Properties

Label 6045.2.a.f.1.1
Level 6045
Weight 2
Character 6045.1
Self dual yes
Analytic conductor 48.270
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6045.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.2695680219\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 6045.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -3.00000 q^{11} -2.00000 q^{12} +1.00000 q^{13} -1.00000 q^{15} +4.00000 q^{16} -3.00000 q^{17} +2.00000 q^{19} +2.00000 q^{20} -1.00000 q^{21} +3.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +2.00000 q^{28} +1.00000 q^{31} -3.00000 q^{33} +1.00000 q^{35} -2.00000 q^{36} -7.00000 q^{37} +1.00000 q^{39} -9.00000 q^{41} -10.0000 q^{43} +6.00000 q^{44} -1.00000 q^{45} +12.0000 q^{47} +4.00000 q^{48} -6.00000 q^{49} -3.00000 q^{51} -2.00000 q^{52} -3.00000 q^{53} +3.00000 q^{55} +2.00000 q^{57} +12.0000 q^{59} +2.00000 q^{60} -1.00000 q^{61} -1.00000 q^{63} -8.00000 q^{64} -1.00000 q^{65} -4.00000 q^{67} +6.00000 q^{68} +3.00000 q^{69} -9.00000 q^{71} +2.00000 q^{73} +1.00000 q^{75} -4.00000 q^{76} +3.00000 q^{77} -1.00000 q^{79} -4.00000 q^{80} +1.00000 q^{81} -18.0000 q^{83} +2.00000 q^{84} +3.00000 q^{85} +15.0000 q^{89} -1.00000 q^{91} -6.00000 q^{92} +1.00000 q^{93} -2.00000 q^{95} +17.0000 q^{97} -3.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 1.00000 0.577350
\(4\) −2.00000 −1.00000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) −2.00000 −0.577350
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 4.00000 1.00000
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 2.00000 0.447214
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 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
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 2.00000 0.377964
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) 0 0
\(33\) −3.00000 −0.522233
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) −2.00000 −0.333333
\(37\) −7.00000 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −9.00000 −1.40556 −0.702782 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 0 0
\(43\) −10.0000 −1.52499 −0.762493 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 6.00000 0.904534
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 4.00000 0.577350
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) −3.00000 −0.420084
\(52\) −2.00000 −0.277350
\(53\) −3.00000 −0.412082 −0.206041 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) 0 0
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 2.00000 0.258199
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) −8.00000 −1.00000
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 6.00000 0.727607
\(69\) 3.00000 0.361158
\(70\) 0 0
\(71\) −9.00000 −1.06810 −0.534052 0.845452i \(-0.679331\pi\)
−0.534052 + 0.845452i \(0.679331\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) −4.00000 −0.458831
\(77\) 3.00000 0.341882
\(78\) 0 0
\(79\) −1.00000 −0.112509 −0.0562544 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) −4.00000 −0.447214
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −18.0000 −1.97576 −0.987878 0.155230i \(-0.950388\pi\)
−0.987878 + 0.155230i \(0.950388\pi\)
\(84\) 2.00000 0.218218
\(85\) 3.00000 0.325396
\(86\) 0 0
\(87\) 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
\(91\) −1.00000 −0.104828
\(92\) −6.00000 −0.625543
\(93\) 1.00000 0.103695
\(94\) 0 0
\(95\) −2.00000 −0.205196
\(96\) 0 0
\(97\) 17.0000 1.72609 0.863044 0.505128i \(-0.168555\pi\)
0.863044 + 0.505128i \(0.168555\pi\)
\(98\) 0 0
\(99\) −3.00000 −0.301511
\(100\) −2.00000 −0.200000
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) 1.00000 0.0975900
\(106\) 0 0
\(107\) 9.00000 0.870063 0.435031 0.900415i \(-0.356737\pi\)
0.435031 + 0.900415i \(0.356737\pi\)
\(108\) −2.00000 −0.192450
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) −7.00000 −0.664411
\(112\) −4.00000 −0.377964
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) −3.00000 −0.279751
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) −9.00000 −0.811503
\(124\) −2.00000 −0.179605
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 20.0000 1.77471 0.887357 0.461084i \(-0.152539\pi\)
0.887357 + 0.461084i \(0.152539\pi\)
\(128\) 0 0
\(129\) −10.0000 −0.880451
\(130\) 0 0
\(131\) 18.0000 1.57267 0.786334 0.617802i \(-0.211977\pi\)
0.786334 + 0.617802i \(0.211977\pi\)
\(132\) 6.00000 0.522233
\(133\) −2.00000 −0.173422
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 12.0000 1.02523 0.512615 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) 0 0
\(139\) 5.00000 0.424094 0.212047 0.977259i \(-0.431987\pi\)
0.212047 + 0.977259i \(0.431987\pi\)
\(140\) −2.00000 −0.169031
\(141\) 12.0000 1.01058
\(142\) 0 0
\(143\) −3.00000 −0.250873
\(144\) 4.00000 0.333333
\(145\) 0 0
\(146\) 0 0
\(147\) −6.00000 −0.494872
\(148\) 14.0000 1.15079
\(149\) 21.0000 1.72039 0.860194 0.509968i \(-0.170343\pi\)
0.860194 + 0.509968i \(0.170343\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 0 0
\(153\) −3.00000 −0.242536
\(154\) 0 0
\(155\) −1.00000 −0.0803219
\(156\) −2.00000 −0.160128
\(157\) −4.00000 −0.319235 −0.159617 0.987179i \(-0.551026\pi\)
−0.159617 + 0.987179i \(0.551026\pi\)
\(158\) 0 0
\(159\) −3.00000 −0.237915
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) 0 0
\(163\) 11.0000 0.861586 0.430793 0.902451i \(-0.358234\pi\)
0.430793 + 0.902451i \(0.358234\pi\)
\(164\) 18.0000 1.40556
\(165\) 3.00000 0.233550
\(166\) 0 0
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) 20.0000 1.52499
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) −12.0000 −0.904534
\(177\) 12.0000 0.901975
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 2.00000 0.149071
\(181\) 11.0000 0.817624 0.408812 0.912619i \(-0.365943\pi\)
0.408812 + 0.912619i \(0.365943\pi\)
\(182\) 0 0
\(183\) −1.00000 −0.0739221
\(184\) 0 0
\(185\) 7.00000 0.514650
\(186\) 0 0
\(187\) 9.00000 0.658145
\(188\) −24.0000 −1.75038
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −6.00000 −0.434145 −0.217072 0.976156i \(-0.569651\pi\)
−0.217072 + 0.976156i \(0.569651\pi\)
\(192\) −8.00000 −0.577350
\(193\) 23.0000 1.65558 0.827788 0.561041i \(-0.189599\pi\)
0.827788 + 0.561041i \(0.189599\pi\)
\(194\) 0 0
\(195\) −1.00000 −0.0716115
\(196\) 12.0000 0.857143
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 0 0
\(203\) 0 0
\(204\) 6.00000 0.420084
\(205\) 9.00000 0.628587
\(206\) 0 0
\(207\) 3.00000 0.208514
\(208\) 4.00000 0.277350
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 6.00000 0.412082
\(213\) −9.00000 −0.616670
\(214\) 0 0
\(215\) 10.0000 0.681994
\(216\) 0 0
\(217\) −1.00000 −0.0678844
\(218\) 0 0
\(219\) 2.00000 0.135147
\(220\) −6.00000 −0.404520
\(221\) −3.00000 −0.201802
\(222\) 0 0
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −6.00000 −0.398234 −0.199117 0.979976i \(-0.563807\pi\)
−0.199117 + 0.979976i \(0.563807\pi\)
\(228\) −4.00000 −0.264906
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) 3.00000 0.197386
\(232\) 0 0
\(233\) 9.00000 0.589610 0.294805 0.955557i \(-0.404745\pi\)
0.294805 + 0.955557i \(0.404745\pi\)
\(234\) 0 0
\(235\) −12.0000 −0.782794
\(236\) −24.0000 −1.56227
\(237\) −1.00000 −0.0649570
\(238\) 0 0
\(239\) −15.0000 −0.970269 −0.485135 0.874439i \(-0.661229\pi\)
−0.485135 + 0.874439i \(0.661229\pi\)
\(240\) −4.00000 −0.258199
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 2.00000 0.128037
\(245\) 6.00000 0.383326
\(246\) 0 0
\(247\) 2.00000 0.127257
\(248\) 0 0
\(249\) −18.0000 −1.14070
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 2.00000 0.125988
\(253\) −9.00000 −0.565825
\(254\) 0 0
\(255\) 3.00000 0.187867
\(256\) 16.0000 1.00000
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) 7.00000 0.434959
\(260\) 2.00000 0.124035
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 3.00000 0.184289
\(266\) 0 0
\(267\) 15.0000 0.917985
\(268\) 8.00000 0.488678
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) −12.0000 −0.727607
\(273\) −1.00000 −0.0605228
\(274\) 0 0
\(275\) −3.00000 −0.180907
\(276\) −6.00000 −0.361158
\(277\) 8.00000 0.480673 0.240337 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) 0 0
\(279\) 1.00000 0.0598684
\(280\) 0 0
\(281\) 30.0000 1.78965 0.894825 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 18.0000 1.06810
\(285\) −2.00000 −0.118470
\(286\) 0 0
\(287\) 9.00000 0.531253
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 17.0000 0.996558
\(292\) −4.00000 −0.234082
\(293\) 12.0000 0.701047 0.350524 0.936554i \(-0.386004\pi\)
0.350524 + 0.936554i \(0.386004\pi\)
\(294\) 0 0
\(295\) −12.0000 −0.698667
\(296\) 0 0
\(297\) −3.00000 −0.174078
\(298\) 0 0
\(299\) 3.00000 0.173494
\(300\) −2.00000 −0.115470
\(301\) 10.0000 0.576390
\(302\) 0 0
\(303\) 18.0000 1.03407
\(304\) 8.00000 0.458831
\(305\) 1.00000 0.0572598
\(306\) 0 0
\(307\) 11.0000 0.627803 0.313902 0.949456i \(-0.398364\pi\)
0.313902 + 0.949456i \(0.398364\pi\)
\(308\) −6.00000 −0.341882
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 0 0
\(315\) 1.00000 0.0563436
\(316\) 2.00000 0.112509
\(317\) 24.0000 1.34797 0.673987 0.738743i \(-0.264580\pi\)
0.673987 + 0.738743i \(0.264580\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 8.00000 0.447214
\(321\) 9.00000 0.502331
\(322\) 0 0
\(323\) −6.00000 −0.333849
\(324\) −2.00000 −0.111111
\(325\) 1.00000 0.0554700
\(326\) 0 0
\(327\) 2.00000 0.110600
\(328\) 0 0
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) 36.0000 1.97576
\(333\) −7.00000 −0.383598
\(334\) 0 0
\(335\) 4.00000 0.218543
\(336\) −4.00000 −0.218218
\(337\) −4.00000 −0.217894 −0.108947 0.994048i \(-0.534748\pi\)
−0.108947 + 0.994048i \(0.534748\pi\)
\(338\) 0 0
\(339\) 6.00000 0.325875
\(340\) −6.00000 −0.325396
\(341\) −3.00000 −0.162459
\(342\) 0 0
\(343\) 13.0000 0.701934
\(344\) 0 0
\(345\) −3.00000 −0.161515
\(346\) 0 0
\(347\) 3.00000 0.161048 0.0805242 0.996753i \(-0.474341\pi\)
0.0805242 + 0.996753i \(0.474341\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) 12.0000 0.638696 0.319348 0.947638i \(-0.396536\pi\)
0.319348 + 0.947638i \(0.396536\pi\)
\(354\) 0 0
\(355\) 9.00000 0.477670
\(356\) −30.0000 −1.59000
\(357\) 3.00000 0.158777
\(358\) 0 0
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) −2.00000 −0.104973
\(364\) 2.00000 0.104828
\(365\) −2.00000 −0.104685
\(366\) 0 0
\(367\) −10.0000 −0.521996 −0.260998 0.965339i \(-0.584052\pi\)
−0.260998 + 0.965339i \(0.584052\pi\)
\(368\) 12.0000 0.625543
\(369\) −9.00000 −0.468521
\(370\) 0 0
\(371\) 3.00000 0.155752
\(372\) −2.00000 −0.103695
\(373\) 14.0000 0.724893 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 2.00000 0.102733 0.0513665 0.998680i \(-0.483642\pi\)
0.0513665 + 0.998680i \(0.483642\pi\)
\(380\) 4.00000 0.205196
\(381\) 20.0000 1.02463
\(382\) 0 0
\(383\) −12.0000 −0.613171 −0.306586 0.951843i \(-0.599187\pi\)
−0.306586 + 0.951843i \(0.599187\pi\)
\(384\) 0 0
\(385\) −3.00000 −0.152894
\(386\) 0 0
\(387\) −10.0000 −0.508329
\(388\) −34.0000 −1.72609
\(389\) −12.0000 −0.608424 −0.304212 0.952604i \(-0.598393\pi\)
−0.304212 + 0.952604i \(0.598393\pi\)
\(390\) 0 0
\(391\) −9.00000 −0.455150
\(392\) 0 0
\(393\) 18.0000 0.907980
\(394\) 0 0
\(395\) 1.00000 0.0503155
\(396\) 6.00000 0.301511
\(397\) 11.0000 0.552074 0.276037 0.961147i \(-0.410979\pi\)
0.276037 + 0.961147i \(0.410979\pi\)
\(398\) 0 0
\(399\) −2.00000 −0.100125
\(400\) 4.00000 0.200000
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 0 0
\(403\) 1.00000 0.0498135
\(404\) −36.0000 −1.79107
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 21.0000 1.04093
\(408\) 0 0
\(409\) −22.0000 −1.08783 −0.543915 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) 0 0
\(411\) 12.0000 0.591916
\(412\) 8.00000 0.394132
\(413\) −12.0000 −0.590481
\(414\) 0 0
\(415\) 18.0000 0.883585
\(416\) 0 0
\(417\) 5.00000 0.244851
\(418\) 0 0
\(419\) −6.00000 −0.293119 −0.146560 0.989202i \(-0.546820\pi\)
−0.146560 + 0.989202i \(0.546820\pi\)
\(420\) −2.00000 −0.0975900
\(421\) 8.00000 0.389896 0.194948 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) 0 0
\(423\) 12.0000 0.583460
\(424\) 0 0
\(425\) −3.00000 −0.145521
\(426\) 0 0
\(427\) 1.00000 0.0483934
\(428\) −18.0000 −0.870063
\(429\) −3.00000 −0.144841
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 4.00000 0.192450
\(433\) −16.0000 −0.768911 −0.384455 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −4.00000 −0.191565
\(437\) 6.00000 0.287019
\(438\) 0 0
\(439\) −1.00000 −0.0477274 −0.0238637 0.999715i \(-0.507597\pi\)
−0.0238637 + 0.999715i \(0.507597\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 0 0
\(443\) 9.00000 0.427603 0.213801 0.976877i \(-0.431415\pi\)
0.213801 + 0.976877i \(0.431415\pi\)
\(444\) 14.0000 0.664411
\(445\) −15.0000 −0.711068
\(446\) 0 0
\(447\) 21.0000 0.993266
\(448\) 8.00000 0.377964
\(449\) −15.0000 −0.707894 −0.353947 0.935266i \(-0.615161\pi\)
−0.353947 + 0.935266i \(0.615161\pi\)
\(450\) 0 0
\(451\) 27.0000 1.27138
\(452\) −12.0000 −0.564433
\(453\) 8.00000 0.375873
\(454\) 0 0
\(455\) 1.00000 0.0468807
\(456\) 0 0
\(457\) −37.0000 −1.73079 −0.865393 0.501093i \(-0.832931\pi\)
−0.865393 + 0.501093i \(0.832931\pi\)
\(458\) 0 0
\(459\) −3.00000 −0.140028
\(460\) 6.00000 0.279751
\(461\) −3.00000 −0.139724 −0.0698620 0.997557i \(-0.522256\pi\)
−0.0698620 + 0.997557i \(0.522256\pi\)
\(462\) 0 0
\(463\) 5.00000 0.232370 0.116185 0.993228i \(-0.462933\pi\)
0.116185 + 0.993228i \(0.462933\pi\)
\(464\) 0 0
\(465\) −1.00000 −0.0463739
\(466\) 0 0
\(467\) 9.00000 0.416470 0.208235 0.978079i \(-0.433228\pi\)
0.208235 + 0.978079i \(0.433228\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) −4.00000 −0.184310
\(472\) 0 0
\(473\) 30.0000 1.37940
\(474\) 0 0
\(475\) 2.00000 0.0917663
\(476\) −6.00000 −0.275010
\(477\) −3.00000 −0.137361
\(478\) 0 0
\(479\) −15.0000 −0.685367 −0.342684 0.939451i \(-0.611336\pi\)
−0.342684 + 0.939451i \(0.611336\pi\)
\(480\) 0 0
\(481\) −7.00000 −0.319173
\(482\) 0 0
\(483\) −3.00000 −0.136505
\(484\) 4.00000 0.181818
\(485\) −17.0000 −0.771930
\(486\) 0 0
\(487\) 11.0000 0.498458 0.249229 0.968445i \(-0.419823\pi\)
0.249229 + 0.968445i \(0.419823\pi\)
\(488\) 0 0
\(489\) 11.0000 0.497437
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 18.0000 0.811503
\(493\) 0 0
\(494\) 0 0
\(495\) 3.00000 0.134840
\(496\) 4.00000 0.179605
\(497\) 9.00000 0.403705
\(498\) 0 0
\(499\) 32.0000 1.43252 0.716258 0.697835i \(-0.245853\pi\)
0.716258 + 0.697835i \(0.245853\pi\)
\(500\) 2.00000 0.0894427
\(501\) −12.0000 −0.536120
\(502\) 0 0
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 0 0
\(505\) −18.0000 −0.800989
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) −40.0000 −1.77471
\(509\) 27.0000 1.19675 0.598377 0.801215i \(-0.295813\pi\)
0.598377 + 0.801215i \(0.295813\pi\)
\(510\) 0 0
\(511\) −2.00000 −0.0884748
\(512\) 0 0
\(513\) 2.00000 0.0883022
\(514\) 0 0
\(515\) 4.00000 0.176261
\(516\) 20.0000 0.880451
\(517\) −36.0000 −1.58328
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 0 0
\(523\) −34.0000 −1.48672 −0.743358 0.668894i \(-0.766768\pi\)
−0.743358 + 0.668894i \(0.766768\pi\)
\(524\) −36.0000 −1.57267
\(525\) −1.00000 −0.0436436
\(526\) 0 0
\(527\) −3.00000 −0.130682
\(528\) −12.0000 −0.522233
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) 4.00000 0.173422
\(533\) −9.00000 −0.389833
\(534\) 0 0
\(535\) −9.00000 −0.389104
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 18.0000 0.775315
\(540\) 2.00000 0.0860663
\(541\) −16.0000 −0.687894 −0.343947 0.938989i \(-0.611764\pi\)
−0.343947 + 0.938989i \(0.611764\pi\)
\(542\) 0 0
\(543\) 11.0000 0.472055
\(544\) 0 0
\(545\) −2.00000 −0.0856706
\(546\) 0 0
\(547\) 44.0000 1.88130 0.940652 0.339372i \(-0.110215\pi\)
0.940652 + 0.339372i \(0.110215\pi\)
\(548\) −24.0000 −1.02523
\(549\) −1.00000 −0.0426790
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 1.00000 0.0425243
\(554\) 0 0
\(555\) 7.00000 0.297133
\(556\) −10.0000 −0.424094
\(557\) −12.0000 −0.508456 −0.254228 0.967144i \(-0.581821\pi\)
−0.254228 + 0.967144i \(0.581821\pi\)
\(558\) 0 0
\(559\) −10.0000 −0.422955
\(560\) 4.00000 0.169031
\(561\) 9.00000 0.379980
\(562\) 0 0
\(563\) −3.00000 −0.126435 −0.0632175 0.998000i \(-0.520136\pi\)
−0.0632175 + 0.998000i \(0.520136\pi\)
\(564\) −24.0000 −1.01058
\(565\) −6.00000 −0.252422
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) −13.0000 −0.544033 −0.272017 0.962293i \(-0.587691\pi\)
−0.272017 + 0.962293i \(0.587691\pi\)
\(572\) 6.00000 0.250873
\(573\) −6.00000 −0.250654
\(574\) 0 0
\(575\) 3.00000 0.125109
\(576\) −8.00000 −0.333333
\(577\) −7.00000 −0.291414 −0.145707 0.989328i \(-0.546546\pi\)
−0.145707 + 0.989328i \(0.546546\pi\)
\(578\) 0 0
\(579\) 23.0000 0.955847
\(580\) 0 0
\(581\) 18.0000 0.746766
\(582\) 0 0
\(583\) 9.00000 0.372742
\(584\) 0 0
\(585\) −1.00000 −0.0413449
\(586\) 0 0
\(587\) 36.0000 1.48588 0.742940 0.669359i \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) 12.0000 0.494872
\(589\) 2.00000 0.0824086
\(590\) 0 0
\(591\) 12.0000 0.493614
\(592\) −28.0000 −1.15079
\(593\) −36.0000 −1.47834 −0.739171 0.673517i \(-0.764783\pi\)
−0.739171 + 0.673517i \(0.764783\pi\)
\(594\) 0 0
\(595\) −3.00000 −0.122988
\(596\) −42.0000 −1.72039
\(597\) −16.0000 −0.654836
\(598\) 0 0
\(599\) 48.0000 1.96123 0.980613 0.195952i \(-0.0627798\pi\)
0.980613 + 0.195952i \(0.0627798\pi\)
\(600\) 0 0
\(601\) −19.0000 −0.775026 −0.387513 0.921864i \(-0.626666\pi\)
−0.387513 + 0.921864i \(0.626666\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) −16.0000 −0.651031
\(605\) 2.00000 0.0813116
\(606\) 0 0
\(607\) 14.0000 0.568242 0.284121 0.958788i \(-0.408298\pi\)
0.284121 + 0.958788i \(0.408298\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 12.0000 0.485468
\(612\) 6.00000 0.242536
\(613\) 11.0000 0.444286 0.222143 0.975014i \(-0.428695\pi\)
0.222143 + 0.975014i \(0.428695\pi\)
\(614\) 0 0
\(615\) 9.00000 0.362915
\(616\) 0 0
\(617\) 12.0000 0.483102 0.241551 0.970388i \(-0.422344\pi\)
0.241551 + 0.970388i \(0.422344\pi\)
\(618\) 0 0
\(619\) 26.0000 1.04503 0.522514 0.852631i \(-0.324994\pi\)
0.522514 + 0.852631i \(0.324994\pi\)
\(620\) 2.00000 0.0803219
\(621\) 3.00000 0.120386
\(622\) 0 0
\(623\) −15.0000 −0.600962
\(624\) 4.00000 0.160128
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −6.00000 −0.239617
\(628\) 8.00000 0.319235
\(629\) 21.0000 0.837325
\(630\) 0 0
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) 0 0
\(633\) −4.00000 −0.158986
\(634\) 0 0
\(635\) −20.0000 −0.793676
\(636\) 6.00000 0.237915
\(637\) −6.00000 −0.237729
\(638\) 0 0
\(639\) −9.00000 −0.356034
\(640\) 0 0
\(641\) −42.0000 −1.65890 −0.829450 0.558581i \(-0.811346\pi\)
−0.829450 + 0.558581i \(0.811346\pi\)
\(642\) 0 0
\(643\) 5.00000 0.197181 0.0985904 0.995128i \(-0.468567\pi\)
0.0985904 + 0.995128i \(0.468567\pi\)
\(644\) 6.00000 0.236433
\(645\) 10.0000 0.393750
\(646\) 0 0
\(647\) 33.0000 1.29736 0.648682 0.761060i \(-0.275321\pi\)
0.648682 + 0.761060i \(0.275321\pi\)
\(648\) 0 0
\(649\) −36.0000 −1.41312
\(650\) 0 0
\(651\) −1.00000 −0.0391931
\(652\) −22.0000 −0.861586
\(653\) 30.0000 1.17399 0.586995 0.809590i \(-0.300311\pi\)
0.586995 + 0.809590i \(0.300311\pi\)
\(654\) 0 0
\(655\) −18.0000 −0.703318
\(656\) −36.0000 −1.40556
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) 6.00000 0.233727 0.116863 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) −6.00000 −0.233550
\(661\) 32.0000 1.24466 0.622328 0.782757i \(-0.286187\pi\)
0.622328 + 0.782757i \(0.286187\pi\)
\(662\) 0 0
\(663\) −3.00000 −0.116510
\(664\) 0 0
\(665\) 2.00000 0.0775567
\(666\) 0 0
\(667\) 0 0
\(668\) 24.0000 0.928588
\(669\) 8.00000 0.309298
\(670\) 0 0
\(671\) 3.00000 0.115814
\(672\) 0 0
\(673\) −28.0000 −1.07932 −0.539660 0.841883i \(-0.681447\pi\)
−0.539660 + 0.841883i \(0.681447\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) −2.00000 −0.0769231
\(677\) 9.00000 0.345898 0.172949 0.984931i \(-0.444670\pi\)
0.172949 + 0.984931i \(0.444670\pi\)
\(678\) 0 0
\(679\) −17.0000 −0.652400
\(680\) 0 0
\(681\) −6.00000 −0.229920
\(682\) 0 0
\(683\) −24.0000 −0.918334 −0.459167 0.888350i \(-0.651852\pi\)
−0.459167 + 0.888350i \(0.651852\pi\)
\(684\) −4.00000 −0.152944
\(685\) −12.0000 −0.458496
\(686\) 0 0
\(687\) 14.0000 0.534133
\(688\) −40.0000 −1.52499
\(689\) −3.00000 −0.114291
\(690\) 0 0
\(691\) −10.0000 −0.380418 −0.190209 0.981744i \(-0.560917\pi\)
−0.190209 + 0.981744i \(0.560917\pi\)
\(692\) 12.0000 0.456172
\(693\) 3.00000 0.113961
\(694\) 0 0
\(695\) −5.00000 −0.189661
\(696\) 0 0
\(697\) 27.0000 1.02270
\(698\) 0 0
\(699\) 9.00000 0.340411
\(700\) 2.00000 0.0755929
\(701\) −24.0000 −0.906467 −0.453234 0.891392i \(-0.649730\pi\)
−0.453234 + 0.891392i \(0.649730\pi\)
\(702\) 0 0
\(703\) −14.0000 −0.528020
\(704\) 24.0000 0.904534
\(705\) −12.0000 −0.451946
\(706\) 0 0
\(707\) −18.0000 −0.676960
\(708\) −24.0000 −0.901975
\(709\) −46.0000 −1.72757 −0.863783 0.503864i \(-0.831911\pi\)
−0.863783 + 0.503864i \(0.831911\pi\)
\(710\) 0 0
\(711\) −1.00000 −0.0375029
\(712\) 0 0
\(713\) 3.00000 0.112351
\(714\) 0 0
\(715\) 3.00000 0.112194
\(716\) 0 0
\(717\) −15.0000 −0.560185
\(718\) 0 0
\(719\) 6.00000 0.223762 0.111881 0.993722i \(-0.464312\pi\)
0.111881 + 0.993722i \(0.464312\pi\)
\(720\) −4.00000 −0.149071
\(721\) 4.00000 0.148968
\(722\) 0 0
\(723\) −10.0000 −0.371904
\(724\) −22.0000 −0.817624
\(725\) 0 0
\(726\) 0 0
\(727\) 44.0000 1.63187 0.815935 0.578144i \(-0.196223\pi\)
0.815935 + 0.578144i \(0.196223\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 30.0000 1.10959
\(732\) 2.00000 0.0739221
\(733\) −49.0000 −1.80986 −0.904928 0.425564i \(-0.860076\pi\)
−0.904928 + 0.425564i \(0.860076\pi\)
\(734\) 0 0
\(735\) 6.00000 0.221313
\(736\) 0 0
\(737\) 12.0000 0.442026
\(738\) 0 0
\(739\) −52.0000 −1.91285 −0.956425 0.291977i \(-0.905687\pi\)
−0.956425 + 0.291977i \(0.905687\pi\)
\(740\) −14.0000 −0.514650
\(741\) 2.00000 0.0734718
\(742\) 0 0
\(743\) 48.0000 1.76095 0.880475 0.474093i \(-0.157224\pi\)
0.880475 + 0.474093i \(0.157224\pi\)
\(744\) 0 0
\(745\) −21.0000 −0.769380
\(746\) 0 0
\(747\) −18.0000 −0.658586
\(748\) −18.0000 −0.658145
\(749\) −9.00000 −0.328853
\(750\) 0 0
\(751\) 23.0000 0.839282 0.419641 0.907690i \(-0.362156\pi\)
0.419641 + 0.907690i \(0.362156\pi\)
\(752\) 48.0000 1.75038
\(753\) −12.0000 −0.437304
\(754\) 0 0
\(755\) −8.00000 −0.291150
\(756\) 2.00000 0.0727393
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) 0 0
\(759\) −9.00000 −0.326679
\(760\) 0 0
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) 0 0
\(763\) −2.00000 −0.0724049
\(764\) 12.0000 0.434145
\(765\) 3.00000 0.108465
\(766\) 0 0
\(767\) 12.0000 0.433295
\(768\) 16.0000 0.577350
\(769\) −40.0000 −1.44244 −0.721218 0.692708i \(-0.756418\pi\)
−0.721218 + 0.692708i \(0.756418\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) −46.0000 −1.65558
\(773\) −24.0000 −0.863220 −0.431610 0.902060i \(-0.642054\pi\)
−0.431610 + 0.902060i \(0.642054\pi\)
\(774\) 0 0
\(775\) 1.00000 0.0359211
\(776\) 0 0
\(777\) 7.00000 0.251124
\(778\) 0 0
\(779\) −18.0000 −0.644917
\(780\) 2.00000 0.0716115
\(781\) 27.0000 0.966136
\(782\) 0 0
\(783\) 0 0
\(784\) −24.0000 −0.857143
\(785\) 4.00000 0.142766
\(786\) 0 0
\(787\) −4.00000 −0.142585 −0.0712923 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) −24.0000 −0.854965
\(789\) 0 0
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) 0 0
\(793\) −1.00000 −0.0355110
\(794\) 0 0
\(795\) 3.00000 0.106399
\(796\) 32.0000 1.13421
\(797\) −21.0000 −0.743858 −0.371929 0.928261i \(-0.621304\pi\)
−0.371929 + 0.928261i \(0.621304\pi\)
\(798\) 0 0
\(799\) −36.0000 −1.27359
\(800\) 0 0
\(801\) 15.0000 0.529999
\(802\) 0 0
\(803\) −6.00000 −0.211735
\(804\) 8.00000 0.282138
\(805\) 3.00000 0.105736
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 36.0000 1.26569 0.632846 0.774277i \(-0.281886\pi\)
0.632846 + 0.774277i \(0.281886\pi\)
\(810\) 0 0
\(811\) 38.0000 1.33436 0.667180 0.744896i \(-0.267501\pi\)
0.667180 + 0.744896i \(0.267501\pi\)
\(812\) 0 0
\(813\) −16.0000 −0.561144
\(814\) 0 0
\(815\) −11.0000 −0.385313
\(816\) −12.0000 −0.420084
\(817\) −20.0000 −0.699711
\(818\) 0 0
\(819\) −1.00000 −0.0349428
\(820\) −18.0000 −0.628587
\(821\) −27.0000 −0.942306 −0.471153 0.882051i \(-0.656162\pi\)
−0.471153 + 0.882051i \(0.656162\pi\)
\(822\) 0 0
\(823\) −40.0000 −1.39431 −0.697156 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) 0 0
\(825\) −3.00000 −0.104447
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) −6.00000 −0.208514
\(829\) 38.0000 1.31979 0.659897 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) 0 0
\(831\) 8.00000 0.277517
\(832\) −8.00000 −0.277350
\(833\) 18.0000 0.623663
\(834\) 0 0
\(835\) 12.0000 0.415277
\(836\) 12.0000 0.415029
\(837\) 1.00000 0.0345651
\(838\) 0 0
\(839\) 15.0000 0.517858 0.258929 0.965896i \(-0.416631\pi\)
0.258929 + 0.965896i \(0.416631\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) 30.0000 1.03325
\(844\) 8.00000 0.275371
\(845\) −1.00000 −0.0344010
\(846\) 0 0
\(847\) 2.00000 0.0687208
\(848\) −12.0000 −0.412082
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) −21.0000 −0.719871
\(852\) 18.0000 0.616670
\(853\) −19.0000 −0.650548 −0.325274 0.945620i \(-0.605456\pi\)
−0.325274 + 0.945620i \(0.605456\pi\)
\(854\) 0 0
\(855\) −2.00000 −0.0683986
\(856\) 0 0
\(857\) 9.00000 0.307434 0.153717 0.988115i \(-0.450876\pi\)
0.153717 + 0.988115i \(0.450876\pi\)
\(858\) 0 0
\(859\) 5.00000 0.170598 0.0852989 0.996355i \(-0.472815\pi\)
0.0852989 + 0.996355i \(0.472815\pi\)
\(860\) −20.0000 −0.681994
\(861\) 9.00000 0.306719
\(862\) 0 0
\(863\) −48.0000 −1.63394 −0.816970 0.576681i \(-0.804348\pi\)
−0.816970 + 0.576681i \(0.804348\pi\)
\(864\) 0 0
\(865\) 6.00000 0.204006
\(866\) 0 0
\(867\) −8.00000 −0.271694
\(868\) 2.00000 0.0678844
\(869\) 3.00000 0.101768
\(870\) 0 0
\(871\) −4.00000 −0.135535
\(872\) 0 0
\(873\) 17.0000 0.575363
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) −4.00000 −0.135147
\(877\) 14.0000 0.472746 0.236373 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) 0 0
\(879\) 12.0000 0.404750
\(880\) 12.0000 0.404520
\(881\) −24.0000 −0.808581 −0.404290 0.914631i \(-0.632481\pi\)
−0.404290 + 0.914631i \(0.632481\pi\)
\(882\) 0 0
\(883\) 2.00000 0.0673054 0.0336527 0.999434i \(-0.489286\pi\)
0.0336527 + 0.999434i \(0.489286\pi\)
\(884\) 6.00000 0.201802
\(885\) −12.0000 −0.403376
\(886\) 0 0
\(887\) 15.0000 0.503651 0.251825 0.967773i \(-0.418969\pi\)
0.251825 + 0.967773i \(0.418969\pi\)
\(888\) 0 0
\(889\) −20.0000 −0.670778
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) −16.0000 −0.535720
\(893\) 24.0000 0.803129
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 3.00000 0.100167
\(898\) 0 0
\(899\) 0 0
\(900\) −2.00000 −0.0666667
\(901\) 9.00000 0.299833
\(902\) 0 0
\(903\) 10.0000 0.332779
\(904\) 0 0
\(905\) −11.0000 −0.365652
\(906\) 0 0
\(907\) −46.0000 −1.52740 −0.763702 0.645568i \(-0.776621\pi\)
−0.763702 + 0.645568i \(0.776621\pi\)
\(908\) 12.0000 0.398234
\(909\) 18.0000 0.597022
\(910\) 0 0
\(911\) −30.0000 −0.993944 −0.496972 0.867766i \(-0.665555\pi\)
−0.496972 + 0.867766i \(0.665555\pi\)
\(912\) 8.00000 0.264906
\(913\) 54.0000 1.78714
\(914\) 0 0
\(915\) 1.00000 0.0330590
\(916\) −28.0000 −0.925146
\(917\) −18.0000 −0.594412
\(918\) 0 0
\(919\) −7.00000 −0.230909 −0.115454 0.993313i \(-0.536832\pi\)
−0.115454 + 0.993313i \(0.536832\pi\)
\(920\) 0 0
\(921\) 11.0000 0.362462
\(922\) 0 0
\(923\) −9.00000 −0.296239
\(924\) −6.00000 −0.197386
\(925\) −7.00000 −0.230159
\(926\) 0 0
\(927\) −4.00000 −0.131377
\(928\) 0 0
\(929\) 15.0000 0.492134 0.246067 0.969253i \(-0.420862\pi\)
0.246067 + 0.969253i \(0.420862\pi\)
\(930\) 0 0
\(931\) −12.0000 −0.393284
\(932\) −18.0000 −0.589610
\(933\) 24.0000 0.785725
\(934\) 0 0
\(935\) −9.00000 −0.294331
\(936\) 0 0
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 0 0
\(939\) −10.0000 −0.326338
\(940\) 24.0000 0.782794
\(941\) 39.0000 1.27136 0.635682 0.771951i \(-0.280719\pi\)
0.635682 + 0.771951i \(0.280719\pi\)
\(942\) 0 0
\(943\) −27.0000 −0.879241
\(944\) 48.0000 1.56227
\(945\) 1.00000 0.0325300
\(946\) 0 0
\(947\) −6.00000 −0.194974 −0.0974869 0.995237i \(-0.531080\pi\)
−0.0974869 + 0.995237i \(0.531080\pi\)
\(948\) 2.00000 0.0649570
\(949\) 2.00000 0.0649227
\(950\) 0 0
\(951\) 24.0000 0.778253
\(952\) 0 0
\(953\) −51.0000 −1.65205 −0.826026 0.563632i \(-0.809404\pi\)
−0.826026 + 0.563632i \(0.809404\pi\)
\(954\) 0 0
\(955\) 6.00000 0.194155
\(956\) 30.0000 0.970269
\(957\) 0 0
\(958\) 0 0
\(959\) −12.0000 −0.387500
\(960\) 8.00000 0.258199
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) 9.00000 0.290021
\(964\) 20.0000 0.644157
\(965\) −23.0000 −0.740396
\(966\) 0 0
\(967\) −40.0000 −1.28631 −0.643157 0.765735i \(-0.722376\pi\)
−0.643157 + 0.765735i \(0.722376\pi\)
\(968\) 0 0
\(969\) −6.00000 −0.192748
\(970\) 0 0
\(971\) 36.0000 1.15529 0.577647 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) −2.00000 −0.0641500
\(973\) −5.00000 −0.160293
\(974\) 0 0
\(975\) 1.00000 0.0320256
\(976\) −4.00000 −0.128037
\(977\) 48.0000 1.53566 0.767828 0.640656i \(-0.221338\pi\)
0.767828 + 0.640656i \(0.221338\pi\)
\(978\) 0 0
\(979\) −45.0000 −1.43821
\(980\) −12.0000 −0.383326
\(981\) 2.00000 0.0638551
\(982\) 0 0
\(983\) −18.0000 −0.574111 −0.287055 0.957914i \(-0.592676\pi\)
−0.287055 + 0.957914i \(0.592676\pi\)
\(984\) 0 0
\(985\) −12.0000 −0.382352
\(986\) 0 0
\(987\) −12.0000 −0.381964
\(988\) −4.00000 −0.127257
\(989\) −30.0000 −0.953945
\(990\) 0 0
\(991\) −7.00000 −0.222362 −0.111181 0.993800i \(-0.535463\pi\)
−0.111181 + 0.993800i \(0.535463\pi\)
\(992\) 0 0
\(993\) 8.00000 0.253872
\(994\) 0 0
\(995\) 16.0000 0.507234
\(996\) 36.0000 1.14070
\(997\) −10.0000 −0.316703 −0.158352 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) 0 0
\(999\) −7.00000 −0.221470
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 6045.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6045.2.a.f.1.1 1 1.1 even 1 trivial