Properties

Label 729.2.a.c.1.2
Level $729$
Weight $2$
Character 729.1
Self dual yes
Analytic conductor $5.821$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 729 = 3^{6} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 729.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.82109430735\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{36})^+\)
Defining polynomial: \( x^{6} - 6x^{4} + 9x^{2} - 3 \) Copy content Toggle raw display
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 \(-1.28558\) of defining polynomial
Character \(\chi\) \(=\) 729.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.28558 q^{2} -0.347296 q^{4} -0.446476 q^{5} -3.53209 q^{7} +3.01763 q^{8} +O(q^{10})\) \(q-1.28558 q^{2} -0.347296 q^{4} -0.446476 q^{5} -3.53209 q^{7} +3.01763 q^{8} +0.573978 q^{10} +2.78006 q^{11} +3.29086 q^{13} +4.54077 q^{14} -3.18479 q^{16} +7.03936 q^{17} -5.18479 q^{19} +0.155059 q^{20} -3.57398 q^{22} -7.27693 q^{23} -4.80066 q^{25} -4.23065 q^{26} +1.22668 q^{28} -3.61916 q^{29} -1.93582 q^{31} -1.94096 q^{32} -9.04963 q^{34} +1.57699 q^{35} -3.22668 q^{37} +6.66544 q^{38} -1.34730 q^{40} +4.86084 q^{41} -5.75877 q^{43} -0.965505 q^{44} +9.35504 q^{46} -3.01763 q^{47} +5.47565 q^{49} +6.17161 q^{50} -1.14290 q^{52} -8.77141 q^{53} -1.24123 q^{55} -10.6585 q^{56} +4.65270 q^{58} -2.96377 q^{59} +7.88713 q^{61} +2.48865 q^{62} +8.86484 q^{64} -1.46929 q^{65} -9.43882 q^{67} -2.44474 q^{68} -2.02734 q^{70} -5.30731 q^{71} -1.55438 q^{73} +4.14814 q^{74} +1.80066 q^{76} -9.81942 q^{77} -11.9017 q^{79} +1.42193 q^{80} -6.24897 q^{82} +16.2573 q^{83} -3.14290 q^{85} +7.40333 q^{86} +8.38919 q^{88} -18.4258 q^{89} -11.6236 q^{91} +2.52725 q^{92} +3.87939 q^{94} +2.31488 q^{95} +10.1138 q^{97} -7.03936 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 12 q^{7} - 12 q^{10} - 12 q^{13} - 12 q^{16} - 24 q^{19} - 6 q^{22} - 6 q^{28} - 30 q^{31} - 6 q^{37} - 6 q^{40} - 12 q^{43} + 6 q^{46} - 6 q^{49} - 6 q^{52} - 30 q^{55} + 30 q^{58} - 12 q^{61} + 6 q^{64} + 6 q^{67} + 30 q^{70} + 12 q^{73} - 18 q^{76} - 48 q^{79} - 12 q^{82} - 18 q^{85} + 42 q^{88} + 12 q^{94} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.28558 −0.909039 −0.454519 0.890737i \(-0.650189\pi\)
−0.454519 + 0.890737i \(0.650189\pi\)
\(3\) 0 0
\(4\) −0.347296 −0.173648
\(5\) −0.446476 −0.199670 −0.0998350 0.995004i \(-0.531831\pi\)
−0.0998350 + 0.995004i \(0.531831\pi\)
\(6\) 0 0
\(7\) −3.53209 −1.33500 −0.667502 0.744608i \(-0.732636\pi\)
−0.667502 + 0.744608i \(0.732636\pi\)
\(8\) 3.01763 1.06689
\(9\) 0 0
\(10\) 0.573978 0.181508
\(11\) 2.78006 0.838220 0.419110 0.907935i \(-0.362342\pi\)
0.419110 + 0.907935i \(0.362342\pi\)
\(12\) 0 0
\(13\) 3.29086 0.912720 0.456360 0.889795i \(-0.349153\pi\)
0.456360 + 0.889795i \(0.349153\pi\)
\(14\) 4.54077 1.21357
\(15\) 0 0
\(16\) −3.18479 −0.796198
\(17\) 7.03936 1.70730 0.853648 0.520850i \(-0.174385\pi\)
0.853648 + 0.520850i \(0.174385\pi\)
\(18\) 0 0
\(19\) −5.18479 −1.18947 −0.594736 0.803921i \(-0.702744\pi\)
−0.594736 + 0.803921i \(0.702744\pi\)
\(20\) 0.155059 0.0346723
\(21\) 0 0
\(22\) −3.57398 −0.761975
\(23\) −7.27693 −1.51734 −0.758672 0.651473i \(-0.774152\pi\)
−0.758672 + 0.651473i \(0.774152\pi\)
\(24\) 0 0
\(25\) −4.80066 −0.960132
\(26\) −4.23065 −0.829698
\(27\) 0 0
\(28\) 1.22668 0.231821
\(29\) −3.61916 −0.672061 −0.336031 0.941851i \(-0.609085\pi\)
−0.336031 + 0.941851i \(0.609085\pi\)
\(30\) 0 0
\(31\) −1.93582 −0.347684 −0.173842 0.984774i \(-0.555618\pi\)
−0.173842 + 0.984774i \(0.555618\pi\)
\(32\) −1.94096 −0.343117
\(33\) 0 0
\(34\) −9.04963 −1.55200
\(35\) 1.57699 0.266560
\(36\) 0 0
\(37\) −3.22668 −0.530463 −0.265232 0.964185i \(-0.585448\pi\)
−0.265232 + 0.964185i \(0.585448\pi\)
\(38\) 6.66544 1.08128
\(39\) 0 0
\(40\) −1.34730 −0.213026
\(41\) 4.86084 0.759135 0.379568 0.925164i \(-0.376073\pi\)
0.379568 + 0.925164i \(0.376073\pi\)
\(42\) 0 0
\(43\) −5.75877 −0.878204 −0.439102 0.898437i \(-0.644703\pi\)
−0.439102 + 0.898437i \(0.644703\pi\)
\(44\) −0.965505 −0.145555
\(45\) 0 0
\(46\) 9.35504 1.37932
\(47\) −3.01763 −0.440166 −0.220083 0.975481i \(-0.570633\pi\)
−0.220083 + 0.975481i \(0.570633\pi\)
\(48\) 0 0
\(49\) 5.47565 0.782236
\(50\) 6.17161 0.872797
\(51\) 0 0
\(52\) −1.14290 −0.158492
\(53\) −8.77141 −1.20485 −0.602423 0.798177i \(-0.705798\pi\)
−0.602423 + 0.798177i \(0.705798\pi\)
\(54\) 0 0
\(55\) −1.24123 −0.167367
\(56\) −10.6585 −1.42431
\(57\) 0 0
\(58\) 4.65270 0.610930
\(59\) −2.96377 −0.385851 −0.192925 0.981213i \(-0.561797\pi\)
−0.192925 + 0.981213i \(0.561797\pi\)
\(60\) 0 0
\(61\) 7.88713 1.00984 0.504922 0.863165i \(-0.331521\pi\)
0.504922 + 0.863165i \(0.331521\pi\)
\(62\) 2.48865 0.316058
\(63\) 0 0
\(64\) 8.86484 1.10810
\(65\) −1.46929 −0.182243
\(66\) 0 0
\(67\) −9.43882 −1.15313 −0.576567 0.817050i \(-0.695608\pi\)
−0.576567 + 0.817050i \(0.695608\pi\)
\(68\) −2.44474 −0.296469
\(69\) 0 0
\(70\) −2.02734 −0.242314
\(71\) −5.30731 −0.629862 −0.314931 0.949115i \(-0.601981\pi\)
−0.314931 + 0.949115i \(0.601981\pi\)
\(72\) 0 0
\(73\) −1.55438 −0.181926 −0.0909631 0.995854i \(-0.528995\pi\)
−0.0909631 + 0.995854i \(0.528995\pi\)
\(74\) 4.14814 0.482212
\(75\) 0 0
\(76\) 1.80066 0.206550
\(77\) −9.81942 −1.11903
\(78\) 0 0
\(79\) −11.9017 −1.33904 −0.669521 0.742793i \(-0.733501\pi\)
−0.669521 + 0.742793i \(0.733501\pi\)
\(80\) 1.42193 0.158977
\(81\) 0 0
\(82\) −6.24897 −0.690083
\(83\) 16.2573 1.78447 0.892233 0.451576i \(-0.149138\pi\)
0.892233 + 0.451576i \(0.149138\pi\)
\(84\) 0 0
\(85\) −3.14290 −0.340896
\(86\) 7.40333 0.798322
\(87\) 0 0
\(88\) 8.38919 0.894290
\(89\) −18.4258 −1.95313 −0.976567 0.215214i \(-0.930955\pi\)
−0.976567 + 0.215214i \(0.930955\pi\)
\(90\) 0 0
\(91\) −11.6236 −1.21849
\(92\) 2.52725 0.263484
\(93\) 0 0
\(94\) 3.87939 0.400128
\(95\) 2.31488 0.237502
\(96\) 0 0
\(97\) 10.1138 1.02690 0.513451 0.858119i \(-0.328367\pi\)
0.513451 + 0.858119i \(0.328367\pi\)
\(98\) −7.03936 −0.711083
\(99\) 0 0
\(100\) 1.66725 0.166725
\(101\) 2.08077 0.207045 0.103522 0.994627i \(-0.466989\pi\)
0.103522 + 0.994627i \(0.466989\pi\)
\(102\) 0 0
\(103\) −1.44831 −0.142706 −0.0713531 0.997451i \(-0.522732\pi\)
−0.0713531 + 0.997451i \(0.522732\pi\)
\(104\) 9.93058 0.973774
\(105\) 0 0
\(106\) 11.2763 1.09525
\(107\) 2.23583 0.216146 0.108073 0.994143i \(-0.465532\pi\)
0.108073 + 0.994143i \(0.465532\pi\)
\(108\) 0 0
\(109\) −11.5030 −1.10179 −0.550893 0.834576i \(-0.685713\pi\)
−0.550893 + 0.834576i \(0.685713\pi\)
\(110\) 1.59569 0.152143
\(111\) 0 0
\(112\) 11.2490 1.06293
\(113\) −1.68815 −0.158808 −0.0794039 0.996843i \(-0.525302\pi\)
−0.0794039 + 0.996843i \(0.525302\pi\)
\(114\) 0 0
\(115\) 3.24897 0.302968
\(116\) 1.25692 0.116702
\(117\) 0 0
\(118\) 3.81016 0.350753
\(119\) −24.8637 −2.27925
\(120\) 0 0
\(121\) −3.27126 −0.297387
\(122\) −10.1395 −0.917987
\(123\) 0 0
\(124\) 0.672304 0.0603747
\(125\) 4.37576 0.391379
\(126\) 0 0
\(127\) −2.67230 −0.237129 −0.118564 0.992946i \(-0.537829\pi\)
−0.118564 + 0.992946i \(0.537829\pi\)
\(128\) −7.51449 −0.664194
\(129\) 0 0
\(130\) 1.88888 0.165666
\(131\) 2.91987 0.255111 0.127555 0.991831i \(-0.459287\pi\)
0.127555 + 0.991831i \(0.459287\pi\)
\(132\) 0 0
\(133\) 18.3131 1.58795
\(134\) 12.1343 1.04824
\(135\) 0 0
\(136\) 21.2422 1.82150
\(137\) 4.65722 0.397893 0.198947 0.980010i \(-0.436248\pi\)
0.198947 + 0.980010i \(0.436248\pi\)
\(138\) 0 0
\(139\) −8.00269 −0.678779 −0.339390 0.940646i \(-0.610220\pi\)
−0.339390 + 0.940646i \(0.610220\pi\)
\(140\) −0.547683 −0.0462877
\(141\) 0 0
\(142\) 6.82295 0.572569
\(143\) 9.14879 0.765060
\(144\) 0 0
\(145\) 1.61587 0.134190
\(146\) 1.99827 0.165378
\(147\) 0 0
\(148\) 1.12061 0.0921140
\(149\) −20.2117 −1.65581 −0.827905 0.560869i \(-0.810467\pi\)
−0.827905 + 0.560869i \(0.810467\pi\)
\(150\) 0 0
\(151\) 6.70233 0.545428 0.272714 0.962095i \(-0.412079\pi\)
0.272714 + 0.962095i \(0.412079\pi\)
\(152\) −15.6458 −1.26904
\(153\) 0 0
\(154\) 12.6236 1.01724
\(155\) 0.864297 0.0694220
\(156\) 0 0
\(157\) 5.32501 0.424982 0.212491 0.977163i \(-0.431842\pi\)
0.212491 + 0.977163i \(0.431842\pi\)
\(158\) 15.3005 1.21724
\(159\) 0 0
\(160\) 0.866592 0.0685101
\(161\) 25.7028 2.02566
\(162\) 0 0
\(163\) 3.81521 0.298830 0.149415 0.988775i \(-0.452261\pi\)
0.149415 + 0.988775i \(0.452261\pi\)
\(164\) −1.68815 −0.131822
\(165\) 0 0
\(166\) −20.8999 −1.62215
\(167\) 9.13538 0.706917 0.353459 0.935450i \(-0.385006\pi\)
0.353459 + 0.935450i \(0.385006\pi\)
\(168\) 0 0
\(169\) −2.17024 −0.166942
\(170\) 4.04044 0.309888
\(171\) 0 0
\(172\) 2.00000 0.152499
\(173\) 6.07386 0.461787 0.230893 0.972979i \(-0.425835\pi\)
0.230893 + 0.972979i \(0.425835\pi\)
\(174\) 0 0
\(175\) 16.9564 1.28178
\(176\) −8.85392 −0.667389
\(177\) 0 0
\(178\) 23.6878 1.77547
\(179\) 10.2811 0.768449 0.384224 0.923240i \(-0.374469\pi\)
0.384224 + 0.923240i \(0.374469\pi\)
\(180\) 0 0
\(181\) −23.1411 −1.72007 −0.860034 0.510237i \(-0.829558\pi\)
−0.860034 + 0.510237i \(0.829558\pi\)
\(182\) 14.9430 1.10765
\(183\) 0 0
\(184\) −21.9590 −1.61884
\(185\) 1.44063 0.105918
\(186\) 0 0
\(187\) 19.5699 1.43109
\(188\) 1.04801 0.0764340
\(189\) 0 0
\(190\) −2.97596 −0.215899
\(191\) 14.6703 1.06151 0.530753 0.847526i \(-0.321909\pi\)
0.530753 + 0.847526i \(0.321909\pi\)
\(192\) 0 0
\(193\) 23.4415 1.68736 0.843678 0.536849i \(-0.180386\pi\)
0.843678 + 0.536849i \(0.180386\pi\)
\(194\) −13.0021 −0.933494
\(195\) 0 0
\(196\) −1.90167 −0.135834
\(197\) −9.02768 −0.643196 −0.321598 0.946876i \(-0.604220\pi\)
−0.321598 + 0.946876i \(0.604220\pi\)
\(198\) 0 0
\(199\) 2.60401 0.184593 0.0922966 0.995732i \(-0.470579\pi\)
0.0922966 + 0.995732i \(0.470579\pi\)
\(200\) −14.4866 −1.02436
\(201\) 0 0
\(202\) −2.67499 −0.188212
\(203\) 12.7832 0.897205
\(204\) 0 0
\(205\) −2.17024 −0.151576
\(206\) 1.86191 0.129726
\(207\) 0 0
\(208\) −10.4807 −0.726706
\(209\) −14.4140 −0.997040
\(210\) 0 0
\(211\) −16.3550 −1.12593 −0.562964 0.826482i \(-0.690339\pi\)
−0.562964 + 0.826482i \(0.690339\pi\)
\(212\) 3.04628 0.209219
\(213\) 0 0
\(214\) −2.87433 −0.196485
\(215\) 2.57115 0.175351
\(216\) 0 0
\(217\) 6.83750 0.464159
\(218\) 14.7880 1.00157
\(219\) 0 0
\(220\) 0.431074 0.0290630
\(221\) 23.1656 1.55828
\(222\) 0 0
\(223\) 3.70233 0.247927 0.123963 0.992287i \(-0.460439\pi\)
0.123963 + 0.992287i \(0.460439\pi\)
\(224\) 6.85565 0.458062
\(225\) 0 0
\(226\) 2.17024 0.144363
\(227\) −10.5608 −0.700943 −0.350472 0.936573i \(-0.613979\pi\)
−0.350472 + 0.936573i \(0.613979\pi\)
\(228\) 0 0
\(229\) 13.5253 0.893776 0.446888 0.894590i \(-0.352532\pi\)
0.446888 + 0.894590i \(0.352532\pi\)
\(230\) −4.17680 −0.275410
\(231\) 0 0
\(232\) −10.9213 −0.717017
\(233\) 12.7007 0.832050 0.416025 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(234\) 0 0
\(235\) 1.34730 0.0878879
\(236\) 1.02931 0.0670022
\(237\) 0 0
\(238\) 31.9641 2.07192
\(239\) −5.85154 −0.378505 −0.189252 0.981928i \(-0.560606\pi\)
−0.189252 + 0.981928i \(0.560606\pi\)
\(240\) 0 0
\(241\) −8.53983 −0.550099 −0.275049 0.961430i \(-0.588694\pi\)
−0.275049 + 0.961430i \(0.588694\pi\)
\(242\) 4.20545 0.270337
\(243\) 0 0
\(244\) −2.73917 −0.175357
\(245\) −2.44474 −0.156189
\(246\) 0 0
\(247\) −17.0624 −1.08566
\(248\) −5.84159 −0.370941
\(249\) 0 0
\(250\) −5.62536 −0.355779
\(251\) −6.75790 −0.426555 −0.213277 0.976992i \(-0.568414\pi\)
−0.213277 + 0.976992i \(0.568414\pi\)
\(252\) 0 0
\(253\) −20.2303 −1.27187
\(254\) 3.43545 0.215559
\(255\) 0 0
\(256\) −8.06923 −0.504327
\(257\) 3.68296 0.229737 0.114868 0.993381i \(-0.463355\pi\)
0.114868 + 0.993381i \(0.463355\pi\)
\(258\) 0 0
\(259\) 11.3969 0.708171
\(260\) 0.510278 0.0316461
\(261\) 0 0
\(262\) −3.75372 −0.231905
\(263\) 3.63786 0.224320 0.112160 0.993690i \(-0.464223\pi\)
0.112160 + 0.993690i \(0.464223\pi\)
\(264\) 0 0
\(265\) 3.91622 0.240572
\(266\) −23.5429 −1.44351
\(267\) 0 0
\(268\) 3.27807 0.200240
\(269\) −7.08672 −0.432085 −0.216042 0.976384i \(-0.569315\pi\)
−0.216042 + 0.976384i \(0.569315\pi\)
\(270\) 0 0
\(271\) −19.0000 −1.15417 −0.577084 0.816685i \(-0.695809\pi\)
−0.577084 + 0.816685i \(0.695809\pi\)
\(272\) −22.4189 −1.35935
\(273\) 0 0
\(274\) −5.98721 −0.361700
\(275\) −13.3461 −0.804802
\(276\) 0 0
\(277\) −13.8922 −0.834700 −0.417350 0.908746i \(-0.637041\pi\)
−0.417350 + 0.908746i \(0.637041\pi\)
\(278\) 10.2881 0.617037
\(279\) 0 0
\(280\) 4.75877 0.284391
\(281\) 22.1275 1.32002 0.660008 0.751259i \(-0.270553\pi\)
0.660008 + 0.751259i \(0.270553\pi\)
\(282\) 0 0
\(283\) −23.7324 −1.41074 −0.705371 0.708838i \(-0.749220\pi\)
−0.705371 + 0.708838i \(0.749220\pi\)
\(284\) 1.84321 0.109374
\(285\) 0 0
\(286\) −11.7615 −0.695470
\(287\) −17.1689 −1.01345
\(288\) 0 0
\(289\) 32.5526 1.91486
\(290\) −2.07732 −0.121984
\(291\) 0 0
\(292\) 0.539830 0.0315911
\(293\) 18.6061 1.08698 0.543489 0.839416i \(-0.317103\pi\)
0.543489 + 0.839416i \(0.317103\pi\)
\(294\) 0 0
\(295\) 1.32325 0.0770428
\(296\) −9.73692 −0.565947
\(297\) 0 0
\(298\) 25.9837 1.50520
\(299\) −23.9473 −1.38491
\(300\) 0 0
\(301\) 20.3405 1.17241
\(302\) −8.61635 −0.495815
\(303\) 0 0
\(304\) 16.5125 0.947056
\(305\) −3.52141 −0.201635
\(306\) 0 0
\(307\) 20.7469 1.18409 0.592044 0.805905i \(-0.298321\pi\)
0.592044 + 0.805905i \(0.298321\pi\)
\(308\) 3.41025 0.194317
\(309\) 0 0
\(310\) −1.11112 −0.0631073
\(311\) 20.3855 1.15596 0.577978 0.816053i \(-0.303842\pi\)
0.577978 + 0.816053i \(0.303842\pi\)
\(312\) 0 0
\(313\) −29.8408 −1.68670 −0.843351 0.537364i \(-0.819420\pi\)
−0.843351 + 0.537364i \(0.819420\pi\)
\(314\) −6.84570 −0.386325
\(315\) 0 0
\(316\) 4.13341 0.232522
\(317\) 4.31661 0.242445 0.121222 0.992625i \(-0.461319\pi\)
0.121222 + 0.992625i \(0.461319\pi\)
\(318\) 0 0
\(319\) −10.0615 −0.563335
\(320\) −3.95793 −0.221255
\(321\) 0 0
\(322\) −33.0428 −1.84140
\(323\) −36.4976 −2.03078
\(324\) 0 0
\(325\) −15.7983 −0.876332
\(326\) −4.90474 −0.271648
\(327\) 0 0
\(328\) 14.6682 0.809915
\(329\) 10.6585 0.587623
\(330\) 0 0
\(331\) −2.91891 −0.160438 −0.0802189 0.996777i \(-0.525562\pi\)
−0.0802189 + 0.996777i \(0.525562\pi\)
\(332\) −5.64608 −0.309869
\(333\) 0 0
\(334\) −11.7442 −0.642615
\(335\) 4.21420 0.230246
\(336\) 0 0
\(337\) 14.4415 0.786679 0.393339 0.919393i \(-0.371320\pi\)
0.393339 + 0.919393i \(0.371320\pi\)
\(338\) 2.79001 0.151757
\(339\) 0 0
\(340\) 1.09152 0.0591959
\(341\) −5.38170 −0.291436
\(342\) 0 0
\(343\) 5.38413 0.290716
\(344\) −17.3778 −0.936949
\(345\) 0 0
\(346\) −7.80840 −0.419782
\(347\) −1.45404 −0.0780571 −0.0390285 0.999238i \(-0.512426\pi\)
−0.0390285 + 0.999238i \(0.512426\pi\)
\(348\) 0 0
\(349\) −27.5817 −1.47642 −0.738208 0.674573i \(-0.764328\pi\)
−0.738208 + 0.674573i \(0.764328\pi\)
\(350\) −21.7987 −1.16519
\(351\) 0 0
\(352\) −5.39599 −0.287607
\(353\) 10.5321 0.560568 0.280284 0.959917i \(-0.409571\pi\)
0.280284 + 0.959917i \(0.409571\pi\)
\(354\) 0 0
\(355\) 2.36959 0.125765
\(356\) 6.39922 0.339158
\(357\) 0 0
\(358\) −13.2172 −0.698550
\(359\) 14.7055 0.776124 0.388062 0.921633i \(-0.373145\pi\)
0.388062 + 0.921633i \(0.373145\pi\)
\(360\) 0 0
\(361\) 7.88207 0.414846
\(362\) 29.7497 1.56361
\(363\) 0 0
\(364\) 4.03684 0.211588
\(365\) 0.693992 0.0363252
\(366\) 0 0
\(367\) −21.3628 −1.11513 −0.557564 0.830134i \(-0.688264\pi\)
−0.557564 + 0.830134i \(0.688264\pi\)
\(368\) 23.1755 1.20811
\(369\) 0 0
\(370\) −1.85204 −0.0962832
\(371\) 30.9814 1.60847
\(372\) 0 0
\(373\) 22.6245 1.17145 0.585727 0.810508i \(-0.300809\pi\)
0.585727 + 0.810508i \(0.300809\pi\)
\(374\) −25.1585 −1.30092
\(375\) 0 0
\(376\) −9.10607 −0.469610
\(377\) −11.9101 −0.613404
\(378\) 0 0
\(379\) −17.0743 −0.877047 −0.438523 0.898720i \(-0.644498\pi\)
−0.438523 + 0.898720i \(0.644498\pi\)
\(380\) −0.803951 −0.0412418
\(381\) 0 0
\(382\) −18.8598 −0.964951
\(383\) −38.9916 −1.99238 −0.996188 0.0872303i \(-0.972198\pi\)
−0.996188 + 0.0872303i \(0.972198\pi\)
\(384\) 0 0
\(385\) 4.38413 0.223436
\(386\) −30.1358 −1.53387
\(387\) 0 0
\(388\) −3.51249 −0.178320
\(389\) 10.2120 0.517771 0.258886 0.965908i \(-0.416645\pi\)
0.258886 + 0.965908i \(0.416645\pi\)
\(390\) 0 0
\(391\) −51.2249 −2.59056
\(392\) 16.5235 0.834561
\(393\) 0 0
\(394\) 11.6058 0.584690
\(395\) 5.31381 0.267367
\(396\) 0 0
\(397\) −1.14290 −0.0573607 −0.0286803 0.999589i \(-0.509130\pi\)
−0.0286803 + 0.999589i \(0.509130\pi\)
\(398\) −3.34765 −0.167802
\(399\) 0 0
\(400\) 15.2891 0.764455
\(401\) −20.5540 −1.02642 −0.513208 0.858264i \(-0.671543\pi\)
−0.513208 + 0.858264i \(0.671543\pi\)
\(402\) 0 0
\(403\) −6.37052 −0.317338
\(404\) −0.722645 −0.0359530
\(405\) 0 0
\(406\) −16.4338 −0.815594
\(407\) −8.97037 −0.444645
\(408\) 0 0
\(409\) −8.12836 −0.401921 −0.200961 0.979599i \(-0.564406\pi\)
−0.200961 + 0.979599i \(0.564406\pi\)
\(410\) 2.79001 0.137789
\(411\) 0 0
\(412\) 0.502993 0.0247807
\(413\) 10.4683 0.515112
\(414\) 0 0
\(415\) −7.25847 −0.356304
\(416\) −6.38743 −0.313170
\(417\) 0 0
\(418\) 18.5303 0.906348
\(419\) 35.7784 1.74789 0.873946 0.486024i \(-0.161553\pi\)
0.873946 + 0.486024i \(0.161553\pi\)
\(420\) 0 0
\(421\) −13.1189 −0.639374 −0.319687 0.947523i \(-0.603578\pi\)
−0.319687 + 0.947523i \(0.603578\pi\)
\(422\) 21.0256 1.02351
\(423\) 0 0
\(424\) −26.4688 −1.28544
\(425\) −33.7936 −1.63923
\(426\) 0 0
\(427\) −27.8580 −1.34814
\(428\) −0.776497 −0.0375334
\(429\) 0 0
\(430\) −3.30541 −0.159401
\(431\) 9.48411 0.456833 0.228417 0.973563i \(-0.426645\pi\)
0.228417 + 0.973563i \(0.426645\pi\)
\(432\) 0 0
\(433\) 17.6628 0.848820 0.424410 0.905470i \(-0.360481\pi\)
0.424410 + 0.905470i \(0.360481\pi\)
\(434\) −8.79012 −0.421939
\(435\) 0 0
\(436\) 3.99495 0.191323
\(437\) 37.7294 1.80484
\(438\) 0 0
\(439\) −17.6655 −0.843128 −0.421564 0.906799i \(-0.638519\pi\)
−0.421564 + 0.906799i \(0.638519\pi\)
\(440\) −3.74557 −0.178563
\(441\) 0 0
\(442\) −29.7811 −1.41654
\(443\) −12.9921 −0.617274 −0.308637 0.951180i \(-0.599873\pi\)
−0.308637 + 0.951180i \(0.599873\pi\)
\(444\) 0 0
\(445\) 8.22668 0.389982
\(446\) −4.75963 −0.225375
\(447\) 0 0
\(448\) −31.3114 −1.47932
\(449\) −4.62857 −0.218436 −0.109218 0.994018i \(-0.534835\pi\)
−0.109218 + 0.994018i \(0.534835\pi\)
\(450\) 0 0
\(451\) 13.5134 0.636322
\(452\) 0.586289 0.0275767
\(453\) 0 0
\(454\) 13.5767 0.637185
\(455\) 5.18966 0.243295
\(456\) 0 0
\(457\) 20.5672 0.962092 0.481046 0.876695i \(-0.340257\pi\)
0.481046 + 0.876695i \(0.340257\pi\)
\(458\) −17.3878 −0.812477
\(459\) 0 0
\(460\) −1.12836 −0.0526098
\(461\) −33.5725 −1.56363 −0.781813 0.623513i \(-0.785705\pi\)
−0.781813 + 0.623513i \(0.785705\pi\)
\(462\) 0 0
\(463\) 13.1310 0.610251 0.305126 0.952312i \(-0.401302\pi\)
0.305126 + 0.952312i \(0.401302\pi\)
\(464\) 11.5263 0.535094
\(465\) 0 0
\(466\) −16.3277 −0.756366
\(467\) 23.6307 1.09350 0.546750 0.837296i \(-0.315865\pi\)
0.546750 + 0.837296i \(0.315865\pi\)
\(468\) 0 0
\(469\) 33.3387 1.53944
\(470\) −1.73205 −0.0798935
\(471\) 0 0
\(472\) −8.94356 −0.411661
\(473\) −16.0097 −0.736128
\(474\) 0 0
\(475\) 24.8904 1.14205
\(476\) 8.63506 0.395787
\(477\) 0 0
\(478\) 7.52259 0.344075
\(479\) −5.88019 −0.268673 −0.134336 0.990936i \(-0.542890\pi\)
−0.134336 + 0.990936i \(0.542890\pi\)
\(480\) 0 0
\(481\) −10.6186 −0.484164
\(482\) 10.9786 0.500061
\(483\) 0 0
\(484\) 1.13610 0.0516407
\(485\) −4.51557 −0.205041
\(486\) 0 0
\(487\) 38.7965 1.75804 0.879020 0.476786i \(-0.158198\pi\)
0.879020 + 0.476786i \(0.158198\pi\)
\(488\) 23.8004 1.07739
\(489\) 0 0
\(490\) 3.14290 0.141982
\(491\) −37.5842 −1.69615 −0.848077 0.529873i \(-0.822239\pi\)
−0.848077 + 0.529873i \(0.822239\pi\)
\(492\) 0 0
\(493\) −25.4766 −1.14741
\(494\) 21.9350 0.986904
\(495\) 0 0
\(496\) 6.16519 0.276825
\(497\) 18.7459 0.840868
\(498\) 0 0
\(499\) 33.7520 1.51095 0.755473 0.655180i \(-0.227407\pi\)
0.755473 + 0.655180i \(0.227407\pi\)
\(500\) −1.51968 −0.0679623
\(501\) 0 0
\(502\) 8.68779 0.387755
\(503\) −18.7119 −0.834324 −0.417162 0.908832i \(-0.636975\pi\)
−0.417162 + 0.908832i \(0.636975\pi\)
\(504\) 0 0
\(505\) −0.929015 −0.0413406
\(506\) 26.0076 1.15618
\(507\) 0 0
\(508\) 0.928081 0.0411770
\(509\) 21.7682 0.964858 0.482429 0.875935i \(-0.339755\pi\)
0.482429 + 0.875935i \(0.339755\pi\)
\(510\) 0 0
\(511\) 5.49020 0.242872
\(512\) 25.4026 1.12265
\(513\) 0 0
\(514\) −4.73473 −0.208840
\(515\) 0.646635 0.0284942
\(516\) 0 0
\(517\) −8.38919 −0.368956
\(518\) −14.6516 −0.643755
\(519\) 0 0
\(520\) −4.43376 −0.194433
\(521\) −6.47643 −0.283738 −0.141869 0.989885i \(-0.545311\pi\)
−0.141869 + 0.989885i \(0.545311\pi\)
\(522\) 0 0
\(523\) −10.8726 −0.475425 −0.237712 0.971336i \(-0.576398\pi\)
−0.237712 + 0.971336i \(0.576398\pi\)
\(524\) −1.01406 −0.0442995
\(525\) 0 0
\(526\) −4.67675 −0.203916
\(527\) −13.6270 −0.593599
\(528\) 0 0
\(529\) 29.9537 1.30233
\(530\) −5.03460 −0.218689
\(531\) 0 0
\(532\) −6.36009 −0.275745
\(533\) 15.9963 0.692878
\(534\) 0 0
\(535\) −0.998245 −0.0431579
\(536\) −28.4828 −1.23027
\(537\) 0 0
\(538\) 9.11051 0.392782
\(539\) 15.2226 0.655686
\(540\) 0 0
\(541\) 24.6459 1.05961 0.529805 0.848120i \(-0.322265\pi\)
0.529805 + 0.848120i \(0.322265\pi\)
\(542\) 24.4259 1.04918
\(543\) 0 0
\(544\) −13.6631 −0.585802
\(545\) 5.13581 0.219994
\(546\) 0 0
\(547\) 30.9273 1.32235 0.661177 0.750230i \(-0.270057\pi\)
0.661177 + 0.750230i \(0.270057\pi\)
\(548\) −1.61744 −0.0690934
\(549\) 0 0
\(550\) 17.1575 0.731596
\(551\) 18.7646 0.799399
\(552\) 0 0
\(553\) 42.0378 1.78763
\(554\) 17.8594 0.758775
\(555\) 0 0
\(556\) 2.77930 0.117869
\(557\) 23.3627 0.989908 0.494954 0.868919i \(-0.335185\pi\)
0.494954 + 0.868919i \(0.335185\pi\)
\(558\) 0 0
\(559\) −18.9513 −0.801555
\(560\) −5.02239 −0.212235
\(561\) 0 0
\(562\) −28.4466 −1.19995
\(563\) 33.6877 1.41977 0.709884 0.704319i \(-0.248747\pi\)
0.709884 + 0.704319i \(0.248747\pi\)
\(564\) 0 0
\(565\) 0.753718 0.0317092
\(566\) 30.5097 1.28242
\(567\) 0 0
\(568\) −16.0155 −0.671995
\(569\) −30.7959 −1.29103 −0.645515 0.763748i \(-0.723357\pi\)
−0.645515 + 0.763748i \(0.723357\pi\)
\(570\) 0 0
\(571\) −29.8753 −1.25024 −0.625120 0.780528i \(-0.714950\pi\)
−0.625120 + 0.780528i \(0.714950\pi\)
\(572\) −3.17734 −0.132851
\(573\) 0 0
\(574\) 22.0719 0.921264
\(575\) 34.9340 1.45685
\(576\) 0 0
\(577\) 4.80747 0.200137 0.100069 0.994981i \(-0.468094\pi\)
0.100069 + 0.994981i \(0.468094\pi\)
\(578\) −41.8488 −1.74068
\(579\) 0 0
\(580\) −0.561185 −0.0233019
\(581\) −57.4221 −2.38227
\(582\) 0 0
\(583\) −24.3851 −1.00993
\(584\) −4.69053 −0.194096
\(585\) 0 0
\(586\) −23.9195 −0.988106
\(587\) 7.93761 0.327620 0.163810 0.986492i \(-0.447622\pi\)
0.163810 + 0.986492i \(0.447622\pi\)
\(588\) 0 0
\(589\) 10.0368 0.413561
\(590\) −1.70114 −0.0700349
\(591\) 0 0
\(592\) 10.2763 0.422354
\(593\) 36.2753 1.48965 0.744824 0.667261i \(-0.232533\pi\)
0.744824 + 0.667261i \(0.232533\pi\)
\(594\) 0 0
\(595\) 11.1010 0.455097
\(596\) 7.01946 0.287528
\(597\) 0 0
\(598\) 30.7861 1.25894
\(599\) 33.6450 1.37470 0.687349 0.726327i \(-0.258774\pi\)
0.687349 + 0.726327i \(0.258774\pi\)
\(600\) 0 0
\(601\) −2.97771 −0.121463 −0.0607317 0.998154i \(-0.519343\pi\)
−0.0607317 + 0.998154i \(0.519343\pi\)
\(602\) −26.1492 −1.06576
\(603\) 0 0
\(604\) −2.32770 −0.0947126
\(605\) 1.46054 0.0593793
\(606\) 0 0
\(607\) −15.6486 −0.635156 −0.317578 0.948232i \(-0.602870\pi\)
−0.317578 + 0.948232i \(0.602870\pi\)
\(608\) 10.0635 0.408128
\(609\) 0 0
\(610\) 4.52704 0.183294
\(611\) −9.93058 −0.401748
\(612\) 0 0
\(613\) −1.06687 −0.0430903 −0.0215452 0.999768i \(-0.506859\pi\)
−0.0215452 + 0.999768i \(0.506859\pi\)
\(614\) −26.6717 −1.07638
\(615\) 0 0
\(616\) −29.6313 −1.19388
\(617\) −13.0700 −0.526177 −0.263088 0.964772i \(-0.584741\pi\)
−0.263088 + 0.964772i \(0.584741\pi\)
\(618\) 0 0
\(619\) 20.5175 0.824670 0.412335 0.911032i \(-0.364713\pi\)
0.412335 + 0.911032i \(0.364713\pi\)
\(620\) −0.300167 −0.0120550
\(621\) 0 0
\(622\) −26.2071 −1.05081
\(623\) 65.0817 2.60744
\(624\) 0 0
\(625\) 22.0496 0.881985
\(626\) 38.3626 1.53328
\(627\) 0 0
\(628\) −1.84936 −0.0737973
\(629\) −22.7138 −0.905658
\(630\) 0 0
\(631\) −10.3122 −0.410523 −0.205261 0.978707i \(-0.565804\pi\)
−0.205261 + 0.978707i \(0.565804\pi\)
\(632\) −35.9148 −1.42861
\(633\) 0 0
\(634\) −5.54933 −0.220392
\(635\) 1.19312 0.0473475
\(636\) 0 0
\(637\) 18.0196 0.713963
\(638\) 12.9348 0.512094
\(639\) 0 0
\(640\) 3.35504 0.132619
\(641\) 3.50616 0.138485 0.0692426 0.997600i \(-0.477942\pi\)
0.0692426 + 0.997600i \(0.477942\pi\)
\(642\) 0 0
\(643\) 31.5517 1.24428 0.622139 0.782907i \(-0.286264\pi\)
0.622139 + 0.782907i \(0.286264\pi\)
\(644\) −8.92647 −0.351752
\(645\) 0 0
\(646\) 46.9205 1.84606
\(647\) −3.04628 −0.119762 −0.0598808 0.998206i \(-0.519072\pi\)
−0.0598808 + 0.998206i \(0.519072\pi\)
\(648\) 0 0
\(649\) −8.23947 −0.323428
\(650\) 20.3099 0.796620
\(651\) 0 0
\(652\) −1.32501 −0.0518913
\(653\) −30.7374 −1.20285 −0.601423 0.798931i \(-0.705399\pi\)
−0.601423 + 0.798931i \(0.705399\pi\)
\(654\) 0 0
\(655\) −1.30365 −0.0509379
\(656\) −15.4808 −0.604422
\(657\) 0 0
\(658\) −13.7023 −0.534173
\(659\) −33.1839 −1.29266 −0.646331 0.763057i \(-0.723698\pi\)
−0.646331 + 0.763057i \(0.723698\pi\)
\(660\) 0 0
\(661\) −14.9077 −0.579841 −0.289920 0.957051i \(-0.593629\pi\)
−0.289920 + 0.957051i \(0.593629\pi\)
\(662\) 3.75248 0.145844
\(663\) 0 0
\(664\) 49.0583 1.90383
\(665\) −8.17637 −0.317066
\(666\) 0 0
\(667\) 26.3364 1.01975
\(668\) −3.17269 −0.122755
\(669\) 0 0
\(670\) −5.41767 −0.209303
\(671\) 21.9267 0.846471
\(672\) 0 0
\(673\) 6.88888 0.265547 0.132773 0.991146i \(-0.457612\pi\)
0.132773 + 0.991146i \(0.457612\pi\)
\(674\) −18.5656 −0.715122
\(675\) 0 0
\(676\) 0.753718 0.0289892
\(677\) −13.1892 −0.506903 −0.253452 0.967348i \(-0.581566\pi\)
−0.253452 + 0.967348i \(0.581566\pi\)
\(678\) 0 0
\(679\) −35.7229 −1.37092
\(680\) −9.48411 −0.363699
\(681\) 0 0
\(682\) 6.91859 0.264926
\(683\) −3.37814 −0.129261 −0.0646305 0.997909i \(-0.520587\pi\)
−0.0646305 + 0.997909i \(0.520587\pi\)
\(684\) 0 0
\(685\) −2.07934 −0.0794473
\(686\) −6.92171 −0.264272
\(687\) 0 0
\(688\) 18.3405 0.699225
\(689\) −28.8655 −1.09969
\(690\) 0 0
\(691\) 23.4151 0.890752 0.445376 0.895344i \(-0.353070\pi\)
0.445376 + 0.895344i \(0.353070\pi\)
\(692\) −2.10943 −0.0801884
\(693\) 0 0
\(694\) 1.86928 0.0709569
\(695\) 3.57300 0.135532
\(696\) 0 0
\(697\) 34.2172 1.29607
\(698\) 35.4584 1.34212
\(699\) 0 0
\(700\) −5.88888 −0.222579
\(701\) 45.5001 1.71852 0.859258 0.511543i \(-0.170926\pi\)
0.859258 + 0.511543i \(0.170926\pi\)
\(702\) 0 0
\(703\) 16.7297 0.630972
\(704\) 24.6448 0.928836
\(705\) 0 0
\(706\) −13.5398 −0.509578
\(707\) −7.34948 −0.276406
\(708\) 0 0
\(709\) 38.6168 1.45028 0.725142 0.688599i \(-0.241774\pi\)
0.725142 + 0.688599i \(0.241774\pi\)
\(710\) −3.04628 −0.114325
\(711\) 0 0
\(712\) −55.6023 −2.08378
\(713\) 14.0868 0.527556
\(714\) 0 0
\(715\) −4.08471 −0.152760
\(716\) −3.57060 −0.133440
\(717\) 0 0
\(718\) −18.9050 −0.705527
\(719\) −49.3182 −1.83926 −0.919630 0.392786i \(-0.871511\pi\)
−0.919630 + 0.392786i \(0.871511\pi\)
\(720\) 0 0
\(721\) 5.11556 0.190513
\(722\) −10.1330 −0.377111
\(723\) 0 0
\(724\) 8.03684 0.298687
\(725\) 17.3744 0.645268
\(726\) 0 0
\(727\) −32.2945 −1.19774 −0.598868 0.800848i \(-0.704383\pi\)
−0.598868 + 0.800848i \(0.704383\pi\)
\(728\) −35.0757 −1.29999
\(729\) 0 0
\(730\) −0.892178 −0.0330210
\(731\) −40.5381 −1.49935
\(732\) 0 0
\(733\) 39.4662 1.45772 0.728858 0.684665i \(-0.240051\pi\)
0.728858 + 0.684665i \(0.240051\pi\)
\(734\) 27.4635 1.01369
\(735\) 0 0
\(736\) 14.1242 0.520626
\(737\) −26.2405 −0.966581
\(738\) 0 0
\(739\) 35.3090 1.29886 0.649432 0.760420i \(-0.275007\pi\)
0.649432 + 0.760420i \(0.275007\pi\)
\(740\) −0.500327 −0.0183924
\(741\) 0 0
\(742\) −39.8289 −1.46217
\(743\) −47.4106 −1.73933 −0.869663 0.493646i \(-0.835664\pi\)
−0.869663 + 0.493646i \(0.835664\pi\)
\(744\) 0 0
\(745\) 9.02404 0.330615
\(746\) −29.0855 −1.06490
\(747\) 0 0
\(748\) −6.79654 −0.248506
\(749\) −7.89716 −0.288556
\(750\) 0 0
\(751\) −8.92808 −0.325790 −0.162895 0.986643i \(-0.552083\pi\)
−0.162895 + 0.986643i \(0.552083\pi\)
\(752\) 9.61051 0.350459
\(753\) 0 0
\(754\) 15.3114 0.557608
\(755\) −2.99243 −0.108906
\(756\) 0 0
\(757\) −3.63816 −0.132231 −0.0661155 0.997812i \(-0.521061\pi\)
−0.0661155 + 0.997812i \(0.521061\pi\)
\(758\) 21.9503 0.797270
\(759\) 0 0
\(760\) 6.98545 0.253389
\(761\) −6.68880 −0.242469 −0.121234 0.992624i \(-0.538685\pi\)
−0.121234 + 0.992624i \(0.538685\pi\)
\(762\) 0 0
\(763\) 40.6296 1.47089
\(764\) −5.09494 −0.184329
\(765\) 0 0
\(766\) 50.1266 1.81115
\(767\) −9.75337 −0.352174
\(768\) 0 0
\(769\) −21.4712 −0.774272 −0.387136 0.922023i \(-0.626536\pi\)
−0.387136 + 0.922023i \(0.626536\pi\)
\(770\) −5.63613 −0.203112
\(771\) 0 0
\(772\) −8.14115 −0.293006
\(773\) −10.2442 −0.368457 −0.184228 0.982883i \(-0.558979\pi\)
−0.184228 + 0.982883i \(0.558979\pi\)
\(774\) 0 0
\(775\) 9.29322 0.333822
\(776\) 30.5197 1.09559
\(777\) 0 0
\(778\) −13.1284 −0.470674
\(779\) −25.2024 −0.902971
\(780\) 0 0
\(781\) −14.7547 −0.527963
\(782\) 65.8535 2.35492
\(783\) 0 0
\(784\) −17.4388 −0.622815
\(785\) −2.37749 −0.0848561
\(786\) 0 0
\(787\) 0.947682 0.0337812 0.0168906 0.999857i \(-0.494623\pi\)
0.0168906 + 0.999857i \(0.494623\pi\)
\(788\) 3.13528 0.111690
\(789\) 0 0
\(790\) −6.83130 −0.243047
\(791\) 5.96270 0.212009
\(792\) 0 0
\(793\) 25.9554 0.921704
\(794\) 1.46929 0.0521431
\(795\) 0 0
\(796\) −0.904362 −0.0320543
\(797\) 7.01071 0.248332 0.124166 0.992261i \(-0.460374\pi\)
0.124166 + 0.992261i \(0.460374\pi\)
\(798\) 0 0
\(799\) −21.2422 −0.751494
\(800\) 9.31790 0.329437
\(801\) 0 0
\(802\) 26.4237 0.933052
\(803\) −4.32127 −0.152494
\(804\) 0 0
\(805\) −11.4757 −0.404464
\(806\) 8.18978 0.288473
\(807\) 0 0
\(808\) 6.27900 0.220894
\(809\) −45.1028 −1.58573 −0.792866 0.609396i \(-0.791412\pi\)
−0.792866 + 0.609396i \(0.791412\pi\)
\(810\) 0 0
\(811\) 8.07285 0.283476 0.141738 0.989904i \(-0.454731\pi\)
0.141738 + 0.989904i \(0.454731\pi\)
\(812\) −4.43956 −0.155798
\(813\) 0 0
\(814\) 11.5321 0.404200
\(815\) −1.70340 −0.0596674
\(816\) 0 0
\(817\) 29.8580 1.04460
\(818\) 10.4496 0.365362
\(819\) 0 0
\(820\) 0.753718 0.0263210
\(821\) 6.26936 0.218802 0.109401 0.993998i \(-0.465107\pi\)
0.109401 + 0.993998i \(0.465107\pi\)
\(822\) 0 0
\(823\) −19.7929 −0.689938 −0.344969 0.938614i \(-0.612111\pi\)
−0.344969 + 0.938614i \(0.612111\pi\)
\(824\) −4.37046 −0.152252
\(825\) 0 0
\(826\) −13.4578 −0.468257
\(827\) −41.8003 −1.45354 −0.726769 0.686882i \(-0.758979\pi\)
−0.726769 + 0.686882i \(0.758979\pi\)
\(828\) 0 0
\(829\) 33.7279 1.17142 0.585710 0.810521i \(-0.300816\pi\)
0.585710 + 0.810521i \(0.300816\pi\)
\(830\) 9.33130 0.323894
\(831\) 0 0
\(832\) 29.1729 1.01139
\(833\) 38.5451 1.33551
\(834\) 0 0
\(835\) −4.07873 −0.141150
\(836\) 5.00594 0.173134
\(837\) 0 0
\(838\) −45.9959 −1.58890
\(839\) 29.2868 1.01109 0.505546 0.862800i \(-0.331291\pi\)
0.505546 + 0.862800i \(0.331291\pi\)
\(840\) 0 0
\(841\) −15.9017 −0.548334
\(842\) 16.8653 0.581216
\(843\) 0 0
\(844\) 5.68004 0.195515
\(845\) 0.968961 0.0333333
\(846\) 0 0
\(847\) 11.5544 0.397013
\(848\) 27.9351 0.959296
\(849\) 0 0
\(850\) 43.4442 1.49012
\(851\) 23.4803 0.804895
\(852\) 0 0
\(853\) 37.5981 1.28734 0.643668 0.765305i \(-0.277412\pi\)
0.643668 + 0.765305i \(0.277412\pi\)
\(854\) 35.8136 1.22552
\(855\) 0 0
\(856\) 6.74691 0.230605
\(857\) 29.0100 0.990961 0.495481 0.868619i \(-0.334992\pi\)
0.495481 + 0.868619i \(0.334992\pi\)
\(858\) 0 0
\(859\) −24.8675 −0.848469 −0.424235 0.905552i \(-0.639457\pi\)
−0.424235 + 0.905552i \(0.639457\pi\)
\(860\) −0.892951 −0.0304494
\(861\) 0 0
\(862\) −12.1925 −0.415279
\(863\) −42.4018 −1.44337 −0.721687 0.692219i \(-0.756633\pi\)
−0.721687 + 0.692219i \(0.756633\pi\)
\(864\) 0 0
\(865\) −2.71183 −0.0922050
\(866\) −22.7069 −0.771611
\(867\) 0 0
\(868\) −2.37464 −0.0806004
\(869\) −33.0874 −1.12241
\(870\) 0 0
\(871\) −31.0618 −1.05249
\(872\) −34.7117 −1.17549
\(873\) 0 0
\(874\) −48.5039 −1.64067
\(875\) −15.4556 −0.522493
\(876\) 0 0
\(877\) 12.0547 0.407058 0.203529 0.979069i \(-0.434759\pi\)
0.203529 + 0.979069i \(0.434759\pi\)
\(878\) 22.7103 0.766436
\(879\) 0 0
\(880\) 3.95306 0.133258
\(881\) 14.7827 0.498041 0.249020 0.968498i \(-0.419891\pi\)
0.249020 + 0.968498i \(0.419891\pi\)
\(882\) 0 0
\(883\) −25.8462 −0.869793 −0.434896 0.900480i \(-0.643215\pi\)
−0.434896 + 0.900480i \(0.643215\pi\)
\(884\) −8.04531 −0.270593
\(885\) 0 0
\(886\) 16.7023 0.561126
\(887\) −45.8560 −1.53969 −0.769846 0.638229i \(-0.779667\pi\)
−0.769846 + 0.638229i \(0.779667\pi\)
\(888\) 0 0
\(889\) 9.43882 0.316568
\(890\) −10.5760 −0.354509
\(891\) 0 0
\(892\) −1.28581 −0.0430520
\(893\) 15.6458 0.523566
\(894\) 0 0
\(895\) −4.59028 −0.153436
\(896\) 26.5419 0.886701
\(897\) 0 0
\(898\) 5.95037 0.198566
\(899\) 7.00605 0.233665
\(900\) 0 0
\(901\) −61.7452 −2.05703
\(902\) −17.3725 −0.578442
\(903\) 0 0
\(904\) −5.09421 −0.169431
\(905\) 10.3320 0.343446
\(906\) 0 0
\(907\) −11.7000 −0.388491 −0.194246 0.980953i \(-0.562226\pi\)
−0.194246 + 0.980953i \(0.562226\pi\)
\(908\) 3.66772 0.121717
\(909\) 0 0
\(910\) −6.67169 −0.221165
\(911\) −28.6753 −0.950054 −0.475027 0.879971i \(-0.657562\pi\)
−0.475027 + 0.879971i \(0.657562\pi\)
\(912\) 0 0
\(913\) 45.1962 1.49577
\(914\) −26.4406 −0.874579
\(915\) 0 0
\(916\) −4.69728 −0.155203
\(917\) −10.3133 −0.340574
\(918\) 0 0
\(919\) 4.33511 0.143002 0.0715011 0.997441i \(-0.477221\pi\)
0.0715011 + 0.997441i \(0.477221\pi\)
\(920\) 9.80418 0.323234
\(921\) 0 0
\(922\) 43.1599 1.42140
\(923\) −17.4656 −0.574888
\(924\) 0 0
\(925\) 15.4902 0.509315
\(926\) −16.8809 −0.554742
\(927\) 0 0
\(928\) 7.02465 0.230596
\(929\) 12.8909 0.422937 0.211468 0.977385i \(-0.432175\pi\)
0.211468 + 0.977385i \(0.432175\pi\)
\(930\) 0 0
\(931\) −28.3901 −0.930449
\(932\) −4.41090 −0.144484
\(933\) 0 0
\(934\) −30.3791 −0.994034
\(935\) −8.73746 −0.285746
\(936\) 0 0
\(937\) −53.2080 −1.73823 −0.869115 0.494610i \(-0.835311\pi\)
−0.869115 + 0.494610i \(0.835311\pi\)
\(938\) −42.8595 −1.39941
\(939\) 0 0
\(940\) −0.467911 −0.0152616
\(941\) −33.4075 −1.08905 −0.544526 0.838744i \(-0.683290\pi\)
−0.544526 + 0.838744i \(0.683290\pi\)
\(942\) 0 0
\(943\) −35.3719 −1.15187
\(944\) 9.43901 0.307214
\(945\) 0 0
\(946\) 20.5817 0.669169
\(947\) −15.5048 −0.503837 −0.251918 0.967748i \(-0.581061\pi\)
−0.251918 + 0.967748i \(0.581061\pi\)
\(948\) 0 0
\(949\) −5.11524 −0.166048
\(950\) −31.9985 −1.03817
\(951\) 0 0
\(952\) −75.0292 −2.43171
\(953\) 22.5049 0.729004 0.364502 0.931203i \(-0.381239\pi\)
0.364502 + 0.931203i \(0.381239\pi\)
\(954\) 0 0
\(955\) −6.54993 −0.211951
\(956\) 2.03222 0.0657266
\(957\) 0 0
\(958\) 7.55943 0.244234
\(959\) −16.4497 −0.531189
\(960\) 0 0
\(961\) −27.2526 −0.879116
\(962\) 13.6510 0.440124
\(963\) 0 0
\(964\) 2.96585 0.0955237
\(965\) −10.4661 −0.336914
\(966\) 0 0
\(967\) −41.8084 −1.34447 −0.672234 0.740339i \(-0.734665\pi\)
−0.672234 + 0.740339i \(0.734665\pi\)
\(968\) −9.87144 −0.317280
\(969\) 0 0
\(970\) 5.80510 0.186391
\(971\) 35.8662 1.15100 0.575501 0.817801i \(-0.304807\pi\)
0.575501 + 0.817801i \(0.304807\pi\)
\(972\) 0 0
\(973\) 28.2662 0.906173
\(974\) −49.8759 −1.59813
\(975\) 0 0
\(976\) −25.1189 −0.804035
\(977\) 14.0501 0.449502 0.224751 0.974416i \(-0.427843\pi\)
0.224751 + 0.974416i \(0.427843\pi\)
\(978\) 0 0
\(979\) −51.2249 −1.63716
\(980\) 0.849051 0.0271219
\(981\) 0 0
\(982\) 48.3174 1.54187
\(983\) −17.6891 −0.564196 −0.282098 0.959386i \(-0.591030\pi\)
−0.282098 + 0.959386i \(0.591030\pi\)
\(984\) 0 0
\(985\) 4.03064 0.128427
\(986\) 32.7521 1.04304
\(987\) 0 0
\(988\) 5.92572 0.188522
\(989\) 41.9062 1.33254
\(990\) 0 0
\(991\) 32.8958 1.04497 0.522485 0.852649i \(-0.325005\pi\)
0.522485 + 0.852649i \(0.325005\pi\)
\(992\) 3.75736 0.119296
\(993\) 0 0
\(994\) −24.0993 −0.764382
\(995\) −1.16263 −0.0368577
\(996\) 0 0
\(997\) 20.0068 0.633622 0.316811 0.948489i \(-0.397388\pi\)
0.316811 + 0.948489i \(0.397388\pi\)
\(998\) −43.3907 −1.37351
\(999\) 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 729.2.a.c.1.2 6
3.2 odd 2 inner 729.2.a.c.1.5 yes 6
9.2 odd 6 729.2.c.c.244.2 12
9.4 even 3 729.2.c.c.487.5 12
9.5 odd 6 729.2.c.c.487.2 12
9.7 even 3 729.2.c.c.244.5 12
27.2 odd 18 729.2.e.q.568.1 12
27.4 even 9 729.2.e.m.406.1 12
27.5 odd 18 729.2.e.r.649.2 12
27.7 even 9 729.2.e.m.325.1 12
27.11 odd 18 729.2.e.r.82.2 12
27.13 even 9 729.2.e.q.163.2 12
27.14 odd 18 729.2.e.q.163.1 12
27.16 even 9 729.2.e.r.82.1 12
27.20 odd 18 729.2.e.m.325.2 12
27.22 even 9 729.2.e.r.649.1 12
27.23 odd 18 729.2.e.m.406.2 12
27.25 even 9 729.2.e.q.568.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
729.2.a.c.1.2 6 1.1 even 1 trivial
729.2.a.c.1.5 yes 6 3.2 odd 2 inner
729.2.c.c.244.2 12 9.2 odd 6
729.2.c.c.244.5 12 9.7 even 3
729.2.c.c.487.2 12 9.5 odd 6
729.2.c.c.487.5 12 9.4 even 3
729.2.e.m.325.1 12 27.7 even 9
729.2.e.m.325.2 12 27.20 odd 18
729.2.e.m.406.1 12 27.4 even 9
729.2.e.m.406.2 12 27.23 odd 18
729.2.e.q.163.1 12 27.14 odd 18
729.2.e.q.163.2 12 27.13 even 9
729.2.e.q.568.1 12 27.2 odd 18
729.2.e.q.568.2 12 27.25 even 9
729.2.e.r.82.1 12 27.16 even 9
729.2.e.r.82.2 12 27.11 odd 18
729.2.e.r.649.1 12 27.22 even 9
729.2.e.r.649.2 12 27.5 odd 18