Properties

Label 6840.2.a.bh.1.1
Level $6840$
Weight $2$
Character 6840.1
Self dual yes
Analytic conductor $54.618$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6840.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(54.6176749826\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1772.1
Defining polynomial: \(x^{3} - x^{2} - 12 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2280)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.654334\) of defining polynomial
Character \(\chi\) \(=\) 6840.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{5} -4.11309 q^{7} +O(q^{10})\) \(q-1.00000 q^{5} -4.11309 q^{7} +2.11309 q^{11} +4.80442 q^{13} -4.00000 q^{17} +1.00000 q^{19} +0.691333 q^{23} +1.00000 q^{25} -4.11309 q^{29} -2.00000 q^{31} +4.11309 q^{35} +3.42176 q^{37} -5.42176 q^{41} -2.80442 q^{43} -0.691333 q^{47} +9.91751 q^{49} +2.69133 q^{53} -2.11309 q^{55} -9.53485 q^{59} -2.91751 q^{61} -4.80442 q^{65} +4.00000 q^{67} -5.60885 q^{71} +6.00000 q^{73} -8.69133 q^{77} +15.1437 q^{79} -6.22618 q^{83} +4.00000 q^{85} +1.42176 q^{89} -19.7610 q^{91} -1.00000 q^{95} +15.4218 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{5} + O(q^{10}) \) \( 3q - 3q^{5} - 6q^{11} + 4q^{13} - 12q^{17} + 3q^{19} + 4q^{23} + 3q^{25} - 6q^{31} - 4q^{37} - 2q^{41} + 2q^{43} - 4q^{47} + 7q^{49} + 10q^{53} + 6q^{55} - 2q^{59} + 14q^{61} - 4q^{65} + 12q^{67} + 4q^{71} + 18q^{73} - 28q^{77} - 2q^{79} + 6q^{83} + 12q^{85} - 10q^{89} - 8q^{91} - 3q^{95} + 32q^{97} + O(q^{100}) \)

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
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −4.11309 −1.55460 −0.777301 0.629129i \(-0.783412\pi\)
−0.777301 + 0.629129i \(0.783412\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.11309 0.637121 0.318560 0.947903i \(-0.396801\pi\)
0.318560 + 0.947903i \(0.396801\pi\)
\(12\) 0 0
\(13\) 4.80442 1.33251 0.666254 0.745725i \(-0.267897\pi\)
0.666254 + 0.745725i \(0.267897\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.691333 0.144153 0.0720764 0.997399i \(-0.477037\pi\)
0.0720764 + 0.997399i \(0.477037\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.11309 −0.763782 −0.381891 0.924207i \(-0.624727\pi\)
−0.381891 + 0.924207i \(0.624727\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.11309 0.695239
\(36\) 0 0
\(37\) 3.42176 0.562533 0.281267 0.959630i \(-0.409245\pi\)
0.281267 + 0.959630i \(0.409245\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −5.42176 −0.846736 −0.423368 0.905958i \(-0.639152\pi\)
−0.423368 + 0.905958i \(0.639152\pi\)
\(42\) 0 0
\(43\) −2.80442 −0.427671 −0.213835 0.976870i \(-0.568596\pi\)
−0.213835 + 0.976870i \(0.568596\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −0.691333 −0.100841 −0.0504206 0.998728i \(-0.516056\pi\)
−0.0504206 + 0.998728i \(0.516056\pi\)
\(48\) 0 0
\(49\) 9.91751 1.41679
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.69133 0.369683 0.184842 0.982768i \(-0.440823\pi\)
0.184842 + 0.982768i \(0.440823\pi\)
\(54\) 0 0
\(55\) −2.11309 −0.284929
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −9.53485 −1.24133 −0.620666 0.784075i \(-0.713138\pi\)
−0.620666 + 0.784075i \(0.713138\pi\)
\(60\) 0 0
\(61\) −2.91751 −0.373549 −0.186775 0.982403i \(-0.559803\pi\)
−0.186775 + 0.982403i \(0.559803\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.80442 −0.595915
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −5.60885 −0.665648 −0.332824 0.942989i \(-0.608001\pi\)
−0.332824 + 0.942989i \(0.608001\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −8.69133 −0.990469
\(78\) 0 0
\(79\) 15.1437 1.70380 0.851899 0.523705i \(-0.175451\pi\)
0.851899 + 0.523705i \(0.175451\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −6.22618 −0.683412 −0.341706 0.939807i \(-0.611005\pi\)
−0.341706 + 0.939807i \(0.611005\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.42176 0.150706 0.0753530 0.997157i \(-0.475992\pi\)
0.0753530 + 0.997157i \(0.475992\pi\)
\(90\) 0 0
\(91\) −19.7610 −2.07152
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) 15.4218 1.56584 0.782921 0.622121i \(-0.213729\pi\)
0.782921 + 0.622121i \(0.213729\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 10.2262 1.01754 0.508772 0.860902i \(-0.330100\pi\)
0.508772 + 0.860902i \(0.330100\pi\)
\(102\) 0 0
\(103\) 16.4524 1.62110 0.810550 0.585670i \(-0.199168\pi\)
0.810550 + 0.585670i \(0.199168\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −16.4524 −1.59051 −0.795255 0.606275i \(-0.792663\pi\)
−0.795255 + 0.606275i \(0.792663\pi\)
\(108\) 0 0
\(109\) −7.53485 −0.721708 −0.360854 0.932622i \(-0.617515\pi\)
−0.360854 + 0.932622i \(0.617515\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 17.5348 1.64954 0.824770 0.565469i \(-0.191305\pi\)
0.824770 + 0.565469i \(0.191305\pi\)
\(114\) 0 0
\(115\) −0.691333 −0.0644671
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 16.4524 1.50819
\(120\) 0 0
\(121\) −6.53485 −0.594077
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −9.83503 −0.872718 −0.436359 0.899773i \(-0.643732\pi\)
−0.436359 + 0.899773i \(0.643732\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −6.33927 −0.553865 −0.276932 0.960889i \(-0.589318\pi\)
−0.276932 + 0.960889i \(0.589318\pi\)
\(132\) 0 0
\(133\) −4.11309 −0.356650
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −7.30867 −0.624422 −0.312211 0.950013i \(-0.601070\pi\)
−0.312211 + 0.950013i \(0.601070\pi\)
\(138\) 0 0
\(139\) −8.22618 −0.697736 −0.348868 0.937172i \(-0.613434\pi\)
−0.348868 + 0.937172i \(0.613434\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 10.1522 0.848968
\(144\) 0 0
\(145\) 4.11309 0.341574
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 7.38267 0.604812 0.302406 0.953179i \(-0.402210\pi\)
0.302406 + 0.953179i \(0.402210\pi\)
\(150\) 0 0
\(151\) −7.38267 −0.600793 −0.300396 0.953814i \(-0.597119\pi\)
−0.300396 + 0.953814i \(0.597119\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2.00000 0.160644
\(156\) 0 0
\(157\) 12.9175 1.03093 0.515465 0.856911i \(-0.327619\pi\)
0.515465 + 0.856911i \(0.327619\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −2.84352 −0.224100
\(162\) 0 0
\(163\) −5.42176 −0.424665 −0.212332 0.977197i \(-0.568106\pi\)
−0.212332 + 0.977197i \(0.568106\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.9175 0.999587 0.499794 0.866145i \(-0.333409\pi\)
0.499794 + 0.866145i \(0.333409\pi\)
\(168\) 0 0
\(169\) 10.0825 0.775576
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 13.5348 1.02904 0.514518 0.857480i \(-0.327971\pi\)
0.514518 + 0.857480i \(0.327971\pi\)
\(174\) 0 0
\(175\) −4.11309 −0.310920
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 10.9175 0.816013 0.408007 0.912979i \(-0.366224\pi\)
0.408007 + 0.912979i \(0.366224\pi\)
\(180\) 0 0
\(181\) −17.3699 −1.29109 −0.645546 0.763721i \(-0.723370\pi\)
−0.645546 + 0.763721i \(0.723370\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −3.42176 −0.251573
\(186\) 0 0
\(187\) −8.45236 −0.618098
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −0.730425 −0.0528517 −0.0264258 0.999651i \(-0.508413\pi\)
−0.0264258 + 0.999651i \(0.508413\pi\)
\(192\) 0 0
\(193\) 25.4830 1.83430 0.917152 0.398537i \(-0.130482\pi\)
0.917152 + 0.398537i \(0.130482\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 21.7610 1.55041 0.775205 0.631710i \(-0.217647\pi\)
0.775205 + 0.631710i \(0.217647\pi\)
\(198\) 0 0
\(199\) 1.60885 0.114048 0.0570241 0.998373i \(-0.481839\pi\)
0.0570241 + 0.998373i \(0.481839\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 16.9175 1.18738
\(204\) 0 0
\(205\) 5.42176 0.378672
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 2.11309 0.146166
\(210\) 0 0
\(211\) 9.83503 0.677071 0.338536 0.940954i \(-0.390068\pi\)
0.338536 + 0.940954i \(0.390068\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 2.80442 0.191260
\(216\) 0 0
\(217\) 8.22618 0.558430
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −19.2177 −1.29272
\(222\) 0 0
\(223\) 13.8350 0.926462 0.463231 0.886238i \(-0.346690\pi\)
0.463231 + 0.886238i \(0.346690\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −10.3002 −0.683647 −0.341823 0.939764i \(-0.611044\pi\)
−0.341823 + 0.939764i \(0.611044\pi\)
\(228\) 0 0
\(229\) 9.08249 0.600188 0.300094 0.953910i \(-0.402982\pi\)
0.300094 + 0.953910i \(0.402982\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 22.5264 1.47575 0.737875 0.674937i \(-0.235829\pi\)
0.737875 + 0.674937i \(0.235829\pi\)
\(234\) 0 0
\(235\) 0.691333 0.0450976
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 23.7219 1.53444 0.767222 0.641381i \(-0.221638\pi\)
0.767222 + 0.641381i \(0.221638\pi\)
\(240\) 0 0
\(241\) 21.2177 1.36675 0.683376 0.730067i \(-0.260511\pi\)
0.683376 + 0.730067i \(0.260511\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −9.91751 −0.633607
\(246\) 0 0
\(247\) 4.80442 0.305698
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 28.1743 1.77835 0.889173 0.457571i \(-0.151280\pi\)
0.889173 + 0.457571i \(0.151280\pi\)
\(252\) 0 0
\(253\) 1.46085 0.0918428
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −12.3002 −0.767264 −0.383632 0.923486i \(-0.625327\pi\)
−0.383632 + 0.923486i \(0.625327\pi\)
\(258\) 0 0
\(259\) −14.0740 −0.874516
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −12.9175 −0.796528 −0.398264 0.917271i \(-0.630387\pi\)
−0.398264 + 0.917271i \(0.630387\pi\)
\(264\) 0 0
\(265\) −2.69133 −0.165327
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 6.73042 0.410361 0.205181 0.978724i \(-0.434222\pi\)
0.205181 + 0.978724i \(0.434222\pi\)
\(270\) 0 0
\(271\) −7.77382 −0.472226 −0.236113 0.971726i \(-0.575874\pi\)
−0.236113 + 0.971726i \(0.575874\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.11309 0.127424
\(276\) 0 0
\(277\) 15.3087 0.919809 0.459904 0.887968i \(-0.347884\pi\)
0.459904 + 0.887968i \(0.347884\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −13.4218 −0.800675 −0.400337 0.916368i \(-0.631107\pi\)
−0.400337 + 0.916368i \(0.631107\pi\)
\(282\) 0 0
\(283\) 15.0306 0.893477 0.446738 0.894665i \(-0.352585\pi\)
0.446738 + 0.894665i \(0.352585\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 22.3002 1.31634
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 9.08249 0.530604 0.265302 0.964165i \(-0.414528\pi\)
0.265302 + 0.964165i \(0.414528\pi\)
\(294\) 0 0
\(295\) 9.53485 0.555140
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 3.32146 0.192085
\(300\) 0 0
\(301\) 11.5348 0.664858
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2.91751 0.167056
\(306\) 0 0
\(307\) 32.2262 1.83925 0.919623 0.392803i \(-0.128495\pi\)
0.919623 + 0.392803i \(0.128495\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 16.7304 0.948695 0.474348 0.880338i \(-0.342684\pi\)
0.474348 + 0.880338i \(0.342684\pi\)
\(312\) 0 0
\(313\) −14.6785 −0.829680 −0.414840 0.909894i \(-0.636163\pi\)
−0.414840 + 0.909894i \(0.636163\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 17.9090 1.00587 0.502936 0.864324i \(-0.332253\pi\)
0.502936 + 0.864324i \(0.332253\pi\)
\(318\) 0 0
\(319\) −8.69133 −0.486621
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −4.00000 −0.222566
\(324\) 0 0
\(325\) 4.80442 0.266501
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 2.84352 0.156768
\(330\) 0 0
\(331\) −1.92600 −0.105863 −0.0529313 0.998598i \(-0.516856\pi\)
−0.0529313 + 0.998598i \(0.516856\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) −17.2568 −0.940037 −0.470019 0.882657i \(-0.655753\pi\)
−0.470019 + 0.882657i \(0.655753\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −4.22618 −0.228861
\(342\) 0 0
\(343\) −12.0000 −0.647939
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 22.6785 1.21745 0.608724 0.793382i \(-0.291682\pi\)
0.608724 + 0.793382i \(0.291682\pi\)
\(348\) 0 0
\(349\) 4.61733 0.247160 0.123580 0.992335i \(-0.460562\pi\)
0.123580 + 0.992335i \(0.460562\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 23.7610 1.26467 0.632336 0.774694i \(-0.282096\pi\)
0.632336 + 0.774694i \(0.282096\pi\)
\(354\) 0 0
\(355\) 5.60885 0.297687
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 12.5042 0.659949 0.329974 0.943990i \(-0.392960\pi\)
0.329974 + 0.943990i \(0.392960\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −6.00000 −0.314054
\(366\) 0 0
\(367\) −6.73042 −0.351325 −0.175663 0.984450i \(-0.556207\pi\)
−0.175663 + 0.984450i \(0.556207\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −11.0697 −0.574710
\(372\) 0 0
\(373\) 8.80442 0.455876 0.227938 0.973676i \(-0.426802\pi\)
0.227938 + 0.973676i \(0.426802\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −19.7610 −1.01774
\(378\) 0 0
\(379\) −31.7610 −1.63145 −0.815727 0.578437i \(-0.803663\pi\)
−0.815727 + 0.578437i \(0.803663\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −23.0697 −1.17881 −0.589403 0.807839i \(-0.700637\pi\)
−0.589403 + 0.807839i \(0.700637\pi\)
\(384\) 0 0
\(385\) 8.69133 0.442951
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 14.0000 0.709828 0.354914 0.934899i \(-0.384510\pi\)
0.354914 + 0.934899i \(0.384510\pi\)
\(390\) 0 0
\(391\) −2.76533 −0.139849
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −15.1437 −0.761962
\(396\) 0 0
\(397\) 38.5264 1.93358 0.966791 0.255567i \(-0.0822622\pi\)
0.966791 + 0.255567i \(0.0822622\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −22.1003 −1.10364 −0.551818 0.833964i \(-0.686066\pi\)
−0.551818 + 0.833964i \(0.686066\pi\)
\(402\) 0 0
\(403\) −9.60885 −0.478651
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 7.23048 0.358402
\(408\) 0 0
\(409\) 30.2262 1.49459 0.747294 0.664493i \(-0.231353\pi\)
0.747294 + 0.664493i \(0.231353\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 39.2177 1.92978
\(414\) 0 0
\(415\) 6.22618 0.305631
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −11.7219 −0.572654 −0.286327 0.958132i \(-0.592434\pi\)
−0.286327 + 0.958132i \(0.592434\pi\)
\(420\) 0 0
\(421\) −30.9787 −1.50981 −0.754905 0.655834i \(-0.772317\pi\)
−0.754905 + 0.655834i \(0.772317\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −4.00000 −0.194029
\(426\) 0 0
\(427\) 12.0000 0.580721
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 19.2959 0.929450 0.464725 0.885455i \(-0.346153\pi\)
0.464725 + 0.885455i \(0.346153\pi\)
\(432\) 0 0
\(433\) −23.4218 −1.12558 −0.562789 0.826601i \(-0.690272\pi\)
−0.562789 + 0.826601i \(0.690272\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0.691333 0.0330709
\(438\) 0 0
\(439\) −8.85630 −0.422688 −0.211344 0.977412i \(-0.567784\pi\)
−0.211344 + 0.977412i \(0.567784\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 7.23467 0.343729 0.171865 0.985121i \(-0.445021\pi\)
0.171865 + 0.985121i \(0.445021\pi\)
\(444\) 0 0
\(445\) −1.42176 −0.0673978
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.19558 0.0564227 0.0282114 0.999602i \(-0.491019\pi\)
0.0282114 + 0.999602i \(0.491019\pi\)
\(450\) 0 0
\(451\) −11.4567 −0.539473
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 19.7610 0.926411
\(456\) 0 0
\(457\) 16.9915 0.794829 0.397415 0.917639i \(-0.369907\pi\)
0.397415 + 0.917639i \(0.369907\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 16.6173 0.773946 0.386973 0.922091i \(-0.373521\pi\)
0.386973 + 0.922091i \(0.373521\pi\)
\(462\) 0 0
\(463\) −12.1131 −0.562943 −0.281472 0.959570i \(-0.590822\pi\)
−0.281472 + 0.959570i \(0.590822\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −37.6701 −1.74316 −0.871581 0.490251i \(-0.836905\pi\)
−0.871581 + 0.490251i \(0.836905\pi\)
\(468\) 0 0
\(469\) −16.4524 −0.759700
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −5.92600 −0.272478
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 34.1131 1.55867 0.779333 0.626609i \(-0.215558\pi\)
0.779333 + 0.626609i \(0.215558\pi\)
\(480\) 0 0
\(481\) 16.4396 0.749580
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −15.4218 −0.700266
\(486\) 0 0
\(487\) −38.5136 −1.74522 −0.872608 0.488421i \(-0.837573\pi\)
−0.872608 + 0.488421i \(0.837573\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −3.49576 −0.157761 −0.0788806 0.996884i \(-0.525135\pi\)
−0.0788806 + 0.996884i \(0.525135\pi\)
\(492\) 0 0
\(493\) 16.4524 0.740977
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 23.0697 1.03482
\(498\) 0 0
\(499\) −16.2262 −0.726384 −0.363192 0.931714i \(-0.618313\pi\)
−0.363192 + 0.931714i \(0.618313\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 33.3699 1.48789 0.743945 0.668241i \(-0.232953\pi\)
0.743945 + 0.668241i \(0.232953\pi\)
\(504\) 0 0
\(505\) −10.2262 −0.455059
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −19.5570 −0.866847 −0.433424 0.901190i \(-0.642695\pi\)
−0.433424 + 0.901190i \(0.642695\pi\)
\(510\) 0 0
\(511\) −24.6785 −1.09171
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −16.4524 −0.724978
\(516\) 0 0
\(517\) −1.46085 −0.0642481
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −4.63945 −0.203258 −0.101629 0.994822i \(-0.532405\pi\)
−0.101629 + 0.994822i \(0.532405\pi\)
\(522\) 0 0
\(523\) −35.8962 −1.56963 −0.784816 0.619728i \(-0.787243\pi\)
−0.784816 + 0.619728i \(0.787243\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8.00000 0.348485
\(528\) 0 0
\(529\) −22.5221 −0.979220
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −26.0484 −1.12828
\(534\) 0 0
\(535\) 16.4524 0.711298
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 20.9566 0.902665
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 7.53485 0.322757
\(546\) 0 0
\(547\) 24.2262 1.03584 0.517918 0.855430i \(-0.326707\pi\)
0.517918 + 0.855430i \(0.326707\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −4.11309 −0.175224
\(552\) 0 0
\(553\) −62.2874 −2.64873
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −6.00000 −0.254228 −0.127114 0.991888i \(-0.540571\pi\)
−0.127114 + 0.991888i \(0.540571\pi\)
\(558\) 0 0
\(559\) −13.4736 −0.569874
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −44.9175 −1.89305 −0.946524 0.322634i \(-0.895432\pi\)
−0.946524 + 0.322634i \(0.895432\pi\)
\(564\) 0 0
\(565\) −17.5348 −0.737697
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 17.1956 0.720876 0.360438 0.932783i \(-0.382627\pi\)
0.360438 + 0.932783i \(0.382627\pi\)
\(570\) 0 0
\(571\) −1.15648 −0.0483974 −0.0241987 0.999707i \(-0.507703\pi\)
−0.0241987 + 0.999707i \(0.507703\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0.691333 0.0288306
\(576\) 0 0
\(577\) −36.5136 −1.52008 −0.760040 0.649876i \(-0.774821\pi\)
−0.760040 + 0.649876i \(0.774821\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 25.6088 1.06243
\(582\) 0 0
\(583\) 5.68703 0.235533
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −4.61733 −0.190578 −0.0952889 0.995450i \(-0.530377\pi\)
−0.0952889 + 0.995450i \(0.530377\pi\)
\(588\) 0 0
\(589\) −2.00000 −0.0824086
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −25.5961 −1.05110 −0.525552 0.850761i \(-0.676141\pi\)
−0.525552 + 0.850761i \(0.676141\pi\)
\(594\) 0 0
\(595\) −16.4524 −0.674481
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −45.3571 −1.85324 −0.926620 0.375999i \(-0.877300\pi\)
−0.926620 + 0.375999i \(0.877300\pi\)
\(600\) 0 0
\(601\) 15.6088 0.636698 0.318349 0.947974i \(-0.396872\pi\)
0.318349 + 0.947974i \(0.396872\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 6.53485 0.265679
\(606\) 0 0
\(607\) 39.8962 1.61934 0.809669 0.586887i \(-0.199647\pi\)
0.809669 + 0.586887i \(0.199647\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −3.32146 −0.134372
\(612\) 0 0
\(613\) 11.6828 0.471866 0.235933 0.971769i \(-0.424185\pi\)
0.235933 + 0.971769i \(0.424185\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 26.2874 1.05829 0.529145 0.848531i \(-0.322513\pi\)
0.529145 + 0.848531i \(0.322513\pi\)
\(618\) 0 0
\(619\) 23.2959 0.936340 0.468170 0.883638i \(-0.344913\pi\)
0.468170 + 0.883638i \(0.344913\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −5.84782 −0.234288
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −13.6870 −0.545738
\(630\) 0 0
\(631\) −21.2347 −0.845339 −0.422669 0.906284i \(-0.638907\pi\)
−0.422669 + 0.906284i \(0.638907\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 9.83503 0.390291
\(636\) 0 0
\(637\) 47.6479 1.88788
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 39.7091 1.56842 0.784209 0.620497i \(-0.213069\pi\)
0.784209 + 0.620497i \(0.213069\pi\)
\(642\) 0 0
\(643\) 26.8044 1.05706 0.528532 0.848914i \(-0.322743\pi\)
0.528532 + 0.848914i \(0.322743\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −10.3002 −0.404942 −0.202471 0.979288i \(-0.564897\pi\)
−0.202471 + 0.979288i \(0.564897\pi\)
\(648\) 0 0
\(649\) −20.1480 −0.790878
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 16.8563 0.659638 0.329819 0.944044i \(-0.393012\pi\)
0.329819 + 0.944044i \(0.393012\pi\)
\(654\) 0 0
\(655\) 6.33927 0.247696
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1.90903 0.0743651 0.0371826 0.999308i \(-0.488162\pi\)
0.0371826 + 0.999308i \(0.488162\pi\)
\(660\) 0 0
\(661\) 2.30018 0.0894666 0.0447333 0.998999i \(-0.485756\pi\)
0.0447333 + 0.998999i \(0.485756\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 4.11309 0.159499
\(666\) 0 0
\(667\) −2.84352 −0.110101
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −6.16497 −0.237996
\(672\) 0 0
\(673\) −32.4745 −1.25180 −0.625900 0.779904i \(-0.715268\pi\)
−0.625900 + 0.779904i \(0.715268\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 8.07400 0.310309 0.155154 0.987890i \(-0.450412\pi\)
0.155154 + 0.987890i \(0.450412\pi\)
\(678\) 0 0
\(679\) −63.4311 −2.43426
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −18.6173 −0.712372 −0.356186 0.934415i \(-0.615923\pi\)
−0.356186 + 0.934415i \(0.615923\pi\)
\(684\) 0 0
\(685\) 7.30867 0.279250
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 12.9303 0.492605
\(690\) 0 0
\(691\) −25.6088 −0.974206 −0.487103 0.873344i \(-0.661946\pi\)
−0.487103 + 0.873344i \(0.661946\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 8.22618 0.312037
\(696\) 0 0
\(697\) 21.6870 0.821455
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −0.765332 −0.0289062 −0.0144531 0.999896i \(-0.504601\pi\)
−0.0144531 + 0.999896i \(0.504601\pi\)
\(702\) 0 0
\(703\) 3.42176 0.129054
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −42.0612 −1.58187
\(708\) 0 0
\(709\) 35.8222 1.34533 0.672666 0.739946i \(-0.265149\pi\)
0.672666 + 0.739946i \(0.265149\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −1.38267 −0.0517812
\(714\) 0 0
\(715\) −10.1522 −0.379670
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 22.7916 0.849985 0.424992 0.905197i \(-0.360277\pi\)
0.424992 + 0.905197i \(0.360277\pi\)
\(720\) 0 0
\(721\) −67.6701 −2.52016
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −4.11309 −0.152756
\(726\) 0 0
\(727\) 35.6351 1.32163 0.660817 0.750547i \(-0.270210\pi\)
0.660817 + 0.750547i \(0.270210\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 11.2177 0.414901
\(732\) 0 0
\(733\) 20.6913 0.764252 0.382126 0.924110i \(-0.375192\pi\)
0.382126 + 0.924110i \(0.375192\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8.45236 0.311347
\(738\) 0 0
\(739\) −23.2177 −0.854077 −0.427038 0.904234i \(-0.640443\pi\)
−0.427038 + 0.904234i \(0.640443\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −7.06970 −0.259362 −0.129681 0.991556i \(-0.541395\pi\)
−0.129681 + 0.991556i \(0.541395\pi\)
\(744\) 0 0
\(745\) −7.38267 −0.270480
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 67.6701 2.47261
\(750\) 0 0
\(751\) −8.61733 −0.314451 −0.157225 0.987563i \(-0.550255\pi\)
−0.157225 + 0.987563i \(0.550255\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 7.38267 0.268683
\(756\) 0 0
\(757\) 47.9872 1.74412 0.872062 0.489395i \(-0.162782\pi\)
0.872062 + 0.489395i \(0.162782\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 23.0527 0.835661 0.417830 0.908525i \(-0.362791\pi\)
0.417830 + 0.908525i \(0.362791\pi\)
\(762\) 0 0
\(763\) 30.9915 1.12197
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −45.8094 −1.65408
\(768\) 0 0
\(769\) 37.0697 1.33677 0.668384 0.743817i \(-0.266986\pi\)
0.668384 + 0.743817i \(0.266986\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 25.3087 0.910289 0.455145 0.890417i \(-0.349588\pi\)
0.455145 + 0.890417i \(0.349588\pi\)
\(774\) 0 0
\(775\) −2.00000 −0.0718421
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −5.42176 −0.194255
\(780\) 0 0
\(781\) −11.8520 −0.424098
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −12.9175 −0.461046
\(786\) 0 0
\(787\) −10.9915 −0.391805 −0.195903 0.980623i \(-0.562764\pi\)
−0.195903 + 0.980623i \(0.562764\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −72.1224 −2.56438
\(792\) 0 0
\(793\) −14.0170 −0.497757
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 7.69982 0.272742 0.136371 0.990658i \(-0.456456\pi\)
0.136371 + 0.990658i \(0.456456\pi\)
\(798\) 0 0
\(799\) 2.76533 0.0978304
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 12.6785 0.447416
\(804\) 0 0
\(805\) 2.84352 0.100221
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −22.4524 −0.789383 −0.394692 0.918814i \(-0.629149\pi\)
−0.394692 + 0.918814i \(0.629149\pi\)
\(810\) 0 0
\(811\) 53.5051 1.87882 0.939409 0.342799i \(-0.111375\pi\)
0.939409 + 0.342799i \(0.111375\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 5.42176 0.189916
\(816\) 0 0
\(817\) −2.80442 −0.0981144
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1.62582 −0.0567415 −0.0283708 0.999597i \(-0.509032\pi\)
−0.0283708 + 0.999597i \(0.509032\pi\)
\(822\) 0 0
\(823\) −29.4958 −1.02816 −0.514079 0.857743i \(-0.671866\pi\)
−0.514079 + 0.857743i \(0.671866\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 4.91751 0.170999 0.0854994 0.996338i \(-0.472751\pi\)
0.0854994 + 0.996338i \(0.472751\pi\)
\(828\) 0 0
\(829\) −16.9175 −0.587570 −0.293785 0.955872i \(-0.594915\pi\)
−0.293785 + 0.955872i \(0.594915\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −39.6701 −1.37449
\(834\) 0 0
\(835\) −12.9175 −0.447029
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 10.8435 0.374360 0.187180 0.982326i \(-0.440065\pi\)
0.187180 + 0.982326i \(0.440065\pi\)
\(840\) 0 0
\(841\) −12.0825 −0.416637
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −10.0825 −0.346848
\(846\) 0 0
\(847\) 26.8784 0.923554
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 2.36557 0.0810908
\(852\) 0 0
\(853\) 28.9175 0.990117 0.495058 0.868860i \(-0.335147\pi\)
0.495058 + 0.868860i \(0.335147\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −27.2917 −0.932266 −0.466133 0.884715i \(-0.654353\pi\)
−0.466133 + 0.884715i \(0.654353\pi\)
\(858\) 0 0
\(859\) −10.6173 −0.362259 −0.181129 0.983459i \(-0.557975\pi\)
−0.181129 + 0.983459i \(0.557975\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 6.61733 0.225257 0.112628 0.993637i \(-0.464073\pi\)
0.112628 + 0.993637i \(0.464073\pi\)
\(864\) 0 0
\(865\) −13.5348 −0.460199
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 32.0000 1.08553
\(870\) 0 0
\(871\) 19.2177 0.651167
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 4.11309 0.139048
\(876\) 0 0
\(877\) 22.4133 0.756842 0.378421 0.925634i \(-0.376467\pi\)
0.378421 + 0.925634i \(0.376467\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 12.5392 0.422455 0.211227 0.977437i \(-0.432254\pi\)
0.211227 + 0.977437i \(0.432254\pi\)
\(882\) 0 0
\(883\) 7.96091 0.267906 0.133953 0.990988i \(-0.457233\pi\)
0.133953 + 0.990988i \(0.457233\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 5.68703 0.190952 0.0954759 0.995432i \(-0.469563\pi\)
0.0954759 + 0.995432i \(0.469563\pi\)
\(888\) 0 0
\(889\) 40.4524 1.35673
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −0.691333 −0.0231346
\(894\) 0 0
\(895\) −10.9175 −0.364932
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 8.22618 0.274359
\(900\) 0 0
\(901\) −10.7653 −0.358645
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 17.3699 0.577394
\(906\) 0 0
\(907\) 17.6088 0.584692 0.292346 0.956313i \(-0.405564\pi\)
0.292346 + 0.956313i \(0.405564\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 23.2177 0.769237 0.384618 0.923076i \(-0.374333\pi\)
0.384618 + 0.923076i \(0.374333\pi\)
\(912\) 0 0
\(913\) −13.1565 −0.435416
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 26.0740 0.861039
\(918\) 0 0
\(919\) 7.06970 0.233208 0.116604 0.993179i \(-0.462799\pi\)
0.116604 + 0.993179i \(0.462799\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −26.9473 −0.886980
\(924\) 0 0
\(925\) 3.42176 0.112507
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 52.9659 1.73776 0.868878 0.495026i \(-0.164842\pi\)
0.868878 + 0.495026i \(0.164842\pi\)
\(930\) 0 0
\(931\) 9.91751 0.325033
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 8.45236 0.276422
\(936\) 0 0
\(937\) 3.75685 0.122731 0.0613654 0.998115i \(-0.480455\pi\)
0.0613654 + 0.998115i \(0.480455\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 13.0178 0.424369 0.212184 0.977230i \(-0.431942\pi\)
0.212184 + 0.977230i \(0.431942\pi\)
\(942\) 0 0
\(943\) −3.74824 −0.122059
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −20.3912 −0.662623 −0.331312 0.943521i \(-0.607491\pi\)
−0.331312 + 0.943521i \(0.607491\pi\)
\(948\) 0 0
\(949\) 28.8265 0.935749
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 48.2746 1.56377 0.781884 0.623424i \(-0.214259\pi\)
0.781884 + 0.623424i \(0.214259\pi\)
\(954\) 0 0
\(955\) 0.730425 0.0236360
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 30.0612 0.970727
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −25.4830 −0.820326
\(966\) 0 0
\(967\) −28.9396 −0.930636 −0.465318 0.885144i \(-0.654060\pi\)
−0.465318 + 0.885144i \(0.654060\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 3.92600 0.125991 0.0629957 0.998014i \(-0.479935\pi\)
0.0629957 + 0.998014i \(0.479935\pi\)
\(972\) 0 0
\(973\) 33.8350 1.08470
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 9.30867 0.297811 0.148905 0.988851i \(-0.452425\pi\)
0.148905 + 0.988851i \(0.452425\pi\)
\(978\) 0 0
\(979\) 3.00430 0.0960179
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 13.8478 0.441677 0.220838 0.975310i \(-0.429121\pi\)
0.220838 + 0.975310i \(0.429121\pi\)
\(984\) 0 0
\(985\) −21.7610 −0.693364
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −1.93879 −0.0616500
\(990\) 0 0
\(991\) 30.5433 0.970241 0.485121 0.874447i \(-0.338776\pi\)
0.485121 + 0.874447i \(0.338776\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −1.60885 −0.0510039
\(996\) 0 0
\(997\) −6.75254 −0.213855 −0.106928 0.994267i \(-0.534101\pi\)
−0.106928 + 0.994267i \(0.534101\pi\)
\(998\) 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 6840.2.a.bh.1.1 3
3.2 odd 2 2280.2.a.u.1.1 3
12.11 even 2 4560.2.a.bs.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2280.2.a.u.1.1 3 3.2 odd 2
4560.2.a.bs.1.3 3 12.11 even 2
6840.2.a.bh.1.1 3 1.1 even 1 trivial