Properties

Label 1638.2.a.q.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

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} +1.00000 q^{8} +5.00000 q^{11} -1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} +2.00000 q^{19} +5.00000 q^{22} -5.00000 q^{23} -5.00000 q^{25} -1.00000 q^{26} +1.00000 q^{28} -4.00000 q^{29} +1.00000 q^{31} +1.00000 q^{32} +4.00000 q^{34} +7.00000 q^{37} +2.00000 q^{38} +9.00000 q^{41} -12.0000 q^{43} +5.00000 q^{44} -5.00000 q^{46} +7.00000 q^{47} +1.00000 q^{49} -5.00000 q^{50} -1.00000 q^{52} +4.00000 q^{53} +1.00000 q^{56} -4.00000 q^{58} +6.00000 q^{59} +13.0000 q^{61} +1.00000 q^{62} +1.00000 q^{64} +11.0000 q^{67} +4.00000 q^{68} +7.00000 q^{73} +7.00000 q^{74} +2.00000 q^{76} +5.00000 q^{77} -17.0000 q^{79} +9.00000 q^{82} -4.00000 q^{83} -12.0000 q^{86} +5.00000 q^{88} -14.0000 q^{89} -1.00000 q^{91} -5.00000 q^{92} +7.00000 q^{94} +5.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 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 \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 5.00000 1.06600
\(23\) −5.00000 −1.04257 −0.521286 0.853382i \(-0.674548\pi\)
−0.521286 + 0.853382i \(0.674548\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) −1.00000 −0.196116
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605 0.0898027 0.995960i \(-0.471376\pi\)
0.0898027 + 0.995960i \(0.471376\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 4.00000 0.685994
\(35\) 0 0
\(36\) 0 0
\(37\) 7.00000 1.15079 0.575396 0.817875i \(-0.304848\pi\)
0.575396 + 0.817875i \(0.304848\pi\)
\(38\) 2.00000 0.324443
\(39\) 0 0
\(40\) 0 0
\(41\) 9.00000 1.40556 0.702782 0.711405i \(-0.251941\pi\)
0.702782 + 0.711405i \(0.251941\pi\)
\(42\) 0 0
\(43\) −12.0000 −1.82998 −0.914991 0.403473i \(-0.867803\pi\)
−0.914991 + 0.403473i \(0.867803\pi\)
\(44\) 5.00000 0.753778
\(45\) 0 0
\(46\) −5.00000 −0.737210
\(47\) 7.00000 1.02105 0.510527 0.859861i \(-0.329450\pi\)
0.510527 + 0.859861i \(0.329450\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −5.00000 −0.707107
\(51\) 0 0
\(52\) −1.00000 −0.138675
\(53\) 4.00000 0.549442 0.274721 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −4.00000 −0.525226
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) 13.0000 1.66448 0.832240 0.554416i \(-0.187058\pi\)
0.832240 + 0.554416i \(0.187058\pi\)
\(62\) 1.00000 0.127000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 11.0000 1.34386 0.671932 0.740613i \(-0.265465\pi\)
0.671932 + 0.740613i \(0.265465\pi\)
\(68\) 4.00000 0.485071
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 7.00000 0.819288 0.409644 0.912245i \(-0.365653\pi\)
0.409644 + 0.912245i \(0.365653\pi\)
\(74\) 7.00000 0.813733
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 5.00000 0.569803
\(78\) 0 0
\(79\) −17.0000 −1.91265 −0.956325 0.292306i \(-0.905577\pi\)
−0.956325 + 0.292306i \(0.905577\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 9.00000 0.993884
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −12.0000 −1.29399
\(87\) 0 0
\(88\) 5.00000 0.533002
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) −5.00000 −0.521286
\(93\) 0 0
\(94\) 7.00000 0.721995
\(95\) 0 0
\(96\) 0 0
\(97\) 5.00000 0.507673 0.253837 0.967247i \(-0.418307\pi\)
0.253837 + 0.967247i \(0.418307\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) −5.00000 −0.500000
\(101\) −15.0000 −1.49256 −0.746278 0.665635i \(-0.768161\pi\)
−0.746278 + 0.665635i \(0.768161\pi\)
\(102\) 0 0
\(103\) 6.00000 0.591198 0.295599 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) 4.00000 0.388514
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 0 0
\(109\) −18.0000 −1.72409 −0.862044 0.506834i \(-0.830816\pi\)
−0.862044 + 0.506834i \(0.830816\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) −1.00000 −0.0940721 −0.0470360 0.998893i \(-0.514978\pi\)
−0.0470360 + 0.998893i \(0.514978\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −4.00000 −0.371391
\(117\) 0 0
\(118\) 6.00000 0.552345
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 13.0000 1.17696
\(123\) 0 0
\(124\) 1.00000 0.0898027
\(125\) 0 0
\(126\) 0 0
\(127\) 9.00000 0.798621 0.399310 0.916816i \(-0.369250\pi\)
0.399310 + 0.916816i \(0.369250\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 0 0
\(133\) 2.00000 0.173422
\(134\) 11.0000 0.950255
\(135\) 0 0
\(136\) 4.00000 0.342997
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −5.00000 −0.418121
\(144\) 0 0
\(145\) 0 0
\(146\) 7.00000 0.579324
\(147\) 0 0
\(148\) 7.00000 0.575396
\(149\) −7.00000 −0.573462 −0.286731 0.958011i \(-0.592569\pi\)
−0.286731 + 0.958011i \(0.592569\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 2.00000 0.162221
\(153\) 0 0
\(154\) 5.00000 0.402911
\(155\) 0 0
\(156\) 0 0
\(157\) 1.00000 0.0798087 0.0399043 0.999204i \(-0.487295\pi\)
0.0399043 + 0.999204i \(0.487295\pi\)
\(158\) −17.0000 −1.35245
\(159\) 0 0
\(160\) 0 0
\(161\) −5.00000 −0.394055
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 9.00000 0.702782
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) −12.0000 −0.914991
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) 0 0
\(175\) −5.00000 −0.377964
\(176\) 5.00000 0.376889
\(177\) 0 0
\(178\) −14.0000 −1.04934
\(179\) 2.00000 0.149487 0.0747435 0.997203i \(-0.476186\pi\)
0.0747435 + 0.997203i \(0.476186\pi\)
\(180\) 0 0
\(181\) −5.00000 −0.371647 −0.185824 0.982583i \(-0.559495\pi\)
−0.185824 + 0.982583i \(0.559495\pi\)
\(182\) −1.00000 −0.0741249
\(183\) 0 0
\(184\) −5.00000 −0.368605
\(185\) 0 0
\(186\) 0 0
\(187\) 20.0000 1.46254
\(188\) 7.00000 0.510527
\(189\) 0 0
\(190\) 0 0
\(191\) 16.0000 1.15772 0.578860 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) 0 0
\(193\) 4.00000 0.287926 0.143963 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) 5.00000 0.358979
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −27.0000 −1.92367 −0.961835 0.273629i \(-0.911776\pi\)
−0.961835 + 0.273629i \(0.911776\pi\)
\(198\) 0 0
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) −5.00000 −0.353553
\(201\) 0 0
\(202\) −15.0000 −1.05540
\(203\) −4.00000 −0.280745
\(204\) 0 0
\(205\) 0 0
\(206\) 6.00000 0.418040
\(207\) 0 0
\(208\) −1.00000 −0.0693375
\(209\) 10.0000 0.691714
\(210\) 0 0
\(211\) −14.0000 −0.963800 −0.481900 0.876226i \(-0.660053\pi\)
−0.481900 + 0.876226i \(0.660053\pi\)
\(212\) 4.00000 0.274721
\(213\) 0 0
\(214\) 8.00000 0.546869
\(215\) 0 0
\(216\) 0 0
\(217\) 1.00000 0.0678844
\(218\) −18.0000 −1.21911
\(219\) 0 0
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) 3.00000 0.200895 0.100447 0.994942i \(-0.467973\pi\)
0.100447 + 0.994942i \(0.467973\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −1.00000 −0.0665190
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) 16.0000 1.05731 0.528655 0.848837i \(-0.322697\pi\)
0.528655 + 0.848837i \(0.322697\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −4.00000 −0.262613
\(233\) 21.0000 1.37576 0.687878 0.725826i \(-0.258542\pi\)
0.687878 + 0.725826i \(0.258542\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 6.00000 0.390567
\(237\) 0 0
\(238\) 4.00000 0.259281
\(239\) 6.00000 0.388108 0.194054 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) 0 0
\(241\) −30.0000 −1.93247 −0.966235 0.257663i \(-0.917048\pi\)
−0.966235 + 0.257663i \(0.917048\pi\)
\(242\) 14.0000 0.899954
\(243\) 0 0
\(244\) 13.0000 0.832240
\(245\) 0 0
\(246\) 0 0
\(247\) −2.00000 −0.127257
\(248\) 1.00000 0.0635001
\(249\) 0 0
\(250\) 0 0
\(251\) 13.0000 0.820553 0.410276 0.911961i \(-0.365432\pi\)
0.410276 + 0.911961i \(0.365432\pi\)
\(252\) 0 0
\(253\) −25.0000 −1.57174
\(254\) 9.00000 0.564710
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −28.0000 −1.74659 −0.873296 0.487190i \(-0.838022\pi\)
−0.873296 + 0.487190i \(0.838022\pi\)
\(258\) 0 0
\(259\) 7.00000 0.434959
\(260\) 0 0
\(261\) 0 0
\(262\) −8.00000 −0.494242
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 2.00000 0.122628
\(267\) 0 0
\(268\) 11.0000 0.671932
\(269\) −13.0000 −0.792624 −0.396312 0.918116i \(-0.629710\pi\)
−0.396312 + 0.918116i \(0.629710\pi\)
\(270\) 0 0
\(271\) −1.00000 −0.0607457 −0.0303728 0.999539i \(-0.509669\pi\)
−0.0303728 + 0.999539i \(0.509669\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) −18.0000 −1.08742
\(275\) −25.0000 −1.50756
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 4.00000 0.239904
\(279\) 0 0
\(280\) 0 0
\(281\) 16.0000 0.954480 0.477240 0.878773i \(-0.341637\pi\)
0.477240 + 0.878773i \(0.341637\pi\)
\(282\) 0 0
\(283\) 3.00000 0.178331 0.0891657 0.996017i \(-0.471580\pi\)
0.0891657 + 0.996017i \(0.471580\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −5.00000 −0.295656
\(287\) 9.00000 0.531253
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 7.00000 0.409644
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 7.00000 0.406867
\(297\) 0 0
\(298\) −7.00000 −0.405499
\(299\) 5.00000 0.289157
\(300\) 0 0
\(301\) −12.0000 −0.691669
\(302\) −12.0000 −0.690522
\(303\) 0 0
\(304\) 2.00000 0.114708
\(305\) 0 0
\(306\) 0 0
\(307\) 32.0000 1.82634 0.913168 0.407583i \(-0.133628\pi\)
0.913168 + 0.407583i \(0.133628\pi\)
\(308\) 5.00000 0.284901
\(309\) 0 0
\(310\) 0 0
\(311\) −26.0000 −1.47432 −0.737162 0.675716i \(-0.763835\pi\)
−0.737162 + 0.675716i \(0.763835\pi\)
\(312\) 0 0
\(313\) −30.0000 −1.69570 −0.847850 0.530236i \(-0.822103\pi\)
−0.847850 + 0.530236i \(0.822103\pi\)
\(314\) 1.00000 0.0564333
\(315\) 0 0
\(316\) −17.0000 −0.956325
\(317\) −11.0000 −0.617822 −0.308911 0.951091i \(-0.599964\pi\)
−0.308911 + 0.951091i \(0.599964\pi\)
\(318\) 0 0
\(319\) −20.0000 −1.11979
\(320\) 0 0
\(321\) 0 0
\(322\) −5.00000 −0.278639
\(323\) 8.00000 0.445132
\(324\) 0 0
\(325\) 5.00000 0.277350
\(326\) 4.00000 0.221540
\(327\) 0 0
\(328\) 9.00000 0.496942
\(329\) 7.00000 0.385922
\(330\) 0 0
\(331\) 3.00000 0.164895 0.0824475 0.996595i \(-0.473726\pi\)
0.0824475 + 0.996595i \(0.473726\pi\)
\(332\) −4.00000 −0.219529
\(333\) 0 0
\(334\) −8.00000 −0.437741
\(335\) 0 0
\(336\) 0 0
\(337\) 9.00000 0.490261 0.245131 0.969490i \(-0.421169\pi\)
0.245131 + 0.969490i \(0.421169\pi\)
\(338\) 1.00000 0.0543928
\(339\) 0 0
\(340\) 0 0
\(341\) 5.00000 0.270765
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −12.0000 −0.646997
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) −16.0000 −0.858925 −0.429463 0.903085i \(-0.641297\pi\)
−0.429463 + 0.903085i \(0.641297\pi\)
\(348\) 0 0
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) −5.00000 −0.267261
\(351\) 0 0
\(352\) 5.00000 0.266501
\(353\) −3.00000 −0.159674 −0.0798369 0.996808i \(-0.525440\pi\)
−0.0798369 + 0.996808i \(0.525440\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −14.0000 −0.741999
\(357\) 0 0
\(358\) 2.00000 0.105703
\(359\) 4.00000 0.211112 0.105556 0.994413i \(-0.466338\pi\)
0.105556 + 0.994413i \(0.466338\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) −5.00000 −0.262794
\(363\) 0 0
\(364\) −1.00000 −0.0524142
\(365\) 0 0
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) −5.00000 −0.260643
\(369\) 0 0
\(370\) 0 0
\(371\) 4.00000 0.207670
\(372\) 0 0
\(373\) 4.00000 0.207112 0.103556 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) 20.0000 1.03418
\(375\) 0 0
\(376\) 7.00000 0.360997
\(377\) 4.00000 0.206010
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 16.0000 0.818631
\(383\) 9.00000 0.459879 0.229939 0.973205i \(-0.426147\pi\)
0.229939 + 0.973205i \(0.426147\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 4.00000 0.203595
\(387\) 0 0
\(388\) 5.00000 0.253837
\(389\) −8.00000 −0.405616 −0.202808 0.979219i \(-0.565007\pi\)
−0.202808 + 0.979219i \(0.565007\pi\)
\(390\) 0 0
\(391\) −20.0000 −1.01144
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) −27.0000 −1.36024
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(398\) −20.0000 −1.00251
\(399\) 0 0
\(400\) −5.00000 −0.250000
\(401\) 28.0000 1.39825 0.699127 0.714998i \(-0.253572\pi\)
0.699127 + 0.714998i \(0.253572\pi\)
\(402\) 0 0
\(403\) −1.00000 −0.0498135
\(404\) −15.0000 −0.746278
\(405\) 0 0
\(406\) −4.00000 −0.198517
\(407\) 35.0000 1.73489
\(408\) 0 0
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 6.00000 0.295599
\(413\) 6.00000 0.295241
\(414\) 0 0
\(415\) 0 0
\(416\) −1.00000 −0.0490290
\(417\) 0 0
\(418\) 10.0000 0.489116
\(419\) 9.00000 0.439679 0.219839 0.975536i \(-0.429447\pi\)
0.219839 + 0.975536i \(0.429447\pi\)
\(420\) 0 0
\(421\) −35.0000 −1.70580 −0.852898 0.522078i \(-0.825157\pi\)
−0.852898 + 0.522078i \(0.825157\pi\)
\(422\) −14.0000 −0.681509
\(423\) 0 0
\(424\) 4.00000 0.194257
\(425\) −20.0000 −0.970143
\(426\) 0 0
\(427\) 13.0000 0.629114
\(428\) 8.00000 0.386695
\(429\) 0 0
\(430\) 0 0
\(431\) 6.00000 0.289010 0.144505 0.989504i \(-0.453841\pi\)
0.144505 + 0.989504i \(0.453841\pi\)
\(432\) 0 0
\(433\) 26.0000 1.24948 0.624740 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) 1.00000 0.0480015
\(435\) 0 0
\(436\) −18.0000 −0.862044
\(437\) −10.0000 −0.478365
\(438\) 0 0
\(439\) −2.00000 −0.0954548 −0.0477274 0.998860i \(-0.515198\pi\)
−0.0477274 + 0.998860i \(0.515198\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −4.00000 −0.190261
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 3.00000 0.142054
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) 45.0000 2.11897
\(452\) −1.00000 −0.0470360
\(453\) 0 0
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) 0 0
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 16.0000 0.747631
\(459\) 0 0
\(460\) 0 0
\(461\) −12.0000 −0.558896 −0.279448 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 0 0
\(463\) 12.0000 0.557687 0.278844 0.960337i \(-0.410049\pi\)
0.278844 + 0.960337i \(0.410049\pi\)
\(464\) −4.00000 −0.185695
\(465\) 0 0
\(466\) 21.0000 0.972806
\(467\) 36.0000 1.66588 0.832941 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) 0 0
\(469\) 11.0000 0.507933
\(470\) 0 0
\(471\) 0 0
\(472\) 6.00000 0.276172
\(473\) −60.0000 −2.75880
\(474\) 0 0
\(475\) −10.0000 −0.458831
\(476\) 4.00000 0.183340
\(477\) 0 0
\(478\) 6.00000 0.274434
\(479\) 16.0000 0.731059 0.365529 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) 0 0
\(481\) −7.00000 −0.319173
\(482\) −30.0000 −1.36646
\(483\) 0 0
\(484\) 14.0000 0.636364
\(485\) 0 0
\(486\) 0 0
\(487\) −26.0000 −1.17817 −0.589086 0.808070i \(-0.700512\pi\)
−0.589086 + 0.808070i \(0.700512\pi\)
\(488\) 13.0000 0.588482
\(489\) 0 0
\(490\) 0 0
\(491\) −28.0000 −1.26362 −0.631811 0.775122i \(-0.717688\pi\)
−0.631811 + 0.775122i \(0.717688\pi\)
\(492\) 0 0
\(493\) −16.0000 −0.720604
\(494\) −2.00000 −0.0899843
\(495\) 0 0
\(496\) 1.00000 0.0449013
\(497\) 0 0
\(498\) 0 0
\(499\) 1.00000 0.0447661 0.0223831 0.999749i \(-0.492875\pi\)
0.0223831 + 0.999749i \(0.492875\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 13.0000 0.580218
\(503\) 40.0000 1.78351 0.891756 0.452517i \(-0.149474\pi\)
0.891756 + 0.452517i \(0.149474\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −25.0000 −1.11139
\(507\) 0 0
\(508\) 9.00000 0.399310
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) 7.00000 0.309662
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −28.0000 −1.23503
\(515\) 0 0
\(516\) 0 0
\(517\) 35.0000 1.53930
\(518\) 7.00000 0.307562
\(519\) 0 0
\(520\) 0 0
\(521\) 14.0000 0.613351 0.306676 0.951814i \(-0.400783\pi\)
0.306676 + 0.951814i \(0.400783\pi\)
\(522\) 0 0
\(523\) −13.0000 −0.568450 −0.284225 0.958758i \(-0.591736\pi\)
−0.284225 + 0.958758i \(0.591736\pi\)
\(524\) −8.00000 −0.349482
\(525\) 0 0
\(526\) 0 0
\(527\) 4.00000 0.174243
\(528\) 0 0
\(529\) 2.00000 0.0869565
\(530\) 0 0
\(531\) 0 0
\(532\) 2.00000 0.0867110
\(533\) −9.00000 −0.389833
\(534\) 0 0
\(535\) 0 0
\(536\) 11.0000 0.475128
\(537\) 0 0
\(538\) −13.0000 −0.560470
\(539\) 5.00000 0.215365
\(540\) 0 0
\(541\) 18.0000 0.773880 0.386940 0.922105i \(-0.373532\pi\)
0.386940 + 0.922105i \(0.373532\pi\)
\(542\) −1.00000 −0.0429537
\(543\) 0 0
\(544\) 4.00000 0.171499
\(545\) 0 0
\(546\) 0 0
\(547\) −32.0000 −1.36822 −0.684111 0.729378i \(-0.739809\pi\)
−0.684111 + 0.729378i \(0.739809\pi\)
\(548\) −18.0000 −0.768922
\(549\) 0 0
\(550\) −25.0000 −1.06600
\(551\) −8.00000 −0.340811
\(552\) 0 0
\(553\) −17.0000 −0.722914
\(554\) −10.0000 −0.424859
\(555\) 0 0
\(556\) 4.00000 0.169638
\(557\) 9.00000 0.381342 0.190671 0.981654i \(-0.438934\pi\)
0.190671 + 0.981654i \(0.438934\pi\)
\(558\) 0 0
\(559\) 12.0000 0.507546
\(560\) 0 0
\(561\) 0 0
\(562\) 16.0000 0.674919
\(563\) −35.0000 −1.47507 −0.737537 0.675307i \(-0.764011\pi\)
−0.737537 + 0.675307i \(0.764011\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 3.00000 0.126099
\(567\) 0 0
\(568\) 0 0
\(569\) −21.0000 −0.880366 −0.440183 0.897908i \(-0.645086\pi\)
−0.440183 + 0.897908i \(0.645086\pi\)
\(570\) 0 0
\(571\) 22.0000 0.920671 0.460336 0.887745i \(-0.347729\pi\)
0.460336 + 0.887745i \(0.347729\pi\)
\(572\) −5.00000 −0.209061
\(573\) 0 0
\(574\) 9.00000 0.375653
\(575\) 25.0000 1.04257
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 0 0
\(580\) 0 0
\(581\) −4.00000 −0.165948
\(582\) 0 0
\(583\) 20.0000 0.828315
\(584\) 7.00000 0.289662
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) −18.0000 −0.742940 −0.371470 0.928445i \(-0.621146\pi\)
−0.371470 + 0.928445i \(0.621146\pi\)
\(588\) 0 0
\(589\) 2.00000 0.0824086
\(590\) 0 0
\(591\) 0 0
\(592\) 7.00000 0.287698
\(593\) 46.0000 1.88899 0.944497 0.328521i \(-0.106550\pi\)
0.944497 + 0.328521i \(0.106550\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −7.00000 −0.286731
\(597\) 0 0
\(598\) 5.00000 0.204465
\(599\) 9.00000 0.367730 0.183865 0.982952i \(-0.441139\pi\)
0.183865 + 0.982952i \(0.441139\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) −12.0000 −0.489083
\(603\) 0 0
\(604\) −12.0000 −0.488273
\(605\) 0 0
\(606\) 0 0
\(607\) 22.0000 0.892952 0.446476 0.894795i \(-0.352679\pi\)
0.446476 + 0.894795i \(0.352679\pi\)
\(608\) 2.00000 0.0811107
\(609\) 0 0
\(610\) 0 0
\(611\) −7.00000 −0.283190
\(612\) 0 0
\(613\) −23.0000 −0.928961 −0.464481 0.885583i \(-0.653759\pi\)
−0.464481 + 0.885583i \(0.653759\pi\)
\(614\) 32.0000 1.29141
\(615\) 0 0
\(616\) 5.00000 0.201456
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 0 0
\(619\) 6.00000 0.241160 0.120580 0.992704i \(-0.461525\pi\)
0.120580 + 0.992704i \(0.461525\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −26.0000 −1.04251
\(623\) −14.0000 −0.560898
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) −30.0000 −1.19904
\(627\) 0 0
\(628\) 1.00000 0.0399043
\(629\) 28.0000 1.11643
\(630\) 0 0
\(631\) −26.0000 −1.03504 −0.517522 0.855670i \(-0.673145\pi\)
−0.517522 + 0.855670i \(0.673145\pi\)
\(632\) −17.0000 −0.676224
\(633\) 0 0
\(634\) −11.0000 −0.436866
\(635\) 0 0
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) −20.0000 −0.791808
\(639\) 0 0
\(640\) 0 0
\(641\) 33.0000 1.30342 0.651711 0.758468i \(-0.274052\pi\)
0.651711 + 0.758468i \(0.274052\pi\)
\(642\) 0 0
\(643\) −36.0000 −1.41970 −0.709851 0.704352i \(-0.751238\pi\)
−0.709851 + 0.704352i \(0.751238\pi\)
\(644\) −5.00000 −0.197028
\(645\) 0 0
\(646\) 8.00000 0.314756
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) 0 0
\(649\) 30.0000 1.17760
\(650\) 5.00000 0.196116
\(651\) 0 0
\(652\) 4.00000 0.156652
\(653\) 12.0000 0.469596 0.234798 0.972044i \(-0.424557\pi\)
0.234798 + 0.972044i \(0.424557\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 9.00000 0.351391
\(657\) 0 0
\(658\) 7.00000 0.272888
\(659\) −26.0000 −1.01282 −0.506408 0.862294i \(-0.669027\pi\)
−0.506408 + 0.862294i \(0.669027\pi\)
\(660\) 0 0
\(661\) −20.0000 −0.777910 −0.388955 0.921257i \(-0.627164\pi\)
−0.388955 + 0.921257i \(0.627164\pi\)
\(662\) 3.00000 0.116598
\(663\) 0 0
\(664\) −4.00000 −0.155230
\(665\) 0 0
\(666\) 0 0
\(667\) 20.0000 0.774403
\(668\) −8.00000 −0.309529
\(669\) 0 0
\(670\) 0 0
\(671\) 65.0000 2.50930
\(672\) 0 0
\(673\) −9.00000 −0.346925 −0.173462 0.984841i \(-0.555495\pi\)
−0.173462 + 0.984841i \(0.555495\pi\)
\(674\) 9.00000 0.346667
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) −3.00000 −0.115299 −0.0576497 0.998337i \(-0.518361\pi\)
−0.0576497 + 0.998337i \(0.518361\pi\)
\(678\) 0 0
\(679\) 5.00000 0.191882
\(680\) 0 0
\(681\) 0 0
\(682\) 5.00000 0.191460
\(683\) −3.00000 −0.114792 −0.0573959 0.998351i \(-0.518280\pi\)
−0.0573959 + 0.998351i \(0.518280\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) −12.0000 −0.457496
\(689\) −4.00000 −0.152388
\(690\) 0 0
\(691\) −36.0000 −1.36950 −0.684752 0.728776i \(-0.740090\pi\)
−0.684752 + 0.728776i \(0.740090\pi\)
\(692\) −18.0000 −0.684257
\(693\) 0 0
\(694\) −16.0000 −0.607352
\(695\) 0 0
\(696\) 0 0
\(697\) 36.0000 1.36360
\(698\) 2.00000 0.0757011
\(699\) 0 0
\(700\) −5.00000 −0.188982
\(701\) 12.0000 0.453234 0.226617 0.973984i \(-0.427233\pi\)
0.226617 + 0.973984i \(0.427233\pi\)
\(702\) 0 0
\(703\) 14.0000 0.528020
\(704\) 5.00000 0.188445
\(705\) 0 0
\(706\) −3.00000 −0.112906
\(707\) −15.0000 −0.564133
\(708\) 0 0
\(709\) 29.0000 1.08912 0.544559 0.838723i \(-0.316697\pi\)
0.544559 + 0.838723i \(0.316697\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −14.0000 −0.524672
\(713\) −5.00000 −0.187251
\(714\) 0 0
\(715\) 0 0
\(716\) 2.00000 0.0747435
\(717\) 0 0
\(718\) 4.00000 0.149279
\(719\) 18.0000 0.671287 0.335643 0.941989i \(-0.391046\pi\)
0.335643 + 0.941989i \(0.391046\pi\)
\(720\) 0 0
\(721\) 6.00000 0.223452
\(722\) −15.0000 −0.558242
\(723\) 0 0
\(724\) −5.00000 −0.185824
\(725\) 20.0000 0.742781
\(726\) 0 0
\(727\) 14.0000 0.519231 0.259616 0.965712i \(-0.416404\pi\)
0.259616 + 0.965712i \(0.416404\pi\)
\(728\) −1.00000 −0.0370625
\(729\) 0 0
\(730\) 0 0
\(731\) −48.0000 −1.77534
\(732\) 0 0
\(733\) 20.0000 0.738717 0.369358 0.929287i \(-0.379577\pi\)
0.369358 + 0.929287i \(0.379577\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) −5.00000 −0.184302
\(737\) 55.0000 2.02595
\(738\) 0 0
\(739\) −16.0000 −0.588570 −0.294285 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 4.00000 0.146845
\(743\) 36.0000 1.32071 0.660356 0.750953i \(-0.270405\pi\)
0.660356 + 0.750953i \(0.270405\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 4.00000 0.146450
\(747\) 0 0
\(748\) 20.0000 0.731272
\(749\) 8.00000 0.292314
\(750\) 0 0
\(751\) 25.0000 0.912263 0.456131 0.889912i \(-0.349235\pi\)
0.456131 + 0.889912i \(0.349235\pi\)
\(752\) 7.00000 0.255264
\(753\) 0 0
\(754\) 4.00000 0.145671
\(755\) 0 0
\(756\) 0 0
\(757\) −18.0000 −0.654221 −0.327111 0.944986i \(-0.606075\pi\)
−0.327111 + 0.944986i \(0.606075\pi\)
\(758\) −16.0000 −0.581146
\(759\) 0 0
\(760\) 0 0
\(761\) 33.0000 1.19625 0.598125 0.801403i \(-0.295913\pi\)
0.598125 + 0.801403i \(0.295913\pi\)
\(762\) 0 0
\(763\) −18.0000 −0.651644
\(764\) 16.0000 0.578860
\(765\) 0 0
\(766\) 9.00000 0.325183
\(767\) −6.00000 −0.216647
\(768\) 0 0
\(769\) −1.00000 −0.0360609 −0.0180305 0.999837i \(-0.505740\pi\)
−0.0180305 + 0.999837i \(0.505740\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 4.00000 0.143963
\(773\) −34.0000 −1.22290 −0.611448 0.791285i \(-0.709412\pi\)
−0.611448 + 0.791285i \(0.709412\pi\)
\(774\) 0 0
\(775\) −5.00000 −0.179605
\(776\) 5.00000 0.179490
\(777\) 0 0
\(778\) −8.00000 −0.286814
\(779\) 18.0000 0.644917
\(780\) 0 0
\(781\) 0 0
\(782\) −20.0000 −0.715199
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) 0 0
\(787\) 8.00000 0.285169 0.142585 0.989783i \(-0.454459\pi\)
0.142585 + 0.989783i \(0.454459\pi\)
\(788\) −27.0000 −0.961835
\(789\) 0 0
\(790\) 0 0
\(791\) −1.00000 −0.0355559
\(792\) 0 0
\(793\) −13.0000 −0.461644
\(794\) 0 0
\(795\) 0 0
\(796\) −20.0000 −0.708881
\(797\) −3.00000 −0.106265 −0.0531327 0.998587i \(-0.516921\pi\)
−0.0531327 + 0.998587i \(0.516921\pi\)
\(798\) 0 0
\(799\) 28.0000 0.990569
\(800\) −5.00000 −0.176777
\(801\) 0 0
\(802\) 28.0000 0.988714
\(803\) 35.0000 1.23512
\(804\) 0 0
\(805\) 0 0
\(806\) −1.00000 −0.0352235
\(807\) 0 0
\(808\) −15.0000 −0.527698
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) 52.0000 1.82597 0.912983 0.407997i \(-0.133772\pi\)
0.912983 + 0.407997i \(0.133772\pi\)
\(812\) −4.00000 −0.140372
\(813\) 0 0
\(814\) 35.0000 1.22675
\(815\) 0 0
\(816\) 0 0
\(817\) −24.0000 −0.839654
\(818\) −6.00000 −0.209785
\(819\) 0 0
\(820\) 0 0
\(821\) −6.00000 −0.209401 −0.104701 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) 0 0
\(823\) −15.0000 −0.522867 −0.261434 0.965221i \(-0.584195\pi\)
−0.261434 + 0.965221i \(0.584195\pi\)
\(824\) 6.00000 0.209020
\(825\) 0 0
\(826\) 6.00000 0.208767
\(827\) 36.0000 1.25184 0.625921 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 0 0
\(829\) −54.0000 −1.87550 −0.937749 0.347314i \(-0.887094\pi\)
−0.937749 + 0.347314i \(0.887094\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −1.00000 −0.0346688
\(833\) 4.00000 0.138592
\(834\) 0 0
\(835\) 0 0
\(836\) 10.0000 0.345857
\(837\) 0 0
\(838\) 9.00000 0.310900
\(839\) −47.0000 −1.62262 −0.811310 0.584616i \(-0.801245\pi\)
−0.811310 + 0.584616i \(0.801245\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) −35.0000 −1.20618
\(843\) 0 0
\(844\) −14.0000 −0.481900
\(845\) 0 0
\(846\) 0 0
\(847\) 14.0000 0.481046
\(848\) 4.00000 0.137361
\(849\) 0 0
\(850\) −20.0000 −0.685994
\(851\) −35.0000 −1.19978
\(852\) 0 0
\(853\) −16.0000 −0.547830 −0.273915 0.961754i \(-0.588319\pi\)
−0.273915 + 0.961754i \(0.588319\pi\)
\(854\) 13.0000 0.444851
\(855\) 0 0
\(856\) 8.00000 0.273434
\(857\) −2.00000 −0.0683187 −0.0341593 0.999416i \(-0.510875\pi\)
−0.0341593 + 0.999416i \(0.510875\pi\)
\(858\) 0 0
\(859\) 29.0000 0.989467 0.494734 0.869045i \(-0.335266\pi\)
0.494734 + 0.869045i \(0.335266\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 6.00000 0.204361
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 26.0000 0.883516
\(867\) 0 0
\(868\) 1.00000 0.0339422
\(869\) −85.0000 −2.88343
\(870\) 0 0
\(871\) −11.0000 −0.372721
\(872\) −18.0000 −0.609557
\(873\) 0 0
\(874\) −10.0000 −0.338255
\(875\) 0 0
\(876\) 0 0
\(877\) 13.0000 0.438979 0.219489 0.975615i \(-0.429561\pi\)
0.219489 + 0.975615i \(0.429561\pi\)
\(878\) −2.00000 −0.0674967
\(879\) 0 0
\(880\) 0 0
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) 0 0
\(883\) 30.0000 1.00958 0.504790 0.863242i \(-0.331570\pi\)
0.504790 + 0.863242i \(0.331570\pi\)
\(884\) −4.00000 −0.134535
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) −54.0000 −1.81314 −0.906571 0.422053i \(-0.861310\pi\)
−0.906571 + 0.422053i \(0.861310\pi\)
\(888\) 0 0
\(889\) 9.00000 0.301850
\(890\) 0 0
\(891\) 0 0
\(892\) 3.00000 0.100447
\(893\) 14.0000 0.468492
\(894\) 0 0
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −18.0000 −0.600668
\(899\) −4.00000 −0.133407
\(900\) 0 0
\(901\) 16.0000 0.533037
\(902\) 45.0000 1.49834
\(903\) 0 0
\(904\) −1.00000 −0.0332595
\(905\) 0 0
\(906\) 0 0
\(907\) 14.0000 0.464862 0.232431 0.972613i \(-0.425332\pi\)
0.232431 + 0.972613i \(0.425332\pi\)
\(908\) 12.0000 0.398234
\(909\) 0 0
\(910\) 0 0
\(911\) −28.0000 −0.927681 −0.463841 0.885919i \(-0.653529\pi\)
−0.463841 + 0.885919i \(0.653529\pi\)
\(912\) 0 0
\(913\) −20.0000 −0.661903
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) 16.0000 0.528655
\(917\) −8.00000 −0.264183
\(918\) 0 0
\(919\) 31.0000 1.02260 0.511298 0.859404i \(-0.329165\pi\)
0.511298 + 0.859404i \(0.329165\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −12.0000 −0.395199
\(923\) 0 0
\(924\) 0 0
\(925\) −35.0000 −1.15079
\(926\) 12.0000 0.394344
\(927\) 0 0
\(928\) −4.00000 −0.131306
\(929\) 29.0000 0.951459 0.475730 0.879592i \(-0.342184\pi\)
0.475730 + 0.879592i \(0.342184\pi\)
\(930\) 0 0
\(931\) 2.00000 0.0655474
\(932\) 21.0000 0.687878
\(933\) 0 0
\(934\) 36.0000 1.17796
\(935\) 0 0
\(936\) 0 0
\(937\) −8.00000 −0.261349 −0.130674 0.991425i \(-0.541714\pi\)
−0.130674 + 0.991425i \(0.541714\pi\)
\(938\) 11.0000 0.359163
\(939\) 0 0
\(940\) 0 0
\(941\) −6.00000 −0.195594 −0.0977972 0.995206i \(-0.531180\pi\)
−0.0977972 + 0.995206i \(0.531180\pi\)
\(942\) 0 0
\(943\) −45.0000 −1.46540
\(944\) 6.00000 0.195283
\(945\) 0 0
\(946\) −60.0000 −1.95077
\(947\) 16.0000 0.519930 0.259965 0.965618i \(-0.416289\pi\)
0.259965 + 0.965618i \(0.416289\pi\)
\(948\) 0 0
\(949\) −7.00000 −0.227230
\(950\) −10.0000 −0.324443
\(951\) 0 0
\(952\) 4.00000 0.129641
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 6.00000 0.194054
\(957\) 0 0
\(958\) 16.0000 0.516937
\(959\) −18.0000 −0.581250
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) −7.00000 −0.225689
\(963\) 0 0
\(964\) −30.0000 −0.966235
\(965\) 0 0
\(966\) 0 0
\(967\) 2.00000 0.0643157 0.0321578 0.999483i \(-0.489762\pi\)
0.0321578 + 0.999483i \(0.489762\pi\)
\(968\) 14.0000 0.449977
\(969\) 0 0
\(970\) 0 0
\(971\) −9.00000 −0.288824 −0.144412 0.989518i \(-0.546129\pi\)
−0.144412 + 0.989518i \(0.546129\pi\)
\(972\) 0 0
\(973\) 4.00000 0.128234
\(974\) −26.0000 −0.833094
\(975\) 0 0
\(976\) 13.0000 0.416120
\(977\) 26.0000 0.831814 0.415907 0.909407i \(-0.363464\pi\)
0.415907 + 0.909407i \(0.363464\pi\)
\(978\) 0 0
\(979\) −70.0000 −2.23721
\(980\) 0 0
\(981\) 0 0
\(982\) −28.0000 −0.893516
\(983\) 56.0000 1.78612 0.893061 0.449935i \(-0.148553\pi\)
0.893061 + 0.449935i \(0.148553\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −16.0000 −0.509544
\(987\) 0 0
\(988\) −2.00000 −0.0636285
\(989\) 60.0000 1.90789
\(990\) 0 0
\(991\) 59.0000 1.87420 0.937098 0.349065i \(-0.113501\pi\)
0.937098 + 0.349065i \(0.113501\pi\)
\(992\) 1.00000 0.0317500
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 33.0000 1.04512 0.522560 0.852602i \(-0.324977\pi\)
0.522560 + 0.852602i \(0.324977\pi\)
\(998\) 1.00000 0.0316544
\(999\) 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.q.1.1 1
3.2 odd 2 182.2.a.b.1.1 1
12.11 even 2 1456.2.a.b.1.1 1
15.14 odd 2 4550.2.a.o.1.1 1
21.2 odd 6 1274.2.f.m.1145.1 2
21.5 even 6 1274.2.f.u.1145.1 2
21.11 odd 6 1274.2.f.m.79.1 2
21.17 even 6 1274.2.f.u.79.1 2
21.20 even 2 1274.2.a.a.1.1 1
24.5 odd 2 5824.2.a.a.1.1 1
24.11 even 2 5824.2.a.be.1.1 1
39.5 even 4 2366.2.d.i.337.2 2
39.8 even 4 2366.2.d.i.337.1 2
39.38 odd 2 2366.2.a.o.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
182.2.a.b.1.1 1 3.2 odd 2
1274.2.a.a.1.1 1 21.20 even 2
1274.2.f.m.79.1 2 21.11 odd 6
1274.2.f.m.1145.1 2 21.2 odd 6
1274.2.f.u.79.1 2 21.17 even 6
1274.2.f.u.1145.1 2 21.5 even 6
1456.2.a.b.1.1 1 12.11 even 2
1638.2.a.q.1.1 1 1.1 even 1 trivial
2366.2.a.o.1.1 1 39.38 odd 2
2366.2.d.i.337.1 2 39.8 even 4
2366.2.d.i.337.2 2 39.5 even 4
4550.2.a.o.1.1 1 15.14 odd 2
5824.2.a.a.1.1 1 24.5 odd 2
5824.2.a.be.1.1 1 24.11 even 2