Properties

Label 3234.2.a.e.1.1
Level $3234$
Weight $2$
Character 3234.1
Self dual yes
Analytic conductor $25.824$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3234.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.8236200137\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3234.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{12} -4.00000 q^{13} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} -1.00000 q^{22} +1.00000 q^{24} -5.00000 q^{25} +4.00000 q^{26} -1.00000 q^{27} +2.00000 q^{29} -4.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} -4.00000 q^{34} +1.00000 q^{36} +10.0000 q^{37} +4.00000 q^{39} -12.0000 q^{41} +4.00000 q^{43} +1.00000 q^{44} +4.00000 q^{47} -1.00000 q^{48} +5.00000 q^{50} -4.00000 q^{51} -4.00000 q^{52} -10.0000 q^{53} +1.00000 q^{54} -2.00000 q^{58} -4.00000 q^{59} +4.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +1.00000 q^{66} +4.00000 q^{67} +4.00000 q^{68} -8.00000 q^{71} -1.00000 q^{72} +4.00000 q^{73} -10.0000 q^{74} +5.00000 q^{75} -4.00000 q^{78} +8.00000 q^{79} +1.00000 q^{81} +12.0000 q^{82} -16.0000 q^{83} -4.00000 q^{86} -2.00000 q^{87} -1.00000 q^{88} +8.00000 q^{89} +4.00000 q^{93} -4.00000 q^{94} +1.00000 q^{96} -8.00000 q^{97} +1.00000 q^{99} +O(q^{100})\)

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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) −1.00000 −0.288675
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) −1.00000 −0.235702
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 1.00000 0.204124
\(25\) −5.00000 −1.00000
\(26\) 4.00000 0.784465
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.00000 −0.174078
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 10.0000 1.64399 0.821995 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 0 0
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) −12.0000 −1.87409 −0.937043 0.349215i \(-0.886448\pi\)
−0.937043 + 0.349215i \(0.886448\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 0 0
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 5.00000 0.707107
\(51\) −4.00000 −0.560112
\(52\) −4.00000 −0.554700
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −2.00000 −0.262613
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 4.00000 0.512148 0.256074 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 1.00000 0.123091
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 4.00000 0.485071
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) −1.00000 −0.117851
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) −10.0000 −1.16248
\(75\) 5.00000 0.577350
\(76\) 0 0
\(77\) 0 0
\(78\) −4.00000 −0.452911
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 12.0000 1.32518
\(83\) −16.0000 −1.75623 −0.878114 0.478451i \(-0.841198\pi\)
−0.878114 + 0.478451i \(0.841198\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) −2.00000 −0.214423
\(88\) −1.00000 −0.106600
\(89\) 8.00000 0.847998 0.423999 0.905663i \(-0.360626\pi\)
0.423999 + 0.905663i \(0.360626\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 4.00000 0.414781
\(94\) −4.00000 −0.412568
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −8.00000 −0.812277 −0.406138 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) −5.00000 −0.500000
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 4.00000 0.396059
\(103\) 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
\(106\) 10.0000 0.971286
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 18.0000 1.72409 0.862044 0.506834i \(-0.169184\pi\)
0.862044 + 0.506834i \(0.169184\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) 0 0
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) −4.00000 −0.369800
\(118\) 4.00000 0.368230
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −4.00000 −0.362143
\(123\) 12.0000 1.08200
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) −16.0000 −1.35710 −0.678551 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 0 0
\(141\) −4.00000 −0.336861
\(142\) 8.00000 0.671345
\(143\) −4.00000 −0.334497
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −4.00000 −0.331042
\(147\) 0 0
\(148\) 10.0000 0.821995
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) −5.00000 −0.408248
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) 4.00000 0.323381
\(154\) 0 0
\(155\) 0 0
\(156\) 4.00000 0.320256
\(157\) 8.00000 0.638470 0.319235 0.947676i \(-0.396574\pi\)
0.319235 + 0.947676i \(0.396574\pi\)
\(158\) −8.00000 −0.636446
\(159\) 10.0000 0.793052
\(160\) 0 0
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) −12.0000 −0.937043
\(165\) 0 0
\(166\) 16.0000 1.24184
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) 4.00000 0.304114 0.152057 0.988372i \(-0.451410\pi\)
0.152057 + 0.988372i \(0.451410\pi\)
\(174\) 2.00000 0.151620
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) 4.00000 0.300658
\(178\) −8.00000 −0.599625
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) −4.00000 −0.295689
\(184\) 0 0
\(185\) 0 0
\(186\) −4.00000 −0.293294
\(187\) 4.00000 0.292509
\(188\) 4.00000 0.291730
\(189\) 0 0
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 8.00000 0.574367
\(195\) 0 0
\(196\) 0 0
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) −1.00000 −0.0710669
\(199\) −12.0000 −0.850657 −0.425329 0.905039i \(-0.639842\pi\)
−0.425329 + 0.905039i \(0.639842\pi\)
\(200\) 5.00000 0.353553
\(201\) −4.00000 −0.282138
\(202\) 12.0000 0.844317
\(203\) 0 0
\(204\) −4.00000 −0.280056
\(205\) 0 0
\(206\) −4.00000 −0.278693
\(207\) 0 0
\(208\) −4.00000 −0.277350
\(209\) 0 0
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) −10.0000 −0.686803
\(213\) 8.00000 0.548151
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −18.0000 −1.21911
\(219\) −4.00000 −0.270295
\(220\) 0 0
\(221\) −16.0000 −1.07628
\(222\) 10.0000 0.671156
\(223\) −12.0000 −0.803579 −0.401790 0.915732i \(-0.631612\pi\)
−0.401790 + 0.915732i \(0.631612\pi\)
\(224\) 0 0
\(225\) −5.00000 −0.333333
\(226\) −2.00000 −0.133038
\(227\) −8.00000 −0.530979 −0.265489 0.964114i \(-0.585534\pi\)
−0.265489 + 0.964114i \(0.585534\pi\)
\(228\) 0 0
\(229\) −16.0000 −1.05731 −0.528655 0.848837i \(-0.677303\pi\)
−0.528655 + 0.848837i \(0.677303\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 4.00000 0.261488
\(235\) 0 0
\(236\) −4.00000 −0.260378
\(237\) −8.00000 −0.519656
\(238\) 0 0
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) −20.0000 −1.28831 −0.644157 0.764894i \(-0.722792\pi\)
−0.644157 + 0.764894i \(0.722792\pi\)
\(242\) −1.00000 −0.0642824
\(243\) −1.00000 −0.0641500
\(244\) 4.00000 0.256074
\(245\) 0 0
\(246\) −12.0000 −0.765092
\(247\) 0 0
\(248\) 4.00000 0.254000
\(249\) 16.0000 1.01396
\(250\) 0 0
\(251\) 4.00000 0.252478 0.126239 0.992000i \(-0.459709\pi\)
0.126239 + 0.992000i \(0.459709\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −24.0000 −1.49708 −0.748539 0.663090i \(-0.769245\pi\)
−0.748539 + 0.663090i \(0.769245\pi\)
\(258\) 4.00000 0.249029
\(259\) 0 0
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) 8.00000 0.494242
\(263\) −8.00000 −0.493301 −0.246651 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) 1.00000 0.0615457
\(265\) 0 0
\(266\) 0 0
\(267\) −8.00000 −0.489592
\(268\) 4.00000 0.244339
\(269\) −8.00000 −0.487769 −0.243884 0.969804i \(-0.578422\pi\)
−0.243884 + 0.969804i \(0.578422\pi\)
\(270\) 0 0
\(271\) −24.0000 −1.45790 −0.728948 0.684569i \(-0.759990\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) −5.00000 −0.301511
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 16.0000 0.959616
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 4.00000 0.238197
\(283\) −24.0000 −1.42665 −0.713326 0.700832i \(-0.752812\pi\)
−0.713326 + 0.700832i \(0.752812\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 8.00000 0.468968
\(292\) 4.00000 0.234082
\(293\) −4.00000 −0.233682 −0.116841 0.993151i \(-0.537277\pi\)
−0.116841 + 0.993151i \(0.537277\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −10.0000 −0.581238
\(297\) −1.00000 −0.0580259
\(298\) 10.0000 0.579284
\(299\) 0 0
\(300\) 5.00000 0.288675
\(301\) 0 0
\(302\) 8.00000 0.460348
\(303\) 12.0000 0.689382
\(304\) 0 0
\(305\) 0 0
\(306\) −4.00000 −0.228665
\(307\) −8.00000 −0.456584 −0.228292 0.973593i \(-0.573314\pi\)
−0.228292 + 0.973593i \(0.573314\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) 4.00000 0.226819 0.113410 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) −4.00000 −0.226455
\(313\) 32.0000 1.80875 0.904373 0.426742i \(-0.140339\pi\)
0.904373 + 0.426742i \(0.140339\pi\)
\(314\) −8.00000 −0.451466
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) −10.0000 −0.560772
\(319\) 2.00000 0.111979
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 20.0000 1.10940
\(326\) −12.0000 −0.664619
\(327\) −18.0000 −0.995402
\(328\) 12.0000 0.662589
\(329\) 0 0
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −16.0000 −0.878114
\(333\) 10.0000 0.547997
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −2.00000 −0.108947 −0.0544735 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) −3.00000 −0.163178
\(339\) −2.00000 −0.108625
\(340\) 0 0
\(341\) −4.00000 −0.216612
\(342\) 0 0
\(343\) 0 0
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) −4.00000 −0.215041
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) −2.00000 −0.107211
\(349\) −28.0000 −1.49881 −0.749403 0.662114i \(-0.769659\pi\)
−0.749403 + 0.662114i \(0.769659\pi\)
\(350\) 0 0
\(351\) 4.00000 0.213504
\(352\) −1.00000 −0.0533002
\(353\) 24.0000 1.27739 0.638696 0.769460i \(-0.279474\pi\)
0.638696 + 0.769460i \(0.279474\pi\)
\(354\) −4.00000 −0.212598
\(355\) 0 0
\(356\) 8.00000 0.423999
\(357\) 0 0
\(358\) 4.00000 0.211407
\(359\) −32.0000 −1.68890 −0.844448 0.535638i \(-0.820071\pi\)
−0.844448 + 0.535638i \(0.820071\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) 0 0
\(366\) 4.00000 0.209083
\(367\) −12.0000 −0.626395 −0.313197 0.949688i \(-0.601400\pi\)
−0.313197 + 0.949688i \(0.601400\pi\)
\(368\) 0 0
\(369\) −12.0000 −0.624695
\(370\) 0 0
\(371\) 0 0
\(372\) 4.00000 0.207390
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) −4.00000 −0.206835
\(375\) 0 0
\(376\) −4.00000 −0.206284
\(377\) −8.00000 −0.412021
\(378\) 0 0
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 16.0000 0.818631
\(383\) −4.00000 −0.204390 −0.102195 0.994764i \(-0.532587\pi\)
−0.102195 + 0.994764i \(0.532587\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 14.0000 0.712581
\(387\) 4.00000 0.203331
\(388\) −8.00000 −0.406138
\(389\) 10.0000 0.507020 0.253510 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 8.00000 0.403547
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) 1.00000 0.0502519
\(397\) −8.00000 −0.401508 −0.200754 0.979642i \(-0.564339\pi\)
−0.200754 + 0.979642i \(0.564339\pi\)
\(398\) 12.0000 0.601506
\(399\) 0 0
\(400\) −5.00000 −0.250000
\(401\) −34.0000 −1.69788 −0.848939 0.528490i \(-0.822758\pi\)
−0.848939 + 0.528490i \(0.822758\pi\)
\(402\) 4.00000 0.199502
\(403\) 16.0000 0.797017
\(404\) −12.0000 −0.597022
\(405\) 0 0
\(406\) 0 0
\(407\) 10.0000 0.495682
\(408\) 4.00000 0.198030
\(409\) 20.0000 0.988936 0.494468 0.869196i \(-0.335363\pi\)
0.494468 + 0.869196i \(0.335363\pi\)
\(410\) 0 0
\(411\) −6.00000 −0.295958
\(412\) 4.00000 0.197066
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 4.00000 0.196116
\(417\) 16.0000 0.783523
\(418\) 0 0
\(419\) −36.0000 −1.75872 −0.879358 0.476162i \(-0.842028\pi\)
−0.879358 + 0.476162i \(0.842028\pi\)
\(420\) 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
\(423\) 4.00000 0.194487
\(424\) 10.0000 0.485643
\(425\) −20.0000 −0.970143
\(426\) −8.00000 −0.387601
\(427\) 0 0
\(428\) −12.0000 −0.580042
\(429\) 4.00000 0.193122
\(430\) 0 0
\(431\) −40.0000 −1.92673 −0.963366 0.268190i \(-0.913575\pi\)
−0.963366 + 0.268190i \(0.913575\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 40.0000 1.92228 0.961139 0.276066i \(-0.0890309\pi\)
0.961139 + 0.276066i \(0.0890309\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 18.0000 0.862044
\(437\) 0 0
\(438\) 4.00000 0.191127
\(439\) −40.0000 −1.90910 −0.954548 0.298057i \(-0.903661\pi\)
−0.954548 + 0.298057i \(0.903661\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 16.0000 0.761042
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) −10.0000 −0.474579
\(445\) 0 0
\(446\) 12.0000 0.568216
\(447\) 10.0000 0.472984
\(448\) 0 0
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 5.00000 0.235702
\(451\) −12.0000 −0.565058
\(452\) 2.00000 0.0940721
\(453\) 8.00000 0.375873
\(454\) 8.00000 0.375459
\(455\) 0 0
\(456\) 0 0
\(457\) 26.0000 1.21623 0.608114 0.793849i \(-0.291926\pi\)
0.608114 + 0.793849i \(0.291926\pi\)
\(458\) 16.0000 0.747631
\(459\) −4.00000 −0.186704
\(460\) 0 0
\(461\) 28.0000 1.30409 0.652045 0.758180i \(-0.273911\pi\)
0.652045 + 0.758180i \(0.273911\pi\)
\(462\) 0 0
\(463\) −40.0000 −1.85896 −0.929479 0.368875i \(-0.879743\pi\)
−0.929479 + 0.368875i \(0.879743\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) 36.0000 1.66588 0.832941 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) −4.00000 −0.184900
\(469\) 0 0
\(470\) 0 0
\(471\) −8.00000 −0.368621
\(472\) 4.00000 0.184115
\(473\) 4.00000 0.183920
\(474\) 8.00000 0.367452
\(475\) 0 0
\(476\) 0 0
\(477\) −10.0000 −0.457869
\(478\) −8.00000 −0.365911
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) −40.0000 −1.82384
\(482\) 20.0000 0.910975
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −8.00000 −0.362515 −0.181257 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) −4.00000 −0.181071
\(489\) −12.0000 −0.542659
\(490\) 0 0
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 12.0000 0.541002
\(493\) 8.00000 0.360302
\(494\) 0 0
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) −16.0000 −0.716977
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −4.00000 −0.178529
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −3.00000 −0.133235
\(508\) 0 0
\(509\) −24.0000 −1.06378 −0.531891 0.846813i \(-0.678518\pi\)
−0.531891 + 0.846813i \(0.678518\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 24.0000 1.05859
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) 4.00000 0.175920
\(518\) 0 0
\(519\) −4.00000 −0.175581
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) −2.00000 −0.0875376
\(523\) 8.00000 0.349816 0.174908 0.984585i \(-0.444037\pi\)
0.174908 + 0.984585i \(0.444037\pi\)
\(524\) −8.00000 −0.349482
\(525\) 0 0
\(526\) 8.00000 0.348817
\(527\) −16.0000 −0.696971
\(528\) −1.00000 −0.0435194
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) 48.0000 2.07911
\(534\) 8.00000 0.346194
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) 4.00000 0.172613
\(538\) 8.00000 0.344904
\(539\) 0 0
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 24.0000 1.03089
\(543\) 0 0
\(544\) −4.00000 −0.171499
\(545\) 0 0
\(546\) 0 0
\(547\) −36.0000 −1.53925 −0.769624 0.638497i \(-0.779557\pi\)
−0.769624 + 0.638497i \(0.779557\pi\)
\(548\) 6.00000 0.256307
\(549\) 4.00000 0.170716
\(550\) 5.00000 0.213201
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) −16.0000 −0.678551
\(557\) 14.0000 0.593199 0.296600 0.955002i \(-0.404147\pi\)
0.296600 + 0.955002i \(0.404147\pi\)
\(558\) 4.00000 0.169334
\(559\) −16.0000 −0.676728
\(560\) 0 0
\(561\) −4.00000 −0.168880
\(562\) 10.0000 0.421825
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) −4.00000 −0.168430
\(565\) 0 0
\(566\) 24.0000 1.00880
\(567\) 0 0
\(568\) 8.00000 0.335673
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) −4.00000 −0.167248
\(573\) 16.0000 0.668410
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 40.0000 1.66522 0.832611 0.553858i \(-0.186845\pi\)
0.832611 + 0.553858i \(0.186845\pi\)
\(578\) 1.00000 0.0415945
\(579\) 14.0000 0.581820
\(580\) 0 0
\(581\) 0 0
\(582\) −8.00000 −0.331611
\(583\) −10.0000 −0.414158
\(584\) −4.00000 −0.165521
\(585\) 0 0
\(586\) 4.00000 0.165238
\(587\) 36.0000 1.48588 0.742940 0.669359i \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 10.0000 0.410997
\(593\) −36.0000 −1.47834 −0.739171 0.673517i \(-0.764783\pi\)
−0.739171 + 0.673517i \(0.764783\pi\)
\(594\) 1.00000 0.0410305
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) 12.0000 0.491127
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) −5.00000 −0.204124
\(601\) 28.0000 1.14214 0.571072 0.820900i \(-0.306528\pi\)
0.571072 + 0.820900i \(0.306528\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) −12.0000 −0.487467
\(607\) 16.0000 0.649420 0.324710 0.945814i \(-0.394733\pi\)
0.324710 + 0.945814i \(0.394733\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −16.0000 −0.647291
\(612\) 4.00000 0.161690
\(613\) 26.0000 1.05013 0.525065 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(614\) 8.00000 0.322854
\(615\) 0 0
\(616\) 0 0
\(617\) 22.0000 0.885687 0.442843 0.896599i \(-0.353970\pi\)
0.442843 + 0.896599i \(0.353970\pi\)
\(618\) 4.00000 0.160904
\(619\) 36.0000 1.44696 0.723481 0.690344i \(-0.242541\pi\)
0.723481 + 0.690344i \(0.242541\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −4.00000 −0.160385
\(623\) 0 0
\(624\) 4.00000 0.160128
\(625\) 25.0000 1.00000
\(626\) −32.0000 −1.27898
\(627\) 0 0
\(628\) 8.00000 0.319235
\(629\) 40.0000 1.59490
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) −8.00000 −0.318223
\(633\) 12.0000 0.476957
\(634\) 18.0000 0.714871
\(635\) 0 0
\(636\) 10.0000 0.396526
\(637\) 0 0
\(638\) −2.00000 −0.0791808
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) −12.0000 −0.473602
\(643\) 20.0000 0.788723 0.394362 0.918955i \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −20.0000 −0.786281 −0.393141 0.919478i \(-0.628611\pi\)
−0.393141 + 0.919478i \(0.628611\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −4.00000 −0.157014
\(650\) −20.0000 −0.784465
\(651\) 0 0
\(652\) 12.0000 0.469956
\(653\) −30.0000 −1.17399 −0.586995 0.809590i \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) 18.0000 0.703856
\(655\) 0 0
\(656\) −12.0000 −0.468521
\(657\) 4.00000 0.156055
\(658\) 0 0
\(659\) 4.00000 0.155818 0.0779089 0.996960i \(-0.475176\pi\)
0.0779089 + 0.996960i \(0.475176\pi\)
\(660\) 0 0
\(661\) 40.0000 1.55582 0.777910 0.628376i \(-0.216280\pi\)
0.777910 + 0.628376i \(0.216280\pi\)
\(662\) −20.0000 −0.777322
\(663\) 16.0000 0.621389
\(664\) 16.0000 0.620920
\(665\) 0 0
\(666\) −10.0000 −0.387492
\(667\) 0 0
\(668\) 0 0
\(669\) 12.0000 0.463947
\(670\) 0 0
\(671\) 4.00000 0.154418
\(672\) 0 0
\(673\) −2.00000 −0.0770943 −0.0385472 0.999257i \(-0.512273\pi\)
−0.0385472 + 0.999257i \(0.512273\pi\)
\(674\) 2.00000 0.0770371
\(675\) 5.00000 0.192450
\(676\) 3.00000 0.115385
\(677\) 4.00000 0.153732 0.0768662 0.997041i \(-0.475509\pi\)
0.0768662 + 0.997041i \(0.475509\pi\)
\(678\) 2.00000 0.0768095
\(679\) 0 0
\(680\) 0 0
\(681\) 8.00000 0.306561
\(682\) 4.00000 0.153168
\(683\) 4.00000 0.153056 0.0765279 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 16.0000 0.610438
\(688\) 4.00000 0.152499
\(689\) 40.0000 1.52388
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 4.00000 0.152057
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) 0 0
\(696\) 2.00000 0.0758098
\(697\) −48.0000 −1.81813
\(698\) 28.0000 1.05982
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) −4.00000 −0.150970
\(703\) 0 0
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) −24.0000 −0.903252
\(707\) 0 0
\(708\) 4.00000 0.150329
\(709\) 26.0000 0.976450 0.488225 0.872718i \(-0.337644\pi\)
0.488225 + 0.872718i \(0.337644\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) −8.00000 −0.299813
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −4.00000 −0.149487
\(717\) −8.00000 −0.298765
\(718\) 32.0000 1.19423
\(719\) −36.0000 −1.34257 −0.671287 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 19.0000 0.707107
\(723\) 20.0000 0.743808
\(724\) 0 0
\(725\) −10.0000 −0.371391
\(726\) 1.00000 0.0371135
\(727\) 12.0000 0.445055 0.222528 0.974926i \(-0.428569\pi\)
0.222528 + 0.974926i \(0.428569\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 16.0000 0.591781
\(732\) −4.00000 −0.147844
\(733\) 4.00000 0.147743 0.0738717 0.997268i \(-0.476464\pi\)
0.0738717 + 0.997268i \(0.476464\pi\)
\(734\) 12.0000 0.442928
\(735\) 0 0
\(736\) 0 0
\(737\) 4.00000 0.147342
\(738\) 12.0000 0.441726
\(739\) 28.0000 1.03000 0.514998 0.857191i \(-0.327793\pi\)
0.514998 + 0.857191i \(0.327793\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 32.0000 1.17397 0.586983 0.809599i \(-0.300316\pi\)
0.586983 + 0.809599i \(0.300316\pi\)
\(744\) −4.00000 −0.146647
\(745\) 0 0
\(746\) −22.0000 −0.805477
\(747\) −16.0000 −0.585409
\(748\) 4.00000 0.146254
\(749\) 0 0
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 4.00000 0.145865
\(753\) −4.00000 −0.145768
\(754\) 8.00000 0.291343
\(755\) 0 0
\(756\) 0 0
\(757\) −22.0000 −0.799604 −0.399802 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) −4.00000 −0.145287
\(759\) 0 0
\(760\) 0 0
\(761\) 44.0000 1.59500 0.797499 0.603320i \(-0.206156\pi\)
0.797499 + 0.603320i \(0.206156\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −16.0000 −0.578860
\(765\) 0 0
\(766\) 4.00000 0.144526
\(767\) 16.0000 0.577727
\(768\) −1.00000 −0.0360844
\(769\) 20.0000 0.721218 0.360609 0.932717i \(-0.382569\pi\)
0.360609 + 0.932717i \(0.382569\pi\)
\(770\) 0 0
\(771\) 24.0000 0.864339
\(772\) −14.0000 −0.503871
\(773\) 40.0000 1.43870 0.719350 0.694648i \(-0.244440\pi\)
0.719350 + 0.694648i \(0.244440\pi\)
\(774\) −4.00000 −0.143777
\(775\) 20.0000 0.718421
\(776\) 8.00000 0.287183
\(777\) 0 0
\(778\) −10.0000 −0.358517
\(779\) 0 0
\(780\) 0 0
\(781\) −8.00000 −0.286263
\(782\) 0 0
\(783\) −2.00000 −0.0714742
\(784\) 0 0
\(785\) 0 0
\(786\) −8.00000 −0.285351
\(787\) 32.0000 1.14068 0.570338 0.821410i \(-0.306812\pi\)
0.570338 + 0.821410i \(0.306812\pi\)
\(788\) 6.00000 0.213741
\(789\) 8.00000 0.284808
\(790\) 0 0
\(791\) 0 0
\(792\) −1.00000 −0.0355335
\(793\) −16.0000 −0.568177
\(794\) 8.00000 0.283909
\(795\) 0 0
\(796\) −12.0000 −0.425329
\(797\) 16.0000 0.566749 0.283375 0.959009i \(-0.408546\pi\)
0.283375 + 0.959009i \(0.408546\pi\)
\(798\) 0 0
\(799\) 16.0000 0.566039
\(800\) 5.00000 0.176777
\(801\) 8.00000 0.282666
\(802\) 34.0000 1.20058
\(803\) 4.00000 0.141157
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) −16.0000 −0.563576
\(807\) 8.00000 0.281613
\(808\) 12.0000 0.422159
\(809\) 10.0000 0.351581 0.175791 0.984428i \(-0.443752\pi\)
0.175791 + 0.984428i \(0.443752\pi\)
\(810\) 0 0
\(811\) −8.00000 −0.280918 −0.140459 0.990086i \(-0.544858\pi\)
−0.140459 + 0.990086i \(0.544858\pi\)
\(812\) 0 0
\(813\) 24.0000 0.841717
\(814\) −10.0000 −0.350500
\(815\) 0 0
\(816\) −4.00000 −0.140028
\(817\) 0 0
\(818\) −20.0000 −0.699284
\(819\) 0 0
\(820\) 0 0
\(821\) 54.0000 1.88461 0.942306 0.334751i \(-0.108652\pi\)
0.942306 + 0.334751i \(0.108652\pi\)
\(822\) 6.00000 0.209274
\(823\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) −4.00000 −0.139347
\(825\) 5.00000 0.174078
\(826\) 0 0
\(827\) −28.0000 −0.973655 −0.486828 0.873498i \(-0.661846\pi\)
−0.486828 + 0.873498i \(0.661846\pi\)
\(828\) 0 0
\(829\) −24.0000 −0.833554 −0.416777 0.909009i \(-0.636840\pi\)
−0.416777 + 0.909009i \(0.636840\pi\)
\(830\) 0 0
\(831\) 10.0000 0.346896
\(832\) −4.00000 −0.138675
\(833\) 0 0
\(834\) −16.0000 −0.554035
\(835\) 0 0
\(836\) 0 0
\(837\) 4.00000 0.138260
\(838\) 36.0000 1.24360
\(839\) −36.0000 −1.24286 −0.621429 0.783470i \(-0.713448\pi\)
−0.621429 + 0.783470i \(0.713448\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −22.0000 −0.758170
\(843\) 10.0000 0.344418
\(844\) −12.0000 −0.413057
\(845\) 0 0
\(846\) −4.00000 −0.137523
\(847\) 0 0
\(848\) −10.0000 −0.343401
\(849\) 24.0000 0.823678
\(850\) 20.0000 0.685994
\(851\) 0 0
\(852\) 8.00000 0.274075
\(853\) −4.00000 −0.136957 −0.0684787 0.997653i \(-0.521815\pi\)
−0.0684787 + 0.997653i \(0.521815\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) −4.00000 −0.136637 −0.0683187 0.997664i \(-0.521763\pi\)
−0.0683187 + 0.997664i \(0.521763\pi\)
\(858\) −4.00000 −0.136558
\(859\) 52.0000 1.77422 0.887109 0.461561i \(-0.152710\pi\)
0.887109 + 0.461561i \(0.152710\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 40.0000 1.36241
\(863\) −32.0000 −1.08929 −0.544646 0.838666i \(-0.683336\pi\)
−0.544646 + 0.838666i \(0.683336\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −40.0000 −1.35926
\(867\) 1.00000 0.0339618
\(868\) 0 0
\(869\) 8.00000 0.271381
\(870\) 0 0
\(871\) −16.0000 −0.542139
\(872\) −18.0000 −0.609557
\(873\) −8.00000 −0.270759
\(874\) 0 0
\(875\) 0 0
\(876\) −4.00000 −0.135147
\(877\) −46.0000 −1.55331 −0.776655 0.629926i \(-0.783085\pi\)
−0.776655 + 0.629926i \(0.783085\pi\)
\(878\) 40.0000 1.34993
\(879\) 4.00000 0.134917
\(880\) 0 0
\(881\) 16.0000 0.539054 0.269527 0.962993i \(-0.413133\pi\)
0.269527 + 0.962993i \(0.413133\pi\)
\(882\) 0 0
\(883\) 4.00000 0.134611 0.0673054 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) −16.0000 −0.538138
\(885\) 0 0
\(886\) −4.00000 −0.134383
\(887\) −8.00000 −0.268614 −0.134307 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) 10.0000 0.335578
\(889\) 0 0
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) −12.0000 −0.401790
\(893\) 0 0
\(894\) −10.0000 −0.334450
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −18.0000 −0.600668
\(899\) −8.00000 −0.266815
\(900\) −5.00000 −0.166667
\(901\) −40.0000 −1.33259
\(902\) 12.0000 0.399556
\(903\) 0 0
\(904\) −2.00000 −0.0665190
\(905\) 0 0
\(906\) −8.00000 −0.265782
\(907\) 12.0000 0.398453 0.199227 0.979953i \(-0.436157\pi\)
0.199227 + 0.979953i \(0.436157\pi\)
\(908\) −8.00000 −0.265489
\(909\) −12.0000 −0.398015
\(910\) 0 0
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) 0 0
\(913\) −16.0000 −0.529523
\(914\) −26.0000 −0.860004
\(915\) 0 0
\(916\) −16.0000 −0.528655
\(917\) 0 0
\(918\) 4.00000 0.132020
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) 8.00000 0.263609
\(922\) −28.0000 −0.922131
\(923\) 32.0000 1.05329
\(924\) 0 0
\(925\) −50.0000 −1.64399
\(926\) 40.0000 1.31448
\(927\) 4.00000 0.131377
\(928\) −2.00000 −0.0656532
\(929\) 48.0000 1.57483 0.787414 0.616424i \(-0.211419\pi\)
0.787414 + 0.616424i \(0.211419\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 6.00000 0.196537
\(933\) −4.00000 −0.130954
\(934\) −36.0000 −1.17796
\(935\) 0 0
\(936\) 4.00000 0.130744
\(937\) 36.0000 1.17607 0.588034 0.808836i \(-0.299902\pi\)
0.588034 + 0.808836i \(0.299902\pi\)
\(938\) 0 0
\(939\) −32.0000 −1.04428
\(940\) 0 0
\(941\) 28.0000 0.912774 0.456387 0.889781i \(-0.349143\pi\)
0.456387 + 0.889781i \(0.349143\pi\)
\(942\) 8.00000 0.260654
\(943\) 0 0
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) −8.00000 −0.259828
\(949\) −16.0000 −0.519382
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) −54.0000 −1.74923 −0.874616 0.484817i \(-0.838886\pi\)
−0.874616 + 0.484817i \(0.838886\pi\)
\(954\) 10.0000 0.323762
\(955\) 0 0
\(956\) 8.00000 0.258738
\(957\) −2.00000 −0.0646508
\(958\) −24.0000 −0.775405
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 40.0000 1.28965
\(963\) −12.0000 −0.386695
\(964\) −20.0000 −0.644157
\(965\) 0 0
\(966\) 0 0
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) 0 0
\(971\) 20.0000 0.641831 0.320915 0.947108i \(-0.396010\pi\)
0.320915 + 0.947108i \(0.396010\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 8.00000 0.256337
\(975\) −20.0000 −0.640513
\(976\) 4.00000 0.128037
\(977\) −50.0000 −1.59964 −0.799821 0.600239i \(-0.795072\pi\)
−0.799821 + 0.600239i \(0.795072\pi\)
\(978\) 12.0000 0.383718
\(979\) 8.00000 0.255681
\(980\) 0 0
\(981\) 18.0000 0.574696
\(982\) −20.0000 −0.638226
\(983\) −60.0000 −1.91370 −0.956851 0.290578i \(-0.906153\pi\)
−0.956851 + 0.290578i \(0.906153\pi\)
\(984\) −12.0000 −0.382546
\(985\) 0 0
\(986\) −8.00000 −0.254772
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 56.0000 1.77890 0.889449 0.457034i \(-0.151088\pi\)
0.889449 + 0.457034i \(0.151088\pi\)
\(992\) 4.00000 0.127000
\(993\) −20.0000 −0.634681
\(994\) 0 0
\(995\) 0 0
\(996\) 16.0000 0.506979
\(997\) 28.0000 0.886769 0.443384 0.896332i \(-0.353778\pi\)
0.443384 + 0.896332i \(0.353778\pi\)
\(998\) −20.0000 −0.633089
\(999\) −10.0000 −0.316386
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 3234.2.a.e.1.1 1
3.2 odd 2 9702.2.a.bk.1.1 1
7.6 odd 2 3234.2.a.m.1.1 yes 1
21.20 even 2 9702.2.a.bo.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3234.2.a.e.1.1 1 1.1 even 1 trivial
3234.2.a.m.1.1 yes 1 7.6 odd 2
9702.2.a.bk.1.1 1 3.2 odd 2
9702.2.a.bo.1.1 1 21.20 even 2