Properties

Label 1638.2.a.j.1.1
Level $1638$
Weight $2$
Character 1638.1
Self dual yes
Analytic conductor $13.079$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.0794958511\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 182)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1638.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +4.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +4.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} -4.00000 q^{10} -1.00000 q^{11} -1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -6.00000 q^{19} +4.00000 q^{20} +1.00000 q^{22} +7.00000 q^{23} +11.0000 q^{25} +1.00000 q^{26} -1.00000 q^{28} +4.00000 q^{29} +7.00000 q^{31} -1.00000 q^{32} -4.00000 q^{35} +9.00000 q^{37} +6.00000 q^{38} -4.00000 q^{40} +3.00000 q^{41} +4.00000 q^{43} -1.00000 q^{44} -7.00000 q^{46} -7.00000 q^{47} +1.00000 q^{49} -11.0000 q^{50} -1.00000 q^{52} -4.00000 q^{55} +1.00000 q^{56} -4.00000 q^{58} +10.0000 q^{59} +1.00000 q^{61} -7.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} +1.00000 q^{67} +4.00000 q^{70} -16.0000 q^{71} +5.00000 q^{73} -9.00000 q^{74} -6.00000 q^{76} +1.00000 q^{77} +11.0000 q^{79} +4.00000 q^{80} -3.00000 q^{82} -4.00000 q^{86} +1.00000 q^{88} +6.00000 q^{89} +1.00000 q^{91} +7.00000 q^{92} +7.00000 q^{94} -24.0000 q^{95} -1.00000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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
\(3\) 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 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 4.00000 0.894427
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 7.00000 1.45960 0.729800 0.683660i \(-0.239613\pi\)
0.729800 + 0.683660i \(0.239613\pi\)
\(24\) 0 0
\(25\) 11.0000 2.20000
\(26\) 1.00000 0.196116
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) 4.00000 0.742781 0.371391 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) 7.00000 1.25724 0.628619 0.777714i \(-0.283621\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 0 0
\(35\) −4.00000 −0.676123
\(36\) 0 0
\(37\) 9.00000 1.47959 0.739795 0.672832i \(-0.234922\pi\)
0.739795 + 0.672832i \(0.234922\pi\)
\(38\) 6.00000 0.973329
\(39\) 0 0
\(40\) −4.00000 −0.632456
\(41\) 3.00000 0.468521 0.234261 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 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
\(46\) −7.00000 −1.03209
\(47\) −7.00000 −1.02105 −0.510527 0.859861i \(-0.670550\pi\)
−0.510527 + 0.859861i \(0.670550\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −11.0000 −1.55563
\(51\) 0 0
\(52\) −1.00000 −0.138675
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −4.00000 −0.525226
\(59\) 10.0000 1.30189 0.650945 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 0 0
\(61\) 1.00000 0.128037 0.0640184 0.997949i \(-0.479608\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) −7.00000 −0.889001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) 1.00000 0.122169 0.0610847 0.998133i \(-0.480544\pi\)
0.0610847 + 0.998133i \(0.480544\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 4.00000 0.478091
\(71\) −16.0000 −1.89885 −0.949425 0.313993i \(-0.898333\pi\)
−0.949425 + 0.313993i \(0.898333\pi\)
\(72\) 0 0
\(73\) 5.00000 0.585206 0.292603 0.956234i \(-0.405479\pi\)
0.292603 + 0.956234i \(0.405479\pi\)
\(74\) −9.00000 −1.04623
\(75\) 0 0
\(76\) −6.00000 −0.688247
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 11.0000 1.23760 0.618798 0.785550i \(-0.287620\pi\)
0.618798 + 0.785550i \(0.287620\pi\)
\(80\) 4.00000 0.447214
\(81\) 0 0
\(82\) −3.00000 −0.331295
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 7.00000 0.729800
\(93\) 0 0
\(94\) 7.00000 0.721995
\(95\) −24.0000 −2.46235
\(96\) 0 0
\(97\) −1.00000 −0.101535 −0.0507673 0.998711i \(-0.516167\pi\)
−0.0507673 + 0.998711i \(0.516167\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 11.0000 1.10000
\(101\) 5.00000 0.497519 0.248759 0.968565i \(-0.419977\pi\)
0.248759 + 0.968565i \(0.419977\pi\)
\(102\) 0 0
\(103\) −14.0000 −1.37946 −0.689730 0.724066i \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) 0 0
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 0 0
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 4.00000 0.381385
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) 7.00000 0.658505 0.329252 0.944242i \(-0.393203\pi\)
0.329252 + 0.944242i \(0.393203\pi\)
\(114\) 0 0
\(115\) 28.0000 2.61101
\(116\) 4.00000 0.371391
\(117\) 0 0
\(118\) −10.0000 −0.920575
\(119\) 0 0
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) −1.00000 −0.0905357
\(123\) 0 0
\(124\) 7.00000 0.628619
\(125\) 24.0000 2.14663
\(126\) 0 0
\(127\) −11.0000 −0.976092 −0.488046 0.872818i \(-0.662290\pi\)
−0.488046 + 0.872818i \(0.662290\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 4.00000 0.350823
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 0 0
\(133\) 6.00000 0.520266
\(134\) −1.00000 −0.0863868
\(135\) 0 0
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) −4.00000 −0.338062
\(141\) 0 0
\(142\) 16.0000 1.34269
\(143\) 1.00000 0.0836242
\(144\) 0 0
\(145\) 16.0000 1.32873
\(146\) −5.00000 −0.413803
\(147\) 0 0
\(148\) 9.00000 0.739795
\(149\) −9.00000 −0.737309 −0.368654 0.929567i \(-0.620181\pi\)
−0.368654 + 0.929567i \(0.620181\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 6.00000 0.486664
\(153\) 0 0
\(154\) −1.00000 −0.0805823
\(155\) 28.0000 2.24901
\(156\) 0 0
\(157\) 5.00000 0.399043 0.199522 0.979893i \(-0.436061\pi\)
0.199522 + 0.979893i \(0.436061\pi\)
\(158\) −11.0000 −0.875113
\(159\) 0 0
\(160\) −4.00000 −0.316228
\(161\) −7.00000 −0.551677
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 3.00000 0.234261
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 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
\(178\) −6.00000 −0.449719
\(179\) 6.00000 0.448461 0.224231 0.974536i \(-0.428013\pi\)
0.224231 + 0.974536i \(0.428013\pi\)
\(180\) 0 0
\(181\) 15.0000 1.11494 0.557471 0.830197i \(-0.311772\pi\)
0.557471 + 0.830197i \(0.311772\pi\)
\(182\) −1.00000 −0.0741249
\(183\) 0 0
\(184\) −7.00000 −0.516047
\(185\) 36.0000 2.64677
\(186\) 0 0
\(187\) 0 0
\(188\) −7.00000 −0.510527
\(189\) 0 0
\(190\) 24.0000 1.74114
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) 20.0000 1.43963 0.719816 0.694165i \(-0.244226\pi\)
0.719816 + 0.694165i \(0.244226\pi\)
\(194\) 1.00000 0.0717958
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 27.0000 1.92367 0.961835 0.273629i \(-0.0882242\pi\)
0.961835 + 0.273629i \(0.0882242\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) −11.0000 −0.777817
\(201\) 0 0
\(202\) −5.00000 −0.351799
\(203\) −4.00000 −0.280745
\(204\) 0 0
\(205\) 12.0000 0.838116
\(206\) 14.0000 0.975426
\(207\) 0 0
\(208\) −1.00000 −0.0693375
\(209\) 6.00000 0.415029
\(210\) 0 0
\(211\) −10.0000 −0.688428 −0.344214 0.938891i \(-0.611855\pi\)
−0.344214 + 0.938891i \(0.611855\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −4.00000 −0.273434
\(215\) 16.0000 1.09119
\(216\) 0 0
\(217\) −7.00000 −0.475191
\(218\) 14.0000 0.948200
\(219\) 0 0
\(220\) −4.00000 −0.269680
\(221\) 0 0
\(222\) 0 0
\(223\) 21.0000 1.40626 0.703132 0.711059i \(-0.251784\pi\)
0.703132 + 0.711059i \(0.251784\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −7.00000 −0.465633
\(227\) −24.0000 −1.59294 −0.796468 0.604681i \(-0.793301\pi\)
−0.796468 + 0.604681i \(0.793301\pi\)
\(228\) 0 0
\(229\) −24.0000 −1.58596 −0.792982 0.609245i \(-0.791473\pi\)
−0.792982 + 0.609245i \(0.791473\pi\)
\(230\) −28.0000 −1.84627
\(231\) 0 0
\(232\) −4.00000 −0.262613
\(233\) 5.00000 0.327561 0.163780 0.986497i \(-0.447631\pi\)
0.163780 + 0.986497i \(0.447631\pi\)
\(234\) 0 0
\(235\) −28.0000 −1.82652
\(236\) 10.0000 0.650945
\(237\) 0 0
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) −18.0000 −1.15948 −0.579741 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) 10.0000 0.642824
\(243\) 0 0
\(244\) 1.00000 0.0640184
\(245\) 4.00000 0.255551
\(246\) 0 0
\(247\) 6.00000 0.381771
\(248\) −7.00000 −0.444500
\(249\) 0 0
\(250\) −24.0000 −1.51789
\(251\) −3.00000 −0.189358 −0.0946792 0.995508i \(-0.530183\pi\)
−0.0946792 + 0.995508i \(0.530183\pi\)
\(252\) 0 0
\(253\) −7.00000 −0.440086
\(254\) 11.0000 0.690201
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −24.0000 −1.49708 −0.748539 0.663090i \(-0.769245\pi\)
−0.748539 + 0.663090i \(0.769245\pi\)
\(258\) 0 0
\(259\) −9.00000 −0.559233
\(260\) −4.00000 −0.248069
\(261\) 0 0
\(262\) 8.00000 0.494242
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −6.00000 −0.367884
\(267\) 0 0
\(268\) 1.00000 0.0610847
\(269\) −9.00000 −0.548740 −0.274370 0.961624i \(-0.588469\pi\)
−0.274370 + 0.961624i \(0.588469\pi\)
\(270\) 0 0
\(271\) −23.0000 −1.39715 −0.698575 0.715537i \(-0.746182\pi\)
−0.698575 + 0.715537i \(0.746182\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) −11.0000 −0.663325
\(276\) 0 0
\(277\) −18.0000 −1.08152 −0.540758 0.841178i \(-0.681862\pi\)
−0.540758 + 0.841178i \(0.681862\pi\)
\(278\) −4.00000 −0.239904
\(279\) 0 0
\(280\) 4.00000 0.239046
\(281\) 8.00000 0.477240 0.238620 0.971113i \(-0.423305\pi\)
0.238620 + 0.971113i \(0.423305\pi\)
\(282\) 0 0
\(283\) 19.0000 1.12943 0.564716 0.825285i \(-0.308986\pi\)
0.564716 + 0.825285i \(0.308986\pi\)
\(284\) −16.0000 −0.949425
\(285\) 0 0
\(286\) −1.00000 −0.0591312
\(287\) −3.00000 −0.177084
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) −16.0000 −0.939552
\(291\) 0 0
\(292\) 5.00000 0.292603
\(293\) 26.0000 1.51894 0.759468 0.650545i \(-0.225459\pi\)
0.759468 + 0.650545i \(0.225459\pi\)
\(294\) 0 0
\(295\) 40.0000 2.32889
\(296\) −9.00000 −0.523114
\(297\) 0 0
\(298\) 9.00000 0.521356
\(299\) −7.00000 −0.404820
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) 0 0
\(303\) 0 0
\(304\) −6.00000 −0.344124
\(305\) 4.00000 0.229039
\(306\) 0 0
\(307\) −4.00000 −0.228292 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) 1.00000 0.0569803
\(309\) 0 0
\(310\) −28.0000 −1.59029
\(311\) −30.0000 −1.70114 −0.850572 0.525859i \(-0.823744\pi\)
−0.850572 + 0.525859i \(0.823744\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) −5.00000 −0.282166
\(315\) 0 0
\(316\) 11.0000 0.618798
\(317\) −21.0000 −1.17948 −0.589739 0.807594i \(-0.700769\pi\)
−0.589739 + 0.807594i \(0.700769\pi\)
\(318\) 0 0
\(319\) −4.00000 −0.223957
\(320\) 4.00000 0.223607
\(321\) 0 0
\(322\) 7.00000 0.390095
\(323\) 0 0
\(324\) 0 0
\(325\) −11.0000 −0.610170
\(326\) 4.00000 0.221540
\(327\) 0 0
\(328\) −3.00000 −0.165647
\(329\) 7.00000 0.385922
\(330\) 0 0
\(331\) −7.00000 −0.384755 −0.192377 0.981321i \(-0.561620\pi\)
−0.192377 + 0.981321i \(0.561620\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) 17.0000 0.926049 0.463025 0.886345i \(-0.346764\pi\)
0.463025 + 0.886345i \(0.346764\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 0 0
\(340\) 0 0
\(341\) −7.00000 −0.379071
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 2.00000 0.107521
\(347\) 24.0000 1.28839 0.644194 0.764862i \(-0.277193\pi\)
0.644194 + 0.764862i \(0.277193\pi\)
\(348\) 0 0
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 11.0000 0.587975
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) −25.0000 −1.33062 −0.665308 0.746569i \(-0.731700\pi\)
−0.665308 + 0.746569i \(0.731700\pi\)
\(354\) 0 0
\(355\) −64.0000 −3.39677
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) −6.00000 −0.317110
\(359\) −12.0000 −0.633336 −0.316668 0.948536i \(-0.602564\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) −15.0000 −0.788382
\(363\) 0 0
\(364\) 1.00000 0.0524142
\(365\) 20.0000 1.04685
\(366\) 0 0
\(367\) 32.0000 1.67039 0.835193 0.549957i \(-0.185356\pi\)
0.835193 + 0.549957i \(0.185356\pi\)
\(368\) 7.00000 0.364900
\(369\) 0 0
\(370\) −36.0000 −1.87155
\(371\) 0 0
\(372\) 0 0
\(373\) −16.0000 −0.828449 −0.414224 0.910175i \(-0.635947\pi\)
−0.414224 + 0.910175i \(0.635947\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 7.00000 0.360997
\(377\) −4.00000 −0.206010
\(378\) 0 0
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) −24.0000 −1.23117
\(381\) 0 0
\(382\) −8.00000 −0.409316
\(383\) 15.0000 0.766464 0.383232 0.923652i \(-0.374811\pi\)
0.383232 + 0.923652i \(0.374811\pi\)
\(384\) 0 0
\(385\) 4.00000 0.203859
\(386\) −20.0000 −1.01797
\(387\) 0 0
\(388\) −1.00000 −0.0507673
\(389\) 36.0000 1.82527 0.912636 0.408773i \(-0.134043\pi\)
0.912636 + 0.408773i \(0.134043\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) −27.0000 −1.36024
\(395\) 44.0000 2.21388
\(396\) 0 0
\(397\) −28.0000 −1.40528 −0.702640 0.711546i \(-0.747995\pi\)
−0.702640 + 0.711546i \(0.747995\pi\)
\(398\) 4.00000 0.200502
\(399\) 0 0
\(400\) 11.0000 0.550000
\(401\) −12.0000 −0.599251 −0.299626 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) 0 0
\(403\) −7.00000 −0.348695
\(404\) 5.00000 0.248759
\(405\) 0 0
\(406\) 4.00000 0.198517
\(407\) −9.00000 −0.446113
\(408\) 0 0
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) −12.0000 −0.592638
\(411\) 0 0
\(412\) −14.0000 −0.689730
\(413\) −10.0000 −0.492068
\(414\) 0 0
\(415\) 0 0
\(416\) 1.00000 0.0490290
\(417\) 0 0
\(418\) −6.00000 −0.293470
\(419\) −15.0000 −0.732798 −0.366399 0.930458i \(-0.619409\pi\)
−0.366399 + 0.930458i \(0.619409\pi\)
\(420\) 0 0
\(421\) 19.0000 0.926003 0.463002 0.886357i \(-0.346772\pi\)
0.463002 + 0.886357i \(0.346772\pi\)
\(422\) 10.0000 0.486792
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −1.00000 −0.0483934
\(428\) 4.00000 0.193347
\(429\) 0 0
\(430\) −16.0000 −0.771589
\(431\) −18.0000 −0.867029 −0.433515 0.901146i \(-0.642727\pi\)
−0.433515 + 0.901146i \(0.642727\pi\)
\(432\) 0 0
\(433\) 34.0000 1.63394 0.816968 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) 7.00000 0.336011
\(435\) 0 0
\(436\) −14.0000 −0.670478
\(437\) −42.0000 −2.00913
\(438\) 0 0
\(439\) 2.00000 0.0954548 0.0477274 0.998860i \(-0.484802\pi\)
0.0477274 + 0.998860i \(0.484802\pi\)
\(440\) 4.00000 0.190693
\(441\) 0 0
\(442\) 0 0
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 0 0
\(445\) 24.0000 1.13771
\(446\) −21.0000 −0.994379
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) 2.00000 0.0943858 0.0471929 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) 0 0
\(451\) −3.00000 −0.141264
\(452\) 7.00000 0.329252
\(453\) 0 0
\(454\) 24.0000 1.12638
\(455\) 4.00000 0.187523
\(456\) 0 0
\(457\) −26.0000 −1.21623 −0.608114 0.793849i \(-0.708074\pi\)
−0.608114 + 0.793849i \(0.708074\pi\)
\(458\) 24.0000 1.12145
\(459\) 0 0
\(460\) 28.0000 1.30551
\(461\) −12.0000 −0.558896 −0.279448 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 4.00000 0.185695
\(465\) 0 0
\(466\) −5.00000 −0.231621
\(467\) 4.00000 0.185098 0.0925490 0.995708i \(-0.470499\pi\)
0.0925490 + 0.995708i \(0.470499\pi\)
\(468\) 0 0
\(469\) −1.00000 −0.0461757
\(470\) 28.0000 1.29154
\(471\) 0 0
\(472\) −10.0000 −0.460287
\(473\) −4.00000 −0.183920
\(474\) 0 0
\(475\) −66.0000 −3.02829
\(476\) 0 0
\(477\) 0 0
\(478\) 6.00000 0.274434
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) −9.00000 −0.410365
\(482\) 18.0000 0.819878
\(483\) 0 0
\(484\) −10.0000 −0.454545
\(485\) −4.00000 −0.181631
\(486\) 0 0
\(487\) 38.0000 1.72194 0.860972 0.508652i \(-0.169856\pi\)
0.860972 + 0.508652i \(0.169856\pi\)
\(488\) −1.00000 −0.0452679
\(489\) 0 0
\(490\) −4.00000 −0.180702
\(491\) −16.0000 −0.722070 −0.361035 0.932552i \(-0.617576\pi\)
−0.361035 + 0.932552i \(0.617576\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −6.00000 −0.269953
\(495\) 0 0
\(496\) 7.00000 0.314309
\(497\) 16.0000 0.717698
\(498\) 0 0
\(499\) −5.00000 −0.223831 −0.111915 0.993718i \(-0.535699\pi\)
−0.111915 + 0.993718i \(0.535699\pi\)
\(500\) 24.0000 1.07331
\(501\) 0 0
\(502\) 3.00000 0.133897
\(503\) 20.0000 0.891756 0.445878 0.895094i \(-0.352892\pi\)
0.445878 + 0.895094i \(0.352892\pi\)
\(504\) 0 0
\(505\) 20.0000 0.889988
\(506\) 7.00000 0.311188
\(507\) 0 0
\(508\) −11.0000 −0.488046
\(509\) −10.0000 −0.443242 −0.221621 0.975133i \(-0.571135\pi\)
−0.221621 + 0.975133i \(0.571135\pi\)
\(510\) 0 0
\(511\) −5.00000 −0.221187
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 24.0000 1.05859
\(515\) −56.0000 −2.46765
\(516\) 0 0
\(517\) 7.00000 0.307860
\(518\) 9.00000 0.395437
\(519\) 0 0
\(520\) 4.00000 0.175412
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 0 0
\(523\) 27.0000 1.18063 0.590314 0.807174i \(-0.299004\pi\)
0.590314 + 0.807174i \(0.299004\pi\)
\(524\) −8.00000 −0.349482
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) 0 0
\(528\) 0 0
\(529\) 26.0000 1.13043
\(530\) 0 0
\(531\) 0 0
\(532\) 6.00000 0.260133
\(533\) −3.00000 −0.129944
\(534\) 0 0
\(535\) 16.0000 0.691740
\(536\) −1.00000 −0.0431934
\(537\) 0 0
\(538\) 9.00000 0.388018
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 23.0000 0.987935
\(543\) 0 0
\(544\) 0 0
\(545\) −56.0000 −2.39878
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 6.00000 0.256307
\(549\) 0 0
\(550\) 11.0000 0.469042
\(551\) −24.0000 −1.02243
\(552\) 0 0
\(553\) −11.0000 −0.467768
\(554\) 18.0000 0.764747
\(555\) 0 0
\(556\) 4.00000 0.169638
\(557\) −9.00000 −0.381342 −0.190671 0.981654i \(-0.561066\pi\)
−0.190671 + 0.981654i \(0.561066\pi\)
\(558\) 0 0
\(559\) −4.00000 −0.169182
\(560\) −4.00000 −0.169031
\(561\) 0 0
\(562\) −8.00000 −0.337460
\(563\) −11.0000 −0.463595 −0.231797 0.972764i \(-0.574461\pi\)
−0.231797 + 0.972764i \(0.574461\pi\)
\(564\) 0 0
\(565\) 28.0000 1.17797
\(566\) −19.0000 −0.798630
\(567\) 0 0
\(568\) 16.0000 0.671345
\(569\) 35.0000 1.46728 0.733638 0.679540i \(-0.237821\pi\)
0.733638 + 0.679540i \(0.237821\pi\)
\(570\) 0 0
\(571\) 6.00000 0.251092 0.125546 0.992088i \(-0.459932\pi\)
0.125546 + 0.992088i \(0.459932\pi\)
\(572\) 1.00000 0.0418121
\(573\) 0 0
\(574\) 3.00000 0.125218
\(575\) 77.0000 3.21112
\(576\) 0 0
\(577\) 30.0000 1.24892 0.624458 0.781058i \(-0.285320\pi\)
0.624458 + 0.781058i \(0.285320\pi\)
\(578\) 17.0000 0.707107
\(579\) 0 0
\(580\) 16.0000 0.664364
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) −5.00000 −0.206901
\(585\) 0 0
\(586\) −26.0000 −1.07405
\(587\) −14.0000 −0.577842 −0.288921 0.957353i \(-0.593296\pi\)
−0.288921 + 0.957353i \(0.593296\pi\)
\(588\) 0 0
\(589\) −42.0000 −1.73058
\(590\) −40.0000 −1.64677
\(591\) 0 0
\(592\) 9.00000 0.369898
\(593\) −14.0000 −0.574911 −0.287456 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −9.00000 −0.368654
\(597\) 0 0
\(598\) 7.00000 0.286251
\(599\) 29.0000 1.18491 0.592454 0.805604i \(-0.298159\pi\)
0.592454 + 0.805604i \(0.298159\pi\)
\(600\) 0 0
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 4.00000 0.163028
\(603\) 0 0
\(604\) 0 0
\(605\) −40.0000 −1.62623
\(606\) 0 0
\(607\) 6.00000 0.243532 0.121766 0.992559i \(-0.461144\pi\)
0.121766 + 0.992559i \(0.461144\pi\)
\(608\) 6.00000 0.243332
\(609\) 0 0
\(610\) −4.00000 −0.161955
\(611\) 7.00000 0.283190
\(612\) 0 0
\(613\) −49.0000 −1.97909 −0.989546 0.144220i \(-0.953933\pi\)
−0.989546 + 0.144220i \(0.953933\pi\)
\(614\) 4.00000 0.161427
\(615\) 0 0
\(616\) −1.00000 −0.0402911
\(617\) −22.0000 −0.885687 −0.442843 0.896599i \(-0.646030\pi\)
−0.442843 + 0.896599i \(0.646030\pi\)
\(618\) 0 0
\(619\) 22.0000 0.884255 0.442127 0.896952i \(-0.354224\pi\)
0.442127 + 0.896952i \(0.354224\pi\)
\(620\) 28.0000 1.12451
\(621\) 0 0
\(622\) 30.0000 1.20289
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) 41.0000 1.64000
\(626\) −14.0000 −0.559553
\(627\) 0 0
\(628\) 5.00000 0.199522
\(629\) 0 0
\(630\) 0 0
\(631\) −2.00000 −0.0796187 −0.0398094 0.999207i \(-0.512675\pi\)
−0.0398094 + 0.999207i \(0.512675\pi\)
\(632\) −11.0000 −0.437557
\(633\) 0 0
\(634\) 21.0000 0.834017
\(635\) −44.0000 −1.74609
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 4.00000 0.158362
\(639\) 0 0
\(640\) −4.00000 −0.158114
\(641\) −7.00000 −0.276483 −0.138242 0.990399i \(-0.544145\pi\)
−0.138242 + 0.990399i \(0.544145\pi\)
\(642\) 0 0
\(643\) −16.0000 −0.630978 −0.315489 0.948929i \(-0.602169\pi\)
−0.315489 + 0.948929i \(0.602169\pi\)
\(644\) −7.00000 −0.275839
\(645\) 0 0
\(646\) 0 0
\(647\) 8.00000 0.314512 0.157256 0.987558i \(-0.449735\pi\)
0.157256 + 0.987558i \(0.449735\pi\)
\(648\) 0 0
\(649\) −10.0000 −0.392534
\(650\) 11.0000 0.431455
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) 36.0000 1.40879 0.704394 0.709809i \(-0.251219\pi\)
0.704394 + 0.709809i \(0.251219\pi\)
\(654\) 0 0
\(655\) −32.0000 −1.25034
\(656\) 3.00000 0.117130
\(657\) 0 0
\(658\) −7.00000 −0.272888
\(659\) 18.0000 0.701180 0.350590 0.936529i \(-0.385981\pi\)
0.350590 + 0.936529i \(0.385981\pi\)
\(660\) 0 0
\(661\) −4.00000 −0.155582 −0.0777910 0.996970i \(-0.524787\pi\)
−0.0777910 + 0.996970i \(0.524787\pi\)
\(662\) 7.00000 0.272063
\(663\) 0 0
\(664\) 0 0
\(665\) 24.0000 0.930680
\(666\) 0 0
\(667\) 28.0000 1.08416
\(668\) 0 0
\(669\) 0 0
\(670\) −4.00000 −0.154533
\(671\) −1.00000 −0.0386046
\(672\) 0 0
\(673\) 7.00000 0.269830 0.134915 0.990857i \(-0.456924\pi\)
0.134915 + 0.990857i \(0.456924\pi\)
\(674\) −17.0000 −0.654816
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) −39.0000 −1.49889 −0.749446 0.662066i \(-0.769680\pi\)
−0.749446 + 0.662066i \(0.769680\pi\)
\(678\) 0 0
\(679\) 1.00000 0.0383765
\(680\) 0 0
\(681\) 0 0
\(682\) 7.00000 0.268044
\(683\) −1.00000 −0.0382639 −0.0191320 0.999817i \(-0.506090\pi\)
−0.0191320 + 0.999817i \(0.506090\pi\)
\(684\) 0 0
\(685\) 24.0000 0.916993
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) 4.00000 0.152499
\(689\) 0 0
\(690\) 0 0
\(691\) −32.0000 −1.21734 −0.608669 0.793424i \(-0.708296\pi\)
−0.608669 + 0.793424i \(0.708296\pi\)
\(692\) −2.00000 −0.0760286
\(693\) 0 0
\(694\) −24.0000 −0.911028
\(695\) 16.0000 0.606915
\(696\) 0 0
\(697\) 0 0
\(698\) 26.0000 0.984115
\(699\) 0 0
\(700\) −11.0000 −0.415761
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) −54.0000 −2.03665
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 25.0000 0.940887
\(707\) −5.00000 −0.188044
\(708\) 0 0
\(709\) −13.0000 −0.488225 −0.244113 0.969747i \(-0.578497\pi\)
−0.244113 + 0.969747i \(0.578497\pi\)
\(710\) 64.0000 2.40188
\(711\) 0 0
\(712\) −6.00000 −0.224860
\(713\) 49.0000 1.83506
\(714\) 0 0
\(715\) 4.00000 0.149592
\(716\) 6.00000 0.224231
\(717\) 0 0
\(718\) 12.0000 0.447836
\(719\) −2.00000 −0.0745874 −0.0372937 0.999304i \(-0.511874\pi\)
−0.0372937 + 0.999304i \(0.511874\pi\)
\(720\) 0 0
\(721\) 14.0000 0.521387
\(722\) −17.0000 −0.632674
\(723\) 0 0
\(724\) 15.0000 0.557471
\(725\) 44.0000 1.63412
\(726\) 0 0
\(727\) −26.0000 −0.964287 −0.482143 0.876092i \(-0.660142\pi\)
−0.482143 + 0.876092i \(0.660142\pi\)
\(728\) −1.00000 −0.0370625
\(729\) 0 0
\(730\) −20.0000 −0.740233
\(731\) 0 0
\(732\) 0 0
\(733\) −4.00000 −0.147743 −0.0738717 0.997268i \(-0.523536\pi\)
−0.0738717 + 0.997268i \(0.523536\pi\)
\(734\) −32.0000 −1.18114
\(735\) 0 0
\(736\) −7.00000 −0.258023
\(737\) −1.00000 −0.0368355
\(738\) 0 0
\(739\) 24.0000 0.882854 0.441427 0.897297i \(-0.354472\pi\)
0.441427 + 0.897297i \(0.354472\pi\)
\(740\) 36.0000 1.32339
\(741\) 0 0
\(742\) 0 0
\(743\) −16.0000 −0.586983 −0.293492 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) 0 0
\(745\) −36.0000 −1.31894
\(746\) 16.0000 0.585802
\(747\) 0 0
\(748\) 0 0
\(749\) −4.00000 −0.146157
\(750\) 0 0
\(751\) −27.0000 −0.985244 −0.492622 0.870243i \(-0.663961\pi\)
−0.492622 + 0.870243i \(0.663961\pi\)
\(752\) −7.00000 −0.255264
\(753\) 0 0
\(754\) 4.00000 0.145671
\(755\) 0 0
\(756\) 0 0
\(757\) 54.0000 1.96266 0.981332 0.192323i \(-0.0616021\pi\)
0.981332 + 0.192323i \(0.0616021\pi\)
\(758\) 8.00000 0.290573
\(759\) 0 0
\(760\) 24.0000 0.870572
\(761\) −45.0000 −1.63125 −0.815624 0.578582i \(-0.803606\pi\)
−0.815624 + 0.578582i \(0.803606\pi\)
\(762\) 0 0
\(763\) 14.0000 0.506834
\(764\) 8.00000 0.289430
\(765\) 0 0
\(766\) −15.0000 −0.541972
\(767\) −10.0000 −0.361079
\(768\) 0 0
\(769\) 21.0000 0.757279 0.378640 0.925544i \(-0.376392\pi\)
0.378640 + 0.925544i \(0.376392\pi\)
\(770\) −4.00000 −0.144150
\(771\) 0 0
\(772\) 20.0000 0.719816
\(773\) −14.0000 −0.503545 −0.251773 0.967786i \(-0.581013\pi\)
−0.251773 + 0.967786i \(0.581013\pi\)
\(774\) 0 0
\(775\) 77.0000 2.76592
\(776\) 1.00000 0.0358979
\(777\) 0 0
\(778\) −36.0000 −1.29066
\(779\) −18.0000 −0.644917
\(780\) 0 0
\(781\) 16.0000 0.572525
\(782\) 0 0
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 20.0000 0.713831
\(786\) 0 0
\(787\) 32.0000 1.14068 0.570338 0.821410i \(-0.306812\pi\)
0.570338 + 0.821410i \(0.306812\pi\)
\(788\) 27.0000 0.961835
\(789\) 0 0
\(790\) −44.0000 −1.56545
\(791\) −7.00000 −0.248891
\(792\) 0 0
\(793\) −1.00000 −0.0355110
\(794\) 28.0000 0.993683
\(795\) 0 0
\(796\) −4.00000 −0.141776
\(797\) −15.0000 −0.531327 −0.265664 0.964066i \(-0.585591\pi\)
−0.265664 + 0.964066i \(0.585591\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −11.0000 −0.388909
\(801\) 0 0
\(802\) 12.0000 0.423735
\(803\) −5.00000 −0.176446
\(804\) 0 0
\(805\) −28.0000 −0.986870
\(806\) 7.00000 0.246564
\(807\) 0 0
\(808\) −5.00000 −0.175899
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) −4.00000 −0.140372
\(813\) 0 0
\(814\) 9.00000 0.315450
\(815\) −16.0000 −0.560456
\(816\) 0 0
\(817\) −24.0000 −0.839654
\(818\) 26.0000 0.909069
\(819\) 0 0
\(820\) 12.0000 0.419058
\(821\) 46.0000 1.60541 0.802706 0.596376i \(-0.203393\pi\)
0.802706 + 0.596376i \(0.203393\pi\)
\(822\) 0 0
\(823\) −3.00000 −0.104573 −0.0522867 0.998632i \(-0.516651\pi\)
−0.0522867 + 0.998632i \(0.516651\pi\)
\(824\) 14.0000 0.487713
\(825\) 0 0
\(826\) 10.0000 0.347945
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 0 0
\(829\) −14.0000 −0.486240 −0.243120 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −1.00000 −0.0346688
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 6.00000 0.207514
\(837\) 0 0
\(838\) 15.0000 0.518166
\(839\) −17.0000 −0.586905 −0.293453 0.955974i \(-0.594804\pi\)
−0.293453 + 0.955974i \(0.594804\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) −19.0000 −0.654783
\(843\) 0 0
\(844\) −10.0000 −0.344214
\(845\) 4.00000 0.137604
\(846\) 0 0
\(847\) 10.0000 0.343604
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 63.0000 2.15961
\(852\) 0 0
\(853\) 16.0000 0.547830 0.273915 0.961754i \(-0.411681\pi\)
0.273915 + 0.961754i \(0.411681\pi\)
\(854\) 1.00000 0.0342193
\(855\) 0 0
\(856\) −4.00000 −0.136717
\(857\) −22.0000 −0.751506 −0.375753 0.926720i \(-0.622616\pi\)
−0.375753 + 0.926720i \(0.622616\pi\)
\(858\) 0 0
\(859\) 5.00000 0.170598 0.0852989 0.996355i \(-0.472815\pi\)
0.0852989 + 0.996355i \(0.472815\pi\)
\(860\) 16.0000 0.545595
\(861\) 0 0
\(862\) 18.0000 0.613082
\(863\) −28.0000 −0.953131 −0.476566 0.879139i \(-0.658119\pi\)
−0.476566 + 0.879139i \(0.658119\pi\)
\(864\) 0 0
\(865\) −8.00000 −0.272008
\(866\) −34.0000 −1.15537
\(867\) 0 0
\(868\) −7.00000 −0.237595
\(869\) −11.0000 −0.373149
\(870\) 0 0
\(871\) −1.00000 −0.0338837
\(872\) 14.0000 0.474100
\(873\) 0 0
\(874\) 42.0000 1.42067
\(875\) −24.0000 −0.811348
\(876\) 0 0
\(877\) −45.0000 −1.51954 −0.759771 0.650191i \(-0.774689\pi\)
−0.759771 + 0.650191i \(0.774689\pi\)
\(878\) −2.00000 −0.0674967
\(879\) 0 0
\(880\) −4.00000 −0.134840
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) −14.0000 −0.471138 −0.235569 0.971858i \(-0.575695\pi\)
−0.235569 + 0.971858i \(0.575695\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) 38.0000 1.27592 0.637958 0.770072i \(-0.279780\pi\)
0.637958 + 0.770072i \(0.279780\pi\)
\(888\) 0 0
\(889\) 11.0000 0.368928
\(890\) −24.0000 −0.804482
\(891\) 0 0
\(892\) 21.0000 0.703132
\(893\) 42.0000 1.40548
\(894\) 0 0
\(895\) 24.0000 0.802232
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −2.00000 −0.0667409
\(899\) 28.0000 0.933852
\(900\) 0 0
\(901\) 0 0
\(902\) 3.00000 0.0998891
\(903\) 0 0
\(904\) −7.00000 −0.232817
\(905\) 60.0000 1.99447
\(906\) 0 0
\(907\) −30.0000 −0.996134 −0.498067 0.867139i \(-0.665957\pi\)
−0.498067 + 0.867139i \(0.665957\pi\)
\(908\) −24.0000 −0.796468
\(909\) 0 0
\(910\) −4.00000 −0.132599
\(911\) −36.0000 −1.19273 −0.596367 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 26.0000 0.860004
\(915\) 0 0
\(916\) −24.0000 −0.792982
\(917\) 8.00000 0.264183
\(918\) 0 0
\(919\) 27.0000 0.890648 0.445324 0.895370i \(-0.353089\pi\)
0.445324 + 0.895370i \(0.353089\pi\)
\(920\) −28.0000 −0.923133
\(921\) 0 0
\(922\) 12.0000 0.395199
\(923\) 16.0000 0.526646
\(924\) 0 0
\(925\) 99.0000 3.25510
\(926\) −16.0000 −0.525793
\(927\) 0 0
\(928\) −4.00000 −0.131306
\(929\) 15.0000 0.492134 0.246067 0.969253i \(-0.420862\pi\)
0.246067 + 0.969253i \(0.420862\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) 5.00000 0.163780
\(933\) 0 0
\(934\) −4.00000 −0.130884
\(935\) 0 0
\(936\) 0 0
\(937\) −36.0000 −1.17607 −0.588034 0.808836i \(-0.700098\pi\)
−0.588034 + 0.808836i \(0.700098\pi\)
\(938\) 1.00000 0.0326512
\(939\) 0 0
\(940\) −28.0000 −0.913259
\(941\) 50.0000 1.62995 0.814977 0.579494i \(-0.196750\pi\)
0.814977 + 0.579494i \(0.196750\pi\)
\(942\) 0 0
\(943\) 21.0000 0.683854
\(944\) 10.0000 0.325472
\(945\) 0 0
\(946\) 4.00000 0.130051
\(947\) −24.0000 −0.779895 −0.389948 0.920837i \(-0.627507\pi\)
−0.389948 + 0.920837i \(0.627507\pi\)
\(948\) 0 0
\(949\) −5.00000 −0.162307
\(950\) 66.0000 2.14132
\(951\) 0 0
\(952\) 0 0
\(953\) 34.0000 1.10137 0.550684 0.834714i \(-0.314367\pi\)
0.550684 + 0.834714i \(0.314367\pi\)
\(954\) 0 0
\(955\) 32.0000 1.03550
\(956\) −6.00000 −0.194054
\(957\) 0 0
\(958\) 24.0000 0.775405
\(959\) −6.00000 −0.193750
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) 9.00000 0.290172
\(963\) 0 0
\(964\) −18.0000 −0.579741
\(965\) 80.0000 2.57529
\(966\) 0 0
\(967\) −22.0000 −0.707472 −0.353736 0.935345i \(-0.615089\pi\)
−0.353736 + 0.935345i \(0.615089\pi\)
\(968\) 10.0000 0.321412
\(969\) 0 0
\(970\) 4.00000 0.128432
\(971\) 15.0000 0.481373 0.240686 0.970603i \(-0.422627\pi\)
0.240686 + 0.970603i \(0.422627\pi\)
\(972\) 0 0
\(973\) −4.00000 −0.128234
\(974\) −38.0000 −1.21760
\(975\) 0 0
\(976\) 1.00000 0.0320092
\(977\) −42.0000 −1.34370 −0.671850 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) 0 0
\(979\) −6.00000 −0.191761
\(980\) 4.00000 0.127775
\(981\) 0 0
\(982\) 16.0000 0.510581
\(983\) 40.0000 1.27580 0.637901 0.770118i \(-0.279803\pi\)
0.637901 + 0.770118i \(0.279803\pi\)
\(984\) 0 0
\(985\) 108.000 3.44117
\(986\) 0 0
\(987\) 0 0
\(988\) 6.00000 0.190885
\(989\) 28.0000 0.890348
\(990\) 0 0
\(991\) −57.0000 −1.81066 −0.905332 0.424704i \(-0.860378\pi\)
−0.905332 + 0.424704i \(0.860378\pi\)
\(992\) −7.00000 −0.222250
\(993\) 0 0
\(994\) −16.0000 −0.507489
\(995\) −16.0000 −0.507234
\(996\) 0 0
\(997\) 5.00000 0.158352 0.0791758 0.996861i \(-0.474771\pi\)
0.0791758 + 0.996861i \(0.474771\pi\)
\(998\) 5.00000 0.158272
\(999\) 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 1638.2.a.j.1.1 1
3.2 odd 2 182.2.a.e.1.1 1
12.11 even 2 1456.2.a.a.1.1 1
15.14 odd 2 4550.2.a.a.1.1 1
21.2 odd 6 1274.2.f.b.1145.1 2
21.5 even 6 1274.2.f.k.1145.1 2
21.11 odd 6 1274.2.f.b.79.1 2
21.17 even 6 1274.2.f.k.79.1 2
21.20 even 2 1274.2.a.h.1.1 1
24.5 odd 2 5824.2.a.b.1.1 1
24.11 even 2 5824.2.a.bf.1.1 1
39.5 even 4 2366.2.d.j.337.1 2
39.8 even 4 2366.2.d.j.337.2 2
39.38 odd 2 2366.2.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
182.2.a.e.1.1 1 3.2 odd 2
1274.2.a.h.1.1 1 21.20 even 2
1274.2.f.b.79.1 2 21.11 odd 6
1274.2.f.b.1145.1 2 21.2 odd 6
1274.2.f.k.79.1 2 21.17 even 6
1274.2.f.k.1145.1 2 21.5 even 6
1456.2.a.a.1.1 1 12.11 even 2
1638.2.a.j.1.1 1 1.1 even 1 trivial
2366.2.a.h.1.1 1 39.38 odd 2
2366.2.d.j.337.1 2 39.5 even 4
2366.2.d.j.337.2 2 39.8 even 4
4550.2.a.a.1.1 1 15.14 odd 2
5824.2.a.b.1.1 1 24.5 odd 2
5824.2.a.bf.1.1 1 24.11 even 2