Properties

Label 1694.2.a.i.1.1
Level $1694$
Weight $2$
Character 1694.1
Self dual yes
Analytic conductor $13.527$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1694 = 2 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1694.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.5266581024\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 154)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1694.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +2.00000 q^{12} +4.00000 q^{13} +1.00000 q^{14} +4.00000 q^{15} +1.00000 q^{16} +1.00000 q^{18} -4.00000 q^{19} +2.00000 q^{20} +2.00000 q^{21} +4.00000 q^{23} +2.00000 q^{24} -1.00000 q^{25} +4.00000 q^{26} -4.00000 q^{27} +1.00000 q^{28} -2.00000 q^{29} +4.00000 q^{30} -10.0000 q^{31} +1.00000 q^{32} +2.00000 q^{35} +1.00000 q^{36} -6.00000 q^{37} -4.00000 q^{38} +8.00000 q^{39} +2.00000 q^{40} +2.00000 q^{42} +4.00000 q^{43} +2.00000 q^{45} +4.00000 q^{46} +10.0000 q^{47} +2.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +4.00000 q^{52} -14.0000 q^{53} -4.00000 q^{54} +1.00000 q^{56} -8.00000 q^{57} -2.00000 q^{58} +10.0000 q^{59} +4.00000 q^{60} +8.00000 q^{61} -10.0000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +8.00000 q^{65} +8.00000 q^{67} +8.00000 q^{69} +2.00000 q^{70} -4.00000 q^{71} +1.00000 q^{72} -4.00000 q^{73} -6.00000 q^{74} -2.00000 q^{75} -4.00000 q^{76} +8.00000 q^{78} -16.0000 q^{79} +2.00000 q^{80} -11.0000 q^{81} -4.00000 q^{83} +2.00000 q^{84} +4.00000 q^{86} -4.00000 q^{87} +10.0000 q^{89} +2.00000 q^{90} +4.00000 q^{91} +4.00000 q^{92} -20.0000 q^{93} +10.0000 q^{94} -8.00000 q^{95} +2.00000 q^{96} +6.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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
\(3\) 2.00000 1.15470 0.577350 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 2.00000 0.816497
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) 0 0
\(12\) 2.00000 0.577350
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 1.00000 0.267261
\(15\) 4.00000 1.03280
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 1.00000 0.235702
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 2.00000 0.447214
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 2.00000 0.408248
\(25\) −1.00000 −0.200000
\(26\) 4.00000 0.784465
\(27\) −4.00000 −0.769800
\(28\) 1.00000 0.188982
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 4.00000 0.730297
\(31\) −10.0000 −1.79605 −0.898027 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 0 0
\(35\) 2.00000 0.338062
\(36\) 1.00000 0.166667
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) −4.00000 −0.648886
\(39\) 8.00000 1.28103
\(40\) 2.00000 0.316228
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 2.00000 0.308607
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 2.00000 0.298142
\(46\) 4.00000 0.589768
\(47\) 10.0000 1.45865 0.729325 0.684167i \(-0.239834\pi\)
0.729325 + 0.684167i \(0.239834\pi\)
\(48\) 2.00000 0.288675
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 4.00000 0.554700
\(53\) −14.0000 −1.92305 −0.961524 0.274721i \(-0.911414\pi\)
−0.961524 + 0.274721i \(0.911414\pi\)
\(54\) −4.00000 −0.544331
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −8.00000 −1.05963
\(58\) −2.00000 −0.262613
\(59\) 10.0000 1.30189 0.650945 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 4.00000 0.516398
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) −10.0000 −1.27000
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 8.00000 0.992278
\(66\) 0 0
\(67\) 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
\(70\) 2.00000 0.239046
\(71\) −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 \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) −6.00000 −0.697486
\(75\) −2.00000 −0.230940
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) 8.00000 0.905822
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 2.00000 0.223607
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) −4.00000 −0.428845
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 2.00000 0.210819
\(91\) 4.00000 0.419314
\(92\) 4.00000 0.417029
\(93\) −20.0000 −2.07390
\(94\) 10.0000 1.03142
\(95\) −8.00000 −0.820783
\(96\) 2.00000 0.204124
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 0 0
\(103\) 2.00000 0.197066 0.0985329 0.995134i \(-0.468585\pi\)
0.0985329 + 0.995134i \(0.468585\pi\)
\(104\) 4.00000 0.392232
\(105\) 4.00000 0.390360
\(106\) −14.0000 −1.35980
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) −4.00000 −0.384900
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) −12.0000 −1.13899
\(112\) 1.00000 0.0944911
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) −8.00000 −0.749269
\(115\) 8.00000 0.746004
\(116\) −2.00000 −0.185695
\(117\) 4.00000 0.369800
\(118\) 10.0000 0.920575
\(119\) 0 0
\(120\) 4.00000 0.365148
\(121\) 0 0
\(122\) 8.00000 0.724286
\(123\) 0 0
\(124\) −10.0000 −0.898027
\(125\) −12.0000 −1.07331
\(126\) 1.00000 0.0890871
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 1.00000 0.0883883
\(129\) 8.00000 0.704361
\(130\) 8.00000 0.701646
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 0 0
\(133\) −4.00000 −0.346844
\(134\) 8.00000 0.691095
\(135\) −8.00000 −0.688530
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 8.00000 0.681005
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 2.00000 0.169031
\(141\) 20.0000 1.68430
\(142\) −4.00000 −0.335673
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −4.00000 −0.332182
\(146\) −4.00000 −0.331042
\(147\) 2.00000 0.164957
\(148\) −6.00000 −0.493197
\(149\) −22.0000 −1.80231 −0.901155 0.433497i \(-0.857280\pi\)
−0.901155 + 0.433497i \(0.857280\pi\)
\(150\) −2.00000 −0.163299
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) −4.00000 −0.324443
\(153\) 0 0
\(154\) 0 0
\(155\) −20.0000 −1.60644
\(156\) 8.00000 0.640513
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) −16.0000 −1.27289
\(159\) −28.0000 −2.22054
\(160\) 2.00000 0.158114
\(161\) 4.00000 0.315244
\(162\) −11.0000 −0.864242
\(163\) 24.0000 1.87983 0.939913 0.341415i \(-0.110906\pi\)
0.939913 + 0.341415i \(0.110906\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 2.00000 0.154303
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 4.00000 0.304997
\(173\) −4.00000 −0.304114 −0.152057 0.988372i \(-0.548590\pi\)
−0.152057 + 0.988372i \(0.548590\pi\)
\(174\) −4.00000 −0.303239
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 20.0000 1.50329
\(178\) 10.0000 0.749532
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 2.00000 0.149071
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 4.00000 0.296500
\(183\) 16.0000 1.18275
\(184\) 4.00000 0.294884
\(185\) −12.0000 −0.882258
\(186\) −20.0000 −1.46647
\(187\) 0 0
\(188\) 10.0000 0.729325
\(189\) −4.00000 −0.290957
\(190\) −8.00000 −0.580381
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 2.00000 0.144338
\(193\) 6.00000 0.431889 0.215945 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) 6.00000 0.430775
\(195\) 16.0000 1.14578
\(196\) 1.00000 0.0714286
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) −14.0000 −0.992434 −0.496217 0.868199i \(-0.665278\pi\)
−0.496217 + 0.868199i \(0.665278\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 16.0000 1.12855
\(202\) −12.0000 −0.844317
\(203\) −2.00000 −0.140372
\(204\) 0 0
\(205\) 0 0
\(206\) 2.00000 0.139347
\(207\) 4.00000 0.278019
\(208\) 4.00000 0.277350
\(209\) 0 0
\(210\) 4.00000 0.276026
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) −14.0000 −0.961524
\(213\) −8.00000 −0.548151
\(214\) 12.0000 0.820303
\(215\) 8.00000 0.545595
\(216\) −4.00000 −0.272166
\(217\) −10.0000 −0.678844
\(218\) 14.0000 0.948200
\(219\) −8.00000 −0.540590
\(220\) 0 0
\(221\) 0 0
\(222\) −12.0000 −0.805387
\(223\) −14.0000 −0.937509 −0.468755 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) 1.00000 0.0668153
\(225\) −1.00000 −0.0666667
\(226\) −14.0000 −0.931266
\(227\) −8.00000 −0.530979 −0.265489 0.964114i \(-0.585534\pi\)
−0.265489 + 0.964114i \(0.585534\pi\)
\(228\) −8.00000 −0.529813
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 8.00000 0.527504
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 4.00000 0.261488
\(235\) 20.0000 1.30466
\(236\) 10.0000 0.650945
\(237\) −32.0000 −2.07862
\(238\) 0 0
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 4.00000 0.258199
\(241\) −8.00000 −0.515325 −0.257663 0.966235i \(-0.582952\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) 0 0
\(243\) −10.0000 −0.641500
\(244\) 8.00000 0.512148
\(245\) 2.00000 0.127775
\(246\) 0 0
\(247\) −16.0000 −1.01806
\(248\) −10.0000 −0.635001
\(249\) −8.00000 −0.506979
\(250\) −12.0000 −0.758947
\(251\) −26.0000 −1.64111 −0.820553 0.571571i \(-0.806334\pi\)
−0.820553 + 0.571571i \(0.806334\pi\)
\(252\) 1.00000 0.0629941
\(253\) 0 0
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 8.00000 0.498058
\(259\) −6.00000 −0.372822
\(260\) 8.00000 0.496139
\(261\) −2.00000 −0.123797
\(262\) −8.00000 −0.494242
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) −28.0000 −1.72003
\(266\) −4.00000 −0.245256
\(267\) 20.0000 1.22398
\(268\) 8.00000 0.488678
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) −8.00000 −0.486864
\(271\) 28.0000 1.70088 0.850439 0.526073i \(-0.176336\pi\)
0.850439 + 0.526073i \(0.176336\pi\)
\(272\) 0 0
\(273\) 8.00000 0.484182
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 8.00000 0.481543
\(277\) −6.00000 −0.360505 −0.180253 0.983620i \(-0.557691\pi\)
−0.180253 + 0.983620i \(0.557691\pi\)
\(278\) −20.0000 −1.19952
\(279\) −10.0000 −0.598684
\(280\) 2.00000 0.119523
\(281\) −30.0000 −1.78965 −0.894825 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 20.0000 1.19098
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) −4.00000 −0.237356
\(285\) −16.0000 −0.947758
\(286\) 0 0
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −17.0000 −1.00000
\(290\) −4.00000 −0.234888
\(291\) 12.0000 0.703452
\(292\) −4.00000 −0.234082
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 2.00000 0.116642
\(295\) 20.0000 1.16445
\(296\) −6.00000 −0.348743
\(297\) 0 0
\(298\) −22.0000 −1.27443
\(299\) 16.0000 0.925304
\(300\) −2.00000 −0.115470
\(301\) 4.00000 0.230556
\(302\) −16.0000 −0.920697
\(303\) −24.0000 −1.37876
\(304\) −4.00000 −0.229416
\(305\) 16.0000 0.916157
\(306\) 0 0
\(307\) −16.0000 −0.913168 −0.456584 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) 0 0
\(309\) 4.00000 0.227552
\(310\) −20.0000 −1.13592
\(311\) −6.00000 −0.340229 −0.170114 0.985424i \(-0.554414\pi\)
−0.170114 + 0.985424i \(0.554414\pi\)
\(312\) 8.00000 0.452911
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 10.0000 0.564333
\(315\) 2.00000 0.112687
\(316\) −16.0000 −0.900070
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) −28.0000 −1.57016
\(319\) 0 0
\(320\) 2.00000 0.111803
\(321\) 24.0000 1.33955
\(322\) 4.00000 0.222911
\(323\) 0 0
\(324\) −11.0000 −0.611111
\(325\) −4.00000 −0.221880
\(326\) 24.0000 1.32924
\(327\) 28.0000 1.54840
\(328\) 0 0
\(329\) 10.0000 0.551318
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) −4.00000 −0.219529
\(333\) −6.00000 −0.328798
\(334\) 8.00000 0.437741
\(335\) 16.0000 0.874173
\(336\) 2.00000 0.109109
\(337\) 34.0000 1.85210 0.926049 0.377403i \(-0.123183\pi\)
0.926049 + 0.377403i \(0.123183\pi\)
\(338\) 3.00000 0.163178
\(339\) −28.0000 −1.52075
\(340\) 0 0
\(341\) 0 0
\(342\) −4.00000 −0.216295
\(343\) 1.00000 0.0539949
\(344\) 4.00000 0.215666
\(345\) 16.0000 0.861411
\(346\) −4.00000 −0.215041
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) −4.00000 −0.214423
\(349\) −32.0000 −1.71292 −0.856460 0.516213i \(-0.827341\pi\)
−0.856460 + 0.516213i \(0.827341\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −16.0000 −0.854017
\(352\) 0 0
\(353\) 2.00000 0.106449 0.0532246 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) 20.0000 1.06299
\(355\) −8.00000 −0.424596
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 2.00000 0.105409
\(361\) −3.00000 −0.157895
\(362\) 14.0000 0.735824
\(363\) 0 0
\(364\) 4.00000 0.209657
\(365\) −8.00000 −0.418739
\(366\) 16.0000 0.836333
\(367\) 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
\(370\) −12.0000 −0.623850
\(371\) −14.0000 −0.726844
\(372\) −20.0000 −1.03695
\(373\) 34.0000 1.76045 0.880227 0.474554i \(-0.157390\pi\)
0.880227 + 0.474554i \(0.157390\pi\)
\(374\) 0 0
\(375\) −24.0000 −1.23935
\(376\) 10.0000 0.515711
\(377\) −8.00000 −0.412021
\(378\) −4.00000 −0.205738
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) −8.00000 −0.410391
\(381\) 32.0000 1.63941
\(382\) 8.00000 0.409316
\(383\) 14.0000 0.715367 0.357683 0.933843i \(-0.383567\pi\)
0.357683 + 0.933843i \(0.383567\pi\)
\(384\) 2.00000 0.102062
\(385\) 0 0
\(386\) 6.00000 0.305392
\(387\) 4.00000 0.203331
\(388\) 6.00000 0.304604
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 16.0000 0.810191
\(391\) 0 0
\(392\) 1.00000 0.0505076
\(393\) −16.0000 −0.807093
\(394\) 18.0000 0.906827
\(395\) −32.0000 −1.61009
\(396\) 0 0
\(397\) 18.0000 0.903394 0.451697 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) −14.0000 −0.701757
\(399\) −8.00000 −0.400501
\(400\) −1.00000 −0.0500000
\(401\) 10.0000 0.499376 0.249688 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) 16.0000 0.798007
\(403\) −40.0000 −1.99254
\(404\) −12.0000 −0.597022
\(405\) −22.0000 −1.09319
\(406\) −2.00000 −0.0992583
\(407\) 0 0
\(408\) 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
\(412\) 2.00000 0.0985329
\(413\) 10.0000 0.492068
\(414\) 4.00000 0.196589
\(415\) −8.00000 −0.392705
\(416\) 4.00000 0.196116
\(417\) −40.0000 −1.95881
\(418\) 0 0
\(419\) −30.0000 −1.46560 −0.732798 0.680446i \(-0.761786\pi\)
−0.732798 + 0.680446i \(0.761786\pi\)
\(420\) 4.00000 0.195180
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 4.00000 0.194717
\(423\) 10.0000 0.486217
\(424\) −14.0000 −0.679900
\(425\) 0 0
\(426\) −8.00000 −0.387601
\(427\) 8.00000 0.387147
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) 8.00000 0.385794
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) −4.00000 −0.192450
\(433\) −10.0000 −0.480569 −0.240285 0.970702i \(-0.577241\pi\)
−0.240285 + 0.970702i \(0.577241\pi\)
\(434\) −10.0000 −0.480015
\(435\) −8.00000 −0.383571
\(436\) 14.0000 0.670478
\(437\) −16.0000 −0.765384
\(438\) −8.00000 −0.382255
\(439\) 28.0000 1.33637 0.668184 0.743996i \(-0.267072\pi\)
0.668184 + 0.743996i \(0.267072\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) −12.0000 −0.569495
\(445\) 20.0000 0.948091
\(446\) −14.0000 −0.662919
\(447\) −44.0000 −2.08113
\(448\) 1.00000 0.0472456
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 0 0
\(452\) −14.0000 −0.658505
\(453\) −32.0000 −1.50349
\(454\) −8.00000 −0.375459
\(455\) 8.00000 0.375046
\(456\) −8.00000 −0.374634
\(457\) 38.0000 1.77757 0.888783 0.458329i \(-0.151552\pi\)
0.888783 + 0.458329i \(0.151552\pi\)
\(458\) −10.0000 −0.467269
\(459\) 0 0
\(460\) 8.00000 0.373002
\(461\) −32.0000 −1.49039 −0.745194 0.666847i \(-0.767643\pi\)
−0.745194 + 0.666847i \(0.767643\pi\)
\(462\) 0 0
\(463\) 12.0000 0.557687 0.278844 0.960337i \(-0.410049\pi\)
0.278844 + 0.960337i \(0.410049\pi\)
\(464\) −2.00000 −0.0928477
\(465\) −40.0000 −1.85496
\(466\) −6.00000 −0.277945
\(467\) 14.0000 0.647843 0.323921 0.946084i \(-0.394999\pi\)
0.323921 + 0.946084i \(0.394999\pi\)
\(468\) 4.00000 0.184900
\(469\) 8.00000 0.369406
\(470\) 20.0000 0.922531
\(471\) 20.0000 0.921551
\(472\) 10.0000 0.460287
\(473\) 0 0
\(474\) −32.0000 −1.46981
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) −14.0000 −0.641016
\(478\) 8.00000 0.365911
\(479\) −12.0000 −0.548294 −0.274147 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) 4.00000 0.182574
\(481\) −24.0000 −1.09431
\(482\) −8.00000 −0.364390
\(483\) 8.00000 0.364013
\(484\) 0 0
\(485\) 12.0000 0.544892
\(486\) −10.0000 −0.453609
\(487\) 12.0000 0.543772 0.271886 0.962329i \(-0.412353\pi\)
0.271886 + 0.962329i \(0.412353\pi\)
\(488\) 8.00000 0.362143
\(489\) 48.0000 2.17064
\(490\) 2.00000 0.0903508
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −16.0000 −0.719874
\(495\) 0 0
\(496\) −10.0000 −0.449013
\(497\) −4.00000 −0.179425
\(498\) −8.00000 −0.358489
\(499\) −16.0000 −0.716258 −0.358129 0.933672i \(-0.616585\pi\)
−0.358129 + 0.933672i \(0.616585\pi\)
\(500\) −12.0000 −0.536656
\(501\) 16.0000 0.714827
\(502\) −26.0000 −1.16044
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 1.00000 0.0445435
\(505\) −24.0000 −1.06799
\(506\) 0 0
\(507\) 6.00000 0.266469
\(508\) 16.0000 0.709885
\(509\) 38.0000 1.68432 0.842160 0.539227i \(-0.181284\pi\)
0.842160 + 0.539227i \(0.181284\pi\)
\(510\) 0 0
\(511\) −4.00000 −0.176950
\(512\) 1.00000 0.0441942
\(513\) 16.0000 0.706417
\(514\) 2.00000 0.0882162
\(515\) 4.00000 0.176261
\(516\) 8.00000 0.352180
\(517\) 0 0
\(518\) −6.00000 −0.263625
\(519\) −8.00000 −0.351161
\(520\) 8.00000 0.350823
\(521\) −42.0000 −1.84005 −0.920027 0.391856i \(-0.871833\pi\)
−0.920027 + 0.391856i \(0.871833\pi\)
\(522\) −2.00000 −0.0875376
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) −8.00000 −0.349482
\(525\) −2.00000 −0.0872872
\(526\) 24.0000 1.04645
\(527\) 0 0
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) −28.0000 −1.21624
\(531\) 10.0000 0.433963
\(532\) −4.00000 −0.173422
\(533\) 0 0
\(534\) 20.0000 0.865485
\(535\) 24.0000 1.03761
\(536\) 8.00000 0.345547
\(537\) 24.0000 1.03568
\(538\) 14.0000 0.603583
\(539\) 0 0
\(540\) −8.00000 −0.344265
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) 28.0000 1.20270
\(543\) 28.0000 1.20160
\(544\) 0 0
\(545\) 28.0000 1.19939
\(546\) 8.00000 0.342368
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) 6.00000 0.256307
\(549\) 8.00000 0.341432
\(550\) 0 0
\(551\) 8.00000 0.340811
\(552\) 8.00000 0.340503
\(553\) −16.0000 −0.680389
\(554\) −6.00000 −0.254916
\(555\) −24.0000 −1.01874
\(556\) −20.0000 −0.848189
\(557\) −30.0000 −1.27114 −0.635570 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) −10.0000 −0.423334
\(559\) 16.0000 0.676728
\(560\) 2.00000 0.0845154
\(561\) 0 0
\(562\) −30.0000 −1.26547
\(563\) −12.0000 −0.505740 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 20.0000 0.842152
\(565\) −28.0000 −1.17797
\(566\) 0 0
\(567\) −11.0000 −0.461957
\(568\) −4.00000 −0.167836
\(569\) −14.0000 −0.586911 −0.293455 0.955973i \(-0.594805\pi\)
−0.293455 + 0.955973i \(0.594805\pi\)
\(570\) −16.0000 −0.670166
\(571\) 28.0000 1.17176 0.585882 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) 0 0
\(573\) 16.0000 0.668410
\(574\) 0 0
\(575\) −4.00000 −0.166812
\(576\) 1.00000 0.0416667
\(577\) −42.0000 −1.74848 −0.874241 0.485491i \(-0.838641\pi\)
−0.874241 + 0.485491i \(0.838641\pi\)
\(578\) −17.0000 −0.707107
\(579\) 12.0000 0.498703
\(580\) −4.00000 −0.166091
\(581\) −4.00000 −0.165948
\(582\) 12.0000 0.497416
\(583\) 0 0
\(584\) −4.00000 −0.165521
\(585\) 8.00000 0.330759
\(586\) 0 0
\(587\) −42.0000 −1.73353 −0.866763 0.498721i \(-0.833803\pi\)
−0.866763 + 0.498721i \(0.833803\pi\)
\(588\) 2.00000 0.0824786
\(589\) 40.0000 1.64817
\(590\) 20.0000 0.823387
\(591\) 36.0000 1.48084
\(592\) −6.00000 −0.246598
\(593\) −12.0000 −0.492781 −0.246390 0.969171i \(-0.579245\pi\)
−0.246390 + 0.969171i \(0.579245\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −22.0000 −0.901155
\(597\) −28.0000 −1.14596
\(598\) 16.0000 0.654289
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) −2.00000 −0.0816497
\(601\) 24.0000 0.978980 0.489490 0.872009i \(-0.337183\pi\)
0.489490 + 0.872009i \(0.337183\pi\)
\(602\) 4.00000 0.163028
\(603\) 8.00000 0.325785
\(604\) −16.0000 −0.651031
\(605\) 0 0
\(606\) −24.0000 −0.974933
\(607\) 24.0000 0.974130 0.487065 0.873366i \(-0.338067\pi\)
0.487065 + 0.873366i \(0.338067\pi\)
\(608\) −4.00000 −0.162221
\(609\) −4.00000 −0.162088
\(610\) 16.0000 0.647821
\(611\) 40.0000 1.61823
\(612\) 0 0
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) −16.0000 −0.645707
\(615\) 0 0
\(616\) 0 0
\(617\) 38.0000 1.52982 0.764911 0.644136i \(-0.222783\pi\)
0.764911 + 0.644136i \(0.222783\pi\)
\(618\) 4.00000 0.160904
\(619\) −2.00000 −0.0803868 −0.0401934 0.999192i \(-0.512797\pi\)
−0.0401934 + 0.999192i \(0.512797\pi\)
\(620\) −20.0000 −0.803219
\(621\) −16.0000 −0.642058
\(622\) −6.00000 −0.240578
\(623\) 10.0000 0.400642
\(624\) 8.00000 0.320256
\(625\) −19.0000 −0.760000
\(626\) −6.00000 −0.239808
\(627\) 0 0
\(628\) 10.0000 0.399043
\(629\) 0 0
\(630\) 2.00000 0.0796819
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) −16.0000 −0.636446
\(633\) 8.00000 0.317971
\(634\) −18.0000 −0.714871
\(635\) 32.0000 1.26988
\(636\) −28.0000 −1.11027
\(637\) 4.00000 0.158486
\(638\) 0 0
\(639\) −4.00000 −0.158238
\(640\) 2.00000 0.0790569
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 24.0000 0.947204
\(643\) 22.0000 0.867595 0.433798 0.901010i \(-0.357173\pi\)
0.433798 + 0.901010i \(0.357173\pi\)
\(644\) 4.00000 0.157622
\(645\) 16.0000 0.629999
\(646\) 0 0
\(647\) 6.00000 0.235884 0.117942 0.993020i \(-0.462370\pi\)
0.117942 + 0.993020i \(0.462370\pi\)
\(648\) −11.0000 −0.432121
\(649\) 0 0
\(650\) −4.00000 −0.156893
\(651\) −20.0000 −0.783862
\(652\) 24.0000 0.939913
\(653\) 46.0000 1.80012 0.900060 0.435767i \(-0.143523\pi\)
0.900060 + 0.435767i \(0.143523\pi\)
\(654\) 28.0000 1.09489
\(655\) −16.0000 −0.625172
\(656\) 0 0
\(657\) −4.00000 −0.156055
\(658\) 10.0000 0.389841
\(659\) −20.0000 −0.779089 −0.389545 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) 0 0
\(661\) −38.0000 −1.47803 −0.739014 0.673690i \(-0.764708\pi\)
−0.739014 + 0.673690i \(0.764708\pi\)
\(662\) −20.0000 −0.777322
\(663\) 0 0
\(664\) −4.00000 −0.155230
\(665\) −8.00000 −0.310227
\(666\) −6.00000 −0.232495
\(667\) −8.00000 −0.309761
\(668\) 8.00000 0.309529
\(669\) −28.0000 −1.08254
\(670\) 16.0000 0.618134
\(671\) 0 0
\(672\) 2.00000 0.0771517
\(673\) 10.0000 0.385472 0.192736 0.981251i \(-0.438264\pi\)
0.192736 + 0.981251i \(0.438264\pi\)
\(674\) 34.0000 1.30963
\(675\) 4.00000 0.153960
\(676\) 3.00000 0.115385
\(677\) −12.0000 −0.461197 −0.230599 0.973049i \(-0.574068\pi\)
−0.230599 + 0.973049i \(0.574068\pi\)
\(678\) −28.0000 −1.07533
\(679\) 6.00000 0.230259
\(680\) 0 0
\(681\) −16.0000 −0.613121
\(682\) 0 0
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) −4.00000 −0.152944
\(685\) 12.0000 0.458496
\(686\) 1.00000 0.0381802
\(687\) −20.0000 −0.763048
\(688\) 4.00000 0.152499
\(689\) −56.0000 −2.13343
\(690\) 16.0000 0.609110
\(691\) 42.0000 1.59776 0.798878 0.601494i \(-0.205427\pi\)
0.798878 + 0.601494i \(0.205427\pi\)
\(692\) −4.00000 −0.152057
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) −40.0000 −1.51729
\(696\) −4.00000 −0.151620
\(697\) 0 0
\(698\) −32.0000 −1.21122
\(699\) −12.0000 −0.453882
\(700\) −1.00000 −0.0377964
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) −16.0000 −0.603881
\(703\) 24.0000 0.905177
\(704\) 0 0
\(705\) 40.0000 1.50649
\(706\) 2.00000 0.0752710
\(707\) −12.0000 −0.451306
\(708\) 20.0000 0.751646
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) −8.00000 −0.300235
\(711\) −16.0000 −0.600047
\(712\) 10.0000 0.374766
\(713\) −40.0000 −1.49801
\(714\) 0 0
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 16.0000 0.597531
\(718\) 0 0
\(719\) 6.00000 0.223762 0.111881 0.993722i \(-0.464312\pi\)
0.111881 + 0.993722i \(0.464312\pi\)
\(720\) 2.00000 0.0745356
\(721\) 2.00000 0.0744839
\(722\) −3.00000 −0.111648
\(723\) −16.0000 −0.595046
\(724\) 14.0000 0.520306
\(725\) 2.00000 0.0742781
\(726\) 0 0
\(727\) 46.0000 1.70605 0.853023 0.521874i \(-0.174767\pi\)
0.853023 + 0.521874i \(0.174767\pi\)
\(728\) 4.00000 0.148250
\(729\) 13.0000 0.481481
\(730\) −8.00000 −0.296093
\(731\) 0 0
\(732\) 16.0000 0.591377
\(733\) 8.00000 0.295487 0.147743 0.989026i \(-0.452799\pi\)
0.147743 + 0.989026i \(0.452799\pi\)
\(734\) 18.0000 0.664392
\(735\) 4.00000 0.147542
\(736\) 4.00000 0.147442
\(737\) 0 0
\(738\) 0 0
\(739\) 52.0000 1.91285 0.956425 0.291977i \(-0.0943129\pi\)
0.956425 + 0.291977i \(0.0943129\pi\)
\(740\) −12.0000 −0.441129
\(741\) −32.0000 −1.17555
\(742\) −14.0000 −0.513956
\(743\) −16.0000 −0.586983 −0.293492 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) −20.0000 −0.733236
\(745\) −44.0000 −1.61204
\(746\) 34.0000 1.24483
\(747\) −4.00000 −0.146352
\(748\) 0 0
\(749\) 12.0000 0.438470
\(750\) −24.0000 −0.876356
\(751\) 20.0000 0.729810 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) 10.0000 0.364662
\(753\) −52.0000 −1.89499
\(754\) −8.00000 −0.291343
\(755\) −32.0000 −1.16460
\(756\) −4.00000 −0.145479
\(757\) 30.0000 1.09037 0.545184 0.838316i \(-0.316460\pi\)
0.545184 + 0.838316i \(0.316460\pi\)
\(758\) 8.00000 0.290573
\(759\) 0 0
\(760\) −8.00000 −0.290191
\(761\) 12.0000 0.435000 0.217500 0.976060i \(-0.430210\pi\)
0.217500 + 0.976060i \(0.430210\pi\)
\(762\) 32.0000 1.15924
\(763\) 14.0000 0.506834
\(764\) 8.00000 0.289430
\(765\) 0 0
\(766\) 14.0000 0.505841
\(767\) 40.0000 1.44432
\(768\) 2.00000 0.0721688
\(769\) −4.00000 −0.144244 −0.0721218 0.997396i \(-0.522977\pi\)
−0.0721218 + 0.997396i \(0.522977\pi\)
\(770\) 0 0
\(771\) 4.00000 0.144056
\(772\) 6.00000 0.215945
\(773\) −34.0000 −1.22290 −0.611448 0.791285i \(-0.709412\pi\)
−0.611448 + 0.791285i \(0.709412\pi\)
\(774\) 4.00000 0.143777
\(775\) 10.0000 0.359211
\(776\) 6.00000 0.215387
\(777\) −12.0000 −0.430498
\(778\) −18.0000 −0.645331
\(779\) 0 0
\(780\) 16.0000 0.572892
\(781\) 0 0
\(782\) 0 0
\(783\) 8.00000 0.285897
\(784\) 1.00000 0.0357143
\(785\) 20.0000 0.713831
\(786\) −16.0000 −0.570701
\(787\) −20.0000 −0.712923 −0.356462 0.934310i \(-0.616017\pi\)
−0.356462 + 0.934310i \(0.616017\pi\)
\(788\) 18.0000 0.641223
\(789\) 48.0000 1.70885
\(790\) −32.0000 −1.13851
\(791\) −14.0000 −0.497783
\(792\) 0 0
\(793\) 32.0000 1.13635
\(794\) 18.0000 0.638796
\(795\) −56.0000 −1.98612
\(796\) −14.0000 −0.496217
\(797\) 26.0000 0.920967 0.460484 0.887668i \(-0.347676\pi\)
0.460484 + 0.887668i \(0.347676\pi\)
\(798\) −8.00000 −0.283197
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) 10.0000 0.353333
\(802\) 10.0000 0.353112
\(803\) 0 0
\(804\) 16.0000 0.564276
\(805\) 8.00000 0.281963
\(806\) −40.0000 −1.40894
\(807\) 28.0000 0.985647
\(808\) −12.0000 −0.422159
\(809\) −26.0000 −0.914111 −0.457056 0.889438i \(-0.651096\pi\)
−0.457056 + 0.889438i \(0.651096\pi\)
\(810\) −22.0000 −0.773001
\(811\) 40.0000 1.40459 0.702295 0.711886i \(-0.252159\pi\)
0.702295 + 0.711886i \(0.252159\pi\)
\(812\) −2.00000 −0.0701862
\(813\) 56.0000 1.96401
\(814\) 0 0
\(815\) 48.0000 1.68137
\(816\) 0 0
\(817\) −16.0000 −0.559769
\(818\) 4.00000 0.139857
\(819\) 4.00000 0.139771
\(820\) 0 0
\(821\) 50.0000 1.74501 0.872506 0.488603i \(-0.162493\pi\)
0.872506 + 0.488603i \(0.162493\pi\)
\(822\) 12.0000 0.418548
\(823\) 8.00000 0.278862 0.139431 0.990232i \(-0.455473\pi\)
0.139431 + 0.990232i \(0.455473\pi\)
\(824\) 2.00000 0.0696733
\(825\) 0 0
\(826\) 10.0000 0.347945
\(827\) 20.0000 0.695468 0.347734 0.937593i \(-0.386951\pi\)
0.347734 + 0.937593i \(0.386951\pi\)
\(828\) 4.00000 0.139010
\(829\) −14.0000 −0.486240 −0.243120 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) −8.00000 −0.277684
\(831\) −12.0000 −0.416275
\(832\) 4.00000 0.138675
\(833\) 0 0
\(834\) −40.0000 −1.38509
\(835\) 16.0000 0.553703
\(836\) 0 0
\(837\) 40.0000 1.38260
\(838\) −30.0000 −1.03633
\(839\) 30.0000 1.03572 0.517858 0.855467i \(-0.326730\pi\)
0.517858 + 0.855467i \(0.326730\pi\)
\(840\) 4.00000 0.138013
\(841\) −25.0000 −0.862069
\(842\) 10.0000 0.344623
\(843\) −60.0000 −2.06651
\(844\) 4.00000 0.137686
\(845\) 6.00000 0.206406
\(846\) 10.0000 0.343807
\(847\) 0 0
\(848\) −14.0000 −0.480762
\(849\) 0 0
\(850\) 0 0
\(851\) −24.0000 −0.822709
\(852\) −8.00000 −0.274075
\(853\) −4.00000 −0.136957 −0.0684787 0.997653i \(-0.521815\pi\)
−0.0684787 + 0.997653i \(0.521815\pi\)
\(854\) 8.00000 0.273754
\(855\) −8.00000 −0.273594
\(856\) 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
\(859\) 14.0000 0.477674 0.238837 0.971060i \(-0.423234\pi\)
0.238837 + 0.971060i \(0.423234\pi\)
\(860\) 8.00000 0.272798
\(861\) 0 0
\(862\) 16.0000 0.544962
\(863\) 40.0000 1.36162 0.680808 0.732462i \(-0.261629\pi\)
0.680808 + 0.732462i \(0.261629\pi\)
\(864\) −4.00000 −0.136083
\(865\) −8.00000 −0.272008
\(866\) −10.0000 −0.339814
\(867\) −34.0000 −1.15470
\(868\) −10.0000 −0.339422
\(869\) 0 0
\(870\) −8.00000 −0.271225
\(871\) 32.0000 1.08428
\(872\) 14.0000 0.474100
\(873\) 6.00000 0.203069
\(874\) −16.0000 −0.541208
\(875\) −12.0000 −0.405674
\(876\) −8.00000 −0.270295
\(877\) 22.0000 0.742887 0.371444 0.928456i \(-0.378863\pi\)
0.371444 + 0.928456i \(0.378863\pi\)
\(878\) 28.0000 0.944954
\(879\) 0 0
\(880\) 0 0
\(881\) −42.0000 −1.41502 −0.707508 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) 1.00000 0.0336718
\(883\) −4.00000 −0.134611 −0.0673054 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) 0 0
\(885\) 40.0000 1.34459
\(886\) 4.00000 0.134383
\(887\) 28.0000 0.940148 0.470074 0.882627i \(-0.344227\pi\)
0.470074 + 0.882627i \(0.344227\pi\)
\(888\) −12.0000 −0.402694
\(889\) 16.0000 0.536623
\(890\) 20.0000 0.670402
\(891\) 0 0
\(892\) −14.0000 −0.468755
\(893\) −40.0000 −1.33855
\(894\) −44.0000 −1.47158
\(895\) 24.0000 0.802232
\(896\) 1.00000 0.0334077
\(897\) 32.0000 1.06845
\(898\) 6.00000 0.200223
\(899\) 20.0000 0.667037
\(900\) −1.00000 −0.0333333
\(901\) 0 0
\(902\) 0 0
\(903\) 8.00000 0.266223
\(904\) −14.0000 −0.465633
\(905\) 28.0000 0.930751
\(906\) −32.0000 −1.06313
\(907\) 48.0000 1.59381 0.796907 0.604102i \(-0.206468\pi\)
0.796907 + 0.604102i \(0.206468\pi\)
\(908\) −8.00000 −0.265489
\(909\) −12.0000 −0.398015
\(910\) 8.00000 0.265197
\(911\) −4.00000 −0.132526 −0.0662630 0.997802i \(-0.521108\pi\)
−0.0662630 + 0.997802i \(0.521108\pi\)
\(912\) −8.00000 −0.264906
\(913\) 0 0
\(914\) 38.0000 1.25693
\(915\) 32.0000 1.05789
\(916\) −10.0000 −0.330409
\(917\) −8.00000 −0.264183
\(918\) 0 0
\(919\) 56.0000 1.84727 0.923635 0.383274i \(-0.125203\pi\)
0.923635 + 0.383274i \(0.125203\pi\)
\(920\) 8.00000 0.263752
\(921\) −32.0000 −1.05444
\(922\) −32.0000 −1.05386
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) 6.00000 0.197279
\(926\) 12.0000 0.394344
\(927\) 2.00000 0.0656886
\(928\) −2.00000 −0.0656532
\(929\) −10.0000 −0.328089 −0.164045 0.986453i \(-0.552454\pi\)
−0.164045 + 0.986453i \(0.552454\pi\)
\(930\) −40.0000 −1.31165
\(931\) −4.00000 −0.131095
\(932\) −6.00000 −0.196537
\(933\) −12.0000 −0.392862
\(934\) 14.0000 0.458094
\(935\) 0 0
\(936\) 4.00000 0.130744
\(937\) 52.0000 1.69877 0.849383 0.527777i \(-0.176974\pi\)
0.849383 + 0.527777i \(0.176974\pi\)
\(938\) 8.00000 0.261209
\(939\) −12.0000 −0.391605
\(940\) 20.0000 0.652328
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 20.0000 0.651635
\(943\) 0 0
\(944\) 10.0000 0.325472
\(945\) −8.00000 −0.260240
\(946\) 0 0
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) −32.0000 −1.03931
\(949\) −16.0000 −0.519382
\(950\) 4.00000 0.129777
\(951\) −36.0000 −1.16738
\(952\) 0 0
\(953\) −34.0000 −1.10137 −0.550684 0.834714i \(-0.685633\pi\)
−0.550684 + 0.834714i \(0.685633\pi\)
\(954\) −14.0000 −0.453267
\(955\) 16.0000 0.517748
\(956\) 8.00000 0.258738
\(957\) 0 0
\(958\) −12.0000 −0.387702
\(959\) 6.00000 0.193750
\(960\) 4.00000 0.129099
\(961\) 69.0000 2.22581
\(962\) −24.0000 −0.773791
\(963\) 12.0000 0.386695
\(964\) −8.00000 −0.257663
\(965\) 12.0000 0.386294
\(966\) 8.00000 0.257396
\(967\) 24.0000 0.771788 0.385894 0.922543i \(-0.373893\pi\)
0.385894 + 0.922543i \(0.373893\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 12.0000 0.385297
\(971\) 30.0000 0.962746 0.481373 0.876516i \(-0.340138\pi\)
0.481373 + 0.876516i \(0.340138\pi\)
\(972\) −10.0000 −0.320750
\(973\) −20.0000 −0.641171
\(974\) 12.0000 0.384505
\(975\) −8.00000 −0.256205
\(976\) 8.00000 0.256074
\(977\) −22.0000 −0.703842 −0.351921 0.936030i \(-0.614471\pi\)
−0.351921 + 0.936030i \(0.614471\pi\)
\(978\) 48.0000 1.53487
\(979\) 0 0
\(980\) 2.00000 0.0638877
\(981\) 14.0000 0.446986
\(982\) 28.0000 0.893516
\(983\) 26.0000 0.829271 0.414636 0.909988i \(-0.363909\pi\)
0.414636 + 0.909988i \(0.363909\pi\)
\(984\) 0 0
\(985\) 36.0000 1.14706
\(986\) 0 0
\(987\) 20.0000 0.636607
\(988\) −16.0000 −0.509028
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) −4.00000 −0.127064 −0.0635321 0.997980i \(-0.520237\pi\)
−0.0635321 + 0.997980i \(0.520237\pi\)
\(992\) −10.0000 −0.317500
\(993\) −40.0000 −1.26936
\(994\) −4.00000 −0.126872
\(995\) −28.0000 −0.887660
\(996\) −8.00000 −0.253490
\(997\) −36.0000 −1.14013 −0.570066 0.821599i \(-0.693082\pi\)
−0.570066 + 0.821599i \(0.693082\pi\)
\(998\) −16.0000 −0.506471
\(999\) 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 1694.2.a.i.1.1 1
11.10 odd 2 154.2.a.b.1.1 1
33.32 even 2 1386.2.a.f.1.1 1
44.43 even 2 1232.2.a.c.1.1 1
55.32 even 4 3850.2.c.d.1849.1 2
55.43 even 4 3850.2.c.d.1849.2 2
55.54 odd 2 3850.2.a.o.1.1 1
77.10 even 6 1078.2.e.l.177.1 2
77.32 odd 6 1078.2.e.h.177.1 2
77.54 even 6 1078.2.e.l.67.1 2
77.65 odd 6 1078.2.e.h.67.1 2
77.76 even 2 1078.2.a.b.1.1 1
88.21 odd 2 4928.2.a.d.1.1 1
88.43 even 2 4928.2.a.bf.1.1 1
231.230 odd 2 9702.2.a.bz.1.1 1
308.307 odd 2 8624.2.a.z.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.a.b.1.1 1 11.10 odd 2
1078.2.a.b.1.1 1 77.76 even 2
1078.2.e.h.67.1 2 77.65 odd 6
1078.2.e.h.177.1 2 77.32 odd 6
1078.2.e.l.67.1 2 77.54 even 6
1078.2.e.l.177.1 2 77.10 even 6
1232.2.a.c.1.1 1 44.43 even 2
1386.2.a.f.1.1 1 33.32 even 2
1694.2.a.i.1.1 1 1.1 even 1 trivial
3850.2.a.o.1.1 1 55.54 odd 2
3850.2.c.d.1849.1 2 55.32 even 4
3850.2.c.d.1849.2 2 55.43 even 4
4928.2.a.d.1.1 1 88.21 odd 2
4928.2.a.bf.1.1 1 88.43 even 2
8624.2.a.z.1.1 1 308.307 odd 2
9702.2.a.bz.1.1 1 231.230 odd 2