Properties

Label 1728.2.c.e.1727.4
Level $1728$
Weight $2$
Character 1728.1727
Analytic conductor $13.798$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.7981494693\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 432)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1727.4
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1727
Dual form 1728.2.c.e.1727.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000i q^{5} +1.73205i q^{7} +O(q^{10})\) \(q+3.00000i q^{5} +1.73205i q^{7} -5.19615 q^{11} +2.00000 q^{13} -6.00000i q^{17} +6.92820i q^{19} -4.00000 q^{25} +6.00000i q^{29} +5.19615i q^{31} -5.19615 q^{35} -8.00000 q^{37} -10.3923i q^{43} -10.3923 q^{47} +4.00000 q^{49} -9.00000i q^{53} -15.5885i q^{55} -10.3923 q^{59} +4.00000 q^{61} +6.00000i q^{65} -3.46410i q^{67} -10.3923 q^{71} +1.00000 q^{73} -9.00000i q^{77} +3.46410i q^{79} +5.19615 q^{83} +18.0000 q^{85} +6.00000i q^{89} +3.46410i q^{91} -20.7846 q^{95} -5.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 8q^{13} - 16q^{25} - 32q^{37} + 16q^{49} + 16q^{61} + 4q^{73} + 72q^{85} - 20q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(5\) 3.00000i 1.34164i 0.741620 + 0.670820i \(0.234058\pi\)
−0.741620 + 0.670820i \(0.765942\pi\)
\(6\) 0 0
\(7\) 1.73205i 0.654654i 0.944911 + 0.327327i \(0.106148\pi\)
−0.944911 + 0.327327i \(0.893852\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.19615 −1.56670 −0.783349 0.621582i \(-0.786490\pi\)
−0.783349 + 0.621582i \(0.786490\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 6.00000i − 1.45521i −0.685994 0.727607i \(-0.740633\pi\)
0.685994 0.727607i \(-0.259367\pi\)
\(18\) 0 0
\(19\) 6.92820i 1.58944i 0.606977 + 0.794719i \(0.292382\pi\)
−0.606977 + 0.794719i \(0.707618\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.00000i 1.11417i 0.830455 + 0.557086i \(0.188081\pi\)
−0.830455 + 0.557086i \(0.811919\pi\)
\(30\) 0 0
\(31\) 5.19615i 0.933257i 0.884454 + 0.466628i \(0.154531\pi\)
−0.884454 + 0.466628i \(0.845469\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −5.19615 −0.878310
\(36\) 0 0
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) − 10.3923i − 1.58481i −0.609994 0.792406i \(-0.708828\pi\)
0.609994 0.792406i \(-0.291172\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −10.3923 −1.51587 −0.757937 0.652328i \(-0.773792\pi\)
−0.757937 + 0.652328i \(0.773792\pi\)
\(48\) 0 0
\(49\) 4.00000 0.571429
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 9.00000i − 1.23625i −0.786082 0.618123i \(-0.787894\pi\)
0.786082 0.618123i \(-0.212106\pi\)
\(54\) 0 0
\(55\) − 15.5885i − 2.10195i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −10.3923 −1.35296 −0.676481 0.736460i \(-0.736496\pi\)
−0.676481 + 0.736460i \(0.736496\pi\)
\(60\) 0 0
\(61\) 4.00000 0.512148 0.256074 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.00000i 0.744208i
\(66\) 0 0
\(67\) − 3.46410i − 0.423207i −0.977356 0.211604i \(-0.932131\pi\)
0.977356 0.211604i \(-0.0678686\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −10.3923 −1.23334 −0.616670 0.787222i \(-0.711519\pi\)
−0.616670 + 0.787222i \(0.711519\pi\)
\(72\) 0 0
\(73\) 1.00000 0.117041 0.0585206 0.998286i \(-0.481362\pi\)
0.0585206 + 0.998286i \(0.481362\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 9.00000i − 1.02565i
\(78\) 0 0
\(79\) 3.46410i 0.389742i 0.980829 + 0.194871i \(0.0624288\pi\)
−0.980829 + 0.194871i \(0.937571\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 5.19615 0.570352 0.285176 0.958475i \(-0.407948\pi\)
0.285176 + 0.958475i \(0.407948\pi\)
\(84\) 0 0
\(85\) 18.0000 1.95237
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000i 0.635999i 0.948091 + 0.317999i \(0.103011\pi\)
−0.948091 + 0.317999i \(0.896989\pi\)
\(90\) 0 0
\(91\) 3.46410i 0.363137i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −20.7846 −2.13246
\(96\) 0 0
\(97\) −5.00000 −0.507673 −0.253837 0.967247i \(-0.581693\pi\)
−0.253837 + 0.967247i \(0.581693\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 9.00000i − 0.895533i −0.894150 0.447767i \(-0.852219\pi\)
0.894150 0.447767i \(-0.147781\pi\)
\(102\) 0 0
\(103\) − 3.46410i − 0.341328i −0.985329 0.170664i \(-0.945409\pi\)
0.985329 0.170664i \(-0.0545913\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −5.19615 −0.502331 −0.251166 0.967944i \(-0.580814\pi\)
−0.251166 + 0.967944i \(0.580814\pi\)
\(108\) 0 0
\(109\) −16.0000 −1.53252 −0.766261 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 12.0000i 1.12887i 0.825479 + 0.564433i \(0.190905\pi\)
−0.825479 + 0.564433i \(0.809095\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 10.3923 0.952661
\(120\) 0 0
\(121\) 16.0000 1.45455
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 3.00000i 0.268328i
\(126\) 0 0
\(127\) 5.19615i 0.461084i 0.973062 + 0.230542i \(0.0740499\pi\)
−0.973062 + 0.230542i \(0.925950\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 5.19615 0.453990 0.226995 0.973896i \(-0.427110\pi\)
0.226995 + 0.973896i \(0.427110\pi\)
\(132\) 0 0
\(133\) −12.0000 −1.04053
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.00000i 0.512615i 0.966595 + 0.256307i \(0.0825059\pi\)
−0.966595 + 0.256307i \(0.917494\pi\)
\(138\) 0 0
\(139\) − 6.92820i − 0.587643i −0.955860 0.293821i \(-0.905073\pi\)
0.955860 0.293821i \(-0.0949270\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −10.3923 −0.869048
\(144\) 0 0
\(145\) −18.0000 −1.49482
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 15.0000i 1.22885i 0.788976 + 0.614424i \(0.210612\pi\)
−0.788976 + 0.614424i \(0.789388\pi\)
\(150\) 0 0
\(151\) 8.66025i 0.704761i 0.935857 + 0.352381i \(0.114628\pi\)
−0.935857 + 0.352381i \(0.885372\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −15.5885 −1.25210
\(156\) 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
\(161\) 0 0
\(162\) 0 0
\(163\) 3.46410i 0.271329i 0.990755 + 0.135665i \(0.0433170\pi\)
−0.990755 + 0.135665i \(0.956683\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 10.3923 0.804181 0.402090 0.915600i \(-0.368284\pi\)
0.402090 + 0.915600i \(0.368284\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 3.00000i − 0.228086i −0.993476 0.114043i \(-0.963620\pi\)
0.993476 0.114043i \(-0.0363801\pi\)
\(174\) 0 0
\(175\) − 6.92820i − 0.523723i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 5.19615 0.388379 0.194189 0.980964i \(-0.437792\pi\)
0.194189 + 0.980964i \(0.437792\pi\)
\(180\) 0 0
\(181\) 8.00000 0.594635 0.297318 0.954779i \(-0.403908\pi\)
0.297318 + 0.954779i \(0.403908\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 24.0000i − 1.76452i
\(186\) 0 0
\(187\) 31.1769i 2.27988i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −10.3923 −0.751961 −0.375980 0.926628i \(-0.622694\pi\)
−0.375980 + 0.926628i \(0.622694\pi\)
\(192\) 0 0
\(193\) 11.0000 0.791797 0.395899 0.918294i \(-0.370433\pi\)
0.395899 + 0.918294i \(0.370433\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 15.0000i 1.06871i 0.845262 + 0.534353i \(0.179445\pi\)
−0.845262 + 0.534353i \(0.820555\pi\)
\(198\) 0 0
\(199\) 22.5167i 1.59616i 0.602549 + 0.798082i \(0.294152\pi\)
−0.602549 + 0.798082i \(0.705848\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −10.3923 −0.729397
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 36.0000i − 2.49017i
\(210\) 0 0
\(211\) 13.8564i 0.953914i 0.878927 + 0.476957i \(0.158260\pi\)
−0.878927 + 0.476957i \(0.841740\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 31.1769 2.12625
\(216\) 0 0
\(217\) −9.00000 −0.610960
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 12.0000i − 0.807207i
\(222\) 0 0
\(223\) − 10.3923i − 0.695920i −0.937509 0.347960i \(-0.886874\pi\)
0.937509 0.347960i \(-0.113126\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 10.3923 0.689761 0.344881 0.938647i \(-0.387919\pi\)
0.344881 + 0.938647i \(0.387919\pi\)
\(228\) 0 0
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.00000i 0.393073i 0.980497 + 0.196537i \(0.0629694\pi\)
−0.980497 + 0.196537i \(0.937031\pi\)
\(234\) 0 0
\(235\) − 31.1769i − 2.03376i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 20.7846 1.34444 0.672222 0.740349i \(-0.265340\pi\)
0.672222 + 0.740349i \(0.265340\pi\)
\(240\) 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
\(245\) 12.0000i 0.766652i
\(246\) 0 0
\(247\) 13.8564i 0.881662i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −10.3923 −0.655956 −0.327978 0.944685i \(-0.606367\pi\)
−0.327978 + 0.944685i \(0.606367\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6.00000i 0.374270i 0.982334 + 0.187135i \(0.0599201\pi\)
−0.982334 + 0.187135i \(0.940080\pi\)
\(258\) 0 0
\(259\) − 13.8564i − 0.860995i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 27.0000 1.65860
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 30.0000i 1.82913i 0.404436 + 0.914566i \(0.367468\pi\)
−0.404436 + 0.914566i \(0.632532\pi\)
\(270\) 0 0
\(271\) − 1.73205i − 0.105215i −0.998615 0.0526073i \(-0.983247\pi\)
0.998615 0.0526073i \(-0.0167532\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 20.7846 1.25336
\(276\) 0 0
\(277\) −28.0000 −1.68236 −0.841178 0.540758i \(-0.818138\pi\)
−0.841178 + 0.540758i \(0.818138\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 6.00000i − 0.357930i −0.983855 0.178965i \(-0.942725\pi\)
0.983855 0.178965i \(-0.0572749\pi\)
\(282\) 0 0
\(283\) 20.7846i 1.23552i 0.786368 + 0.617758i \(0.211959\pi\)
−0.786368 + 0.617758i \(0.788041\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 6.00000i − 0.350524i −0.984522 0.175262i \(-0.943923\pi\)
0.984522 0.175262i \(-0.0560772\pi\)
\(294\) 0 0
\(295\) − 31.1769i − 1.81519i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 18.0000 1.03750
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 12.0000i 0.687118i
\(306\) 0 0
\(307\) − 31.1769i − 1.77936i −0.456584 0.889680i \(-0.650927\pi\)
0.456584 0.889680i \(-0.349073\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 20.7846 1.17859 0.589294 0.807919i \(-0.299406\pi\)
0.589294 + 0.807919i \(0.299406\pi\)
\(312\) 0 0
\(313\) −31.0000 −1.75222 −0.876112 0.482108i \(-0.839871\pi\)
−0.876112 + 0.482108i \(0.839871\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 15.0000i − 0.842484i −0.906948 0.421242i \(-0.861594\pi\)
0.906948 0.421242i \(-0.138406\pi\)
\(318\) 0 0
\(319\) − 31.1769i − 1.74557i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 41.5692 2.31297
\(324\) 0 0
\(325\) −8.00000 −0.443760
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 18.0000i − 0.992372i
\(330\) 0 0
\(331\) 3.46410i 0.190404i 0.995458 + 0.0952021i \(0.0303497\pi\)
−0.995458 + 0.0952021i \(0.969650\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 10.3923 0.567792
\(336\) 0 0
\(337\) −2.00000 −0.108947 −0.0544735 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 27.0000i − 1.46213i
\(342\) 0 0
\(343\) 19.0526i 1.02874i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 15.5885 0.836832 0.418416 0.908255i \(-0.362585\pi\)
0.418416 + 0.908255i \(0.362585\pi\)
\(348\) 0 0
\(349\) 4.00000 0.214115 0.107058 0.994253i \(-0.465857\pi\)
0.107058 + 0.994253i \(0.465857\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 36.0000i 1.91609i 0.286623 + 0.958043i \(0.407467\pi\)
−0.286623 + 0.958043i \(0.592533\pi\)
\(354\) 0 0
\(355\) − 31.1769i − 1.65470i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 10.3923 0.548485 0.274242 0.961661i \(-0.411573\pi\)
0.274242 + 0.961661i \(0.411573\pi\)
\(360\) 0 0
\(361\) −29.0000 −1.52632
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 3.00000i 0.157027i
\(366\) 0 0
\(367\) − 8.66025i − 0.452062i −0.974120 0.226031i \(-0.927425\pi\)
0.974120 0.226031i \(-0.0725750\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 15.5885 0.809312
\(372\) 0 0
\(373\) −2.00000 −0.103556 −0.0517780 0.998659i \(-0.516489\pi\)
−0.0517780 + 0.998659i \(0.516489\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000i 0.618031i
\(378\) 0 0
\(379\) 27.7128i 1.42351i 0.702427 + 0.711756i \(0.252100\pi\)
−0.702427 + 0.711756i \(0.747900\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −31.1769 −1.59307 −0.796533 0.604595i \(-0.793335\pi\)
−0.796533 + 0.604595i \(0.793335\pi\)
\(384\) 0 0
\(385\) 27.0000 1.37605
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 9.00000i − 0.456318i −0.973624 0.228159i \(-0.926729\pi\)
0.973624 0.228159i \(-0.0732706\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −10.3923 −0.522894
\(396\) 0 0
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 24.0000i 1.19850i 0.800561 + 0.599251i \(0.204535\pi\)
−0.800561 + 0.599251i \(0.795465\pi\)
\(402\) 0 0
\(403\) 10.3923i 0.517678i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 41.5692 2.06051
\(408\) 0 0
\(409\) 25.0000 1.23617 0.618085 0.786111i \(-0.287909\pi\)
0.618085 + 0.786111i \(0.287909\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 18.0000i − 0.885722i
\(414\) 0 0
\(415\) 15.5885i 0.765207i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 10.3923 0.507697 0.253849 0.967244i \(-0.418303\pi\)
0.253849 + 0.967244i \(0.418303\pi\)
\(420\) 0 0
\(421\) −4.00000 −0.194948 −0.0974740 0.995238i \(-0.531076\pi\)
−0.0974740 + 0.995238i \(0.531076\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 24.0000i 1.16417i
\(426\) 0 0
\(427\) 6.92820i 0.335279i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −31.1769 −1.50174 −0.750870 0.660451i \(-0.770365\pi\)
−0.750870 + 0.660451i \(0.770365\pi\)
\(432\) 0 0
\(433\) 11.0000 0.528626 0.264313 0.964437i \(-0.414855\pi\)
0.264313 + 0.964437i \(0.414855\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 15.5885i 0.743996i 0.928233 + 0.371998i \(0.121327\pi\)
−0.928233 + 0.371998i \(0.878673\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 31.1769 1.48126 0.740630 0.671913i \(-0.234527\pi\)
0.740630 + 0.671913i \(0.234527\pi\)
\(444\) 0 0
\(445\) −18.0000 −0.853282
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 36.0000i − 1.69895i −0.527633 0.849473i \(-0.676920\pi\)
0.527633 0.849473i \(-0.323080\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −10.3923 −0.487199
\(456\) 0 0
\(457\) 1.00000 0.0467780 0.0233890 0.999726i \(-0.492554\pi\)
0.0233890 + 0.999726i \(0.492554\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 9.00000i 0.419172i 0.977790 + 0.209586i \(0.0672116\pi\)
−0.977790 + 0.209586i \(0.932788\pi\)
\(462\) 0 0
\(463\) 32.9090i 1.52941i 0.644381 + 0.764705i \(0.277115\pi\)
−0.644381 + 0.764705i \(0.722885\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 25.9808 1.20225 0.601123 0.799156i \(-0.294720\pi\)
0.601123 + 0.799156i \(0.294720\pi\)
\(468\) 0 0
\(469\) 6.00000 0.277054
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 54.0000i 2.48292i
\(474\) 0 0
\(475\) − 27.7128i − 1.27155i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 10.3923 0.474837 0.237418 0.971408i \(-0.423699\pi\)
0.237418 + 0.971408i \(0.423699\pi\)
\(480\) 0 0
\(481\) −16.0000 −0.729537
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 15.0000i − 0.681115i
\(486\) 0 0
\(487\) − 24.2487i − 1.09881i −0.835555 0.549407i \(-0.814854\pi\)
0.835555 0.549407i \(-0.185146\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −5.19615 −0.234499 −0.117250 0.993102i \(-0.537408\pi\)
−0.117250 + 0.993102i \(0.537408\pi\)
\(492\) 0 0
\(493\) 36.0000 1.62136
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 18.0000i − 0.807410i
\(498\) 0 0
\(499\) 31.1769i 1.39567i 0.716258 + 0.697835i \(0.245853\pi\)
−0.716258 + 0.697835i \(0.754147\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 41.5692 1.85348 0.926740 0.375703i \(-0.122599\pi\)
0.926740 + 0.375703i \(0.122599\pi\)
\(504\) 0 0
\(505\) 27.0000 1.20148
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 9.00000i 0.398918i 0.979906 + 0.199459i \(0.0639185\pi\)
−0.979906 + 0.199459i \(0.936082\pi\)
\(510\) 0 0
\(511\) 1.73205i 0.0766214i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 10.3923 0.457940
\(516\) 0 0
\(517\) 54.0000 2.37492
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 18.0000i − 0.788594i −0.918983 0.394297i \(-0.870988\pi\)
0.918983 0.394297i \(-0.129012\pi\)
\(522\) 0 0
\(523\) 17.3205i 0.757373i 0.925525 + 0.378686i \(0.123624\pi\)
−0.925525 + 0.378686i \(0.876376\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 31.1769 1.35809
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) − 15.5885i − 0.673948i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −20.7846 −0.895257
\(540\) 0 0
\(541\) −28.0000 −1.20381 −0.601907 0.798566i \(-0.705592\pi\)
−0.601907 + 0.798566i \(0.705592\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 48.0000i − 2.05609i
\(546\) 0 0
\(547\) 24.2487i 1.03680i 0.855138 + 0.518400i \(0.173472\pi\)
−0.855138 + 0.518400i \(0.826528\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −41.5692 −1.77091
\(552\) 0 0
\(553\) −6.00000 −0.255146
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 39.0000i − 1.65248i −0.563316 0.826242i \(-0.690475\pi\)
0.563316 0.826242i \(-0.309525\pi\)
\(558\) 0 0
\(559\) − 20.7846i − 0.879095i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −15.5885 −0.656975 −0.328488 0.944508i \(-0.606539\pi\)
−0.328488 + 0.944508i \(0.606539\pi\)
\(564\) 0 0
\(565\) −36.0000 −1.51453
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 24.0000i − 1.00613i −0.864248 0.503066i \(-0.832205\pi\)
0.864248 0.503066i \(-0.167795\pi\)
\(570\) 0 0
\(571\) − 24.2487i − 1.01478i −0.861717 0.507388i \(-0.830611\pi\)
0.861717 0.507388i \(-0.169389\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −10.0000 −0.416305 −0.208153 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 9.00000i 0.373383i
\(582\) 0 0
\(583\) 46.7654i 1.93682i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −5.19615 −0.214468 −0.107234 0.994234i \(-0.534199\pi\)
−0.107234 + 0.994234i \(0.534199\pi\)
\(588\) 0 0
\(589\) −36.0000 −1.48335
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 6.00000i 0.246390i 0.992382 + 0.123195i \(0.0393141\pi\)
−0.992382 + 0.123195i \(0.960686\pi\)
\(594\) 0 0
\(595\) 31.1769i 1.27813i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 31.1769 1.27385 0.636927 0.770924i \(-0.280205\pi\)
0.636927 + 0.770924i \(0.280205\pi\)
\(600\) 0 0
\(601\) −19.0000 −0.775026 −0.387513 0.921864i \(-0.626666\pi\)
−0.387513 + 0.921864i \(0.626666\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 48.0000i 1.95148i
\(606\) 0 0
\(607\) 38.1051i 1.54664i 0.634017 + 0.773320i \(0.281405\pi\)
−0.634017 + 0.773320i \(0.718595\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −20.7846 −0.840855
\(612\) 0 0
\(613\) 10.0000 0.403896 0.201948 0.979396i \(-0.435273\pi\)
0.201948 + 0.979396i \(0.435273\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 12.0000i − 0.483102i −0.970388 0.241551i \(-0.922344\pi\)
0.970388 0.241551i \(-0.0776561\pi\)
\(618\) 0 0
\(619\) − 24.2487i − 0.974638i −0.873224 0.487319i \(-0.837975\pi\)
0.873224 0.487319i \(-0.162025\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −10.3923 −0.416359
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 48.0000i 1.91389i
\(630\) 0 0
\(631\) − 12.1244i − 0.482663i −0.970443 0.241331i \(-0.922416\pi\)
0.970443 0.241331i \(-0.0775841\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −15.5885 −0.618609
\(636\) 0 0
\(637\) 8.00000 0.316972
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 24.0000i 0.947943i 0.880540 + 0.473972i \(0.157180\pi\)
−0.880540 + 0.473972i \(0.842820\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −20.7846 −0.817127 −0.408564 0.912730i \(-0.633970\pi\)
−0.408564 + 0.912730i \(0.633970\pi\)
\(648\) 0 0
\(649\) 54.0000 2.11969
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 9.00000i 0.352197i 0.984373 + 0.176099i \(0.0563478\pi\)
−0.984373 + 0.176099i \(0.943652\pi\)
\(654\) 0 0
\(655\) 15.5885i 0.609091i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −36.3731 −1.41689 −0.708447 0.705764i \(-0.750604\pi\)
−0.708447 + 0.705764i \(0.750604\pi\)
\(660\) 0 0
\(661\) −20.0000 −0.777910 −0.388955 0.921257i \(-0.627164\pi\)
−0.388955 + 0.921257i \(0.627164\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 36.0000i − 1.39602i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −20.7846 −0.802381
\(672\) 0 0
\(673\) 23.0000 0.886585 0.443292 0.896377i \(-0.353810\pi\)
0.443292 + 0.896377i \(0.353810\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 18.0000i 0.691796i 0.938272 + 0.345898i \(0.112426\pi\)
−0.938272 + 0.345898i \(0.887574\pi\)
\(678\) 0 0
\(679\) − 8.66025i − 0.332350i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 10.3923 0.397650 0.198825 0.980035i \(-0.436287\pi\)
0.198825 + 0.980035i \(0.436287\pi\)
\(684\) 0 0
\(685\) −18.0000 −0.687745
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 18.0000i − 0.685745i
\(690\) 0 0
\(691\) − 20.7846i − 0.790684i −0.918534 0.395342i \(-0.870626\pi\)
0.918534 0.395342i \(-0.129374\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 20.7846 0.788405
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 21.0000i 0.793159i 0.918000 + 0.396580i \(0.129803\pi\)
−0.918000 + 0.396580i \(0.870197\pi\)
\(702\) 0 0
\(703\) − 55.4256i − 2.09042i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 15.5885 0.586264
\(708\) 0 0
\(709\) 32.0000 1.20179 0.600893 0.799330i \(-0.294812\pi\)
0.600893 + 0.799330i \(0.294812\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) − 31.1769i − 1.16595i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 20.7846 0.775135 0.387568 0.921841i \(-0.373315\pi\)
0.387568 + 0.921841i \(0.373315\pi\)
\(720\) 0 0
\(721\) 6.00000 0.223452
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 24.0000i − 0.891338i
\(726\) 0 0
\(727\) − 46.7654i − 1.73443i −0.497932 0.867216i \(-0.665907\pi\)
0.497932 0.867216i \(-0.334093\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −62.3538 −2.30624
\(732\) 0 0
\(733\) −4.00000 −0.147743 −0.0738717 0.997268i \(-0.523536\pi\)
−0.0738717 + 0.997268i \(0.523536\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 18.0000i 0.663039i
\(738\) 0 0
\(739\) 10.3923i 0.382287i 0.981562 + 0.191144i \(0.0612196\pi\)
−0.981562 + 0.191144i \(0.938780\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −31.1769 −1.14377 −0.571885 0.820334i \(-0.693788\pi\)
−0.571885 + 0.820334i \(0.693788\pi\)
\(744\) 0 0
\(745\) −45.0000 −1.64867
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 9.00000i − 0.328853i
\(750\) 0 0
\(751\) − 22.5167i − 0.821645i −0.911715 0.410822i \(-0.865242\pi\)
0.911715 0.410822i \(-0.134758\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −25.9808 −0.945537
\(756\) 0 0
\(757\) 26.0000 0.944986 0.472493 0.881334i \(-0.343354\pi\)
0.472493 + 0.881334i \(0.343354\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 24.0000i 0.869999i 0.900431 + 0.435000i \(0.143252\pi\)
−0.900431 + 0.435000i \(0.856748\pi\)
\(762\) 0 0
\(763\) − 27.7128i − 1.00327i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −20.7846 −0.750489
\(768\) 0 0
\(769\) −25.0000 −0.901523 −0.450762 0.892644i \(-0.648848\pi\)
−0.450762 + 0.892644i \(0.648848\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 6.00000i 0.215805i 0.994161 + 0.107903i \(0.0344134\pi\)
−0.994161 + 0.107903i \(0.965587\pi\)
\(774\) 0 0
\(775\) − 20.7846i − 0.746605i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 54.0000 1.93227
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 12.0000i 0.428298i
\(786\) 0 0
\(787\) − 34.6410i − 1.23482i −0.786642 0.617409i \(-0.788182\pi\)
0.786642 0.617409i \(-0.211818\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −20.7846 −0.739016
\(792\) 0 0
\(793\) 8.00000 0.284088
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 33.0000i 1.16892i 0.811423 + 0.584460i \(0.198694\pi\)
−0.811423 + 0.584460i \(0.801306\pi\)
\(798\) 0 0
\(799\) 62.3538i 2.20592i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −5.19615 −0.183368
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 6.00000i 0.210949i 0.994422 + 0.105474i \(0.0336361\pi\)
−0.994422 + 0.105474i \(0.966364\pi\)
\(810\) 0 0
\(811\) 20.7846i 0.729846i 0.931038 + 0.364923i \(0.118905\pi\)
−0.931038 + 0.364923i \(0.881095\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −10.3923 −0.364027
\(816\) 0 0
\(817\) 72.0000 2.51896
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 42.0000i − 1.46581i −0.680331 0.732905i \(-0.738164\pi\)
0.680331 0.732905i \(-0.261836\pi\)
\(822\) 0 0
\(823\) − 32.9090i − 1.14713i −0.819159 0.573567i \(-0.805559\pi\)
0.819159 0.573567i \(-0.194441\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 31.1769 1.08413 0.542064 0.840337i \(-0.317643\pi\)
0.542064 + 0.840337i \(0.317643\pi\)
\(828\) 0 0
\(829\) 26.0000 0.903017 0.451509 0.892267i \(-0.350886\pi\)
0.451509 + 0.892267i \(0.350886\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 24.0000i − 0.831551i
\(834\) 0 0
\(835\) 31.1769i 1.07892i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −31.1769 −1.07635 −0.538173 0.842834i \(-0.680885\pi\)
−0.538173 + 0.842834i \(0.680885\pi\)
\(840\) 0 0
\(841\) −7.00000 −0.241379
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 27.0000i − 0.928828i
\(846\) 0 0
\(847\) 27.7128i 0.952224i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −50.0000 −1.71197 −0.855984 0.517003i \(-0.827048\pi\)
−0.855984 + 0.517003i \(0.827048\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 30.0000i − 1.02478i −0.858753 0.512390i \(-0.828760\pi\)
0.858753 0.512390i \(-0.171240\pi\)
\(858\) 0 0
\(859\) − 6.92820i − 0.236387i −0.992991 0.118194i \(-0.962290\pi\)
0.992991 0.118194i \(-0.0377103\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −20.7846 −0.707516 −0.353758 0.935337i \(-0.615096\pi\)
−0.353758 + 0.935337i \(0.615096\pi\)
\(864\) 0 0
\(865\) 9.00000 0.306009
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 18.0000i − 0.610608i
\(870\) 0 0
\(871\) − 6.92820i − 0.234753i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −5.19615 −0.175662
\(876\) 0 0
\(877\) −38.0000 −1.28317 −0.641584 0.767052i \(-0.721723\pi\)
−0.641584 + 0.767052i \(0.721723\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 54.0000i 1.81931i 0.415369 + 0.909653i \(0.363653\pi\)
−0.415369 + 0.909653i \(0.636347\pi\)
\(882\) 0 0
\(883\) − 6.92820i − 0.233153i −0.993182 0.116576i \(-0.962808\pi\)
0.993182 0.116576i \(-0.0371920\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −10.3923 −0.348939 −0.174470 0.984663i \(-0.555821\pi\)
−0.174470 + 0.984663i \(0.555821\pi\)
\(888\) 0 0
\(889\) −9.00000 −0.301850
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 72.0000i − 2.40939i
\(894\) 0 0
\(895\) 15.5885i 0.521065i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −31.1769 −1.03981
\(900\) 0 0
\(901\) −54.0000 −1.79900
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 24.0000i 0.797787i
\(906\) 0 0
\(907\) 27.7128i 0.920189i 0.887870 + 0.460094i \(0.152184\pi\)
−0.887870 + 0.460094i \(0.847816\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −10.3923 −0.344312 −0.172156 0.985070i \(-0.555073\pi\)
−0.172156 + 0.985070i \(0.555073\pi\)
\(912\) 0 0
\(913\) −27.0000 −0.893570
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 9.00000i 0.297206i
\(918\) 0 0
\(919\) 15.5885i 0.514216i 0.966383 + 0.257108i \(0.0827696\pi\)
−0.966383 + 0.257108i \(0.917230\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −20.7846 −0.684134
\(924\) 0 0
\(925\) 32.0000 1.05215
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 12.0000i 0.393707i 0.980433 + 0.196854i \(0.0630724\pi\)
−0.980433 + 0.196854i \(0.936928\pi\)
\(930\) 0 0
\(931\) 27.7128i 0.908251i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −93.5307 −3.05878
\(936\) 0 0
\(937\) 17.0000 0.555366 0.277683 0.960673i \(-0.410434\pi\)
0.277683 + 0.960673i \(0.410434\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 3.00000i − 0.0977972i −0.998804 0.0488986i \(-0.984429\pi\)
0.998804 0.0488986i \(-0.0155711\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 46.7654 1.51967 0.759835 0.650116i \(-0.225280\pi\)
0.759835 + 0.650116i \(0.225280\pi\)
\(948\) 0 0
\(949\) 2.00000 0.0649227
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 24.0000i 0.777436i 0.921357 + 0.388718i \(0.127082\pi\)
−0.921357 + 0.388718i \(0.872918\pi\)
\(954\) 0 0
\(955\) − 31.1769i − 1.00886i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −10.3923 −0.335585
\(960\) 0 0
\(961\) 4.00000 0.129032
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 33.0000i 1.06231i
\(966\) 0 0
\(967\) 50.2295i 1.61527i 0.589682 + 0.807635i \(0.299253\pi\)
−0.589682 + 0.807635i \(0.700747\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −25.9808 −0.833762 −0.416881 0.908961i \(-0.636877\pi\)
−0.416881 + 0.908961i \(0.636877\pi\)
\(972\) 0 0
\(973\) 12.0000 0.384702
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 54.0000i 1.72761i 0.503824 + 0.863807i \(0.331926\pi\)
−0.503824 + 0.863807i \(0.668074\pi\)
\(978\) 0 0
\(979\) − 31.1769i − 0.996419i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 10.3923 0.331463 0.165732 0.986171i \(-0.447001\pi\)
0.165732 + 0.986171i \(0.447001\pi\)
\(984\) 0 0
\(985\) −45.0000 −1.43382
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 32.9090i 1.04539i 0.852520 + 0.522694i \(0.175073\pi\)
−0.852520 + 0.522694i \(0.824927\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −67.5500 −2.14148
\(996\) 0 0
\(997\) −26.0000 −0.823428 −0.411714 0.911313i \(-0.635070\pi\)
−0.411714 + 0.911313i \(0.635070\pi\)
\(998\) 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.c.e.1727.4 4
3.2 odd 2 inner 1728.2.c.e.1727.2 4
4.3 odd 2 inner 1728.2.c.e.1727.3 4
8.3 odd 2 432.2.c.c.431.1 4
8.5 even 2 432.2.c.c.431.2 yes 4
12.11 even 2 inner 1728.2.c.e.1727.1 4
24.5 odd 2 432.2.c.c.431.4 yes 4
24.11 even 2 432.2.c.c.431.3 yes 4
72.5 odd 6 1296.2.s.g.431.2 4
72.11 even 6 1296.2.s.g.863.1 4
72.13 even 6 1296.2.s.g.431.1 4
72.29 odd 6 1296.2.s.i.863.1 4
72.43 odd 6 1296.2.s.g.863.2 4
72.59 even 6 1296.2.s.i.431.2 4
72.61 even 6 1296.2.s.i.863.2 4
72.67 odd 6 1296.2.s.i.431.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
432.2.c.c.431.1 4 8.3 odd 2
432.2.c.c.431.2 yes 4 8.5 even 2
432.2.c.c.431.3 yes 4 24.11 even 2
432.2.c.c.431.4 yes 4 24.5 odd 2
1296.2.s.g.431.1 4 72.13 even 6
1296.2.s.g.431.2 4 72.5 odd 6
1296.2.s.g.863.1 4 72.11 even 6
1296.2.s.g.863.2 4 72.43 odd 6
1296.2.s.i.431.1 4 72.67 odd 6
1296.2.s.i.431.2 4 72.59 even 6
1296.2.s.i.863.1 4 72.29 odd 6
1296.2.s.i.863.2 4 72.61 even 6
1728.2.c.e.1727.1 4 12.11 even 2 inner
1728.2.c.e.1727.2 4 3.2 odd 2 inner
1728.2.c.e.1727.3 4 4.3 odd 2 inner
1728.2.c.e.1727.4 4 1.1 even 1 trivial