Properties

Label 585.2.a.i.1.1
Level $585$
Weight $2$
Character 585.1
Self dual yes
Analytic conductor $4.671$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.67124851824\)
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 \(\chi\) \(=\) 585.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} -3.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} -3.00000 q^{8} +1.00000 q^{10} +4.00000 q^{11} -1.00000 q^{13} +2.00000 q^{14} -1.00000 q^{16} +4.00000 q^{17} +6.00000 q^{19} -1.00000 q^{20} +4.00000 q^{22} +1.00000 q^{25} -1.00000 q^{26} -2.00000 q^{28} +4.00000 q^{29} -10.0000 q^{31} +5.00000 q^{32} +4.00000 q^{34} +2.00000 q^{35} -2.00000 q^{37} +6.00000 q^{38} -3.00000 q^{40} +6.00000 q^{41} -8.00000 q^{43} -4.00000 q^{44} +8.00000 q^{47} -3.00000 q^{49} +1.00000 q^{50} +1.00000 q^{52} +4.00000 q^{53} +4.00000 q^{55} -6.00000 q^{56} +4.00000 q^{58} -12.0000 q^{59} +2.00000 q^{61} -10.0000 q^{62} +7.00000 q^{64} -1.00000 q^{65} -10.0000 q^{67} -4.00000 q^{68} +2.00000 q^{70} -6.00000 q^{73} -2.00000 q^{74} -6.00000 q^{76} +8.00000 q^{77} +12.0000 q^{79} -1.00000 q^{80} +6.00000 q^{82} +4.00000 q^{83} +4.00000 q^{85} -8.00000 q^{86} -12.0000 q^{88} -14.0000 q^{89} -2.00000 q^{91} +8.00000 q^{94} +6.00000 q^{95} -14.0000 q^{97} -3.00000 q^{98} +O(q^{100})\)

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 0.353553 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −3.00000 −1.06066
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −1.00000 −0.196116
\(27\) 0 0
\(28\) −2.00000 −0.377964
\(29\) 4.00000 0.742781 0.371391 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) −10.0000 −1.79605 −0.898027 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) 5.00000 0.883883
\(33\) 0 0
\(34\) 4.00000 0.685994
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 6.00000 0.973329
\(39\) 0 0
\(40\) −3.00000 −0.474342
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 1.00000 0.138675
\(53\) 4.00000 0.549442 0.274721 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) −6.00000 −0.801784
\(57\) 0 0
\(58\) 4.00000 0.525226
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −10.0000 −1.27000
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) −10.0000 −1.22169 −0.610847 0.791748i \(-0.709171\pi\)
−0.610847 + 0.791748i \(0.709171\pi\)
\(68\) −4.00000 −0.485071
\(69\) 0 0
\(70\) 2.00000 0.239046
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) −6.00000 −0.688247
\(77\) 8.00000 0.911685
\(78\) 0 0
\(79\) 12.0000 1.35011 0.675053 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 6.00000 0.662589
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) −8.00000 −0.862662
\(87\) 0 0
\(88\) −12.0000 −1.27920
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 0 0
\(93\) 0 0
\(94\) 8.00000 0.825137
\(95\) 6.00000 0.615587
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) −3.00000 −0.303046
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 3.00000 0.294174
\(105\) 0 0
\(106\) 4.00000 0.388514
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 4.00000 0.381385
\(111\) 0 0
\(112\) −2.00000 −0.188982
\(113\) −16.0000 −1.50515 −0.752577 0.658505i \(-0.771189\pi\)
−0.752577 + 0.658505i \(0.771189\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −4.00000 −0.371391
\(117\) 0 0
\(118\) −12.0000 −1.10469
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 2.00000 0.181071
\(123\) 0 0
\(124\) 10.0000 0.898027
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) −3.00000 −0.265165
\(129\) 0 0
\(130\) −1.00000 −0.0877058
\(131\) 16.0000 1.39793 0.698963 0.715158i \(-0.253645\pi\)
0.698963 + 0.715158i \(0.253645\pi\)
\(132\) 0 0
\(133\) 12.0000 1.04053
\(134\) −10.0000 −0.863868
\(135\) 0 0
\(136\) −12.0000 −1.02899
\(137\) −10.0000 −0.854358 −0.427179 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) −2.00000 −0.169031
\(141\) 0 0
\(142\) 0 0
\(143\) −4.00000 −0.334497
\(144\) 0 0
\(145\) 4.00000 0.332182
\(146\) −6.00000 −0.496564
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 0 0
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) −18.0000 −1.45999
\(153\) 0 0
\(154\) 8.00000 0.644658
\(155\) −10.0000 −0.803219
\(156\) 0 0
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 12.0000 0.954669
\(159\) 0 0
\(160\) 5.00000 0.395285
\(161\) 0 0
\(162\) 0 0
\(163\) 6.00000 0.469956 0.234978 0.972001i \(-0.424498\pi\)
0.234978 + 0.972001i \(0.424498\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 24.0000 1.85718 0.928588 0.371113i \(-0.121024\pi\)
0.928588 + 0.371113i \(0.121024\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 4.00000 0.306786
\(171\) 0 0
\(172\) 8.00000 0.609994
\(173\) −24.0000 −1.82469 −0.912343 0.409426i \(-0.865729\pi\)
−0.912343 + 0.409426i \(0.865729\pi\)
\(174\) 0 0
\(175\) 2.00000 0.151186
\(176\) −4.00000 −0.301511
\(177\) 0 0
\(178\) −14.0000 −1.04934
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) −2.00000 −0.148250
\(183\) 0 0
\(184\) 0 0
\(185\) −2.00000 −0.147043
\(186\) 0 0
\(187\) 16.0000 1.17004
\(188\) −8.00000 −0.583460
\(189\) 0 0
\(190\) 6.00000 0.435286
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) −26.0000 −1.87152 −0.935760 0.352636i \(-0.885285\pi\)
−0.935760 + 0.352636i \(0.885285\pi\)
\(194\) −14.0000 −1.00514
\(195\) 0 0
\(196\) 3.00000 0.214286
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) −3.00000 −0.212132
\(201\) 0 0
\(202\) −12.0000 −0.844317
\(203\) 8.00000 0.561490
\(204\) 0 0
\(205\) 6.00000 0.419058
\(206\) −4.00000 −0.278693
\(207\) 0 0
\(208\) 1.00000 0.0693375
\(209\) 24.0000 1.66011
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) −4.00000 −0.274721
\(213\) 0 0
\(214\) −4.00000 −0.273434
\(215\) −8.00000 −0.545595
\(216\) 0 0
\(217\) −20.0000 −1.35769
\(218\) −2.00000 −0.135457
\(219\) 0 0
\(220\) −4.00000 −0.269680
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) −14.0000 −0.937509 −0.468755 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) 10.0000 0.668153
\(225\) 0 0
\(226\) −16.0000 −1.06430
\(227\) 20.0000 1.32745 0.663723 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −12.0000 −0.787839
\(233\) 8.00000 0.524097 0.262049 0.965055i \(-0.415602\pi\)
0.262049 + 0.965055i \(0.415602\pi\)
\(234\) 0 0
\(235\) 8.00000 0.521862
\(236\) 12.0000 0.781133
\(237\) 0 0
\(238\) 8.00000 0.518563
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 22.0000 1.41714 0.708572 0.705638i \(-0.249340\pi\)
0.708572 + 0.705638i \(0.249340\pi\)
\(242\) 5.00000 0.321412
\(243\) 0 0
\(244\) −2.00000 −0.128037
\(245\) −3.00000 −0.191663
\(246\) 0 0
\(247\) −6.00000 −0.381771
\(248\) 30.0000 1.90500
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) −8.00000 −0.499026 −0.249513 0.968371i \(-0.580271\pi\)
−0.249513 + 0.968371i \(0.580271\pi\)
\(258\) 0 0
\(259\) −4.00000 −0.248548
\(260\) 1.00000 0.0620174
\(261\) 0 0
\(262\) 16.0000 0.988483
\(263\) 4.00000 0.246651 0.123325 0.992366i \(-0.460644\pi\)
0.123325 + 0.992366i \(0.460644\pi\)
\(264\) 0 0
\(265\) 4.00000 0.245718
\(266\) 12.0000 0.735767
\(267\) 0 0
\(268\) 10.0000 0.610847
\(269\) −12.0000 −0.731653 −0.365826 0.930683i \(-0.619214\pi\)
−0.365826 + 0.930683i \(0.619214\pi\)
\(270\) 0 0
\(271\) 14.0000 0.850439 0.425220 0.905090i \(-0.360197\pi\)
0.425220 + 0.905090i \(0.360197\pi\)
\(272\) −4.00000 −0.242536
\(273\) 0 0
\(274\) −10.0000 −0.604122
\(275\) 4.00000 0.241209
\(276\) 0 0
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 12.0000 0.719712
\(279\) 0 0
\(280\) −6.00000 −0.358569
\(281\) 30.0000 1.78965 0.894825 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) 12.0000 0.708338
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 4.00000 0.234888
\(291\) 0 0
\(292\) 6.00000 0.351123
\(293\) 2.00000 0.116841 0.0584206 0.998292i \(-0.481394\pi\)
0.0584206 + 0.998292i \(0.481394\pi\)
\(294\) 0 0
\(295\) −12.0000 −0.698667
\(296\) 6.00000 0.348743
\(297\) 0 0
\(298\) 18.0000 1.04271
\(299\) 0 0
\(300\) 0 0
\(301\) −16.0000 −0.922225
\(302\) 10.0000 0.575435
\(303\) 0 0
\(304\) −6.00000 −0.344124
\(305\) 2.00000 0.114520
\(306\) 0 0
\(307\) −22.0000 −1.25561 −0.627803 0.778372i \(-0.716046\pi\)
−0.627803 + 0.778372i \(0.716046\pi\)
\(308\) −8.00000 −0.455842
\(309\) 0 0
\(310\) −10.0000 −0.567962
\(311\) 4.00000 0.226819 0.113410 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) 0 0
\(313\) 2.00000 0.113047 0.0565233 0.998401i \(-0.481998\pi\)
0.0565233 + 0.998401i \(0.481998\pi\)
\(314\) 18.0000 1.01580
\(315\) 0 0
\(316\) −12.0000 −0.675053
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 0 0
\(319\) 16.0000 0.895828
\(320\) 7.00000 0.391312
\(321\) 0 0
\(322\) 0 0
\(323\) 24.0000 1.33540
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) 6.00000 0.332309
\(327\) 0 0
\(328\) −18.0000 −0.993884
\(329\) 16.0000 0.882109
\(330\) 0 0
\(331\) 18.0000 0.989369 0.494685 0.869072i \(-0.335284\pi\)
0.494685 + 0.869072i \(0.335284\pi\)
\(332\) −4.00000 −0.219529
\(333\) 0 0
\(334\) 24.0000 1.31322
\(335\) −10.0000 −0.546358
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 1.00000 0.0543928
\(339\) 0 0
\(340\) −4.00000 −0.216930
\(341\) −40.0000 −2.16612
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 24.0000 1.29399
\(345\) 0 0
\(346\) −24.0000 −1.29025
\(347\) −32.0000 −1.71785 −0.858925 0.512101i \(-0.828867\pi\)
−0.858925 + 0.512101i \(0.828867\pi\)
\(348\) 0 0
\(349\) −6.00000 −0.321173 −0.160586 0.987022i \(-0.551338\pi\)
−0.160586 + 0.987022i \(0.551338\pi\)
\(350\) 2.00000 0.106904
\(351\) 0 0
\(352\) 20.0000 1.06600
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 14.0000 0.741999
\(357\) 0 0
\(358\) 0 0
\(359\) 16.0000 0.844448 0.422224 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) −10.0000 −0.525588
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) −6.00000 −0.314054
\(366\) 0 0
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) −2.00000 −0.103975
\(371\) 8.00000 0.415339
\(372\) 0 0
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) 16.0000 0.827340
\(375\) 0 0
\(376\) −24.0000 −1.23771
\(377\) −4.00000 −0.206010
\(378\) 0 0
\(379\) −2.00000 −0.102733 −0.0513665 0.998680i \(-0.516358\pi\)
−0.0513665 + 0.998680i \(0.516358\pi\)
\(380\) −6.00000 −0.307794
\(381\) 0 0
\(382\) 12.0000 0.613973
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 0 0
\(385\) 8.00000 0.407718
\(386\) −26.0000 −1.32337
\(387\) 0 0
\(388\) 14.0000 0.710742
\(389\) −8.00000 −0.405616 −0.202808 0.979219i \(-0.565007\pi\)
−0.202808 + 0.979219i \(0.565007\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 9.00000 0.454569
\(393\) 0 0
\(394\) −18.0000 −0.906827
\(395\) 12.0000 0.603786
\(396\) 0 0
\(397\) −18.0000 −0.903394 −0.451697 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) 8.00000 0.401004
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −22.0000 −1.09863 −0.549314 0.835616i \(-0.685111\pi\)
−0.549314 + 0.835616i \(0.685111\pi\)
\(402\) 0 0
\(403\) 10.0000 0.498135
\(404\) 12.0000 0.597022
\(405\) 0 0
\(406\) 8.00000 0.397033
\(407\) −8.00000 −0.396545
\(408\) 0 0
\(409\) 30.0000 1.48340 0.741702 0.670729i \(-0.234019\pi\)
0.741702 + 0.670729i \(0.234019\pi\)
\(410\) 6.00000 0.296319
\(411\) 0 0
\(412\) 4.00000 0.197066
\(413\) −24.0000 −1.18096
\(414\) 0 0
\(415\) 4.00000 0.196352
\(416\) −5.00000 −0.245145
\(417\) 0 0
\(418\) 24.0000 1.17388
\(419\) 16.0000 0.781651 0.390826 0.920465i \(-0.372190\pi\)
0.390826 + 0.920465i \(0.372190\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) −12.0000 −0.582772
\(425\) 4.00000 0.194029
\(426\) 0 0
\(427\) 4.00000 0.193574
\(428\) 4.00000 0.193347
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) 40.0000 1.92673 0.963366 0.268190i \(-0.0864254\pi\)
0.963366 + 0.268190i \(0.0864254\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) −20.0000 −0.960031
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 0 0
\(438\) 0 0
\(439\) −36.0000 −1.71819 −0.859093 0.511819i \(-0.828972\pi\)
−0.859093 + 0.511819i \(0.828972\pi\)
\(440\) −12.0000 −0.572078
\(441\) 0 0
\(442\) −4.00000 −0.190261
\(443\) −20.0000 −0.950229 −0.475114 0.879924i \(-0.657593\pi\)
−0.475114 + 0.879924i \(0.657593\pi\)
\(444\) 0 0
\(445\) −14.0000 −0.663664
\(446\) −14.0000 −0.662919
\(447\) 0 0
\(448\) 14.0000 0.661438
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) 24.0000 1.13012
\(452\) 16.0000 0.752577
\(453\) 0 0
\(454\) 20.0000 0.938647
\(455\) −2.00000 −0.0937614
\(456\) 0 0
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) −10.0000 −0.467269
\(459\) 0 0
\(460\) 0 0
\(461\) 14.0000 0.652045 0.326023 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) 0 0
\(463\) 18.0000 0.836531 0.418265 0.908325i \(-0.362638\pi\)
0.418265 + 0.908325i \(0.362638\pi\)
\(464\) −4.00000 −0.185695
\(465\) 0 0
\(466\) 8.00000 0.370593
\(467\) −8.00000 −0.370196 −0.185098 0.982720i \(-0.559260\pi\)
−0.185098 + 0.982720i \(0.559260\pi\)
\(468\) 0 0
\(469\) −20.0000 −0.923514
\(470\) 8.00000 0.369012
\(471\) 0 0
\(472\) 36.0000 1.65703
\(473\) −32.0000 −1.47136
\(474\) 0 0
\(475\) 6.00000 0.275299
\(476\) −8.00000 −0.366679
\(477\) 0 0
\(478\) 0 0
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) 22.0000 1.00207
\(483\) 0 0
\(484\) −5.00000 −0.227273
\(485\) −14.0000 −0.635707
\(486\) 0 0
\(487\) −22.0000 −0.996915 −0.498458 0.866914i \(-0.666100\pi\)
−0.498458 + 0.866914i \(0.666100\pi\)
\(488\) −6.00000 −0.271607
\(489\) 0 0
\(490\) −3.00000 −0.135526
\(491\) −36.0000 −1.62466 −0.812329 0.583200i \(-0.801800\pi\)
−0.812329 + 0.583200i \(0.801800\pi\)
\(492\) 0 0
\(493\) 16.0000 0.720604
\(494\) −6.00000 −0.269953
\(495\) 0 0
\(496\) 10.0000 0.449013
\(497\) 0 0
\(498\) 0 0
\(499\) −38.0000 −1.70111 −0.850557 0.525883i \(-0.823735\pi\)
−0.850557 + 0.525883i \(0.823735\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −12.0000 −0.535586
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) 0 0
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) −42.0000 −1.86162 −0.930809 0.365507i \(-0.880896\pi\)
−0.930809 + 0.365507i \(0.880896\pi\)
\(510\) 0 0
\(511\) −12.0000 −0.530849
\(512\) −11.0000 −0.486136
\(513\) 0 0
\(514\) −8.00000 −0.352865
\(515\) −4.00000 −0.176261
\(516\) 0 0
\(517\) 32.0000 1.40736
\(518\) −4.00000 −0.175750
\(519\) 0 0
\(520\) 3.00000 0.131559
\(521\) −12.0000 −0.525730 −0.262865 0.964833i \(-0.584667\pi\)
−0.262865 + 0.964833i \(0.584667\pi\)
\(522\) 0 0
\(523\) −12.0000 −0.524723 −0.262362 0.964970i \(-0.584501\pi\)
−0.262362 + 0.964970i \(0.584501\pi\)
\(524\) −16.0000 −0.698963
\(525\) 0 0
\(526\) 4.00000 0.174408
\(527\) −40.0000 −1.74243
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 4.00000 0.173749
\(531\) 0 0
\(532\) −12.0000 −0.520266
\(533\) −6.00000 −0.259889
\(534\) 0 0
\(535\) −4.00000 −0.172935
\(536\) 30.0000 1.29580
\(537\) 0 0
\(538\) −12.0000 −0.517357
\(539\) −12.0000 −0.516877
\(540\) 0 0
\(541\) 14.0000 0.601907 0.300954 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) 14.0000 0.601351
\(543\) 0 0
\(544\) 20.0000 0.857493
\(545\) −2.00000 −0.0856706
\(546\) 0 0
\(547\) −36.0000 −1.53925 −0.769624 0.638497i \(-0.779557\pi\)
−0.769624 + 0.638497i \(0.779557\pi\)
\(548\) 10.0000 0.427179
\(549\) 0 0
\(550\) 4.00000 0.170561
\(551\) 24.0000 1.02243
\(552\) 0 0
\(553\) 24.0000 1.02058
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) −12.0000 −0.508913
\(557\) 26.0000 1.10166 0.550828 0.834619i \(-0.314312\pi\)
0.550828 + 0.834619i \(0.314312\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) −2.00000 −0.0845154
\(561\) 0 0
\(562\) 30.0000 1.26547
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 0 0
\(565\) −16.0000 −0.673125
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) 0 0
\(569\) −28.0000 −1.17382 −0.586911 0.809652i \(-0.699656\pi\)
−0.586911 + 0.809652i \(0.699656\pi\)
\(570\) 0 0
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 4.00000 0.167248
\(573\) 0 0
\(574\) 12.0000 0.500870
\(575\) 0 0
\(576\) 0 0
\(577\) 38.0000 1.58196 0.790980 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 0 0
\(580\) −4.00000 −0.166091
\(581\) 8.00000 0.331896
\(582\) 0 0
\(583\) 16.0000 0.662652
\(584\) 18.0000 0.744845
\(585\) 0 0
\(586\) 2.00000 0.0826192
\(587\) 20.0000 0.825488 0.412744 0.910847i \(-0.364570\pi\)
0.412744 + 0.910847i \(0.364570\pi\)
\(588\) 0 0
\(589\) −60.0000 −2.47226
\(590\) −12.0000 −0.494032
\(591\) 0 0
\(592\) 2.00000 0.0821995
\(593\) −14.0000 −0.574911 −0.287456 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) 0 0
\(595\) 8.00000 0.327968
\(596\) −18.0000 −0.737309
\(597\) 0 0
\(598\) 0 0
\(599\) −8.00000 −0.326871 −0.163436 0.986554i \(-0.552258\pi\)
−0.163436 + 0.986554i \(0.552258\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) −16.0000 −0.652111
\(603\) 0 0
\(604\) −10.0000 −0.406894
\(605\) 5.00000 0.203279
\(606\) 0 0
\(607\) −8.00000 −0.324710 −0.162355 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) 30.0000 1.21666
\(609\) 0 0
\(610\) 2.00000 0.0809776
\(611\) −8.00000 −0.323645
\(612\) 0 0
\(613\) 6.00000 0.242338 0.121169 0.992632i \(-0.461336\pi\)
0.121169 + 0.992632i \(0.461336\pi\)
\(614\) −22.0000 −0.887848
\(615\) 0 0
\(616\) −24.0000 −0.966988
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 0 0
\(619\) 22.0000 0.884255 0.442127 0.896952i \(-0.354224\pi\)
0.442127 + 0.896952i \(0.354224\pi\)
\(620\) 10.0000 0.401610
\(621\) 0 0
\(622\) 4.00000 0.160385
\(623\) −28.0000 −1.12180
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 2.00000 0.0799361
\(627\) 0 0
\(628\) −18.0000 −0.718278
\(629\) −8.00000 −0.318981
\(630\) 0 0
\(631\) −14.0000 −0.557331 −0.278666 0.960388i \(-0.589892\pi\)
−0.278666 + 0.960388i \(0.589892\pi\)
\(632\) −36.0000 −1.43200
\(633\) 0 0
\(634\) −18.0000 −0.714871
\(635\) −8.00000 −0.317470
\(636\) 0 0
\(637\) 3.00000 0.118864
\(638\) 16.0000 0.633446
\(639\) 0 0
\(640\) −3.00000 −0.118585
\(641\) 40.0000 1.57991 0.789953 0.613168i \(-0.210105\pi\)
0.789953 + 0.613168i \(0.210105\pi\)
\(642\) 0 0
\(643\) 14.0000 0.552106 0.276053 0.961142i \(-0.410973\pi\)
0.276053 + 0.961142i \(0.410973\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 24.0000 0.944267
\(647\) 28.0000 1.10079 0.550397 0.834903i \(-0.314476\pi\)
0.550397 + 0.834903i \(0.314476\pi\)
\(648\) 0 0
\(649\) −48.0000 −1.88416
\(650\) −1.00000 −0.0392232
\(651\) 0 0
\(652\) −6.00000 −0.234978
\(653\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) 16.0000 0.625172
\(656\) −6.00000 −0.234261
\(657\) 0 0
\(658\) 16.0000 0.623745
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) 26.0000 1.01128 0.505641 0.862744i \(-0.331256\pi\)
0.505641 + 0.862744i \(0.331256\pi\)
\(662\) 18.0000 0.699590
\(663\) 0 0
\(664\) −12.0000 −0.465690
\(665\) 12.0000 0.465340
\(666\) 0 0
\(667\) 0 0
\(668\) −24.0000 −0.928588
\(669\) 0 0
\(670\) −10.0000 −0.386334
\(671\) 8.00000 0.308837
\(672\) 0 0
\(673\) −18.0000 −0.693849 −0.346925 0.937893i \(-0.612774\pi\)
−0.346925 + 0.937893i \(0.612774\pi\)
\(674\) 14.0000 0.539260
\(675\) 0 0
\(676\) −1.00000 −0.0384615
\(677\) 36.0000 1.38359 0.691796 0.722093i \(-0.256820\pi\)
0.691796 + 0.722093i \(0.256820\pi\)
\(678\) 0 0
\(679\) −28.0000 −1.07454
\(680\) −12.0000 −0.460179
\(681\) 0 0
\(682\) −40.0000 −1.53168
\(683\) −4.00000 −0.153056 −0.0765279 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) 0 0
\(685\) −10.0000 −0.382080
\(686\) −20.0000 −0.763604
\(687\) 0 0
\(688\) 8.00000 0.304997
\(689\) −4.00000 −0.152388
\(690\) 0 0
\(691\) 34.0000 1.29342 0.646710 0.762736i \(-0.276144\pi\)
0.646710 + 0.762736i \(0.276144\pi\)
\(692\) 24.0000 0.912343
\(693\) 0 0
\(694\) −32.0000 −1.21470
\(695\) 12.0000 0.455186
\(696\) 0 0
\(697\) 24.0000 0.909065
\(698\) −6.00000 −0.227103
\(699\) 0 0
\(700\) −2.00000 −0.0755929
\(701\) −40.0000 −1.51078 −0.755390 0.655276i \(-0.772552\pi\)
−0.755390 + 0.655276i \(0.772552\pi\)
\(702\) 0 0
\(703\) −12.0000 −0.452589
\(704\) 28.0000 1.05529
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) −24.0000 −0.902613
\(708\) 0 0
\(709\) −50.0000 −1.87779 −0.938895 0.344204i \(-0.888149\pi\)
−0.938895 + 0.344204i \(0.888149\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 42.0000 1.57402
\(713\) 0 0
\(714\) 0 0
\(715\) −4.00000 −0.149592
\(716\) 0 0
\(717\) 0 0
\(718\) 16.0000 0.597115
\(719\) 20.0000 0.745874 0.372937 0.927857i \(-0.378351\pi\)
0.372937 + 0.927857i \(0.378351\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 17.0000 0.632674
\(723\) 0 0
\(724\) 10.0000 0.371647
\(725\) 4.00000 0.148556
\(726\) 0 0
\(727\) −24.0000 −0.890111 −0.445055 0.895503i \(-0.646816\pi\)
−0.445055 + 0.895503i \(0.646816\pi\)
\(728\) 6.00000 0.222375
\(729\) 0 0
\(730\) −6.00000 −0.222070
\(731\) −32.0000 −1.18356
\(732\) 0 0
\(733\) 34.0000 1.25582 0.627909 0.778287i \(-0.283911\pi\)
0.627909 + 0.778287i \(0.283911\pi\)
\(734\) −8.00000 −0.295285
\(735\) 0 0
\(736\) 0 0
\(737\) −40.0000 −1.47342
\(738\) 0 0
\(739\) 30.0000 1.10357 0.551784 0.833987i \(-0.313947\pi\)
0.551784 + 0.833987i \(0.313947\pi\)
\(740\) 2.00000 0.0735215
\(741\) 0 0
\(742\) 8.00000 0.293689
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) 18.0000 0.659469
\(746\) 22.0000 0.805477
\(747\) 0 0
\(748\) −16.0000 −0.585018
\(749\) −8.00000 −0.292314
\(750\) 0 0
\(751\) 20.0000 0.729810 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) −8.00000 −0.291730
\(753\) 0 0
\(754\) −4.00000 −0.145671
\(755\) 10.0000 0.363937
\(756\) 0 0
\(757\) −34.0000 −1.23575 −0.617876 0.786276i \(-0.712006\pi\)
−0.617876 + 0.786276i \(0.712006\pi\)
\(758\) −2.00000 −0.0726433
\(759\) 0 0
\(760\) −18.0000 −0.652929
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 0 0
\(763\) −4.00000 −0.144810
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) 12.0000 0.433295
\(768\) 0 0
\(769\) 2.00000 0.0721218 0.0360609 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 8.00000 0.288300
\(771\) 0 0
\(772\) 26.0000 0.935760
\(773\) 50.0000 1.79838 0.899188 0.437564i \(-0.144158\pi\)
0.899188 + 0.437564i \(0.144158\pi\)
\(774\) 0 0
\(775\) −10.0000 −0.359211
\(776\) 42.0000 1.50771
\(777\) 0 0
\(778\) −8.00000 −0.286814
\(779\) 36.0000 1.28983
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 3.00000 0.107143
\(785\) 18.0000 0.642448
\(786\) 0 0
\(787\) −22.0000 −0.784215 −0.392108 0.919919i \(-0.628254\pi\)
−0.392108 + 0.919919i \(0.628254\pi\)
\(788\) 18.0000 0.641223
\(789\) 0 0
\(790\) 12.0000 0.426941
\(791\) −32.0000 −1.13779
\(792\) 0 0
\(793\) −2.00000 −0.0710221
\(794\) −18.0000 −0.638796
\(795\) 0 0
\(796\) −8.00000 −0.283552
\(797\) 12.0000 0.425062 0.212531 0.977154i \(-0.431829\pi\)
0.212531 + 0.977154i \(0.431829\pi\)
\(798\) 0 0
\(799\) 32.0000 1.13208
\(800\) 5.00000 0.176777
\(801\) 0 0
\(802\) −22.0000 −0.776847
\(803\) −24.0000 −0.846942
\(804\) 0 0
\(805\) 0 0
\(806\) 10.0000 0.352235
\(807\) 0 0
\(808\) 36.0000 1.26648
\(809\) −28.0000 −0.984428 −0.492214 0.870474i \(-0.663812\pi\)
−0.492214 + 0.870474i \(0.663812\pi\)
\(810\) 0 0
\(811\) 30.0000 1.05344 0.526721 0.850038i \(-0.323421\pi\)
0.526721 + 0.850038i \(0.323421\pi\)
\(812\) −8.00000 −0.280745
\(813\) 0 0
\(814\) −8.00000 −0.280400
\(815\) 6.00000 0.210171
\(816\) 0 0
\(817\) −48.0000 −1.67931
\(818\) 30.0000 1.04893
\(819\) 0 0
\(820\) −6.00000 −0.209529
\(821\) 10.0000 0.349002 0.174501 0.984657i \(-0.444169\pi\)
0.174501 + 0.984657i \(0.444169\pi\)
\(822\) 0 0
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) 12.0000 0.418040
\(825\) 0 0
\(826\) −24.0000 −0.835067
\(827\) 20.0000 0.695468 0.347734 0.937593i \(-0.386951\pi\)
0.347734 + 0.937593i \(0.386951\pi\)
\(828\) 0 0
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 4.00000 0.138842
\(831\) 0 0
\(832\) −7.00000 −0.242681
\(833\) −12.0000 −0.415775
\(834\) 0 0
\(835\) 24.0000 0.830554
\(836\) −24.0000 −0.830057
\(837\) 0 0
\(838\) 16.0000 0.552711
\(839\) 40.0000 1.38095 0.690477 0.723355i \(-0.257401\pi\)
0.690477 + 0.723355i \(0.257401\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 22.0000 0.758170
\(843\) 0 0
\(844\) 0 0
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) 10.0000 0.343604
\(848\) −4.00000 −0.137361
\(849\) 0 0
\(850\) 4.00000 0.137199
\(851\) 0 0
\(852\) 0 0
\(853\) 30.0000 1.02718 0.513590 0.858036i \(-0.328315\pi\)
0.513590 + 0.858036i \(0.328315\pi\)
\(854\) 4.00000 0.136877
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 48.0000 1.63965 0.819824 0.572615i \(-0.194071\pi\)
0.819824 + 0.572615i \(0.194071\pi\)
\(858\) 0 0
\(859\) 16.0000 0.545913 0.272956 0.962026i \(-0.411998\pi\)
0.272956 + 0.962026i \(0.411998\pi\)
\(860\) 8.00000 0.272798
\(861\) 0 0
\(862\) 40.0000 1.36241
\(863\) 48.0000 1.63394 0.816970 0.576681i \(-0.195652\pi\)
0.816970 + 0.576681i \(0.195652\pi\)
\(864\) 0 0
\(865\) −24.0000 −0.816024
\(866\) −2.00000 −0.0679628
\(867\) 0 0
\(868\) 20.0000 0.678844
\(869\) 48.0000 1.62829
\(870\) 0 0
\(871\) 10.0000 0.338837
\(872\) 6.00000 0.203186
\(873\) 0 0
\(874\) 0 0
\(875\) 2.00000 0.0676123
\(876\) 0 0
\(877\) −42.0000 −1.41824 −0.709120 0.705088i \(-0.750907\pi\)
−0.709120 + 0.705088i \(0.750907\pi\)
\(878\) −36.0000 −1.21494
\(879\) 0 0
\(880\) −4.00000 −0.134840
\(881\) 28.0000 0.943344 0.471672 0.881774i \(-0.343651\pi\)
0.471672 + 0.881774i \(0.343651\pi\)
\(882\) 0 0
\(883\) −4.00000 −0.134611 −0.0673054 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) 4.00000 0.134535
\(885\) 0 0
\(886\) −20.0000 −0.671913
\(887\) 56.0000 1.88030 0.940148 0.340766i \(-0.110687\pi\)
0.940148 + 0.340766i \(0.110687\pi\)
\(888\) 0 0
\(889\) −16.0000 −0.536623
\(890\) −14.0000 −0.469281
\(891\) 0 0
\(892\) 14.0000 0.468755
\(893\) 48.0000 1.60626
\(894\) 0 0
\(895\) 0 0
\(896\) −6.00000 −0.200446
\(897\) 0 0
\(898\) −30.0000 −1.00111
\(899\) −40.0000 −1.33407
\(900\) 0 0
\(901\) 16.0000 0.533037
\(902\) 24.0000 0.799113
\(903\) 0 0
\(904\) 48.0000 1.59646
\(905\) −10.0000 −0.332411
\(906\) 0 0
\(907\) 44.0000 1.46100 0.730498 0.682915i \(-0.239288\pi\)
0.730498 + 0.682915i \(0.239288\pi\)
\(908\) −20.0000 −0.663723
\(909\) 0 0
\(910\) −2.00000 −0.0662994
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 0 0
\(913\) 16.0000 0.529523
\(914\) −22.0000 −0.727695
\(915\) 0 0
\(916\) 10.0000 0.330409
\(917\) 32.0000 1.05673
\(918\) 0 0
\(919\) 44.0000 1.45143 0.725713 0.687998i \(-0.241510\pi\)
0.725713 + 0.687998i \(0.241510\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 14.0000 0.461065
\(923\) 0 0
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) 18.0000 0.591517
\(927\) 0 0
\(928\) 20.0000 0.656532
\(929\) −10.0000 −0.328089 −0.164045 0.986453i \(-0.552454\pi\)
−0.164045 + 0.986453i \(0.552454\pi\)
\(930\) 0 0
\(931\) −18.0000 −0.589926
\(932\) −8.00000 −0.262049
\(933\) 0 0
\(934\) −8.00000 −0.261768
\(935\) 16.0000 0.523256
\(936\) 0 0
\(937\) 54.0000 1.76410 0.882052 0.471153i \(-0.156162\pi\)
0.882052 + 0.471153i \(0.156162\pi\)
\(938\) −20.0000 −0.653023
\(939\) 0 0
\(940\) −8.00000 −0.260931
\(941\) −6.00000 −0.195594 −0.0977972 0.995206i \(-0.531180\pi\)
−0.0977972 + 0.995206i \(0.531180\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) −32.0000 −1.04041
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 0 0
\(949\) 6.00000 0.194768
\(950\) 6.00000 0.194666
\(951\) 0 0
\(952\) −24.0000 −0.777844
\(953\) 48.0000 1.55487 0.777436 0.628962i \(-0.216520\pi\)
0.777436 + 0.628962i \(0.216520\pi\)
\(954\) 0 0
\(955\) 12.0000 0.388311
\(956\) 0 0
\(957\) 0 0
\(958\) 24.0000 0.775405
\(959\) −20.0000 −0.645834
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) 2.00000 0.0644826
\(963\) 0 0
\(964\) −22.0000 −0.708572
\(965\) −26.0000 −0.836970
\(966\) 0 0
\(967\) 58.0000 1.86515 0.932577 0.360971i \(-0.117555\pi\)
0.932577 + 0.360971i \(0.117555\pi\)
\(968\) −15.0000 −0.482118
\(969\) 0 0
\(970\) −14.0000 −0.449513
\(971\) 52.0000 1.66876 0.834380 0.551190i \(-0.185826\pi\)
0.834380 + 0.551190i \(0.185826\pi\)
\(972\) 0 0
\(973\) 24.0000 0.769405
\(974\) −22.0000 −0.704925
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) −54.0000 −1.72761 −0.863807 0.503824i \(-0.831926\pi\)
−0.863807 + 0.503824i \(0.831926\pi\)
\(978\) 0 0
\(979\) −56.0000 −1.78977
\(980\) 3.00000 0.0958315
\(981\) 0 0
\(982\) −36.0000 −1.14881
\(983\) 16.0000 0.510321 0.255160 0.966899i \(-0.417872\pi\)
0.255160 + 0.966899i \(0.417872\pi\)
\(984\) 0 0
\(985\) −18.0000 −0.573528
\(986\) 16.0000 0.509544
\(987\) 0 0
\(988\) 6.00000 0.190885
\(989\) 0 0
\(990\) 0 0
\(991\) −48.0000 −1.52477 −0.762385 0.647124i \(-0.775972\pi\)
−0.762385 + 0.647124i \(0.775972\pi\)
\(992\) −50.0000 −1.58750
\(993\) 0 0
\(994\) 0 0
\(995\) 8.00000 0.253617
\(996\) 0 0
\(997\) −58.0000 −1.83688 −0.918439 0.395562i \(-0.870550\pi\)
−0.918439 + 0.395562i \(0.870550\pi\)
\(998\) −38.0000 −1.20287
\(999\) 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 585.2.a.i.1.1 yes 1
3.2 odd 2 585.2.a.d.1.1 1
4.3 odd 2 9360.2.a.be.1.1 1
5.2 odd 4 2925.2.c.k.2224.2 2
5.3 odd 4 2925.2.c.k.2224.1 2
5.4 even 2 2925.2.a.c.1.1 1
12.11 even 2 9360.2.a.i.1.1 1
13.12 even 2 7605.2.a.c.1.1 1
15.2 even 4 2925.2.c.g.2224.1 2
15.8 even 4 2925.2.c.g.2224.2 2
15.14 odd 2 2925.2.a.m.1.1 1
39.38 odd 2 7605.2.a.q.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
585.2.a.d.1.1 1 3.2 odd 2
585.2.a.i.1.1 yes 1 1.1 even 1 trivial
2925.2.a.c.1.1 1 5.4 even 2
2925.2.a.m.1.1 1 15.14 odd 2
2925.2.c.g.2224.1 2 15.2 even 4
2925.2.c.g.2224.2 2 15.8 even 4
2925.2.c.k.2224.1 2 5.3 odd 4
2925.2.c.k.2224.2 2 5.2 odd 4
7605.2.a.c.1.1 1 13.12 even 2
7605.2.a.q.1.1 1 39.38 odd 2
9360.2.a.i.1.1 1 12.11 even 2
9360.2.a.be.1.1 1 4.3 odd 2