Properties

Label 585.2.a.b.1.1
Level $585$
Weight $2$
Character 585.1
Self dual yes
Analytic conductor $4.671$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 195)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 585.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} +2.00000 q^{4} -1.00000 q^{5} +3.00000 q^{7} +O(q^{10})\) \(q-2.00000 q^{2} +2.00000 q^{4} -1.00000 q^{5} +3.00000 q^{7} +2.00000 q^{10} +1.00000 q^{11} -1.00000 q^{13} -6.00000 q^{14} -4.00000 q^{16} +1.00000 q^{17} -2.00000 q^{19} -2.00000 q^{20} -2.00000 q^{22} +3.00000 q^{23} +1.00000 q^{25} +2.00000 q^{26} +6.00000 q^{28} +2.00000 q^{29} -6.00000 q^{31} +8.00000 q^{32} -2.00000 q^{34} -3.00000 q^{35} +11.0000 q^{37} +4.00000 q^{38} +5.00000 q^{41} +4.00000 q^{43} +2.00000 q^{44} -6.00000 q^{46} +10.0000 q^{47} +2.00000 q^{49} -2.00000 q^{50} -2.00000 q^{52} -11.0000 q^{53} -1.00000 q^{55} -4.00000 q^{58} -8.00000 q^{59} +13.0000 q^{61} +12.0000 q^{62} -8.00000 q^{64} +1.00000 q^{65} +12.0000 q^{67} +2.00000 q^{68} +6.00000 q^{70} +5.00000 q^{71} +10.0000 q^{73} -22.0000 q^{74} -4.00000 q^{76} +3.00000 q^{77} -3.00000 q^{79} +4.00000 q^{80} -10.0000 q^{82} +12.0000 q^{83} -1.00000 q^{85} -8.00000 q^{86} +15.0000 q^{89} -3.00000 q^{91} +6.00000 q^{92} -20.0000 q^{94} +2.00000 q^{95} +17.0000 q^{97} -4.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 0 0
\(4\) 2.00000 1.00000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 2.00000 0.632456
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) −6.00000 −1.60357
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 1.00000 0.242536 0.121268 0.992620i \(-0.461304\pi\)
0.121268 + 0.992620i \(0.461304\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) 3.00000 0.625543 0.312772 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) 6.00000 1.13389
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) 8.00000 1.41421
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) 11.0000 1.80839 0.904194 0.427121i \(-0.140472\pi\)
0.904194 + 0.427121i \(0.140472\pi\)
\(38\) 4.00000 0.648886
\(39\) 0 0
\(40\) 0 0
\(41\) 5.00000 0.780869 0.390434 0.920631i \(-0.372325\pi\)
0.390434 + 0.920631i \(0.372325\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) 10.0000 1.45865 0.729325 0.684167i \(-0.239834\pi\)
0.729325 + 0.684167i \(0.239834\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) −2.00000 −0.282843
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) −11.0000 −1.51097 −0.755483 0.655168i \(-0.772598\pi\)
−0.755483 + 0.655168i \(0.772598\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) 0 0
\(58\) −4.00000 −0.525226
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) 13.0000 1.66448 0.832240 0.554416i \(-0.187058\pi\)
0.832240 + 0.554416i \(0.187058\pi\)
\(62\) 12.0000 1.52400
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) 6.00000 0.717137
\(71\) 5.00000 0.593391 0.296695 0.954972i \(-0.404115\pi\)
0.296695 + 0.954972i \(0.404115\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) −22.0000 −2.55745
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 3.00000 0.341882
\(78\) 0 0
\(79\) −3.00000 −0.337526 −0.168763 0.985657i \(-0.553977\pi\)
−0.168763 + 0.985657i \(0.553977\pi\)
\(80\) 4.00000 0.447214
\(81\) 0 0
\(82\) −10.0000 −1.10432
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) −1.00000 −0.108465
\(86\) −8.00000 −0.862662
\(87\) 0 0
\(88\) 0 0
\(89\) 15.0000 1.59000 0.794998 0.606612i \(-0.207472\pi\)
0.794998 + 0.606612i \(0.207472\pi\)
\(90\) 0 0
\(91\) −3.00000 −0.314485
\(92\) 6.00000 0.625543
\(93\) 0 0
\(94\) −20.0000 −2.06284
\(95\) 2.00000 0.205196
\(96\) 0 0
\(97\) 17.0000 1.72609 0.863044 0.505128i \(-0.168555\pi\)
0.863044 + 0.505128i \(0.168555\pi\)
\(98\) −4.00000 −0.404061
\(99\) 0 0
\(100\) 2.00000 0.200000
\(101\) 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 \(0.789020\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 22.0000 2.13683
\(107\) −9.00000 −0.870063 −0.435031 0.900415i \(-0.643263\pi\)
−0.435031 + 0.900415i \(0.643263\pi\)
\(108\) 0 0
\(109\) −16.0000 −1.53252 −0.766261 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 2.00000 0.190693
\(111\) 0 0
\(112\) −12.0000 −1.13389
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 0 0
\(115\) −3.00000 −0.279751
\(116\) 4.00000 0.371391
\(117\) 0 0
\(118\) 16.0000 1.47292
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) −26.0000 −2.35393
\(123\) 0 0
\(124\) −12.0000 −1.07763
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 10.0000 0.887357 0.443678 0.896186i \(-0.353673\pi\)
0.443678 + 0.896186i \(0.353673\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) −2.00000 −0.175412
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 0 0
\(133\) −6.00000 −0.520266
\(134\) −24.0000 −2.07328
\(135\) 0 0
\(136\) 0 0
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) −1.00000 −0.0848189 −0.0424094 0.999100i \(-0.513503\pi\)
−0.0424094 + 0.999100i \(0.513503\pi\)
\(140\) −6.00000 −0.507093
\(141\) 0 0
\(142\) −10.0000 −0.839181
\(143\) −1.00000 −0.0836242
\(144\) 0 0
\(145\) −2.00000 −0.166091
\(146\) −20.0000 −1.65521
\(147\) 0 0
\(148\) 22.0000 1.80839
\(149\) −13.0000 −1.06500 −0.532501 0.846430i \(-0.678748\pi\)
−0.532501 + 0.846430i \(0.678748\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −6.00000 −0.483494
\(155\) 6.00000 0.481932
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 6.00000 0.477334
\(159\) 0 0
\(160\) −8.00000 −0.632456
\(161\) 9.00000 0.709299
\(162\) 0 0
\(163\) −13.0000 −1.01824 −0.509119 0.860696i \(-0.670029\pi\)
−0.509119 + 0.860696i \(0.670029\pi\)
\(164\) 10.0000 0.780869
\(165\) 0 0
\(166\) −24.0000 −1.86276
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 2.00000 0.153393
\(171\) 0 0
\(172\) 8.00000 0.609994
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 3.00000 0.226779
\(176\) −4.00000 −0.301511
\(177\) 0 0
\(178\) −30.0000 −2.24860
\(179\) −2.00000 −0.149487 −0.0747435 0.997203i \(-0.523814\pi\)
−0.0747435 + 0.997203i \(0.523814\pi\)
\(180\) 0 0
\(181\) −7.00000 −0.520306 −0.260153 0.965567i \(-0.583773\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) 6.00000 0.444750
\(183\) 0 0
\(184\) 0 0
\(185\) −11.0000 −0.808736
\(186\) 0 0
\(187\) 1.00000 0.0731272
\(188\) 20.0000 1.45865
\(189\) 0 0
\(190\) −4.00000 −0.290191
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) −13.0000 −0.935760 −0.467880 0.883792i \(-0.654982\pi\)
−0.467880 + 0.883792i \(0.654982\pi\)
\(194\) −34.0000 −2.44106
\(195\) 0 0
\(196\) 4.00000 0.285714
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 6.00000 0.421117
\(204\) 0 0
\(205\) −5.00000 −0.349215
\(206\) 32.0000 2.22955
\(207\) 0 0
\(208\) 4.00000 0.277350
\(209\) −2.00000 −0.138343
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) −22.0000 −1.51097
\(213\) 0 0
\(214\) 18.0000 1.23045
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) −18.0000 −1.22192
\(218\) 32.0000 2.16731
\(219\) 0 0
\(220\) −2.00000 −0.134840
\(221\) −1.00000 −0.0672673
\(222\) 0 0
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 24.0000 1.60357
\(225\) 0 0
\(226\) 28.0000 1.86253
\(227\) −22.0000 −1.46019 −0.730096 0.683345i \(-0.760525\pi\)
−0.730096 + 0.683345i \(0.760525\pi\)
\(228\) 0 0
\(229\) 18.0000 1.18947 0.594737 0.803921i \(-0.297256\pi\)
0.594737 + 0.803921i \(0.297256\pi\)
\(230\) 6.00000 0.395628
\(231\) 0 0
\(232\) 0 0
\(233\) 27.0000 1.76883 0.884414 0.466702i \(-0.154558\pi\)
0.884414 + 0.466702i \(0.154558\pi\)
\(234\) 0 0
\(235\) −10.0000 −0.652328
\(236\) −16.0000 −1.04151
\(237\) 0 0
\(238\) −6.00000 −0.388922
\(239\) 13.0000 0.840900 0.420450 0.907316i \(-0.361872\pi\)
0.420450 + 0.907316i \(0.361872\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) 20.0000 1.28565
\(243\) 0 0
\(244\) 26.0000 1.66448
\(245\) −2.00000 −0.127775
\(246\) 0 0
\(247\) 2.00000 0.127257
\(248\) 0 0
\(249\) 0 0
\(250\) 2.00000 0.126491
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 3.00000 0.188608
\(254\) −20.0000 −1.25491
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) 33.0000 2.05052
\(260\) 2.00000 0.124035
\(261\) 0 0
\(262\) −12.0000 −0.741362
\(263\) 8.00000 0.493301 0.246651 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(264\) 0 0
\(265\) 11.0000 0.675725
\(266\) 12.0000 0.735767
\(267\) 0 0
\(268\) 24.0000 1.46603
\(269\) 4.00000 0.243884 0.121942 0.992537i \(-0.461088\pi\)
0.121942 + 0.992537i \(0.461088\pi\)
\(270\) 0 0
\(271\) 22.0000 1.33640 0.668202 0.743980i \(-0.267064\pi\)
0.668202 + 0.743980i \(0.267064\pi\)
\(272\) −4.00000 −0.242536
\(273\) 0 0
\(274\) 36.0000 2.17484
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) −18.0000 −1.08152 −0.540758 0.841178i \(-0.681862\pi\)
−0.540758 + 0.841178i \(0.681862\pi\)
\(278\) 2.00000 0.119952
\(279\) 0 0
\(280\) 0 0
\(281\) −30.0000 −1.78965 −0.894825 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 0 0
\(283\) −12.0000 −0.713326 −0.356663 0.934233i \(-0.616086\pi\)
−0.356663 + 0.934233i \(0.616086\pi\)
\(284\) 10.0000 0.593391
\(285\) 0 0
\(286\) 2.00000 0.118262
\(287\) 15.0000 0.885422
\(288\) 0 0
\(289\) −16.0000 −0.941176
\(290\) 4.00000 0.234888
\(291\) 0 0
\(292\) 20.0000 1.17041
\(293\) −24.0000 −1.40209 −0.701047 0.713115i \(-0.747284\pi\)
−0.701047 + 0.713115i \(0.747284\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) 0 0
\(297\) 0 0
\(298\) 26.0000 1.50614
\(299\) −3.00000 −0.173494
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) −32.0000 −1.84139
\(303\) 0 0
\(304\) 8.00000 0.458831
\(305\) −13.0000 −0.744378
\(306\) 0 0
\(307\) −5.00000 −0.285365 −0.142683 0.989769i \(-0.545573\pi\)
−0.142683 + 0.989769i \(0.545573\pi\)
\(308\) 6.00000 0.341882
\(309\) 0 0
\(310\) −12.0000 −0.681554
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 20.0000 1.12867
\(315\) 0 0
\(316\) −6.00000 −0.337526
\(317\) −28.0000 −1.57264 −0.786318 0.617822i \(-0.788015\pi\)
−0.786318 + 0.617822i \(0.788015\pi\)
\(318\) 0 0
\(319\) 2.00000 0.111979
\(320\) 8.00000 0.447214
\(321\) 0 0
\(322\) −18.0000 −1.00310
\(323\) −2.00000 −0.111283
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) 26.0000 1.44001
\(327\) 0 0
\(328\) 0 0
\(329\) 30.0000 1.65395
\(330\) 0 0
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) 24.0000 1.31717
\(333\) 0 0
\(334\) −24.0000 −1.31322
\(335\) −12.0000 −0.655630
\(336\) 0 0
\(337\) 4.00000 0.217894 0.108947 0.994048i \(-0.465252\pi\)
0.108947 + 0.994048i \(0.465252\pi\)
\(338\) −2.00000 −0.108786
\(339\) 0 0
\(340\) −2.00000 −0.108465
\(341\) −6.00000 −0.324918
\(342\) 0 0
\(343\) −15.0000 −0.809924
\(344\) 0 0
\(345\) 0 0
\(346\) −12.0000 −0.645124
\(347\) 19.0000 1.01997 0.509987 0.860182i \(-0.329650\pi\)
0.509987 + 0.860182i \(0.329650\pi\)
\(348\) 0 0
\(349\) −8.00000 −0.428230 −0.214115 0.976808i \(-0.568687\pi\)
−0.214115 + 0.976808i \(0.568687\pi\)
\(350\) −6.00000 −0.320713
\(351\) 0 0
\(352\) 8.00000 0.426401
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) −5.00000 −0.265372
\(356\) 30.0000 1.59000
\(357\) 0 0
\(358\) 4.00000 0.211407
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 14.0000 0.735824
\(363\) 0 0
\(364\) −6.00000 −0.314485
\(365\) −10.0000 −0.523424
\(366\) 0 0
\(367\) 36.0000 1.87918 0.939592 0.342296i \(-0.111204\pi\)
0.939592 + 0.342296i \(0.111204\pi\)
\(368\) −12.0000 −0.625543
\(369\) 0 0
\(370\) 22.0000 1.14373
\(371\) −33.0000 −1.71327
\(372\) 0 0
\(373\) 4.00000 0.207112 0.103556 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) −2.00000 −0.103418
\(375\) 0 0
\(376\) 0 0
\(377\) −2.00000 −0.103005
\(378\) 0 0
\(379\) −14.0000 −0.719132 −0.359566 0.933120i \(-0.617075\pi\)
−0.359566 + 0.933120i \(0.617075\pi\)
\(380\) 4.00000 0.205196
\(381\) 0 0
\(382\) −16.0000 −0.818631
\(383\) 30.0000 1.53293 0.766464 0.642287i \(-0.222014\pi\)
0.766464 + 0.642287i \(0.222014\pi\)
\(384\) 0 0
\(385\) −3.00000 −0.152894
\(386\) 26.0000 1.32337
\(387\) 0 0
\(388\) 34.0000 1.72609
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 3.00000 0.151717
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 3.00000 0.150946
\(396\) 0 0
\(397\) −29.0000 −1.45547 −0.727734 0.685859i \(-0.759427\pi\)
−0.727734 + 0.685859i \(0.759427\pi\)
\(398\) −8.00000 −0.401004
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 0 0
\(403\) 6.00000 0.298881
\(404\) 0 0
\(405\) 0 0
\(406\) −12.0000 −0.595550
\(407\) 11.0000 0.545250
\(408\) 0 0
\(409\) 2.00000 0.0988936 0.0494468 0.998777i \(-0.484254\pi\)
0.0494468 + 0.998777i \(0.484254\pi\)
\(410\) 10.0000 0.493865
\(411\) 0 0
\(412\) −32.0000 −1.57653
\(413\) −24.0000 −1.18096
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) −8.00000 −0.392232
\(417\) 0 0
\(418\) 4.00000 0.195646
\(419\) 26.0000 1.27018 0.635092 0.772437i \(-0.280962\pi\)
0.635092 + 0.772437i \(0.280962\pi\)
\(420\) 0 0
\(421\) 4.00000 0.194948 0.0974740 0.995238i \(-0.468924\pi\)
0.0974740 + 0.995238i \(0.468924\pi\)
\(422\) 8.00000 0.389434
\(423\) 0 0
\(424\) 0 0
\(425\) 1.00000 0.0485071
\(426\) 0 0
\(427\) 39.0000 1.88734
\(428\) −18.0000 −0.870063
\(429\) 0 0
\(430\) 8.00000 0.385794
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 0 0
\(433\) 4.00000 0.192228 0.0961139 0.995370i \(-0.469359\pi\)
0.0961139 + 0.995370i \(0.469359\pi\)
\(434\) 36.0000 1.72806
\(435\) 0 0
\(436\) −32.0000 −1.53252
\(437\) −6.00000 −0.287019
\(438\) 0 0
\(439\) −17.0000 −0.811366 −0.405683 0.914014i \(-0.632966\pi\)
−0.405683 + 0.914014i \(0.632966\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 2.00000 0.0951303
\(443\) −9.00000 −0.427603 −0.213801 0.976877i \(-0.568585\pi\)
−0.213801 + 0.976877i \(0.568585\pi\)
\(444\) 0 0
\(445\) −15.0000 −0.711068
\(446\) −16.0000 −0.757622
\(447\) 0 0
\(448\) −24.0000 −1.13389
\(449\) 13.0000 0.613508 0.306754 0.951789i \(-0.400757\pi\)
0.306754 + 0.951789i \(0.400757\pi\)
\(450\) 0 0
\(451\) 5.00000 0.235441
\(452\) −28.0000 −1.31701
\(453\) 0 0
\(454\) 44.0000 2.06502
\(455\) 3.00000 0.140642
\(456\) 0 0
\(457\) −11.0000 −0.514558 −0.257279 0.966337i \(-0.582826\pi\)
−0.257279 + 0.966337i \(0.582826\pi\)
\(458\) −36.0000 −1.68217
\(459\) 0 0
\(460\) −6.00000 −0.279751
\(461\) −15.0000 −0.698620 −0.349310 0.937007i \(-0.613584\pi\)
−0.349310 + 0.937007i \(0.613584\pi\)
\(462\) 0 0
\(463\) 27.0000 1.25480 0.627398 0.778699i \(-0.284120\pi\)
0.627398 + 0.778699i \(0.284120\pi\)
\(464\) −8.00000 −0.371391
\(465\) 0 0
\(466\) −54.0000 −2.50150
\(467\) 23.0000 1.06431 0.532157 0.846646i \(-0.321382\pi\)
0.532157 + 0.846646i \(0.321382\pi\)
\(468\) 0 0
\(469\) 36.0000 1.66233
\(470\) 20.0000 0.922531
\(471\) 0 0
\(472\) 0 0
\(473\) 4.00000 0.183920
\(474\) 0 0
\(475\) −2.00000 −0.0917663
\(476\) 6.00000 0.275010
\(477\) 0 0
\(478\) −26.0000 −1.18921
\(479\) −9.00000 −0.411220 −0.205610 0.978634i \(-0.565918\pi\)
−0.205610 + 0.978634i \(0.565918\pi\)
\(480\) 0 0
\(481\) −11.0000 −0.501557
\(482\) 4.00000 0.182195
\(483\) 0 0
\(484\) −20.0000 −0.909091
\(485\) −17.0000 −0.771930
\(486\) 0 0
\(487\) −7.00000 −0.317200 −0.158600 0.987343i \(-0.550698\pi\)
−0.158600 + 0.987343i \(0.550698\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 4.00000 0.180702
\(491\) −28.0000 −1.26362 −0.631811 0.775122i \(-0.717688\pi\)
−0.631811 + 0.775122i \(0.717688\pi\)
\(492\) 0 0
\(493\) 2.00000 0.0900755
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) 24.0000 1.07763
\(497\) 15.0000 0.672842
\(498\) 0 0
\(499\) −14.0000 −0.626726 −0.313363 0.949633i \(-0.601456\pi\)
−0.313363 + 0.949633i \(0.601456\pi\)
\(500\) −2.00000 −0.0894427
\(501\) 0 0
\(502\) 0 0
\(503\) −28.0000 −1.24846 −0.624229 0.781241i \(-0.714587\pi\)
−0.624229 + 0.781241i \(0.714587\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −6.00000 −0.266733
\(507\) 0 0
\(508\) 20.0000 0.887357
\(509\) 7.00000 0.310270 0.155135 0.987893i \(-0.450419\pi\)
0.155135 + 0.987893i \(0.450419\pi\)
\(510\) 0 0
\(511\) 30.0000 1.32712
\(512\) −32.0000 −1.41421
\(513\) 0 0
\(514\) −36.0000 −1.58789
\(515\) 16.0000 0.705044
\(516\) 0 0
\(517\) 10.0000 0.439799
\(518\) −66.0000 −2.89987
\(519\) 0 0
\(520\) 0 0
\(521\) 14.0000 0.613351 0.306676 0.951814i \(-0.400783\pi\)
0.306676 + 0.951814i \(0.400783\pi\)
\(522\) 0 0
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) −6.00000 −0.261364
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) −22.0000 −0.955619
\(531\) 0 0
\(532\) −12.0000 −0.520266
\(533\) −5.00000 −0.216574
\(534\) 0 0
\(535\) 9.00000 0.389104
\(536\) 0 0
\(537\) 0 0
\(538\) −8.00000 −0.344904
\(539\) 2.00000 0.0861461
\(540\) 0 0
\(541\) −30.0000 −1.28980 −0.644900 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) −44.0000 −1.88996
\(543\) 0 0
\(544\) 8.00000 0.342997
\(545\) 16.0000 0.685365
\(546\) 0 0
\(547\) −32.0000 −1.36822 −0.684111 0.729378i \(-0.739809\pi\)
−0.684111 + 0.729378i \(0.739809\pi\)
\(548\) −36.0000 −1.53784
\(549\) 0 0
\(550\) −2.00000 −0.0852803
\(551\) −4.00000 −0.170406
\(552\) 0 0
\(553\) −9.00000 −0.382719
\(554\) 36.0000 1.52949
\(555\) 0 0
\(556\) −2.00000 −0.0848189
\(557\) 30.0000 1.27114 0.635570 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) 0 0
\(559\) −4.00000 −0.169182
\(560\) 12.0000 0.507093
\(561\) 0 0
\(562\) 60.0000 2.53095
\(563\) −21.0000 −0.885044 −0.442522 0.896758i \(-0.645916\pi\)
−0.442522 + 0.896758i \(0.645916\pi\)
\(564\) 0 0
\(565\) 14.0000 0.588984
\(566\) 24.0000 1.00880
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) −31.0000 −1.29731 −0.648655 0.761083i \(-0.724668\pi\)
−0.648655 + 0.761083i \(0.724668\pi\)
\(572\) −2.00000 −0.0836242
\(573\) 0 0
\(574\) −30.0000 −1.25218
\(575\) 3.00000 0.125109
\(576\) 0 0
\(577\) −19.0000 −0.790980 −0.395490 0.918470i \(-0.629425\pi\)
−0.395490 + 0.918470i \(0.629425\pi\)
\(578\) 32.0000 1.33102
\(579\) 0 0
\(580\) −4.00000 −0.166091
\(581\) 36.0000 1.49353
\(582\) 0 0
\(583\) −11.0000 −0.455573
\(584\) 0 0
\(585\) 0 0
\(586\) 48.0000 1.98286
\(587\) 18.0000 0.742940 0.371470 0.928445i \(-0.378854\pi\)
0.371470 + 0.928445i \(0.378854\pi\)
\(588\) 0 0
\(589\) 12.0000 0.494451
\(590\) −16.0000 −0.658710
\(591\) 0 0
\(592\) −44.0000 −1.80839
\(593\) −4.00000 −0.164260 −0.0821302 0.996622i \(-0.526172\pi\)
−0.0821302 + 0.996622i \(0.526172\pi\)
\(594\) 0 0
\(595\) −3.00000 −0.122988
\(596\) −26.0000 −1.06500
\(597\) 0 0
\(598\) 6.00000 0.245358
\(599\) −16.0000 −0.653742 −0.326871 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) 0 0
\(601\) −37.0000 −1.50926 −0.754631 0.656150i \(-0.772184\pi\)
−0.754631 + 0.656150i \(0.772184\pi\)
\(602\) −24.0000 −0.978167
\(603\) 0 0
\(604\) 32.0000 1.30206
\(605\) 10.0000 0.406558
\(606\) 0 0
\(607\) −32.0000 −1.29884 −0.649420 0.760430i \(-0.724988\pi\)
−0.649420 + 0.760430i \(0.724988\pi\)
\(608\) −16.0000 −0.648886
\(609\) 0 0
\(610\) 26.0000 1.05271
\(611\) −10.0000 −0.404557
\(612\) 0 0
\(613\) 13.0000 0.525065 0.262533 0.964923i \(-0.415442\pi\)
0.262533 + 0.964923i \(0.415442\pi\)
\(614\) 10.0000 0.403567
\(615\) 0 0
\(616\) 0 0
\(617\) 42.0000 1.69086 0.845428 0.534089i \(-0.179345\pi\)
0.845428 + 0.534089i \(0.179345\pi\)
\(618\) 0 0
\(619\) −34.0000 −1.36658 −0.683288 0.730149i \(-0.739451\pi\)
−0.683288 + 0.730149i \(0.739451\pi\)
\(620\) 12.0000 0.481932
\(621\) 0 0
\(622\) 48.0000 1.92462
\(623\) 45.0000 1.80289
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 20.0000 0.799361
\(627\) 0 0
\(628\) −20.0000 −0.798087
\(629\) 11.0000 0.438599
\(630\) 0 0
\(631\) 40.0000 1.59237 0.796187 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 56.0000 2.22404
\(635\) −10.0000 −0.396838
\(636\) 0 0
\(637\) −2.00000 −0.0792429
\(638\) −4.00000 −0.158362
\(639\) 0 0
\(640\) 0 0
\(641\) 12.0000 0.473972 0.236986 0.971513i \(-0.423841\pi\)
0.236986 + 0.971513i \(0.423841\pi\)
\(642\) 0 0
\(643\) 15.0000 0.591542 0.295771 0.955259i \(-0.404423\pi\)
0.295771 + 0.955259i \(0.404423\pi\)
\(644\) 18.0000 0.709299
\(645\) 0 0
\(646\) 4.00000 0.157378
\(647\) −47.0000 −1.84776 −0.923880 0.382682i \(-0.875001\pi\)
−0.923880 + 0.382682i \(0.875001\pi\)
\(648\) 0 0
\(649\) −8.00000 −0.314027
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) −26.0000 −1.01824
\(653\) −22.0000 −0.860927 −0.430463 0.902608i \(-0.641650\pi\)
−0.430463 + 0.902608i \(0.641650\pi\)
\(654\) 0 0
\(655\) −6.00000 −0.234439
\(656\) −20.0000 −0.780869
\(657\) 0 0
\(658\) −60.0000 −2.33904
\(659\) −20.0000 −0.779089 −0.389545 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) 0 0
\(661\) 4.00000 0.155582 0.0777910 0.996970i \(-0.475213\pi\)
0.0777910 + 0.996970i \(0.475213\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 6.00000 0.232670
\(666\) 0 0
\(667\) 6.00000 0.232321
\(668\) 24.0000 0.928588
\(669\) 0 0
\(670\) 24.0000 0.927201
\(671\) 13.0000 0.501859
\(672\) 0 0
\(673\) −6.00000 −0.231283 −0.115642 0.993291i \(-0.536892\pi\)
−0.115642 + 0.993291i \(0.536892\pi\)
\(674\) −8.00000 −0.308148
\(675\) 0 0
\(676\) 2.00000 0.0769231
\(677\) −3.00000 −0.115299 −0.0576497 0.998337i \(-0.518361\pi\)
−0.0576497 + 0.998337i \(0.518361\pi\)
\(678\) 0 0
\(679\) 51.0000 1.95720
\(680\) 0 0
\(681\) 0 0
\(682\) 12.0000 0.459504
\(683\) 24.0000 0.918334 0.459167 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) 0 0
\(685\) 18.0000 0.687745
\(686\) 30.0000 1.14541
\(687\) 0 0
\(688\) −16.0000 −0.609994
\(689\) 11.0000 0.419067
\(690\) 0 0
\(691\) 22.0000 0.836919 0.418460 0.908235i \(-0.362570\pi\)
0.418460 + 0.908235i \(0.362570\pi\)
\(692\) 12.0000 0.456172
\(693\) 0 0
\(694\) −38.0000 −1.44246
\(695\) 1.00000 0.0379322
\(696\) 0 0
\(697\) 5.00000 0.189389
\(698\) 16.0000 0.605609
\(699\) 0 0
\(700\) 6.00000 0.226779
\(701\) 20.0000 0.755390 0.377695 0.925930i \(-0.376717\pi\)
0.377695 + 0.925930i \(0.376717\pi\)
\(702\) 0 0
\(703\) −22.0000 −0.829746
\(704\) −8.00000 −0.301511
\(705\) 0 0
\(706\) 12.0000 0.451626
\(707\) 0 0
\(708\) 0 0
\(709\) 4.00000 0.150223 0.0751116 0.997175i \(-0.476069\pi\)
0.0751116 + 0.997175i \(0.476069\pi\)
\(710\) 10.0000 0.375293
\(711\) 0 0
\(712\) 0 0
\(713\) −18.0000 −0.674105
\(714\) 0 0
\(715\) 1.00000 0.0373979
\(716\) −4.00000 −0.149487
\(717\) 0 0
\(718\) 48.0000 1.79134
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) −48.0000 −1.78761
\(722\) 30.0000 1.11648
\(723\) 0 0
\(724\) −14.0000 −0.520306
\(725\) 2.00000 0.0742781
\(726\) 0 0
\(727\) 38.0000 1.40934 0.704671 0.709534i \(-0.251095\pi\)
0.704671 + 0.709534i \(0.251095\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 20.0000 0.740233
\(731\) 4.00000 0.147945
\(732\) 0 0
\(733\) −49.0000 −1.80986 −0.904928 0.425564i \(-0.860076\pi\)
−0.904928 + 0.425564i \(0.860076\pi\)
\(734\) −72.0000 −2.65757
\(735\) 0 0
\(736\) 24.0000 0.884652
\(737\) 12.0000 0.442026
\(738\) 0 0
\(739\) 10.0000 0.367856 0.183928 0.982940i \(-0.441119\pi\)
0.183928 + 0.982940i \(0.441119\pi\)
\(740\) −22.0000 −0.808736
\(741\) 0 0
\(742\) 66.0000 2.42294
\(743\) 34.0000 1.24734 0.623670 0.781688i \(-0.285641\pi\)
0.623670 + 0.781688i \(0.285641\pi\)
\(744\) 0 0
\(745\) 13.0000 0.476283
\(746\) −8.00000 −0.292901
\(747\) 0 0
\(748\) 2.00000 0.0731272
\(749\) −27.0000 −0.986559
\(750\) 0 0
\(751\) −5.00000 −0.182453 −0.0912263 0.995830i \(-0.529079\pi\)
−0.0912263 + 0.995830i \(0.529079\pi\)
\(752\) −40.0000 −1.45865
\(753\) 0 0
\(754\) 4.00000 0.145671
\(755\) −16.0000 −0.582300
\(756\) 0 0
\(757\) −8.00000 −0.290765 −0.145382 0.989376i \(-0.546441\pi\)
−0.145382 + 0.989376i \(0.546441\pi\)
\(758\) 28.0000 1.01701
\(759\) 0 0
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 0 0
\(763\) −48.0000 −1.73772
\(764\) 16.0000 0.578860
\(765\) 0 0
\(766\) −60.0000 −2.16789
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) −40.0000 −1.44244 −0.721218 0.692708i \(-0.756418\pi\)
−0.721218 + 0.692708i \(0.756418\pi\)
\(770\) 6.00000 0.216225
\(771\) 0 0
\(772\) −26.0000 −0.935760
\(773\) −36.0000 −1.29483 −0.647415 0.762138i \(-0.724150\pi\)
−0.647415 + 0.762138i \(0.724150\pi\)
\(774\) 0 0
\(775\) −6.00000 −0.215526
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −10.0000 −0.358287
\(780\) 0 0
\(781\) 5.00000 0.178914
\(782\) −6.00000 −0.214560
\(783\) 0 0
\(784\) −8.00000 −0.285714
\(785\) 10.0000 0.356915
\(786\) 0 0
\(787\) 52.0000 1.85360 0.926800 0.375555i \(-0.122548\pi\)
0.926800 + 0.375555i \(0.122548\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) −6.00000 −0.213470
\(791\) −42.0000 −1.49335
\(792\) 0 0
\(793\) −13.0000 −0.461644
\(794\) 58.0000 2.05834
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) 47.0000 1.66483 0.832413 0.554156i \(-0.186959\pi\)
0.832413 + 0.554156i \(0.186959\pi\)
\(798\) 0 0
\(799\) 10.0000 0.353775
\(800\) 8.00000 0.282843
\(801\) 0 0
\(802\) 60.0000 2.11867
\(803\) 10.0000 0.352892
\(804\) 0 0
\(805\) −9.00000 −0.317208
\(806\) −12.0000 −0.422682
\(807\) 0 0
\(808\) 0 0
\(809\) 26.0000 0.914111 0.457056 0.889438i \(-0.348904\pi\)
0.457056 + 0.889438i \(0.348904\pi\)
\(810\) 0 0
\(811\) 36.0000 1.26413 0.632065 0.774915i \(-0.282207\pi\)
0.632065 + 0.774915i \(0.282207\pi\)
\(812\) 12.0000 0.421117
\(813\) 0 0
\(814\) −22.0000 −0.771100
\(815\) 13.0000 0.455370
\(816\) 0 0
\(817\) −8.00000 −0.279885
\(818\) −4.00000 −0.139857
\(819\) 0 0
\(820\) −10.0000 −0.349215
\(821\) −27.0000 −0.942306 −0.471153 0.882051i \(-0.656162\pi\)
−0.471153 + 0.882051i \(0.656162\pi\)
\(822\) 0 0
\(823\) 20.0000 0.697156 0.348578 0.937280i \(-0.386665\pi\)
0.348578 + 0.937280i \(0.386665\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 48.0000 1.67013
\(827\) −26.0000 −0.904109 −0.452054 0.891990i \(-0.649309\pi\)
−0.452054 + 0.891990i \(0.649309\pi\)
\(828\) 0 0
\(829\) 30.0000 1.04194 0.520972 0.853574i \(-0.325570\pi\)
0.520972 + 0.853574i \(0.325570\pi\)
\(830\) 24.0000 0.833052
\(831\) 0 0
\(832\) 8.00000 0.277350
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) −12.0000 −0.415277
\(836\) −4.00000 −0.138343
\(837\) 0 0
\(838\) −52.0000 −1.79631
\(839\) 5.00000 0.172619 0.0863096 0.996268i \(-0.472493\pi\)
0.0863096 + 0.996268i \(0.472493\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −8.00000 −0.275698
\(843\) 0 0
\(844\) −8.00000 −0.275371
\(845\) −1.00000 −0.0344010
\(846\) 0 0
\(847\) −30.0000 −1.03081
\(848\) 44.0000 1.51097
\(849\) 0 0
\(850\) −2.00000 −0.0685994
\(851\) 33.0000 1.13123
\(852\) 0 0
\(853\) 45.0000 1.54077 0.770385 0.637579i \(-0.220064\pi\)
0.770385 + 0.637579i \(0.220064\pi\)
\(854\) −78.0000 −2.66911
\(855\) 0 0
\(856\) 0 0
\(857\) −29.0000 −0.990621 −0.495311 0.868716i \(-0.664946\pi\)
−0.495311 + 0.868716i \(0.664946\pi\)
\(858\) 0 0
\(859\) −29.0000 −0.989467 −0.494734 0.869045i \(-0.664734\pi\)
−0.494734 + 0.869045i \(0.664734\pi\)
\(860\) −8.00000 −0.272798
\(861\) 0 0
\(862\) −48.0000 −1.63489
\(863\) −34.0000 −1.15737 −0.578687 0.815550i \(-0.696435\pi\)
−0.578687 + 0.815550i \(0.696435\pi\)
\(864\) 0 0
\(865\) −6.00000 −0.204006
\(866\) −8.00000 −0.271851
\(867\) 0 0
\(868\) −36.0000 −1.22192
\(869\) −3.00000 −0.101768
\(870\) 0 0
\(871\) −12.0000 −0.406604
\(872\) 0 0
\(873\) 0 0
\(874\) 12.0000 0.405906
\(875\) −3.00000 −0.101419
\(876\) 0 0
\(877\) 18.0000 0.607817 0.303908 0.952701i \(-0.401708\pi\)
0.303908 + 0.952701i \(0.401708\pi\)
\(878\) 34.0000 1.14744
\(879\) 0 0
\(880\) 4.00000 0.134840
\(881\) −2.00000 −0.0673817 −0.0336909 0.999432i \(-0.510726\pi\)
−0.0336909 + 0.999432i \(0.510726\pi\)
\(882\) 0 0
\(883\) −16.0000 −0.538443 −0.269221 0.963078i \(-0.586766\pi\)
−0.269221 + 0.963078i \(0.586766\pi\)
\(884\) −2.00000 −0.0672673
\(885\) 0 0
\(886\) 18.0000 0.604722
\(887\) 21.0000 0.705111 0.352555 0.935791i \(-0.385313\pi\)
0.352555 + 0.935791i \(0.385313\pi\)
\(888\) 0 0
\(889\) 30.0000 1.00617
\(890\) 30.0000 1.00560
\(891\) 0 0
\(892\) 16.0000 0.535720
\(893\) −20.0000 −0.669274
\(894\) 0 0
\(895\) 2.00000 0.0668526
\(896\) 0 0
\(897\) 0 0
\(898\) −26.0000 −0.867631
\(899\) −12.0000 −0.400222
\(900\) 0 0
\(901\) −11.0000 −0.366463
\(902\) −10.0000 −0.332964
\(903\) 0 0
\(904\) 0 0
\(905\) 7.00000 0.232688
\(906\) 0 0
\(907\) 6.00000 0.199227 0.0996134 0.995026i \(-0.468239\pi\)
0.0996134 + 0.995026i \(0.468239\pi\)
\(908\) −44.0000 −1.46019
\(909\) 0 0
\(910\) −6.00000 −0.198898
\(911\) −44.0000 −1.45779 −0.728893 0.684628i \(-0.759965\pi\)
−0.728893 + 0.684628i \(0.759965\pi\)
\(912\) 0 0
\(913\) 12.0000 0.397142
\(914\) 22.0000 0.727695
\(915\) 0 0
\(916\) 36.0000 1.18947
\(917\) 18.0000 0.594412
\(918\) 0 0
\(919\) 37.0000 1.22052 0.610259 0.792202i \(-0.291065\pi\)
0.610259 + 0.792202i \(0.291065\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 30.0000 0.987997
\(923\) −5.00000 −0.164577
\(924\) 0 0
\(925\) 11.0000 0.361678
\(926\) −54.0000 −1.77455
\(927\) 0 0
\(928\) 16.0000 0.525226
\(929\) −1.00000 −0.0328089 −0.0164045 0.999865i \(-0.505222\pi\)
−0.0164045 + 0.999865i \(0.505222\pi\)
\(930\) 0 0
\(931\) −4.00000 −0.131095
\(932\) 54.0000 1.76883
\(933\) 0 0
\(934\) −46.0000 −1.50517
\(935\) −1.00000 −0.0327035
\(936\) 0 0
\(937\) 30.0000 0.980057 0.490029 0.871706i \(-0.336986\pi\)
0.490029 + 0.871706i \(0.336986\pi\)
\(938\) −72.0000 −2.35088
\(939\) 0 0
\(940\) −20.0000 −0.652328
\(941\) −37.0000 −1.20617 −0.603083 0.797679i \(-0.706061\pi\)
−0.603083 + 0.797679i \(0.706061\pi\)
\(942\) 0 0
\(943\) 15.0000 0.488467
\(944\) 32.0000 1.04151
\(945\) 0 0
\(946\) −8.00000 −0.260102
\(947\) 24.0000 0.779895 0.389948 0.920837i \(-0.372493\pi\)
0.389948 + 0.920837i \(0.372493\pi\)
\(948\) 0 0
\(949\) −10.0000 −0.324614
\(950\) 4.00000 0.129777
\(951\) 0 0
\(952\) 0 0
\(953\) 1.00000 0.0323932 0.0161966 0.999869i \(-0.494844\pi\)
0.0161966 + 0.999869i \(0.494844\pi\)
\(954\) 0 0
\(955\) −8.00000 −0.258874
\(956\) 26.0000 0.840900
\(957\) 0 0
\(958\) 18.0000 0.581554
\(959\) −54.0000 −1.74375
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 22.0000 0.709308
\(963\) 0 0
\(964\) −4.00000 −0.128831
\(965\) 13.0000 0.418485
\(966\) 0 0
\(967\) −16.0000 −0.514525 −0.257263 0.966342i \(-0.582821\pi\)
−0.257263 + 0.966342i \(0.582821\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 34.0000 1.09167
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 0 0
\(973\) −3.00000 −0.0961756
\(974\) 14.0000 0.448589
\(975\) 0 0
\(976\) −52.0000 −1.66448
\(977\) −32.0000 −1.02377 −0.511885 0.859054i \(-0.671053\pi\)
−0.511885 + 0.859054i \(0.671053\pi\)
\(978\) 0 0
\(979\) 15.0000 0.479402
\(980\) −4.00000 −0.127775
\(981\) 0 0
\(982\) 56.0000 1.78703
\(983\) 12.0000 0.382741 0.191370 0.981518i \(-0.438707\pi\)
0.191370 + 0.981518i \(0.438707\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −4.00000 −0.127386
\(987\) 0 0
\(988\) 4.00000 0.127257
\(989\) 12.0000 0.381578
\(990\) 0 0
\(991\) −25.0000 −0.794151 −0.397076 0.917786i \(-0.629975\pi\)
−0.397076 + 0.917786i \(0.629975\pi\)
\(992\) −48.0000 −1.52400
\(993\) 0 0
\(994\) −30.0000 −0.951542
\(995\) −4.00000 −0.126809
\(996\) 0 0
\(997\) −36.0000 −1.14013 −0.570066 0.821599i \(-0.693082\pi\)
−0.570066 + 0.821599i \(0.693082\pi\)
\(998\) 28.0000 0.886325
\(999\) 0 0
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 585.2.a.b.1.1 1
3.2 odd 2 195.2.a.b.1.1 1
4.3 odd 2 9360.2.a.d.1.1 1
5.2 odd 4 2925.2.c.c.2224.1 2
5.3 odd 4 2925.2.c.c.2224.2 2
5.4 even 2 2925.2.a.q.1.1 1
12.11 even 2 3120.2.a.u.1.1 1
13.12 even 2 7605.2.a.u.1.1 1
15.2 even 4 975.2.c.a.274.2 2
15.8 even 4 975.2.c.a.274.1 2
15.14 odd 2 975.2.a.c.1.1 1
21.20 even 2 9555.2.a.v.1.1 1
39.38 odd 2 2535.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.a.b.1.1 1 3.2 odd 2
585.2.a.b.1.1 1 1.1 even 1 trivial
975.2.a.c.1.1 1 15.14 odd 2
975.2.c.a.274.1 2 15.8 even 4
975.2.c.a.274.2 2 15.2 even 4
2535.2.a.a.1.1 1 39.38 odd 2
2925.2.a.q.1.1 1 5.4 even 2
2925.2.c.c.2224.1 2 5.2 odd 4
2925.2.c.c.2224.2 2 5.3 odd 4
3120.2.a.u.1.1 1 12.11 even 2
7605.2.a.u.1.1 1 13.12 even 2
9360.2.a.d.1.1 1 4.3 odd 2
9555.2.a.v.1.1 1 21.20 even 2