Properties

Label 729.2.a.b.1.6
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: 6.6.7459857.1
Defining polynomial: \( x^{6} - 3x^{5} - 6x^{4} + 13x^{3} + 12x^{2} - 12x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-0.578404\) of defining polynomial
Character \(\chi\) \(=\) 729.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.45779 q^{2} +4.04073 q^{4} +3.08026 q^{5} -2.65867 q^{7} +5.01568 q^{8} +O(q^{10})\) \(q+2.45779 q^{2} +4.04073 q^{4} +3.08026 q^{5} -2.65867 q^{7} +5.01568 q^{8} +7.57064 q^{10} +3.43434 q^{11} -3.34396 q^{13} -6.53444 q^{14} +4.24603 q^{16} -2.57282 q^{17} -2.09676 q^{19} +12.4465 q^{20} +8.44089 q^{22} -0.534444 q^{23} +4.48802 q^{25} -8.21874 q^{26} -10.7430 q^{28} -2.53089 q^{29} +7.71470 q^{31} +0.404491 q^{32} -6.32344 q^{34} -8.18939 q^{35} -10.2957 q^{37} -5.15340 q^{38} +15.4496 q^{40} +4.88501 q^{41} +2.74149 q^{43} +13.8772 q^{44} -1.31355 q^{46} +5.65800 q^{47} +0.0685109 q^{49} +11.0306 q^{50} -13.5120 q^{52} -6.42657 q^{53} +10.5787 q^{55} -13.3350 q^{56} -6.22040 q^{58} -1.65495 q^{59} +14.3722 q^{61} +18.9611 q^{62} -7.49791 q^{64} -10.3003 q^{65} -5.87898 q^{67} -10.3961 q^{68} -20.1278 q^{70} -14.8163 q^{71} +1.88140 q^{73} -25.3046 q^{74} -8.47245 q^{76} -9.13077 q^{77} +17.1935 q^{79} +13.0789 q^{80} +12.0063 q^{82} -3.96878 q^{83} -7.92496 q^{85} +6.73801 q^{86} +17.2256 q^{88} -5.09880 q^{89} +8.89047 q^{91} -2.15954 q^{92} +13.9062 q^{94} -6.45858 q^{95} -10.6319 q^{97} +0.168385 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{2} + 9 q^{4} + 3 q^{5} + 6 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{2} + 9 q^{4} + 3 q^{5} + 6 q^{7} - 6 q^{8} + 6 q^{10} + 6 q^{11} + 6 q^{13} - 24 q^{14} + 15 q^{16} + 9 q^{17} + 12 q^{19} + 21 q^{20} + 3 q^{22} + 12 q^{23} + 9 q^{25} - 24 q^{26} + 3 q^{28} - 21 q^{29} + 15 q^{31} - 30 q^{35} + 3 q^{37} - 15 q^{38} + 3 q^{40} + 12 q^{41} + 6 q^{43} + 33 q^{44} - 3 q^{46} + 15 q^{47} + 12 q^{49} + 24 q^{50} + 3 q^{52} + 9 q^{53} + 15 q^{55} - 12 q^{56} - 15 q^{58} - 6 q^{59} + 24 q^{61} + 30 q^{62} + 6 q^{64} + 15 q^{65} + 15 q^{67} - 36 q^{68} - 15 q^{70} + 12 q^{73} - 24 q^{74} + 9 q^{76} - 15 q^{77} + 24 q^{79} + 21 q^{80} - 21 q^{82} + 6 q^{83} - 18 q^{85} + 30 q^{86} - 21 q^{88} + 9 q^{89} + 18 q^{91} - 6 q^{92} - 6 q^{94} + 33 q^{95} - 21 q^{97} - 18 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.45779 1.73792 0.868960 0.494883i \(-0.164789\pi\)
0.868960 + 0.494883i \(0.164789\pi\)
\(3\) 0 0
\(4\) 4.04073 2.02036
\(5\) 3.08026 1.37754 0.688768 0.724982i \(-0.258152\pi\)
0.688768 + 0.724982i \(0.258152\pi\)
\(6\) 0 0
\(7\) −2.65867 −1.00488 −0.502441 0.864612i \(-0.667565\pi\)
−0.502441 + 0.864612i \(0.667565\pi\)
\(8\) 5.01568 1.77331
\(9\) 0 0
\(10\) 7.57064 2.39405
\(11\) 3.43434 1.03549 0.517746 0.855534i \(-0.326771\pi\)
0.517746 + 0.855534i \(0.326771\pi\)
\(12\) 0 0
\(13\) −3.34396 −0.927447 −0.463723 0.885980i \(-0.653487\pi\)
−0.463723 + 0.885980i \(0.653487\pi\)
\(14\) −6.53444 −1.74640
\(15\) 0 0
\(16\) 4.24603 1.06151
\(17\) −2.57282 −0.624000 −0.312000 0.950082i \(-0.600999\pi\)
−0.312000 + 0.950082i \(0.600999\pi\)
\(18\) 0 0
\(19\) −2.09676 −0.481030 −0.240515 0.970645i \(-0.577316\pi\)
−0.240515 + 0.970645i \(0.577316\pi\)
\(20\) 12.4465 2.78312
\(21\) 0 0
\(22\) 8.44089 1.79960
\(23\) −0.534444 −0.111439 −0.0557196 0.998446i \(-0.517745\pi\)
−0.0557196 + 0.998446i \(0.517745\pi\)
\(24\) 0 0
\(25\) 4.48802 0.897604
\(26\) −8.21874 −1.61183
\(27\) 0 0
\(28\) −10.7430 −2.03023
\(29\) −2.53089 −0.469975 −0.234988 0.971998i \(-0.575505\pi\)
−0.234988 + 0.971998i \(0.575505\pi\)
\(30\) 0 0
\(31\) 7.71470 1.38560 0.692801 0.721129i \(-0.256377\pi\)
0.692801 + 0.721129i \(0.256377\pi\)
\(32\) 0.404491 0.0715045
\(33\) 0 0
\(34\) −6.32344 −1.08446
\(35\) −8.18939 −1.38426
\(36\) 0 0
\(37\) −10.2957 −1.69260 −0.846298 0.532709i \(-0.821174\pi\)
−0.846298 + 0.532709i \(0.821174\pi\)
\(38\) −5.15340 −0.835992
\(39\) 0 0
\(40\) 15.4496 2.44280
\(41\) 4.88501 0.762910 0.381455 0.924387i \(-0.375423\pi\)
0.381455 + 0.924387i \(0.375423\pi\)
\(42\) 0 0
\(43\) 2.74149 0.418073 0.209037 0.977908i \(-0.432967\pi\)
0.209037 + 0.977908i \(0.432967\pi\)
\(44\) 13.8772 2.09207
\(45\) 0 0
\(46\) −1.31355 −0.193673
\(47\) 5.65800 0.825304 0.412652 0.910889i \(-0.364603\pi\)
0.412652 + 0.910889i \(0.364603\pi\)
\(48\) 0 0
\(49\) 0.0685109 0.00978728
\(50\) 11.0306 1.55996
\(51\) 0 0
\(52\) −13.5120 −1.87378
\(53\) −6.42657 −0.882758 −0.441379 0.897321i \(-0.645511\pi\)
−0.441379 + 0.897321i \(0.645511\pi\)
\(54\) 0 0
\(55\) 10.5787 1.42643
\(56\) −13.3350 −1.78197
\(57\) 0 0
\(58\) −6.22040 −0.816779
\(59\) −1.65495 −0.215456 −0.107728 0.994180i \(-0.534358\pi\)
−0.107728 + 0.994180i \(0.534358\pi\)
\(60\) 0 0
\(61\) 14.3722 1.84017 0.920086 0.391716i \(-0.128118\pi\)
0.920086 + 0.391716i \(0.128118\pi\)
\(62\) 18.9611 2.40806
\(63\) 0 0
\(64\) −7.49791 −0.937239
\(65\) −10.3003 −1.27759
\(66\) 0 0
\(67\) −5.87898 −0.718232 −0.359116 0.933293i \(-0.616922\pi\)
−0.359116 + 0.933293i \(0.616922\pi\)
\(68\) −10.3961 −1.26071
\(69\) 0 0
\(70\) −20.1278 −2.40573
\(71\) −14.8163 −1.75837 −0.879184 0.476483i \(-0.841911\pi\)
−0.879184 + 0.476483i \(0.841911\pi\)
\(72\) 0 0
\(73\) 1.88140 0.220201 0.110101 0.993920i \(-0.464883\pi\)
0.110101 + 0.993920i \(0.464883\pi\)
\(74\) −25.3046 −2.94160
\(75\) 0 0
\(76\) −8.47245 −0.971857
\(77\) −9.13077 −1.04055
\(78\) 0 0
\(79\) 17.1935 1.93442 0.967209 0.253981i \(-0.0817400\pi\)
0.967209 + 0.253981i \(0.0817400\pi\)
\(80\) 13.0789 1.46227
\(81\) 0 0
\(82\) 12.0063 1.32588
\(83\) −3.96878 −0.435631 −0.217815 0.975990i \(-0.569893\pi\)
−0.217815 + 0.975990i \(0.569893\pi\)
\(84\) 0 0
\(85\) −7.92496 −0.859582
\(86\) 6.73801 0.726578
\(87\) 0 0
\(88\) 17.2256 1.83625
\(89\) −5.09880 −0.540471 −0.270236 0.962794i \(-0.587102\pi\)
−0.270236 + 0.962794i \(0.587102\pi\)
\(90\) 0 0
\(91\) 8.89047 0.931974
\(92\) −2.15954 −0.225148
\(93\) 0 0
\(94\) 13.9062 1.43431
\(95\) −6.45858 −0.662637
\(96\) 0 0
\(97\) −10.6319 −1.07950 −0.539752 0.841824i \(-0.681482\pi\)
−0.539752 + 0.841824i \(0.681482\pi\)
\(98\) 0.168385 0.0170095
\(99\) 0 0
\(100\) 18.1349 1.81349
\(101\) −5.65809 −0.563001 −0.281500 0.959561i \(-0.590832\pi\)
−0.281500 + 0.959561i \(0.590832\pi\)
\(102\) 0 0
\(103\) 10.3705 1.02184 0.510920 0.859628i \(-0.329305\pi\)
0.510920 + 0.859628i \(0.329305\pi\)
\(104\) −16.7722 −1.64465
\(105\) 0 0
\(106\) −15.7952 −1.53416
\(107\) 14.2457 1.37719 0.688594 0.725147i \(-0.258228\pi\)
0.688594 + 0.725147i \(0.258228\pi\)
\(108\) 0 0
\(109\) 5.76064 0.551769 0.275884 0.961191i \(-0.411029\pi\)
0.275884 + 0.961191i \(0.411029\pi\)
\(110\) 26.0001 2.47902
\(111\) 0 0
\(112\) −11.2888 −1.06669
\(113\) 11.4167 1.07399 0.536996 0.843585i \(-0.319559\pi\)
0.536996 + 0.843585i \(0.319559\pi\)
\(114\) 0 0
\(115\) −1.64623 −0.153512
\(116\) −10.2267 −0.949521
\(117\) 0 0
\(118\) −4.06752 −0.374445
\(119\) 6.84027 0.627046
\(120\) 0 0
\(121\) 0.794696 0.0722451
\(122\) 35.3239 3.19807
\(123\) 0 0
\(124\) 31.1730 2.79942
\(125\) −1.57703 −0.141054
\(126\) 0 0
\(127\) 2.59019 0.229843 0.114921 0.993375i \(-0.463338\pi\)
0.114921 + 0.993375i \(0.463338\pi\)
\(128\) −19.2373 −1.70035
\(129\) 0 0
\(130\) −25.3159 −2.22035
\(131\) 4.41099 0.385390 0.192695 0.981259i \(-0.438277\pi\)
0.192695 + 0.981259i \(0.438277\pi\)
\(132\) 0 0
\(133\) 5.57460 0.483379
\(134\) −14.4493 −1.24823
\(135\) 0 0
\(136\) −12.9044 −1.10655
\(137\) −18.1677 −1.55217 −0.776085 0.630628i \(-0.782797\pi\)
−0.776085 + 0.630628i \(0.782797\pi\)
\(138\) 0 0
\(139\) 1.95794 0.166071 0.0830353 0.996547i \(-0.473539\pi\)
0.0830353 + 0.996547i \(0.473539\pi\)
\(140\) −33.0911 −2.79671
\(141\) 0 0
\(142\) −36.4153 −3.05590
\(143\) −11.4843 −0.960364
\(144\) 0 0
\(145\) −7.79582 −0.647407
\(146\) 4.62408 0.382692
\(147\) 0 0
\(148\) −41.6020 −3.41966
\(149\) −7.31519 −0.599284 −0.299642 0.954052i \(-0.596867\pi\)
−0.299642 + 0.954052i \(0.596867\pi\)
\(150\) 0 0
\(151\) 4.85099 0.394768 0.197384 0.980326i \(-0.436755\pi\)
0.197384 + 0.980326i \(0.436755\pi\)
\(152\) −10.5167 −0.853017
\(153\) 0 0
\(154\) −22.4415 −1.80839
\(155\) 23.7633 1.90871
\(156\) 0 0
\(157\) 1.47670 0.117854 0.0589269 0.998262i \(-0.481232\pi\)
0.0589269 + 0.998262i \(0.481232\pi\)
\(158\) 42.2580 3.36186
\(159\) 0 0
\(160\) 1.24594 0.0985000
\(161\) 1.42091 0.111983
\(162\) 0 0
\(163\) −17.2536 −1.35141 −0.675703 0.737174i \(-0.736160\pi\)
−0.675703 + 0.737174i \(0.736160\pi\)
\(164\) 19.7390 1.54136
\(165\) 0 0
\(166\) −9.75444 −0.757091
\(167\) −6.84036 −0.529323 −0.264662 0.964341i \(-0.585260\pi\)
−0.264662 + 0.964341i \(0.585260\pi\)
\(168\) 0 0
\(169\) −1.81796 −0.139843
\(170\) −19.4779 −1.49388
\(171\) 0 0
\(172\) 11.0776 0.844661
\(173\) 23.9674 1.82221 0.911103 0.412179i \(-0.135232\pi\)
0.911103 + 0.412179i \(0.135232\pi\)
\(174\) 0 0
\(175\) −11.9322 −0.901986
\(176\) 14.5823 1.09918
\(177\) 0 0
\(178\) −12.5318 −0.939296
\(179\) 20.5722 1.53764 0.768820 0.639466i \(-0.220844\pi\)
0.768820 + 0.639466i \(0.220844\pi\)
\(180\) 0 0
\(181\) −15.4701 −1.14989 −0.574943 0.818194i \(-0.694976\pi\)
−0.574943 + 0.818194i \(0.694976\pi\)
\(182\) 21.8509 1.61970
\(183\) 0 0
\(184\) −2.68060 −0.197617
\(185\) −31.7133 −2.33161
\(186\) 0 0
\(187\) −8.83593 −0.646147
\(188\) 22.8624 1.66741
\(189\) 0 0
\(190\) −15.8738 −1.15161
\(191\) 10.9508 0.792376 0.396188 0.918170i \(-0.370333\pi\)
0.396188 + 0.918170i \(0.370333\pi\)
\(192\) 0 0
\(193\) 1.14485 0.0824082 0.0412041 0.999151i \(-0.486881\pi\)
0.0412041 + 0.999151i \(0.486881\pi\)
\(194\) −26.1309 −1.87609
\(195\) 0 0
\(196\) 0.276834 0.0197739
\(197\) 5.04195 0.359224 0.179612 0.983738i \(-0.442516\pi\)
0.179612 + 0.983738i \(0.442516\pi\)
\(198\) 0 0
\(199\) 13.7258 0.972997 0.486499 0.873681i \(-0.338274\pi\)
0.486499 + 0.873681i \(0.338274\pi\)
\(200\) 22.5105 1.59173
\(201\) 0 0
\(202\) −13.9064 −0.978450
\(203\) 6.72880 0.472269
\(204\) 0 0
\(205\) 15.0471 1.05094
\(206\) 25.4886 1.77588
\(207\) 0 0
\(208\) −14.1985 −0.984492
\(209\) −7.20100 −0.498104
\(210\) 0 0
\(211\) −5.83513 −0.401707 −0.200854 0.979621i \(-0.564372\pi\)
−0.200854 + 0.979621i \(0.564372\pi\)
\(212\) −25.9680 −1.78349
\(213\) 0 0
\(214\) 35.0130 2.39344
\(215\) 8.44451 0.575911
\(216\) 0 0
\(217\) −20.5108 −1.39237
\(218\) 14.1584 0.958930
\(219\) 0 0
\(220\) 42.7455 2.88190
\(221\) 8.60339 0.578727
\(222\) 0 0
\(223\) 8.72868 0.584516 0.292258 0.956340i \(-0.405593\pi\)
0.292258 + 0.956340i \(0.405593\pi\)
\(224\) −1.07541 −0.0718536
\(225\) 0 0
\(226\) 28.0598 1.86651
\(227\) −24.2825 −1.61169 −0.805844 0.592128i \(-0.798288\pi\)
−0.805844 + 0.592128i \(0.798288\pi\)
\(228\) 0 0
\(229\) 18.1514 1.19948 0.599740 0.800195i \(-0.295271\pi\)
0.599740 + 0.800195i \(0.295271\pi\)
\(230\) −4.04608 −0.266791
\(231\) 0 0
\(232\) −12.6942 −0.833412
\(233\) −10.5380 −0.690367 −0.345183 0.938535i \(-0.612183\pi\)
−0.345183 + 0.938535i \(0.612183\pi\)
\(234\) 0 0
\(235\) 17.4281 1.13688
\(236\) −6.68720 −0.435300
\(237\) 0 0
\(238\) 16.8119 1.08976
\(239\) 9.53660 0.616871 0.308436 0.951245i \(-0.400195\pi\)
0.308436 + 0.951245i \(0.400195\pi\)
\(240\) 0 0
\(241\) −7.03728 −0.453311 −0.226656 0.973975i \(-0.572779\pi\)
−0.226656 + 0.973975i \(0.572779\pi\)
\(242\) 1.95320 0.125556
\(243\) 0 0
\(244\) 58.0742 3.71782
\(245\) 0.211032 0.0134823
\(246\) 0 0
\(247\) 7.01148 0.446130
\(248\) 38.6945 2.45710
\(249\) 0 0
\(250\) −3.87602 −0.245141
\(251\) 15.5870 0.983843 0.491921 0.870640i \(-0.336295\pi\)
0.491921 + 0.870640i \(0.336295\pi\)
\(252\) 0 0
\(253\) −1.83546 −0.115395
\(254\) 6.36615 0.399448
\(255\) 0 0
\(256\) −32.2853 −2.01783
\(257\) 12.1898 0.760377 0.380189 0.924909i \(-0.375859\pi\)
0.380189 + 0.924909i \(0.375859\pi\)
\(258\) 0 0
\(259\) 27.3727 1.70086
\(260\) −41.6206 −2.58120
\(261\) 0 0
\(262\) 10.8413 0.669776
\(263\) −6.88963 −0.424833 −0.212417 0.977179i \(-0.568133\pi\)
−0.212417 + 0.977179i \(0.568133\pi\)
\(264\) 0 0
\(265\) −19.7955 −1.21603
\(266\) 13.7012 0.840073
\(267\) 0 0
\(268\) −23.7554 −1.45109
\(269\) 7.05875 0.430380 0.215190 0.976572i \(-0.430963\pi\)
0.215190 + 0.976572i \(0.430963\pi\)
\(270\) 0 0
\(271\) 23.7575 1.44316 0.721581 0.692330i \(-0.243416\pi\)
0.721581 + 0.692330i \(0.243416\pi\)
\(272\) −10.9243 −0.662381
\(273\) 0 0
\(274\) −44.6523 −2.69755
\(275\) 15.4134 0.929462
\(276\) 0 0
\(277\) −0.104464 −0.00627662 −0.00313831 0.999995i \(-0.500999\pi\)
−0.00313831 + 0.999995i \(0.500999\pi\)
\(278\) 4.81221 0.288617
\(279\) 0 0
\(280\) −41.0754 −2.45472
\(281\) 8.15220 0.486319 0.243160 0.969986i \(-0.421816\pi\)
0.243160 + 0.969986i \(0.421816\pi\)
\(282\) 0 0
\(283\) 23.6160 1.40382 0.701912 0.712263i \(-0.252330\pi\)
0.701912 + 0.712263i \(0.252330\pi\)
\(284\) −59.8685 −3.55254
\(285\) 0 0
\(286\) −28.2260 −1.66904
\(287\) −12.9876 −0.766634
\(288\) 0 0
\(289\) −10.3806 −0.610624
\(290\) −19.1605 −1.12514
\(291\) 0 0
\(292\) 7.60222 0.444886
\(293\) 21.6258 1.26339 0.631695 0.775217i \(-0.282360\pi\)
0.631695 + 0.775217i \(0.282360\pi\)
\(294\) 0 0
\(295\) −5.09768 −0.296798
\(296\) −51.6398 −3.00150
\(297\) 0 0
\(298\) −17.9792 −1.04151
\(299\) 1.78716 0.103354
\(300\) 0 0
\(301\) −7.28871 −0.420114
\(302\) 11.9227 0.686074
\(303\) 0 0
\(304\) −8.90293 −0.510618
\(305\) 44.2702 2.53490
\(306\) 0 0
\(307\) 16.3599 0.933711 0.466855 0.884334i \(-0.345387\pi\)
0.466855 + 0.884334i \(0.345387\pi\)
\(308\) −36.8950 −2.10229
\(309\) 0 0
\(310\) 58.4052 3.31719
\(311\) 7.75486 0.439738 0.219869 0.975529i \(-0.429437\pi\)
0.219869 + 0.975529i \(0.429437\pi\)
\(312\) 0 0
\(313\) 15.4181 0.871480 0.435740 0.900073i \(-0.356487\pi\)
0.435740 + 0.900073i \(0.356487\pi\)
\(314\) 3.62942 0.204820
\(315\) 0 0
\(316\) 69.4742 3.90823
\(317\) −8.63621 −0.485058 −0.242529 0.970144i \(-0.577977\pi\)
−0.242529 + 0.970144i \(0.577977\pi\)
\(318\) 0 0
\(319\) −8.69195 −0.486656
\(320\) −23.0955 −1.29108
\(321\) 0 0
\(322\) 3.49229 0.194618
\(323\) 5.39459 0.300163
\(324\) 0 0
\(325\) −15.0077 −0.832480
\(326\) −42.4057 −2.34864
\(327\) 0 0
\(328\) 24.5016 1.35288
\(329\) −15.0427 −0.829332
\(330\) 0 0
\(331\) 20.5445 1.12923 0.564614 0.825355i \(-0.309025\pi\)
0.564614 + 0.825355i \(0.309025\pi\)
\(332\) −16.0368 −0.880133
\(333\) 0 0
\(334\) −16.8122 −0.919921
\(335\) −18.1088 −0.989390
\(336\) 0 0
\(337\) 15.9782 0.870388 0.435194 0.900337i \(-0.356680\pi\)
0.435194 + 0.900337i \(0.356680\pi\)
\(338\) −4.46816 −0.243036
\(339\) 0 0
\(340\) −32.0226 −1.73667
\(341\) 26.4949 1.43478
\(342\) 0 0
\(343\) 18.4285 0.995047
\(344\) 13.7504 0.741374
\(345\) 0 0
\(346\) 58.9068 3.16685
\(347\) 9.80427 0.526321 0.263161 0.964752i \(-0.415235\pi\)
0.263161 + 0.964752i \(0.415235\pi\)
\(348\) 0 0
\(349\) −9.43497 −0.505043 −0.252521 0.967591i \(-0.581260\pi\)
−0.252521 + 0.967591i \(0.581260\pi\)
\(350\) −29.3267 −1.56758
\(351\) 0 0
\(352\) 1.38916 0.0740424
\(353\) 3.38516 0.180174 0.0900870 0.995934i \(-0.471285\pi\)
0.0900870 + 0.995934i \(0.471285\pi\)
\(354\) 0 0
\(355\) −45.6380 −2.42221
\(356\) −20.6029 −1.09195
\(357\) 0 0
\(358\) 50.5622 2.67229
\(359\) −35.2273 −1.85923 −0.929614 0.368535i \(-0.879859\pi\)
−0.929614 + 0.368535i \(0.879859\pi\)
\(360\) 0 0
\(361\) −14.6036 −0.768610
\(362\) −38.0223 −1.99841
\(363\) 0 0
\(364\) 35.9240 1.88293
\(365\) 5.79520 0.303335
\(366\) 0 0
\(367\) −31.5454 −1.64666 −0.823329 0.567564i \(-0.807886\pi\)
−0.823329 + 0.567564i \(0.807886\pi\)
\(368\) −2.26927 −0.118294
\(369\) 0 0
\(370\) −77.9447 −4.05215
\(371\) 17.0861 0.887067
\(372\) 0 0
\(373\) 14.4423 0.747795 0.373897 0.927470i \(-0.378021\pi\)
0.373897 + 0.927470i \(0.378021\pi\)
\(374\) −21.7169 −1.12295
\(375\) 0 0
\(376\) 28.3787 1.46352
\(377\) 8.46320 0.435877
\(378\) 0 0
\(379\) 1.00099 0.0514176 0.0257088 0.999669i \(-0.491816\pi\)
0.0257088 + 0.999669i \(0.491816\pi\)
\(380\) −26.0974 −1.33877
\(381\) 0 0
\(382\) 26.9149 1.37709
\(383\) −4.11734 −0.210386 −0.105193 0.994452i \(-0.533546\pi\)
−0.105193 + 0.994452i \(0.533546\pi\)
\(384\) 0 0
\(385\) −28.1252 −1.43339
\(386\) 2.81380 0.143219
\(387\) 0 0
\(388\) −42.9605 −2.18099
\(389\) 15.5199 0.786891 0.393445 0.919348i \(-0.371283\pi\)
0.393445 + 0.919348i \(0.371283\pi\)
\(390\) 0 0
\(391\) 1.37503 0.0695381
\(392\) 0.343629 0.0173559
\(393\) 0 0
\(394\) 12.3920 0.624302
\(395\) 52.9605 2.66473
\(396\) 0 0
\(397\) −1.54893 −0.0777384 −0.0388692 0.999244i \(-0.512376\pi\)
−0.0388692 + 0.999244i \(0.512376\pi\)
\(398\) 33.7352 1.69099
\(399\) 0 0
\(400\) 19.0563 0.952814
\(401\) −7.94738 −0.396873 −0.198437 0.980114i \(-0.563586\pi\)
−0.198437 + 0.980114i \(0.563586\pi\)
\(402\) 0 0
\(403\) −25.7976 −1.28507
\(404\) −22.8628 −1.13747
\(405\) 0 0
\(406\) 16.5380 0.820766
\(407\) −35.3588 −1.75267
\(408\) 0 0
\(409\) 26.4554 1.30814 0.654068 0.756436i \(-0.273061\pi\)
0.654068 + 0.756436i \(0.273061\pi\)
\(410\) 36.9826 1.82644
\(411\) 0 0
\(412\) 41.9045 2.06449
\(413\) 4.39996 0.216508
\(414\) 0 0
\(415\) −12.2249 −0.600097
\(416\) −1.35260 −0.0663166
\(417\) 0 0
\(418\) −17.6985 −0.865664
\(419\) −38.3400 −1.87303 −0.936516 0.350626i \(-0.885969\pi\)
−0.936516 + 0.350626i \(0.885969\pi\)
\(420\) 0 0
\(421\) 7.34450 0.357949 0.178974 0.983854i \(-0.442722\pi\)
0.178974 + 0.983854i \(0.442722\pi\)
\(422\) −14.3415 −0.698134
\(423\) 0 0
\(424\) −32.2337 −1.56540
\(425\) −11.5469 −0.560105
\(426\) 0 0
\(427\) −38.2109 −1.84916
\(428\) 57.5632 2.78242
\(429\) 0 0
\(430\) 20.7548 1.00089
\(431\) 15.8463 0.763289 0.381644 0.924309i \(-0.375358\pi\)
0.381644 + 0.924309i \(0.375358\pi\)
\(432\) 0 0
\(433\) −23.8507 −1.14619 −0.573097 0.819488i \(-0.694258\pi\)
−0.573097 + 0.819488i \(0.694258\pi\)
\(434\) −50.4113 −2.41982
\(435\) 0 0
\(436\) 23.2772 1.11477
\(437\) 1.12060 0.0536057
\(438\) 0 0
\(439\) −21.4699 −1.02470 −0.512352 0.858776i \(-0.671226\pi\)
−0.512352 + 0.858776i \(0.671226\pi\)
\(440\) 53.0593 2.52950
\(441\) 0 0
\(442\) 21.1453 1.00578
\(443\) −31.5479 −1.49889 −0.749443 0.662068i \(-0.769679\pi\)
−0.749443 + 0.662068i \(0.769679\pi\)
\(444\) 0 0
\(445\) −15.7056 −0.744518
\(446\) 21.4533 1.01584
\(447\) 0 0
\(448\) 19.9345 0.941815
\(449\) −20.7898 −0.981130 −0.490565 0.871405i \(-0.663210\pi\)
−0.490565 + 0.871405i \(0.663210\pi\)
\(450\) 0 0
\(451\) 16.7768 0.789988
\(452\) 46.1318 2.16986
\(453\) 0 0
\(454\) −59.6813 −2.80098
\(455\) 27.3850 1.28383
\(456\) 0 0
\(457\) −17.4277 −0.815235 −0.407618 0.913153i \(-0.633640\pi\)
−0.407618 + 0.913153i \(0.633640\pi\)
\(458\) 44.6124 2.08460
\(459\) 0 0
\(460\) −6.65196 −0.310149
\(461\) 31.0045 1.44402 0.722011 0.691882i \(-0.243218\pi\)
0.722011 + 0.691882i \(0.243218\pi\)
\(462\) 0 0
\(463\) 6.48795 0.301521 0.150760 0.988570i \(-0.451828\pi\)
0.150760 + 0.988570i \(0.451828\pi\)
\(464\) −10.7463 −0.498883
\(465\) 0 0
\(466\) −25.9002 −1.19980
\(467\) −1.94390 −0.0899530 −0.0449765 0.998988i \(-0.514321\pi\)
−0.0449765 + 0.998988i \(0.514321\pi\)
\(468\) 0 0
\(469\) 15.6303 0.721738
\(470\) 42.8346 1.97581
\(471\) 0 0
\(472\) −8.30070 −0.382071
\(473\) 9.41521 0.432912
\(474\) 0 0
\(475\) −9.41031 −0.431775
\(476\) 27.6397 1.26686
\(477\) 0 0
\(478\) 23.4390 1.07207
\(479\) −1.32939 −0.0607414 −0.0303707 0.999539i \(-0.509669\pi\)
−0.0303707 + 0.999539i \(0.509669\pi\)
\(480\) 0 0
\(481\) 34.4282 1.56979
\(482\) −17.2962 −0.787818
\(483\) 0 0
\(484\) 3.21115 0.145961
\(485\) −32.7490 −1.48705
\(486\) 0 0
\(487\) 21.2040 0.960844 0.480422 0.877037i \(-0.340484\pi\)
0.480422 + 0.877037i \(0.340484\pi\)
\(488\) 72.0864 3.26320
\(489\) 0 0
\(490\) 0.518671 0.0234312
\(491\) −27.0101 −1.21895 −0.609475 0.792805i \(-0.708620\pi\)
−0.609475 + 0.792805i \(0.708620\pi\)
\(492\) 0 0
\(493\) 6.51153 0.293265
\(494\) 17.2328 0.775338
\(495\) 0 0
\(496\) 32.7569 1.47083
\(497\) 39.3915 1.76695
\(498\) 0 0
\(499\) 15.0389 0.673234 0.336617 0.941642i \(-0.390717\pi\)
0.336617 + 0.941642i \(0.390717\pi\)
\(500\) −6.37237 −0.284981
\(501\) 0 0
\(502\) 38.3096 1.70984
\(503\) −4.60650 −0.205394 −0.102697 0.994713i \(-0.532747\pi\)
−0.102697 + 0.994713i \(0.532747\pi\)
\(504\) 0 0
\(505\) −17.4284 −0.775554
\(506\) −4.51118 −0.200546
\(507\) 0 0
\(508\) 10.4663 0.464366
\(509\) −29.7967 −1.32071 −0.660357 0.750952i \(-0.729595\pi\)
−0.660357 + 0.750952i \(0.729595\pi\)
\(510\) 0 0
\(511\) −5.00201 −0.221276
\(512\) −40.8760 −1.80648
\(513\) 0 0
\(514\) 29.9599 1.32147
\(515\) 31.9440 1.40762
\(516\) 0 0
\(517\) 19.4315 0.854596
\(518\) 67.2764 2.95596
\(519\) 0 0
\(520\) −51.6629 −2.26557
\(521\) 11.7621 0.515306 0.257653 0.966237i \(-0.417051\pi\)
0.257653 + 0.966237i \(0.417051\pi\)
\(522\) 0 0
\(523\) 29.3853 1.28493 0.642464 0.766316i \(-0.277912\pi\)
0.642464 + 0.766316i \(0.277912\pi\)
\(524\) 17.8236 0.778627
\(525\) 0 0
\(526\) −16.9333 −0.738326
\(527\) −19.8485 −0.864615
\(528\) 0 0
\(529\) −22.7144 −0.987581
\(530\) −48.6533 −2.11336
\(531\) 0 0
\(532\) 22.5254 0.976601
\(533\) −16.3352 −0.707558
\(534\) 0 0
\(535\) 43.8806 1.89712
\(536\) −29.4871 −1.27365
\(537\) 0 0
\(538\) 17.3489 0.747965
\(539\) 0.235290 0.0101347
\(540\) 0 0
\(541\) −22.9116 −0.985046 −0.492523 0.870300i \(-0.663925\pi\)
−0.492523 + 0.870300i \(0.663925\pi\)
\(542\) 58.3908 2.50810
\(543\) 0 0
\(544\) −1.04068 −0.0446188
\(545\) 17.7443 0.760081
\(546\) 0 0
\(547\) −13.7577 −0.588237 −0.294118 0.955769i \(-0.595026\pi\)
−0.294118 + 0.955769i \(0.595026\pi\)
\(548\) −73.4107 −3.13595
\(549\) 0 0
\(550\) 37.8829 1.61533
\(551\) 5.30668 0.226072
\(552\) 0 0
\(553\) −45.7118 −1.94386
\(554\) −0.256750 −0.0109083
\(555\) 0 0
\(556\) 7.91152 0.335523
\(557\) −21.9041 −0.928105 −0.464053 0.885808i \(-0.653605\pi\)
−0.464053 + 0.885808i \(0.653605\pi\)
\(558\) 0 0
\(559\) −9.16742 −0.387741
\(560\) −34.7724 −1.46940
\(561\) 0 0
\(562\) 20.0364 0.845184
\(563\) −14.5684 −0.613986 −0.306993 0.951712i \(-0.599323\pi\)
−0.306993 + 0.951712i \(0.599323\pi\)
\(564\) 0 0
\(565\) 35.1664 1.47946
\(566\) 58.0431 2.43973
\(567\) 0 0
\(568\) −74.3137 −3.11813
\(569\) 22.3570 0.937255 0.468628 0.883396i \(-0.344749\pi\)
0.468628 + 0.883396i \(0.344749\pi\)
\(570\) 0 0
\(571\) −15.4261 −0.645564 −0.322782 0.946473i \(-0.604618\pi\)
−0.322782 + 0.946473i \(0.604618\pi\)
\(572\) −46.4049 −1.94029
\(573\) 0 0
\(574\) −31.9208 −1.33235
\(575\) −2.39860 −0.100028
\(576\) 0 0
\(577\) 32.8081 1.36582 0.682909 0.730503i \(-0.260714\pi\)
0.682909 + 0.730503i \(0.260714\pi\)
\(578\) −25.5133 −1.06122
\(579\) 0 0
\(580\) −31.5008 −1.30800
\(581\) 10.5517 0.437757
\(582\) 0 0
\(583\) −22.0710 −0.914089
\(584\) 9.43650 0.390485
\(585\) 0 0
\(586\) 53.1515 2.19567
\(587\) −30.2343 −1.24790 −0.623951 0.781464i \(-0.714473\pi\)
−0.623951 + 0.781464i \(0.714473\pi\)
\(588\) 0 0
\(589\) −16.1759 −0.666516
\(590\) −12.5290 −0.515812
\(591\) 0 0
\(592\) −43.7157 −1.79671
\(593\) −41.1023 −1.68787 −0.843935 0.536446i \(-0.819766\pi\)
−0.843935 + 0.536446i \(0.819766\pi\)
\(594\) 0 0
\(595\) 21.0698 0.863778
\(596\) −29.5587 −1.21077
\(597\) 0 0
\(598\) 4.39246 0.179621
\(599\) −36.5505 −1.49341 −0.746707 0.665153i \(-0.768366\pi\)
−0.746707 + 0.665153i \(0.768366\pi\)
\(600\) 0 0
\(601\) 3.94580 0.160953 0.0804764 0.996757i \(-0.474356\pi\)
0.0804764 + 0.996757i \(0.474356\pi\)
\(602\) −17.9141 −0.730125
\(603\) 0 0
\(604\) 19.6015 0.797575
\(605\) 2.44787 0.0995202
\(606\) 0 0
\(607\) 10.0115 0.406355 0.203178 0.979142i \(-0.434873\pi\)
0.203178 + 0.979142i \(0.434873\pi\)
\(608\) −0.848121 −0.0343958
\(609\) 0 0
\(610\) 108.807 4.40546
\(611\) −18.9201 −0.765425
\(612\) 0 0
\(613\) −18.7568 −0.757578 −0.378789 0.925483i \(-0.623659\pi\)
−0.378789 + 0.925483i \(0.623659\pi\)
\(614\) 40.2093 1.62271
\(615\) 0 0
\(616\) −45.7970 −1.84522
\(617\) 23.0101 0.926350 0.463175 0.886267i \(-0.346710\pi\)
0.463175 + 0.886267i \(0.346710\pi\)
\(618\) 0 0
\(619\) 7.41402 0.297995 0.148997 0.988838i \(-0.452395\pi\)
0.148997 + 0.988838i \(0.452395\pi\)
\(620\) 96.0211 3.85630
\(621\) 0 0
\(622\) 19.0598 0.764229
\(623\) 13.5560 0.543110
\(624\) 0 0
\(625\) −27.2978 −1.09191
\(626\) 37.8943 1.51456
\(627\) 0 0
\(628\) 5.96696 0.238107
\(629\) 26.4889 1.05618
\(630\) 0 0
\(631\) −30.9924 −1.23379 −0.616894 0.787046i \(-0.711609\pi\)
−0.616894 + 0.787046i \(0.711609\pi\)
\(632\) 86.2371 3.43033
\(633\) 0 0
\(634\) −21.2260 −0.842991
\(635\) 7.97848 0.316616
\(636\) 0 0
\(637\) −0.229098 −0.00907717
\(638\) −21.3630 −0.845769
\(639\) 0 0
\(640\) −59.2559 −2.34229
\(641\) 35.3461 1.39609 0.698044 0.716055i \(-0.254054\pi\)
0.698044 + 0.716055i \(0.254054\pi\)
\(642\) 0 0
\(643\) −19.7009 −0.776929 −0.388465 0.921464i \(-0.626994\pi\)
−0.388465 + 0.921464i \(0.626994\pi\)
\(644\) 5.74151 0.226247
\(645\) 0 0
\(646\) 13.2588 0.521659
\(647\) −46.8317 −1.84114 −0.920572 0.390572i \(-0.872277\pi\)
−0.920572 + 0.390572i \(0.872277\pi\)
\(648\) 0 0
\(649\) −5.68366 −0.223103
\(650\) −36.8859 −1.44678
\(651\) 0 0
\(652\) −69.7171 −2.73033
\(653\) −17.3267 −0.678047 −0.339023 0.940778i \(-0.610097\pi\)
−0.339023 + 0.940778i \(0.610097\pi\)
\(654\) 0 0
\(655\) 13.5870 0.530888
\(656\) 20.7419 0.809835
\(657\) 0 0
\(658\) −36.9719 −1.44131
\(659\) −43.7342 −1.70364 −0.851821 0.523833i \(-0.824502\pi\)
−0.851821 + 0.523833i \(0.824502\pi\)
\(660\) 0 0
\(661\) −9.08686 −0.353438 −0.176719 0.984261i \(-0.556548\pi\)
−0.176719 + 0.984261i \(0.556548\pi\)
\(662\) 50.4941 1.96251
\(663\) 0 0
\(664\) −19.9062 −0.772509
\(665\) 17.1712 0.665871
\(666\) 0 0
\(667\) 1.35262 0.0523737
\(668\) −27.6401 −1.06943
\(669\) 0 0
\(670\) −44.5076 −1.71948
\(671\) 49.3591 1.90549
\(672\) 0 0
\(673\) 7.49668 0.288976 0.144488 0.989507i \(-0.453846\pi\)
0.144488 + 0.989507i \(0.453846\pi\)
\(674\) 39.2711 1.51266
\(675\) 0 0
\(676\) −7.34587 −0.282534
\(677\) −15.3626 −0.590431 −0.295215 0.955431i \(-0.595391\pi\)
−0.295215 + 0.955431i \(0.595391\pi\)
\(678\) 0 0
\(679\) 28.2666 1.08477
\(680\) −39.7491 −1.52431
\(681\) 0 0
\(682\) 65.1189 2.49353
\(683\) −6.62157 −0.253367 −0.126684 0.991943i \(-0.540433\pi\)
−0.126684 + 0.991943i \(0.540433\pi\)
\(684\) 0 0
\(685\) −55.9612 −2.13817
\(686\) 45.2934 1.72931
\(687\) 0 0
\(688\) 11.6405 0.443788
\(689\) 21.4902 0.818711
\(690\) 0 0
\(691\) −18.6569 −0.709740 −0.354870 0.934916i \(-0.615475\pi\)
−0.354870 + 0.934916i \(0.615475\pi\)
\(692\) 96.8457 3.68152
\(693\) 0 0
\(694\) 24.0968 0.914704
\(695\) 6.03098 0.228768
\(696\) 0 0
\(697\) −12.5682 −0.476056
\(698\) −23.1892 −0.877723
\(699\) 0 0
\(700\) −48.2146 −1.82234
\(701\) −24.8903 −0.940092 −0.470046 0.882642i \(-0.655763\pi\)
−0.470046 + 0.882642i \(0.655763\pi\)
\(702\) 0 0
\(703\) 21.5876 0.814190
\(704\) −25.7504 −0.970504
\(705\) 0 0
\(706\) 8.32002 0.313128
\(707\) 15.0430 0.565749
\(708\) 0 0
\(709\) −26.1787 −0.983163 −0.491582 0.870831i \(-0.663581\pi\)
−0.491582 + 0.870831i \(0.663581\pi\)
\(710\) −112.169 −4.20961
\(711\) 0 0
\(712\) −25.5739 −0.958424
\(713\) −4.12308 −0.154410
\(714\) 0 0
\(715\) −35.3746 −1.32294
\(716\) 83.1267 3.10659
\(717\) 0 0
\(718\) −86.5814 −3.23119
\(719\) −21.0290 −0.784251 −0.392125 0.919912i \(-0.628260\pi\)
−0.392125 + 0.919912i \(0.628260\pi\)
\(720\) 0 0
\(721\) −27.5718 −1.02683
\(722\) −35.8925 −1.33578
\(723\) 0 0
\(724\) −62.5106 −2.32319
\(725\) −11.3587 −0.421852
\(726\) 0 0
\(727\) 41.3957 1.53528 0.767640 0.640881i \(-0.221431\pi\)
0.767640 + 0.640881i \(0.221431\pi\)
\(728\) 44.5918 1.65268
\(729\) 0 0
\(730\) 14.2434 0.527171
\(731\) −7.05336 −0.260878
\(732\) 0 0
\(733\) 33.6267 1.24203 0.621016 0.783798i \(-0.286720\pi\)
0.621016 + 0.783798i \(0.286720\pi\)
\(734\) −77.5320 −2.86176
\(735\) 0 0
\(736\) −0.216178 −0.00796841
\(737\) −20.1904 −0.743724
\(738\) 0 0
\(739\) −12.9454 −0.476202 −0.238101 0.971240i \(-0.576525\pi\)
−0.238101 + 0.971240i \(0.576525\pi\)
\(740\) −128.145 −4.71071
\(741\) 0 0
\(742\) 41.9941 1.54165
\(743\) 0.0879292 0.00322581 0.00161290 0.999999i \(-0.499487\pi\)
0.00161290 + 0.999999i \(0.499487\pi\)
\(744\) 0 0
\(745\) −22.5327 −0.825534
\(746\) 35.4962 1.29961
\(747\) 0 0
\(748\) −35.7036 −1.30545
\(749\) −37.8747 −1.38391
\(750\) 0 0
\(751\) 49.8960 1.82073 0.910366 0.413804i \(-0.135800\pi\)
0.910366 + 0.413804i \(0.135800\pi\)
\(752\) 24.0240 0.876067
\(753\) 0 0
\(754\) 20.8008 0.757519
\(755\) 14.9423 0.543806
\(756\) 0 0
\(757\) −11.8679 −0.431348 −0.215674 0.976465i \(-0.569195\pi\)
−0.215674 + 0.976465i \(0.569195\pi\)
\(758\) 2.46023 0.0893596
\(759\) 0 0
\(760\) −32.3942 −1.17506
\(761\) −3.79030 −0.137398 −0.0686992 0.997637i \(-0.521885\pi\)
−0.0686992 + 0.997637i \(0.521885\pi\)
\(762\) 0 0
\(763\) −15.3156 −0.554462
\(764\) 44.2494 1.60089
\(765\) 0 0
\(766\) −10.1196 −0.365634
\(767\) 5.53408 0.199824
\(768\) 0 0
\(769\) 6.26950 0.226084 0.113042 0.993590i \(-0.463941\pi\)
0.113042 + 0.993590i \(0.463941\pi\)
\(770\) −69.1257 −2.49112
\(771\) 0 0
\(772\) 4.62603 0.166495
\(773\) 1.29536 0.0465907 0.0232954 0.999729i \(-0.492584\pi\)
0.0232954 + 0.999729i \(0.492584\pi\)
\(774\) 0 0
\(775\) 34.6237 1.24372
\(776\) −53.3261 −1.91430
\(777\) 0 0
\(778\) 38.1447 1.36755
\(779\) −10.2427 −0.366983
\(780\) 0 0
\(781\) −50.8841 −1.82078
\(782\) 3.37953 0.120852
\(783\) 0 0
\(784\) 0.290900 0.0103893
\(785\) 4.54863 0.162348
\(786\) 0 0
\(787\) −12.3922 −0.441735 −0.220867 0.975304i \(-0.570889\pi\)
−0.220867 + 0.975304i \(0.570889\pi\)
\(788\) 20.3731 0.725763
\(789\) 0 0
\(790\) 130.166 4.63109
\(791\) −30.3532 −1.07924
\(792\) 0 0
\(793\) −48.0600 −1.70666
\(794\) −3.80693 −0.135103
\(795\) 0 0
\(796\) 55.4623 1.96581
\(797\) −24.3827 −0.863680 −0.431840 0.901950i \(-0.642135\pi\)
−0.431840 + 0.901950i \(0.642135\pi\)
\(798\) 0 0
\(799\) −14.5570 −0.514989
\(800\) 1.81536 0.0641827
\(801\) 0 0
\(802\) −19.5330 −0.689734
\(803\) 6.46136 0.228017
\(804\) 0 0
\(805\) 4.37677 0.154261
\(806\) −63.4051 −2.23335
\(807\) 0 0
\(808\) −28.3792 −0.998376
\(809\) −24.1156 −0.847861 −0.423930 0.905695i \(-0.639350\pi\)
−0.423930 + 0.905695i \(0.639350\pi\)
\(810\) 0 0
\(811\) 48.1121 1.68944 0.844721 0.535206i \(-0.179766\pi\)
0.844721 + 0.535206i \(0.179766\pi\)
\(812\) 27.1893 0.954156
\(813\) 0 0
\(814\) −86.9045 −3.04600
\(815\) −53.1456 −1.86161
\(816\) 0 0
\(817\) −5.74826 −0.201106
\(818\) 65.0218 2.27343
\(819\) 0 0
\(820\) 60.8013 2.12327
\(821\) 16.9388 0.591169 0.295585 0.955317i \(-0.404486\pi\)
0.295585 + 0.955317i \(0.404486\pi\)
\(822\) 0 0
\(823\) 11.5559 0.402815 0.201407 0.979508i \(-0.435449\pi\)
0.201407 + 0.979508i \(0.435449\pi\)
\(824\) 52.0153 1.81204
\(825\) 0 0
\(826\) 10.8142 0.376273
\(827\) 10.2374 0.355988 0.177994 0.984032i \(-0.443039\pi\)
0.177994 + 0.984032i \(0.443039\pi\)
\(828\) 0 0
\(829\) 9.35244 0.324824 0.162412 0.986723i \(-0.448073\pi\)
0.162412 + 0.986723i \(0.448073\pi\)
\(830\) −30.0462 −1.04292
\(831\) 0 0
\(832\) 25.0727 0.869239
\(833\) −0.176266 −0.00610726
\(834\) 0 0
\(835\) −21.0701 −0.729161
\(836\) −29.0973 −1.00635
\(837\) 0 0
\(838\) −94.2316 −3.25518
\(839\) −11.8451 −0.408939 −0.204469 0.978873i \(-0.565547\pi\)
−0.204469 + 0.978873i \(0.565547\pi\)
\(840\) 0 0
\(841\) −22.5946 −0.779123
\(842\) 18.0512 0.622086
\(843\) 0 0
\(844\) −23.5782 −0.811595
\(845\) −5.59979 −0.192639
\(846\) 0 0
\(847\) −2.11283 −0.0725978
\(848\) −27.2874 −0.937055
\(849\) 0 0
\(850\) −28.3797 −0.973417
\(851\) 5.50246 0.188622
\(852\) 0 0
\(853\) −37.2713 −1.27614 −0.638072 0.769976i \(-0.720268\pi\)
−0.638072 + 0.769976i \(0.720268\pi\)
\(854\) −93.9144 −3.21368
\(855\) 0 0
\(856\) 71.4521 2.44218
\(857\) −41.4318 −1.41528 −0.707642 0.706571i \(-0.750241\pi\)
−0.707642 + 0.706571i \(0.750241\pi\)
\(858\) 0 0
\(859\) −9.25851 −0.315896 −0.157948 0.987447i \(-0.550488\pi\)
−0.157948 + 0.987447i \(0.550488\pi\)
\(860\) 34.1220 1.16355
\(861\) 0 0
\(862\) 38.9468 1.32653
\(863\) 51.4748 1.75222 0.876110 0.482110i \(-0.160130\pi\)
0.876110 + 0.482110i \(0.160130\pi\)
\(864\) 0 0
\(865\) 73.8258 2.51015
\(866\) −58.6201 −1.99199
\(867\) 0 0
\(868\) −82.8787 −2.81309
\(869\) 59.0483 2.00308
\(870\) 0 0
\(871\) 19.6591 0.666122
\(872\) 28.8935 0.978458
\(873\) 0 0
\(874\) 2.75421 0.0931624
\(875\) 4.19281 0.141743
\(876\) 0 0
\(877\) −24.8990 −0.840781 −0.420390 0.907343i \(-0.638107\pi\)
−0.420390 + 0.907343i \(0.638107\pi\)
\(878\) −52.7686 −1.78085
\(879\) 0 0
\(880\) 44.9174 1.51416
\(881\) −1.43361 −0.0482997 −0.0241498 0.999708i \(-0.507688\pi\)
−0.0241498 + 0.999708i \(0.507688\pi\)
\(882\) 0 0
\(883\) 26.2046 0.881855 0.440928 0.897543i \(-0.354650\pi\)
0.440928 + 0.897543i \(0.354650\pi\)
\(884\) 34.7640 1.16924
\(885\) 0 0
\(886\) −77.5381 −2.60494
\(887\) 43.3726 1.45631 0.728154 0.685414i \(-0.240379\pi\)
0.728154 + 0.685414i \(0.240379\pi\)
\(888\) 0 0
\(889\) −6.88646 −0.230965
\(890\) −38.6011 −1.29391
\(891\) 0 0
\(892\) 35.2702 1.18093
\(893\) −11.8635 −0.396996
\(894\) 0 0
\(895\) 63.3678 2.11815
\(896\) 51.1455 1.70865
\(897\) 0 0
\(898\) −51.0969 −1.70513
\(899\) −19.5251 −0.651198
\(900\) 0 0
\(901\) 16.5344 0.550841
\(902\) 41.2338 1.37293
\(903\) 0 0
\(904\) 57.2625 1.90452
\(905\) −47.6521 −1.58401
\(906\) 0 0
\(907\) 23.8718 0.792649 0.396325 0.918110i \(-0.370286\pi\)
0.396325 + 0.918110i \(0.370286\pi\)
\(908\) −98.1191 −3.25620
\(909\) 0 0
\(910\) 67.3065 2.23119
\(911\) 4.32052 0.143145 0.0715726 0.997435i \(-0.477198\pi\)
0.0715726 + 0.997435i \(0.477198\pi\)
\(912\) 0 0
\(913\) −13.6302 −0.451092
\(914\) −42.8337 −1.41681
\(915\) 0 0
\(916\) 73.3450 2.42339
\(917\) −11.7273 −0.387271
\(918\) 0 0
\(919\) 3.56151 0.117483 0.0587417 0.998273i \(-0.481291\pi\)
0.0587417 + 0.998273i \(0.481291\pi\)
\(920\) −8.25696 −0.272224
\(921\) 0 0
\(922\) 76.2025 2.50959
\(923\) 49.5449 1.63079
\(924\) 0 0
\(925\) −46.2071 −1.51928
\(926\) 15.9460 0.524019
\(927\) 0 0
\(928\) −1.02372 −0.0336053
\(929\) −33.3258 −1.09338 −0.546692 0.837334i \(-0.684113\pi\)
−0.546692 + 0.837334i \(0.684113\pi\)
\(930\) 0 0
\(931\) −0.143651 −0.00470798
\(932\) −42.5812 −1.39479
\(933\) 0 0
\(934\) −4.77770 −0.156331
\(935\) −27.2170 −0.890091
\(936\) 0 0
\(937\) 26.0594 0.851322 0.425661 0.904883i \(-0.360042\pi\)
0.425661 + 0.904883i \(0.360042\pi\)
\(938\) 38.4159 1.25432
\(939\) 0 0
\(940\) 70.4223 2.29692
\(941\) −23.9936 −0.782167 −0.391084 0.920355i \(-0.627900\pi\)
−0.391084 + 0.920355i \(0.627900\pi\)
\(942\) 0 0
\(943\) −2.61076 −0.0850181
\(944\) −7.02697 −0.228708
\(945\) 0 0
\(946\) 23.1406 0.752366
\(947\) 29.8847 0.971121 0.485561 0.874203i \(-0.338615\pi\)
0.485561 + 0.874203i \(0.338615\pi\)
\(948\) 0 0
\(949\) −6.29131 −0.204225
\(950\) −23.1286 −0.750390
\(951\) 0 0
\(952\) 34.3086 1.11195
\(953\) −10.1934 −0.330196 −0.165098 0.986277i \(-0.552794\pi\)
−0.165098 + 0.986277i \(0.552794\pi\)
\(954\) 0 0
\(955\) 33.7315 1.09153
\(956\) 38.5348 1.24631
\(957\) 0 0
\(958\) −3.26736 −0.105564
\(959\) 48.3018 1.55975
\(960\) 0 0
\(961\) 28.5166 0.919891
\(962\) 84.6174 2.72817
\(963\) 0 0
\(964\) −28.4357 −0.915854
\(965\) 3.52644 0.113520
\(966\) 0 0
\(967\) −18.7666 −0.603493 −0.301747 0.953388i \(-0.597570\pi\)
−0.301747 + 0.953388i \(0.597570\pi\)
\(968\) 3.98594 0.128113
\(969\) 0 0
\(970\) −80.4901 −2.58438
\(971\) −51.9535 −1.66727 −0.833633 0.552319i \(-0.813743\pi\)
−0.833633 + 0.552319i \(0.813743\pi\)
\(972\) 0 0
\(973\) −5.20552 −0.166881
\(974\) 52.1149 1.66987
\(975\) 0 0
\(976\) 61.0249 1.95336
\(977\) 32.5841 1.04246 0.521229 0.853417i \(-0.325474\pi\)
0.521229 + 0.853417i \(0.325474\pi\)
\(978\) 0 0
\(979\) −17.5110 −0.559654
\(980\) 0.852722 0.0272392
\(981\) 0 0
\(982\) −66.3852 −2.11844
\(983\) 35.0795 1.11886 0.559431 0.828877i \(-0.311020\pi\)
0.559431 + 0.828877i \(0.311020\pi\)
\(984\) 0 0
\(985\) 15.5305 0.494844
\(986\) 16.0040 0.509670
\(987\) 0 0
\(988\) 28.3315 0.901345
\(989\) −1.46517 −0.0465898
\(990\) 0 0
\(991\) −54.7635 −1.73962 −0.869810 0.493386i \(-0.835759\pi\)
−0.869810 + 0.493386i \(0.835759\pi\)
\(992\) 3.12052 0.0990767
\(993\) 0 0
\(994\) 96.8161 3.07082
\(995\) 42.2791 1.34034
\(996\) 0 0
\(997\) 50.3301 1.59397 0.796984 0.604000i \(-0.206427\pi\)
0.796984 + 0.604000i \(0.206427\pi\)
\(998\) 36.9625 1.17003
\(999\) 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.b.1.6 6
3.2 odd 2 729.2.a.e.1.1 yes 6
9.2 odd 6 729.2.c.a.244.6 12
9.4 even 3 729.2.c.d.487.1 12
9.5 odd 6 729.2.c.a.487.6 12
9.7 even 3 729.2.c.d.244.1 12
27.2 odd 18 729.2.e.u.568.2 12
27.4 even 9 729.2.e.s.406.2 12
27.5 odd 18 729.2.e.k.649.1 12
27.7 even 9 729.2.e.s.325.2 12
27.11 odd 18 729.2.e.k.82.1 12
27.13 even 9 729.2.e.j.163.1 12
27.14 odd 18 729.2.e.u.163.2 12
27.16 even 9 729.2.e.t.82.2 12
27.20 odd 18 729.2.e.l.325.1 12
27.22 even 9 729.2.e.t.649.2 12
27.23 odd 18 729.2.e.l.406.1 12
27.25 even 9 729.2.e.j.568.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
729.2.a.b.1.6 6 1.1 even 1 trivial
729.2.a.e.1.1 yes 6 3.2 odd 2
729.2.c.a.244.6 12 9.2 odd 6
729.2.c.a.487.6 12 9.5 odd 6
729.2.c.d.244.1 12 9.7 even 3
729.2.c.d.487.1 12 9.4 even 3
729.2.e.j.163.1 12 27.13 even 9
729.2.e.j.568.1 12 27.25 even 9
729.2.e.k.82.1 12 27.11 odd 18
729.2.e.k.649.1 12 27.5 odd 18
729.2.e.l.325.1 12 27.20 odd 18
729.2.e.l.406.1 12 27.23 odd 18
729.2.e.s.325.2 12 27.7 even 9
729.2.e.s.406.2 12 27.4 even 9
729.2.e.t.82.2 12 27.16 even 9
729.2.e.t.649.2 12 27.22 even 9
729.2.e.u.163.2 12 27.14 odd 18
729.2.e.u.568.2 12 27.2 odd 18