Properties

Label 3450.2.a.p.1.1
Level $3450$
Weight $2$
Character 3450.1
Self dual yes
Analytic conductor $27.548$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(27.5483886973\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 690)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3450.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -4.00000 q^{11} -1.00000 q^{12} +6.00000 q^{13} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} -4.00000 q^{22} -1.00000 q^{23} -1.00000 q^{24} +6.00000 q^{26} -1.00000 q^{27} -6.00000 q^{29} -8.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} -6.00000 q^{37} +4.00000 q^{38} -6.00000 q^{39} +10.0000 q^{41} -4.00000 q^{43} -4.00000 q^{44} -1.00000 q^{46} +8.00000 q^{47} -1.00000 q^{48} -7.00000 q^{49} -6.00000 q^{51} +6.00000 q^{52} +14.0000 q^{53} -1.00000 q^{54} -4.00000 q^{57} -6.00000 q^{58} +10.0000 q^{61} -8.00000 q^{62} +1.00000 q^{64} +4.00000 q^{66} -4.00000 q^{67} +6.00000 q^{68} +1.00000 q^{69} +8.00000 q^{71} +1.00000 q^{72} -2.00000 q^{73} -6.00000 q^{74} +4.00000 q^{76} -6.00000 q^{78} -12.0000 q^{79} +1.00000 q^{81} +10.0000 q^{82} +16.0000 q^{83} -4.00000 q^{86} +6.00000 q^{87} -4.00000 q^{88} -2.00000 q^{89} -1.00000 q^{92} +8.00000 q^{93} +8.00000 q^{94} -1.00000 q^{96} +14.0000 q^{97} -7.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) −1.00000 −0.288675
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 0 0
\(15\) 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
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) −1.00000 −0.208514
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 6.00000 1.17670
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 1.00000 0.176777
\(33\) 4.00000 0.696311
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 4.00000 0.648886
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) −1.00000 −0.144338
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 6.00000 0.832050
\(53\) 14.0000 1.92305 0.961524 0.274721i \(-0.0885855\pi\)
0.961524 + 0.274721i \(0.0885855\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) −6.00000 −0.787839
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) −8.00000 −1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 6.00000 0.727607
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 1.00000 0.117851
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) −6.00000 −0.679366
\(79\) −12.0000 −1.35011 −0.675053 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 10.0000 1.10432
\(83\) 16.0000 1.75623 0.878114 0.478451i \(-0.158802\pi\)
0.878114 + 0.478451i \(0.158802\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 6.00000 0.643268
\(88\) −4.00000 −0.426401
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) 8.00000 0.829561
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 14.0000 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) −7.00000 −0.707107
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) −6.00000 −0.594089
\(103\) 16.0000 1.57653 0.788263 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) 14.0000 1.35980
\(107\) −8.00000 −0.773389 −0.386695 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) 0 0
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) 6.00000 0.554700
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 10.0000 0.905357
\(123\) −10.0000 −0.901670
\(124\) −8.00000 −0.718421
\(125\) 0 0
\(126\) 0 0
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) 1.00000 0.0883883
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) 4.00000 0.348155
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 1.00000 0.0851257
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 8.00000 0.671345
\(143\) −24.0000 −2.00698
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −2.00000 −0.165521
\(147\) 7.00000 0.577350
\(148\) −6.00000 −0.493197
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 4.00000 0.324443
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) −6.00000 −0.480384
\(157\) −6.00000 −0.478852 −0.239426 0.970915i \(-0.576959\pi\)
−0.239426 + 0.970915i \(0.576959\pi\)
\(158\) −12.0000 −0.954669
\(159\) −14.0000 −1.11027
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 10.0000 0.780869
\(165\) 0 0
\(166\) 16.0000 1.24184
\(167\) 16.0000 1.23812 0.619059 0.785345i \(-0.287514\pi\)
0.619059 + 0.785345i \(0.287514\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) −4.00000 −0.304997
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) 6.00000 0.454859
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) 0 0
\(178\) −2.00000 −0.149906
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) −10.0000 −0.739221
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) 8.00000 0.586588
\(187\) −24.0000 −1.75505
\(188\) 8.00000 0.583460
\(189\) 0 0
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) −26.0000 −1.85242 −0.926212 0.377004i \(-0.876954\pi\)
−0.926212 + 0.377004i \(0.876954\pi\)
\(198\) −4.00000 −0.284268
\(199\) −12.0000 −0.850657 −0.425329 0.905039i \(-0.639842\pi\)
−0.425329 + 0.905039i \(0.639842\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 18.0000 1.26648
\(203\) 0 0
\(204\) −6.00000 −0.420084
\(205\) 0 0
\(206\) 16.0000 1.11477
\(207\) −1.00000 −0.0695048
\(208\) 6.00000 0.416025
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 14.0000 0.961524
\(213\) −8.00000 −0.548151
\(214\) −8.00000 −0.546869
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 2.00000 0.135457
\(219\) 2.00000 0.135147
\(220\) 0 0
\(221\) 36.0000 2.42162
\(222\) 6.00000 0.402694
\(223\) 20.0000 1.33930 0.669650 0.742677i \(-0.266444\pi\)
0.669650 + 0.742677i \(0.266444\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 14.0000 0.931266
\(227\) −8.00000 −0.530979 −0.265489 0.964114i \(-0.585534\pi\)
−0.265489 + 0.964114i \(0.585534\pi\)
\(228\) −4.00000 −0.264906
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) −22.0000 −1.44127 −0.720634 0.693316i \(-0.756149\pi\)
−0.720634 + 0.693316i \(0.756149\pi\)
\(234\) 6.00000 0.392232
\(235\) 0 0
\(236\) 0 0
\(237\) 12.0000 0.779484
\(238\) 0 0
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 5.00000 0.321412
\(243\) −1.00000 −0.0641500
\(244\) 10.0000 0.640184
\(245\) 0 0
\(246\) −10.0000 −0.637577
\(247\) 24.0000 1.52708
\(248\) −8.00000 −0.508001
\(249\) −16.0000 −1.01396
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) −12.0000 −0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 4.00000 0.249029
\(259\) 0 0
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 8.00000 0.494242
\(263\) 32.0000 1.97320 0.986602 0.163144i \(-0.0521635\pi\)
0.986602 + 0.163144i \(0.0521635\pi\)
\(264\) 4.00000 0.246183
\(265\) 0 0
\(266\) 0 0
\(267\) 2.00000 0.122398
\(268\) −4.00000 −0.244339
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 6.00000 0.363803
\(273\) 0 0
\(274\) −2.00000 −0.120824
\(275\) 0 0
\(276\) 1.00000 0.0601929
\(277\) 30.0000 1.80253 0.901263 0.433273i \(-0.142641\pi\)
0.901263 + 0.433273i \(0.142641\pi\)
\(278\) 12.0000 0.719712
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) 30.0000 1.78965 0.894825 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) −8.00000 −0.476393
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) −24.0000 −1.41915
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) −14.0000 −0.820695
\(292\) −2.00000 −0.117041
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 7.00000 0.408248
\(295\) 0 0
\(296\) −6.00000 −0.348743
\(297\) 4.00000 0.232104
\(298\) 10.0000 0.579284
\(299\) −6.00000 −0.346989
\(300\) 0 0
\(301\) 0 0
\(302\) 16.0000 0.920697
\(303\) −18.0000 −1.03407
\(304\) 4.00000 0.229416
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 0 0
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) −6.00000 −0.339683
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) −6.00000 −0.338600
\(315\) 0 0
\(316\) −12.0000 −0.675053
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) −14.0000 −0.785081
\(319\) 24.0000 1.34374
\(320\) 0 0
\(321\) 8.00000 0.446516
\(322\) 0 0
\(323\) 24.0000 1.33540
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) −2.00000 −0.110600
\(328\) 10.0000 0.552158
\(329\) 0 0
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 16.0000 0.878114
\(333\) −6.00000 −0.328798
\(334\) 16.0000 0.875481
\(335\) 0 0
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 23.0000 1.25104
\(339\) −14.0000 −0.760376
\(340\) 0 0
\(341\) 32.0000 1.73290
\(342\) 4.00000 0.216295
\(343\) 0 0
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) 28.0000 1.50312 0.751559 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) 6.00000 0.321634
\(349\) −34.0000 −1.81998 −0.909989 0.414632i \(-0.863910\pi\)
−0.909989 + 0.414632i \(0.863910\pi\)
\(350\) 0 0
\(351\) −6.00000 −0.320256
\(352\) −4.00000 −0.213201
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −2.00000 −0.106000
\(357\) 0 0
\(358\) 0 0
\(359\) −8.00000 −0.422224 −0.211112 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) −6.00000 −0.315353
\(363\) −5.00000 −0.262432
\(364\) 0 0
\(365\) 0 0
\(366\) −10.0000 −0.522708
\(367\) 32.0000 1.67039 0.835193 0.549957i \(-0.185356\pi\)
0.835193 + 0.549957i \(0.185356\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 10.0000 0.520579
\(370\) 0 0
\(371\) 0 0
\(372\) 8.00000 0.414781
\(373\) 18.0000 0.932005 0.466002 0.884783i \(-0.345694\pi\)
0.466002 + 0.884783i \(0.345694\pi\)
\(374\) −24.0000 −1.24101
\(375\) 0 0
\(376\) 8.00000 0.412568
\(377\) −36.0000 −1.85409
\(378\) 0 0
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 0 0
\(381\) 12.0000 0.614779
\(382\) −8.00000 −0.409316
\(383\) −16.0000 −0.817562 −0.408781 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) −4.00000 −0.203331
\(388\) 14.0000 0.710742
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) −6.00000 −0.303433
\(392\) −7.00000 −0.353553
\(393\) −8.00000 −0.403547
\(394\) −26.0000 −1.30986
\(395\) 0 0
\(396\) −4.00000 −0.201008
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) −12.0000 −0.601506
\(399\) 0 0
\(400\) 0 0
\(401\) 6.00000 0.299626 0.149813 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) 4.00000 0.199502
\(403\) −48.0000 −2.39105
\(404\) 18.0000 0.895533
\(405\) 0 0
\(406\) 0 0
\(407\) 24.0000 1.18964
\(408\) −6.00000 −0.297044
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 0 0
\(411\) 2.00000 0.0986527
\(412\) 16.0000 0.788263
\(413\) 0 0
\(414\) −1.00000 −0.0491473
\(415\) 0 0
\(416\) 6.00000 0.294174
\(417\) −12.0000 −0.587643
\(418\) −16.0000 −0.782586
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) −38.0000 −1.85201 −0.926003 0.377515i \(-0.876779\pi\)
−0.926003 + 0.377515i \(0.876779\pi\)
\(422\) −20.0000 −0.973585
\(423\) 8.00000 0.388973
\(424\) 14.0000 0.679900
\(425\) 0 0
\(426\) −8.00000 −0.387601
\(427\) 0 0
\(428\) −8.00000 −0.386695
\(429\) 24.0000 1.15873
\(430\) 0 0
\(431\) 8.00000 0.385346 0.192673 0.981263i \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) −4.00000 −0.191346
\(438\) 2.00000 0.0955637
\(439\) −24.0000 −1.14546 −0.572729 0.819745i \(-0.694115\pi\)
−0.572729 + 0.819745i \(0.694115\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) 36.0000 1.71235
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 6.00000 0.284747
\(445\) 0 0
\(446\) 20.0000 0.947027
\(447\) −10.0000 −0.472984
\(448\) 0 0
\(449\) 10.0000 0.471929 0.235965 0.971762i \(-0.424175\pi\)
0.235965 + 0.971762i \(0.424175\pi\)
\(450\) 0 0
\(451\) −40.0000 −1.88353
\(452\) 14.0000 0.658505
\(453\) −16.0000 −0.751746
\(454\) −8.00000 −0.375459
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) −14.0000 −0.654177
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) 2.00000 0.0931493 0.0465746 0.998915i \(-0.485169\pi\)
0.0465746 + 0.998915i \(0.485169\pi\)
\(462\) 0 0
\(463\) 20.0000 0.929479 0.464739 0.885448i \(-0.346148\pi\)
0.464739 + 0.885448i \(0.346148\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) −22.0000 −1.01913
\(467\) −8.00000 −0.370196 −0.185098 0.982720i \(-0.559260\pi\)
−0.185098 + 0.982720i \(0.559260\pi\)
\(468\) 6.00000 0.277350
\(469\) 0 0
\(470\) 0 0
\(471\) 6.00000 0.276465
\(472\) 0 0
\(473\) 16.0000 0.735681
\(474\) 12.0000 0.551178
\(475\) 0 0
\(476\) 0 0
\(477\) 14.0000 0.641016
\(478\) −8.00000 −0.365911
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) −36.0000 −1.64146
\(482\) 2.00000 0.0910975
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −20.0000 −0.906287 −0.453143 0.891438i \(-0.649697\pi\)
−0.453143 + 0.891438i \(0.649697\pi\)
\(488\) 10.0000 0.452679
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) −10.0000 −0.450835
\(493\) −36.0000 −1.62136
\(494\) 24.0000 1.07981
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) 0 0
\(498\) −16.0000 −0.716977
\(499\) 12.0000 0.537194 0.268597 0.963253i \(-0.413440\pi\)
0.268597 + 0.963253i \(0.413440\pi\)
\(500\) 0 0
\(501\) −16.0000 −0.714827
\(502\) −12.0000 −0.535586
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 4.00000 0.177822
\(507\) −23.0000 −1.02147
\(508\) −12.0000 −0.532414
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −4.00000 −0.176604
\(514\) −6.00000 −0.264649
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) −32.0000 −1.40736
\(518\) 0 0
\(519\) 18.0000 0.790112
\(520\) 0 0
\(521\) −42.0000 −1.84005 −0.920027 0.391856i \(-0.871833\pi\)
−0.920027 + 0.391856i \(0.871833\pi\)
\(522\) −6.00000 −0.262613
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 8.00000 0.349482
\(525\) 0 0
\(526\) 32.0000 1.39527
\(527\) −48.0000 −2.09091
\(528\) 4.00000 0.174078
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 60.0000 2.59889
\(534\) 2.00000 0.0865485
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) 0 0
\(538\) 10.0000 0.431131
\(539\) 28.0000 1.20605
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) −8.00000 −0.343629
\(543\) 6.00000 0.257485
\(544\) 6.00000 0.257248
\(545\) 0 0
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) −2.00000 −0.0854358
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) −24.0000 −1.02243
\(552\) 1.00000 0.0425628
\(553\) 0 0
\(554\) 30.0000 1.27458
\(555\) 0 0
\(556\) 12.0000 0.508913
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) −8.00000 −0.338667
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) 24.0000 1.01328
\(562\) 30.0000 1.26547
\(563\) 8.00000 0.337160 0.168580 0.985688i \(-0.446082\pi\)
0.168580 + 0.985688i \(0.446082\pi\)
\(564\) −8.00000 −0.336861
\(565\) 0 0
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) 8.00000 0.335673
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) −24.0000 −1.00349
\(573\) 8.00000 0.334205
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −34.0000 −1.41544 −0.707719 0.706494i \(-0.750276\pi\)
−0.707719 + 0.706494i \(0.750276\pi\)
\(578\) 19.0000 0.790296
\(579\) 10.0000 0.415586
\(580\) 0 0
\(581\) 0 0
\(582\) −14.0000 −0.580319
\(583\) −56.0000 −2.31928
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) 14.0000 0.578335
\(587\) 4.00000 0.165098 0.0825488 0.996587i \(-0.473694\pi\)
0.0825488 + 0.996587i \(0.473694\pi\)
\(588\) 7.00000 0.288675
\(589\) −32.0000 −1.31854
\(590\) 0 0
\(591\) 26.0000 1.06950
\(592\) −6.00000 −0.246598
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) 12.0000 0.491127
\(598\) −6.00000 −0.245358
\(599\) −40.0000 −1.63436 −0.817178 0.576386i \(-0.804463\pi\)
−0.817178 + 0.576386i \(0.804463\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) 16.0000 0.651031
\(605\) 0 0
\(606\) −18.0000 −0.731200
\(607\) 28.0000 1.13648 0.568242 0.822861i \(-0.307624\pi\)
0.568242 + 0.822861i \(0.307624\pi\)
\(608\) 4.00000 0.162221
\(609\) 0 0
\(610\) 0 0
\(611\) 48.0000 1.94187
\(612\) 6.00000 0.242536
\(613\) −14.0000 −0.565455 −0.282727 0.959200i \(-0.591239\pi\)
−0.282727 + 0.959200i \(0.591239\pi\)
\(614\) −12.0000 −0.484281
\(615\) 0 0
\(616\) 0 0
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) −16.0000 −0.643614
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) −8.00000 −0.320771
\(623\) 0 0
\(624\) −6.00000 −0.240192
\(625\) 0 0
\(626\) −10.0000 −0.399680
\(627\) 16.0000 0.638978
\(628\) −6.00000 −0.239426
\(629\) −36.0000 −1.43541
\(630\) 0 0
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) −12.0000 −0.477334
\(633\) 20.0000 0.794929
\(634\) −18.0000 −0.714871
\(635\) 0 0
\(636\) −14.0000 −0.555136
\(637\) −42.0000 −1.66410
\(638\) 24.0000 0.950169
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) 8.00000 0.315735
\(643\) 44.0000 1.73519 0.867595 0.497271i \(-0.165665\pi\)
0.867595 + 0.497271i \(0.165665\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 24.0000 0.944267
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) 14.0000 0.547862 0.273931 0.961749i \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(654\) −2.00000 −0.0782062
\(655\) 0 0
\(656\) 10.0000 0.390434
\(657\) −2.00000 −0.0780274
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) 34.0000 1.32245 0.661223 0.750189i \(-0.270038\pi\)
0.661223 + 0.750189i \(0.270038\pi\)
\(662\) −20.0000 −0.777322
\(663\) −36.0000 −1.39812
\(664\) 16.0000 0.620920
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) 6.00000 0.232321
\(668\) 16.0000 0.619059
\(669\) −20.0000 −0.773245
\(670\) 0 0
\(671\) −40.0000 −1.54418
\(672\) 0 0
\(673\) 22.0000 0.848038 0.424019 0.905653i \(-0.360619\pi\)
0.424019 + 0.905653i \(0.360619\pi\)
\(674\) 14.0000 0.539260
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) −14.0000 −0.537667
\(679\) 0 0
\(680\) 0 0
\(681\) 8.00000 0.306561
\(682\) 32.0000 1.22534
\(683\) −44.0000 −1.68361 −0.841807 0.539779i \(-0.818508\pi\)
−0.841807 + 0.539779i \(0.818508\pi\)
\(684\) 4.00000 0.152944
\(685\) 0 0
\(686\) 0 0
\(687\) 14.0000 0.534133
\(688\) −4.00000 −0.152499
\(689\) 84.0000 3.20015
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) −18.0000 −0.684257
\(693\) 0 0
\(694\) 28.0000 1.06287
\(695\) 0 0
\(696\) 6.00000 0.227429
\(697\) 60.0000 2.27266
\(698\) −34.0000 −1.28692
\(699\) 22.0000 0.832116
\(700\) 0 0
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) −6.00000 −0.226455
\(703\) −24.0000 −0.905177
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) 0 0
\(708\) 0 0
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) −12.0000 −0.450035
\(712\) −2.00000 −0.0749532
\(713\) 8.00000 0.299602
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 8.00000 0.298765
\(718\) −8.00000 −0.298557
\(719\) −40.0000 −1.49175 −0.745874 0.666087i \(-0.767968\pi\)
−0.745874 + 0.666087i \(0.767968\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −3.00000 −0.111648
\(723\) −2.00000 −0.0743808
\(724\) −6.00000 −0.222988
\(725\) 0 0
\(726\) −5.00000 −0.185567
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −24.0000 −0.887672
\(732\) −10.0000 −0.369611
\(733\) −6.00000 −0.221615 −0.110808 0.993842i \(-0.535344\pi\)
−0.110808 + 0.993842i \(0.535344\pi\)
\(734\) 32.0000 1.18114
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 16.0000 0.589368
\(738\) 10.0000 0.368105
\(739\) −12.0000 −0.441427 −0.220714 0.975339i \(-0.570839\pi\)
−0.220714 + 0.975339i \(0.570839\pi\)
\(740\) 0 0
\(741\) −24.0000 −0.881662
\(742\) 0 0
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 8.00000 0.293294
\(745\) 0 0
\(746\) 18.0000 0.659027
\(747\) 16.0000 0.585409
\(748\) −24.0000 −0.877527
\(749\) 0 0
\(750\) 0 0
\(751\) −28.0000 −1.02173 −0.510867 0.859660i \(-0.670676\pi\)
−0.510867 + 0.859660i \(0.670676\pi\)
\(752\) 8.00000 0.291730
\(753\) 12.0000 0.437304
\(754\) −36.0000 −1.31104
\(755\) 0 0
\(756\) 0 0
\(757\) −54.0000 −1.96266 −0.981332 0.192323i \(-0.938398\pi\)
−0.981332 + 0.192323i \(0.938398\pi\)
\(758\) 4.00000 0.145287
\(759\) −4.00000 −0.145191
\(760\) 0 0
\(761\) 10.0000 0.362500 0.181250 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) 12.0000 0.434714
\(763\) 0 0
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) −16.0000 −0.578103
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) −6.00000 −0.216366 −0.108183 0.994131i \(-0.534503\pi\)
−0.108183 + 0.994131i \(0.534503\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) −10.0000 −0.359908
\(773\) −2.00000 −0.0719350 −0.0359675 0.999353i \(-0.511451\pi\)
−0.0359675 + 0.999353i \(0.511451\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) 14.0000 0.502571
\(777\) 0 0
\(778\) −6.00000 −0.215110
\(779\) 40.0000 1.43315
\(780\) 0 0
\(781\) −32.0000 −1.14505
\(782\) −6.00000 −0.214560
\(783\) 6.00000 0.214423
\(784\) −7.00000 −0.250000
\(785\) 0 0
\(786\) −8.00000 −0.285351
\(787\) −36.0000 −1.28326 −0.641631 0.767014i \(-0.721742\pi\)
−0.641631 + 0.767014i \(0.721742\pi\)
\(788\) −26.0000 −0.926212
\(789\) −32.0000 −1.13923
\(790\) 0 0
\(791\) 0 0
\(792\) −4.00000 −0.142134
\(793\) 60.0000 2.13066
\(794\) −2.00000 −0.0709773
\(795\) 0 0
\(796\) −12.0000 −0.425329
\(797\) −18.0000 −0.637593 −0.318796 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) 0 0
\(799\) 48.0000 1.69812
\(800\) 0 0
\(801\) −2.00000 −0.0706665
\(802\) 6.00000 0.211867
\(803\) 8.00000 0.282314
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) −48.0000 −1.69073
\(807\) −10.0000 −0.352017
\(808\) 18.0000 0.633238
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 0 0
\(813\) 8.00000 0.280572
\(814\) 24.0000 0.841200
\(815\) 0 0
\(816\) −6.00000 −0.210042
\(817\) −16.0000 −0.559769
\(818\) 10.0000 0.349642
\(819\) 0 0
\(820\) 0 0
\(821\) −46.0000 −1.60541 −0.802706 0.596376i \(-0.796607\pi\)
−0.802706 + 0.596376i \(0.796607\pi\)
\(822\) 2.00000 0.0697580
\(823\) −20.0000 −0.697156 −0.348578 0.937280i \(-0.613335\pi\)
−0.348578 + 0.937280i \(0.613335\pi\)
\(824\) 16.0000 0.557386
\(825\) 0 0
\(826\) 0 0
\(827\) 8.00000 0.278187 0.139094 0.990279i \(-0.455581\pi\)
0.139094 + 0.990279i \(0.455581\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −18.0000 −0.625166 −0.312583 0.949890i \(-0.601194\pi\)
−0.312583 + 0.949890i \(0.601194\pi\)
\(830\) 0 0
\(831\) −30.0000 −1.04069
\(832\) 6.00000 0.208013
\(833\) −42.0000 −1.45521
\(834\) −12.0000 −0.415526
\(835\) 0 0
\(836\) −16.0000 −0.553372
\(837\) 8.00000 0.276520
\(838\) −12.0000 −0.414533
\(839\) −48.0000 −1.65714 −0.828572 0.559883i \(-0.810846\pi\)
−0.828572 + 0.559883i \(0.810846\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −38.0000 −1.30957
\(843\) −30.0000 −1.03325
\(844\) −20.0000 −0.688428
\(845\) 0 0
\(846\) 8.00000 0.275046
\(847\) 0 0
\(848\) 14.0000 0.480762
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) 6.00000 0.205677
\(852\) −8.00000 −0.274075
\(853\) 22.0000 0.753266 0.376633 0.926363i \(-0.377082\pi\)
0.376633 + 0.926363i \(0.377082\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −8.00000 −0.273434
\(857\) 42.0000 1.43469 0.717346 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) 24.0000 0.819346
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 8.00000 0.272481
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −2.00000 −0.0679628
\(867\) −19.0000 −0.645274
\(868\) 0 0
\(869\) 48.0000 1.62829
\(870\) 0 0
\(871\) −24.0000 −0.813209
\(872\) 2.00000 0.0677285
\(873\) 14.0000 0.473828
\(874\) −4.00000 −0.135302
\(875\) 0 0
\(876\) 2.00000 0.0675737
\(877\) −2.00000 −0.0675352 −0.0337676 0.999430i \(-0.510751\pi\)
−0.0337676 + 0.999430i \(0.510751\pi\)
\(878\) −24.0000 −0.809961
\(879\) −14.0000 −0.472208
\(880\) 0 0
\(881\) 6.00000 0.202145 0.101073 0.994879i \(-0.467773\pi\)
0.101073 + 0.994879i \(0.467773\pi\)
\(882\) −7.00000 −0.235702
\(883\) −44.0000 −1.48072 −0.740359 0.672212i \(-0.765344\pi\)
−0.740359 + 0.672212i \(0.765344\pi\)
\(884\) 36.0000 1.21081
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) −8.00000 −0.268614 −0.134307 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) 6.00000 0.201347
\(889\) 0 0
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) 20.0000 0.669650
\(893\) 32.0000 1.07084
\(894\) −10.0000 −0.334450
\(895\) 0 0
\(896\) 0 0
\(897\) 6.00000 0.200334
\(898\) 10.0000 0.333704
\(899\) 48.0000 1.60089
\(900\) 0 0
\(901\) 84.0000 2.79845
\(902\) −40.0000 −1.33185
\(903\) 0 0
\(904\) 14.0000 0.465633
\(905\) 0 0
\(906\) −16.0000 −0.531564
\(907\) 4.00000 0.132818 0.0664089 0.997792i \(-0.478846\pi\)
0.0664089 + 0.997792i \(0.478846\pi\)
\(908\) −8.00000 −0.265489
\(909\) 18.0000 0.597022
\(910\) 0 0
\(911\) 8.00000 0.265052 0.132526 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) −4.00000 −0.132453
\(913\) −64.0000 −2.11809
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) −14.0000 −0.462573
\(917\) 0 0
\(918\) −6.00000 −0.198030
\(919\) 12.0000 0.395843 0.197922 0.980218i \(-0.436581\pi\)
0.197922 + 0.980218i \(0.436581\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) 2.00000 0.0658665
\(923\) 48.0000 1.57994
\(924\) 0 0
\(925\) 0 0
\(926\) 20.0000 0.657241
\(927\) 16.0000 0.525509
\(928\) −6.00000 −0.196960
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 0 0
\(931\) −28.0000 −0.917663
\(932\) −22.0000 −0.720634
\(933\) 8.00000 0.261908
\(934\) −8.00000 −0.261768
\(935\) 0 0
\(936\) 6.00000 0.196116
\(937\) −10.0000 −0.326686 −0.163343 0.986569i \(-0.552228\pi\)
−0.163343 + 0.986569i \(0.552228\pi\)
\(938\) 0 0
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) −14.0000 −0.456387 −0.228193 0.973616i \(-0.573282\pi\)
−0.228193 + 0.973616i \(0.573282\pi\)
\(942\) 6.00000 0.195491
\(943\) −10.0000 −0.325645
\(944\) 0 0
\(945\) 0 0
\(946\) 16.0000 0.520205
\(947\) 20.0000 0.649913 0.324956 0.945729i \(-0.394650\pi\)
0.324956 + 0.945729i \(0.394650\pi\)
\(948\) 12.0000 0.389742
\(949\) −12.0000 −0.389536
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) 22.0000 0.712650 0.356325 0.934362i \(-0.384030\pi\)
0.356325 + 0.934362i \(0.384030\pi\)
\(954\) 14.0000 0.453267
\(955\) 0 0
\(956\) −8.00000 −0.258738
\(957\) −24.0000 −0.775810
\(958\) 24.0000 0.775405
\(959\) 0 0
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) −36.0000 −1.16069
\(963\) −8.00000 −0.257796
\(964\) 2.00000 0.0644157
\(965\) 0 0
\(966\) 0 0
\(967\) −28.0000 −0.900419 −0.450210 0.892923i \(-0.648651\pi\)
−0.450210 + 0.892923i \(0.648651\pi\)
\(968\) 5.00000 0.160706
\(969\) −24.0000 −0.770991
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −20.0000 −0.640841
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) −2.00000 −0.0639857 −0.0319928 0.999488i \(-0.510185\pi\)
−0.0319928 + 0.999488i \(0.510185\pi\)
\(978\) 4.00000 0.127906
\(979\) 8.00000 0.255681
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) 0 0
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) −10.0000 −0.318788
\(985\) 0 0
\(986\) −36.0000 −1.14647
\(987\) 0 0
\(988\) 24.0000 0.763542
\(989\) 4.00000 0.127193
\(990\) 0 0
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) −8.00000 −0.254000
\(993\) 20.0000 0.634681
\(994\) 0 0
\(995\) 0 0
\(996\) −16.0000 −0.506979
\(997\) 22.0000 0.696747 0.348373 0.937356i \(-0.386734\pi\)
0.348373 + 0.937356i \(0.386734\pi\)
\(998\) 12.0000 0.379853
\(999\) 6.00000 0.189832
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3450.2.a.p.1.1 1
5.2 odd 4 3450.2.d.a.2899.2 2
5.3 odd 4 3450.2.d.a.2899.1 2
5.4 even 2 690.2.a.e.1.1 1
15.14 odd 2 2070.2.a.r.1.1 1
20.19 odd 2 5520.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.e.1.1 1 5.4 even 2
2070.2.a.r.1.1 1 15.14 odd 2
3450.2.a.p.1.1 1 1.1 even 1 trivial
3450.2.d.a.2899.1 2 5.3 odd 4
3450.2.d.a.2899.2 2 5.2 odd 4
5520.2.a.f.1.1 1 20.19 odd 2