Properties

Label 3381.2.a.l.1.1
Level $3381$
Weight $2$
Character 3381.1
Self dual yes
Analytic conductor $26.997$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3381.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.9974209234\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 483)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3381.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} -4.00000 q^{5} -2.00000 q^{6} +1.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} -4.00000 q^{5} -2.00000 q^{6} +1.00000 q^{9} -8.00000 q^{10} -5.00000 q^{11} -2.00000 q^{12} +2.00000 q^{13} +4.00000 q^{15} -4.00000 q^{16} +2.00000 q^{18} +5.00000 q^{19} -8.00000 q^{20} -10.0000 q^{22} -1.00000 q^{23} +11.0000 q^{25} +4.00000 q^{26} -1.00000 q^{27} -2.00000 q^{29} +8.00000 q^{30} -6.00000 q^{31} -8.00000 q^{32} +5.00000 q^{33} +2.00000 q^{36} +6.00000 q^{37} +10.0000 q^{38} -2.00000 q^{39} -5.00000 q^{41} +8.00000 q^{43} -10.0000 q^{44} -4.00000 q^{45} -2.00000 q^{46} +9.00000 q^{47} +4.00000 q^{48} +22.0000 q^{50} +4.00000 q^{52} +9.00000 q^{53} -2.00000 q^{54} +20.0000 q^{55} -5.00000 q^{57} -4.00000 q^{58} -9.00000 q^{59} +8.00000 q^{60} +5.00000 q^{61} -12.0000 q^{62} -8.00000 q^{64} -8.00000 q^{65} +10.0000 q^{66} +4.00000 q^{67} +1.00000 q^{69} +12.0000 q^{71} +12.0000 q^{74} -11.0000 q^{75} +10.0000 q^{76} -4.00000 q^{78} -10.0000 q^{79} +16.0000 q^{80} +1.00000 q^{81} -10.0000 q^{82} +18.0000 q^{83} +16.0000 q^{86} +2.00000 q^{87} -10.0000 q^{89} -8.00000 q^{90} -2.00000 q^{92} +6.00000 q^{93} +18.0000 q^{94} -20.0000 q^{95} +8.00000 q^{96} +18.0000 q^{97} -5.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) −1.00000 −0.577350
\(4\) 2.00000 1.00000
\(5\) −4.00000 −1.78885 −0.894427 0.447214i \(-0.852416\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) −2.00000 −0.816497
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) −8.00000 −2.52982
\(11\) −5.00000 −1.50756 −0.753778 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) −2.00000 −0.577350
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 4.00000 1.03280
\(16\) −4.00000 −1.00000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 2.00000 0.471405
\(19\) 5.00000 1.14708 0.573539 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) −8.00000 −1.78885
\(21\) 0 0
\(22\) −10.0000 −2.13201
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 11.0000 2.20000
\(26\) 4.00000 0.784465
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 8.00000 1.46059
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) −8.00000 −1.41421
\(33\) 5.00000 0.870388
\(34\) 0 0
\(35\) 0 0
\(36\) 2.00000 0.333333
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 10.0000 1.62221
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −5.00000 −0.780869 −0.390434 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) −10.0000 −1.50756
\(45\) −4.00000 −0.596285
\(46\) −2.00000 −0.294884
\(47\) 9.00000 1.31278 0.656392 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) 4.00000 0.577350
\(49\) 0 0
\(50\) 22.0000 3.11127
\(51\) 0 0
\(52\) 4.00000 0.554700
\(53\) 9.00000 1.23625 0.618123 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) −2.00000 −0.272166
\(55\) 20.0000 2.69680
\(56\) 0 0
\(57\) −5.00000 −0.662266
\(58\) −4.00000 −0.525226
\(59\) −9.00000 −1.17170 −0.585850 0.810419i \(-0.699239\pi\)
−0.585850 + 0.810419i \(0.699239\pi\)
\(60\) 8.00000 1.03280
\(61\) 5.00000 0.640184 0.320092 0.947386i \(-0.396286\pi\)
0.320092 + 0.947386i \(0.396286\pi\)
\(62\) −12.0000 −1.52400
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) −8.00000 −0.992278
\(66\) 10.0000 1.23091
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 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 \(0.5\pi\)
\(74\) 12.0000 1.39497
\(75\) −11.0000 −1.27017
\(76\) 10.0000 1.14708
\(77\) 0 0
\(78\) −4.00000 −0.452911
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 16.0000 1.78885
\(81\) 1.00000 0.111111
\(82\) −10.0000 −1.10432
\(83\) 18.0000 1.97576 0.987878 0.155230i \(-0.0496119\pi\)
0.987878 + 0.155230i \(0.0496119\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 16.0000 1.72532
\(87\) 2.00000 0.214423
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) −8.00000 −0.843274
\(91\) 0 0
\(92\) −2.00000 −0.208514
\(93\) 6.00000 0.622171
\(94\) 18.0000 1.85656
\(95\) −20.0000 −2.05196
\(96\) 8.00000 0.816497
\(97\) 18.0000 1.82762 0.913812 0.406138i \(-0.133125\pi\)
0.913812 + 0.406138i \(0.133125\pi\)
\(98\) 0 0
\(99\) −5.00000 −0.502519
\(100\) 22.0000 2.20000
\(101\) −5.00000 −0.497519 −0.248759 0.968565i \(-0.580023\pi\)
−0.248759 + 0.968565i \(0.580023\pi\)
\(102\) 0 0
\(103\) 19.0000 1.87213 0.936063 0.351833i \(-0.114441\pi\)
0.936063 + 0.351833i \(0.114441\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 18.0000 1.74831
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) −2.00000 −0.192450
\(109\) −12.0000 −1.14939 −0.574696 0.818367i \(-0.694880\pi\)
−0.574696 + 0.818367i \(0.694880\pi\)
\(110\) 40.0000 3.81385
\(111\) −6.00000 −0.569495
\(112\) 0 0
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) −10.0000 −0.936586
\(115\) 4.00000 0.373002
\(116\) −4.00000 −0.371391
\(117\) 2.00000 0.184900
\(118\) −18.0000 −1.65703
\(119\) 0 0
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 10.0000 0.905357
\(123\) 5.00000 0.450835
\(124\) −12.0000 −1.07763
\(125\) −24.0000 −2.14663
\(126\) 0 0
\(127\) 9.00000 0.798621 0.399310 0.916816i \(-0.369250\pi\)
0.399310 + 0.916816i \(0.369250\pi\)
\(128\) 0 0
\(129\) −8.00000 −0.704361
\(130\) −16.0000 −1.40329
\(131\) 5.00000 0.436852 0.218426 0.975854i \(-0.429908\pi\)
0.218426 + 0.975854i \(0.429908\pi\)
\(132\) 10.0000 0.870388
\(133\) 0 0
\(134\) 8.00000 0.691095
\(135\) 4.00000 0.344265
\(136\) 0 0
\(137\) 9.00000 0.768922 0.384461 0.923141i \(-0.374387\pi\)
0.384461 + 0.923141i \(0.374387\pi\)
\(138\) 2.00000 0.170251
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) −9.00000 −0.757937
\(142\) 24.0000 2.01404
\(143\) −10.0000 −0.836242
\(144\) −4.00000 −0.333333
\(145\) 8.00000 0.664364
\(146\) 0 0
\(147\) 0 0
\(148\) 12.0000 0.986394
\(149\) 3.00000 0.245770 0.122885 0.992421i \(-0.460785\pi\)
0.122885 + 0.992421i \(0.460785\pi\)
\(150\) −22.0000 −1.79629
\(151\) −19.0000 −1.54620 −0.773099 0.634285i \(-0.781294\pi\)
−0.773099 + 0.634285i \(0.781294\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 24.0000 1.92773
\(156\) −4.00000 −0.320256
\(157\) −7.00000 −0.558661 −0.279330 0.960195i \(-0.590112\pi\)
−0.279330 + 0.960195i \(0.590112\pi\)
\(158\) −20.0000 −1.59111
\(159\) −9.00000 −0.713746
\(160\) 32.0000 2.52982
\(161\) 0 0
\(162\) 2.00000 0.157135
\(163\) 13.0000 1.01824 0.509119 0.860696i \(-0.329971\pi\)
0.509119 + 0.860696i \(0.329971\pi\)
\(164\) −10.0000 −0.780869
\(165\) −20.0000 −1.55700
\(166\) 36.0000 2.79414
\(167\) −19.0000 −1.47026 −0.735132 0.677924i \(-0.762880\pi\)
−0.735132 + 0.677924i \(0.762880\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 5.00000 0.382360
\(172\) 16.0000 1.21999
\(173\) −2.00000 −0.152057 −0.0760286 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) 4.00000 0.303239
\(175\) 0 0
\(176\) 20.0000 1.50756
\(177\) 9.00000 0.676481
\(178\) −20.0000 −1.49906
\(179\) 8.00000 0.597948 0.298974 0.954261i \(-0.403356\pi\)
0.298974 + 0.954261i \(0.403356\pi\)
\(180\) −8.00000 −0.596285
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) −5.00000 −0.369611
\(184\) 0 0
\(185\) −24.0000 −1.76452
\(186\) 12.0000 0.879883
\(187\) 0 0
\(188\) 18.0000 1.31278
\(189\) 0 0
\(190\) −40.0000 −2.90191
\(191\) −23.0000 −1.66422 −0.832111 0.554609i \(-0.812868\pi\)
−0.832111 + 0.554609i \(0.812868\pi\)
\(192\) 8.00000 0.577350
\(193\) −19.0000 −1.36765 −0.683825 0.729646i \(-0.739685\pi\)
−0.683825 + 0.729646i \(0.739685\pi\)
\(194\) 36.0000 2.58465
\(195\) 8.00000 0.572892
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) −10.0000 −0.710669
\(199\) −3.00000 −0.212664 −0.106332 0.994331i \(-0.533911\pi\)
−0.106332 + 0.994331i \(0.533911\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) −10.0000 −0.703598
\(203\) 0 0
\(204\) 0 0
\(205\) 20.0000 1.39686
\(206\) 38.0000 2.64759
\(207\) −1.00000 −0.0695048
\(208\) −8.00000 −0.554700
\(209\) −25.0000 −1.72929
\(210\) 0 0
\(211\) 13.0000 0.894957 0.447478 0.894295i \(-0.352322\pi\)
0.447478 + 0.894295i \(0.352322\pi\)
\(212\) 18.0000 1.23625
\(213\) −12.0000 −0.822226
\(214\) 8.00000 0.546869
\(215\) −32.0000 −2.18238
\(216\) 0 0
\(217\) 0 0
\(218\) −24.0000 −1.62549
\(219\) 0 0
\(220\) 40.0000 2.69680
\(221\) 0 0
\(222\) −12.0000 −0.805387
\(223\) −10.0000 −0.669650 −0.334825 0.942280i \(-0.608677\pi\)
−0.334825 + 0.942280i \(0.608677\pi\)
\(224\) 0 0
\(225\) 11.0000 0.733333
\(226\) −4.00000 −0.266076
\(227\) 18.0000 1.19470 0.597351 0.801980i \(-0.296220\pi\)
0.597351 + 0.801980i \(0.296220\pi\)
\(228\) −10.0000 −0.662266
\(229\) 13.0000 0.859064 0.429532 0.903052i \(-0.358679\pi\)
0.429532 + 0.903052i \(0.358679\pi\)
\(230\) 8.00000 0.527504
\(231\) 0 0
\(232\) 0 0
\(233\) −12.0000 −0.786146 −0.393073 0.919507i \(-0.628588\pi\)
−0.393073 + 0.919507i \(0.628588\pi\)
\(234\) 4.00000 0.261488
\(235\) −36.0000 −2.34838
\(236\) −18.0000 −1.17170
\(237\) 10.0000 0.649570
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) −16.0000 −1.03280
\(241\) −17.0000 −1.09507 −0.547533 0.836784i \(-0.684433\pi\)
−0.547533 + 0.836784i \(0.684433\pi\)
\(242\) 28.0000 1.79991
\(243\) −1.00000 −0.0641500
\(244\) 10.0000 0.640184
\(245\) 0 0
\(246\) 10.0000 0.637577
\(247\) 10.0000 0.636285
\(248\) 0 0
\(249\) −18.0000 −1.14070
\(250\) −48.0000 −3.03579
\(251\) −10.0000 −0.631194 −0.315597 0.948893i \(-0.602205\pi\)
−0.315597 + 0.948893i \(0.602205\pi\)
\(252\) 0 0
\(253\) 5.00000 0.314347
\(254\) 18.0000 1.12942
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) −17.0000 −1.06043 −0.530215 0.847863i \(-0.677889\pi\)
−0.530215 + 0.847863i \(0.677889\pi\)
\(258\) −16.0000 −0.996116
\(259\) 0 0
\(260\) −16.0000 −0.992278
\(261\) −2.00000 −0.123797
\(262\) 10.0000 0.617802
\(263\) 7.00000 0.431638 0.215819 0.976433i \(-0.430758\pi\)
0.215819 + 0.976433i \(0.430758\pi\)
\(264\) 0 0
\(265\) −36.0000 −2.21146
\(266\) 0 0
\(267\) 10.0000 0.611990
\(268\) 8.00000 0.488678
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 8.00000 0.486864
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 18.0000 1.08742
\(275\) −55.0000 −3.31662
\(276\) 2.00000 0.120386
\(277\) 7.00000 0.420589 0.210295 0.977638i \(-0.432558\pi\)
0.210295 + 0.977638i \(0.432558\pi\)
\(278\) −24.0000 −1.43942
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) 2.00000 0.119310 0.0596550 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) −18.0000 −1.07188
\(283\) 20.0000 1.18888 0.594438 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) 24.0000 1.42414
\(285\) 20.0000 1.18470
\(286\) −20.0000 −1.18262
\(287\) 0 0
\(288\) −8.00000 −0.471405
\(289\) −17.0000 −1.00000
\(290\) 16.0000 0.939552
\(291\) −18.0000 −1.05518
\(292\) 0 0
\(293\) 4.00000 0.233682 0.116841 0.993151i \(-0.462723\pi\)
0.116841 + 0.993151i \(0.462723\pi\)
\(294\) 0 0
\(295\) 36.0000 2.09600
\(296\) 0 0
\(297\) 5.00000 0.290129
\(298\) 6.00000 0.347571
\(299\) −2.00000 −0.115663
\(300\) −22.0000 −1.27017
\(301\) 0 0
\(302\) −38.0000 −2.18665
\(303\) 5.00000 0.287242
\(304\) −20.0000 −1.14708
\(305\) −20.0000 −1.14520
\(306\) 0 0
\(307\) 8.00000 0.456584 0.228292 0.973593i \(-0.426686\pi\)
0.228292 + 0.973593i \(0.426686\pi\)
\(308\) 0 0
\(309\) −19.0000 −1.08087
\(310\) 48.0000 2.72622
\(311\) 15.0000 0.850572 0.425286 0.905059i \(-0.360174\pi\)
0.425286 + 0.905059i \(0.360174\pi\)
\(312\) 0 0
\(313\) 25.0000 1.41308 0.706542 0.707671i \(-0.250254\pi\)
0.706542 + 0.707671i \(0.250254\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) −20.0000 −1.12509
\(317\) 30.0000 1.68497 0.842484 0.538721i \(-0.181092\pi\)
0.842484 + 0.538721i \(0.181092\pi\)
\(318\) −18.0000 −1.00939
\(319\) 10.0000 0.559893
\(320\) 32.0000 1.78885
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) 0 0
\(324\) 2.00000 0.111111
\(325\) 22.0000 1.22034
\(326\) 26.0000 1.44001
\(327\) 12.0000 0.663602
\(328\) 0 0
\(329\) 0 0
\(330\) −40.0000 −2.20193
\(331\) 29.0000 1.59398 0.796992 0.603990i \(-0.206423\pi\)
0.796992 + 0.603990i \(0.206423\pi\)
\(332\) 36.0000 1.97576
\(333\) 6.00000 0.328798
\(334\) −38.0000 −2.07927
\(335\) −16.0000 −0.874173
\(336\) 0 0
\(337\) 22.0000 1.19842 0.599208 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) −18.0000 −0.979071
\(339\) 2.00000 0.108625
\(340\) 0 0
\(341\) 30.0000 1.62459
\(342\) 10.0000 0.540738
\(343\) 0 0
\(344\) 0 0
\(345\) −4.00000 −0.215353
\(346\) −4.00000 −0.215041
\(347\) 2.00000 0.107366 0.0536828 0.998558i \(-0.482904\pi\)
0.0536828 + 0.998558i \(0.482904\pi\)
\(348\) 4.00000 0.214423
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) 40.0000 2.13201
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 18.0000 0.956689
\(355\) −48.0000 −2.54758
\(356\) −20.0000 −1.06000
\(357\) 0 0
\(358\) 16.0000 0.845626
\(359\) 32.0000 1.68890 0.844448 0.535638i \(-0.179929\pi\)
0.844448 + 0.535638i \(0.179929\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 28.0000 1.47165
\(363\) −14.0000 −0.734809
\(364\) 0 0
\(365\) 0 0
\(366\) −10.0000 −0.522708
\(367\) −27.0000 −1.40939 −0.704694 0.709511i \(-0.748916\pi\)
−0.704694 + 0.709511i \(0.748916\pi\)
\(368\) 4.00000 0.208514
\(369\) −5.00000 −0.260290
\(370\) −48.0000 −2.49540
\(371\) 0 0
\(372\) 12.0000 0.622171
\(373\) −34.0000 −1.76045 −0.880227 0.474554i \(-0.842610\pi\)
−0.880227 + 0.474554i \(0.842610\pi\)
\(374\) 0 0
\(375\) 24.0000 1.23935
\(376\) 0 0
\(377\) −4.00000 −0.206010
\(378\) 0 0
\(379\) −30.0000 −1.54100 −0.770498 0.637442i \(-0.779993\pi\)
−0.770498 + 0.637442i \(0.779993\pi\)
\(380\) −40.0000 −2.05196
\(381\) −9.00000 −0.461084
\(382\) −46.0000 −2.35356
\(383\) −6.00000 −0.306586 −0.153293 0.988181i \(-0.548988\pi\)
−0.153293 + 0.988181i \(0.548988\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −38.0000 −1.93415
\(387\) 8.00000 0.406663
\(388\) 36.0000 1.82762
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 16.0000 0.810191
\(391\) 0 0
\(392\) 0 0
\(393\) −5.00000 −0.252217
\(394\) −4.00000 −0.201517
\(395\) 40.0000 2.01262
\(396\) −10.0000 −0.502519
\(397\) −30.0000 −1.50566 −0.752828 0.658217i \(-0.771311\pi\)
−0.752828 + 0.658217i \(0.771311\pi\)
\(398\) −6.00000 −0.300753
\(399\) 0 0
\(400\) −44.0000 −2.20000
\(401\) 3.00000 0.149813 0.0749064 0.997191i \(-0.476134\pi\)
0.0749064 + 0.997191i \(0.476134\pi\)
\(402\) −8.00000 −0.399004
\(403\) −12.0000 −0.597763
\(404\) −10.0000 −0.497519
\(405\) −4.00000 −0.198762
\(406\) 0 0
\(407\) −30.0000 −1.48704
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 40.0000 1.97546
\(411\) −9.00000 −0.443937
\(412\) 38.0000 1.87213
\(413\) 0 0
\(414\) −2.00000 −0.0982946
\(415\) −72.0000 −3.53434
\(416\) −16.0000 −0.784465
\(417\) 12.0000 0.587643
\(418\) −50.0000 −2.44558
\(419\) 18.0000 0.879358 0.439679 0.898155i \(-0.355092\pi\)
0.439679 + 0.898155i \(0.355092\pi\)
\(420\) 0 0
\(421\) 8.00000 0.389896 0.194948 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) 26.0000 1.26566
\(423\) 9.00000 0.437595
\(424\) 0 0
\(425\) 0 0
\(426\) −24.0000 −1.16280
\(427\) 0 0
\(428\) 8.00000 0.386695
\(429\) 10.0000 0.482805
\(430\) −64.0000 −3.08635
\(431\) 27.0000 1.30054 0.650272 0.759701i \(-0.274655\pi\)
0.650272 + 0.759701i \(0.274655\pi\)
\(432\) 4.00000 0.192450
\(433\) 11.0000 0.528626 0.264313 0.964437i \(-0.414855\pi\)
0.264313 + 0.964437i \(0.414855\pi\)
\(434\) 0 0
\(435\) −8.00000 −0.383571
\(436\) −24.0000 −1.14939
\(437\) −5.00000 −0.239182
\(438\) 0 0
\(439\) 36.0000 1.71819 0.859093 0.511819i \(-0.171028\pi\)
0.859093 + 0.511819i \(0.171028\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −36.0000 −1.71041 −0.855206 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) −12.0000 −0.569495
\(445\) 40.0000 1.89618
\(446\) −20.0000 −0.947027
\(447\) −3.00000 −0.141895
\(448\) 0 0
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 22.0000 1.03709
\(451\) 25.0000 1.17720
\(452\) −4.00000 −0.188144
\(453\) 19.0000 0.892698
\(454\) 36.0000 1.68956
\(455\) 0 0
\(456\) 0 0
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 26.0000 1.21490
\(459\) 0 0
\(460\) 8.00000 0.373002
\(461\) −2.00000 −0.0931493 −0.0465746 0.998915i \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) 0 0
\(463\) −35.0000 −1.62659 −0.813294 0.581853i \(-0.802328\pi\)
−0.813294 + 0.581853i \(0.802328\pi\)
\(464\) 8.00000 0.371391
\(465\) −24.0000 −1.11297
\(466\) −24.0000 −1.11178
\(467\) 6.00000 0.277647 0.138823 0.990317i \(-0.455668\pi\)
0.138823 + 0.990317i \(0.455668\pi\)
\(468\) 4.00000 0.184900
\(469\) 0 0
\(470\) −72.0000 −3.32111
\(471\) 7.00000 0.322543
\(472\) 0 0
\(473\) −40.0000 −1.83920
\(474\) 20.0000 0.918630
\(475\) 55.0000 2.52357
\(476\) 0 0
\(477\) 9.00000 0.412082
\(478\) 24.0000 1.09773
\(479\) −32.0000 −1.46212 −0.731059 0.682315i \(-0.760973\pi\)
−0.731059 + 0.682315i \(0.760973\pi\)
\(480\) −32.0000 −1.46059
\(481\) 12.0000 0.547153
\(482\) −34.0000 −1.54866
\(483\) 0 0
\(484\) 28.0000 1.27273
\(485\) −72.0000 −3.26935
\(486\) −2.00000 −0.0907218
\(487\) −20.0000 −0.906287 −0.453143 0.891438i \(-0.649697\pi\)
−0.453143 + 0.891438i \(0.649697\pi\)
\(488\) 0 0
\(489\) −13.0000 −0.587880
\(490\) 0 0
\(491\) 14.0000 0.631811 0.315906 0.948791i \(-0.397692\pi\)
0.315906 + 0.948791i \(0.397692\pi\)
\(492\) 10.0000 0.450835
\(493\) 0 0
\(494\) 20.0000 0.899843
\(495\) 20.0000 0.898933
\(496\) 24.0000 1.07763
\(497\) 0 0
\(498\) −36.0000 −1.61320
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) −48.0000 −2.14663
\(501\) 19.0000 0.848857
\(502\) −20.0000 −0.892644
\(503\) 36.0000 1.60516 0.802580 0.596544i \(-0.203460\pi\)
0.802580 + 0.596544i \(0.203460\pi\)
\(504\) 0 0
\(505\) 20.0000 0.889988
\(506\) 10.0000 0.444554
\(507\) 9.00000 0.399704
\(508\) 18.0000 0.798621
\(509\) 19.0000 0.842160 0.421080 0.907023i \(-0.361651\pi\)
0.421080 + 0.907023i \(0.361651\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 32.0000 1.41421
\(513\) −5.00000 −0.220755
\(514\) −34.0000 −1.49968
\(515\) −76.0000 −3.34896
\(516\) −16.0000 −0.704361
\(517\) −45.0000 −1.97910
\(518\) 0 0
\(519\) 2.00000 0.0877903
\(520\) 0 0
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) −4.00000 −0.175075
\(523\) −19.0000 −0.830812 −0.415406 0.909636i \(-0.636360\pi\)
−0.415406 + 0.909636i \(0.636360\pi\)
\(524\) 10.0000 0.436852
\(525\) 0 0
\(526\) 14.0000 0.610429
\(527\) 0 0
\(528\) −20.0000 −0.870388
\(529\) 1.00000 0.0434783
\(530\) −72.0000 −3.12748
\(531\) −9.00000 −0.390567
\(532\) 0 0
\(533\) −10.0000 −0.433148
\(534\) 20.0000 0.865485
\(535\) −16.0000 −0.691740
\(536\) 0 0
\(537\) −8.00000 −0.345225
\(538\) 36.0000 1.55207
\(539\) 0 0
\(540\) 8.00000 0.344265
\(541\) 11.0000 0.472927 0.236463 0.971640i \(-0.424012\pi\)
0.236463 + 0.971640i \(0.424012\pi\)
\(542\) 16.0000 0.687259
\(543\) −14.0000 −0.600798
\(544\) 0 0
\(545\) 48.0000 2.05609
\(546\) 0 0
\(547\) −8.00000 −0.342055 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) 18.0000 0.768922
\(549\) 5.00000 0.213395
\(550\) −110.000 −4.69042
\(551\) −10.0000 −0.426014
\(552\) 0 0
\(553\) 0 0
\(554\) 14.0000 0.594803
\(555\) 24.0000 1.01874
\(556\) −24.0000 −1.01783
\(557\) 46.0000 1.94908 0.974541 0.224208i \(-0.0719796\pi\)
0.974541 + 0.224208i \(0.0719796\pi\)
\(558\) −12.0000 −0.508001
\(559\) 16.0000 0.676728
\(560\) 0 0
\(561\) 0 0
\(562\) 4.00000 0.168730
\(563\) −40.0000 −1.68580 −0.842900 0.538071i \(-0.819153\pi\)
−0.842900 + 0.538071i \(0.819153\pi\)
\(564\) −18.0000 −0.757937
\(565\) 8.00000 0.336563
\(566\) 40.0000 1.68133
\(567\) 0 0
\(568\) 0 0
\(569\) −17.0000 −0.712677 −0.356339 0.934357i \(-0.615975\pi\)
−0.356339 + 0.934357i \(0.615975\pi\)
\(570\) 40.0000 1.67542
\(571\) 8.00000 0.334790 0.167395 0.985890i \(-0.446465\pi\)
0.167395 + 0.985890i \(0.446465\pi\)
\(572\) −20.0000 −0.836242
\(573\) 23.0000 0.960839
\(574\) 0 0
\(575\) −11.0000 −0.458732
\(576\) −8.00000 −0.333333
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) −34.0000 −1.41421
\(579\) 19.0000 0.789613
\(580\) 16.0000 0.664364
\(581\) 0 0
\(582\) −36.0000 −1.49225
\(583\) −45.0000 −1.86371
\(584\) 0 0
\(585\) −8.00000 −0.330759
\(586\) 8.00000 0.330477
\(587\) 17.0000 0.701665 0.350833 0.936438i \(-0.385899\pi\)
0.350833 + 0.936438i \(0.385899\pi\)
\(588\) 0 0
\(589\) −30.0000 −1.23613
\(590\) 72.0000 2.96419
\(591\) 2.00000 0.0822690
\(592\) −24.0000 −0.986394
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 10.0000 0.410305
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) 3.00000 0.122782
\(598\) −4.00000 −0.163572
\(599\) 42.0000 1.71607 0.858037 0.513588i \(-0.171684\pi\)
0.858037 + 0.513588i \(0.171684\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) −38.0000 −1.54620
\(605\) −56.0000 −2.27672
\(606\) 10.0000 0.406222
\(607\) −8.00000 −0.324710 −0.162355 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) −40.0000 −1.62221
\(609\) 0 0
\(610\) −40.0000 −1.61955
\(611\) 18.0000 0.728202
\(612\) 0 0
\(613\) 34.0000 1.37325 0.686624 0.727013i \(-0.259092\pi\)
0.686624 + 0.727013i \(0.259092\pi\)
\(614\) 16.0000 0.645707
\(615\) −20.0000 −0.806478
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) −38.0000 −1.52858
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 48.0000 1.92773
\(621\) 1.00000 0.0401286
\(622\) 30.0000 1.20289
\(623\) 0 0
\(624\) 8.00000 0.320256
\(625\) 41.0000 1.64000
\(626\) 50.0000 1.99840
\(627\) 25.0000 0.998404
\(628\) −14.0000 −0.558661
\(629\) 0 0
\(630\) 0 0
\(631\) −10.0000 −0.398094 −0.199047 0.979990i \(-0.563785\pi\)
−0.199047 + 0.979990i \(0.563785\pi\)
\(632\) 0 0
\(633\) −13.0000 −0.516704
\(634\) 60.0000 2.38290
\(635\) −36.0000 −1.42862
\(636\) −18.0000 −0.713746
\(637\) 0 0
\(638\) 20.0000 0.791808
\(639\) 12.0000 0.474713
\(640\) 0 0
\(641\) −9.00000 −0.355479 −0.177739 0.984078i \(-0.556878\pi\)
−0.177739 + 0.984078i \(0.556878\pi\)
\(642\) −8.00000 −0.315735
\(643\) 31.0000 1.22252 0.611260 0.791430i \(-0.290663\pi\)
0.611260 + 0.791430i \(0.290663\pi\)
\(644\) 0 0
\(645\) 32.0000 1.26000
\(646\) 0 0
\(647\) 28.0000 1.10079 0.550397 0.834903i \(-0.314476\pi\)
0.550397 + 0.834903i \(0.314476\pi\)
\(648\) 0 0
\(649\) 45.0000 1.76640
\(650\) 44.0000 1.72582
\(651\) 0 0
\(652\) 26.0000 1.01824
\(653\) 24.0000 0.939193 0.469596 0.882881i \(-0.344399\pi\)
0.469596 + 0.882881i \(0.344399\pi\)
\(654\) 24.0000 0.938474
\(655\) −20.0000 −0.781465
\(656\) 20.0000 0.780869
\(657\) 0 0
\(658\) 0 0
\(659\) 24.0000 0.934907 0.467454 0.884018i \(-0.345171\pi\)
0.467454 + 0.884018i \(0.345171\pi\)
\(660\) −40.0000 −1.55700
\(661\) 5.00000 0.194477 0.0972387 0.995261i \(-0.468999\pi\)
0.0972387 + 0.995261i \(0.468999\pi\)
\(662\) 58.0000 2.25423
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 12.0000 0.464991
\(667\) 2.00000 0.0774403
\(668\) −38.0000 −1.47026
\(669\) 10.0000 0.386622
\(670\) −32.0000 −1.23627
\(671\) −25.0000 −0.965114
\(672\) 0 0
\(673\) 49.0000 1.88881 0.944406 0.328783i \(-0.106638\pi\)
0.944406 + 0.328783i \(0.106638\pi\)
\(674\) 44.0000 1.69482
\(675\) −11.0000 −0.423390
\(676\) −18.0000 −0.692308
\(677\) −36.0000 −1.38359 −0.691796 0.722093i \(-0.743180\pi\)
−0.691796 + 0.722093i \(0.743180\pi\)
\(678\) 4.00000 0.153619
\(679\) 0 0
\(680\) 0 0
\(681\) −18.0000 −0.689761
\(682\) 60.0000 2.29752
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 10.0000 0.382360
\(685\) −36.0000 −1.37549
\(686\) 0 0
\(687\) −13.0000 −0.495981
\(688\) −32.0000 −1.21999
\(689\) 18.0000 0.685745
\(690\) −8.00000 −0.304555
\(691\) 4.00000 0.152167 0.0760836 0.997101i \(-0.475758\pi\)
0.0760836 + 0.997101i \(0.475758\pi\)
\(692\) −4.00000 −0.152057
\(693\) 0 0
\(694\) 4.00000 0.151838
\(695\) 48.0000 1.82074
\(696\) 0 0
\(697\) 0 0
\(698\) 4.00000 0.151402
\(699\) 12.0000 0.453882
\(700\) 0 0
\(701\) −15.0000 −0.566542 −0.283271 0.959040i \(-0.591420\pi\)
−0.283271 + 0.959040i \(0.591420\pi\)
\(702\) −4.00000 −0.150970
\(703\) 30.0000 1.13147
\(704\) 40.0000 1.50756
\(705\) 36.0000 1.35584
\(706\) 28.0000 1.05379
\(707\) 0 0
\(708\) 18.0000 0.676481
\(709\) −6.00000 −0.225335 −0.112667 0.993633i \(-0.535939\pi\)
−0.112667 + 0.993633i \(0.535939\pi\)
\(710\) −96.0000 −3.60282
\(711\) −10.0000 −0.375029
\(712\) 0 0
\(713\) 6.00000 0.224702
\(714\) 0 0
\(715\) 40.0000 1.49592
\(716\) 16.0000 0.597948
\(717\) −12.0000 −0.448148
\(718\) 64.0000 2.38846
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 16.0000 0.596285
\(721\) 0 0
\(722\) 12.0000 0.446594
\(723\) 17.0000 0.632237
\(724\) 28.0000 1.04061
\(725\) −22.0000 −0.817059
\(726\) −28.0000 −1.03918
\(727\) −41.0000 −1.52061 −0.760303 0.649569i \(-0.774949\pi\)
−0.760303 + 0.649569i \(0.774949\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) −10.0000 −0.369611
\(733\) 6.00000 0.221615 0.110808 0.993842i \(-0.464656\pi\)
0.110808 + 0.993842i \(0.464656\pi\)
\(734\) −54.0000 −1.99318
\(735\) 0 0
\(736\) 8.00000 0.294884
\(737\) −20.0000 −0.736709
\(738\) −10.0000 −0.368105
\(739\) 12.0000 0.441427 0.220714 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) −48.0000 −1.76452
\(741\) −10.0000 −0.367359
\(742\) 0 0
\(743\) 15.0000 0.550297 0.275148 0.961402i \(-0.411273\pi\)
0.275148 + 0.961402i \(0.411273\pi\)
\(744\) 0 0
\(745\) −12.0000 −0.439646
\(746\) −68.0000 −2.48966
\(747\) 18.0000 0.658586
\(748\) 0 0
\(749\) 0 0
\(750\) 48.0000 1.75271
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) −36.0000 −1.31278
\(753\) 10.0000 0.364420
\(754\) −8.00000 −0.291343
\(755\) 76.0000 2.76592
\(756\) 0 0
\(757\) 8.00000 0.290765 0.145382 0.989376i \(-0.453559\pi\)
0.145382 + 0.989376i \(0.453559\pi\)
\(758\) −60.0000 −2.17930
\(759\) −5.00000 −0.181489
\(760\) 0 0
\(761\) 21.0000 0.761249 0.380625 0.924730i \(-0.375709\pi\)
0.380625 + 0.924730i \(0.375709\pi\)
\(762\) −18.0000 −0.652071
\(763\) 0 0
\(764\) −46.0000 −1.66422
\(765\) 0 0
\(766\) −12.0000 −0.433578
\(767\) −18.0000 −0.649942
\(768\) −16.0000 −0.577350
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) 17.0000 0.612240
\(772\) −38.0000 −1.36765
\(773\) −20.0000 −0.719350 −0.359675 0.933078i \(-0.617112\pi\)
−0.359675 + 0.933078i \(0.617112\pi\)
\(774\) 16.0000 0.575108
\(775\) −66.0000 −2.37079
\(776\) 0 0
\(777\) 0 0
\(778\) 12.0000 0.430221
\(779\) −25.0000 −0.895718
\(780\) 16.0000 0.572892
\(781\) −60.0000 −2.14697
\(782\) 0 0
\(783\) 2.00000 0.0714742
\(784\) 0 0
\(785\) 28.0000 0.999363
\(786\) −10.0000 −0.356688
\(787\) 23.0000 0.819861 0.409931 0.912117i \(-0.365553\pi\)
0.409931 + 0.912117i \(0.365553\pi\)
\(788\) −4.00000 −0.142494
\(789\) −7.00000 −0.249207
\(790\) 80.0000 2.84627
\(791\) 0 0
\(792\) 0 0
\(793\) 10.0000 0.355110
\(794\) −60.0000 −2.12932
\(795\) 36.0000 1.27679
\(796\) −6.00000 −0.212664
\(797\) −46.0000 −1.62940 −0.814702 0.579880i \(-0.803099\pi\)
−0.814702 + 0.579880i \(0.803099\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −88.0000 −3.11127
\(801\) −10.0000 −0.353333
\(802\) 6.00000 0.211867
\(803\) 0 0
\(804\) −8.00000 −0.282138
\(805\) 0 0
\(806\) −24.0000 −0.845364
\(807\) −18.0000 −0.633630
\(808\) 0 0
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) −8.00000 −0.281091
\(811\) −2.00000 −0.0702295 −0.0351147 0.999383i \(-0.511180\pi\)
−0.0351147 + 0.999383i \(0.511180\pi\)
\(812\) 0 0
\(813\) −8.00000 −0.280572
\(814\) −60.0000 −2.10300
\(815\) −52.0000 −1.82148
\(816\) 0 0
\(817\) 40.0000 1.39942
\(818\) −28.0000 −0.978997
\(819\) 0 0
\(820\) 40.0000 1.39686
\(821\) −4.00000 −0.139601 −0.0698005 0.997561i \(-0.522236\pi\)
−0.0698005 + 0.997561i \(0.522236\pi\)
\(822\) −18.0000 −0.627822
\(823\) 27.0000 0.941161 0.470580 0.882357i \(-0.344045\pi\)
0.470580 + 0.882357i \(0.344045\pi\)
\(824\) 0 0
\(825\) 55.0000 1.91485
\(826\) 0 0
\(827\) −21.0000 −0.730242 −0.365121 0.930960i \(-0.618972\pi\)
−0.365121 + 0.930960i \(0.618972\pi\)
\(828\) −2.00000 −0.0695048
\(829\) −26.0000 −0.903017 −0.451509 0.892267i \(-0.649114\pi\)
−0.451509 + 0.892267i \(0.649114\pi\)
\(830\) −144.000 −4.99831
\(831\) −7.00000 −0.242827
\(832\) −16.0000 −0.554700
\(833\) 0 0
\(834\) 24.0000 0.831052
\(835\) 76.0000 2.63009
\(836\) −50.0000 −1.72929
\(837\) 6.00000 0.207390
\(838\) 36.0000 1.24360
\(839\) −46.0000 −1.58810 −0.794048 0.607855i \(-0.792030\pi\)
−0.794048 + 0.607855i \(0.792030\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 16.0000 0.551396
\(843\) −2.00000 −0.0688837
\(844\) 26.0000 0.894957
\(845\) 36.0000 1.23844
\(846\) 18.0000 0.618853
\(847\) 0 0
\(848\) −36.0000 −1.23625
\(849\) −20.0000 −0.686398
\(850\) 0 0
\(851\) −6.00000 −0.205677
\(852\) −24.0000 −0.822226
\(853\) −8.00000 −0.273915 −0.136957 0.990577i \(-0.543732\pi\)
−0.136957 + 0.990577i \(0.543732\pi\)
\(854\) 0 0
\(855\) −20.0000 −0.683986
\(856\) 0 0
\(857\) −1.00000 −0.0341593 −0.0170797 0.999854i \(-0.505437\pi\)
−0.0170797 + 0.999854i \(0.505437\pi\)
\(858\) 20.0000 0.682789
\(859\) 40.0000 1.36478 0.682391 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) −64.0000 −2.18238
\(861\) 0 0
\(862\) 54.0000 1.83925
\(863\) −30.0000 −1.02121 −0.510606 0.859815i \(-0.670579\pi\)
−0.510606 + 0.859815i \(0.670579\pi\)
\(864\) 8.00000 0.272166
\(865\) 8.00000 0.272008
\(866\) 22.0000 0.747590
\(867\) 17.0000 0.577350
\(868\) 0 0
\(869\) 50.0000 1.69613
\(870\) −16.0000 −0.542451
\(871\) 8.00000 0.271070
\(872\) 0 0
\(873\) 18.0000 0.609208
\(874\) −10.0000 −0.338255
\(875\) 0 0
\(876\) 0 0
\(877\) −17.0000 −0.574049 −0.287025 0.957923i \(-0.592666\pi\)
−0.287025 + 0.957923i \(0.592666\pi\)
\(878\) 72.0000 2.42988
\(879\) −4.00000 −0.134917
\(880\) −80.0000 −2.69680
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) 0 0
\(883\) 32.0000 1.07689 0.538443 0.842662i \(-0.319013\pi\)
0.538443 + 0.842662i \(0.319013\pi\)
\(884\) 0 0
\(885\) −36.0000 −1.21013
\(886\) −72.0000 −2.41889
\(887\) 16.0000 0.537227 0.268614 0.963248i \(-0.413434\pi\)
0.268614 + 0.963248i \(0.413434\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 80.0000 2.68161
\(891\) −5.00000 −0.167506
\(892\) −20.0000 −0.669650
\(893\) 45.0000 1.50587
\(894\) −6.00000 −0.200670
\(895\) −32.0000 −1.06964
\(896\) 0 0
\(897\) 2.00000 0.0667781
\(898\) 12.0000 0.400445
\(899\) 12.0000 0.400222
\(900\) 22.0000 0.733333
\(901\) 0 0
\(902\) 50.0000 1.66482
\(903\) 0 0
\(904\) 0 0
\(905\) −56.0000 −1.86150
\(906\) 38.0000 1.26247
\(907\) 44.0000 1.46100 0.730498 0.682915i \(-0.239288\pi\)
0.730498 + 0.682915i \(0.239288\pi\)
\(908\) 36.0000 1.19470
\(909\) −5.00000 −0.165840
\(910\) 0 0
\(911\) 24.0000 0.795155 0.397578 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) 20.0000 0.662266
\(913\) −90.0000 −2.97857
\(914\) 20.0000 0.661541
\(915\) 20.0000 0.661180
\(916\) 26.0000 0.859064
\(917\) 0 0
\(918\) 0 0
\(919\) 2.00000 0.0659739 0.0329870 0.999456i \(-0.489498\pi\)
0.0329870 + 0.999456i \(0.489498\pi\)
\(920\) 0 0
\(921\) −8.00000 −0.263609
\(922\) −4.00000 −0.131733
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) 66.0000 2.17007
\(926\) −70.0000 −2.30034
\(927\) 19.0000 0.624042
\(928\) 16.0000 0.525226
\(929\) −26.0000 −0.853032 −0.426516 0.904480i \(-0.640259\pi\)
−0.426516 + 0.904480i \(0.640259\pi\)
\(930\) −48.0000 −1.57398
\(931\) 0 0
\(932\) −24.0000 −0.786146
\(933\) −15.0000 −0.491078
\(934\) 12.0000 0.392652
\(935\) 0 0
\(936\) 0 0
\(937\) 27.0000 0.882052 0.441026 0.897494i \(-0.354615\pi\)
0.441026 + 0.897494i \(0.354615\pi\)
\(938\) 0 0
\(939\) −25.0000 −0.815844
\(940\) −72.0000 −2.34838
\(941\) −20.0000 −0.651981 −0.325991 0.945373i \(-0.605698\pi\)
−0.325991 + 0.945373i \(0.605698\pi\)
\(942\) 14.0000 0.456145
\(943\) 5.00000 0.162822
\(944\) 36.0000 1.17170
\(945\) 0 0
\(946\) −80.0000 −2.60102
\(947\) 32.0000 1.03986 0.519930 0.854209i \(-0.325958\pi\)
0.519930 + 0.854209i \(0.325958\pi\)
\(948\) 20.0000 0.649570
\(949\) 0 0
\(950\) 110.000 3.56887
\(951\) −30.0000 −0.972817
\(952\) 0 0
\(953\) −55.0000 −1.78162 −0.890812 0.454371i \(-0.849864\pi\)
−0.890812 + 0.454371i \(0.849864\pi\)
\(954\) 18.0000 0.582772
\(955\) 92.0000 2.97705
\(956\) 24.0000 0.776215
\(957\) −10.0000 −0.323254
\(958\) −64.0000 −2.06775
\(959\) 0 0
\(960\) −32.0000 −1.03280
\(961\) 5.00000 0.161290
\(962\) 24.0000 0.773791
\(963\) 4.00000 0.128898
\(964\) −34.0000 −1.09507
\(965\) 76.0000 2.44653
\(966\) 0 0
\(967\) 28.0000 0.900419 0.450210 0.892923i \(-0.351349\pi\)
0.450210 + 0.892923i \(0.351349\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −144.000 −4.62356
\(971\) 18.0000 0.577647 0.288824 0.957382i \(-0.406736\pi\)
0.288824 + 0.957382i \(0.406736\pi\)
\(972\) −2.00000 −0.0641500
\(973\) 0 0
\(974\) −40.0000 −1.28168
\(975\) −22.0000 −0.704564
\(976\) −20.0000 −0.640184
\(977\) −3.00000 −0.0959785 −0.0479893 0.998848i \(-0.515281\pi\)
−0.0479893 + 0.998848i \(0.515281\pi\)
\(978\) −26.0000 −0.831388
\(979\) 50.0000 1.59801
\(980\) 0 0
\(981\) −12.0000 −0.383131
\(982\) 28.0000 0.893516
\(983\) −42.0000 −1.33959 −0.669796 0.742545i \(-0.733618\pi\)
−0.669796 + 0.742545i \(0.733618\pi\)
\(984\) 0 0
\(985\) 8.00000 0.254901
\(986\) 0 0
\(987\) 0 0
\(988\) 20.0000 0.636285
\(989\) −8.00000 −0.254385
\(990\) 40.0000 1.27128
\(991\) 5.00000 0.158830 0.0794151 0.996842i \(-0.474695\pi\)
0.0794151 + 0.996842i \(0.474695\pi\)
\(992\) 48.0000 1.52400
\(993\) −29.0000 −0.920287
\(994\) 0 0
\(995\) 12.0000 0.380426
\(996\) −36.0000 −1.14070
\(997\) 56.0000 1.77354 0.886769 0.462213i \(-0.152944\pi\)
0.886769 + 0.462213i \(0.152944\pi\)
\(998\) −40.0000 −1.26618
\(999\) −6.00000 −0.189832
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 3381.2.a.l.1.1 1
7.6 odd 2 483.2.a.b.1.1 1
21.20 even 2 1449.2.a.a.1.1 1
28.27 even 2 7728.2.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.a.b.1.1 1 7.6 odd 2
1449.2.a.a.1.1 1 21.20 even 2
3381.2.a.l.1.1 1 1.1 even 1 trivial
7728.2.a.l.1.1 1 28.27 even 2