Properties

Label 729.2.a.d.1.3
Level $729$
Weight $2$
Character 729.1
Self dual yes
Analytic conductor $5.821$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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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: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.1397493.1
Defining polynomial: \( x^{6} - 3x^{5} - 3x^{4} + 10x^{3} + 3x^{2} - 6x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 27)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.11662\) of defining polynomial
Character \(\chi\) \(=\) 729.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.415466 q^{2} -1.82739 q^{4} -2.21519 q^{5} -1.31963 q^{7} -1.59015 q^{8} +O(q^{10})\) \(q+0.415466 q^{2} -1.82739 q^{4} -2.21519 q^{5} -1.31963 q^{7} -1.59015 q^{8} -0.920335 q^{10} +5.21519 q^{11} -0.0180585 q^{13} -0.548261 q^{14} +2.99412 q^{16} +3.13280 q^{17} +0.417352 q^{19} +4.04801 q^{20} +2.16673 q^{22} +1.03439 q^{23} -0.0929475 q^{25} -0.00750270 q^{26} +2.41147 q^{28} +7.80722 q^{29} +3.72966 q^{31} +4.42426 q^{32} +1.30157 q^{34} +2.92322 q^{35} +4.42476 q^{37} +0.173396 q^{38} +3.52248 q^{40} +3.67494 q^{41} -8.30787 q^{43} -9.53017 q^{44} +0.429753 q^{46} +7.09791 q^{47} -5.25858 q^{49} -0.0386165 q^{50} +0.0329999 q^{52} +1.30057 q^{53} -11.5526 q^{55} +2.09841 q^{56} +3.24364 q^{58} +3.70181 q^{59} +6.91424 q^{61} +1.54955 q^{62} -4.15011 q^{64} +0.0400030 q^{65} -11.0268 q^{67} -5.72483 q^{68} +1.21450 q^{70} +6.08428 q^{71} -0.546973 q^{73} +1.83834 q^{74} -0.762665 q^{76} -6.88211 q^{77} +0.489144 q^{79} -6.63254 q^{80} +1.52681 q^{82} -4.61367 q^{83} -6.93973 q^{85} -3.45164 q^{86} -8.29293 q^{88} +3.37307 q^{89} +0.0238305 q^{91} -1.89023 q^{92} +2.94894 q^{94} -0.924513 q^{95} +9.94136 q^{97} -2.18476 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{2} + 3 q^{4} + 6 q^{5} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{2} + 3 q^{4} + 6 q^{5} + 6 q^{8} + 3 q^{10} + 12 q^{11} + 6 q^{14} - 3 q^{16} + 9 q^{17} + 3 q^{19} + 6 q^{20} + 6 q^{22} + 15 q^{23} - 6 q^{25} + 15 q^{26} - 6 q^{28} + 12 q^{29} + 12 q^{35} + 3 q^{37} - 3 q^{38} + 6 q^{40} + 15 q^{41} + 3 q^{44} + 3 q^{46} + 21 q^{47} - 12 q^{49} + 3 q^{50} + 12 q^{52} + 9 q^{53} - 6 q^{55} - 6 q^{56} - 12 q^{58} + 24 q^{59} - 9 q^{61} - 12 q^{62} - 12 q^{64} - 6 q^{65} - 9 q^{67} - 9 q^{68} + 15 q^{70} + 27 q^{71} - 6 q^{73} - 12 q^{74} + 6 q^{76} - 12 q^{77} - 21 q^{80} - 6 q^{82} + 12 q^{83} - 21 q^{86} + 12 q^{88} + 9 q^{89} - 6 q^{91} + 6 q^{92} + 6 q^{94} + 12 q^{95} - 45 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.415466 0.293779 0.146889 0.989153i \(-0.453074\pi\)
0.146889 + 0.989153i \(0.453074\pi\)
\(3\) 0 0
\(4\) −1.82739 −0.913694
\(5\) −2.21519 −0.990662 −0.495331 0.868704i \(-0.664953\pi\)
−0.495331 + 0.868704i \(0.664953\pi\)
\(6\) 0 0
\(7\) −1.31963 −0.498773 −0.249386 0.968404i \(-0.580229\pi\)
−0.249386 + 0.968404i \(0.580229\pi\)
\(8\) −1.59015 −0.562203
\(9\) 0 0
\(10\) −0.920335 −0.291036
\(11\) 5.21519 1.57244 0.786219 0.617948i \(-0.212036\pi\)
0.786219 + 0.617948i \(0.212036\pi\)
\(12\) 0 0
\(13\) −0.0180585 −0.00500853 −0.00250427 0.999997i \(-0.500797\pi\)
−0.00250427 + 0.999997i \(0.500797\pi\)
\(14\) −0.548261 −0.146529
\(15\) 0 0
\(16\) 2.99412 0.748530
\(17\) 3.13280 0.759814 0.379907 0.925025i \(-0.375956\pi\)
0.379907 + 0.925025i \(0.375956\pi\)
\(18\) 0 0
\(19\) 0.417352 0.0957472 0.0478736 0.998853i \(-0.484756\pi\)
0.0478736 + 0.998853i \(0.484756\pi\)
\(20\) 4.04801 0.905162
\(21\) 0 0
\(22\) 2.16673 0.461949
\(23\) 1.03439 0.215684 0.107842 0.994168i \(-0.465606\pi\)
0.107842 + 0.994168i \(0.465606\pi\)
\(24\) 0 0
\(25\) −0.0929475 −0.0185895
\(26\) −0.00750270 −0.00147140
\(27\) 0 0
\(28\) 2.41147 0.455726
\(29\) 7.80722 1.44976 0.724882 0.688873i \(-0.241894\pi\)
0.724882 + 0.688873i \(0.241894\pi\)
\(30\) 0 0
\(31\) 3.72966 0.669868 0.334934 0.942242i \(-0.391286\pi\)
0.334934 + 0.942242i \(0.391286\pi\)
\(32\) 4.42426 0.782106
\(33\) 0 0
\(34\) 1.30157 0.223218
\(35\) 2.92322 0.494115
\(36\) 0 0
\(37\) 4.42476 0.727426 0.363713 0.931511i \(-0.381509\pi\)
0.363713 + 0.931511i \(0.381509\pi\)
\(38\) 0.173396 0.0281285
\(39\) 0 0
\(40\) 3.52248 0.556953
\(41\) 3.67494 0.573929 0.286965 0.957941i \(-0.407354\pi\)
0.286965 + 0.957941i \(0.407354\pi\)
\(42\) 0 0
\(43\) −8.30787 −1.26694 −0.633469 0.773768i \(-0.718370\pi\)
−0.633469 + 0.773768i \(0.718370\pi\)
\(44\) −9.53017 −1.43673
\(45\) 0 0
\(46\) 0.429753 0.0633636
\(47\) 7.09791 1.03534 0.517668 0.855581i \(-0.326800\pi\)
0.517668 + 0.855581i \(0.326800\pi\)
\(48\) 0 0
\(49\) −5.25858 −0.751226
\(50\) −0.0386165 −0.00546120
\(51\) 0 0
\(52\) 0.0329999 0.00457627
\(53\) 1.30057 0.178648 0.0893238 0.996003i \(-0.471529\pi\)
0.0893238 + 0.996003i \(0.471529\pi\)
\(54\) 0 0
\(55\) −11.5526 −1.55775
\(56\) 2.09841 0.280412
\(57\) 0 0
\(58\) 3.24364 0.425910
\(59\) 3.70181 0.481935 0.240967 0.970533i \(-0.422535\pi\)
0.240967 + 0.970533i \(0.422535\pi\)
\(60\) 0 0
\(61\) 6.91424 0.885277 0.442639 0.896700i \(-0.354042\pi\)
0.442639 + 0.896700i \(0.354042\pi\)
\(62\) 1.54955 0.196793
\(63\) 0 0
\(64\) −4.15011 −0.518764
\(65\) 0.0400030 0.00496176
\(66\) 0 0
\(67\) −11.0268 −1.34714 −0.673569 0.739125i \(-0.735239\pi\)
−0.673569 + 0.739125i \(0.735239\pi\)
\(68\) −5.72483 −0.694238
\(69\) 0 0
\(70\) 1.21450 0.145161
\(71\) 6.08428 0.722071 0.361035 0.932552i \(-0.382423\pi\)
0.361035 + 0.932552i \(0.382423\pi\)
\(72\) 0 0
\(73\) −0.546973 −0.0640183 −0.0320092 0.999488i \(-0.510191\pi\)
−0.0320092 + 0.999488i \(0.510191\pi\)
\(74\) 1.83834 0.213702
\(75\) 0 0
\(76\) −0.762665 −0.0874836
\(77\) −6.88211 −0.784289
\(78\) 0 0
\(79\) 0.489144 0.0550330 0.0275165 0.999621i \(-0.491240\pi\)
0.0275165 + 0.999621i \(0.491240\pi\)
\(80\) −6.63254 −0.741540
\(81\) 0 0
\(82\) 1.52681 0.168608
\(83\) −4.61367 −0.506416 −0.253208 0.967412i \(-0.581486\pi\)
−0.253208 + 0.967412i \(0.581486\pi\)
\(84\) 0 0
\(85\) −6.93973 −0.752719
\(86\) −3.45164 −0.372200
\(87\) 0 0
\(88\) −8.29293 −0.884029
\(89\) 3.37307 0.357544 0.178772 0.983891i \(-0.442788\pi\)
0.178772 + 0.983891i \(0.442788\pi\)
\(90\) 0 0
\(91\) 0.0238305 0.00249812
\(92\) −1.89023 −0.197070
\(93\) 0 0
\(94\) 2.94894 0.304160
\(95\) −0.924513 −0.0948531
\(96\) 0 0
\(97\) 9.94136 1.00939 0.504696 0.863297i \(-0.331605\pi\)
0.504696 + 0.863297i \(0.331605\pi\)
\(98\) −2.18476 −0.220694
\(99\) 0 0
\(100\) 0.169851 0.0169851
\(101\) −13.7995 −1.37310 −0.686550 0.727082i \(-0.740876\pi\)
−0.686550 + 0.727082i \(0.740876\pi\)
\(102\) 0 0
\(103\) 4.56512 0.449815 0.224907 0.974380i \(-0.427792\pi\)
0.224907 + 0.974380i \(0.427792\pi\)
\(104\) 0.0287158 0.00281581
\(105\) 0 0
\(106\) 0.540345 0.0524829
\(107\) 11.2965 1.09207 0.546035 0.837762i \(-0.316136\pi\)
0.546035 + 0.837762i \(0.316136\pi\)
\(108\) 0 0
\(109\) 14.5032 1.38915 0.694577 0.719419i \(-0.255592\pi\)
0.694577 + 0.719419i \(0.255592\pi\)
\(110\) −4.79972 −0.457635
\(111\) 0 0
\(112\) −3.95113 −0.373347
\(113\) −12.5584 −1.18140 −0.590699 0.806892i \(-0.701148\pi\)
−0.590699 + 0.806892i \(0.701148\pi\)
\(114\) 0 0
\(115\) −2.29136 −0.213670
\(116\) −14.2668 −1.32464
\(117\) 0 0
\(118\) 1.53798 0.141582
\(119\) −4.13413 −0.378975
\(120\) 0 0
\(121\) 16.1982 1.47256
\(122\) 2.87263 0.260076
\(123\) 0 0
\(124\) −6.81554 −0.612054
\(125\) 11.2818 1.00908
\(126\) 0 0
\(127\) −8.39499 −0.744935 −0.372467 0.928045i \(-0.621488\pi\)
−0.372467 + 0.928045i \(0.621488\pi\)
\(128\) −10.5727 −0.934508
\(129\) 0 0
\(130\) 0.0166199 0.00145766
\(131\) 15.5349 1.35729 0.678645 0.734466i \(-0.262567\pi\)
0.678645 + 0.734466i \(0.262567\pi\)
\(132\) 0 0
\(133\) −0.550750 −0.0477561
\(134\) −4.58126 −0.395761
\(135\) 0 0
\(136\) −4.98162 −0.427170
\(137\) −12.0074 −1.02586 −0.512930 0.858430i \(-0.671440\pi\)
−0.512930 + 0.858430i \(0.671440\pi\)
\(138\) 0 0
\(139\) 6.14512 0.521222 0.260611 0.965444i \(-0.416076\pi\)
0.260611 + 0.965444i \(0.416076\pi\)
\(140\) −5.34187 −0.451470
\(141\) 0 0
\(142\) 2.52781 0.212129
\(143\) −0.0941785 −0.00787561
\(144\) 0 0
\(145\) −17.2945 −1.43623
\(146\) −0.227249 −0.0188072
\(147\) 0 0
\(148\) −8.08575 −0.664644
\(149\) −0.882820 −0.0723235 −0.0361617 0.999346i \(-0.511513\pi\)
−0.0361617 + 0.999346i \(0.511513\pi\)
\(150\) 0 0
\(151\) 8.22547 0.669379 0.334690 0.942328i \(-0.391368\pi\)
0.334690 + 0.942328i \(0.391368\pi\)
\(152\) −0.663653 −0.0538294
\(153\) 0 0
\(154\) −2.85929 −0.230408
\(155\) −8.26190 −0.663612
\(156\) 0 0
\(157\) 12.5598 1.00238 0.501192 0.865336i \(-0.332895\pi\)
0.501192 + 0.865336i \(0.332895\pi\)
\(158\) 0.203223 0.0161675
\(159\) 0 0
\(160\) −9.80056 −0.774802
\(161\) −1.36501 −0.107578
\(162\) 0 0
\(163\) 3.31466 0.259624 0.129812 0.991539i \(-0.458563\pi\)
0.129812 + 0.991539i \(0.458563\pi\)
\(164\) −6.71554 −0.524396
\(165\) 0 0
\(166\) −1.91682 −0.148774
\(167\) 20.5630 1.59121 0.795606 0.605815i \(-0.207153\pi\)
0.795606 + 0.605815i \(0.207153\pi\)
\(168\) 0 0
\(169\) −12.9997 −0.999975
\(170\) −2.88322 −0.221133
\(171\) 0 0
\(172\) 15.1817 1.15759
\(173\) −14.0333 −1.06693 −0.533465 0.845822i \(-0.679110\pi\)
−0.533465 + 0.845822i \(0.679110\pi\)
\(174\) 0 0
\(175\) 0.122656 0.00927194
\(176\) 15.6149 1.17702
\(177\) 0 0
\(178\) 1.40139 0.105039
\(179\) −10.1900 −0.761636 −0.380818 0.924650i \(-0.624358\pi\)
−0.380818 + 0.924650i \(0.624358\pi\)
\(180\) 0 0
\(181\) 24.0547 1.78797 0.893987 0.448093i \(-0.147897\pi\)
0.893987 + 0.448093i \(0.147897\pi\)
\(182\) 0.00990079 0.000733895 0
\(183\) 0 0
\(184\) −1.64483 −0.121258
\(185\) −9.80166 −0.720633
\(186\) 0 0
\(187\) 16.3381 1.19476
\(188\) −12.9706 −0.945981
\(189\) 0 0
\(190\) −0.384104 −0.0278658
\(191\) −10.9464 −0.792052 −0.396026 0.918239i \(-0.629611\pi\)
−0.396026 + 0.918239i \(0.629611\pi\)
\(192\) 0 0
\(193\) −10.8060 −0.777830 −0.388915 0.921274i \(-0.627150\pi\)
−0.388915 + 0.921274i \(0.627150\pi\)
\(194\) 4.13030 0.296538
\(195\) 0 0
\(196\) 9.60946 0.686390
\(197\) 22.0734 1.57266 0.786331 0.617806i \(-0.211978\pi\)
0.786331 + 0.617806i \(0.211978\pi\)
\(198\) 0 0
\(199\) 12.8868 0.913518 0.456759 0.889590i \(-0.349010\pi\)
0.456759 + 0.889590i \(0.349010\pi\)
\(200\) 0.147801 0.0104511
\(201\) 0 0
\(202\) −5.73322 −0.403388
\(203\) −10.3026 −0.723103
\(204\) 0 0
\(205\) −8.14068 −0.568570
\(206\) 1.89665 0.132146
\(207\) 0 0
\(208\) −0.0540694 −0.00374904
\(209\) 2.17657 0.150557
\(210\) 0 0
\(211\) −23.9956 −1.65193 −0.825964 0.563723i \(-0.809368\pi\)
−0.825964 + 0.563723i \(0.809368\pi\)
\(212\) −2.37665 −0.163229
\(213\) 0 0
\(214\) 4.69330 0.320827
\(215\) 18.4035 1.25511
\(216\) 0 0
\(217\) −4.92177 −0.334112
\(218\) 6.02558 0.408104
\(219\) 0 0
\(220\) 21.1111 1.42331
\(221\) −0.0565736 −0.00380555
\(222\) 0 0
\(223\) 21.6622 1.45061 0.725303 0.688430i \(-0.241700\pi\)
0.725303 + 0.688430i \(0.241700\pi\)
\(224\) −5.83838 −0.390093
\(225\) 0 0
\(226\) −5.21760 −0.347070
\(227\) −21.6419 −1.43642 −0.718211 0.695826i \(-0.755039\pi\)
−0.718211 + 0.695826i \(0.755039\pi\)
\(228\) 0 0
\(229\) −10.8054 −0.714038 −0.357019 0.934097i \(-0.616207\pi\)
−0.357019 + 0.934097i \(0.616207\pi\)
\(230\) −0.951982 −0.0627719
\(231\) 0 0
\(232\) −12.4147 −0.815062
\(233\) −7.63900 −0.500447 −0.250224 0.968188i \(-0.580504\pi\)
−0.250224 + 0.968188i \(0.580504\pi\)
\(234\) 0 0
\(235\) −15.7232 −1.02567
\(236\) −6.76465 −0.440341
\(237\) 0 0
\(238\) −1.71759 −0.111335
\(239\) 3.23149 0.209028 0.104514 0.994523i \(-0.466671\pi\)
0.104514 + 0.994523i \(0.466671\pi\)
\(240\) 0 0
\(241\) −26.5449 −1.70991 −0.854955 0.518702i \(-0.826415\pi\)
−0.854955 + 0.518702i \(0.826415\pi\)
\(242\) 6.72979 0.432608
\(243\) 0 0
\(244\) −12.6350 −0.808873
\(245\) 11.6487 0.744210
\(246\) 0 0
\(247\) −0.00753676 −0.000479553 0
\(248\) −5.93073 −0.376602
\(249\) 0 0
\(250\) 4.68722 0.296446
\(251\) 4.49930 0.283993 0.141997 0.989867i \(-0.454648\pi\)
0.141997 + 0.989867i \(0.454648\pi\)
\(252\) 0 0
\(253\) 5.39452 0.339150
\(254\) −3.48783 −0.218846
\(255\) 0 0
\(256\) 3.90761 0.244226
\(257\) −13.7354 −0.856792 −0.428396 0.903591i \(-0.640921\pi\)
−0.428396 + 0.903591i \(0.640921\pi\)
\(258\) 0 0
\(259\) −5.83904 −0.362820
\(260\) −0.0731010 −0.00453353
\(261\) 0 0
\(262\) 6.45423 0.398743
\(263\) 24.2026 1.49239 0.746197 0.665725i \(-0.231878\pi\)
0.746197 + 0.665725i \(0.231878\pi\)
\(264\) 0 0
\(265\) −2.88101 −0.176979
\(266\) −0.228818 −0.0140297
\(267\) 0 0
\(268\) 20.1502 1.23087
\(269\) −12.0062 −0.732032 −0.366016 0.930609i \(-0.619278\pi\)
−0.366016 + 0.930609i \(0.619278\pi\)
\(270\) 0 0
\(271\) 3.71777 0.225839 0.112919 0.993604i \(-0.463980\pi\)
0.112919 + 0.993604i \(0.463980\pi\)
\(272\) 9.37997 0.568744
\(273\) 0 0
\(274\) −4.98867 −0.301376
\(275\) −0.484739 −0.0292308
\(276\) 0 0
\(277\) −23.4831 −1.41096 −0.705482 0.708728i \(-0.749269\pi\)
−0.705482 + 0.708728i \(0.749269\pi\)
\(278\) 2.55309 0.153124
\(279\) 0 0
\(280\) −4.64837 −0.277793
\(281\) 20.3717 1.21528 0.607638 0.794214i \(-0.292117\pi\)
0.607638 + 0.794214i \(0.292117\pi\)
\(282\) 0 0
\(283\) 11.5999 0.689545 0.344772 0.938686i \(-0.387956\pi\)
0.344772 + 0.938686i \(0.387956\pi\)
\(284\) −11.1183 −0.659752
\(285\) 0 0
\(286\) −0.0391280 −0.00231369
\(287\) −4.84956 −0.286260
\(288\) 0 0
\(289\) −7.18559 −0.422682
\(290\) −7.18526 −0.421933
\(291\) 0 0
\(292\) 0.999532 0.0584932
\(293\) −31.5742 −1.84458 −0.922291 0.386496i \(-0.873685\pi\)
−0.922291 + 0.386496i \(0.873685\pi\)
\(294\) 0 0
\(295\) −8.20020 −0.477434
\(296\) −7.03603 −0.408961
\(297\) 0 0
\(298\) −0.366782 −0.0212471
\(299\) −0.0186795 −0.00108026
\(300\) 0 0
\(301\) 10.9633 0.631915
\(302\) 3.41741 0.196650
\(303\) 0 0
\(304\) 1.24960 0.0716697
\(305\) −15.3163 −0.877010
\(306\) 0 0
\(307\) −8.12054 −0.463464 −0.231732 0.972780i \(-0.574439\pi\)
−0.231732 + 0.972780i \(0.574439\pi\)
\(308\) 12.5763 0.716601
\(309\) 0 0
\(310\) −3.43254 −0.194955
\(311\) 23.8486 1.35233 0.676164 0.736751i \(-0.263641\pi\)
0.676164 + 0.736751i \(0.263641\pi\)
\(312\) 0 0
\(313\) 26.9105 1.52107 0.760535 0.649297i \(-0.224937\pi\)
0.760535 + 0.649297i \(0.224937\pi\)
\(314\) 5.21818 0.294479
\(315\) 0 0
\(316\) −0.893856 −0.0502833
\(317\) 8.33233 0.467990 0.233995 0.972238i \(-0.424820\pi\)
0.233995 + 0.972238i \(0.424820\pi\)
\(318\) 0 0
\(319\) 40.7161 2.27967
\(320\) 9.19328 0.513920
\(321\) 0 0
\(322\) −0.567114 −0.0316040
\(323\) 1.30748 0.0727501
\(324\) 0 0
\(325\) 0.00167849 9.31061e−5 0
\(326\) 1.37713 0.0762721
\(327\) 0 0
\(328\) −5.84371 −0.322665
\(329\) −9.36661 −0.516398
\(330\) 0 0
\(331\) −6.42026 −0.352889 −0.176445 0.984311i \(-0.556460\pi\)
−0.176445 + 0.984311i \(0.556460\pi\)
\(332\) 8.43096 0.462709
\(333\) 0 0
\(334\) 8.54322 0.467464
\(335\) 24.4264 1.33456
\(336\) 0 0
\(337\) 7.47489 0.407183 0.203592 0.979056i \(-0.434739\pi\)
0.203592 + 0.979056i \(0.434739\pi\)
\(338\) −5.40093 −0.293772
\(339\) 0 0
\(340\) 12.6816 0.687755
\(341\) 19.4509 1.05333
\(342\) 0 0
\(343\) 16.1768 0.873464
\(344\) 13.2108 0.712277
\(345\) 0 0
\(346\) −5.83035 −0.313442
\(347\) 31.4545 1.68857 0.844283 0.535898i \(-0.180027\pi\)
0.844283 + 0.535898i \(0.180027\pi\)
\(348\) 0 0
\(349\) 11.8529 0.634474 0.317237 0.948346i \(-0.397245\pi\)
0.317237 + 0.948346i \(0.397245\pi\)
\(350\) 0.0509595 0.00272390
\(351\) 0 0
\(352\) 23.0733 1.22981
\(353\) 8.20708 0.436819 0.218409 0.975857i \(-0.429913\pi\)
0.218409 + 0.975857i \(0.429913\pi\)
\(354\) 0 0
\(355\) −13.4778 −0.715328
\(356\) −6.16390 −0.326686
\(357\) 0 0
\(358\) −4.23360 −0.223753
\(359\) −17.7273 −0.935611 −0.467806 0.883831i \(-0.654955\pi\)
−0.467806 + 0.883831i \(0.654955\pi\)
\(360\) 0 0
\(361\) −18.8258 −0.990832
\(362\) 9.99393 0.525269
\(363\) 0 0
\(364\) −0.0435477 −0.00228252
\(365\) 1.21165 0.0634205
\(366\) 0 0
\(367\) −20.3195 −1.06067 −0.530335 0.847788i \(-0.677934\pi\)
−0.530335 + 0.847788i \(0.677934\pi\)
\(368\) 3.09708 0.161446
\(369\) 0 0
\(370\) −4.07226 −0.211707
\(371\) −1.71628 −0.0891046
\(372\) 0 0
\(373\) −9.68144 −0.501286 −0.250643 0.968080i \(-0.580642\pi\)
−0.250643 + 0.968080i \(0.580642\pi\)
\(374\) 6.78793 0.350996
\(375\) 0 0
\(376\) −11.2867 −0.582069
\(377\) −0.140987 −0.00726119
\(378\) 0 0
\(379\) −4.12905 −0.212095 −0.106048 0.994361i \(-0.533820\pi\)
−0.106048 + 0.994361i \(0.533820\pi\)
\(380\) 1.68944 0.0866667
\(381\) 0 0
\(382\) −4.54785 −0.232688
\(383\) −4.75018 −0.242723 −0.121362 0.992608i \(-0.538726\pi\)
−0.121362 + 0.992608i \(0.538726\pi\)
\(384\) 0 0
\(385\) 15.2452 0.776966
\(386\) −4.48951 −0.228510
\(387\) 0 0
\(388\) −18.1667 −0.922275
\(389\) 21.8133 1.10598 0.552990 0.833188i \(-0.313487\pi\)
0.552990 + 0.833188i \(0.313487\pi\)
\(390\) 0 0
\(391\) 3.24052 0.163880
\(392\) 8.36193 0.422341
\(393\) 0 0
\(394\) 9.17074 0.462015
\(395\) −1.08355 −0.0545191
\(396\) 0 0
\(397\) −34.8490 −1.74902 −0.874512 0.485005i \(-0.838818\pi\)
−0.874512 + 0.485005i \(0.838818\pi\)
\(398\) 5.35401 0.268373
\(399\) 0 0
\(400\) −0.278296 −0.0139148
\(401\) 18.8261 0.940130 0.470065 0.882632i \(-0.344231\pi\)
0.470065 + 0.882632i \(0.344231\pi\)
\(402\) 0 0
\(403\) −0.0673522 −0.00335505
\(404\) 25.2170 1.25459
\(405\) 0 0
\(406\) −4.28040 −0.212433
\(407\) 23.0759 1.14383
\(408\) 0 0
\(409\) −6.35996 −0.314480 −0.157240 0.987560i \(-0.550260\pi\)
−0.157240 + 0.987560i \(0.550260\pi\)
\(410\) −3.38218 −0.167034
\(411\) 0 0
\(412\) −8.34224 −0.410993
\(413\) −4.88502 −0.240376
\(414\) 0 0
\(415\) 10.2201 0.501687
\(416\) −0.0798955 −0.00391720
\(417\) 0 0
\(418\) 0.904291 0.0442303
\(419\) 24.3180 1.18801 0.594005 0.804461i \(-0.297546\pi\)
0.594005 + 0.804461i \(0.297546\pi\)
\(420\) 0 0
\(421\) −7.99004 −0.389411 −0.194705 0.980862i \(-0.562375\pi\)
−0.194705 + 0.980862i \(0.562375\pi\)
\(422\) −9.96937 −0.485302
\(423\) 0 0
\(424\) −2.06811 −0.100436
\(425\) −0.291185 −0.0141246
\(426\) 0 0
\(427\) −9.12423 −0.441552
\(428\) −20.6430 −0.997818
\(429\) 0 0
\(430\) 7.64603 0.368724
\(431\) −9.87124 −0.475481 −0.237740 0.971329i \(-0.576407\pi\)
−0.237740 + 0.971329i \(0.576407\pi\)
\(432\) 0 0
\(433\) −6.10369 −0.293325 −0.146662 0.989187i \(-0.546853\pi\)
−0.146662 + 0.989187i \(0.546853\pi\)
\(434\) −2.04483 −0.0981550
\(435\) 0 0
\(436\) −26.5029 −1.26926
\(437\) 0.431704 0.0206512
\(438\) 0 0
\(439\) −15.1340 −0.722308 −0.361154 0.932506i \(-0.617617\pi\)
−0.361154 + 0.932506i \(0.617617\pi\)
\(440\) 18.3704 0.875774
\(441\) 0 0
\(442\) −0.0235044 −0.00111799
\(443\) 0.722793 0.0343410 0.0171705 0.999853i \(-0.494534\pi\)
0.0171705 + 0.999853i \(0.494534\pi\)
\(444\) 0 0
\(445\) −7.47197 −0.354205
\(446\) 8.99990 0.426158
\(447\) 0 0
\(448\) 5.47661 0.258746
\(449\) −1.66845 −0.0787389 −0.0393695 0.999225i \(-0.512535\pi\)
−0.0393695 + 0.999225i \(0.512535\pi\)
\(450\) 0 0
\(451\) 19.1655 0.902468
\(452\) 22.9491 1.07944
\(453\) 0 0
\(454\) −8.99147 −0.421991
\(455\) −0.0527891 −0.00247479
\(456\) 0 0
\(457\) 11.0834 0.518462 0.259231 0.965815i \(-0.416531\pi\)
0.259231 + 0.965815i \(0.416531\pi\)
\(458\) −4.48926 −0.209769
\(459\) 0 0
\(460\) 4.18720 0.195229
\(461\) 21.8844 1.01926 0.509629 0.860394i \(-0.329783\pi\)
0.509629 + 0.860394i \(0.329783\pi\)
\(462\) 0 0
\(463\) 24.8517 1.15496 0.577479 0.816406i \(-0.304037\pi\)
0.577479 + 0.816406i \(0.304037\pi\)
\(464\) 23.3758 1.08519
\(465\) 0 0
\(466\) −3.17375 −0.147021
\(467\) 11.8355 0.547683 0.273842 0.961775i \(-0.411706\pi\)
0.273842 + 0.961775i \(0.411706\pi\)
\(468\) 0 0
\(469\) 14.5513 0.671916
\(470\) −6.53246 −0.301320
\(471\) 0 0
\(472\) −5.88644 −0.270945
\(473\) −43.3271 −1.99218
\(474\) 0 0
\(475\) −0.0387919 −0.00177989
\(476\) 7.55465 0.346267
\(477\) 0 0
\(478\) 1.34258 0.0614080
\(479\) −2.88735 −0.131926 −0.0659632 0.997822i \(-0.521012\pi\)
−0.0659632 + 0.997822i \(0.521012\pi\)
\(480\) 0 0
\(481\) −0.0799046 −0.00364333
\(482\) −11.0285 −0.502336
\(483\) 0 0
\(484\) −29.6003 −1.34547
\(485\) −22.0220 −0.999966
\(486\) 0 0
\(487\) 8.75903 0.396910 0.198455 0.980110i \(-0.436408\pi\)
0.198455 + 0.980110i \(0.436408\pi\)
\(488\) −10.9947 −0.497706
\(489\) 0 0
\(490\) 4.83966 0.218633
\(491\) 22.5730 1.01871 0.509354 0.860557i \(-0.329885\pi\)
0.509354 + 0.860557i \(0.329885\pi\)
\(492\) 0 0
\(493\) 24.4584 1.10155
\(494\) −0.00313127 −0.000140883 0
\(495\) 0 0
\(496\) 11.1671 0.501416
\(497\) −8.02899 −0.360149
\(498\) 0 0
\(499\) −25.3328 −1.13405 −0.567026 0.823700i \(-0.691906\pi\)
−0.567026 + 0.823700i \(0.691906\pi\)
\(500\) −20.6163 −0.921988
\(501\) 0 0
\(502\) 1.86931 0.0834312
\(503\) −3.74414 −0.166943 −0.0834714 0.996510i \(-0.526601\pi\)
−0.0834714 + 0.996510i \(0.526601\pi\)
\(504\) 0 0
\(505\) 30.5684 1.36028
\(506\) 2.24124 0.0996353
\(507\) 0 0
\(508\) 15.3409 0.680642
\(509\) −24.3499 −1.07929 −0.539645 0.841893i \(-0.681441\pi\)
−0.539645 + 0.841893i \(0.681441\pi\)
\(510\) 0 0
\(511\) 0.721801 0.0319306
\(512\) 22.7690 1.00626
\(513\) 0 0
\(514\) −5.70660 −0.251707
\(515\) −10.1126 −0.445614
\(516\) 0 0
\(517\) 37.0169 1.62800
\(518\) −2.42592 −0.106589
\(519\) 0 0
\(520\) −0.0636108 −0.00278952
\(521\) −19.6209 −0.859608 −0.429804 0.902922i \(-0.641417\pi\)
−0.429804 + 0.902922i \(0.641417\pi\)
\(522\) 0 0
\(523\) 20.8154 0.910194 0.455097 0.890442i \(-0.349605\pi\)
0.455097 + 0.890442i \(0.349605\pi\)
\(524\) −28.3883 −1.24015
\(525\) 0 0
\(526\) 10.0553 0.438434
\(527\) 11.6843 0.508975
\(528\) 0 0
\(529\) −21.9300 −0.953480
\(530\) −1.19696 −0.0519928
\(531\) 0 0
\(532\) 1.00643 0.0436345
\(533\) −0.0663640 −0.00287454
\(534\) 0 0
\(535\) −25.0238 −1.08187
\(536\) 17.5343 0.757365
\(537\) 0 0
\(538\) −4.98818 −0.215056
\(539\) −27.4245 −1.18126
\(540\) 0 0
\(541\) −30.6272 −1.31676 −0.658382 0.752684i \(-0.728759\pi\)
−0.658382 + 0.752684i \(0.728759\pi\)
\(542\) 1.54461 0.0663467
\(543\) 0 0
\(544\) 13.8603 0.594255
\(545\) −32.1273 −1.37618
\(546\) 0 0
\(547\) 22.6477 0.968345 0.484172 0.874973i \(-0.339121\pi\)
0.484172 + 0.874973i \(0.339121\pi\)
\(548\) 21.9422 0.937323
\(549\) 0 0
\(550\) −0.201393 −0.00858741
\(551\) 3.25836 0.138811
\(552\) 0 0
\(553\) −0.645489 −0.0274490
\(554\) −9.75644 −0.414512
\(555\) 0 0
\(556\) −11.2295 −0.476237
\(557\) −36.4518 −1.54451 −0.772256 0.635311i \(-0.780872\pi\)
−0.772256 + 0.635311i \(0.780872\pi\)
\(558\) 0 0
\(559\) 0.150028 0.00634550
\(560\) 8.75249 0.369860
\(561\) 0 0
\(562\) 8.46377 0.357023
\(563\) −26.5162 −1.11753 −0.558763 0.829327i \(-0.688724\pi\)
−0.558763 + 0.829327i \(0.688724\pi\)
\(564\) 0 0
\(565\) 27.8193 1.17037
\(566\) 4.81938 0.202574
\(567\) 0 0
\(568\) −9.67492 −0.405950
\(569\) −22.9674 −0.962844 −0.481422 0.876489i \(-0.659879\pi\)
−0.481422 + 0.876489i \(0.659879\pi\)
\(570\) 0 0
\(571\) −4.79801 −0.200790 −0.100395 0.994948i \(-0.532011\pi\)
−0.100395 + 0.994948i \(0.532011\pi\)
\(572\) 0.172101 0.00719589
\(573\) 0 0
\(574\) −2.01483 −0.0840973
\(575\) −0.0961436 −0.00400947
\(576\) 0 0
\(577\) −4.31333 −0.179566 −0.0897831 0.995961i \(-0.528617\pi\)
−0.0897831 + 0.995961i \(0.528617\pi\)
\(578\) −2.98537 −0.124175
\(579\) 0 0
\(580\) 31.6037 1.31227
\(581\) 6.08833 0.252587
\(582\) 0 0
\(583\) 6.78274 0.280912
\(584\) 0.869769 0.0359913
\(585\) 0 0
\(586\) −13.1180 −0.541899
\(587\) −41.8222 −1.72619 −0.863094 0.505044i \(-0.831476\pi\)
−0.863094 + 0.505044i \(0.831476\pi\)
\(588\) 0 0
\(589\) 1.55658 0.0641379
\(590\) −3.40691 −0.140260
\(591\) 0 0
\(592\) 13.2483 0.544500
\(593\) 31.5370 1.29507 0.647536 0.762035i \(-0.275800\pi\)
0.647536 + 0.762035i \(0.275800\pi\)
\(594\) 0 0
\(595\) 9.15786 0.375436
\(596\) 1.61326 0.0660815
\(597\) 0 0
\(598\) −0.00776070 −0.000317358 0
\(599\) 12.6303 0.516060 0.258030 0.966137i \(-0.416927\pi\)
0.258030 + 0.966137i \(0.416927\pi\)
\(600\) 0 0
\(601\) 20.5430 0.837965 0.418983 0.907994i \(-0.362387\pi\)
0.418983 + 0.907994i \(0.362387\pi\)
\(602\) 4.55489 0.185643
\(603\) 0 0
\(604\) −15.0311 −0.611608
\(605\) −35.8820 −1.45881
\(606\) 0 0
\(607\) 12.9126 0.524105 0.262052 0.965054i \(-0.415601\pi\)
0.262052 + 0.965054i \(0.415601\pi\)
\(608\) 1.84647 0.0748844
\(609\) 0 0
\(610\) −6.36342 −0.257647
\(611\) −0.128178 −0.00518552
\(612\) 0 0
\(613\) −31.1598 −1.25853 −0.629265 0.777191i \(-0.716644\pi\)
−0.629265 + 0.777191i \(0.716644\pi\)
\(614\) −3.37381 −0.136156
\(615\) 0 0
\(616\) 10.9436 0.440930
\(617\) 7.14078 0.287477 0.143739 0.989616i \(-0.454088\pi\)
0.143739 + 0.989616i \(0.454088\pi\)
\(618\) 0 0
\(619\) −10.0309 −0.403176 −0.201588 0.979470i \(-0.564610\pi\)
−0.201588 + 0.979470i \(0.564610\pi\)
\(620\) 15.0977 0.606338
\(621\) 0 0
\(622\) 9.90827 0.397285
\(623\) −4.45119 −0.178333
\(624\) 0 0
\(625\) −24.5266 −0.981065
\(626\) 11.1804 0.446858
\(627\) 0 0
\(628\) −22.9517 −0.915871
\(629\) 13.8619 0.552708
\(630\) 0 0
\(631\) −7.07560 −0.281675 −0.140838 0.990033i \(-0.544980\pi\)
−0.140838 + 0.990033i \(0.544980\pi\)
\(632\) −0.777813 −0.0309397
\(633\) 0 0
\(634\) 3.46180 0.137486
\(635\) 18.5965 0.737978
\(636\) 0 0
\(637\) 0.0949621 0.00376254
\(638\) 16.9162 0.669718
\(639\) 0 0
\(640\) 23.4206 0.925781
\(641\) 5.01121 0.197931 0.0989655 0.995091i \(-0.468447\pi\)
0.0989655 + 0.995091i \(0.468447\pi\)
\(642\) 0 0
\(643\) 1.63840 0.0646123 0.0323062 0.999478i \(-0.489715\pi\)
0.0323062 + 0.999478i \(0.489715\pi\)
\(644\) 2.49440 0.0982930
\(645\) 0 0
\(646\) 0.543213 0.0213724
\(647\) −34.4927 −1.35605 −0.678024 0.735040i \(-0.737164\pi\)
−0.678024 + 0.735040i \(0.737164\pi\)
\(648\) 0 0
\(649\) 19.3056 0.757813
\(650\) 0.000697358 0 2.73526e−5 0
\(651\) 0 0
\(652\) −6.05717 −0.237217
\(653\) −38.7606 −1.51682 −0.758410 0.651778i \(-0.774023\pi\)
−0.758410 + 0.651778i \(0.774023\pi\)
\(654\) 0 0
\(655\) −34.4127 −1.34462
\(656\) 11.0032 0.429604
\(657\) 0 0
\(658\) −3.89151 −0.151707
\(659\) −9.39192 −0.365857 −0.182929 0.983126i \(-0.558558\pi\)
−0.182929 + 0.983126i \(0.558558\pi\)
\(660\) 0 0
\(661\) −24.1474 −0.939226 −0.469613 0.882872i \(-0.655607\pi\)
−0.469613 + 0.882872i \(0.655607\pi\)
\(662\) −2.66740 −0.103671
\(663\) 0 0
\(664\) 7.33643 0.284709
\(665\) 1.22001 0.0473101
\(666\) 0 0
\(667\) 8.07569 0.312692
\(668\) −37.5765 −1.45388
\(669\) 0 0
\(670\) 10.1483 0.392065
\(671\) 36.0590 1.39204
\(672\) 0 0
\(673\) −26.4661 −1.02019 −0.510097 0.860117i \(-0.670391\pi\)
−0.510097 + 0.860117i \(0.670391\pi\)
\(674\) 3.10557 0.119622
\(675\) 0 0
\(676\) 23.7554 0.913671
\(677\) −31.0668 −1.19400 −0.596998 0.802243i \(-0.703640\pi\)
−0.596998 + 0.802243i \(0.703640\pi\)
\(678\) 0 0
\(679\) −13.1189 −0.503457
\(680\) 11.0352 0.423181
\(681\) 0 0
\(682\) 8.08119 0.309445
\(683\) −38.1361 −1.45924 −0.729619 0.683854i \(-0.760303\pi\)
−0.729619 + 0.683854i \(0.760303\pi\)
\(684\) 0 0
\(685\) 26.5986 1.01628
\(686\) 6.72090 0.256605
\(687\) 0 0
\(688\) −24.8748 −0.948342
\(689\) −0.0234864 −0.000894762 0
\(690\) 0 0
\(691\) 32.9295 1.25270 0.626349 0.779543i \(-0.284548\pi\)
0.626349 + 0.779543i \(0.284548\pi\)
\(692\) 25.6442 0.974847
\(693\) 0 0
\(694\) 13.0683 0.496065
\(695\) −13.6126 −0.516355
\(696\) 0 0
\(697\) 11.5128 0.436080
\(698\) 4.92450 0.186395
\(699\) 0 0
\(700\) −0.224140 −0.00847171
\(701\) 2.30710 0.0871381 0.0435690 0.999050i \(-0.486127\pi\)
0.0435690 + 0.999050i \(0.486127\pi\)
\(702\) 0 0
\(703\) 1.84668 0.0696490
\(704\) −21.6436 −0.815725
\(705\) 0 0
\(706\) 3.40977 0.128328
\(707\) 18.2102 0.684865
\(708\) 0 0
\(709\) −11.1521 −0.418825 −0.209412 0.977827i \(-0.567155\pi\)
−0.209412 + 0.977827i \(0.567155\pi\)
\(710\) −5.59958 −0.210148
\(711\) 0 0
\(712\) −5.36368 −0.201012
\(713\) 3.85791 0.144480
\(714\) 0 0
\(715\) 0.208623 0.00780206
\(716\) 18.6211 0.695902
\(717\) 0 0
\(718\) −7.36510 −0.274863
\(719\) 32.1700 1.19974 0.599869 0.800098i \(-0.295219\pi\)
0.599869 + 0.800098i \(0.295219\pi\)
\(720\) 0 0
\(721\) −6.02426 −0.224355
\(722\) −7.82149 −0.291086
\(723\) 0 0
\(724\) −43.9573 −1.63366
\(725\) −0.725662 −0.0269504
\(726\) 0 0
\(727\) −5.36551 −0.198996 −0.0994979 0.995038i \(-0.531724\pi\)
−0.0994979 + 0.995038i \(0.531724\pi\)
\(728\) −0.0378942 −0.00140445
\(729\) 0 0
\(730\) 0.503399 0.0186316
\(731\) −26.0269 −0.962638
\(732\) 0 0
\(733\) −14.5964 −0.539129 −0.269564 0.962982i \(-0.586880\pi\)
−0.269564 + 0.962982i \(0.586880\pi\)
\(734\) −8.44208 −0.311603
\(735\) 0 0
\(736\) 4.57639 0.168688
\(737\) −57.5068 −2.11829
\(738\) 0 0
\(739\) −43.2165 −1.58975 −0.794873 0.606776i \(-0.792463\pi\)
−0.794873 + 0.606776i \(0.792463\pi\)
\(740\) 17.9114 0.658438
\(741\) 0 0
\(742\) −0.713055 −0.0261771
\(743\) −8.11221 −0.297608 −0.148804 0.988867i \(-0.547542\pi\)
−0.148804 + 0.988867i \(0.547542\pi\)
\(744\) 0 0
\(745\) 1.95561 0.0716481
\(746\) −4.02231 −0.147267
\(747\) 0 0
\(748\) −29.8561 −1.09165
\(749\) −14.9071 −0.544695
\(750\) 0 0
\(751\) 8.75545 0.319491 0.159746 0.987158i \(-0.448933\pi\)
0.159746 + 0.987158i \(0.448933\pi\)
\(752\) 21.2520 0.774981
\(753\) 0 0
\(754\) −0.0585753 −0.00213319
\(755\) −18.2210 −0.663129
\(756\) 0 0
\(757\) −32.1511 −1.16855 −0.584276 0.811555i \(-0.698622\pi\)
−0.584276 + 0.811555i \(0.698622\pi\)
\(758\) −1.71548 −0.0623091
\(759\) 0 0
\(760\) 1.47012 0.0533267
\(761\) 24.5459 0.889789 0.444894 0.895583i \(-0.353241\pi\)
0.444894 + 0.895583i \(0.353241\pi\)
\(762\) 0 0
\(763\) −19.1388 −0.692872
\(764\) 20.0033 0.723693
\(765\) 0 0
\(766\) −1.97354 −0.0713069
\(767\) −0.0668492 −0.00241379
\(768\) 0 0
\(769\) 31.4144 1.13283 0.566416 0.824119i \(-0.308330\pi\)
0.566416 + 0.824119i \(0.308330\pi\)
\(770\) 6.33385 0.228256
\(771\) 0 0
\(772\) 19.7467 0.710699
\(773\) −28.7145 −1.03279 −0.516395 0.856351i \(-0.672726\pi\)
−0.516395 + 0.856351i \(0.672726\pi\)
\(774\) 0 0
\(775\) −0.346663 −0.0124525
\(776\) −15.8083 −0.567483
\(777\) 0 0
\(778\) 9.06270 0.324913
\(779\) 1.53374 0.0549521
\(780\) 0 0
\(781\) 31.7306 1.13541
\(782\) 1.34633 0.0481446
\(783\) 0 0
\(784\) −15.7448 −0.562315
\(785\) −27.8224 −0.993022
\(786\) 0 0
\(787\) −38.8160 −1.38364 −0.691821 0.722069i \(-0.743191\pi\)
−0.691821 + 0.722069i \(0.743191\pi\)
\(788\) −40.3366 −1.43693
\(789\) 0 0
\(790\) −0.450177 −0.0160166
\(791\) 16.5725 0.589249
\(792\) 0 0
\(793\) −0.124861 −0.00443394
\(794\) −14.4786 −0.513826
\(795\) 0 0
\(796\) −23.5491 −0.834676
\(797\) 4.03410 0.142895 0.0714476 0.997444i \(-0.477238\pi\)
0.0714476 + 0.997444i \(0.477238\pi\)
\(798\) 0 0
\(799\) 22.2363 0.786664
\(800\) −0.411224 −0.0145390
\(801\) 0 0
\(802\) 7.82160 0.276190
\(803\) −2.85257 −0.100665
\(804\) 0 0
\(805\) 3.02374 0.106573
\(806\) −0.0279826 −0.000985644 0
\(807\) 0 0
\(808\) 21.9433 0.771961
\(809\) 29.9454 1.05283 0.526413 0.850229i \(-0.323537\pi\)
0.526413 + 0.850229i \(0.323537\pi\)
\(810\) 0 0
\(811\) 20.2173 0.709927 0.354963 0.934880i \(-0.384493\pi\)
0.354963 + 0.934880i \(0.384493\pi\)
\(812\) 18.8269 0.660695
\(813\) 0 0
\(814\) 9.58727 0.336034
\(815\) −7.34259 −0.257200
\(816\) 0 0
\(817\) −3.46731 −0.121306
\(818\) −2.64235 −0.0923875
\(819\) 0 0
\(820\) 14.8762 0.519499
\(821\) −26.8736 −0.937896 −0.468948 0.883226i \(-0.655367\pi\)
−0.468948 + 0.883226i \(0.655367\pi\)
\(822\) 0 0
\(823\) 23.0543 0.803623 0.401812 0.915722i \(-0.368381\pi\)
0.401812 + 0.915722i \(0.368381\pi\)
\(824\) −7.25923 −0.252887
\(825\) 0 0
\(826\) −2.02956 −0.0706174
\(827\) 5.10953 0.177676 0.0888378 0.996046i \(-0.471685\pi\)
0.0888378 + 0.996046i \(0.471685\pi\)
\(828\) 0 0
\(829\) −30.5982 −1.06272 −0.531360 0.847146i \(-0.678319\pi\)
−0.531360 + 0.847146i \(0.678319\pi\)
\(830\) 4.24612 0.147385
\(831\) 0 0
\(832\) 0.0749449 0.00259825
\(833\) −16.4741 −0.570792
\(834\) 0 0
\(835\) −45.5508 −1.57635
\(836\) −3.97744 −0.137563
\(837\) 0 0
\(838\) 10.1033 0.349013
\(839\) 56.3087 1.94399 0.971996 0.234997i \(-0.0755081\pi\)
0.971996 + 0.234997i \(0.0755081\pi\)
\(840\) 0 0
\(841\) 31.9527 1.10182
\(842\) −3.31959 −0.114401
\(843\) 0 0
\(844\) 43.8493 1.50936
\(845\) 28.7967 0.990637
\(846\) 0 0
\(847\) −21.3756 −0.734474
\(848\) 3.89408 0.133723
\(849\) 0 0
\(850\) −0.120978 −0.00414950
\(851\) 4.57691 0.156894
\(852\) 0 0
\(853\) −45.5450 −1.55943 −0.779715 0.626134i \(-0.784636\pi\)
−0.779715 + 0.626134i \(0.784636\pi\)
\(854\) −3.79081 −0.129719
\(855\) 0 0
\(856\) −17.9631 −0.613966
\(857\) 17.4892 0.597419 0.298709 0.954344i \(-0.403444\pi\)
0.298709 + 0.954344i \(0.403444\pi\)
\(858\) 0 0
\(859\) 18.3460 0.625958 0.312979 0.949760i \(-0.398673\pi\)
0.312979 + 0.949760i \(0.398673\pi\)
\(860\) −33.6303 −1.14678
\(861\) 0 0
\(862\) −4.10117 −0.139686
\(863\) −4.65373 −0.158415 −0.0792073 0.996858i \(-0.525239\pi\)
−0.0792073 + 0.996858i \(0.525239\pi\)
\(864\) 0 0
\(865\) 31.0863 1.05697
\(866\) −2.53588 −0.0861726
\(867\) 0 0
\(868\) 8.99399 0.305276
\(869\) 2.55098 0.0865360
\(870\) 0 0
\(871\) 0.199128 0.00674718
\(872\) −23.0622 −0.780986
\(873\) 0 0
\(874\) 0.179358 0.00606688
\(875\) −14.8878 −0.503301
\(876\) 0 0
\(877\) −3.66710 −0.123829 −0.0619145 0.998081i \(-0.519721\pi\)
−0.0619145 + 0.998081i \(0.519721\pi\)
\(878\) −6.28768 −0.212199
\(879\) 0 0
\(880\) −34.5899 −1.16603
\(881\) −38.3008 −1.29039 −0.645193 0.764020i \(-0.723223\pi\)
−0.645193 + 0.764020i \(0.723223\pi\)
\(882\) 0 0
\(883\) 22.6142 0.761027 0.380513 0.924775i \(-0.375747\pi\)
0.380513 + 0.924775i \(0.375747\pi\)
\(884\) 0.103382 0.00347711
\(885\) 0 0
\(886\) 0.300296 0.0100887
\(887\) 1.89656 0.0636802 0.0318401 0.999493i \(-0.489863\pi\)
0.0318401 + 0.999493i \(0.489863\pi\)
\(888\) 0 0
\(889\) 11.0783 0.371553
\(890\) −3.10435 −0.104058
\(891\) 0 0
\(892\) −39.5852 −1.32541
\(893\) 2.96233 0.0991306
\(894\) 0 0
\(895\) 22.5727 0.754523
\(896\) 13.9521 0.466107
\(897\) 0 0
\(898\) −0.693184 −0.0231318
\(899\) 29.1183 0.971150
\(900\) 0 0
\(901\) 4.07443 0.135739
\(902\) 7.96262 0.265126
\(903\) 0 0
\(904\) 19.9698 0.664185
\(905\) −53.2857 −1.77128
\(906\) 0 0
\(907\) 6.53094 0.216856 0.108428 0.994104i \(-0.465418\pi\)
0.108428 + 0.994104i \(0.465418\pi\)
\(908\) 39.5481 1.31245
\(909\) 0 0
\(910\) −0.0219321 −0.000727042 0
\(911\) 43.0371 1.42588 0.712942 0.701223i \(-0.247362\pi\)
0.712942 + 0.701223i \(0.247362\pi\)
\(912\) 0 0
\(913\) −24.0612 −0.796308
\(914\) 4.60480 0.152313
\(915\) 0 0
\(916\) 19.7456 0.652412
\(917\) −20.5003 −0.676980
\(918\) 0 0
\(919\) −16.7911 −0.553887 −0.276943 0.960886i \(-0.589321\pi\)
−0.276943 + 0.960886i \(0.589321\pi\)
\(920\) 3.64361 0.120126
\(921\) 0 0
\(922\) 9.09222 0.299436
\(923\) −0.109873 −0.00361652
\(924\) 0 0
\(925\) −0.411270 −0.0135225
\(926\) 10.3251 0.339302
\(927\) 0 0
\(928\) 34.5412 1.13387
\(929\) 11.6000 0.380584 0.190292 0.981728i \(-0.439057\pi\)
0.190292 + 0.981728i \(0.439057\pi\)
\(930\) 0 0
\(931\) −2.19468 −0.0719277
\(932\) 13.9594 0.457256
\(933\) 0 0
\(934\) 4.91727 0.160898
\(935\) −36.1920 −1.18360
\(936\) 0 0
\(937\) 47.7953 1.56140 0.780702 0.624904i \(-0.214862\pi\)
0.780702 + 0.624904i \(0.214862\pi\)
\(938\) 6.04557 0.197395
\(939\) 0 0
\(940\) 28.7324 0.937147
\(941\) −11.2604 −0.367077 −0.183538 0.983013i \(-0.558755\pi\)
−0.183538 + 0.983013i \(0.558755\pi\)
\(942\) 0 0
\(943\) 3.80131 0.123788
\(944\) 11.0837 0.360743
\(945\) 0 0
\(946\) −18.0010 −0.585261
\(947\) −7.33189 −0.238255 −0.119127 0.992879i \(-0.538010\pi\)
−0.119127 + 0.992879i \(0.538010\pi\)
\(948\) 0 0
\(949\) 0.00987752 0.000320638 0
\(950\) −0.0161167 −0.000522895 0
\(951\) 0 0
\(952\) 6.57388 0.213061
\(953\) 24.8753 0.805791 0.402895 0.915246i \(-0.368004\pi\)
0.402895 + 0.915246i \(0.368004\pi\)
\(954\) 0 0
\(955\) 24.2483 0.784655
\(956\) −5.90519 −0.190987
\(957\) 0 0
\(958\) −1.19960 −0.0387572
\(959\) 15.8453 0.511672
\(960\) 0 0
\(961\) −17.0896 −0.551277
\(962\) −0.0331976 −0.00107034
\(963\) 0 0
\(964\) 48.5079 1.56233
\(965\) 23.9372 0.770567
\(966\) 0 0
\(967\) −34.0300 −1.09433 −0.547165 0.837025i \(-0.684293\pi\)
−0.547165 + 0.837025i \(0.684293\pi\)
\(968\) −25.7575 −0.827878
\(969\) 0 0
\(970\) −9.14938 −0.293769
\(971\) 34.2476 1.09906 0.549530 0.835474i \(-0.314807\pi\)
0.549530 + 0.835474i \(0.314807\pi\)
\(972\) 0 0
\(973\) −8.10928 −0.259971
\(974\) 3.63908 0.116604
\(975\) 0 0
\(976\) 20.7021 0.662657
\(977\) 23.4173 0.749186 0.374593 0.927189i \(-0.377782\pi\)
0.374593 + 0.927189i \(0.377782\pi\)
\(978\) 0 0
\(979\) 17.5912 0.562216
\(980\) −21.2868 −0.679980
\(981\) 0 0
\(982\) 9.37834 0.299275
\(983\) 33.2031 1.05902 0.529508 0.848305i \(-0.322377\pi\)
0.529508 + 0.848305i \(0.322377\pi\)
\(984\) 0 0
\(985\) −48.8966 −1.55798
\(986\) 10.1617 0.323613
\(987\) 0 0
\(988\) 0.0137726 0.000438165 0
\(989\) −8.59355 −0.273259
\(990\) 0 0
\(991\) −28.1806 −0.895187 −0.447594 0.894237i \(-0.647719\pi\)
−0.447594 + 0.894237i \(0.647719\pi\)
\(992\) 16.5010 0.523907
\(993\) 0 0
\(994\) −3.33577 −0.105804
\(995\) −28.5466 −0.904988
\(996\) 0 0
\(997\) −44.9507 −1.42360 −0.711802 0.702381i \(-0.752120\pi\)
−0.711802 + 0.702381i \(0.752120\pi\)
\(998\) −10.5249 −0.333161
\(999\) 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.d.1.3 6
3.2 odd 2 729.2.a.a.1.4 6
9.2 odd 6 729.2.c.e.244.3 12
9.4 even 3 729.2.c.b.487.4 12
9.5 odd 6 729.2.c.e.487.3 12
9.7 even 3 729.2.c.b.244.4 12
27.2 odd 18 243.2.e.c.190.1 12
27.4 even 9 81.2.e.a.46.1 12
27.5 odd 18 243.2.e.d.217.2 12
27.7 even 9 81.2.e.a.37.1 12
27.11 odd 18 243.2.e.d.28.2 12
27.13 even 9 243.2.e.b.55.2 12
27.14 odd 18 243.2.e.c.55.1 12
27.16 even 9 243.2.e.a.28.1 12
27.20 odd 18 27.2.e.a.22.2 yes 12
27.22 even 9 243.2.e.a.217.1 12
27.23 odd 18 27.2.e.a.16.2 12
27.25 even 9 243.2.e.b.190.2 12
108.23 even 18 432.2.u.c.97.1 12
108.47 even 18 432.2.u.c.49.1 12
135.23 even 36 675.2.u.b.124.2 24
135.47 even 36 675.2.u.b.49.2 24
135.74 odd 18 675.2.l.c.76.1 12
135.77 even 36 675.2.u.b.124.3 24
135.104 odd 18 675.2.l.c.151.1 12
135.128 even 36 675.2.u.b.49.3 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.2.e.a.16.2 12 27.23 odd 18
27.2.e.a.22.2 yes 12 27.20 odd 18
81.2.e.a.37.1 12 27.7 even 9
81.2.e.a.46.1 12 27.4 even 9
243.2.e.a.28.1 12 27.16 even 9
243.2.e.a.217.1 12 27.22 even 9
243.2.e.b.55.2 12 27.13 even 9
243.2.e.b.190.2 12 27.25 even 9
243.2.e.c.55.1 12 27.14 odd 18
243.2.e.c.190.1 12 27.2 odd 18
243.2.e.d.28.2 12 27.11 odd 18
243.2.e.d.217.2 12 27.5 odd 18
432.2.u.c.49.1 12 108.47 even 18
432.2.u.c.97.1 12 108.23 even 18
675.2.l.c.76.1 12 135.74 odd 18
675.2.l.c.151.1 12 135.104 odd 18
675.2.u.b.49.2 24 135.47 even 36
675.2.u.b.49.3 24 135.128 even 36
675.2.u.b.124.2 24 135.23 even 36
675.2.u.b.124.3 24 135.77 even 36
729.2.a.a.1.4 6 3.2 odd 2
729.2.a.d.1.3 6 1.1 even 1 trivial
729.2.c.b.244.4 12 9.7 even 3
729.2.c.b.487.4 12 9.4 even 3
729.2.c.e.244.3 12 9.2 odd 6
729.2.c.e.487.3 12 9.5 odd 6