Properties

Label 9408.2.a.p.1.1
Level 9408
Weight 2
Character 9408.1
Self dual yes
Analytic conductor 75.123
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 9408 = 2^{6} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9408.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(75.1232582216\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 9408.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{9} +3.00000 q^{11} +4.00000 q^{13} +1.00000 q^{15} +4.00000 q^{19} -8.00000 q^{23} -4.00000 q^{25} -1.00000 q^{27} +3.00000 q^{29} -5.00000 q^{31} -3.00000 q^{33} -8.00000 q^{37} -4.00000 q^{39} -8.00000 q^{41} +6.00000 q^{43} -1.00000 q^{45} +10.0000 q^{47} -9.00000 q^{53} -3.00000 q^{55} -4.00000 q^{57} +5.00000 q^{59} -10.0000 q^{61} -4.00000 q^{65} +6.00000 q^{67} +8.00000 q^{69} -10.0000 q^{71} -2.00000 q^{73} +4.00000 q^{75} -11.0000 q^{79} +1.00000 q^{81} -7.00000 q^{83} -3.00000 q^{87} +18.0000 q^{89} +5.00000 q^{93} -4.00000 q^{95} +17.0000 q^{97} +3.00000 q^{99} +O(q^{100})\)

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
\(5\) −1.00000 −0.447214 −0.223607 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) −5.00000 −0.898027 −0.449013 0.893525i \(-0.648224\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) 0 0
\(33\) −3.00000 −0.522233
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 0 0
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 10.0000 1.45865 0.729325 0.684167i \(-0.239834\pi\)
0.729325 + 0.684167i \(0.239834\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −9.00000 −1.23625 −0.618123 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) 0 0
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) 5.00000 0.650945 0.325472 0.945552i \(-0.394477\pi\)
0.325472 + 0.945552i \(0.394477\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) 6.00000 0.733017 0.366508 0.930415i \(-0.380553\pi\)
0.366508 + 0.930415i \(0.380553\pi\)
\(68\) 0 0
\(69\) 8.00000 0.963087
\(70\) 0 0
\(71\) −10.0000 −1.18678 −0.593391 0.804914i \(-0.702211\pi\)
−0.593391 + 0.804914i \(0.702211\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 0 0
\(75\) 4.00000 0.461880
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −11.0000 −1.23760 −0.618798 0.785550i \(-0.712380\pi\)
−0.618798 + 0.785550i \(0.712380\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −7.00000 −0.768350 −0.384175 0.923260i \(-0.625514\pi\)
−0.384175 + 0.923260i \(0.625514\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −3.00000 −0.321634
\(88\) 0 0
\(89\) 18.0000 1.90800 0.953998 0.299813i \(-0.0969242\pi\)
0.953998 + 0.299813i \(0.0969242\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 5.00000 0.518476
\(94\) 0 0
\(95\) −4.00000 −0.410391
\(96\) 0 0
\(97\) 17.0000 1.72609 0.863044 0.505128i \(-0.168555\pi\)
0.863044 + 0.505128i \(0.168555\pi\)
\(98\) 0 0
\(99\) 3.00000 0.301511
\(100\) 0 0
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −11.0000 −1.06341 −0.531705 0.846930i \(-0.678449\pi\)
−0.531705 + 0.846930i \(0.678449\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 8.00000 0.759326
\(112\) 0 0
\(113\) −8.00000 −0.752577 −0.376288 0.926503i \(-0.622800\pi\)
−0.376288 + 0.926503i \(0.622800\pi\)
\(114\) 0 0
\(115\) 8.00000 0.746004
\(116\) 0 0
\(117\) 4.00000 0.369800
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) 8.00000 0.721336
\(124\) 0 0
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) 7.00000 0.621150 0.310575 0.950549i \(-0.399478\pi\)
0.310575 + 0.950549i \(0.399478\pi\)
\(128\) 0 0
\(129\) −6.00000 −0.528271
\(130\) 0 0
\(131\) −15.0000 −1.31056 −0.655278 0.755388i \(-0.727449\pi\)
−0.655278 + 0.755388i \(0.727449\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −14.0000 −1.19610 −0.598050 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) 0 0
\(139\) 22.0000 1.86602 0.933008 0.359856i \(-0.117174\pi\)
0.933008 + 0.359856i \(0.117174\pi\)
\(140\) 0 0
\(141\) −10.0000 −0.842152
\(142\) 0 0
\(143\) 12.0000 1.00349
\(144\) 0 0
\(145\) −3.00000 −0.249136
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) 5.00000 0.406894 0.203447 0.979086i \(-0.434786\pi\)
0.203447 + 0.979086i \(0.434786\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 5.00000 0.401610
\(156\) 0 0
\(157\) −4.00000 −0.319235 −0.159617 0.987179i \(-0.551026\pi\)
−0.159617 + 0.987179i \(0.551026\pi\)
\(158\) 0 0
\(159\) 9.00000 0.713746
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 8.00000 0.626608 0.313304 0.949653i \(-0.398564\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) 0 0
\(165\) 3.00000 0.233550
\(166\) 0 0
\(167\) −2.00000 −0.154765 −0.0773823 0.997001i \(-0.524656\pi\)
−0.0773823 + 0.997001i \(0.524656\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 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
\(175\) 0 0
\(176\) 0 0
\(177\) −5.00000 −0.375823
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 10.0000 0.739221
\(184\) 0 0
\(185\) 8.00000 0.588172
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 16.0000 1.15772 0.578860 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) 0 0
\(193\) −27.0000 −1.94350 −0.971751 0.236007i \(-0.924161\pi\)
−0.971751 + 0.236007i \(0.924161\pi\)
\(194\) 0 0
\(195\) 4.00000 0.286446
\(196\) 0 0
\(197\) 26.0000 1.85242 0.926212 0.377004i \(-0.123046\pi\)
0.926212 + 0.377004i \(0.123046\pi\)
\(198\) 0 0
\(199\) 12.0000 0.850657 0.425329 0.905039i \(-0.360158\pi\)
0.425329 + 0.905039i \(0.360158\pi\)
\(200\) 0 0
\(201\) −6.00000 −0.423207
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 8.00000 0.558744
\(206\) 0 0
\(207\) −8.00000 −0.556038
\(208\) 0 0
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) 6.00000 0.413057 0.206529 0.978441i \(-0.433783\pi\)
0.206529 + 0.978441i \(0.433783\pi\)
\(212\) 0 0
\(213\) 10.0000 0.685189
\(214\) 0 0
\(215\) −6.00000 −0.409197
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 2.00000 0.135147
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 1.00000 0.0669650 0.0334825 0.999439i \(-0.489340\pi\)
0.0334825 + 0.999439i \(0.489340\pi\)
\(224\) 0 0
\(225\) −4.00000 −0.266667
\(226\) 0 0
\(227\) −21.0000 −1.39382 −0.696909 0.717159i \(-0.745442\pi\)
−0.696909 + 0.717159i \(0.745442\pi\)
\(228\) 0 0
\(229\) −12.0000 −0.792982 −0.396491 0.918039i \(-0.629772\pi\)
−0.396491 + 0.918039i \(0.629772\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −24.0000 −1.57229 −0.786146 0.618041i \(-0.787927\pi\)
−0.786146 + 0.618041i \(0.787927\pi\)
\(234\) 0 0
\(235\) −10.0000 −0.652328
\(236\) 0 0
\(237\) 11.0000 0.714527
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −23.0000 −1.48156 −0.740780 0.671748i \(-0.765544\pi\)
−0.740780 + 0.671748i \(0.765544\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 16.0000 1.01806
\(248\) 0 0
\(249\) 7.00000 0.443607
\(250\) 0 0
\(251\) −11.0000 −0.694314 −0.347157 0.937807i \(-0.612853\pi\)
−0.347157 + 0.937807i \(0.612853\pi\)
\(252\) 0 0
\(253\) −24.0000 −1.50887
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 3.00000 0.185695
\(262\) 0 0
\(263\) 14.0000 0.863277 0.431638 0.902047i \(-0.357936\pi\)
0.431638 + 0.902047i \(0.357936\pi\)
\(264\) 0 0
\(265\) 9.00000 0.552866
\(266\) 0 0
\(267\) −18.0000 −1.10158
\(268\) 0 0
\(269\) 1.00000 0.0609711 0.0304855 0.999535i \(-0.490295\pi\)
0.0304855 + 0.999535i \(0.490295\pi\)
\(270\) 0 0
\(271\) −17.0000 −1.03268 −0.516338 0.856385i \(-0.672705\pi\)
−0.516338 + 0.856385i \(0.672705\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −12.0000 −0.723627
\(276\) 0 0
\(277\) −8.00000 −0.480673 −0.240337 0.970690i \(-0.577258\pi\)
−0.240337 + 0.970690i \(0.577258\pi\)
\(278\) 0 0
\(279\) −5.00000 −0.299342
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 18.0000 1.06999 0.534994 0.844856i \(-0.320314\pi\)
0.534994 + 0.844856i \(0.320314\pi\)
\(284\) 0 0
\(285\) 4.00000 0.236940
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) −17.0000 −0.996558
\(292\) 0 0
\(293\) −19.0000 −1.10999 −0.554996 0.831853i \(-0.687280\pi\)
−0.554996 + 0.831853i \(0.687280\pi\)
\(294\) 0 0
\(295\) −5.00000 −0.291111
\(296\) 0 0
\(297\) −3.00000 −0.174078
\(298\) 0 0
\(299\) −32.0000 −1.85061
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 2.00000 0.114897
\(304\) 0 0
\(305\) 10.0000 0.572598
\(306\) 0 0
\(307\) 16.0000 0.913168 0.456584 0.889680i \(-0.349073\pi\)
0.456584 + 0.889680i \(0.349073\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −9.00000 −0.508710 −0.254355 0.967111i \(-0.581863\pi\)
−0.254355 + 0.967111i \(0.581863\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −13.0000 −0.730153 −0.365076 0.930978i \(-0.618957\pi\)
−0.365076 + 0.930978i \(0.618957\pi\)
\(318\) 0 0
\(319\) 9.00000 0.503903
\(320\) 0 0
\(321\) 11.0000 0.613960
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −16.0000 −0.887520
\(326\) 0 0
\(327\) −10.0000 −0.553001
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 0 0
\(333\) −8.00000 −0.438397
\(334\) 0 0
\(335\) −6.00000 −0.327815
\(336\) 0 0
\(337\) −15.0000 −0.817102 −0.408551 0.912735i \(-0.633966\pi\)
−0.408551 + 0.912735i \(0.633966\pi\)
\(338\) 0 0
\(339\) 8.00000 0.434500
\(340\) 0 0
\(341\) −15.0000 −0.812296
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −8.00000 −0.430706
\(346\) 0 0
\(347\) −20.0000 −1.07366 −0.536828 0.843692i \(-0.680378\pi\)
−0.536828 + 0.843692i \(0.680378\pi\)
\(348\) 0 0
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) 0 0
\(353\) −8.00000 −0.425797 −0.212899 0.977074i \(-0.568290\pi\)
−0.212899 + 0.977074i \(0.568290\pi\)
\(354\) 0 0
\(355\) 10.0000 0.530745
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −6.00000 −0.316668 −0.158334 0.987386i \(-0.550612\pi\)
−0.158334 + 0.987386i \(0.550612\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 2.00000 0.104973
\(364\) 0 0
\(365\) 2.00000 0.104685
\(366\) 0 0
\(367\) −7.00000 −0.365397 −0.182699 0.983169i \(-0.558483\pi\)
−0.182699 + 0.983169i \(0.558483\pi\)
\(368\) 0 0
\(369\) −8.00000 −0.416463
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −16.0000 −0.828449 −0.414224 0.910175i \(-0.635947\pi\)
−0.414224 + 0.910175i \(0.635947\pi\)
\(374\) 0 0
\(375\) −9.00000 −0.464758
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) −7.00000 −0.358621
\(382\) 0 0
\(383\) 2.00000 0.102195 0.0510976 0.998694i \(-0.483728\pi\)
0.0510976 + 0.998694i \(0.483728\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 6.00000 0.304997
\(388\) 0 0
\(389\) −26.0000 −1.31825 −0.659126 0.752032i \(-0.729074\pi\)
−0.659126 + 0.752032i \(0.729074\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 15.0000 0.756650
\(394\) 0 0
\(395\) 11.0000 0.553470
\(396\) 0 0
\(397\) 28.0000 1.40528 0.702640 0.711546i \(-0.252005\pi\)
0.702640 + 0.711546i \(0.252005\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −8.00000 −0.399501 −0.199750 0.979847i \(-0.564013\pi\)
−0.199750 + 0.979847i \(0.564013\pi\)
\(402\) 0 0
\(403\) −20.0000 −0.996271
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −24.0000 −1.18964
\(408\) 0 0
\(409\) −7.00000 −0.346128 −0.173064 0.984911i \(-0.555367\pi\)
−0.173064 + 0.984911i \(0.555367\pi\)
\(410\) 0 0
\(411\) 14.0000 0.690569
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 7.00000 0.343616
\(416\) 0 0
\(417\) −22.0000 −1.07734
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 6.00000 0.292422 0.146211 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) 0 0
\(423\) 10.0000 0.486217
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −12.0000 −0.579365
\(430\) 0 0
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) 3.00000 0.143839
\(436\) 0 0
\(437\) −32.0000 −1.53077
\(438\) 0 0
\(439\) −9.00000 −0.429547 −0.214773 0.976664i \(-0.568901\pi\)
−0.214773 + 0.976664i \(0.568901\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −1.00000 −0.0475114 −0.0237557 0.999718i \(-0.507562\pi\)
−0.0237557 + 0.999718i \(0.507562\pi\)
\(444\) 0 0
\(445\) −18.0000 −0.853282
\(446\) 0 0
\(447\) 18.0000 0.851371
\(448\) 0 0
\(449\) 4.00000 0.188772 0.0943858 0.995536i \(-0.469911\pi\)
0.0943858 + 0.995536i \(0.469911\pi\)
\(450\) 0 0
\(451\) −24.0000 −1.13012
\(452\) 0 0
\(453\) −5.00000 −0.234920
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −17.0000 −0.795226 −0.397613 0.917553i \(-0.630161\pi\)
−0.397613 + 0.917553i \(0.630161\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 0 0
\(465\) −5.00000 −0.231869
\(466\) 0 0
\(467\) 28.0000 1.29569 0.647843 0.761774i \(-0.275671\pi\)
0.647843 + 0.761774i \(0.275671\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 4.00000 0.184310
\(472\) 0 0
\(473\) 18.0000 0.827641
\(474\) 0 0
\(475\) −16.0000 −0.734130
\(476\) 0 0
\(477\) −9.00000 −0.412082
\(478\) 0 0
\(479\) 6.00000 0.274147 0.137073 0.990561i \(-0.456230\pi\)
0.137073 + 0.990561i \(0.456230\pi\)
\(480\) 0 0
\(481\) −32.0000 −1.45907
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −17.0000 −0.771930
\(486\) 0 0
\(487\) −37.0000 −1.67663 −0.838315 0.545186i \(-0.816459\pi\)
−0.838315 + 0.545186i \(0.816459\pi\)
\(488\) 0 0
\(489\) −8.00000 −0.361773
\(490\) 0 0
\(491\) −9.00000 −0.406164 −0.203082 0.979162i \(-0.565096\pi\)
−0.203082 + 0.979162i \(0.565096\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −3.00000 −0.134840
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 10.0000 0.447661 0.223831 0.974628i \(-0.428144\pi\)
0.223831 + 0.974628i \(0.428144\pi\)
\(500\) 0 0
\(501\) 2.00000 0.0893534
\(502\) 0 0
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 0 0
\(505\) 2.00000 0.0889988
\(506\) 0 0
\(507\) −3.00000 −0.133235
\(508\) 0 0
\(509\) −39.0000 −1.72864 −0.864322 0.502938i \(-0.832252\pi\)
−0.864322 + 0.502938i \(0.832252\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −4.00000 −0.176604
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 30.0000 1.31940
\(518\) 0 0
\(519\) −2.00000 −0.0877903
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 0 0
\(523\) 12.0000 0.524723 0.262362 0.964970i \(-0.415499\pi\)
0.262362 + 0.964970i \(0.415499\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) 5.00000 0.216982
\(532\) 0 0
\(533\) −32.0000 −1.38607
\(534\) 0 0
\(535\) 11.0000 0.475571
\(536\) 0 0
\(537\) −12.0000 −0.517838
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −10.0000 −0.428353
\(546\) 0 0
\(547\) −8.00000 −0.342055 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) 0 0
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −8.00000 −0.339581
\(556\) 0 0
\(557\) −15.0000 −0.635570 −0.317785 0.948163i \(-0.602939\pi\)
−0.317785 + 0.948163i \(0.602939\pi\)
\(558\) 0 0
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −39.0000 −1.64365 −0.821827 0.569737i \(-0.807045\pi\)
−0.821827 + 0.569737i \(0.807045\pi\)
\(564\) 0 0
\(565\) 8.00000 0.336563
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 12.0000 0.503066 0.251533 0.967849i \(-0.419065\pi\)
0.251533 + 0.967849i \(0.419065\pi\)
\(570\) 0 0
\(571\) 22.0000 0.920671 0.460336 0.887745i \(-0.347729\pi\)
0.460336 + 0.887745i \(0.347729\pi\)
\(572\) 0 0
\(573\) −16.0000 −0.668410
\(574\) 0 0
\(575\) 32.0000 1.33449
\(576\) 0 0
\(577\) 33.0000 1.37381 0.686904 0.726748i \(-0.258969\pi\)
0.686904 + 0.726748i \(0.258969\pi\)
\(578\) 0 0
\(579\) 27.0000 1.12208
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −27.0000 −1.11823
\(584\) 0 0
\(585\) −4.00000 −0.165380
\(586\) 0 0
\(587\) −45.0000 −1.85735 −0.928674 0.370896i \(-0.879051\pi\)
−0.928674 + 0.370896i \(0.879051\pi\)
\(588\) 0 0
\(589\) −20.0000 −0.824086
\(590\) 0 0
\(591\) −26.0000 −1.06950
\(592\) 0 0
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −12.0000 −0.491127
\(598\) 0 0
\(599\) 42.0000 1.71607 0.858037 0.513588i \(-0.171684\pi\)
0.858037 + 0.513588i \(0.171684\pi\)
\(600\) 0 0
\(601\) 13.0000 0.530281 0.265141 0.964210i \(-0.414582\pi\)
0.265141 + 0.964210i \(0.414582\pi\)
\(602\) 0 0
\(603\) 6.00000 0.244339
\(604\) 0 0
\(605\) 2.00000 0.0813116
\(606\) 0 0
\(607\) 13.0000 0.527654 0.263827 0.964570i \(-0.415015\pi\)
0.263827 + 0.964570i \(0.415015\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 40.0000 1.61823
\(612\) 0 0
\(613\) −8.00000 −0.323117 −0.161558 0.986863i \(-0.551652\pi\)
−0.161558 + 0.986863i \(0.551652\pi\)
\(614\) 0 0
\(615\) −8.00000 −0.322591
\(616\) 0 0
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) 0 0
\(619\) −6.00000 −0.241160 −0.120580 0.992704i \(-0.538475\pi\)
−0.120580 + 0.992704i \(0.538475\pi\)
\(620\) 0 0
\(621\) 8.00000 0.321029
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) −12.0000 −0.479234
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 11.0000 0.437903 0.218952 0.975736i \(-0.429736\pi\)
0.218952 + 0.975736i \(0.429736\pi\)
\(632\) 0 0
\(633\) −6.00000 −0.238479
\(634\) 0 0
\(635\) −7.00000 −0.277787
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −10.0000 −0.395594
\(640\) 0 0
\(641\) −22.0000 −0.868948 −0.434474 0.900684i \(-0.643066\pi\)
−0.434474 + 0.900684i \(0.643066\pi\)
\(642\) 0 0
\(643\) −46.0000 −1.81406 −0.907031 0.421063i \(-0.861657\pi\)
−0.907031 + 0.421063i \(0.861657\pi\)
\(644\) 0 0
\(645\) 6.00000 0.236250
\(646\) 0 0
\(647\) −50.0000 −1.96570 −0.982851 0.184399i \(-0.940966\pi\)
−0.982851 + 0.184399i \(0.940966\pi\)
\(648\) 0 0
\(649\) 15.0000 0.588802
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −15.0000 −0.586995 −0.293498 0.955960i \(-0.594819\pi\)
−0.293498 + 0.955960i \(0.594819\pi\)
\(654\) 0 0
\(655\) 15.0000 0.586098
\(656\) 0 0
\(657\) −2.00000 −0.0780274
\(658\) 0 0
\(659\) 24.0000 0.934907 0.467454 0.884018i \(-0.345171\pi\)
0.467454 + 0.884018i \(0.345171\pi\)
\(660\) 0 0
\(661\) 38.0000 1.47803 0.739014 0.673690i \(-0.235292\pi\)
0.739014 + 0.673690i \(0.235292\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −24.0000 −0.929284
\(668\) 0 0
\(669\) −1.00000 −0.0386622
\(670\) 0 0
\(671\) −30.0000 −1.15814
\(672\) 0 0
\(673\) 29.0000 1.11787 0.558934 0.829212i \(-0.311211\pi\)
0.558934 + 0.829212i \(0.311211\pi\)
\(674\) 0 0
\(675\) 4.00000 0.153960
\(676\) 0 0
\(677\) 19.0000 0.730229 0.365115 0.930963i \(-0.381030\pi\)
0.365115 + 0.930963i \(0.381030\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 21.0000 0.804722
\(682\) 0 0
\(683\) 17.0000 0.650487 0.325243 0.945630i \(-0.394554\pi\)
0.325243 + 0.945630i \(0.394554\pi\)
\(684\) 0 0
\(685\) 14.0000 0.534913
\(686\) 0 0
\(687\) 12.0000 0.457829
\(688\) 0 0
\(689\) −36.0000 −1.37149
\(690\) 0 0
\(691\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −22.0000 −0.834508
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 24.0000 0.907763
\(700\) 0 0
\(701\) −5.00000 −0.188847 −0.0944237 0.995532i \(-0.530101\pi\)
−0.0944237 + 0.995532i \(0.530101\pi\)
\(702\) 0 0
\(703\) −32.0000 −1.20690
\(704\) 0 0
\(705\) 10.0000 0.376622
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −6.00000 −0.225335 −0.112667 0.993633i \(-0.535939\pi\)
−0.112667 + 0.993633i \(0.535939\pi\)
\(710\) 0 0
\(711\) −11.0000 −0.412532
\(712\) 0 0
\(713\) 40.0000 1.49801
\(714\) 0 0
\(715\) −12.0000 −0.448775
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −34.0000 −1.26799 −0.633993 0.773339i \(-0.718585\pi\)
−0.633993 + 0.773339i \(0.718585\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 23.0000 0.855379
\(724\) 0 0
\(725\) −12.0000 −0.445669
\(726\) 0 0
\(727\) 47.0000 1.74313 0.871567 0.490277i \(-0.163104\pi\)
0.871567 + 0.490277i \(0.163104\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 14.0000 0.517102 0.258551 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 18.0000 0.663039
\(738\) 0 0
\(739\) 10.0000 0.367856 0.183928 0.982940i \(-0.441119\pi\)
0.183928 + 0.982940i \(0.441119\pi\)
\(740\) 0 0
\(741\) −16.0000 −0.587775
\(742\) 0 0
\(743\) −34.0000 −1.24734 −0.623670 0.781688i \(-0.714359\pi\)
−0.623670 + 0.781688i \(0.714359\pi\)
\(744\) 0 0
\(745\) 18.0000 0.659469
\(746\) 0 0
\(747\) −7.00000 −0.256117
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 27.0000 0.985244 0.492622 0.870243i \(-0.336039\pi\)
0.492622 + 0.870243i \(0.336039\pi\)
\(752\) 0 0
\(753\) 11.0000 0.400862
\(754\) 0 0
\(755\) −5.00000 −0.181969
\(756\) 0 0
\(757\) 10.0000 0.363456 0.181728 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) 0 0
\(759\) 24.0000 0.871145
\(760\) 0 0
\(761\) 12.0000 0.435000 0.217500 0.976060i \(-0.430210\pi\)
0.217500 + 0.976060i \(0.430210\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 20.0000 0.722158
\(768\) 0 0
\(769\) −37.0000 −1.33425 −0.667127 0.744944i \(-0.732476\pi\)
−0.667127 + 0.744944i \(0.732476\pi\)
\(770\) 0 0
\(771\) 14.0000 0.504198
\(772\) 0 0
\(773\) −26.0000 −0.935155 −0.467578 0.883952i \(-0.654873\pi\)
−0.467578 + 0.883952i \(0.654873\pi\)
\(774\) 0 0
\(775\) 20.0000 0.718421
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −32.0000 −1.14652
\(780\) 0 0
\(781\) −30.0000 −1.07348
\(782\) 0 0
\(783\) −3.00000 −0.107211
\(784\) 0 0
\(785\) 4.00000 0.142766
\(786\) 0 0
\(787\) 22.0000 0.784215 0.392108 0.919919i \(-0.371746\pi\)
0.392108 + 0.919919i \(0.371746\pi\)
\(788\) 0 0
\(789\) −14.0000 −0.498413
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −40.0000 −1.42044
\(794\) 0 0
\(795\) −9.00000 −0.319197
\(796\) 0 0
\(797\) 3.00000 0.106265 0.0531327 0.998587i \(-0.483079\pi\)
0.0531327 + 0.998587i \(0.483079\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 18.0000 0.635999
\(802\) 0 0
\(803\) −6.00000 −0.211735
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −1.00000 −0.0352017
\(808\) 0 0
\(809\) −16.0000 −0.562530 −0.281265 0.959630i \(-0.590754\pi\)
−0.281265 + 0.959630i \(0.590754\pi\)
\(810\) 0 0
\(811\) −2.00000 −0.0702295 −0.0351147 0.999383i \(-0.511180\pi\)
−0.0351147 + 0.999383i \(0.511180\pi\)
\(812\) 0 0
\(813\) 17.0000 0.596216
\(814\) 0 0
\(815\) −8.00000 −0.280228
\(816\) 0 0
\(817\) 24.0000 0.839654
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 39.0000 1.36111 0.680555 0.732697i \(-0.261739\pi\)
0.680555 + 0.732697i \(0.261739\pi\)
\(822\) 0 0
\(823\) −32.0000 −1.11545 −0.557725 0.830026i \(-0.688326\pi\)
−0.557725 + 0.830026i \(0.688326\pi\)
\(824\) 0 0
\(825\) 12.0000 0.417786
\(826\) 0 0
\(827\) −41.0000 −1.42571 −0.712855 0.701312i \(-0.752598\pi\)
−0.712855 + 0.701312i \(0.752598\pi\)
\(828\) 0 0
\(829\) −4.00000 −0.138926 −0.0694629 0.997585i \(-0.522129\pi\)
−0.0694629 + 0.997585i \(0.522129\pi\)
\(830\) 0 0
\(831\) 8.00000 0.277517
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 2.00000 0.0692129
\(836\) 0 0
\(837\) 5.00000 0.172825
\(838\) 0 0
\(839\) −24.0000 −0.828572 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) −6.00000 −0.206651
\(844\) 0 0
\(845\) −3.00000 −0.103203
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −18.0000 −0.617758
\(850\) 0 0
\(851\) 64.0000 2.19389
\(852\) 0 0
\(853\) 14.0000 0.479351 0.239675 0.970853i \(-0.422959\pi\)
0.239675 + 0.970853i \(0.422959\pi\)
\(854\) 0 0
\(855\) −4.00000 −0.136797
\(856\) 0 0
\(857\) −14.0000 −0.478231 −0.239115 0.970991i \(-0.576857\pi\)
−0.239115 + 0.970991i \(0.576857\pi\)
\(858\) 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
\(863\) −22.0000 −0.748889 −0.374444 0.927249i \(-0.622167\pi\)
−0.374444 + 0.927249i \(0.622167\pi\)
\(864\) 0 0
\(865\) −2.00000 −0.0680020
\(866\) 0 0
\(867\) 17.0000 0.577350
\(868\) 0 0
\(869\) −33.0000 −1.11945
\(870\) 0 0
\(871\) 24.0000 0.813209
\(872\) 0 0
\(873\) 17.0000 0.575363
\(874\) 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
\(879\) 19.0000 0.640854
\(880\) 0 0
\(881\) 54.0000 1.81931 0.909653 0.415369i \(-0.136347\pi\)
0.909653 + 0.415369i \(0.136347\pi\)
\(882\) 0 0
\(883\) 40.0000 1.34611 0.673054 0.739594i \(-0.264982\pi\)
0.673054 + 0.739594i \(0.264982\pi\)
\(884\) 0 0
\(885\) 5.00000 0.168073
\(886\) 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
\(891\) 3.00000 0.100504
\(892\) 0 0
\(893\) 40.0000 1.33855
\(894\) 0 0
\(895\) −12.0000 −0.401116
\(896\) 0 0
\(897\) 32.0000 1.06845
\(898\) 0 0
\(899\) −15.0000 −0.500278
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −16.0000 −0.531271 −0.265636 0.964073i \(-0.585582\pi\)
−0.265636 + 0.964073i \(0.585582\pi\)
\(908\) 0 0
\(909\) −2.00000 −0.0663358
\(910\) 0 0
\(911\) −6.00000 −0.198789 −0.0993944 0.995048i \(-0.531691\pi\)
−0.0993944 + 0.995048i \(0.531691\pi\)
\(912\) 0 0
\(913\) −21.0000 −0.694999
\(914\) 0 0
\(915\) −10.0000 −0.330590
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) −16.0000 −0.527218
\(922\) 0 0
\(923\) −40.0000 −1.31662
\(924\) 0 0
\(925\) 32.0000 1.05215
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −42.0000 −1.37798 −0.688988 0.724773i \(-0.741945\pi\)
−0.688988 + 0.724773i \(0.741945\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −19.0000 −0.620703 −0.310351 0.950622i \(-0.600447\pi\)
−0.310351 + 0.950622i \(0.600447\pi\)
\(938\) 0 0
\(939\) 9.00000 0.293704
\(940\) 0 0
\(941\) −13.0000 −0.423788 −0.211894 0.977293i \(-0.567963\pi\)
−0.211894 + 0.977293i \(0.567963\pi\)
\(942\) 0 0
\(943\) 64.0000 2.08413
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −48.0000 −1.55979 −0.779895 0.625910i \(-0.784728\pi\)
−0.779895 + 0.625910i \(0.784728\pi\)
\(948\) 0 0
\(949\) −8.00000 −0.259691
\(950\) 0 0
\(951\) 13.0000 0.421554
\(952\) 0 0
\(953\) −38.0000 −1.23094 −0.615470 0.788160i \(-0.711034\pi\)
−0.615470 + 0.788160i \(0.711034\pi\)
\(954\) 0 0
\(955\) −16.0000 −0.517748
\(956\) 0 0
\(957\) −9.00000 −0.290929
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) 0 0
\(963\) −11.0000 −0.354470
\(964\) 0 0
\(965\) 27.0000 0.869161
\(966\) 0 0
\(967\) −3.00000 −0.0964735 −0.0482367 0.998836i \(-0.515360\pi\)
−0.0482367 + 0.998836i \(0.515360\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −25.0000 −0.802288 −0.401144 0.916015i \(-0.631387\pi\)
−0.401144 + 0.916015i \(0.631387\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 16.0000 0.512410
\(976\) 0 0
\(977\) −22.0000 −0.703842 −0.351921 0.936030i \(-0.614471\pi\)
−0.351921 + 0.936030i \(0.614471\pi\)
\(978\) 0 0
\(979\) 54.0000 1.72585
\(980\) 0 0
\(981\) 10.0000 0.319275
\(982\) 0 0
\(983\) 44.0000 1.40338 0.701691 0.712481i \(-0.252429\pi\)
0.701691 + 0.712481i \(0.252429\pi\)
\(984\) 0 0
\(985\) −26.0000 −0.828429
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −48.0000 −1.52631
\(990\) 0 0
\(991\) −15.0000 −0.476491 −0.238245 0.971205i \(-0.576572\pi\)
−0.238245 + 0.971205i \(0.576572\pi\)
\(992\) 0 0
\(993\) −4.00000 −0.126936
\(994\) 0 0
\(995\) −12.0000 −0.380426
\(996\) 0 0
\(997\) −26.0000 −0.823428 −0.411714 0.911313i \(-0.635070\pi\)
−0.411714 + 0.911313i \(0.635070\pi\)
\(998\) 0 0
\(999\) 8.00000 0.253109
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 9408.2.a.p.1.1 1
4.3 odd 2 9408.2.a.cf.1.1 1
7.3 odd 6 1344.2.q.d.961.1 2
7.5 odd 6 1344.2.q.d.193.1 2
7.6 odd 2 9408.2.a.cq.1.1 1
8.3 odd 2 1176.2.a.c.1.1 1
8.5 even 2 2352.2.a.u.1.1 1
24.5 odd 2 7056.2.a.t.1.1 1
24.11 even 2 3528.2.a.i.1.1 1
28.3 even 6 1344.2.q.o.961.1 2
28.19 even 6 1344.2.q.o.193.1 2
28.27 even 2 9408.2.a.ba.1.1 1
56.3 even 6 168.2.q.a.121.1 yes 2
56.5 odd 6 336.2.q.e.193.1 2
56.11 odd 6 1176.2.q.g.961.1 2
56.13 odd 2 2352.2.a.g.1.1 1
56.19 even 6 168.2.q.a.25.1 2
56.27 even 2 1176.2.a.g.1.1 1
56.37 even 6 2352.2.q.f.1537.1 2
56.45 odd 6 336.2.q.e.289.1 2
56.51 odd 6 1176.2.q.g.361.1 2
56.53 even 6 2352.2.q.f.961.1 2
168.5 even 6 1008.2.s.f.865.1 2
168.11 even 6 3528.2.s.p.3313.1 2
168.59 odd 6 504.2.s.d.289.1 2
168.83 odd 2 3528.2.a.q.1.1 1
168.101 even 6 1008.2.s.f.289.1 2
168.107 even 6 3528.2.s.p.361.1 2
168.125 even 2 7056.2.a.bk.1.1 1
168.131 odd 6 504.2.s.d.361.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.q.a.25.1 2 56.19 even 6
168.2.q.a.121.1 yes 2 56.3 even 6
336.2.q.e.193.1 2 56.5 odd 6
336.2.q.e.289.1 2 56.45 odd 6
504.2.s.d.289.1 2 168.59 odd 6
504.2.s.d.361.1 2 168.131 odd 6
1008.2.s.f.289.1 2 168.101 even 6
1008.2.s.f.865.1 2 168.5 even 6
1176.2.a.c.1.1 1 8.3 odd 2
1176.2.a.g.1.1 1 56.27 even 2
1176.2.q.g.361.1 2 56.51 odd 6
1176.2.q.g.961.1 2 56.11 odd 6
1344.2.q.d.193.1 2 7.5 odd 6
1344.2.q.d.961.1 2 7.3 odd 6
1344.2.q.o.193.1 2 28.19 even 6
1344.2.q.o.961.1 2 28.3 even 6
2352.2.a.g.1.1 1 56.13 odd 2
2352.2.a.u.1.1 1 8.5 even 2
2352.2.q.f.961.1 2 56.53 even 6
2352.2.q.f.1537.1 2 56.37 even 6
3528.2.a.i.1.1 1 24.11 even 2
3528.2.a.q.1.1 1 168.83 odd 2
3528.2.s.p.361.1 2 168.107 even 6
3528.2.s.p.3313.1 2 168.11 even 6
7056.2.a.t.1.1 1 24.5 odd 2
7056.2.a.bk.1.1 1 168.125 even 2
9408.2.a.p.1.1 1 1.1 even 1 trivial
9408.2.a.ba.1.1 1 28.27 even 2
9408.2.a.cf.1.1 1 4.3 odd 2
9408.2.a.cq.1.1 1 7.6 odd 2