p.1.3 3 21.20 even 2
9072.2.a.bq.1.2 3 4.3 odd 2
9072.2.a.cd.1.2 3 12.11 even 2