Properties

Label 729.2.a.a.1.3
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 1.3
Root \(2.68091\) of defining polynomial
Character \(\chi\) \(=\) 729.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.801527 q^{2} -1.35755 q^{4} -2.74984 q^{5} +2.37683 q^{7} +2.69117 q^{8} +O(q^{10})\) \(q-0.801527 q^{2} -1.35755 q^{4} -2.74984 q^{5} +2.37683 q^{7} +2.69117 q^{8} +2.20407 q^{10} -0.250159 q^{11} +2.61198 q^{13} -1.90510 q^{14} +0.558064 q^{16} -0.293377 q^{17} -2.78475 q^{19} +3.73306 q^{20} +0.200509 q^{22} -6.68984 q^{23} +2.56163 q^{25} -2.09357 q^{26} -3.22668 q^{28} -0.355057 q^{29} +2.76547 q^{31} -5.82964 q^{32} +0.235149 q^{34} -6.53592 q^{35} -6.99238 q^{37} +2.23205 q^{38} -7.40029 q^{40} -9.71761 q^{41} +0.260706 q^{43} +0.339604 q^{44} +5.36209 q^{46} -11.4256 q^{47} -1.35066 q^{49} -2.05321 q^{50} -3.54591 q^{52} +5.43137 q^{53} +0.687897 q^{55} +6.39646 q^{56} +0.284588 q^{58} +5.97693 q^{59} -11.8468 q^{61} -2.21660 q^{62} +3.55649 q^{64} -7.18254 q^{65} +1.81030 q^{67} +0.398275 q^{68} +5.23871 q^{70} -0.370510 q^{71} +5.02679 q^{73} +5.60458 q^{74} +3.78044 q^{76} -0.594586 q^{77} +0.802822 q^{79} -1.53459 q^{80} +7.78892 q^{82} -2.75565 q^{83} +0.806740 q^{85} -0.208963 q^{86} -0.673220 q^{88} +10.4507 q^{89} +6.20825 q^{91} +9.08183 q^{92} +9.15789 q^{94} +7.65761 q^{95} -14.8346 q^{97} +1.08259 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{2} + 3 q^{4} - 6 q^{5} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{2} + 3 q^{4} - 6 q^{5} - 6 q^{8} + 3 q^{10} - 12 q^{11} - 6 q^{14} - 3 q^{16} - 9 q^{17} + 3 q^{19} - 6 q^{20} + 6 q^{22} - 15 q^{23} - 6 q^{25} - 15 q^{26} - 6 q^{28} - 12 q^{29} - 12 q^{35} + 3 q^{37} + 3 q^{38} + 6 q^{40} - 15 q^{41} - 3 q^{44} + 3 q^{46} - 21 q^{47} - 12 q^{49} - 3 q^{50} + 12 q^{52} - 9 q^{53} - 6 q^{55} + 6 q^{56} - 12 q^{58} - 24 q^{59} - 9 q^{61} + 12 q^{62} - 12 q^{64} + 6 q^{65} - 9 q^{67} + 9 q^{68} + 15 q^{70} - 27 q^{71} - 6 q^{73} + 12 q^{74} + 6 q^{76} + 12 q^{77} + 21 q^{80} - 6 q^{82} - 12 q^{83} + 21 q^{86} + 12 q^{88} - 9 q^{89} - 6 q^{91} - 6 q^{92} + 6 q^{94} - 12 q^{95} + 45 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.801527 −0.566765 −0.283383 0.959007i \(-0.591457\pi\)
−0.283383 + 0.959007i \(0.591457\pi\)
\(3\) 0 0
\(4\) −1.35755 −0.678777
\(5\) −2.74984 −1.22977 −0.614883 0.788618i \(-0.710797\pi\)
−0.614883 + 0.788618i \(0.710797\pi\)
\(6\) 0 0
\(7\) 2.37683 0.898359 0.449179 0.893442i \(-0.351716\pi\)
0.449179 + 0.893442i \(0.351716\pi\)
\(8\) 2.69117 0.951472
\(9\) 0 0
\(10\) 2.20407 0.696989
\(11\) −0.250159 −0.0754257 −0.0377129 0.999289i \(-0.512007\pi\)
−0.0377129 + 0.999289i \(0.512007\pi\)
\(12\) 0 0
\(13\) 2.61198 0.724434 0.362217 0.932094i \(-0.382020\pi\)
0.362217 + 0.932094i \(0.382020\pi\)
\(14\) −1.90510 −0.509158
\(15\) 0 0
\(16\) 0.558064 0.139516
\(17\) −0.293377 −0.0711543 −0.0355772 0.999367i \(-0.511327\pi\)
−0.0355772 + 0.999367i \(0.511327\pi\)
\(18\) 0 0
\(19\) −2.78475 −0.638864 −0.319432 0.947609i \(-0.603492\pi\)
−0.319432 + 0.947609i \(0.603492\pi\)
\(20\) 3.73306 0.834737
\(21\) 0 0
\(22\) 0.200509 0.0427487
\(23\) −6.68984 −1.39493 −0.697464 0.716620i \(-0.745688\pi\)
−0.697464 + 0.716620i \(0.745688\pi\)
\(24\) 0 0
\(25\) 2.56163 0.512325
\(26\) −2.09357 −0.410584
\(27\) 0 0
\(28\) −3.22668 −0.609786
\(29\) −0.355057 −0.0659324 −0.0329662 0.999456i \(-0.510495\pi\)
−0.0329662 + 0.999456i \(0.510495\pi\)
\(30\) 0 0
\(31\) 2.76547 0.496692 0.248346 0.968671i \(-0.420113\pi\)
0.248346 + 0.968671i \(0.420113\pi\)
\(32\) −5.82964 −1.03055
\(33\) 0 0
\(34\) 0.235149 0.0403278
\(35\) −6.53592 −1.10477
\(36\) 0 0
\(37\) −6.99238 −1.14954 −0.574770 0.818315i \(-0.694909\pi\)
−0.574770 + 0.818315i \(0.694909\pi\)
\(38\) 2.23205 0.362086
\(39\) 0 0
\(40\) −7.40029 −1.17009
\(41\) −9.71761 −1.51764 −0.758818 0.651303i \(-0.774223\pi\)
−0.758818 + 0.651303i \(0.774223\pi\)
\(42\) 0 0
\(43\) 0.260706 0.0397574 0.0198787 0.999802i \(-0.493672\pi\)
0.0198787 + 0.999802i \(0.493672\pi\)
\(44\) 0.339604 0.0511973
\(45\) 0 0
\(46\) 5.36209 0.790597
\(47\) −11.4256 −1.66659 −0.833295 0.552829i \(-0.813548\pi\)
−0.833295 + 0.552829i \(0.813548\pi\)
\(48\) 0 0
\(49\) −1.35066 −0.192952
\(50\) −2.05321 −0.290368
\(51\) 0 0
\(52\) −3.54591 −0.491729
\(53\) 5.43137 0.746056 0.373028 0.927820i \(-0.378320\pi\)
0.373028 + 0.927820i \(0.378320\pi\)
\(54\) 0 0
\(55\) 0.687897 0.0927560
\(56\) 6.39646 0.854764
\(57\) 0 0
\(58\) 0.284588 0.0373682
\(59\) 5.97693 0.778130 0.389065 0.921210i \(-0.372798\pi\)
0.389065 + 0.921210i \(0.372798\pi\)
\(60\) 0 0
\(61\) −11.8468 −1.51682 −0.758411 0.651776i \(-0.774024\pi\)
−0.758411 + 0.651776i \(0.774024\pi\)
\(62\) −2.21660 −0.281508
\(63\) 0 0
\(64\) 3.55649 0.444561
\(65\) −7.18254 −0.890884
\(66\) 0 0
\(67\) 1.81030 0.221164 0.110582 0.993867i \(-0.464729\pi\)
0.110582 + 0.993867i \(0.464729\pi\)
\(68\) 0.398275 0.0482979
\(69\) 0 0
\(70\) 5.23871 0.626146
\(71\) −0.370510 −0.0439714 −0.0219857 0.999758i \(-0.506999\pi\)
−0.0219857 + 0.999758i \(0.506999\pi\)
\(72\) 0 0
\(73\) 5.02679 0.588341 0.294171 0.955753i \(-0.404957\pi\)
0.294171 + 0.955753i \(0.404957\pi\)
\(74\) 5.60458 0.651519
\(75\) 0 0
\(76\) 3.78044 0.433647
\(77\) −0.594586 −0.0677594
\(78\) 0 0
\(79\) 0.802822 0.0903245 0.0451622 0.998980i \(-0.485620\pi\)
0.0451622 + 0.998980i \(0.485620\pi\)
\(80\) −1.53459 −0.171572
\(81\) 0 0
\(82\) 7.78892 0.860143
\(83\) −2.75565 −0.302472 −0.151236 0.988498i \(-0.548325\pi\)
−0.151236 + 0.988498i \(0.548325\pi\)
\(84\) 0 0
\(85\) 0.806740 0.0875032
\(86\) −0.208963 −0.0225331
\(87\) 0 0
\(88\) −0.673220 −0.0717655
\(89\) 10.4507 1.10777 0.553884 0.832594i \(-0.313145\pi\)
0.553884 + 0.832594i \(0.313145\pi\)
\(90\) 0 0
\(91\) 6.20825 0.650801
\(92\) 9.08183 0.946846
\(93\) 0 0
\(94\) 9.15789 0.944565
\(95\) 7.65761 0.785654
\(96\) 0 0
\(97\) −14.8346 −1.50623 −0.753113 0.657891i \(-0.771449\pi\)
−0.753113 + 0.657891i \(0.771449\pi\)
\(98\) 1.08259 0.109358
\(99\) 0 0
\(100\) −3.47755 −0.347755
\(101\) 4.00533 0.398545 0.199272 0.979944i \(-0.436142\pi\)
0.199272 + 0.979944i \(0.436142\pi\)
\(102\) 0 0
\(103\) −5.91936 −0.583252 −0.291626 0.956532i \(-0.594196\pi\)
−0.291626 + 0.956532i \(0.594196\pi\)
\(104\) 7.02929 0.689279
\(105\) 0 0
\(106\) −4.35339 −0.422838
\(107\) −0.258978 −0.0250364 −0.0125182 0.999922i \(-0.503985\pi\)
−0.0125182 + 0.999922i \(0.503985\pi\)
\(108\) 0 0
\(109\) −8.55787 −0.819695 −0.409848 0.912154i \(-0.634418\pi\)
−0.409848 + 0.912154i \(0.634418\pi\)
\(110\) −0.551368 −0.0525709
\(111\) 0 0
\(112\) 1.32642 0.125335
\(113\) −3.11918 −0.293428 −0.146714 0.989179i \(-0.546870\pi\)
−0.146714 + 0.989179i \(0.546870\pi\)
\(114\) 0 0
\(115\) 18.3960 1.71544
\(116\) 0.482009 0.0447534
\(117\) 0 0
\(118\) −4.79067 −0.441017
\(119\) −0.697308 −0.0639221
\(120\) 0 0
\(121\) −10.9374 −0.994311
\(122\) 9.49550 0.859682
\(123\) 0 0
\(124\) −3.75427 −0.337144
\(125\) 6.70514 0.599726
\(126\) 0 0
\(127\) −18.4545 −1.63757 −0.818787 0.574097i \(-0.805353\pi\)
−0.818787 + 0.574097i \(0.805353\pi\)
\(128\) 8.80867 0.778583
\(129\) 0 0
\(130\) 5.75700 0.504922
\(131\) 14.2255 1.24289 0.621443 0.783460i \(-0.286547\pi\)
0.621443 + 0.783460i \(0.286547\pi\)
\(132\) 0 0
\(133\) −6.61888 −0.573929
\(134\) −1.45101 −0.125348
\(135\) 0 0
\(136\) −0.789527 −0.0677014
\(137\) −19.6856 −1.68185 −0.840926 0.541149i \(-0.817989\pi\)
−0.840926 + 0.541149i \(0.817989\pi\)
\(138\) 0 0
\(139\) −17.9110 −1.51919 −0.759594 0.650398i \(-0.774602\pi\)
−0.759594 + 0.650398i \(0.774602\pi\)
\(140\) 8.87286 0.749894
\(141\) 0 0
\(142\) 0.296974 0.0249215
\(143\) −0.653411 −0.0546410
\(144\) 0 0
\(145\) 0.976351 0.0810815
\(146\) −4.02911 −0.333451
\(147\) 0 0
\(148\) 9.49254 0.780282
\(149\) 16.2895 1.33448 0.667242 0.744841i \(-0.267475\pi\)
0.667242 + 0.744841i \(0.267475\pi\)
\(150\) 0 0
\(151\) 14.2749 1.16167 0.580836 0.814021i \(-0.302726\pi\)
0.580836 + 0.814021i \(0.302726\pi\)
\(152\) −7.49422 −0.607862
\(153\) 0 0
\(154\) 0.476577 0.0384036
\(155\) −7.60459 −0.610816
\(156\) 0 0
\(157\) 0.763354 0.0609223 0.0304612 0.999536i \(-0.490302\pi\)
0.0304612 + 0.999536i \(0.490302\pi\)
\(158\) −0.643483 −0.0511928
\(159\) 0 0
\(160\) 16.0306 1.26733
\(161\) −15.9006 −1.25315
\(162\) 0 0
\(163\) 5.12834 0.401682 0.200841 0.979624i \(-0.435632\pi\)
0.200841 + 0.979624i \(0.435632\pi\)
\(164\) 13.1922 1.03014
\(165\) 0 0
\(166\) 2.20873 0.171431
\(167\) 8.90112 0.688790 0.344395 0.938825i \(-0.388084\pi\)
0.344395 + 0.938825i \(0.388084\pi\)
\(168\) 0 0
\(169\) −6.17754 −0.475196
\(170\) −0.646624 −0.0495938
\(171\) 0 0
\(172\) −0.353923 −0.0269864
\(173\) −6.81124 −0.517849 −0.258924 0.965898i \(-0.583368\pi\)
−0.258924 + 0.965898i \(0.583368\pi\)
\(174\) 0 0
\(175\) 6.08856 0.460252
\(176\) −0.139605 −0.0105231
\(177\) 0 0
\(178\) −8.37649 −0.627844
\(179\) −18.3476 −1.37137 −0.685684 0.727900i \(-0.740497\pi\)
−0.685684 + 0.727900i \(0.740497\pi\)
\(180\) 0 0
\(181\) 11.3256 0.841829 0.420914 0.907100i \(-0.361709\pi\)
0.420914 + 0.907100i \(0.361709\pi\)
\(182\) −4.97608 −0.368852
\(183\) 0 0
\(184\) −18.0035 −1.32724
\(185\) 19.2279 1.41367
\(186\) 0 0
\(187\) 0.0733908 0.00536687
\(188\) 15.5108 1.13124
\(189\) 0 0
\(190\) −6.13778 −0.445281
\(191\) 6.85841 0.496257 0.248129 0.968727i \(-0.420184\pi\)
0.248129 + 0.968727i \(0.420184\pi\)
\(192\) 0 0
\(193\) 20.4128 1.46935 0.734673 0.678422i \(-0.237336\pi\)
0.734673 + 0.678422i \(0.237336\pi\)
\(194\) 11.8903 0.853676
\(195\) 0 0
\(196\) 1.83360 0.130971
\(197\) 3.03573 0.216287 0.108143 0.994135i \(-0.465509\pi\)
0.108143 + 0.994135i \(0.465509\pi\)
\(198\) 0 0
\(199\) −2.26247 −0.160382 −0.0801912 0.996779i \(-0.525553\pi\)
−0.0801912 + 0.996779i \(0.525553\pi\)
\(200\) 6.89377 0.487463
\(201\) 0 0
\(202\) −3.21038 −0.225881
\(203\) −0.843912 −0.0592310
\(204\) 0 0
\(205\) 26.7219 1.86634
\(206\) 4.74453 0.330567
\(207\) 0 0
\(208\) 1.45765 0.101070
\(209\) 0.696629 0.0481868
\(210\) 0 0
\(211\) 25.4308 1.75073 0.875364 0.483464i \(-0.160621\pi\)
0.875364 + 0.483464i \(0.160621\pi\)
\(212\) −7.37338 −0.506406
\(213\) 0 0
\(214\) 0.207578 0.0141898
\(215\) −0.716901 −0.0488923
\(216\) 0 0
\(217\) 6.57305 0.446208
\(218\) 6.85936 0.464575
\(219\) 0 0
\(220\) −0.933858 −0.0629607
\(221\) −0.766295 −0.0515466
\(222\) 0 0
\(223\) −3.83134 −0.256565 −0.128283 0.991738i \(-0.540946\pi\)
−0.128283 + 0.991738i \(0.540946\pi\)
\(224\) −13.8561 −0.925799
\(225\) 0 0
\(226\) 2.50011 0.166305
\(227\) 2.51599 0.166992 0.0834961 0.996508i \(-0.473391\pi\)
0.0834961 + 0.996508i \(0.473391\pi\)
\(228\) 0 0
\(229\) −15.9396 −1.05332 −0.526660 0.850076i \(-0.676556\pi\)
−0.526660 + 0.850076i \(0.676556\pi\)
\(230\) −14.7449 −0.972249
\(231\) 0 0
\(232\) −0.955519 −0.0627329
\(233\) 28.1283 1.84274 0.921372 0.388682i \(-0.127070\pi\)
0.921372 + 0.388682i \(0.127070\pi\)
\(234\) 0 0
\(235\) 31.4185 2.04952
\(236\) −8.11400 −0.528177
\(237\) 0 0
\(238\) 0.558911 0.0362288
\(239\) −14.7058 −0.951238 −0.475619 0.879651i \(-0.657776\pi\)
−0.475619 + 0.879651i \(0.657776\pi\)
\(240\) 0 0
\(241\) 8.44295 0.543858 0.271929 0.962317i \(-0.412338\pi\)
0.271929 + 0.962317i \(0.412338\pi\)
\(242\) 8.76664 0.563541
\(243\) 0 0
\(244\) 16.0826 1.02958
\(245\) 3.71410 0.237285
\(246\) 0 0
\(247\) −7.27371 −0.462815
\(248\) 7.44234 0.472589
\(249\) 0 0
\(250\) −5.37435 −0.339904
\(251\) −23.2205 −1.46566 −0.732832 0.680409i \(-0.761802\pi\)
−0.732832 + 0.680409i \(0.761802\pi\)
\(252\) 0 0
\(253\) 1.67352 0.105213
\(254\) 14.7918 0.928120
\(255\) 0 0
\(256\) −14.1734 −0.885835
\(257\) 6.86520 0.428239 0.214120 0.976807i \(-0.431312\pi\)
0.214120 + 0.976807i \(0.431312\pi\)
\(258\) 0 0
\(259\) −16.6197 −1.03270
\(260\) 9.75069 0.604712
\(261\) 0 0
\(262\) −11.4021 −0.704424
\(263\) 3.35294 0.206751 0.103376 0.994642i \(-0.467036\pi\)
0.103376 + 0.994642i \(0.467036\pi\)
\(264\) 0 0
\(265\) −14.9354 −0.917474
\(266\) 5.30521 0.325283
\(267\) 0 0
\(268\) −2.45758 −0.150121
\(269\) 12.7416 0.776869 0.388434 0.921476i \(-0.373016\pi\)
0.388434 + 0.921476i \(0.373016\pi\)
\(270\) 0 0
\(271\) −23.5566 −1.43096 −0.715481 0.698632i \(-0.753792\pi\)
−0.715481 + 0.698632i \(0.753792\pi\)
\(272\) −0.163723 −0.00992716
\(273\) 0 0
\(274\) 15.7785 0.953216
\(275\) −0.640814 −0.0386425
\(276\) 0 0
\(277\) 4.18122 0.251225 0.125613 0.992079i \(-0.459910\pi\)
0.125613 + 0.992079i \(0.459910\pi\)
\(278\) 14.3561 0.861023
\(279\) 0 0
\(280\) −17.5893 −1.05116
\(281\) −21.6360 −1.29070 −0.645348 0.763888i \(-0.723288\pi\)
−0.645348 + 0.763888i \(0.723288\pi\)
\(282\) 0 0
\(283\) 5.22734 0.310733 0.155366 0.987857i \(-0.450344\pi\)
0.155366 + 0.987857i \(0.450344\pi\)
\(284\) 0.502987 0.0298468
\(285\) 0 0
\(286\) 0.523726 0.0309686
\(287\) −23.0971 −1.36338
\(288\) 0 0
\(289\) −16.9139 −0.994937
\(290\) −0.782571 −0.0459542
\(291\) 0 0
\(292\) −6.82414 −0.399353
\(293\) −6.14217 −0.358829 −0.179415 0.983774i \(-0.557420\pi\)
−0.179415 + 0.983774i \(0.557420\pi\)
\(294\) 0 0
\(295\) −16.4356 −0.956918
\(296\) −18.8177 −1.09376
\(297\) 0 0
\(298\) −13.0564 −0.756339
\(299\) −17.4738 −1.01053
\(300\) 0 0
\(301\) 0.619656 0.0357164
\(302\) −11.4417 −0.658395
\(303\) 0 0
\(304\) −1.55407 −0.0891317
\(305\) 32.5767 1.86534
\(306\) 0 0
\(307\) 19.0039 1.08461 0.542304 0.840182i \(-0.317552\pi\)
0.542304 + 0.840182i \(0.317552\pi\)
\(308\) 0.807183 0.0459935
\(309\) 0 0
\(310\) 6.09529 0.346189
\(311\) −21.5469 −1.22181 −0.610907 0.791703i \(-0.709195\pi\)
−0.610907 + 0.791703i \(0.709195\pi\)
\(312\) 0 0
\(313\) −3.81362 −0.215558 −0.107779 0.994175i \(-0.534374\pi\)
−0.107779 + 0.994175i \(0.534374\pi\)
\(314\) −0.611849 −0.0345286
\(315\) 0 0
\(316\) −1.08987 −0.0613102
\(317\) 4.25449 0.238956 0.119478 0.992837i \(-0.461878\pi\)
0.119478 + 0.992837i \(0.461878\pi\)
\(318\) 0 0
\(319\) 0.0888207 0.00497300
\(320\) −9.77978 −0.546706
\(321\) 0 0
\(322\) 12.7448 0.710239
\(323\) 0.816980 0.0454580
\(324\) 0 0
\(325\) 6.69092 0.371146
\(326\) −4.11050 −0.227660
\(327\) 0 0
\(328\) −26.1517 −1.44399
\(329\) −27.1567 −1.49719
\(330\) 0 0
\(331\) −14.2938 −0.785658 −0.392829 0.919612i \(-0.628504\pi\)
−0.392829 + 0.919612i \(0.628504\pi\)
\(332\) 3.74095 0.205311
\(333\) 0 0
\(334\) −7.13449 −0.390382
\(335\) −4.97804 −0.271980
\(336\) 0 0
\(337\) 35.7185 1.94571 0.972855 0.231414i \(-0.0743352\pi\)
0.972855 + 0.231414i \(0.0743352\pi\)
\(338\) 4.95147 0.269324
\(339\) 0 0
\(340\) −1.09519 −0.0593952
\(341\) −0.691806 −0.0374634
\(342\) 0 0
\(343\) −19.8481 −1.07170
\(344\) 0.701606 0.0378280
\(345\) 0 0
\(346\) 5.45939 0.293499
\(347\) −19.4415 −1.04368 −0.521838 0.853045i \(-0.674753\pi\)
−0.521838 + 0.853045i \(0.674753\pi\)
\(348\) 0 0
\(349\) −8.02070 −0.429338 −0.214669 0.976687i \(-0.568867\pi\)
−0.214669 + 0.976687i \(0.568867\pi\)
\(350\) −4.88014 −0.260855
\(351\) 0 0
\(352\) 1.45834 0.0777296
\(353\) 8.75381 0.465919 0.232959 0.972486i \(-0.425159\pi\)
0.232959 + 0.972486i \(0.425159\pi\)
\(354\) 0 0
\(355\) 1.01884 0.0540746
\(356\) −14.1873 −0.751928
\(357\) 0 0
\(358\) 14.7061 0.777243
\(359\) −8.27791 −0.436892 −0.218446 0.975849i \(-0.570099\pi\)
−0.218446 + 0.975849i \(0.570099\pi\)
\(360\) 0 0
\(361\) −11.2452 −0.591852
\(362\) −9.07781 −0.477119
\(363\) 0 0
\(364\) −8.42804 −0.441749
\(365\) −13.8229 −0.723522
\(366\) 0 0
\(367\) −14.7999 −0.772546 −0.386273 0.922384i \(-0.626238\pi\)
−0.386273 + 0.922384i \(0.626238\pi\)
\(368\) −3.73336 −0.194615
\(369\) 0 0
\(370\) −15.4117 −0.801216
\(371\) 12.9095 0.670226
\(372\) 0 0
\(373\) 25.5334 1.32207 0.661035 0.750355i \(-0.270118\pi\)
0.661035 + 0.750355i \(0.270118\pi\)
\(374\) −0.0588247 −0.00304175
\(375\) 0 0
\(376\) −30.7481 −1.58571
\(377\) −0.927403 −0.0477637
\(378\) 0 0
\(379\) 20.1244 1.03372 0.516861 0.856070i \(-0.327101\pi\)
0.516861 + 0.856070i \(0.327101\pi\)
\(380\) −10.3956 −0.533284
\(381\) 0 0
\(382\) −5.49720 −0.281261
\(383\) −23.8613 −1.21925 −0.609627 0.792689i \(-0.708681\pi\)
−0.609627 + 0.792689i \(0.708681\pi\)
\(384\) 0 0
\(385\) 1.63502 0.0833282
\(386\) −16.3614 −0.832774
\(387\) 0 0
\(388\) 20.1388 1.02239
\(389\) 37.9733 1.92532 0.962662 0.270708i \(-0.0872577\pi\)
0.962662 + 0.270708i \(0.0872577\pi\)
\(390\) 0 0
\(391\) 1.96264 0.0992552
\(392\) −3.63486 −0.183588
\(393\) 0 0
\(394\) −2.43322 −0.122584
\(395\) −2.20763 −0.111078
\(396\) 0 0
\(397\) 20.3493 1.02130 0.510651 0.859788i \(-0.329404\pi\)
0.510651 + 0.859788i \(0.329404\pi\)
\(398\) 1.81343 0.0908992
\(399\) 0 0
\(400\) 1.42955 0.0714775
\(401\) 6.94497 0.346815 0.173408 0.984850i \(-0.444522\pi\)
0.173408 + 0.984850i \(0.444522\pi\)
\(402\) 0 0
\(403\) 7.22335 0.359821
\(404\) −5.43745 −0.270523
\(405\) 0 0
\(406\) 0.676418 0.0335701
\(407\) 1.74921 0.0867049
\(408\) 0 0
\(409\) −10.9060 −0.539265 −0.269632 0.962963i \(-0.586902\pi\)
−0.269632 + 0.962963i \(0.586902\pi\)
\(410\) −21.4183 −1.05777
\(411\) 0 0
\(412\) 8.03586 0.395898
\(413\) 14.2062 0.699040
\(414\) 0 0
\(415\) 7.57760 0.371970
\(416\) −15.2269 −0.746562
\(417\) 0 0
\(418\) −0.558367 −0.0273106
\(419\) −10.0692 −0.491912 −0.245956 0.969281i \(-0.579102\pi\)
−0.245956 + 0.969281i \(0.579102\pi\)
\(420\) 0 0
\(421\) −3.10756 −0.151453 −0.0757267 0.997129i \(-0.524128\pi\)
−0.0757267 + 0.997129i \(0.524128\pi\)
\(422\) −20.3835 −0.992252
\(423\) 0 0
\(424\) 14.6167 0.709852
\(425\) −0.751522 −0.0364542
\(426\) 0 0
\(427\) −28.1578 −1.36265
\(428\) 0.351577 0.0169941
\(429\) 0 0
\(430\) 0.574616 0.0277104
\(431\) −28.0701 −1.35209 −0.676044 0.736862i \(-0.736307\pi\)
−0.676044 + 0.736862i \(0.736307\pi\)
\(432\) 0 0
\(433\) 19.5251 0.938317 0.469158 0.883114i \(-0.344557\pi\)
0.469158 + 0.883114i \(0.344557\pi\)
\(434\) −5.26848 −0.252895
\(435\) 0 0
\(436\) 11.6178 0.556391
\(437\) 18.6295 0.891170
\(438\) 0 0
\(439\) 14.6296 0.698232 0.349116 0.937080i \(-0.386482\pi\)
0.349116 + 0.937080i \(0.386482\pi\)
\(440\) 1.85125 0.0882548
\(441\) 0 0
\(442\) 0.614206 0.0292148
\(443\) 18.3559 0.872117 0.436059 0.899918i \(-0.356374\pi\)
0.436059 + 0.899918i \(0.356374\pi\)
\(444\) 0 0
\(445\) −28.7377 −1.36230
\(446\) 3.07092 0.145412
\(447\) 0 0
\(448\) 8.45318 0.399375
\(449\) 13.8594 0.654065 0.327032 0.945013i \(-0.393951\pi\)
0.327032 + 0.945013i \(0.393951\pi\)
\(450\) 0 0
\(451\) 2.43095 0.114469
\(452\) 4.23445 0.199172
\(453\) 0 0
\(454\) −2.01663 −0.0946454
\(455\) −17.0717 −0.800334
\(456\) 0 0
\(457\) 17.6481 0.825545 0.412772 0.910834i \(-0.364561\pi\)
0.412772 + 0.910834i \(0.364561\pi\)
\(458\) 12.7760 0.596985
\(459\) 0 0
\(460\) −24.9736 −1.16440
\(461\) 25.6380 1.19408 0.597041 0.802211i \(-0.296343\pi\)
0.597041 + 0.802211i \(0.296343\pi\)
\(462\) 0 0
\(463\) 18.3474 0.852677 0.426339 0.904564i \(-0.359803\pi\)
0.426339 + 0.904564i \(0.359803\pi\)
\(464\) −0.198144 −0.00919863
\(465\) 0 0
\(466\) −22.5456 −1.04440
\(467\) −16.2618 −0.752509 −0.376254 0.926516i \(-0.622788\pi\)
−0.376254 + 0.926516i \(0.622788\pi\)
\(468\) 0 0
\(469\) 4.30279 0.198684
\(470\) −25.1828 −1.16159
\(471\) 0 0
\(472\) 16.0849 0.740369
\(473\) −0.0652180 −0.00299873
\(474\) 0 0
\(475\) −7.13348 −0.327306
\(476\) 0.946634 0.0433889
\(477\) 0 0
\(478\) 11.7871 0.539129
\(479\) −9.47171 −0.432773 −0.216387 0.976308i \(-0.569427\pi\)
−0.216387 + 0.976308i \(0.569427\pi\)
\(480\) 0 0
\(481\) −18.2640 −0.832766
\(482\) −6.76725 −0.308240
\(483\) 0 0
\(484\) 14.8481 0.674916
\(485\) 40.7928 1.85231
\(486\) 0 0
\(487\) −0.467564 −0.0211874 −0.0105937 0.999944i \(-0.503372\pi\)
−0.0105937 + 0.999944i \(0.503372\pi\)
\(488\) −31.8817 −1.44321
\(489\) 0 0
\(490\) −2.97695 −0.134485
\(491\) 25.0470 1.13035 0.565177 0.824969i \(-0.308808\pi\)
0.565177 + 0.824969i \(0.308808\pi\)
\(492\) 0 0
\(493\) 0.104166 0.00469138
\(494\) 5.83007 0.262307
\(495\) 0 0
\(496\) 1.54331 0.0692965
\(497\) −0.880640 −0.0395021
\(498\) 0 0
\(499\) −14.0342 −0.628255 −0.314128 0.949381i \(-0.601712\pi\)
−0.314128 + 0.949381i \(0.601712\pi\)
\(500\) −9.10259 −0.407080
\(501\) 0 0
\(502\) 18.6119 0.830688
\(503\) 28.3116 1.26235 0.631176 0.775640i \(-0.282573\pi\)
0.631176 + 0.775640i \(0.282573\pi\)
\(504\) 0 0
\(505\) −11.0140 −0.490117
\(506\) −1.34137 −0.0596313
\(507\) 0 0
\(508\) 25.0530 1.11155
\(509\) −28.6875 −1.27155 −0.635774 0.771875i \(-0.719319\pi\)
−0.635774 + 0.771875i \(0.719319\pi\)
\(510\) 0 0
\(511\) 11.9478 0.528541
\(512\) −6.25700 −0.276523
\(513\) 0 0
\(514\) −5.50264 −0.242711
\(515\) 16.2773 0.717264
\(516\) 0 0
\(517\) 2.85820 0.125704
\(518\) 13.3212 0.585298
\(519\) 0 0
\(520\) −19.3294 −0.847652
\(521\) −24.9096 −1.09131 −0.545655 0.838010i \(-0.683719\pi\)
−0.545655 + 0.838010i \(0.683719\pi\)
\(522\) 0 0
\(523\) −25.8648 −1.13099 −0.565494 0.824753i \(-0.691314\pi\)
−0.565494 + 0.824753i \(0.691314\pi\)
\(524\) −19.3119 −0.843642
\(525\) 0 0
\(526\) −2.68747 −0.117179
\(527\) −0.811324 −0.0353418
\(528\) 0 0
\(529\) 21.7540 0.945825
\(530\) 11.9711 0.519992
\(531\) 0 0
\(532\) 8.98549 0.389570
\(533\) −25.3822 −1.09943
\(534\) 0 0
\(535\) 0.712150 0.0307889
\(536\) 4.87183 0.210431
\(537\) 0 0
\(538\) −10.2127 −0.440302
\(539\) 0.337880 0.0145535
\(540\) 0 0
\(541\) −21.9158 −0.942232 −0.471116 0.882071i \(-0.656149\pi\)
−0.471116 + 0.882071i \(0.656149\pi\)
\(542\) 18.8813 0.811019
\(543\) 0 0
\(544\) 1.71028 0.0733278
\(545\) 23.5328 1.00803
\(546\) 0 0
\(547\) −9.97605 −0.426545 −0.213273 0.976993i \(-0.568412\pi\)
−0.213273 + 0.976993i \(0.568412\pi\)
\(548\) 26.7243 1.14160
\(549\) 0 0
\(550\) 0.513629 0.0219012
\(551\) 0.988744 0.0421219
\(552\) 0 0
\(553\) 1.90817 0.0811438
\(554\) −3.35136 −0.142386
\(555\) 0 0
\(556\) 24.3151 1.03119
\(557\) −18.5330 −0.785268 −0.392634 0.919695i \(-0.628436\pi\)
−0.392634 + 0.919695i \(0.628436\pi\)
\(558\) 0 0
\(559\) 0.680961 0.0288016
\(560\) −3.64746 −0.154133
\(561\) 0 0
\(562\) 17.3418 0.731522
\(563\) 43.6831 1.84102 0.920511 0.390716i \(-0.127772\pi\)
0.920511 + 0.390716i \(0.127772\pi\)
\(564\) 0 0
\(565\) 8.57724 0.360847
\(566\) −4.18985 −0.176113
\(567\) 0 0
\(568\) −0.997105 −0.0418376
\(569\) 13.5667 0.568745 0.284373 0.958714i \(-0.408215\pi\)
0.284373 + 0.958714i \(0.408215\pi\)
\(570\) 0 0
\(571\) 23.7487 0.993853 0.496926 0.867793i \(-0.334462\pi\)
0.496926 + 0.867793i \(0.334462\pi\)
\(572\) 0.887041 0.0370890
\(573\) 0 0
\(574\) 18.5130 0.772717
\(575\) −17.1369 −0.714657
\(576\) 0 0
\(577\) −8.11902 −0.337999 −0.169000 0.985616i \(-0.554054\pi\)
−0.169000 + 0.985616i \(0.554054\pi\)
\(578\) 13.5570 0.563896
\(579\) 0 0
\(580\) −1.32545 −0.0550363
\(581\) −6.54972 −0.271728
\(582\) 0 0
\(583\) −1.35871 −0.0562718
\(584\) 13.5279 0.559790
\(585\) 0 0
\(586\) 4.92311 0.203372
\(587\) −3.69199 −0.152385 −0.0761925 0.997093i \(-0.524276\pi\)
−0.0761925 + 0.997093i \(0.524276\pi\)
\(588\) 0 0
\(589\) −7.70112 −0.317319
\(590\) 13.1736 0.542348
\(591\) 0 0
\(592\) −3.90219 −0.160379
\(593\) 29.4590 1.20974 0.604869 0.796325i \(-0.293226\pi\)
0.604869 + 0.796325i \(0.293226\pi\)
\(594\) 0 0
\(595\) 1.91749 0.0786093
\(596\) −22.1138 −0.905818
\(597\) 0 0
\(598\) 14.0057 0.572735
\(599\) −21.8754 −0.893804 −0.446902 0.894583i \(-0.647473\pi\)
−0.446902 + 0.894583i \(0.647473\pi\)
\(600\) 0 0
\(601\) 36.5207 1.48971 0.744854 0.667227i \(-0.232519\pi\)
0.744854 + 0.667227i \(0.232519\pi\)
\(602\) −0.496671 −0.0202428
\(603\) 0 0
\(604\) −19.3789 −0.788517
\(605\) 30.0762 1.22277
\(606\) 0 0
\(607\) −6.58082 −0.267107 −0.133554 0.991042i \(-0.542639\pi\)
−0.133554 + 0.991042i \(0.542639\pi\)
\(608\) 16.2341 0.658379
\(609\) 0 0
\(610\) −26.1111 −1.05721
\(611\) −29.8434 −1.20733
\(612\) 0 0
\(613\) −7.14867 −0.288732 −0.144366 0.989524i \(-0.546114\pi\)
−0.144366 + 0.989524i \(0.546114\pi\)
\(614\) −15.2321 −0.614718
\(615\) 0 0
\(616\) −1.60013 −0.0644712
\(617\) 16.5375 0.665774 0.332887 0.942967i \(-0.391977\pi\)
0.332887 + 0.942967i \(0.391977\pi\)
\(618\) 0 0
\(619\) −1.49935 −0.0602640 −0.0301320 0.999546i \(-0.509593\pi\)
−0.0301320 + 0.999546i \(0.509593\pi\)
\(620\) 10.3236 0.414608
\(621\) 0 0
\(622\) 17.2704 0.692481
\(623\) 24.8395 0.995173
\(624\) 0 0
\(625\) −31.2462 −1.24985
\(626\) 3.05672 0.122171
\(627\) 0 0
\(628\) −1.03630 −0.0413527
\(629\) 2.05140 0.0817948
\(630\) 0 0
\(631\) −35.8913 −1.42881 −0.714404 0.699733i \(-0.753302\pi\)
−0.714404 + 0.699733i \(0.753302\pi\)
\(632\) 2.16053 0.0859413
\(633\) 0 0
\(634\) −3.41009 −0.135432
\(635\) 50.7470 2.01383
\(636\) 0 0
\(637\) −3.52790 −0.139781
\(638\) −0.0711922 −0.00281852
\(639\) 0 0
\(640\) −24.2224 −0.957476
\(641\) 39.2279 1.54941 0.774705 0.632322i \(-0.217898\pi\)
0.774705 + 0.632322i \(0.217898\pi\)
\(642\) 0 0
\(643\) −10.4185 −0.410866 −0.205433 0.978671i \(-0.565860\pi\)
−0.205433 + 0.978671i \(0.565860\pi\)
\(644\) 21.5860 0.850607
\(645\) 0 0
\(646\) −0.654831 −0.0257640
\(647\) −39.1517 −1.53921 −0.769606 0.638519i \(-0.779547\pi\)
−0.769606 + 0.638519i \(0.779547\pi\)
\(648\) 0 0
\(649\) −1.49518 −0.0586910
\(650\) −5.36296 −0.210352
\(651\) 0 0
\(652\) −6.96200 −0.272653
\(653\) 32.9099 1.28786 0.643932 0.765083i \(-0.277302\pi\)
0.643932 + 0.765083i \(0.277302\pi\)
\(654\) 0 0
\(655\) −39.1178 −1.52846
\(656\) −5.42304 −0.211734
\(657\) 0 0
\(658\) 21.7668 0.848558
\(659\) −21.5684 −0.840186 −0.420093 0.907481i \(-0.638003\pi\)
−0.420093 + 0.907481i \(0.638003\pi\)
\(660\) 0 0
\(661\) 26.2964 1.02281 0.511405 0.859340i \(-0.329125\pi\)
0.511405 + 0.859340i \(0.329125\pi\)
\(662\) 11.4569 0.445283
\(663\) 0 0
\(664\) −7.41593 −0.287794
\(665\) 18.2009 0.705799
\(666\) 0 0
\(667\) 2.37528 0.0919710
\(668\) −12.0838 −0.467535
\(669\) 0 0
\(670\) 3.99004 0.154149
\(671\) 2.96357 0.114407
\(672\) 0 0
\(673\) 11.5221 0.444145 0.222073 0.975030i \(-0.428718\pi\)
0.222073 + 0.975030i \(0.428718\pi\)
\(674\) −28.6293 −1.10276
\(675\) 0 0
\(676\) 8.38635 0.322552
\(677\) −33.9250 −1.30384 −0.651922 0.758286i \(-0.726037\pi\)
−0.651922 + 0.758286i \(0.726037\pi\)
\(678\) 0 0
\(679\) −35.2594 −1.35313
\(680\) 2.17107 0.0832569
\(681\) 0 0
\(682\) 0.554501 0.0212329
\(683\) 36.7553 1.40640 0.703201 0.710991i \(-0.251753\pi\)
0.703201 + 0.710991i \(0.251753\pi\)
\(684\) 0 0
\(685\) 54.1322 2.06829
\(686\) 15.9088 0.607401
\(687\) 0 0
\(688\) 0.145491 0.00554678
\(689\) 14.1866 0.540468
\(690\) 0 0
\(691\) 13.3781 0.508928 0.254464 0.967082i \(-0.418101\pi\)
0.254464 + 0.967082i \(0.418101\pi\)
\(692\) 9.24663 0.351504
\(693\) 0 0
\(694\) 15.5829 0.591519
\(695\) 49.2523 1.86825
\(696\) 0 0
\(697\) 2.85092 0.107986
\(698\) 6.42881 0.243334
\(699\) 0 0
\(700\) −8.26555 −0.312409
\(701\) 5.00452 0.189018 0.0945091 0.995524i \(-0.469872\pi\)
0.0945091 + 0.995524i \(0.469872\pi\)
\(702\) 0 0
\(703\) 19.4720 0.734400
\(704\) −0.889687 −0.0335314
\(705\) 0 0
\(706\) −7.01642 −0.264066
\(707\) 9.52000 0.358036
\(708\) 0 0
\(709\) 17.1439 0.643851 0.321925 0.946765i \(-0.395670\pi\)
0.321925 + 0.946765i \(0.395670\pi\)
\(710\) −0.816630 −0.0306476
\(711\) 0 0
\(712\) 28.1245 1.05401
\(713\) −18.5005 −0.692850
\(714\) 0 0
\(715\) 1.79678 0.0671956
\(716\) 24.9079 0.930853
\(717\) 0 0
\(718\) 6.63497 0.247615
\(719\) −43.3519 −1.61675 −0.808377 0.588665i \(-0.799654\pi\)
−0.808377 + 0.588665i \(0.799654\pi\)
\(720\) 0 0
\(721\) −14.0693 −0.523969
\(722\) 9.01333 0.335441
\(723\) 0 0
\(724\) −15.3752 −0.571414
\(725\) −0.909524 −0.0337789
\(726\) 0 0
\(727\) −36.3439 −1.34792 −0.673960 0.738768i \(-0.735408\pi\)
−0.673960 + 0.738768i \(0.735408\pi\)
\(728\) 16.7075 0.619220
\(729\) 0 0
\(730\) 11.0794 0.410067
\(731\) −0.0764852 −0.00282891
\(732\) 0 0
\(733\) 3.87561 0.143149 0.0715744 0.997435i \(-0.477198\pi\)
0.0715744 + 0.997435i \(0.477198\pi\)
\(734\) 11.8625 0.437852
\(735\) 0 0
\(736\) 38.9994 1.43754
\(737\) −0.452863 −0.0166814
\(738\) 0 0
\(739\) 26.4482 0.972913 0.486456 0.873705i \(-0.338289\pi\)
0.486456 + 0.873705i \(0.338289\pi\)
\(740\) −26.1030 −0.959564
\(741\) 0 0
\(742\) −10.3473 −0.379861
\(743\) −13.4634 −0.493923 −0.246961 0.969025i \(-0.579432\pi\)
−0.246961 + 0.969025i \(0.579432\pi\)
\(744\) 0 0
\(745\) −44.7934 −1.64110
\(746\) −20.4657 −0.749303
\(747\) 0 0
\(748\) −0.0996320 −0.00364291
\(749\) −0.615549 −0.0224917
\(750\) 0 0
\(751\) 3.76830 0.137507 0.0687537 0.997634i \(-0.478098\pi\)
0.0687537 + 0.997634i \(0.478098\pi\)
\(752\) −6.37619 −0.232516
\(753\) 0 0
\(754\) 0.743339 0.0270708
\(755\) −39.2536 −1.42859
\(756\) 0 0
\(757\) −33.7073 −1.22511 −0.612556 0.790427i \(-0.709859\pi\)
−0.612556 + 0.790427i \(0.709859\pi\)
\(758\) −16.1303 −0.585877
\(759\) 0 0
\(760\) 20.6079 0.747528
\(761\) −9.65543 −0.350009 −0.175005 0.984568i \(-0.555994\pi\)
−0.175005 + 0.984568i \(0.555994\pi\)
\(762\) 0 0
\(763\) −20.3406 −0.736381
\(764\) −9.31067 −0.336848
\(765\) 0 0
\(766\) 19.1254 0.691030
\(767\) 15.6116 0.563703
\(768\) 0 0
\(769\) 38.7110 1.39595 0.697977 0.716120i \(-0.254084\pi\)
0.697977 + 0.716120i \(0.254084\pi\)
\(770\) −1.31051 −0.0472275
\(771\) 0 0
\(772\) −27.7115 −0.997358
\(773\) 24.3039 0.874150 0.437075 0.899425i \(-0.356014\pi\)
0.437075 + 0.899425i \(0.356014\pi\)
\(774\) 0 0
\(775\) 7.08409 0.254468
\(776\) −39.9225 −1.43313
\(777\) 0 0
\(778\) −30.4366 −1.09121
\(779\) 27.0611 0.969563
\(780\) 0 0
\(781\) 0.0926863 0.00331658
\(782\) −1.57311 −0.0562544
\(783\) 0 0
\(784\) −0.753755 −0.0269198
\(785\) −2.09910 −0.0749202
\(786\) 0 0
\(787\) 20.9406 0.746452 0.373226 0.927740i \(-0.378252\pi\)
0.373226 + 0.927740i \(0.378252\pi\)
\(788\) −4.12117 −0.146810
\(789\) 0 0
\(790\) 1.76948 0.0629551
\(791\) −7.41377 −0.263603
\(792\) 0 0
\(793\) −30.9435 −1.09884
\(794\) −16.3105 −0.578839
\(795\) 0 0
\(796\) 3.07143 0.108864
\(797\) 11.9368 0.422822 0.211411 0.977397i \(-0.432194\pi\)
0.211411 + 0.977397i \(0.432194\pi\)
\(798\) 0 0
\(799\) 3.35199 0.118585
\(800\) −14.9334 −0.527974
\(801\) 0 0
\(802\) −5.56658 −0.196563
\(803\) −1.25750 −0.0443761
\(804\) 0 0
\(805\) 43.7242 1.54108
\(806\) −5.78971 −0.203934
\(807\) 0 0
\(808\) 10.7790 0.379205
\(809\) 8.60808 0.302644 0.151322 0.988485i \(-0.451647\pi\)
0.151322 + 0.988485i \(0.451647\pi\)
\(810\) 0 0
\(811\) 1.53770 0.0539958 0.0269979 0.999635i \(-0.491405\pi\)
0.0269979 + 0.999635i \(0.491405\pi\)
\(812\) 1.14566 0.0402046
\(813\) 0 0
\(814\) −1.40204 −0.0491413
\(815\) −14.1021 −0.493975
\(816\) 0 0
\(817\) −0.726001 −0.0253996
\(818\) 8.74141 0.305636
\(819\) 0 0
\(820\) −36.2764 −1.26683
\(821\) 28.9763 1.01128 0.505639 0.862745i \(-0.331257\pi\)
0.505639 + 0.862745i \(0.331257\pi\)
\(822\) 0 0
\(823\) −11.2568 −0.392386 −0.196193 0.980565i \(-0.562858\pi\)
−0.196193 + 0.980565i \(0.562858\pi\)
\(824\) −15.9300 −0.554948
\(825\) 0 0
\(826\) −11.3866 −0.396191
\(827\) 30.9279 1.07547 0.537734 0.843114i \(-0.319280\pi\)
0.537734 + 0.843114i \(0.319280\pi\)
\(828\) 0 0
\(829\) −9.83524 −0.341592 −0.170796 0.985306i \(-0.554634\pi\)
−0.170796 + 0.985306i \(0.554634\pi\)
\(830\) −6.07365 −0.210820
\(831\) 0 0
\(832\) 9.28949 0.322055
\(833\) 0.396253 0.0137293
\(834\) 0 0
\(835\) −24.4767 −0.847050
\(836\) −0.945711 −0.0327081
\(837\) 0 0
\(838\) 8.07072 0.278798
\(839\) −13.1432 −0.453754 −0.226877 0.973923i \(-0.572852\pi\)
−0.226877 + 0.973923i \(0.572852\pi\)
\(840\) 0 0
\(841\) −28.8739 −0.995653
\(842\) 2.49080 0.0858385
\(843\) 0 0
\(844\) −34.5237 −1.18835
\(845\) 16.9873 0.584380
\(846\) 0 0
\(847\) −25.9964 −0.893248
\(848\) 3.03105 0.104087
\(849\) 0 0
\(850\) 0.602365 0.0206609
\(851\) 46.7779 1.60353
\(852\) 0 0
\(853\) 15.4423 0.528735 0.264368 0.964422i \(-0.414837\pi\)
0.264368 + 0.964422i \(0.414837\pi\)
\(854\) 22.5692 0.772303
\(855\) 0 0
\(856\) −0.696955 −0.0238214
\(857\) 21.9827 0.750916 0.375458 0.926839i \(-0.377485\pi\)
0.375458 + 0.926839i \(0.377485\pi\)
\(858\) 0 0
\(859\) 19.5667 0.667606 0.333803 0.942643i \(-0.391668\pi\)
0.333803 + 0.942643i \(0.391668\pi\)
\(860\) 0.973233 0.0331870
\(861\) 0 0
\(862\) 22.4989 0.766316
\(863\) 21.8676 0.744383 0.372191 0.928156i \(-0.378607\pi\)
0.372191 + 0.928156i \(0.378607\pi\)
\(864\) 0 0
\(865\) 18.7298 0.636833
\(866\) −15.6499 −0.531805
\(867\) 0 0
\(868\) −8.92328 −0.302876
\(869\) −0.200833 −0.00681279
\(870\) 0 0
\(871\) 4.72848 0.160218
\(872\) −23.0307 −0.779918
\(873\) 0 0
\(874\) −14.9320 −0.505084
\(875\) 15.9370 0.538769
\(876\) 0 0
\(877\) −39.0999 −1.32031 −0.660155 0.751130i \(-0.729509\pi\)
−0.660155 + 0.751130i \(0.729509\pi\)
\(878\) −11.7260 −0.395733
\(879\) 0 0
\(880\) 0.383890 0.0129409
\(881\) −7.30508 −0.246115 −0.123057 0.992400i \(-0.539270\pi\)
−0.123057 + 0.992400i \(0.539270\pi\)
\(882\) 0 0
\(883\) −3.49293 −0.117546 −0.0587732 0.998271i \(-0.518719\pi\)
−0.0587732 + 0.998271i \(0.518719\pi\)
\(884\) 1.04029 0.0349887
\(885\) 0 0
\(886\) −14.7128 −0.494286
\(887\) 28.4213 0.954293 0.477147 0.878824i \(-0.341671\pi\)
0.477147 + 0.878824i \(0.341671\pi\)
\(888\) 0 0
\(889\) −43.8633 −1.47113
\(890\) 23.0340 0.772102
\(891\) 0 0
\(892\) 5.20125 0.174151
\(893\) 31.8173 1.06472
\(894\) 0 0
\(895\) 50.4531 1.68646
\(896\) 20.9367 0.699447
\(897\) 0 0
\(898\) −11.1087 −0.370701
\(899\) −0.981898 −0.0327481
\(900\) 0 0
\(901\) −1.59344 −0.0530851
\(902\) −1.94847 −0.0648769
\(903\) 0 0
\(904\) −8.39424 −0.279188
\(905\) −31.1437 −1.03525
\(906\) 0 0
\(907\) 53.1167 1.76371 0.881855 0.471520i \(-0.156295\pi\)
0.881855 + 0.471520i \(0.156295\pi\)
\(908\) −3.41560 −0.113351
\(909\) 0 0
\(910\) 13.6834 0.453601
\(911\) −8.07589 −0.267566 −0.133783 0.991011i \(-0.542713\pi\)
−0.133783 + 0.991011i \(0.542713\pi\)
\(912\) 0 0
\(913\) 0.689351 0.0228142
\(914\) −14.1455 −0.467890
\(915\) 0 0
\(916\) 21.6389 0.714970
\(917\) 33.8116 1.11656
\(918\) 0 0
\(919\) −47.9961 −1.58325 −0.791623 0.611009i \(-0.790764\pi\)
−0.791623 + 0.611009i \(0.790764\pi\)
\(920\) 49.5068 1.63219
\(921\) 0 0
\(922\) −20.5496 −0.676764
\(923\) −0.967766 −0.0318544
\(924\) 0 0
\(925\) −17.9119 −0.588938
\(926\) −14.7060 −0.483268
\(927\) 0 0
\(928\) 2.06986 0.0679464
\(929\) −28.9939 −0.951260 −0.475630 0.879645i \(-0.657780\pi\)
−0.475630 + 0.879645i \(0.657780\pi\)
\(930\) 0 0
\(931\) 3.76125 0.123270
\(932\) −38.1857 −1.25081
\(933\) 0 0
\(934\) 13.0343 0.426496
\(935\) −0.201813 −0.00659999
\(936\) 0 0
\(937\) 5.02850 0.164274 0.0821369 0.996621i \(-0.473826\pi\)
0.0821369 + 0.996621i \(0.473826\pi\)
\(938\) −3.44880 −0.112607
\(939\) 0 0
\(940\) −42.6523 −1.39116
\(941\) −55.8411 −1.82037 −0.910183 0.414206i \(-0.864059\pi\)
−0.910183 + 0.414206i \(0.864059\pi\)
\(942\) 0 0
\(943\) 65.0092 2.11699
\(944\) 3.33551 0.108561
\(945\) 0 0
\(946\) 0.0522740 0.00169957
\(947\) 42.5294 1.38202 0.691009 0.722846i \(-0.257166\pi\)
0.691009 + 0.722846i \(0.257166\pi\)
\(948\) 0 0
\(949\) 13.1299 0.426214
\(950\) 5.71767 0.185506
\(951\) 0 0
\(952\) −1.87657 −0.0608201
\(953\) 21.8148 0.706651 0.353325 0.935501i \(-0.385051\pi\)
0.353325 + 0.935501i \(0.385051\pi\)
\(954\) 0 0
\(955\) −18.8595 −0.610280
\(956\) 19.9639 0.645679
\(957\) 0 0
\(958\) 7.59183 0.245281
\(959\) −46.7894 −1.51091
\(960\) 0 0
\(961\) −23.3522 −0.753297
\(962\) 14.6391 0.471983
\(963\) 0 0
\(964\) −11.4618 −0.369158
\(965\) −56.1319 −1.80695
\(966\) 0 0
\(967\) 4.62859 0.148845 0.0744227 0.997227i \(-0.476289\pi\)
0.0744227 + 0.997227i \(0.476289\pi\)
\(968\) −29.4345 −0.946059
\(969\) 0 0
\(970\) −32.6965 −1.04982
\(971\) −21.6509 −0.694809 −0.347405 0.937715i \(-0.612937\pi\)
−0.347405 + 0.937715i \(0.612937\pi\)
\(972\) 0 0
\(973\) −42.5714 −1.36478
\(974\) 0.374766 0.0120083
\(975\) 0 0
\(976\) −6.61125 −0.211621
\(977\) −22.0848 −0.706556 −0.353278 0.935518i \(-0.614933\pi\)
−0.353278 + 0.935518i \(0.614933\pi\)
\(978\) 0 0
\(979\) −2.61433 −0.0835542
\(980\) −5.04210 −0.161064
\(981\) 0 0
\(982\) −20.0758 −0.640646
\(983\) −13.8630 −0.442163 −0.221081 0.975255i \(-0.570959\pi\)
−0.221081 + 0.975255i \(0.570959\pi\)
\(984\) 0 0
\(985\) −8.34777 −0.265982
\(986\) −0.0834915 −0.00265891
\(987\) 0 0
\(988\) 9.87446 0.314148
\(989\) −1.74408 −0.0554587
\(990\) 0 0
\(991\) 34.8224 1.10617 0.553084 0.833125i \(-0.313451\pi\)
0.553084 + 0.833125i \(0.313451\pi\)
\(992\) −16.1217 −0.511864
\(993\) 0 0
\(994\) 0.705857 0.0223884
\(995\) 6.22144 0.197233
\(996\) 0 0
\(997\) −24.7498 −0.783833 −0.391916 0.920001i \(-0.628188\pi\)
−0.391916 + 0.920001i \(0.628188\pi\)
\(998\) 11.2488 0.356073
\(999\) 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.1.3 6
3.2 odd 2 729.2.a.d.1.4 6
9.2 odd 6 729.2.c.b.244.3 12
9.4 even 3 729.2.c.e.487.4 12
9.5 odd 6 729.2.c.b.487.3 12
9.7 even 3 729.2.c.e.244.4 12
27.2 odd 18 81.2.e.a.64.1 12
27.4 even 9 243.2.e.d.136.1 12
27.5 odd 18 243.2.e.b.217.2 12
27.7 even 9 243.2.e.d.109.1 12
27.11 odd 18 243.2.e.b.28.2 12
27.13 even 9 27.2.e.a.7.2 yes 12
27.14 odd 18 81.2.e.a.19.1 12
27.16 even 9 243.2.e.c.28.1 12
27.20 odd 18 243.2.e.a.109.2 12
27.22 even 9 243.2.e.c.217.1 12
27.23 odd 18 243.2.e.a.136.2 12
27.25 even 9 27.2.e.a.4.2 12
108.67 odd 18 432.2.u.c.385.2 12
108.79 odd 18 432.2.u.c.193.2 12
135.13 odd 36 675.2.u.b.574.2 24
135.52 odd 36 675.2.u.b.274.2 24
135.67 odd 36 675.2.u.b.574.3 24
135.79 even 18 675.2.l.c.301.1 12
135.94 even 18 675.2.l.c.601.1 12
135.133 odd 36 675.2.u.b.274.3 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.2.e.a.4.2 12 27.25 even 9
27.2.e.a.7.2 yes 12 27.13 even 9
81.2.e.a.19.1 12 27.14 odd 18
81.2.e.a.64.1 12 27.2 odd 18
243.2.e.a.109.2 12 27.20 odd 18
243.2.e.a.136.2 12 27.23 odd 18
243.2.e.b.28.2 12 27.11 odd 18
243.2.e.b.217.2 12 27.5 odd 18
243.2.e.c.28.1 12 27.16 even 9
243.2.e.c.217.1 12 27.22 even 9
243.2.e.d.109.1 12 27.7 even 9
243.2.e.d.136.1 12 27.4 even 9
432.2.u.c.193.2 12 108.79 odd 18
432.2.u.c.385.2 12 108.67 odd 18
675.2.l.c.301.1 12 135.79 even 18
675.2.l.c.601.1 12 135.94 even 18
675.2.u.b.274.2 24 135.52 odd 36
675.2.u.b.274.3 24 135.133 odd 36
675.2.u.b.574.2 24 135.13 odd 36
675.2.u.b.574.3 24 135.67 odd 36
729.2.a.a.1.3 6 1.1 even 1 trivial
729.2.a.d.1.4 6 3.2 odd 2
729.2.c.b.244.3 12 9.2 odd 6
729.2.c.b.487.3 12 9.5 odd 6
729.2.c.e.244.4 12 9.7 even 3
729.2.c.e.487.4 12 9.4 even 3