Properties

Label 729.2.a.e.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: 6.6.7459857.1
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 1.3
Root \(-1.70506\) of defining polynomial
Character \(\chi\) \(=\) 729.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.172976 q^{2} -1.97008 q^{4} +3.73656 q^{5} +3.03150 q^{7} +0.686728 q^{8} +O(q^{10})\) \(q-0.172976 q^{2} -1.97008 q^{4} +3.73656 q^{5} +3.03150 q^{7} +0.686728 q^{8} -0.646335 q^{10} +2.49170 q^{11} -0.765139 q^{13} -0.524376 q^{14} +3.82137 q^{16} -4.62278 q^{17} -0.611844 q^{19} -7.36132 q^{20} -0.431003 q^{22} -6.52438 q^{23} +8.96190 q^{25} +0.132351 q^{26} -5.97229 q^{28} +6.55089 q^{29} +6.55043 q^{31} -2.03446 q^{32} +0.799630 q^{34} +11.3274 q^{35} +4.95969 q^{37} +0.105834 q^{38} +2.56600 q^{40} -5.26024 q^{41} +5.57057 q^{43} -4.90884 q^{44} +1.12856 q^{46} -1.10762 q^{47} +2.18998 q^{49} -1.55019 q^{50} +1.50738 q^{52} -8.84310 q^{53} +9.31038 q^{55} +2.08181 q^{56} -1.13315 q^{58} +11.8518 q^{59} +8.18700 q^{61} -1.13307 q^{62} -7.29083 q^{64} -2.85899 q^{65} -1.21234 q^{67} +9.10725 q^{68} -1.95936 q^{70} +4.91946 q^{71} +4.29945 q^{73} -0.857907 q^{74} +1.20538 q^{76} +7.55357 q^{77} -11.7946 q^{79} +14.2788 q^{80} +0.909895 q^{82} -9.01607 q^{83} -17.2733 q^{85} -0.963575 q^{86} +1.71112 q^{88} -7.53885 q^{89} -2.31952 q^{91} +12.8535 q^{92} +0.191591 q^{94} -2.28619 q^{95} -0.948354 q^{97} -0.378814 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{2} + 9 q^{4} - 3 q^{5} + 6 q^{7} + 6 q^{8}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.172976 −0.122312 −0.0611562 0.998128i \(-0.519479\pi\)
−0.0611562 + 0.998128i \(0.519479\pi\)
\(3\) 0 0
\(4\) −1.97008 −0.985040
\(5\) 3.73656 1.67104 0.835521 0.549459i \(-0.185166\pi\)
0.835521 + 0.549459i \(0.185166\pi\)
\(6\) 0 0
\(7\) 3.03150 1.14580 0.572899 0.819626i \(-0.305819\pi\)
0.572899 + 0.819626i \(0.305819\pi\)
\(8\) 0.686728 0.242795
\(9\) 0 0
\(10\) −0.646335 −0.204389
\(11\) 2.49170 0.751275 0.375637 0.926767i \(-0.377424\pi\)
0.375637 + 0.926767i \(0.377424\pi\)
\(12\) 0 0
\(13\) −0.765139 −0.212211 −0.106106 0.994355i \(-0.533838\pi\)
−0.106106 + 0.994355i \(0.533838\pi\)
\(14\) −0.524376 −0.140145
\(15\) 0 0
\(16\) 3.82137 0.955343
\(17\) −4.62278 −1.12119 −0.560595 0.828090i \(-0.689427\pi\)
−0.560595 + 0.828090i \(0.689427\pi\)
\(18\) 0 0
\(19\) −0.611844 −0.140367 −0.0701833 0.997534i \(-0.522358\pi\)
−0.0701833 + 0.997534i \(0.522358\pi\)
\(20\) −7.36132 −1.64604
\(21\) 0 0
\(22\) −0.431003 −0.0918902
\(23\) −6.52438 −1.36043 −0.680213 0.733014i \(-0.738113\pi\)
−0.680213 + 0.733014i \(0.738113\pi\)
\(24\) 0 0
\(25\) 8.96190 1.79238
\(26\) 0.132351 0.0259561
\(27\) 0 0
\(28\) −5.97229 −1.12866
\(29\) 6.55089 1.21647 0.608235 0.793757i \(-0.291878\pi\)
0.608235 + 0.793757i \(0.291878\pi\)
\(30\) 0 0
\(31\) 6.55043 1.17649 0.588246 0.808682i \(-0.299819\pi\)
0.588246 + 0.808682i \(0.299819\pi\)
\(32\) −2.03446 −0.359645
\(33\) 0 0
\(34\) 0.799630 0.137135
\(35\) 11.3274 1.91468
\(36\) 0 0
\(37\) 4.95969 0.815368 0.407684 0.913123i \(-0.366337\pi\)
0.407684 + 0.913123i \(0.366337\pi\)
\(38\) 0.105834 0.0171686
\(39\) 0 0
\(40\) 2.56600 0.405721
\(41\) −5.26024 −0.821511 −0.410756 0.911745i \(-0.634735\pi\)
−0.410756 + 0.911745i \(0.634735\pi\)
\(42\) 0 0
\(43\) 5.57057 0.849505 0.424752 0.905310i \(-0.360361\pi\)
0.424752 + 0.905310i \(0.360361\pi\)
\(44\) −4.90884 −0.740035
\(45\) 0 0
\(46\) 1.12856 0.166397
\(47\) −1.10762 −0.161562 −0.0807812 0.996732i \(-0.525741\pi\)
−0.0807812 + 0.996732i \(0.525741\pi\)
\(48\) 0 0
\(49\) 2.18998 0.312854
\(50\) −1.55019 −0.219230
\(51\) 0 0
\(52\) 1.50738 0.209037
\(53\) −8.84310 −1.21469 −0.607346 0.794437i \(-0.707766\pi\)
−0.607346 + 0.794437i \(0.707766\pi\)
\(54\) 0 0
\(55\) 9.31038 1.25541
\(56\) 2.08181 0.278194
\(57\) 0 0
\(58\) −1.13315 −0.148789
\(59\) 11.8518 1.54297 0.771484 0.636249i \(-0.219515\pi\)
0.771484 + 0.636249i \(0.219515\pi\)
\(60\) 0 0
\(61\) 8.18700 1.04824 0.524119 0.851645i \(-0.324395\pi\)
0.524119 + 0.851645i \(0.324395\pi\)
\(62\) −1.13307 −0.143900
\(63\) 0 0
\(64\) −7.29083 −0.911354
\(65\) −2.85899 −0.354614
\(66\) 0 0
\(67\) −1.21234 −0.148111 −0.0740553 0.997254i \(-0.523594\pi\)
−0.0740553 + 0.997254i \(0.523594\pi\)
\(68\) 9.10725 1.10442
\(69\) 0 0
\(70\) −1.95936 −0.234189
\(71\) 4.91946 0.583833 0.291916 0.956444i \(-0.405707\pi\)
0.291916 + 0.956444i \(0.405707\pi\)
\(72\) 0 0
\(73\) 4.29945 0.503213 0.251606 0.967830i \(-0.419041\pi\)
0.251606 + 0.967830i \(0.419041\pi\)
\(74\) −0.857907 −0.0997296
\(75\) 0 0
\(76\) 1.20538 0.138267
\(77\) 7.55357 0.860809
\(78\) 0 0
\(79\) −11.7946 −1.32700 −0.663498 0.748178i \(-0.730929\pi\)
−0.663498 + 0.748178i \(0.730929\pi\)
\(80\) 14.2788 1.59642
\(81\) 0 0
\(82\) 0.909895 0.100481
\(83\) −9.01607 −0.989642 −0.494821 0.868995i \(-0.664766\pi\)
−0.494821 + 0.868995i \(0.664766\pi\)
\(84\) 0 0
\(85\) −17.2733 −1.87355
\(86\) −0.963575 −0.103905
\(87\) 0 0
\(88\) 1.71112 0.182406
\(89\) −7.53885 −0.799117 −0.399558 0.916708i \(-0.630837\pi\)
−0.399558 + 0.916708i \(0.630837\pi\)
\(90\) 0 0
\(91\) −2.31952 −0.243151
\(92\) 12.8535 1.34007
\(93\) 0 0
\(94\) 0.191591 0.0197611
\(95\) −2.28619 −0.234558
\(96\) 0 0
\(97\) −0.948354 −0.0962908 −0.0481454 0.998840i \(-0.515331\pi\)
−0.0481454 + 0.998840i \(0.515331\pi\)
\(98\) −0.378814 −0.0382659
\(99\) 0 0
\(100\) −17.6557 −1.76557
\(101\) 5.60815 0.558032 0.279016 0.960286i \(-0.409992\pi\)
0.279016 + 0.960286i \(0.409992\pi\)
\(102\) 0 0
\(103\) −9.42502 −0.928675 −0.464337 0.885658i \(-0.653707\pi\)
−0.464337 + 0.885658i \(0.653707\pi\)
\(104\) −0.525442 −0.0515239
\(105\) 0 0
\(106\) 1.52964 0.148572
\(107\) 1.27825 0.123573 0.0617864 0.998089i \(-0.480320\pi\)
0.0617864 + 0.998089i \(0.480320\pi\)
\(108\) 0 0
\(109\) −7.40689 −0.709451 −0.354726 0.934970i \(-0.615426\pi\)
−0.354726 + 0.934970i \(0.615426\pi\)
\(110\) −1.61047 −0.153552
\(111\) 0 0
\(112\) 11.5845 1.09463
\(113\) −9.35196 −0.879759 −0.439879 0.898057i \(-0.644979\pi\)
−0.439879 + 0.898057i \(0.644979\pi\)
\(114\) 0 0
\(115\) −24.3787 −2.27333
\(116\) −12.9058 −1.19827
\(117\) 0 0
\(118\) −2.05007 −0.188724
\(119\) −14.0140 −1.28466
\(120\) 0 0
\(121\) −4.79145 −0.435586
\(122\) −1.41615 −0.128213
\(123\) 0 0
\(124\) −12.9049 −1.15889
\(125\) 14.8039 1.32410
\(126\) 0 0
\(127\) 20.7968 1.84542 0.922710 0.385496i \(-0.125970\pi\)
0.922710 + 0.385496i \(0.125970\pi\)
\(128\) 5.33006 0.471115
\(129\) 0 0
\(130\) 0.494536 0.0433737
\(131\) 0.655830 0.0573001 0.0286501 0.999590i \(-0.490879\pi\)
0.0286501 + 0.999590i \(0.490879\pi\)
\(132\) 0 0
\(133\) −1.85480 −0.160832
\(134\) 0.209705 0.0181158
\(135\) 0 0
\(136\) −3.17460 −0.272219
\(137\) −8.58760 −0.733689 −0.366844 0.930282i \(-0.619562\pi\)
−0.366844 + 0.930282i \(0.619562\pi\)
\(138\) 0 0
\(139\) 13.4461 1.14049 0.570243 0.821476i \(-0.306849\pi\)
0.570243 + 0.821476i \(0.306849\pi\)
\(140\) −22.3158 −1.88603
\(141\) 0 0
\(142\) −0.850949 −0.0714100
\(143\) −1.90649 −0.159429
\(144\) 0 0
\(145\) 24.4778 2.03277
\(146\) −0.743701 −0.0615492
\(147\) 0 0
\(148\) −9.77098 −0.803170
\(149\) −9.62207 −0.788270 −0.394135 0.919052i \(-0.628956\pi\)
−0.394135 + 0.919052i \(0.628956\pi\)
\(150\) 0 0
\(151\) −7.12820 −0.580085 −0.290042 0.957014i \(-0.593669\pi\)
−0.290042 + 0.957014i \(0.593669\pi\)
\(152\) −0.420170 −0.0340803
\(153\) 0 0
\(154\) −1.30659 −0.105288
\(155\) 24.4761 1.96597
\(156\) 0 0
\(157\) 7.68577 0.613391 0.306696 0.951808i \(-0.400777\pi\)
0.306696 + 0.951808i \(0.400777\pi\)
\(158\) 2.04018 0.162308
\(159\) 0 0
\(160\) −7.60189 −0.600982
\(161\) −19.7786 −1.55877
\(162\) 0 0
\(163\) 1.04750 0.0820465 0.0410232 0.999158i \(-0.486938\pi\)
0.0410232 + 0.999158i \(0.486938\pi\)
\(164\) 10.3631 0.809221
\(165\) 0 0
\(166\) 1.55956 0.121046
\(167\) −8.35408 −0.646458 −0.323229 0.946321i \(-0.604768\pi\)
−0.323229 + 0.946321i \(0.604768\pi\)
\(168\) 0 0
\(169\) −12.4146 −0.954966
\(170\) 2.98787 0.229159
\(171\) 0 0
\(172\) −10.9745 −0.836796
\(173\) −21.8458 −1.66090 −0.830452 0.557090i \(-0.811918\pi\)
−0.830452 + 0.557090i \(0.811918\pi\)
\(174\) 0 0
\(175\) 27.1680 2.05371
\(176\) 9.52170 0.717725
\(177\) 0 0
\(178\) 1.30404 0.0977419
\(179\) 9.08866 0.679319 0.339659 0.940549i \(-0.389688\pi\)
0.339659 + 0.940549i \(0.389688\pi\)
\(180\) 0 0
\(181\) −7.13077 −0.530026 −0.265013 0.964245i \(-0.585376\pi\)
−0.265013 + 0.964245i \(0.585376\pi\)
\(182\) 0.401221 0.0297404
\(183\) 0 0
\(184\) −4.48047 −0.330305
\(185\) 18.5322 1.36251
\(186\) 0 0
\(187\) −11.5186 −0.842321
\(188\) 2.18209 0.159145
\(189\) 0 0
\(190\) 0.395456 0.0286894
\(191\) −11.9556 −0.865079 −0.432539 0.901615i \(-0.642382\pi\)
−0.432539 + 0.901615i \(0.642382\pi\)
\(192\) 0 0
\(193\) −8.87364 −0.638738 −0.319369 0.947630i \(-0.603471\pi\)
−0.319369 + 0.947630i \(0.603471\pi\)
\(194\) 0.164042 0.0117776
\(195\) 0 0
\(196\) −4.31443 −0.308174
\(197\) −7.39790 −0.527079 −0.263539 0.964649i \(-0.584890\pi\)
−0.263539 + 0.964649i \(0.584890\pi\)
\(198\) 0 0
\(199\) −10.3837 −0.736084 −0.368042 0.929809i \(-0.619972\pi\)
−0.368042 + 0.929809i \(0.619972\pi\)
\(200\) 6.15439 0.435181
\(201\) 0 0
\(202\) −0.970076 −0.0682543
\(203\) 19.8590 1.39383
\(204\) 0 0
\(205\) −19.6552 −1.37278
\(206\) 1.63030 0.113588
\(207\) 0 0
\(208\) −2.92388 −0.202735
\(209\) −1.52453 −0.105454
\(210\) 0 0
\(211\) 20.8611 1.43614 0.718070 0.695971i \(-0.245026\pi\)
0.718070 + 0.695971i \(0.245026\pi\)
\(212\) 17.4216 1.19652
\(213\) 0 0
\(214\) −0.221106 −0.0151145
\(215\) 20.8148 1.41956
\(216\) 0 0
\(217\) 19.8576 1.34802
\(218\) 1.28121 0.0867747
\(219\) 0 0
\(220\) −18.3422 −1.23663
\(221\) 3.53707 0.237929
\(222\) 0 0
\(223\) 23.5785 1.57893 0.789466 0.613794i \(-0.210357\pi\)
0.789466 + 0.613794i \(0.210357\pi\)
\(224\) −6.16747 −0.412081
\(225\) 0 0
\(226\) 1.61766 0.107605
\(227\) 10.4841 0.695856 0.347928 0.937521i \(-0.386885\pi\)
0.347928 + 0.937521i \(0.386885\pi\)
\(228\) 0 0
\(229\) −13.8824 −0.917376 −0.458688 0.888597i \(-0.651680\pi\)
−0.458688 + 0.888597i \(0.651680\pi\)
\(230\) 4.21694 0.278056
\(231\) 0 0
\(232\) 4.49868 0.295353
\(233\) −7.59964 −0.497869 −0.248935 0.968520i \(-0.580080\pi\)
−0.248935 + 0.968520i \(0.580080\pi\)
\(234\) 0 0
\(235\) −4.13868 −0.269977
\(236\) −23.3489 −1.51988
\(237\) 0 0
\(238\) 2.42408 0.157130
\(239\) −16.5587 −1.07109 −0.535546 0.844506i \(-0.679894\pi\)
−0.535546 + 0.844506i \(0.679894\pi\)
\(240\) 0 0
\(241\) −14.5185 −0.935218 −0.467609 0.883935i \(-0.654884\pi\)
−0.467609 + 0.883935i \(0.654884\pi\)
\(242\) 0.828806 0.0532776
\(243\) 0 0
\(244\) −16.1290 −1.03256
\(245\) 8.18299 0.522792
\(246\) 0 0
\(247\) 0.468145 0.0297874
\(248\) 4.49836 0.285646
\(249\) 0 0
\(250\) −2.56072 −0.161954
\(251\) −9.05181 −0.571345 −0.285673 0.958327i \(-0.592217\pi\)
−0.285673 + 0.958327i \(0.592217\pi\)
\(252\) 0 0
\(253\) −16.2568 −1.02205
\(254\) −3.59735 −0.225718
\(255\) 0 0
\(256\) 13.6597 0.853730
\(257\) −9.69988 −0.605062 −0.302531 0.953140i \(-0.597832\pi\)
−0.302531 + 0.953140i \(0.597832\pi\)
\(258\) 0 0
\(259\) 15.0353 0.934247
\(260\) 5.63243 0.349309
\(261\) 0 0
\(262\) −0.113443 −0.00700852
\(263\) 26.8552 1.65597 0.827983 0.560754i \(-0.189489\pi\)
0.827983 + 0.560754i \(0.189489\pi\)
\(264\) 0 0
\(265\) −33.0428 −2.02980
\(266\) 0.320836 0.0196717
\(267\) 0 0
\(268\) 2.38840 0.145895
\(269\) −11.7388 −0.715729 −0.357865 0.933774i \(-0.616495\pi\)
−0.357865 + 0.933774i \(0.616495\pi\)
\(270\) 0 0
\(271\) 0.144576 0.00878238 0.00439119 0.999990i \(-0.498602\pi\)
0.00439119 + 0.999990i \(0.498602\pi\)
\(272\) −17.6654 −1.07112
\(273\) 0 0
\(274\) 1.48545 0.0897392
\(275\) 22.3303 1.34657
\(276\) 0 0
\(277\) −1.01442 −0.0609509 −0.0304754 0.999536i \(-0.509702\pi\)
−0.0304754 + 0.999536i \(0.509702\pi\)
\(278\) −2.32586 −0.139496
\(279\) 0 0
\(280\) 7.77883 0.464874
\(281\) 27.5295 1.64227 0.821135 0.570734i \(-0.193341\pi\)
0.821135 + 0.570734i \(0.193341\pi\)
\(282\) 0 0
\(283\) −26.8029 −1.59327 −0.796633 0.604463i \(-0.793388\pi\)
−0.796633 + 0.604463i \(0.793388\pi\)
\(284\) −9.69173 −0.575098
\(285\) 0 0
\(286\) 0.329777 0.0195001
\(287\) −15.9464 −0.941286
\(288\) 0 0
\(289\) 4.37012 0.257066
\(290\) −4.23407 −0.248633
\(291\) 0 0
\(292\) −8.47026 −0.495684
\(293\) −18.6573 −1.08997 −0.544984 0.838446i \(-0.683464\pi\)
−0.544984 + 0.838446i \(0.683464\pi\)
\(294\) 0 0
\(295\) 44.2848 2.57836
\(296\) 3.40596 0.197967
\(297\) 0 0
\(298\) 1.66439 0.0964153
\(299\) 4.99205 0.288698
\(300\) 0 0
\(301\) 16.8872 0.973361
\(302\) 1.23301 0.0709516
\(303\) 0 0
\(304\) −2.33808 −0.134098
\(305\) 30.5913 1.75165
\(306\) 0 0
\(307\) 33.7893 1.92845 0.964227 0.265077i \(-0.0853973\pi\)
0.964227 + 0.265077i \(0.0853973\pi\)
\(308\) −14.8811 −0.847931
\(309\) 0 0
\(310\) −4.23377 −0.240462
\(311\) −34.6866 −1.96690 −0.983448 0.181193i \(-0.942004\pi\)
−0.983448 + 0.181193i \(0.942004\pi\)
\(312\) 0 0
\(313\) 3.34038 0.188809 0.0944047 0.995534i \(-0.469905\pi\)
0.0944047 + 0.995534i \(0.469905\pi\)
\(314\) −1.32945 −0.0750254
\(315\) 0 0
\(316\) 23.2363 1.30714
\(317\) −31.0328 −1.74298 −0.871488 0.490417i \(-0.836844\pi\)
−0.871488 + 0.490417i \(0.836844\pi\)
\(318\) 0 0
\(319\) 16.3228 0.913903
\(320\) −27.2426 −1.52291
\(321\) 0 0
\(322\) 3.42123 0.190658
\(323\) 2.82842 0.157378
\(324\) 0 0
\(325\) −6.85710 −0.380363
\(326\) −0.181192 −0.0100353
\(327\) 0 0
\(328\) −3.61235 −0.199459
\(329\) −3.35773 −0.185118
\(330\) 0 0
\(331\) 3.27168 0.179828 0.0899138 0.995950i \(-0.471341\pi\)
0.0899138 + 0.995950i \(0.471341\pi\)
\(332\) 17.7624 0.974837
\(333\) 0 0
\(334\) 1.44505 0.0790699
\(335\) −4.52998 −0.247499
\(336\) 0 0
\(337\) 6.36581 0.346768 0.173384 0.984854i \(-0.444530\pi\)
0.173384 + 0.984854i \(0.444530\pi\)
\(338\) 2.14742 0.116804
\(339\) 0 0
\(340\) 34.0298 1.84553
\(341\) 16.3217 0.883868
\(342\) 0 0
\(343\) −14.5816 −0.787331
\(344\) 3.82547 0.206256
\(345\) 0 0
\(346\) 3.77880 0.203149
\(347\) −8.79241 −0.472001 −0.236001 0.971753i \(-0.575837\pi\)
−0.236001 + 0.971753i \(0.575837\pi\)
\(348\) 0 0
\(349\) 14.4002 0.770826 0.385413 0.922744i \(-0.374059\pi\)
0.385413 + 0.922744i \(0.374059\pi\)
\(350\) −4.69941 −0.251194
\(351\) 0 0
\(352\) −5.06926 −0.270192
\(353\) 33.2005 1.76708 0.883542 0.468353i \(-0.155152\pi\)
0.883542 + 0.468353i \(0.155152\pi\)
\(354\) 0 0
\(355\) 18.3819 0.975609
\(356\) 14.8521 0.787162
\(357\) 0 0
\(358\) −1.57212 −0.0830891
\(359\) 4.94514 0.260995 0.130497 0.991449i \(-0.458343\pi\)
0.130497 + 0.991449i \(0.458343\pi\)
\(360\) 0 0
\(361\) −18.6256 −0.980297
\(362\) 1.23345 0.0648288
\(363\) 0 0
\(364\) 4.56963 0.239514
\(365\) 16.0652 0.840889
\(366\) 0 0
\(367\) −2.49245 −0.130105 −0.0650525 0.997882i \(-0.520721\pi\)
−0.0650525 + 0.997882i \(0.520721\pi\)
\(368\) −24.9321 −1.29967
\(369\) 0 0
\(370\) −3.20562 −0.166652
\(371\) −26.8078 −1.39179
\(372\) 0 0
\(373\) −28.0476 −1.45225 −0.726124 0.687563i \(-0.758680\pi\)
−0.726124 + 0.687563i \(0.758680\pi\)
\(374\) 1.99244 0.103026
\(375\) 0 0
\(376\) −0.760631 −0.0392265
\(377\) −5.01234 −0.258149
\(378\) 0 0
\(379\) 5.13991 0.264019 0.132010 0.991248i \(-0.457857\pi\)
0.132010 + 0.991248i \(0.457857\pi\)
\(380\) 4.50398 0.231049
\(381\) 0 0
\(382\) 2.06804 0.105810
\(383\) −0.0446729 −0.00228268 −0.00114134 0.999999i \(-0.500363\pi\)
−0.00114134 + 0.999999i \(0.500363\pi\)
\(384\) 0 0
\(385\) 28.2244 1.43845
\(386\) 1.53493 0.0781256
\(387\) 0 0
\(388\) 1.86833 0.0948502
\(389\) 20.9823 1.06384 0.531921 0.846794i \(-0.321470\pi\)
0.531921 + 0.846794i \(0.321470\pi\)
\(390\) 0 0
\(391\) 30.1608 1.52530
\(392\) 1.50392 0.0759594
\(393\) 0 0
\(394\) 1.27966 0.0644683
\(395\) −44.0713 −2.21747
\(396\) 0 0
\(397\) −0.00245641 −0.000123284 0 −6.16419e−5 1.00000i \(-0.500020\pi\)
−6.16419e−5 1.00000i \(0.500020\pi\)
\(398\) 1.79614 0.0900323
\(399\) 0 0
\(400\) 34.2467 1.71234
\(401\) 25.2563 1.26124 0.630620 0.776091i \(-0.282801\pi\)
0.630620 + 0.776091i \(0.282801\pi\)
\(402\) 0 0
\(403\) −5.01198 −0.249665
\(404\) −11.0485 −0.549684
\(405\) 0 0
\(406\) −3.43513 −0.170483
\(407\) 12.3580 0.612565
\(408\) 0 0
\(409\) −23.2885 −1.15154 −0.575772 0.817610i \(-0.695298\pi\)
−0.575772 + 0.817610i \(0.695298\pi\)
\(410\) 3.39988 0.167908
\(411\) 0 0
\(412\) 18.5680 0.914781
\(413\) 35.9286 1.76793
\(414\) 0 0
\(415\) −33.6891 −1.65373
\(416\) 1.55665 0.0763208
\(417\) 0 0
\(418\) 0.263707 0.0128983
\(419\) 31.2884 1.52854 0.764268 0.644898i \(-0.223100\pi\)
0.764268 + 0.644898i \(0.223100\pi\)
\(420\) 0 0
\(421\) 30.5296 1.48792 0.743960 0.668224i \(-0.232945\pi\)
0.743960 + 0.668224i \(0.232945\pi\)
\(422\) −3.60848 −0.175658
\(423\) 0 0
\(424\) −6.07280 −0.294921
\(425\) −41.4289 −2.00960
\(426\) 0 0
\(427\) 24.8189 1.20107
\(428\) −2.51825 −0.121724
\(429\) 0 0
\(430\) −3.60046 −0.173630
\(431\) 12.4246 0.598474 0.299237 0.954179i \(-0.403268\pi\)
0.299237 + 0.954179i \(0.403268\pi\)
\(432\) 0 0
\(433\) −0.760649 −0.0365545 −0.0182772 0.999833i \(-0.505818\pi\)
−0.0182772 + 0.999833i \(0.505818\pi\)
\(434\) −3.43489 −0.164880
\(435\) 0 0
\(436\) 14.5922 0.698838
\(437\) 3.99190 0.190958
\(438\) 0 0
\(439\) 30.1094 1.43704 0.718521 0.695506i \(-0.244820\pi\)
0.718521 + 0.695506i \(0.244820\pi\)
\(440\) 6.39370 0.304808
\(441\) 0 0
\(442\) −0.611828 −0.0291017
\(443\) −13.6616 −0.649081 −0.324541 0.945872i \(-0.605210\pi\)
−0.324541 + 0.945872i \(0.605210\pi\)
\(444\) 0 0
\(445\) −28.1694 −1.33536
\(446\) −4.07851 −0.193123
\(447\) 0 0
\(448\) −22.1021 −1.04423
\(449\) 21.9989 1.03819 0.519097 0.854715i \(-0.326268\pi\)
0.519097 + 0.854715i \(0.326268\pi\)
\(450\) 0 0
\(451\) −13.1069 −0.617181
\(452\) 18.4241 0.866597
\(453\) 0 0
\(454\) −1.81350 −0.0851119
\(455\) −8.66702 −0.406316
\(456\) 0 0
\(457\) −1.48883 −0.0696444 −0.0348222 0.999394i \(-0.511086\pi\)
−0.0348222 + 0.999394i \(0.511086\pi\)
\(458\) 2.40132 0.112206
\(459\) 0 0
\(460\) 48.0281 2.23932
\(461\) 7.11334 0.331301 0.165651 0.986185i \(-0.447028\pi\)
0.165651 + 0.986185i \(0.447028\pi\)
\(462\) 0 0
\(463\) −26.5407 −1.23345 −0.616726 0.787178i \(-0.711541\pi\)
−0.616726 + 0.787178i \(0.711541\pi\)
\(464\) 25.0334 1.16215
\(465\) 0 0
\(466\) 1.31456 0.0608956
\(467\) −26.1519 −1.21017 −0.605084 0.796162i \(-0.706860\pi\)
−0.605084 + 0.796162i \(0.706860\pi\)
\(468\) 0 0
\(469\) −3.67520 −0.169705
\(470\) 0.715891 0.0330216
\(471\) 0 0
\(472\) 8.13894 0.374625
\(473\) 13.8802 0.638211
\(474\) 0 0
\(475\) −5.48328 −0.251590
\(476\) 27.6086 1.26544
\(477\) 0 0
\(478\) 2.86425 0.131008
\(479\) 10.4065 0.475487 0.237744 0.971328i \(-0.423592\pi\)
0.237744 + 0.971328i \(0.423592\pi\)
\(480\) 0 0
\(481\) −3.79485 −0.173030
\(482\) 2.51135 0.114389
\(483\) 0 0
\(484\) 9.43954 0.429070
\(485\) −3.54358 −0.160906
\(486\) 0 0
\(487\) −18.4664 −0.836791 −0.418396 0.908265i \(-0.637407\pi\)
−0.418396 + 0.908265i \(0.637407\pi\)
\(488\) 5.62225 0.254507
\(489\) 0 0
\(490\) −1.41546 −0.0639440
\(491\) 16.9739 0.766021 0.383011 0.923744i \(-0.374887\pi\)
0.383011 + 0.923744i \(0.374887\pi\)
\(492\) 0 0
\(493\) −30.2834 −1.36389
\(494\) −0.0809779 −0.00364337
\(495\) 0 0
\(496\) 25.0316 1.12395
\(497\) 14.9133 0.668955
\(498\) 0 0
\(499\) 24.6462 1.10331 0.551657 0.834071i \(-0.313996\pi\)
0.551657 + 0.834071i \(0.313996\pi\)
\(500\) −29.1648 −1.30429
\(501\) 0 0
\(502\) 1.56575 0.0698826
\(503\) −40.1137 −1.78858 −0.894291 0.447485i \(-0.852320\pi\)
−0.894291 + 0.447485i \(0.852320\pi\)
\(504\) 0 0
\(505\) 20.9552 0.932495
\(506\) 2.81203 0.125010
\(507\) 0 0
\(508\) −40.9714 −1.81781
\(509\) −4.94852 −0.219339 −0.109670 0.993968i \(-0.534979\pi\)
−0.109670 + 0.993968i \(0.534979\pi\)
\(510\) 0 0
\(511\) 13.0338 0.576580
\(512\) −13.0229 −0.575537
\(513\) 0 0
\(514\) 1.67785 0.0740066
\(515\) −35.2172 −1.55185
\(516\) 0 0
\(517\) −2.75984 −0.121378
\(518\) −2.60074 −0.114270
\(519\) 0 0
\(520\) −1.96335 −0.0860985
\(521\) −7.73958 −0.339077 −0.169539 0.985524i \(-0.554228\pi\)
−0.169539 + 0.985524i \(0.554228\pi\)
\(522\) 0 0
\(523\) 36.0140 1.57478 0.787391 0.616453i \(-0.211431\pi\)
0.787391 + 0.616453i \(0.211431\pi\)
\(524\) −1.29204 −0.0564429
\(525\) 0 0
\(526\) −4.64531 −0.202545
\(527\) −30.2812 −1.31907
\(528\) 0 0
\(529\) 19.5675 0.850760
\(530\) 5.71561 0.248270
\(531\) 0 0
\(532\) 3.65411 0.158426
\(533\) 4.02481 0.174334
\(534\) 0 0
\(535\) 4.77625 0.206495
\(536\) −0.832547 −0.0359605
\(537\) 0 0
\(538\) 2.03053 0.0875426
\(539\) 5.45676 0.235039
\(540\) 0 0
\(541\) −24.4147 −1.04967 −0.524834 0.851204i \(-0.675873\pi\)
−0.524834 + 0.851204i \(0.675873\pi\)
\(542\) −0.0250082 −0.00107419
\(543\) 0 0
\(544\) 9.40487 0.403231
\(545\) −27.6763 −1.18552
\(546\) 0 0
\(547\) −28.3618 −1.21266 −0.606331 0.795212i \(-0.707359\pi\)
−0.606331 + 0.795212i \(0.707359\pi\)
\(548\) 16.9183 0.722712
\(549\) 0 0
\(550\) −3.86261 −0.164702
\(551\) −4.00812 −0.170752
\(552\) 0 0
\(553\) −35.7553 −1.52047
\(554\) 0.175471 0.00745505
\(555\) 0 0
\(556\) −26.4900 −1.12342
\(557\) 36.9373 1.56508 0.782542 0.622598i \(-0.213923\pi\)
0.782542 + 0.622598i \(0.213923\pi\)
\(558\) 0 0
\(559\) −4.26226 −0.180274
\(560\) 43.2861 1.82917
\(561\) 0 0
\(562\) −4.76193 −0.200870
\(563\) −22.7754 −0.959869 −0.479935 0.877304i \(-0.659340\pi\)
−0.479935 + 0.877304i \(0.659340\pi\)
\(564\) 0 0
\(565\) −34.9442 −1.47011
\(566\) 4.63625 0.194876
\(567\) 0 0
\(568\) 3.37833 0.141752
\(569\) −30.8688 −1.29409 −0.647043 0.762454i \(-0.723995\pi\)
−0.647043 + 0.762454i \(0.723995\pi\)
\(570\) 0 0
\(571\) 12.8205 0.536521 0.268260 0.963346i \(-0.413551\pi\)
0.268260 + 0.963346i \(0.413551\pi\)
\(572\) 3.75594 0.157044
\(573\) 0 0
\(574\) 2.75834 0.115131
\(575\) −58.4708 −2.43840
\(576\) 0 0
\(577\) −23.5264 −0.979417 −0.489708 0.871886i \(-0.662897\pi\)
−0.489708 + 0.871886i \(0.662897\pi\)
\(578\) −0.755926 −0.0314424
\(579\) 0 0
\(580\) −48.2232 −2.00236
\(581\) −27.3322 −1.13393
\(582\) 0 0
\(583\) −22.0343 −0.912568
\(584\) 2.95255 0.122178
\(585\) 0 0
\(586\) 3.22726 0.133317
\(587\) −11.3874 −0.470010 −0.235005 0.971994i \(-0.575511\pi\)
−0.235005 + 0.971994i \(0.575511\pi\)
\(588\) 0 0
\(589\) −4.00784 −0.165140
\(590\) −7.66021 −0.315366
\(591\) 0 0
\(592\) 18.9528 0.778956
\(593\) 37.7324 1.54948 0.774742 0.632277i \(-0.217880\pi\)
0.774742 + 0.632277i \(0.217880\pi\)
\(594\) 0 0
\(595\) −52.3640 −2.14672
\(596\) 18.9562 0.776478
\(597\) 0 0
\(598\) −0.863505 −0.0353113
\(599\) −47.3582 −1.93500 −0.967502 0.252865i \(-0.918627\pi\)
−0.967502 + 0.252865i \(0.918627\pi\)
\(600\) 0 0
\(601\) −31.1074 −1.26890 −0.634449 0.772964i \(-0.718773\pi\)
−0.634449 + 0.772964i \(0.718773\pi\)
\(602\) −2.92108 −0.119054
\(603\) 0 0
\(604\) 14.0431 0.571407
\(605\) −17.9036 −0.727883
\(606\) 0 0
\(607\) 29.4864 1.19682 0.598409 0.801191i \(-0.295800\pi\)
0.598409 + 0.801191i \(0.295800\pi\)
\(608\) 1.24477 0.0504822
\(609\) 0 0
\(610\) −5.29155 −0.214249
\(611\) 0.847480 0.0342854
\(612\) 0 0
\(613\) −6.10428 −0.246550 −0.123275 0.992373i \(-0.539340\pi\)
−0.123275 + 0.992373i \(0.539340\pi\)
\(614\) −5.84473 −0.235874
\(615\) 0 0
\(616\) 5.18725 0.209000
\(617\) 19.1201 0.769747 0.384873 0.922969i \(-0.374245\pi\)
0.384873 + 0.922969i \(0.374245\pi\)
\(618\) 0 0
\(619\) −6.75385 −0.271460 −0.135730 0.990746i \(-0.543338\pi\)
−0.135730 + 0.990746i \(0.543338\pi\)
\(620\) −48.2198 −1.93655
\(621\) 0 0
\(622\) 5.99994 0.240576
\(623\) −22.8540 −0.915627
\(624\) 0 0
\(625\) 10.5061 0.420246
\(626\) −0.577805 −0.0230937
\(627\) 0 0
\(628\) −15.1416 −0.604215
\(629\) −22.9276 −0.914182
\(630\) 0 0
\(631\) −0.456907 −0.0181892 −0.00909458 0.999959i \(-0.502895\pi\)
−0.00909458 + 0.999959i \(0.502895\pi\)
\(632\) −8.09969 −0.322188
\(633\) 0 0
\(634\) 5.36793 0.213188
\(635\) 77.7086 3.08377
\(636\) 0 0
\(637\) −1.67564 −0.0663912
\(638\) −2.82346 −0.111782
\(639\) 0 0
\(640\) 19.9161 0.787253
\(641\) −2.87103 −0.113399 −0.0566994 0.998391i \(-0.518058\pi\)
−0.0566994 + 0.998391i \(0.518058\pi\)
\(642\) 0 0
\(643\) −1.70284 −0.0671536 −0.0335768 0.999436i \(-0.510690\pi\)
−0.0335768 + 0.999436i \(0.510690\pi\)
\(644\) 38.9655 1.53545
\(645\) 0 0
\(646\) −0.489249 −0.0192492
\(647\) 36.1004 1.41925 0.709626 0.704579i \(-0.248864\pi\)
0.709626 + 0.704579i \(0.248864\pi\)
\(648\) 0 0
\(649\) 29.5310 1.15919
\(650\) 1.18611 0.0465232
\(651\) 0 0
\(652\) −2.06366 −0.0808191
\(653\) −43.5680 −1.70495 −0.852473 0.522771i \(-0.824898\pi\)
−0.852473 + 0.522771i \(0.824898\pi\)
\(654\) 0 0
\(655\) 2.45055 0.0957509
\(656\) −20.1013 −0.784825
\(657\) 0 0
\(658\) 0.580807 0.0226422
\(659\) 25.4810 0.992598 0.496299 0.868152i \(-0.334692\pi\)
0.496299 + 0.868152i \(0.334692\pi\)
\(660\) 0 0
\(661\) 34.1672 1.32895 0.664475 0.747310i \(-0.268655\pi\)
0.664475 + 0.747310i \(0.268655\pi\)
\(662\) −0.565921 −0.0219952
\(663\) 0 0
\(664\) −6.19159 −0.240280
\(665\) −6.93059 −0.268757
\(666\) 0 0
\(667\) −42.7405 −1.65492
\(668\) 16.4582 0.636787
\(669\) 0 0
\(670\) 0.783577 0.0302722
\(671\) 20.3995 0.787515
\(672\) 0 0
\(673\) −29.5437 −1.13883 −0.569413 0.822051i \(-0.692830\pi\)
−0.569413 + 0.822051i \(0.692830\pi\)
\(674\) −1.10113 −0.0424140
\(675\) 0 0
\(676\) 24.4577 0.940680
\(677\) 40.7802 1.56731 0.783656 0.621195i \(-0.213353\pi\)
0.783656 + 0.621195i \(0.213353\pi\)
\(678\) 0 0
\(679\) −2.87493 −0.110330
\(680\) −11.8621 −0.454890
\(681\) 0 0
\(682\) −2.82326 −0.108108
\(683\) 31.6426 1.21077 0.605384 0.795933i \(-0.293019\pi\)
0.605384 + 0.795933i \(0.293019\pi\)
\(684\) 0 0
\(685\) −32.0881 −1.22602
\(686\) 2.52226 0.0963004
\(687\) 0 0
\(688\) 21.2872 0.811568
\(689\) 6.76620 0.257772
\(690\) 0 0
\(691\) −28.5848 −1.08742 −0.543708 0.839275i \(-0.682980\pi\)
−0.543708 + 0.839275i \(0.682980\pi\)
\(692\) 43.0379 1.63606
\(693\) 0 0
\(694\) 1.52088 0.0577316
\(695\) 50.2423 1.90580
\(696\) 0 0
\(697\) 24.3169 0.921070
\(698\) −2.49089 −0.0942817
\(699\) 0 0
\(700\) −53.5231 −2.02298
\(701\) −7.52982 −0.284397 −0.142199 0.989838i \(-0.545417\pi\)
−0.142199 + 0.989838i \(0.545417\pi\)
\(702\) 0 0
\(703\) −3.03455 −0.114450
\(704\) −18.1665 −0.684677
\(705\) 0 0
\(706\) −5.74288 −0.216136
\(707\) 17.0011 0.639392
\(708\) 0 0
\(709\) 8.01399 0.300972 0.150486 0.988612i \(-0.451916\pi\)
0.150486 + 0.988612i \(0.451916\pi\)
\(710\) −3.17962 −0.119329
\(711\) 0 0
\(712\) −5.17714 −0.194022
\(713\) −42.7374 −1.60053
\(714\) 0 0
\(715\) −7.12373 −0.266412
\(716\) −17.9054 −0.669156
\(717\) 0 0
\(718\) −0.855391 −0.0319229
\(719\) 26.9826 1.00628 0.503140 0.864205i \(-0.332178\pi\)
0.503140 + 0.864205i \(0.332178\pi\)
\(720\) 0 0
\(721\) −28.5719 −1.06407
\(722\) 3.22179 0.119903
\(723\) 0 0
\(724\) 14.0482 0.522097
\(725\) 58.7084 2.18038
\(726\) 0 0
\(727\) −14.6943 −0.544983 −0.272491 0.962158i \(-0.587848\pi\)
−0.272491 + 0.962158i \(0.587848\pi\)
\(728\) −1.59288 −0.0590360
\(729\) 0 0
\(730\) −2.77889 −0.102851
\(731\) −25.7516 −0.952456
\(732\) 0 0
\(733\) −31.3438 −1.15771 −0.578854 0.815431i \(-0.696500\pi\)
−0.578854 + 0.815431i \(0.696500\pi\)
\(734\) 0.431134 0.0159135
\(735\) 0 0
\(736\) 13.2736 0.489271
\(737\) −3.02078 −0.111272
\(738\) 0 0
\(739\) 0.482909 0.0177641 0.00888205 0.999961i \(-0.497173\pi\)
0.00888205 + 0.999961i \(0.497173\pi\)
\(740\) −36.5099 −1.34213
\(741\) 0 0
\(742\) 4.63711 0.170234
\(743\) 43.0507 1.57938 0.789689 0.613507i \(-0.210242\pi\)
0.789689 + 0.613507i \(0.210242\pi\)
\(744\) 0 0
\(745\) −35.9535 −1.31723
\(746\) 4.85156 0.177628
\(747\) 0 0
\(748\) 22.6925 0.829720
\(749\) 3.87500 0.141589
\(750\) 0 0
\(751\) 43.9216 1.60272 0.801361 0.598181i \(-0.204109\pi\)
0.801361 + 0.598181i \(0.204109\pi\)
\(752\) −4.23261 −0.154347
\(753\) 0 0
\(754\) 0.867014 0.0315748
\(755\) −26.6350 −0.969346
\(756\) 0 0
\(757\) 22.4143 0.814661 0.407331 0.913281i \(-0.366460\pi\)
0.407331 + 0.913281i \(0.366460\pi\)
\(758\) −0.889081 −0.0322929
\(759\) 0 0
\(760\) −1.56999 −0.0569496
\(761\) −9.99674 −0.362382 −0.181191 0.983448i \(-0.557995\pi\)
−0.181191 + 0.983448i \(0.557995\pi\)
\(762\) 0 0
\(763\) −22.4540 −0.812888
\(764\) 23.5535 0.852137
\(765\) 0 0
\(766\) 0.00772733 0.000279200 0
\(767\) −9.06824 −0.327435
\(768\) 0 0
\(769\) −7.49619 −0.270320 −0.135160 0.990824i \(-0.543155\pi\)
−0.135160 + 0.990824i \(0.543155\pi\)
\(770\) −4.88214 −0.175940
\(771\) 0 0
\(772\) 17.4818 0.629183
\(773\) −19.8391 −0.713562 −0.356781 0.934188i \(-0.616126\pi\)
−0.356781 + 0.934188i \(0.616126\pi\)
\(774\) 0 0
\(775\) 58.7043 2.10872
\(776\) −0.651262 −0.0233789
\(777\) 0 0
\(778\) −3.62943 −0.130121
\(779\) 3.21845 0.115313
\(780\) 0 0
\(781\) 12.2578 0.438619
\(782\) −5.21709 −0.186563
\(783\) 0 0
\(784\) 8.36872 0.298883
\(785\) 28.7184 1.02500
\(786\) 0 0
\(787\) 39.7283 1.41616 0.708080 0.706133i \(-0.249562\pi\)
0.708080 + 0.706133i \(0.249562\pi\)
\(788\) 14.5745 0.519193
\(789\) 0 0
\(790\) 7.62327 0.271224
\(791\) −28.3505 −1.00803
\(792\) 0 0
\(793\) −6.26419 −0.222448
\(794\) 0.000424900 0 1.50791e−5 0
\(795\) 0 0
\(796\) 20.4568 0.725072
\(797\) −9.10595 −0.322549 −0.161275 0.986910i \(-0.551561\pi\)
−0.161275 + 0.986910i \(0.551561\pi\)
\(798\) 0 0
\(799\) 5.12027 0.181142
\(800\) −18.2326 −0.644621
\(801\) 0 0
\(802\) −4.36874 −0.154265
\(803\) 10.7129 0.378051
\(804\) 0 0
\(805\) −73.9041 −2.60478
\(806\) 0.866953 0.0305371
\(807\) 0 0
\(808\) 3.85128 0.135487
\(809\) 3.01910 0.106146 0.0530730 0.998591i \(-0.483098\pi\)
0.0530730 + 0.998591i \(0.483098\pi\)
\(810\) 0 0
\(811\) 33.1722 1.16483 0.582416 0.812891i \(-0.302107\pi\)
0.582416 + 0.812891i \(0.302107\pi\)
\(812\) −39.1238 −1.37298
\(813\) 0 0
\(814\) −2.13764 −0.0749243
\(815\) 3.91405 0.137103
\(816\) 0 0
\(817\) −3.40832 −0.119242
\(818\) 4.02836 0.140848
\(819\) 0 0
\(820\) 38.7223 1.35224
\(821\) 42.7620 1.49240 0.746201 0.665720i \(-0.231876\pi\)
0.746201 + 0.665720i \(0.231876\pi\)
\(822\) 0 0
\(823\) −35.2289 −1.22800 −0.614000 0.789306i \(-0.710441\pi\)
−0.614000 + 0.789306i \(0.710441\pi\)
\(824\) −6.47243 −0.225478
\(825\) 0 0
\(826\) −6.21478 −0.216240
\(827\) 26.0380 0.905429 0.452714 0.891656i \(-0.350456\pi\)
0.452714 + 0.891656i \(0.350456\pi\)
\(828\) 0 0
\(829\) −7.90268 −0.274471 −0.137236 0.990538i \(-0.543822\pi\)
−0.137236 + 0.990538i \(0.543822\pi\)
\(830\) 5.82741 0.202272
\(831\) 0 0
\(832\) 5.57850 0.193400
\(833\) −10.1238 −0.350769
\(834\) 0 0
\(835\) −31.2155 −1.08026
\(836\) 3.00344 0.103876
\(837\) 0 0
\(838\) −5.41213 −0.186959
\(839\) 5.14839 0.177742 0.0888711 0.996043i \(-0.471674\pi\)
0.0888711 + 0.996043i \(0.471674\pi\)
\(840\) 0 0
\(841\) 13.9142 0.479800
\(842\) −5.28088 −0.181991
\(843\) 0 0
\(844\) −41.0981 −1.41466
\(845\) −46.3878 −1.59579
\(846\) 0 0
\(847\) −14.5253 −0.499094
\(848\) −33.7928 −1.16045
\(849\) 0 0
\(850\) 7.16621 0.245799
\(851\) −32.3589 −1.10925
\(852\) 0 0
\(853\) 25.9905 0.889896 0.444948 0.895556i \(-0.353222\pi\)
0.444948 + 0.895556i \(0.353222\pi\)
\(854\) −4.29307 −0.146906
\(855\) 0 0
\(856\) 0.877808 0.0300028
\(857\) −4.11913 −0.140707 −0.0703535 0.997522i \(-0.522413\pi\)
−0.0703535 + 0.997522i \(0.522413\pi\)
\(858\) 0 0
\(859\) −6.42943 −0.219369 −0.109685 0.993966i \(-0.534984\pi\)
−0.109685 + 0.993966i \(0.534984\pi\)
\(860\) −41.0068 −1.39832
\(861\) 0 0
\(862\) −2.14916 −0.0732008
\(863\) 29.6195 1.00826 0.504129 0.863628i \(-0.331813\pi\)
0.504129 + 0.863628i \(0.331813\pi\)
\(864\) 0 0
\(865\) −81.6282 −2.77544
\(866\) 0.131574 0.00447107
\(867\) 0 0
\(868\) −39.1210 −1.32785
\(869\) −29.3886 −0.996939
\(870\) 0 0
\(871\) 0.927607 0.0314308
\(872\) −5.08652 −0.172251
\(873\) 0 0
\(874\) −0.690503 −0.0233566
\(875\) 44.8779 1.51715
\(876\) 0 0
\(877\) −31.8677 −1.07610 −0.538048 0.842914i \(-0.680838\pi\)
−0.538048 + 0.842914i \(0.680838\pi\)
\(878\) −5.20819 −0.175768
\(879\) 0 0
\(880\) 35.5784 1.19935
\(881\) 34.7864 1.17198 0.585991 0.810317i \(-0.300705\pi\)
0.585991 + 0.810317i \(0.300705\pi\)
\(882\) 0 0
\(883\) −30.3764 −1.02225 −0.511124 0.859507i \(-0.670771\pi\)
−0.511124 + 0.859507i \(0.670771\pi\)
\(884\) −6.96831 −0.234370
\(885\) 0 0
\(886\) 2.36312 0.0793907
\(887\) −50.0276 −1.67976 −0.839882 0.542769i \(-0.817376\pi\)
−0.839882 + 0.542769i \(0.817376\pi\)
\(888\) 0 0
\(889\) 63.0455 2.11448
\(890\) 4.87263 0.163331
\(891\) 0 0
\(892\) −46.4515 −1.55531
\(893\) 0.677688 0.0226780
\(894\) 0 0
\(895\) 33.9604 1.13517
\(896\) 16.1581 0.539803
\(897\) 0 0
\(898\) −3.80529 −0.126984
\(899\) 42.9111 1.43117
\(900\) 0 0
\(901\) 40.8797 1.36190
\(902\) 2.26718 0.0754889
\(903\) 0 0
\(904\) −6.42226 −0.213601
\(905\) −26.6446 −0.885696
\(906\) 0 0
\(907\) −44.0643 −1.46313 −0.731566 0.681771i \(-0.761210\pi\)
−0.731566 + 0.681771i \(0.761210\pi\)
\(908\) −20.6546 −0.685446
\(909\) 0 0
\(910\) 1.49919 0.0496975
\(911\) −37.1783 −1.23177 −0.615885 0.787836i \(-0.711202\pi\)
−0.615885 + 0.787836i \(0.711202\pi\)
\(912\) 0 0
\(913\) −22.4653 −0.743493
\(914\) 0.257531 0.00851838
\(915\) 0 0
\(916\) 27.3495 0.903652
\(917\) 1.98815 0.0656544
\(918\) 0 0
\(919\) 12.3976 0.408958 0.204479 0.978871i \(-0.434450\pi\)
0.204479 + 0.978871i \(0.434450\pi\)
\(920\) −16.7416 −0.551953
\(921\) 0 0
\(922\) −1.23044 −0.0405223
\(923\) −3.76407 −0.123896
\(924\) 0 0
\(925\) 44.4482 1.46145
\(926\) 4.59091 0.150867
\(927\) 0 0
\(928\) −13.3275 −0.437498
\(929\) 33.3882 1.09543 0.547716 0.836664i \(-0.315498\pi\)
0.547716 + 0.836664i \(0.315498\pi\)
\(930\) 0 0
\(931\) −1.33993 −0.0439143
\(932\) 14.9719 0.490421
\(933\) 0 0
\(934\) 4.52366 0.148019
\(935\) −43.0399 −1.40755
\(936\) 0 0
\(937\) −24.8441 −0.811620 −0.405810 0.913957i \(-0.633011\pi\)
−0.405810 + 0.913957i \(0.633011\pi\)
\(938\) 0.635721 0.0207570
\(939\) 0 0
\(940\) 8.15352 0.265938
\(941\) 25.4299 0.828990 0.414495 0.910051i \(-0.363958\pi\)
0.414495 + 0.910051i \(0.363958\pi\)
\(942\) 0 0
\(943\) 34.3198 1.11761
\(944\) 45.2900 1.47406
\(945\) 0 0
\(946\) −2.40094 −0.0780612
\(947\) 16.6301 0.540407 0.270204 0.962803i \(-0.412909\pi\)
0.270204 + 0.962803i \(0.412909\pi\)
\(948\) 0 0
\(949\) −3.28968 −0.106787
\(950\) 0.948476 0.0307726
\(951\) 0 0
\(952\) −9.62378 −0.311908
\(953\) 14.2671 0.462158 0.231079 0.972935i \(-0.425774\pi\)
0.231079 + 0.972935i \(0.425774\pi\)
\(954\) 0 0
\(955\) −44.6730 −1.44558
\(956\) 32.6219 1.05507
\(957\) 0 0
\(958\) −1.80008 −0.0581580
\(959\) −26.0333 −0.840659
\(960\) 0 0
\(961\) 11.9081 0.384131
\(962\) 0.656418 0.0211638
\(963\) 0 0
\(964\) 28.6026 0.921227
\(965\) −33.1569 −1.06736
\(966\) 0 0
\(967\) −39.0848 −1.25688 −0.628440 0.777858i \(-0.716306\pi\)
−0.628440 + 0.777858i \(0.716306\pi\)
\(968\) −3.29042 −0.105758
\(969\) 0 0
\(970\) 0.612955 0.0196808
\(971\) −4.40370 −0.141321 −0.0706607 0.997500i \(-0.522511\pi\)
−0.0706607 + 0.997500i \(0.522511\pi\)
\(972\) 0 0
\(973\) 40.7619 1.30677
\(974\) 3.19424 0.102350
\(975\) 0 0
\(976\) 31.2856 1.00143
\(977\) 42.1266 1.34775 0.673874 0.738846i \(-0.264629\pi\)
0.673874 + 0.738846i \(0.264629\pi\)
\(978\) 0 0
\(979\) −18.7845 −0.600356
\(980\) −16.1211 −0.514971
\(981\) 0 0
\(982\) −2.93608 −0.0936939
\(983\) −20.4261 −0.651493 −0.325746 0.945457i \(-0.605616\pi\)
−0.325746 + 0.945457i \(0.605616\pi\)
\(984\) 0 0
\(985\) −27.6427 −0.880770
\(986\) 5.23829 0.166821
\(987\) 0 0
\(988\) −0.922284 −0.0293418
\(989\) −36.3445 −1.15569
\(990\) 0 0
\(991\) 0.0680712 0.00216235 0.00108118 0.999999i \(-0.499656\pi\)
0.00108118 + 0.999999i \(0.499656\pi\)
\(992\) −13.3266 −0.423120
\(993\) 0 0
\(994\) −2.57965 −0.0818215
\(995\) −38.7995 −1.23003
\(996\) 0 0
\(997\) 10.6461 0.337166 0.168583 0.985688i \(-0.446081\pi\)
0.168583 + 0.985688i \(0.446081\pi\)
\(998\) −4.26319 −0.134949
\(999\) 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.e.1.3 yes 6
3.2 odd 2 729.2.a.b.1.4 6
9.2 odd 6 729.2.c.d.244.3 12
9.4 even 3 729.2.c.a.487.4 12
9.5 odd 6 729.2.c.d.487.3 12
9.7 even 3 729.2.c.a.244.4 12
27.2 odd 18 729.2.e.t.568.1 12
27.4 even 9 729.2.e.u.406.1 12
27.5 odd 18 729.2.e.s.649.2 12
27.7 even 9 729.2.e.u.325.1 12
27.11 odd 18 729.2.e.s.82.2 12
27.13 even 9 729.2.e.k.163.2 12
27.14 odd 18 729.2.e.t.163.1 12
27.16 even 9 729.2.e.l.82.1 12
27.20 odd 18 729.2.e.j.325.2 12
27.22 even 9 729.2.e.l.649.1 12
27.23 odd 18 729.2.e.j.406.2 12
27.25 even 9 729.2.e.k.568.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
729.2.a.b.1.4 6 3.2 odd 2
729.2.a.e.1.3 yes 6 1.1 even 1 trivial
729.2.c.a.244.4 12 9.7 even 3
729.2.c.a.487.4 12 9.4 even 3
729.2.c.d.244.3 12 9.2 odd 6
729.2.c.d.487.3 12 9.5 odd 6
729.2.e.j.325.2 12 27.20 odd 18
729.2.e.j.406.2 12 27.23 odd 18
729.2.e.k.163.2 12 27.13 even 9
729.2.e.k.568.2 12 27.25 even 9
729.2.e.l.82.1 12 27.16 even 9
729.2.e.l.649.1 12 27.22 even 9
729.2.e.s.82.2 12 27.11 odd 18
729.2.e.s.649.2 12 27.5 odd 18
729.2.e.t.163.1 12 27.14 odd 18
729.2.e.t.568.1 12 27.2 odd 18
729.2.e.u.325.1 12 27.7 even 9
729.2.e.u.406.1 12 27.4 even 9