Properties

Label 3850.2.a.bb.1.1
Level $3850$
Weight $2$
Character 3850.1
Self dual yes
Analytic conductor $30.742$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.7424047782\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3850.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{11} +2.00000 q^{12} +1.00000 q^{14} +1.00000 q^{16} +1.00000 q^{18} +2.00000 q^{21} +1.00000 q^{22} +4.00000 q^{23} +2.00000 q^{24} -4.00000 q^{27} +1.00000 q^{28} +2.00000 q^{29} -2.00000 q^{31} +1.00000 q^{32} +2.00000 q^{33} +1.00000 q^{36} +6.00000 q^{37} +8.00000 q^{41} +2.00000 q^{42} +12.0000 q^{43} +1.00000 q^{44} +4.00000 q^{46} +6.00000 q^{47} +2.00000 q^{48} +1.00000 q^{49} +6.00000 q^{53} -4.00000 q^{54} +1.00000 q^{56} +2.00000 q^{58} -10.0000 q^{59} -4.00000 q^{61} -2.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +2.00000 q^{66} +8.00000 q^{67} +8.00000 q^{69} -4.00000 q^{71} +1.00000 q^{72} +4.00000 q^{73} +6.00000 q^{74} +1.00000 q^{77} -16.0000 q^{79} -11.0000 q^{81} +8.00000 q^{82} +2.00000 q^{84} +12.0000 q^{86} +4.00000 q^{87} +1.00000 q^{88} -6.00000 q^{89} +4.00000 q^{92} -4.00000 q^{93} +6.00000 q^{94} +2.00000 q^{96} -14.0000 q^{97} +1.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 2.00000 1.15470 0.577350 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 2.00000 0.577350
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 1.00000 0.235702
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 1.00000 0.213201
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 2.00000 0.408248
\(25\) 0 0
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 1.00000 0.188982
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.00000 0.348155
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 8.00000 1.24939 0.624695 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 2.00000 0.308607
\(43\) 12.0000 1.82998 0.914991 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 2.00000 0.288675
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −4.00000 −0.544331
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) 2.00000 0.262613
\(59\) −10.0000 −1.30189 −0.650945 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) −2.00000 −0.254000
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 0 0
\(69\) 8.00000 0.963087
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 1.00000 0.117851
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 6.00000 0.697486
\(75\) 0 0
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 8.00000 0.883452
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) 12.0000 1.29399
\(87\) 4.00000 0.428845
\(88\) 1.00000 0.106600
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 4.00000 0.417029
\(93\) −4.00000 −0.414781
\(94\) 6.00000 0.618853
\(95\) 0 0
\(96\) 2.00000 0.204124
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 1.00000 0.101015
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) −4.00000 −0.384900
\(109\) −6.00000 −0.574696 −0.287348 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) 0 0
\(111\) 12.0000 1.13899
\(112\) 1.00000 0.0944911
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) 0 0
\(118\) −10.0000 −0.920575
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −4.00000 −0.362143
\(123\) 16.0000 1.44267
\(124\) −2.00000 −0.179605
\(125\) 0 0
\(126\) 1.00000 0.0890871
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 1.00000 0.0883883
\(129\) 24.0000 2.11308
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 2.00000 0.174078
\(133\) 0 0
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) 0 0
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 8.00000 0.681005
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 0 0
\(141\) 12.0000 1.01058
\(142\) −4.00000 −0.335673
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 4.00000 0.331042
\(147\) 2.00000 0.164957
\(148\) 6.00000 0.493197
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 1.00000 0.0805823
\(155\) 0 0
\(156\) 0 0
\(157\) −6.00000 −0.478852 −0.239426 0.970915i \(-0.576959\pi\)
−0.239426 + 0.970915i \(0.576959\pi\)
\(158\) −16.0000 −1.27289
\(159\) 12.0000 0.951662
\(160\) 0 0
\(161\) 4.00000 0.315244
\(162\) −11.0000 −0.864242
\(163\) 8.00000 0.626608 0.313304 0.949653i \(-0.398564\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) 8.00000 0.624695
\(165\) 0 0
\(166\) 0 0
\(167\) −16.0000 −1.23812 −0.619059 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 2.00000 0.154303
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 12.0000 0.914991
\(173\) −24.0000 −1.82469 −0.912343 0.409426i \(-0.865729\pi\)
−0.912343 + 0.409426i \(0.865729\pi\)
\(174\) 4.00000 0.303239
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) −20.0000 −1.50329
\(178\) −6.00000 −0.449719
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) −8.00000 −0.591377
\(184\) 4.00000 0.294884
\(185\) 0 0
\(186\) −4.00000 −0.293294
\(187\) 0 0
\(188\) 6.00000 0.437595
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 2.00000 0.144338
\(193\) 22.0000 1.58359 0.791797 0.610784i \(-0.209146\pi\)
0.791797 + 0.610784i \(0.209146\pi\)
\(194\) −14.0000 −1.00514
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −14.0000 −0.997459 −0.498729 0.866758i \(-0.666200\pi\)
−0.498729 + 0.866758i \(0.666200\pi\)
\(198\) 1.00000 0.0710669
\(199\) −6.00000 −0.425329 −0.212664 0.977125i \(-0.568214\pi\)
−0.212664 + 0.977125i \(0.568214\pi\)
\(200\) 0 0
\(201\) 16.0000 1.12855
\(202\) 0 0
\(203\) 2.00000 0.140372
\(204\) 0 0
\(205\) 0 0
\(206\) 14.0000 0.975426
\(207\) 4.00000 0.278019
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 6.00000 0.412082
\(213\) −8.00000 −0.548151
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) −4.00000 −0.272166
\(217\) −2.00000 −0.135769
\(218\) −6.00000 −0.406371
\(219\) 8.00000 0.540590
\(220\) 0 0
\(221\) 0 0
\(222\) 12.0000 0.805387
\(223\) 6.00000 0.401790 0.200895 0.979613i \(-0.435615\pi\)
0.200895 + 0.979613i \(0.435615\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −2.00000 −0.133038
\(227\) −4.00000 −0.265489 −0.132745 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) 0 0
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 0 0
\(231\) 2.00000 0.131590
\(232\) 2.00000 0.131306
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −10.0000 −0.650945
\(237\) −32.0000 −2.07862
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −16.0000 −1.03065 −0.515325 0.856995i \(-0.672329\pi\)
−0.515325 + 0.856995i \(0.672329\pi\)
\(242\) 1.00000 0.0642824
\(243\) −10.0000 −0.641500
\(244\) −4.00000 −0.256074
\(245\) 0 0
\(246\) 16.0000 1.02012
\(247\) 0 0
\(248\) −2.00000 −0.127000
\(249\) 0 0
\(250\) 0 0
\(251\) 10.0000 0.631194 0.315597 0.948893i \(-0.397795\pi\)
0.315597 + 0.948893i \(0.397795\pi\)
\(252\) 1.00000 0.0629941
\(253\) 4.00000 0.251478
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 14.0000 0.873296 0.436648 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) 24.0000 1.49417
\(259\) 6.00000 0.372822
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) 4.00000 0.247121
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 2.00000 0.123091
\(265\) 0 0
\(266\) 0 0
\(267\) −12.0000 −0.734388
\(268\) 8.00000 0.488678
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) 8.00000 0.481543
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 8.00000 0.479808
\(279\) −2.00000 −0.119737
\(280\) 0 0
\(281\) −2.00000 −0.119310 −0.0596550 0.998219i \(-0.519000\pi\)
−0.0596550 + 0.998219i \(0.519000\pi\)
\(282\) 12.0000 0.714590
\(283\) −28.0000 −1.66443 −0.832214 0.554455i \(-0.812927\pi\)
−0.832214 + 0.554455i \(0.812927\pi\)
\(284\) −4.00000 −0.237356
\(285\) 0 0
\(286\) 0 0
\(287\) 8.00000 0.472225
\(288\) 1.00000 0.0589256
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) −28.0000 −1.64139
\(292\) 4.00000 0.234082
\(293\) 4.00000 0.233682 0.116841 0.993151i \(-0.462723\pi\)
0.116841 + 0.993151i \(0.462723\pi\)
\(294\) 2.00000 0.116642
\(295\) 0 0
\(296\) 6.00000 0.348743
\(297\) −4.00000 −0.232104
\(298\) 6.00000 0.347571
\(299\) 0 0
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) −8.00000 −0.460348
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −28.0000 −1.59804 −0.799022 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) 1.00000 0.0569803
\(309\) 28.0000 1.59286
\(310\) 0 0
\(311\) −6.00000 −0.340229 −0.170114 0.985424i \(-0.554414\pi\)
−0.170114 + 0.985424i \(0.554414\pi\)
\(312\) 0 0
\(313\) −2.00000 −0.113047 −0.0565233 0.998401i \(-0.518002\pi\)
−0.0565233 + 0.998401i \(0.518002\pi\)
\(314\) −6.00000 −0.338600
\(315\) 0 0
\(316\) −16.0000 −0.900070
\(317\) 2.00000 0.112331 0.0561656 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) 12.0000 0.672927
\(319\) 2.00000 0.111979
\(320\) 0 0
\(321\) −24.0000 −1.33955
\(322\) 4.00000 0.222911
\(323\) 0 0
\(324\) −11.0000 −0.611111
\(325\) 0 0
\(326\) 8.00000 0.443079
\(327\) −12.0000 −0.663602
\(328\) 8.00000 0.441726
\(329\) 6.00000 0.330791
\(330\) 0 0
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) 0 0
\(333\) 6.00000 0.328798
\(334\) −16.0000 −0.875481
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) −13.0000 −0.707107
\(339\) −4.00000 −0.217250
\(340\) 0 0
\(341\) −2.00000 −0.108306
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 12.0000 0.646997
\(345\) 0 0
\(346\) −24.0000 −1.29025
\(347\) −28.0000 −1.50312 −0.751559 0.659665i \(-0.770698\pi\)
−0.751559 + 0.659665i \(0.770698\pi\)
\(348\) 4.00000 0.214423
\(349\) −20.0000 −1.07058 −0.535288 0.844670i \(-0.679797\pi\)
−0.535288 + 0.844670i \(0.679797\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) −20.0000 −1.06299
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) −8.00000 −0.422224 −0.211112 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) −6.00000 −0.315353
\(363\) 2.00000 0.104973
\(364\) 0 0
\(365\) 0 0
\(366\) −8.00000 −0.418167
\(367\) 22.0000 1.14839 0.574195 0.818718i \(-0.305315\pi\)
0.574195 + 0.818718i \(0.305315\pi\)
\(368\) 4.00000 0.208514
\(369\) 8.00000 0.416463
\(370\) 0 0
\(371\) 6.00000 0.311504
\(372\) −4.00000 −0.207390
\(373\) −14.0000 −0.724893 −0.362446 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 6.00000 0.309426
\(377\) 0 0
\(378\) −4.00000 −0.205738
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) −8.00000 −0.409316
\(383\) 2.00000 0.102195 0.0510976 0.998694i \(-0.483728\pi\)
0.0510976 + 0.998694i \(0.483728\pi\)
\(384\) 2.00000 0.102062
\(385\) 0 0
\(386\) 22.0000 1.11977
\(387\) 12.0000 0.609994
\(388\) −14.0000 −0.710742
\(389\) 22.0000 1.11544 0.557722 0.830028i \(-0.311675\pi\)
0.557722 + 0.830028i \(0.311675\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 1.00000 0.0505076
\(393\) 8.00000 0.403547
\(394\) −14.0000 −0.705310
\(395\) 0 0
\(396\) 1.00000 0.0502519
\(397\) 34.0000 1.70641 0.853206 0.521575i \(-0.174655\pi\)
0.853206 + 0.521575i \(0.174655\pi\)
\(398\) −6.00000 −0.300753
\(399\) 0 0
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 16.0000 0.798007
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 2.00000 0.0992583
\(407\) 6.00000 0.297409
\(408\) 0 0
\(409\) 4.00000 0.197787 0.0988936 0.995098i \(-0.468470\pi\)
0.0988936 + 0.995098i \(0.468470\pi\)
\(410\) 0 0
\(411\) −12.0000 −0.591916
\(412\) 14.0000 0.689730
\(413\) −10.0000 −0.492068
\(414\) 4.00000 0.196589
\(415\) 0 0
\(416\) 0 0
\(417\) 16.0000 0.783523
\(418\) 0 0
\(419\) −26.0000 −1.27018 −0.635092 0.772437i \(-0.719038\pi\)
−0.635092 + 0.772437i \(0.719038\pi\)
\(420\) 0 0
\(421\) 18.0000 0.877266 0.438633 0.898666i \(-0.355463\pi\)
0.438633 + 0.898666i \(0.355463\pi\)
\(422\) −12.0000 −0.584151
\(423\) 6.00000 0.291730
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) −8.00000 −0.387601
\(427\) −4.00000 −0.193574
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 0 0
\(431\) 8.00000 0.385346 0.192673 0.981263i \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) −4.00000 −0.192450
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) −2.00000 −0.0960031
\(435\) 0 0
\(436\) −6.00000 −0.287348
\(437\) 0 0
\(438\) 8.00000 0.382255
\(439\) 4.00000 0.190910 0.0954548 0.995434i \(-0.469569\pi\)
0.0954548 + 0.995434i \(0.469569\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 20.0000 0.950229 0.475114 0.879924i \(-0.342407\pi\)
0.475114 + 0.879924i \(0.342407\pi\)
\(444\) 12.0000 0.569495
\(445\) 0 0
\(446\) 6.00000 0.284108
\(447\) 12.0000 0.567581
\(448\) 1.00000 0.0472456
\(449\) −10.0000 −0.471929 −0.235965 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) 0 0
\(451\) 8.00000 0.376705
\(452\) −2.00000 −0.0940721
\(453\) −16.0000 −0.751746
\(454\) −4.00000 −0.187729
\(455\) 0 0
\(456\) 0 0
\(457\) −26.0000 −1.21623 −0.608114 0.793849i \(-0.708074\pi\)
−0.608114 + 0.793849i \(0.708074\pi\)
\(458\) 2.00000 0.0934539
\(459\) 0 0
\(460\) 0 0
\(461\) −20.0000 −0.931493 −0.465746 0.884918i \(-0.654214\pi\)
−0.465746 + 0.884918i \(0.654214\pi\)
\(462\) 2.00000 0.0930484
\(463\) −4.00000 −0.185896 −0.0929479 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) 18.0000 0.833834
\(467\) 30.0000 1.38823 0.694117 0.719862i \(-0.255795\pi\)
0.694117 + 0.719862i \(0.255795\pi\)
\(468\) 0 0
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) −12.0000 −0.552931
\(472\) −10.0000 −0.460287
\(473\) 12.0000 0.551761
\(474\) −32.0000 −1.46981
\(475\) 0 0
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 0 0
\(479\) 20.0000 0.913823 0.456912 0.889512i \(-0.348956\pi\)
0.456912 + 0.889512i \(0.348956\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −16.0000 −0.728780
\(483\) 8.00000 0.364013
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) −10.0000 −0.453609
\(487\) −20.0000 −0.906287 −0.453143 0.891438i \(-0.649697\pi\)
−0.453143 + 0.891438i \(0.649697\pi\)
\(488\) −4.00000 −0.181071
\(489\) 16.0000 0.723545
\(490\) 0 0
\(491\) 36.0000 1.62466 0.812329 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) 16.0000 0.721336
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −2.00000 −0.0898027
\(497\) −4.00000 −0.179425
\(498\) 0 0
\(499\) −40.0000 −1.79065 −0.895323 0.445418i \(-0.853055\pi\)
−0.895323 + 0.445418i \(0.853055\pi\)
\(500\) 0 0
\(501\) −32.0000 −1.42965
\(502\) 10.0000 0.446322
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 1.00000 0.0445435
\(505\) 0 0
\(506\) 4.00000 0.177822
\(507\) −26.0000 −1.15470
\(508\) −8.00000 −0.354943
\(509\) 18.0000 0.797836 0.398918 0.916987i \(-0.369386\pi\)
0.398918 + 0.916987i \(0.369386\pi\)
\(510\) 0 0
\(511\) 4.00000 0.176950
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 14.0000 0.617514
\(515\) 0 0
\(516\) 24.0000 1.05654
\(517\) 6.00000 0.263880
\(518\) 6.00000 0.263625
\(519\) −48.0000 −2.10697
\(520\) 0 0
\(521\) −34.0000 −1.48957 −0.744784 0.667306i \(-0.767447\pi\)
−0.744784 + 0.667306i \(0.767447\pi\)
\(522\) 2.00000 0.0875376
\(523\) −28.0000 −1.22435 −0.612177 0.790721i \(-0.709706\pi\)
−0.612177 + 0.790721i \(0.709706\pi\)
\(524\) 4.00000 0.174741
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) 0 0
\(528\) 2.00000 0.0870388
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −10.0000 −0.433963
\(532\) 0 0
\(533\) 0 0
\(534\) −12.0000 −0.519291
\(535\) 0 0
\(536\) 8.00000 0.345547
\(537\) 24.0000 1.03568
\(538\) 10.0000 0.431131
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 20.0000 0.859074
\(543\) −12.0000 −0.514969
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 44.0000 1.88130 0.940652 0.339372i \(-0.110215\pi\)
0.940652 + 0.339372i \(0.110215\pi\)
\(548\) −6.00000 −0.256307
\(549\) −4.00000 −0.170716
\(550\) 0 0
\(551\) 0 0
\(552\) 8.00000 0.340503
\(553\) −16.0000 −0.680389
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) 8.00000 0.339276
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) −2.00000 −0.0846668
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −2.00000 −0.0843649
\(563\) −32.0000 −1.34864 −0.674320 0.738440i \(-0.735563\pi\)
−0.674320 + 0.738440i \(0.735563\pi\)
\(564\) 12.0000 0.505291
\(565\) 0 0
\(566\) −28.0000 −1.17693
\(567\) −11.0000 −0.461957
\(568\) −4.00000 −0.167836
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 0 0
\(573\) −16.0000 −0.668410
\(574\) 8.00000 0.333914
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 26.0000 1.08239 0.541197 0.840896i \(-0.317971\pi\)
0.541197 + 0.840896i \(0.317971\pi\)
\(578\) −17.0000 −0.707107
\(579\) 44.0000 1.82858
\(580\) 0 0
\(581\) 0 0
\(582\) −28.0000 −1.16064
\(583\) 6.00000 0.248495
\(584\) 4.00000 0.165521
\(585\) 0 0
\(586\) 4.00000 0.165238
\(587\) −2.00000 −0.0825488 −0.0412744 0.999148i \(-0.513142\pi\)
−0.0412744 + 0.999148i \(0.513142\pi\)
\(588\) 2.00000 0.0824786
\(589\) 0 0
\(590\) 0 0
\(591\) −28.0000 −1.15177
\(592\) 6.00000 0.246598
\(593\) 12.0000 0.492781 0.246390 0.969171i \(-0.420755\pi\)
0.246390 + 0.969171i \(0.420755\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) −12.0000 −0.491127
\(598\) 0 0
\(599\) 20.0000 0.817178 0.408589 0.912719i \(-0.366021\pi\)
0.408589 + 0.912719i \(0.366021\pi\)
\(600\) 0 0
\(601\) −8.00000 −0.326327 −0.163163 0.986599i \(-0.552170\pi\)
−0.163163 + 0.986599i \(0.552170\pi\)
\(602\) 12.0000 0.489083
\(603\) 8.00000 0.325785
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) 0 0
\(607\) −24.0000 −0.974130 −0.487065 0.873366i \(-0.661933\pi\)
−0.487065 + 0.873366i \(0.661933\pi\)
\(608\) 0 0
\(609\) 4.00000 0.162088
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 6.00000 0.242338 0.121169 0.992632i \(-0.461336\pi\)
0.121169 + 0.992632i \(0.461336\pi\)
\(614\) −28.0000 −1.12999
\(615\) 0 0
\(616\) 1.00000 0.0402911
\(617\) 42.0000 1.69086 0.845428 0.534089i \(-0.179345\pi\)
0.845428 + 0.534089i \(0.179345\pi\)
\(618\) 28.0000 1.12633
\(619\) −22.0000 −0.884255 −0.442127 0.896952i \(-0.645776\pi\)
−0.442127 + 0.896952i \(0.645776\pi\)
\(620\) 0 0
\(621\) −16.0000 −0.642058
\(622\) −6.00000 −0.240578
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) 0 0
\(626\) −2.00000 −0.0799361
\(627\) 0 0
\(628\) −6.00000 −0.239426
\(629\) 0 0
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) −16.0000 −0.636446
\(633\) −24.0000 −0.953914
\(634\) 2.00000 0.0794301
\(635\) 0 0
\(636\) 12.0000 0.475831
\(637\) 0 0
\(638\) 2.00000 0.0791808
\(639\) −4.00000 −0.158238
\(640\) 0 0
\(641\) 46.0000 1.81689 0.908445 0.418004i \(-0.137270\pi\)
0.908445 + 0.418004i \(0.137270\pi\)
\(642\) −24.0000 −0.947204
\(643\) −34.0000 −1.34083 −0.670415 0.741987i \(-0.733884\pi\)
−0.670415 + 0.741987i \(0.733884\pi\)
\(644\) 4.00000 0.157622
\(645\) 0 0
\(646\) 0 0
\(647\) 42.0000 1.65119 0.825595 0.564263i \(-0.190840\pi\)
0.825595 + 0.564263i \(0.190840\pi\)
\(648\) −11.0000 −0.432121
\(649\) −10.0000 −0.392534
\(650\) 0 0
\(651\) −4.00000 −0.156772
\(652\) 8.00000 0.313304
\(653\) −22.0000 −0.860927 −0.430463 0.902608i \(-0.641650\pi\)
−0.430463 + 0.902608i \(0.641650\pi\)
\(654\) −12.0000 −0.469237
\(655\) 0 0
\(656\) 8.00000 0.312348
\(657\) 4.00000 0.156055
\(658\) 6.00000 0.233904
\(659\) −20.0000 −0.779089 −0.389545 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) 0 0
\(661\) 46.0000 1.78919 0.894596 0.446875i \(-0.147463\pi\)
0.894596 + 0.446875i \(0.147463\pi\)
\(662\) 28.0000 1.08825
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 6.00000 0.232495
\(667\) 8.00000 0.309761
\(668\) −16.0000 −0.619059
\(669\) 12.0000 0.463947
\(670\) 0 0
\(671\) −4.00000 −0.154418
\(672\) 2.00000 0.0771517
\(673\) −14.0000 −0.539660 −0.269830 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) −14.0000 −0.539260
\(675\) 0 0
\(676\) −13.0000 −0.500000
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) −4.00000 −0.153619
\(679\) −14.0000 −0.537271
\(680\) 0 0
\(681\) −8.00000 −0.306561
\(682\) −2.00000 −0.0765840
\(683\) −20.0000 −0.765279 −0.382639 0.923898i \(-0.624985\pi\)
−0.382639 + 0.923898i \(0.624985\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) 4.00000 0.152610
\(688\) 12.0000 0.457496
\(689\) 0 0
\(690\) 0 0
\(691\) 6.00000 0.228251 0.114125 0.993466i \(-0.463593\pi\)
0.114125 + 0.993466i \(0.463593\pi\)
\(692\) −24.0000 −0.912343
\(693\) 1.00000 0.0379869
\(694\) −28.0000 −1.06287
\(695\) 0 0
\(696\) 4.00000 0.151620
\(697\) 0 0
\(698\) −20.0000 −0.757011
\(699\) 36.0000 1.36165
\(700\) 0 0
\(701\) −14.0000 −0.528773 −0.264386 0.964417i \(-0.585169\pi\)
−0.264386 + 0.964417i \(0.585169\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) 0 0
\(708\) −20.0000 −0.751646
\(709\) 30.0000 1.12667 0.563337 0.826227i \(-0.309517\pi\)
0.563337 + 0.826227i \(0.309517\pi\)
\(710\) 0 0
\(711\) −16.0000 −0.600047
\(712\) −6.00000 −0.224860
\(713\) −8.00000 −0.299602
\(714\) 0 0
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) −8.00000 −0.298557
\(719\) −34.0000 −1.26799 −0.633993 0.773339i \(-0.718585\pi\)
−0.633993 + 0.773339i \(0.718585\pi\)
\(720\) 0 0
\(721\) 14.0000 0.521387
\(722\) −19.0000 −0.707107
\(723\) −32.0000 −1.19009
\(724\) −6.00000 −0.222988
\(725\) 0 0
\(726\) 2.00000 0.0742270
\(727\) −14.0000 −0.519231 −0.259616 0.965712i \(-0.583596\pi\)
−0.259616 + 0.965712i \(0.583596\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 0 0
\(732\) −8.00000 −0.295689
\(733\) −4.00000 −0.147743 −0.0738717 0.997268i \(-0.523536\pi\)
−0.0738717 + 0.997268i \(0.523536\pi\)
\(734\) 22.0000 0.812035
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) 8.00000 0.294684
\(738\) 8.00000 0.294484
\(739\) 12.0000 0.441427 0.220714 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 6.00000 0.220267
\(743\) 8.00000 0.293492 0.146746 0.989174i \(-0.453120\pi\)
0.146746 + 0.989174i \(0.453120\pi\)
\(744\) −4.00000 −0.146647
\(745\) 0 0
\(746\) −14.0000 −0.512576
\(747\) 0 0
\(748\) 0 0
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) 52.0000 1.89751 0.948753 0.316017i \(-0.102346\pi\)
0.948753 + 0.316017i \(0.102346\pi\)
\(752\) 6.00000 0.218797
\(753\) 20.0000 0.728841
\(754\) 0 0
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) 26.0000 0.944986 0.472493 0.881334i \(-0.343354\pi\)
0.472493 + 0.881334i \(0.343354\pi\)
\(758\) −16.0000 −0.581146
\(759\) 8.00000 0.290382
\(760\) 0 0
\(761\) 12.0000 0.435000 0.217500 0.976060i \(-0.430210\pi\)
0.217500 + 0.976060i \(0.430210\pi\)
\(762\) −16.0000 −0.579619
\(763\) −6.00000 −0.217215
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) 2.00000 0.0722629
\(767\) 0 0
\(768\) 2.00000 0.0721688
\(769\) 12.0000 0.432731 0.216366 0.976312i \(-0.430580\pi\)
0.216366 + 0.976312i \(0.430580\pi\)
\(770\) 0 0
\(771\) 28.0000 1.00840
\(772\) 22.0000 0.791797
\(773\) −18.0000 −0.647415 −0.323708 0.946157i \(-0.604929\pi\)
−0.323708 + 0.946157i \(0.604929\pi\)
\(774\) 12.0000 0.431331
\(775\) 0 0
\(776\) −14.0000 −0.502571
\(777\) 12.0000 0.430498
\(778\) 22.0000 0.788738
\(779\) 0 0
\(780\) 0 0
\(781\) −4.00000 −0.143131
\(782\) 0 0
\(783\) −8.00000 −0.285897
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) 8.00000 0.285351
\(787\) −48.0000 −1.71102 −0.855508 0.517790i \(-0.826755\pi\)
−0.855508 + 0.517790i \(0.826755\pi\)
\(788\) −14.0000 −0.498729
\(789\) 48.0000 1.70885
\(790\) 0 0
\(791\) −2.00000 −0.0711118
\(792\) 1.00000 0.0355335
\(793\) 0 0
\(794\) 34.0000 1.20661
\(795\) 0 0
\(796\) −6.00000 −0.212664
\(797\) −6.00000 −0.212531 −0.106265 0.994338i \(-0.533889\pi\)
−0.106265 + 0.994338i \(0.533889\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) −6.00000 −0.211867
\(803\) 4.00000 0.141157
\(804\) 16.0000 0.564276
\(805\) 0 0
\(806\) 0 0
\(807\) 20.0000 0.704033
\(808\) 0 0
\(809\) 18.0000 0.632846 0.316423 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 0 0
\(811\) −44.0000 −1.54505 −0.772524 0.634985i \(-0.781006\pi\)
−0.772524 + 0.634985i \(0.781006\pi\)
\(812\) 2.00000 0.0701862
\(813\) 40.0000 1.40286
\(814\) 6.00000 0.210300
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 4.00000 0.139857
\(819\) 0 0
\(820\) 0 0
\(821\) −34.0000 −1.18661 −0.593304 0.804978i \(-0.702177\pi\)
−0.593304 + 0.804978i \(0.702177\pi\)
\(822\) −12.0000 −0.418548
\(823\) −32.0000 −1.11545 −0.557725 0.830026i \(-0.688326\pi\)
−0.557725 + 0.830026i \(0.688326\pi\)
\(824\) 14.0000 0.487713
\(825\) 0 0
\(826\) −10.0000 −0.347945
\(827\) 36.0000 1.25184 0.625921 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 4.00000 0.139010
\(829\) −10.0000 −0.347314 −0.173657 0.984806i \(-0.555558\pi\)
−0.173657 + 0.984806i \(0.555558\pi\)
\(830\) 0 0
\(831\) 4.00000 0.138758
\(832\) 0 0
\(833\) 0 0
\(834\) 16.0000 0.554035
\(835\) 0 0
\(836\) 0 0
\(837\) 8.00000 0.276520
\(838\) −26.0000 −0.898155
\(839\) −10.0000 −0.345238 −0.172619 0.984989i \(-0.555223\pi\)
−0.172619 + 0.984989i \(0.555223\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 18.0000 0.620321
\(843\) −4.00000 −0.137767
\(844\) −12.0000 −0.413057
\(845\) 0 0
\(846\) 6.00000 0.206284
\(847\) 1.00000 0.0343604
\(848\) 6.00000 0.206041
\(849\) −56.0000 −1.92192
\(850\) 0 0
\(851\) 24.0000 0.822709
\(852\) −8.00000 −0.274075
\(853\) 8.00000 0.273915 0.136957 0.990577i \(-0.456268\pi\)
0.136957 + 0.990577i \(0.456268\pi\)
\(854\) −4.00000 −0.136877
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) −12.0000 −0.409912 −0.204956 0.978771i \(-0.565705\pi\)
−0.204956 + 0.978771i \(0.565705\pi\)
\(858\) 0 0
\(859\) 2.00000 0.0682391 0.0341196 0.999418i \(-0.489137\pi\)
0.0341196 + 0.999418i \(0.489137\pi\)
\(860\) 0 0
\(861\) 16.0000 0.545279
\(862\) 8.00000 0.272481
\(863\) 32.0000 1.08929 0.544646 0.838666i \(-0.316664\pi\)
0.544646 + 0.838666i \(0.316664\pi\)
\(864\) −4.00000 −0.136083
\(865\) 0 0
\(866\) −14.0000 −0.475739
\(867\) −34.0000 −1.15470
\(868\) −2.00000 −0.0678844
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) 0 0
\(872\) −6.00000 −0.203186
\(873\) −14.0000 −0.473828
\(874\) 0 0
\(875\) 0 0
\(876\) 8.00000 0.270295
\(877\) −10.0000 −0.337676 −0.168838 0.985644i \(-0.554001\pi\)
−0.168838 + 0.985644i \(0.554001\pi\)
\(878\) 4.00000 0.134993
\(879\) 8.00000 0.269833
\(880\) 0 0
\(881\) −26.0000 −0.875962 −0.437981 0.898984i \(-0.644306\pi\)
−0.437981 + 0.898984i \(0.644306\pi\)
\(882\) 1.00000 0.0336718
\(883\) 44.0000 1.48072 0.740359 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 20.0000 0.671913
\(887\) 52.0000 1.74599 0.872995 0.487730i \(-0.162175\pi\)
0.872995 + 0.487730i \(0.162175\pi\)
\(888\) 12.0000 0.402694
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) −11.0000 −0.368514
\(892\) 6.00000 0.200895
\(893\) 0 0
\(894\) 12.0000 0.401340
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −10.0000 −0.333704
\(899\) −4.00000 −0.133407
\(900\) 0 0
\(901\) 0 0
\(902\) 8.00000 0.266371
\(903\) 24.0000 0.798670
\(904\) −2.00000 −0.0665190
\(905\) 0 0
\(906\) −16.0000 −0.531564
\(907\) −48.0000 −1.59381 −0.796907 0.604102i \(-0.793532\pi\)
−0.796907 + 0.604102i \(0.793532\pi\)
\(908\) −4.00000 −0.132745
\(909\) 0 0
\(910\) 0 0
\(911\) 28.0000 0.927681 0.463841 0.885919i \(-0.346471\pi\)
0.463841 + 0.885919i \(0.346471\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −26.0000 −0.860004
\(915\) 0 0
\(916\) 2.00000 0.0660819
\(917\) 4.00000 0.132092
\(918\) 0 0
\(919\) −8.00000 −0.263896 −0.131948 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 0 0
\(921\) −56.0000 −1.84526
\(922\) −20.0000 −0.658665
\(923\) 0 0
\(924\) 2.00000 0.0657952
\(925\) 0 0
\(926\) −4.00000 −0.131448
\(927\) 14.0000 0.459820
\(928\) 2.00000 0.0656532
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 18.0000 0.589610
\(933\) −12.0000 −0.392862
\(934\) 30.0000 0.981630
\(935\) 0 0
\(936\) 0 0
\(937\) −36.0000 −1.17607 −0.588034 0.808836i \(-0.700098\pi\)
−0.588034 + 0.808836i \(0.700098\pi\)
\(938\) 8.00000 0.261209
\(939\) −4.00000 −0.130535
\(940\) 0 0
\(941\) −60.0000 −1.95594 −0.977972 0.208736i \(-0.933065\pi\)
−0.977972 + 0.208736i \(0.933065\pi\)
\(942\) −12.0000 −0.390981
\(943\) 32.0000 1.04206
\(944\) −10.0000 −0.325472
\(945\) 0 0
\(946\) 12.0000 0.390154
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) −32.0000 −1.03931
\(949\) 0 0
\(950\) 0 0
\(951\) 4.00000 0.129709
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) 0 0
\(957\) 4.00000 0.129302
\(958\) 20.0000 0.646171
\(959\) −6.00000 −0.193750
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) −12.0000 −0.386695
\(964\) −16.0000 −0.515325
\(965\) 0 0
\(966\) 8.00000 0.257396
\(967\) 48.0000 1.54358 0.771788 0.635880i \(-0.219363\pi\)
0.771788 + 0.635880i \(0.219363\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) 0 0
\(971\) 26.0000 0.834380 0.417190 0.908819i \(-0.363015\pi\)
0.417190 + 0.908819i \(0.363015\pi\)
\(972\) −10.0000 −0.320750
\(973\) 8.00000 0.256468
\(974\) −20.0000 −0.640841
\(975\) 0 0
\(976\) −4.00000 −0.128037
\(977\) −42.0000 −1.34370 −0.671850 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) 16.0000 0.511624
\(979\) −6.00000 −0.191761
\(980\) 0 0
\(981\) −6.00000 −0.191565
\(982\) 36.0000 1.14881
\(983\) 22.0000 0.701691 0.350846 0.936433i \(-0.385894\pi\)
0.350846 + 0.936433i \(0.385894\pi\)
\(984\) 16.0000 0.510061
\(985\) 0 0
\(986\) 0 0
\(987\) 12.0000 0.381964
\(988\) 0 0
\(989\) 48.0000 1.52631
\(990\) 0 0
\(991\) −4.00000 −0.127064 −0.0635321 0.997980i \(-0.520237\pi\)
−0.0635321 + 0.997980i \(0.520237\pi\)
\(992\) −2.00000 −0.0635001
\(993\) 56.0000 1.77711
\(994\) −4.00000 −0.126872
\(995\) 0 0
\(996\) 0 0
\(997\) 48.0000 1.52018 0.760088 0.649821i \(-0.225156\pi\)
0.760088 + 0.649821i \(0.225156\pi\)
\(998\) −40.0000 −1.26618
\(999\) −24.0000 −0.759326
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 3850.2.a.bb.1.1 1
5.2 odd 4 3850.2.c.p.1849.2 2
5.3 odd 4 3850.2.c.p.1849.1 2
5.4 even 2 770.2.a.b.1.1 1
15.14 odd 2 6930.2.a.s.1.1 1
20.19 odd 2 6160.2.a.p.1.1 1
35.34 odd 2 5390.2.a.q.1.1 1
55.54 odd 2 8470.2.a.v.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.b.1.1 1 5.4 even 2
3850.2.a.bb.1.1 1 1.1 even 1 trivial
3850.2.c.p.1849.1 2 5.3 odd 4
3850.2.c.p.1849.2 2 5.2 odd 4
5390.2.a.q.1.1 1 35.34 odd 2
6160.2.a.p.1.1 1 20.19 odd 2
6930.2.a.s.1.1 1 15.14 odd 2
8470.2.a.v.1.1 1 55.54 odd 2