Properties

Label 729.2.a.a.1.6
Level $729$
Weight $2$
Character 729.1
Self dual yes
Analytic conductor $5.821$
Analytic rank $1$
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: \(1\)
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 1.6
Root \(0.198473\) of defining polynomial
Character \(\chi\) \(=\) 729.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.68091 q^{2} +0.825466 q^{4} -1.12954 q^{5} -3.90892 q^{7} -1.97429 q^{8} +O(q^{10})\) \(q+1.68091 q^{2} +0.825466 q^{4} -1.12954 q^{5} -3.90892 q^{7} -1.97429 q^{8} -1.89866 q^{10} -1.87046 q^{11} -0.732598 q^{13} -6.57056 q^{14} -4.96954 q^{16} +1.88964 q^{17} +2.74286 q^{19} -0.932400 q^{20} -3.14407 q^{22} -5.82770 q^{23} -3.72413 q^{25} -1.23143 q^{26} -3.22668 q^{28} -5.31994 q^{29} +1.34060 q^{31} -4.40478 q^{32} +3.17632 q^{34} +4.41530 q^{35} +3.39611 q^{37} +4.61050 q^{38} +2.23005 q^{40} +1.79633 q^{41} +5.03015 q^{43} -1.54400 q^{44} -9.79585 q^{46} +1.70868 q^{47} +8.27968 q^{49} -6.25994 q^{50} -0.604734 q^{52} +2.84494 q^{53} +2.11276 q^{55} +7.71734 q^{56} -8.94234 q^{58} -11.2600 q^{59} +5.23089 q^{61} +2.25343 q^{62} +2.53503 q^{64} +0.827502 q^{65} -1.88903 q^{67} +1.55984 q^{68} +7.42173 q^{70} -12.1839 q^{71} +9.88768 q^{73} +5.70857 q^{74} +2.26413 q^{76} +7.31147 q^{77} +12.3529 q^{79} +5.61331 q^{80} +3.01948 q^{82} -11.6832 q^{83} -2.13444 q^{85} +8.45525 q^{86} +3.69282 q^{88} -5.72873 q^{89} +2.86367 q^{91} -4.81057 q^{92} +2.87214 q^{94} -3.09818 q^{95} -0.343378 q^{97} +13.9174 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{2} + 3 q^{4} - 6 q^{5} - 6 q^{8}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.68091 1.18858 0.594292 0.804249i \(-0.297432\pi\)
0.594292 + 0.804249i \(0.297432\pi\)
\(3\) 0 0
\(4\) 0.825466 0.412733
\(5\) −1.12954 −0.505147 −0.252574 0.967578i \(-0.581277\pi\)
−0.252574 + 0.967578i \(0.581277\pi\)
\(6\) 0 0
\(7\) −3.90892 −1.47743 −0.738717 0.674016i \(-0.764568\pi\)
−0.738717 + 0.674016i \(0.764568\pi\)
\(8\) −1.97429 −0.698017
\(9\) 0 0
\(10\) −1.89866 −0.600410
\(11\) −1.87046 −0.563964 −0.281982 0.959420i \(-0.590992\pi\)
−0.281982 + 0.959420i \(0.590992\pi\)
\(12\) 0 0
\(13\) −0.732598 −0.203186 −0.101593 0.994826i \(-0.532394\pi\)
−0.101593 + 0.994826i \(0.532394\pi\)
\(14\) −6.57056 −1.75605
\(15\) 0 0
\(16\) −4.96954 −1.24238
\(17\) 1.88964 0.458306 0.229153 0.973390i \(-0.426404\pi\)
0.229153 + 0.973390i \(0.426404\pi\)
\(18\) 0 0
\(19\) 2.74286 0.629254 0.314627 0.949215i \(-0.398121\pi\)
0.314627 + 0.949215i \(0.398121\pi\)
\(20\) −0.932400 −0.208491
\(21\) 0 0
\(22\) −3.14407 −0.670318
\(23\) −5.82770 −1.21516 −0.607580 0.794259i \(-0.707859\pi\)
−0.607580 + 0.794259i \(0.707859\pi\)
\(24\) 0 0
\(25\) −3.72413 −0.744826
\(26\) −1.23143 −0.241504
\(27\) 0 0
\(28\) −3.22668 −0.609786
\(29\) −5.31994 −0.987887 −0.493944 0.869494i \(-0.664445\pi\)
−0.493944 + 0.869494i \(0.664445\pi\)
\(30\) 0 0
\(31\) 1.34060 0.240779 0.120390 0.992727i \(-0.461586\pi\)
0.120390 + 0.992727i \(0.461586\pi\)
\(32\) −4.40478 −0.778662
\(33\) 0 0
\(34\) 3.17632 0.544735
\(35\) 4.41530 0.746322
\(36\) 0 0
\(37\) 3.39611 0.558318 0.279159 0.960245i \(-0.409944\pi\)
0.279159 + 0.960245i \(0.409944\pi\)
\(38\) 4.61050 0.747922
\(39\) 0 0
\(40\) 2.23005 0.352601
\(41\) 1.79633 0.280540 0.140270 0.990113i \(-0.455203\pi\)
0.140270 + 0.990113i \(0.455203\pi\)
\(42\) 0 0
\(43\) 5.03015 0.767091 0.383546 0.923522i \(-0.374703\pi\)
0.383546 + 0.923522i \(0.374703\pi\)
\(44\) −1.54400 −0.232766
\(45\) 0 0
\(46\) −9.79585 −1.44432
\(47\) 1.70868 0.249236 0.124618 0.992205i \(-0.460229\pi\)
0.124618 + 0.992205i \(0.460229\pi\)
\(48\) 0 0
\(49\) 8.27968 1.18281
\(50\) −6.25994 −0.885289
\(51\) 0 0
\(52\) −0.604734 −0.0838616
\(53\) 2.84494 0.390783 0.195391 0.980725i \(-0.437402\pi\)
0.195391 + 0.980725i \(0.437402\pi\)
\(54\) 0 0
\(55\) 2.11276 0.284885
\(56\) 7.71734 1.03127
\(57\) 0 0
\(58\) −8.94234 −1.17419
\(59\) −11.2600 −1.46593 −0.732967 0.680265i \(-0.761865\pi\)
−0.732967 + 0.680265i \(0.761865\pi\)
\(60\) 0 0
\(61\) 5.23089 0.669747 0.334874 0.942263i \(-0.391306\pi\)
0.334874 + 0.942263i \(0.391306\pi\)
\(62\) 2.25343 0.286186
\(63\) 0 0
\(64\) 2.53503 0.316879
\(65\) 0.827502 0.102639
\(66\) 0 0
\(67\) −1.88903 −0.230781 −0.115391 0.993320i \(-0.536812\pi\)
−0.115391 + 0.993320i \(0.536812\pi\)
\(68\) 1.55984 0.189158
\(69\) 0 0
\(70\) 7.42173 0.887067
\(71\) −12.1839 −1.44596 −0.722980 0.690869i \(-0.757228\pi\)
−0.722980 + 0.690869i \(0.757228\pi\)
\(72\) 0 0
\(73\) 9.88768 1.15727 0.578633 0.815588i \(-0.303587\pi\)
0.578633 + 0.815588i \(0.303587\pi\)
\(74\) 5.70857 0.663608
\(75\) 0 0
\(76\) 2.26413 0.259714
\(77\) 7.31147 0.833219
\(78\) 0 0
\(79\) 12.3529 1.38981 0.694904 0.719103i \(-0.255447\pi\)
0.694904 + 0.719103i \(0.255447\pi\)
\(80\) 5.61331 0.627587
\(81\) 0 0
\(82\) 3.01948 0.333445
\(83\) −11.6832 −1.28239 −0.641197 0.767376i \(-0.721562\pi\)
−0.641197 + 0.767376i \(0.721562\pi\)
\(84\) 0 0
\(85\) −2.13444 −0.231512
\(86\) 8.45525 0.911753
\(87\) 0 0
\(88\) 3.69282 0.393656
\(89\) −5.72873 −0.607244 −0.303622 0.952793i \(-0.598196\pi\)
−0.303622 + 0.952793i \(0.598196\pi\)
\(90\) 0 0
\(91\) 2.86367 0.300194
\(92\) −4.81057 −0.501536
\(93\) 0 0
\(94\) 2.87214 0.296238
\(95\) −3.09818 −0.317866
\(96\) 0 0
\(97\) −0.343378 −0.0348648 −0.0174324 0.999848i \(-0.505549\pi\)
−0.0174324 + 0.999848i \(0.505549\pi\)
\(98\) 13.9174 1.40587
\(99\) 0 0
\(100\) −3.07414 −0.307414
\(101\) −17.4023 −1.73159 −0.865794 0.500400i \(-0.833186\pi\)
−0.865794 + 0.500400i \(0.833186\pi\)
\(102\) 0 0
\(103\) −15.8130 −1.55810 −0.779051 0.626961i \(-0.784299\pi\)
−0.779051 + 0.626961i \(0.784299\pi\)
\(104\) 1.44636 0.141827
\(105\) 0 0
\(106\) 4.78210 0.464478
\(107\) −16.5298 −1.59800 −0.798999 0.601332i \(-0.794637\pi\)
−0.798999 + 0.601332i \(0.794637\pi\)
\(108\) 0 0
\(109\) −4.71844 −0.451945 −0.225972 0.974134i \(-0.572556\pi\)
−0.225972 + 0.974134i \(0.572556\pi\)
\(110\) 3.55137 0.338610
\(111\) 0 0
\(112\) 19.4255 1.83554
\(113\) 19.9448 1.87625 0.938125 0.346296i \(-0.112561\pi\)
0.938125 + 0.346296i \(0.112561\pi\)
\(114\) 0 0
\(115\) 6.58264 0.613835
\(116\) −4.39142 −0.407733
\(117\) 0 0
\(118\) −18.9271 −1.74239
\(119\) −7.38647 −0.677117
\(120\) 0 0
\(121\) −7.50139 −0.681945
\(122\) 8.79267 0.796051
\(123\) 0 0
\(124\) 1.10662 0.0993774
\(125\) 9.85429 0.881394
\(126\) 0 0
\(127\) 1.06946 0.0948989 0.0474495 0.998874i \(-0.484891\pi\)
0.0474495 + 0.998874i \(0.484891\pi\)
\(128\) 13.0707 1.15530
\(129\) 0 0
\(130\) 1.39096 0.121995
\(131\) 7.64030 0.667536 0.333768 0.942655i \(-0.391680\pi\)
0.333768 + 0.942655i \(0.391680\pi\)
\(132\) 0 0
\(133\) −10.7216 −0.929682
\(134\) −3.17529 −0.274303
\(135\) 0 0
\(136\) −3.73070 −0.319905
\(137\) 15.6505 1.33711 0.668556 0.743662i \(-0.266913\pi\)
0.668556 + 0.743662i \(0.266913\pi\)
\(138\) 0 0
\(139\) 8.65693 0.734272 0.367136 0.930167i \(-0.380338\pi\)
0.367136 + 0.930167i \(0.380338\pi\)
\(140\) 3.64468 0.308032
\(141\) 0 0
\(142\) −20.4800 −1.71864
\(143\) 1.37029 0.114590
\(144\) 0 0
\(145\) 6.00910 0.499029
\(146\) 16.6203 1.37551
\(147\) 0 0
\(148\) 2.80337 0.230436
\(149\) −2.46382 −0.201844 −0.100922 0.994894i \(-0.532179\pi\)
−0.100922 + 0.994894i \(0.532179\pi\)
\(150\) 0 0
\(151\) −10.5134 −0.855569 −0.427785 0.903881i \(-0.640706\pi\)
−0.427785 + 0.903881i \(0.640706\pi\)
\(152\) −5.41519 −0.439230
\(153\) 0 0
\(154\) 12.2899 0.990351
\(155\) −1.51427 −0.121629
\(156\) 0 0
\(157\) 0.359949 0.0287270 0.0143635 0.999897i \(-0.495428\pi\)
0.0143635 + 0.999897i \(0.495428\pi\)
\(158\) 20.7641 1.65190
\(159\) 0 0
\(160\) 4.97539 0.393339
\(161\) 22.7800 1.79532
\(162\) 0 0
\(163\) 14.6186 1.14502 0.572508 0.819899i \(-0.305971\pi\)
0.572508 + 0.819899i \(0.305971\pi\)
\(164\) 1.48281 0.115788
\(165\) 0 0
\(166\) −19.6384 −1.52423
\(167\) 2.17585 0.168372 0.0841861 0.996450i \(-0.473171\pi\)
0.0841861 + 0.996450i \(0.473171\pi\)
\(168\) 0 0
\(169\) −12.4633 −0.958715
\(170\) −3.58780 −0.275172
\(171\) 0 0
\(172\) 4.15222 0.316604
\(173\) −17.5670 −1.33560 −0.667798 0.744343i \(-0.732763\pi\)
−0.667798 + 0.744343i \(0.732763\pi\)
\(174\) 0 0
\(175\) 14.5573 1.10043
\(176\) 9.29530 0.700660
\(177\) 0 0
\(178\) −9.62950 −0.721761
\(179\) 1.00447 0.0750777 0.0375388 0.999295i \(-0.488048\pi\)
0.0375388 + 0.999295i \(0.488048\pi\)
\(180\) 0 0
\(181\) −21.1732 −1.57380 −0.786898 0.617084i \(-0.788314\pi\)
−0.786898 + 0.617084i \(0.788314\pi\)
\(182\) 4.81358 0.356806
\(183\) 0 0
\(184\) 11.5056 0.848201
\(185\) −3.83606 −0.282033
\(186\) 0 0
\(187\) −3.53449 −0.258468
\(188\) 1.41045 0.102868
\(189\) 0 0
\(190\) −5.20776 −0.377811
\(191\) −9.82838 −0.711157 −0.355578 0.934646i \(-0.615716\pi\)
−0.355578 + 0.934646i \(0.615716\pi\)
\(192\) 0 0
\(193\) −11.1561 −0.803033 −0.401516 0.915852i \(-0.631517\pi\)
−0.401516 + 0.915852i \(0.631517\pi\)
\(194\) −0.577189 −0.0414397
\(195\) 0 0
\(196\) 6.83459 0.488185
\(197\) 9.08994 0.647631 0.323816 0.946120i \(-0.395034\pi\)
0.323816 + 0.946120i \(0.395034\pi\)
\(198\) 0 0
\(199\) −14.6939 −1.04162 −0.520811 0.853672i \(-0.674370\pi\)
−0.520811 + 0.853672i \(0.674370\pi\)
\(200\) 7.35251 0.519901
\(201\) 0 0
\(202\) −29.2517 −2.05814
\(203\) 20.7952 1.45954
\(204\) 0 0
\(205\) −2.02904 −0.141714
\(206\) −26.5803 −1.85194
\(207\) 0 0
\(208\) 3.64067 0.252435
\(209\) −5.13039 −0.354877
\(210\) 0 0
\(211\) 7.85830 0.540988 0.270494 0.962722i \(-0.412813\pi\)
0.270494 + 0.962722i \(0.412813\pi\)
\(212\) 2.34840 0.161289
\(213\) 0 0
\(214\) −27.7852 −1.89936
\(215\) −5.68178 −0.387494
\(216\) 0 0
\(217\) −5.24030 −0.355735
\(218\) −7.93128 −0.537174
\(219\) 0 0
\(220\) 1.74401 0.117581
\(221\) −1.38435 −0.0931214
\(222\) 0 0
\(223\) −8.74089 −0.585333 −0.292666 0.956215i \(-0.594543\pi\)
−0.292666 + 0.956215i \(0.594543\pi\)
\(224\) 17.2179 1.15042
\(225\) 0 0
\(226\) 33.5255 2.23008
\(227\) 4.06842 0.270030 0.135015 0.990844i \(-0.456892\pi\)
0.135015 + 0.990844i \(0.456892\pi\)
\(228\) 0 0
\(229\) 16.1559 1.06761 0.533805 0.845608i \(-0.320762\pi\)
0.533805 + 0.845608i \(0.320762\pi\)
\(230\) 11.0648 0.729594
\(231\) 0 0
\(232\) 10.5031 0.689562
\(233\) 17.2132 1.12767 0.563836 0.825887i \(-0.309325\pi\)
0.563836 + 0.825887i \(0.309325\pi\)
\(234\) 0 0
\(235\) −1.93003 −0.125901
\(236\) −9.29478 −0.605039
\(237\) 0 0
\(238\) −12.4160 −0.804810
\(239\) −1.53544 −0.0993193 −0.0496597 0.998766i \(-0.515814\pi\)
−0.0496597 + 0.998766i \(0.515814\pi\)
\(240\) 0 0
\(241\) −5.52167 −0.355682 −0.177841 0.984059i \(-0.556911\pi\)
−0.177841 + 0.984059i \(0.556911\pi\)
\(242\) −12.6092 −0.810549
\(243\) 0 0
\(244\) 4.31792 0.276427
\(245\) −9.35226 −0.597494
\(246\) 0 0
\(247\) −2.00941 −0.127856
\(248\) −2.64673 −0.168068
\(249\) 0 0
\(250\) 16.5642 1.04761
\(251\) −21.4409 −1.35334 −0.676668 0.736288i \(-0.736577\pi\)
−0.676668 + 0.736288i \(0.736577\pi\)
\(252\) 0 0
\(253\) 10.9005 0.685306
\(254\) 1.79766 0.112795
\(255\) 0 0
\(256\) 16.9007 1.05629
\(257\) −14.7865 −0.922355 −0.461177 0.887308i \(-0.652573\pi\)
−0.461177 + 0.887308i \(0.652573\pi\)
\(258\) 0 0
\(259\) −13.2751 −0.824877
\(260\) 0.683074 0.0423625
\(261\) 0 0
\(262\) 12.8427 0.793423
\(263\) 2.80276 0.172825 0.0864127 0.996259i \(-0.472460\pi\)
0.0864127 + 0.996259i \(0.472460\pi\)
\(264\) 0 0
\(265\) −3.21349 −0.197403
\(266\) −18.0221 −1.10501
\(267\) 0 0
\(268\) −1.55933 −0.0952511
\(269\) −0.356528 −0.0217379 −0.0108689 0.999941i \(-0.503460\pi\)
−0.0108689 + 0.999941i \(0.503460\pi\)
\(270\) 0 0
\(271\) −12.1467 −0.737857 −0.368928 0.929458i \(-0.620275\pi\)
−0.368928 + 0.929458i \(0.620275\pi\)
\(272\) −9.39065 −0.569392
\(273\) 0 0
\(274\) 26.3071 1.58927
\(275\) 6.96582 0.420055
\(276\) 0 0
\(277\) 24.9900 1.50150 0.750751 0.660586i \(-0.229692\pi\)
0.750751 + 0.660586i \(0.229692\pi\)
\(278\) 14.5515 0.872744
\(279\) 0 0
\(280\) −8.71708 −0.520945
\(281\) 7.32287 0.436846 0.218423 0.975854i \(-0.429909\pi\)
0.218423 + 0.975854i \(0.429909\pi\)
\(282\) 0 0
\(283\) −14.3881 −0.855283 −0.427641 0.903948i \(-0.640655\pi\)
−0.427641 + 0.903948i \(0.640655\pi\)
\(284\) −10.0574 −0.596795
\(285\) 0 0
\(286\) 2.30334 0.136199
\(287\) −7.02172 −0.414479
\(288\) 0 0
\(289\) −13.4292 −0.789956
\(290\) 10.1008 0.593138
\(291\) 0 0
\(292\) 8.16194 0.477641
\(293\) −14.4155 −0.842163 −0.421082 0.907023i \(-0.638349\pi\)
−0.421082 + 0.907023i \(0.638349\pi\)
\(294\) 0 0
\(295\) 12.7187 0.740512
\(296\) −6.70491 −0.389715
\(297\) 0 0
\(298\) −4.14147 −0.239909
\(299\) 4.26936 0.246904
\(300\) 0 0
\(301\) −19.6625 −1.13333
\(302\) −17.6721 −1.01692
\(303\) 0 0
\(304\) −13.6307 −0.781776
\(305\) −5.90853 −0.338321
\(306\) 0 0
\(307\) −30.4326 −1.73688 −0.868440 0.495795i \(-0.834877\pi\)
−0.868440 + 0.495795i \(0.834877\pi\)
\(308\) 6.03537 0.343897
\(309\) 0 0
\(310\) −2.54535 −0.144566
\(311\) −13.9956 −0.793618 −0.396809 0.917901i \(-0.629882\pi\)
−0.396809 + 0.917901i \(0.629882\pi\)
\(312\) 0 0
\(313\) 22.0876 1.24846 0.624231 0.781239i \(-0.285412\pi\)
0.624231 + 0.781239i \(0.285412\pi\)
\(314\) 0.605042 0.0341445
\(315\) 0 0
\(316\) 10.1969 0.573619
\(317\) 17.3887 0.976648 0.488324 0.872662i \(-0.337608\pi\)
0.488324 + 0.872662i \(0.337608\pi\)
\(318\) 0 0
\(319\) 9.95070 0.557132
\(320\) −2.86343 −0.160070
\(321\) 0 0
\(322\) 38.2912 2.13389
\(323\) 5.18302 0.288391
\(324\) 0 0
\(325\) 2.72829 0.151338
\(326\) 24.5725 1.36095
\(327\) 0 0
\(328\) −3.54648 −0.195822
\(329\) −6.67909 −0.368230
\(330\) 0 0
\(331\) 0.864147 0.0474978 0.0237489 0.999718i \(-0.492440\pi\)
0.0237489 + 0.999718i \(0.492440\pi\)
\(332\) −9.64405 −0.529286
\(333\) 0 0
\(334\) 3.65741 0.200124
\(335\) 2.13374 0.116579
\(336\) 0 0
\(337\) 0.418529 0.0227987 0.0113994 0.999935i \(-0.496371\pi\)
0.0113994 + 0.999935i \(0.496371\pi\)
\(338\) −20.9497 −1.13951
\(339\) 0 0
\(340\) −1.76190 −0.0955526
\(341\) −2.50753 −0.135791
\(342\) 0 0
\(343\) −5.00216 −0.270091
\(344\) −9.93098 −0.535442
\(345\) 0 0
\(346\) −29.5286 −1.58747
\(347\) 23.4097 1.25670 0.628351 0.777930i \(-0.283730\pi\)
0.628351 + 0.777930i \(0.283730\pi\)
\(348\) 0 0
\(349\) 21.1764 1.13355 0.566773 0.823874i \(-0.308192\pi\)
0.566773 + 0.823874i \(0.308192\pi\)
\(350\) 24.4696 1.30796
\(351\) 0 0
\(352\) 8.23894 0.439137
\(353\) 23.5445 1.25315 0.626573 0.779363i \(-0.284457\pi\)
0.626573 + 0.779363i \(0.284457\pi\)
\(354\) 0 0
\(355\) 13.7622 0.730423
\(356\) −4.72887 −0.250630
\(357\) 0 0
\(358\) 1.68843 0.0892362
\(359\) −10.4609 −0.552107 −0.276053 0.961142i \(-0.589027\pi\)
−0.276053 + 0.961142i \(0.589027\pi\)
\(360\) 0 0
\(361\) −11.4767 −0.604039
\(362\) −35.5904 −1.87059
\(363\) 0 0
\(364\) 2.36386 0.123900
\(365\) −11.1686 −0.584590
\(366\) 0 0
\(367\) 21.2583 1.10967 0.554836 0.831959i \(-0.312781\pi\)
0.554836 + 0.831959i \(0.312781\pi\)
\(368\) 28.9610 1.50970
\(369\) 0 0
\(370\) −6.44808 −0.335220
\(371\) −11.1207 −0.577356
\(372\) 0 0
\(373\) 2.31009 0.119612 0.0598059 0.998210i \(-0.480952\pi\)
0.0598059 + 0.998210i \(0.480952\pi\)
\(374\) −5.94118 −0.307211
\(375\) 0 0
\(376\) −3.37342 −0.173971
\(377\) 3.89737 0.200725
\(378\) 0 0
\(379\) 12.5539 0.644850 0.322425 0.946595i \(-0.395502\pi\)
0.322425 + 0.946595i \(0.395502\pi\)
\(380\) −2.55744 −0.131194
\(381\) 0 0
\(382\) −16.5206 −0.845270
\(383\) −19.6545 −1.00430 −0.502150 0.864781i \(-0.667457\pi\)
−0.502150 + 0.864781i \(0.667457\pi\)
\(384\) 0 0
\(385\) −8.25862 −0.420898
\(386\) −18.7524 −0.954472
\(387\) 0 0
\(388\) −0.283447 −0.0143898
\(389\) −18.8426 −0.955357 −0.477679 0.878535i \(-0.658522\pi\)
−0.477679 + 0.878535i \(0.658522\pi\)
\(390\) 0 0
\(391\) −11.0123 −0.556915
\(392\) −16.3465 −0.825622
\(393\) 0 0
\(394\) 15.2794 0.769764
\(395\) −13.9531 −0.702058
\(396\) 0 0
\(397\) 20.1178 1.00968 0.504841 0.863212i \(-0.331551\pi\)
0.504841 + 0.863212i \(0.331551\pi\)
\(398\) −24.6991 −1.23806
\(399\) 0 0
\(400\) 18.5072 0.925360
\(401\) −30.8057 −1.53836 −0.769181 0.639031i \(-0.779336\pi\)
−0.769181 + 0.639031i \(0.779336\pi\)
\(402\) 0 0
\(403\) −0.982121 −0.0489230
\(404\) −14.3650 −0.714684
\(405\) 0 0
\(406\) 34.9549 1.73478
\(407\) −6.35228 −0.314871
\(408\) 0 0
\(409\) 39.6921 1.96265 0.981323 0.192367i \(-0.0616164\pi\)
0.981323 + 0.192367i \(0.0616164\pi\)
\(410\) −3.41063 −0.168439
\(411\) 0 0
\(412\) −13.0531 −0.643080
\(413\) 44.0146 2.16582
\(414\) 0 0
\(415\) 13.1966 0.647798
\(416\) 3.22693 0.158213
\(417\) 0 0
\(418\) −8.62374 −0.421801
\(419\) 15.4411 0.754349 0.377174 0.926142i \(-0.376896\pi\)
0.377174 + 0.926142i \(0.376896\pi\)
\(420\) 0 0
\(421\) −18.0936 −0.881830 −0.440915 0.897549i \(-0.645346\pi\)
−0.440915 + 0.897549i \(0.645346\pi\)
\(422\) 13.2091 0.643009
\(423\) 0 0
\(424\) −5.61674 −0.272773
\(425\) −7.03728 −0.341358
\(426\) 0 0
\(427\) −20.4472 −0.989507
\(428\) −13.6448 −0.659546
\(429\) 0 0
\(430\) −9.55057 −0.460569
\(431\) −28.9683 −1.39535 −0.697677 0.716412i \(-0.745783\pi\)
−0.697677 + 0.716412i \(0.745783\pi\)
\(432\) 0 0
\(433\) −37.5902 −1.80647 −0.903235 0.429146i \(-0.858815\pi\)
−0.903235 + 0.429146i \(0.858815\pi\)
\(434\) −8.80849 −0.422821
\(435\) 0 0
\(436\) −3.89491 −0.186532
\(437\) −15.9845 −0.764644
\(438\) 0 0
\(439\) 10.2635 0.489852 0.244926 0.969542i \(-0.421236\pi\)
0.244926 + 0.969542i \(0.421236\pi\)
\(440\) −4.17120 −0.198854
\(441\) 0 0
\(442\) −2.32697 −0.110683
\(443\) 21.9085 1.04090 0.520452 0.853891i \(-0.325763\pi\)
0.520452 + 0.853891i \(0.325763\pi\)
\(444\) 0 0
\(445\) 6.47086 0.306748
\(446\) −14.6927 −0.695718
\(447\) 0 0
\(448\) −9.90924 −0.468167
\(449\) 9.97130 0.470575 0.235287 0.971926i \(-0.424397\pi\)
0.235287 + 0.971926i \(0.424397\pi\)
\(450\) 0 0
\(451\) −3.35996 −0.158214
\(452\) 16.4638 0.774390
\(453\) 0 0
\(454\) 6.83865 0.320954
\(455\) −3.23464 −0.151642
\(456\) 0 0
\(457\) −7.61810 −0.356360 −0.178180 0.983998i \(-0.557021\pi\)
−0.178180 + 0.983998i \(0.557021\pi\)
\(458\) 27.1566 1.26894
\(459\) 0 0
\(460\) 5.43375 0.253350
\(461\) −22.8005 −1.06193 −0.530963 0.847395i \(-0.678170\pi\)
−0.530963 + 0.847395i \(0.678170\pi\)
\(462\) 0 0
\(463\) −8.79573 −0.408772 −0.204386 0.978890i \(-0.565520\pi\)
−0.204386 + 0.978890i \(0.565520\pi\)
\(464\) 26.4376 1.22734
\(465\) 0 0
\(466\) 28.9338 1.34033
\(467\) −10.9976 −0.508906 −0.254453 0.967085i \(-0.581895\pi\)
−0.254453 + 0.967085i \(0.581895\pi\)
\(468\) 0 0
\(469\) 7.38406 0.340964
\(470\) −3.24420 −0.149644
\(471\) 0 0
\(472\) 22.2306 1.02325
\(473\) −9.40868 −0.432612
\(474\) 0 0
\(475\) −10.2148 −0.468685
\(476\) −6.09728 −0.279468
\(477\) 0 0
\(478\) −2.58094 −0.118049
\(479\) 25.6925 1.17392 0.586961 0.809616i \(-0.300324\pi\)
0.586961 + 0.809616i \(0.300324\pi\)
\(480\) 0 0
\(481\) −2.48799 −0.113442
\(482\) −9.28145 −0.422758
\(483\) 0 0
\(484\) −6.19214 −0.281461
\(485\) 0.387861 0.0176119
\(486\) 0 0
\(487\) −30.3800 −1.37665 −0.688325 0.725402i \(-0.741654\pi\)
−0.688325 + 0.725402i \(0.741654\pi\)
\(488\) −10.3273 −0.467495
\(489\) 0 0
\(490\) −15.7203 −0.710172
\(491\) 8.70827 0.392999 0.196499 0.980504i \(-0.437043\pi\)
0.196499 + 0.980504i \(0.437043\pi\)
\(492\) 0 0
\(493\) −10.0528 −0.452754
\(494\) −3.37764 −0.151967
\(495\) 0 0
\(496\) −6.66217 −0.299140
\(497\) 47.6258 2.13631
\(498\) 0 0
\(499\) 11.6355 0.520875 0.260438 0.965491i \(-0.416133\pi\)
0.260438 + 0.965491i \(0.416133\pi\)
\(500\) 8.13438 0.363780
\(501\) 0 0
\(502\) −36.0402 −1.60855
\(503\) −37.7991 −1.68538 −0.842689 0.538400i \(-0.819029\pi\)
−0.842689 + 0.538400i \(0.819029\pi\)
\(504\) 0 0
\(505\) 19.6566 0.874708
\(506\) 18.3227 0.814544
\(507\) 0 0
\(508\) 0.882800 0.0391679
\(509\) 23.5736 1.04488 0.522442 0.852675i \(-0.325021\pi\)
0.522442 + 0.852675i \(0.325021\pi\)
\(510\) 0 0
\(511\) −38.6502 −1.70978
\(512\) 2.26711 0.100193
\(513\) 0 0
\(514\) −24.8548 −1.09630
\(515\) 17.8615 0.787071
\(516\) 0 0
\(517\) −3.19601 −0.140560
\(518\) −22.3143 −0.980436
\(519\) 0 0
\(520\) −1.63373 −0.0716437
\(521\) 7.86948 0.344768 0.172384 0.985030i \(-0.444853\pi\)
0.172384 + 0.985030i \(0.444853\pi\)
\(522\) 0 0
\(523\) 33.2935 1.45582 0.727911 0.685672i \(-0.240491\pi\)
0.727911 + 0.685672i \(0.240491\pi\)
\(524\) 6.30680 0.275514
\(525\) 0 0
\(526\) 4.71119 0.205418
\(527\) 2.53326 0.110350
\(528\) 0 0
\(529\) 10.9621 0.476612
\(530\) −5.40159 −0.234630
\(531\) 0 0
\(532\) −8.85032 −0.383710
\(533\) −1.31599 −0.0570018
\(534\) 0 0
\(535\) 18.6712 0.807225
\(536\) 3.72949 0.161089
\(537\) 0 0
\(538\) −0.599292 −0.0258373
\(539\) −15.4868 −0.667062
\(540\) 0 0
\(541\) 22.4283 0.964266 0.482133 0.876098i \(-0.339862\pi\)
0.482133 + 0.876098i \(0.339862\pi\)
\(542\) −20.4175 −0.877005
\(543\) 0 0
\(544\) −8.32346 −0.356865
\(545\) 5.32969 0.228299
\(546\) 0 0
\(547\) −20.8912 −0.893241 −0.446621 0.894723i \(-0.647373\pi\)
−0.446621 + 0.894723i \(0.647373\pi\)
\(548\) 12.9190 0.551870
\(549\) 0 0
\(550\) 11.7089 0.499271
\(551\) −14.5918 −0.621632
\(552\) 0 0
\(553\) −48.2864 −2.05335
\(554\) 42.0059 1.78466
\(555\) 0 0
\(556\) 7.14600 0.303058
\(557\) 8.57840 0.363478 0.181739 0.983347i \(-0.441827\pi\)
0.181739 + 0.983347i \(0.441827\pi\)
\(558\) 0 0
\(559\) −3.68508 −0.155862
\(560\) −21.9420 −0.927219
\(561\) 0 0
\(562\) 12.3091 0.519228
\(563\) −15.7149 −0.662303 −0.331152 0.943578i \(-0.607437\pi\)
−0.331152 + 0.943578i \(0.607437\pi\)
\(564\) 0 0
\(565\) −22.5285 −0.947783
\(566\) −24.1851 −1.01658
\(567\) 0 0
\(568\) 24.0545 1.00930
\(569\) 12.6728 0.531271 0.265635 0.964074i \(-0.414418\pi\)
0.265635 + 0.964074i \(0.414418\pi\)
\(570\) 0 0
\(571\) −26.3227 −1.10157 −0.550785 0.834647i \(-0.685672\pi\)
−0.550785 + 0.834647i \(0.685672\pi\)
\(572\) 1.13113 0.0472949
\(573\) 0 0
\(574\) −11.8029 −0.492644
\(575\) 21.7031 0.905082
\(576\) 0 0
\(577\) −22.1154 −0.920676 −0.460338 0.887744i \(-0.652272\pi\)
−0.460338 + 0.887744i \(0.652272\pi\)
\(578\) −22.5734 −0.938929
\(579\) 0 0
\(580\) 4.96031 0.205966
\(581\) 45.6686 1.89465
\(582\) 0 0
\(583\) −5.32134 −0.220387
\(584\) −19.5211 −0.807790
\(585\) 0 0
\(586\) −24.2312 −1.00098
\(587\) 14.3031 0.590353 0.295177 0.955443i \(-0.404621\pi\)
0.295177 + 0.955443i \(0.404621\pi\)
\(588\) 0 0
\(589\) 3.67708 0.151511
\(590\) 21.3790 0.880161
\(591\) 0 0
\(592\) −16.8771 −0.693645
\(593\) 47.7300 1.96004 0.980018 0.198908i \(-0.0637397\pi\)
0.980018 + 0.198908i \(0.0637397\pi\)
\(594\) 0 0
\(595\) 8.34334 0.342044
\(596\) −2.03380 −0.0833077
\(597\) 0 0
\(598\) 7.17642 0.293466
\(599\) −0.499570 −0.0204119 −0.0102059 0.999948i \(-0.503249\pi\)
−0.0102059 + 0.999948i \(0.503249\pi\)
\(600\) 0 0
\(601\) −16.9271 −0.690470 −0.345235 0.938516i \(-0.612201\pi\)
−0.345235 + 0.938516i \(0.612201\pi\)
\(602\) −33.0509 −1.34705
\(603\) 0 0
\(604\) −8.67846 −0.353121
\(605\) 8.47316 0.344483
\(606\) 0 0
\(607\) 0.694633 0.0281943 0.0140971 0.999901i \(-0.495513\pi\)
0.0140971 + 0.999901i \(0.495513\pi\)
\(608\) −12.0817 −0.489977
\(609\) 0 0
\(610\) −9.93171 −0.402123
\(611\) −1.25177 −0.0506413
\(612\) 0 0
\(613\) −32.6633 −1.31926 −0.659630 0.751590i \(-0.729287\pi\)
−0.659630 + 0.751590i \(0.729287\pi\)
\(614\) −51.1545 −2.06443
\(615\) 0 0
\(616\) −14.4349 −0.581601
\(617\) 22.8844 0.921292 0.460646 0.887584i \(-0.347618\pi\)
0.460646 + 0.887584i \(0.347618\pi\)
\(618\) 0 0
\(619\) 8.48574 0.341071 0.170535 0.985352i \(-0.445450\pi\)
0.170535 + 0.985352i \(0.445450\pi\)
\(620\) −1.24998 −0.0502002
\(621\) 0 0
\(622\) −23.5254 −0.943282
\(623\) 22.3932 0.897163
\(624\) 0 0
\(625\) 7.48980 0.299592
\(626\) 37.1273 1.48390
\(627\) 0 0
\(628\) 0.297125 0.0118566
\(629\) 6.41744 0.255880
\(630\) 0 0
\(631\) 1.59173 0.0633657 0.0316829 0.999498i \(-0.489913\pi\)
0.0316829 + 0.999498i \(0.489913\pi\)
\(632\) −24.3881 −0.970108
\(633\) 0 0
\(634\) 29.2289 1.16083
\(635\) −1.20800 −0.0479379
\(636\) 0 0
\(637\) −6.06567 −0.240331
\(638\) 16.7263 0.662199
\(639\) 0 0
\(640\) −14.7640 −0.583597
\(641\) 21.6747 0.856098 0.428049 0.903756i \(-0.359201\pi\)
0.428049 + 0.903756i \(0.359201\pi\)
\(642\) 0 0
\(643\) −6.90206 −0.272191 −0.136095 0.990696i \(-0.543455\pi\)
−0.136095 + 0.990696i \(0.543455\pi\)
\(644\) 18.8041 0.740987
\(645\) 0 0
\(646\) 8.71220 0.342777
\(647\) −6.18972 −0.243343 −0.121671 0.992570i \(-0.538825\pi\)
−0.121671 + 0.992570i \(0.538825\pi\)
\(648\) 0 0
\(649\) 21.0614 0.826733
\(650\) 4.58602 0.179878
\(651\) 0 0
\(652\) 12.0671 0.472585
\(653\) −27.2145 −1.06498 −0.532492 0.846435i \(-0.678744\pi\)
−0.532492 + 0.846435i \(0.678744\pi\)
\(654\) 0 0
\(655\) −8.63005 −0.337204
\(656\) −8.92694 −0.348539
\(657\) 0 0
\(658\) −11.2270 −0.437672
\(659\) 5.28200 0.205757 0.102879 0.994694i \(-0.467195\pi\)
0.102879 + 0.994694i \(0.467195\pi\)
\(660\) 0 0
\(661\) 11.6650 0.453717 0.226859 0.973928i \(-0.427154\pi\)
0.226859 + 0.973928i \(0.427154\pi\)
\(662\) 1.45256 0.0564552
\(663\) 0 0
\(664\) 23.0659 0.895132
\(665\) 12.1105 0.469626
\(666\) 0 0
\(667\) 31.0030 1.20044
\(668\) 1.79609 0.0694927
\(669\) 0 0
\(670\) 3.58663 0.138564
\(671\) −9.78416 −0.377713
\(672\) 0 0
\(673\) 49.1024 1.89276 0.946380 0.323057i \(-0.104710\pi\)
0.946380 + 0.323057i \(0.104710\pi\)
\(674\) 0.703510 0.0270982
\(675\) 0 0
\(676\) −10.2880 −0.395693
\(677\) −22.6190 −0.869321 −0.434660 0.900594i \(-0.643132\pi\)
−0.434660 + 0.900594i \(0.643132\pi\)
\(678\) 0 0
\(679\) 1.34224 0.0515104
\(680\) 4.21399 0.161599
\(681\) 0 0
\(682\) −4.21495 −0.161399
\(683\) −17.1386 −0.655791 −0.327896 0.944714i \(-0.606339\pi\)
−0.327896 + 0.944714i \(0.606339\pi\)
\(684\) 0 0
\(685\) −17.6779 −0.675439
\(686\) −8.40818 −0.321026
\(687\) 0 0
\(688\) −24.9975 −0.953022
\(689\) −2.08420 −0.0794017
\(690\) 0 0
\(691\) 18.2559 0.694488 0.347244 0.937775i \(-0.387118\pi\)
0.347244 + 0.937775i \(0.387118\pi\)
\(692\) −14.5010 −0.551244
\(693\) 0 0
\(694\) 39.3497 1.49370
\(695\) −9.77839 −0.370915
\(696\) 0 0
\(697\) 3.39443 0.128573
\(698\) 35.5957 1.34732
\(699\) 0 0
\(700\) 12.0166 0.454184
\(701\) −2.92075 −0.110315 −0.0551575 0.998478i \(-0.517566\pi\)
−0.0551575 + 0.998478i \(0.517566\pi\)
\(702\) 0 0
\(703\) 9.31505 0.351324
\(704\) −4.74166 −0.178708
\(705\) 0 0
\(706\) 39.5762 1.48947
\(707\) 68.0241 2.55831
\(708\) 0 0
\(709\) 30.0273 1.12770 0.563850 0.825877i \(-0.309320\pi\)
0.563850 + 0.825877i \(0.309320\pi\)
\(710\) 23.1331 0.868169
\(711\) 0 0
\(712\) 11.3102 0.423867
\(713\) −7.81262 −0.292585
\(714\) 0 0
\(715\) −1.54781 −0.0578846
\(716\) 0.829156 0.0309870
\(717\) 0 0
\(718\) −17.5839 −0.656226
\(719\) 40.0569 1.49387 0.746936 0.664896i \(-0.231524\pi\)
0.746936 + 0.664896i \(0.231524\pi\)
\(720\) 0 0
\(721\) 61.8118 2.30199
\(722\) −19.2914 −0.717951
\(723\) 0 0
\(724\) −17.4778 −0.649557
\(725\) 19.8121 0.735804
\(726\) 0 0
\(727\) −1.53878 −0.0570702 −0.0285351 0.999593i \(-0.509084\pi\)
−0.0285351 + 0.999593i \(0.509084\pi\)
\(728\) −5.65371 −0.209540
\(729\) 0 0
\(730\) −18.7734 −0.694834
\(731\) 9.50520 0.351562
\(732\) 0 0
\(733\) −1.51189 −0.0558431 −0.0279215 0.999610i \(-0.508889\pi\)
−0.0279215 + 0.999610i \(0.508889\pi\)
\(734\) 35.7333 1.31894
\(735\) 0 0
\(736\) 25.6697 0.946199
\(737\) 3.53334 0.130152
\(738\) 0 0
\(739\) 21.3558 0.785584 0.392792 0.919627i \(-0.371509\pi\)
0.392792 + 0.919627i \(0.371509\pi\)
\(740\) −3.16654 −0.116404
\(741\) 0 0
\(742\) −18.6929 −0.686236
\(743\) −20.3287 −0.745789 −0.372895 0.927874i \(-0.621635\pi\)
−0.372895 + 0.927874i \(0.621635\pi\)
\(744\) 0 0
\(745\) 2.78300 0.101961
\(746\) 3.88305 0.142169
\(747\) 0 0
\(748\) −2.91760 −0.106678
\(749\) 64.6138 2.36094
\(750\) 0 0
\(751\) 4.26316 0.155565 0.0777824 0.996970i \(-0.475216\pi\)
0.0777824 + 0.996970i \(0.475216\pi\)
\(752\) −8.49134 −0.309647
\(753\) 0 0
\(754\) 6.55114 0.238579
\(755\) 11.8754 0.432189
\(756\) 0 0
\(757\) 54.3419 1.97509 0.987546 0.157332i \(-0.0502892\pi\)
0.987546 + 0.157332i \(0.0502892\pi\)
\(758\) 21.1020 0.766458
\(759\) 0 0
\(760\) 6.11670 0.221876
\(761\) 18.1611 0.658340 0.329170 0.944271i \(-0.393231\pi\)
0.329170 + 0.944271i \(0.393231\pi\)
\(762\) 0 0
\(763\) 18.4440 0.667718
\(764\) −8.11299 −0.293518
\(765\) 0 0
\(766\) −33.0375 −1.19369
\(767\) 8.24909 0.297857
\(768\) 0 0
\(769\) 24.8547 0.896285 0.448142 0.893962i \(-0.352086\pi\)
0.448142 + 0.893962i \(0.352086\pi\)
\(770\) −13.8820 −0.500273
\(771\) 0 0
\(772\) −9.20897 −0.331438
\(773\) 38.4832 1.38414 0.692071 0.721829i \(-0.256698\pi\)
0.692071 + 0.721829i \(0.256698\pi\)
\(774\) 0 0
\(775\) −4.99257 −0.179338
\(776\) 0.677928 0.0243362
\(777\) 0 0
\(778\) −31.6727 −1.13552
\(779\) 4.92708 0.176531
\(780\) 0 0
\(781\) 22.7894 0.815468
\(782\) −18.5107 −0.661940
\(783\) 0 0
\(784\) −41.1462 −1.46951
\(785\) −0.406578 −0.0145114
\(786\) 0 0
\(787\) −1.07248 −0.0382297 −0.0191149 0.999817i \(-0.506085\pi\)
−0.0191149 + 0.999817i \(0.506085\pi\)
\(788\) 7.50343 0.267299
\(789\) 0 0
\(790\) −23.4540 −0.834455
\(791\) −77.9627 −2.77204
\(792\) 0 0
\(793\) −3.83214 −0.136083
\(794\) 33.8162 1.20009
\(795\) 0 0
\(796\) −12.1293 −0.429912
\(797\) 25.2472 0.894302 0.447151 0.894458i \(-0.352439\pi\)
0.447151 + 0.894458i \(0.352439\pi\)
\(798\) 0 0
\(799\) 3.22879 0.114226
\(800\) 16.4040 0.579968
\(801\) 0 0
\(802\) −51.7817 −1.82847
\(803\) −18.4945 −0.652656
\(804\) 0 0
\(805\) −25.7310 −0.906900
\(806\) −1.65086 −0.0581491
\(807\) 0 0
\(808\) 34.3571 1.20868
\(809\) −11.7337 −0.412536 −0.206268 0.978495i \(-0.566132\pi\)
−0.206268 + 0.978495i \(0.566132\pi\)
\(810\) 0 0
\(811\) −38.4085 −1.34871 −0.674353 0.738409i \(-0.735577\pi\)
−0.674353 + 0.738409i \(0.735577\pi\)
\(812\) 17.1657 0.602399
\(813\) 0 0
\(814\) −10.6776 −0.374251
\(815\) −16.5123 −0.578401
\(816\) 0 0
\(817\) 13.7970 0.482695
\(818\) 66.7189 2.33277
\(819\) 0 0
\(820\) −1.67490 −0.0584901
\(821\) 26.7321 0.932955 0.466477 0.884533i \(-0.345523\pi\)
0.466477 + 0.884533i \(0.345523\pi\)
\(822\) 0 0
\(823\) 4.36426 0.152129 0.0760643 0.997103i \(-0.475765\pi\)
0.0760643 + 0.997103i \(0.475765\pi\)
\(824\) 31.2194 1.08758
\(825\) 0 0
\(826\) 73.9848 2.57426
\(827\) −39.1454 −1.36122 −0.680610 0.732646i \(-0.738285\pi\)
−0.680610 + 0.732646i \(0.738285\pi\)
\(828\) 0 0
\(829\) −32.8867 −1.14220 −0.571101 0.820880i \(-0.693484\pi\)
−0.571101 + 0.820880i \(0.693484\pi\)
\(830\) 22.1824 0.769963
\(831\) 0 0
\(832\) −1.85716 −0.0643854
\(833\) 15.6456 0.542089
\(834\) 0 0
\(835\) −2.45772 −0.0850527
\(836\) −4.23496 −0.146469
\(837\) 0 0
\(838\) 25.9552 0.896607
\(839\) 31.6403 1.09235 0.546173 0.837673i \(-0.316084\pi\)
0.546173 + 0.837673i \(0.316084\pi\)
\(840\) 0 0
\(841\) −0.698289 −0.0240789
\(842\) −30.4138 −1.04813
\(843\) 0 0
\(844\) 6.48676 0.223283
\(845\) 14.0778 0.484293
\(846\) 0 0
\(847\) 29.3224 1.00753
\(848\) −14.1380 −0.485503
\(849\) 0 0
\(850\) −11.8290 −0.405733
\(851\) −19.7915 −0.678445
\(852\) 0 0
\(853\) −5.65765 −0.193714 −0.0968571 0.995298i \(-0.530879\pi\)
−0.0968571 + 0.995298i \(0.530879\pi\)
\(854\) −34.3699 −1.17611
\(855\) 0 0
\(856\) 32.6346 1.11543
\(857\) −41.2104 −1.40772 −0.703859 0.710339i \(-0.748542\pi\)
−0.703859 + 0.710339i \(0.748542\pi\)
\(858\) 0 0
\(859\) 35.8544 1.22334 0.611668 0.791114i \(-0.290499\pi\)
0.611668 + 0.791114i \(0.290499\pi\)
\(860\) −4.69011 −0.159932
\(861\) 0 0
\(862\) −48.6932 −1.65850
\(863\) 20.9694 0.713806 0.356903 0.934142i \(-0.383833\pi\)
0.356903 + 0.934142i \(0.383833\pi\)
\(864\) 0 0
\(865\) 19.8427 0.674673
\(866\) −63.1858 −2.14714
\(867\) 0 0
\(868\) −4.32569 −0.146824
\(869\) −23.1055 −0.783801
\(870\) 0 0
\(871\) 1.38390 0.0468916
\(872\) 9.31556 0.315465
\(873\) 0 0
\(874\) −26.8686 −0.908844
\(875\) −38.5197 −1.30220
\(876\) 0 0
\(877\) 20.5104 0.692588 0.346294 0.938126i \(-0.387440\pi\)
0.346294 + 0.938126i \(0.387440\pi\)
\(878\) 17.2521 0.582230
\(879\) 0 0
\(880\) −10.4995 −0.353936
\(881\) −38.2877 −1.28994 −0.644972 0.764206i \(-0.723131\pi\)
−0.644972 + 0.764206i \(0.723131\pi\)
\(882\) 0 0
\(883\) −17.4465 −0.587122 −0.293561 0.955940i \(-0.594840\pi\)
−0.293561 + 0.955940i \(0.594840\pi\)
\(884\) −1.14273 −0.0384343
\(885\) 0 0
\(886\) 36.8263 1.23720
\(887\) −54.2169 −1.82043 −0.910213 0.414141i \(-0.864082\pi\)
−0.910213 + 0.414141i \(0.864082\pi\)
\(888\) 0 0
\(889\) −4.18042 −0.140207
\(890\) 10.8769 0.364596
\(891\) 0 0
\(892\) −7.21530 −0.241586
\(893\) 4.68666 0.156833
\(894\) 0 0
\(895\) −1.13459 −0.0379253
\(896\) −51.0924 −1.70688
\(897\) 0 0
\(898\) 16.7609 0.559318
\(899\) −7.13191 −0.237862
\(900\) 0 0
\(901\) 5.37593 0.179098
\(902\) −5.64780 −0.188051
\(903\) 0 0
\(904\) −39.3768 −1.30965
\(905\) 23.9161 0.794999
\(906\) 0 0
\(907\) 28.6832 0.952409 0.476204 0.879335i \(-0.342012\pi\)
0.476204 + 0.879335i \(0.342012\pi\)
\(908\) 3.35834 0.111450
\(909\) 0 0
\(910\) −5.43715 −0.180240
\(911\) −28.3816 −0.940324 −0.470162 0.882580i \(-0.655805\pi\)
−0.470162 + 0.882580i \(0.655805\pi\)
\(912\) 0 0
\(913\) 21.8528 0.723224
\(914\) −12.8054 −0.423564
\(915\) 0 0
\(916\) 13.3361 0.440638
\(917\) −29.8653 −0.986240
\(918\) 0 0
\(919\) 37.7786 1.24620 0.623101 0.782141i \(-0.285873\pi\)
0.623101 + 0.782141i \(0.285873\pi\)
\(920\) −12.9960 −0.428467
\(921\) 0 0
\(922\) −38.3257 −1.26219
\(923\) 8.92588 0.293799
\(924\) 0 0
\(925\) −12.6476 −0.415849
\(926\) −14.7849 −0.485861
\(927\) 0 0
\(928\) 23.4331 0.769230
\(929\) −34.5769 −1.13443 −0.567215 0.823570i \(-0.691979\pi\)
−0.567215 + 0.823570i \(0.691979\pi\)
\(930\) 0 0
\(931\) 22.7100 0.744289
\(932\) 14.2089 0.465427
\(933\) 0 0
\(934\) −18.4859 −0.604878
\(935\) 3.99237 0.130564
\(936\) 0 0
\(937\) −19.4368 −0.634972 −0.317486 0.948263i \(-0.602839\pi\)
−0.317486 + 0.948263i \(0.602839\pi\)
\(938\) 12.4120 0.405265
\(939\) 0 0
\(940\) −1.59317 −0.0519635
\(941\) 9.93846 0.323984 0.161992 0.986792i \(-0.448208\pi\)
0.161992 + 0.986792i \(0.448208\pi\)
\(942\) 0 0
\(943\) −10.4685 −0.340901
\(944\) 55.9572 1.82125
\(945\) 0 0
\(946\) −15.8152 −0.514195
\(947\) 16.9107 0.549524 0.274762 0.961512i \(-0.411401\pi\)
0.274762 + 0.961512i \(0.411401\pi\)
\(948\) 0 0
\(949\) −7.24369 −0.235140
\(950\) −17.1701 −0.557072
\(951\) 0 0
\(952\) 14.5830 0.472639
\(953\) 17.3573 0.562259 0.281129 0.959670i \(-0.409291\pi\)
0.281129 + 0.959670i \(0.409291\pi\)
\(954\) 0 0
\(955\) 11.1016 0.359239
\(956\) −1.26745 −0.0409923
\(957\) 0 0
\(958\) 43.1869 1.39530
\(959\) −61.1766 −1.97550
\(960\) 0 0
\(961\) −29.2028 −0.942025
\(962\) −4.18208 −0.134836
\(963\) 0 0
\(964\) −4.55795 −0.146802
\(965\) 12.6013 0.405650
\(966\) 0 0
\(967\) −16.7834 −0.539716 −0.269858 0.962900i \(-0.586977\pi\)
−0.269858 + 0.962900i \(0.586977\pi\)
\(968\) 14.8099 0.476009
\(969\) 0 0
\(970\) 0.651960 0.0209332
\(971\) −33.4811 −1.07446 −0.537230 0.843436i \(-0.680529\pi\)
−0.537230 + 0.843436i \(0.680529\pi\)
\(972\) 0 0
\(973\) −33.8393 −1.08484
\(974\) −51.0662 −1.63627
\(975\) 0 0
\(976\) −25.9951 −0.832084
\(977\) 22.7348 0.727352 0.363676 0.931526i \(-0.381522\pi\)
0.363676 + 0.931526i \(0.381522\pi\)
\(978\) 0 0
\(979\) 10.7153 0.342464
\(980\) −7.71997 −0.246605
\(981\) 0 0
\(982\) 14.6378 0.467112
\(983\) 47.1596 1.50416 0.752078 0.659074i \(-0.229051\pi\)
0.752078 + 0.659074i \(0.229051\pi\)
\(984\) 0 0
\(985\) −10.2675 −0.327149
\(986\) −16.8978 −0.538137
\(987\) 0 0
\(988\) −1.65870 −0.0527703
\(989\) −29.3142 −0.932138
\(990\) 0 0
\(991\) −4.37673 −0.139032 −0.0695158 0.997581i \(-0.522145\pi\)
−0.0695158 + 0.997581i \(0.522145\pi\)
\(992\) −5.90505 −0.187486
\(993\) 0 0
\(994\) 80.0548 2.53918
\(995\) 16.5974 0.526173
\(996\) 0 0
\(997\) −2.14450 −0.0679170 −0.0339585 0.999423i \(-0.510811\pi\)
−0.0339585 + 0.999423i \(0.510811\pi\)
\(998\) 19.5582 0.619104
\(999\) 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.a.1.6 6
3.2 odd 2 729.2.a.d.1.1 6
9.2 odd 6 729.2.c.b.244.6 12
9.4 even 3 729.2.c.e.487.1 12
9.5 odd 6 729.2.c.b.487.6 12
9.7 even 3 729.2.c.e.244.1 12
27.2 odd 18 81.2.e.a.64.2 12
27.4 even 9 243.2.e.d.136.2 12
27.5 odd 18 243.2.e.b.217.1 12
27.7 even 9 243.2.e.d.109.2 12
27.11 odd 18 243.2.e.b.28.1 12
27.13 even 9 27.2.e.a.7.1 yes 12
27.14 odd 18 81.2.e.a.19.2 12
27.16 even 9 243.2.e.c.28.2 12
27.20 odd 18 243.2.e.a.109.1 12
27.22 even 9 243.2.e.c.217.2 12
27.23 odd 18 243.2.e.a.136.1 12
27.25 even 9 27.2.e.a.4.1 12
108.67 odd 18 432.2.u.c.385.1 12
108.79 odd 18 432.2.u.c.193.1 12
135.13 odd 36 675.2.u.b.574.4 24
135.52 odd 36 675.2.u.b.274.4 24
135.67 odd 36 675.2.u.b.574.1 24
135.79 even 18 675.2.l.c.301.2 12
135.94 even 18 675.2.l.c.601.2 12
135.133 odd 36 675.2.u.b.274.1 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.2.e.a.4.1 12 27.25 even 9
27.2.e.a.7.1 yes 12 27.13 even 9
81.2.e.a.19.2 12 27.14 odd 18
81.2.e.a.64.2 12 27.2 odd 18
243.2.e.a.109.1 12 27.20 odd 18
243.2.e.a.136.1 12 27.23 odd 18
243.2.e.b.28.1 12 27.11 odd 18
243.2.e.b.217.1 12 27.5 odd 18
243.2.e.c.28.2 12 27.16 even 9
243.2.e.c.217.2 12 27.22 even 9
243.2.e.d.109.2 12 27.7 even 9
243.2.e.d.136.2 12 27.4 even 9
432.2.u.c.193.1 12 108.79 odd 18
432.2.u.c.385.1 12 108.67 odd 18
675.2.l.c.301.2 12 135.79 even 18
675.2.l.c.601.2 12 135.94 even 18
675.2.u.b.274.1 24 135.133 odd 36
675.2.u.b.274.4 24 135.52 odd 36
675.2.u.b.574.1 24 135.67 odd 36
675.2.u.b.574.4 24 135.13 odd 36
729.2.a.a.1.6 6 1.1 even 1 trivial
729.2.a.d.1.1 6 3.2 odd 2
729.2.c.b.244.6 12 9.2 odd 6
729.2.c.b.487.6 12 9.5 odd 6
729.2.c.e.244.1 12 9.7 even 3
729.2.c.e.487.1 12 9.4 even 3