Properties

Label 1638.2.a.r.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 546)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{10} +1.00000 q^{11} +1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} +7.00000 q^{19} +1.00000 q^{20} +1.00000 q^{22} -3.00000 q^{23} -4.00000 q^{25} +1.00000 q^{26} -1.00000 q^{28} +3.00000 q^{29} +8.00000 q^{31} +1.00000 q^{32} +1.00000 q^{34} -1.00000 q^{35} +7.00000 q^{37} +7.00000 q^{38} +1.00000 q^{40} -8.00000 q^{41} +7.00000 q^{43} +1.00000 q^{44} -3.00000 q^{46} -8.00000 q^{47} +1.00000 q^{49} -4.00000 q^{50} +1.00000 q^{52} +10.0000 q^{53} +1.00000 q^{55} -1.00000 q^{56} +3.00000 q^{58} -4.00000 q^{59} +7.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} +1.00000 q^{65} +2.00000 q^{67} +1.00000 q^{68} -1.00000 q^{70} -4.00000 q^{71} -1.00000 q^{73} +7.00000 q^{74} +7.00000 q^{76} -1.00000 q^{77} +2.00000 q^{79} +1.00000 q^{80} -8.00000 q^{82} +6.00000 q^{83} +1.00000 q^{85} +7.00000 q^{86} +1.00000 q^{88} -14.0000 q^{89} -1.00000 q^{91} -3.00000 q^{92} -8.00000 q^{94} +7.00000 q^{95} -14.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.00000 0.242536 0.121268 0.992620i \(-0.461304\pi\)
0.121268 + 0.992620i \(0.461304\pi\)
\(18\) 0 0
\(19\) 7.00000 1.60591 0.802955 0.596040i \(-0.203260\pi\)
0.802955 + 0.596040i \(0.203260\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −3.00000 −0.625543 −0.312772 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 1.00000 0.196116
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 1.00000 0.171499
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) 7.00000 1.15079 0.575396 0.817875i \(-0.304848\pi\)
0.575396 + 0.817875i \(0.304848\pi\)
\(38\) 7.00000 1.13555
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) −8.00000 −1.24939 −0.624695 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) 7.00000 1.06749 0.533745 0.845645i \(-0.320784\pi\)
0.533745 + 0.845645i \(0.320784\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −3.00000 −0.442326
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −4.00000 −0.565685
\(51\) 0 0
\(52\) 1.00000 0.138675
\(53\) 10.0000 1.37361 0.686803 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 3.00000 0.393919
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 7.00000 0.896258 0.448129 0.893969i \(-0.352090\pi\)
0.448129 + 0.893969i \(0.352090\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 1.00000 0.121268
\(69\) 0 0
\(70\) −1.00000 −0.119523
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 0 0
\(73\) −1.00000 −0.117041 −0.0585206 0.998286i \(-0.518638\pi\)
−0.0585206 + 0.998286i \(0.518638\pi\)
\(74\) 7.00000 0.813733
\(75\) 0 0
\(76\) 7.00000 0.802955
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) 2.00000 0.225018 0.112509 0.993651i \(-0.464111\pi\)
0.112509 + 0.993651i \(0.464111\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) −8.00000 −0.883452
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) 1.00000 0.108465
\(86\) 7.00000 0.754829
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) −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
\(92\) −3.00000 −0.312772
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) 7.00000 0.718185
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) −4.00000 −0.400000
\(101\) −4.00000 −0.398015 −0.199007 0.979998i \(-0.563772\pi\)
−0.199007 + 0.979998i \(0.563772\pi\)
\(102\) 0 0
\(103\) −5.00000 −0.492665 −0.246332 0.969185i \(-0.579225\pi\)
−0.246332 + 0.969185i \(0.579225\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) 10.0000 0.971286
\(107\) 18.0000 1.74013 0.870063 0.492941i \(-0.164078\pi\)
0.870063 + 0.492941i \(0.164078\pi\)
\(108\) 0 0
\(109\) −11.0000 −1.05361 −0.526804 0.849987i \(-0.676610\pi\)
−0.526804 + 0.849987i \(0.676610\pi\)
\(110\) 1.00000 0.0953463
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) −3.00000 −0.279751
\(116\) 3.00000 0.278543
\(117\) 0 0
\(118\) −4.00000 −0.368230
\(119\) −1.00000 −0.0916698
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 7.00000 0.633750
\(123\) 0 0
\(124\) 8.00000 0.718421
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) 18.0000 1.59724 0.798621 0.601834i \(-0.205563\pi\)
0.798621 + 0.601834i \(0.205563\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 1.00000 0.0877058
\(131\) 5.00000 0.436852 0.218426 0.975854i \(-0.429908\pi\)
0.218426 + 0.975854i \(0.429908\pi\)
\(132\) 0 0
\(133\) −7.00000 −0.606977
\(134\) 2.00000 0.172774
\(135\) 0 0
\(136\) 1.00000 0.0857493
\(137\) 19.0000 1.62328 0.811640 0.584158i \(-0.198575\pi\)
0.811640 + 0.584158i \(0.198575\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 0 0
\(142\) −4.00000 −0.335673
\(143\) 1.00000 0.0836242
\(144\) 0 0
\(145\) 3.00000 0.249136
\(146\) −1.00000 −0.0827606
\(147\) 0 0
\(148\) 7.00000 0.575396
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −9.00000 −0.732410 −0.366205 0.930534i \(-0.619343\pi\)
−0.366205 + 0.930534i \(0.619343\pi\)
\(152\) 7.00000 0.567775
\(153\) 0 0
\(154\) −1.00000 −0.0805823
\(155\) 8.00000 0.642575
\(156\) 0 0
\(157\) −13.0000 −1.03751 −0.518756 0.854922i \(-0.673605\pi\)
−0.518756 + 0.854922i \(0.673605\pi\)
\(158\) 2.00000 0.159111
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) 3.00000 0.236433
\(162\) 0 0
\(163\) 2.00000 0.156652 0.0783260 0.996928i \(-0.475042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) −8.00000 −0.624695
\(165\) 0 0
\(166\) 6.00000 0.465690
\(167\) −15.0000 −1.16073 −0.580367 0.814355i \(-0.697091\pi\)
−0.580367 + 0.814355i \(0.697091\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 1.00000 0.0766965
\(171\) 0 0
\(172\) 7.00000 0.533745
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) −14.0000 −1.04934
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) −1.00000 −0.0741249
\(183\) 0 0
\(184\) −3.00000 −0.221163
\(185\) 7.00000 0.514650
\(186\) 0 0
\(187\) 1.00000 0.0731272
\(188\) −8.00000 −0.583460
\(189\) 0 0
\(190\) 7.00000 0.507833
\(191\) −19.0000 −1.37479 −0.687396 0.726283i \(-0.741246\pi\)
−0.687396 + 0.726283i \(0.741246\pi\)
\(192\) 0 0
\(193\) 16.0000 1.15171 0.575853 0.817554i \(-0.304670\pi\)
0.575853 + 0.817554i \(0.304670\pi\)
\(194\) −14.0000 −1.00514
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) 1.00000 0.0708881 0.0354441 0.999372i \(-0.488715\pi\)
0.0354441 + 0.999372i \(0.488715\pi\)
\(200\) −4.00000 −0.282843
\(201\) 0 0
\(202\) −4.00000 −0.281439
\(203\) −3.00000 −0.210559
\(204\) 0 0
\(205\) −8.00000 −0.558744
\(206\) −5.00000 −0.348367
\(207\) 0 0
\(208\) 1.00000 0.0693375
\(209\) 7.00000 0.484200
\(210\) 0 0
\(211\) −3.00000 −0.206529 −0.103264 0.994654i \(-0.532929\pi\)
−0.103264 + 0.994654i \(0.532929\pi\)
\(212\) 10.0000 0.686803
\(213\) 0 0
\(214\) 18.0000 1.23045
\(215\) 7.00000 0.477396
\(216\) 0 0
\(217\) −8.00000 −0.543075
\(218\) −11.0000 −0.745014
\(219\) 0 0
\(220\) 1.00000 0.0674200
\(221\) 1.00000 0.0672673
\(222\) 0 0
\(223\) 14.0000 0.937509 0.468755 0.883328i \(-0.344703\pi\)
0.468755 + 0.883328i \(0.344703\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 22.0000 1.46019 0.730096 0.683345i \(-0.239475\pi\)
0.730096 + 0.683345i \(0.239475\pi\)
\(228\) 0 0
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) −3.00000 −0.197814
\(231\) 0 0
\(232\) 3.00000 0.196960
\(233\) −12.0000 −0.786146 −0.393073 0.919507i \(-0.628588\pi\)
−0.393073 + 0.919507i \(0.628588\pi\)
\(234\) 0 0
\(235\) −8.00000 −0.521862
\(236\) −4.00000 −0.260378
\(237\) 0 0
\(238\) −1.00000 −0.0648204
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) −10.0000 −0.642824
\(243\) 0 0
\(244\) 7.00000 0.448129
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 7.00000 0.445399
\(248\) 8.00000 0.508001
\(249\) 0 0
\(250\) −9.00000 −0.569210
\(251\) −7.00000 −0.441836 −0.220918 0.975292i \(-0.570905\pi\)
−0.220918 + 0.975292i \(0.570905\pi\)
\(252\) 0 0
\(253\) −3.00000 −0.188608
\(254\) 18.0000 1.12942
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) −7.00000 −0.434959
\(260\) 1.00000 0.0620174
\(261\) 0 0
\(262\) 5.00000 0.308901
\(263\) 4.00000 0.246651 0.123325 0.992366i \(-0.460644\pi\)
0.123325 + 0.992366i \(0.460644\pi\)
\(264\) 0 0
\(265\) 10.0000 0.614295
\(266\) −7.00000 −0.429198
\(267\) 0 0
\(268\) 2.00000 0.122169
\(269\) 2.00000 0.121942 0.0609711 0.998140i \(-0.480580\pi\)
0.0609711 + 0.998140i \(0.480580\pi\)
\(270\) 0 0
\(271\) 2.00000 0.121491 0.0607457 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) 1.00000 0.0606339
\(273\) 0 0
\(274\) 19.0000 1.14783
\(275\) −4.00000 −0.241209
\(276\) 0 0
\(277\) −14.0000 −0.841178 −0.420589 0.907251i \(-0.638177\pi\)
−0.420589 + 0.907251i \(0.638177\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) −1.00000 −0.0597614
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 0 0
\(283\) 32.0000 1.90220 0.951101 0.308879i \(-0.0999539\pi\)
0.951101 + 0.308879i \(0.0999539\pi\)
\(284\) −4.00000 −0.237356
\(285\) 0 0
\(286\) 1.00000 0.0591312
\(287\) 8.00000 0.472225
\(288\) 0 0
\(289\) −16.0000 −0.941176
\(290\) 3.00000 0.176166
\(291\) 0 0
\(292\) −1.00000 −0.0585206
\(293\) −14.0000 −0.817889 −0.408944 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) 0 0
\(295\) −4.00000 −0.232889
\(296\) 7.00000 0.406867
\(297\) 0 0
\(298\) 0 0
\(299\) −3.00000 −0.173494
\(300\) 0 0
\(301\) −7.00000 −0.403473
\(302\) −9.00000 −0.517892
\(303\) 0 0
\(304\) 7.00000 0.401478
\(305\) 7.00000 0.400819
\(306\) 0 0
\(307\) −32.0000 −1.82634 −0.913168 0.407583i \(-0.866372\pi\)
−0.913168 + 0.407583i \(0.866372\pi\)
\(308\) −1.00000 −0.0569803
\(309\) 0 0
\(310\) 8.00000 0.454369
\(311\) −30.0000 −1.70114 −0.850572 0.525859i \(-0.823744\pi\)
−0.850572 + 0.525859i \(0.823744\pi\)
\(312\) 0 0
\(313\) −24.0000 −1.35656 −0.678280 0.734803i \(-0.737274\pi\)
−0.678280 + 0.734803i \(0.737274\pi\)
\(314\) −13.0000 −0.733632
\(315\) 0 0
\(316\) 2.00000 0.112509
\(317\) −12.0000 −0.673987 −0.336994 0.941507i \(-0.609410\pi\)
−0.336994 + 0.941507i \(0.609410\pi\)
\(318\) 0 0
\(319\) 3.00000 0.167968
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 3.00000 0.167183
\(323\) 7.00000 0.389490
\(324\) 0 0
\(325\) −4.00000 −0.221880
\(326\) 2.00000 0.110770
\(327\) 0 0
\(328\) −8.00000 −0.441726
\(329\) 8.00000 0.441054
\(330\) 0 0
\(331\) −10.0000 −0.549650 −0.274825 0.961494i \(-0.588620\pi\)
−0.274825 + 0.961494i \(0.588620\pi\)
\(332\) 6.00000 0.329293
\(333\) 0 0
\(334\) −15.0000 −0.820763
\(335\) 2.00000 0.109272
\(336\) 0 0
\(337\) −21.0000 −1.14394 −0.571971 0.820274i \(-0.693821\pi\)
−0.571971 + 0.820274i \(0.693821\pi\)
\(338\) 1.00000 0.0543928
\(339\) 0 0
\(340\) 1.00000 0.0542326
\(341\) 8.00000 0.433224
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 7.00000 0.377415
\(345\) 0 0
\(346\) −14.0000 −0.752645
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) −18.0000 −0.963518 −0.481759 0.876304i \(-0.660002\pi\)
−0.481759 + 0.876304i \(0.660002\pi\)
\(350\) 4.00000 0.213809
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 0 0
\(355\) −4.00000 −0.212298
\(356\) −14.0000 −0.741999
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) −22.0000 −1.15629
\(363\) 0 0
\(364\) −1.00000 −0.0524142
\(365\) −1.00000 −0.0523424
\(366\) 0 0
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) −3.00000 −0.156386
\(369\) 0 0
\(370\) 7.00000 0.363913
\(371\) −10.0000 −0.519174
\(372\) 0 0
\(373\) −12.0000 −0.621336 −0.310668 0.950518i \(-0.600553\pi\)
−0.310668 + 0.950518i \(0.600553\pi\)
\(374\) 1.00000 0.0517088
\(375\) 0 0
\(376\) −8.00000 −0.412568
\(377\) 3.00000 0.154508
\(378\) 0 0
\(379\) 12.0000 0.616399 0.308199 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) 7.00000 0.359092
\(381\) 0 0
\(382\) −19.0000 −0.972125
\(383\) 9.00000 0.459879 0.229939 0.973205i \(-0.426147\pi\)
0.229939 + 0.973205i \(0.426147\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) 16.0000 0.814379
\(387\) 0 0
\(388\) −14.0000 −0.710742
\(389\) 2.00000 0.101404 0.0507020 0.998714i \(-0.483854\pi\)
0.0507020 + 0.998714i \(0.483854\pi\)
\(390\) 0 0
\(391\) −3.00000 −0.151717
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) 12.0000 0.604551
\(395\) 2.00000 0.100631
\(396\) 0 0
\(397\) 18.0000 0.903394 0.451697 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) 1.00000 0.0501255
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) 8.00000 0.398508
\(404\) −4.00000 −0.199007
\(405\) 0 0
\(406\) −3.00000 −0.148888
\(407\) 7.00000 0.346977
\(408\) 0 0
\(409\) 5.00000 0.247234 0.123617 0.992330i \(-0.460551\pi\)
0.123617 + 0.992330i \(0.460551\pi\)
\(410\) −8.00000 −0.395092
\(411\) 0 0
\(412\) −5.00000 −0.246332
\(413\) 4.00000 0.196827
\(414\) 0 0
\(415\) 6.00000 0.294528
\(416\) 1.00000 0.0490290
\(417\) 0 0
\(418\) 7.00000 0.342381
\(419\) −1.00000 −0.0488532 −0.0244266 0.999702i \(-0.507776\pi\)
−0.0244266 + 0.999702i \(0.507776\pi\)
\(420\) 0 0
\(421\) 34.0000 1.65706 0.828529 0.559946i \(-0.189178\pi\)
0.828529 + 0.559946i \(0.189178\pi\)
\(422\) −3.00000 −0.146038
\(423\) 0 0
\(424\) 10.0000 0.485643
\(425\) −4.00000 −0.194029
\(426\) 0 0
\(427\) −7.00000 −0.338754
\(428\) 18.0000 0.870063
\(429\) 0 0
\(430\) 7.00000 0.337570
\(431\) 38.0000 1.83040 0.915198 0.403005i \(-0.132034\pi\)
0.915198 + 0.403005i \(0.132034\pi\)
\(432\) 0 0
\(433\) 12.0000 0.576683 0.288342 0.957528i \(-0.406896\pi\)
0.288342 + 0.957528i \(0.406896\pi\)
\(434\) −8.00000 −0.384012
\(435\) 0 0
\(436\) −11.0000 −0.526804
\(437\) −21.0000 −1.00457
\(438\) 0 0
\(439\) −17.0000 −0.811366 −0.405683 0.914014i \(-0.632966\pi\)
−0.405683 + 0.914014i \(0.632966\pi\)
\(440\) 1.00000 0.0476731
\(441\) 0 0
\(442\) 1.00000 0.0475651
\(443\) 10.0000 0.475114 0.237557 0.971374i \(-0.423653\pi\)
0.237557 + 0.971374i \(0.423653\pi\)
\(444\) 0 0
\(445\) −14.0000 −0.663664
\(446\) 14.0000 0.662919
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) 39.0000 1.84052 0.920262 0.391303i \(-0.127976\pi\)
0.920262 + 0.391303i \(0.127976\pi\)
\(450\) 0 0
\(451\) −8.00000 −0.376705
\(452\) −6.00000 −0.282216
\(453\) 0 0
\(454\) 22.0000 1.03251
\(455\) −1.00000 −0.0468807
\(456\) 0 0
\(457\) −12.0000 −0.561336 −0.280668 0.959805i \(-0.590556\pi\)
−0.280668 + 0.959805i \(0.590556\pi\)
\(458\) 10.0000 0.467269
\(459\) 0 0
\(460\) −3.00000 −0.139876
\(461\) −21.0000 −0.978068 −0.489034 0.872265i \(-0.662651\pi\)
−0.489034 + 0.872265i \(0.662651\pi\)
\(462\) 0 0
\(463\) −17.0000 −0.790057 −0.395029 0.918669i \(-0.629265\pi\)
−0.395029 + 0.918669i \(0.629265\pi\)
\(464\) 3.00000 0.139272
\(465\) 0 0
\(466\) −12.0000 −0.555889
\(467\) 25.0000 1.15686 0.578431 0.815731i \(-0.303665\pi\)
0.578431 + 0.815731i \(0.303665\pi\)
\(468\) 0 0
\(469\) −2.00000 −0.0923514
\(470\) −8.00000 −0.369012
\(471\) 0 0
\(472\) −4.00000 −0.184115
\(473\) 7.00000 0.321860
\(474\) 0 0
\(475\) −28.0000 −1.28473
\(476\) −1.00000 −0.0458349
\(477\) 0 0
\(478\) −24.0000 −1.09773
\(479\) −5.00000 −0.228456 −0.114228 0.993455i \(-0.536439\pi\)
−0.114228 + 0.993455i \(0.536439\pi\)
\(480\) 0 0
\(481\) 7.00000 0.319173
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) −10.0000 −0.454545
\(485\) −14.0000 −0.635707
\(486\) 0 0
\(487\) 16.0000 0.725029 0.362515 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) 7.00000 0.316875
\(489\) 0 0
\(490\) 1.00000 0.0451754
\(491\) 30.0000 1.35388 0.676941 0.736038i \(-0.263305\pi\)
0.676941 + 0.736038i \(0.263305\pi\)
\(492\) 0 0
\(493\) 3.00000 0.135113
\(494\) 7.00000 0.314945
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) 4.00000 0.179425
\(498\) 0 0
\(499\) 12.0000 0.537194 0.268597 0.963253i \(-0.413440\pi\)
0.268597 + 0.963253i \(0.413440\pi\)
\(500\) −9.00000 −0.402492
\(501\) 0 0
\(502\) −7.00000 −0.312425
\(503\) −34.0000 −1.51599 −0.757993 0.652263i \(-0.773820\pi\)
−0.757993 + 0.652263i \(0.773820\pi\)
\(504\) 0 0
\(505\) −4.00000 −0.177998
\(506\) −3.00000 −0.133366
\(507\) 0 0
\(508\) 18.0000 0.798621
\(509\) 3.00000 0.132973 0.0664863 0.997787i \(-0.478821\pi\)
0.0664863 + 0.997787i \(0.478821\pi\)
\(510\) 0 0
\(511\) 1.00000 0.0442374
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −18.0000 −0.793946
\(515\) −5.00000 −0.220326
\(516\) 0 0
\(517\) −8.00000 −0.351840
\(518\) −7.00000 −0.307562
\(519\) 0 0
\(520\) 1.00000 0.0438529
\(521\) 35.0000 1.53338 0.766689 0.642019i \(-0.221903\pi\)
0.766689 + 0.642019i \(0.221903\pi\)
\(522\) 0 0
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 5.00000 0.218426
\(525\) 0 0
\(526\) 4.00000 0.174408
\(527\) 8.00000 0.348485
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 10.0000 0.434372
\(531\) 0 0
\(532\) −7.00000 −0.303488
\(533\) −8.00000 −0.346518
\(534\) 0 0
\(535\) 18.0000 0.778208
\(536\) 2.00000 0.0863868
\(537\) 0 0
\(538\) 2.00000 0.0862261
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) −43.0000 −1.84871 −0.924357 0.381528i \(-0.875398\pi\)
−0.924357 + 0.381528i \(0.875398\pi\)
\(542\) 2.00000 0.0859074
\(543\) 0 0
\(544\) 1.00000 0.0428746
\(545\) −11.0000 −0.471188
\(546\) 0 0
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) 19.0000 0.811640
\(549\) 0 0
\(550\) −4.00000 −0.170561
\(551\) 21.0000 0.894630
\(552\) 0 0
\(553\) −2.00000 −0.0850487
\(554\) −14.0000 −0.594803
\(555\) 0 0
\(556\) 0 0
\(557\) 24.0000 1.01691 0.508456 0.861088i \(-0.330216\pi\)
0.508456 + 0.861088i \(0.330216\pi\)
\(558\) 0 0
\(559\) 7.00000 0.296068
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −10.0000 −0.421825
\(563\) −19.0000 −0.800755 −0.400377 0.916350i \(-0.631121\pi\)
−0.400377 + 0.916350i \(0.631121\pi\)
\(564\) 0 0
\(565\) −6.00000 −0.252422
\(566\) 32.0000 1.34506
\(567\) 0 0
\(568\) −4.00000 −0.167836
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 16.0000 0.669579 0.334790 0.942293i \(-0.391335\pi\)
0.334790 + 0.942293i \(0.391335\pi\)
\(572\) 1.00000 0.0418121
\(573\) 0 0
\(574\) 8.00000 0.333914
\(575\) 12.0000 0.500435
\(576\) 0 0
\(577\) 42.0000 1.74848 0.874241 0.485491i \(-0.161359\pi\)
0.874241 + 0.485491i \(0.161359\pi\)
\(578\) −16.0000 −0.665512
\(579\) 0 0
\(580\) 3.00000 0.124568
\(581\) −6.00000 −0.248922
\(582\) 0 0
\(583\) 10.0000 0.414158
\(584\) −1.00000 −0.0413803
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) 38.0000 1.56843 0.784214 0.620491i \(-0.213066\pi\)
0.784214 + 0.620491i \(0.213066\pi\)
\(588\) 0 0
\(589\) 56.0000 2.30744
\(590\) −4.00000 −0.164677
\(591\) 0 0
\(592\) 7.00000 0.287698
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 0 0
\(595\) −1.00000 −0.0409960
\(596\) 0 0
\(597\) 0 0
\(598\) −3.00000 −0.122679
\(599\) 3.00000 0.122577 0.0612883 0.998120i \(-0.480479\pi\)
0.0612883 + 0.998120i \(0.480479\pi\)
\(600\) 0 0
\(601\) −28.0000 −1.14214 −0.571072 0.820900i \(-0.693472\pi\)
−0.571072 + 0.820900i \(0.693472\pi\)
\(602\) −7.00000 −0.285299
\(603\) 0 0
\(604\) −9.00000 −0.366205
\(605\) −10.0000 −0.406558
\(606\) 0 0
\(607\) −1.00000 −0.0405887 −0.0202944 0.999794i \(-0.506460\pi\)
−0.0202944 + 0.999794i \(0.506460\pi\)
\(608\) 7.00000 0.283887
\(609\) 0 0
\(610\) 7.00000 0.283422
\(611\) −8.00000 −0.323645
\(612\) 0 0
\(613\) 37.0000 1.49442 0.747208 0.664590i \(-0.231394\pi\)
0.747208 + 0.664590i \(0.231394\pi\)
\(614\) −32.0000 −1.29141
\(615\) 0 0
\(616\) −1.00000 −0.0402911
\(617\) 33.0000 1.32853 0.664265 0.747497i \(-0.268745\pi\)
0.664265 + 0.747497i \(0.268745\pi\)
\(618\) 0 0
\(619\) −1.00000 −0.0401934 −0.0200967 0.999798i \(-0.506397\pi\)
−0.0200967 + 0.999798i \(0.506397\pi\)
\(620\) 8.00000 0.321288
\(621\) 0 0
\(622\) −30.0000 −1.20289
\(623\) 14.0000 0.560898
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) −24.0000 −0.959233
\(627\) 0 0
\(628\) −13.0000 −0.518756
\(629\) 7.00000 0.279108
\(630\) 0 0
\(631\) 27.0000 1.07485 0.537427 0.843311i \(-0.319397\pi\)
0.537427 + 0.843311i \(0.319397\pi\)
\(632\) 2.00000 0.0795557
\(633\) 0 0
\(634\) −12.0000 −0.476581
\(635\) 18.0000 0.714308
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 3.00000 0.118771
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) −50.0000 −1.97488 −0.987441 0.157991i \(-0.949498\pi\)
−0.987441 + 0.157991i \(0.949498\pi\)
\(642\) 0 0
\(643\) 17.0000 0.670415 0.335207 0.942144i \(-0.391194\pi\)
0.335207 + 0.942144i \(0.391194\pi\)
\(644\) 3.00000 0.118217
\(645\) 0 0
\(646\) 7.00000 0.275411
\(647\) −12.0000 −0.471769 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 0 0
\(649\) −4.00000 −0.157014
\(650\) −4.00000 −0.156893
\(651\) 0 0
\(652\) 2.00000 0.0783260
\(653\) −35.0000 −1.36966 −0.684828 0.728705i \(-0.740123\pi\)
−0.684828 + 0.728705i \(0.740123\pi\)
\(654\) 0 0
\(655\) 5.00000 0.195366
\(656\) −8.00000 −0.312348
\(657\) 0 0
\(658\) 8.00000 0.311872
\(659\) −6.00000 −0.233727 −0.116863 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) −10.0000 −0.388661
\(663\) 0 0
\(664\) 6.00000 0.232845
\(665\) −7.00000 −0.271448
\(666\) 0 0
\(667\) −9.00000 −0.348481
\(668\) −15.0000 −0.580367
\(669\) 0 0
\(670\) 2.00000 0.0772667
\(671\) 7.00000 0.270232
\(672\) 0 0
\(673\) 7.00000 0.269830 0.134915 0.990857i \(-0.456924\pi\)
0.134915 + 0.990857i \(0.456924\pi\)
\(674\) −21.0000 −0.808890
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) 22.0000 0.845529 0.422764 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) 0 0
\(679\) 14.0000 0.537271
\(680\) 1.00000 0.0383482
\(681\) 0 0
\(682\) 8.00000 0.306336
\(683\) 19.0000 0.727015 0.363507 0.931591i \(-0.381579\pi\)
0.363507 + 0.931591i \(0.381579\pi\)
\(684\) 0 0
\(685\) 19.0000 0.725953
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 7.00000 0.266872
\(689\) 10.0000 0.380970
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) −14.0000 −0.532200
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) 0 0
\(697\) −8.00000 −0.303022
\(698\) −18.0000 −0.681310
\(699\) 0 0
\(700\) 4.00000 0.151186
\(701\) −10.0000 −0.377695 −0.188847 0.982006i \(-0.560475\pi\)
−0.188847 + 0.982006i \(0.560475\pi\)
\(702\) 0 0
\(703\) 49.0000 1.84807
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) −14.0000 −0.526897
\(707\) 4.00000 0.150435
\(708\) 0 0
\(709\) −6.00000 −0.225335 −0.112667 0.993633i \(-0.535939\pi\)
−0.112667 + 0.993633i \(0.535939\pi\)
\(710\) −4.00000 −0.150117
\(711\) 0 0
\(712\) −14.0000 −0.524672
\(713\) −24.0000 −0.898807
\(714\) 0 0
\(715\) 1.00000 0.0373979
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) 0 0
\(719\) 42.0000 1.56634 0.783168 0.621810i \(-0.213603\pi\)
0.783168 + 0.621810i \(0.213603\pi\)
\(720\) 0 0
\(721\) 5.00000 0.186210
\(722\) 30.0000 1.11648
\(723\) 0 0
\(724\) −22.0000 −0.817624
\(725\) −12.0000 −0.445669
\(726\) 0 0
\(727\) 23.0000 0.853023 0.426511 0.904482i \(-0.359742\pi\)
0.426511 + 0.904482i \(0.359742\pi\)
\(728\) −1.00000 −0.0370625
\(729\) 0 0
\(730\) −1.00000 −0.0370117
\(731\) 7.00000 0.258904
\(732\) 0 0
\(733\) −32.0000 −1.18195 −0.590973 0.806691i \(-0.701256\pi\)
−0.590973 + 0.806691i \(0.701256\pi\)
\(734\) −8.00000 −0.295285
\(735\) 0 0
\(736\) −3.00000 −0.110581
\(737\) 2.00000 0.0736709
\(738\) 0 0
\(739\) 22.0000 0.809283 0.404642 0.914475i \(-0.367396\pi\)
0.404642 + 0.914475i \(0.367396\pi\)
\(740\) 7.00000 0.257325
\(741\) 0 0
\(742\) −10.0000 −0.367112
\(743\) −6.00000 −0.220119 −0.110059 0.993925i \(-0.535104\pi\)
−0.110059 + 0.993925i \(0.535104\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −12.0000 −0.439351
\(747\) 0 0
\(748\) 1.00000 0.0365636
\(749\) −18.0000 −0.657706
\(750\) 0 0
\(751\) −28.0000 −1.02173 −0.510867 0.859660i \(-0.670676\pi\)
−0.510867 + 0.859660i \(0.670676\pi\)
\(752\) −8.00000 −0.291730
\(753\) 0 0
\(754\) 3.00000 0.109254
\(755\) −9.00000 −0.327544
\(756\) 0 0
\(757\) 18.0000 0.654221 0.327111 0.944986i \(-0.393925\pi\)
0.327111 + 0.944986i \(0.393925\pi\)
\(758\) 12.0000 0.435860
\(759\) 0 0
\(760\) 7.00000 0.253917
\(761\) −16.0000 −0.580000 −0.290000 0.957027i \(-0.593655\pi\)
−0.290000 + 0.957027i \(0.593655\pi\)
\(762\) 0 0
\(763\) 11.0000 0.398227
\(764\) −19.0000 −0.687396
\(765\) 0 0
\(766\) 9.00000 0.325183
\(767\) −4.00000 −0.144432
\(768\) 0 0
\(769\) 5.00000 0.180305 0.0901523 0.995928i \(-0.471265\pi\)
0.0901523 + 0.995928i \(0.471265\pi\)
\(770\) −1.00000 −0.0360375
\(771\) 0 0
\(772\) 16.0000 0.575853
\(773\) 27.0000 0.971123 0.485561 0.874203i \(-0.338615\pi\)
0.485561 + 0.874203i \(0.338615\pi\)
\(774\) 0 0
\(775\) −32.0000 −1.14947
\(776\) −14.0000 −0.502571
\(777\) 0 0
\(778\) 2.00000 0.0717035
\(779\) −56.0000 −2.00641
\(780\) 0 0
\(781\) −4.00000 −0.143131
\(782\) −3.00000 −0.107280
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −13.0000 −0.463990
\(786\) 0 0
\(787\) 55.0000 1.96054 0.980269 0.197667i \(-0.0633366\pi\)
0.980269 + 0.197667i \(0.0633366\pi\)
\(788\) 12.0000 0.427482
\(789\) 0 0
\(790\) 2.00000 0.0711568
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) 7.00000 0.248577
\(794\) 18.0000 0.638796
\(795\) 0 0
\(796\) 1.00000 0.0354441
\(797\) −16.0000 −0.566749 −0.283375 0.959009i \(-0.591454\pi\)
−0.283375 + 0.959009i \(0.591454\pi\)
\(798\) 0 0
\(799\) −8.00000 −0.283020
\(800\) −4.00000 −0.141421
\(801\) 0 0
\(802\) −18.0000 −0.635602
\(803\) −1.00000 −0.0352892
\(804\) 0 0
\(805\) 3.00000 0.105736
\(806\) 8.00000 0.281788
\(807\) 0 0
\(808\) −4.00000 −0.140720
\(809\) 10.0000 0.351581 0.175791 0.984428i \(-0.443752\pi\)
0.175791 + 0.984428i \(0.443752\pi\)
\(810\) 0 0
\(811\) −45.0000 −1.58016 −0.790082 0.613001i \(-0.789962\pi\)
−0.790082 + 0.613001i \(0.789962\pi\)
\(812\) −3.00000 −0.105279
\(813\) 0 0
\(814\) 7.00000 0.245350
\(815\) 2.00000 0.0700569
\(816\) 0 0
\(817\) 49.0000 1.71429
\(818\) 5.00000 0.174821
\(819\) 0 0
\(820\) −8.00000 −0.279372
\(821\) 24.0000 0.837606 0.418803 0.908077i \(-0.362450\pi\)
0.418803 + 0.908077i \(0.362450\pi\)
\(822\) 0 0
\(823\) 20.0000 0.697156 0.348578 0.937280i \(-0.386665\pi\)
0.348578 + 0.937280i \(0.386665\pi\)
\(824\) −5.00000 −0.174183
\(825\) 0 0
\(826\) 4.00000 0.139178
\(827\) 41.0000 1.42571 0.712855 0.701312i \(-0.247402\pi\)
0.712855 + 0.701312i \(0.247402\pi\)
\(828\) 0 0
\(829\) 7.00000 0.243120 0.121560 0.992584i \(-0.461210\pi\)
0.121560 + 0.992584i \(0.461210\pi\)
\(830\) 6.00000 0.208263
\(831\) 0 0
\(832\) 1.00000 0.0346688
\(833\) 1.00000 0.0346479
\(834\) 0 0
\(835\) −15.0000 −0.519096
\(836\) 7.00000 0.242100
\(837\) 0 0
\(838\) −1.00000 −0.0345444
\(839\) 4.00000 0.138095 0.0690477 0.997613i \(-0.478004\pi\)
0.0690477 + 0.997613i \(0.478004\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 34.0000 1.17172
\(843\) 0 0
\(844\) −3.00000 −0.103264
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) 10.0000 0.343604
\(848\) 10.0000 0.343401
\(849\) 0 0
\(850\) −4.00000 −0.137199
\(851\) −21.0000 −0.719871
\(852\) 0 0
\(853\) −44.0000 −1.50653 −0.753266 0.657716i \(-0.771523\pi\)
−0.753266 + 0.657716i \(0.771523\pi\)
\(854\) −7.00000 −0.239535
\(855\) 0 0
\(856\) 18.0000 0.615227
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) −44.0000 −1.50126 −0.750630 0.660722i \(-0.770250\pi\)
−0.750630 + 0.660722i \(0.770250\pi\)
\(860\) 7.00000 0.238698
\(861\) 0 0
\(862\) 38.0000 1.29429
\(863\) −46.0000 −1.56586 −0.782929 0.622111i \(-0.786275\pi\)
−0.782929 + 0.622111i \(0.786275\pi\)
\(864\) 0 0
\(865\) −14.0000 −0.476014
\(866\) 12.0000 0.407777
\(867\) 0 0
\(868\) −8.00000 −0.271538
\(869\) 2.00000 0.0678454
\(870\) 0 0
\(871\) 2.00000 0.0677674
\(872\) −11.0000 −0.372507
\(873\) 0 0
\(874\) −21.0000 −0.710336
\(875\) 9.00000 0.304256
\(876\) 0 0
\(877\) 34.0000 1.14810 0.574049 0.818821i \(-0.305372\pi\)
0.574049 + 0.818821i \(0.305372\pi\)
\(878\) −17.0000 −0.573722
\(879\) 0 0
\(880\) 1.00000 0.0337100
\(881\) 49.0000 1.65085 0.825426 0.564510i \(-0.190935\pi\)
0.825426 + 0.564510i \(0.190935\pi\)
\(882\) 0 0
\(883\) 31.0000 1.04323 0.521617 0.853180i \(-0.325329\pi\)
0.521617 + 0.853180i \(0.325329\pi\)
\(884\) 1.00000 0.0336336
\(885\) 0 0
\(886\) 10.0000 0.335957
\(887\) 36.0000 1.20876 0.604381 0.796696i \(-0.293421\pi\)
0.604381 + 0.796696i \(0.293421\pi\)
\(888\) 0 0
\(889\) −18.0000 −0.603701
\(890\) −14.0000 −0.469281
\(891\) 0 0
\(892\) 14.0000 0.468755
\(893\) −56.0000 −1.87397
\(894\) 0 0
\(895\) −12.0000 −0.401116
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 39.0000 1.30145
\(899\) 24.0000 0.800445
\(900\) 0 0
\(901\) 10.0000 0.333148
\(902\) −8.00000 −0.266371
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) −22.0000 −0.731305
\(906\) 0 0
\(907\) 4.00000 0.132818 0.0664089 0.997792i \(-0.478846\pi\)
0.0664089 + 0.997792i \(0.478846\pi\)
\(908\) 22.0000 0.730096
\(909\) 0 0
\(910\) −1.00000 −0.0331497
\(911\) −13.0000 −0.430709 −0.215355 0.976536i \(-0.569091\pi\)
−0.215355 + 0.976536i \(0.569091\pi\)
\(912\) 0 0
\(913\) 6.00000 0.198571
\(914\) −12.0000 −0.396925
\(915\) 0 0
\(916\) 10.0000 0.330409
\(917\) −5.00000 −0.165115
\(918\) 0 0
\(919\) 38.0000 1.25350 0.626752 0.779219i \(-0.284384\pi\)
0.626752 + 0.779219i \(0.284384\pi\)
\(920\) −3.00000 −0.0989071
\(921\) 0 0
\(922\) −21.0000 −0.691598
\(923\) −4.00000 −0.131662
\(924\) 0 0
\(925\) −28.0000 −0.920634
\(926\) −17.0000 −0.558655
\(927\) 0 0
\(928\) 3.00000 0.0984798
\(929\) −28.0000 −0.918650 −0.459325 0.888268i \(-0.651909\pi\)
−0.459325 + 0.888268i \(0.651909\pi\)
\(930\) 0 0
\(931\) 7.00000 0.229416
\(932\) −12.0000 −0.393073
\(933\) 0 0
\(934\) 25.0000 0.818025
\(935\) 1.00000 0.0327035
\(936\) 0 0
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) −2.00000 −0.0653023
\(939\) 0 0
\(940\) −8.00000 −0.260931
\(941\) 46.0000 1.49956 0.749779 0.661689i \(-0.230160\pi\)
0.749779 + 0.661689i \(0.230160\pi\)
\(942\) 0 0
\(943\) 24.0000 0.781548
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 7.00000 0.227590
\(947\) 13.0000 0.422443 0.211222 0.977438i \(-0.432256\pi\)
0.211222 + 0.977438i \(0.432256\pi\)
\(948\) 0 0
\(949\) −1.00000 −0.0324614
\(950\) −28.0000 −0.908440
\(951\) 0 0
\(952\) −1.00000 −0.0324102
\(953\) −36.0000 −1.16615 −0.583077 0.812417i \(-0.698151\pi\)
−0.583077 + 0.812417i \(0.698151\pi\)
\(954\) 0 0
\(955\) −19.0000 −0.614826
\(956\) −24.0000 −0.776215
\(957\) 0 0
\(958\) −5.00000 −0.161543
\(959\) −19.0000 −0.613542
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 7.00000 0.225689
\(963\) 0 0
\(964\) 10.0000 0.322078
\(965\) 16.0000 0.515058
\(966\) 0 0
\(967\) 13.0000 0.418052 0.209026 0.977910i \(-0.432971\pi\)
0.209026 + 0.977910i \(0.432971\pi\)
\(968\) −10.0000 −0.321412
\(969\) 0 0
\(970\) −14.0000 −0.449513
\(971\) 40.0000 1.28366 0.641831 0.766846i \(-0.278175\pi\)
0.641831 + 0.766846i \(0.278175\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 16.0000 0.512673
\(975\) 0 0
\(976\) 7.00000 0.224065
\(977\) 45.0000 1.43968 0.719839 0.694141i \(-0.244216\pi\)
0.719839 + 0.694141i \(0.244216\pi\)
\(978\) 0 0
\(979\) −14.0000 −0.447442
\(980\) 1.00000 0.0319438
\(981\) 0 0
\(982\) 30.0000 0.957338
\(983\) 3.00000 0.0956851 0.0478426 0.998855i \(-0.484765\pi\)
0.0478426 + 0.998855i \(0.484765\pi\)
\(984\) 0 0
\(985\) 12.0000 0.382352
\(986\) 3.00000 0.0955395
\(987\) 0 0
\(988\) 7.00000 0.222700
\(989\) −21.0000 −0.667761
\(990\) 0 0
\(991\) 42.0000 1.33417 0.667087 0.744980i \(-0.267541\pi\)
0.667087 + 0.744980i \(0.267541\pi\)
\(992\) 8.00000 0.254000
\(993\) 0 0
\(994\) 4.00000 0.126872
\(995\) 1.00000 0.0317021
\(996\) 0 0
\(997\) −14.0000 −0.443384 −0.221692 0.975117i \(-0.571158\pi\)
−0.221692 + 0.975117i \(0.571158\pi\)
\(998\) 12.0000 0.379853
\(999\) 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.r.1.1 1
3.2 odd 2 546.2.a.a.1.1 1
12.11 even 2 4368.2.a.s.1.1 1
21.20 even 2 3822.2.a.n.1.1 1
39.38 odd 2 7098.2.a.t.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.a.a.1.1 1 3.2 odd 2
1638.2.a.r.1.1 1 1.1 even 1 trivial
3822.2.a.n.1.1 1 21.20 even 2
4368.2.a.s.1.1 1 12.11 even 2
7098.2.a.t.1.1 1 39.38 odd 2