Properties

Label 168.2.a.a.1.1
Level $168$
Weight $2$
Character 168.1
Self dual yes
Analytic conductor $1.341$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 168.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.34148675396\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 168.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +2.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +2.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +6.00000 q^{13} -2.00000 q^{15} -2.00000 q^{17} +4.00000 q^{19} -1.00000 q^{21} -4.00000 q^{23} -1.00000 q^{25} -1.00000 q^{27} -10.0000 q^{29} -8.00000 q^{31} +2.00000 q^{35} +6.00000 q^{37} -6.00000 q^{39} -2.00000 q^{41} -4.00000 q^{43} +2.00000 q^{45} +8.00000 q^{47} +1.00000 q^{49} +2.00000 q^{51} -10.0000 q^{53} -4.00000 q^{57} +12.0000 q^{59} -2.00000 q^{61} +1.00000 q^{63} +12.0000 q^{65} +12.0000 q^{67} +4.00000 q^{69} -12.0000 q^{71} -14.0000 q^{73} +1.00000 q^{75} -8.00000 q^{79} +1.00000 q^{81} +12.0000 q^{83} -4.00000 q^{85} +10.0000 q^{87} -2.00000 q^{89} +6.00000 q^{91} +8.00000 q^{93} +8.00000 q^{95} +10.0000 q^{97} +O(q^{100})\)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −10.0000 −1.85695 −0.928477 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 0 0
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 2.00000 0.298142
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 12.0000 1.48842
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 0 0
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) −14.0000 −1.63858 −0.819288 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) 0 0
\(87\) 10.0000 1.07211
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 6.00000 0.628971
\(92\) 0 0
\(93\) 8.00000 0.829561
\(94\) 0 0
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) −2.00000 −0.195180
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) −6.00000 −0.569495
\(112\) 0 0
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 0 0
\(115\) −8.00000 −0.746004
\(116\) 0 0
\(117\) 6.00000 0.554700
\(118\) 0 0
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 2.00000 0.180334
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) 0 0
\(135\) −2.00000 −0.172133
\(136\) 0 0
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −20.0000 −1.66091
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) 22.0000 1.80231 0.901155 0.433497i \(-0.142720\pi\)
0.901155 + 0.433497i \(0.142720\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) −16.0000 −1.28515
\(156\) 0 0
\(157\) −18.0000 −1.43656 −0.718278 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) 0 0
\(159\) 10.0000 0.793052
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) 0 0
\(163\) −20.0000 −1.56652 −0.783260 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) 0 0
\(173\) 2.00000 0.152057 0.0760286 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −12.0000 −0.901975
\(178\) 0 0
\(179\) −16.0000 −1.19590 −0.597948 0.801535i \(-0.704017\pi\)
−0.597948 + 0.801535i \(0.704017\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) 12.0000 0.882258
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −20.0000 −1.44715 −0.723575 0.690246i \(-0.757502\pi\)
−0.723575 + 0.690246i \(0.757502\pi\)
\(192\) 0 0
\(193\) 18.0000 1.29567 0.647834 0.761781i \(-0.275675\pi\)
0.647834 + 0.761781i \(0.275675\pi\)
\(194\) 0 0
\(195\) −12.0000 −0.859338
\(196\) 0 0
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 0 0
\(201\) −12.0000 −0.846415
\(202\) 0 0
\(203\) −10.0000 −0.701862
\(204\) 0 0
\(205\) −4.00000 −0.279372
\(206\) 0 0
\(207\) −4.00000 −0.278019
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 0 0
\(213\) 12.0000 0.822226
\(214\) 0 0
\(215\) −8.00000 −0.545595
\(216\) 0 0
\(217\) −8.00000 −0.543075
\(218\) 0 0
\(219\) 14.0000 0.946032
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 0 0
\(227\) 20.0000 1.32745 0.663723 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) 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
\(233\) 2.00000 0.131024 0.0655122 0.997852i \(-0.479132\pi\)
0.0655122 + 0.997852i \(0.479132\pi\)
\(234\) 0 0
\(235\) 16.0000 1.04372
\(236\) 0 0
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) 20.0000 1.29369 0.646846 0.762620i \(-0.276088\pi\)
0.646846 + 0.762620i \(0.276088\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 2.00000 0.127775
\(246\) 0 0
\(247\) 24.0000 1.52708
\(248\) 0 0
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) −20.0000 −1.26239 −0.631194 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 4.00000 0.250490
\(256\) 0 0
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) 6.00000 0.372822
\(260\) 0 0
\(261\) −10.0000 −0.618984
\(262\) 0 0
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 0 0
\(265\) −20.0000 −1.22859
\(266\) 0 0
\(267\) 2.00000 0.122398
\(268\) 0 0
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 0 0
\(273\) −6.00000 −0.363137
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) 0 0
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 0 0
\(285\) −8.00000 −0.473879
\(286\) 0 0
\(287\) −2.00000 −0.118056
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) 0 0
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) 24.0000 1.39733
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −24.0000 −1.38796
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) 0 0
\(303\) 6.00000 0.344691
\(304\) 0 0
\(305\) −4.00000 −0.229039
\(306\) 0 0
\(307\) 4.00000 0.228292 0.114146 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −22.0000 −1.24351 −0.621757 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) 0 0
\(315\) 2.00000 0.112687
\(316\) 0 0
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −8.00000 −0.445132
\(324\) 0 0
\(325\) −6.00000 −0.332820
\(326\) 0 0
\(327\) 2.00000 0.110600
\(328\) 0 0
\(329\) 8.00000 0.441054
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 0 0
\(333\) 6.00000 0.328798
\(334\) 0 0
\(335\) 24.0000 1.31126
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 0 0
\(339\) −18.0000 −0.977626
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 8.00000 0.430706
\(346\) 0 0
\(347\) 24.0000 1.28839 0.644194 0.764862i \(-0.277193\pi\)
0.644194 + 0.764862i \(0.277193\pi\)
\(348\) 0 0
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) −6.00000 −0.320256
\(352\) 0 0
\(353\) 30.0000 1.59674 0.798369 0.602168i \(-0.205696\pi\)
0.798369 + 0.602168i \(0.205696\pi\)
\(354\) 0 0
\(355\) −24.0000 −1.27379
\(356\) 0 0
\(357\) 2.00000 0.105851
\(358\) 0 0
\(359\) −12.0000 −0.633336 −0.316668 0.948536i \(-0.602564\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 11.0000 0.577350
\(364\) 0 0
\(365\) −28.0000 −1.46559
\(366\) 0 0
\(367\) −16.0000 −0.835193 −0.417597 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) 0 0
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) −10.0000 −0.519174
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 0 0
\(375\) 12.0000 0.619677
\(376\) 0 0
\(377\) −60.0000 −3.09016
\(378\) 0 0
\(379\) 12.0000 0.616399 0.308199 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4.00000 −0.203331
\(388\) 0 0
\(389\) −2.00000 −0.101404 −0.0507020 0.998714i \(-0.516146\pi\)
−0.0507020 + 0.998714i \(0.516146\pi\)
\(390\) 0 0
\(391\) 8.00000 0.404577
\(392\) 0 0
\(393\) 4.00000 0.201773
\(394\) 0 0
\(395\) −16.0000 −0.805047
\(396\) 0 0
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 0 0
\(399\) −4.00000 −0.200250
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 0 0
\(403\) −48.0000 −2.39105
\(404\) 0 0
\(405\) 2.00000 0.0993808
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 18.0000 0.890043 0.445021 0.895520i \(-0.353196\pi\)
0.445021 + 0.895520i \(0.353196\pi\)
\(410\) 0 0
\(411\) −2.00000 −0.0986527
\(412\) 0 0
\(413\) 12.0000 0.590481
\(414\) 0 0
\(415\) 24.0000 1.17811
\(416\) 0 0
\(417\) 12.0000 0.587643
\(418\) 0 0
\(419\) 20.0000 0.977064 0.488532 0.872546i \(-0.337533\pi\)
0.488532 + 0.872546i \(0.337533\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) 0 0
\(423\) 8.00000 0.388973
\(424\) 0 0
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) −2.00000 −0.0967868
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 0 0
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) 20.0000 0.958927
\(436\) 0 0
\(437\) −16.0000 −0.765384
\(438\) 0 0
\(439\) −24.0000 −1.14546 −0.572729 0.819745i \(-0.694115\pi\)
−0.572729 + 0.819745i \(0.694115\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 8.00000 0.380091 0.190046 0.981775i \(-0.439136\pi\)
0.190046 + 0.981775i \(0.439136\pi\)
\(444\) 0 0
\(445\) −4.00000 −0.189618
\(446\) 0 0
\(447\) −22.0000 −1.04056
\(448\) 0 0
\(449\) 34.0000 1.60456 0.802280 0.596948i \(-0.203620\pi\)
0.802280 + 0.596948i \(0.203620\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 8.00000 0.375873
\(454\) 0 0
\(455\) 12.0000 0.562569
\(456\) 0 0
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 0 0
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) −40.0000 −1.85896 −0.929479 0.368875i \(-0.879743\pi\)
−0.929479 + 0.368875i \(0.879743\pi\)
\(464\) 0 0
\(465\) 16.0000 0.741982
\(466\) 0 0
\(467\) 36.0000 1.66588 0.832941 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) 0 0
\(469\) 12.0000 0.554109
\(470\) 0 0
\(471\) 18.0000 0.829396
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) −10.0000 −0.457869
\(478\) 0 0
\(479\) −8.00000 −0.365529 −0.182765 0.983157i \(-0.558505\pi\)
−0.182765 + 0.983157i \(0.558505\pi\)
\(480\) 0 0
\(481\) 36.0000 1.64146
\(482\) 0 0
\(483\) 4.00000 0.182006
\(484\) 0 0
\(485\) 20.0000 0.908153
\(486\) 0 0
\(487\) −8.00000 −0.362515 −0.181257 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) 0 0
\(489\) 20.0000 0.904431
\(490\) 0 0
\(491\) −24.0000 −1.08310 −0.541552 0.840667i \(-0.682163\pi\)
−0.541552 + 0.840667i \(0.682163\pi\)
\(492\) 0 0
\(493\) 20.0000 0.900755
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −12.0000 −0.538274
\(498\) 0 0
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 40.0000 1.78351 0.891756 0.452517i \(-0.149474\pi\)
0.891756 + 0.452517i \(0.149474\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) 0 0
\(507\) −23.0000 −1.02147
\(508\) 0 0
\(509\) 42.0000 1.86162 0.930809 0.365507i \(-0.119104\pi\)
0.930809 + 0.365507i \(0.119104\pi\)
\(510\) 0 0
\(511\) −14.0000 −0.619324
\(512\) 0 0
\(513\) −4.00000 −0.176604
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −2.00000 −0.0877903
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 0 0
\(525\) 1.00000 0.0436436
\(526\) 0 0
\(527\) 16.0000 0.696971
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 16.0000 0.690451
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 0 0
\(543\) −14.0000 −0.600798
\(544\) 0 0
\(545\) −4.00000 −0.171341
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 0 0
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) −40.0000 −1.70406
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) 0 0
\(555\) −12.0000 −0.509372
\(556\) 0 0
\(557\) −2.00000 −0.0847427 −0.0423714 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) 0 0
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 0 0
\(565\) 36.0000 1.51453
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 0 0
\(573\) 20.0000 0.835512
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) 18.0000 0.749350 0.374675 0.927156i \(-0.377754\pi\)
0.374675 + 0.927156i \(0.377754\pi\)
\(578\) 0 0
\(579\) −18.0000 −0.748054
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 12.0000 0.496139
\(586\) 0 0
\(587\) 28.0000 1.15568 0.577842 0.816149i \(-0.303895\pi\)
0.577842 + 0.816149i \(0.303895\pi\)
\(588\) 0 0
\(589\) −32.0000 −1.31854
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 0 0
\(593\) −18.0000 −0.739171 −0.369586 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) 0 0
\(595\) −4.00000 −0.163984
\(596\) 0 0
\(597\) −8.00000 −0.327418
\(598\) 0 0
\(599\) −44.0000 −1.79779 −0.898896 0.438163i \(-0.855629\pi\)
−0.898896 + 0.438163i \(0.855629\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 0 0
\(603\) 12.0000 0.488678
\(604\) 0 0
\(605\) −22.0000 −0.894427
\(606\) 0 0
\(607\) 16.0000 0.649420 0.324710 0.945814i \(-0.394733\pi\)
0.324710 + 0.945814i \(0.394733\pi\)
\(608\) 0 0
\(609\) 10.0000 0.405220
\(610\) 0 0
\(611\) 48.0000 1.94187
\(612\) 0 0
\(613\) −26.0000 −1.05013 −0.525065 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) 0 0
\(615\) 4.00000 0.161296
\(616\) 0 0
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) 0 0
\(619\) −12.0000 −0.482321 −0.241160 0.970485i \(-0.577528\pi\)
−0.241160 + 0.970485i \(0.577528\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 0 0
\(623\) −2.00000 −0.0801283
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 0 0
\(633\) −20.0000 −0.794929
\(634\) 0 0
\(635\) 16.0000 0.634941
\(636\) 0 0
\(637\) 6.00000 0.237729
\(638\) 0 0
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) 10.0000 0.394976 0.197488 0.980305i \(-0.436722\pi\)
0.197488 + 0.980305i \(0.436722\pi\)
\(642\) 0 0
\(643\) 4.00000 0.157745 0.0788723 0.996885i \(-0.474868\pi\)
0.0788723 + 0.996885i \(0.474868\pi\)
\(644\) 0 0
\(645\) 8.00000 0.315000
\(646\) 0 0
\(647\) −48.0000 −1.88707 −0.943537 0.331266i \(-0.892524\pi\)
−0.943537 + 0.331266i \(0.892524\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 8.00000 0.313545
\(652\) 0 0
\(653\) −10.0000 −0.391330 −0.195665 0.980671i \(-0.562687\pi\)
−0.195665 + 0.980671i \(0.562687\pi\)
\(654\) 0 0
\(655\) −8.00000 −0.312586
\(656\) 0 0
\(657\) −14.0000 −0.546192
\(658\) 0 0
\(659\) 8.00000 0.311636 0.155818 0.987786i \(-0.450199\pi\)
0.155818 + 0.987786i \(0.450199\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) 0 0
\(663\) 12.0000 0.466041
\(664\) 0 0
\(665\) 8.00000 0.310227
\(666\) 0 0
\(667\) 40.0000 1.54881
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −14.0000 −0.539660 −0.269830 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 0 0
\(679\) 10.0000 0.383765
\(680\) 0 0
\(681\) −20.0000 −0.766402
\(682\) 0 0
\(683\) −24.0000 −0.918334 −0.459167 0.888350i \(-0.651852\pi\)
−0.459167 + 0.888350i \(0.651852\pi\)
\(684\) 0 0
\(685\) 4.00000 0.152832
\(686\) 0 0
\(687\) −14.0000 −0.534133
\(688\) 0 0
\(689\) −60.0000 −2.28582
\(690\) 0 0
\(691\) 44.0000 1.67384 0.836919 0.547326i \(-0.184354\pi\)
0.836919 + 0.547326i \(0.184354\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −24.0000 −0.910372
\(696\) 0 0
\(697\) 4.00000 0.151511
\(698\) 0 0
\(699\) −2.00000 −0.0756469
\(700\) 0 0
\(701\) 22.0000 0.830929 0.415464 0.909610i \(-0.363619\pi\)
0.415464 + 0.909610i \(0.363619\pi\)
\(702\) 0 0
\(703\) 24.0000 0.905177
\(704\) 0 0
\(705\) −16.0000 −0.602595
\(706\) 0 0
\(707\) −6.00000 −0.225653
\(708\) 0 0
\(709\) 22.0000 0.826227 0.413114 0.910679i \(-0.364441\pi\)
0.413114 + 0.910679i \(0.364441\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) 32.0000 1.19841
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −20.0000 −0.746914
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −10.0000 −0.371904
\(724\) 0 0
\(725\) 10.0000 0.371391
\(726\) 0 0
\(727\) −16.0000 −0.593407 −0.296704 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 8.00000 0.295891
\(732\) 0 0
\(733\) −26.0000 −0.960332 −0.480166 0.877178i \(-0.659424\pi\)
−0.480166 + 0.877178i \(0.659424\pi\)
\(734\) 0 0
\(735\) −2.00000 −0.0737711
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) −24.0000 −0.881662
\(742\) 0 0
\(743\) 20.0000 0.733729 0.366864 0.930274i \(-0.380431\pi\)
0.366864 + 0.930274i \(0.380431\pi\)
\(744\) 0 0
\(745\) 44.0000 1.61204
\(746\) 0 0
\(747\) 12.0000 0.439057
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 0 0
\(753\) 20.0000 0.728841
\(754\) 0 0
\(755\) −16.0000 −0.582300
\(756\) 0 0
\(757\) −26.0000 −0.944986 −0.472493 0.881334i \(-0.656646\pi\)
−0.472493 + 0.881334i \(0.656646\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 0 0
\(763\) −2.00000 −0.0724049
\(764\) 0 0
\(765\) −4.00000 −0.144620
\(766\) 0 0
\(767\) 72.0000 2.59977
\(768\) 0 0
\(769\) 34.0000 1.22607 0.613036 0.790055i \(-0.289948\pi\)
0.613036 + 0.790055i \(0.289948\pi\)
\(770\) 0 0
\(771\) −6.00000 −0.216085
\(772\) 0 0
\(773\) 10.0000 0.359675 0.179838 0.983696i \(-0.442443\pi\)
0.179838 + 0.983696i \(0.442443\pi\)
\(774\) 0 0
\(775\) 8.00000 0.287368
\(776\) 0 0
\(777\) −6.00000 −0.215249
\(778\) 0 0
\(779\) −8.00000 −0.286630
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 10.0000 0.357371
\(784\) 0 0
\(785\) −36.0000 −1.28490
\(786\) 0 0
\(787\) 28.0000 0.998092 0.499046 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) 0 0
\(789\) −12.0000 −0.427211
\(790\) 0 0
\(791\) 18.0000 0.640006
\(792\) 0 0
\(793\) −12.0000 −0.426132
\(794\) 0 0
\(795\) 20.0000 0.709327
\(796\) 0 0
\(797\) 18.0000 0.637593 0.318796 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) 0 0
\(801\) −2.00000 −0.0706665
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −8.00000 −0.281963
\(806\) 0 0
\(807\) 6.00000 0.211210
\(808\) 0 0
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) 36.0000 1.26413 0.632065 0.774915i \(-0.282207\pi\)
0.632065 + 0.774915i \(0.282207\pi\)
\(812\) 0 0
\(813\) −8.00000 −0.280572
\(814\) 0 0
\(815\) −40.0000 −1.40114
\(816\) 0 0
\(817\) −16.0000 −0.559769
\(818\) 0 0
\(819\) 6.00000 0.209657
\(820\) 0 0
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) 0 0
\(823\) −16.0000 −0.557725 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 38.0000 1.31979 0.659897 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) 0 0
\(831\) −22.0000 −0.763172
\(832\) 0 0
\(833\) −2.00000 −0.0692959
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 8.00000 0.276520
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) 0 0
\(843\) −18.0000 −0.619953
\(844\) 0 0
\(845\) 46.0000 1.58245
\(846\) 0 0
\(847\) −11.0000 −0.377964
\(848\) 0 0
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) −24.0000 −0.822709
\(852\) 0 0
\(853\) −26.0000 −0.890223 −0.445112 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) 0 0
\(855\) 8.00000 0.273594
\(856\) 0 0
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 0 0
\(861\) 2.00000 0.0681598
\(862\) 0 0
\(863\) 20.0000 0.680808 0.340404 0.940279i \(-0.389436\pi\)
0.340404 + 0.940279i \(0.389436\pi\)
\(864\) 0 0
\(865\) 4.00000 0.136004
\(866\) 0 0
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 72.0000 2.43963
\(872\) 0 0
\(873\) 10.0000 0.338449
\(874\) 0 0
\(875\) −12.0000 −0.405674
\(876\) 0 0
\(877\) 14.0000 0.472746 0.236373 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) 0 0
\(879\) 6.00000 0.202375
\(880\) 0 0
\(881\) −2.00000 −0.0673817 −0.0336909 0.999432i \(-0.510726\pi\)
−0.0336909 + 0.999432i \(0.510726\pi\)
\(882\) 0 0
\(883\) 4.00000 0.134611 0.0673054 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) 0 0
\(885\) −24.0000 −0.806751
\(886\) 0 0
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 32.0000 1.07084
\(894\) 0 0
\(895\) −32.0000 −1.06964
\(896\) 0 0
\(897\) 24.0000 0.801337
\(898\) 0 0
\(899\) 80.0000 2.66815
\(900\) 0 0
\(901\) 20.0000 0.666297
\(902\) 0 0
\(903\) 4.00000 0.133112
\(904\) 0 0
\(905\) 28.0000 0.930751
\(906\) 0 0
\(907\) 44.0000 1.46100 0.730498 0.682915i \(-0.239288\pi\)
0.730498 + 0.682915i \(0.239288\pi\)
\(908\) 0 0
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) −20.0000 −0.662630 −0.331315 0.943520i \(-0.607492\pi\)
−0.331315 + 0.943520i \(0.607492\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 4.00000 0.132236
\(916\) 0 0
\(917\) −4.00000 −0.132092
\(918\) 0 0
\(919\) −8.00000 −0.263896 −0.131948 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 0 0
\(921\) −4.00000 −0.131804
\(922\) 0 0
\(923\) −72.0000 −2.36991
\(924\) 0 0
\(925\) −6.00000 −0.197279
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −26.0000 −0.853032 −0.426516 0.904480i \(-0.640259\pi\)
−0.426516 + 0.904480i \(0.640259\pi\)
\(930\) 0 0
\(931\) 4.00000 0.131095
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −54.0000 −1.76410 −0.882052 0.471153i \(-0.843838\pi\)
−0.882052 + 0.471153i \(0.843838\pi\)
\(938\) 0 0
\(939\) 22.0000 0.717943
\(940\) 0 0
\(941\) 42.0000 1.36916 0.684580 0.728937i \(-0.259985\pi\)
0.684580 + 0.728937i \(0.259985\pi\)
\(942\) 0 0
\(943\) 8.00000 0.260516
\(944\) 0 0
\(945\) −2.00000 −0.0650600
\(946\) 0 0
\(947\) −56.0000 −1.81976 −0.909878 0.414876i \(-0.863825\pi\)
−0.909878 + 0.414876i \(0.863825\pi\)
\(948\) 0 0
\(949\) −84.0000 −2.72676
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) 58.0000 1.87880 0.939402 0.342817i \(-0.111381\pi\)
0.939402 + 0.342817i \(0.111381\pi\)
\(954\) 0 0
\(955\) −40.0000 −1.29437
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 2.00000 0.0645834
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 36.0000 1.15888
\(966\) 0 0
\(967\) −48.0000 −1.54358 −0.771788 0.635880i \(-0.780637\pi\)
−0.771788 + 0.635880i \(0.780637\pi\)
\(968\) 0 0
\(969\) 8.00000 0.256997
\(970\) 0 0
\(971\) 4.00000 0.128366 0.0641831 0.997938i \(-0.479556\pi\)
0.0641831 + 0.997938i \(0.479556\pi\)
\(972\) 0 0
\(973\) −12.0000 −0.384702
\(974\) 0 0
\(975\) 6.00000 0.192154
\(976\) 0 0
\(977\) −6.00000 −0.191957 −0.0959785 0.995383i \(-0.530598\pi\)
−0.0959785 + 0.995383i \(0.530598\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −2.00000 −0.0638551
\(982\) 0 0
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 0 0
\(985\) 12.0000 0.382352
\(986\) 0 0
\(987\) −8.00000 −0.254643
\(988\) 0 0
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) 4.00000 0.126936
\(994\) 0 0
\(995\) 16.0000 0.507234
\(996\) 0 0
\(997\) −26.0000 −0.823428 −0.411714 0.911313i \(-0.635070\pi\)
−0.411714 + 0.911313i \(0.635070\pi\)
\(998\) 0 0
\(999\) −6.00000 −0.189832
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 168.2.a.a.1.1 1
3.2 odd 2 504.2.a.e.1.1 1
4.3 odd 2 336.2.a.e.1.1 1
5.2 odd 4 4200.2.t.j.1849.2 2
5.3 odd 4 4200.2.t.j.1849.1 2
5.4 even 2 4200.2.a.t.1.1 1
7.2 even 3 1176.2.q.f.361.1 2
7.3 odd 6 1176.2.q.d.961.1 2
7.4 even 3 1176.2.q.f.961.1 2
7.5 odd 6 1176.2.q.d.361.1 2
7.6 odd 2 1176.2.a.f.1.1 1
8.3 odd 2 1344.2.a.b.1.1 1
8.5 even 2 1344.2.a.m.1.1 1
12.11 even 2 1008.2.a.b.1.1 1
16.3 odd 4 5376.2.c.bb.2689.2 2
16.5 even 4 5376.2.c.d.2689.2 2
16.11 odd 4 5376.2.c.bb.2689.1 2
16.13 even 4 5376.2.c.d.2689.1 2
20.19 odd 2 8400.2.a.y.1.1 1
21.2 odd 6 3528.2.s.w.361.1 2
21.5 even 6 3528.2.s.g.361.1 2
21.11 odd 6 3528.2.s.w.3313.1 2
21.17 even 6 3528.2.s.g.3313.1 2
21.20 even 2 3528.2.a.v.1.1 1
24.5 odd 2 4032.2.a.bh.1.1 1
24.11 even 2 4032.2.a.bc.1.1 1
28.3 even 6 2352.2.q.w.961.1 2
28.11 odd 6 2352.2.q.d.961.1 2
28.19 even 6 2352.2.q.w.1537.1 2
28.23 odd 6 2352.2.q.d.1537.1 2
28.27 even 2 2352.2.a.c.1.1 1
56.13 odd 2 9408.2.a.be.1.1 1
56.27 even 2 9408.2.a.da.1.1 1
84.83 odd 2 7056.2.a.bq.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.a.a.1.1 1 1.1 even 1 trivial
336.2.a.e.1.1 1 4.3 odd 2
504.2.a.e.1.1 1 3.2 odd 2
1008.2.a.b.1.1 1 12.11 even 2
1176.2.a.f.1.1 1 7.6 odd 2
1176.2.q.d.361.1 2 7.5 odd 6
1176.2.q.d.961.1 2 7.3 odd 6
1176.2.q.f.361.1 2 7.2 even 3
1176.2.q.f.961.1 2 7.4 even 3
1344.2.a.b.1.1 1 8.3 odd 2
1344.2.a.m.1.1 1 8.5 even 2
2352.2.a.c.1.1 1 28.27 even 2
2352.2.q.d.961.1 2 28.11 odd 6
2352.2.q.d.1537.1 2 28.23 odd 6
2352.2.q.w.961.1 2 28.3 even 6
2352.2.q.w.1537.1 2 28.19 even 6
3528.2.a.v.1.1 1 21.20 even 2
3528.2.s.g.361.1 2 21.5 even 6
3528.2.s.g.3313.1 2 21.17 even 6
3528.2.s.w.361.1 2 21.2 odd 6
3528.2.s.w.3313.1 2 21.11 odd 6
4032.2.a.bc.1.1 1 24.11 even 2
4032.2.a.bh.1.1 1 24.5 odd 2
4200.2.a.t.1.1 1 5.4 even 2
4200.2.t.j.1849.1 2 5.3 odd 4
4200.2.t.j.1849.2 2 5.2 odd 4
5376.2.c.d.2689.1 2 16.13 even 4
5376.2.c.d.2689.2 2 16.5 even 4
5376.2.c.bb.2689.1 2 16.11 odd 4
5376.2.c.bb.2689.2 2 16.3 odd 4
7056.2.a.bq.1.1 1 84.83 odd 2
8400.2.a.y.1.1 1 20.19 odd 2
9408.2.a.be.1.1 1 56.13 odd 2
9408.2.a.da.1.1 1 56.27 even 2