Properties

Label 1470.2.a.a.1.1
Level 1470
Weight 2
Character 1470.1
Self dual yes
Analytic conductor 11.738
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.7380090971\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 210)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} +1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} -3.00000 q^{19} -1.00000 q^{20} +1.00000 q^{22} +7.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{26} -1.00000 q^{27} -8.00000 q^{29} -1.00000 q^{30} -2.00000 q^{31} -1.00000 q^{32} +1.00000 q^{33} +1.00000 q^{36} +11.0000 q^{37} +3.00000 q^{38} -1.00000 q^{39} +1.00000 q^{40} -11.0000 q^{41} +8.00000 q^{43} -1.00000 q^{44} -1.00000 q^{45} -7.00000 q^{46} -5.00000 q^{47} -1.00000 q^{48} -1.00000 q^{50} +1.00000 q^{52} -11.0000 q^{53} +1.00000 q^{54} +1.00000 q^{55} +3.00000 q^{57} +8.00000 q^{58} +4.00000 q^{59} +1.00000 q^{60} +2.00000 q^{62} +1.00000 q^{64} -1.00000 q^{65} -1.00000 q^{66} -7.00000 q^{69} -6.00000 q^{71} -1.00000 q^{72} -6.00000 q^{73} -11.0000 q^{74} -1.00000 q^{75} -3.00000 q^{76} +1.00000 q^{78} -8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +11.0000 q^{82} +8.00000 q^{83} -8.00000 q^{86} +8.00000 q^{87} +1.00000 q^{88} -10.0000 q^{89} +1.00000 q^{90} +7.00000 q^{92} +2.00000 q^{93} +5.00000 q^{94} +3.00000 q^{95} +1.00000 q^{96} -16.0000 q^{97} -1.00000 q^{99} +O(q^{100})\)

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
\(5\) −1.00000 −0.447214
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) −1.00000 −0.288675
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −1.00000 −0.235702
\(19\) −3.00000 −0.688247 −0.344124 0.938924i \(-0.611824\pi\)
−0.344124 + 0.938924i \(0.611824\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 7.00000 1.45960 0.729800 0.683660i \(-0.239613\pi\)
0.729800 + 0.683660i \(0.239613\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −8.00000 −1.48556 −0.742781 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) −1.00000 −0.182574
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.00000 0.174078
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 11.0000 1.80839 0.904194 0.427121i \(-0.140472\pi\)
0.904194 + 0.427121i \(0.140472\pi\)
\(38\) 3.00000 0.486664
\(39\) −1.00000 −0.160128
\(40\) 1.00000 0.158114
\(41\) −11.0000 −1.71791 −0.858956 0.512050i \(-0.828886\pi\)
−0.858956 + 0.512050i \(0.828886\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) −1.00000 −0.150756
\(45\) −1.00000 −0.149071
\(46\) −7.00000 −1.03209
\(47\) −5.00000 −0.729325 −0.364662 0.931140i \(-0.618816\pi\)
−0.364662 + 0.931140i \(0.618816\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 1.00000 0.138675
\(53\) −11.0000 −1.51097 −0.755483 0.655168i \(-0.772598\pi\)
−0.755483 + 0.655168i \(0.772598\pi\)
\(54\) 1.00000 0.136083
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) 3.00000 0.397360
\(58\) 8.00000 1.05045
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 1.00000 0.129099
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 2.00000 0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.00000 −0.124035
\(66\) −1.00000 −0.123091
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) −7.00000 −0.842701
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) −1.00000 −0.117851
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) −11.0000 −1.27872
\(75\) −1.00000 −0.115470
\(76\) −3.00000 −0.344124
\(77\) 0 0
\(78\) 1.00000 0.113228
\(79\) −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
\(82\) 11.0000 1.21475
\(83\) 8.00000 0.878114 0.439057 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −8.00000 −0.862662
\(87\) 8.00000 0.857690
\(88\) 1.00000 0.106600
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) 7.00000 0.729800
\(93\) 2.00000 0.207390
\(94\) 5.00000 0.515711
\(95\) 3.00000 0.307794
\(96\) 1.00000 0.102062
\(97\) −16.0000 −1.62455 −0.812277 0.583272i \(-0.801772\pi\)
−0.812277 + 0.583272i \(0.801772\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 1.00000 0.100000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) 11.0000 1.06841
\(107\) −10.0000 −0.966736 −0.483368 0.875417i \(-0.660587\pi\)
−0.483368 + 0.875417i \(0.660587\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) −1.00000 −0.0953463
\(111\) −11.0000 −1.04407
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) −3.00000 −0.280976
\(115\) −7.00000 −0.652753
\(116\) −8.00000 −0.742781
\(117\) 1.00000 0.0924500
\(118\) −4.00000 −0.368230
\(119\) 0 0
\(120\) −1.00000 −0.0912871
\(121\) −10.0000 −0.909091
\(122\) 0 0
\(123\) 11.0000 0.991837
\(124\) −2.00000 −0.179605
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −17.0000 −1.50851 −0.754253 0.656584i \(-0.772001\pi\)
−0.754253 + 0.656584i \(0.772001\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −8.00000 −0.704361
\(130\) 1.00000 0.0877058
\(131\) −5.00000 −0.436852 −0.218426 0.975854i \(-0.570092\pi\)
−0.218426 + 0.975854i \(0.570092\pi\)
\(132\) 1.00000 0.0870388
\(133\) 0 0
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 7.00000 0.595880
\(139\) 20.0000 1.69638 0.848189 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) 0 0
\(141\) 5.00000 0.421076
\(142\) 6.00000 0.503509
\(143\) −1.00000 −0.0836242
\(144\) 1.00000 0.0833333
\(145\) 8.00000 0.664364
\(146\) 6.00000 0.496564
\(147\) 0 0
\(148\) 11.0000 0.904194
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 1.00000 0.0816497
\(151\) 6.00000 0.488273 0.244137 0.969741i \(-0.421495\pi\)
0.244137 + 0.969741i \(0.421495\pi\)
\(152\) 3.00000 0.243332
\(153\) 0 0
\(154\) 0 0
\(155\) 2.00000 0.160644
\(156\) −1.00000 −0.0800641
\(157\) −7.00000 −0.558661 −0.279330 0.960195i \(-0.590112\pi\)
−0.279330 + 0.960195i \(0.590112\pi\)
\(158\) 8.00000 0.636446
\(159\) 11.0000 0.872357
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 16.0000 1.25322 0.626608 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) −11.0000 −0.858956
\(165\) −1.00000 −0.0778499
\(166\) −8.00000 −0.620920
\(167\) −3.00000 −0.232147 −0.116073 0.993241i \(-0.537031\pi\)
−0.116073 + 0.993241i \(0.537031\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) −3.00000 −0.229416
\(172\) 8.00000 0.609994
\(173\) −15.0000 −1.14043 −0.570214 0.821496i \(-0.693140\pi\)
−0.570214 + 0.821496i \(0.693140\pi\)
\(174\) −8.00000 −0.606478
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) −4.00000 −0.300658
\(178\) 10.0000 0.749532
\(179\) −19.0000 −1.42013 −0.710063 0.704138i \(-0.751334\pi\)
−0.710063 + 0.704138i \(0.751334\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −24.0000 −1.78391 −0.891953 0.452128i \(-0.850665\pi\)
−0.891953 + 0.452128i \(0.850665\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −7.00000 −0.516047
\(185\) −11.0000 −0.808736
\(186\) −2.00000 −0.146647
\(187\) 0 0
\(188\) −5.00000 −0.364662
\(189\) 0 0
\(190\) −3.00000 −0.217643
\(191\) −6.00000 −0.434145 −0.217072 0.976156i \(-0.569651\pi\)
−0.217072 + 0.976156i \(0.569651\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 22.0000 1.58359 0.791797 0.610784i \(-0.209146\pi\)
0.791797 + 0.610784i \(0.209146\pi\)
\(194\) 16.0000 1.14873
\(195\) 1.00000 0.0716115
\(196\) 0 0
\(197\) −1.00000 −0.0712470 −0.0356235 0.999365i \(-0.511342\pi\)
−0.0356235 + 0.999365i \(0.511342\pi\)
\(198\) 1.00000 0.0710669
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 11.0000 0.768273
\(206\) 16.0000 1.11477
\(207\) 7.00000 0.486534
\(208\) 1.00000 0.0693375
\(209\) 3.00000 0.207514
\(210\) 0 0
\(211\) 5.00000 0.344214 0.172107 0.985078i \(-0.444942\pi\)
0.172107 + 0.985078i \(0.444942\pi\)
\(212\) −11.0000 −0.755483
\(213\) 6.00000 0.411113
\(214\) 10.0000 0.683586
\(215\) −8.00000 −0.545595
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −6.00000 −0.406371
\(219\) 6.00000 0.405442
\(220\) 1.00000 0.0674200
\(221\) 0 0
\(222\) 11.0000 0.738272
\(223\) 12.0000 0.803579 0.401790 0.915732i \(-0.368388\pi\)
0.401790 + 0.915732i \(0.368388\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) −6.00000 −0.399114
\(227\) 8.00000 0.530979 0.265489 0.964114i \(-0.414466\pi\)
0.265489 + 0.964114i \(0.414466\pi\)
\(228\) 3.00000 0.198680
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 7.00000 0.461566
\(231\) 0 0
\(232\) 8.00000 0.525226
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 5.00000 0.326164
\(236\) 4.00000 0.260378
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) −18.0000 −1.16432 −0.582162 0.813073i \(-0.697793\pi\)
−0.582162 + 0.813073i \(0.697793\pi\)
\(240\) 1.00000 0.0645497
\(241\) 7.00000 0.450910 0.225455 0.974254i \(-0.427613\pi\)
0.225455 + 0.974254i \(0.427613\pi\)
\(242\) 10.0000 0.642824
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) −11.0000 −0.701334
\(247\) −3.00000 −0.190885
\(248\) 2.00000 0.127000
\(249\) −8.00000 −0.506979
\(250\) 1.00000 0.0632456
\(251\) 13.0000 0.820553 0.410276 0.911961i \(-0.365432\pi\)
0.410276 + 0.911961i \(0.365432\pi\)
\(252\) 0 0
\(253\) −7.00000 −0.440086
\(254\) 17.0000 1.06667
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −20.0000 −1.24757 −0.623783 0.781598i \(-0.714405\pi\)
−0.623783 + 0.781598i \(0.714405\pi\)
\(258\) 8.00000 0.498058
\(259\) 0 0
\(260\) −1.00000 −0.0620174
\(261\) −8.00000 −0.495188
\(262\) 5.00000 0.308901
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) −1.00000 −0.0615457
\(265\) 11.0000 0.675725
\(266\) 0 0
\(267\) 10.0000 0.611990
\(268\) 0 0
\(269\) 20.0000 1.21942 0.609711 0.792624i \(-0.291286\pi\)
0.609711 + 0.792624i \(0.291286\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 32.0000 1.94386 0.971931 0.235267i \(-0.0755965\pi\)
0.971931 + 0.235267i \(0.0755965\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 18.0000 1.08742
\(275\) −1.00000 −0.0603023
\(276\) −7.00000 −0.421350
\(277\) −22.0000 −1.32185 −0.660926 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) −20.0000 −1.19952
\(279\) −2.00000 −0.119737
\(280\) 0 0
\(281\) −1.00000 −0.0596550 −0.0298275 0.999555i \(-0.509496\pi\)
−0.0298275 + 0.999555i \(0.509496\pi\)
\(282\) −5.00000 −0.297746
\(283\) −14.0000 −0.832214 −0.416107 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) −6.00000 −0.356034
\(285\) −3.00000 −0.177705
\(286\) 1.00000 0.0591312
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −17.0000 −1.00000
\(290\) −8.00000 −0.469776
\(291\) 16.0000 0.937937
\(292\) −6.00000 −0.351123
\(293\) 27.0000 1.57736 0.788678 0.614806i \(-0.210766\pi\)
0.788678 + 0.614806i \(0.210766\pi\)
\(294\) 0 0
\(295\) −4.00000 −0.232889
\(296\) −11.0000 −0.639362
\(297\) 1.00000 0.0580259
\(298\) 0 0
\(299\) 7.00000 0.404820
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) −6.00000 −0.345261
\(303\) 0 0
\(304\) −3.00000 −0.172062
\(305\) 0 0
\(306\) 0 0
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 0 0
\(309\) 16.0000 0.910208
\(310\) −2.00000 −0.113592
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 1.00000 0.0566139
\(313\) −12.0000 −0.678280 −0.339140 0.940736i \(-0.610136\pi\)
−0.339140 + 0.940736i \(0.610136\pi\)
\(314\) 7.00000 0.395033
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) −11.0000 −0.616849
\(319\) 8.00000 0.447914
\(320\) −1.00000 −0.0559017
\(321\) 10.0000 0.558146
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 1.00000 0.0554700
\(326\) −16.0000 −0.886158
\(327\) −6.00000 −0.331801
\(328\) 11.0000 0.607373
\(329\) 0 0
\(330\) 1.00000 0.0550482
\(331\) −13.0000 −0.714545 −0.357272 0.934000i \(-0.616293\pi\)
−0.357272 + 0.934000i \(0.616293\pi\)
\(332\) 8.00000 0.439057
\(333\) 11.0000 0.602796
\(334\) 3.00000 0.164153
\(335\) 0 0
\(336\) 0 0
\(337\) −12.0000 −0.653682 −0.326841 0.945079i \(-0.605984\pi\)
−0.326841 + 0.945079i \(0.605984\pi\)
\(338\) 12.0000 0.652714
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 2.00000 0.108306
\(342\) 3.00000 0.162221
\(343\) 0 0
\(344\) −8.00000 −0.431331
\(345\) 7.00000 0.376867
\(346\) 15.0000 0.806405
\(347\) 14.0000 0.751559 0.375780 0.926709i \(-0.377375\pi\)
0.375780 + 0.926709i \(0.377375\pi\)
\(348\) 8.00000 0.428845
\(349\) 12.0000 0.642345 0.321173 0.947021i \(-0.395923\pi\)
0.321173 + 0.947021i \(0.395923\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 1.00000 0.0533002
\(353\) 24.0000 1.27739 0.638696 0.769460i \(-0.279474\pi\)
0.638696 + 0.769460i \(0.279474\pi\)
\(354\) 4.00000 0.212598
\(355\) 6.00000 0.318447
\(356\) −10.0000 −0.529999
\(357\) 0 0
\(358\) 19.0000 1.00418
\(359\) 4.00000 0.211112 0.105556 0.994413i \(-0.466338\pi\)
0.105556 + 0.994413i \(0.466338\pi\)
\(360\) 1.00000 0.0527046
\(361\) −10.0000 −0.526316
\(362\) 24.0000 1.26141
\(363\) 10.0000 0.524864
\(364\) 0 0
\(365\) 6.00000 0.314054
\(366\) 0 0
\(367\) 25.0000 1.30499 0.652495 0.757793i \(-0.273722\pi\)
0.652495 + 0.757793i \(0.273722\pi\)
\(368\) 7.00000 0.364900
\(369\) −11.0000 −0.572637
\(370\) 11.0000 0.571863
\(371\) 0 0
\(372\) 2.00000 0.103695
\(373\) 6.00000 0.310668 0.155334 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 5.00000 0.257855
\(377\) −8.00000 −0.412021
\(378\) 0 0
\(379\) −19.0000 −0.975964 −0.487982 0.872854i \(-0.662267\pi\)
−0.487982 + 0.872854i \(0.662267\pi\)
\(380\) 3.00000 0.153897
\(381\) 17.0000 0.870936
\(382\) 6.00000 0.306987
\(383\) 35.0000 1.78842 0.894208 0.447651i \(-0.147739\pi\)
0.894208 + 0.447651i \(0.147739\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −22.0000 −1.11977
\(387\) 8.00000 0.406663
\(388\) −16.0000 −0.812277
\(389\) 10.0000 0.507020 0.253510 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) −1.00000 −0.0506370
\(391\) 0 0
\(392\) 0 0
\(393\) 5.00000 0.252217
\(394\) 1.00000 0.0503793
\(395\) 8.00000 0.402524
\(396\) −1.00000 −0.0502519
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 24.0000 1.20301
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −5.00000 −0.249688 −0.124844 0.992176i \(-0.539843\pi\)
−0.124844 + 0.992176i \(0.539843\pi\)
\(402\) 0 0
\(403\) −2.00000 −0.0996271
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −11.0000 −0.545250
\(408\) 0 0
\(409\) 2.00000 0.0988936 0.0494468 0.998777i \(-0.484254\pi\)
0.0494468 + 0.998777i \(0.484254\pi\)
\(410\) −11.0000 −0.543251
\(411\) 18.0000 0.887875
\(412\) −16.0000 −0.788263
\(413\) 0 0
\(414\) −7.00000 −0.344031
\(415\) −8.00000 −0.392705
\(416\) −1.00000 −0.0490290
\(417\) −20.0000 −0.979404
\(418\) −3.00000 −0.146735
\(419\) −5.00000 −0.244266 −0.122133 0.992514i \(-0.538973\pi\)
−0.122133 + 0.992514i \(0.538973\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) −5.00000 −0.243396
\(423\) −5.00000 −0.243108
\(424\) 11.0000 0.534207
\(425\) 0 0
\(426\) −6.00000 −0.290701
\(427\) 0 0
\(428\) −10.0000 −0.483368
\(429\) 1.00000 0.0482805
\(430\) 8.00000 0.385794
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −32.0000 −1.53782 −0.768911 0.639356i \(-0.779201\pi\)
−0.768911 + 0.639356i \(0.779201\pi\)
\(434\) 0 0
\(435\) −8.00000 −0.383571
\(436\) 6.00000 0.287348
\(437\) −21.0000 −1.00457
\(438\) −6.00000 −0.286691
\(439\) 32.0000 1.52728 0.763638 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 0 0
\(442\) 0 0
\(443\) 16.0000 0.760183 0.380091 0.924949i \(-0.375893\pi\)
0.380091 + 0.924949i \(0.375893\pi\)
\(444\) −11.0000 −0.522037
\(445\) 10.0000 0.474045
\(446\) −12.0000 −0.568216
\(447\) 0 0
\(448\) 0 0
\(449\) 11.0000 0.519122 0.259561 0.965727i \(-0.416422\pi\)
0.259561 + 0.965727i \(0.416422\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 11.0000 0.517970
\(452\) 6.00000 0.282216
\(453\) −6.00000 −0.281905
\(454\) −8.00000 −0.375459
\(455\) 0 0
\(456\) −3.00000 −0.140488
\(457\) −18.0000 −0.842004 −0.421002 0.907060i \(-0.638322\pi\)
−0.421002 + 0.907060i \(0.638322\pi\)
\(458\) −14.0000 −0.654177
\(459\) 0 0
\(460\) −7.00000 −0.326377
\(461\) −12.0000 −0.558896 −0.279448 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 0 0
\(463\) −13.0000 −0.604161 −0.302081 0.953282i \(-0.597681\pi\)
−0.302081 + 0.953282i \(0.597681\pi\)
\(464\) −8.00000 −0.371391
\(465\) −2.00000 −0.0927478
\(466\) −18.0000 −0.833834
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 1.00000 0.0462250
\(469\) 0 0
\(470\) −5.00000 −0.230633
\(471\) 7.00000 0.322543
\(472\) −4.00000 −0.184115
\(473\) −8.00000 −0.367840
\(474\) −8.00000 −0.367452
\(475\) −3.00000 −0.137649
\(476\) 0 0
\(477\) −11.0000 −0.503655
\(478\) 18.0000 0.823301
\(479\) 22.0000 1.00521 0.502603 0.864517i \(-0.332376\pi\)
0.502603 + 0.864517i \(0.332376\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 11.0000 0.501557
\(482\) −7.00000 −0.318841
\(483\) 0 0
\(484\) −10.0000 −0.454545
\(485\) 16.0000 0.726523
\(486\) 1.00000 0.0453609
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) 0 0
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 11.0000 0.495918
\(493\) 0 0
\(494\) 3.00000 0.134976
\(495\) 1.00000 0.0449467
\(496\) −2.00000 −0.0898027
\(497\) 0 0
\(498\) 8.00000 0.358489
\(499\) −32.0000 −1.43252 −0.716258 0.697835i \(-0.754147\pi\)
−0.716258 + 0.697835i \(0.754147\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 3.00000 0.134030
\(502\) −13.0000 −0.580218
\(503\) 36.0000 1.60516 0.802580 0.596544i \(-0.203460\pi\)
0.802580 + 0.596544i \(0.203460\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 7.00000 0.311188
\(507\) 12.0000 0.532939
\(508\) −17.0000 −0.754253
\(509\) −34.0000 −1.50702 −0.753512 0.657434i \(-0.771642\pi\)
−0.753512 + 0.657434i \(0.771642\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 3.00000 0.132453
\(514\) 20.0000 0.882162
\(515\) 16.0000 0.705044
\(516\) −8.00000 −0.352180
\(517\) 5.00000 0.219900
\(518\) 0 0
\(519\) 15.0000 0.658427
\(520\) 1.00000 0.0438529
\(521\) −33.0000 −1.44576 −0.722878 0.690976i \(-0.757181\pi\)
−0.722878 + 0.690976i \(0.757181\pi\)
\(522\) 8.00000 0.350150
\(523\) 2.00000 0.0874539 0.0437269 0.999044i \(-0.486077\pi\)
0.0437269 + 0.999044i \(0.486077\pi\)
\(524\) −5.00000 −0.218426
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) 0 0
\(528\) 1.00000 0.0435194
\(529\) 26.0000 1.13043
\(530\) −11.0000 −0.477809
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) −11.0000 −0.476463
\(534\) −10.0000 −0.432742
\(535\) 10.0000 0.432338
\(536\) 0 0
\(537\) 19.0000 0.819911
\(538\) −20.0000 −0.862261
\(539\) 0 0
\(540\) 1.00000 0.0430331
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) −32.0000 −1.37452
\(543\) 24.0000 1.02994
\(544\) 0 0
\(545\) −6.00000 −0.257012
\(546\) 0 0
\(547\) 16.0000 0.684111 0.342055 0.939680i \(-0.388877\pi\)
0.342055 + 0.939680i \(0.388877\pi\)
\(548\) −18.0000 −0.768922
\(549\) 0 0
\(550\) 1.00000 0.0426401
\(551\) 24.0000 1.02243
\(552\) 7.00000 0.297940
\(553\) 0 0
\(554\) 22.0000 0.934690
\(555\) 11.0000 0.466924
\(556\) 20.0000 0.848189
\(557\) −33.0000 −1.39825 −0.699127 0.714997i \(-0.746428\pi\)
−0.699127 + 0.714997i \(0.746428\pi\)
\(558\) 2.00000 0.0846668
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) 1.00000 0.0421825
\(563\) −38.0000 −1.60151 −0.800755 0.598993i \(-0.795568\pi\)
−0.800755 + 0.598993i \(0.795568\pi\)
\(564\) 5.00000 0.210538
\(565\) −6.00000 −0.252422
\(566\) 14.0000 0.588464
\(567\) 0 0
\(568\) 6.00000 0.251754
\(569\) −9.00000 −0.377300 −0.188650 0.982044i \(-0.560411\pi\)
−0.188650 + 0.982044i \(0.560411\pi\)
\(570\) 3.00000 0.125656
\(571\) −32.0000 −1.33916 −0.669579 0.742741i \(-0.733526\pi\)
−0.669579 + 0.742741i \(0.733526\pi\)
\(572\) −1.00000 −0.0418121
\(573\) 6.00000 0.250654
\(574\) 0 0
\(575\) 7.00000 0.291920
\(576\) 1.00000 0.0416667
\(577\) −14.0000 −0.582828 −0.291414 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) 17.0000 0.707107
\(579\) −22.0000 −0.914289
\(580\) 8.00000 0.332182
\(581\) 0 0
\(582\) −16.0000 −0.663221
\(583\) 11.0000 0.455573
\(584\) 6.00000 0.248282
\(585\) −1.00000 −0.0413449
\(586\) −27.0000 −1.11536
\(587\) −18.0000 −0.742940 −0.371470 0.928445i \(-0.621146\pi\)
−0.371470 + 0.928445i \(0.621146\pi\)
\(588\) 0 0
\(589\) 6.00000 0.247226
\(590\) 4.00000 0.164677
\(591\) 1.00000 0.0411345
\(592\) 11.0000 0.452097
\(593\) 16.0000 0.657041 0.328521 0.944497i \(-0.393450\pi\)
0.328521 + 0.944497i \(0.393450\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 0 0
\(596\) 0 0
\(597\) 24.0000 0.982255
\(598\) −7.00000 −0.286251
\(599\) 2.00000 0.0817178 0.0408589 0.999165i \(-0.486991\pi\)
0.0408589 + 0.999165i \(0.486991\pi\)
\(600\) 1.00000 0.0408248
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 6.00000 0.244137
\(605\) 10.0000 0.406558
\(606\) 0 0
\(607\) −37.0000 −1.50178 −0.750892 0.660425i \(-0.770376\pi\)
−0.750892 + 0.660425i \(0.770376\pi\)
\(608\) 3.00000 0.121666
\(609\) 0 0
\(610\) 0 0
\(611\) −5.00000 −0.202278
\(612\) 0 0
\(613\) −41.0000 −1.65597 −0.827987 0.560747i \(-0.810514\pi\)
−0.827987 + 0.560747i \(0.810514\pi\)
\(614\) −20.0000 −0.807134
\(615\) −11.0000 −0.443563
\(616\) 0 0
\(617\) 28.0000 1.12724 0.563619 0.826035i \(-0.309409\pi\)
0.563619 + 0.826035i \(0.309409\pi\)
\(618\) −16.0000 −0.643614
\(619\) 29.0000 1.16561 0.582804 0.812613i \(-0.301955\pi\)
0.582804 + 0.812613i \(0.301955\pi\)
\(620\) 2.00000 0.0803219
\(621\) −7.00000 −0.280900
\(622\) −12.0000 −0.481156
\(623\) 0 0
\(624\) −1.00000 −0.0400320
\(625\) 1.00000 0.0400000
\(626\) 12.0000 0.479616
\(627\) −3.00000 −0.119808
\(628\) −7.00000 −0.279330
\(629\) 0 0
\(630\) 0 0
\(631\) −26.0000 −1.03504 −0.517522 0.855670i \(-0.673145\pi\)
−0.517522 + 0.855670i \(0.673145\pi\)
\(632\) 8.00000 0.318223
\(633\) −5.00000 −0.198732
\(634\) 18.0000 0.714871
\(635\) 17.0000 0.674624
\(636\) 11.0000 0.436178
\(637\) 0 0
\(638\) −8.00000 −0.316723
\(639\) −6.00000 −0.237356
\(640\) 1.00000 0.0395285
\(641\) −3.00000 −0.118493 −0.0592464 0.998243i \(-0.518870\pi\)
−0.0592464 + 0.998243i \(0.518870\pi\)
\(642\) −10.0000 −0.394669
\(643\) −10.0000 −0.394362 −0.197181 0.980367i \(-0.563179\pi\)
−0.197181 + 0.980367i \(0.563179\pi\)
\(644\) 0 0
\(645\) 8.00000 0.315000
\(646\) 0 0
\(647\) −17.0000 −0.668339 −0.334169 0.942513i \(-0.608456\pi\)
−0.334169 + 0.942513i \(0.608456\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −4.00000 −0.157014
\(650\) −1.00000 −0.0392232
\(651\) 0 0
\(652\) 16.0000 0.626608
\(653\) 7.00000 0.273931 0.136966 0.990576i \(-0.456265\pi\)
0.136966 + 0.990576i \(0.456265\pi\)
\(654\) 6.00000 0.234619
\(655\) 5.00000 0.195366
\(656\) −11.0000 −0.429478
\(657\) −6.00000 −0.234082
\(658\) 0 0
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) −1.00000 −0.0389249
\(661\) −24.0000 −0.933492 −0.466746 0.884391i \(-0.654574\pi\)
−0.466746 + 0.884391i \(0.654574\pi\)
\(662\) 13.0000 0.505259
\(663\) 0 0
\(664\) −8.00000 −0.310460
\(665\) 0 0
\(666\) −11.0000 −0.426241
\(667\) −56.0000 −2.16833
\(668\) −3.00000 −0.116073
\(669\) −12.0000 −0.463947
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 28.0000 1.07932 0.539660 0.841883i \(-0.318553\pi\)
0.539660 + 0.841883i \(0.318553\pi\)
\(674\) 12.0000 0.462223
\(675\) −1.00000 −0.0384900
\(676\) −12.0000 −0.461538
\(677\) −13.0000 −0.499631 −0.249815 0.968294i \(-0.580370\pi\)
−0.249815 + 0.968294i \(0.580370\pi\)
\(678\) 6.00000 0.230429
\(679\) 0 0
\(680\) 0 0
\(681\) −8.00000 −0.306561
\(682\) −2.00000 −0.0765840
\(683\) 4.00000 0.153056 0.0765279 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) −3.00000 −0.114708
\(685\) 18.0000 0.687745
\(686\) 0 0
\(687\) −14.0000 −0.534133
\(688\) 8.00000 0.304997
\(689\) −11.0000 −0.419067
\(690\) −7.00000 −0.266485
\(691\) −12.0000 −0.456502 −0.228251 0.973602i \(-0.573301\pi\)
−0.228251 + 0.973602i \(0.573301\pi\)
\(692\) −15.0000 −0.570214
\(693\) 0 0
\(694\) −14.0000 −0.531433
\(695\) −20.0000 −0.758643
\(696\) −8.00000 −0.303239
\(697\) 0 0
\(698\) −12.0000 −0.454207
\(699\) −18.0000 −0.680823
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 1.00000 0.0377426
\(703\) −33.0000 −1.24462
\(704\) −1.00000 −0.0376889
\(705\) −5.00000 −0.188311
\(706\) −24.0000 −0.903252
\(707\) 0 0
\(708\) −4.00000 −0.150329
\(709\) 44.0000 1.65245 0.826227 0.563337i \(-0.190483\pi\)
0.826227 + 0.563337i \(0.190483\pi\)
\(710\) −6.00000 −0.225176
\(711\) −8.00000 −0.300023
\(712\) 10.0000 0.374766
\(713\) −14.0000 −0.524304
\(714\) 0 0
\(715\) 1.00000 0.0373979
\(716\) −19.0000 −0.710063
\(717\) 18.0000 0.672222
\(718\) −4.00000 −0.149279
\(719\) 2.00000 0.0745874 0.0372937 0.999304i \(-0.488126\pi\)
0.0372937 + 0.999304i \(0.488126\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 0 0
\(722\) 10.0000 0.372161
\(723\) −7.00000 −0.260333
\(724\) −24.0000 −0.891953
\(725\) −8.00000 −0.297113
\(726\) −10.0000 −0.371135
\(727\) 11.0000 0.407967 0.203984 0.978974i \(-0.434611\pi\)
0.203984 + 0.978974i \(0.434611\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −6.00000 −0.222070
\(731\) 0 0
\(732\) 0 0
\(733\) 11.0000 0.406294 0.203147 0.979148i \(-0.434883\pi\)
0.203147 + 0.979148i \(0.434883\pi\)
\(734\) −25.0000 −0.922767
\(735\) 0 0
\(736\) −7.00000 −0.258023
\(737\) 0 0
\(738\) 11.0000 0.404916
\(739\) −19.0000 −0.698926 −0.349463 0.936950i \(-0.613636\pi\)
−0.349463 + 0.936950i \(0.613636\pi\)
\(740\) −11.0000 −0.404368
\(741\) 3.00000 0.110208
\(742\) 0 0
\(743\) 49.0000 1.79764 0.898818 0.438322i \(-0.144427\pi\)
0.898818 + 0.438322i \(0.144427\pi\)
\(744\) −2.00000 −0.0733236
\(745\) 0 0
\(746\) −6.00000 −0.219676
\(747\) 8.00000 0.292705
\(748\) 0 0
\(749\) 0 0
\(750\) −1.00000 −0.0365148
\(751\) 26.0000 0.948753 0.474377 0.880322i \(-0.342673\pi\)
0.474377 + 0.880322i \(0.342673\pi\)
\(752\) −5.00000 −0.182331
\(753\) −13.0000 −0.473746
\(754\) 8.00000 0.291343
\(755\) −6.00000 −0.218362
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 19.0000 0.690111
\(759\) 7.00000 0.254084
\(760\) −3.00000 −0.108821
\(761\) 27.0000 0.978749 0.489375 0.872074i \(-0.337225\pi\)
0.489375 + 0.872074i \(0.337225\pi\)
\(762\) −17.0000 −0.615845
\(763\) 0 0
\(764\) −6.00000 −0.217072
\(765\) 0 0
\(766\) −35.0000 −1.26460
\(767\) 4.00000 0.144432
\(768\) −1.00000 −0.0360844
\(769\) −5.00000 −0.180305 −0.0901523 0.995928i \(-0.528735\pi\)
−0.0901523 + 0.995928i \(0.528735\pi\)
\(770\) 0 0
\(771\) 20.0000 0.720282
\(772\) 22.0000 0.791797
\(773\) 33.0000 1.18693 0.593464 0.804861i \(-0.297760\pi\)
0.593464 + 0.804861i \(0.297760\pi\)
\(774\) −8.00000 −0.287554
\(775\) −2.00000 −0.0718421
\(776\) 16.0000 0.574367
\(777\) 0 0
\(778\) −10.0000 −0.358517
\(779\) 33.0000 1.18235
\(780\) 1.00000 0.0358057
\(781\) 6.00000 0.214697
\(782\) 0 0
\(783\) 8.00000 0.285897
\(784\) 0 0
\(785\) 7.00000 0.249841
\(786\) −5.00000 −0.178344
\(787\) 22.0000 0.784215 0.392108 0.919919i \(-0.371746\pi\)
0.392108 + 0.919919i \(0.371746\pi\)
\(788\) −1.00000 −0.0356235
\(789\) −24.0000 −0.854423
\(790\) −8.00000 −0.284627
\(791\) 0 0
\(792\) 1.00000 0.0355335
\(793\) 0 0
\(794\) −14.0000 −0.496841
\(795\) −11.0000 −0.390130
\(796\) −24.0000 −0.850657
\(797\) −18.0000 −0.637593 −0.318796 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) −10.0000 −0.353333
\(802\) 5.00000 0.176556
\(803\) 6.00000 0.211735
\(804\) 0 0
\(805\) 0 0
\(806\) 2.00000 0.0704470
\(807\) −20.0000 −0.704033
\(808\) 0 0
\(809\) −39.0000 −1.37117 −0.685583 0.727994i \(-0.740453\pi\)
−0.685583 + 0.727994i \(0.740453\pi\)
\(810\) 1.00000 0.0351364
\(811\) 9.00000 0.316033 0.158016 0.987436i \(-0.449490\pi\)
0.158016 + 0.987436i \(0.449490\pi\)
\(812\) 0 0
\(813\) −32.0000 −1.12229
\(814\) 11.0000 0.385550
\(815\) −16.0000 −0.560456
\(816\) 0 0
\(817\) −24.0000 −0.839654
\(818\) −2.00000 −0.0699284
\(819\) 0 0
\(820\) 11.0000 0.384137
\(821\) 42.0000 1.46581 0.732905 0.680331i \(-0.238164\pi\)
0.732905 + 0.680331i \(0.238164\pi\)
\(822\) −18.0000 −0.627822
\(823\) −24.0000 −0.836587 −0.418294 0.908312i \(-0.637372\pi\)
−0.418294 + 0.908312i \(0.637372\pi\)
\(824\) 16.0000 0.557386
\(825\) 1.00000 0.0348155
\(826\) 0 0
\(827\) −10.0000 −0.347734 −0.173867 0.984769i \(-0.555626\pi\)
−0.173867 + 0.984769i \(0.555626\pi\)
\(828\) 7.00000 0.243267
\(829\) −8.00000 −0.277851 −0.138926 0.990303i \(-0.544365\pi\)
−0.138926 + 0.990303i \(0.544365\pi\)
\(830\) 8.00000 0.277684
\(831\) 22.0000 0.763172
\(832\) 1.00000 0.0346688
\(833\) 0 0
\(834\) 20.0000 0.692543
\(835\) 3.00000 0.103819
\(836\) 3.00000 0.103757
\(837\) 2.00000 0.0691301
\(838\) 5.00000 0.172722
\(839\) −40.0000 −1.38095 −0.690477 0.723355i \(-0.742599\pi\)
−0.690477 + 0.723355i \(0.742599\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) −22.0000 −0.758170
\(843\) 1.00000 0.0344418
\(844\) 5.00000 0.172107
\(845\) 12.0000 0.412813
\(846\) 5.00000 0.171904
\(847\) 0 0
\(848\) −11.0000 −0.377742
\(849\) 14.0000 0.480479
\(850\) 0 0
\(851\) 77.0000 2.63953
\(852\) 6.00000 0.205557
\(853\) −41.0000 −1.40381 −0.701907 0.712269i \(-0.747668\pi\)
−0.701907 + 0.712269i \(0.747668\pi\)
\(854\) 0 0
\(855\) 3.00000 0.102598
\(856\) 10.0000 0.341793
\(857\) −14.0000 −0.478231 −0.239115 0.970991i \(-0.576857\pi\)
−0.239115 + 0.970991i \(0.576857\pi\)
\(858\) −1.00000 −0.0341394
\(859\) 12.0000 0.409435 0.204717 0.978821i \(-0.434372\pi\)
0.204717 + 0.978821i \(0.434372\pi\)
\(860\) −8.00000 −0.272798
\(861\) 0 0
\(862\) 0 0
\(863\) −53.0000 −1.80414 −0.902070 0.431589i \(-0.857953\pi\)
−0.902070 + 0.431589i \(0.857953\pi\)
\(864\) 1.00000 0.0340207
\(865\) 15.0000 0.510015
\(866\) 32.0000 1.08740
\(867\) 17.0000 0.577350
\(868\) 0 0
\(869\) 8.00000 0.271381
\(870\) 8.00000 0.271225
\(871\) 0 0
\(872\) −6.00000 −0.203186
\(873\) −16.0000 −0.541518
\(874\) 21.0000 0.710336
\(875\) 0 0
\(876\) 6.00000 0.202721
\(877\) 17.0000 0.574049 0.287025 0.957923i \(-0.407334\pi\)
0.287025 + 0.957923i \(0.407334\pi\)
\(878\) −32.0000 −1.07995
\(879\) −27.0000 −0.910687
\(880\) 1.00000 0.0337100
\(881\) −53.0000 −1.78562 −0.892808 0.450438i \(-0.851268\pi\)
−0.892808 + 0.450438i \(0.851268\pi\)
\(882\) 0 0
\(883\) −42.0000 −1.41341 −0.706706 0.707507i \(-0.749820\pi\)
−0.706706 + 0.707507i \(0.749820\pi\)
\(884\) 0 0
\(885\) 4.00000 0.134459
\(886\) −16.0000 −0.537531
\(887\) 44.0000 1.47738 0.738688 0.674048i \(-0.235446\pi\)
0.738688 + 0.674048i \(0.235446\pi\)
\(888\) 11.0000 0.369136
\(889\) 0 0
\(890\) −10.0000 −0.335201
\(891\) −1.00000 −0.0335013
\(892\) 12.0000 0.401790
\(893\) 15.0000 0.501956
\(894\) 0 0
\(895\) 19.0000 0.635100
\(896\) 0 0
\(897\) −7.00000 −0.233723
\(898\) −11.0000 −0.367075
\(899\) 16.0000 0.533630
\(900\) 1.00000 0.0333333
\(901\) 0 0
\(902\) −11.0000 −0.366260
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) 24.0000 0.797787
\(906\) 6.00000 0.199337
\(907\) −10.0000 −0.332045 −0.166022 0.986122i \(-0.553092\pi\)
−0.166022 + 0.986122i \(0.553092\pi\)
\(908\) 8.00000 0.265489
\(909\) 0 0
\(910\) 0 0
\(911\) −2.00000 −0.0662630 −0.0331315 0.999451i \(-0.510548\pi\)
−0.0331315 + 0.999451i \(0.510548\pi\)
\(912\) 3.00000 0.0993399
\(913\) −8.00000 −0.264761
\(914\) 18.0000 0.595387
\(915\) 0 0
\(916\) 14.0000 0.462573
\(917\) 0 0
\(918\) 0 0
\(919\) −36.0000 −1.18753 −0.593765 0.804638i \(-0.702359\pi\)
−0.593765 + 0.804638i \(0.702359\pi\)
\(920\) 7.00000 0.230783
\(921\) −20.0000 −0.659022
\(922\) 12.0000 0.395199
\(923\) −6.00000 −0.197492
\(924\) 0 0
\(925\) 11.0000 0.361678
\(926\) 13.0000 0.427207
\(927\) −16.0000 −0.525509
\(928\) 8.00000 0.262613
\(929\) −21.0000 −0.688988 −0.344494 0.938789i \(-0.611949\pi\)
−0.344494 + 0.938789i \(0.611949\pi\)
\(930\) 2.00000 0.0655826
\(931\) 0 0
\(932\) 18.0000 0.589610
\(933\) −12.0000 −0.392862
\(934\) 0 0
\(935\) 0 0
\(936\) −1.00000 −0.0326860
\(937\) 12.0000 0.392023 0.196011 0.980602i \(-0.437201\pi\)
0.196011 + 0.980602i \(0.437201\pi\)
\(938\) 0 0
\(939\) 12.0000 0.391605
\(940\) 5.00000 0.163082
\(941\) 54.0000 1.76035 0.880175 0.474650i \(-0.157425\pi\)
0.880175 + 0.474650i \(0.157425\pi\)
\(942\) −7.00000 −0.228072
\(943\) −77.0000 −2.50746
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) 8.00000 0.260102
\(947\) 22.0000 0.714904 0.357452 0.933932i \(-0.383646\pi\)
0.357452 + 0.933932i \(0.383646\pi\)
\(948\) 8.00000 0.259828
\(949\) −6.00000 −0.194768
\(950\) 3.00000 0.0973329
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) −20.0000 −0.647864 −0.323932 0.946080i \(-0.605005\pi\)
−0.323932 + 0.946080i \(0.605005\pi\)
\(954\) 11.0000 0.356138
\(955\) 6.00000 0.194155
\(956\) −18.0000 −0.582162
\(957\) −8.00000 −0.258603
\(958\) −22.0000 −0.710788
\(959\) 0 0
\(960\) 1.00000 0.0322749
\(961\) −27.0000 −0.870968
\(962\) −11.0000 −0.354654
\(963\) −10.0000 −0.322245
\(964\) 7.00000 0.225455
\(965\) −22.0000 −0.708205
\(966\) 0 0
\(967\) −20.0000 −0.643157 −0.321578 0.946883i \(-0.604213\pi\)
−0.321578 + 0.946883i \(0.604213\pi\)
\(968\) 10.0000 0.321412
\(969\) 0 0
\(970\) −16.0000 −0.513729
\(971\) −45.0000 −1.44412 −0.722059 0.691831i \(-0.756804\pi\)
−0.722059 + 0.691831i \(0.756804\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 16.0000 0.512673
\(975\) −1.00000 −0.0320256
\(976\) 0 0
\(977\) 54.0000 1.72761 0.863807 0.503824i \(-0.168074\pi\)
0.863807 + 0.503824i \(0.168074\pi\)
\(978\) 16.0000 0.511624
\(979\) 10.0000 0.319601
\(980\) 0 0
\(981\) 6.00000 0.191565
\(982\) −20.0000 −0.638226
\(983\) 49.0000 1.56286 0.781429 0.623995i \(-0.214491\pi\)
0.781429 + 0.623995i \(0.214491\pi\)
\(984\) −11.0000 −0.350667
\(985\) 1.00000 0.0318626
\(986\) 0 0
\(987\) 0 0
\(988\) −3.00000 −0.0954427
\(989\) 56.0000 1.78070
\(990\) −1.00000 −0.0317821
\(991\) 22.0000 0.698853 0.349427 0.936964i \(-0.386376\pi\)
0.349427 + 0.936964i \(0.386376\pi\)
\(992\) 2.00000 0.0635001
\(993\) 13.0000 0.412543
\(994\) 0 0
\(995\) 24.0000 0.760851
\(996\) −8.00000 −0.253490
\(997\) 14.0000 0.443384 0.221692 0.975117i \(-0.428842\pi\)
0.221692 + 0.975117i \(0.428842\pi\)
\(998\) 32.0000 1.01294
\(999\) −11.0000 −0.348025
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 1470.2.a.a.1.1 1
3.2 odd 2 4410.2.a.bj.1.1 1
5.4 even 2 7350.2.a.cp.1.1 1
7.2 even 3 210.2.i.d.151.1 yes 2
7.3 odd 6 1470.2.i.m.961.1 2
7.4 even 3 210.2.i.d.121.1 2
7.5 odd 6 1470.2.i.m.361.1 2
7.6 odd 2 1470.2.a.h.1.1 1
21.2 odd 6 630.2.k.c.361.1 2
21.11 odd 6 630.2.k.c.541.1 2
21.20 even 2 4410.2.a.ba.1.1 1
28.11 odd 6 1680.2.bg.g.961.1 2
28.23 odd 6 1680.2.bg.g.1201.1 2
35.2 odd 12 1050.2.o.i.949.1 4
35.4 even 6 1050.2.i.b.751.1 2
35.9 even 6 1050.2.i.b.151.1 2
35.18 odd 12 1050.2.o.i.499.1 4
35.23 odd 12 1050.2.o.i.949.2 4
35.32 odd 12 1050.2.o.i.499.2 4
35.34 odd 2 7350.2.a.bu.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
210.2.i.d.121.1 2 7.4 even 3
210.2.i.d.151.1 yes 2 7.2 even 3
630.2.k.c.361.1 2 21.2 odd 6
630.2.k.c.541.1 2 21.11 odd 6
1050.2.i.b.151.1 2 35.9 even 6
1050.2.i.b.751.1 2 35.4 even 6
1050.2.o.i.499.1 4 35.18 odd 12
1050.2.o.i.499.2 4 35.32 odd 12
1050.2.o.i.949.1 4 35.2 odd 12
1050.2.o.i.949.2 4 35.23 odd 12
1470.2.a.a.1.1 1 1.1 even 1 trivial
1470.2.a.h.1.1 1 7.6 odd 2
1470.2.i.m.361.1 2 7.5 odd 6
1470.2.i.m.961.1 2 7.3 odd 6
1680.2.bg.g.961.1 2 28.11 odd 6
1680.2.bg.g.1201.1 2 28.23 odd 6
4410.2.a.ba.1.1 1 21.20 even 2
4410.2.a.bj.1.1 1 3.2 odd 2
7350.2.a.bu.1.1 1 35.34 odd 2
7350.2.a.cp.1.1 1 5.4 even 2