Properties

Label 1710.2.a.b.1.1
Level $1710$
Weight $2$
Character 1710.1
Self dual yes
Analytic conductor $13.654$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1710.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.6544187456\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 570)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1710.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} -2.00000 q^{11} +4.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{19} -1.00000 q^{20} +2.00000 q^{22} -4.00000 q^{23} +1.00000 q^{25} -4.00000 q^{26} -2.00000 q^{28} -8.00000 q^{31} -1.00000 q^{32} -2.00000 q^{34} +2.00000 q^{35} +8.00000 q^{37} +1.00000 q^{38} +1.00000 q^{40} +8.00000 q^{41} -6.00000 q^{43} -2.00000 q^{44} +4.00000 q^{46} +12.0000 q^{47} -3.00000 q^{49} -1.00000 q^{50} +4.00000 q^{52} +6.00000 q^{53} +2.00000 q^{55} +2.00000 q^{56} +2.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} +8.00000 q^{67} +2.00000 q^{68} -2.00000 q^{70} +8.00000 q^{71} +14.0000 q^{73} -8.00000 q^{74} -1.00000 q^{76} +4.00000 q^{77} -1.00000 q^{80} -8.00000 q^{82} -4.00000 q^{83} -2.00000 q^{85} +6.00000 q^{86} +2.00000 q^{88} -8.00000 q^{91} -4.00000 q^{92} -12.0000 q^{94} +1.00000 q^{95} -12.0000 q^{97} +3.00000 q^{98} +O(q^{100})\)

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 \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −4.00000 −0.784465
\(27\) 0 0
\(28\) −2.00000 −0.377964
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 8.00000 1.24939 0.624695 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 0 0
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 4.00000 0.554700
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) 2.00000 0.267261
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 2.00000 0.242536
\(69\) 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
\(73\) 14.0000 1.63858 0.819288 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) −8.00000 −0.929981
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) −8.00000 −0.883452
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) 6.00000 0.646997
\(87\) 0 0
\(88\) 2.00000 0.213201
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) −4.00000 −0.417029
\(93\) 0 0
\(94\) −12.0000 −1.23771
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) −12.0000 −1.21842 −0.609208 0.793011i \(-0.708512\pi\)
−0.609208 + 0.793011i \(0.708512\pi\)
\(98\) 3.00000 0.303046
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) −4.00000 −0.392232
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) −2.00000 −0.190693
\(111\) 0 0
\(112\) −2.00000 −0.188982
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) −2.00000 −0.181071
\(123\) 0 0
\(124\) −8.00000 −0.718421
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 4.00000 0.350823
\(131\) 18.0000 1.57267 0.786334 0.617802i \(-0.211977\pi\)
0.786334 + 0.617802i \(0.211977\pi\)
\(132\) 0 0
\(133\) 2.00000 0.173422
\(134\) −8.00000 −0.691095
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) 22.0000 1.87959 0.939793 0.341743i \(-0.111017\pi\)
0.939793 + 0.341743i \(0.111017\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 2.00000 0.169031
\(141\) 0 0
\(142\) −8.00000 −0.671345
\(143\) −8.00000 −0.668994
\(144\) 0 0
\(145\) 0 0
\(146\) −14.0000 −1.15865
\(147\) 0 0
\(148\) 8.00000 0.657596
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 1.00000 0.0811107
\(153\) 0 0
\(154\) −4.00000 −0.322329
\(155\) 8.00000 0.642575
\(156\) 0 0
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) 8.00000 0.630488
\(162\) 0 0
\(163\) 14.0000 1.09656 0.548282 0.836293i \(-0.315282\pi\)
0.548282 + 0.836293i \(0.315282\pi\)
\(164\) 8.00000 0.624695
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 2.00000 0.153393
\(171\) 0 0
\(172\) −6.00000 −0.457496
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 0 0
\(175\) −2.00000 −0.151186
\(176\) −2.00000 −0.150756
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 8.00000 0.592999
\(183\) 0 0
\(184\) 4.00000 0.294884
\(185\) −8.00000 −0.588172
\(186\) 0 0
\(187\) −4.00000 −0.292509
\(188\) 12.0000 0.875190
\(189\) 0 0
\(190\) −1.00000 −0.0725476
\(191\) 18.0000 1.30243 0.651217 0.758891i \(-0.274259\pi\)
0.651217 + 0.758891i \(0.274259\pi\)
\(192\) 0 0
\(193\) 24.0000 1.72756 0.863779 0.503871i \(-0.168091\pi\)
0.863779 + 0.503871i \(0.168091\pi\)
\(194\) 12.0000 0.861550
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) 2.00000 0.140720
\(203\) 0 0
\(204\) 0 0
\(205\) −8.00000 −0.558744
\(206\) −4.00000 −0.278693
\(207\) 0 0
\(208\) 4.00000 0.277350
\(209\) 2.00000 0.138343
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) 6.00000 0.409197
\(216\) 0 0
\(217\) 16.0000 1.08615
\(218\) −10.0000 −0.677285
\(219\) 0 0
\(220\) 2.00000 0.134840
\(221\) 8.00000 0.538138
\(222\) 0 0
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) 2.00000 0.133631
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) −4.00000 −0.263752
\(231\) 0 0
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) −12.0000 −0.782794
\(236\) 0 0
\(237\) 0 0
\(238\) 4.00000 0.259281
\(239\) −10.0000 −0.646846 −0.323423 0.946254i \(-0.604834\pi\)
−0.323423 + 0.946254i \(0.604834\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 7.00000 0.449977
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) 3.00000 0.191663
\(246\) 0 0
\(247\) −4.00000 −0.254514
\(248\) 8.00000 0.508001
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) −2.00000 −0.126239 −0.0631194 0.998006i \(-0.520105\pi\)
−0.0631194 + 0.998006i \(0.520105\pi\)
\(252\) 0 0
\(253\) 8.00000 0.502956
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) −16.0000 −0.994192
\(260\) −4.00000 −0.248069
\(261\) 0 0
\(262\) −18.0000 −1.11204
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) −2.00000 −0.122628
\(267\) 0 0
\(268\) 8.00000 0.488678
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) −22.0000 −1.32907
\(275\) −2.00000 −0.120605
\(276\) 0 0
\(277\) 18.0000 1.08152 0.540758 0.841178i \(-0.318138\pi\)
0.540758 + 0.841178i \(0.318138\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) −2.00000 −0.119523
\(281\) −12.0000 −0.715860 −0.357930 0.933748i \(-0.616517\pi\)
−0.357930 + 0.933748i \(0.616517\pi\)
\(282\) 0 0
\(283\) 14.0000 0.832214 0.416107 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) 8.00000 0.473050
\(287\) −16.0000 −0.944450
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) 14.0000 0.819288
\(293\) −14.0000 −0.817889 −0.408944 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −8.00000 −0.464991
\(297\) 0 0
\(298\) 10.0000 0.579284
\(299\) −16.0000 −0.925304
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) 8.00000 0.460348
\(303\) 0 0
\(304\) −1.00000 −0.0573539
\(305\) −2.00000 −0.114520
\(306\) 0 0
\(307\) −32.0000 −1.82634 −0.913168 0.407583i \(-0.866372\pi\)
−0.913168 + 0.407583i \(0.866372\pi\)
\(308\) 4.00000 0.227921
\(309\) 0 0
\(310\) −8.00000 −0.454369
\(311\) 18.0000 1.02069 0.510343 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) −18.0000 −1.01580
\(315\) 0 0
\(316\) 0 0
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) −8.00000 −0.445823
\(323\) −2.00000 −0.111283
\(324\) 0 0
\(325\) 4.00000 0.221880
\(326\) −14.0000 −0.775388
\(327\) 0 0
\(328\) −8.00000 −0.441726
\(329\) −24.0000 −1.32316
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) −4.00000 −0.219529
\(333\) 0 0
\(334\) 8.00000 0.437741
\(335\) −8.00000 −0.437087
\(336\) 0 0
\(337\) 28.0000 1.52526 0.762629 0.646837i \(-0.223908\pi\)
0.762629 + 0.646837i \(0.223908\pi\)
\(338\) −3.00000 −0.163178
\(339\) 0 0
\(340\) −2.00000 −0.108465
\(341\) 16.0000 0.866449
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 6.00000 0.323498
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 2.00000 0.106904
\(351\) 0 0
\(352\) 2.00000 0.106600
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) −8.00000 −0.424596
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 10.0000 0.527780 0.263890 0.964553i \(-0.414994\pi\)
0.263890 + 0.964553i \(0.414994\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −2.00000 −0.105118
\(363\) 0 0
\(364\) −8.00000 −0.419314
\(365\) −14.0000 −0.732793
\(366\) 0 0
\(367\) 18.0000 0.939592 0.469796 0.882775i \(-0.344327\pi\)
0.469796 + 0.882775i \(0.344327\pi\)
\(368\) −4.00000 −0.208514
\(369\) 0 0
\(370\) 8.00000 0.415900
\(371\) −12.0000 −0.623009
\(372\) 0 0
\(373\) −36.0000 −1.86401 −0.932005 0.362446i \(-0.881942\pi\)
−0.932005 + 0.362446i \(0.881942\pi\)
\(374\) 4.00000 0.206835
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 0 0
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 1.00000 0.0512989
\(381\) 0 0
\(382\) −18.0000 −0.920960
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 0 0
\(385\) −4.00000 −0.203859
\(386\) −24.0000 −1.22157
\(387\) 0 0
\(388\) −12.0000 −0.609208
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) 3.00000 0.151523
\(393\) 0 0
\(394\) 18.0000 0.906827
\(395\) 0 0
\(396\) 0 0
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) 20.0000 1.00251
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −12.0000 −0.599251 −0.299626 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) 0 0
\(403\) −32.0000 −1.59403
\(404\) −2.00000 −0.0995037
\(405\) 0 0
\(406\) 0 0
\(407\) −16.0000 −0.793091
\(408\) 0 0
\(409\) 30.0000 1.48340 0.741702 0.670729i \(-0.234019\pi\)
0.741702 + 0.670729i \(0.234019\pi\)
\(410\) 8.00000 0.395092
\(411\) 0 0
\(412\) 4.00000 0.197066
\(413\) 0 0
\(414\) 0 0
\(415\) 4.00000 0.196352
\(416\) −4.00000 −0.196116
\(417\) 0 0
\(418\) −2.00000 −0.0978232
\(419\) 30.0000 1.46560 0.732798 0.680446i \(-0.238214\pi\)
0.732798 + 0.680446i \(0.238214\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) −12.0000 −0.584151
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) −4.00000 −0.193574
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) −6.00000 −0.289346
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) −16.0000 −0.768911 −0.384455 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) −16.0000 −0.768025
\(435\) 0 0
\(436\) 10.0000 0.478913
\(437\) 4.00000 0.191346
\(438\) 0 0
\(439\) −40.0000 −1.90910 −0.954548 0.298057i \(-0.903661\pi\)
−0.954548 + 0.298057i \(0.903661\pi\)
\(440\) −2.00000 −0.0953463
\(441\) 0 0
\(442\) −8.00000 −0.380521
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −4.00000 −0.189405
\(447\) 0 0
\(448\) −2.00000 −0.0944911
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) −16.0000 −0.753411
\(452\) 6.00000 0.282216
\(453\) 0 0
\(454\) −12.0000 −0.563188
\(455\) 8.00000 0.375046
\(456\) 0 0
\(457\) 18.0000 0.842004 0.421002 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) −10.0000 −0.467269
\(459\) 0 0
\(460\) 4.00000 0.186501
\(461\) −2.00000 −0.0931493 −0.0465746 0.998915i \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) 0 0
\(463\) −6.00000 −0.278844 −0.139422 0.990233i \(-0.544524\pi\)
−0.139422 + 0.990233i \(0.544524\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) −8.00000 −0.370196 −0.185098 0.982720i \(-0.559260\pi\)
−0.185098 + 0.982720i \(0.559260\pi\)
\(468\) 0 0
\(469\) −16.0000 −0.738811
\(470\) 12.0000 0.553519
\(471\) 0 0
\(472\) 0 0
\(473\) 12.0000 0.551761
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) −4.00000 −0.183340
\(477\) 0 0
\(478\) 10.0000 0.457389
\(479\) 10.0000 0.456912 0.228456 0.973554i \(-0.426632\pi\)
0.228456 + 0.973554i \(0.426632\pi\)
\(480\) 0 0
\(481\) 32.0000 1.45907
\(482\) −2.00000 −0.0910975
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 12.0000 0.544892
\(486\) 0 0
\(487\) 28.0000 1.26880 0.634401 0.773004i \(-0.281247\pi\)
0.634401 + 0.773004i \(0.281247\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 0 0
\(490\) −3.00000 −0.135526
\(491\) −22.0000 −0.992846 −0.496423 0.868081i \(-0.665354\pi\)
−0.496423 + 0.868081i \(0.665354\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 4.00000 0.179969
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) −16.0000 −0.717698
\(498\) 0 0
\(499\) −40.0000 −1.79065 −0.895323 0.445418i \(-0.853055\pi\)
−0.895323 + 0.445418i \(0.853055\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 2.00000 0.0892644
\(503\) 16.0000 0.713405 0.356702 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) 0 0
\(505\) 2.00000 0.0889988
\(506\) −8.00000 −0.355643
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) −20.0000 −0.886484 −0.443242 0.896402i \(-0.646172\pi\)
−0.443242 + 0.896402i \(0.646172\pi\)
\(510\) 0 0
\(511\) −28.0000 −1.23865
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 18.0000 0.793946
\(515\) −4.00000 −0.176261
\(516\) 0 0
\(517\) −24.0000 −1.05552
\(518\) 16.0000 0.703000
\(519\) 0 0
\(520\) 4.00000 0.175412
\(521\) 28.0000 1.22670 0.613351 0.789810i \(-0.289821\pi\)
0.613351 + 0.789810i \(0.289821\pi\)
\(522\) 0 0
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) 18.0000 0.786334
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) −16.0000 −0.696971
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 6.00000 0.260623
\(531\) 0 0
\(532\) 2.00000 0.0867110
\(533\) 32.0000 1.38607
\(534\) 0 0
\(535\) −12.0000 −0.518805
\(536\) −8.00000 −0.345547
\(537\) 0 0
\(538\) 0 0
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) −12.0000 −0.515444
\(543\) 0 0
\(544\) −2.00000 −0.0857493
\(545\) −10.0000 −0.428353
\(546\) 0 0
\(547\) −32.0000 −1.36822 −0.684111 0.729378i \(-0.739809\pi\)
−0.684111 + 0.729378i \(0.739809\pi\)
\(548\) 22.0000 0.939793
\(549\) 0 0
\(550\) 2.00000 0.0852803
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) −18.0000 −0.764747
\(555\) 0 0
\(556\) 0 0
\(557\) 42.0000 1.77960 0.889799 0.456354i \(-0.150845\pi\)
0.889799 + 0.456354i \(0.150845\pi\)
\(558\) 0 0
\(559\) −24.0000 −1.01509
\(560\) 2.00000 0.0845154
\(561\) 0 0
\(562\) 12.0000 0.506189
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 0 0
\(565\) −6.00000 −0.252422
\(566\) −14.0000 −0.588464
\(567\) 0 0
\(568\) −8.00000 −0.335673
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 32.0000 1.33916 0.669579 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) −8.00000 −0.334497
\(573\) 0 0
\(574\) 16.0000 0.667827
\(575\) −4.00000 −0.166812
\(576\) 0 0
\(577\) −22.0000 −0.915872 −0.457936 0.888985i \(-0.651411\pi\)
−0.457936 + 0.888985i \(0.651411\pi\)
\(578\) 13.0000 0.540729
\(579\) 0 0
\(580\) 0 0
\(581\) 8.00000 0.331896
\(582\) 0 0
\(583\) −12.0000 −0.496989
\(584\) −14.0000 −0.579324
\(585\) 0 0
\(586\) 14.0000 0.578335
\(587\) −8.00000 −0.330195 −0.165098 0.986277i \(-0.552794\pi\)
−0.165098 + 0.986277i \(0.552794\pi\)
\(588\) 0 0
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) 0 0
\(592\) 8.00000 0.328798
\(593\) 46.0000 1.88899 0.944497 0.328521i \(-0.106550\pi\)
0.944497 + 0.328521i \(0.106550\pi\)
\(594\) 0 0
\(595\) 4.00000 0.163984
\(596\) −10.0000 −0.409616
\(597\) 0 0
\(598\) 16.0000 0.654289
\(599\) −20.0000 −0.817178 −0.408589 0.912719i \(-0.633979\pi\)
−0.408589 + 0.912719i \(0.633979\pi\)
\(600\) 0 0
\(601\) −18.0000 −0.734235 −0.367118 0.930175i \(-0.619655\pi\)
−0.367118 + 0.930175i \(0.619655\pi\)
\(602\) −12.0000 −0.489083
\(603\) 0 0
\(604\) −8.00000 −0.325515
\(605\) 7.00000 0.284590
\(606\) 0 0
\(607\) −12.0000 −0.487065 −0.243532 0.969893i \(-0.578306\pi\)
−0.243532 + 0.969893i \(0.578306\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) 2.00000 0.0809776
\(611\) 48.0000 1.94187
\(612\) 0 0
\(613\) −6.00000 −0.242338 −0.121169 0.992632i \(-0.538664\pi\)
−0.121169 + 0.992632i \(0.538664\pi\)
\(614\) 32.0000 1.29141
\(615\) 0 0
\(616\) −4.00000 −0.161165
\(617\) 42.0000 1.69086 0.845428 0.534089i \(-0.179345\pi\)
0.845428 + 0.534089i \(0.179345\pi\)
\(618\) 0 0
\(619\) −40.0000 −1.60774 −0.803868 0.594808i \(-0.797228\pi\)
−0.803868 + 0.594808i \(0.797228\pi\)
\(620\) 8.00000 0.321288
\(621\) 0 0
\(622\) −18.0000 −0.721734
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 6.00000 0.239808
\(627\) 0 0
\(628\) 18.0000 0.718278
\(629\) 16.0000 0.637962
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 18.0000 0.714871
\(635\) −8.00000 −0.317470
\(636\) 0 0
\(637\) −12.0000 −0.475457
\(638\) 0 0
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) −32.0000 −1.26392 −0.631962 0.774999i \(-0.717750\pi\)
−0.631962 + 0.774999i \(0.717750\pi\)
\(642\) 0 0
\(643\) 14.0000 0.552106 0.276053 0.961142i \(-0.410973\pi\)
0.276053 + 0.961142i \(0.410973\pi\)
\(644\) 8.00000 0.315244
\(645\) 0 0
\(646\) 2.00000 0.0786889
\(647\) 32.0000 1.25805 0.629025 0.777385i \(-0.283454\pi\)
0.629025 + 0.777385i \(0.283454\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −4.00000 −0.156893
\(651\) 0 0
\(652\) 14.0000 0.548282
\(653\) 46.0000 1.80012 0.900060 0.435767i \(-0.143523\pi\)
0.900060 + 0.435767i \(0.143523\pi\)
\(654\) 0 0
\(655\) −18.0000 −0.703318
\(656\) 8.00000 0.312348
\(657\) 0 0
\(658\) 24.0000 0.935617
\(659\) −20.0000 −0.779089 −0.389545 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) 0 0
\(661\) 42.0000 1.63361 0.816805 0.576913i \(-0.195743\pi\)
0.816805 + 0.576913i \(0.195743\pi\)
\(662\) −12.0000 −0.466393
\(663\) 0 0
\(664\) 4.00000 0.155230
\(665\) −2.00000 −0.0775567
\(666\) 0 0
\(667\) 0 0
\(668\) −8.00000 −0.309529
\(669\) 0 0
\(670\) 8.00000 0.309067
\(671\) −4.00000 −0.154418
\(672\) 0 0
\(673\) 4.00000 0.154189 0.0770943 0.997024i \(-0.475436\pi\)
0.0770943 + 0.997024i \(0.475436\pi\)
\(674\) −28.0000 −1.07852
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 0 0
\(679\) 24.0000 0.921035
\(680\) 2.00000 0.0766965
\(681\) 0 0
\(682\) −16.0000 −0.612672
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 0 0
\(685\) −22.0000 −0.840577
\(686\) −20.0000 −0.763604
\(687\) 0 0
\(688\) −6.00000 −0.228748
\(689\) 24.0000 0.914327
\(690\) 0 0
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) −14.0000 −0.532200
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) 0 0
\(696\) 0 0
\(697\) 16.0000 0.606043
\(698\) 10.0000 0.378506
\(699\) 0 0
\(700\) −2.00000 −0.0755929
\(701\) −2.00000 −0.0755390 −0.0377695 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) 0 0
\(703\) −8.00000 −0.301726
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) 4.00000 0.150435
\(708\) 0 0
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 8.00000 0.300235
\(711\) 0 0
\(712\) 0 0
\(713\) 32.0000 1.19841
\(714\) 0 0
\(715\) 8.00000 0.299183
\(716\) 0 0
\(717\) 0 0
\(718\) −10.0000 −0.373197
\(719\) −50.0000 −1.86469 −0.932343 0.361576i \(-0.882239\pi\)
−0.932343 + 0.361576i \(0.882239\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) −1.00000 −0.0372161
\(723\) 0 0
\(724\) 2.00000 0.0743294
\(725\) 0 0
\(726\) 0 0
\(727\) 18.0000 0.667583 0.333792 0.942647i \(-0.391672\pi\)
0.333792 + 0.942647i \(0.391672\pi\)
\(728\) 8.00000 0.296500
\(729\) 0 0
\(730\) 14.0000 0.518163
\(731\) −12.0000 −0.443836
\(732\) 0 0
\(733\) −26.0000 −0.960332 −0.480166 0.877178i \(-0.659424\pi\)
−0.480166 + 0.877178i \(0.659424\pi\)
\(734\) −18.0000 −0.664392
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) −16.0000 −0.589368
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) −8.00000 −0.294086
\(741\) 0 0
\(742\) 12.0000 0.440534
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) 10.0000 0.366372
\(746\) 36.0000 1.31805
\(747\) 0 0
\(748\) −4.00000 −0.146254
\(749\) −24.0000 −0.876941
\(750\) 0 0
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) 12.0000 0.437595
\(753\) 0 0
\(754\) 0 0
\(755\) 8.00000 0.291150
\(756\) 0 0
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) 20.0000 0.726433
\(759\) 0 0
\(760\) −1.00000 −0.0362738
\(761\) 18.0000 0.652499 0.326250 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) 0 0
\(763\) −20.0000 −0.724049
\(764\) 18.0000 0.651217
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) 0 0
\(768\) 0 0
\(769\) 50.0000 1.80305 0.901523 0.432731i \(-0.142450\pi\)
0.901523 + 0.432731i \(0.142450\pi\)
\(770\) 4.00000 0.144150
\(771\) 0 0
\(772\) 24.0000 0.863779
\(773\) −34.0000 −1.22290 −0.611448 0.791285i \(-0.709412\pi\)
−0.611448 + 0.791285i \(0.709412\pi\)
\(774\) 0 0
\(775\) −8.00000 −0.287368
\(776\) 12.0000 0.430775
\(777\) 0 0
\(778\) 30.0000 1.07555
\(779\) −8.00000 −0.286630
\(780\) 0 0
\(781\) −16.0000 −0.572525
\(782\) 8.00000 0.286079
\(783\) 0 0
\(784\) −3.00000 −0.107143
\(785\) −18.0000 −0.642448
\(786\) 0 0
\(787\) 28.0000 0.998092 0.499046 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) −18.0000 −0.641223
\(789\) 0 0
\(790\) 0 0
\(791\) −12.0000 −0.426671
\(792\) 0 0
\(793\) 8.00000 0.284088
\(794\) 22.0000 0.780751
\(795\) 0 0
\(796\) −20.0000 −0.708881
\(797\) 2.00000 0.0708436 0.0354218 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 12.0000 0.423735
\(803\) −28.0000 −0.988099
\(804\) 0 0
\(805\) −8.00000 −0.281963
\(806\) 32.0000 1.12715
\(807\) 0 0
\(808\) 2.00000 0.0703598
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) 12.0000 0.421377 0.210688 0.977553i \(-0.432429\pi\)
0.210688 + 0.977553i \(0.432429\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 16.0000 0.560800
\(815\) −14.0000 −0.490399
\(816\) 0 0
\(817\) 6.00000 0.209913
\(818\) −30.0000 −1.04893
\(819\) 0 0
\(820\) −8.00000 −0.279372
\(821\) 38.0000 1.32621 0.663105 0.748527i \(-0.269238\pi\)
0.663105 + 0.748527i \(0.269238\pi\)
\(822\) 0 0
\(823\) 34.0000 1.18517 0.592583 0.805510i \(-0.298108\pi\)
0.592583 + 0.805510i \(0.298108\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) 0 0
\(827\) −28.0000 −0.973655 −0.486828 0.873498i \(-0.661846\pi\)
−0.486828 + 0.873498i \(0.661846\pi\)
\(828\) 0 0
\(829\) 10.0000 0.347314 0.173657 0.984806i \(-0.444442\pi\)
0.173657 + 0.984806i \(0.444442\pi\)
\(830\) −4.00000 −0.138842
\(831\) 0 0
\(832\) 4.00000 0.138675
\(833\) −6.00000 −0.207888
\(834\) 0 0
\(835\) 8.00000 0.276851
\(836\) 2.00000 0.0691714
\(837\) 0 0
\(838\) −30.0000 −1.03633
\(839\) 40.0000 1.38095 0.690477 0.723355i \(-0.257401\pi\)
0.690477 + 0.723355i \(0.257401\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) −22.0000 −0.758170
\(843\) 0 0
\(844\) 12.0000 0.413057
\(845\) −3.00000 −0.103203
\(846\) 0 0
\(847\) 14.0000 0.481046
\(848\) 6.00000 0.206041
\(849\) 0 0
\(850\) −2.00000 −0.0685994
\(851\) −32.0000 −1.09695
\(852\) 0 0
\(853\) 14.0000 0.479351 0.239675 0.970853i \(-0.422959\pi\)
0.239675 + 0.970853i \(0.422959\pi\)
\(854\) 4.00000 0.136877
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) 42.0000 1.43469 0.717346 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 6.00000 0.204598
\(861\) 0 0
\(862\) 12.0000 0.408722
\(863\) 16.0000 0.544646 0.272323 0.962206i \(-0.412208\pi\)
0.272323 + 0.962206i \(0.412208\pi\)
\(864\) 0 0
\(865\) 14.0000 0.476014
\(866\) 16.0000 0.543702
\(867\) 0 0
\(868\) 16.0000 0.543075
\(869\) 0 0
\(870\) 0 0
\(871\) 32.0000 1.08428
\(872\) −10.0000 −0.338643
\(873\) 0 0
\(874\) −4.00000 −0.135302
\(875\) 2.00000 0.0676123
\(876\) 0 0
\(877\) −12.0000 −0.405211 −0.202606 0.979260i \(-0.564941\pi\)
−0.202606 + 0.979260i \(0.564941\pi\)
\(878\) 40.0000 1.34993
\(879\) 0 0
\(880\) 2.00000 0.0674200
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 0 0
\(883\) −26.0000 −0.874970 −0.437485 0.899226i \(-0.644131\pi\)
−0.437485 + 0.899226i \(0.644131\pi\)
\(884\) 8.00000 0.269069
\(885\) 0 0
\(886\) 24.0000 0.806296
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 0 0
\(889\) −16.0000 −0.536623
\(890\) 0 0
\(891\) 0 0
\(892\) 4.00000 0.133930
\(893\) −12.0000 −0.401565
\(894\) 0 0
\(895\) 0 0
\(896\) 2.00000 0.0668153
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) 16.0000 0.532742
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) −2.00000 −0.0664822
\(906\) 0 0
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 12.0000 0.398234
\(909\) 0 0
\(910\) −8.00000 −0.265197
\(911\) −32.0000 −1.06021 −0.530104 0.847933i \(-0.677847\pi\)
−0.530104 + 0.847933i \(0.677847\pi\)
\(912\) 0 0
\(913\) 8.00000 0.264761
\(914\) −18.0000 −0.595387
\(915\) 0 0
\(916\) 10.0000 0.330409
\(917\) −36.0000 −1.18882
\(918\) 0 0
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) −4.00000 −0.131876
\(921\) 0 0
\(922\) 2.00000 0.0658665
\(923\) 32.0000 1.05329
\(924\) 0 0
\(925\) 8.00000 0.263038
\(926\) 6.00000 0.197172
\(927\) 0 0
\(928\) 0 0
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) 3.00000 0.0983210
\(932\) 6.00000 0.196537
\(933\) 0 0
\(934\) 8.00000 0.261768
\(935\) 4.00000 0.130814
\(936\) 0 0
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 16.0000 0.522419
\(939\) 0 0
\(940\) −12.0000 −0.391397
\(941\) 8.00000 0.260793 0.130396 0.991462i \(-0.458375\pi\)
0.130396 + 0.991462i \(0.458375\pi\)
\(942\) 0 0
\(943\) −32.0000 −1.04206
\(944\) 0 0
\(945\) 0 0
\(946\) −12.0000 −0.390154
\(947\) 52.0000 1.68977 0.844886 0.534946i \(-0.179668\pi\)
0.844886 + 0.534946i \(0.179668\pi\)
\(948\) 0 0
\(949\) 56.0000 1.81784
\(950\) 1.00000 0.0324443
\(951\) 0 0
\(952\) 4.00000 0.129641
\(953\) −34.0000 −1.10137 −0.550684 0.834714i \(-0.685633\pi\)
−0.550684 + 0.834714i \(0.685633\pi\)
\(954\) 0 0
\(955\) −18.0000 −0.582466
\(956\) −10.0000 −0.323423
\(957\) 0 0
\(958\) −10.0000 −0.323085
\(959\) −44.0000 −1.42083
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) −32.0000 −1.03172
\(963\) 0 0
\(964\) 2.00000 0.0644157
\(965\) −24.0000 −0.772587
\(966\) 0 0
\(967\) 58.0000 1.86515 0.932577 0.360971i \(-0.117555\pi\)
0.932577 + 0.360971i \(0.117555\pi\)
\(968\) 7.00000 0.224989
\(969\) 0 0
\(970\) −12.0000 −0.385297
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −28.0000 −0.897178
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) −38.0000 −1.21573 −0.607864 0.794041i \(-0.707973\pi\)
−0.607864 + 0.794041i \(0.707973\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 3.00000 0.0958315
\(981\) 0 0
\(982\) 22.0000 0.702048
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 0 0
\(985\) 18.0000 0.573528
\(986\) 0 0
\(987\) 0 0
\(988\) −4.00000 −0.127257
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) 8.00000 0.254000
\(993\) 0 0
\(994\) 16.0000 0.507489
\(995\) 20.0000 0.634043
\(996\) 0 0
\(997\) −62.0000 −1.96356 −0.981780 0.190022i \(-0.939144\pi\)
−0.981780 + 0.190022i \(0.939144\pi\)
\(998\) 40.0000 1.26618
\(999\) 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 1710.2.a.b.1.1 1
3.2 odd 2 570.2.a.l.1.1 1
5.4 even 2 8550.2.a.bf.1.1 1
12.11 even 2 4560.2.a.o.1.1 1
15.2 even 4 2850.2.d.q.799.2 2
15.8 even 4 2850.2.d.q.799.1 2
15.14 odd 2 2850.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.a.l.1.1 1 3.2 odd 2
1710.2.a.b.1.1 1 1.1 even 1 trivial
2850.2.a.e.1.1 1 15.14 odd 2
2850.2.d.q.799.1 2 15.8 even 4
2850.2.d.q.799.2 2 15.2 even 4
4560.2.a.o.1.1 1 12.11 even 2
8550.2.a.bf.1.1 1 5.4 even 2