Properties

Label 2646.2.a.z.1.1
Level $2646$
Weight $2$
Character 2646.1
Self dual yes
Analytic conductor $21.128$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2646.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(21.1284163748\)
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 \(\chi\) \(=\) 2646.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{8} +1.00000 q^{10} +1.00000 q^{11} +2.00000 q^{13} +1.00000 q^{16} +6.00000 q^{17} +5.00000 q^{19} +1.00000 q^{20} +1.00000 q^{22} -3.00000 q^{23} -4.00000 q^{25} +2.00000 q^{26} -2.00000 q^{29} -5.00000 q^{31} +1.00000 q^{32} +6.00000 q^{34} +3.00000 q^{37} +5.00000 q^{38} +1.00000 q^{40} -3.00000 q^{41} -2.00000 q^{43} +1.00000 q^{44} -3.00000 q^{46} +10.0000 q^{47} -4.00000 q^{50} +2.00000 q^{52} -8.00000 q^{53} +1.00000 q^{55} -2.00000 q^{58} +10.0000 q^{59} +4.00000 q^{61} -5.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} +10.0000 q^{67} +6.00000 q^{68} +13.0000 q^{71} -14.0000 q^{73} +3.00000 q^{74} +5.00000 q^{76} -2.00000 q^{79} +1.00000 q^{80} -3.00000 q^{82} -6.00000 q^{83} +6.00000 q^{85} -2.00000 q^{86} +1.00000 q^{88} +17.0000 q^{89} -3.00000 q^{92} +10.0000 q^{94} +5.00000 q^{95} +4.00000 q^{97} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) 5.00000 1.14708 0.573539 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −3.00000 −0.625543 −0.312772 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −5.00000 −0.898027 −0.449013 0.893525i \(-0.648224\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 0 0
\(37\) 3.00000 0.493197 0.246598 0.969118i \(-0.420687\pi\)
0.246598 + 0.969118i \(0.420687\pi\)
\(38\) 5.00000 0.811107
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) −3.00000 −0.468521 −0.234261 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −3.00000 −0.442326
\(47\) 10.0000 1.45865 0.729325 0.684167i \(-0.239834\pi\)
0.729325 + 0.684167i \(0.239834\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −4.00000 −0.565685
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) −8.00000 −1.09888 −0.549442 0.835532i \(-0.685160\pi\)
−0.549442 + 0.835532i \(0.685160\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) 0 0
\(58\) −2.00000 −0.262613
\(59\) 10.0000 1.30189 0.650945 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 0 0
\(61\) 4.00000 0.512148 0.256074 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) −5.00000 −0.635001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) 10.0000 1.22169 0.610847 0.791748i \(-0.290829\pi\)
0.610847 + 0.791748i \(0.290829\pi\)
\(68\) 6.00000 0.727607
\(69\) 0 0
\(70\) 0 0
\(71\) 13.0000 1.54282 0.771408 0.636341i \(-0.219553\pi\)
0.771408 + 0.636341i \(0.219553\pi\)
\(72\) 0 0
\(73\) −14.0000 −1.63858 −0.819288 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 3.00000 0.348743
\(75\) 0 0
\(76\) 5.00000 0.573539
\(77\) 0 0
\(78\) 0 0
\(79\) −2.00000 −0.225018 −0.112509 0.993651i \(-0.535889\pi\)
−0.112509 + 0.993651i \(0.535889\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) −3.00000 −0.331295
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) 6.00000 0.650791
\(86\) −2.00000 −0.215666
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) 17.0000 1.80200 0.900998 0.433823i \(-0.142836\pi\)
0.900998 + 0.433823i \(0.142836\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −3.00000 −0.312772
\(93\) 0 0
\(94\) 10.0000 1.03142
\(95\) 5.00000 0.512989
\(96\) 0 0
\(97\) 4.00000 0.406138 0.203069 0.979164i \(-0.434908\pi\)
0.203069 + 0.979164i \(0.434908\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −4.00000 −0.400000
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) 11.0000 1.08386 0.541931 0.840423i \(-0.317693\pi\)
0.541931 + 0.840423i \(0.317693\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) −8.00000 −0.777029
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 0 0
\(109\) 1.00000 0.0957826 0.0478913 0.998853i \(-0.484750\pi\)
0.0478913 + 0.998853i \(0.484750\pi\)
\(110\) 1.00000 0.0953463
\(111\) 0 0
\(112\) 0 0
\(113\) −16.0000 −1.50515 −0.752577 0.658505i \(-0.771189\pi\)
−0.752577 + 0.658505i \(0.771189\pi\)
\(114\) 0 0
\(115\) −3.00000 −0.279751
\(116\) −2.00000 −0.185695
\(117\) 0 0
\(118\) 10.0000 0.920575
\(119\) 0 0
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 4.00000 0.362143
\(123\) 0 0
\(124\) −5.00000 −0.449013
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) −10.0000 −0.887357 −0.443678 0.896186i \(-0.646327\pi\)
−0.443678 + 0.896186i \(0.646327\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 2.00000 0.175412
\(131\) 18.0000 1.57267 0.786334 0.617802i \(-0.211977\pi\)
0.786334 + 0.617802i \(0.211977\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 10.0000 0.863868
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 13.0000 1.09094
\(143\) 2.00000 0.167248
\(144\) 0 0
\(145\) −2.00000 −0.166091
\(146\) −14.0000 −1.15865
\(147\) 0 0
\(148\) 3.00000 0.246598
\(149\) −20.0000 −1.63846 −0.819232 0.573462i \(-0.805600\pi\)
−0.819232 + 0.573462i \(0.805600\pi\)
\(150\) 0 0
\(151\) −20.0000 −1.62758 −0.813788 0.581161i \(-0.802599\pi\)
−0.813788 + 0.581161i \(0.802599\pi\)
\(152\) 5.00000 0.405554
\(153\) 0 0
\(154\) 0 0
\(155\) −5.00000 −0.401610
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) −2.00000 −0.159111
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) 0 0
\(163\) 6.00000 0.469956 0.234978 0.972001i \(-0.424498\pi\)
0.234978 + 0.972001i \(0.424498\pi\)
\(164\) −3.00000 −0.234261
\(165\) 0 0
\(166\) −6.00000 −0.465690
\(167\) −16.0000 −1.23812 −0.619059 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 6.00000 0.460179
\(171\) 0 0
\(172\) −2.00000 −0.152499
\(173\) 11.0000 0.836315 0.418157 0.908375i \(-0.362676\pi\)
0.418157 + 0.908375i \(0.362676\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) 17.0000 1.27420
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −20.0000 −1.48659 −0.743294 0.668965i \(-0.766738\pi\)
−0.743294 + 0.668965i \(0.766738\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −3.00000 −0.221163
\(185\) 3.00000 0.220564
\(186\) 0 0
\(187\) 6.00000 0.438763
\(188\) 10.0000 0.729325
\(189\) 0 0
\(190\) 5.00000 0.362738
\(191\) 3.00000 0.217072 0.108536 0.994092i \(-0.465384\pi\)
0.108536 + 0.994092i \(0.465384\pi\)
\(192\) 0 0
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 4.00000 0.287183
\(195\) 0 0
\(196\) 0 0
\(197\) 24.0000 1.70993 0.854965 0.518686i \(-0.173579\pi\)
0.854965 + 0.518686i \(0.173579\pi\)
\(198\) 0 0
\(199\) 7.00000 0.496217 0.248108 0.968732i \(-0.420191\pi\)
0.248108 + 0.968732i \(0.420191\pi\)
\(200\) −4.00000 −0.282843
\(201\) 0 0
\(202\) 10.0000 0.703598
\(203\) 0 0
\(204\) 0 0
\(205\) −3.00000 −0.209529
\(206\) 11.0000 0.766406
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) 5.00000 0.345857
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) −8.00000 −0.549442
\(213\) 0 0
\(214\) 8.00000 0.546869
\(215\) −2.00000 −0.136399
\(216\) 0 0
\(217\) 0 0
\(218\) 1.00000 0.0677285
\(219\) 0 0
\(220\) 1.00000 0.0674200
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) −23.0000 −1.54019 −0.770097 0.637927i \(-0.779792\pi\)
−0.770097 + 0.637927i \(0.779792\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −16.0000 −1.06430
\(227\) 8.00000 0.530979 0.265489 0.964114i \(-0.414466\pi\)
0.265489 + 0.964114i \(0.414466\pi\)
\(228\) 0 0
\(229\) −16.0000 −1.05731 −0.528655 0.848837i \(-0.677303\pi\)
−0.528655 + 0.848837i \(0.677303\pi\)
\(230\) −3.00000 −0.197814
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) 28.0000 1.83434 0.917170 0.398495i \(-0.130467\pi\)
0.917170 + 0.398495i \(0.130467\pi\)
\(234\) 0 0
\(235\) 10.0000 0.652328
\(236\) 10.0000 0.650945
\(237\) 0 0
\(238\) 0 0
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) −26.0000 −1.67481 −0.837404 0.546585i \(-0.815928\pi\)
−0.837404 + 0.546585i \(0.815928\pi\)
\(242\) −10.0000 −0.642824
\(243\) 0 0
\(244\) 4.00000 0.256074
\(245\) 0 0
\(246\) 0 0
\(247\) 10.0000 0.636285
\(248\) −5.00000 −0.317500
\(249\) 0 0
\(250\) −9.00000 −0.569210
\(251\) 4.00000 0.252478 0.126239 0.992000i \(-0.459709\pi\)
0.126239 + 0.992000i \(0.459709\pi\)
\(252\) 0 0
\(253\) −3.00000 −0.188608
\(254\) −10.0000 −0.627456
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −7.00000 −0.436648 −0.218324 0.975876i \(-0.570059\pi\)
−0.218324 + 0.975876i \(0.570059\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 2.00000 0.124035
\(261\) 0 0
\(262\) 18.0000 1.11204
\(263\) −11.0000 −0.678289 −0.339145 0.940734i \(-0.610138\pi\)
−0.339145 + 0.940734i \(0.610138\pi\)
\(264\) 0 0
\(265\) −8.00000 −0.491436
\(266\) 0 0
\(267\) 0 0
\(268\) 10.0000 0.610847
\(269\) 1.00000 0.0609711 0.0304855 0.999535i \(-0.490295\pi\)
0.0304855 + 0.999535i \(0.490295\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 6.00000 0.363803
\(273\) 0 0
\(274\) 10.0000 0.604122
\(275\) −4.00000 −0.241209
\(276\) 0 0
\(277\) −17.0000 −1.02143 −0.510716 0.859750i \(-0.670619\pi\)
−0.510716 + 0.859750i \(0.670619\pi\)
\(278\) −12.0000 −0.719712
\(279\) 0 0
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 13.0000 0.771408
\(285\) 0 0
\(286\) 2.00000 0.118262
\(287\) 0 0
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) −2.00000 −0.117444
\(291\) 0 0
\(292\) −14.0000 −0.819288
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) 10.0000 0.582223
\(296\) 3.00000 0.174371
\(297\) 0 0
\(298\) −20.0000 −1.15857
\(299\) −6.00000 −0.346989
\(300\) 0 0
\(301\) 0 0
\(302\) −20.0000 −1.15087
\(303\) 0 0
\(304\) 5.00000 0.286770
\(305\) 4.00000 0.229039
\(306\) 0 0
\(307\) −11.0000 −0.627803 −0.313902 0.949456i \(-0.601636\pi\)
−0.313902 + 0.949456i \(0.601636\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −5.00000 −0.283981
\(311\) 2.00000 0.113410 0.0567048 0.998391i \(-0.481941\pi\)
0.0567048 + 0.998391i \(0.481941\pi\)
\(312\) 0 0
\(313\) −18.0000 −1.01742 −0.508710 0.860938i \(-0.669877\pi\)
−0.508710 + 0.860938i \(0.669877\pi\)
\(314\) −10.0000 −0.564333
\(315\) 0 0
\(316\) −2.00000 −0.112509
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 0 0
\(319\) −2.00000 −0.111979
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 0 0
\(323\) 30.0000 1.66924
\(324\) 0 0
\(325\) −8.00000 −0.443760
\(326\) 6.00000 0.332309
\(327\) 0 0
\(328\) −3.00000 −0.165647
\(329\) 0 0
\(330\) 0 0
\(331\) −16.0000 −0.879440 −0.439720 0.898135i \(-0.644922\pi\)
−0.439720 + 0.898135i \(0.644922\pi\)
\(332\) −6.00000 −0.329293
\(333\) 0 0
\(334\) −16.0000 −0.875481
\(335\) 10.0000 0.546358
\(336\) 0 0
\(337\) −13.0000 −0.708155 −0.354078 0.935216i \(-0.615205\pi\)
−0.354078 + 0.935216i \(0.615205\pi\)
\(338\) −9.00000 −0.489535
\(339\) 0 0
\(340\) 6.00000 0.325396
\(341\) −5.00000 −0.270765
\(342\) 0 0
\(343\) 0 0
\(344\) −2.00000 −0.107833
\(345\) 0 0
\(346\) 11.0000 0.591364
\(347\) 9.00000 0.483145 0.241573 0.970383i \(-0.422337\pi\)
0.241573 + 0.970383i \(0.422337\pi\)
\(348\) 0 0
\(349\) −12.0000 −0.642345 −0.321173 0.947021i \(-0.604077\pi\)
−0.321173 + 0.947021i \(0.604077\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) −9.00000 −0.479022 −0.239511 0.970894i \(-0.576987\pi\)
−0.239511 + 0.970894i \(0.576987\pi\)
\(354\) 0 0
\(355\) 13.0000 0.689968
\(356\) 17.0000 0.900998
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 32.0000 1.68890 0.844448 0.535638i \(-0.179929\pi\)
0.844448 + 0.535638i \(0.179929\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) −20.0000 −1.05118
\(363\) 0 0
\(364\) 0 0
\(365\) −14.0000 −0.732793
\(366\) 0 0
\(367\) 7.00000 0.365397 0.182699 0.983169i \(-0.441517\pi\)
0.182699 + 0.983169i \(0.441517\pi\)
\(368\) −3.00000 −0.156386
\(369\) 0 0
\(370\) 3.00000 0.155963
\(371\) 0 0
\(372\) 0 0
\(373\) 1.00000 0.0517780 0.0258890 0.999665i \(-0.491758\pi\)
0.0258890 + 0.999665i \(0.491758\pi\)
\(374\) 6.00000 0.310253
\(375\) 0 0
\(376\) 10.0000 0.515711
\(377\) −4.00000 −0.206010
\(378\) 0 0
\(379\) −6.00000 −0.308199 −0.154100 0.988055i \(-0.549248\pi\)
−0.154100 + 0.988055i \(0.549248\pi\)
\(380\) 5.00000 0.256495
\(381\) 0 0
\(382\) 3.00000 0.153493
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 14.0000 0.712581
\(387\) 0 0
\(388\) 4.00000 0.203069
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) −18.0000 −0.910299
\(392\) 0 0
\(393\) 0 0
\(394\) 24.0000 1.20910
\(395\) −2.00000 −0.100631
\(396\) 0 0
\(397\) −28.0000 −1.40528 −0.702640 0.711546i \(-0.747995\pi\)
−0.702640 + 0.711546i \(0.747995\pi\)
\(398\) 7.00000 0.350878
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) −10.0000 −0.498135
\(404\) 10.0000 0.497519
\(405\) 0 0
\(406\) 0 0
\(407\) 3.00000 0.148704
\(408\) 0 0
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) −3.00000 −0.148159
\(411\) 0 0
\(412\) 11.0000 0.541931
\(413\) 0 0
\(414\) 0 0
\(415\) −6.00000 −0.294528
\(416\) 2.00000 0.0980581
\(417\) 0 0
\(418\) 5.00000 0.244558
\(419\) 28.0000 1.36789 0.683945 0.729534i \(-0.260263\pi\)
0.683945 + 0.729534i \(0.260263\pi\)
\(420\) 0 0
\(421\) 15.0000 0.731055 0.365528 0.930800i \(-0.380889\pi\)
0.365528 + 0.930800i \(0.380889\pi\)
\(422\) 20.0000 0.973585
\(423\) 0 0
\(424\) −8.00000 −0.388514
\(425\) −24.0000 −1.16417
\(426\) 0 0
\(427\) 0 0
\(428\) 8.00000 0.386695
\(429\) 0 0
\(430\) −2.00000 −0.0964486
\(431\) −13.0000 −0.626188 −0.313094 0.949722i \(-0.601365\pi\)
−0.313094 + 0.949722i \(0.601365\pi\)
\(432\) 0 0
\(433\) 26.0000 1.24948 0.624740 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 1.00000 0.0478913
\(437\) −15.0000 −0.717547
\(438\) 0 0
\(439\) 24.0000 1.14546 0.572729 0.819745i \(-0.305885\pi\)
0.572729 + 0.819745i \(0.305885\pi\)
\(440\) 1.00000 0.0476731
\(441\) 0 0
\(442\) 12.0000 0.570782
\(443\) −21.0000 −0.997740 −0.498870 0.866677i \(-0.666252\pi\)
−0.498870 + 0.866677i \(0.666252\pi\)
\(444\) 0 0
\(445\) 17.0000 0.805877
\(446\) −23.0000 −1.08908
\(447\) 0 0
\(448\) 0 0
\(449\) 12.0000 0.566315 0.283158 0.959073i \(-0.408618\pi\)
0.283158 + 0.959073i \(0.408618\pi\)
\(450\) 0 0
\(451\) −3.00000 −0.141264
\(452\) −16.0000 −0.752577
\(453\) 0 0
\(454\) 8.00000 0.375459
\(455\) 0 0
\(456\) 0 0
\(457\) −31.0000 −1.45012 −0.725059 0.688686i \(-0.758188\pi\)
−0.725059 + 0.688686i \(0.758188\pi\)
\(458\) −16.0000 −0.747631
\(459\) 0 0
\(460\) −3.00000 −0.139876
\(461\) 3.00000 0.139724 0.0698620 0.997557i \(-0.477744\pi\)
0.0698620 + 0.997557i \(0.477744\pi\)
\(462\) 0 0
\(463\) −4.00000 −0.185896 −0.0929479 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) 28.0000 1.29707
\(467\) −22.0000 −1.01804 −0.509019 0.860755i \(-0.669992\pi\)
−0.509019 + 0.860755i \(0.669992\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 10.0000 0.461266
\(471\) 0 0
\(472\) 10.0000 0.460287
\(473\) −2.00000 −0.0919601
\(474\) 0 0
\(475\) −20.0000 −0.917663
\(476\) 0 0
\(477\) 0 0
\(478\) −24.0000 −1.09773
\(479\) −36.0000 −1.64488 −0.822441 0.568850i \(-0.807388\pi\)
−0.822441 + 0.568850i \(0.807388\pi\)
\(480\) 0 0
\(481\) 6.00000 0.273576
\(482\) −26.0000 −1.18427
\(483\) 0 0
\(484\) −10.0000 −0.454545
\(485\) 4.00000 0.181631
\(486\) 0 0
\(487\) 22.0000 0.996915 0.498458 0.866914i \(-0.333900\pi\)
0.498458 + 0.866914i \(0.333900\pi\)
\(488\) 4.00000 0.181071
\(489\) 0 0
\(490\) 0 0
\(491\) −27.0000 −1.21849 −0.609246 0.792981i \(-0.708528\pi\)
−0.609246 + 0.792981i \(0.708528\pi\)
\(492\) 0 0
\(493\) −12.0000 −0.540453
\(494\) 10.0000 0.449921
\(495\) 0 0
\(496\) −5.00000 −0.224507
\(497\) 0 0
\(498\) 0 0
\(499\) −6.00000 −0.268597 −0.134298 0.990941i \(-0.542878\pi\)
−0.134298 + 0.990941i \(0.542878\pi\)
\(500\) −9.00000 −0.402492
\(501\) 0 0
\(502\) 4.00000 0.178529
\(503\) 26.0000 1.15928 0.579641 0.814872i \(-0.303193\pi\)
0.579641 + 0.814872i \(0.303193\pi\)
\(504\) 0 0
\(505\) 10.0000 0.444994
\(506\) −3.00000 −0.133366
\(507\) 0 0
\(508\) −10.0000 −0.443678
\(509\) −10.0000 −0.443242 −0.221621 0.975133i \(-0.571135\pi\)
−0.221621 + 0.975133i \(0.571135\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −7.00000 −0.308757
\(515\) 11.0000 0.484718
\(516\) 0 0
\(517\) 10.0000 0.439799
\(518\) 0 0
\(519\) 0 0
\(520\) 2.00000 0.0877058
\(521\) −25.0000 −1.09527 −0.547635 0.836717i \(-0.684472\pi\)
−0.547635 + 0.836717i \(0.684472\pi\)
\(522\) 0 0
\(523\) 29.0000 1.26808 0.634041 0.773300i \(-0.281395\pi\)
0.634041 + 0.773300i \(0.281395\pi\)
\(524\) 18.0000 0.786334
\(525\) 0 0
\(526\) −11.0000 −0.479623
\(527\) −30.0000 −1.30682
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) −8.00000 −0.347498
\(531\) 0 0
\(532\) 0 0
\(533\) −6.00000 −0.259889
\(534\) 0 0
\(535\) 8.00000 0.345870
\(536\) 10.0000 0.431934
\(537\) 0 0
\(538\) 1.00000 0.0431131
\(539\) 0 0
\(540\) 0 0
\(541\) 43.0000 1.84871 0.924357 0.381528i \(-0.124602\pi\)
0.924357 + 0.381528i \(0.124602\pi\)
\(542\) 8.00000 0.343629
\(543\) 0 0
\(544\) 6.00000 0.257248
\(545\) 1.00000 0.0428353
\(546\) 0 0
\(547\) −8.00000 −0.342055 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) 10.0000 0.427179
\(549\) 0 0
\(550\) −4.00000 −0.170561
\(551\) −10.0000 −0.426014
\(552\) 0 0
\(553\) 0 0
\(554\) −17.0000 −0.722261
\(555\) 0 0
\(556\) −12.0000 −0.508913
\(557\) −12.0000 −0.508456 −0.254228 0.967144i \(-0.581821\pi\)
−0.254228 + 0.967144i \(0.581821\pi\)
\(558\) 0 0
\(559\) −4.00000 −0.169182
\(560\) 0 0
\(561\) 0 0
\(562\) −6.00000 −0.253095
\(563\) −34.0000 −1.43293 −0.716465 0.697623i \(-0.754241\pi\)
−0.716465 + 0.697623i \(0.754241\pi\)
\(564\) 0 0
\(565\) −16.0000 −0.673125
\(566\) 4.00000 0.168133
\(567\) 0 0
\(568\) 13.0000 0.545468
\(569\) 12.0000 0.503066 0.251533 0.967849i \(-0.419065\pi\)
0.251533 + 0.967849i \(0.419065\pi\)
\(570\) 0 0
\(571\) −24.0000 −1.00437 −0.502184 0.864761i \(-0.667470\pi\)
−0.502184 + 0.864761i \(0.667470\pi\)
\(572\) 2.00000 0.0836242
\(573\) 0 0
\(574\) 0 0
\(575\) 12.0000 0.500435
\(576\) 0 0
\(577\) −8.00000 −0.333044 −0.166522 0.986038i \(-0.553254\pi\)
−0.166522 + 0.986038i \(0.553254\pi\)
\(578\) 19.0000 0.790296
\(579\) 0 0
\(580\) −2.00000 −0.0830455
\(581\) 0 0
\(582\) 0 0
\(583\) −8.00000 −0.331326
\(584\) −14.0000 −0.579324
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) −32.0000 −1.32078 −0.660391 0.750922i \(-0.729609\pi\)
−0.660391 + 0.750922i \(0.729609\pi\)
\(588\) 0 0
\(589\) −25.0000 −1.03011
\(590\) 10.0000 0.411693
\(591\) 0 0
\(592\) 3.00000 0.123299
\(593\) −33.0000 −1.35515 −0.677574 0.735455i \(-0.736969\pi\)
−0.677574 + 0.735455i \(0.736969\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −20.0000 −0.819232
\(597\) 0 0
\(598\) −6.00000 −0.245358
\(599\) 15.0000 0.612883 0.306442 0.951889i \(-0.400862\pi\)
0.306442 + 0.951889i \(0.400862\pi\)
\(600\) 0 0
\(601\) 38.0000 1.55005 0.775026 0.631929i \(-0.217737\pi\)
0.775026 + 0.631929i \(0.217737\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −20.0000 −0.813788
\(605\) −10.0000 −0.406558
\(606\) 0 0
\(607\) 12.0000 0.487065 0.243532 0.969893i \(-0.421694\pi\)
0.243532 + 0.969893i \(0.421694\pi\)
\(608\) 5.00000 0.202777
\(609\) 0 0
\(610\) 4.00000 0.161955
\(611\) 20.0000 0.809113
\(612\) 0 0
\(613\) −1.00000 −0.0403896 −0.0201948 0.999796i \(-0.506429\pi\)
−0.0201948 + 0.999796i \(0.506429\pi\)
\(614\) −11.0000 −0.443924
\(615\) 0 0
\(616\) 0 0
\(617\) −10.0000 −0.402585 −0.201292 0.979531i \(-0.564514\pi\)
−0.201292 + 0.979531i \(0.564514\pi\)
\(618\) 0 0
\(619\) 31.0000 1.24600 0.622998 0.782224i \(-0.285915\pi\)
0.622998 + 0.782224i \(0.285915\pi\)
\(620\) −5.00000 −0.200805
\(621\) 0 0
\(622\) 2.00000 0.0801927
\(623\) 0 0
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) −18.0000 −0.719425
\(627\) 0 0
\(628\) −10.0000 −0.399043
\(629\) 18.0000 0.717707
\(630\) 0 0
\(631\) −42.0000 −1.67199 −0.835997 0.548734i \(-0.815110\pi\)
−0.835997 + 0.548734i \(0.815110\pi\)
\(632\) −2.00000 −0.0795557
\(633\) 0 0
\(634\) −18.0000 −0.714871
\(635\) −10.0000 −0.396838
\(636\) 0 0
\(637\) 0 0
\(638\) −2.00000 −0.0791808
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) −40.0000 −1.57991 −0.789953 0.613168i \(-0.789895\pi\)
−0.789953 + 0.613168i \(0.789895\pi\)
\(642\) 0 0
\(643\) 33.0000 1.30139 0.650696 0.759338i \(-0.274477\pi\)
0.650696 + 0.759338i \(0.274477\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 30.0000 1.18033
\(647\) 6.00000 0.235884 0.117942 0.993020i \(-0.462370\pi\)
0.117942 + 0.993020i \(0.462370\pi\)
\(648\) 0 0
\(649\) 10.0000 0.392534
\(650\) −8.00000 −0.313786
\(651\) 0 0
\(652\) 6.00000 0.234978
\(653\) 22.0000 0.860927 0.430463 0.902608i \(-0.358350\pi\)
0.430463 + 0.902608i \(0.358350\pi\)
\(654\) 0 0
\(655\) 18.0000 0.703318
\(656\) −3.00000 −0.117130
\(657\) 0 0
\(658\) 0 0
\(659\) 17.0000 0.662226 0.331113 0.943591i \(-0.392576\pi\)
0.331113 + 0.943591i \(0.392576\pi\)
\(660\) 0 0
\(661\) −46.0000 −1.78919 −0.894596 0.446875i \(-0.852537\pi\)
−0.894596 + 0.446875i \(0.852537\pi\)
\(662\) −16.0000 −0.621858
\(663\) 0 0
\(664\) −6.00000 −0.232845
\(665\) 0 0
\(666\) 0 0
\(667\) 6.00000 0.232321
\(668\) −16.0000 −0.619059
\(669\) 0 0
\(670\) 10.0000 0.386334
\(671\) 4.00000 0.154418
\(672\) 0 0
\(673\) 26.0000 1.00223 0.501113 0.865382i \(-0.332924\pi\)
0.501113 + 0.865382i \(0.332924\pi\)
\(674\) −13.0000 −0.500741
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −33.0000 −1.26829 −0.634147 0.773213i \(-0.718648\pi\)
−0.634147 + 0.773213i \(0.718648\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 6.00000 0.230089
\(681\) 0 0
\(682\) −5.00000 −0.191460
\(683\) 11.0000 0.420903 0.210452 0.977604i \(-0.432507\pi\)
0.210452 + 0.977604i \(0.432507\pi\)
\(684\) 0 0
\(685\) 10.0000 0.382080
\(686\) 0 0
\(687\) 0 0
\(688\) −2.00000 −0.0762493
\(689\) −16.0000 −0.609551
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 11.0000 0.418157
\(693\) 0 0
\(694\) 9.00000 0.341635
\(695\) −12.0000 −0.455186
\(696\) 0 0
\(697\) −18.0000 −0.681799
\(698\) −12.0000 −0.454207
\(699\) 0 0
\(700\) 0 0
\(701\) 26.0000 0.982006 0.491003 0.871158i \(-0.336630\pi\)
0.491003 + 0.871158i \(0.336630\pi\)
\(702\) 0 0
\(703\) 15.0000 0.565736
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) −9.00000 −0.338719
\(707\) 0 0
\(708\) 0 0
\(709\) −35.0000 −1.31445 −0.657226 0.753693i \(-0.728270\pi\)
−0.657226 + 0.753693i \(0.728270\pi\)
\(710\) 13.0000 0.487881
\(711\) 0 0
\(712\) 17.0000 0.637102
\(713\) 15.0000 0.561754
\(714\) 0 0
\(715\) 2.00000 0.0747958
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) 32.0000 1.19423
\(719\) 6.00000 0.223762 0.111881 0.993722i \(-0.464312\pi\)
0.111881 + 0.993722i \(0.464312\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 6.00000 0.223297
\(723\) 0 0
\(724\) −20.0000 −0.743294
\(725\) 8.00000 0.297113
\(726\) 0 0
\(727\) 44.0000 1.63187 0.815935 0.578144i \(-0.196223\pi\)
0.815935 + 0.578144i \(0.196223\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −14.0000 −0.518163
\(731\) −12.0000 −0.443836
\(732\) 0 0
\(733\) −50.0000 −1.84679 −0.923396 0.383849i \(-0.874598\pi\)
−0.923396 + 0.383849i \(0.874598\pi\)
\(734\) 7.00000 0.258375
\(735\) 0 0
\(736\) −3.00000 −0.110581
\(737\) 10.0000 0.368355
\(738\) 0 0
\(739\) 16.0000 0.588570 0.294285 0.955718i \(-0.404919\pi\)
0.294285 + 0.955718i \(0.404919\pi\)
\(740\) 3.00000 0.110282
\(741\) 0 0
\(742\) 0 0
\(743\) 21.0000 0.770415 0.385208 0.922830i \(-0.374130\pi\)
0.385208 + 0.922830i \(0.374130\pi\)
\(744\) 0 0
\(745\) −20.0000 −0.732743
\(746\) 1.00000 0.0366126
\(747\) 0 0
\(748\) 6.00000 0.219382
\(749\) 0 0
\(750\) 0 0
\(751\) 30.0000 1.09472 0.547358 0.836899i \(-0.315634\pi\)
0.547358 + 0.836899i \(0.315634\pi\)
\(752\) 10.0000 0.364662
\(753\) 0 0
\(754\) −4.00000 −0.145671
\(755\) −20.0000 −0.727875
\(756\) 0 0
\(757\) 46.0000 1.67190 0.835949 0.548807i \(-0.184918\pi\)
0.835949 + 0.548807i \(0.184918\pi\)
\(758\) −6.00000 −0.217930
\(759\) 0 0
\(760\) 5.00000 0.181369
\(761\) 22.0000 0.797499 0.398750 0.917060i \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 3.00000 0.108536
\(765\) 0 0
\(766\) −24.0000 −0.867155
\(767\) 20.0000 0.722158
\(768\) 0 0
\(769\) 26.0000 0.937584 0.468792 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 14.0000 0.503871
\(773\) −45.0000 −1.61854 −0.809269 0.587439i \(-0.800136\pi\)
−0.809269 + 0.587439i \(0.800136\pi\)
\(774\) 0 0
\(775\) 20.0000 0.718421
\(776\) 4.00000 0.143592
\(777\) 0 0
\(778\) 0 0
\(779\) −15.0000 −0.537431
\(780\) 0 0
\(781\) 13.0000 0.465177
\(782\) −18.0000 −0.643679
\(783\) 0 0
\(784\) 0 0
\(785\) −10.0000 −0.356915
\(786\) 0 0
\(787\) 52.0000 1.85360 0.926800 0.375555i \(-0.122548\pi\)
0.926800 + 0.375555i \(0.122548\pi\)
\(788\) 24.0000 0.854965
\(789\) 0 0
\(790\) −2.00000 −0.0711568
\(791\) 0 0
\(792\) 0 0
\(793\) 8.00000 0.284088
\(794\) −28.0000 −0.993683
\(795\) 0 0
\(796\) 7.00000 0.248108
\(797\) −9.00000 −0.318796 −0.159398 0.987214i \(-0.550955\pi\)
−0.159398 + 0.987214i \(0.550955\pi\)
\(798\) 0 0
\(799\) 60.0000 2.12265
\(800\) −4.00000 −0.141421
\(801\) 0 0
\(802\) −18.0000 −0.635602
\(803\) −14.0000 −0.494049
\(804\) 0 0
\(805\) 0 0
\(806\) −10.0000 −0.352235
\(807\) 0 0
\(808\) 10.0000 0.351799
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) −25.0000 −0.877869 −0.438934 0.898519i \(-0.644644\pi\)
−0.438934 + 0.898519i \(0.644644\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 3.00000 0.105150
\(815\) 6.00000 0.210171
\(816\) 0 0
\(817\) −10.0000 −0.349856
\(818\) −6.00000 −0.209785
\(819\) 0 0
\(820\) −3.00000 −0.104765
\(821\) 10.0000 0.349002 0.174501 0.984657i \(-0.444169\pi\)
0.174501 + 0.984657i \(0.444169\pi\)
\(822\) 0 0
\(823\) −20.0000 −0.697156 −0.348578 0.937280i \(-0.613335\pi\)
−0.348578 + 0.937280i \(0.613335\pi\)
\(824\) 11.0000 0.383203
\(825\) 0 0
\(826\) 0 0
\(827\) −33.0000 −1.14752 −0.573761 0.819023i \(-0.694516\pi\)
−0.573761 + 0.819023i \(0.694516\pi\)
\(828\) 0 0
\(829\) 10.0000 0.347314 0.173657 0.984806i \(-0.444442\pi\)
0.173657 + 0.984806i \(0.444442\pi\)
\(830\) −6.00000 −0.208263
\(831\) 0 0
\(832\) 2.00000 0.0693375
\(833\) 0 0
\(834\) 0 0
\(835\) −16.0000 −0.553703
\(836\) 5.00000 0.172929
\(837\) 0 0
\(838\) 28.0000 0.967244
\(839\) 30.0000 1.03572 0.517858 0.855467i \(-0.326730\pi\)
0.517858 + 0.855467i \(0.326730\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 15.0000 0.516934
\(843\) 0 0
\(844\) 20.0000 0.688428
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) 0 0
\(848\) −8.00000 −0.274721
\(849\) 0 0
\(850\) −24.0000 −0.823193
\(851\) −9.00000 −0.308516
\(852\) 0 0
\(853\) −4.00000 −0.136957 −0.0684787 0.997653i \(-0.521815\pi\)
−0.0684787 + 0.997653i \(0.521815\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 8.00000 0.273434
\(857\) 45.0000 1.53717 0.768585 0.639747i \(-0.220961\pi\)
0.768585 + 0.639747i \(0.220961\pi\)
\(858\) 0 0
\(859\) 21.0000 0.716511 0.358255 0.933624i \(-0.383372\pi\)
0.358255 + 0.933624i \(0.383372\pi\)
\(860\) −2.00000 −0.0681994
\(861\) 0 0
\(862\) −13.0000 −0.442782
\(863\) −48.0000 −1.63394 −0.816970 0.576681i \(-0.804348\pi\)
−0.816970 + 0.576681i \(0.804348\pi\)
\(864\) 0 0
\(865\) 11.0000 0.374011
\(866\) 26.0000 0.883516
\(867\) 0 0
\(868\) 0 0
\(869\) −2.00000 −0.0678454
\(870\) 0 0
\(871\) 20.0000 0.677674
\(872\) 1.00000 0.0338643
\(873\) 0 0
\(874\) −15.0000 −0.507383
\(875\) 0 0
\(876\) 0 0
\(877\) −18.0000 −0.607817 −0.303908 0.952701i \(-0.598292\pi\)
−0.303908 + 0.952701i \(0.598292\pi\)
\(878\) 24.0000 0.809961
\(879\) 0 0
\(880\) 1.00000 0.0337100
\(881\) −5.00000 −0.168454 −0.0842271 0.996447i \(-0.526842\pi\)
−0.0842271 + 0.996447i \(0.526842\pi\)
\(882\) 0 0
\(883\) −32.0000 −1.07689 −0.538443 0.842662i \(-0.680987\pi\)
−0.538443 + 0.842662i \(0.680987\pi\)
\(884\) 12.0000 0.403604
\(885\) 0 0
\(886\) −21.0000 −0.705509
\(887\) −42.0000 −1.41022 −0.705111 0.709097i \(-0.749103\pi\)
−0.705111 + 0.709097i \(0.749103\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 17.0000 0.569841
\(891\) 0 0
\(892\) −23.0000 −0.770097
\(893\) 50.0000 1.67319
\(894\) 0 0
\(895\) −12.0000 −0.401116
\(896\) 0 0
\(897\) 0 0
\(898\) 12.0000 0.400445
\(899\) 10.0000 0.333519
\(900\) 0 0
\(901\) −48.0000 −1.59911
\(902\) −3.00000 −0.0998891
\(903\) 0 0
\(904\) −16.0000 −0.532152
\(905\) −20.0000 −0.664822
\(906\) 0 0
\(907\) −32.0000 −1.06254 −0.531271 0.847202i \(-0.678286\pi\)
−0.531271 + 0.847202i \(0.678286\pi\)
\(908\) 8.00000 0.265489
\(909\) 0 0
\(910\) 0 0
\(911\) −28.0000 −0.927681 −0.463841 0.885919i \(-0.653529\pi\)
−0.463841 + 0.885919i \(0.653529\pi\)
\(912\) 0 0
\(913\) −6.00000 −0.198571
\(914\) −31.0000 −1.02539
\(915\) 0 0
\(916\) −16.0000 −0.528655
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) −3.00000 −0.0989071
\(921\) 0 0
\(922\) 3.00000 0.0987997
\(923\) 26.0000 0.855800
\(924\) 0 0
\(925\) −12.0000 −0.394558
\(926\) −4.00000 −0.131448
\(927\) 0 0
\(928\) −2.00000 −0.0656532
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 28.0000 0.917170
\(933\) 0 0
\(934\) −22.0000 −0.719862
\(935\) 6.00000 0.196221
\(936\) 0 0
\(937\) 48.0000 1.56809 0.784046 0.620703i \(-0.213153\pi\)
0.784046 + 0.620703i \(0.213153\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 10.0000 0.326164
\(941\) 33.0000 1.07577 0.537885 0.843018i \(-0.319224\pi\)
0.537885 + 0.843018i \(0.319224\pi\)
\(942\) 0 0
\(943\) 9.00000 0.293080
\(944\) 10.0000 0.325472
\(945\) 0 0
\(946\) −2.00000 −0.0650256
\(947\) 43.0000 1.39731 0.698656 0.715458i \(-0.253782\pi\)
0.698656 + 0.715458i \(0.253782\pi\)
\(948\) 0 0
\(949\) −28.0000 −0.908918
\(950\) −20.0000 −0.648886
\(951\) 0 0
\(952\) 0 0
\(953\) −18.0000 −0.583077 −0.291539 0.956559i \(-0.594167\pi\)
−0.291539 + 0.956559i \(0.594167\pi\)
\(954\) 0 0
\(955\) 3.00000 0.0970777
\(956\) −24.0000 −0.776215
\(957\) 0 0
\(958\) −36.0000 −1.16311
\(959\) 0 0
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) 6.00000 0.193448
\(963\) 0 0
\(964\) −26.0000 −0.837404
\(965\) 14.0000 0.450676
\(966\) 0 0
\(967\) 16.0000 0.514525 0.257263 0.966342i \(-0.417179\pi\)
0.257263 + 0.966342i \(0.417179\pi\)
\(968\) −10.0000 −0.321412
\(969\) 0 0
\(970\) 4.00000 0.128432
\(971\) 22.0000 0.706014 0.353007 0.935621i \(-0.385159\pi\)
0.353007 + 0.935621i \(0.385159\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 22.0000 0.704925
\(975\) 0 0
\(976\) 4.00000 0.128037
\(977\) −8.00000 −0.255943 −0.127971 0.991778i \(-0.540847\pi\)
−0.127971 + 0.991778i \(0.540847\pi\)
\(978\) 0 0
\(979\) 17.0000 0.543322
\(980\) 0 0
\(981\) 0 0
\(982\) −27.0000 −0.861605
\(983\) 10.0000 0.318950 0.159475 0.987202i \(-0.449020\pi\)
0.159475 + 0.987202i \(0.449020\pi\)
\(984\) 0 0
\(985\) 24.0000 0.764704
\(986\) −12.0000 −0.382158
\(987\) 0 0
\(988\) 10.0000 0.318142
\(989\) 6.00000 0.190789
\(990\) 0 0
\(991\) 50.0000 1.58830 0.794151 0.607720i \(-0.207916\pi\)
0.794151 + 0.607720i \(0.207916\pi\)
\(992\) −5.00000 −0.158750
\(993\) 0 0
\(994\) 0 0
\(995\) 7.00000 0.221915
\(996\) 0 0
\(997\) 44.0000 1.39349 0.696747 0.717317i \(-0.254630\pi\)
0.696747 + 0.717317i \(0.254630\pi\)
\(998\) −6.00000 −0.189927
\(999\) 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 2646.2.a.z.1.1 yes 1
3.2 odd 2 2646.2.a.e.1.1 1
7.6 odd 2 2646.2.a.u.1.1 yes 1
21.20 even 2 2646.2.a.j.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2646.2.a.e.1.1 1 3.2 odd 2
2646.2.a.j.1.1 yes 1 21.20 even 2
2646.2.a.u.1.1 yes 1 7.6 odd 2
2646.2.a.z.1.1 yes 1 1.1 even 1 trivial