Properties

Label 1848.2.a.k.1.1
Level $1848$
Weight $2$
Character 1848.1
Self dual yes
Analytic conductor $14.756$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1848 = 2^{3} \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1848.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.7563542935\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1848.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} +3.00000 q^{13} +1.00000 q^{15} +7.00000 q^{19} -1.00000 q^{21} +6.00000 q^{23} -4.00000 q^{25} +1.00000 q^{27} -9.00000 q^{29} -1.00000 q^{33} -1.00000 q^{35} -3.00000 q^{37} +3.00000 q^{39} +8.00000 q^{41} +10.0000 q^{43} +1.00000 q^{45} +3.00000 q^{47} +1.00000 q^{49} +6.00000 q^{53} -1.00000 q^{55} +7.00000 q^{57} +7.00000 q^{59} +10.0000 q^{61} -1.00000 q^{63} +3.00000 q^{65} -3.00000 q^{67} +6.00000 q^{69} -8.00000 q^{71} -7.00000 q^{73} -4.00000 q^{75} +1.00000 q^{77} +8.00000 q^{79} +1.00000 q^{81} -9.00000 q^{87} -6.00000 q^{89} -3.00000 q^{91} +7.00000 q^{95} -10.0000 q^{97} -1.00000 q^{99} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 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
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 3.00000 0.832050 0.416025 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 7.00000 1.60591 0.802955 0.596040i \(-0.203260\pi\)
0.802955 + 0.596040i \(0.203260\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −9.00000 −1.67126 −0.835629 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) −3.00000 −0.493197 −0.246598 0.969118i \(-0.579313\pi\)
−0.246598 + 0.969118i \(0.579313\pi\)
\(38\) 0 0
\(39\) 3.00000 0.480384
\(40\) 0 0
\(41\) 8.00000 1.24939 0.624695 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 0 0
\(43\) 10.0000 1.52499 0.762493 0.646997i \(-0.223975\pi\)
0.762493 + 0.646997i \(0.223975\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 3.00000 0.437595 0.218797 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) 7.00000 0.927173
\(58\) 0 0
\(59\) 7.00000 0.911322 0.455661 0.890153i \(-0.349403\pi\)
0.455661 + 0.890153i \(0.349403\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 3.00000 0.372104
\(66\) 0 0
\(67\) −3.00000 −0.366508 −0.183254 0.983066i \(-0.558663\pi\)
−0.183254 + 0.983066i \(0.558663\pi\)
\(68\) 0 0
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −7.00000 −0.819288 −0.409644 0.912245i \(-0.634347\pi\)
−0.409644 + 0.912245i \(0.634347\pi\)
\(74\) 0 0
\(75\) −4.00000 −0.461880
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −9.00000 −0.964901
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −3.00000 −0.314485
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 7.00000 0.718185
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −4.00000 −0.398015 −0.199007 0.979998i \(-0.563772\pi\)
−0.199007 + 0.979998i \(0.563772\pi\)
\(102\) 0 0
\(103\) 6.00000 0.591198 0.295599 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(104\) 0 0
\(105\) −1.00000 −0.0975900
\(106\) 0 0
\(107\) −9.00000 −0.870063 −0.435031 0.900415i \(-0.643263\pi\)
−0.435031 + 0.900415i \(0.643263\pi\)
\(108\) 0 0
\(109\) 12.0000 1.14939 0.574696 0.818367i \(-0.305120\pi\)
0.574696 + 0.818367i \(0.305120\pi\)
\(110\) 0 0
\(111\) −3.00000 −0.284747
\(112\) 0 0
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) 6.00000 0.559503
\(116\) 0 0
\(117\) 3.00000 0.277350
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 8.00000 0.721336
\(124\) 0 0
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) 0 0
\(129\) 10.0000 0.880451
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) −7.00000 −0.606977
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 12.0000 1.02523 0.512615 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) 0 0
\(139\) 20.0000 1.69638 0.848189 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) 0 0
\(141\) 3.00000 0.252646
\(142\) 0 0
\(143\) −3.00000 −0.250873
\(144\) 0 0
\(145\) −9.00000 −0.747409
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) −21.0000 −1.72039 −0.860194 0.509968i \(-0.829657\pi\)
−0.860194 + 0.509968i \(0.829657\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 4.00000 0.319235 0.159617 0.987179i \(-0.448974\pi\)
0.159617 + 0.987179i \(0.448974\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) −6.00000 −0.472866
\(162\) 0 0
\(163\) 5.00000 0.391630 0.195815 0.980641i \(-0.437265\pi\)
0.195815 + 0.980641i \(0.437265\pi\)
\(164\) 0 0
\(165\) −1.00000 −0.0778499
\(166\) 0 0
\(167\) −6.00000 −0.464294 −0.232147 0.972681i \(-0.574575\pi\)
−0.232147 + 0.972681i \(0.574575\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) 0 0
\(171\) 7.00000 0.535303
\(172\) 0 0
\(173\) −16.0000 −1.21646 −0.608229 0.793762i \(-0.708120\pi\)
−0.608229 + 0.793762i \(0.708120\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) 0 0
\(177\) 7.00000 0.526152
\(178\) 0 0
\(179\) −8.00000 −0.597948 −0.298974 0.954261i \(-0.596644\pi\)
−0.298974 + 0.954261i \(0.596644\pi\)
\(180\) 0 0
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) 0 0
\(183\) 10.0000 0.739221
\(184\) 0 0
\(185\) −3.00000 −0.220564
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −20.0000 −1.44715 −0.723575 0.690246i \(-0.757502\pi\)
−0.723575 + 0.690246i \(0.757502\pi\)
\(192\) 0 0
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 0 0
\(195\) 3.00000 0.214834
\(196\) 0 0
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) −14.0000 −0.992434 −0.496217 0.868199i \(-0.665278\pi\)
−0.496217 + 0.868199i \(0.665278\pi\)
\(200\) 0 0
\(201\) −3.00000 −0.211604
\(202\) 0 0
\(203\) 9.00000 0.631676
\(204\) 0 0
\(205\) 8.00000 0.558744
\(206\) 0 0
\(207\) 6.00000 0.417029
\(208\) 0 0
\(209\) −7.00000 −0.484200
\(210\) 0 0
\(211\) 10.0000 0.688428 0.344214 0.938891i \(-0.388145\pi\)
0.344214 + 0.938891i \(0.388145\pi\)
\(212\) 0 0
\(213\) −8.00000 −0.548151
\(214\) 0 0
\(215\) 10.0000 0.681994
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −7.00000 −0.473016
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 26.0000 1.74109 0.870544 0.492090i \(-0.163767\pi\)
0.870544 + 0.492090i \(0.163767\pi\)
\(224\) 0 0
\(225\) −4.00000 −0.266667
\(226\) 0 0
\(227\) −14.0000 −0.929213 −0.464606 0.885517i \(-0.653804\pi\)
−0.464606 + 0.885517i \(0.653804\pi\)
\(228\) 0 0
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 0 0
\(231\) 1.00000 0.0657952
\(232\) 0 0
\(233\) 22.0000 1.44127 0.720634 0.693316i \(-0.243851\pi\)
0.720634 + 0.693316i \(0.243851\pi\)
\(234\) 0 0
\(235\) 3.00000 0.195698
\(236\) 0 0
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) −15.0000 −0.970269 −0.485135 0.874439i \(-0.661229\pi\)
−0.485135 + 0.874439i \(0.661229\pi\)
\(240\) 0 0
\(241\) 23.0000 1.48156 0.740780 0.671748i \(-0.234456\pi\)
0.740780 + 0.671748i \(0.234456\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 21.0000 1.33620
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −7.00000 −0.441836 −0.220918 0.975292i \(-0.570905\pi\)
−0.220918 + 0.975292i \(0.570905\pi\)
\(252\) 0 0
\(253\) −6.00000 −0.377217
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −27.0000 −1.68421 −0.842107 0.539311i \(-0.818685\pi\)
−0.842107 + 0.539311i \(0.818685\pi\)
\(258\) 0 0
\(259\) 3.00000 0.186411
\(260\) 0 0
\(261\) −9.00000 −0.557086
\(262\) 0 0
\(263\) −9.00000 −0.554964 −0.277482 0.960731i \(-0.589500\pi\)
−0.277482 + 0.960731i \(0.589500\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) −6.00000 −0.367194
\(268\) 0 0
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) −1.00000 −0.0607457 −0.0303728 0.999539i \(-0.509669\pi\)
−0.0303728 + 0.999539i \(0.509669\pi\)
\(272\) 0 0
\(273\) −3.00000 −0.181568
\(274\) 0 0
\(275\) 4.00000 0.241209
\(276\) 0 0
\(277\) −24.0000 −1.44202 −0.721010 0.692925i \(-0.756322\pi\)
−0.721010 + 0.692925i \(0.756322\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 11.0000 0.656205 0.328102 0.944642i \(-0.393591\pi\)
0.328102 + 0.944642i \(0.393591\pi\)
\(282\) 0 0
\(283\) −9.00000 −0.534994 −0.267497 0.963559i \(-0.586197\pi\)
−0.267497 + 0.963559i \(0.586197\pi\)
\(284\) 0 0
\(285\) 7.00000 0.414644
\(286\) 0 0
\(287\) −8.00000 −0.472225
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 7.00000 0.407556
\(296\) 0 0
\(297\) −1.00000 −0.0580259
\(298\) 0 0
\(299\) 18.0000 1.04097
\(300\) 0 0
\(301\) −10.0000 −0.576390
\(302\) 0 0
\(303\) −4.00000 −0.229794
\(304\) 0 0
\(305\) 10.0000 0.572598
\(306\) 0 0
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 0 0
\(309\) 6.00000 0.341328
\(310\) 0 0
\(311\) −16.0000 −0.907277 −0.453638 0.891186i \(-0.649874\pi\)
−0.453638 + 0.891186i \(0.649874\pi\)
\(312\) 0 0
\(313\) 20.0000 1.13047 0.565233 0.824931i \(-0.308786\pi\)
0.565233 + 0.824931i \(0.308786\pi\)
\(314\) 0 0
\(315\) −1.00000 −0.0563436
\(316\) 0 0
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) 0 0
\(319\) 9.00000 0.503903
\(320\) 0 0
\(321\) −9.00000 −0.502331
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −12.0000 −0.665640
\(326\) 0 0
\(327\) 12.0000 0.663602
\(328\) 0 0
\(329\) −3.00000 −0.165395
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 0 0
\(333\) −3.00000 −0.164399
\(334\) 0 0
\(335\) −3.00000 −0.163908
\(336\) 0 0
\(337\) −30.0000 −1.63420 −0.817102 0.576493i \(-0.804421\pi\)
−0.817102 + 0.576493i \(0.804421\pi\)
\(338\) 0 0
\(339\) −2.00000 −0.108625
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 6.00000 0.323029
\(346\) 0 0
\(347\) 32.0000 1.71785 0.858925 0.512101i \(-0.171133\pi\)
0.858925 + 0.512101i \(0.171133\pi\)
\(348\) 0 0
\(349\) −9.00000 −0.481759 −0.240879 0.970555i \(-0.577436\pi\)
−0.240879 + 0.970555i \(0.577436\pi\)
\(350\) 0 0
\(351\) 3.00000 0.160128
\(352\) 0 0
\(353\) −29.0000 −1.54351 −0.771757 0.635917i \(-0.780622\pi\)
−0.771757 + 0.635917i \(0.780622\pi\)
\(354\) 0 0
\(355\) −8.00000 −0.424596
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −16.0000 −0.844448 −0.422224 0.906492i \(-0.638750\pi\)
−0.422224 + 0.906492i \(0.638750\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) 0 0
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) −7.00000 −0.366397
\(366\) 0 0
\(367\) −22.0000 −1.14839 −0.574195 0.818718i \(-0.694685\pi\)
−0.574195 + 0.818718i \(0.694685\pi\)
\(368\) 0 0
\(369\) 8.00000 0.416463
\(370\) 0 0
\(371\) −6.00000 −0.311504
\(372\) 0 0
\(373\) −8.00000 −0.414224 −0.207112 0.978317i \(-0.566407\pi\)
−0.207112 + 0.978317i \(0.566407\pi\)
\(374\) 0 0
\(375\) −9.00000 −0.464758
\(376\) 0 0
\(377\) −27.0000 −1.39057
\(378\) 0 0
\(379\) −5.00000 −0.256833 −0.128416 0.991720i \(-0.540989\pi\)
−0.128416 + 0.991720i \(0.540989\pi\)
\(380\) 0 0
\(381\) −12.0000 −0.614779
\(382\) 0 0
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 0 0
\(385\) 1.00000 0.0509647
\(386\) 0 0
\(387\) 10.0000 0.508329
\(388\) 0 0
\(389\) 8.00000 0.405616 0.202808 0.979219i \(-0.434993\pi\)
0.202808 + 0.979219i \(0.434993\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 12.0000 0.605320
\(394\) 0 0
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) 8.00000 0.401508 0.200754 0.979642i \(-0.435661\pi\)
0.200754 + 0.979642i \(0.435661\pi\)
\(398\) 0 0
\(399\) −7.00000 −0.350438
\(400\) 0 0
\(401\) −2.00000 −0.0998752 −0.0499376 0.998752i \(-0.515902\pi\)
−0.0499376 + 0.998752i \(0.515902\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 3.00000 0.148704
\(408\) 0 0
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 0 0
\(411\) 12.0000 0.591916
\(412\) 0 0
\(413\) −7.00000 −0.344447
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 20.0000 0.979404
\(418\) 0 0
\(419\) −27.0000 −1.31904 −0.659518 0.751689i \(-0.729240\pi\)
−0.659518 + 0.751689i \(0.729240\pi\)
\(420\) 0 0
\(421\) −29.0000 −1.41337 −0.706687 0.707527i \(-0.749811\pi\)
−0.706687 + 0.707527i \(0.749811\pi\)
\(422\) 0 0
\(423\) 3.00000 0.145865
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −10.0000 −0.483934
\(428\) 0 0
\(429\) −3.00000 −0.144841
\(430\) 0 0
\(431\) 31.0000 1.49322 0.746609 0.665263i \(-0.231681\pi\)
0.746609 + 0.665263i \(0.231681\pi\)
\(432\) 0 0
\(433\) −28.0000 −1.34559 −0.672797 0.739827i \(-0.734907\pi\)
−0.672797 + 0.739827i \(0.734907\pi\)
\(434\) 0 0
\(435\) −9.00000 −0.431517
\(436\) 0 0
\(437\) 42.0000 2.00913
\(438\) 0 0
\(439\) 31.0000 1.47955 0.739775 0.672855i \(-0.234932\pi\)
0.739775 + 0.672855i \(0.234932\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 20.0000 0.950229 0.475114 0.879924i \(-0.342407\pi\)
0.475114 + 0.879924i \(0.342407\pi\)
\(444\) 0 0
\(445\) −6.00000 −0.284427
\(446\) 0 0
\(447\) −21.0000 −0.993266
\(448\) 0 0
\(449\) 8.00000 0.377543 0.188772 0.982021i \(-0.439549\pi\)
0.188772 + 0.982021i \(0.439549\pi\)
\(450\) 0 0
\(451\) −8.00000 −0.376705
\(452\) 0 0
\(453\) 16.0000 0.751746
\(454\) 0 0
\(455\) −3.00000 −0.140642
\(456\) 0 0
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −32.0000 −1.49039 −0.745194 0.666847i \(-0.767643\pi\)
−0.745194 + 0.666847i \(0.767643\pi\)
\(462\) 0 0
\(463\) −15.0000 −0.697109 −0.348555 0.937288i \(-0.613327\pi\)
−0.348555 + 0.937288i \(0.613327\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −9.00000 −0.416470 −0.208235 0.978079i \(-0.566772\pi\)
−0.208235 + 0.978079i \(0.566772\pi\)
\(468\) 0 0
\(469\) 3.00000 0.138527
\(470\) 0 0
\(471\) 4.00000 0.184310
\(472\) 0 0
\(473\) −10.0000 −0.459800
\(474\) 0 0
\(475\) −28.0000 −1.28473
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 0 0
\(479\) −16.0000 −0.731059 −0.365529 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) 0 0
\(481\) −9.00000 −0.410365
\(482\) 0 0
\(483\) −6.00000 −0.273009
\(484\) 0 0
\(485\) −10.0000 −0.454077
\(486\) 0 0
\(487\) 24.0000 1.08754 0.543772 0.839233i \(-0.316996\pi\)
0.543772 + 0.839233i \(0.316996\pi\)
\(488\) 0 0
\(489\) 5.00000 0.226108
\(490\) 0 0
\(491\) −41.0000 −1.85030 −0.925152 0.379597i \(-0.876063\pi\)
−0.925152 + 0.379597i \(0.876063\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −1.00000 −0.0449467
\(496\) 0 0
\(497\) 8.00000 0.358849
\(498\) 0 0
\(499\) 19.0000 0.850557 0.425278 0.905063i \(-0.360176\pi\)
0.425278 + 0.905063i \(0.360176\pi\)
\(500\) 0 0
\(501\) −6.00000 −0.268060
\(502\) 0 0
\(503\) −14.0000 −0.624229 −0.312115 0.950044i \(-0.601037\pi\)
−0.312115 + 0.950044i \(0.601037\pi\)
\(504\) 0 0
\(505\) −4.00000 −0.177998
\(506\) 0 0
\(507\) −4.00000 −0.177646
\(508\) 0 0
\(509\) −38.0000 −1.68432 −0.842160 0.539227i \(-0.818716\pi\)
−0.842160 + 0.539227i \(0.818716\pi\)
\(510\) 0 0
\(511\) 7.00000 0.309662
\(512\) 0 0
\(513\) 7.00000 0.309058
\(514\) 0 0
\(515\) 6.00000 0.264392
\(516\) 0 0
\(517\) −3.00000 −0.131940
\(518\) 0 0
\(519\) −16.0000 −0.702322
\(520\) 0 0
\(521\) 25.0000 1.09527 0.547635 0.836717i \(-0.315528\pi\)
0.547635 + 0.836717i \(0.315528\pi\)
\(522\) 0 0
\(523\) 37.0000 1.61790 0.808949 0.587879i \(-0.200037\pi\)
0.808949 + 0.587879i \(0.200037\pi\)
\(524\) 0 0
\(525\) 4.00000 0.174574
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 7.00000 0.303774
\(532\) 0 0
\(533\) 24.0000 1.03956
\(534\) 0 0
\(535\) −9.00000 −0.389104
\(536\) 0 0
\(537\) −8.00000 −0.345225
\(538\) 0 0
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −32.0000 −1.37579 −0.687894 0.725811i \(-0.741464\pi\)
−0.687894 + 0.725811i \(0.741464\pi\)
\(542\) 0 0
\(543\) −22.0000 −0.944110
\(544\) 0 0
\(545\) 12.0000 0.514024
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 0 0
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) −63.0000 −2.68389
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) 0 0
\(555\) −3.00000 −0.127343
\(556\) 0 0
\(557\) 15.0000 0.635570 0.317785 0.948163i \(-0.397061\pi\)
0.317785 + 0.948163i \(0.397061\pi\)
\(558\) 0 0
\(559\) 30.0000 1.26886
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −18.0000 −0.758610 −0.379305 0.925272i \(-0.623837\pi\)
−0.379305 + 0.925272i \(0.623837\pi\)
\(564\) 0 0
\(565\) −2.00000 −0.0841406
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 42.0000 1.76073 0.880366 0.474295i \(-0.157297\pi\)
0.880366 + 0.474295i \(0.157297\pi\)
\(570\) 0 0
\(571\) −16.0000 −0.669579 −0.334790 0.942293i \(-0.608665\pi\)
−0.334790 + 0.942293i \(0.608665\pi\)
\(572\) 0 0
\(573\) −20.0000 −0.835512
\(574\) 0 0
\(575\) −24.0000 −1.00087
\(576\) 0 0
\(577\) 16.0000 0.666089 0.333044 0.942911i \(-0.391924\pi\)
0.333044 + 0.942911i \(0.391924\pi\)
\(578\) 0 0
\(579\) 14.0000 0.581820
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −6.00000 −0.248495
\(584\) 0 0
\(585\) 3.00000 0.124035
\(586\) 0 0
\(587\) 5.00000 0.206372 0.103186 0.994662i \(-0.467096\pi\)
0.103186 + 0.994662i \(0.467096\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 0 0
\(593\) 20.0000 0.821302 0.410651 0.911793i \(-0.365302\pi\)
0.410651 + 0.911793i \(0.365302\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −14.0000 −0.572982
\(598\) 0 0
\(599\) 16.0000 0.653742 0.326871 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) 0 0
\(601\) −23.0000 −0.938190 −0.469095 0.883148i \(-0.655420\pi\)
−0.469095 + 0.883148i \(0.655420\pi\)
\(602\) 0 0
\(603\) −3.00000 −0.122169
\(604\) 0 0
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) 35.0000 1.42061 0.710303 0.703896i \(-0.248558\pi\)
0.710303 + 0.703896i \(0.248558\pi\)
\(608\) 0 0
\(609\) 9.00000 0.364698
\(610\) 0 0
\(611\) 9.00000 0.364101
\(612\) 0 0
\(613\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(614\) 0 0
\(615\) 8.00000 0.322591
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 0 0
\(619\) 16.0000 0.643094 0.321547 0.946894i \(-0.395797\pi\)
0.321547 + 0.946894i \(0.395797\pi\)
\(620\) 0 0
\(621\) 6.00000 0.240772
\(622\) 0 0
\(623\) 6.00000 0.240385
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) −7.00000 −0.279553
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 0 0
\(633\) 10.0000 0.397464
\(634\) 0 0
\(635\) −12.0000 −0.476205
\(636\) 0 0
\(637\) 3.00000 0.118864
\(638\) 0 0
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) 28.0000 1.10593 0.552967 0.833203i \(-0.313496\pi\)
0.552967 + 0.833203i \(0.313496\pi\)
\(642\) 0 0
\(643\) −34.0000 −1.34083 −0.670415 0.741987i \(-0.733884\pi\)
−0.670415 + 0.741987i \(0.733884\pi\)
\(644\) 0 0
\(645\) 10.0000 0.393750
\(646\) 0 0
\(647\) −3.00000 −0.117942 −0.0589711 0.998260i \(-0.518782\pi\)
−0.0589711 + 0.998260i \(0.518782\pi\)
\(648\) 0 0
\(649\) −7.00000 −0.274774
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 48.0000 1.87839 0.939193 0.343391i \(-0.111576\pi\)
0.939193 + 0.343391i \(0.111576\pi\)
\(654\) 0 0
\(655\) 12.0000 0.468879
\(656\) 0 0
\(657\) −7.00000 −0.273096
\(658\) 0 0
\(659\) −21.0000 −0.818044 −0.409022 0.912525i \(-0.634130\pi\)
−0.409022 + 0.912525i \(0.634130\pi\)
\(660\) 0 0
\(661\) −14.0000 −0.544537 −0.272268 0.962221i \(-0.587774\pi\)
−0.272268 + 0.962221i \(0.587774\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −7.00000 −0.271448
\(666\) 0 0
\(667\) −54.0000 −2.09089
\(668\) 0 0
\(669\) 26.0000 1.00522
\(670\) 0 0
\(671\) −10.0000 −0.386046
\(672\) 0 0
\(673\) 46.0000 1.77317 0.886585 0.462566i \(-0.153071\pi\)
0.886585 + 0.462566i \(0.153071\pi\)
\(674\) 0 0
\(675\) −4.00000 −0.153960
\(676\) 0 0
\(677\) −38.0000 −1.46046 −0.730229 0.683202i \(-0.760587\pi\)
−0.730229 + 0.683202i \(0.760587\pi\)
\(678\) 0 0
\(679\) 10.0000 0.383765
\(680\) 0 0
\(681\) −14.0000 −0.536481
\(682\) 0 0
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 0 0
\(685\) 12.0000 0.458496
\(686\) 0 0
\(687\) −2.00000 −0.0763048
\(688\) 0 0
\(689\) 18.0000 0.685745
\(690\) 0 0
\(691\) 6.00000 0.228251 0.114125 0.993466i \(-0.463593\pi\)
0.114125 + 0.993466i \(0.463593\pi\)
\(692\) 0 0
\(693\) 1.00000 0.0379869
\(694\) 0 0
\(695\) 20.0000 0.758643
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 22.0000 0.832116
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 0 0
\(703\) −21.0000 −0.792030
\(704\) 0 0
\(705\) 3.00000 0.112987
\(706\) 0 0
\(707\) 4.00000 0.150435
\(708\) 0 0
\(709\) 15.0000 0.563337 0.281668 0.959512i \(-0.409112\pi\)
0.281668 + 0.959512i \(0.409112\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −3.00000 −0.112194
\(716\) 0 0
\(717\) −15.0000 −0.560185
\(718\) 0 0
\(719\) 21.0000 0.783168 0.391584 0.920142i \(-0.371927\pi\)
0.391584 + 0.920142i \(0.371927\pi\)
\(720\) 0 0
\(721\) −6.00000 −0.223452
\(722\) 0 0
\(723\) 23.0000 0.855379
\(724\) 0 0
\(725\) 36.0000 1.33701
\(726\) 0 0
\(727\) −10.0000 −0.370879 −0.185440 0.982656i \(-0.559371\pi\)
−0.185440 + 0.982656i \(0.559371\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −46.0000 −1.69905 −0.849524 0.527549i \(-0.823111\pi\)
−0.849524 + 0.527549i \(0.823111\pi\)
\(734\) 0 0
\(735\) 1.00000 0.0368856
\(736\) 0 0
\(737\) 3.00000 0.110506
\(738\) 0 0
\(739\) −22.0000 −0.809283 −0.404642 0.914475i \(-0.632604\pi\)
−0.404642 + 0.914475i \(0.632604\pi\)
\(740\) 0 0
\(741\) 21.0000 0.771454
\(742\) 0 0
\(743\) 15.0000 0.550297 0.275148 0.961402i \(-0.411273\pi\)
0.275148 + 0.961402i \(0.411273\pi\)
\(744\) 0 0
\(745\) −21.0000 −0.769380
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 9.00000 0.328853
\(750\) 0 0
\(751\) 37.0000 1.35015 0.675075 0.737749i \(-0.264111\pi\)
0.675075 + 0.737749i \(0.264111\pi\)
\(752\) 0 0
\(753\) −7.00000 −0.255094
\(754\) 0 0
\(755\) 16.0000 0.582300
\(756\) 0 0
\(757\) 21.0000 0.763258 0.381629 0.924316i \(-0.375363\pi\)
0.381629 + 0.924316i \(0.375363\pi\)
\(758\) 0 0
\(759\) −6.00000 −0.217786
\(760\) 0 0
\(761\) 4.00000 0.145000 0.0724999 0.997368i \(-0.476902\pi\)
0.0724999 + 0.997368i \(0.476902\pi\)
\(762\) 0 0
\(763\) −12.0000 −0.434429
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 21.0000 0.758266
\(768\) 0 0
\(769\) 7.00000 0.252426 0.126213 0.992003i \(-0.459718\pi\)
0.126213 + 0.992003i \(0.459718\pi\)
\(770\) 0 0
\(771\) −27.0000 −0.972381
\(772\) 0 0
\(773\) −15.0000 −0.539513 −0.269756 0.962929i \(-0.586943\pi\)
−0.269756 + 0.962929i \(0.586943\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 3.00000 0.107624
\(778\) 0 0
\(779\) 56.0000 2.00641
\(780\) 0 0
\(781\) 8.00000 0.286263
\(782\) 0 0
\(783\) −9.00000 −0.321634
\(784\) 0 0
\(785\) 4.00000 0.142766
\(786\) 0 0
\(787\) −23.0000 −0.819861 −0.409931 0.912117i \(-0.634447\pi\)
−0.409931 + 0.912117i \(0.634447\pi\)
\(788\) 0 0
\(789\) −9.00000 −0.320408
\(790\) 0 0
\(791\) 2.00000 0.0711118
\(792\) 0 0
\(793\) 30.0000 1.06533
\(794\) 0 0
\(795\) 6.00000 0.212798
\(796\) 0 0
\(797\) 15.0000 0.531327 0.265664 0.964066i \(-0.414409\pi\)
0.265664 + 0.964066i \(0.414409\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) 7.00000 0.247025
\(804\) 0 0
\(805\) −6.00000 −0.211472
\(806\) 0 0
\(807\) −6.00000 −0.211210
\(808\) 0 0
\(809\) 15.0000 0.527372 0.263686 0.964609i \(-0.415062\pi\)
0.263686 + 0.964609i \(0.415062\pi\)
\(810\) 0 0
\(811\) 5.00000 0.175574 0.0877869 0.996139i \(-0.472021\pi\)
0.0877869 + 0.996139i \(0.472021\pi\)
\(812\) 0 0
\(813\) −1.00000 −0.0350715
\(814\) 0 0
\(815\) 5.00000 0.175142
\(816\) 0 0
\(817\) 70.0000 2.44899
\(818\) 0 0
\(819\) −3.00000 −0.104828
\(820\) 0 0
\(821\) −31.0000 −1.08191 −0.540954 0.841052i \(-0.681937\pi\)
−0.540954 + 0.841052i \(0.681937\pi\)
\(822\) 0 0
\(823\) −19.0000 −0.662298 −0.331149 0.943578i \(-0.607436\pi\)
−0.331149 + 0.943578i \(0.607436\pi\)
\(824\) 0 0
\(825\) 4.00000 0.139262
\(826\) 0 0
\(827\) −21.0000 −0.730242 −0.365121 0.930960i \(-0.618972\pi\)
−0.365121 + 0.930960i \(0.618972\pi\)
\(828\) 0 0
\(829\) −40.0000 −1.38926 −0.694629 0.719368i \(-0.744431\pi\)
−0.694629 + 0.719368i \(0.744431\pi\)
\(830\) 0 0
\(831\) −24.0000 −0.832551
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −6.00000 −0.207639
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −37.0000 −1.27738 −0.638691 0.769463i \(-0.720524\pi\)
−0.638691 + 0.769463i \(0.720524\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 0 0
\(843\) 11.0000 0.378860
\(844\) 0 0
\(845\) −4.00000 −0.137604
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) 0 0
\(849\) −9.00000 −0.308879
\(850\) 0 0
\(851\) −18.0000 −0.617032
\(852\) 0 0
\(853\) −6.00000 −0.205436 −0.102718 0.994711i \(-0.532754\pi\)
−0.102718 + 0.994711i \(0.532754\pi\)
\(854\) 0 0
\(855\) 7.00000 0.239395
\(856\) 0 0
\(857\) 14.0000 0.478231 0.239115 0.970991i \(-0.423143\pi\)
0.239115 + 0.970991i \(0.423143\pi\)
\(858\) 0 0
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 0 0
\(861\) −8.00000 −0.272639
\(862\) 0 0
\(863\) 30.0000 1.02121 0.510606 0.859815i \(-0.329421\pi\)
0.510606 + 0.859815i \(0.329421\pi\)
\(864\) 0 0
\(865\) −16.0000 −0.544016
\(866\) 0 0
\(867\) −17.0000 −0.577350
\(868\) 0 0
\(869\) −8.00000 −0.271381
\(870\) 0 0
\(871\) −9.00000 −0.304953
\(872\) 0 0
\(873\) −10.0000 −0.338449
\(874\) 0 0
\(875\) 9.00000 0.304256
\(876\) 0 0
\(877\) 36.0000 1.21563 0.607817 0.794077i \(-0.292045\pi\)
0.607817 + 0.794077i \(0.292045\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −19.0000 −0.640126 −0.320063 0.947396i \(-0.603704\pi\)
−0.320063 + 0.947396i \(0.603704\pi\)
\(882\) 0 0
\(883\) −17.0000 −0.572096 −0.286048 0.958215i \(-0.592342\pi\)
−0.286048 + 0.958215i \(0.592342\pi\)
\(884\) 0 0
\(885\) 7.00000 0.235302
\(886\) 0 0
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) 0 0
\(889\) 12.0000 0.402467
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 0 0
\(893\) 21.0000 0.702738
\(894\) 0 0
\(895\) −8.00000 −0.267411
\(896\) 0 0
\(897\) 18.0000 0.601003
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −10.0000 −0.332779
\(904\) 0 0
\(905\) −22.0000 −0.731305
\(906\) 0 0
\(907\) 52.0000 1.72663 0.863316 0.504664i \(-0.168384\pi\)
0.863316 + 0.504664i \(0.168384\pi\)
\(908\) 0 0
\(909\) −4.00000 −0.132672
\(910\) 0 0
\(911\) −42.0000 −1.39152 −0.695761 0.718273i \(-0.744933\pi\)
−0.695761 + 0.718273i \(0.744933\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 10.0000 0.330590
\(916\) 0 0
\(917\) −12.0000 −0.396275
\(918\) 0 0
\(919\) 46.0000 1.51740 0.758700 0.651440i \(-0.225835\pi\)
0.758700 + 0.651440i \(0.225835\pi\)
\(920\) 0 0
\(921\) 20.0000 0.659022
\(922\) 0 0
\(923\) −24.0000 −0.789970
\(924\) 0 0
\(925\) 12.0000 0.394558
\(926\) 0 0
\(927\) 6.00000 0.197066
\(928\) 0 0
\(929\) 39.0000 1.27955 0.639774 0.768563i \(-0.279028\pi\)
0.639774 + 0.768563i \(0.279028\pi\)
\(930\) 0 0
\(931\) 7.00000 0.229416
\(932\) 0 0
\(933\) −16.0000 −0.523816
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 0 0
\(939\) 20.0000 0.652675
\(940\) 0 0
\(941\) −12.0000 −0.391189 −0.195594 0.980685i \(-0.562664\pi\)
−0.195594 + 0.980685i \(0.562664\pi\)
\(942\) 0 0
\(943\) 48.0000 1.56310
\(944\) 0 0
\(945\) −1.00000 −0.0325300
\(946\) 0 0
\(947\) −18.0000 −0.584921 −0.292461 0.956278i \(-0.594474\pi\)
−0.292461 + 0.956278i \(0.594474\pi\)
\(948\) 0 0
\(949\) −21.0000 −0.681689
\(950\) 0 0
\(951\) −2.00000 −0.0648544
\(952\) 0 0
\(953\) 19.0000 0.615470 0.307735 0.951472i \(-0.400429\pi\)
0.307735 + 0.951472i \(0.400429\pi\)
\(954\) 0 0
\(955\) −20.0000 −0.647185
\(956\) 0 0
\(957\) 9.00000 0.290929
\(958\) 0 0
\(959\) −12.0000 −0.387500
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) −9.00000 −0.290021
\(964\) 0 0
\(965\) 14.0000 0.450676
\(966\) 0 0
\(967\) 46.0000 1.47926 0.739630 0.673014i \(-0.235000\pi\)
0.739630 + 0.673014i \(0.235000\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 21.0000 0.673922 0.336961 0.941519i \(-0.390601\pi\)
0.336961 + 0.941519i \(0.390601\pi\)
\(972\) 0 0
\(973\) −20.0000 −0.641171
\(974\) 0 0
\(975\) −12.0000 −0.384308
\(976\) 0 0
\(977\) 30.0000 0.959785 0.479893 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) 0 0
\(979\) 6.00000 0.191761
\(980\) 0 0
\(981\) 12.0000 0.383131
\(982\) 0 0
\(983\) 36.0000 1.14822 0.574111 0.818778i \(-0.305348\pi\)
0.574111 + 0.818778i \(0.305348\pi\)
\(984\) 0 0
\(985\) −6.00000 −0.191176
\(986\) 0 0
\(987\) −3.00000 −0.0954911
\(988\) 0 0
\(989\) 60.0000 1.90789
\(990\) 0 0
\(991\) −11.0000 −0.349427 −0.174713 0.984619i \(-0.555900\pi\)
−0.174713 + 0.984619i \(0.555900\pi\)
\(992\) 0 0
\(993\) −20.0000 −0.634681
\(994\) 0 0
\(995\) −14.0000 −0.443830
\(996\) 0 0
\(997\) 46.0000 1.45683 0.728417 0.685134i \(-0.240256\pi\)
0.728417 + 0.685134i \(0.240256\pi\)
\(998\) 0 0
\(999\) −3.00000 −0.0949158
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 1848.2.a.k.1.1 1
3.2 odd 2 5544.2.a.h.1.1 1
4.3 odd 2 3696.2.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1848.2.a.k.1.1 1 1.1 even 1 trivial
3696.2.a.l.1.1 1 4.3 odd 2
5544.2.a.h.1.1 1 3.2 odd 2