Properties

Label 1728.2.a.bd.1.1
Level $1728$
Weight $2$
Character 1728.1
Self dual yes
Analytic conductor $13.798$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.7981494693\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 864)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 1728.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.60555 q^{5} -3.60555 q^{7} +O(q^{10})\) \(q-3.60555 q^{5} -3.60555 q^{7} +1.00000 q^{11} -4.00000 q^{13} -7.21110 q^{19} +6.00000 q^{23} +8.00000 q^{25} +7.21110 q^{29} +3.60555 q^{31} +13.0000 q^{35} -10.0000 q^{37} -7.21110 q^{41} +7.21110 q^{43} +10.0000 q^{47} +6.00000 q^{49} +3.60555 q^{53} -3.60555 q^{55} -4.00000 q^{59} +14.4222 q^{65} +7.21110 q^{67} -8.00000 q^{71} -3.00000 q^{73} -3.60555 q^{77} +14.4222 q^{79} +9.00000 q^{83} +7.21110 q^{89} +14.4222 q^{91} +26.0000 q^{95} +7.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q + 2q^{11} - 8q^{13} + 12q^{23} + 16q^{25} + 26q^{35} - 20q^{37} + 20q^{47} + 12q^{49} - 8q^{59} - 16q^{71} - 6q^{73} + 18q^{83} + 52q^{95} + 14q^{97} + O(q^{100}) \)

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 0
\(3\) 0 0
\(4\) 0 0
\(5\) −3.60555 −1.61245 −0.806226 0.591608i \(-0.798493\pi\)
−0.806226 + 0.591608i \(0.798493\pi\)
\(6\) 0 0
\(7\) −3.60555 −1.36277 −0.681385 0.731925i \(-0.738622\pi\)
−0.681385 + 0.731925i \(0.738622\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −7.21110 −1.65434 −0.827170 0.561951i \(-0.810051\pi\)
−0.827170 + 0.561951i \(0.810051\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) 8.00000 1.60000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7.21110 1.33907 0.669534 0.742781i \(-0.266494\pi\)
0.669534 + 0.742781i \(0.266494\pi\)
\(30\) 0 0
\(31\) 3.60555 0.647576 0.323788 0.946130i \(-0.395044\pi\)
0.323788 + 0.946130i \(0.395044\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 13.0000 2.19740
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −7.21110 −1.12619 −0.563093 0.826394i \(-0.690389\pi\)
−0.563093 + 0.826394i \(0.690389\pi\)
\(42\) 0 0
\(43\) 7.21110 1.09968 0.549841 0.835269i \(-0.314688\pi\)
0.549841 + 0.835269i \(0.314688\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 10.0000 1.45865 0.729325 0.684167i \(-0.239834\pi\)
0.729325 + 0.684167i \(0.239834\pi\)
\(48\) 0 0
\(49\) 6.00000 0.857143
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.60555 0.495261 0.247630 0.968855i \(-0.420348\pi\)
0.247630 + 0.968855i \(0.420348\pi\)
\(54\) 0 0
\(55\) −3.60555 −0.486172
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 14.4222 1.78885
\(66\) 0 0
\(67\) 7.21110 0.880976 0.440488 0.897758i \(-0.354805\pi\)
0.440488 + 0.897758i \(0.354805\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −3.00000 −0.351123 −0.175562 0.984468i \(-0.556174\pi\)
−0.175562 + 0.984468i \(0.556174\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.60555 −0.410891
\(78\) 0 0
\(79\) 14.4222 1.62262 0.811312 0.584613i \(-0.198754\pi\)
0.811312 + 0.584613i \(0.198754\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 9.00000 0.987878 0.493939 0.869496i \(-0.335557\pi\)
0.493939 + 0.869496i \(0.335557\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 7.21110 0.764375 0.382188 0.924085i \(-0.375171\pi\)
0.382188 + 0.924085i \(0.375171\pi\)
\(90\) 0 0
\(91\) 14.4222 1.51186
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 26.0000 2.66754
\(96\) 0 0
\(97\) 7.00000 0.710742 0.355371 0.934725i \(-0.384354\pi\)
0.355371 + 0.934725i \(0.384354\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −10.8167 −1.07630 −0.538149 0.842850i \(-0.680876\pi\)
−0.538149 + 0.842850i \(0.680876\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 17.0000 1.64345 0.821726 0.569883i \(-0.193011\pi\)
0.821726 + 0.569883i \(0.193011\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −7.21110 −0.678363 −0.339182 0.940721i \(-0.610150\pi\)
−0.339182 + 0.940721i \(0.610150\pi\)
\(114\) 0 0
\(115\) −21.6333 −2.01732
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −10.8167 −0.967471
\(126\) 0 0
\(127\) 3.60555 0.319941 0.159970 0.987122i \(-0.448860\pi\)
0.159970 + 0.987122i \(0.448860\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1.00000 0.0873704 0.0436852 0.999045i \(-0.486090\pi\)
0.0436852 + 0.999045i \(0.486090\pi\)
\(132\) 0 0
\(133\) 26.0000 2.25449
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −21.6333 −1.84826 −0.924129 0.382080i \(-0.875208\pi\)
−0.924129 + 0.382080i \(0.875208\pi\)
\(138\) 0 0
\(139\) −14.4222 −1.22328 −0.611638 0.791138i \(-0.709489\pi\)
−0.611638 + 0.791138i \(0.709489\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −4.00000 −0.334497
\(144\) 0 0
\(145\) −26.0000 −2.15918
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 10.8167 0.886135 0.443067 0.896488i \(-0.353890\pi\)
0.443067 + 0.896488i \(0.353890\pi\)
\(150\) 0 0
\(151\) −10.8167 −0.880247 −0.440123 0.897937i \(-0.645065\pi\)
−0.440123 + 0.897937i \(0.645065\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −13.0000 −1.04419
\(156\) 0 0
\(157\) 4.00000 0.319235 0.159617 0.987179i \(-0.448974\pi\)
0.159617 + 0.987179i \(0.448974\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −21.6333 −1.70494
\(162\) 0 0
\(163\) 14.4222 1.12963 0.564817 0.825216i \(-0.308947\pi\)
0.564817 + 0.825216i \(0.308947\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2.00000 −0.154765 −0.0773823 0.997001i \(-0.524656\pi\)
−0.0773823 + 0.997001i \(0.524656\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 10.8167 0.822375 0.411187 0.911551i \(-0.365114\pi\)
0.411187 + 0.911551i \(0.365114\pi\)
\(174\) 0 0
\(175\) −28.8444 −2.18043
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 15.0000 1.12115 0.560576 0.828103i \(-0.310580\pi\)
0.560576 + 0.828103i \(0.310580\pi\)
\(180\) 0 0
\(181\) −8.00000 −0.594635 −0.297318 0.954779i \(-0.596092\pi\)
−0.297318 + 0.954779i \(0.596092\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 36.0555 2.65085
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) −15.0000 −1.07972 −0.539862 0.841754i \(-0.681524\pi\)
−0.539862 + 0.841754i \(0.681524\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 18.0278 1.28442 0.642212 0.766527i \(-0.278017\pi\)
0.642212 + 0.766527i \(0.278017\pi\)
\(198\) 0 0
\(199\) 3.60555 0.255591 0.127795 0.991801i \(-0.459210\pi\)
0.127795 + 0.991801i \(0.459210\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −26.0000 −1.82484
\(204\) 0 0
\(205\) 26.0000 1.81592
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −7.21110 −0.498802
\(210\) 0 0
\(211\) −7.21110 −0.496433 −0.248216 0.968705i \(-0.579844\pi\)
−0.248216 + 0.968705i \(0.579844\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −26.0000 −1.77319
\(216\) 0 0
\(217\) −13.0000 −0.882498
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −14.4222 −0.965782 −0.482891 0.875680i \(-0.660413\pi\)
−0.482891 + 0.875680i \(0.660413\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 21.6333 1.41725 0.708623 0.705588i \(-0.249317\pi\)
0.708623 + 0.705588i \(0.249317\pi\)
\(234\) 0 0
\(235\) −36.0555 −2.35200
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) 22.0000 1.41714 0.708572 0.705638i \(-0.249340\pi\)
0.708572 + 0.705638i \(0.249340\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −21.6333 −1.38210
\(246\) 0 0
\(247\) 28.8444 1.83533
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −24.0000 −1.51487 −0.757433 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) 0 0
\(253\) 6.00000 0.377217
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 14.4222 0.899632 0.449816 0.893121i \(-0.351489\pi\)
0.449816 + 0.893121i \(0.351489\pi\)
\(258\) 0 0
\(259\) 36.0555 2.24038
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −18.0000 −1.10993 −0.554964 0.831875i \(-0.687268\pi\)
−0.554964 + 0.831875i \(0.687268\pi\)
\(264\) 0 0
\(265\) −13.0000 −0.798584
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −21.6333 −1.31901 −0.659503 0.751702i \(-0.729233\pi\)
−0.659503 + 0.751702i \(0.729233\pi\)
\(270\) 0 0
\(271\) 10.8167 0.657065 0.328532 0.944493i \(-0.393446\pi\)
0.328532 + 0.944493i \(0.393446\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 8.00000 0.482418
\(276\) 0 0
\(277\) 8.00000 0.480673 0.240337 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 7.21110 0.428656 0.214328 0.976762i \(-0.431244\pi\)
0.214328 + 0.976762i \(0.431244\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 26.0000 1.53473
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 21.6333 1.26383 0.631916 0.775037i \(-0.282269\pi\)
0.631916 + 0.775037i \(0.282269\pi\)
\(294\) 0 0
\(295\) 14.4222 0.839693
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −24.0000 −1.38796
\(300\) 0 0
\(301\) −26.0000 −1.49862
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −18.0000 −1.02069 −0.510343 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) 0 0
\(313\) 1.00000 0.0565233 0.0282617 0.999601i \(-0.491003\pi\)
0.0282617 + 0.999601i \(0.491003\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −10.8167 −0.607524 −0.303762 0.952748i \(-0.598243\pi\)
−0.303762 + 0.952748i \(0.598243\pi\)
\(318\) 0 0
\(319\) 7.21110 0.403744
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −32.0000 −1.77504
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −36.0555 −1.98780
\(330\) 0 0
\(331\) 7.21110 0.396358 0.198179 0.980166i \(-0.436497\pi\)
0.198179 + 0.980166i \(0.436497\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −26.0000 −1.42053
\(336\) 0 0
\(337\) 26.0000 1.41631 0.708155 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 3.60555 0.195252
\(342\) 0 0
\(343\) 3.60555 0.194681
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −1.00000 −0.0536828 −0.0268414 0.999640i \(-0.508545\pi\)
−0.0268414 + 0.999640i \(0.508545\pi\)
\(348\) 0 0
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 7.21110 0.383808 0.191904 0.981414i \(-0.438534\pi\)
0.191904 + 0.981414i \(0.438534\pi\)
\(354\) 0 0
\(355\) 28.8444 1.53090
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −10.0000 −0.527780 −0.263890 0.964553i \(-0.585006\pi\)
−0.263890 + 0.964553i \(0.585006\pi\)
\(360\) 0 0
\(361\) 33.0000 1.73684
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 10.8167 0.566170
\(366\) 0 0
\(367\) −10.8167 −0.564625 −0.282312 0.959323i \(-0.591101\pi\)
−0.282312 + 0.959323i \(0.591101\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −13.0000 −0.674926
\(372\) 0 0
\(373\) 16.0000 0.828449 0.414224 0.910175i \(-0.364053\pi\)
0.414224 + 0.910175i \(0.364053\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −28.8444 −1.48556
\(378\) 0 0
\(379\) 14.4222 0.740819 0.370409 0.928869i \(-0.379217\pi\)
0.370409 + 0.928869i \(0.379217\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −8.00000 −0.408781 −0.204390 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) 0 0
\(385\) 13.0000 0.662541
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 10.8167 0.548426 0.274213 0.961669i \(-0.411583\pi\)
0.274213 + 0.961669i \(0.411583\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −52.0000 −2.61640
\(396\) 0 0
\(397\) −4.00000 −0.200754 −0.100377 0.994949i \(-0.532005\pi\)
−0.100377 + 0.994949i \(0.532005\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −14.4222 −0.720211 −0.360105 0.932912i \(-0.617259\pi\)
−0.360105 + 0.932912i \(0.617259\pi\)
\(402\) 0 0
\(403\) −14.4222 −0.718421
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −10.0000 −0.495682
\(408\) 0 0
\(409\) 7.00000 0.346128 0.173064 0.984911i \(-0.444633\pi\)
0.173064 + 0.984911i \(0.444633\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 14.4222 0.709670
\(414\) 0 0
\(415\) −32.4500 −1.59291
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 4.00000 0.195413 0.0977064 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 0 0
\(421\) −32.0000 −1.55958 −0.779792 0.626038i \(-0.784675\pi\)
−0.779792 + 0.626038i \(0.784675\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −18.0000 −0.867029 −0.433515 0.901146i \(-0.642727\pi\)
−0.433515 + 0.901146i \(0.642727\pi\)
\(432\) 0 0
\(433\) −35.0000 −1.68199 −0.840996 0.541041i \(-0.818030\pi\)
−0.840996 + 0.541041i \(0.818030\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −43.2666 −2.06972
\(438\) 0 0
\(439\) 18.0278 0.860418 0.430209 0.902729i \(-0.358440\pi\)
0.430209 + 0.902729i \(0.358440\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 0 0
\(445\) −26.0000 −1.23252
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 7.21110 0.340313 0.170156 0.985417i \(-0.445573\pi\)
0.170156 + 0.985417i \(0.445573\pi\)
\(450\) 0 0
\(451\) −7.21110 −0.339558
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −52.0000 −2.43780
\(456\) 0 0
\(457\) 23.0000 1.07589 0.537947 0.842978i \(-0.319200\pi\)
0.537947 + 0.842978i \(0.319200\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 10.8167 0.503782 0.251891 0.967756i \(-0.418948\pi\)
0.251891 + 0.967756i \(0.418948\pi\)
\(462\) 0 0
\(463\) 25.2389 1.17295 0.586475 0.809968i \(-0.300515\pi\)
0.586475 + 0.809968i \(0.300515\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −23.0000 −1.06431 −0.532157 0.846646i \(-0.678618\pi\)
−0.532157 + 0.846646i \(0.678618\pi\)
\(468\) 0 0
\(469\) −26.0000 −1.20057
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 7.21110 0.331567
\(474\) 0 0
\(475\) −57.6888 −2.64694
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 42.0000 1.91903 0.959514 0.281659i \(-0.0908848\pi\)
0.959514 + 0.281659i \(0.0908848\pi\)
\(480\) 0 0
\(481\) 40.0000 1.82384
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −25.2389 −1.14604
\(486\) 0 0
\(487\) −14.4222 −0.653532 −0.326766 0.945105i \(-0.605959\pi\)
−0.326766 + 0.945105i \(0.605959\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 7.00000 0.315906 0.157953 0.987447i \(-0.449511\pi\)
0.157953 + 0.987447i \(0.449511\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 28.8444 1.29385
\(498\) 0 0
\(499\) −21.6333 −0.968440 −0.484220 0.874946i \(-0.660897\pi\)
−0.484220 + 0.874946i \(0.660897\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 42.0000 1.87269 0.936344 0.351085i \(-0.114187\pi\)
0.936344 + 0.351085i \(0.114187\pi\)
\(504\) 0 0
\(505\) 39.0000 1.73548
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −10.8167 −0.479440 −0.239720 0.970842i \(-0.577056\pi\)
−0.239720 + 0.970842i \(0.577056\pi\)
\(510\) 0 0
\(511\) 10.8167 0.478501
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 10.0000 0.439799
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 43.2666 1.89554 0.947772 0.318947i \(-0.103329\pi\)
0.947772 + 0.318947i \(0.103329\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 28.8444 1.24939
\(534\) 0 0
\(535\) −61.2944 −2.64999
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) −4.00000 −0.171973 −0.0859867 0.996296i \(-0.527404\pi\)
−0.0859867 + 0.996296i \(0.527404\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 7.21110 0.308890
\(546\) 0 0
\(547\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −52.0000 −2.21527
\(552\) 0 0
\(553\) −52.0000 −2.21126
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 39.6611 1.68049 0.840247 0.542204i \(-0.182410\pi\)
0.840247 + 0.542204i \(0.182410\pi\)
\(558\) 0 0
\(559\) −28.8444 −1.21999
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 39.0000 1.64365 0.821827 0.569737i \(-0.192955\pi\)
0.821827 + 0.569737i \(0.192955\pi\)
\(564\) 0 0
\(565\) 26.0000 1.09383
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 14.4222 0.604610 0.302305 0.953211i \(-0.402244\pi\)
0.302305 + 0.953211i \(0.402244\pi\)
\(570\) 0 0
\(571\) 14.4222 0.603550 0.301775 0.953379i \(-0.402421\pi\)
0.301775 + 0.953379i \(0.402421\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 48.0000 2.00174
\(576\) 0 0
\(577\) 6.00000 0.249783 0.124892 0.992170i \(-0.460142\pi\)
0.124892 + 0.992170i \(0.460142\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −32.4500 −1.34625
\(582\) 0 0
\(583\) 3.60555 0.149327
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −3.00000 −0.123823 −0.0619116 0.998082i \(-0.519720\pi\)
−0.0619116 + 0.998082i \(0.519720\pi\)
\(588\) 0 0
\(589\) −26.0000 −1.07131
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 21.6333 0.888373 0.444187 0.895934i \(-0.353493\pi\)
0.444187 + 0.895934i \(0.353493\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −14.0000 −0.572024 −0.286012 0.958226i \(-0.592330\pi\)
−0.286012 + 0.958226i \(0.592330\pi\)
\(600\) 0 0
\(601\) −17.0000 −0.693444 −0.346722 0.937968i \(-0.612705\pi\)
−0.346722 + 0.937968i \(0.612705\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 36.0555 1.46587
\(606\) 0 0
\(607\) 14.4222 0.585379 0.292690 0.956208i \(-0.405450\pi\)
0.292690 + 0.956208i \(0.405450\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −40.0000 −1.61823
\(612\) 0 0
\(613\) 18.0000 0.727013 0.363507 0.931592i \(-0.381579\pi\)
0.363507 + 0.931592i \(0.381579\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −21.6333 −0.870924 −0.435462 0.900207i \(-0.643415\pi\)
−0.435462 + 0.900207i \(0.643415\pi\)
\(618\) 0 0
\(619\) 43.2666 1.73903 0.869516 0.493905i \(-0.164431\pi\)
0.869516 + 0.493905i \(0.164431\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −26.0000 −1.04167
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 25.2389 1.00474 0.502372 0.864652i \(-0.332461\pi\)
0.502372 + 0.864652i \(0.332461\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −13.0000 −0.515889
\(636\) 0 0
\(637\) −24.0000 −0.950915
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −36.0555 −1.42411 −0.712054 0.702125i \(-0.752235\pi\)
−0.712054 + 0.702125i \(0.752235\pi\)
\(642\) 0 0
\(643\) −43.2666 −1.70627 −0.853134 0.521691i \(-0.825301\pi\)
−0.853134 + 0.521691i \(0.825301\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) −4.00000 −0.157014
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −18.0278 −0.705481 −0.352740 0.935721i \(-0.614750\pi\)
−0.352740 + 0.935721i \(0.614750\pi\)
\(654\) 0 0
\(655\) −3.60555 −0.140881
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 27.0000 1.05177 0.525885 0.850555i \(-0.323734\pi\)
0.525885 + 0.850555i \(0.323734\pi\)
\(660\) 0 0
\(661\) −14.0000 −0.544537 −0.272268 0.962221i \(-0.587774\pi\)
−0.272268 + 0.962221i \(0.587774\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −93.7443 −3.63525
\(666\) 0 0
\(667\) 43.2666 1.67529
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −19.0000 −0.732396 −0.366198 0.930537i \(-0.619341\pi\)
−0.366198 + 0.930537i \(0.619341\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −7.21110 −0.277145 −0.138573 0.990352i \(-0.544251\pi\)
−0.138573 + 0.990352i \(0.544251\pi\)
\(678\) 0 0
\(679\) −25.2389 −0.968579
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) 78.0000 2.98023
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −14.4222 −0.549442
\(690\) 0 0
\(691\) 14.4222 0.548647 0.274323 0.961638i \(-0.411546\pi\)
0.274323 + 0.961638i \(0.411546\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 52.0000 1.97247
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −10.8167 −0.408539 −0.204270 0.978915i \(-0.565482\pi\)
−0.204270 + 0.978915i \(0.565482\pi\)
\(702\) 0 0
\(703\) 72.1110 2.71972
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 39.0000 1.46675
\(708\) 0 0
\(709\) −4.00000 −0.150223 −0.0751116 0.997175i \(-0.523931\pi\)
−0.0751116 + 0.997175i \(0.523931\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 21.6333 0.810174
\(714\) 0 0
\(715\) 14.4222 0.539360
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 4.00000 0.149175 0.0745874 0.997214i \(-0.476236\pi\)
0.0745874 + 0.997214i \(0.476236\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 57.6888 2.14251
\(726\) 0 0
\(727\) −32.4500 −1.20350 −0.601751 0.798684i \(-0.705530\pi\)
−0.601751 + 0.798684i \(0.705530\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 46.0000 1.69905 0.849524 0.527549i \(-0.176889\pi\)
0.849524 + 0.527549i \(0.176889\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 7.21110 0.265624
\(738\) 0 0
\(739\) −28.8444 −1.06106 −0.530529 0.847667i \(-0.678007\pi\)
−0.530529 + 0.847667i \(0.678007\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 36.0000 1.32071 0.660356 0.750953i \(-0.270405\pi\)
0.660356 + 0.750953i \(0.270405\pi\)
\(744\) 0 0
\(745\) −39.0000 −1.42885
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −61.2944 −2.23965
\(750\) 0 0
\(751\) 46.8722 1.71039 0.855195 0.518307i \(-0.173437\pi\)
0.855195 + 0.518307i \(0.173437\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 39.0000 1.41936
\(756\) 0 0
\(757\) −42.0000 −1.52652 −0.763258 0.646094i \(-0.776401\pi\)
−0.763258 + 0.646094i \(0.776401\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 7.21110 0.261059
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 16.0000 0.577727
\(768\) 0 0
\(769\) −11.0000 −0.396670 −0.198335 0.980134i \(-0.563553\pi\)
−0.198335 + 0.980134i \(0.563553\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −21.6333 −0.778096 −0.389048 0.921217i \(-0.627196\pi\)
−0.389048 + 0.921217i \(0.627196\pi\)
\(774\) 0 0
\(775\) 28.8444 1.03612
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 52.0000 1.86309
\(780\) 0 0
\(781\) −8.00000 −0.286263
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −14.4222 −0.514751
\(786\) 0 0
\(787\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 26.0000 0.924454
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 10.8167 0.383146 0.191573 0.981478i \(-0.438641\pi\)
0.191573 + 0.981478i \(0.438641\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −3.00000 −0.105868
\(804\) 0 0
\(805\) 78.0000 2.74914
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 50.4777 1.77470 0.887351 0.461095i \(-0.152543\pi\)
0.887351 + 0.461095i \(0.152543\pi\)
\(810\) 0 0
\(811\) −7.21110 −0.253216 −0.126608 0.991953i \(-0.540409\pi\)
−0.126608 + 0.991953i \(0.540409\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −52.0000 −1.82148
\(816\) 0 0
\(817\) −52.0000 −1.81925
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 7.21110 0.251669 0.125835 0.992051i \(-0.459839\pi\)
0.125835 + 0.992051i \(0.459839\pi\)
\(822\) 0 0
\(823\) 18.0278 0.628408 0.314204 0.949355i \(-0.398262\pi\)
0.314204 + 0.949355i \(0.398262\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 16.0000 0.556375 0.278187 0.960527i \(-0.410266\pi\)
0.278187 + 0.960527i \(0.410266\pi\)
\(828\) 0 0
\(829\) 46.0000 1.59765 0.798823 0.601566i \(-0.205456\pi\)
0.798823 + 0.601566i \(0.205456\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 7.21110 0.249550
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 40.0000 1.38095 0.690477 0.723355i \(-0.257401\pi\)
0.690477 + 0.723355i \(0.257401\pi\)
\(840\) 0 0
\(841\) 23.0000 0.793103
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −10.8167 −0.372104
\(846\) 0 0
\(847\) 36.0555 1.23888
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −60.0000 −2.05677
\(852\) 0 0
\(853\) 42.0000 1.43805 0.719026 0.694983i \(-0.244588\pi\)
0.719026 + 0.694983i \(0.244588\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 28.8444 0.985306 0.492653 0.870226i \(-0.336027\pi\)
0.492653 + 0.870226i \(0.336027\pi\)
\(858\) 0 0
\(859\) 28.8444 0.984159 0.492079 0.870550i \(-0.336237\pi\)
0.492079 + 0.870550i \(0.336237\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 28.0000 0.953131 0.476566 0.879139i \(-0.341881\pi\)
0.476566 + 0.879139i \(0.341881\pi\)
\(864\) 0 0
\(865\) −39.0000 −1.32604
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 14.4222 0.489240
\(870\) 0 0
\(871\) −28.8444 −0.977356
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 39.0000 1.31844
\(876\) 0 0
\(877\) −34.0000 −1.14810 −0.574049 0.818821i \(-0.694628\pi\)
−0.574049 + 0.818821i \(0.694628\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 43.2666 1.45769 0.728845 0.684679i \(-0.240058\pi\)
0.728845 + 0.684679i \(0.240058\pi\)
\(882\) 0 0
\(883\) 50.4777 1.69871 0.849355 0.527822i \(-0.176991\pi\)
0.849355 + 0.527822i \(0.176991\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 36.0000 1.20876 0.604381 0.796696i \(-0.293421\pi\)
0.604381 + 0.796696i \(0.293421\pi\)
\(888\) 0 0
\(889\) −13.0000 −0.436006
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −72.1110 −2.41310
\(894\) 0 0
\(895\) −54.0833 −1.80780
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 26.0000 0.867149
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 28.8444 0.958821
\(906\) 0 0
\(907\) −7.21110 −0.239441 −0.119720 0.992808i \(-0.538200\pi\)
−0.119720 + 0.992808i \(0.538200\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −30.0000 −0.993944 −0.496972 0.867766i \(-0.665555\pi\)
−0.496972 + 0.867766i \(0.665555\pi\)
\(912\) 0 0
\(913\) 9.00000 0.297857
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −3.60555 −0.119066
\(918\) 0 0
\(919\) −54.0833 −1.78404 −0.892021 0.451994i \(-0.850713\pi\)
−0.892021 + 0.451994i \(0.850713\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 32.0000 1.05329
\(924\) 0 0
\(925\) −80.0000 −2.63038
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −43.2666 −1.41953 −0.709766 0.704438i \(-0.751199\pi\)
−0.709766 + 0.704438i \(0.751199\pi\)
\(930\) 0 0
\(931\) −43.2666 −1.41801
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 35.0000 1.14340 0.571700 0.820463i \(-0.306284\pi\)
0.571700 + 0.820463i \(0.306284\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 25.2389 0.822763 0.411382 0.911463i \(-0.365046\pi\)
0.411382 + 0.911463i \(0.365046\pi\)
\(942\) 0 0
\(943\) −43.2666 −1.40895
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 27.0000 0.877382 0.438691 0.898638i \(-0.355442\pi\)
0.438691 + 0.898638i \(0.355442\pi\)
\(948\) 0 0
\(949\) 12.0000 0.389536
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −43.2666 −1.40154 −0.700772 0.713386i \(-0.747161\pi\)
−0.700772 + 0.713386i \(0.747161\pi\)
\(954\) 0 0
\(955\) −43.2666 −1.40007
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 78.0000 2.51875
\(960\) 0 0
\(961\) −18.0000 −0.580645
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 54.0833 1.74100
\(966\) 0 0
\(967\) 10.8167 0.347840 0.173920 0.984760i \(-0.444357\pi\)
0.173920 + 0.984760i \(0.444357\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 25.0000 0.802288 0.401144 0.916015i \(-0.368613\pi\)
0.401144 + 0.916015i \(0.368613\pi\)
\(972\) 0 0
\(973\) 52.0000 1.66704
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 36.0555 1.15352 0.576759 0.816914i \(-0.304317\pi\)
0.576759 + 0.816914i \(0.304317\pi\)
\(978\) 0 0
\(979\) 7.21110 0.230468
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −6.00000 −0.191370 −0.0956851 0.995412i \(-0.530504\pi\)
−0.0956851 + 0.995412i \(0.530504\pi\)
\(984\) 0 0
\(985\) −65.0000 −2.07107
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 43.2666 1.37580
\(990\) 0 0
\(991\) −61.2944 −1.94708 −0.973540 0.228517i \(-0.926612\pi\)
−0.973540 + 0.228517i \(0.926612\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −13.0000 −0.412128
\(996\) 0 0
\(997\) −36.0000 −1.14013 −0.570066 0.821599i \(-0.693082\pi\)
−0.570066 + 0.821599i \(0.693082\pi\)
\(998\) 0 0
\(999\) 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 1728.2.a.bd.1.1 2
3.2 odd 2 1728.2.a.bc.1.2 2
4.3 odd 2 1728.2.a.bc.1.1 2
8.3 odd 2 864.2.a.n.1.2 yes 2
8.5 even 2 864.2.a.m.1.2 yes 2
12.11 even 2 inner 1728.2.a.bd.1.2 2
24.5 odd 2 864.2.a.n.1.1 yes 2
24.11 even 2 864.2.a.m.1.1 2
72.5 odd 6 2592.2.i.bb.865.2 4
72.11 even 6 2592.2.i.bc.1729.2 4
72.13 even 6 2592.2.i.bc.865.1 4
72.29 odd 6 2592.2.i.bb.1729.2 4
72.43 odd 6 2592.2.i.bb.1729.1 4
72.59 even 6 2592.2.i.bc.865.2 4
72.61 even 6 2592.2.i.bc.1729.1 4
72.67 odd 6 2592.2.i.bb.865.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.a.m.1.1 2 24.11 even 2
864.2.a.m.1.2 yes 2 8.5 even 2
864.2.a.n.1.1 yes 2 24.5 odd 2
864.2.a.n.1.2 yes 2 8.3 odd 2
1728.2.a.bc.1.1 2 4.3 odd 2
1728.2.a.bc.1.2 2 3.2 odd 2
1728.2.a.bd.1.1 2 1.1 even 1 trivial
1728.2.a.bd.1.2 2 12.11 even 2 inner
2592.2.i.bb.865.1 4 72.67 odd 6
2592.2.i.bb.865.2 4 72.5 odd 6
2592.2.i.bb.1729.1 4 72.43 odd 6
2592.2.i.bb.1729.2 4 72.29 odd 6
2592.2.i.bc.865.1 4 72.13 even 6
2592.2.i.bc.865.2 4 72.59 even 6
2592.2.i.bc.1729.1 4 72.61 even 6
2592.2.i.bc.1729.2 4 72.11 even 6