Properties

Label 1425.2.a.g.1.1
Level $1425$
Weight $2$
Character 1425.1
Self dual yes
Analytic conductor $11.379$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1425.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.3786822880\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 285)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1425.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} -6.00000 q^{11} +1.00000 q^{12} +2.00000 q^{14} -1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} +1.00000 q^{19} -2.00000 q^{21} -6.00000 q^{22} +8.00000 q^{23} +3.00000 q^{24} -1.00000 q^{27} -2.00000 q^{28} +4.00000 q^{29} +5.00000 q^{32} +6.00000 q^{33} +6.00000 q^{34} -1.00000 q^{36} -4.00000 q^{37} +1.00000 q^{38} -2.00000 q^{42} +2.00000 q^{43} +6.00000 q^{44} +8.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} -6.00000 q^{51} -2.00000 q^{53} -1.00000 q^{54} -6.00000 q^{56} -1.00000 q^{57} +4.00000 q^{58} +12.0000 q^{59} +2.00000 q^{61} +2.00000 q^{63} +7.00000 q^{64} +6.00000 q^{66} +8.00000 q^{67} -6.00000 q^{68} -8.00000 q^{69} +16.0000 q^{71} -3.00000 q^{72} -14.0000 q^{73} -4.00000 q^{74} -1.00000 q^{76} -12.0000 q^{77} +8.00000 q^{79} +1.00000 q^{81} +2.00000 q^{84} +2.00000 q^{86} -4.00000 q^{87} +18.0000 q^{88} -8.00000 q^{92} +8.00000 q^{94} -5.00000 q^{96} +12.0000 q^{97} -3.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

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
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −3.00000 −1.06066
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −6.00000 −1.80907 −0.904534 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 2.00000 0.534522
\(15\) 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
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) −6.00000 −1.27920
\(23\) 8.00000 1.66812 0.834058 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 3.00000 0.612372
\(25\) 0 0
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) −2.00000 −0.377964
\(29\) 4.00000 0.742781 0.371391 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 5.00000 0.883883
\(33\) 6.00000 1.04447
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) −2.00000 −0.308607
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 6.00000 0.904534
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 0 0
\(53\) −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
\(56\) −6.00000 −0.801784
\(57\) −1.00000 −0.132453
\(58\) 4.00000 0.525226
\(59\) 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 \(0.459132\pi\)
\(62\) 0 0
\(63\) 2.00000 0.251976
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) 6.00000 0.738549
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −6.00000 −0.727607
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) 16.0000 1.89885 0.949425 0.313993i \(-0.101667\pi\)
0.949425 + 0.313993i \(0.101667\pi\)
\(72\) −3.00000 −0.353553
\(73\) −14.0000 −1.63858 −0.819288 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) −4.00000 −0.464991
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) −12.0000 −1.36753
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 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
\(85\) 0 0
\(86\) 2.00000 0.215666
\(87\) −4.00000 −0.428845
\(88\) 18.0000 1.91881
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −8.00000 −0.834058
\(93\) 0 0
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) −5.00000 −0.510310
\(97\) 12.0000 1.21842 0.609208 0.793011i \(-0.291488\pi\)
0.609208 + 0.793011i \(0.291488\pi\)
\(98\) −3.00000 −0.303046
\(99\) −6.00000 −0.603023
\(100\) 0 0
\(101\) −18.0000 −1.79107 −0.895533 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) −6.00000 −0.594089
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) 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
\(109\) −6.00000 −0.574696 −0.287348 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) 0 0
\(111\) 4.00000 0.379663
\(112\) −2.00000 −0.188982
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 0 0
\(116\) −4.00000 −0.371391
\(117\) 0 0
\(118\) 12.0000 1.10469
\(119\) 12.0000 1.10004
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 2.00000 0.181071
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 2.00000 0.178174
\(127\) −4.00000 −0.354943 −0.177471 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) −3.00000 −0.265165
\(129\) −2.00000 −0.176090
\(130\) 0 0
\(131\) −2.00000 −0.174741 −0.0873704 0.996176i \(-0.527846\pi\)
−0.0873704 + 0.996176i \(0.527846\pi\)
\(132\) −6.00000 −0.522233
\(133\) 2.00000 0.173422
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) −18.0000 −1.54349
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) −8.00000 −0.681005
\(139\) −16.0000 −1.35710 −0.678551 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 16.0000 1.34269
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −14.0000 −1.15865
\(147\) 3.00000 0.247436
\(148\) 4.00000 0.328798
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) −3.00000 −0.243332
\(153\) 6.00000 0.485071
\(154\) −12.0000 −0.966988
\(155\) 0 0
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 8.00000 0.636446
\(159\) 2.00000 0.158610
\(160\) 0 0
\(161\) 16.0000 1.26098
\(162\) 1.00000 0.0785674
\(163\) 22.0000 1.72317 0.861586 0.507611i \(-0.169471\pi\)
0.861586 + 0.507611i \(0.169471\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −16.0000 −1.23812 −0.619059 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 6.00000 0.462910
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) −2.00000 −0.152499
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) −4.00000 −0.303239
\(175\) 0 0
\(176\) 6.00000 0.452267
\(177\) −12.0000 −0.901975
\(178\) 0 0
\(179\) −20.0000 −1.49487 −0.747435 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) −24.0000 −1.76930
\(185\) 0 0
\(186\) 0 0
\(187\) −36.0000 −2.63258
\(188\) −8.00000 −0.583460
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) 10.0000 0.723575 0.361787 0.932261i \(-0.382167\pi\)
0.361787 + 0.932261i \(0.382167\pi\)
\(192\) −7.00000 −0.505181
\(193\) −24.0000 −1.72756 −0.863779 0.503871i \(-0.831909\pi\)
−0.863779 + 0.503871i \(0.831909\pi\)
\(194\) 12.0000 0.861550
\(195\) 0 0
\(196\) 3.00000 0.214286
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) −6.00000 −0.426401
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 0 0
\(201\) −8.00000 −0.564276
\(202\) −18.0000 −1.26648
\(203\) 8.00000 0.561490
\(204\) 6.00000 0.420084
\(205\) 0 0
\(206\) 8.00000 0.557386
\(207\) 8.00000 0.556038
\(208\) 0 0
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 2.00000 0.137361
\(213\) −16.0000 −1.09630
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) 3.00000 0.204124
\(217\) 0 0
\(218\) −6.00000 −0.406371
\(219\) 14.0000 0.946032
\(220\) 0 0
\(221\) 0 0
\(222\) 4.00000 0.268462
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 10.0000 0.668153
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) 28.0000 1.85843 0.929213 0.369546i \(-0.120487\pi\)
0.929213 + 0.369546i \(0.120487\pi\)
\(228\) 1.00000 0.0662266
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 0 0
\(231\) 12.0000 0.789542
\(232\) −12.0000 −0.787839
\(233\) −22.0000 −1.44127 −0.720634 0.693316i \(-0.756149\pi\)
−0.720634 + 0.693316i \(0.756149\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) −8.00000 −0.519656
\(238\) 12.0000 0.777844
\(239\) 6.00000 0.388108 0.194054 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) 0 0
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) 25.0000 1.60706
\(243\) −1.00000 −0.0641500
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −6.00000 −0.378717 −0.189358 0.981908i \(-0.560641\pi\)
−0.189358 + 0.981908i \(0.560641\pi\)
\(252\) −2.00000 −0.125988
\(253\) −48.0000 −3.01773
\(254\) −4.00000 −0.250982
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 22.0000 1.37232 0.686161 0.727450i \(-0.259294\pi\)
0.686161 + 0.727450i \(0.259294\pi\)
\(258\) −2.00000 −0.124515
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) 4.00000 0.247594
\(262\) −2.00000 −0.123560
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) −18.0000 −1.10782
\(265\) 0 0
\(266\) 2.00000 0.122628
\(267\) 0 0
\(268\) −8.00000 −0.488678
\(269\) 12.0000 0.731653 0.365826 0.930683i \(-0.380786\pi\)
0.365826 + 0.930683i \(0.380786\pi\)
\(270\) 0 0
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) −6.00000 −0.363803
\(273\) 0 0
\(274\) 10.0000 0.604122
\(275\) 0 0
\(276\) 8.00000 0.481543
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) −16.0000 −0.959616
\(279\) 0 0
\(280\) 0 0
\(281\) −4.00000 −0.238620 −0.119310 0.992857i \(-0.538068\pi\)
−0.119310 + 0.992857i \(0.538068\pi\)
\(282\) −8.00000 −0.476393
\(283\) 14.0000 0.832214 0.416107 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) −16.0000 −0.949425
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 5.00000 0.294628
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) −12.0000 −0.703452
\(292\) 14.0000 0.819288
\(293\) −30.0000 −1.75262 −0.876309 0.481749i \(-0.840002\pi\)
−0.876309 + 0.481749i \(0.840002\pi\)
\(294\) 3.00000 0.174964
\(295\) 0 0
\(296\) 12.0000 0.697486
\(297\) 6.00000 0.348155
\(298\) 6.00000 0.347571
\(299\) 0 0
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) −16.0000 −0.920697
\(303\) 18.0000 1.03407
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) 8.00000 0.456584 0.228292 0.973593i \(-0.426686\pi\)
0.228292 + 0.973593i \(0.426686\pi\)
\(308\) 12.0000 0.683763
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) 34.0000 1.92796 0.963982 0.265969i \(-0.0856919\pi\)
0.963982 + 0.265969i \(0.0856919\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 2.00000 0.112867
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 6.00000 0.336994 0.168497 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) 2.00000 0.112154
\(319\) −24.0000 −1.34374
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 16.0000 0.891645
\(323\) 6.00000 0.333849
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 22.0000 1.21847
\(327\) 6.00000 0.331801
\(328\) 0 0
\(329\) 16.0000 0.882109
\(330\) 0 0
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) 0 0
\(333\) −4.00000 −0.219199
\(334\) −16.0000 −0.875481
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) 12.0000 0.653682 0.326841 0.945079i \(-0.394016\pi\)
0.326841 + 0.945079i \(0.394016\pi\)
\(338\) −13.0000 −0.707107
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 0 0
\(342\) 1.00000 0.0540738
\(343\) −20.0000 −1.07990
\(344\) −6.00000 −0.323498
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) −8.00000 −0.429463 −0.214731 0.976673i \(-0.568888\pi\)
−0.214731 + 0.976673i \(0.568888\pi\)
\(348\) 4.00000 0.214423
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −30.0000 −1.59901
\(353\) 34.0000 1.80964 0.904819 0.425797i \(-0.140006\pi\)
0.904819 + 0.425797i \(0.140006\pi\)
\(354\) −12.0000 −0.637793
\(355\) 0 0
\(356\) 0 0
\(357\) −12.0000 −0.635107
\(358\) −20.0000 −1.05703
\(359\) 10.0000 0.527780 0.263890 0.964553i \(-0.414994\pi\)
0.263890 + 0.964553i \(0.414994\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 10.0000 0.525588
\(363\) −25.0000 −1.31216
\(364\) 0 0
\(365\) 0 0
\(366\) −2.00000 −0.104542
\(367\) 14.0000 0.730794 0.365397 0.930852i \(-0.380933\pi\)
0.365397 + 0.930852i \(0.380933\pi\)
\(368\) −8.00000 −0.417029
\(369\) 0 0
\(370\) 0 0
\(371\) −4.00000 −0.207670
\(372\) 0 0
\(373\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(374\) −36.0000 −1.86152
\(375\) 0 0
\(376\) −24.0000 −1.23771
\(377\) 0 0
\(378\) −2.00000 −0.102869
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) 4.00000 0.204926
\(382\) 10.0000 0.511645
\(383\) −8.00000 −0.408781 −0.204390 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) 3.00000 0.153093
\(385\) 0 0
\(386\) −24.0000 −1.22157
\(387\) 2.00000 0.101666
\(388\) −12.0000 −0.609208
\(389\) −14.0000 −0.709828 −0.354914 0.934899i \(-0.615490\pi\)
−0.354914 + 0.934899i \(0.615490\pi\)
\(390\) 0 0
\(391\) 48.0000 2.42746
\(392\) 9.00000 0.454569
\(393\) 2.00000 0.100887
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) 6.00000 0.301511
\(397\) −6.00000 −0.301131 −0.150566 0.988600i \(-0.548110\pi\)
−0.150566 + 0.988600i \(0.548110\pi\)
\(398\) −20.0000 −1.00251
\(399\) −2.00000 −0.100125
\(400\) 0 0
\(401\) 4.00000 0.199750 0.0998752 0.995000i \(-0.468156\pi\)
0.0998752 + 0.995000i \(0.468156\pi\)
\(402\) −8.00000 −0.399004
\(403\) 0 0
\(404\) 18.0000 0.895533
\(405\) 0 0
\(406\) 8.00000 0.397033
\(407\) 24.0000 1.18964
\(408\) 18.0000 0.891133
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) −10.0000 −0.493264
\(412\) −8.00000 −0.394132
\(413\) 24.0000 1.18096
\(414\) 8.00000 0.393179
\(415\) 0 0
\(416\) 0 0
\(417\) 16.0000 0.783523
\(418\) −6.00000 −0.293470
\(419\) −30.0000 −1.46560 −0.732798 0.680446i \(-0.761786\pi\)
−0.732798 + 0.680446i \(0.761786\pi\)
\(420\) 0 0
\(421\) 6.00000 0.292422 0.146211 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) −20.0000 −0.973585
\(423\) 8.00000 0.388973
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) −16.0000 −0.775203
\(427\) 4.00000 0.193574
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 0 0
\(431\) 36.0000 1.73406 0.867029 0.498257i \(-0.166026\pi\)
0.867029 + 0.498257i \(0.166026\pi\)
\(432\) 1.00000 0.0481125
\(433\) 16.0000 0.768911 0.384455 0.923144i \(-0.374389\pi\)
0.384455 + 0.923144i \(0.374389\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 6.00000 0.287348
\(437\) 8.00000 0.382692
\(438\) 14.0000 0.668946
\(439\) 24.0000 1.14546 0.572729 0.819745i \(-0.305885\pi\)
0.572729 + 0.819745i \(0.305885\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) −4.00000 −0.189832
\(445\) 0 0
\(446\) −8.00000 −0.378811
\(447\) −6.00000 −0.283790
\(448\) 14.0000 0.661438
\(449\) −8.00000 −0.377543 −0.188772 0.982021i \(-0.560451\pi\)
−0.188772 + 0.982021i \(0.560451\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −6.00000 −0.282216
\(453\) 16.0000 0.751746
\(454\) 28.0000 1.31411
\(455\) 0 0
\(456\) 3.00000 0.140488
\(457\) 6.00000 0.280668 0.140334 0.990104i \(-0.455182\pi\)
0.140334 + 0.990104i \(0.455182\pi\)
\(458\) 2.00000 0.0934539
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 12.0000 0.558291
\(463\) 14.0000 0.650635 0.325318 0.945605i \(-0.394529\pi\)
0.325318 + 0.945605i \(0.394529\pi\)
\(464\) −4.00000 −0.185695
\(465\) 0 0
\(466\) −22.0000 −1.01913
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 0 0
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) −2.00000 −0.0921551
\(472\) −36.0000 −1.65703
\(473\) −12.0000 −0.551761
\(474\) −8.00000 −0.367452
\(475\) 0 0
\(476\) −12.0000 −0.550019
\(477\) −2.00000 −0.0915737
\(478\) 6.00000 0.274434
\(479\) 18.0000 0.822441 0.411220 0.911536i \(-0.365103\pi\)
0.411220 + 0.911536i \(0.365103\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 18.0000 0.819878
\(483\) −16.0000 −0.728025
\(484\) −25.0000 −1.13636
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −8.00000 −0.362515 −0.181257 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) −6.00000 −0.271607
\(489\) −22.0000 −0.994874
\(490\) 0 0
\(491\) −26.0000 −1.17336 −0.586682 0.809818i \(-0.699566\pi\)
−0.586682 + 0.809818i \(0.699566\pi\)
\(492\) 0 0
\(493\) 24.0000 1.08091
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 32.0000 1.43540
\(498\) 0 0
\(499\) −24.0000 −1.07439 −0.537194 0.843459i \(-0.680516\pi\)
−0.537194 + 0.843459i \(0.680516\pi\)
\(500\) 0 0
\(501\) 16.0000 0.714827
\(502\) −6.00000 −0.267793
\(503\) 36.0000 1.60516 0.802580 0.596544i \(-0.203460\pi\)
0.802580 + 0.596544i \(0.203460\pi\)
\(504\) −6.00000 −0.267261
\(505\) 0 0
\(506\) −48.0000 −2.13386
\(507\) 13.0000 0.577350
\(508\) 4.00000 0.177471
\(509\) −24.0000 −1.06378 −0.531891 0.846813i \(-0.678518\pi\)
−0.531891 + 0.846813i \(0.678518\pi\)
\(510\) 0 0
\(511\) −28.0000 −1.23865
\(512\) −11.0000 −0.486136
\(513\) −1.00000 −0.0441511
\(514\) 22.0000 0.970378
\(515\) 0 0
\(516\) 2.00000 0.0880451
\(517\) −48.0000 −2.11104
\(518\) −8.00000 −0.351500
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) −4.00000 −0.175243 −0.0876216 0.996154i \(-0.527927\pi\)
−0.0876216 + 0.996154i \(0.527927\pi\)
\(522\) 4.00000 0.175075
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) 2.00000 0.0873704
\(525\) 0 0
\(526\) 12.0000 0.523225
\(527\) 0 0
\(528\) −6.00000 −0.261116
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) −2.00000 −0.0867110
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) −24.0000 −1.03664
\(537\) 20.0000 0.863064
\(538\) 12.0000 0.517357
\(539\) 18.0000 0.775315
\(540\) 0 0
\(541\) −42.0000 −1.80572 −0.902861 0.429934i \(-0.858537\pi\)
−0.902861 + 0.429934i \(0.858537\pi\)
\(542\) −20.0000 −0.859074
\(543\) −10.0000 −0.429141
\(544\) 30.0000 1.28624
\(545\) 0 0
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) −10.0000 −0.427179
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) 4.00000 0.170406
\(552\) 24.0000 1.02151
\(553\) 16.0000 0.680389
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) 16.0000 0.678551
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 36.0000 1.51992
\(562\) −4.00000 −0.168730
\(563\) −36.0000 −1.51722 −0.758610 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) 8.00000 0.336861
\(565\) 0 0
\(566\) 14.0000 0.588464
\(567\) 2.00000 0.0839921
\(568\) −48.0000 −2.01404
\(569\) −8.00000 −0.335377 −0.167689 0.985840i \(-0.553630\pi\)
−0.167689 + 0.985840i \(0.553630\pi\)
\(570\) 0 0
\(571\) −24.0000 −1.00437 −0.502184 0.864761i \(-0.667470\pi\)
−0.502184 + 0.864761i \(0.667470\pi\)
\(572\) 0 0
\(573\) −10.0000 −0.417756
\(574\) 0 0
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) −2.00000 −0.0832611 −0.0416305 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) 19.0000 0.790296
\(579\) 24.0000 0.997406
\(580\) 0 0
\(581\) 0 0
\(582\) −12.0000 −0.497416
\(583\) 12.0000 0.496989
\(584\) 42.0000 1.73797
\(585\) 0 0
\(586\) −30.0000 −1.23929
\(587\) 20.0000 0.825488 0.412744 0.910847i \(-0.364570\pi\)
0.412744 + 0.910847i \(0.364570\pi\)
\(588\) −3.00000 −0.123718
\(589\) 0 0
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) 4.00000 0.164399
\(593\) 10.0000 0.410651 0.205325 0.978694i \(-0.434175\pi\)
0.205325 + 0.978694i \(0.434175\pi\)
\(594\) 6.00000 0.246183
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) 20.0000 0.818546
\(598\) 0 0
\(599\) −20.0000 −0.817178 −0.408589 0.912719i \(-0.633979\pi\)
−0.408589 + 0.912719i \(0.633979\pi\)
\(600\) 0 0
\(601\) 22.0000 0.897399 0.448699 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 4.00000 0.163028
\(603\) 8.00000 0.325785
\(604\) 16.0000 0.651031
\(605\) 0 0
\(606\) 18.0000 0.731200
\(607\) −16.0000 −0.649420 −0.324710 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) 5.00000 0.202777
\(609\) −8.00000 −0.324176
\(610\) 0 0
\(611\) 0 0
\(612\) −6.00000 −0.242536
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 8.00000 0.322854
\(615\) 0 0
\(616\) 36.0000 1.45048
\(617\) −34.0000 −1.36879 −0.684394 0.729112i \(-0.739933\pi\)
−0.684394 + 0.729112i \(0.739933\pi\)
\(618\) −8.00000 −0.321807
\(619\) 8.00000 0.321547 0.160774 0.986991i \(-0.448601\pi\)
0.160774 + 0.986991i \(0.448601\pi\)
\(620\) 0 0
\(621\) −8.00000 −0.321029
\(622\) 34.0000 1.36328
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) −10.0000 −0.399680
\(627\) 6.00000 0.239617
\(628\) −2.00000 −0.0798087
\(629\) −24.0000 −0.956943
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) −24.0000 −0.954669
\(633\) 20.0000 0.794929
\(634\) 6.00000 0.238290
\(635\) 0 0
\(636\) −2.00000 −0.0793052
\(637\) 0 0
\(638\) −24.0000 −0.950169
\(639\) 16.0000 0.632950
\(640\) 0 0
\(641\) 40.0000 1.57991 0.789953 0.613168i \(-0.210105\pi\)
0.789953 + 0.613168i \(0.210105\pi\)
\(642\) −12.0000 −0.473602
\(643\) −2.00000 −0.0788723 −0.0394362 0.999222i \(-0.512556\pi\)
−0.0394362 + 0.999222i \(0.512556\pi\)
\(644\) −16.0000 −0.630488
\(645\) 0 0
\(646\) 6.00000 0.236067
\(647\) −20.0000 −0.786281 −0.393141 0.919478i \(-0.628611\pi\)
−0.393141 + 0.919478i \(0.628611\pi\)
\(648\) −3.00000 −0.117851
\(649\) −72.0000 −2.82625
\(650\) 0 0
\(651\) 0 0
\(652\) −22.0000 −0.861586
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) 6.00000 0.234619
\(655\) 0 0
\(656\) 0 0
\(657\) −14.0000 −0.546192
\(658\) 16.0000 0.623745
\(659\) 48.0000 1.86981 0.934907 0.354892i \(-0.115482\pi\)
0.934907 + 0.354892i \(0.115482\pi\)
\(660\) 0 0
\(661\) −22.0000 −0.855701 −0.427850 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) 28.0000 1.08825
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) −4.00000 −0.154997
\(667\) 32.0000 1.23904
\(668\) 16.0000 0.619059
\(669\) 8.00000 0.309298
\(670\) 0 0
\(671\) −12.0000 −0.463255
\(672\) −10.0000 −0.385758
\(673\) −36.0000 −1.38770 −0.693849 0.720121i \(-0.744086\pi\)
−0.693849 + 0.720121i \(0.744086\pi\)
\(674\) 12.0000 0.462223
\(675\) 0 0
\(676\) 13.0000 0.500000
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) −6.00000 −0.230429
\(679\) 24.0000 0.921035
\(680\) 0 0
\(681\) −28.0000 −1.07296
\(682\) 0 0
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 0 0
\(686\) −20.0000 −0.763604
\(687\) −2.00000 −0.0763048
\(688\) −2.00000 −0.0762493
\(689\) 0 0
\(690\) 0 0
\(691\) 32.0000 1.21734 0.608669 0.793424i \(-0.291704\pi\)
0.608669 + 0.793424i \(0.291704\pi\)
\(692\) 6.00000 0.228086
\(693\) −12.0000 −0.455842
\(694\) −8.00000 −0.303676
\(695\) 0 0
\(696\) 12.0000 0.454859
\(697\) 0 0
\(698\) −2.00000 −0.0757011
\(699\) 22.0000 0.832116
\(700\) 0 0
\(701\) −42.0000 −1.58632 −0.793159 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) 0 0
\(703\) −4.00000 −0.150863
\(704\) −42.0000 −1.58293
\(705\) 0 0
\(706\) 34.0000 1.27961
\(707\) −36.0000 −1.35392
\(708\) 12.0000 0.450988
\(709\) −14.0000 −0.525781 −0.262891 0.964826i \(-0.584676\pi\)
−0.262891 + 0.964826i \(0.584676\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) 0 0
\(714\) −12.0000 −0.449089
\(715\) 0 0
\(716\) 20.0000 0.747435
\(717\) −6.00000 −0.224074
\(718\) 10.0000 0.373197
\(719\) −18.0000 −0.671287 −0.335643 0.941989i \(-0.608954\pi\)
−0.335643 + 0.941989i \(0.608954\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 1.00000 0.0372161
\(723\) −18.0000 −0.669427
\(724\) −10.0000 −0.371647
\(725\) 0 0
\(726\) −25.0000 −0.927837
\(727\) −26.0000 −0.964287 −0.482143 0.876092i \(-0.660142\pi\)
−0.482143 + 0.876092i \(0.660142\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 12.0000 0.443836
\(732\) 2.00000 0.0739221
\(733\) 22.0000 0.812589 0.406294 0.913742i \(-0.366821\pi\)
0.406294 + 0.913742i \(0.366821\pi\)
\(734\) 14.0000 0.516749
\(735\) 0 0
\(736\) 40.0000 1.47442
\(737\) −48.0000 −1.76810
\(738\) 0 0
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −4.00000 −0.146845
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 36.0000 1.31629
\(749\) 24.0000 0.876941
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) −8.00000 −0.291730
\(753\) 6.00000 0.218652
\(754\) 0 0
\(755\) 0 0
\(756\) 2.00000 0.0727393
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) 20.0000 0.726433
\(759\) 48.0000 1.74229
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 4.00000 0.144905
\(763\) −12.0000 −0.434429
\(764\) −10.0000 −0.361787
\(765\) 0 0
\(766\) −8.00000 −0.289052
\(767\) 0 0
\(768\) 17.0000 0.613435
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) −22.0000 −0.792311
\(772\) 24.0000 0.863779
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 2.00000 0.0718885
\(775\) 0 0
\(776\) −36.0000 −1.29232
\(777\) 8.00000 0.286998
\(778\) −14.0000 −0.501924
\(779\) 0 0
\(780\) 0 0
\(781\) −96.0000 −3.43515
\(782\) 48.0000 1.71648
\(783\) −4.00000 −0.142948
\(784\) 3.00000 0.107143
\(785\) 0 0
\(786\) 2.00000 0.0713376
\(787\) 28.0000 0.998092 0.499046 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) 6.00000 0.213741
\(789\) −12.0000 −0.427211
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) 18.0000 0.639602
\(793\) 0 0
\(794\) −6.00000 −0.212932
\(795\) 0 0
\(796\) 20.0000 0.708881
\(797\) 26.0000 0.920967 0.460484 0.887668i \(-0.347676\pi\)
0.460484 + 0.887668i \(0.347676\pi\)
\(798\) −2.00000 −0.0707992
\(799\) 48.0000 1.69812
\(800\) 0 0
\(801\) 0 0
\(802\) 4.00000 0.141245
\(803\) 84.0000 2.96430
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) 0 0
\(807\) −12.0000 −0.422420
\(808\) 54.0000 1.89971
\(809\) 2.00000 0.0703163 0.0351581 0.999382i \(-0.488807\pi\)
0.0351581 + 0.999382i \(0.488807\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) −8.00000 −0.280745
\(813\) 20.0000 0.701431
\(814\) 24.0000 0.841200
\(815\) 0 0
\(816\) 6.00000 0.210042
\(817\) 2.00000 0.0699711
\(818\) −10.0000 −0.349642
\(819\) 0 0
\(820\) 0 0
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) −10.0000 −0.348790
\(823\) 14.0000 0.488009 0.244005 0.969774i \(-0.421539\pi\)
0.244005 + 0.969774i \(0.421539\pi\)
\(824\) −24.0000 −0.836080
\(825\) 0 0
\(826\) 24.0000 0.835067
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) −8.00000 −0.278019
\(829\) −6.00000 −0.208389 −0.104194 0.994557i \(-0.533226\pi\)
−0.104194 + 0.994557i \(0.533226\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) 0 0
\(833\) −18.0000 −0.623663
\(834\) 16.0000 0.554035
\(835\) 0 0
\(836\) 6.00000 0.207514
\(837\) 0 0
\(838\) −30.0000 −1.03633
\(839\) 8.00000 0.276191 0.138095 0.990419i \(-0.455902\pi\)
0.138095 + 0.990419i \(0.455902\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 6.00000 0.206774
\(843\) 4.00000 0.137767
\(844\) 20.0000 0.688428
\(845\) 0 0
\(846\) 8.00000 0.275046
\(847\) 50.0000 1.71802
\(848\) 2.00000 0.0686803
\(849\) −14.0000 −0.480479
\(850\) 0 0
\(851\) −32.0000 −1.09695
\(852\) 16.0000 0.548151
\(853\) −26.0000 −0.890223 −0.445112 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) 4.00000 0.136877
\(855\) 0 0
\(856\) −36.0000 −1.23045
\(857\) −46.0000 −1.57133 −0.785665 0.618652i \(-0.787679\pi\)
−0.785665 + 0.618652i \(0.787679\pi\)
\(858\) 0 0
\(859\) 44.0000 1.50126 0.750630 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 36.0000 1.22616
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) −5.00000 −0.170103
\(865\) 0 0
\(866\) 16.0000 0.543702
\(867\) −19.0000 −0.645274
\(868\) 0 0
\(869\) −48.0000 −1.62829
\(870\) 0 0
\(871\) 0 0
\(872\) 18.0000 0.609557
\(873\) 12.0000 0.406138
\(874\) 8.00000 0.270604
\(875\) 0 0
\(876\) −14.0000 −0.473016
\(877\) 24.0000 0.810422 0.405211 0.914223i \(-0.367198\pi\)
0.405211 + 0.914223i \(0.367198\pi\)
\(878\) 24.0000 0.809961
\(879\) 30.0000 1.01187
\(880\) 0 0
\(881\) −42.0000 −1.41502 −0.707508 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) −3.00000 −0.101015
\(883\) −2.00000 −0.0673054 −0.0336527 0.999434i \(-0.510714\pi\)
−0.0336527 + 0.999434i \(0.510714\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) −12.0000 −0.402694
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) −6.00000 −0.201008
\(892\) 8.00000 0.267860
\(893\) 8.00000 0.267710
\(894\) −6.00000 −0.200670
\(895\) 0 0
\(896\) −6.00000 −0.200446
\(897\) 0 0
\(898\) −8.00000 −0.266963
\(899\) 0 0
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) 0 0
\(903\) −4.00000 −0.133112
\(904\) −18.0000 −0.598671
\(905\) 0 0
\(906\) 16.0000 0.531564
\(907\) 52.0000 1.72663 0.863316 0.504664i \(-0.168384\pi\)
0.863316 + 0.504664i \(0.168384\pi\)
\(908\) −28.0000 −0.929213
\(909\) −18.0000 −0.597022
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 1.00000 0.0331133
\(913\) 0 0
\(914\) 6.00000 0.198462
\(915\) 0 0
\(916\) −2.00000 −0.0660819
\(917\) −4.00000 −0.132092
\(918\) −6.00000 −0.198030
\(919\) −40.0000 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) 0 0
\(921\) −8.00000 −0.263609
\(922\) 30.0000 0.987997
\(923\) 0 0
\(924\) −12.0000 −0.394771
\(925\) 0 0
\(926\) 14.0000 0.460069
\(927\) 8.00000 0.262754
\(928\) 20.0000 0.656532
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 0 0
\(931\) −3.00000 −0.0983210
\(932\) 22.0000 0.720634
\(933\) −34.0000 −1.11311
\(934\) −12.0000 −0.392652
\(935\) 0 0
\(936\) 0 0
\(937\) −30.0000 −0.980057 −0.490029 0.871706i \(-0.663014\pi\)
−0.490029 + 0.871706i \(0.663014\pi\)
\(938\) 16.0000 0.522419
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) 44.0000 1.43436 0.717180 0.696888i \(-0.245433\pi\)
0.717180 + 0.696888i \(0.245433\pi\)
\(942\) −2.00000 −0.0651635
\(943\) 0 0
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) −12.0000 −0.390154
\(947\) −48.0000 −1.55979 −0.779895 0.625910i \(-0.784728\pi\)
−0.779895 + 0.625910i \(0.784728\pi\)
\(948\) 8.00000 0.259828
\(949\) 0 0
\(950\) 0 0
\(951\) −6.00000 −0.194563
\(952\) −36.0000 −1.16677
\(953\) 30.0000 0.971795 0.485898 0.874016i \(-0.338493\pi\)
0.485898 + 0.874016i \(0.338493\pi\)
\(954\) −2.00000 −0.0647524
\(955\) 0 0
\(956\) −6.00000 −0.194054
\(957\) 24.0000 0.775810
\(958\) 18.0000 0.581554
\(959\) 20.0000 0.645834
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 12.0000 0.386695
\(964\) −18.0000 −0.579741
\(965\) 0 0
\(966\) −16.0000 −0.514792
\(967\) 14.0000 0.450210 0.225105 0.974335i \(-0.427728\pi\)
0.225105 + 0.974335i \(0.427728\pi\)
\(968\) −75.0000 −2.41059
\(969\) −6.00000 −0.192748
\(970\) 0 0
\(971\) −8.00000 −0.256732 −0.128366 0.991727i \(-0.540973\pi\)
−0.128366 + 0.991727i \(0.540973\pi\)
\(972\) 1.00000 0.0320750
\(973\) −32.0000 −1.02587
\(974\) −8.00000 −0.256337
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) −22.0000 −0.703482
\(979\) 0 0
\(980\) 0 0
\(981\) −6.00000 −0.191565
\(982\) −26.0000 −0.829693
\(983\) −8.00000 −0.255160 −0.127580 0.991828i \(-0.540721\pi\)
−0.127580 + 0.991828i \(0.540721\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 24.0000 0.764316
\(987\) −16.0000 −0.509286
\(988\) 0 0
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 0 0
\(993\) −28.0000 −0.888553
\(994\) 32.0000 1.01498
\(995\) 0 0
\(996\) 0 0
\(997\) −14.0000 −0.443384 −0.221692 0.975117i \(-0.571158\pi\)
−0.221692 + 0.975117i \(0.571158\pi\)
\(998\) −24.0000 −0.759707
\(999\) 4.00000 0.126554
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 1425.2.a.g.1.1 1
3.2 odd 2 4275.2.a.h.1.1 1
5.2 odd 4 1425.2.c.c.799.2 2
5.3 odd 4 1425.2.c.c.799.1 2
5.4 even 2 285.2.a.a.1.1 1
15.14 odd 2 855.2.a.c.1.1 1
20.19 odd 2 4560.2.a.h.1.1 1
95.94 odd 2 5415.2.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.a.1.1 1 5.4 even 2
855.2.a.c.1.1 1 15.14 odd 2
1425.2.a.g.1.1 1 1.1 even 1 trivial
1425.2.c.c.799.1 2 5.3 odd 4
1425.2.c.c.799.2 2 5.2 odd 4
4275.2.a.h.1.1 1 3.2 odd 2
4560.2.a.h.1.1 1 20.19 odd 2
5415.2.a.h.1.1 1 95.94 odd 2