Properties

Label 3525.2.a.k.1.1
Level $3525$
Weight $2$
Character 3525.1
Self dual yes
Analytic conductor $28.147$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3525 = 3 \cdot 5^{2} \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3525.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.1472667125\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 141)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3525.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{12} -6.00000 q^{13} -4.00000 q^{14} -1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} +2.00000 q^{19} -4.00000 q^{21} -4.00000 q^{23} -3.00000 q^{24} -6.00000 q^{26} +1.00000 q^{27} +4.00000 q^{28} +8.00000 q^{29} +6.00000 q^{31} +5.00000 q^{32} +6.00000 q^{34} -1.00000 q^{36} +6.00000 q^{37} +2.00000 q^{38} -6.00000 q^{39} -8.00000 q^{41} -4.00000 q^{42} +6.00000 q^{43} -4.00000 q^{46} -1.00000 q^{47} -1.00000 q^{48} +9.00000 q^{49} +6.00000 q^{51} +6.00000 q^{52} -2.00000 q^{53} +1.00000 q^{54} +12.0000 q^{56} +2.00000 q^{57} +8.00000 q^{58} +12.0000 q^{59} +2.00000 q^{61} +6.00000 q^{62} -4.00000 q^{63} +7.00000 q^{64} +2.00000 q^{67} -6.00000 q^{68} -4.00000 q^{69} -3.00000 q^{72} +10.0000 q^{73} +6.00000 q^{74} -2.00000 q^{76} -6.00000 q^{78} -4.00000 q^{79} +1.00000 q^{81} -8.00000 q^{82} -4.00000 q^{83} +4.00000 q^{84} +6.00000 q^{86} +8.00000 q^{87} -10.0000 q^{89} +24.0000 q^{91} +4.00000 q^{92} +6.00000 q^{93} -1.00000 q^{94} +5.00000 q^{96} +18.0000 q^{97} +9.00000 q^{98} +O(q^{100})\)

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 \(0.384973\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) −4.00000 −1.51186 −0.755929 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) −3.00000 −1.06066
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −1.00000 −0.288675
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 1.00000 0.235702
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) −3.00000 −0.612372
\(25\) 0 0
\(26\) −6.00000 −1.17670
\(27\) 1.00000 0.192450
\(28\) 4.00000 0.755929
\(29\) 8.00000 1.48556 0.742781 0.669534i \(-0.233506\pi\)
0.742781 + 0.669534i \(0.233506\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 5.00000 0.883883
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 2.00000 0.324443
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) −4.00000 −0.617213
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) −1.00000 −0.145865
\(48\) −1.00000 −0.144338
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) 6.00000 0.832050
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 12.0000 1.60357
\(57\) 2.00000 0.264906
\(58\) 8.00000 1.05045
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 6.00000 0.762001
\(63\) −4.00000 −0.503953
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) −6.00000 −0.727607
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −3.00000 −0.353553
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 6.00000 0.697486
\(75\) 0 0
\(76\) −2.00000 −0.229416
\(77\) 0 0
\(78\) −6.00000 −0.679366
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −8.00000 −0.883452
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 4.00000 0.436436
\(85\) 0 0
\(86\) 6.00000 0.646997
\(87\) 8.00000 0.857690
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 24.0000 2.51588
\(92\) 4.00000 0.417029
\(93\) 6.00000 0.622171
\(94\) −1.00000 −0.103142
\(95\) 0 0
\(96\) 5.00000 0.510310
\(97\) 18.0000 1.82762 0.913812 0.406138i \(-0.133125\pi\)
0.913812 + 0.406138i \(0.133125\pi\)
\(98\) 9.00000 0.909137
\(99\) 0 0
\(100\) 0 0
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 6.00000 0.594089
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 18.0000 1.76505
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) 4.00000 0.377964
\(113\) −8.00000 −0.752577 −0.376288 0.926503i \(-0.622800\pi\)
−0.376288 + 0.926503i \(0.622800\pi\)
\(114\) 2.00000 0.187317
\(115\) 0 0
\(116\) −8.00000 −0.742781
\(117\) −6.00000 −0.554700
\(118\) 12.0000 1.10469
\(119\) −24.0000 −2.20008
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 2.00000 0.181071
\(123\) −8.00000 −0.721336
\(124\) −6.00000 −0.538816
\(125\) 0 0
\(126\) −4.00000 −0.356348
\(127\) 6.00000 0.532414 0.266207 0.963916i \(-0.414230\pi\)
0.266207 + 0.963916i \(0.414230\pi\)
\(128\) −3.00000 −0.265165
\(129\) 6.00000 0.528271
\(130\) 0 0
\(131\) 20.0000 1.74741 0.873704 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) 0 0
\(133\) −8.00000 −0.693688
\(134\) 2.00000 0.172774
\(135\) 0 0
\(136\) −18.0000 −1.54349
\(137\) 16.0000 1.36697 0.683486 0.729964i \(-0.260463\pi\)
0.683486 + 0.729964i \(0.260463\pi\)
\(138\) −4.00000 −0.340503
\(139\) −2.00000 −0.169638 −0.0848189 0.996396i \(-0.527031\pi\)
−0.0848189 + 0.996396i \(0.527031\pi\)
\(140\) 0 0
\(141\) −1.00000 −0.0842152
\(142\) 0 0
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 10.0000 0.827606
\(147\) 9.00000 0.742307
\(148\) −6.00000 −0.493197
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) −2.00000 −0.162758 −0.0813788 0.996683i \(-0.525932\pi\)
−0.0813788 + 0.996683i \(0.525932\pi\)
\(152\) −6.00000 −0.486664
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) 6.00000 0.480384
\(157\) −22.0000 −1.75579 −0.877896 0.478852i \(-0.841053\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) −4.00000 −0.318223
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) 16.0000 1.26098
\(162\) 1.00000 0.0785674
\(163\) 10.0000 0.783260 0.391630 0.920123i \(-0.371911\pi\)
0.391630 + 0.920123i \(0.371911\pi\)
\(164\) 8.00000 0.624695
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 12.0000 0.925820
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) −6.00000 −0.457496
\(173\) −2.00000 −0.152057 −0.0760286 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) 8.00000 0.606478
\(175\) 0 0
\(176\) 0 0
\(177\) 12.0000 0.901975
\(178\) −10.0000 −0.749532
\(179\) −20.0000 −1.49487 −0.747435 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 24.0000 1.77900
\(183\) 2.00000 0.147844
\(184\) 12.0000 0.884652
\(185\) 0 0
\(186\) 6.00000 0.439941
\(187\) 0 0
\(188\) 1.00000 0.0729325
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) 24.0000 1.73658 0.868290 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) 7.00000 0.505181
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) 18.0000 1.29232
\(195\) 0 0
\(196\) −9.00000 −0.642857
\(197\) −14.0000 −0.997459 −0.498729 0.866758i \(-0.666200\pi\)
−0.498729 + 0.866758i \(0.666200\pi\)
\(198\) 0 0
\(199\) 2.00000 0.141776 0.0708881 0.997484i \(-0.477417\pi\)
0.0708881 + 0.997484i \(0.477417\pi\)
\(200\) 0 0
\(201\) 2.00000 0.141069
\(202\) −10.0000 −0.703598
\(203\) −32.0000 −2.24596
\(204\) −6.00000 −0.420084
\(205\) 0 0
\(206\) −4.00000 −0.278693
\(207\) −4.00000 −0.278019
\(208\) 6.00000 0.416025
\(209\) 0 0
\(210\) 0 0
\(211\) −10.0000 −0.688428 −0.344214 0.938891i \(-0.611855\pi\)
−0.344214 + 0.938891i \(0.611855\pi\)
\(212\) 2.00000 0.137361
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) −3.00000 −0.204124
\(217\) −24.0000 −1.62923
\(218\) −2.00000 −0.135457
\(219\) 10.0000 0.675737
\(220\) 0 0
\(221\) −36.0000 −2.42162
\(222\) 6.00000 0.402694
\(223\) 22.0000 1.47323 0.736614 0.676313i \(-0.236423\pi\)
0.736614 + 0.676313i \(0.236423\pi\)
\(224\) −20.0000 −1.33631
\(225\) 0 0
\(226\) −8.00000 −0.532152
\(227\) −20.0000 −1.32745 −0.663723 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(228\) −2.00000 −0.132453
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −24.0000 −1.57568
\(233\) 16.0000 1.04819 0.524097 0.851658i \(-0.324403\pi\)
0.524097 + 0.851658i \(0.324403\pi\)
\(234\) −6.00000 −0.392232
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) −4.00000 −0.259828
\(238\) −24.0000 −1.55569
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) −11.0000 −0.707107
\(243\) 1.00000 0.0641500
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) −8.00000 −0.510061
\(247\) −12.0000 −0.763542
\(248\) −18.0000 −1.14300
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 4.00000 0.251976
\(253\) 0 0
\(254\) 6.00000 0.376473
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 12.0000 0.748539 0.374270 0.927320i \(-0.377893\pi\)
0.374270 + 0.927320i \(0.377893\pi\)
\(258\) 6.00000 0.373544
\(259\) −24.0000 −1.49129
\(260\) 0 0
\(261\) 8.00000 0.495188
\(262\) 20.0000 1.23560
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −8.00000 −0.490511
\(267\) −10.0000 −0.611990
\(268\) −2.00000 −0.122169
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) −6.00000 −0.363803
\(273\) 24.0000 1.45255
\(274\) 16.0000 0.966595
\(275\) 0 0
\(276\) 4.00000 0.240772
\(277\) 14.0000 0.841178 0.420589 0.907251i \(-0.361823\pi\)
0.420589 + 0.907251i \(0.361823\pi\)
\(278\) −2.00000 −0.119952
\(279\) 6.00000 0.359211
\(280\) 0 0
\(281\) 16.0000 0.954480 0.477240 0.878773i \(-0.341637\pi\)
0.477240 + 0.878773i \(0.341637\pi\)
\(282\) −1.00000 −0.0595491
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 32.0000 1.88890
\(288\) 5.00000 0.294628
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 18.0000 1.05518
\(292\) −10.0000 −0.585206
\(293\) 8.00000 0.467365 0.233682 0.972313i \(-0.424922\pi\)
0.233682 + 0.972313i \(0.424922\pi\)
\(294\) 9.00000 0.524891
\(295\) 0 0
\(296\) −18.0000 −1.04623
\(297\) 0 0
\(298\) −10.0000 −0.579284
\(299\) 24.0000 1.38796
\(300\) 0 0
\(301\) −24.0000 −1.38334
\(302\) −2.00000 −0.115087
\(303\) −10.0000 −0.574485
\(304\) −2.00000 −0.114708
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) −28.0000 −1.58773 −0.793867 0.608091i \(-0.791935\pi\)
−0.793867 + 0.608091i \(0.791935\pi\)
\(312\) 18.0000 1.01905
\(313\) −34.0000 −1.92179 −0.960897 0.276907i \(-0.910691\pi\)
−0.960897 + 0.276907i \(0.910691\pi\)
\(314\) −22.0000 −1.24153
\(315\) 0 0
\(316\) 4.00000 0.225018
\(317\) −12.0000 −0.673987 −0.336994 0.941507i \(-0.609410\pi\)
−0.336994 + 0.941507i \(0.609410\pi\)
\(318\) −2.00000 −0.112154
\(319\) 0 0
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 16.0000 0.891645
\(323\) 12.0000 0.667698
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 10.0000 0.553849
\(327\) −2.00000 −0.110600
\(328\) 24.0000 1.32518
\(329\) 4.00000 0.220527
\(330\) 0 0
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) 4.00000 0.219529
\(333\) 6.00000 0.328798
\(334\) 12.0000 0.656611
\(335\) 0 0
\(336\) 4.00000 0.218218
\(337\) 6.00000 0.326841 0.163420 0.986557i \(-0.447747\pi\)
0.163420 + 0.986557i \(0.447747\pi\)
\(338\) 23.0000 1.25104
\(339\) −8.00000 −0.434500
\(340\) 0 0
\(341\) 0 0
\(342\) 2.00000 0.108148
\(343\) −8.00000 −0.431959
\(344\) −18.0000 −0.970495
\(345\) 0 0
\(346\) −2.00000 −0.107521
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) −8.00000 −0.428845
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) −6.00000 −0.320256
\(352\) 0 0
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 12.0000 0.637793
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) −24.0000 −1.27021
\(358\) −20.0000 −1.05703
\(359\) 20.0000 1.05556 0.527780 0.849381i \(-0.323025\pi\)
0.527780 + 0.849381i \(0.323025\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 14.0000 0.735824
\(363\) −11.0000 −0.577350
\(364\) −24.0000 −1.25794
\(365\) 0 0
\(366\) 2.00000 0.104542
\(367\) 10.0000 0.521996 0.260998 0.965339i \(-0.415948\pi\)
0.260998 + 0.965339i \(0.415948\pi\)
\(368\) 4.00000 0.208514
\(369\) −8.00000 −0.416463
\(370\) 0 0
\(371\) 8.00000 0.415339
\(372\) −6.00000 −0.311086
\(373\) 38.0000 1.96757 0.983783 0.179364i \(-0.0574041\pi\)
0.983783 + 0.179364i \(0.0574041\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 3.00000 0.154713
\(377\) −48.0000 −2.47213
\(378\) −4.00000 −0.205738
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 6.00000 0.307389
\(382\) 24.0000 1.22795
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) −3.00000 −0.153093
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) 6.00000 0.304997
\(388\) −18.0000 −0.913812
\(389\) 20.0000 1.01404 0.507020 0.861934i \(-0.330747\pi\)
0.507020 + 0.861934i \(0.330747\pi\)
\(390\) 0 0
\(391\) −24.0000 −1.21373
\(392\) −27.0000 −1.36371
\(393\) 20.0000 1.00887
\(394\) −14.0000 −0.705310
\(395\) 0 0
\(396\) 0 0
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) 2.00000 0.100251
\(399\) −8.00000 −0.400501
\(400\) 0 0
\(401\) −10.0000 −0.499376 −0.249688 0.968326i \(-0.580328\pi\)
−0.249688 + 0.968326i \(0.580328\pi\)
\(402\) 2.00000 0.0997509
\(403\) −36.0000 −1.79329
\(404\) 10.0000 0.497519
\(405\) 0 0
\(406\) −32.0000 −1.58813
\(407\) 0 0
\(408\) −18.0000 −0.891133
\(409\) 38.0000 1.87898 0.939490 0.342578i \(-0.111300\pi\)
0.939490 + 0.342578i \(0.111300\pi\)
\(410\) 0 0
\(411\) 16.0000 0.789222
\(412\) 4.00000 0.197066
\(413\) −48.0000 −2.36193
\(414\) −4.00000 −0.196589
\(415\) 0 0
\(416\) −30.0000 −1.47087
\(417\) −2.00000 −0.0979404
\(418\) 0 0
\(419\) 28.0000 1.36789 0.683945 0.729534i \(-0.260263\pi\)
0.683945 + 0.729534i \(0.260263\pi\)
\(420\) 0 0
\(421\) 6.00000 0.292422 0.146211 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) −10.0000 −0.486792
\(423\) −1.00000 −0.0486217
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) −8.00000 −0.387147
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 0 0
\(431\) −16.0000 −0.770693 −0.385346 0.922772i \(-0.625918\pi\)
−0.385346 + 0.922772i \(0.625918\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −26.0000 −1.24948 −0.624740 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) −24.0000 −1.15204
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) −8.00000 −0.382692
\(438\) 10.0000 0.477818
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) −36.0000 −1.71235
\(443\) 36.0000 1.71041 0.855206 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) −6.00000 −0.284747
\(445\) 0 0
\(446\) 22.0000 1.04173
\(447\) −10.0000 −0.472984
\(448\) −28.0000 −1.32288
\(449\) −40.0000 −1.88772 −0.943858 0.330350i \(-0.892833\pi\)
−0.943858 + 0.330350i \(0.892833\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 8.00000 0.376288
\(453\) −2.00000 −0.0939682
\(454\) −20.0000 −0.938647
\(455\) 0 0
\(456\) −6.00000 −0.280976
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) −2.00000 −0.0934539
\(459\) 6.00000 0.280056
\(460\) 0 0
\(461\) 24.0000 1.11779 0.558896 0.829238i \(-0.311225\pi\)
0.558896 + 0.829238i \(0.311225\pi\)
\(462\) 0 0
\(463\) 18.0000 0.836531 0.418265 0.908325i \(-0.362638\pi\)
0.418265 + 0.908325i \(0.362638\pi\)
\(464\) −8.00000 −0.371391
\(465\) 0 0
\(466\) 16.0000 0.741186
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 6.00000 0.277350
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) −22.0000 −1.01371
\(472\) −36.0000 −1.65703
\(473\) 0 0
\(474\) −4.00000 −0.183726
\(475\) 0 0
\(476\) 24.0000 1.10004
\(477\) −2.00000 −0.0915737
\(478\) 8.00000 0.365911
\(479\) 32.0000 1.46212 0.731059 0.682315i \(-0.239027\pi\)
0.731059 + 0.682315i \(0.239027\pi\)
\(480\) 0 0
\(481\) −36.0000 −1.64146
\(482\) 10.0000 0.455488
\(483\) 16.0000 0.728025
\(484\) 11.0000 0.500000
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 20.0000 0.906287 0.453143 0.891438i \(-0.350303\pi\)
0.453143 + 0.891438i \(0.350303\pi\)
\(488\) −6.00000 −0.271607
\(489\) 10.0000 0.452216
\(490\) 0 0
\(491\) 4.00000 0.180517 0.0902587 0.995918i \(-0.471231\pi\)
0.0902587 + 0.995918i \(0.471231\pi\)
\(492\) 8.00000 0.360668
\(493\) 48.0000 2.16181
\(494\) −12.0000 −0.539906
\(495\) 0 0
\(496\) −6.00000 −0.269408
\(497\) 0 0
\(498\) −4.00000 −0.179244
\(499\) −14.0000 −0.626726 −0.313363 0.949633i \(-0.601456\pi\)
−0.313363 + 0.949633i \(0.601456\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 12.0000 0.535586
\(503\) 8.00000 0.356702 0.178351 0.983967i \(-0.442924\pi\)
0.178351 + 0.983967i \(0.442924\pi\)
\(504\) 12.0000 0.534522
\(505\) 0 0
\(506\) 0 0
\(507\) 23.0000 1.02147
\(508\) −6.00000 −0.266207
\(509\) −28.0000 −1.24108 −0.620539 0.784176i \(-0.713086\pi\)
−0.620539 + 0.784176i \(0.713086\pi\)
\(510\) 0 0
\(511\) −40.0000 −1.76950
\(512\) −11.0000 −0.486136
\(513\) 2.00000 0.0883022
\(514\) 12.0000 0.529297
\(515\) 0 0
\(516\) −6.00000 −0.264135
\(517\) 0 0
\(518\) −24.0000 −1.05450
\(519\) −2.00000 −0.0877903
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 8.00000 0.350150
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) −20.0000 −0.873704
\(525\) 0 0
\(526\) 0 0
\(527\) 36.0000 1.56818
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) 8.00000 0.346844
\(533\) 48.0000 2.07911
\(534\) −10.0000 −0.432742
\(535\) 0 0
\(536\) −6.00000 −0.259161
\(537\) −20.0000 −0.863064
\(538\) −6.00000 −0.258678
\(539\) 0 0
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 12.0000 0.515444
\(543\) 14.0000 0.600798
\(544\) 30.0000 1.28624
\(545\) 0 0
\(546\) 24.0000 1.02711
\(547\) −22.0000 −0.940652 −0.470326 0.882493i \(-0.655864\pi\)
−0.470326 + 0.882493i \(0.655864\pi\)
\(548\) −16.0000 −0.683486
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) 16.0000 0.681623
\(552\) 12.0000 0.510754
\(553\) 16.0000 0.680389
\(554\) 14.0000 0.594803
\(555\) 0 0
\(556\) 2.00000 0.0848189
\(557\) −24.0000 −1.01691 −0.508456 0.861088i \(-0.669784\pi\)
−0.508456 + 0.861088i \(0.669784\pi\)
\(558\) 6.00000 0.254000
\(559\) −36.0000 −1.52264
\(560\) 0 0
\(561\) 0 0
\(562\) 16.0000 0.674919
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 1.00000 0.0421076
\(565\) 0 0
\(566\) −4.00000 −0.168133
\(567\) −4.00000 −0.167984
\(568\) 0 0
\(569\) 24.0000 1.00613 0.503066 0.864248i \(-0.332205\pi\)
0.503066 + 0.864248i \(0.332205\pi\)
\(570\) 0 0
\(571\) 44.0000 1.84134 0.920671 0.390339i \(-0.127642\pi\)
0.920671 + 0.390339i \(0.127642\pi\)
\(572\) 0 0
\(573\) 24.0000 1.00261
\(574\) 32.0000 1.33565
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) −42.0000 −1.74848 −0.874241 0.485491i \(-0.838641\pi\)
−0.874241 + 0.485491i \(0.838641\pi\)
\(578\) 19.0000 0.790296
\(579\) −10.0000 −0.415586
\(580\) 0 0
\(581\) 16.0000 0.663792
\(582\) 18.0000 0.746124
\(583\) 0 0
\(584\) −30.0000 −1.24141
\(585\) 0 0
\(586\) 8.00000 0.330477
\(587\) −16.0000 −0.660391 −0.330195 0.943913i \(-0.607115\pi\)
−0.330195 + 0.943913i \(0.607115\pi\)
\(588\) −9.00000 −0.371154
\(589\) 12.0000 0.494451
\(590\) 0 0
\(591\) −14.0000 −0.575883
\(592\) −6.00000 −0.246598
\(593\) 48.0000 1.97112 0.985562 0.169316i \(-0.0541557\pi\)
0.985562 + 0.169316i \(0.0541557\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) 2.00000 0.0818546
\(598\) 24.0000 0.981433
\(599\) −32.0000 −1.30748 −0.653742 0.756717i \(-0.726802\pi\)
−0.653742 + 0.756717i \(0.726802\pi\)
\(600\) 0 0
\(601\) −34.0000 −1.38689 −0.693444 0.720510i \(-0.743908\pi\)
−0.693444 + 0.720510i \(0.743908\pi\)
\(602\) −24.0000 −0.978167
\(603\) 2.00000 0.0814463
\(604\) 2.00000 0.0813788
\(605\) 0 0
\(606\) −10.0000 −0.406222
\(607\) −22.0000 −0.892952 −0.446476 0.894795i \(-0.647321\pi\)
−0.446476 + 0.894795i \(0.647321\pi\)
\(608\) 10.0000 0.405554
\(609\) −32.0000 −1.29671
\(610\) 0 0
\(611\) 6.00000 0.242734
\(612\) −6.00000 −0.242536
\(613\) −6.00000 −0.242338 −0.121169 0.992632i \(-0.538664\pi\)
−0.121169 + 0.992632i \(0.538664\pi\)
\(614\) 20.0000 0.807134
\(615\) 0 0
\(616\) 0 0
\(617\) 38.0000 1.52982 0.764911 0.644136i \(-0.222783\pi\)
0.764911 + 0.644136i \(0.222783\pi\)
\(618\) −4.00000 −0.160904
\(619\) 24.0000 0.964641 0.482321 0.875995i \(-0.339794\pi\)
0.482321 + 0.875995i \(0.339794\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) −28.0000 −1.12270
\(623\) 40.0000 1.60257
\(624\) 6.00000 0.240192
\(625\) 0 0
\(626\) −34.0000 −1.35891
\(627\) 0 0
\(628\) 22.0000 0.877896
\(629\) 36.0000 1.43541
\(630\) 0 0
\(631\) 34.0000 1.35352 0.676759 0.736204i \(-0.263384\pi\)
0.676759 + 0.736204i \(0.263384\pi\)
\(632\) 12.0000 0.477334
\(633\) −10.0000 −0.397464
\(634\) −12.0000 −0.476581
\(635\) 0 0
\(636\) 2.00000 0.0793052
\(637\) −54.0000 −2.13956
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −12.0000 −0.473972 −0.236986 0.971513i \(-0.576159\pi\)
−0.236986 + 0.971513i \(0.576159\pi\)
\(642\) 12.0000 0.473602
\(643\) −4.00000 −0.157745 −0.0788723 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) −16.0000 −0.630488
\(645\) 0 0
\(646\) 12.0000 0.472134
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) −3.00000 −0.117851
\(649\) 0 0
\(650\) 0 0
\(651\) −24.0000 −0.940634
\(652\) −10.0000 −0.391630
\(653\) −34.0000 −1.33052 −0.665261 0.746611i \(-0.731680\pi\)
−0.665261 + 0.746611i \(0.731680\pi\)
\(654\) −2.00000 −0.0782062
\(655\) 0 0
\(656\) 8.00000 0.312348
\(657\) 10.0000 0.390137
\(658\) 4.00000 0.155936
\(659\) 4.00000 0.155818 0.0779089 0.996960i \(-0.475176\pi\)
0.0779089 + 0.996960i \(0.475176\pi\)
\(660\) 0 0
\(661\) 10.0000 0.388955 0.194477 0.980907i \(-0.437699\pi\)
0.194477 + 0.980907i \(0.437699\pi\)
\(662\) −8.00000 −0.310929
\(663\) −36.0000 −1.39812
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) 6.00000 0.232495
\(667\) −32.0000 −1.23904
\(668\) −12.0000 −0.464294
\(669\) 22.0000 0.850569
\(670\) 0 0
\(671\) 0 0
\(672\) −20.0000 −0.771517
\(673\) −34.0000 −1.31060 −0.655302 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) 6.00000 0.231111
\(675\) 0 0
\(676\) −23.0000 −0.884615
\(677\) 24.0000 0.922395 0.461197 0.887298i \(-0.347420\pi\)
0.461197 + 0.887298i \(0.347420\pi\)
\(678\) −8.00000 −0.307238
\(679\) −72.0000 −2.76311
\(680\) 0 0
\(681\) −20.0000 −0.766402
\(682\) 0 0
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) −2.00000 −0.0764719
\(685\) 0 0
\(686\) −8.00000 −0.305441
\(687\) −2.00000 −0.0763048
\(688\) −6.00000 −0.228748
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) −2.00000 −0.0760836 −0.0380418 0.999276i \(-0.512112\pi\)
−0.0380418 + 0.999276i \(0.512112\pi\)
\(692\) 2.00000 0.0760286
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) −24.0000 −0.909718
\(697\) −48.0000 −1.81813
\(698\) 10.0000 0.378506
\(699\) 16.0000 0.605176
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) −6.00000 −0.226455
\(703\) 12.0000 0.452589
\(704\) 0 0
\(705\) 0 0
\(706\) 18.0000 0.677439
\(707\) 40.0000 1.50435
\(708\) −12.0000 −0.450988
\(709\) −30.0000 −1.12667 −0.563337 0.826227i \(-0.690483\pi\)
−0.563337 + 0.826227i \(0.690483\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 30.0000 1.12430
\(713\) −24.0000 −0.898807
\(714\) −24.0000 −0.898177
\(715\) 0 0
\(716\) 20.0000 0.747435
\(717\) 8.00000 0.298765
\(718\) 20.0000 0.746393
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) −15.0000 −0.558242
\(723\) 10.0000 0.371904
\(724\) −14.0000 −0.520306
\(725\) 0 0
\(726\) −11.0000 −0.408248
\(727\) −50.0000 −1.85440 −0.927199 0.374570i \(-0.877790\pi\)
−0.927199 + 0.374570i \(0.877790\pi\)
\(728\) −72.0000 −2.66850
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 36.0000 1.33151
\(732\) −2.00000 −0.0739221
\(733\) −30.0000 −1.10808 −0.554038 0.832492i \(-0.686914\pi\)
−0.554038 + 0.832492i \(0.686914\pi\)
\(734\) 10.0000 0.369107
\(735\) 0 0
\(736\) −20.0000 −0.737210
\(737\) 0 0
\(738\) −8.00000 −0.294484
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) −12.0000 −0.440831
\(742\) 8.00000 0.293689
\(743\) 20.0000 0.733729 0.366864 0.930274i \(-0.380431\pi\)
0.366864 + 0.930274i \(0.380431\pi\)
\(744\) −18.0000 −0.659912
\(745\) 0 0
\(746\) 38.0000 1.39128
\(747\) −4.00000 −0.146352
\(748\) 0 0
\(749\) −48.0000 −1.75388
\(750\) 0 0
\(751\) −22.0000 −0.802791 −0.401396 0.915905i \(-0.631475\pi\)
−0.401396 + 0.915905i \(0.631475\pi\)
\(752\) 1.00000 0.0364662
\(753\) 12.0000 0.437304
\(754\) −48.0000 −1.74806
\(755\) 0 0
\(756\) 4.00000 0.145479
\(757\) 42.0000 1.52652 0.763258 0.646094i \(-0.223599\pi\)
0.763258 + 0.646094i \(0.223599\pi\)
\(758\) −20.0000 −0.726433
\(759\) 0 0
\(760\) 0 0
\(761\) 22.0000 0.797499 0.398750 0.917060i \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) 6.00000 0.217357
\(763\) 8.00000 0.289619
\(764\) −24.0000 −0.868290
\(765\) 0 0
\(766\) −24.0000 −0.867155
\(767\) −72.0000 −2.59977
\(768\) −17.0000 −0.613435
\(769\) −2.00000 −0.0721218 −0.0360609 0.999350i \(-0.511481\pi\)
−0.0360609 + 0.999350i \(0.511481\pi\)
\(770\) 0 0
\(771\) 12.0000 0.432169
\(772\) 10.0000 0.359908
\(773\) −42.0000 −1.51064 −0.755318 0.655359i \(-0.772517\pi\)
−0.755318 + 0.655359i \(0.772517\pi\)
\(774\) 6.00000 0.215666
\(775\) 0 0
\(776\) −54.0000 −1.93849
\(777\) −24.0000 −0.860995
\(778\) 20.0000 0.717035
\(779\) −16.0000 −0.573259
\(780\) 0 0
\(781\) 0 0
\(782\) −24.0000 −0.858238
\(783\) 8.00000 0.285897
\(784\) −9.00000 −0.321429
\(785\) 0 0
\(786\) 20.0000 0.713376
\(787\) −46.0000 −1.63972 −0.819861 0.572562i \(-0.805950\pi\)
−0.819861 + 0.572562i \(0.805950\pi\)
\(788\) 14.0000 0.498729
\(789\) 0 0
\(790\) 0 0
\(791\) 32.0000 1.13779
\(792\) 0 0
\(793\) −12.0000 −0.426132
\(794\) −22.0000 −0.780751
\(795\) 0 0
\(796\) −2.00000 −0.0708881
\(797\) −12.0000 −0.425062 −0.212531 0.977154i \(-0.568171\pi\)
−0.212531 + 0.977154i \(0.568171\pi\)
\(798\) −8.00000 −0.283197
\(799\) −6.00000 −0.212265
\(800\) 0 0
\(801\) −10.0000 −0.353333
\(802\) −10.0000 −0.353112
\(803\) 0 0
\(804\) −2.00000 −0.0705346
\(805\) 0 0
\(806\) −36.0000 −1.26805
\(807\) −6.00000 −0.211210
\(808\) 30.0000 1.05540
\(809\) 12.0000 0.421898 0.210949 0.977497i \(-0.432345\pi\)
0.210949 + 0.977497i \(0.432345\pi\)
\(810\) 0 0
\(811\) 24.0000 0.842754 0.421377 0.906886i \(-0.361547\pi\)
0.421377 + 0.906886i \(0.361547\pi\)
\(812\) 32.0000 1.12298
\(813\) 12.0000 0.420858
\(814\) 0 0
\(815\) 0 0
\(816\) −6.00000 −0.210042
\(817\) 12.0000 0.419827
\(818\) 38.0000 1.32864
\(819\) 24.0000 0.838628
\(820\) 0 0
\(821\) −12.0000 −0.418803 −0.209401 0.977830i \(-0.567152\pi\)
−0.209401 + 0.977830i \(0.567152\pi\)
\(822\) 16.0000 0.558064
\(823\) −40.0000 −1.39431 −0.697156 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) 12.0000 0.418040
\(825\) 0 0
\(826\) −48.0000 −1.67013
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 4.00000 0.139010
\(829\) 14.0000 0.486240 0.243120 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) 0 0
\(831\) 14.0000 0.485655
\(832\) −42.0000 −1.45609
\(833\) 54.0000 1.87099
\(834\) −2.00000 −0.0692543
\(835\) 0 0
\(836\) 0 0
\(837\) 6.00000 0.207390
\(838\) 28.0000 0.967244
\(839\) 32.0000 1.10476 0.552381 0.833592i \(-0.313719\pi\)
0.552381 + 0.833592i \(0.313719\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 6.00000 0.206774
\(843\) 16.0000 0.551069
\(844\) 10.0000 0.344214
\(845\) 0 0
\(846\) −1.00000 −0.0343807
\(847\) 44.0000 1.51186
\(848\) 2.00000 0.0686803
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) −24.0000 −0.822709
\(852\) 0 0
\(853\) 2.00000 0.0684787 0.0342393 0.999414i \(-0.489099\pi\)
0.0342393 + 0.999414i \(0.489099\pi\)
\(854\) −8.00000 −0.273754
\(855\) 0 0
\(856\) −36.0000 −1.23045
\(857\) 32.0000 1.09310 0.546550 0.837427i \(-0.315941\pi\)
0.546550 + 0.837427i \(0.315941\pi\)
\(858\) 0 0
\(859\) −34.0000 −1.16007 −0.580033 0.814593i \(-0.696960\pi\)
−0.580033 + 0.814593i \(0.696960\pi\)
\(860\) 0 0
\(861\) 32.0000 1.09056
\(862\) −16.0000 −0.544962
\(863\) −48.0000 −1.63394 −0.816970 0.576681i \(-0.804348\pi\)
−0.816970 + 0.576681i \(0.804348\pi\)
\(864\) 5.00000 0.170103
\(865\) 0 0
\(866\) −26.0000 −0.883516
\(867\) 19.0000 0.645274
\(868\) 24.0000 0.814613
\(869\) 0 0
\(870\) 0 0
\(871\) −12.0000 −0.406604
\(872\) 6.00000 0.203186
\(873\) 18.0000 0.609208
\(874\) −8.00000 −0.270604
\(875\) 0 0
\(876\) −10.0000 −0.337869
\(877\) 22.0000 0.742887 0.371444 0.928456i \(-0.378863\pi\)
0.371444 + 0.928456i \(0.378863\pi\)
\(878\) 0 0
\(879\) 8.00000 0.269833
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 9.00000 0.303046
\(883\) 36.0000 1.21150 0.605748 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) 36.0000 1.21081
\(885\) 0 0
\(886\) 36.0000 1.20944
\(887\) 8.00000 0.268614 0.134307 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) −18.0000 −0.604040
\(889\) −24.0000 −0.804934
\(890\) 0 0
\(891\) 0 0
\(892\) −22.0000 −0.736614
\(893\) −2.00000 −0.0669274
\(894\) −10.0000 −0.334450
\(895\) 0 0
\(896\) 12.0000 0.400892
\(897\) 24.0000 0.801337
\(898\) −40.0000 −1.33482
\(899\) 48.0000 1.60089
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) 0 0
\(903\) −24.0000 −0.798670
\(904\) 24.0000 0.798228
\(905\) 0 0
\(906\) −2.00000 −0.0664455
\(907\) −24.0000 −0.796907 −0.398453 0.917189i \(-0.630453\pi\)
−0.398453 + 0.917189i \(0.630453\pi\)
\(908\) 20.0000 0.663723
\(909\) −10.0000 −0.331679
\(910\) 0 0
\(911\) 24.0000 0.795155 0.397578 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) −2.00000 −0.0662266
\(913\) 0 0
\(914\) 10.0000 0.330771
\(915\) 0 0
\(916\) 2.00000 0.0660819
\(917\) −80.0000 −2.64183
\(918\) 6.00000 0.198030
\(919\) 10.0000 0.329870 0.164935 0.986304i \(-0.447259\pi\)
0.164935 + 0.986304i \(0.447259\pi\)
\(920\) 0 0
\(921\) 20.0000 0.659022
\(922\) 24.0000 0.790398
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 18.0000 0.591517
\(927\) −4.00000 −0.131377
\(928\) 40.0000 1.31306
\(929\) 6.00000 0.196854 0.0984268 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) 18.0000 0.589926
\(932\) −16.0000 −0.524097
\(933\) −28.0000 −0.916679
\(934\) 8.00000 0.261768
\(935\) 0 0
\(936\) 18.0000 0.588348
\(937\) −10.0000 −0.326686 −0.163343 0.986569i \(-0.552228\pi\)
−0.163343 + 0.986569i \(0.552228\pi\)
\(938\) −8.00000 −0.261209
\(939\) −34.0000 −1.10955
\(940\) 0 0
\(941\) −6.00000 −0.195594 −0.0977972 0.995206i \(-0.531180\pi\)
−0.0977972 + 0.995206i \(0.531180\pi\)
\(942\) −22.0000 −0.716799
\(943\) 32.0000 1.04206
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) 0 0
\(947\) 52.0000 1.68977 0.844886 0.534946i \(-0.179668\pi\)
0.844886 + 0.534946i \(0.179668\pi\)
\(948\) 4.00000 0.129914
\(949\) −60.0000 −1.94768
\(950\) 0 0
\(951\) −12.0000 −0.389127
\(952\) 72.0000 2.33353
\(953\) 24.0000 0.777436 0.388718 0.921357i \(-0.372918\pi\)
0.388718 + 0.921357i \(0.372918\pi\)
\(954\) −2.00000 −0.0647524
\(955\) 0 0
\(956\) −8.00000 −0.258738
\(957\) 0 0
\(958\) 32.0000 1.03387
\(959\) −64.0000 −2.06667
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) −36.0000 −1.16069
\(963\) 12.0000 0.386695
\(964\) −10.0000 −0.322078
\(965\) 0 0
\(966\) 16.0000 0.514792
\(967\) −20.0000 −0.643157 −0.321578 0.946883i \(-0.604213\pi\)
−0.321578 + 0.946883i \(0.604213\pi\)
\(968\) 33.0000 1.06066
\(969\) 12.0000 0.385496
\(970\) 0 0
\(971\) −28.0000 −0.898563 −0.449281 0.893390i \(-0.648320\pi\)
−0.449281 + 0.893390i \(0.648320\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 8.00000 0.256468
\(974\) 20.0000 0.640841
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) −2.00000 −0.0639857 −0.0319928 0.999488i \(-0.510185\pi\)
−0.0319928 + 0.999488i \(0.510185\pi\)
\(978\) 10.0000 0.319765
\(979\) 0 0
\(980\) 0 0
\(981\) −2.00000 −0.0638551
\(982\) 4.00000 0.127645
\(983\) −12.0000 −0.382741 −0.191370 0.981518i \(-0.561293\pi\)
−0.191370 + 0.981518i \(0.561293\pi\)
\(984\) 24.0000 0.765092
\(985\) 0 0
\(986\) 48.0000 1.52863
\(987\) 4.00000 0.127321
\(988\) 12.0000 0.381771
\(989\) −24.0000 −0.763156
\(990\) 0 0
\(991\) 48.0000 1.52477 0.762385 0.647124i \(-0.224028\pi\)
0.762385 + 0.647124i \(0.224028\pi\)
\(992\) 30.0000 0.952501
\(993\) −8.00000 −0.253872
\(994\) 0 0
\(995\) 0 0
\(996\) 4.00000 0.126745
\(997\) 14.0000 0.443384 0.221692 0.975117i \(-0.428842\pi\)
0.221692 + 0.975117i \(0.428842\pi\)
\(998\) −14.0000 −0.443162
\(999\) 6.00000 0.189832
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 3525.2.a.k.1.1 1
5.4 even 2 141.2.a.b.1.1 1
15.14 odd 2 423.2.a.e.1.1 1
20.19 odd 2 2256.2.a.l.1.1 1
35.34 odd 2 6909.2.a.e.1.1 1
40.19 odd 2 9024.2.a.i.1.1 1
40.29 even 2 9024.2.a.bk.1.1 1
60.59 even 2 6768.2.a.h.1.1 1
235.234 odd 2 6627.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
141.2.a.b.1.1 1 5.4 even 2
423.2.a.e.1.1 1 15.14 odd 2
2256.2.a.l.1.1 1 20.19 odd 2
3525.2.a.k.1.1 1 1.1 even 1 trivial
6627.2.a.b.1.1 1 235.234 odd 2
6768.2.a.h.1.1 1 60.59 even 2
6909.2.a.e.1.1 1 35.34 odd 2
9024.2.a.i.1.1 1 40.19 odd 2
9024.2.a.bk.1.1 1 40.29 even 2