Properties

Label 1170.2.a.e.1.1
Level $1170$
Weight $2$
Character 1170.1
Self dual yes
Analytic conductor $9.342$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1170.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.34249703649\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1170.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} -4.00000 q^{11} -1.00000 q^{13} -2.00000 q^{14} +1.00000 q^{16} -8.00000 q^{17} -6.00000 q^{19} +1.00000 q^{20} +4.00000 q^{22} -6.00000 q^{23} +1.00000 q^{25} +1.00000 q^{26} +2.00000 q^{28} +4.00000 q^{29} -1.00000 q^{32} +8.00000 q^{34} +2.00000 q^{35} -2.00000 q^{37} +6.00000 q^{38} -1.00000 q^{40} +2.00000 q^{41} -4.00000 q^{43} -4.00000 q^{44} +6.00000 q^{46} -3.00000 q^{49} -1.00000 q^{50} -1.00000 q^{52} +10.0000 q^{53} -4.00000 q^{55} -2.00000 q^{56} -4.00000 q^{58} -4.00000 q^{59} -10.0000 q^{61} +1.00000 q^{64} -1.00000 q^{65} +12.0000 q^{67} -8.00000 q^{68} -2.00000 q^{70} +8.00000 q^{71} -8.00000 q^{73} +2.00000 q^{74} -6.00000 q^{76} -8.00000 q^{77} +8.00000 q^{79} +1.00000 q^{80} -2.00000 q^{82} -12.0000 q^{83} -8.00000 q^{85} +4.00000 q^{86} +4.00000 q^{88} +14.0000 q^{89} -2.00000 q^{91} -6.00000 q^{92} -6.00000 q^{95} -16.0000 q^{97} +3.00000 q^{98} +O(q^{100})\)

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
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −8.00000 −1.94029 −0.970143 0.242536i \(-0.922021\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 1.00000 0.196116
\(27\) 0 0
\(28\) 2.00000 0.377964
\(29\) 4.00000 0.742781 0.371391 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 8.00000 1.37199
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 6.00000 0.973329
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −1.00000 −0.138675
\(53\) 10.0000 1.37361 0.686803 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) −2.00000 −0.267261
\(57\) 0 0
\(58\) −4.00000 −0.525226
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) −8.00000 −0.970143
\(69\) 0 0
\(70\) −2.00000 −0.239046
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) −8.00000 −0.936329 −0.468165 0.883641i \(-0.655085\pi\)
−0.468165 + 0.883641i \(0.655085\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) −6.00000 −0.688247
\(77\) −8.00000 −0.911685
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) −2.00000 −0.220863
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) −8.00000 −0.867722
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) 4.00000 0.426401
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) −6.00000 −0.625543
\(93\) 0 0
\(94\) 0 0
\(95\) −6.00000 −0.615587
\(96\) 0 0
\(97\) −16.0000 −1.62455 −0.812277 0.583272i \(-0.801772\pi\)
−0.812277 + 0.583272i \(0.801772\pi\)
\(98\) 3.00000 0.303046
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 16.0000 1.59206 0.796030 0.605257i \(-0.206930\pi\)
0.796030 + 0.605257i \(0.206930\pi\)
\(102\) 0 0
\(103\) −12.0000 −1.18240 −0.591198 0.806527i \(-0.701345\pi\)
−0.591198 + 0.806527i \(0.701345\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) −10.0000 −0.971286
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) 12.0000 1.14939 0.574696 0.818367i \(-0.305120\pi\)
0.574696 + 0.818367i \(0.305120\pi\)
\(110\) 4.00000 0.381385
\(111\) 0 0
\(112\) 2.00000 0.188982
\(113\) −20.0000 −1.88144 −0.940721 0.339182i \(-0.889850\pi\)
−0.940721 + 0.339182i \(0.889850\pi\)
\(114\) 0 0
\(115\) −6.00000 −0.559503
\(116\) 4.00000 0.371391
\(117\) 0 0
\(118\) 4.00000 0.368230
\(119\) −16.0000 −1.46672
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 10.0000 0.905357
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 1.00000 0.0877058
\(131\) −10.0000 −0.873704 −0.436852 0.899533i \(-0.643907\pi\)
−0.436852 + 0.899533i \(0.643907\pi\)
\(132\) 0 0
\(133\) −12.0000 −1.04053
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) 8.00000 0.685994
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 2.00000 0.169031
\(141\) 0 0
\(142\) −8.00000 −0.671345
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) 4.00000 0.332182
\(146\) 8.00000 0.662085
\(147\) 0 0
\(148\) −2.00000 −0.164399
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 6.00000 0.486664
\(153\) 0 0
\(154\) 8.00000 0.644658
\(155\) 0 0
\(156\) 0 0
\(157\) 22.0000 1.75579 0.877896 0.478852i \(-0.158947\pi\)
0.877896 + 0.478852i \(0.158947\pi\)
\(158\) −8.00000 −0.636446
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) −12.0000 −0.945732
\(162\) 0 0
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 4.00000 0.309529 0.154765 0.987951i \(-0.450538\pi\)
0.154765 + 0.987951i \(0.450538\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 8.00000 0.613572
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) −22.0000 −1.67263 −0.836315 0.548250i \(-0.815294\pi\)
−0.836315 + 0.548250i \(0.815294\pi\)
\(174\) 0 0
\(175\) 2.00000 0.151186
\(176\) −4.00000 −0.301511
\(177\) 0 0
\(178\) −14.0000 −1.04934
\(179\) 10.0000 0.747435 0.373718 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 2.00000 0.148250
\(183\) 0 0
\(184\) 6.00000 0.442326
\(185\) −2.00000 −0.147043
\(186\) 0 0
\(187\) 32.0000 2.34007
\(188\) 0 0
\(189\) 0 0
\(190\) 6.00000 0.435286
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 16.0000 1.14873
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −16.0000 −1.12576
\(203\) 8.00000 0.561490
\(204\) 0 0
\(205\) 2.00000 0.139686
\(206\) 12.0000 0.836080
\(207\) 0 0
\(208\) −1.00000 −0.0693375
\(209\) 24.0000 1.66011
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 10.0000 0.686803
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) 0 0
\(218\) −12.0000 −0.812743
\(219\) 0 0
\(220\) −4.00000 −0.269680
\(221\) 8.00000 0.538138
\(222\) 0 0
\(223\) −2.00000 −0.133930 −0.0669650 0.997755i \(-0.521332\pi\)
−0.0669650 + 0.997755i \(0.521332\pi\)
\(224\) −2.00000 −0.133631
\(225\) 0 0
\(226\) 20.0000 1.33038
\(227\) 4.00000 0.265489 0.132745 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) 0 0
\(229\) 4.00000 0.264327 0.132164 0.991228i \(-0.457808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) 6.00000 0.395628
\(231\) 0 0
\(232\) −4.00000 −0.262613
\(233\) −24.0000 −1.57229 −0.786146 0.618041i \(-0.787927\pi\)
−0.786146 + 0.618041i \(0.787927\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −4.00000 −0.260378
\(237\) 0 0
\(238\) 16.0000 1.03713
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) −5.00000 −0.321412
\(243\) 0 0
\(244\) −10.0000 −0.640184
\(245\) −3.00000 −0.191663
\(246\) 0 0
\(247\) 6.00000 0.381771
\(248\) 0 0
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) −6.00000 −0.378717 −0.189358 0.981908i \(-0.560641\pi\)
−0.189358 + 0.981908i \(0.560641\pi\)
\(252\) 0 0
\(253\) 24.0000 1.50887
\(254\) −4.00000 −0.250982
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −12.0000 −0.748539 −0.374270 0.927320i \(-0.622107\pi\)
−0.374270 + 0.927320i \(0.622107\pi\)
\(258\) 0 0
\(259\) −4.00000 −0.248548
\(260\) −1.00000 −0.0620174
\(261\) 0 0
\(262\) 10.0000 0.617802
\(263\) −2.00000 −0.123325 −0.0616626 0.998097i \(-0.519640\pi\)
−0.0616626 + 0.998097i \(0.519640\pi\)
\(264\) 0 0
\(265\) 10.0000 0.614295
\(266\) 12.0000 0.735767
\(267\) 0 0
\(268\) 12.0000 0.733017
\(269\) 24.0000 1.46331 0.731653 0.681677i \(-0.238749\pi\)
0.731653 + 0.681677i \(0.238749\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) −8.00000 −0.485071
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) −4.00000 −0.241209
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 8.00000 0.479808
\(279\) 0 0
\(280\) −2.00000 −0.119523
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) 4.00000 0.236113
\(288\) 0 0
\(289\) 47.0000 2.76471
\(290\) −4.00000 −0.234888
\(291\) 0 0
\(292\) −8.00000 −0.468165
\(293\) 26.0000 1.51894 0.759468 0.650545i \(-0.225459\pi\)
0.759468 + 0.650545i \(0.225459\pi\)
\(294\) 0 0
\(295\) −4.00000 −0.232889
\(296\) 2.00000 0.116248
\(297\) 0 0
\(298\) −10.0000 −0.579284
\(299\) 6.00000 0.346989
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) 0 0
\(303\) 0 0
\(304\) −6.00000 −0.344124
\(305\) −10.0000 −0.572598
\(306\) 0 0
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) −8.00000 −0.455842
\(309\) 0 0
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) −22.0000 −1.24153
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 30.0000 1.68497 0.842484 0.538721i \(-0.181092\pi\)
0.842484 + 0.538721i \(0.181092\pi\)
\(318\) 0 0
\(319\) −16.0000 −0.895828
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 12.0000 0.668734
\(323\) 48.0000 2.67079
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) 16.0000 0.886158
\(327\) 0 0
\(328\) −2.00000 −0.110432
\(329\) 0 0
\(330\) 0 0
\(331\) 10.0000 0.549650 0.274825 0.961494i \(-0.411380\pi\)
0.274825 + 0.961494i \(0.411380\pi\)
\(332\) −12.0000 −0.658586
\(333\) 0 0
\(334\) −4.00000 −0.218870
\(335\) 12.0000 0.655630
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 0 0
\(340\) −8.00000 −0.433861
\(341\) 0 0
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 22.0000 1.18273
\(347\) 24.0000 1.28839 0.644194 0.764862i \(-0.277193\pi\)
0.644194 + 0.764862i \(0.277193\pi\)
\(348\) 0 0
\(349\) −28.0000 −1.49881 −0.749403 0.662114i \(-0.769659\pi\)
−0.749403 + 0.662114i \(0.769659\pi\)
\(350\) −2.00000 −0.106904
\(351\) 0 0
\(352\) 4.00000 0.213201
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 0 0
\(355\) 8.00000 0.424596
\(356\) 14.0000 0.741999
\(357\) 0 0
\(358\) −10.0000 −0.528516
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) −10.0000 −0.525588
\(363\) 0 0
\(364\) −2.00000 −0.104828
\(365\) −8.00000 −0.418739
\(366\) 0 0
\(367\) −36.0000 −1.87918 −0.939592 0.342296i \(-0.888796\pi\)
−0.939592 + 0.342296i \(0.888796\pi\)
\(368\) −6.00000 −0.312772
\(369\) 0 0
\(370\) 2.00000 0.103975
\(371\) 20.0000 1.03835
\(372\) 0 0
\(373\) −2.00000 −0.103556 −0.0517780 0.998659i \(-0.516489\pi\)
−0.0517780 + 0.998659i \(0.516489\pi\)
\(374\) −32.0000 −1.65468
\(375\) 0 0
\(376\) 0 0
\(377\) −4.00000 −0.206010
\(378\) 0 0
\(379\) −18.0000 −0.924598 −0.462299 0.886724i \(-0.652975\pi\)
−0.462299 + 0.886724i \(0.652975\pi\)
\(380\) −6.00000 −0.307794
\(381\) 0 0
\(382\) 0 0
\(383\) 12.0000 0.613171 0.306586 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) 0 0
\(385\) −8.00000 −0.407718
\(386\) 4.00000 0.203595
\(387\) 0 0
\(388\) −16.0000 −0.812277
\(389\) −8.00000 −0.405616 −0.202808 0.979219i \(-0.565007\pi\)
−0.202808 + 0.979219i \(0.565007\pi\)
\(390\) 0 0
\(391\) 48.0000 2.42746
\(392\) 3.00000 0.151523
\(393\) 0 0
\(394\) −18.0000 −0.906827
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 16.0000 0.796030
\(405\) 0 0
\(406\) −8.00000 −0.397033
\(407\) 8.00000 0.396545
\(408\) 0 0
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) −2.00000 −0.0987730
\(411\) 0 0
\(412\) −12.0000 −0.591198
\(413\) −8.00000 −0.393654
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) 1.00000 0.0490290
\(417\) 0 0
\(418\) −24.0000 −1.17388
\(419\) −10.0000 −0.488532 −0.244266 0.969708i \(-0.578547\pi\)
−0.244266 + 0.969708i \(0.578547\pi\)
\(420\) 0 0
\(421\) −8.00000 −0.389896 −0.194948 0.980814i \(-0.562454\pi\)
−0.194948 + 0.980814i \(0.562454\pi\)
\(422\) 20.0000 0.973585
\(423\) 0 0
\(424\) −10.0000 −0.485643
\(425\) −8.00000 −0.388057
\(426\) 0 0
\(427\) −20.0000 −0.967868
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 4.00000 0.192897
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 12.0000 0.574696
\(437\) 36.0000 1.72211
\(438\) 0 0
\(439\) −32.0000 −1.52728 −0.763638 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) 4.00000 0.190693
\(441\) 0 0
\(442\) −8.00000 −0.380521
\(443\) −16.0000 −0.760183 −0.380091 0.924949i \(-0.624107\pi\)
−0.380091 + 0.924949i \(0.624107\pi\)
\(444\) 0 0
\(445\) 14.0000 0.663664
\(446\) 2.00000 0.0947027
\(447\) 0 0
\(448\) 2.00000 0.0944911
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) −8.00000 −0.376705
\(452\) −20.0000 −0.940721
\(453\) 0 0
\(454\) −4.00000 −0.187729
\(455\) −2.00000 −0.0937614
\(456\) 0 0
\(457\) 8.00000 0.374224 0.187112 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) −4.00000 −0.186908
\(459\) 0 0
\(460\) −6.00000 −0.279751
\(461\) 6.00000 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 0 0
\(463\) −26.0000 −1.20832 −0.604161 0.796862i \(-0.706492\pi\)
−0.604161 + 0.796862i \(0.706492\pi\)
\(464\) 4.00000 0.185695
\(465\) 0 0
\(466\) 24.0000 1.11178
\(467\) −28.0000 −1.29569 −0.647843 0.761774i \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(468\) 0 0
\(469\) 24.0000 1.10822
\(470\) 0 0
\(471\) 0 0
\(472\) 4.00000 0.184115
\(473\) 16.0000 0.735681
\(474\) 0 0
\(475\) −6.00000 −0.275299
\(476\) −16.0000 −0.733359
\(477\) 0 0
\(478\) 16.0000 0.731823
\(479\) −32.0000 −1.46212 −0.731059 0.682315i \(-0.760973\pi\)
−0.731059 + 0.682315i \(0.760973\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) −2.00000 −0.0910975
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) −16.0000 −0.726523
\(486\) 0 0
\(487\) 26.0000 1.17817 0.589086 0.808070i \(-0.299488\pi\)
0.589086 + 0.808070i \(0.299488\pi\)
\(488\) 10.0000 0.452679
\(489\) 0 0
\(490\) 3.00000 0.135526
\(491\) −42.0000 −1.89543 −0.947717 0.319113i \(-0.896615\pi\)
−0.947717 + 0.319113i \(0.896615\pi\)
\(492\) 0 0
\(493\) −32.0000 −1.44121
\(494\) −6.00000 −0.269953
\(495\) 0 0
\(496\) 0 0
\(497\) 16.0000 0.717698
\(498\) 0 0
\(499\) 38.0000 1.70111 0.850557 0.525883i \(-0.176265\pi\)
0.850557 + 0.525883i \(0.176265\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 6.00000 0.267793
\(503\) −10.0000 −0.445878 −0.222939 0.974832i \(-0.571565\pi\)
−0.222939 + 0.974832i \(0.571565\pi\)
\(504\) 0 0
\(505\) 16.0000 0.711991
\(506\) −24.0000 −1.06693
\(507\) 0 0
\(508\) 4.00000 0.177471
\(509\) −18.0000 −0.797836 −0.398918 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 0 0
\(511\) −16.0000 −0.707798
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 12.0000 0.529297
\(515\) −12.0000 −0.528783
\(516\) 0 0
\(517\) 0 0
\(518\) 4.00000 0.175750
\(519\) 0 0
\(520\) 1.00000 0.0438529
\(521\) 26.0000 1.13908 0.569540 0.821963i \(-0.307121\pi\)
0.569540 + 0.821963i \(0.307121\pi\)
\(522\) 0 0
\(523\) 36.0000 1.57417 0.787085 0.616844i \(-0.211589\pi\)
0.787085 + 0.616844i \(0.211589\pi\)
\(524\) −10.0000 −0.436852
\(525\) 0 0
\(526\) 2.00000 0.0872041
\(527\) 0 0
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) −10.0000 −0.434372
\(531\) 0 0
\(532\) −12.0000 −0.520266
\(533\) −2.00000 −0.0866296
\(534\) 0 0
\(535\) −12.0000 −0.518805
\(536\) −12.0000 −0.518321
\(537\) 0 0
\(538\) −24.0000 −1.03471
\(539\) 12.0000 0.516877
\(540\) 0 0
\(541\) −8.00000 −0.343947 −0.171973 0.985102i \(-0.555014\pi\)
−0.171973 + 0.985102i \(0.555014\pi\)
\(542\) −16.0000 −0.687259
\(543\) 0 0
\(544\) 8.00000 0.342997
\(545\) 12.0000 0.514024
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 6.00000 0.256307
\(549\) 0 0
\(550\) 4.00000 0.170561
\(551\) −24.0000 −1.02243
\(552\) 0 0
\(553\) 16.0000 0.680389
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) −8.00000 −0.339276
\(557\) −2.00000 −0.0847427 −0.0423714 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 2.00000 0.0845154
\(561\) 0 0
\(562\) −6.00000 −0.253095
\(563\) −24.0000 −1.01148 −0.505740 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) 0 0
\(565\) −20.0000 −0.841406
\(566\) 4.00000 0.168133
\(567\) 0 0
\(568\) −8.00000 −0.335673
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) 40.0000 1.67395 0.836974 0.547243i \(-0.184323\pi\)
0.836974 + 0.547243i \(0.184323\pi\)
\(572\) 4.00000 0.167248
\(573\) 0 0
\(574\) −4.00000 −0.166957
\(575\) −6.00000 −0.250217
\(576\) 0 0
\(577\) 12.0000 0.499567 0.249783 0.968302i \(-0.419641\pi\)
0.249783 + 0.968302i \(0.419641\pi\)
\(578\) −47.0000 −1.95494
\(579\) 0 0
\(580\) 4.00000 0.166091
\(581\) −24.0000 −0.995688
\(582\) 0 0
\(583\) −40.0000 −1.65663
\(584\) 8.00000 0.331042
\(585\) 0 0
\(586\) −26.0000 −1.07405
\(587\) −36.0000 −1.48588 −0.742940 0.669359i \(-0.766569\pi\)
−0.742940 + 0.669359i \(0.766569\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 4.00000 0.164677
\(591\) 0 0
\(592\) −2.00000 −0.0821995
\(593\) −22.0000 −0.903432 −0.451716 0.892162i \(-0.649188\pi\)
−0.451716 + 0.892162i \(0.649188\pi\)
\(594\) 0 0
\(595\) −16.0000 −0.655936
\(596\) 10.0000 0.409616
\(597\) 0 0
\(598\) −6.00000 −0.245358
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) 22.0000 0.897399 0.448699 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 8.00000 0.326056
\(603\) 0 0
\(604\) 0 0
\(605\) 5.00000 0.203279
\(606\) 0 0
\(607\) −28.0000 −1.13648 −0.568242 0.822861i \(-0.692376\pi\)
−0.568242 + 0.822861i \(0.692376\pi\)
\(608\) 6.00000 0.243332
\(609\) 0 0
\(610\) 10.0000 0.404888
\(611\) 0 0
\(612\) 0 0
\(613\) −10.0000 −0.403896 −0.201948 0.979396i \(-0.564727\pi\)
−0.201948 + 0.979396i \(0.564727\pi\)
\(614\) −12.0000 −0.484281
\(615\) 0 0
\(616\) 8.00000 0.322329
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) −10.0000 −0.401934 −0.200967 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 24.0000 0.962312
\(623\) 28.0000 1.12180
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −6.00000 −0.239808
\(627\) 0 0
\(628\) 22.0000 0.877896
\(629\) 16.0000 0.637962
\(630\) 0 0
\(631\) 12.0000 0.477712 0.238856 0.971055i \(-0.423228\pi\)
0.238856 + 0.971055i \(0.423228\pi\)
\(632\) −8.00000 −0.318223
\(633\) 0 0
\(634\) −30.0000 −1.19145
\(635\) 4.00000 0.158735
\(636\) 0 0
\(637\) 3.00000 0.118864
\(638\) 16.0000 0.633446
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) 44.0000 1.73519 0.867595 0.497271i \(-0.165665\pi\)
0.867595 + 0.497271i \(0.165665\pi\)
\(644\) −12.0000 −0.472866
\(645\) 0 0
\(646\) −48.0000 −1.88853
\(647\) 30.0000 1.17942 0.589711 0.807614i \(-0.299242\pi\)
0.589711 + 0.807614i \(0.299242\pi\)
\(648\) 0 0
\(649\) 16.0000 0.628055
\(650\) 1.00000 0.0392232
\(651\) 0 0
\(652\) −16.0000 −0.626608
\(653\) −42.0000 −1.64359 −0.821794 0.569785i \(-0.807026\pi\)
−0.821794 + 0.569785i \(0.807026\pi\)
\(654\) 0 0
\(655\) −10.0000 −0.390732
\(656\) 2.00000 0.0780869
\(657\) 0 0
\(658\) 0 0
\(659\) −2.00000 −0.0779089 −0.0389545 0.999241i \(-0.512403\pi\)
−0.0389545 + 0.999241i \(0.512403\pi\)
\(660\) 0 0
\(661\) 48.0000 1.86698 0.933492 0.358599i \(-0.116745\pi\)
0.933492 + 0.358599i \(0.116745\pi\)
\(662\) −10.0000 −0.388661
\(663\) 0 0
\(664\) 12.0000 0.465690
\(665\) −12.0000 −0.465340
\(666\) 0 0
\(667\) −24.0000 −0.929284
\(668\) 4.00000 0.154765
\(669\) 0 0
\(670\) −12.0000 −0.463600
\(671\) 40.0000 1.54418
\(672\) 0 0
\(673\) 26.0000 1.00223 0.501113 0.865382i \(-0.332924\pi\)
0.501113 + 0.865382i \(0.332924\pi\)
\(674\) −14.0000 −0.539260
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) 30.0000 1.15299 0.576497 0.817099i \(-0.304419\pi\)
0.576497 + 0.817099i \(0.304419\pi\)
\(678\) 0 0
\(679\) −32.0000 −1.22805
\(680\) 8.00000 0.306786
\(681\) 0 0
\(682\) 0 0
\(683\) 44.0000 1.68361 0.841807 0.539779i \(-0.181492\pi\)
0.841807 + 0.539779i \(0.181492\pi\)
\(684\) 0 0
\(685\) 6.00000 0.229248
\(686\) 20.0000 0.763604
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) −10.0000 −0.380970
\(690\) 0 0
\(691\) −14.0000 −0.532585 −0.266293 0.963892i \(-0.585799\pi\)
−0.266293 + 0.963892i \(0.585799\pi\)
\(692\) −22.0000 −0.836315
\(693\) 0 0
\(694\) −24.0000 −0.911028
\(695\) −8.00000 −0.303457
\(696\) 0 0
\(697\) −16.0000 −0.606043
\(698\) 28.0000 1.05982
\(699\) 0 0
\(700\) 2.00000 0.0755929
\(701\) −32.0000 −1.20862 −0.604312 0.796748i \(-0.706552\pi\)
−0.604312 + 0.796748i \(0.706552\pi\)
\(702\) 0 0
\(703\) 12.0000 0.452589
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) −14.0000 −0.526897
\(707\) 32.0000 1.20348
\(708\) 0 0
\(709\) −4.00000 −0.150223 −0.0751116 0.997175i \(-0.523931\pi\)
−0.0751116 + 0.997175i \(0.523931\pi\)
\(710\) −8.00000 −0.300235
\(711\) 0 0
\(712\) −14.0000 −0.524672
\(713\) 0 0
\(714\) 0 0
\(715\) 4.00000 0.149592
\(716\) 10.0000 0.373718
\(717\) 0 0
\(718\) −24.0000 −0.895672
\(719\) −36.0000 −1.34257 −0.671287 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) 0 0
\(721\) −24.0000 −0.893807
\(722\) −17.0000 −0.632674
\(723\) 0 0
\(724\) 10.0000 0.371647
\(725\) 4.00000 0.148556
\(726\) 0 0
\(727\) 16.0000 0.593407 0.296704 0.954970i \(-0.404113\pi\)
0.296704 + 0.954970i \(0.404113\pi\)
\(728\) 2.00000 0.0741249
\(729\) 0 0
\(730\) 8.00000 0.296093
\(731\) 32.0000 1.18356
\(732\) 0 0
\(733\) 38.0000 1.40356 0.701781 0.712393i \(-0.252388\pi\)
0.701781 + 0.712393i \(0.252388\pi\)
\(734\) 36.0000 1.32878
\(735\) 0 0
\(736\) 6.00000 0.221163
\(737\) −48.0000 −1.76810
\(738\) 0 0
\(739\) 30.0000 1.10357 0.551784 0.833987i \(-0.313947\pi\)
0.551784 + 0.833987i \(0.313947\pi\)
\(740\) −2.00000 −0.0735215
\(741\) 0 0
\(742\) −20.0000 −0.734223
\(743\) −12.0000 −0.440237 −0.220119 0.975473i \(-0.570644\pi\)
−0.220119 + 0.975473i \(0.570644\pi\)
\(744\) 0 0
\(745\) 10.0000 0.366372
\(746\) 2.00000 0.0732252
\(747\) 0 0
\(748\) 32.0000 1.17004
\(749\) −24.0000 −0.876941
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 4.00000 0.145671
\(755\) 0 0
\(756\) 0 0
\(757\) −6.00000 −0.218074 −0.109037 0.994038i \(-0.534777\pi\)
−0.109037 + 0.994038i \(0.534777\pi\)
\(758\) 18.0000 0.653789
\(759\) 0 0
\(760\) 6.00000 0.217643
\(761\) −22.0000 −0.797499 −0.398750 0.917060i \(-0.630556\pi\)
−0.398750 + 0.917060i \(0.630556\pi\)
\(762\) 0 0
\(763\) 24.0000 0.868858
\(764\) 0 0
\(765\) 0 0
\(766\) −12.0000 −0.433578
\(767\) 4.00000 0.144432
\(768\) 0 0
\(769\) 10.0000 0.360609 0.180305 0.983611i \(-0.442292\pi\)
0.180305 + 0.983611i \(0.442292\pi\)
\(770\) 8.00000 0.288300
\(771\) 0 0
\(772\) −4.00000 −0.143963
\(773\) −38.0000 −1.36677 −0.683383 0.730061i \(-0.739492\pi\)
−0.683383 + 0.730061i \(0.739492\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 16.0000 0.574367
\(777\) 0 0
\(778\) 8.00000 0.286814
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) −32.0000 −1.14505
\(782\) −48.0000 −1.71648
\(783\) 0 0
\(784\) −3.00000 −0.107143
\(785\) 22.0000 0.785214
\(786\) 0 0
\(787\) −32.0000 −1.14068 −0.570338 0.821410i \(-0.693188\pi\)
−0.570338 + 0.821410i \(0.693188\pi\)
\(788\) 18.0000 0.641223
\(789\) 0 0
\(790\) −8.00000 −0.284627
\(791\) −40.0000 −1.42224
\(792\) 0 0
\(793\) 10.0000 0.355110
\(794\) −14.0000 −0.496841
\(795\) 0 0
\(796\) 0 0
\(797\) 10.0000 0.354218 0.177109 0.984191i \(-0.443325\pi\)
0.177109 + 0.984191i \(0.443325\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 30.0000 1.05934
\(803\) 32.0000 1.12926
\(804\) 0 0
\(805\) −12.0000 −0.422944
\(806\) 0 0
\(807\) 0 0
\(808\) −16.0000 −0.562878
\(809\) −2.00000 −0.0703163 −0.0351581 0.999382i \(-0.511193\pi\)
−0.0351581 + 0.999382i \(0.511193\pi\)
\(810\) 0 0
\(811\) −38.0000 −1.33436 −0.667180 0.744896i \(-0.732499\pi\)
−0.667180 + 0.744896i \(0.732499\pi\)
\(812\) 8.00000 0.280745
\(813\) 0 0
\(814\) −8.00000 −0.280400
\(815\) −16.0000 −0.560456
\(816\) 0 0
\(817\) 24.0000 0.839654
\(818\) 26.0000 0.909069
\(819\) 0 0
\(820\) 2.00000 0.0698430
\(821\) 10.0000 0.349002 0.174501 0.984657i \(-0.444169\pi\)
0.174501 + 0.984657i \(0.444169\pi\)
\(822\) 0 0
\(823\) −52.0000 −1.81261 −0.906303 0.422628i \(-0.861108\pi\)
−0.906303 + 0.422628i \(0.861108\pi\)
\(824\) 12.0000 0.418040
\(825\) 0 0
\(826\) 8.00000 0.278356
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 0 0
\(829\) −10.0000 −0.347314 −0.173657 0.984806i \(-0.555558\pi\)
−0.173657 + 0.984806i \(0.555558\pi\)
\(830\) 12.0000 0.416526
\(831\) 0 0
\(832\) −1.00000 −0.0346688
\(833\) 24.0000 0.831551
\(834\) 0 0
\(835\) 4.00000 0.138426
\(836\) 24.0000 0.830057
\(837\) 0 0
\(838\) 10.0000 0.345444
\(839\) 40.0000 1.38095 0.690477 0.723355i \(-0.257401\pi\)
0.690477 + 0.723355i \(0.257401\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 8.00000 0.275698
\(843\) 0 0
\(844\) −20.0000 −0.688428
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) 10.0000 0.343604
\(848\) 10.0000 0.343401
\(849\) 0 0
\(850\) 8.00000 0.274398
\(851\) 12.0000 0.411355
\(852\) 0 0
\(853\) 46.0000 1.57501 0.787505 0.616308i \(-0.211372\pi\)
0.787505 + 0.616308i \(0.211372\pi\)
\(854\) 20.0000 0.684386
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) −4.00000 −0.136637 −0.0683187 0.997664i \(-0.521763\pi\)
−0.0683187 + 0.997664i \(0.521763\pi\)
\(858\) 0 0
\(859\) 36.0000 1.22830 0.614152 0.789188i \(-0.289498\pi\)
0.614152 + 0.789188i \(0.289498\pi\)
\(860\) −4.00000 −0.136399
\(861\) 0 0
\(862\) 0 0
\(863\) 12.0000 0.408485 0.204242 0.978920i \(-0.434527\pi\)
0.204242 + 0.978920i \(0.434527\pi\)
\(864\) 0 0
\(865\) −22.0000 −0.748022
\(866\) 2.00000 0.0679628
\(867\) 0 0
\(868\) 0 0
\(869\) −32.0000 −1.08553
\(870\) 0 0
\(871\) −12.0000 −0.406604
\(872\) −12.0000 −0.406371
\(873\) 0 0
\(874\) −36.0000 −1.21772
\(875\) 2.00000 0.0676123
\(876\) 0 0
\(877\) −14.0000 −0.472746 −0.236373 0.971662i \(-0.575959\pi\)
−0.236373 + 0.971662i \(0.575959\pi\)
\(878\) 32.0000 1.07995
\(879\) 0 0
\(880\) −4.00000 −0.134840
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 0 0
\(883\) −28.0000 −0.942275 −0.471138 0.882060i \(-0.656156\pi\)
−0.471138 + 0.882060i \(0.656156\pi\)
\(884\) 8.00000 0.269069
\(885\) 0 0
\(886\) 16.0000 0.537531
\(887\) 2.00000 0.0671534 0.0335767 0.999436i \(-0.489310\pi\)
0.0335767 + 0.999436i \(0.489310\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) −14.0000 −0.469281
\(891\) 0 0
\(892\) −2.00000 −0.0669650
\(893\) 0 0
\(894\) 0 0
\(895\) 10.0000 0.334263
\(896\) −2.00000 −0.0668153
\(897\) 0 0
\(898\) −6.00000 −0.200223
\(899\) 0 0
\(900\) 0 0
\(901\) −80.0000 −2.66519
\(902\) 8.00000 0.266371
\(903\) 0 0
\(904\) 20.0000 0.665190
\(905\) 10.0000 0.332411
\(906\) 0 0
\(907\) 52.0000 1.72663 0.863316 0.504664i \(-0.168384\pi\)
0.863316 + 0.504664i \(0.168384\pi\)
\(908\) 4.00000 0.132745
\(909\) 0 0
\(910\) 2.00000 0.0662994
\(911\) 20.0000 0.662630 0.331315 0.943520i \(-0.392508\pi\)
0.331315 + 0.943520i \(0.392508\pi\)
\(912\) 0 0
\(913\) 48.0000 1.58857
\(914\) −8.00000 −0.264616
\(915\) 0 0
\(916\) 4.00000 0.132164
\(917\) −20.0000 −0.660458
\(918\) 0 0
\(919\) 40.0000 1.31948 0.659739 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) 6.00000 0.197814
\(921\) 0 0
\(922\) −6.00000 −0.197599
\(923\) −8.00000 −0.263323
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) 26.0000 0.854413
\(927\) 0 0
\(928\) −4.00000 −0.131306
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) 18.0000 0.589926
\(932\) −24.0000 −0.786146
\(933\) 0 0
\(934\) 28.0000 0.916188
\(935\) 32.0000 1.04651
\(936\) 0 0
\(937\) 18.0000 0.588034 0.294017 0.955800i \(-0.405008\pi\)
0.294017 + 0.955800i \(0.405008\pi\)
\(938\) −24.0000 −0.783628
\(939\) 0 0
\(940\) 0 0
\(941\) 34.0000 1.10837 0.554184 0.832394i \(-0.313030\pi\)
0.554184 + 0.832394i \(0.313030\pi\)
\(942\) 0 0
\(943\) −12.0000 −0.390774
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) −16.0000 −0.520205
\(947\) −4.00000 −0.129983 −0.0649913 0.997886i \(-0.520702\pi\)
−0.0649913 + 0.997886i \(0.520702\pi\)
\(948\) 0 0
\(949\) 8.00000 0.259691
\(950\) 6.00000 0.194666
\(951\) 0 0
\(952\) 16.0000 0.518563
\(953\) −24.0000 −0.777436 −0.388718 0.921357i \(-0.627082\pi\)
−0.388718 + 0.921357i \(0.627082\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −16.0000 −0.517477
\(957\) 0 0
\(958\) 32.0000 1.03387
\(959\) 12.0000 0.387500
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) −2.00000 −0.0644826
\(963\) 0 0
\(964\) 2.00000 0.0644157
\(965\) −4.00000 −0.128765
\(966\) 0 0
\(967\) −14.0000 −0.450210 −0.225105 0.974335i \(-0.572272\pi\)
−0.225105 + 0.974335i \(0.572272\pi\)
\(968\) −5.00000 −0.160706
\(969\) 0 0
\(970\) 16.0000 0.513729
\(971\) −38.0000 −1.21948 −0.609739 0.792602i \(-0.708726\pi\)
−0.609739 + 0.792602i \(0.708726\pi\)
\(972\) 0 0
\(973\) −16.0000 −0.512936
\(974\) −26.0000 −0.833094
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) −30.0000 −0.959785 −0.479893 0.877327i \(-0.659324\pi\)
−0.479893 + 0.877327i \(0.659324\pi\)
\(978\) 0 0
\(979\) −56.0000 −1.78977
\(980\) −3.00000 −0.0958315
\(981\) 0 0
\(982\) 42.0000 1.34027
\(983\) 52.0000 1.65854 0.829271 0.558846i \(-0.188756\pi\)
0.829271 + 0.558846i \(0.188756\pi\)
\(984\) 0 0
\(985\) 18.0000 0.573528
\(986\) 32.0000 1.01909
\(987\) 0 0
\(988\) 6.00000 0.190885
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) −16.0000 −0.507489
\(995\) 0 0
\(996\) 0 0
\(997\) −26.0000 −0.823428 −0.411714 0.911313i \(-0.635070\pi\)
−0.411714 + 0.911313i \(0.635070\pi\)
\(998\) −38.0000 −1.20287
\(999\) 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 1170.2.a.e.1.1 1
3.2 odd 2 390.2.a.e.1.1 1
4.3 odd 2 9360.2.a.bh.1.1 1
5.2 odd 4 5850.2.e.i.5149.1 2
5.3 odd 4 5850.2.e.i.5149.2 2
5.4 even 2 5850.2.a.bi.1.1 1
12.11 even 2 3120.2.a.o.1.1 1
15.2 even 4 1950.2.e.f.1249.2 2
15.8 even 4 1950.2.e.f.1249.1 2
15.14 odd 2 1950.2.a.h.1.1 1
39.5 even 4 5070.2.b.e.1351.1 2
39.8 even 4 5070.2.b.e.1351.2 2
39.38 odd 2 5070.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.e.1.1 1 3.2 odd 2
1170.2.a.e.1.1 1 1.1 even 1 trivial
1950.2.a.h.1.1 1 15.14 odd 2
1950.2.e.f.1249.1 2 15.8 even 4
1950.2.e.f.1249.2 2 15.2 even 4
3120.2.a.o.1.1 1 12.11 even 2
5070.2.a.e.1.1 1 39.38 odd 2
5070.2.b.e.1351.1 2 39.5 even 4
5070.2.b.e.1351.2 2 39.8 even 4
5850.2.a.bi.1.1 1 5.4 even 2
5850.2.e.i.5149.1 2 5.2 odd 4
5850.2.e.i.5149.2 2 5.3 odd 4
9360.2.a.bh.1.1 1 4.3 odd 2