Properties

Label 270.2.a.c.1.1
Level 270
Weight 2
Character 270.1
Self dual yes
Analytic conductor 2.156
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 270.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.15596085457\)
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
Root \(0\)
Character \(\chi\) = 270.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} +3.00000 q^{11} +5.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} -4.00000 q^{19} -1.00000 q^{20} +3.00000 q^{22} -9.00000 q^{23} +1.00000 q^{25} +5.00000 q^{26} +2.00000 q^{28} -3.00000 q^{29} +5.00000 q^{31} +1.00000 q^{32} -3.00000 q^{34} -2.00000 q^{35} -10.0000 q^{37} -4.00000 q^{38} -1.00000 q^{40} -1.00000 q^{43} +3.00000 q^{44} -9.00000 q^{46} +9.00000 q^{47} -3.00000 q^{49} +1.00000 q^{50} +5.00000 q^{52} -12.0000 q^{53} -3.00000 q^{55} +2.00000 q^{56} -3.00000 q^{58} +12.0000 q^{59} +2.00000 q^{61} +5.00000 q^{62} +1.00000 q^{64} -5.00000 q^{65} -4.00000 q^{67} -3.00000 q^{68} -2.00000 q^{70} +12.0000 q^{71} -10.0000 q^{73} -10.0000 q^{74} -4.00000 q^{76} +6.00000 q^{77} -13.0000 q^{79} -1.00000 q^{80} +6.00000 q^{83} +3.00000 q^{85} -1.00000 q^{86} +3.00000 q^{88} -12.0000 q^{89} +10.0000 q^{91} -9.00000 q^{92} +9.00000 q^{94} +4.00000 q^{95} +2.00000 q^{97} -3.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 0 0
\(13\) 5.00000 1.38675 0.693375 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 3.00000 0.639602
\(23\) −9.00000 −1.87663 −0.938315 0.345782i \(-0.887614\pi\)
−0.938315 + 0.345782i \(0.887614\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 5.00000 0.980581
\(27\) 0 0
\(28\) 2.00000 0.377964
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) 5.00000 0.898027 0.449013 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −3.00000 −0.514496
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) −4.00000 −0.648886
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 3.00000 0.452267
\(45\) 0 0
\(46\) −9.00000 −1.32698
\(47\) 9.00000 1.31278 0.656392 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 5.00000 0.693375
\(53\) −12.0000 −1.64833 −0.824163 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) 0 0
\(55\) −3.00000 −0.404520
\(56\) 2.00000 0.267261
\(57\) 0 0
\(58\) −3.00000 −0.393919
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 5.00000 0.635001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −5.00000 −0.620174
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −3.00000 −0.363803
\(69\) 0 0
\(70\) −2.00000 −0.239046
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) −10.0000 −1.16248
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 6.00000 0.683763
\(78\) 0 0
\(79\) −13.0000 −1.46261 −0.731307 0.682048i \(-0.761089\pi\)
−0.731307 + 0.682048i \(0.761089\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 0 0
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) 3.00000 0.325396
\(86\) −1.00000 −0.107833
\(87\) 0 0
\(88\) 3.00000 0.319801
\(89\) −12.0000 −1.27200 −0.635999 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) 0 0
\(91\) 10.0000 1.04828
\(92\) −9.00000 −0.938315
\(93\) 0 0
\(94\) 9.00000 0.928279
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −3.00000 −0.303046
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 15.0000 1.49256 0.746278 0.665635i \(-0.231839\pi\)
0.746278 + 0.665635i \(0.231839\pi\)
\(102\) 0 0
\(103\) 2.00000 0.197066 0.0985329 0.995134i \(-0.468585\pi\)
0.0985329 + 0.995134i \(0.468585\pi\)
\(104\) 5.00000 0.490290
\(105\) 0 0
\(106\) −12.0000 −1.16554
\(107\) 6.00000 0.580042 0.290021 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) 0 0
\(109\) 20.0000 1.91565 0.957826 0.287348i \(-0.0927736\pi\)
0.957826 + 0.287348i \(0.0927736\pi\)
\(110\) −3.00000 −0.286039
\(111\) 0 0
\(112\) 2.00000 0.188982
\(113\) −15.0000 −1.41108 −0.705541 0.708669i \(-0.749296\pi\)
−0.705541 + 0.708669i \(0.749296\pi\)
\(114\) 0 0
\(115\) 9.00000 0.839254
\(116\) −3.00000 −0.278543
\(117\) 0 0
\(118\) 12.0000 1.10469
\(119\) −6.00000 −0.550019
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 2.00000 0.181071
\(123\) 0 0
\(124\) 5.00000 0.449013
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −5.00000 −0.438529
\(131\) −3.00000 −0.262111 −0.131056 0.991375i \(-0.541837\pi\)
−0.131056 + 0.991375i \(0.541837\pi\)
\(132\) 0 0
\(133\) −8.00000 −0.693688
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) −3.00000 −0.257248
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) 2.00000 0.169638 0.0848189 0.996396i \(-0.472969\pi\)
0.0848189 + 0.996396i \(0.472969\pi\)
\(140\) −2.00000 −0.169031
\(141\) 0 0
\(142\) 12.0000 1.00702
\(143\) 15.0000 1.25436
\(144\) 0 0
\(145\) 3.00000 0.249136
\(146\) −10.0000 −0.827606
\(147\) 0 0
\(148\) −10.0000 −0.821995
\(149\) 21.0000 1.72039 0.860194 0.509968i \(-0.170343\pi\)
0.860194 + 0.509968i \(0.170343\pi\)
\(150\) 0 0
\(151\) 11.0000 0.895167 0.447584 0.894242i \(-0.352285\pi\)
0.447584 + 0.894242i \(0.352285\pi\)
\(152\) −4.00000 −0.324443
\(153\) 0 0
\(154\) 6.00000 0.483494
\(155\) −5.00000 −0.401610
\(156\) 0 0
\(157\) −13.0000 −1.03751 −0.518756 0.854922i \(-0.673605\pi\)
−0.518756 + 0.854922i \(0.673605\pi\)
\(158\) −13.0000 −1.03422
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) −18.0000 −1.41860
\(162\) 0 0
\(163\) 11.0000 0.861586 0.430793 0.902451i \(-0.358234\pi\)
0.430793 + 0.902451i \(0.358234\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 6.00000 0.465690
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 3.00000 0.230089
\(171\) 0 0
\(172\) −1.00000 −0.0762493
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 0 0
\(175\) 2.00000 0.151186
\(176\) 3.00000 0.226134
\(177\) 0 0
\(178\) −12.0000 −0.899438
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 20.0000 1.48659 0.743294 0.668965i \(-0.233262\pi\)
0.743294 + 0.668965i \(0.233262\pi\)
\(182\) 10.0000 0.741249
\(183\) 0 0
\(184\) −9.00000 −0.663489
\(185\) 10.0000 0.735215
\(186\) 0 0
\(187\) −9.00000 −0.658145
\(188\) 9.00000 0.656392
\(189\) 0 0
\(190\) 4.00000 0.290191
\(191\) 6.00000 0.434145 0.217072 0.976156i \(-0.430349\pi\)
0.217072 + 0.976156i \(0.430349\pi\)
\(192\) 0 0
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) −7.00000 −0.496217 −0.248108 0.968732i \(-0.579809\pi\)
−0.248108 + 0.968732i \(0.579809\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 15.0000 1.05540
\(203\) −6.00000 −0.421117
\(204\) 0 0
\(205\) 0 0
\(206\) 2.00000 0.139347
\(207\) 0 0
\(208\) 5.00000 0.346688
\(209\) −12.0000 −0.830057
\(210\) 0 0
\(211\) −22.0000 −1.51454 −0.757271 0.653101i \(-0.773468\pi\)
−0.757271 + 0.653101i \(0.773468\pi\)
\(212\) −12.0000 −0.824163
\(213\) 0 0
\(214\) 6.00000 0.410152
\(215\) 1.00000 0.0681994
\(216\) 0 0
\(217\) 10.0000 0.678844
\(218\) 20.0000 1.35457
\(219\) 0 0
\(220\) −3.00000 −0.202260
\(221\) −15.0000 −1.00901
\(222\) 0 0
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 2.00000 0.133631
\(225\) 0 0
\(226\) −15.0000 −0.997785
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −4.00000 −0.264327 −0.132164 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) 9.00000 0.593442
\(231\) 0 0
\(232\) −3.00000 −0.196960
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) −9.00000 −0.587095
\(236\) 12.0000 0.781133
\(237\) 0 0
\(238\) −6.00000 −0.388922
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) −19.0000 −1.22390 −0.611949 0.790897i \(-0.709614\pi\)
−0.611949 + 0.790897i \(0.709614\pi\)
\(242\) −2.00000 −0.128565
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) 3.00000 0.191663
\(246\) 0 0
\(247\) −20.0000 −1.27257
\(248\) 5.00000 0.317500
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) −9.00000 −0.568075 −0.284037 0.958813i \(-0.591674\pi\)
−0.284037 + 0.958813i \(0.591674\pi\)
\(252\) 0 0
\(253\) −27.0000 −1.69748
\(254\) 2.00000 0.125491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −15.0000 −0.935674 −0.467837 0.883815i \(-0.654967\pi\)
−0.467837 + 0.883815i \(0.654967\pi\)
\(258\) 0 0
\(259\) −20.0000 −1.24274
\(260\) −5.00000 −0.310087
\(261\) 0 0
\(262\) −3.00000 −0.185341
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 0 0
\(265\) 12.0000 0.737154
\(266\) −8.00000 −0.490511
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) −9.00000 −0.548740 −0.274370 0.961624i \(-0.588469\pi\)
−0.274370 + 0.961624i \(0.588469\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) −3.00000 −0.181902
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) 3.00000 0.180907
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 2.00000 0.119952
\(279\) 0 0
\(280\) −2.00000 −0.119523
\(281\) −30.0000 −1.78965 −0.894825 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 0 0
\(283\) 32.0000 1.90220 0.951101 0.308879i \(-0.0999539\pi\)
0.951101 + 0.308879i \(0.0999539\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) 15.0000 0.886969
\(287\) 0 0
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 3.00000 0.176166
\(291\) 0 0
\(292\) −10.0000 −0.585206
\(293\) 18.0000 1.05157 0.525786 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) 0 0
\(295\) −12.0000 −0.698667
\(296\) −10.0000 −0.581238
\(297\) 0 0
\(298\) 21.0000 1.21650
\(299\) −45.0000 −2.60242
\(300\) 0 0
\(301\) −2.00000 −0.115278
\(302\) 11.0000 0.632979
\(303\) 0 0
\(304\) −4.00000 −0.229416
\(305\) −2.00000 −0.114520
\(306\) 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
\(309\) 0 0
\(310\) −5.00000 −0.283981
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) 8.00000 0.452187 0.226093 0.974106i \(-0.427405\pi\)
0.226093 + 0.974106i \(0.427405\pi\)
\(314\) −13.0000 −0.733632
\(315\) 0 0
\(316\) −13.0000 −0.731307
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) −9.00000 −0.503903
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) −18.0000 −1.00310
\(323\) 12.0000 0.667698
\(324\) 0 0
\(325\) 5.00000 0.277350
\(326\) 11.0000 0.609234
\(327\) 0 0
\(328\) 0 0
\(329\) 18.0000 0.992372
\(330\) 0 0
\(331\) 14.0000 0.769510 0.384755 0.923019i \(-0.374286\pi\)
0.384755 + 0.923019i \(0.374286\pi\)
\(332\) 6.00000 0.329293
\(333\) 0 0
\(334\) −12.0000 −0.656611
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) 32.0000 1.74315 0.871576 0.490261i \(-0.163099\pi\)
0.871576 + 0.490261i \(0.163099\pi\)
\(338\) 12.0000 0.652714
\(339\) 0 0
\(340\) 3.00000 0.162698
\(341\) 15.0000 0.812296
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) −1.00000 −0.0539164
\(345\) 0 0
\(346\) 18.0000 0.967686
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 2.00000 0.106904
\(351\) 0 0
\(352\) 3.00000 0.159901
\(353\) 3.00000 0.159674 0.0798369 0.996808i \(-0.474560\pi\)
0.0798369 + 0.996808i \(0.474560\pi\)
\(354\) 0 0
\(355\) −12.0000 −0.636894
\(356\) −12.0000 −0.635999
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) 12.0000 0.633336 0.316668 0.948536i \(-0.397436\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 20.0000 1.05118
\(363\) 0 0
\(364\) 10.0000 0.524142
\(365\) 10.0000 0.523424
\(366\) 0 0
\(367\) 26.0000 1.35719 0.678594 0.734513i \(-0.262589\pi\)
0.678594 + 0.734513i \(0.262589\pi\)
\(368\) −9.00000 −0.469157
\(369\) 0 0
\(370\) 10.0000 0.519875
\(371\) −24.0000 −1.24602
\(372\) 0 0
\(373\) −1.00000 −0.0517780 −0.0258890 0.999665i \(-0.508242\pi\)
−0.0258890 + 0.999665i \(0.508242\pi\)
\(374\) −9.00000 −0.465379
\(375\) 0 0
\(376\) 9.00000 0.464140
\(377\) −15.0000 −0.772539
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 4.00000 0.205196
\(381\) 0 0
\(382\) 6.00000 0.306987
\(383\) −9.00000 −0.459879 −0.229939 0.973205i \(-0.573853\pi\)
−0.229939 + 0.973205i \(0.573853\pi\)
\(384\) 0 0
\(385\) −6.00000 −0.305788
\(386\) −10.0000 −0.508987
\(387\) 0 0
\(388\) 2.00000 0.101535
\(389\) 9.00000 0.456318 0.228159 0.973624i \(-0.426729\pi\)
0.228159 + 0.973624i \(0.426729\pi\)
\(390\) 0 0
\(391\) 27.0000 1.36545
\(392\) −3.00000 −0.151523
\(393\) 0 0
\(394\) 0 0
\(395\) 13.0000 0.654101
\(396\) 0 0
\(397\) −7.00000 −0.351320 −0.175660 0.984451i \(-0.556206\pi\)
−0.175660 + 0.984451i \(0.556206\pi\)
\(398\) −7.00000 −0.350878
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 36.0000 1.79775 0.898877 0.438201i \(-0.144384\pi\)
0.898877 + 0.438201i \(0.144384\pi\)
\(402\) 0 0
\(403\) 25.0000 1.24534
\(404\) 15.0000 0.746278
\(405\) 0 0
\(406\) −6.00000 −0.297775
\(407\) −30.0000 −1.48704
\(408\) 0 0
\(409\) 23.0000 1.13728 0.568638 0.822588i \(-0.307470\pi\)
0.568638 + 0.822588i \(0.307470\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 2.00000 0.0985329
\(413\) 24.0000 1.18096
\(414\) 0 0
\(415\) −6.00000 −0.294528
\(416\) 5.00000 0.245145
\(417\) 0 0
\(418\) −12.0000 −0.586939
\(419\) −3.00000 −0.146560 −0.0732798 0.997311i \(-0.523347\pi\)
−0.0732798 + 0.997311i \(0.523347\pi\)
\(420\) 0 0
\(421\) −4.00000 −0.194948 −0.0974740 0.995238i \(-0.531076\pi\)
−0.0974740 + 0.995238i \(0.531076\pi\)
\(422\) −22.0000 −1.07094
\(423\) 0 0
\(424\) −12.0000 −0.582772
\(425\) −3.00000 −0.145521
\(426\) 0 0
\(427\) 4.00000 0.193574
\(428\) 6.00000 0.290021
\(429\) 0 0
\(430\) 1.00000 0.0482243
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 10.0000 0.480015
\(435\) 0 0
\(436\) 20.0000 0.957826
\(437\) 36.0000 1.72211
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) −3.00000 −0.143019
\(441\) 0 0
\(442\) −15.0000 −0.713477
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 0 0
\(445\) 12.0000 0.568855
\(446\) 8.00000 0.378811
\(447\) 0 0
\(448\) 2.00000 0.0944911
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −15.0000 −0.705541
\(453\) 0 0
\(454\) 0 0
\(455\) −10.0000 −0.468807
\(456\) 0 0
\(457\) −28.0000 −1.30978 −0.654892 0.755722i \(-0.727286\pi\)
−0.654892 + 0.755722i \(0.727286\pi\)
\(458\) −4.00000 −0.186908
\(459\) 0 0
\(460\) 9.00000 0.419627
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) 38.0000 1.76601 0.883005 0.469364i \(-0.155517\pi\)
0.883005 + 0.469364i \(0.155517\pi\)
\(464\) −3.00000 −0.139272
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) −9.00000 −0.415139
\(471\) 0 0
\(472\) 12.0000 0.552345
\(473\) −3.00000 −0.137940
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) −6.00000 −0.275010
\(477\) 0 0
\(478\) −6.00000 −0.274434
\(479\) 36.0000 1.64488 0.822441 0.568850i \(-0.192612\pi\)
0.822441 + 0.568850i \(0.192612\pi\)
\(480\) 0 0
\(481\) −50.0000 −2.27980
\(482\) −19.0000 −0.865426
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) −2.00000 −0.0908153
\(486\) 0 0
\(487\) 20.0000 0.906287 0.453143 0.891438i \(-0.350303\pi\)
0.453143 + 0.891438i \(0.350303\pi\)
\(488\) 2.00000 0.0905357
\(489\) 0 0
\(490\) 3.00000 0.135526
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) 9.00000 0.405340
\(494\) −20.0000 −0.899843
\(495\) 0 0
\(496\) 5.00000 0.224507
\(497\) 24.0000 1.07655
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −9.00000 −0.401690
\(503\) −9.00000 −0.401290 −0.200645 0.979664i \(-0.564304\pi\)
−0.200645 + 0.979664i \(0.564304\pi\)
\(504\) 0 0
\(505\) −15.0000 −0.667491
\(506\) −27.0000 −1.20030
\(507\) 0 0
\(508\) 2.00000 0.0887357
\(509\) 9.00000 0.398918 0.199459 0.979906i \(-0.436082\pi\)
0.199459 + 0.979906i \(0.436082\pi\)
\(510\) 0 0
\(511\) −20.0000 −0.884748
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −15.0000 −0.661622
\(515\) −2.00000 −0.0881305
\(516\) 0 0
\(517\) 27.0000 1.18746
\(518\) −20.0000 −0.878750
\(519\) 0 0
\(520\) −5.00000 −0.219265
\(521\) −12.0000 −0.525730 −0.262865 0.964833i \(-0.584667\pi\)
−0.262865 + 0.964833i \(0.584667\pi\)
\(522\) 0 0
\(523\) −31.0000 −1.35554 −0.677768 0.735276i \(-0.737052\pi\)
−0.677768 + 0.735276i \(0.737052\pi\)
\(524\) −3.00000 −0.131056
\(525\) 0 0
\(526\) 12.0000 0.523225
\(527\) −15.0000 −0.653410
\(528\) 0 0
\(529\) 58.0000 2.52174
\(530\) 12.0000 0.521247
\(531\) 0 0
\(532\) −8.00000 −0.346844
\(533\) 0 0
\(534\) 0 0
\(535\) −6.00000 −0.259403
\(536\) −4.00000 −0.172774
\(537\) 0 0
\(538\) −9.00000 −0.388018
\(539\) −9.00000 −0.387657
\(540\) 0 0
\(541\) −22.0000 −0.945854 −0.472927 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) 8.00000 0.343629
\(543\) 0 0
\(544\) −3.00000 −0.128624
\(545\) −20.0000 −0.856706
\(546\) 0 0
\(547\) −13.0000 −0.555840 −0.277920 0.960604i \(-0.589645\pi\)
−0.277920 + 0.960604i \(0.589645\pi\)
\(548\) −6.00000 −0.256307
\(549\) 0 0
\(550\) 3.00000 0.127920
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) −26.0000 −1.10563
\(554\) −10.0000 −0.424859
\(555\) 0 0
\(556\) 2.00000 0.0848189
\(557\) 12.0000 0.508456 0.254228 0.967144i \(-0.418179\pi\)
0.254228 + 0.967144i \(0.418179\pi\)
\(558\) 0 0
\(559\) −5.00000 −0.211477
\(560\) −2.00000 −0.0845154
\(561\) 0 0
\(562\) −30.0000 −1.26547
\(563\) 6.00000 0.252870 0.126435 0.991975i \(-0.459647\pi\)
0.126435 + 0.991975i \(0.459647\pi\)
\(564\) 0 0
\(565\) 15.0000 0.631055
\(566\) 32.0000 1.34506
\(567\) 0 0
\(568\) 12.0000 0.503509
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) −10.0000 −0.418487 −0.209243 0.977864i \(-0.567100\pi\)
−0.209243 + 0.977864i \(0.567100\pi\)
\(572\) 15.0000 0.627182
\(573\) 0 0
\(574\) 0 0
\(575\) −9.00000 −0.375326
\(576\) 0 0
\(577\) 32.0000 1.33218 0.666089 0.745873i \(-0.267967\pi\)
0.666089 + 0.745873i \(0.267967\pi\)
\(578\) −8.00000 −0.332756
\(579\) 0 0
\(580\) 3.00000 0.124568
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) −36.0000 −1.49097
\(584\) −10.0000 −0.413803
\(585\) 0 0
\(586\) 18.0000 0.743573
\(587\) −18.0000 −0.742940 −0.371470 0.928445i \(-0.621146\pi\)
−0.371470 + 0.928445i \(0.621146\pi\)
\(588\) 0 0
\(589\) −20.0000 −0.824086
\(590\) −12.0000 −0.494032
\(591\) 0 0
\(592\) −10.0000 −0.410997
\(593\) −27.0000 −1.10876 −0.554379 0.832265i \(-0.687044\pi\)
−0.554379 + 0.832265i \(0.687044\pi\)
\(594\) 0 0
\(595\) 6.00000 0.245976
\(596\) 21.0000 0.860194
\(597\) 0 0
\(598\) −45.0000 −1.84019
\(599\) −30.0000 −1.22577 −0.612883 0.790173i \(-0.709990\pi\)
−0.612883 + 0.790173i \(0.709990\pi\)
\(600\) 0 0
\(601\) −19.0000 −0.775026 −0.387513 0.921864i \(-0.626666\pi\)
−0.387513 + 0.921864i \(0.626666\pi\)
\(602\) −2.00000 −0.0815139
\(603\) 0 0
\(604\) 11.0000 0.447584
\(605\) 2.00000 0.0813116
\(606\) 0 0
\(607\) −34.0000 −1.38002 −0.690009 0.723801i \(-0.742393\pi\)
−0.690009 + 0.723801i \(0.742393\pi\)
\(608\) −4.00000 −0.162221
\(609\) 0 0
\(610\) −2.00000 −0.0809776
\(611\) 45.0000 1.82051
\(612\) 0 0
\(613\) −19.0000 −0.767403 −0.383701 0.923457i \(-0.625351\pi\)
−0.383701 + 0.923457i \(0.625351\pi\)
\(614\) 11.0000 0.443924
\(615\) 0 0
\(616\) 6.00000 0.241747
\(617\) 3.00000 0.120775 0.0603877 0.998175i \(-0.480766\pi\)
0.0603877 + 0.998175i \(0.480766\pi\)
\(618\) 0 0
\(619\) −34.0000 −1.36658 −0.683288 0.730149i \(-0.739451\pi\)
−0.683288 + 0.730149i \(0.739451\pi\)
\(620\) −5.00000 −0.200805
\(621\) 0 0
\(622\) −24.0000 −0.962312
\(623\) −24.0000 −0.961540
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 8.00000 0.319744
\(627\) 0 0
\(628\) −13.0000 −0.518756
\(629\) 30.0000 1.19618
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) −13.0000 −0.517112
\(633\) 0 0
\(634\) 0 0
\(635\) −2.00000 −0.0793676
\(636\) 0 0
\(637\) −15.0000 −0.594322
\(638\) −9.00000 −0.356313
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 0 0
\(643\) 23.0000 0.907031 0.453516 0.891248i \(-0.350170\pi\)
0.453516 + 0.891248i \(0.350170\pi\)
\(644\) −18.0000 −0.709299
\(645\) 0 0
\(646\) 12.0000 0.472134
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) 0 0
\(649\) 36.0000 1.41312
\(650\) 5.00000 0.196116
\(651\) 0 0
\(652\) 11.0000 0.430793
\(653\) 36.0000 1.40879 0.704394 0.709809i \(-0.251219\pi\)
0.704394 + 0.709809i \(0.251219\pi\)
\(654\) 0 0
\(655\) 3.00000 0.117220
\(656\) 0 0
\(657\) 0 0
\(658\) 18.0000 0.701713
\(659\) 48.0000 1.86981 0.934907 0.354892i \(-0.115482\pi\)
0.934907 + 0.354892i \(0.115482\pi\)
\(660\) 0 0
\(661\) 2.00000 0.0777910 0.0388955 0.999243i \(-0.487616\pi\)
0.0388955 + 0.999243i \(0.487616\pi\)
\(662\) 14.0000 0.544125
\(663\) 0 0
\(664\) 6.00000 0.232845
\(665\) 8.00000 0.310227
\(666\) 0 0
\(667\) 27.0000 1.04544
\(668\) −12.0000 −0.464294
\(669\) 0 0
\(670\) 4.00000 0.154533
\(671\) 6.00000 0.231627
\(672\) 0 0
\(673\) −22.0000 −0.848038 −0.424019 0.905653i \(-0.639381\pi\)
−0.424019 + 0.905653i \(0.639381\pi\)
\(674\) 32.0000 1.23259
\(675\) 0 0
\(676\) 12.0000 0.461538
\(677\) 12.0000 0.461197 0.230599 0.973049i \(-0.425932\pi\)
0.230599 + 0.973049i \(0.425932\pi\)
\(678\) 0 0
\(679\) 4.00000 0.153506
\(680\) 3.00000 0.115045
\(681\) 0 0
\(682\) 15.0000 0.574380
\(683\) −48.0000 −1.83667 −0.918334 0.395805i \(-0.870466\pi\)
−0.918334 + 0.395805i \(0.870466\pi\)
\(684\) 0 0
\(685\) 6.00000 0.229248
\(686\) −20.0000 −0.763604
\(687\) 0 0
\(688\) −1.00000 −0.0381246
\(689\) −60.0000 −2.28582
\(690\) 0 0
\(691\) −10.0000 −0.380418 −0.190209 0.981744i \(-0.560917\pi\)
−0.190209 + 0.981744i \(0.560917\pi\)
\(692\) 18.0000 0.684257
\(693\) 0 0
\(694\) 0 0
\(695\) −2.00000 −0.0758643
\(696\) 0 0
\(697\) 0 0
\(698\) 14.0000 0.529908
\(699\) 0 0
\(700\) 2.00000 0.0755929
\(701\) 21.0000 0.793159 0.396580 0.918000i \(-0.370197\pi\)
0.396580 + 0.918000i \(0.370197\pi\)
\(702\) 0 0
\(703\) 40.0000 1.50863
\(704\) 3.00000 0.113067
\(705\) 0 0
\(706\) 3.00000 0.112906
\(707\) 30.0000 1.12827
\(708\) 0 0
\(709\) 20.0000 0.751116 0.375558 0.926799i \(-0.377451\pi\)
0.375558 + 0.926799i \(0.377451\pi\)
\(710\) −12.0000 −0.450352
\(711\) 0 0
\(712\) −12.0000 −0.449719
\(713\) −45.0000 −1.68526
\(714\) 0 0
\(715\) −15.0000 −0.560968
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) 12.0000 0.447836
\(719\) −30.0000 −1.11881 −0.559406 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) −3.00000 −0.111648
\(723\) 0 0
\(724\) 20.0000 0.743294
\(725\) −3.00000 −0.111417
\(726\) 0 0
\(727\) 44.0000 1.63187 0.815935 0.578144i \(-0.196223\pi\)
0.815935 + 0.578144i \(0.196223\pi\)
\(728\) 10.0000 0.370625
\(729\) 0 0
\(730\) 10.0000 0.370117
\(731\) 3.00000 0.110959
\(732\) 0 0
\(733\) −22.0000 −0.812589 −0.406294 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) 26.0000 0.959678
\(735\) 0 0
\(736\) −9.00000 −0.331744
\(737\) −12.0000 −0.442026
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 10.0000 0.367607
\(741\) 0 0
\(742\) −24.0000 −0.881068
\(743\) −27.0000 −0.990534 −0.495267 0.868741i \(-0.664930\pi\)
−0.495267 + 0.868741i \(0.664930\pi\)
\(744\) 0 0
\(745\) −21.0000 −0.769380
\(746\) −1.00000 −0.0366126
\(747\) 0 0
\(748\) −9.00000 −0.329073
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) −25.0000 −0.912263 −0.456131 0.889912i \(-0.650765\pi\)
−0.456131 + 0.889912i \(0.650765\pi\)
\(752\) 9.00000 0.328196
\(753\) 0 0
\(754\) −15.0000 −0.546268
\(755\) −11.0000 −0.400331
\(756\) 0 0
\(757\) −13.0000 −0.472493 −0.236247 0.971693i \(-0.575917\pi\)
−0.236247 + 0.971693i \(0.575917\pi\)
\(758\) −16.0000 −0.581146
\(759\) 0 0
\(760\) 4.00000 0.145095
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 0 0
\(763\) 40.0000 1.44810
\(764\) 6.00000 0.217072
\(765\) 0 0
\(766\) −9.00000 −0.325183
\(767\) 60.0000 2.16647
\(768\) 0 0
\(769\) 23.0000 0.829401 0.414701 0.909958i \(-0.363886\pi\)
0.414701 + 0.909958i \(0.363886\pi\)
\(770\) −6.00000 −0.216225
\(771\) 0 0
\(772\) −10.0000 −0.359908
\(773\) −42.0000 −1.51064 −0.755318 0.655359i \(-0.772517\pi\)
−0.755318 + 0.655359i \(0.772517\pi\)
\(774\) 0 0
\(775\) 5.00000 0.179605
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) 9.00000 0.322666
\(779\) 0 0
\(780\) 0 0
\(781\) 36.0000 1.28818
\(782\) 27.0000 0.965518
\(783\) 0 0
\(784\) −3.00000 −0.107143
\(785\) 13.0000 0.463990
\(786\) 0 0
\(787\) −1.00000 −0.0356462 −0.0178231 0.999841i \(-0.505674\pi\)
−0.0178231 + 0.999841i \(0.505674\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 13.0000 0.462519
\(791\) −30.0000 −1.06668
\(792\) 0 0
\(793\) 10.0000 0.355110
\(794\) −7.00000 −0.248421
\(795\) 0 0
\(796\) −7.00000 −0.248108
\(797\) −36.0000 −1.27519 −0.637593 0.770374i \(-0.720070\pi\)
−0.637593 + 0.770374i \(0.720070\pi\)
\(798\) 0 0
\(799\) −27.0000 −0.955191
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 36.0000 1.27120
\(803\) −30.0000 −1.05868
\(804\) 0 0
\(805\) 18.0000 0.634417
\(806\) 25.0000 0.880587
\(807\) 0 0
\(808\) 15.0000 0.527698
\(809\) 24.0000 0.843795 0.421898 0.906644i \(-0.361364\pi\)
0.421898 + 0.906644i \(0.361364\pi\)
\(810\) 0 0
\(811\) 38.0000 1.33436 0.667180 0.744896i \(-0.267501\pi\)
0.667180 + 0.744896i \(0.267501\pi\)
\(812\) −6.00000 −0.210559
\(813\) 0 0
\(814\) −30.0000 −1.05150
\(815\) −11.0000 −0.385313
\(816\) 0 0
\(817\) 4.00000 0.139942
\(818\) 23.0000 0.804176
\(819\) 0 0
\(820\) 0 0
\(821\) −42.0000 −1.46581 −0.732905 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) 0 0
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) 2.00000 0.0696733
\(825\) 0 0
\(826\) 24.0000 0.835067
\(827\) −42.0000 −1.46048 −0.730242 0.683189i \(-0.760592\pi\)
−0.730242 + 0.683189i \(0.760592\pi\)
\(828\) 0 0
\(829\) 44.0000 1.52818 0.764092 0.645108i \(-0.223188\pi\)
0.764092 + 0.645108i \(0.223188\pi\)
\(830\) −6.00000 −0.208263
\(831\) 0 0
\(832\) 5.00000 0.173344
\(833\) 9.00000 0.311832
\(834\) 0 0
\(835\) 12.0000 0.415277
\(836\) −12.0000 −0.415029
\(837\) 0 0
\(838\) −3.00000 −0.103633
\(839\) 6.00000 0.207143 0.103572 0.994622i \(-0.466973\pi\)
0.103572 + 0.994622i \(0.466973\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) −4.00000 −0.137849
\(843\) 0 0
\(844\) −22.0000 −0.757271
\(845\) −12.0000 −0.412813
\(846\) 0 0
\(847\) −4.00000 −0.137442
\(848\) −12.0000 −0.412082
\(849\) 0 0
\(850\) −3.00000 −0.102899
\(851\) 90.0000 3.08516
\(852\) 0 0
\(853\) −19.0000 −0.650548 −0.325274 0.945620i \(-0.605456\pi\)
−0.325274 + 0.945620i \(0.605456\pi\)
\(854\) 4.00000 0.136877
\(855\) 0 0
\(856\) 6.00000 0.205076
\(857\) −6.00000 −0.204956 −0.102478 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) 0 0
\(859\) 2.00000 0.0682391 0.0341196 0.999418i \(-0.489137\pi\)
0.0341196 + 0.999418i \(0.489137\pi\)
\(860\) 1.00000 0.0340997
\(861\) 0 0
\(862\) 0 0
\(863\) 21.0000 0.714848 0.357424 0.933942i \(-0.383655\pi\)
0.357424 + 0.933942i \(0.383655\pi\)
\(864\) 0 0
\(865\) −18.0000 −0.612018
\(866\) 2.00000 0.0679628
\(867\) 0 0
\(868\) 10.0000 0.339422
\(869\) −39.0000 −1.32298
\(870\) 0 0
\(871\) −20.0000 −0.677674
\(872\) 20.0000 0.677285
\(873\) 0 0
\(874\) 36.0000 1.21772
\(875\) −2.00000 −0.0676123
\(876\) 0 0
\(877\) −1.00000 −0.0337676 −0.0168838 0.999857i \(-0.505375\pi\)
−0.0168838 + 0.999857i \(0.505375\pi\)
\(878\) 8.00000 0.269987
\(879\) 0 0
\(880\) −3.00000 −0.101130
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) −4.00000 −0.134611 −0.0673054 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) −15.0000 −0.504505
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) −3.00000 −0.100730 −0.0503651 0.998731i \(-0.516038\pi\)
−0.0503651 + 0.998731i \(0.516038\pi\)
\(888\) 0 0
\(889\) 4.00000 0.134156
\(890\) 12.0000 0.402241
\(891\) 0 0
\(892\) 8.00000 0.267860
\(893\) −36.0000 −1.20469
\(894\) 0 0
\(895\) −12.0000 −0.401116
\(896\) 2.00000 0.0668153
\(897\) 0 0
\(898\) −6.00000 −0.200223
\(899\) −15.0000 −0.500278
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) 0 0
\(903\) 0 0
\(904\) −15.0000 −0.498893
\(905\) −20.0000 −0.664822
\(906\) 0 0
\(907\) 59.0000 1.95906 0.979531 0.201291i \(-0.0645138\pi\)
0.979531 + 0.201291i \(0.0645138\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) −10.0000 −0.331497
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 18.0000 0.595713
\(914\) −28.0000 −0.926158
\(915\) 0 0
\(916\) −4.00000 −0.132164
\(917\) −6.00000 −0.198137
\(918\) 0 0
\(919\) −25.0000 −0.824674 −0.412337 0.911031i \(-0.635287\pi\)
−0.412337 + 0.911031i \(0.635287\pi\)
\(920\) 9.00000 0.296721
\(921\) 0 0
\(922\) 18.0000 0.592798
\(923\) 60.0000 1.97492
\(924\) 0 0
\(925\) −10.0000 −0.328798
\(926\) 38.0000 1.24876
\(927\) 0 0
\(928\) −3.00000 −0.0984798
\(929\) 36.0000 1.18112 0.590561 0.806993i \(-0.298907\pi\)
0.590561 + 0.806993i \(0.298907\pi\)
\(930\) 0 0
\(931\) 12.0000 0.393284
\(932\) 6.00000 0.196537
\(933\) 0 0
\(934\) 12.0000 0.392652
\(935\) 9.00000 0.294331
\(936\) 0 0
\(937\) −22.0000 −0.718709 −0.359354 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) −8.00000 −0.261209
\(939\) 0 0
\(940\) −9.00000 −0.293548
\(941\) 33.0000 1.07577 0.537885 0.843018i \(-0.319224\pi\)
0.537885 + 0.843018i \(0.319224\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) −3.00000 −0.0975384
\(947\) −42.0000 −1.36482 −0.682408 0.730971i \(-0.739067\pi\)
−0.682408 + 0.730971i \(0.739067\pi\)
\(948\) 0 0
\(949\) −50.0000 −1.62307
\(950\) −4.00000 −0.129777
\(951\) 0 0
\(952\) −6.00000 −0.194461
\(953\) 15.0000 0.485898 0.242949 0.970039i \(-0.421885\pi\)
0.242949 + 0.970039i \(0.421885\pi\)
\(954\) 0 0
\(955\) −6.00000 −0.194155
\(956\) −6.00000 −0.194054
\(957\) 0 0
\(958\) 36.0000 1.16311
\(959\) −12.0000 −0.387500
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) −50.0000 −1.61206
\(963\) 0 0
\(964\) −19.0000 −0.611949
\(965\) 10.0000 0.321911
\(966\) 0 0
\(967\) 2.00000 0.0643157 0.0321578 0.999483i \(-0.489762\pi\)
0.0321578 + 0.999483i \(0.489762\pi\)
\(968\) −2.00000 −0.0642824
\(969\) 0 0
\(970\) −2.00000 −0.0642161
\(971\) −57.0000 −1.82922 −0.914609 0.404341i \(-0.867501\pi\)
−0.914609 + 0.404341i \(0.867501\pi\)
\(972\) 0 0
\(973\) 4.00000 0.128234
\(974\) 20.0000 0.640841
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) 33.0000 1.05576 0.527882 0.849318i \(-0.322986\pi\)
0.527882 + 0.849318i \(0.322986\pi\)
\(978\) 0 0
\(979\) −36.0000 −1.15056
\(980\) 3.00000 0.0958315
\(981\) 0 0
\(982\) −12.0000 −0.382935
\(983\) 39.0000 1.24391 0.621953 0.783054i \(-0.286339\pi\)
0.621953 + 0.783054i \(0.286339\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 9.00000 0.286618
\(987\) 0 0
\(988\) −20.0000 −0.636285
\(989\) 9.00000 0.286183
\(990\) 0 0
\(991\) 5.00000 0.158830 0.0794151 0.996842i \(-0.474695\pi\)
0.0794151 + 0.996842i \(0.474695\pi\)
\(992\) 5.00000 0.158750
\(993\) 0 0
\(994\) 24.0000 0.761234
\(995\) 7.00000 0.221915
\(996\) 0 0
\(997\) −37.0000 −1.17180 −0.585901 0.810383i \(-0.699259\pi\)
−0.585901 + 0.810383i \(0.699259\pi\)
\(998\) −4.00000 −0.126618
\(999\) 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 270.2.a.c.1.1 yes 1
3.2 odd 2 270.2.a.b.1.1 1
4.3 odd 2 2160.2.a.b.1.1 1
5.2 odd 4 1350.2.c.j.649.2 2
5.3 odd 4 1350.2.c.j.649.1 2
5.4 even 2 1350.2.a.d.1.1 1
8.3 odd 2 8640.2.a.bn.1.1 1
8.5 even 2 8640.2.a.bx.1.1 1
9.2 odd 6 810.2.e.i.271.1 2
9.4 even 3 810.2.e.d.541.1 2
9.5 odd 6 810.2.e.i.541.1 2
9.7 even 3 810.2.e.d.271.1 2
12.11 even 2 2160.2.a.q.1.1 1
15.2 even 4 1350.2.c.c.649.1 2
15.8 even 4 1350.2.c.c.649.2 2
15.14 odd 2 1350.2.a.o.1.1 1
24.5 odd 2 8640.2.a.y.1.1 1
24.11 even 2 8640.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
270.2.a.b.1.1 1 3.2 odd 2
270.2.a.c.1.1 yes 1 1.1 even 1 trivial
810.2.e.d.271.1 2 9.7 even 3
810.2.e.d.541.1 2 9.4 even 3
810.2.e.i.271.1 2 9.2 odd 6
810.2.e.i.541.1 2 9.5 odd 6
1350.2.a.d.1.1 1 5.4 even 2
1350.2.a.o.1.1 1 15.14 odd 2
1350.2.c.c.649.1 2 15.2 even 4
1350.2.c.c.649.2 2 15.8 even 4
1350.2.c.j.649.1 2 5.3 odd 4
1350.2.c.j.649.2 2 5.2 odd 4
2160.2.a.b.1.1 1 4.3 odd 2
2160.2.a.q.1.1 1 12.11 even 2
8640.2.a.e.1.1 1 24.11 even 2
8640.2.a.y.1.1 1 24.5 odd 2
8640.2.a.bn.1.1 1 8.3 odd 2
8640.2.a.bx.1.1 1 8.5 even 2