Properties

Label 5415.2.a.h.1.1
Level $5415$
Weight $2$
Character 5415.1
Self dual yes
Analytic conductor $43.239$
Analytic rank $2$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5415 = 3 \cdot 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5415.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.2389926945\)
Analytic rank: \(2\)
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 \(\chi\) \(=\) 5415.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -2.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -2.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -6.00000 q^{11} +1.00000 q^{12} -2.00000 q^{14} +1.00000 q^{15} -1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} +1.00000 q^{20} +2.00000 q^{21} -6.00000 q^{22} -8.00000 q^{23} +3.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} +2.00000 q^{28} -4.00000 q^{29} +1.00000 q^{30} +5.00000 q^{32} +6.00000 q^{33} -6.00000 q^{34} +2.00000 q^{35} -1.00000 q^{36} -4.00000 q^{37} +3.00000 q^{40} +2.00000 q^{42} -2.00000 q^{43} +6.00000 q^{44} -1.00000 q^{45} -8.00000 q^{46} -8.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} +6.00000 q^{51} -2.00000 q^{53} -1.00000 q^{54} +6.00000 q^{55} +6.00000 q^{56} -4.00000 q^{58} -12.0000 q^{59} -1.00000 q^{60} +2.00000 q^{61} -2.00000 q^{63} +7.00000 q^{64} +6.00000 q^{66} +8.00000 q^{67} +6.00000 q^{68} +8.00000 q^{69} +2.00000 q^{70} -16.0000 q^{71} -3.00000 q^{72} +14.0000 q^{73} -4.00000 q^{74} -1.00000 q^{75} +12.0000 q^{77} -8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{84} +6.00000 q^{85} -2.00000 q^{86} +4.00000 q^{87} +18.0000 q^{88} -1.00000 q^{90} +8.00000 q^{92} -8.00000 q^{94} -5.00000 q^{96} +12.0000 q^{97} -3.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

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
\(5\) −1.00000 −0.447214
\(6\) −1.00000 −0.408248
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) −3.00000 −1.06066
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −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
\(15\) 1.00000 0.258199
\(16\) −1.00000 −0.250000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 1.00000 0.235702
\(19\) 0 0
\(20\) 1.00000 0.223607
\(21\) 2.00000 0.436436
\(22\) −6.00000 −1.27920
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 3.00000 0.612372
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 2.00000 0.377964
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 1.00000 0.182574
\(31\) 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
\(35\) 2.00000 0.338062
\(36\) −1.00000 −0.166667
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 3.00000 0.474342
\(41\) 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 \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 6.00000 0.904534
\(45\) −1.00000 −0.149071
\(46\) −8.00000 −1.17954
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) 1.00000 0.141421
\(51\) 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
\(55\) 6.00000 0.809040
\(56\) 6.00000 0.801784
\(57\) 0 0
\(58\) −4.00000 −0.525226
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) −1.00000 −0.129099
\(61\) 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
\(70\) 2.00000 0.239046
\(71\) −16.0000 −1.89885 −0.949425 0.313993i \(-0.898333\pi\)
−0.949425 + 0.313993i \(0.898333\pi\)
\(72\) −3.00000 −0.353553
\(73\) 14.0000 1.63858 0.819288 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) −4.00000 −0.464991
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 12.0000 1.36753
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 1.00000 0.111803
\(81\) 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
\(86\) −2.00000 −0.215666
\(87\) 4.00000 0.428845
\(88\) 18.0000 1.91881
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) −1.00000 −0.105409
\(91\) 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
\(100\) −1.00000 −0.100000
\(101\) −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
\(105\) −2.00000 −0.195180
\(106\) −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 \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 6.00000 0.572078
\(111\) 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 \(0.408930\pi\)
\(114\) 0 0
\(115\) 8.00000 0.746004
\(116\) 4.00000 0.371391
\(117\) 0 0
\(118\) −12.0000 −1.10469
\(119\) 12.0000 1.10004
\(120\) −3.00000 −0.273861
\(121\) 25.0000 2.27273
\(122\) 2.00000 0.181071
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) −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
\(133\) 0 0
\(134\) 8.00000 0.691095
\(135\) 1.00000 0.0860663
\(136\) 18.0000 1.54349
\(137\) −10.0000 −0.854358 −0.427179 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 8.00000 0.681005
\(139\) −16.0000 −1.35710 −0.678551 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) −2.00000 −0.169031
\(141\) 8.00000 0.673722
\(142\) −16.0000 −1.34269
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 4.00000 0.332182
\(146\) 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 \(0.420959\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 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
\(154\) 12.0000 0.966988
\(155\) 0 0
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) −8.00000 −0.636446
\(159\) 2.00000 0.158610
\(160\) −5.00000 −0.395285
\(161\) 16.0000 1.26098
\(162\) 1.00000 0.0785674
\(163\) −22.0000 −1.72317 −0.861586 0.507611i \(-0.830529\pi\)
−0.861586 + 0.507611i \(0.830529\pi\)
\(164\) 0 0
\(165\) −6.00000 −0.467099
\(166\) 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
\(170\) 6.00000 0.460179
\(171\) 0 0
\(172\) 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
\(175\) −2.00000 −0.151186
\(176\) 6.00000 0.452267
\(177\) 12.0000 0.901975
\(178\) 0 0
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 1.00000 0.0745356
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) 24.0000 1.76930
\(185\) 4.00000 0.294086
\(186\) 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 \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) −6.00000 −0.426401
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) −3.00000 −0.212132
\(201\) −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
\(209\) 0 0
\(210\) −2.00000 −0.138013
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 2.00000 0.137361
\(213\) 16.0000 1.09630
\(214\) 12.0000 0.820303
\(215\) 2.00000 0.136399
\(216\) 3.00000 0.204124
\(217\) 0 0
\(218\) 6.00000 0.406371
\(219\) −14.0000 −0.946032
\(220\) −6.00000 −0.404520
\(221\) 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
\(225\) 1.00000 0.0666667
\(226\) 6.00000 0.399114
\(227\) 28.0000 1.85843 0.929213 0.369546i \(-0.120487\pi\)
0.929213 + 0.369546i \(0.120487\pi\)
\(228\) 0 0
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 8.00000 0.527504
\(231\) −12.0000 −0.789542
\(232\) 12.0000 0.787839
\(233\) 22.0000 1.44127 0.720634 0.693316i \(-0.243851\pi\)
0.720634 + 0.693316i \(0.243851\pi\)
\(234\) 0 0
\(235\) 8.00000 0.521862
\(236\) 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 \(0.437836\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −18.0000 −1.15948 −0.579741 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) 25.0000 1.60706
\(243\) −1.00000 −0.0641500
\(244\) −2.00000 −0.128037
\(245\) 3.00000 0.191663
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) −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
\(255\) −6.00000 −0.375735
\(256\) −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 \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) −18.0000 −1.10782
\(265\) 2.00000 0.122859
\(266\) 0 0
\(267\) 0 0
\(268\) −8.00000 −0.488678
\(269\) −12.0000 −0.731653 −0.365826 0.930683i \(-0.619214\pi\)
−0.365826 + 0.930683i \(0.619214\pi\)
\(270\) 1.00000 0.0608581
\(271\) −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
\(275\) −6.00000 −0.361814
\(276\) −8.00000 −0.481543
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) −16.0000 −0.959616
\(279\) 0 0
\(280\) −6.00000 −0.358569
\(281\) 4.00000 0.238620 0.119310 0.992857i \(-0.461932\pi\)
0.119310 + 0.992857i \(0.461932\pi\)
\(282\) 8.00000 0.476393
\(283\) −14.0000 −0.832214 −0.416107 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) 16.0000 0.949425
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 5.00000 0.294628
\(289\) 19.0000 1.11765
\(290\) 4.00000 0.234888
\(291\) −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
\(295\) 12.0000 0.698667
\(296\) 12.0000 0.697486
\(297\) 6.00000 0.348155
\(298\) 6.00000 0.347571
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) 4.00000 0.230556
\(302\) 16.0000 0.920697
\(303\) 18.0000 1.03407
\(304\) 0 0
\(305\) −2.00000 −0.114520
\(306\) −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 \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) −2.00000 −0.112867
\(315\) 2.00000 0.112687
\(316\) 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
\(320\) −7.00000 −0.391312
\(321\) −12.0000 −0.669775
\(322\) 16.0000 0.891645
\(323\) 0 0
\(324\) −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
\(330\) −6.00000 −0.330289
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) 0 0
\(333\) −4.00000 −0.219199
\(334\) −16.0000 −0.875481
\(335\) −8.00000 −0.437087
\(336\) −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
\(340\) −6.00000 −0.325396
\(341\) 0 0
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 6.00000 0.323498
\(345\) −8.00000 −0.430706
\(346\) −6.00000 −0.322562
\(347\) 8.00000 0.429463 0.214731 0.976673i \(-0.431112\pi\)
0.214731 + 0.976673i \(0.431112\pi\)
\(348\) −4.00000 −0.214423
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) −2.00000 −0.106904
\(351\) 0 0
\(352\) −30.0000 −1.59901
\(353\) −34.0000 −1.80964 −0.904819 0.425797i \(-0.859994\pi\)
−0.904819 + 0.425797i \(0.859994\pi\)
\(354\) 12.0000 0.637793
\(355\) 16.0000 0.849192
\(356\) 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 \(0.414994\pi\)
\(360\) 3.00000 0.158114
\(361\) 0 0
\(362\) −10.0000 −0.525588
\(363\) −25.0000 −1.31216
\(364\) 0 0
\(365\) −14.0000 −0.732793
\(366\) −2.00000 −0.104542
\(367\) −14.0000 −0.730794 −0.365397 0.930852i \(-0.619067\pi\)
−0.365397 + 0.930852i \(0.619067\pi\)
\(368\) 8.00000 0.417029
\(369\) 0 0
\(370\) 4.00000 0.207950
\(371\) 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
\(375\) 1.00000 0.0516398
\(376\) 24.0000 1.23771
\(377\) 0 0
\(378\) 2.00000 0.102869
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 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
\(385\) −12.0000 −0.611577
\(386\) −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
\(395\) 8.00000 0.402524
\(396\) 6.00000 0.301511
\(397\) 6.00000 0.301131 0.150566 0.988600i \(-0.451890\pi\)
0.150566 + 0.988600i \(0.451890\pi\)
\(398\) −20.0000 −1.00251
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −4.00000 −0.199750 −0.0998752 0.995000i \(-0.531844\pi\)
−0.0998752 + 0.995000i \(0.531844\pi\)
\(402\) −8.00000 −0.399004
\(403\) 0 0
\(404\) 18.0000 0.895533
\(405\) −1.00000 −0.0496904
\(406\) 8.00000 0.397033
\(407\) 24.0000 1.18964
\(408\) −18.0000 −0.891133
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 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
\(418\) 0 0
\(419\) −30.0000 −1.46560 −0.732798 0.680446i \(-0.761786\pi\)
−0.732798 + 0.680446i \(0.761786\pi\)
\(420\) 2.00000 0.0975900
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 20.0000 0.973585
\(423\) −8.00000 −0.388973
\(424\) 6.00000 0.291386
\(425\) −6.00000 −0.291043
\(426\) 16.0000 0.775203
\(427\) −4.00000 −0.193574
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 2.00000 0.0964486
\(431\) −36.0000 −1.73406 −0.867029 0.498257i \(-0.833974\pi\)
−0.867029 + 0.498257i \(0.833974\pi\)
\(432\) 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
\(435\) −4.00000 −0.191785
\(436\) −6.00000 −0.287348
\(437\) 0 0
\(438\) −14.0000 −0.668946
\(439\) −24.0000 −1.14546 −0.572729 0.819745i \(-0.694115\pi\)
−0.572729 + 0.819745i \(0.694115\pi\)
\(440\) −18.0000 −0.858116
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) −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 \(-0.439549\pi\)
0.188772 + 0.982021i \(0.439549\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0 0
\(452\) −6.00000 −0.282216
\(453\) −16.0000 −0.751746
\(454\) 28.0000 1.31411
\(455\) 0 0
\(456\) 0 0
\(457\) −6.00000 −0.280668 −0.140334 0.990104i \(-0.544818\pi\)
−0.140334 + 0.990104i \(0.544818\pi\)
\(458\) 2.00000 0.0934539
\(459\) 6.00000 0.280056
\(460\) −8.00000 −0.373002
\(461\) 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 \(-0.605471\pi\)
−0.325318 + 0.945605i \(0.605471\pi\)
\(464\) 4.00000 0.185695
\(465\) 0 0
\(466\) 22.0000 1.01913
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) −16.0000 −0.738811
\(470\) 8.00000 0.369012
\(471\) 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 \(0.365103\pi\)
\(480\) 5.00000 0.228218
\(481\) 0 0
\(482\) −18.0000 −0.819878
\(483\) −16.0000 −0.728025
\(484\) −25.0000 −1.13636
\(485\) −12.0000 −0.544892
\(486\) −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
\(490\) 3.00000 0.135526
\(491\) −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
\(495\) 6.00000 0.269680
\(496\) 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 \(0.680516\pi\)
\(500\) 1.00000 0.0447214
\(501\) 16.0000 0.714827
\(502\) −6.00000 −0.267793
\(503\) −36.0000 −1.60516 −0.802580 0.596544i \(-0.796540\pi\)
−0.802580 + 0.596544i \(0.796540\pi\)
\(504\) 6.00000 0.267261
\(505\) 18.0000 0.800989
\(506\) 48.0000 2.13386
\(507\) 13.0000 0.577350
\(508\) 4.00000 0.177471
\(509\) 24.0000 1.06378 0.531891 0.846813i \(-0.321482\pi\)
0.531891 + 0.846813i \(0.321482\pi\)
\(510\) −6.00000 −0.265684
\(511\) −28.0000 −1.23865
\(512\) −11.0000 −0.486136
\(513\) 0 0
\(514\) 22.0000 0.970378
\(515\) −8.00000 −0.352522
\(516\) −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 \(-0.472073\pi\)
0.0876216 + 0.996154i \(0.472073\pi\)
\(522\) −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
\(525\) 2.00000 0.0872872
\(526\) −12.0000 −0.523225
\(527\) 0 0
\(528\) −6.00000 −0.261116
\(529\) 41.0000 1.78261
\(530\) 2.00000 0.0868744
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −12.0000 −0.518805
\(536\) −24.0000 −1.03664
\(537\) −20.0000 −0.863064
\(538\) −12.0000 −0.517357
\(539\) 18.0000 0.775315
\(540\) −1.00000 −0.0430331
\(541\) −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
\(545\) −6.00000 −0.257012
\(546\) 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
\(550\) −6.00000 −0.255841
\(551\) 0 0
\(552\) −24.0000 −1.02151
\(553\) 16.0000 0.680389
\(554\) −2.00000 −0.0849719
\(555\) −4.00000 −0.169791
\(556\) 16.0000 0.678551
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −2.00000 −0.0845154
\(561\) −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
\(565\) −6.00000 −0.252422
\(566\) −14.0000 −0.588464
\(567\) −2.00000 −0.0839921
\(568\) 48.0000 2.01404
\(569\) 8.00000 0.335377 0.167689 0.985840i \(-0.446370\pi\)
0.167689 + 0.985840i \(0.446370\pi\)
\(570\) 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
\(575\) −8.00000 −0.333623
\(576\) 7.00000 0.291667
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 19.0000 0.790296
\(579\) 24.0000 0.997406
\(580\) −4.00000 −0.166091
\(581\) 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 \(-0.635430\pi\)
−0.412744 + 0.910847i \(0.635430\pi\)
\(588\) −3.00000 −0.123718
\(589\) 0 0
\(590\) 12.0000 0.494032
\(591\) −6.00000 −0.246807
\(592\) 4.00000 0.164399
\(593\) −10.0000 −0.410651 −0.205325 0.978694i \(-0.565825\pi\)
−0.205325 + 0.978694i \(0.565825\pi\)
\(594\) 6.00000 0.246183
\(595\) −12.0000 −0.491952
\(596\) −6.00000 −0.245770
\(597\) 20.0000 0.818546
\(598\) 0 0
\(599\) 20.0000 0.817178 0.408589 0.912719i \(-0.366021\pi\)
0.408589 + 0.912719i \(0.366021\pi\)
\(600\) 3.00000 0.122474
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 4.00000 0.163028
\(603\) 8.00000 0.325785
\(604\) −16.0000 −0.651031
\(605\) −25.0000 −1.01639
\(606\) 18.0000 0.731200
\(607\) −16.0000 −0.649420 −0.324710 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) 0 0
\(609\) −8.00000 −0.324176
\(610\) −2.00000 −0.0809776
\(611\) 0 0
\(612\) 6.00000 0.242536
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) 8.00000 0.322854
\(615\) 0 0
\(616\) −36.0000 −1.45048
\(617\) 34.0000 1.36879 0.684394 0.729112i \(-0.260067\pi\)
0.684394 + 0.729112i \(0.260067\pi\)
\(618\) −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
\(625\) 1.00000 0.0400000
\(626\) 10.0000 0.399680
\(627\) 0 0
\(628\) 2.00000 0.0798087
\(629\) 24.0000 0.956943
\(630\) 2.00000 0.0796819
\(631\) −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
\(635\) 4.00000 0.158735
\(636\) −2.00000 −0.0793052
\(637\) 0 0
\(638\) 24.0000 0.950169
\(639\) −16.0000 −0.632950
\(640\) 3.00000 0.118585
\(641\) −40.0000 −1.57991 −0.789953 0.613168i \(-0.789895\pi\)
−0.789953 + 0.613168i \(0.789895\pi\)
\(642\) −12.0000 −0.473602
\(643\) 2.00000 0.0788723 0.0394362 0.999222i \(-0.487444\pi\)
0.0394362 + 0.999222i \(0.487444\pi\)
\(644\) −16.0000 −0.630488
\(645\) −2.00000 −0.0787499
\(646\) 0 0
\(647\) 20.0000 0.786281 0.393141 0.919478i \(-0.371389\pi\)
0.393141 + 0.919478i \(0.371389\pi\)
\(648\) −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 \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) −6.00000 −0.234619
\(655\) 2.00000 0.0781465
\(656\) 0 0
\(657\) 14.0000 0.546192
\(658\) 16.0000 0.623745
\(659\) −48.0000 −1.86981 −0.934907 0.354892i \(-0.884518\pi\)
−0.934907 + 0.354892i \(0.884518\pi\)
\(660\) 6.00000 0.233550
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) −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
\(670\) −8.00000 −0.309067
\(671\) −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
\(675\) −1.00000 −0.0384900
\(676\) 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
\(680\) −18.0000 −0.690268
\(681\) −28.0000 −1.07296
\(682\) 0 0
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 0 0
\(685\) 10.0000 0.382080
\(686\) 20.0000 0.763604
\(687\) −2.00000 −0.0763048
\(688\) 2.00000 0.0762493
\(689\) 0 0
\(690\) −8.00000 −0.304555
\(691\) 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
\(695\) 16.0000 0.606915
\(696\) −12.0000 −0.454859
\(697\) 0 0
\(698\) −2.00000 −0.0757011
\(699\) −22.0000 −0.832116
\(700\) 2.00000 0.0755929
\(701\) −42.0000 −1.58632 −0.793159 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −42.0000 −1.58293
\(705\) −8.00000 −0.301297
\(706\) −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 \(0.584676\pi\)
\(710\) 16.0000 0.600469
\(711\) −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 \(0.608954\pi\)
\(720\) 1.00000 0.0372678
\(721\) −16.0000 −0.595871
\(722\) 0 0
\(723\) 18.0000 0.669427
\(724\) 10.0000 0.371647
\(725\) −4.00000 −0.148556
\(726\) −25.0000 −0.927837
\(727\) 26.0000 0.964287 0.482143 0.876092i \(-0.339858\pi\)
0.482143 + 0.876092i \(0.339858\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −14.0000 −0.518163
\(731\) 12.0000 0.443836
\(732\) 2.00000 0.0739221
\(733\) −22.0000 −0.812589 −0.406294 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) −14.0000 −0.516749
\(735\) −3.00000 −0.110657
\(736\) −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 \(0.619908\pi\)
\(740\) −4.00000 −0.147043
\(741\) 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
\(745\) −6.00000 −0.219823
\(746\) 0 0
\(747\) 0 0
\(748\) −36.0000 −1.31629
\(749\) −24.0000 −0.876941
\(750\) 1.00000 0.0365148
\(751\) 40.0000 1.45962 0.729810 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 8.00000 0.291730
\(753\) 6.00000 0.218652
\(754\) 0 0
\(755\) −16.0000 −0.582300
\(756\) −2.00000 −0.0727393
\(757\) −38.0000 −1.38113 −0.690567 0.723269i \(-0.742639\pi\)
−0.690567 + 0.723269i \(0.742639\pi\)
\(758\) −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
\(765\) 6.00000 0.216930
\(766\) −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 \(0.581229\pi\)
\(770\) −12.0000 −0.432450
\(771\) −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
\(785\) 2.00000 0.0713831
\(786\) 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
\(790\) 8.00000 0.284627
\(791\) −12.0000 −0.426671
\(792\) 18.0000 0.639602
\(793\) 0 0
\(794\) 6.00000 0.212932
\(795\) −2.00000 −0.0709327
\(796\) 20.0000 0.708881
\(797\) 26.0000 0.920967 0.460484 0.887668i \(-0.347676\pi\)
0.460484 + 0.887668i \(0.347676\pi\)
\(798\) 0 0
\(799\) 48.0000 1.69812
\(800\) 5.00000 0.176777
\(801\) 0 0
\(802\) −4.00000 −0.141245
\(803\) −84.0000 −2.96430
\(804\) 8.00000 0.282138
\(805\) −16.0000 −0.563926
\(806\) 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 \(0.488807\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −20.0000 −0.702295 −0.351147 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(812\) −8.00000 −0.280745
\(813\) 20.0000 0.701431
\(814\) 24.0000 0.841200
\(815\) 22.0000 0.770626
\(816\) −6.00000 −0.210042
\(817\) 0 0
\(818\) 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 \(-0.578461\pi\)
−0.244005 + 0.969774i \(0.578461\pi\)
\(824\) −24.0000 −0.836080
\(825\) 6.00000 0.208893
\(826\) 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 \(-0.466774\pi\)
0.104194 + 0.994557i \(0.466774\pi\)
\(830\) 0 0
\(831\) 2.00000 0.0693792
\(832\) 0 0
\(833\) 18.0000 0.623663
\(834\) 16.0000 0.554035
\(835\) 16.0000 0.553703
\(836\) 0 0
\(837\) 0 0
\(838\) −30.0000 −1.03633
\(839\) −8.00000 −0.276191 −0.138095 0.990419i \(-0.544098\pi\)
−0.138095 + 0.990419i \(0.544098\pi\)
\(840\) 6.00000 0.207020
\(841\) −13.0000 −0.448276
\(842\) −6.00000 −0.206774
\(843\) −4.00000 −0.137767
\(844\) −20.0000 −0.688428
\(845\) 13.0000 0.447214
\(846\) −8.00000 −0.275046
\(847\) −50.0000 −1.71802
\(848\) 2.00000 0.0686803
\(849\) 14.0000 0.480479
\(850\) −6.00000 −0.205798
\(851\) 32.0000 1.09695
\(852\) −16.0000 −0.548151
\(853\) 26.0000 0.890223 0.445112 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) −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 \(0.229750\pi\)
\(860\) −2.00000 −0.0681994
\(861\) 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
\(865\) 6.00000 0.204006
\(866\) 16.0000 0.543702
\(867\) −19.0000 −0.645274
\(868\) 0 0
\(869\) 48.0000 1.62829
\(870\) −4.00000 −0.135613
\(871\) 0 0
\(872\) −18.0000 −0.609557
\(873\) 12.0000 0.406138
\(874\) 0 0
\(875\) 2.00000 0.0676123
\(876\) 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
\(880\) −6.00000 −0.202260
\(881\) −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 \(-0.489286\pi\)
0.0336527 + 0.999434i \(0.489286\pi\)
\(884\) 0 0
\(885\) −12.0000 −0.403376
\(886\) −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
\(893\) 0 0
\(894\) −6.00000 −0.200670
\(895\) −20.0000 −0.668526
\(896\) 6.00000 0.200446
\(897\) 0 0
\(898\) 8.00000 0.266963
\(899\) 0 0
\(900\) −1.00000 −0.0333333
\(901\) 12.0000 0.399778
\(902\) 0 0
\(903\) −4.00000 −0.133112
\(904\) −18.0000 −0.598671
\(905\) 10.0000 0.332411
\(906\) −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 \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −6.00000 −0.198462
\(915\) 2.00000 0.0661180
\(916\) −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 \(0.729333\pi\)
\(920\) −24.0000 −0.791257
\(921\) −8.00000 −0.263609
\(922\) 30.0000 0.987997
\(923\) 0 0
\(924\) 12.0000 0.394771
\(925\) −4.00000 −0.131519
\(926\) −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
\(931\) 0 0
\(932\) −22.0000 −0.720634
\(933\) −34.0000 −1.11311
\(934\) 12.0000 0.392652
\(935\) −36.0000 −1.17733
\(936\) 0 0
\(937\) 30.0000 0.980057 0.490029 0.871706i \(-0.336986\pi\)
0.490029 + 0.871706i \(0.336986\pi\)
\(938\) −16.0000 −0.522419
\(939\) −10.0000 −0.326338
\(940\) −8.00000 −0.260931
\(941\) −44.0000 −1.43436 −0.717180 0.696888i \(-0.754567\pi\)
−0.717180 + 0.696888i \(0.754567\pi\)
\(942\) 2.00000 0.0651635
\(943\) 0 0
\(944\) 12.0000 0.390567
\(945\) −2.00000 −0.0650600
\(946\) 12.0000 0.390154
\(947\) 48.0000 1.55979 0.779895 0.625910i \(-0.215272\pi\)
0.779895 + 0.625910i \(0.215272\pi\)
\(948\) −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
\(955\) −10.0000 −0.323592
\(956\) −6.00000 −0.194054
\(957\) −24.0000 −0.775810
\(958\) 18.0000 0.581554
\(959\) 20.0000 0.645834
\(960\) 7.00000 0.225924
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 12.0000 0.386695
\(964\) 18.0000 0.579741
\(965\) 24.0000 0.772587
\(966\) −16.0000 −0.514792
\(967\) −14.0000 −0.450210 −0.225105 0.974335i \(-0.572272\pi\)
−0.225105 + 0.974335i \(0.572272\pi\)
\(968\) −75.0000 −2.41059
\(969\) 0 0
\(970\) −12.0000 −0.385297
\(971\) 8.00000 0.256732 0.128366 0.991727i \(-0.459027\pi\)
0.128366 + 0.991727i \(0.459027\pi\)
\(972\) 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
\(980\) −3.00000 −0.0958315
\(981\) 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
\(985\) −6.00000 −0.191176
\(986\) 24.0000 0.764316
\(987\) −16.0000 −0.509286
\(988\) 0 0
\(989\) 16.0000 0.508770
\(990\) 6.00000 0.190693
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 0 0
\(993\) 28.0000 0.888553
\(994\) 32.0000 1.01498
\(995\) 20.0000 0.634043
\(996\) 0 0
\(997\) 14.0000 0.443384 0.221692 0.975117i \(-0.428842\pi\)
0.221692 + 0.975117i \(0.428842\pi\)
\(998\) −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 5415.2.a.h.1.1 1
19.18 odd 2 285.2.a.a.1.1 1
57.56 even 2 855.2.a.c.1.1 1
76.75 even 2 4560.2.a.h.1.1 1
95.18 even 4 1425.2.c.c.799.2 2
95.37 even 4 1425.2.c.c.799.1 2
95.94 odd 2 1425.2.a.g.1.1 1
285.284 even 2 4275.2.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.a.1.1 1 19.18 odd 2
855.2.a.c.1.1 1 57.56 even 2
1425.2.a.g.1.1 1 95.94 odd 2
1425.2.c.c.799.1 2 95.37 even 4
1425.2.c.c.799.2 2 95.18 even 4
4275.2.a.h.1.1 1 285.284 even 2
4560.2.a.h.1.1 1 76.75 even 2
5415.2.a.h.1.1 1 1.1 even 1 trivial