Properties

Label 1386.2.a.e.1.1
Level $1386$
Weight $2$
Character 1386.1
Self dual yes
Analytic conductor $11.067$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1386.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.0672657201\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 462)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1386.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +4.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +4.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} -4.00000 q^{10} +1.00000 q^{11} -6.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} -2.00000 q^{19} +4.00000 q^{20} -1.00000 q^{22} +8.00000 q^{23} +11.0000 q^{25} +6.00000 q^{26} +1.00000 q^{28} +6.00000 q^{29} +6.00000 q^{31} -1.00000 q^{32} -4.00000 q^{34} +4.00000 q^{35} -6.00000 q^{37} +2.00000 q^{38} -4.00000 q^{40} -12.0000 q^{41} +4.00000 q^{43} +1.00000 q^{44} -8.00000 q^{46} -6.00000 q^{47} +1.00000 q^{49} -11.0000 q^{50} -6.00000 q^{52} -2.00000 q^{53} +4.00000 q^{55} -1.00000 q^{56} -6.00000 q^{58} +10.0000 q^{61} -6.00000 q^{62} +1.00000 q^{64} -24.0000 q^{65} +4.00000 q^{67} +4.00000 q^{68} -4.00000 q^{70} +12.0000 q^{71} +6.00000 q^{74} -2.00000 q^{76} +1.00000 q^{77} -16.0000 q^{79} +4.00000 q^{80} +12.0000 q^{82} +14.0000 q^{83} +16.0000 q^{85} -4.00000 q^{86} -1.00000 q^{88} +14.0000 q^{89} -6.00000 q^{91} +8.00000 q^{92} +6.00000 q^{94} -8.00000 q^{95} -14.0000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 4.00000 1.78885 0.894427 0.447214i \(-0.147584\pi\)
0.894427 + 0.447214i \(0.147584\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −4.00000 −1.26491
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 4.00000 0.894427
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) 8.00000 1.66812 0.834058 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 0 0
\(25\) 11.0000 2.20000
\(26\) 6.00000 1.17670
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −4.00000 −0.685994
\(35\) 4.00000 0.676123
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 2.00000 0.324443
\(39\) 0 0
\(40\) −4.00000 −0.632456
\(41\) −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
\(46\) −8.00000 −1.17954
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −11.0000 −1.55563
\(51\) 0 0
\(52\) −6.00000 −0.832050
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) −6.00000 −0.762001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −24.0000 −2.97683
\(66\) 0 0
\(67\) 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
\(70\) −4.00000 −0.478091
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 6.00000 0.697486
\(75\) 0 0
\(76\) −2.00000 −0.229416
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 4.00000 0.447214
\(81\) 0 0
\(82\) 12.0000 1.32518
\(83\) 14.0000 1.53670 0.768350 0.640030i \(-0.221078\pi\)
0.768350 + 0.640030i \(0.221078\pi\)
\(84\) 0 0
\(85\) 16.0000 1.73544
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) 8.00000 0.834058
\(93\) 0 0
\(94\) 6.00000 0.618853
\(95\) −8.00000 −0.820783
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 11.0000 1.10000
\(101\) −14.0000 −1.39305 −0.696526 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) 0 0
\(103\) −6.00000 −0.591198 −0.295599 0.955312i \(-0.595519\pi\)
−0.295599 + 0.955312i \(0.595519\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) 2.00000 0.194257
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 0 0
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) −4.00000 −0.381385
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 0 0
\(115\) 32.0000 2.98402
\(116\) 6.00000 0.557086
\(117\) 0 0
\(118\) 0 0
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −10.0000 −0.905357
\(123\) 0 0
\(124\) 6.00000 0.538816
\(125\) 24.0000 2.14663
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 24.0000 2.10494
\(131\) 2.00000 0.174741 0.0873704 0.996176i \(-0.472154\pi\)
0.0873704 + 0.996176i \(0.472154\pi\)
\(132\) 0 0
\(133\) −2.00000 −0.173422
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) −14.0000 −1.19610 −0.598050 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) 0 0
\(139\) 2.00000 0.169638 0.0848189 0.996396i \(-0.472969\pi\)
0.0848189 + 0.996396i \(0.472969\pi\)
\(140\) 4.00000 0.338062
\(141\) 0 0
\(142\) −12.0000 −1.00702
\(143\) −6.00000 −0.501745
\(144\) 0 0
\(145\) 24.0000 1.99309
\(146\) 0 0
\(147\) 0 0
\(148\) −6.00000 −0.493197
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 2.00000 0.162221
\(153\) 0 0
\(154\) −1.00000 −0.0805823
\(155\) 24.0000 1.92773
\(156\) 0 0
\(157\) −4.00000 −0.319235 −0.159617 0.987179i \(-0.551026\pi\)
−0.159617 + 0.987179i \(0.551026\pi\)
\(158\) 16.0000 1.27289
\(159\) 0 0
\(160\) −4.00000 −0.316228
\(161\) 8.00000 0.630488
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) −12.0000 −0.937043
\(165\) 0 0
\(166\) −14.0000 −1.08661
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) −16.0000 −1.22714
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) −2.00000 −0.152057 −0.0760286 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) 0 0
\(175\) 11.0000 0.831522
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) −14.0000 −1.04934
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) −8.00000 −0.594635 −0.297318 0.954779i \(-0.596092\pi\)
−0.297318 + 0.954779i \(0.596092\pi\)
\(182\) 6.00000 0.444750
\(183\) 0 0
\(184\) −8.00000 −0.589768
\(185\) −24.0000 −1.76452
\(186\) 0 0
\(187\) 4.00000 0.292509
\(188\) −6.00000 −0.437595
\(189\) 0 0
\(190\) 8.00000 0.580381
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −10.0000 −0.712470 −0.356235 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) 0 0
\(199\) 2.00000 0.141776 0.0708881 0.997484i \(-0.477417\pi\)
0.0708881 + 0.997484i \(0.477417\pi\)
\(200\) −11.0000 −0.777817
\(201\) 0 0
\(202\) 14.0000 0.985037
\(203\) 6.00000 0.421117
\(204\) 0 0
\(205\) −48.0000 −3.35247
\(206\) 6.00000 0.418040
\(207\) 0 0
\(208\) −6.00000 −0.416025
\(209\) −2.00000 −0.138343
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) −2.00000 −0.137361
\(213\) 0 0
\(214\) −8.00000 −0.546869
\(215\) 16.0000 1.09119
\(216\) 0 0
\(217\) 6.00000 0.407307
\(218\) −6.00000 −0.406371
\(219\) 0 0
\(220\) 4.00000 0.269680
\(221\) −24.0000 −1.61441
\(222\) 0 0
\(223\) 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
\(226\) 14.0000 0.931266
\(227\) −6.00000 −0.398234 −0.199117 0.979976i \(-0.563807\pi\)
−0.199117 + 0.979976i \(0.563807\pi\)
\(228\) 0 0
\(229\) 8.00000 0.528655 0.264327 0.964433i \(-0.414850\pi\)
0.264327 + 0.964433i \(0.414850\pi\)
\(230\) −32.0000 −2.11002
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) 14.0000 0.917170 0.458585 0.888650i \(-0.348356\pi\)
0.458585 + 0.888650i \(0.348356\pi\)
\(234\) 0 0
\(235\) −24.0000 −1.56559
\(236\) 0 0
\(237\) 0 0
\(238\) −4.00000 −0.259281
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 0 0
\(241\) −8.00000 −0.515325 −0.257663 0.966235i \(-0.582952\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) 10.0000 0.640184
\(245\) 4.00000 0.255551
\(246\) 0 0
\(247\) 12.0000 0.763542
\(248\) −6.00000 −0.381000
\(249\) 0 0
\(250\) −24.0000 −1.51789
\(251\) 16.0000 1.00991 0.504956 0.863145i \(-0.331509\pi\)
0.504956 + 0.863145i \(0.331509\pi\)
\(252\) 0 0
\(253\) 8.00000 0.502956
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −26.0000 −1.62184 −0.810918 0.585160i \(-0.801032\pi\)
−0.810918 + 0.585160i \(0.801032\pi\)
\(258\) 0 0
\(259\) −6.00000 −0.372822
\(260\) −24.0000 −1.48842
\(261\) 0 0
\(262\) −2.00000 −0.123560
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) −8.00000 −0.491436
\(266\) 2.00000 0.122628
\(267\) 0 0
\(268\) 4.00000 0.244339
\(269\) 4.00000 0.243884 0.121942 0.992537i \(-0.461088\pi\)
0.121942 + 0.992537i \(0.461088\pi\)
\(270\) 0 0
\(271\) −4.00000 −0.242983 −0.121491 0.992592i \(-0.538768\pi\)
−0.121491 + 0.992592i \(0.538768\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) 14.0000 0.845771
\(275\) 11.0000 0.663325
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) −2.00000 −0.119952
\(279\) 0 0
\(280\) −4.00000 −0.239046
\(281\) −26.0000 −1.55103 −0.775515 0.631329i \(-0.782510\pi\)
−0.775515 + 0.631329i \(0.782510\pi\)
\(282\) 0 0
\(283\) −14.0000 −0.832214 −0.416107 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) 6.00000 0.354787
\(287\) −12.0000 −0.708338
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) −24.0000 −1.40933
\(291\) 0 0
\(292\) 0 0
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 6.00000 0.348743
\(297\) 0 0
\(298\) −10.0000 −0.579284
\(299\) −48.0000 −2.77591
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 8.00000 0.460348
\(303\) 0 0
\(304\) −2.00000 −0.114708
\(305\) 40.0000 2.29039
\(306\) 0 0
\(307\) −14.0000 −0.799022 −0.399511 0.916728i \(-0.630820\pi\)
−0.399511 + 0.916728i \(0.630820\pi\)
\(308\) 1.00000 0.0569803
\(309\) 0 0
\(310\) −24.0000 −1.36311
\(311\) −10.0000 −0.567048 −0.283524 0.958965i \(-0.591504\pi\)
−0.283524 + 0.958965i \(0.591504\pi\)
\(312\) 0 0
\(313\) −34.0000 −1.92179 −0.960897 0.276907i \(-0.910691\pi\)
−0.960897 + 0.276907i \(0.910691\pi\)
\(314\) 4.00000 0.225733
\(315\) 0 0
\(316\) −16.0000 −0.900070
\(317\) 6.00000 0.336994 0.168497 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) 0 0
\(319\) 6.00000 0.335936
\(320\) 4.00000 0.223607
\(321\) 0 0
\(322\) −8.00000 −0.445823
\(323\) −8.00000 −0.445132
\(324\) 0 0
\(325\) −66.0000 −3.66102
\(326\) −4.00000 −0.221540
\(327\) 0 0
\(328\) 12.0000 0.662589
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 14.0000 0.768350
\(333\) 0 0
\(334\) −12.0000 −0.656611
\(335\) 16.0000 0.874173
\(336\) 0 0
\(337\) 6.00000 0.326841 0.163420 0.986557i \(-0.447747\pi\)
0.163420 + 0.986557i \(0.447747\pi\)
\(338\) −23.0000 −1.25104
\(339\) 0 0
\(340\) 16.0000 0.867722
\(341\) 6.00000 0.324918
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 2.00000 0.107521
\(347\) −8.00000 −0.429463 −0.214731 0.976673i \(-0.568888\pi\)
−0.214731 + 0.976673i \(0.568888\pi\)
\(348\) 0 0
\(349\) 6.00000 0.321173 0.160586 0.987022i \(-0.448662\pi\)
0.160586 + 0.987022i \(0.448662\pi\)
\(350\) −11.0000 −0.587975
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) 48.0000 2.54758
\(356\) 14.0000 0.741999
\(357\) 0 0
\(358\) −4.00000 −0.211407
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 8.00000 0.420471
\(363\) 0 0
\(364\) −6.00000 −0.314485
\(365\) 0 0
\(366\) 0 0
\(367\) −34.0000 −1.77479 −0.887393 0.461014i \(-0.847486\pi\)
−0.887393 + 0.461014i \(0.847486\pi\)
\(368\) 8.00000 0.417029
\(369\) 0 0
\(370\) 24.0000 1.24770
\(371\) −2.00000 −0.103835
\(372\) 0 0
\(373\) −14.0000 −0.724893 −0.362446 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) −4.00000 −0.206835
\(375\) 0 0
\(376\) 6.00000 0.309426
\(377\) −36.0000 −1.85409
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) −8.00000 −0.410391
\(381\) 0 0
\(382\) 0 0
\(383\) 22.0000 1.12415 0.562074 0.827087i \(-0.310004\pi\)
0.562074 + 0.827087i \(0.310004\pi\)
\(384\) 0 0
\(385\) 4.00000 0.203859
\(386\) 14.0000 0.712581
\(387\) 0 0
\(388\) −14.0000 −0.710742
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 32.0000 1.61831
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) 10.0000 0.503793
\(395\) −64.0000 −3.22019
\(396\) 0 0
\(397\) −20.0000 −1.00377 −0.501886 0.864934i \(-0.667360\pi\)
−0.501886 + 0.864934i \(0.667360\pi\)
\(398\) −2.00000 −0.100251
\(399\) 0 0
\(400\) 11.0000 0.550000
\(401\) 26.0000 1.29838 0.649189 0.760627i \(-0.275108\pi\)
0.649189 + 0.760627i \(0.275108\pi\)
\(402\) 0 0
\(403\) −36.0000 −1.79329
\(404\) −14.0000 −0.696526
\(405\) 0 0
\(406\) −6.00000 −0.297775
\(407\) −6.00000 −0.297409
\(408\) 0 0
\(409\) 4.00000 0.197787 0.0988936 0.995098i \(-0.468470\pi\)
0.0988936 + 0.995098i \(0.468470\pi\)
\(410\) 48.0000 2.37055
\(411\) 0 0
\(412\) −6.00000 −0.295599
\(413\) 0 0
\(414\) 0 0
\(415\) 56.0000 2.74893
\(416\) 6.00000 0.294174
\(417\) 0 0
\(418\) 2.00000 0.0978232
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 38.0000 1.85201 0.926003 0.377515i \(-0.123221\pi\)
0.926003 + 0.377515i \(0.123221\pi\)
\(422\) 4.00000 0.194717
\(423\) 0 0
\(424\) 2.00000 0.0971286
\(425\) 44.0000 2.13431
\(426\) 0 0
\(427\) 10.0000 0.483934
\(428\) 8.00000 0.386695
\(429\) 0 0
\(430\) −16.0000 −0.771589
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 0 0
\(433\) 26.0000 1.24948 0.624740 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) −6.00000 −0.288009
\(435\) 0 0
\(436\) 6.00000 0.287348
\(437\) −16.0000 −0.765384
\(438\) 0 0
\(439\) −20.0000 −0.954548 −0.477274 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) −4.00000 −0.190693
\(441\) 0 0
\(442\) 24.0000 1.14156
\(443\) −28.0000 −1.33032 −0.665160 0.746701i \(-0.731637\pi\)
−0.665160 + 0.746701i \(0.731637\pi\)
\(444\) 0 0
\(445\) 56.0000 2.65465
\(446\) −6.00000 −0.284108
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) −14.0000 −0.660701 −0.330350 0.943858i \(-0.607167\pi\)
−0.330350 + 0.943858i \(0.607167\pi\)
\(450\) 0 0
\(451\) −12.0000 −0.565058
\(452\) −14.0000 −0.658505
\(453\) 0 0
\(454\) 6.00000 0.281594
\(455\) −24.0000 −1.12514
\(456\) 0 0
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) −8.00000 −0.373815
\(459\) 0 0
\(460\) 32.0000 1.49201
\(461\) −14.0000 −0.652045 −0.326023 0.945362i \(-0.605709\pi\)
−0.326023 + 0.945362i \(0.605709\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) −14.0000 −0.648537
\(467\) −36.0000 −1.66588 −0.832941 0.553362i \(-0.813345\pi\)
−0.832941 + 0.553362i \(0.813345\pi\)
\(468\) 0 0
\(469\) 4.00000 0.184703
\(470\) 24.0000 1.10704
\(471\) 0 0
\(472\) 0 0
\(473\) 4.00000 0.183920
\(474\) 0 0
\(475\) −22.0000 −1.00943
\(476\) 4.00000 0.183340
\(477\) 0 0
\(478\) −16.0000 −0.731823
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) 36.0000 1.64146
\(482\) 8.00000 0.364390
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −56.0000 −2.54283
\(486\) 0 0
\(487\) 20.0000 0.906287 0.453143 0.891438i \(-0.350303\pi\)
0.453143 + 0.891438i \(0.350303\pi\)
\(488\) −10.0000 −0.452679
\(489\) 0 0
\(490\) −4.00000 −0.180702
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) 0 0
\(493\) 24.0000 1.08091
\(494\) −12.0000 −0.539906
\(495\) 0 0
\(496\) 6.00000 0.269408
\(497\) 12.0000 0.538274
\(498\) 0 0
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 24.0000 1.07331
\(501\) 0 0
\(502\) −16.0000 −0.714115
\(503\) −20.0000 −0.891756 −0.445878 0.895094i \(-0.647108\pi\)
−0.445878 + 0.895094i \(0.647108\pi\)
\(504\) 0 0
\(505\) −56.0000 −2.49197
\(506\) −8.00000 −0.355643
\(507\) 0 0
\(508\) 0 0
\(509\) −20.0000 −0.886484 −0.443242 0.896402i \(-0.646172\pi\)
−0.443242 + 0.896402i \(0.646172\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 26.0000 1.14681
\(515\) −24.0000 −1.05757
\(516\) 0 0
\(517\) −6.00000 −0.263880
\(518\) 6.00000 0.263625
\(519\) 0 0
\(520\) 24.0000 1.05247
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) −22.0000 −0.961993 −0.480996 0.876723i \(-0.659725\pi\)
−0.480996 + 0.876723i \(0.659725\pi\)
\(524\) 2.00000 0.0873704
\(525\) 0 0
\(526\) 0 0
\(527\) 24.0000 1.04546
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 8.00000 0.347498
\(531\) 0 0
\(532\) −2.00000 −0.0867110
\(533\) 72.0000 3.11867
\(534\) 0 0
\(535\) 32.0000 1.38348
\(536\) −4.00000 −0.172774
\(537\) 0 0
\(538\) −4.00000 −0.172452
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) −14.0000 −0.601907 −0.300954 0.953639i \(-0.597305\pi\)
−0.300954 + 0.953639i \(0.597305\pi\)
\(542\) 4.00000 0.171815
\(543\) 0 0
\(544\) −4.00000 −0.171499
\(545\) 24.0000 1.02805
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) −14.0000 −0.598050
\(549\) 0 0
\(550\) −11.0000 −0.469042
\(551\) −12.0000 −0.511217
\(552\) 0 0
\(553\) −16.0000 −0.680389
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) 2.00000 0.0848189
\(557\) 30.0000 1.27114 0.635570 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) 0 0
\(559\) −24.0000 −1.01509
\(560\) 4.00000 0.169031
\(561\) 0 0
\(562\) 26.0000 1.09674
\(563\) 26.0000 1.09577 0.547885 0.836554i \(-0.315433\pi\)
0.547885 + 0.836554i \(0.315433\pi\)
\(564\) 0 0
\(565\) −56.0000 −2.35594
\(566\) 14.0000 0.588464
\(567\) 0 0
\(568\) −12.0000 −0.503509
\(569\) −10.0000 −0.419222 −0.209611 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) 0 0
\(571\) −32.0000 −1.33916 −0.669579 0.742741i \(-0.733526\pi\)
−0.669579 + 0.742741i \(0.733526\pi\)
\(572\) −6.00000 −0.250873
\(573\) 0 0
\(574\) 12.0000 0.500870
\(575\) 88.0000 3.66985
\(576\) 0 0
\(577\) −34.0000 −1.41544 −0.707719 0.706494i \(-0.750276\pi\)
−0.707719 + 0.706494i \(0.750276\pi\)
\(578\) 1.00000 0.0415945
\(579\) 0 0
\(580\) 24.0000 0.996546
\(581\) 14.0000 0.580818
\(582\) 0 0
\(583\) −2.00000 −0.0828315
\(584\) 0 0
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) −24.0000 −0.990586 −0.495293 0.868726i \(-0.664939\pi\)
−0.495293 + 0.868726i \(0.664939\pi\)
\(588\) 0 0
\(589\) −12.0000 −0.494451
\(590\) 0 0
\(591\) 0 0
\(592\) −6.00000 −0.246598
\(593\) 12.0000 0.492781 0.246390 0.969171i \(-0.420755\pi\)
0.246390 + 0.969171i \(0.420755\pi\)
\(594\) 0 0
\(595\) 16.0000 0.655936
\(596\) 10.0000 0.409616
\(597\) 0 0
\(598\) 48.0000 1.96287
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 0 0
\(601\) 4.00000 0.163163 0.0815817 0.996667i \(-0.474003\pi\)
0.0815817 + 0.996667i \(0.474003\pi\)
\(602\) −4.00000 −0.163028
\(603\) 0 0
\(604\) −8.00000 −0.325515
\(605\) 4.00000 0.162623
\(606\) 0 0
\(607\) −8.00000 −0.324710 −0.162355 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) 2.00000 0.0811107
\(609\) 0 0
\(610\) −40.0000 −1.61955
\(611\) 36.0000 1.45640
\(612\) 0 0
\(613\) −6.00000 −0.242338 −0.121169 0.992632i \(-0.538664\pi\)
−0.121169 + 0.992632i \(0.538664\pi\)
\(614\) 14.0000 0.564994
\(615\) 0 0
\(616\) −1.00000 −0.0402911
\(617\) −26.0000 −1.04672 −0.523360 0.852111i \(-0.675322\pi\)
−0.523360 + 0.852111i \(0.675322\pi\)
\(618\) 0 0
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 24.0000 0.963863
\(621\) 0 0
\(622\) 10.0000 0.400963
\(623\) 14.0000 0.560898
\(624\) 0 0
\(625\) 41.0000 1.64000
\(626\) 34.0000 1.35891
\(627\) 0 0
\(628\) −4.00000 −0.159617
\(629\) −24.0000 −0.956943
\(630\) 0 0
\(631\) 28.0000 1.11466 0.557331 0.830290i \(-0.311825\pi\)
0.557331 + 0.830290i \(0.311825\pi\)
\(632\) 16.0000 0.636446
\(633\) 0 0
\(634\) −6.00000 −0.238290
\(635\) 0 0
\(636\) 0 0
\(637\) −6.00000 −0.237729
\(638\) −6.00000 −0.237542
\(639\) 0 0
\(640\) −4.00000 −0.158114
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) 32.0000 1.26196 0.630978 0.775800i \(-0.282654\pi\)
0.630978 + 0.775800i \(0.282654\pi\)
\(644\) 8.00000 0.315244
\(645\) 0 0
\(646\) 8.00000 0.314756
\(647\) −26.0000 −1.02217 −0.511083 0.859532i \(-0.670755\pi\)
−0.511083 + 0.859532i \(0.670755\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 66.0000 2.58873
\(651\) 0 0
\(652\) 4.00000 0.156652
\(653\) −30.0000 −1.17399 −0.586995 0.809590i \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) 0 0
\(655\) 8.00000 0.312586
\(656\) −12.0000 −0.468521
\(657\) 0 0
\(658\) 6.00000 0.233904
\(659\) 24.0000 0.934907 0.467454 0.884018i \(-0.345171\pi\)
0.467454 + 0.884018i \(0.345171\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) −20.0000 −0.777322
\(663\) 0 0
\(664\) −14.0000 −0.543305
\(665\) −8.00000 −0.310227
\(666\) 0 0
\(667\) 48.0000 1.85857
\(668\) 12.0000 0.464294
\(669\) 0 0
\(670\) −16.0000 −0.618134
\(671\) 10.0000 0.386046
\(672\) 0 0
\(673\) −10.0000 −0.385472 −0.192736 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) −6.00000 −0.231111
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) −34.0000 −1.30673 −0.653363 0.757045i \(-0.726642\pi\)
−0.653363 + 0.757045i \(0.726642\pi\)
\(678\) 0 0
\(679\) −14.0000 −0.537271
\(680\) −16.0000 −0.613572
\(681\) 0 0
\(682\) −6.00000 −0.229752
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 0 0
\(685\) −56.0000 −2.13965
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 4.00000 0.152499
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) 24.0000 0.913003 0.456502 0.889723i \(-0.349102\pi\)
0.456502 + 0.889723i \(0.349102\pi\)
\(692\) −2.00000 −0.0760286
\(693\) 0 0
\(694\) 8.00000 0.303676
\(695\) 8.00000 0.303457
\(696\) 0 0
\(697\) −48.0000 −1.81813
\(698\) −6.00000 −0.227103
\(699\) 0 0
\(700\) 11.0000 0.415761
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 0 0
\(703\) 12.0000 0.452589
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) −14.0000 −0.526524
\(708\) 0 0
\(709\) −30.0000 −1.12667 −0.563337 0.826227i \(-0.690483\pi\)
−0.563337 + 0.826227i \(0.690483\pi\)
\(710\) −48.0000 −1.80141
\(711\) 0 0
\(712\) −14.0000 −0.524672
\(713\) 48.0000 1.79761
\(714\) 0 0
\(715\) −24.0000 −0.897549
\(716\) 4.00000 0.149487
\(717\) 0 0
\(718\) 24.0000 0.895672
\(719\) −18.0000 −0.671287 −0.335643 0.941989i \(-0.608954\pi\)
−0.335643 + 0.941989i \(0.608954\pi\)
\(720\) 0 0
\(721\) −6.00000 −0.223452
\(722\) 15.0000 0.558242
\(723\) 0 0
\(724\) −8.00000 −0.297318
\(725\) 66.0000 2.45118
\(726\) 0 0
\(727\) −42.0000 −1.55769 −0.778847 0.627214i \(-0.784195\pi\)
−0.778847 + 0.627214i \(0.784195\pi\)
\(728\) 6.00000 0.222375
\(729\) 0 0
\(730\) 0 0
\(731\) 16.0000 0.591781
\(732\) 0 0
\(733\) 50.0000 1.84679 0.923396 0.383849i \(-0.125402\pi\)
0.923396 + 0.383849i \(0.125402\pi\)
\(734\) 34.0000 1.25496
\(735\) 0 0
\(736\) −8.00000 −0.294884
\(737\) 4.00000 0.147342
\(738\) 0 0
\(739\) −16.0000 −0.588570 −0.294285 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) −24.0000 −0.882258
\(741\) 0 0
\(742\) 2.00000 0.0734223
\(743\) −16.0000 −0.586983 −0.293492 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) 0 0
\(745\) 40.0000 1.46549
\(746\) 14.0000 0.512576
\(747\) 0 0
\(748\) 4.00000 0.146254
\(749\) 8.00000 0.292314
\(750\) 0 0
\(751\) −28.0000 −1.02173 −0.510867 0.859660i \(-0.670676\pi\)
−0.510867 + 0.859660i \(0.670676\pi\)
\(752\) −6.00000 −0.218797
\(753\) 0 0
\(754\) 36.0000 1.31104
\(755\) −32.0000 −1.16460
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) −20.0000 −0.726433
\(759\) 0 0
\(760\) 8.00000 0.290191
\(761\) −24.0000 −0.869999 −0.435000 0.900431i \(-0.643252\pi\)
−0.435000 + 0.900431i \(0.643252\pi\)
\(762\) 0 0
\(763\) 6.00000 0.217215
\(764\) 0 0
\(765\) 0 0
\(766\) −22.0000 −0.794892
\(767\) 0 0
\(768\) 0 0
\(769\) 28.0000 1.00971 0.504853 0.863205i \(-0.331547\pi\)
0.504853 + 0.863205i \(0.331547\pi\)
\(770\) −4.00000 −0.144150
\(771\) 0 0
\(772\) −14.0000 −0.503871
\(773\) 24.0000 0.863220 0.431610 0.902060i \(-0.357946\pi\)
0.431610 + 0.902060i \(0.357946\pi\)
\(774\) 0 0
\(775\) 66.0000 2.37079
\(776\) 14.0000 0.502571
\(777\) 0 0
\(778\) 6.00000 0.215110
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) 12.0000 0.429394
\(782\) −32.0000 −1.14432
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −16.0000 −0.571064
\(786\) 0 0
\(787\) −50.0000 −1.78231 −0.891154 0.453701i \(-0.850103\pi\)
−0.891154 + 0.453701i \(0.850103\pi\)
\(788\) −10.0000 −0.356235
\(789\) 0 0
\(790\) 64.0000 2.27702
\(791\) −14.0000 −0.497783
\(792\) 0 0
\(793\) −60.0000 −2.13066
\(794\) 20.0000 0.709773
\(795\) 0 0
\(796\) 2.00000 0.0708881
\(797\) −24.0000 −0.850124 −0.425062 0.905164i \(-0.639748\pi\)
−0.425062 + 0.905164i \(0.639748\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) −11.0000 −0.388909
\(801\) 0 0
\(802\) −26.0000 −0.918092
\(803\) 0 0
\(804\) 0 0
\(805\) 32.0000 1.12785
\(806\) 36.0000 1.26805
\(807\) 0 0
\(808\) 14.0000 0.492518
\(809\) 2.00000 0.0703163 0.0351581 0.999382i \(-0.488807\pi\)
0.0351581 + 0.999382i \(0.488807\pi\)
\(810\) 0 0
\(811\) 26.0000 0.912983 0.456492 0.889728i \(-0.349106\pi\)
0.456492 + 0.889728i \(0.349106\pi\)
\(812\) 6.00000 0.210559
\(813\) 0 0
\(814\) 6.00000 0.210300
\(815\) 16.0000 0.560456
\(816\) 0 0
\(817\) −8.00000 −0.279885
\(818\) −4.00000 −0.139857
\(819\) 0 0
\(820\) −48.0000 −1.67623
\(821\) 26.0000 0.907406 0.453703 0.891153i \(-0.350103\pi\)
0.453703 + 0.891153i \(0.350103\pi\)
\(822\) 0 0
\(823\) 44.0000 1.53374 0.766872 0.641800i \(-0.221812\pi\)
0.766872 + 0.641800i \(0.221812\pi\)
\(824\) 6.00000 0.209020
\(825\) 0 0
\(826\) 0 0
\(827\) 28.0000 0.973655 0.486828 0.873498i \(-0.338154\pi\)
0.486828 + 0.873498i \(0.338154\pi\)
\(828\) 0 0
\(829\) 16.0000 0.555703 0.277851 0.960624i \(-0.410378\pi\)
0.277851 + 0.960624i \(0.410378\pi\)
\(830\) −56.0000 −1.94379
\(831\) 0 0
\(832\) −6.00000 −0.208013
\(833\) 4.00000 0.138592
\(834\) 0 0
\(835\) 48.0000 1.66111
\(836\) −2.00000 −0.0691714
\(837\) 0 0
\(838\) 12.0000 0.414533
\(839\) 54.0000 1.86429 0.932144 0.362089i \(-0.117936\pi\)
0.932144 + 0.362089i \(0.117936\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −38.0000 −1.30957
\(843\) 0 0
\(844\) −4.00000 −0.137686
\(845\) 92.0000 3.16490
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) −2.00000 −0.0686803
\(849\) 0 0
\(850\) −44.0000 −1.50919
\(851\) −48.0000 −1.64542
\(852\) 0 0
\(853\) −6.00000 −0.205436 −0.102718 0.994711i \(-0.532754\pi\)
−0.102718 + 0.994711i \(0.532754\pi\)
\(854\) −10.0000 −0.342193
\(855\) 0 0
\(856\) −8.00000 −0.273434
\(857\) 40.0000 1.36637 0.683187 0.730243i \(-0.260593\pi\)
0.683187 + 0.730243i \(0.260593\pi\)
\(858\) 0 0
\(859\) 40.0000 1.36478 0.682391 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) 16.0000 0.545595
\(861\) 0 0
\(862\) −24.0000 −0.817443
\(863\) 16.0000 0.544646 0.272323 0.962206i \(-0.412208\pi\)
0.272323 + 0.962206i \(0.412208\pi\)
\(864\) 0 0
\(865\) −8.00000 −0.272008
\(866\) −26.0000 −0.883516
\(867\) 0 0
\(868\) 6.00000 0.203653
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) −24.0000 −0.813209
\(872\) −6.00000 −0.203186
\(873\) 0 0
\(874\) 16.0000 0.541208
\(875\) 24.0000 0.811348
\(876\) 0 0
\(877\) −18.0000 −0.607817 −0.303908 0.952701i \(-0.598292\pi\)
−0.303908 + 0.952701i \(0.598292\pi\)
\(878\) 20.0000 0.674967
\(879\) 0 0
\(880\) 4.00000 0.134840
\(881\) 50.0000 1.68454 0.842271 0.539054i \(-0.181218\pi\)
0.842271 + 0.539054i \(0.181218\pi\)
\(882\) 0 0
\(883\) 28.0000 0.942275 0.471138 0.882060i \(-0.343844\pi\)
0.471138 + 0.882060i \(0.343844\pi\)
\(884\) −24.0000 −0.807207
\(885\) 0 0
\(886\) 28.0000 0.940678
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −56.0000 −1.87712
\(891\) 0 0
\(892\) 6.00000 0.200895
\(893\) 12.0000 0.401565
\(894\) 0 0
\(895\) 16.0000 0.534821
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 14.0000 0.467186
\(899\) 36.0000 1.20067
\(900\) 0 0
\(901\) −8.00000 −0.266519
\(902\) 12.0000 0.399556
\(903\) 0 0
\(904\) 14.0000 0.465633
\(905\) −32.0000 −1.06372
\(906\) 0 0
\(907\) 52.0000 1.72663 0.863316 0.504664i \(-0.168384\pi\)
0.863316 + 0.504664i \(0.168384\pi\)
\(908\) −6.00000 −0.199117
\(909\) 0 0
\(910\) 24.0000 0.795592
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 0 0
\(913\) 14.0000 0.463332
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) 8.00000 0.264327
\(917\) 2.00000 0.0660458
\(918\) 0 0
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) −32.0000 −1.05501
\(921\) 0 0
\(922\) 14.0000 0.461065
\(923\) −72.0000 −2.36991
\(924\) 0 0
\(925\) −66.0000 −2.17007
\(926\) 24.0000 0.788689
\(927\) 0 0
\(928\) −6.00000 −0.196960
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 0 0
\(931\) −2.00000 −0.0655474
\(932\) 14.0000 0.458585
\(933\) 0 0
\(934\) 36.0000 1.17796
\(935\) 16.0000 0.523256
\(936\) 0 0
\(937\) −32.0000 −1.04539 −0.522697 0.852518i \(-0.675074\pi\)
−0.522697 + 0.852518i \(0.675074\pi\)
\(938\) −4.00000 −0.130605
\(939\) 0 0
\(940\) −24.0000 −0.782794
\(941\) 30.0000 0.977972 0.488986 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) 0 0
\(943\) −96.0000 −3.12619
\(944\) 0 0
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) −36.0000 −1.16984 −0.584921 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 22.0000 0.713774
\(951\) 0 0
\(952\) −4.00000 −0.129641
\(953\) −30.0000 −0.971795 −0.485898 0.874016i \(-0.661507\pi\)
−0.485898 + 0.874016i \(0.661507\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 16.0000 0.517477
\(957\) 0 0
\(958\) 24.0000 0.775405
\(959\) −14.0000 −0.452084
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) −36.0000 −1.16069
\(963\) 0 0
\(964\) −8.00000 −0.257663
\(965\) −56.0000 −1.80270
\(966\) 0 0
\(967\) −32.0000 −1.02905 −0.514525 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) 56.0000 1.79805
\(971\) 36.0000 1.15529 0.577647 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) 0 0
\(973\) 2.00000 0.0641171
\(974\) −20.0000 −0.640841
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) −2.00000 −0.0639857 −0.0319928 0.999488i \(-0.510185\pi\)
−0.0319928 + 0.999488i \(0.510185\pi\)
\(978\) 0 0
\(979\) 14.0000 0.447442
\(980\) 4.00000 0.127775
\(981\) 0 0
\(982\) −28.0000 −0.893516
\(983\) 42.0000 1.33959 0.669796 0.742545i \(-0.266382\pi\)
0.669796 + 0.742545i \(0.266382\pi\)
\(984\) 0 0
\(985\) −40.0000 −1.27451
\(986\) −24.0000 −0.764316
\(987\) 0 0
\(988\) 12.0000 0.381771
\(989\) 32.0000 1.01754
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) −6.00000 −0.190500
\(993\) 0 0
\(994\) −12.0000 −0.380617
\(995\) 8.00000 0.253617
\(996\) 0 0
\(997\) −26.0000 −0.823428 −0.411714 0.911313i \(-0.635070\pi\)
−0.411714 + 0.911313i \(0.635070\pi\)
\(998\) 28.0000 0.886325
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1386.2.a.e.1.1 1
3.2 odd 2 462.2.a.e.1.1 1
7.6 odd 2 9702.2.a.b.1.1 1
12.11 even 2 3696.2.a.p.1.1 1
21.20 even 2 3234.2.a.v.1.1 1
33.32 even 2 5082.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.a.e.1.1 1 3.2 odd 2
1386.2.a.e.1.1 1 1.1 even 1 trivial
3234.2.a.v.1.1 1 21.20 even 2
3696.2.a.p.1.1 1 12.11 even 2
5082.2.a.a.1.1 1 33.32 even 2
9702.2.a.b.1.1 1 7.6 odd 2