Properties

Label 7056.2.b.i.1567.2
Level $7056$
Weight $2$
Character 7056.1567
Analytic conductor $56.342$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7056.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(56.3424436662\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 336)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1567.2
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 7056.1567
Dual form 7056.2.b.i.1567.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.73205i q^{5} +O(q^{10})\) \(q+1.73205i q^{5} +1.73205i q^{11} -3.46410i q^{17} +2.00000 q^{19} +2.00000 q^{25} -9.00000 q^{29} -5.00000 q^{31} +10.0000 q^{37} +10.3923i q^{41} -3.46410i q^{43} +12.0000 q^{47} +9.00000 q^{53} -3.00000 q^{55} -9.00000 q^{59} +13.8564i q^{67} +13.8564i q^{71} -6.92820i q^{73} +5.19615i q^{79} +3.00000 q^{83} +6.00000 q^{85} -3.46410i q^{89} +3.46410i q^{95} -19.0526i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q + 4q^{19} + 4q^{25} - 18q^{29} - 10q^{31} + 20q^{37} + 24q^{47} + 18q^{53} - 6q^{55} - 18q^{59} + 6q^{83} + 12q^{85} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7056\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1765\) \(4609\) \(6175\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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
\(5\) 1.73205i 0.774597i 0.921954 + 0.387298i \(0.126592\pi\)
−0.921954 + 0.387298i \(0.873408\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.73205i 0.522233i 0.965307 + 0.261116i \(0.0840907\pi\)
−0.965307 + 0.261116i \(0.915909\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 3.46410i − 0.840168i −0.907485 0.420084i \(-0.862001\pi\)
0.907485 0.420084i \(-0.137999\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 2.00000 0.400000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −9.00000 −1.67126 −0.835629 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 0 0
\(31\) −5.00000 −0.898027 −0.449013 0.893525i \(-0.648224\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 10.0000 1.64399 0.821995 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 10.3923i 1.62301i 0.584349 + 0.811503i \(0.301350\pi\)
−0.584349 + 0.811503i \(0.698650\pi\)
\(42\) 0 0
\(43\) − 3.46410i − 0.528271i −0.964486 0.264135i \(-0.914913\pi\)
0.964486 0.264135i \(-0.0850865\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 9.00000 1.23625 0.618123 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) 0 0
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −9.00000 −1.17170 −0.585850 0.810419i \(-0.699239\pi\)
−0.585850 + 0.810419i \(0.699239\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 13.8564i 1.69283i 0.532524 + 0.846415i \(0.321244\pi\)
−0.532524 + 0.846415i \(0.678756\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 13.8564i 1.64445i 0.569160 + 0.822226i \(0.307268\pi\)
−0.569160 + 0.822226i \(0.692732\pi\)
\(72\) 0 0
\(73\) − 6.92820i − 0.810885i −0.914121 0.405442i \(-0.867117\pi\)
0.914121 0.405442i \(-0.132883\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 5.19615i 0.584613i 0.956325 + 0.292306i \(0.0944227\pi\)
−0.956325 + 0.292306i \(0.905577\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.00000 0.329293 0.164646 0.986353i \(-0.447352\pi\)
0.164646 + 0.986353i \(0.447352\pi\)
\(84\) 0 0
\(85\) 6.00000 0.650791
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 3.46410i − 0.367194i −0.983002 0.183597i \(-0.941226\pi\)
0.983002 0.183597i \(-0.0587741\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.46410i 0.355409i
\(96\) 0 0
\(97\) − 19.0526i − 1.93449i −0.253837 0.967247i \(-0.581693\pi\)
0.253837 0.967247i \(-0.418307\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 13.8564i 1.37876i 0.724398 + 0.689382i \(0.242118\pi\)
−0.724398 + 0.689382i \(0.757882\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.1244i 1.17211i 0.810273 + 0.586053i \(0.199319\pi\)
−0.810273 + 0.586053i \(0.800681\pi\)
\(108\) 0 0
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 8.00000 0.727273
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 12.1244i 1.08444i
\(126\) 0 0
\(127\) − 5.19615i − 0.461084i −0.973062 0.230542i \(-0.925950\pi\)
0.973062 0.230542i \(-0.0740499\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −9.00000 −0.786334 −0.393167 0.919467i \(-0.628621\pi\)
−0.393167 + 0.919467i \(0.628621\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 12.0000 1.02523 0.512615 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) 0 0
\(139\) −14.0000 −1.18746 −0.593732 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) − 15.5885i − 1.29455i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) − 5.19615i − 0.422857i −0.977393 0.211428i \(-0.932188\pi\)
0.977393 0.211428i \(-0.0678115\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 8.66025i − 0.695608i
\(156\) 0 0
\(157\) 6.92820i 0.552931i 0.961024 + 0.276465i \(0.0891631\pi\)
−0.961024 + 0.276465i \(0.910837\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) − 13.8564i − 1.08532i −0.839953 0.542659i \(-0.817418\pi\)
0.839953 0.542659i \(-0.182582\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 24.2487i 1.81243i 0.422813 + 0.906217i \(0.361043\pi\)
−0.422813 + 0.906217i \(0.638957\pi\)
\(180\) 0 0
\(181\) 10.3923i 0.772454i 0.922404 + 0.386227i \(0.126222\pi\)
−0.922404 + 0.386227i \(0.873778\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 17.3205i 1.27343i
\(186\) 0 0
\(187\) 6.00000 0.438763
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 13.8564i 1.00261i 0.865269 + 0.501307i \(0.167147\pi\)
−0.865269 + 0.501307i \(0.832853\pi\)
\(192\) 0 0
\(193\) −5.00000 −0.359908 −0.179954 0.983675i \(-0.557595\pi\)
−0.179954 + 0.983675i \(0.557595\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −18.0000 −1.25717
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 3.46410i 0.239617i
\(210\) 0 0
\(211\) 6.92820i 0.476957i 0.971148 + 0.238479i \(0.0766487\pi\)
−0.971148 + 0.238479i \(0.923351\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 6.00000 0.409197
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −19.0000 −1.27233 −0.636167 0.771551i \(-0.719481\pi\)
−0.636167 + 0.771551i \(0.719481\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −21.0000 −1.39382 −0.696909 0.717159i \(-0.745442\pi\)
−0.696909 + 0.717159i \(0.745442\pi\)
\(228\) 0 0
\(229\) 3.46410i 0.228914i 0.993428 + 0.114457i \(0.0365129\pi\)
−0.993428 + 0.114457i \(0.963487\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) 20.7846i 1.35584i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 6.92820i 0.448148i 0.974572 + 0.224074i \(0.0719358\pi\)
−0.974572 + 0.224074i \(0.928064\pi\)
\(240\) 0 0
\(241\) 22.5167i 1.45043i 0.688525 + 0.725213i \(0.258259\pi\)
−0.688525 + 0.725213i \(0.741741\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −15.0000 −0.946792 −0.473396 0.880850i \(-0.656972\pi\)
−0.473396 + 0.880850i \(0.656972\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 10.3923i − 0.648254i −0.946014 0.324127i \(-0.894929\pi\)
0.946014 0.324127i \(-0.105071\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 17.3205i 1.06803i 0.845476 + 0.534014i \(0.179317\pi\)
−0.845476 + 0.534014i \(0.820683\pi\)
\(264\) 0 0
\(265\) 15.5885i 0.957591i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 15.5885i 0.950445i 0.879866 + 0.475223i \(0.157632\pi\)
−0.879866 + 0.475223i \(0.842368\pi\)
\(270\) 0 0
\(271\) −11.0000 −0.668202 −0.334101 0.942537i \(-0.608433\pi\)
−0.334101 + 0.942537i \(0.608433\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.46410i 0.208893i
\(276\) 0 0
\(277\) 28.0000 1.68236 0.841178 0.540758i \(-0.181862\pi\)
0.841178 + 0.540758i \(0.181862\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 24.0000 1.43172 0.715860 0.698244i \(-0.246035\pi\)
0.715860 + 0.698244i \(0.246035\pi\)
\(282\) 0 0
\(283\) 14.0000 0.832214 0.416107 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 5.00000 0.294118
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 5.19615i − 0.303562i −0.988414 0.151781i \(-0.951499\pi\)
0.988414 0.151781i \(-0.0485009\pi\)
\(294\) 0 0
\(295\) − 15.5885i − 0.907595i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 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
\(313\) 5.19615i 0.293704i 0.989158 + 0.146852i \(0.0469141\pi\)
−0.989158 + 0.146852i \(0.953086\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 21.0000 1.17948 0.589739 0.807594i \(-0.299231\pi\)
0.589739 + 0.807594i \(0.299231\pi\)
\(318\) 0 0
\(319\) − 15.5885i − 0.872786i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 6.92820i − 0.385496i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 13.8564i 0.761617i 0.924654 + 0.380808i \(0.124354\pi\)
−0.924654 + 0.380808i \(0.875646\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −24.0000 −1.31126
\(336\) 0 0
\(337\) −1.00000 −0.0544735 −0.0272367 0.999629i \(-0.508671\pi\)
−0.0272367 + 0.999629i \(0.508671\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 8.66025i − 0.468979i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 17.3205i − 0.929814i −0.885360 0.464907i \(-0.846088\pi\)
0.885360 0.464907i \(-0.153912\pi\)
\(348\) 0 0
\(349\) − 6.92820i − 0.370858i −0.982658 0.185429i \(-0.940632\pi\)
0.982658 0.185429i \(-0.0593675\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 6.92820i − 0.368751i −0.982856 0.184376i \(-0.940974\pi\)
0.982856 0.184376i \(-0.0590263\pi\)
\(354\) 0 0
\(355\) −24.0000 −1.27379
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 27.7128i − 1.46263i −0.682042 0.731313i \(-0.738908\pi\)
0.682042 0.731313i \(-0.261092\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 12.0000 0.628109
\(366\) 0 0
\(367\) −19.0000 −0.991792 −0.495896 0.868382i \(-0.665160\pi\)
−0.495896 + 0.868382i \(0.665160\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 2.00000 0.103556 0.0517780 0.998659i \(-0.483511\pi\)
0.0517780 + 0.998659i \(0.483511\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 24.2487i 1.24557i 0.782392 + 0.622786i \(0.213999\pi\)
−0.782392 + 0.622786i \(0.786001\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 18.0000 0.919757 0.459879 0.887982i \(-0.347893\pi\)
0.459879 + 0.887982i \(0.347893\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −9.00000 −0.452839
\(396\) 0 0
\(397\) 20.7846i 1.04315i 0.853206 + 0.521575i \(0.174655\pi\)
−0.853206 + 0.521575i \(0.825345\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 12.0000 0.599251 0.299626 0.954057i \(-0.403138\pi\)
0.299626 + 0.954057i \(0.403138\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 17.3205i 0.858546i
\(408\) 0 0
\(409\) − 22.5167i − 1.11338i −0.830721 0.556689i \(-0.812072\pi\)
0.830721 0.556689i \(-0.187928\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 5.19615i 0.255069i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) −20.0000 −0.974740 −0.487370 0.873195i \(-0.662044\pi\)
−0.487370 + 0.873195i \(0.662044\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 6.92820i − 0.336067i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 38.1051i 1.83546i 0.397206 + 0.917729i \(0.369980\pi\)
−0.397206 + 0.917729i \(0.630020\pi\)
\(432\) 0 0
\(433\) 34.6410i 1.66474i 0.554220 + 0.832370i \(0.313017\pi\)
−0.554220 + 0.832370i \(0.686983\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 29.0000 1.38409 0.692047 0.721852i \(-0.256709\pi\)
0.692047 + 0.721852i \(0.256709\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 12.1244i − 0.576046i −0.957624 0.288023i \(-0.907002\pi\)
0.957624 0.288023i \(-0.0929979\pi\)
\(444\) 0 0
\(445\) 6.00000 0.284427
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −12.0000 −0.566315 −0.283158 0.959073i \(-0.591382\pi\)
−0.283158 + 0.959073i \(0.591382\pi\)
\(450\) 0 0
\(451\) −18.0000 −0.847587
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −31.0000 −1.45012 −0.725059 0.688686i \(-0.758188\pi\)
−0.725059 + 0.688686i \(0.758188\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 6.92820i 0.322679i 0.986899 + 0.161339i \(0.0515813\pi\)
−0.986899 + 0.161339i \(0.948419\pi\)
\(462\) 0 0
\(463\) 38.1051i 1.77090i 0.464739 + 0.885448i \(0.346148\pi\)
−0.464739 + 0.885448i \(0.653852\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 6.00000 0.275880
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 6.00000 0.274147 0.137073 0.990561i \(-0.456230\pi\)
0.137073 + 0.990561i \(0.456230\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 33.0000 1.49845
\(486\) 0 0
\(487\) 8.66025i 0.392434i 0.980561 + 0.196217i \(0.0628656\pi\)
−0.980561 + 0.196217i \(0.937134\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 25.9808i − 1.17250i −0.810132 0.586248i \(-0.800605\pi\)
0.810132 0.586248i \(-0.199395\pi\)
\(492\) 0 0
\(493\) 31.1769i 1.40414i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 3.46410i 0.155074i 0.996989 + 0.0775372i \(0.0247057\pi\)
−0.996989 + 0.0775372i \(0.975294\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 6.00000 0.267527 0.133763 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 0 0
\(505\) −24.0000 −1.06799
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 32.9090i 1.45866i 0.684160 + 0.729332i \(0.260169\pi\)
−0.684160 + 0.729332i \(0.739831\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 6.92820i 0.305293i
\(516\) 0 0
\(517\) 20.7846i 0.914106i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 38.1051i − 1.66942i −0.550693 0.834708i \(-0.685637\pi\)
0.550693 0.834708i \(-0.314363\pi\)
\(522\) 0 0
\(523\) 28.0000 1.22435 0.612177 0.790721i \(-0.290294\pi\)
0.612177 + 0.790721i \(0.290294\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 17.3205i 0.754493i
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −21.0000 −0.907909
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 22.0000 0.945854 0.472927 0.881102i \(-0.343197\pi\)
0.472927 + 0.881102i \(0.343197\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 6.92820i − 0.296772i
\(546\) 0 0
\(547\) 10.3923i 0.444343i 0.975008 + 0.222171i \(0.0713145\pi\)
−0.975008 + 0.222171i \(0.928686\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −18.0000 −0.766826
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 27.0000 1.14403 0.572013 0.820244i \(-0.306163\pi\)
0.572013 + 0.820244i \(0.306163\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −45.0000 −1.89652 −0.948262 0.317489i \(-0.897160\pi\)
−0.948262 + 0.317489i \(0.897160\pi\)
\(564\) 0 0
\(565\) − 10.3923i − 0.437208i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 20.7846i 0.869809i 0.900477 + 0.434904i \(0.143218\pi\)
−0.900477 + 0.434904i \(0.856782\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 1.73205i − 0.0721062i −0.999350 0.0360531i \(-0.988521\pi\)
0.999350 0.0360531i \(-0.0114785\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 15.5885i 0.645608i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −33.0000 −1.36206 −0.681028 0.732257i \(-0.738467\pi\)
−0.681028 + 0.732257i \(0.738467\pi\)
\(588\) 0 0
\(589\) −10.0000 −0.412043
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 27.7128i − 1.13803i −0.822328 0.569014i \(-0.807325\pi\)
0.822328 0.569014i \(-0.192675\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 27.7128i 1.13231i 0.824297 + 0.566157i \(0.191571\pi\)
−0.824297 + 0.566157i \(0.808429\pi\)
\(600\) 0 0
\(601\) − 15.5885i − 0.635866i −0.948113 0.317933i \(-0.897011\pi\)
0.948113 0.317933i \(-0.102989\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 13.8564i 0.563343i
\(606\) 0 0
\(607\) −19.0000 −0.771186 −0.385593 0.922669i \(-0.626003\pi\)
−0.385593 + 0.922669i \(0.626003\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −4.00000 −0.161558 −0.0807792 0.996732i \(-0.525741\pi\)
−0.0807792 + 0.996732i \(0.525741\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −12.0000 −0.483102 −0.241551 0.970388i \(-0.577656\pi\)
−0.241551 + 0.970388i \(0.577656\pi\)
\(618\) 0 0
\(619\) 28.0000 1.12542 0.562708 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 34.6410i − 1.38123i
\(630\) 0 0
\(631\) − 32.9090i − 1.31009i −0.755592 0.655043i \(-0.772651\pi\)
0.755592 0.655043i \(-0.227349\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 9.00000 0.357154
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −36.0000 −1.42191 −0.710957 0.703235i \(-0.751738\pi\)
−0.710957 + 0.703235i \(0.751738\pi\)
\(642\) 0 0
\(643\) −16.0000 −0.630978 −0.315489 0.948929i \(-0.602169\pi\)
−0.315489 + 0.948929i \(0.602169\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) − 15.5885i − 0.611900i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 27.0000 1.05659 0.528296 0.849060i \(-0.322831\pi\)
0.528296 + 0.849060i \(0.322831\pi\)
\(654\) 0 0
\(655\) − 15.5885i − 0.609091i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 10.3923i 0.404827i 0.979300 + 0.202413i \(0.0648785\pi\)
−0.979300 + 0.202413i \(0.935122\pi\)
\(660\) 0 0
\(661\) 3.46410i 0.134738i 0.997728 + 0.0673690i \(0.0214605\pi\)
−0.997728 + 0.0673690i \(0.978540\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −35.0000 −1.34915 −0.674575 0.738206i \(-0.735673\pi\)
−0.674575 + 0.738206i \(0.735673\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 29.4449i 1.13166i 0.824523 + 0.565829i \(0.191444\pi\)
−0.824523 + 0.565829i \(0.808556\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 25.9808i − 0.994126i −0.867714 0.497063i \(-0.834412\pi\)
0.867714 0.497063i \(-0.165588\pi\)
\(684\) 0 0
\(685\) 20.7846i 0.794139i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 22.0000 0.836919 0.418460 0.908235i \(-0.362570\pi\)
0.418460 + 0.908235i \(0.362570\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 24.2487i − 0.919806i
\(696\) 0 0
\(697\) 36.0000 1.36360
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 45.0000 1.69963 0.849813 0.527084i \(-0.176715\pi\)
0.849813 + 0.527084i \(0.176715\pi\)
\(702\) 0 0
\(703\) 20.0000 0.754314
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 18.0000 0.671287 0.335643 0.941989i \(-0.391046\pi\)
0.335643 + 0.941989i \(0.391046\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −18.0000 −0.668503
\(726\) 0 0
\(727\) 37.0000 1.37225 0.686127 0.727482i \(-0.259309\pi\)
0.686127 + 0.727482i \(0.259309\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −12.0000 −0.443836
\(732\) 0 0
\(733\) 10.3923i 0.383849i 0.981410 + 0.191924i \(0.0614728\pi\)
−0.981410 + 0.191924i \(0.938527\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −24.0000 −0.884051
\(738\) 0 0
\(739\) − 10.3923i − 0.382287i −0.981562 0.191144i \(-0.938780\pi\)
0.981562 0.191144i \(-0.0612196\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 13.8564i − 0.508342i −0.967159 0.254171i \(-0.918197\pi\)
0.967159 0.254171i \(-0.0818026\pi\)
\(744\) 0 0
\(745\) − 10.3923i − 0.380745i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 5.19615i 0.189610i 0.995496 + 0.0948051i \(0.0302228\pi\)
−0.995496 + 0.0948051i \(0.969777\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 9.00000 0.327544
\(756\) 0 0
\(757\) −4.00000 −0.145382 −0.0726912 0.997354i \(-0.523159\pi\)
−0.0726912 + 0.997354i \(0.523159\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 48.4974i − 1.75803i −0.476794 0.879015i \(-0.658201\pi\)
0.476794 0.879015i \(-0.341799\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 19.0526i 0.687053i 0.939143 + 0.343526i \(0.111621\pi\)
−0.939143 + 0.343526i \(0.888379\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) −10.0000 −0.359211
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 20.7846i 0.744686i
\(780\) 0 0
\(781\) −24.0000 −0.858788
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −12.0000 −0.428298
\(786\) 0 0
\(787\) 28.0000 0.998092 0.499046 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 5.19615i − 0.184057i −0.995756 0.0920286i \(-0.970665\pi\)
0.995756 0.0920286i \(-0.0293351\pi\)
\(798\) 0 0
\(799\) − 41.5692i − 1.47061i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 12.0000 0.423471
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) 16.0000 0.561836 0.280918 0.959732i \(-0.409361\pi\)
0.280918 + 0.959732i \(0.409361\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 24.0000 0.840683
\(816\) 0 0
\(817\) − 6.92820i − 0.242387i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −3.00000 −0.104701 −0.0523504 0.998629i \(-0.516671\pi\)
−0.0523504 + 0.998629i \(0.516671\pi\)
\(822\) 0 0
\(823\) 38.1051i 1.32826i 0.747617 + 0.664130i \(0.231198\pi\)
−0.747617 + 0.664130i \(0.768802\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 39.8372i 1.38527i 0.721286 + 0.692637i \(0.243551\pi\)
−0.721286 + 0.692637i \(0.756449\pi\)
\(828\) 0 0
\(829\) − 17.3205i − 0.601566i −0.953693 0.300783i \(-0.902752\pi\)
0.953693 0.300783i \(-0.0972480\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) − 20.7846i − 0.719281i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 30.0000 1.03572 0.517858 0.855467i \(-0.326730\pi\)
0.517858 + 0.855467i \(0.326730\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 22.5167i 0.774597i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) − 48.4974i − 1.66052i −0.557376 0.830260i \(-0.688192\pi\)
0.557376 0.830260i \(-0.311808\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 14.0000 0.477674 0.238837 0.971060i \(-0.423234\pi\)
0.238837 + 0.971060i \(0.423234\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 31.1769i 1.06127i 0.847599 + 0.530637i \(0.178047\pi\)
−0.847599 + 0.530637i \(0.821953\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −9.00000 −0.305304
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 32.0000 1.08056 0.540282 0.841484i \(-0.318318\pi\)
0.540282 + 0.841484i \(0.318318\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 13.8564i 0.466834i 0.972377 + 0.233417i \(0.0749907\pi\)
−0.972377 + 0.233417i \(0.925009\pi\)
\(882\) 0 0
\(883\) 24.2487i 0.816034i 0.912974 + 0.408017i \(0.133780\pi\)
−0.912974 + 0.408017i \(0.866220\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 24.0000 0.803129
\(894\) 0 0
\(895\) −42.0000 −1.40391
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 45.0000 1.50083
\(900\) 0 0
\(901\) − 31.1769i − 1.03865i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −18.0000 −0.598340
\(906\) 0 0
\(907\) − 20.7846i − 0.690142i −0.938577 0.345071i \(-0.887855\pi\)
0.938577 0.345071i \(-0.112145\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 41.5692i − 1.37725i −0.725118 0.688625i \(-0.758215\pi\)
0.725118 0.688625i \(-0.241785\pi\)
\(912\) 0 0
\(913\) 5.19615i 0.171968i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 17.3205i 0.571351i 0.958326 + 0.285675i \(0.0922179\pi\)
−0.958326 + 0.285675i \(0.907782\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 20.0000 0.657596
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 17.3205i − 0.568267i −0.958785 0.284134i \(-0.908294\pi\)
0.958785 0.284134i \(-0.0917060\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 10.3923i 0.339865i
\(936\) 0 0
\(937\) − 19.0526i − 0.622420i −0.950341 0.311210i \(-0.899266\pi\)
0.950341 0.311210i \(-0.100734\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 39.8372i 1.29865i 0.760509 + 0.649327i \(0.224949\pi\)
−0.760509 + 0.649327i \(0.775051\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 38.1051i 1.23825i 0.785292 + 0.619125i \(0.212513\pi\)
−0.785292 + 0.619125i \(0.787487\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 18.0000 0.583077 0.291539 0.956559i \(-0.405833\pi\)
0.291539 + 0.956559i \(0.405833\pi\)
\(954\) 0 0
\(955\) −24.0000 −0.776622
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 8.66025i − 0.278783i
\(966\) 0 0
\(967\) 39.8372i 1.28108i 0.767926 + 0.640538i \(0.221289\pi\)
−0.767926 + 0.640538i \(0.778711\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −33.0000 −1.05902 −0.529510 0.848304i \(-0.677624\pi\)
−0.529510 + 0.848304i \(0.677624\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 0 0
\(979\) 6.00000 0.191761
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 18.0000 0.574111 0.287055 0.957914i \(-0.407324\pi\)
0.287055 + 0.957914i \(0.407324\pi\)
\(984\) 0 0
\(985\) 10.3923i 0.331126i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) − 12.1244i − 0.385143i −0.981283 0.192571i \(-0.938317\pi\)
0.981283 0.192571i \(-0.0616827\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 27.7128i − 0.878555i
\(996\) 0 0
\(997\) 27.7128i 0.877674i 0.898567 + 0.438837i \(0.144609\pi\)
−0.898567 + 0.438837i \(0.855391\pi\)
\(998\) 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 7056.2.b.i.1567.2 2
3.2 odd 2 2352.2.b.c.1567.1 2
4.3 odd 2 7056.2.b.e.1567.2 2
7.4 even 3 1008.2.cs.d.271.1 2
7.5 odd 6 1008.2.cs.e.703.1 2
7.6 odd 2 7056.2.b.e.1567.1 2
12.11 even 2 2352.2.b.g.1567.1 2
21.2 odd 6 2352.2.bl.h.31.1 2
21.5 even 6 336.2.bl.c.31.1 2
21.11 odd 6 336.2.bl.g.271.1 yes 2
21.17 even 6 2352.2.bl.b.607.1 2
21.20 even 2 2352.2.b.g.1567.2 2
28.11 odd 6 1008.2.cs.e.271.1 2
28.19 even 6 1008.2.cs.d.703.1 2
28.27 even 2 inner 7056.2.b.i.1567.1 2
84.11 even 6 336.2.bl.c.271.1 yes 2
84.23 even 6 2352.2.bl.b.31.1 2
84.47 odd 6 336.2.bl.g.31.1 yes 2
84.59 odd 6 2352.2.bl.h.607.1 2
84.83 odd 2 2352.2.b.c.1567.2 2
168.5 even 6 1344.2.bl.f.703.1 2
168.11 even 6 1344.2.bl.f.1279.1 2
168.53 odd 6 1344.2.bl.b.1279.1 2
168.131 odd 6 1344.2.bl.b.703.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
336.2.bl.c.31.1 2 21.5 even 6
336.2.bl.c.271.1 yes 2 84.11 even 6
336.2.bl.g.31.1 yes 2 84.47 odd 6
336.2.bl.g.271.1 yes 2 21.11 odd 6
1008.2.cs.d.271.1 2 7.4 even 3
1008.2.cs.d.703.1 2 28.19 even 6
1008.2.cs.e.271.1 2 28.11 odd 6
1008.2.cs.e.703.1 2 7.5 odd 6
1344.2.bl.b.703.1 2 168.131 odd 6
1344.2.bl.b.1279.1 2 168.53 odd 6
1344.2.bl.f.703.1 2 168.5 even 6
1344.2.bl.f.1279.1 2 168.11 even 6
2352.2.b.c.1567.1 2 3.2 odd 2
2352.2.b.c.1567.2 2 84.83 odd 2
2352.2.b.g.1567.1 2 12.11 even 2
2352.2.b.g.1567.2 2 21.20 even 2
2352.2.bl.b.31.1 2 84.23 even 6
2352.2.bl.b.607.1 2 21.17 even 6
2352.2.bl.h.31.1 2 21.2 odd 6
2352.2.bl.h.607.1 2 84.59 odd 6
7056.2.b.e.1567.1 2 7.6 odd 2
7056.2.b.e.1567.2 2 4.3 odd 2
7056.2.b.i.1567.1 2 28.27 even 2 inner
7056.2.b.i.1567.2 2 1.1 even 1 trivial