Properties

Label 285.2.a.a.1.1
Level $285$
Weight $2$
Character 285.1
Self dual yes
Analytic conductor $2.276$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 285 = 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 285.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.27573645761\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 285.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -2.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -2.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -6.00000 q^{11} -1.00000 q^{12} +2.00000 q^{14} -1.00000 q^{15} -1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} +1.00000 q^{19} +1.00000 q^{20} -2.00000 q^{21} +6.00000 q^{22} -8.00000 q^{23} +3.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} +2.00000 q^{28} +4.00000 q^{29} +1.00000 q^{30} -5.00000 q^{32} -6.00000 q^{33} +6.00000 q^{34} +2.00000 q^{35} -1.00000 q^{36} +4.00000 q^{37} -1.00000 q^{38} -3.00000 q^{40} +2.00000 q^{42} -2.00000 q^{43} +6.00000 q^{44} -1.00000 q^{45} +8.00000 q^{46} -8.00000 q^{47} -1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} -6.00000 q^{51} +2.00000 q^{53} -1.00000 q^{54} +6.00000 q^{55} -6.00000 q^{56} +1.00000 q^{57} -4.00000 q^{58} +12.0000 q^{59} +1.00000 q^{60} +2.00000 q^{61} -2.00000 q^{63} +7.00000 q^{64} +6.00000 q^{66} -8.00000 q^{67} +6.00000 q^{68} -8.00000 q^{69} -2.00000 q^{70} +16.0000 q^{71} +3.00000 q^{72} +14.0000 q^{73} -4.00000 q^{74} +1.00000 q^{75} -1.00000 q^{76} +12.0000 q^{77} +8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{84} +6.00000 q^{85} +2.00000 q^{86} +4.00000 q^{87} -18.0000 q^{88} +1.00000 q^{90} +8.00000 q^{92} +8.00000 q^{94} -1.00000 q^{95} -5.00000 q^{96} -12.0000 q^{97} +3.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

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 \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 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
\(19\) 1.00000 0.229416
\(20\) 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 \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 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 \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) −1.00000 −0.162221
\(39\) 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 \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) −1.00000 −0.136083
\(55\) 6.00000 0.809040
\(56\) −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 \(0.214642\pi\)
\(60\) 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 \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 6.00000 0.727607
\(69\) −8.00000 −0.963087
\(70\) −2.00000 −0.239046
\(71\) 16.0000 1.89885 0.949425 0.313993i \(-0.101667\pi\)
0.949425 + 0.313993i \(0.101667\pi\)
\(72\) 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
\(76\) −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 \(0.351411\pi\)
\(80\) 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
\(95\) −1.00000 −0.102598
\(96\) −5.00000 −0.510310
\(97\) −12.0000 −1.21842 −0.609208 0.793011i \(-0.708512\pi\)
−0.609208 + 0.793011i \(0.708512\pi\)
\(98\) 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 \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 0 0
\(105\) 2.00000 0.195180
\(106\) −2.00000 −0.194257
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −6.00000 −0.574696 −0.287348 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) −6.00000 −0.572078
\(111\) 4.00000 0.379663
\(112\) 2.00000 0.188982
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 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 \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 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
\(135\) −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 \(-0.725663\pi\)
−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 \(-0.525431\pi\)
−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 \(-0.287514\pi\)
0.619059 + 0.785345i \(0.287514\pi\)
\(168\) −6.00000 −0.462910
\(169\) −13.0000 −1.00000
\(170\) −6.00000 −0.460179
\(171\) 1.00000 0.0764719
\(172\) 2.00000 0.152499
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) −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 \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 1.00000 0.0745356
\(181\) 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
\(185\) −4.00000 −0.294086
\(186\) 0 0
\(187\) 36.0000 2.63258
\(188\) 8.00000 0.583460
\(189\) −2.00000 −0.145479
\(190\) 1.00000 0.0725476
\(191\) 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 \(-0.168091\pi\)
0.863779 + 0.503871i \(0.168091\pi\)
\(194\) 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
\(209\) −6.00000 −0.415029
\(210\) −2.00000 −0.138013
\(211\) −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
\(215\) 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 \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 10.0000 0.668153
\(225\) 1.00000 0.0666667
\(226\) 6.00000 0.399114
\(227\) −28.0000 −1.85843 −0.929213 0.369546i \(-0.879513\pi\)
−0.929213 + 0.369546i \(0.879513\pi\)
\(228\) −1.00000 −0.0662266
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) −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 \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) −25.0000 −1.60706
\(243\) 1.00000 0.0641500
\(244\) −2.00000 −0.128037
\(245\) 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 \(-0.740706\pi\)
−0.686161 + 0.727450i \(0.740706\pi\)
\(258\) 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
\(266\) 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 \(0.380786\pi\)
\(270\) 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 \(-0.538068\pi\)
−0.119310 + 0.992857i \(0.538068\pi\)
\(282\) 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
\(285\) −1.00000 −0.0592349
\(286\) 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 \(-0.159998\pi\)
0.876309 + 0.481749i \(0.159998\pi\)
\(294\) 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
\(304\) −1.00000 −0.0573539
\(305\) −2.00000 −0.114520
\(306\) 6.00000 0.342997
\(307\) −8.00000 −0.456584 −0.228292 0.973593i \(-0.573314\pi\)
−0.228292 + 0.973593i \(0.573314\pi\)
\(308\) −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 \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) −2.00000 −0.112154
\(319\) −24.0000 −1.34374
\(320\) −7.00000 −0.391312
\(321\) −12.0000 −0.669775
\(322\) −16.0000 −0.891645
\(323\) −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
\(330\) −6.00000 −0.330289
\(331\) 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
\(335\) 8.00000 0.437087
\(336\) 2.00000 0.109109
\(337\) −12.0000 −0.653682 −0.326841 0.945079i \(-0.605984\pi\)
−0.326841 + 0.945079i \(0.605984\pi\)
\(338\) 13.0000 0.707107
\(339\) −6.00000 −0.325875
\(340\) −6.00000 −0.325396
\(341\) 0 0
\(342\) −1.00000 −0.0540738
\(343\) 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
\(361\) 1.00000 0.0526316
\(362\) −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 \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 1.00000 0.0512989
\(381\) 4.00000 0.204926
\(382\) −10.0000 −0.511645
\(383\) 8.00000 0.408781 0.204390 0.978889i \(-0.434479\pi\)
0.204390 + 0.978889i \(0.434479\pi\)
\(384\) 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
\(399\) −2.00000 −0.100125
\(400\) −1.00000 −0.0500000
\(401\) 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
\(405\) −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 \(-0.579522\pi\)
−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 \(0.761786\pi\)
\(420\) −2.00000 −0.0975900
\(421\) 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
\(425\) −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 \(-0.166026\pi\)
0.867029 + 0.498257i \(0.166026\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −16.0000 −0.768911 −0.384455 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) 0 0
\(435\) −4.00000 −0.191785
\(436\) 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 \(0.305885\pi\)
\(440\) 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 \(-0.560451\pi\)
−0.188772 + 0.982021i \(0.560451\pi\)
\(450\) −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
\(456\) 3.00000 0.140488
\(457\) −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
\(475\) 1.00000 0.0458831
\(476\) −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 \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 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 \(-0.678518\pi\)
−0.531891 + 0.846813i \(0.678518\pi\)
\(510\) −6.00000 −0.265684
\(511\) −28.0000 −1.23865
\(512\) 11.0000 0.486136
\(513\) 1.00000 0.0441511
\(514\) 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 \(-0.527927\pi\)
−0.0876216 + 0.996154i \(0.527927\pi\)
\(522\) −4.00000 −0.175075
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) 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
\(532\) 2.00000 0.0867110
\(533\) 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 \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) 10.0000 0.427179
\(549\) 2.00000 0.0853579
\(550\) 6.00000 0.255841
\(551\) 4.00000 0.170406
\(552\) −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 \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 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 \(-0.553630\pi\)
−0.167689 + 0.985840i \(0.553630\pi\)
\(570\) 1.00000 0.0418854
\(571\) −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 \(-0.633979\pi\)
−0.408589 + 0.912719i \(0.633979\pi\)
\(600\) 3.00000 0.122474
\(601\) 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
\(605\) −25.0000 −1.01639
\(606\) 18.0000 0.731200
\(607\) 16.0000 0.649420 0.324710 0.945814i \(-0.394733\pi\)
0.324710 + 0.945814i \(0.394733\pi\)
\(608\) −5.00000 −0.202777
\(609\) −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
\(627\) −6.00000 −0.239617
\(628\) 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 \(-0.210105\pi\)
0.789953 + 0.613168i \(0.210105\pi\)
\(642\) 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
\(646\) 6.00000 0.236067
\(647\) 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 \(-0.115482\pi\)
0.934907 + 0.354892i \(0.115482\pi\)
\(660\) −6.00000 −0.233550
\(661\) −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
\(665\) 2.00000 0.0775567
\(666\) −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 \(-0.255914\pi\)
0.693849 + 0.720121i \(0.255914\pi\)
\(674\) 12.0000 0.462223
\(675\) 1.00000 0.0384900
\(676\) 13.0000 0.500000
\(677\) 18.0000 0.691796 0.345898 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) 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 \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 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
\(703\) 4.00000 0.150863
\(704\) −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
\(722\) −1.00000 −0.0372161
\(723\) 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 \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 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 \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 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
\(760\) −3.00000 −0.108821
\(761\) 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 \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) 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 \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) −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 \(-0.652324\pi\)
−0.460484 + 0.887668i \(0.652324\pi\)
\(798\) 2.00000 0.0707992
\(799\) 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 \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 8.00000 0.280745
\(813\) −20.0000 −0.701431
\(814\) 24.0000 0.841200
\(815\) 22.0000 0.770626
\(816\) 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 \(-0.578461\pi\)
−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 \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 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
\(835\) −16.0000 −0.553703
\(836\) 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 \(0.455902\pi\)
\(840\) 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
\(855\) −1.00000 −0.0341993
\(856\) −36.0000 −1.23045
\(857\) 46.0000 1.57133 0.785665 0.618652i \(-0.212321\pi\)
0.785665 + 0.618652i \(0.212321\pi\)
\(858\) 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 \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) −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
\(874\) 8.00000 0.270604
\(875\) 2.00000 0.0676123
\(876\) −14.0000 −0.473016
\(877\) −24.0000 −0.810422 −0.405211 0.914223i \(-0.632802\pi\)
−0.405211 + 0.914223i \(0.632802\pi\)
\(878\) −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 \(-0.201714\pi\)
0.805841 + 0.592132i \(0.201714\pi\)
\(888\) 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
\(895\) 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 \(-0.831616\pi\)
−0.863316 + 0.504664i \(0.831616\pi\)
\(908\) 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
\(915\) −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
\(931\) −3.00000 −0.0983210
\(932\) −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 \(-0.245433\pi\)
0.717180 + 0.696888i \(0.245433\pi\)
\(942\) 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
\(950\) −1.00000 −0.0324443
\(951\) −6.00000 −0.194563
\(952\) 36.0000 1.16677
\(953\) −30.0000 −0.971795 −0.485898 0.874016i \(-0.661507\pi\)
−0.485898 + 0.874016i \(0.661507\pi\)
\(954\) −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
\(969\) −6.00000 −0.192748
\(970\) −12.0000 −0.385297
\(971\) −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 \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) 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 \(-0.459279\pi\)
0.127580 + 0.991828i \(0.459279\pi\)
\(984\) 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 \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 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 285.2.a.a.1.1 1
3.2 odd 2 855.2.a.c.1.1 1
4.3 odd 2 4560.2.a.h.1.1 1
5.2 odd 4 1425.2.c.c.799.1 2
5.3 odd 4 1425.2.c.c.799.2 2
5.4 even 2 1425.2.a.g.1.1 1
15.14 odd 2 4275.2.a.h.1.1 1
19.18 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 1.1 even 1 trivial
855.2.a.c.1.1 1 3.2 odd 2
1425.2.a.g.1.1 1 5.4 even 2
1425.2.c.c.799.1 2 5.2 odd 4
1425.2.c.c.799.2 2 5.3 odd 4
4275.2.a.h.1.1 1 15.14 odd 2
4560.2.a.h.1.1 1 4.3 odd 2
5415.2.a.h.1.1 1 19.18 odd 2