Properties

Label 5265.2.a.m.1.1
Level $5265$
Weight $2$
Character 5265.1
Self dual yes
Analytic conductor $42.041$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5265 = 3^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5265.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(42.0412366642\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 585)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5265.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} -3.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} -3.00000 q^{8} +1.00000 q^{10} +1.00000 q^{11} -1.00000 q^{13} +2.00000 q^{14} -1.00000 q^{16} -2.00000 q^{17} -3.00000 q^{19} -1.00000 q^{20} +1.00000 q^{22} +1.00000 q^{25} -1.00000 q^{26} -2.00000 q^{28} -5.00000 q^{29} -1.00000 q^{31} +5.00000 q^{32} -2.00000 q^{34} +2.00000 q^{35} -5.00000 q^{37} -3.00000 q^{38} -3.00000 q^{40} -8.00000 q^{43} -1.00000 q^{44} +2.00000 q^{47} -3.00000 q^{49} +1.00000 q^{50} +1.00000 q^{52} -14.0000 q^{53} +1.00000 q^{55} -6.00000 q^{56} -5.00000 q^{58} +9.00000 q^{59} -1.00000 q^{61} -1.00000 q^{62} +7.00000 q^{64} -1.00000 q^{65} +14.0000 q^{67} +2.00000 q^{68} +2.00000 q^{70} +6.00000 q^{73} -5.00000 q^{74} +3.00000 q^{76} +2.00000 q^{77} -12.0000 q^{79} -1.00000 q^{80} +10.0000 q^{83} -2.00000 q^{85} -8.00000 q^{86} -3.00000 q^{88} -2.00000 q^{89} -2.00000 q^{91} +2.00000 q^{94} -3.00000 q^{95} +1.00000 q^{97} -3.00000 q^{98} +O(q^{100})\)

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
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) −3.00000 −0.688247 −0.344124 0.938924i \(-0.611824\pi\)
−0.344124 + 0.938924i \(0.611824\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 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
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605 −0.0898027 0.995960i \(-0.528624\pi\)
−0.0898027 + 0.995960i \(0.528624\pi\)
\(32\) 5.00000 0.883883
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) −5.00000 −0.821995 −0.410997 0.911636i \(-0.634819\pi\)
−0.410997 + 0.911636i \(0.634819\pi\)
\(38\) −3.00000 −0.486664
\(39\) 0 0
\(40\) −3.00000 −0.474342
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 0 0
\(47\) 2.00000 0.291730 0.145865 0.989305i \(-0.453403\pi\)
0.145865 + 0.989305i \(0.453403\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 1.00000 0.138675
\(53\) −14.0000 −1.92305 −0.961524 0.274721i \(-0.911414\pi\)
−0.961524 + 0.274721i \(0.911414\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) −6.00000 −0.801784
\(57\) 0 0
\(58\) −5.00000 −0.656532
\(59\) 9.00000 1.17170 0.585850 0.810419i \(-0.300761\pi\)
0.585850 + 0.810419i \(0.300761\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) −1.00000 −0.127000
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) 14.0000 1.71037 0.855186 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) 2.00000 0.242536
\(69\) 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 \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) −5.00000 −0.581238
\(75\) 0 0
\(76\) 3.00000 0.344124
\(77\) 2.00000 0.227921
\(78\) 0 0
\(79\) −12.0000 −1.35011 −0.675053 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 0 0
\(83\) 10.0000 1.09764 0.548821 0.835940i \(-0.315077\pi\)
0.548821 + 0.835940i \(0.315077\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) −8.00000 −0.862662
\(87\) 0 0
\(88\) −3.00000 −0.319801
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 0 0
\(93\) 0 0
\(94\) 2.00000 0.206284
\(95\) −3.00000 −0.307794
\(96\) 0 0
\(97\) 1.00000 0.101535 0.0507673 0.998711i \(-0.483833\pi\)
0.0507673 + 0.998711i \(0.483833\pi\)
\(98\) −3.00000 −0.303046
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) −3.00000 −0.298511 −0.149256 0.988799i \(-0.547688\pi\)
−0.149256 + 0.988799i \(0.547688\pi\)
\(102\) 0 0
\(103\) −1.00000 −0.0985329 −0.0492665 0.998786i \(-0.515688\pi\)
−0.0492665 + 0.998786i \(0.515688\pi\)
\(104\) 3.00000 0.294174
\(105\) 0 0
\(106\) −14.0000 −1.35980
\(107\) −19.0000 −1.83680 −0.918400 0.395654i \(-0.870518\pi\)
−0.918400 + 0.395654i \(0.870518\pi\)
\(108\) 0 0
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 1.00000 0.0953463
\(111\) 0 0
\(112\) −2.00000 −0.188982
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 5.00000 0.464238
\(117\) 0 0
\(118\) 9.00000 0.828517
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) −1.00000 −0.0905357
\(123\) 0 0
\(124\) 1.00000 0.0898027
\(125\) 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
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) −6.00000 −0.520266
\(134\) 14.0000 1.20942
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) 5.00000 0.427179 0.213589 0.976924i \(-0.431485\pi\)
0.213589 + 0.976924i \(0.431485\pi\)
\(138\) 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
\(143\) −1.00000 −0.0836242
\(144\) 0 0
\(145\) −5.00000 −0.415227
\(146\) 6.00000 0.496564
\(147\) 0 0
\(148\) 5.00000 0.410997
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 9.00000 0.729996
\(153\) 0 0
\(154\) 2.00000 0.161165
\(155\) −1.00000 −0.0803219
\(156\) 0 0
\(157\) −6.00000 −0.478852 −0.239426 0.970915i \(-0.576959\pi\)
−0.239426 + 0.970915i \(0.576959\pi\)
\(158\) −12.0000 −0.954669
\(159\) 0 0
\(160\) 5.00000 0.395285
\(161\) 0 0
\(162\) 0 0
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 10.0000 0.776151
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −2.00000 −0.153393
\(171\) 0 0
\(172\) 8.00000 0.609994
\(173\) −12.0000 −0.912343 −0.456172 0.889892i \(-0.650780\pi\)
−0.456172 + 0.889892i \(0.650780\pi\)
\(174\) 0 0
\(175\) 2.00000 0.151186
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) −2.00000 −0.149906
\(179\) 18.0000 1.34538 0.672692 0.739923i \(-0.265138\pi\)
0.672692 + 0.739923i \(0.265138\pi\)
\(180\) 0 0
\(181\) 17.0000 1.26360 0.631800 0.775131i \(-0.282316\pi\)
0.631800 + 0.775131i \(0.282316\pi\)
\(182\) −2.00000 −0.148250
\(183\) 0 0
\(184\) 0 0
\(185\) −5.00000 −0.367607
\(186\) 0 0
\(187\) −2.00000 −0.146254
\(188\) −2.00000 −0.145865
\(189\) 0 0
\(190\) −3.00000 −0.217643
\(191\) −18.0000 −1.30243 −0.651217 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) 0 0
\(193\) 13.0000 0.935760 0.467880 0.883792i \(-0.345018\pi\)
0.467880 + 0.883792i \(0.345018\pi\)
\(194\) 1.00000 0.0717958
\(195\) 0 0
\(196\) 3.00000 0.214286
\(197\) −15.0000 −1.06871 −0.534353 0.845262i \(-0.679445\pi\)
−0.534353 + 0.845262i \(0.679445\pi\)
\(198\) 0 0
\(199\) −22.0000 −1.55954 −0.779769 0.626067i \(-0.784664\pi\)
−0.779769 + 0.626067i \(0.784664\pi\)
\(200\) −3.00000 −0.212132
\(201\) 0 0
\(202\) −3.00000 −0.211079
\(203\) −10.0000 −0.701862
\(204\) 0 0
\(205\) 0 0
\(206\) −1.00000 −0.0696733
\(207\) 0 0
\(208\) 1.00000 0.0693375
\(209\) −3.00000 −0.207514
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 14.0000 0.961524
\(213\) 0 0
\(214\) −19.0000 −1.29881
\(215\) −8.00000 −0.545595
\(216\) 0 0
\(217\) −2.00000 −0.135769
\(218\) −14.0000 −0.948200
\(219\) 0 0
\(220\) −1.00000 −0.0674200
\(221\) 2.00000 0.134535
\(222\) 0 0
\(223\) −26.0000 −1.74109 −0.870544 0.492090i \(-0.836233\pi\)
−0.870544 + 0.492090i \(0.836233\pi\)
\(224\) 10.0000 0.668153
\(225\) 0 0
\(226\) −10.0000 −0.665190
\(227\) 26.0000 1.72568 0.862840 0.505477i \(-0.168683\pi\)
0.862840 + 0.505477i \(0.168683\pi\)
\(228\) 0 0
\(229\) −16.0000 −1.05731 −0.528655 0.848837i \(-0.677303\pi\)
−0.528655 + 0.848837i \(0.677303\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 15.0000 0.984798
\(233\) 14.0000 0.917170 0.458585 0.888650i \(-0.348356\pi\)
0.458585 + 0.888650i \(0.348356\pi\)
\(234\) 0 0
\(235\) 2.00000 0.130466
\(236\) −9.00000 −0.585850
\(237\) 0 0
\(238\) −4.00000 −0.259281
\(239\) −15.0000 −0.970269 −0.485135 0.874439i \(-0.661229\pi\)
−0.485135 + 0.874439i \(0.661229\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) −10.0000 −0.642824
\(243\) 0 0
\(244\) 1.00000 0.0640184
\(245\) −3.00000 −0.191663
\(246\) 0 0
\(247\) 3.00000 0.190885
\(248\) 3.00000 0.190500
\(249\) 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
\(257\) 28.0000 1.74659 0.873296 0.487190i \(-0.161978\pi\)
0.873296 + 0.487190i \(0.161978\pi\)
\(258\) 0 0
\(259\) −10.0000 −0.621370
\(260\) 1.00000 0.0620174
\(261\) 0 0
\(262\) 4.00000 0.247121
\(263\) 16.0000 0.986602 0.493301 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 0 0
\(265\) −14.0000 −0.860013
\(266\) −6.00000 −0.367884
\(267\) 0 0
\(268\) −14.0000 −0.855186
\(269\) −3.00000 −0.182913 −0.0914566 0.995809i \(-0.529152\pi\)
−0.0914566 + 0.995809i \(0.529152\pi\)
\(270\) 0 0
\(271\) −7.00000 −0.425220 −0.212610 0.977137i \(-0.568196\pi\)
−0.212610 + 0.977137i \(0.568196\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) 5.00000 0.302061
\(275\) 1.00000 0.0603023
\(276\) 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
\(281\) 12.0000 0.715860 0.357930 0.933748i \(-0.383483\pi\)
0.357930 + 0.933748i \(0.383483\pi\)
\(282\) 0 0
\(283\) −25.0000 −1.48610 −0.743048 0.669238i \(-0.766621\pi\)
−0.743048 + 0.669238i \(0.766621\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −1.00000 −0.0591312
\(287\) 0 0
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) −5.00000 −0.293610
\(291\) 0 0
\(292\) −6.00000 −0.351123
\(293\) −1.00000 −0.0584206 −0.0292103 0.999573i \(-0.509299\pi\)
−0.0292103 + 0.999573i \(0.509299\pi\)
\(294\) 0 0
\(295\) 9.00000 0.524000
\(296\) 15.0000 0.871857
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) 0 0
\(300\) 0 0
\(301\) −16.0000 −0.922225
\(302\) −8.00000 −0.460348
\(303\) 0 0
\(304\) 3.00000 0.172062
\(305\) −1.00000 −0.0572598
\(306\) 0 0
\(307\) 2.00000 0.114146 0.0570730 0.998370i \(-0.481823\pi\)
0.0570730 + 0.998370i \(0.481823\pi\)
\(308\) −2.00000 −0.113961
\(309\) 0 0
\(310\) −1.00000 −0.0567962
\(311\) 4.00000 0.226819 0.113410 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) 0 0
\(313\) −28.0000 −1.58265 −0.791327 0.611393i \(-0.790609\pi\)
−0.791327 + 0.611393i \(0.790609\pi\)
\(314\) −6.00000 −0.338600
\(315\) 0 0
\(316\) 12.0000 0.675053
\(317\) 27.0000 1.51647 0.758236 0.651981i \(-0.226062\pi\)
0.758236 + 0.651981i \(0.226062\pi\)
\(318\) 0 0
\(319\) −5.00000 −0.279946
\(320\) 7.00000 0.391312
\(321\) 0 0
\(322\) 0 0
\(323\) 6.00000 0.333849
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) −12.0000 −0.664619
\(327\) 0 0
\(328\) 0 0
\(329\) 4.00000 0.220527
\(330\) 0 0
\(331\) 27.0000 1.48405 0.742027 0.670370i \(-0.233865\pi\)
0.742027 + 0.670370i \(0.233865\pi\)
\(332\) −10.0000 −0.548821
\(333\) 0 0
\(334\) 0 0
\(335\) 14.0000 0.764902
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 1.00000 0.0543928
\(339\) 0 0
\(340\) 2.00000 0.108465
\(341\) −1.00000 −0.0541530
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 24.0000 1.29399
\(345\) 0 0
\(346\) −12.0000 −0.645124
\(347\) −17.0000 −0.912608 −0.456304 0.889824i \(-0.650827\pi\)
−0.456304 + 0.889824i \(0.650827\pi\)
\(348\) 0 0
\(349\) 12.0000 0.642345 0.321173 0.947021i \(-0.395923\pi\)
0.321173 + 0.947021i \(0.395923\pi\)
\(350\) 2.00000 0.106904
\(351\) 0 0
\(352\) 5.00000 0.266501
\(353\) 3.00000 0.159674 0.0798369 0.996808i \(-0.474560\pi\)
0.0798369 + 0.996808i \(0.474560\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 2.00000 0.106000
\(357\) 0 0
\(358\) 18.0000 0.951330
\(359\) 31.0000 1.63612 0.818059 0.575135i \(-0.195050\pi\)
0.818059 + 0.575135i \(0.195050\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) 17.0000 0.893500
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) 6.00000 0.314054
\(366\) 0 0
\(367\) 25.0000 1.30499 0.652495 0.757793i \(-0.273722\pi\)
0.652495 + 0.757793i \(0.273722\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) −5.00000 −0.259938
\(371\) −28.0000 −1.45369
\(372\) 0 0
\(373\) −20.0000 −1.03556 −0.517780 0.855514i \(-0.673242\pi\)
−0.517780 + 0.855514i \(0.673242\pi\)
\(374\) −2.00000 −0.103418
\(375\) 0 0
\(376\) −6.00000 −0.309426
\(377\) 5.00000 0.257513
\(378\) 0 0
\(379\) −23.0000 −1.18143 −0.590715 0.806880i \(-0.701154\pi\)
−0.590715 + 0.806880i \(0.701154\pi\)
\(380\) 3.00000 0.153897
\(381\) 0 0
\(382\) −18.0000 −0.920960
\(383\) 12.0000 0.613171 0.306586 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) 0 0
\(385\) 2.00000 0.101929
\(386\) 13.0000 0.661683
\(387\) 0 0
\(388\) −1.00000 −0.0507673
\(389\) 22.0000 1.11544 0.557722 0.830028i \(-0.311675\pi\)
0.557722 + 0.830028i \(0.311675\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 9.00000 0.454569
\(393\) 0 0
\(394\) −15.0000 −0.755689
\(395\) −12.0000 −0.603786
\(396\) 0 0
\(397\) −30.0000 −1.50566 −0.752828 0.658217i \(-0.771311\pi\)
−0.752828 + 0.658217i \(0.771311\pi\)
\(398\) −22.0000 −1.10276
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −28.0000 −1.39825 −0.699127 0.714998i \(-0.746428\pi\)
−0.699127 + 0.714998i \(0.746428\pi\)
\(402\) 0 0
\(403\) 1.00000 0.0498135
\(404\) 3.00000 0.149256
\(405\) 0 0
\(406\) −10.0000 −0.496292
\(407\) −5.00000 −0.247841
\(408\) 0 0
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 1.00000 0.0492665
\(413\) 18.0000 0.885722
\(414\) 0 0
\(415\) 10.0000 0.490881
\(416\) −5.00000 −0.245145
\(417\) 0 0
\(418\) −3.00000 −0.146735
\(419\) −20.0000 −0.977064 −0.488532 0.872546i \(-0.662467\pi\)
−0.488532 + 0.872546i \(0.662467\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 42.0000 2.03970
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) −2.00000 −0.0967868
\(428\) 19.0000 0.918400
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) −17.0000 −0.818861 −0.409431 0.912341i \(-0.634273\pi\)
−0.409431 + 0.912341i \(0.634273\pi\)
\(432\) 0 0
\(433\) −38.0000 −1.82616 −0.913082 0.407777i \(-0.866304\pi\)
−0.913082 + 0.407777i \(0.866304\pi\)
\(434\) −2.00000 −0.0960031
\(435\) 0 0
\(436\) 14.0000 0.670478
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) −3.00000 −0.143019
\(441\) 0 0
\(442\) 2.00000 0.0951303
\(443\) 7.00000 0.332580 0.166290 0.986077i \(-0.446821\pi\)
0.166290 + 0.986077i \(0.446821\pi\)
\(444\) 0 0
\(445\) −2.00000 −0.0948091
\(446\) −26.0000 −1.23114
\(447\) 0 0
\(448\) 14.0000 0.661438
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 10.0000 0.470360
\(453\) 0 0
\(454\) 26.0000 1.22024
\(455\) −2.00000 −0.0937614
\(456\) 0 0
\(457\) 26.0000 1.21623 0.608114 0.793849i \(-0.291926\pi\)
0.608114 + 0.793849i \(0.291926\pi\)
\(458\) −16.0000 −0.747631
\(459\) 0 0
\(460\) 0 0
\(461\) 26.0000 1.21094 0.605470 0.795868i \(-0.292985\pi\)
0.605470 + 0.795868i \(0.292985\pi\)
\(462\) 0 0
\(463\) −6.00000 −0.278844 −0.139422 0.990233i \(-0.544524\pi\)
−0.139422 + 0.990233i \(0.544524\pi\)
\(464\) 5.00000 0.232119
\(465\) 0 0
\(466\) 14.0000 0.648537
\(467\) 28.0000 1.29569 0.647843 0.761774i \(-0.275671\pi\)
0.647843 + 0.761774i \(0.275671\pi\)
\(468\) 0 0
\(469\) 28.0000 1.29292
\(470\) 2.00000 0.0922531
\(471\) 0 0
\(472\) −27.0000 −1.24278
\(473\) −8.00000 −0.367840
\(474\) 0 0
\(475\) −3.00000 −0.137649
\(476\) 4.00000 0.183340
\(477\) 0 0
\(478\) −15.0000 −0.686084
\(479\) −15.0000 −0.685367 −0.342684 0.939451i \(-0.611336\pi\)
−0.342684 + 0.939451i \(0.611336\pi\)
\(480\) 0 0
\(481\) 5.00000 0.227980
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) 10.0000 0.454545
\(485\) 1.00000 0.0454077
\(486\) 0 0
\(487\) −34.0000 −1.54069 −0.770344 0.637629i \(-0.779915\pi\)
−0.770344 + 0.637629i \(0.779915\pi\)
\(488\) 3.00000 0.135804
\(489\) 0 0
\(490\) −3.00000 −0.135526
\(491\) 6.00000 0.270776 0.135388 0.990793i \(-0.456772\pi\)
0.135388 + 0.990793i \(0.456772\pi\)
\(492\) 0 0
\(493\) 10.0000 0.450377
\(494\) 3.00000 0.134976
\(495\) 0 0
\(496\) 1.00000 0.0449013
\(497\) 0 0
\(498\) 0 0
\(499\) 28.0000 1.25345 0.626726 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −12.0000 −0.535586
\(503\) 9.00000 0.401290 0.200645 0.979664i \(-0.435696\pi\)
0.200645 + 0.979664i \(0.435696\pi\)
\(504\) 0 0
\(505\) −3.00000 −0.133498
\(506\) 0 0
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) 24.0000 1.06378 0.531891 0.846813i \(-0.321482\pi\)
0.531891 + 0.846813i \(0.321482\pi\)
\(510\) 0 0
\(511\) 12.0000 0.530849
\(512\) −11.0000 −0.486136
\(513\) 0 0
\(514\) 28.0000 1.23503
\(515\) −1.00000 −0.0440653
\(516\) 0 0
\(517\) 2.00000 0.0879599
\(518\) −10.0000 −0.439375
\(519\) 0 0
\(520\) 3.00000 0.131559
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) −15.0000 −0.655904 −0.327952 0.944694i \(-0.606358\pi\)
−0.327952 + 0.944694i \(0.606358\pi\)
\(524\) −4.00000 −0.174741
\(525\) 0 0
\(526\) 16.0000 0.697633
\(527\) 2.00000 0.0871214
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) −14.0000 −0.608121
\(531\) 0 0
\(532\) 6.00000 0.260133
\(533\) 0 0
\(534\) 0 0
\(535\) −19.0000 −0.821442
\(536\) −42.0000 −1.81412
\(537\) 0 0
\(538\) −3.00000 −0.129339
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) 32.0000 1.37579 0.687894 0.725811i \(-0.258536\pi\)
0.687894 + 0.725811i \(0.258536\pi\)
\(542\) −7.00000 −0.300676
\(543\) 0 0
\(544\) −10.0000 −0.428746
\(545\) −14.0000 −0.599694
\(546\) 0 0
\(547\) −15.0000 −0.641354 −0.320677 0.947189i \(-0.603910\pi\)
−0.320677 + 0.947189i \(0.603910\pi\)
\(548\) −5.00000 −0.213589
\(549\) 0 0
\(550\) 1.00000 0.0426401
\(551\) 15.0000 0.639021
\(552\) 0 0
\(553\) −24.0000 −1.02058
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) −12.0000 −0.508913
\(557\) 5.00000 0.211857 0.105928 0.994374i \(-0.466219\pi\)
0.105928 + 0.994374i \(0.466219\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) −2.00000 −0.0845154
\(561\) 0 0
\(562\) 12.0000 0.506189
\(563\) 19.0000 0.800755 0.400377 0.916350i \(-0.368879\pi\)
0.400377 + 0.916350i \(0.368879\pi\)
\(564\) 0 0
\(565\) −10.0000 −0.420703
\(566\) −25.0000 −1.05083
\(567\) 0 0
\(568\) 0 0
\(569\) 11.0000 0.461144 0.230572 0.973055i \(-0.425940\pi\)
0.230572 + 0.973055i \(0.425940\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) 1.00000 0.0418121
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 11.0000 0.457936 0.228968 0.973434i \(-0.426465\pi\)
0.228968 + 0.973434i \(0.426465\pi\)
\(578\) −13.0000 −0.540729
\(579\) 0 0
\(580\) 5.00000 0.207614
\(581\) 20.0000 0.829740
\(582\) 0 0
\(583\) −14.0000 −0.579821
\(584\) −18.0000 −0.744845
\(585\) 0 0
\(586\) −1.00000 −0.0413096
\(587\) 14.0000 0.577842 0.288921 0.957353i \(-0.406704\pi\)
0.288921 + 0.957353i \(0.406704\pi\)
\(588\) 0 0
\(589\) 3.00000 0.123613
\(590\) 9.00000 0.370524
\(591\) 0 0
\(592\) 5.00000 0.205499
\(593\) 31.0000 1.27302 0.636509 0.771270i \(-0.280378\pi\)
0.636509 + 0.771270i \(0.280378\pi\)
\(594\) 0 0
\(595\) −4.00000 −0.163984
\(596\) 6.00000 0.245770
\(597\) 0 0
\(598\) 0 0
\(599\) −14.0000 −0.572024 −0.286012 0.958226i \(-0.592330\pi\)
−0.286012 + 0.958226i \(0.592330\pi\)
\(600\) 0 0
\(601\) −17.0000 −0.693444 −0.346722 0.937968i \(-0.612705\pi\)
−0.346722 + 0.937968i \(0.612705\pi\)
\(602\) −16.0000 −0.652111
\(603\) 0 0
\(604\) 8.00000 0.325515
\(605\) −10.0000 −0.406558
\(606\) 0 0
\(607\) −8.00000 −0.324710 −0.162355 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) −15.0000 −0.608330
\(609\) 0 0
\(610\) −1.00000 −0.0404888
\(611\) −2.00000 −0.0809113
\(612\) 0 0
\(613\) 30.0000 1.21169 0.605844 0.795583i \(-0.292835\pi\)
0.605844 + 0.795583i \(0.292835\pi\)
\(614\) 2.00000 0.0807134
\(615\) 0 0
\(616\) −6.00000 −0.241747
\(617\) −9.00000 −0.362326 −0.181163 0.983453i \(-0.557986\pi\)
−0.181163 + 0.983453i \(0.557986\pi\)
\(618\) 0 0
\(619\) −11.0000 −0.442127 −0.221064 0.975259i \(-0.570953\pi\)
−0.221064 + 0.975259i \(0.570953\pi\)
\(620\) 1.00000 0.0401610
\(621\) 0 0
\(622\) 4.00000 0.160385
\(623\) −4.00000 −0.160257
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −28.0000 −1.11911
\(627\) 0 0
\(628\) 6.00000 0.239426
\(629\) 10.0000 0.398726
\(630\) 0 0
\(631\) −5.00000 −0.199047 −0.0995234 0.995035i \(-0.531732\pi\)
−0.0995234 + 0.995035i \(0.531732\pi\)
\(632\) 36.0000 1.43200
\(633\) 0 0
\(634\) 27.0000 1.07231
\(635\) −8.00000 −0.317470
\(636\) 0 0
\(637\) 3.00000 0.118864
\(638\) −5.00000 −0.197952
\(639\) 0 0
\(640\) −3.00000 −0.118585
\(641\) −17.0000 −0.671460 −0.335730 0.941958i \(-0.608983\pi\)
−0.335730 + 0.941958i \(0.608983\pi\)
\(642\) 0 0
\(643\) 32.0000 1.26196 0.630978 0.775800i \(-0.282654\pi\)
0.630978 + 0.775800i \(0.282654\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 6.00000 0.236067
\(647\) −41.0000 −1.61188 −0.805938 0.592000i \(-0.798339\pi\)
−0.805938 + 0.592000i \(0.798339\pi\)
\(648\) 0 0
\(649\) 9.00000 0.353281
\(650\) −1.00000 −0.0392232
\(651\) 0 0
\(652\) 12.0000 0.469956
\(653\) −48.0000 −1.87839 −0.939193 0.343391i \(-0.888424\pi\)
−0.939193 + 0.343391i \(0.888424\pi\)
\(654\) 0 0
\(655\) 4.00000 0.156293
\(656\) 0 0
\(657\) 0 0
\(658\) 4.00000 0.155936
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) 38.0000 1.47803 0.739014 0.673690i \(-0.235292\pi\)
0.739014 + 0.673690i \(0.235292\pi\)
\(662\) 27.0000 1.04938
\(663\) 0 0
\(664\) −30.0000 −1.16423
\(665\) −6.00000 −0.232670
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 14.0000 0.540867
\(671\) −1.00000 −0.0386046
\(672\) 0 0
\(673\) 24.0000 0.925132 0.462566 0.886585i \(-0.346929\pi\)
0.462566 + 0.886585i \(0.346929\pi\)
\(674\) −22.0000 −0.847408
\(675\) 0 0
\(676\) −1.00000 −0.0384615
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 0 0
\(679\) 2.00000 0.0767530
\(680\) 6.00000 0.230089
\(681\) 0 0
\(682\) −1.00000 −0.0382920
\(683\) 20.0000 0.765279 0.382639 0.923898i \(-0.375015\pi\)
0.382639 + 0.923898i \(0.375015\pi\)
\(684\) 0 0
\(685\) 5.00000 0.191040
\(686\) −20.0000 −0.763604
\(687\) 0 0
\(688\) 8.00000 0.304997
\(689\) 14.0000 0.533358
\(690\) 0 0
\(691\) 1.00000 0.0380418 0.0190209 0.999819i \(-0.493945\pi\)
0.0190209 + 0.999819i \(0.493945\pi\)
\(692\) 12.0000 0.456172
\(693\) 0 0
\(694\) −17.0000 −0.645311
\(695\) 12.0000 0.455186
\(696\) 0 0
\(697\) 0 0
\(698\) 12.0000 0.454207
\(699\) 0 0
\(700\) −2.00000 −0.0755929
\(701\) 5.00000 0.188847 0.0944237 0.995532i \(-0.469899\pi\)
0.0944237 + 0.995532i \(0.469899\pi\)
\(702\) 0 0
\(703\) 15.0000 0.565736
\(704\) 7.00000 0.263822
\(705\) 0 0
\(706\) 3.00000 0.112906
\(707\) −6.00000 −0.225653
\(708\) 0 0
\(709\) 4.00000 0.150223 0.0751116 0.997175i \(-0.476069\pi\)
0.0751116 + 0.997175i \(0.476069\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 6.00000 0.224860
\(713\) 0 0
\(714\) 0 0
\(715\) −1.00000 −0.0373979
\(716\) −18.0000 −0.672692
\(717\) 0 0
\(718\) 31.0000 1.15691
\(719\) 2.00000 0.0745874 0.0372937 0.999304i \(-0.488126\pi\)
0.0372937 + 0.999304i \(0.488126\pi\)
\(720\) 0 0
\(721\) −2.00000 −0.0744839
\(722\) −10.0000 −0.372161
\(723\) 0 0
\(724\) −17.0000 −0.631800
\(725\) −5.00000 −0.185695
\(726\) 0 0
\(727\) 48.0000 1.78022 0.890111 0.455744i \(-0.150627\pi\)
0.890111 + 0.455744i \(0.150627\pi\)
\(728\) 6.00000 0.222375
\(729\) 0 0
\(730\) 6.00000 0.222070
\(731\) 16.0000 0.591781
\(732\) 0 0
\(733\) 13.0000 0.480166 0.240083 0.970752i \(-0.422825\pi\)
0.240083 + 0.970752i \(0.422825\pi\)
\(734\) 25.0000 0.922767
\(735\) 0 0
\(736\) 0 0
\(737\) 14.0000 0.515697
\(738\) 0 0
\(739\) 9.00000 0.331070 0.165535 0.986204i \(-0.447065\pi\)
0.165535 + 0.986204i \(0.447065\pi\)
\(740\) 5.00000 0.183804
\(741\) 0 0
\(742\) −28.0000 −1.02791
\(743\) 48.0000 1.76095 0.880475 0.474093i \(-0.157224\pi\)
0.880475 + 0.474093i \(0.157224\pi\)
\(744\) 0 0
\(745\) −6.00000 −0.219823
\(746\) −20.0000 −0.732252
\(747\) 0 0
\(748\) 2.00000 0.0731272
\(749\) −38.0000 −1.38849
\(750\) 0 0
\(751\) 2.00000 0.0729810 0.0364905 0.999334i \(-0.488382\pi\)
0.0364905 + 0.999334i \(0.488382\pi\)
\(752\) −2.00000 −0.0729325
\(753\) 0 0
\(754\) 5.00000 0.182089
\(755\) −8.00000 −0.291150
\(756\) 0 0
\(757\) −16.0000 −0.581530 −0.290765 0.956795i \(-0.593910\pi\)
−0.290765 + 0.956795i \(0.593910\pi\)
\(758\) −23.0000 −0.835398
\(759\) 0 0
\(760\) 9.00000 0.326464
\(761\) 36.0000 1.30500 0.652499 0.757789i \(-0.273720\pi\)
0.652499 + 0.757789i \(0.273720\pi\)
\(762\) 0 0
\(763\) −28.0000 −1.01367
\(764\) 18.0000 0.651217
\(765\) 0 0
\(766\) 12.0000 0.433578
\(767\) −9.00000 −0.324971
\(768\) 0 0
\(769\) −40.0000 −1.44244 −0.721218 0.692708i \(-0.756418\pi\)
−0.721218 + 0.692708i \(0.756418\pi\)
\(770\) 2.00000 0.0720750
\(771\) 0 0
\(772\) −13.0000 −0.467880
\(773\) −31.0000 −1.11499 −0.557496 0.830179i \(-0.688238\pi\)
−0.557496 + 0.830179i \(0.688238\pi\)
\(774\) 0 0
\(775\) −1.00000 −0.0359211
\(776\) −3.00000 −0.107694
\(777\) 0 0
\(778\) 22.0000 0.788738
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 3.00000 0.107143
\(785\) −6.00000 −0.214149
\(786\) 0 0
\(787\) 38.0000 1.35455 0.677277 0.735728i \(-0.263160\pi\)
0.677277 + 0.735728i \(0.263160\pi\)
\(788\) 15.0000 0.534353
\(789\) 0 0
\(790\) −12.0000 −0.426941
\(791\) −20.0000 −0.711118
\(792\) 0 0
\(793\) 1.00000 0.0355110
\(794\) −30.0000 −1.06466
\(795\) 0 0
\(796\) 22.0000 0.779769
\(797\) 42.0000 1.48772 0.743858 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) 0 0
\(799\) −4.00000 −0.141510
\(800\) 5.00000 0.176777
\(801\) 0 0
\(802\) −28.0000 −0.988714
\(803\) 6.00000 0.211735
\(804\) 0 0
\(805\) 0 0
\(806\) 1.00000 0.0352235
\(807\) 0 0
\(808\) 9.00000 0.316619
\(809\) 5.00000 0.175791 0.0878953 0.996130i \(-0.471986\pi\)
0.0878953 + 0.996130i \(0.471986\pi\)
\(810\) 0 0
\(811\) −36.0000 −1.26413 −0.632065 0.774915i \(-0.717793\pi\)
−0.632065 + 0.774915i \(0.717793\pi\)
\(812\) 10.0000 0.350931
\(813\) 0 0
\(814\) −5.00000 −0.175250
\(815\) −12.0000 −0.420342
\(816\) 0 0
\(817\) 24.0000 0.839654
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −32.0000 −1.11681 −0.558404 0.829569i \(-0.688586\pi\)
−0.558404 + 0.829569i \(0.688586\pi\)
\(822\) 0 0
\(823\) 11.0000 0.383436 0.191718 0.981450i \(-0.438594\pi\)
0.191718 + 0.981450i \(0.438594\pi\)
\(824\) 3.00000 0.104510
\(825\) 0 0
\(826\) 18.0000 0.626300
\(827\) 44.0000 1.53003 0.765015 0.644013i \(-0.222732\pi\)
0.765015 + 0.644013i \(0.222732\pi\)
\(828\) 0 0
\(829\) 34.0000 1.18087 0.590434 0.807086i \(-0.298956\pi\)
0.590434 + 0.807086i \(0.298956\pi\)
\(830\) 10.0000 0.347105
\(831\) 0 0
\(832\) −7.00000 −0.242681
\(833\) 6.00000 0.207888
\(834\) 0 0
\(835\) 0 0
\(836\) 3.00000 0.103757
\(837\) 0 0
\(838\) −20.0000 −0.690889
\(839\) −5.00000 −0.172619 −0.0863096 0.996268i \(-0.527507\pi\)
−0.0863096 + 0.996268i \(0.527507\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 10.0000 0.344623
\(843\) 0 0
\(844\) 0 0
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) −20.0000 −0.687208
\(848\) 14.0000 0.480762
\(849\) 0 0
\(850\) −2.00000 −0.0685994
\(851\) 0 0
\(852\) 0 0
\(853\) −54.0000 −1.84892 −0.924462 0.381273i \(-0.875486\pi\)
−0.924462 + 0.381273i \(0.875486\pi\)
\(854\) −2.00000 −0.0684386
\(855\) 0 0
\(856\) 57.0000 1.94822
\(857\) −24.0000 −0.819824 −0.409912 0.912125i \(-0.634441\pi\)
−0.409912 + 0.912125i \(0.634441\pi\)
\(858\) 0 0
\(859\) 28.0000 0.955348 0.477674 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) 8.00000 0.272798
\(861\) 0 0
\(862\) −17.0000 −0.579022
\(863\) 30.0000 1.02121 0.510606 0.859815i \(-0.329421\pi\)
0.510606 + 0.859815i \(0.329421\pi\)
\(864\) 0 0
\(865\) −12.0000 −0.408012
\(866\) −38.0000 −1.29129
\(867\) 0 0
\(868\) 2.00000 0.0678844
\(869\) −12.0000 −0.407072
\(870\) 0 0
\(871\) −14.0000 −0.474372
\(872\) 42.0000 1.42230
\(873\) 0 0
\(874\) 0 0
\(875\) 2.00000 0.0676123
\(876\) 0 0
\(877\) 15.0000 0.506514 0.253257 0.967399i \(-0.418498\pi\)
0.253257 + 0.967399i \(0.418498\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) −1.00000 −0.0337100
\(881\) −23.0000 −0.774890 −0.387445 0.921893i \(-0.626642\pi\)
−0.387445 + 0.921893i \(0.626642\pi\)
\(882\) 0 0
\(883\) 47.0000 1.58168 0.790838 0.612026i \(-0.209645\pi\)
0.790838 + 0.612026i \(0.209645\pi\)
\(884\) −2.00000 −0.0672673
\(885\) 0 0
\(886\) 7.00000 0.235170
\(887\) 23.0000 0.772264 0.386132 0.922443i \(-0.373811\pi\)
0.386132 + 0.922443i \(0.373811\pi\)
\(888\) 0 0
\(889\) −16.0000 −0.536623
\(890\) −2.00000 −0.0670402
\(891\) 0 0
\(892\) 26.0000 0.870544
\(893\) −6.00000 −0.200782
\(894\) 0 0
\(895\) 18.0000 0.601674
\(896\) −6.00000 −0.200446
\(897\) 0 0
\(898\) 30.0000 1.00111
\(899\) 5.00000 0.166759
\(900\) 0 0
\(901\) 28.0000 0.932815
\(902\) 0 0
\(903\) 0 0
\(904\) 30.0000 0.997785
\(905\) 17.0000 0.565099
\(906\) 0 0
\(907\) −1.00000 −0.0332045 −0.0166022 0.999862i \(-0.505285\pi\)
−0.0166022 + 0.999862i \(0.505285\pi\)
\(908\) −26.0000 −0.862840
\(909\) 0 0
\(910\) −2.00000 −0.0662994
\(911\) −42.0000 −1.39152 −0.695761 0.718273i \(-0.744933\pi\)
−0.695761 + 0.718273i \(0.744933\pi\)
\(912\) 0 0
\(913\) 10.0000 0.330952
\(914\) 26.0000 0.860004
\(915\) 0 0
\(916\) 16.0000 0.528655
\(917\) 8.00000 0.264183
\(918\) 0 0
\(919\) 38.0000 1.25350 0.626752 0.779219i \(-0.284384\pi\)
0.626752 + 0.779219i \(0.284384\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 26.0000 0.856264
\(923\) 0 0
\(924\) 0 0
\(925\) −5.00000 −0.164399
\(926\) −6.00000 −0.197172
\(927\) 0 0
\(928\) −25.0000 −0.820665
\(929\) 56.0000 1.83730 0.918650 0.395072i \(-0.129280\pi\)
0.918650 + 0.395072i \(0.129280\pi\)
\(930\) 0 0
\(931\) 9.00000 0.294963
\(932\) −14.0000 −0.458585
\(933\) 0 0
\(934\) 28.0000 0.916188
\(935\) −2.00000 −0.0654070
\(936\) 0 0
\(937\) 60.0000 1.96011 0.980057 0.198715i \(-0.0636769\pi\)
0.980057 + 0.198715i \(0.0636769\pi\)
\(938\) 28.0000 0.914232
\(939\) 0 0
\(940\) −2.00000 −0.0652328
\(941\) 36.0000 1.17357 0.586783 0.809744i \(-0.300394\pi\)
0.586783 + 0.809744i \(0.300394\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −9.00000 −0.292925
\(945\) 0 0
\(946\) −8.00000 −0.260102
\(947\) −24.0000 −0.779895 −0.389948 0.920837i \(-0.627507\pi\)
−0.389948 + 0.920837i \(0.627507\pi\)
\(948\) 0 0
\(949\) −6.00000 −0.194768
\(950\) −3.00000 −0.0973329
\(951\) 0 0
\(952\) 12.0000 0.388922
\(953\) 48.0000 1.55487 0.777436 0.628962i \(-0.216520\pi\)
0.777436 + 0.628962i \(0.216520\pi\)
\(954\) 0 0
\(955\) −18.0000 −0.582466
\(956\) 15.0000 0.485135
\(957\) 0 0
\(958\) −15.0000 −0.484628
\(959\) 10.0000 0.322917
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) 5.00000 0.161206
\(963\) 0 0
\(964\) −10.0000 −0.322078
\(965\) 13.0000 0.418485
\(966\) 0 0
\(967\) −50.0000 −1.60789 −0.803946 0.594703i \(-0.797270\pi\)
−0.803946 + 0.594703i \(0.797270\pi\)
\(968\) 30.0000 0.964237
\(969\) 0 0
\(970\) 1.00000 0.0321081
\(971\) −50.0000 −1.60458 −0.802288 0.596937i \(-0.796384\pi\)
−0.802288 + 0.596937i \(0.796384\pi\)
\(972\) 0 0
\(973\) 24.0000 0.769405
\(974\) −34.0000 −1.08943
\(975\) 0 0
\(976\) 1.00000 0.0320092
\(977\) 9.00000 0.287936 0.143968 0.989582i \(-0.454014\pi\)
0.143968 + 0.989582i \(0.454014\pi\)
\(978\) 0 0
\(979\) −2.00000 −0.0639203
\(980\) 3.00000 0.0958315
\(981\) 0 0
\(982\) 6.00000 0.191468
\(983\) 4.00000 0.127580 0.0637901 0.997963i \(-0.479681\pi\)
0.0637901 + 0.997963i \(0.479681\pi\)
\(984\) 0 0
\(985\) −15.0000 −0.477940
\(986\) 10.0000 0.318465
\(987\) 0 0
\(988\) −3.00000 −0.0954427
\(989\) 0 0
\(990\) 0 0
\(991\) −18.0000 −0.571789 −0.285894 0.958261i \(-0.592291\pi\)
−0.285894 + 0.958261i \(0.592291\pi\)
\(992\) −5.00000 −0.158750
\(993\) 0 0
\(994\) 0 0
\(995\) −22.0000 −0.697447
\(996\) 0 0
\(997\) −40.0000 −1.26681 −0.633406 0.773819i \(-0.718344\pi\)
−0.633406 + 0.773819i \(0.718344\pi\)
\(998\) 28.0000 0.886325
\(999\) 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 5265.2.a.m.1.1 1
3.2 odd 2 5265.2.a.e.1.1 1
9.2 odd 6 585.2.i.b.391.1 yes 2
9.4 even 3 1755.2.i.b.586.1 2
9.5 odd 6 585.2.i.b.196.1 2
9.7 even 3 1755.2.i.b.1171.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
585.2.i.b.196.1 2 9.5 odd 6
585.2.i.b.391.1 yes 2 9.2 odd 6
1755.2.i.b.586.1 2 9.4 even 3
1755.2.i.b.1171.1 2 9.7 even 3
5265.2.a.e.1.1 1 3.2 odd 2
5265.2.a.m.1.1 1 1.1 even 1 trivial