Properties

Label 1815.2.a.c.1.1
Level $1815$
Weight $2$
Character 1815.1
Self dual yes
Analytic conductor $14.493$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1815.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.4928479669\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 1815.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{12} -2.00000 q^{13} -1.00000 q^{15} +4.00000 q^{16} -6.00000 q^{17} +7.00000 q^{19} +2.00000 q^{20} +1.00000 q^{21} -6.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -2.00000 q^{28} -1.00000 q^{31} -1.00000 q^{35} -2.00000 q^{36} -7.00000 q^{37} -2.00000 q^{39} +6.00000 q^{41} -8.00000 q^{43} -1.00000 q^{45} +4.00000 q^{48} -6.00000 q^{49} -6.00000 q^{51} +4.00000 q^{52} -6.00000 q^{53} +7.00000 q^{57} -12.0000 q^{59} +2.00000 q^{60} +1.00000 q^{61} +1.00000 q^{63} -8.00000 q^{64} +2.00000 q^{65} -7.00000 q^{67} +12.0000 q^{68} -6.00000 q^{69} +6.00000 q^{71} +13.0000 q^{73} +1.00000 q^{75} -14.0000 q^{76} -11.0000 q^{79} -4.00000 q^{80} +1.00000 q^{81} -2.00000 q^{84} +6.00000 q^{85} -18.0000 q^{89} -2.00000 q^{91} +12.0000 q^{92} -1.00000 q^{93} -7.00000 q^{95} -1.00000 q^{97} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 1.00000 0.577350
\(4\) −2.00000 −1.00000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964 0.188982 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0
\(12\) −2.00000 −0.577350
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 4.00000 1.00000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) 7.00000 1.60591 0.802955 0.596040i \(-0.203260\pi\)
0.802955 + 0.596040i \(0.203260\pi\)
\(20\) 2.00000 0.447214
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −2.00000 −0.377964
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605 −0.0898027 0.995960i \(-0.528624\pi\)
−0.0898027 + 0.995960i \(0.528624\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) −2.00000 −0.333333
\(37\) −7.00000 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) 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 \(0.708828\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 4.00000 0.577350
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 4.00000 0.554700
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 7.00000 0.927173
\(58\) 0 0
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 2.00000 0.258199
\(61\) 1.00000 0.128037 0.0640184 0.997949i \(-0.479608\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) −8.00000 −1.00000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) −7.00000 −0.855186 −0.427593 0.903971i \(-0.640638\pi\)
−0.427593 + 0.903971i \(0.640638\pi\)
\(68\) 12.0000 1.45521
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) 13.0000 1.52153 0.760767 0.649025i \(-0.224823\pi\)
0.760767 + 0.649025i \(0.224823\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) −14.0000 −1.60591
\(77\) 0 0
\(78\) 0 0
\(79\) −11.0000 −1.23760 −0.618798 0.785550i \(-0.712380\pi\)
−0.618798 + 0.785550i \(0.712380\pi\)
\(80\) −4.00000 −0.447214
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) −2.00000 −0.218218
\(85\) 6.00000 0.650791
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −18.0000 −1.90800 −0.953998 0.299813i \(-0.903076\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 12.0000 1.25109
\(93\) −1.00000 −0.103695
\(94\) 0 0
\(95\) −7.00000 −0.718185
\(96\) 0 0
\(97\) −1.00000 −0.101535 −0.0507673 0.998711i \(-0.516167\pi\)
−0.0507673 + 0.998711i \(0.516167\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −2.00000 −0.200000
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) 0 0
\(103\) −1.00000 −0.0985329 −0.0492665 0.998786i \(-0.515688\pi\)
−0.0492665 + 0.998786i \(0.515688\pi\)
\(104\) 0 0
\(105\) −1.00000 −0.0975900
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) −2.00000 −0.192450
\(109\) 7.00000 0.670478 0.335239 0.942133i \(-0.391183\pi\)
0.335239 + 0.942133i \(0.391183\pi\)
\(110\) 0 0
\(111\) −7.00000 −0.664411
\(112\) 4.00000 0.377964
\(113\) −12.0000 −1.12887 −0.564433 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) 0 0
\(115\) 6.00000 0.559503
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) −6.00000 −0.550019
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 6.00000 0.541002
\(124\) 2.00000 0.179605
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −17.0000 −1.50851 −0.754253 0.656584i \(-0.772001\pi\)
−0.754253 + 0.656584i \(0.772001\pi\)
\(128\) 0 0
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) −18.0000 −1.57267 −0.786334 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) 0 0
\(133\) 7.00000 0.606977
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 2.00000 0.169031
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 4.00000 0.333333
\(145\) 0 0
\(146\) 0 0
\(147\) −6.00000 −0.494872
\(148\) 14.0000 1.15079
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 0 0
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) 1.00000 0.0803219
\(156\) 4.00000 0.320256
\(157\) 17.0000 1.35675 0.678374 0.734717i \(-0.262685\pi\)
0.678374 + 0.734717i \(0.262685\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) −6.00000 −0.472866
\(162\) 0 0
\(163\) −1.00000 −0.0783260 −0.0391630 0.999233i \(-0.512469\pi\)
−0.0391630 + 0.999233i \(0.512469\pi\)
\(164\) −12.0000 −0.937043
\(165\) 0 0
\(166\) 0 0
\(167\) 6.00000 0.464294 0.232147 0.972681i \(-0.425425\pi\)
0.232147 + 0.972681i \(0.425425\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 7.00000 0.535303
\(172\) 16.0000 1.21999
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −12.0000 −0.901975
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 2.00000 0.149071
\(181\) 17.0000 1.26360 0.631800 0.775131i \(-0.282316\pi\)
0.631800 + 0.775131i \(0.282316\pi\)
\(182\) 0 0
\(183\) 1.00000 0.0739221
\(184\) 0 0
\(185\) 7.00000 0.514650
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) −8.00000 −0.577350
\(193\) −5.00000 −0.359908 −0.179954 0.983675i \(-0.557595\pi\)
−0.179954 + 0.983675i \(0.557595\pi\)
\(194\) 0 0
\(195\) 2.00000 0.143223
\(196\) 12.0000 0.857143
\(197\) 24.0000 1.70993 0.854965 0.518686i \(-0.173579\pi\)
0.854965 + 0.518686i \(0.173579\pi\)
\(198\) 0 0
\(199\) 11.0000 0.779769 0.389885 0.920864i \(-0.372515\pi\)
0.389885 + 0.920864i \(0.372515\pi\)
\(200\) 0 0
\(201\) −7.00000 −0.493742
\(202\) 0 0
\(203\) 0 0
\(204\) 12.0000 0.840168
\(205\) −6.00000 −0.419058
\(206\) 0 0
\(207\) −6.00000 −0.417029
\(208\) −8.00000 −0.554700
\(209\) 0 0
\(210\) 0 0
\(211\) −23.0000 −1.58339 −0.791693 0.610920i \(-0.790800\pi\)
−0.791693 + 0.610920i \(0.790800\pi\)
\(212\) 12.0000 0.824163
\(213\) 6.00000 0.411113
\(214\) 0 0
\(215\) 8.00000 0.545595
\(216\) 0 0
\(217\) −1.00000 −0.0678844
\(218\) 0 0
\(219\) 13.0000 0.878459
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) 29.0000 1.94198 0.970992 0.239113i \(-0.0768565\pi\)
0.970992 + 0.239113i \(0.0768565\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) −14.0000 −0.927173
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −24.0000 −1.57229 −0.786146 0.618041i \(-0.787927\pi\)
−0.786146 + 0.618041i \(0.787927\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 24.0000 1.56227
\(237\) −11.0000 −0.714527
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) −4.00000 −0.258199
\(241\) 22.0000 1.41714 0.708572 0.705638i \(-0.249340\pi\)
0.708572 + 0.705638i \(0.249340\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) −2.00000 −0.128037
\(245\) 6.00000 0.383326
\(246\) 0 0
\(247\) −14.0000 −0.890799
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 6.00000 0.378717 0.189358 0.981908i \(-0.439359\pi\)
0.189358 + 0.981908i \(0.439359\pi\)
\(252\) −2.00000 −0.125988
\(253\) 0 0
\(254\) 0 0
\(255\) 6.00000 0.375735
\(256\) 16.0000 1.00000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 0 0
\(259\) −7.00000 −0.434959
\(260\) −4.00000 −0.248069
\(261\) 0 0
\(262\) 0 0
\(263\) 6.00000 0.369976 0.184988 0.982741i \(-0.440775\pi\)
0.184988 + 0.982741i \(0.440775\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) −18.0000 −1.10158
\(268\) 14.0000 0.855186
\(269\) −24.0000 −1.46331 −0.731653 0.681677i \(-0.761251\pi\)
−0.731653 + 0.681677i \(0.761251\pi\)
\(270\) 0 0
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) −24.0000 −1.45521
\(273\) −2.00000 −0.121046
\(274\) 0 0
\(275\) 0 0
\(276\) 12.0000 0.722315
\(277\) −11.0000 −0.660926 −0.330463 0.943819i \(-0.607205\pi\)
−0.330463 + 0.943819i \(0.607205\pi\)
\(278\) 0 0
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) 30.0000 1.78965 0.894825 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 0 0
\(283\) 1.00000 0.0594438 0.0297219 0.999558i \(-0.490538\pi\)
0.0297219 + 0.999558i \(0.490538\pi\)
\(284\) −12.0000 −0.712069
\(285\) −7.00000 −0.414644
\(286\) 0 0
\(287\) 6.00000 0.354169
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) −1.00000 −0.0586210
\(292\) −26.0000 −1.52153
\(293\) −18.0000 −1.05157 −0.525786 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) 0 0
\(295\) 12.0000 0.698667
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 12.0000 0.693978
\(300\) −2.00000 −0.115470
\(301\) −8.00000 −0.461112
\(302\) 0 0
\(303\) 12.0000 0.689382
\(304\) 28.0000 1.60591
\(305\) −1.00000 −0.0572598
\(306\) 0 0
\(307\) 13.0000 0.741949 0.370975 0.928643i \(-0.379024\pi\)
0.370975 + 0.928643i \(0.379024\pi\)
\(308\) 0 0
\(309\) −1.00000 −0.0568880
\(310\) 0 0
\(311\) −30.0000 −1.70114 −0.850572 0.525859i \(-0.823744\pi\)
−0.850572 + 0.525859i \(0.823744\pi\)
\(312\) 0 0
\(313\) −22.0000 −1.24351 −0.621757 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) 0 0
\(315\) −1.00000 −0.0563436
\(316\) 22.0000 1.23760
\(317\) −6.00000 −0.336994 −0.168497 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 8.00000 0.447214
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) −42.0000 −2.33694
\(324\) −2.00000 −0.111111
\(325\) −2.00000 −0.110940
\(326\) 0 0
\(327\) 7.00000 0.387101
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 29.0000 1.59398 0.796992 0.603990i \(-0.206423\pi\)
0.796992 + 0.603990i \(0.206423\pi\)
\(332\) 0 0
\(333\) −7.00000 −0.383598
\(334\) 0 0
\(335\) 7.00000 0.382451
\(336\) 4.00000 0.218218
\(337\) 13.0000 0.708155 0.354078 0.935216i \(-0.384795\pi\)
0.354078 + 0.935216i \(0.384795\pi\)
\(338\) 0 0
\(339\) −12.0000 −0.651751
\(340\) −12.0000 −0.650791
\(341\) 0 0
\(342\) 0 0
\(343\) −13.0000 −0.701934
\(344\) 0 0
\(345\) 6.00000 0.323029
\(346\) 0 0
\(347\) 30.0000 1.61048 0.805242 0.592946i \(-0.202035\pi\)
0.805242 + 0.592946i \(0.202035\pi\)
\(348\) 0 0
\(349\) −35.0000 −1.87351 −0.936754 0.349990i \(-0.886185\pi\)
−0.936754 + 0.349990i \(0.886185\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) −6.00000 −0.318447
\(356\) 36.0000 1.90800
\(357\) −6.00000 −0.317554
\(358\) 0 0
\(359\) −6.00000 −0.316668 −0.158334 0.987386i \(-0.550612\pi\)
−0.158334 + 0.987386i \(0.550612\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) 0 0
\(363\) 0 0
\(364\) 4.00000 0.209657
\(365\) −13.0000 −0.680451
\(366\) 0 0
\(367\) −4.00000 −0.208798 −0.104399 0.994535i \(-0.533292\pi\)
−0.104399 + 0.994535i \(0.533292\pi\)
\(368\) −24.0000 −1.25109
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) −6.00000 −0.311504
\(372\) 2.00000 0.103695
\(373\) 31.0000 1.60512 0.802560 0.596572i \(-0.203471\pi\)
0.802560 + 0.596572i \(0.203471\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 14.0000 0.718185
\(381\) −17.0000 −0.870936
\(382\) 0 0
\(383\) 6.00000 0.306586 0.153293 0.988181i \(-0.451012\pi\)
0.153293 + 0.988181i \(0.451012\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −8.00000 −0.406663
\(388\) 2.00000 0.101535
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) 36.0000 1.82060
\(392\) 0 0
\(393\) −18.0000 −0.907980
\(394\) 0 0
\(395\) 11.0000 0.553470
\(396\) 0 0
\(397\) −7.00000 −0.351320 −0.175660 0.984451i \(-0.556206\pi\)
−0.175660 + 0.984451i \(0.556206\pi\)
\(398\) 0 0
\(399\) 7.00000 0.350438
\(400\) 4.00000 0.200000
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 0 0
\(403\) 2.00000 0.0996271
\(404\) −24.0000 −1.19404
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −5.00000 −0.247234 −0.123617 0.992330i \(-0.539449\pi\)
−0.123617 + 0.992330i \(0.539449\pi\)
\(410\) 0 0
\(411\) −12.0000 −0.591916
\(412\) 2.00000 0.0985329
\(413\) −12.0000 −0.590481
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 4.00000 0.195881
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 2.00000 0.0975900
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −6.00000 −0.291043
\(426\) 0 0
\(427\) 1.00000 0.0483934
\(428\) 24.0000 1.16008
\(429\) 0 0
\(430\) 0 0
\(431\) 6.00000 0.289010 0.144505 0.989504i \(-0.453841\pi\)
0.144505 + 0.989504i \(0.453841\pi\)
\(432\) 4.00000 0.192450
\(433\) 5.00000 0.240285 0.120142 0.992757i \(-0.461665\pi\)
0.120142 + 0.992757i \(0.461665\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −14.0000 −0.670478
\(437\) −42.0000 −2.00913
\(438\) 0 0
\(439\) 19.0000 0.906821 0.453410 0.891302i \(-0.350207\pi\)
0.453410 + 0.891302i \(0.350207\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 0 0
\(443\) −18.0000 −0.855206 −0.427603 0.903967i \(-0.640642\pi\)
−0.427603 + 0.903967i \(0.640642\pi\)
\(444\) 14.0000 0.664411
\(445\) 18.0000 0.853282
\(446\) 0 0
\(447\) −6.00000 −0.283790
\(448\) −8.00000 −0.377964
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 24.0000 1.12887
\(453\) 16.0000 0.751746
\(454\) 0 0
\(455\) 2.00000 0.0937614
\(456\) 0 0
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 0 0
\(459\) −6.00000 −0.280056
\(460\) −12.0000 −0.559503
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) 32.0000 1.48717 0.743583 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) 0 0
\(465\) 1.00000 0.0463739
\(466\) 0 0
\(467\) −42.0000 −1.94353 −0.971764 0.235954i \(-0.924178\pi\)
−0.971764 + 0.235954i \(0.924178\pi\)
\(468\) 4.00000 0.184900
\(469\) −7.00000 −0.323230
\(470\) 0 0
\(471\) 17.0000 0.783319
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 7.00000 0.321182
\(476\) 12.0000 0.550019
\(477\) −6.00000 −0.274721
\(478\) 0 0
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) 14.0000 0.638345
\(482\) 0 0
\(483\) −6.00000 −0.273009
\(484\) 0 0
\(485\) 1.00000 0.0454077
\(486\) 0 0
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 0 0
\(489\) −1.00000 −0.0452216
\(490\) 0 0
\(491\) −42.0000 −1.89543 −0.947717 0.319113i \(-0.896615\pi\)
−0.947717 + 0.319113i \(0.896615\pi\)
\(492\) −12.0000 −0.541002
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 6.00000 0.269137
\(498\) 0 0
\(499\) 23.0000 1.02962 0.514811 0.857304i \(-0.327862\pi\)
0.514811 + 0.857304i \(0.327862\pi\)
\(500\) 2.00000 0.0894427
\(501\) 6.00000 0.268060
\(502\) 0 0
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) 34.0000 1.50851
\(509\) 24.0000 1.06378 0.531891 0.846813i \(-0.321482\pi\)
0.531891 + 0.846813i \(0.321482\pi\)
\(510\) 0 0
\(511\) 13.0000 0.575086
\(512\) 0 0
\(513\) 7.00000 0.309058
\(514\) 0 0
\(515\) 1.00000 0.0440653
\(516\) 16.0000 0.704361
\(517\) 0 0
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) 25.0000 1.09317 0.546587 0.837402i \(-0.315927\pi\)
0.546587 + 0.837402i \(0.315927\pi\)
\(524\) 36.0000 1.57267
\(525\) 1.00000 0.0436436
\(526\) 0 0
\(527\) 6.00000 0.261364
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) −14.0000 −0.606977
\(533\) −12.0000 −0.519778
\(534\) 0 0
\(535\) 12.0000 0.518805
\(536\) 0 0
\(537\) −12.0000 −0.517838
\(538\) 0 0
\(539\) 0 0
\(540\) 2.00000 0.0860663
\(541\) −14.0000 −0.601907 −0.300954 0.953639i \(-0.597305\pi\)
−0.300954 + 0.953639i \(0.597305\pi\)
\(542\) 0 0
\(543\) 17.0000 0.729540
\(544\) 0 0
\(545\) −7.00000 −0.299847
\(546\) 0 0
\(547\) −8.00000 −0.342055 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) 24.0000 1.02523
\(549\) 1.00000 0.0426790
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −11.0000 −0.467768
\(554\) 0 0
\(555\) 7.00000 0.297133
\(556\) −8.00000 −0.339276
\(557\) 42.0000 1.77960 0.889799 0.456354i \(-0.150845\pi\)
0.889799 + 0.456354i \(0.150845\pi\)
\(558\) 0 0
\(559\) 16.0000 0.676728
\(560\) −4.00000 −0.169031
\(561\) 0 0
\(562\) 0 0
\(563\) −36.0000 −1.51722 −0.758610 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) 0 0
\(565\) 12.0000 0.504844
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) −23.0000 −0.962520 −0.481260 0.876578i \(-0.659821\pi\)
−0.481260 + 0.876578i \(0.659821\pi\)
\(572\) 0 0
\(573\) 12.0000 0.501307
\(574\) 0 0
\(575\) −6.00000 −0.250217
\(576\) −8.00000 −0.333333
\(577\) −43.0000 −1.79011 −0.895057 0.445952i \(-0.852865\pi\)
−0.895057 + 0.445952i \(0.852865\pi\)
\(578\) 0 0
\(579\) −5.00000 −0.207793
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 2.00000 0.0826898
\(586\) 0 0
\(587\) 36.0000 1.48588 0.742940 0.669359i \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) 12.0000 0.494872
\(589\) −7.00000 −0.288430
\(590\) 0 0
\(591\) 24.0000 0.987228
\(592\) −28.0000 −1.15079
\(593\) 42.0000 1.72473 0.862367 0.506284i \(-0.168981\pi\)
0.862367 + 0.506284i \(0.168981\pi\)
\(594\) 0 0
\(595\) 6.00000 0.245976
\(596\) 12.0000 0.491539
\(597\) 11.0000 0.450200
\(598\) 0 0
\(599\) 42.0000 1.71607 0.858037 0.513588i \(-0.171684\pi\)
0.858037 + 0.513588i \(0.171684\pi\)
\(600\) 0 0
\(601\) −11.0000 −0.448699 −0.224350 0.974509i \(-0.572026\pi\)
−0.224350 + 0.974509i \(0.572026\pi\)
\(602\) 0 0
\(603\) −7.00000 −0.285062
\(604\) −32.0000 −1.30206
\(605\) 0 0
\(606\) 0 0
\(607\) −32.0000 −1.29884 −0.649420 0.760430i \(-0.724988\pi\)
−0.649420 + 0.760430i \(0.724988\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 12.0000 0.485071
\(613\) −5.00000 −0.201948 −0.100974 0.994889i \(-0.532196\pi\)
−0.100974 + 0.994889i \(0.532196\pi\)
\(614\) 0 0
\(615\) −6.00000 −0.241943
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 0 0
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) −2.00000 −0.0803219
\(621\) −6.00000 −0.240772
\(622\) 0 0
\(623\) −18.0000 −0.721155
\(624\) −8.00000 −0.320256
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) −34.0000 −1.35675
\(629\) 42.0000 1.67465
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 0 0
\(633\) −23.0000 −0.914168
\(634\) 0 0
\(635\) 17.0000 0.674624
\(636\) 12.0000 0.475831
\(637\) 12.0000 0.475457
\(638\) 0 0
\(639\) 6.00000 0.237356
\(640\) 0 0
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 0 0
\(643\) 29.0000 1.14365 0.571824 0.820376i \(-0.306236\pi\)
0.571824 + 0.820376i \(0.306236\pi\)
\(644\) 12.0000 0.472866
\(645\) 8.00000 0.315000
\(646\) 0 0
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −1.00000 −0.0391931
\(652\) 2.00000 0.0783260
\(653\) −42.0000 −1.64359 −0.821794 0.569785i \(-0.807026\pi\)
−0.821794 + 0.569785i \(0.807026\pi\)
\(654\) 0 0
\(655\) 18.0000 0.703318
\(656\) 24.0000 0.937043
\(657\) 13.0000 0.507178
\(658\) 0 0
\(659\) −6.00000 −0.233727 −0.116863 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) 0 0
\(661\) −1.00000 −0.0388955 −0.0194477 0.999811i \(-0.506191\pi\)
−0.0194477 + 0.999811i \(0.506191\pi\)
\(662\) 0 0
\(663\) 12.0000 0.466041
\(664\) 0 0
\(665\) −7.00000 −0.271448
\(666\) 0 0
\(667\) 0 0
\(668\) −12.0000 −0.464294
\(669\) 29.0000 1.12120
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −29.0000 −1.11787 −0.558934 0.829212i \(-0.688789\pi\)
−0.558934 + 0.829212i \(0.688789\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 18.0000 0.692308
\(677\) −48.0000 −1.84479 −0.922395 0.386248i \(-0.873771\pi\)
−0.922395 + 0.386248i \(0.873771\pi\)
\(678\) 0 0
\(679\) −1.00000 −0.0383765
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) −30.0000 −1.14792 −0.573959 0.818884i \(-0.694593\pi\)
−0.573959 + 0.818884i \(0.694593\pi\)
\(684\) −14.0000 −0.535303
\(685\) 12.0000 0.458496
\(686\) 0 0
\(687\) 14.0000 0.534133
\(688\) −32.0000 −1.21999
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) −13.0000 −0.494543 −0.247272 0.968946i \(-0.579534\pi\)
−0.247272 + 0.968946i \(0.579534\pi\)
\(692\) −12.0000 −0.456172
\(693\) 0 0
\(694\) 0 0
\(695\) −4.00000 −0.151729
\(696\) 0 0
\(697\) −36.0000 −1.36360
\(698\) 0 0
\(699\) −24.0000 −0.907763
\(700\) −2.00000 −0.0755929
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) −49.0000 −1.84807
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 12.0000 0.451306
\(708\) 24.0000 0.901975
\(709\) −34.0000 −1.27690 −0.638448 0.769665i \(-0.720423\pi\)
−0.638448 + 0.769665i \(0.720423\pi\)
\(710\) 0 0
\(711\) −11.0000 −0.412532
\(712\) 0 0
\(713\) 6.00000 0.224702
\(714\) 0 0
\(715\) 0 0
\(716\) 24.0000 0.896922
\(717\) 12.0000 0.448148
\(718\) 0 0
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) −4.00000 −0.149071
\(721\) −1.00000 −0.0372419
\(722\) 0 0
\(723\) 22.0000 0.818189
\(724\) −34.0000 −1.26360
\(725\) 0 0
\(726\) 0 0
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 48.0000 1.77534
\(732\) −2.00000 −0.0739221
\(733\) 22.0000 0.812589 0.406294 0.913742i \(-0.366821\pi\)
0.406294 + 0.913742i \(0.366821\pi\)
\(734\) 0 0
\(735\) 6.00000 0.221313
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 19.0000 0.698926 0.349463 0.936950i \(-0.386364\pi\)
0.349463 + 0.936950i \(0.386364\pi\)
\(740\) −14.0000 −0.514650
\(741\) −14.0000 −0.514303
\(742\) 0 0
\(743\) 36.0000 1.32071 0.660356 0.750953i \(-0.270405\pi\)
0.660356 + 0.750953i \(0.270405\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) −37.0000 −1.35015 −0.675075 0.737749i \(-0.735889\pi\)
−0.675075 + 0.737749i \(0.735889\pi\)
\(752\) 0 0
\(753\) 6.00000 0.218652
\(754\) 0 0
\(755\) −16.0000 −0.582300
\(756\) −2.00000 −0.0727393
\(757\) −25.0000 −0.908640 −0.454320 0.890838i \(-0.650118\pi\)
−0.454320 + 0.890838i \(0.650118\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 0 0
\(763\) 7.00000 0.253417
\(764\) −24.0000 −0.868290
\(765\) 6.00000 0.216930
\(766\) 0 0
\(767\) 24.0000 0.866590
\(768\) 16.0000 0.577350
\(769\) 25.0000 0.901523 0.450762 0.892644i \(-0.351152\pi\)
0.450762 + 0.892644i \(0.351152\pi\)
\(770\) 0 0
\(771\) −6.00000 −0.216085
\(772\) 10.0000 0.359908
\(773\) 12.0000 0.431610 0.215805 0.976436i \(-0.430762\pi\)
0.215805 + 0.976436i \(0.430762\pi\)
\(774\) 0 0
\(775\) −1.00000 −0.0359211
\(776\) 0 0
\(777\) −7.00000 −0.251124
\(778\) 0 0
\(779\) 42.0000 1.50481
\(780\) −4.00000 −0.143223
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −24.0000 −0.857143
\(785\) −17.0000 −0.606756
\(786\) 0 0
\(787\) 28.0000 0.998092 0.499046 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) −48.0000 −1.70993
\(789\) 6.00000 0.213606
\(790\) 0 0
\(791\) −12.0000 −0.426671
\(792\) 0 0
\(793\) −2.00000 −0.0710221
\(794\) 0 0
\(795\) 6.00000 0.212798
\(796\) −22.0000 −0.779769
\(797\) 18.0000 0.637593 0.318796 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −18.0000 −0.635999
\(802\) 0 0
\(803\) 0 0
\(804\) 14.0000 0.493742
\(805\) 6.00000 0.211472
\(806\) 0 0
\(807\) −24.0000 −0.844840
\(808\) 0 0
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) −11.0000 −0.386262 −0.193131 0.981173i \(-0.561864\pi\)
−0.193131 + 0.981173i \(0.561864\pi\)
\(812\) 0 0
\(813\) −20.0000 −0.701431
\(814\) 0 0
\(815\) 1.00000 0.0350285
\(816\) −24.0000 −0.840168
\(817\) −56.0000 −1.95919
\(818\) 0 0
\(819\) −2.00000 −0.0698857
\(820\) 12.0000 0.419058
\(821\) 42.0000 1.46581 0.732905 0.680331i \(-0.238164\pi\)
0.732905 + 0.680331i \(0.238164\pi\)
\(822\) 0 0
\(823\) −13.0000 −0.453152 −0.226576 0.973994i \(-0.572753\pi\)
−0.226576 + 0.973994i \(0.572753\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 12.0000 0.417029
\(829\) −7.00000 −0.243120 −0.121560 0.992584i \(-0.538790\pi\)
−0.121560 + 0.992584i \(0.538790\pi\)
\(830\) 0 0
\(831\) −11.0000 −0.381586
\(832\) 16.0000 0.554700
\(833\) 36.0000 1.24733
\(834\) 0 0
\(835\) −6.00000 −0.207639
\(836\) 0 0
\(837\) −1.00000 −0.0345651
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) 30.0000 1.03325
\(844\) 46.0000 1.58339
\(845\) 9.00000 0.309609
\(846\) 0 0
\(847\) 0 0
\(848\) −24.0000 −0.824163
\(849\) 1.00000 0.0343199
\(850\) 0 0
\(851\) 42.0000 1.43974
\(852\) −12.0000 −0.411113
\(853\) −5.00000 −0.171197 −0.0855984 0.996330i \(-0.527280\pi\)
−0.0855984 + 0.996330i \(0.527280\pi\)
\(854\) 0 0
\(855\) −7.00000 −0.239395
\(856\) 0 0
\(857\) −54.0000 −1.84460 −0.922302 0.386469i \(-0.873695\pi\)
−0.922302 + 0.386469i \(0.873695\pi\)
\(858\) 0 0
\(859\) −49.0000 −1.67186 −0.835929 0.548837i \(-0.815071\pi\)
−0.835929 + 0.548837i \(0.815071\pi\)
\(860\) −16.0000 −0.545595
\(861\) 6.00000 0.204479
\(862\) 0 0
\(863\) −36.0000 −1.22545 −0.612727 0.790295i \(-0.709928\pi\)
−0.612727 + 0.790295i \(0.709928\pi\)
\(864\) 0 0
\(865\) −6.00000 −0.204006
\(866\) 0 0
\(867\) 19.0000 0.645274
\(868\) 2.00000 0.0678844
\(869\) 0 0
\(870\) 0 0
\(871\) 14.0000 0.474372
\(872\) 0 0
\(873\) −1.00000 −0.0338449
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) −26.0000 −0.878459
\(877\) −5.00000 −0.168838 −0.0844190 0.996430i \(-0.526903\pi\)
−0.0844190 + 0.996430i \(0.526903\pi\)
\(878\) 0 0
\(879\) −18.0000 −0.607125
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) −25.0000 −0.841317 −0.420658 0.907219i \(-0.638201\pi\)
−0.420658 + 0.907219i \(0.638201\pi\)
\(884\) −24.0000 −0.807207
\(885\) 12.0000 0.403376
\(886\) 0 0
\(887\) 6.00000 0.201460 0.100730 0.994914i \(-0.467882\pi\)
0.100730 + 0.994914i \(0.467882\pi\)
\(888\) 0 0
\(889\) −17.0000 −0.570162
\(890\) 0 0
\(891\) 0 0
\(892\) −58.0000 −1.94198
\(893\) 0 0
\(894\) 0 0
\(895\) 12.0000 0.401116
\(896\) 0 0
\(897\) 12.0000 0.400668
\(898\) 0 0
\(899\) 0 0
\(900\) −2.00000 −0.0666667
\(901\) 36.0000 1.19933
\(902\) 0 0
\(903\) −8.00000 −0.266223
\(904\) 0 0
\(905\) −17.0000 −0.565099
\(906\) 0 0
\(907\) 35.0000 1.16216 0.581078 0.813848i \(-0.302631\pi\)
0.581078 + 0.813848i \(0.302631\pi\)
\(908\) −24.0000 −0.796468
\(909\) 12.0000 0.398015
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 28.0000 0.927173
\(913\) 0 0
\(914\) 0 0
\(915\) −1.00000 −0.0330590
\(916\) −28.0000 −0.925146
\(917\) −18.0000 −0.594412
\(918\) 0 0
\(919\) −11.0000 −0.362857 −0.181428 0.983404i \(-0.558072\pi\)
−0.181428 + 0.983404i \(0.558072\pi\)
\(920\) 0 0
\(921\) 13.0000 0.428365
\(922\) 0 0
\(923\) −12.0000 −0.394985
\(924\) 0 0
\(925\) −7.00000 −0.230159
\(926\) 0 0
\(927\) −1.00000 −0.0328443
\(928\) 0 0
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) −42.0000 −1.37649
\(932\) 48.0000 1.57229
\(933\) −30.0000 −0.982156
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 43.0000 1.40475 0.702374 0.711808i \(-0.252123\pi\)
0.702374 + 0.711808i \(0.252123\pi\)
\(938\) 0 0
\(939\) −22.0000 −0.717943
\(940\) 0 0
\(941\) −30.0000 −0.977972 −0.488986 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) 0 0
\(943\) −36.0000 −1.17232
\(944\) −48.0000 −1.56227
\(945\) −1.00000 −0.0325300
\(946\) 0 0
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 22.0000 0.714527
\(949\) −26.0000 −0.843996
\(950\) 0 0
\(951\) −6.00000 −0.194563
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 0 0
\(955\) −12.0000 −0.388311
\(956\) −24.0000 −0.776215
\(957\) 0 0
\(958\) 0 0
\(959\) −12.0000 −0.387500
\(960\) 8.00000 0.258199
\(961\) −30.0000 −0.967742
\(962\) 0 0
\(963\) −12.0000 −0.386695
\(964\) −44.0000 −1.41714
\(965\) 5.00000 0.160956
\(966\) 0 0
\(967\) −23.0000 −0.739630 −0.369815 0.929105i \(-0.620579\pi\)
−0.369815 + 0.929105i \(0.620579\pi\)
\(968\) 0 0
\(969\) −42.0000 −1.34923
\(970\) 0 0
\(971\) 18.0000 0.577647 0.288824 0.957382i \(-0.406736\pi\)
0.288824 + 0.957382i \(0.406736\pi\)
\(972\) −2.00000 −0.0641500
\(973\) 4.00000 0.128234
\(974\) 0 0
\(975\) −2.00000 −0.0640513
\(976\) 4.00000 0.128037
\(977\) −18.0000 −0.575871 −0.287936 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −12.0000 −0.383326
\(981\) 7.00000 0.223493
\(982\) 0 0
\(983\) −30.0000 −0.956851 −0.478426 0.878128i \(-0.658792\pi\)
−0.478426 + 0.878128i \(0.658792\pi\)
\(984\) 0 0
\(985\) −24.0000 −0.764704
\(986\) 0 0
\(987\) 0 0
\(988\) 28.0000 0.890799
\(989\) 48.0000 1.52631
\(990\) 0 0
\(991\) −40.0000 −1.27064 −0.635321 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) 0 0
\(993\) 29.0000 0.920287
\(994\) 0 0
\(995\) −11.0000 −0.348723
\(996\) 0 0
\(997\) −41.0000 −1.29848 −0.649242 0.760582i \(-0.724914\pi\)
−0.649242 + 0.760582i \(0.724914\pi\)
\(998\) 0 0
\(999\) −7.00000 −0.221470
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 1815.2.a.c.1.1 yes 1
3.2 odd 2 5445.2.a.g.1.1 1
5.4 even 2 9075.2.a.j.1.1 1
11.10 odd 2 1815.2.a.b.1.1 1
33.32 even 2 5445.2.a.f.1.1 1
55.54 odd 2 9075.2.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1815.2.a.b.1.1 1 11.10 odd 2
1815.2.a.c.1.1 yes 1 1.1 even 1 trivial
5445.2.a.f.1.1 1 33.32 even 2
5445.2.a.g.1.1 1 3.2 odd 2
9075.2.a.j.1.1 1 5.4 even 2
9075.2.a.k.1.1 1 55.54 odd 2