Properties

Label 1620.2.a.h.1.2
Level $1620$
Weight $2$
Character 1620.1
Self dual yes
Analytic conductor $12.936$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1620.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.9357651274\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 1620.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{5} +0.732051 q^{7} +O(q^{10})\) \(q+1.00000 q^{5} +0.732051 q^{7} +1.73205 q^{11} -1.46410 q^{13} +1.26795 q^{17} +2.46410 q^{19} +3.46410 q^{23} +1.00000 q^{25} +4.26795 q^{29} -7.92820 q^{31} +0.732051 q^{35} +4.19615 q^{37} +0.803848 q^{41} +6.73205 q^{43} +4.73205 q^{47} -6.46410 q^{49} +10.7321 q^{53} +1.73205 q^{55} +4.26795 q^{59} -4.00000 q^{61} -1.46410 q^{65} -14.3923 q^{67} +0.803848 q^{71} +10.1962 q^{73} +1.26795 q^{77} +6.39230 q^{79} -9.12436 q^{83} +1.26795 q^{85} +5.19615 q^{89} -1.07180 q^{91} +2.46410 q^{95} -2.73205 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 2 q^{7} + O(q^{10}) \) \( 2 q + 2 q^{5} - 2 q^{7} + 4 q^{13} + 6 q^{17} - 2 q^{19} + 2 q^{25} + 12 q^{29} - 2 q^{31} - 2 q^{35} - 2 q^{37} + 12 q^{41} + 10 q^{43} + 6 q^{47} - 6 q^{49} + 18 q^{53} + 12 q^{59} - 8 q^{61} + 4 q^{65} - 8 q^{67} + 12 q^{71} + 10 q^{73} + 6 q^{77} - 8 q^{79} + 6 q^{83} + 6 q^{85} - 16 q^{91} - 2 q^{95} - 2 q^{97} + O(q^{100}) \)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0.732051 0.276689 0.138345 0.990384i \(-0.455822\pi\)
0.138345 + 0.990384i \(0.455822\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.73205 0.522233 0.261116 0.965307i \(-0.415909\pi\)
0.261116 + 0.965307i \(0.415909\pi\)
\(12\) 0 0
\(13\) −1.46410 −0.406069 −0.203034 0.979172i \(-0.565080\pi\)
−0.203034 + 0.979172i \(0.565080\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.26795 0.307523 0.153761 0.988108i \(-0.450861\pi\)
0.153761 + 0.988108i \(0.450861\pi\)
\(18\) 0 0
\(19\) 2.46410 0.565304 0.282652 0.959223i \(-0.408786\pi\)
0.282652 + 0.959223i \(0.408786\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.46410 0.722315 0.361158 0.932505i \(-0.382382\pi\)
0.361158 + 0.932505i \(0.382382\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.26795 0.792538 0.396269 0.918134i \(-0.370305\pi\)
0.396269 + 0.918134i \(0.370305\pi\)
\(30\) 0 0
\(31\) −7.92820 −1.42395 −0.711974 0.702206i \(-0.752198\pi\)
−0.711974 + 0.702206i \(0.752198\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.732051 0.123739
\(36\) 0 0
\(37\) 4.19615 0.689843 0.344922 0.938631i \(-0.387905\pi\)
0.344922 + 0.938631i \(0.387905\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.803848 0.125540 0.0627700 0.998028i \(-0.480007\pi\)
0.0627700 + 0.998028i \(0.480007\pi\)
\(42\) 0 0
\(43\) 6.73205 1.02663 0.513314 0.858201i \(-0.328418\pi\)
0.513314 + 0.858201i \(0.328418\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.73205 0.690241 0.345120 0.938558i \(-0.387838\pi\)
0.345120 + 0.938558i \(0.387838\pi\)
\(48\) 0 0
\(49\) −6.46410 −0.923443
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 10.7321 1.47416 0.737080 0.675805i \(-0.236204\pi\)
0.737080 + 0.675805i \(0.236204\pi\)
\(54\) 0 0
\(55\) 1.73205 0.233550
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.26795 0.555640 0.277820 0.960633i \(-0.410388\pi\)
0.277820 + 0.960633i \(0.410388\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.46410 −0.181599
\(66\) 0 0
\(67\) −14.3923 −1.75830 −0.879150 0.476545i \(-0.841889\pi\)
−0.879150 + 0.476545i \(0.841889\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0.803848 0.0953992 0.0476996 0.998862i \(-0.484811\pi\)
0.0476996 + 0.998862i \(0.484811\pi\)
\(72\) 0 0
\(73\) 10.1962 1.19337 0.596685 0.802476i \(-0.296484\pi\)
0.596685 + 0.802476i \(0.296484\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.26795 0.144496
\(78\) 0 0
\(79\) 6.39230 0.719190 0.359595 0.933108i \(-0.382915\pi\)
0.359595 + 0.933108i \(0.382915\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −9.12436 −1.00153 −0.500764 0.865584i \(-0.666948\pi\)
−0.500764 + 0.865584i \(0.666948\pi\)
\(84\) 0 0
\(85\) 1.26795 0.137528
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 5.19615 0.550791 0.275396 0.961331i \(-0.411191\pi\)
0.275396 + 0.961331i \(0.411191\pi\)
\(90\) 0 0
\(91\) −1.07180 −0.112355
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.46410 0.252811
\(96\) 0 0
\(97\) −2.73205 −0.277398 −0.138699 0.990335i \(-0.544292\pi\)
−0.138699 + 0.990335i \(0.544292\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 18.1244 1.80344 0.901720 0.432320i \(-0.142305\pi\)
0.901720 + 0.432320i \(0.142305\pi\)
\(102\) 0 0
\(103\) −2.39230 −0.235721 −0.117860 0.993030i \(-0.537604\pi\)
−0.117860 + 0.993030i \(0.537604\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3.46410 0.334887 0.167444 0.985882i \(-0.446449\pi\)
0.167444 + 0.985882i \(0.446449\pi\)
\(108\) 0 0
\(109\) 5.92820 0.567819 0.283909 0.958851i \(-0.408368\pi\)
0.283909 + 0.958851i \(0.408368\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 17.6603 1.66134 0.830668 0.556767i \(-0.187959\pi\)
0.830668 + 0.556767i \(0.187959\pi\)
\(114\) 0 0
\(115\) 3.46410 0.323029
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.928203 0.0850883
\(120\) 0 0
\(121\) −8.00000 −0.727273
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −6.19615 −0.549820 −0.274910 0.961470i \(-0.588648\pi\)
−0.274910 + 0.961470i \(0.588648\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −13.7321 −1.19977 −0.599887 0.800084i \(-0.704788\pi\)
−0.599887 + 0.800084i \(0.704788\pi\)
\(132\) 0 0
\(133\) 1.80385 0.156413
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 16.3923 1.40049 0.700245 0.713903i \(-0.253074\pi\)
0.700245 + 0.713903i \(0.253074\pi\)
\(138\) 0 0
\(139\) −5.39230 −0.457369 −0.228685 0.973501i \(-0.573442\pi\)
−0.228685 + 0.973501i \(0.573442\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −2.53590 −0.212062
\(144\) 0 0
\(145\) 4.26795 0.354434
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 3.39230 0.276062 0.138031 0.990428i \(-0.455923\pi\)
0.138031 + 0.990428i \(0.455923\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −7.92820 −0.636809
\(156\) 0 0
\(157\) 6.73205 0.537276 0.268638 0.963241i \(-0.413426\pi\)
0.268638 + 0.963241i \(0.413426\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 2.53590 0.199857
\(162\) 0 0
\(163\) 15.2679 1.19588 0.597939 0.801542i \(-0.295986\pi\)
0.597939 + 0.801542i \(0.295986\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −3.12436 −0.241770 −0.120885 0.992667i \(-0.538573\pi\)
−0.120885 + 0.992667i \(0.538573\pi\)
\(168\) 0 0
\(169\) −10.8564 −0.835108
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −24.2487 −1.84360 −0.921798 0.387671i \(-0.873280\pi\)
−0.921798 + 0.387671i \(0.873280\pi\)
\(174\) 0 0
\(175\) 0.732051 0.0553378
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −12.1244 −0.906217 −0.453108 0.891455i \(-0.649685\pi\)
−0.453108 + 0.891455i \(0.649685\pi\)
\(180\) 0 0
\(181\) −9.53590 −0.708798 −0.354399 0.935094i \(-0.615314\pi\)
−0.354399 + 0.935094i \(0.615314\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 4.19615 0.308507
\(186\) 0 0
\(187\) 2.19615 0.160599
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 19.0526 1.37859 0.689297 0.724479i \(-0.257919\pi\)
0.689297 + 0.724479i \(0.257919\pi\)
\(192\) 0 0
\(193\) 20.5885 1.48199 0.740995 0.671511i \(-0.234354\pi\)
0.740995 + 0.671511i \(0.234354\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −13.8564 −0.987228 −0.493614 0.869681i \(-0.664324\pi\)
−0.493614 + 0.869681i \(0.664324\pi\)
\(198\) 0 0
\(199\) −11.8564 −0.840478 −0.420239 0.907413i \(-0.638054\pi\)
−0.420239 + 0.907413i \(0.638054\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 3.12436 0.219287
\(204\) 0 0
\(205\) 0.803848 0.0561432
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 4.26795 0.295220
\(210\) 0 0
\(211\) −6.07180 −0.418000 −0.209000 0.977916i \(-0.567021\pi\)
−0.209000 + 0.977916i \(0.567021\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 6.73205 0.459122
\(216\) 0 0
\(217\) −5.80385 −0.393991
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −1.85641 −0.124875
\(222\) 0 0
\(223\) 21.8564 1.46361 0.731807 0.681512i \(-0.238677\pi\)
0.731807 + 0.681512i \(0.238677\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −15.8038 −1.04894 −0.524469 0.851429i \(-0.675736\pi\)
−0.524469 + 0.851429i \(0.675736\pi\)
\(228\) 0 0
\(229\) 9.85641 0.651330 0.325665 0.945485i \(-0.394412\pi\)
0.325665 + 0.945485i \(0.394412\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.58846 0.431624 0.215812 0.976435i \(-0.430760\pi\)
0.215812 + 0.976435i \(0.430760\pi\)
\(234\) 0 0
\(235\) 4.73205 0.308685
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −15.4641 −1.00029 −0.500145 0.865942i \(-0.666720\pi\)
−0.500145 + 0.865942i \(0.666720\pi\)
\(240\) 0 0
\(241\) −24.3205 −1.56662 −0.783311 0.621630i \(-0.786471\pi\)
−0.783311 + 0.621630i \(0.786471\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −6.46410 −0.412976
\(246\) 0 0
\(247\) −3.60770 −0.229552
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −27.4641 −1.73352 −0.866759 0.498727i \(-0.833801\pi\)
−0.866759 + 0.498727i \(0.833801\pi\)
\(252\) 0 0
\(253\) 6.00000 0.377217
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −8.53590 −0.532455 −0.266227 0.963910i \(-0.585777\pi\)
−0.266227 + 0.963910i \(0.585777\pi\)
\(258\) 0 0
\(259\) 3.07180 0.190872
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −24.2487 −1.49524 −0.747620 0.664127i \(-0.768803\pi\)
−0.747620 + 0.664127i \(0.768803\pi\)
\(264\) 0 0
\(265\) 10.7321 0.659265
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 19.7321 1.20308 0.601542 0.798841i \(-0.294553\pi\)
0.601542 + 0.798841i \(0.294553\pi\)
\(270\) 0 0
\(271\) −24.7846 −1.50556 −0.752779 0.658273i \(-0.771287\pi\)
−0.752779 + 0.658273i \(0.771287\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.73205 0.104447
\(276\) 0 0
\(277\) −19.1244 −1.14907 −0.574536 0.818480i \(-0.694817\pi\)
−0.574536 + 0.818480i \(0.694817\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 5.32051 0.317395 0.158697 0.987327i \(-0.449271\pi\)
0.158697 + 0.987327i \(0.449271\pi\)
\(282\) 0 0
\(283\) 23.4641 1.39480 0.697398 0.716684i \(-0.254341\pi\)
0.697398 + 0.716684i \(0.254341\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0.588457 0.0347355
\(288\) 0 0
\(289\) −15.3923 −0.905430
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 6.58846 0.384902 0.192451 0.981307i \(-0.438356\pi\)
0.192451 + 0.981307i \(0.438356\pi\)
\(294\) 0 0
\(295\) 4.26795 0.248490
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −5.07180 −0.293310
\(300\) 0 0
\(301\) 4.92820 0.284057
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −4.00000 −0.229039
\(306\) 0 0
\(307\) −12.1962 −0.696071 −0.348036 0.937481i \(-0.613151\pi\)
−0.348036 + 0.937481i \(0.613151\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −28.5167 −1.61703 −0.808516 0.588475i \(-0.799729\pi\)
−0.808516 + 0.588475i \(0.799729\pi\)
\(312\) 0 0
\(313\) 1.07180 0.0605815 0.0302908 0.999541i \(-0.490357\pi\)
0.0302908 + 0.999541i \(0.490357\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 21.1244 1.18646 0.593231 0.805032i \(-0.297852\pi\)
0.593231 + 0.805032i \(0.297852\pi\)
\(318\) 0 0
\(319\) 7.39230 0.413890
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3.12436 0.173844
\(324\) 0 0
\(325\) −1.46410 −0.0812137
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 3.46410 0.190982
\(330\) 0 0
\(331\) −8.60770 −0.473122 −0.236561 0.971617i \(-0.576020\pi\)
−0.236561 + 0.971617i \(0.576020\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −14.3923 −0.786336
\(336\) 0 0
\(337\) 14.2487 0.776177 0.388088 0.921622i \(-0.373136\pi\)
0.388088 + 0.921622i \(0.373136\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −13.7321 −0.743632
\(342\) 0 0
\(343\) −9.85641 −0.532196
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −32.1962 −1.72838 −0.864190 0.503166i \(-0.832169\pi\)
−0.864190 + 0.503166i \(0.832169\pi\)
\(348\) 0 0
\(349\) −25.0000 −1.33822 −0.669110 0.743164i \(-0.733324\pi\)
−0.669110 + 0.743164i \(0.733324\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −22.9808 −1.22314 −0.611571 0.791189i \(-0.709462\pi\)
−0.611571 + 0.791189i \(0.709462\pi\)
\(354\) 0 0
\(355\) 0.803848 0.0426638
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −26.6603 −1.40707 −0.703537 0.710658i \(-0.748397\pi\)
−0.703537 + 0.710658i \(0.748397\pi\)
\(360\) 0 0
\(361\) −12.9282 −0.680432
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 10.1962 0.533691
\(366\) 0 0
\(367\) −5.60770 −0.292719 −0.146360 0.989231i \(-0.546756\pi\)
−0.146360 + 0.989231i \(0.546756\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 7.85641 0.407884
\(372\) 0 0
\(373\) 24.0526 1.24539 0.622697 0.782463i \(-0.286037\pi\)
0.622697 + 0.782463i \(0.286037\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −6.24871 −0.321825
\(378\) 0 0
\(379\) 11.4641 0.588871 0.294436 0.955671i \(-0.404868\pi\)
0.294436 + 0.955671i \(0.404868\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 30.2487 1.54564 0.772818 0.634627i \(-0.218846\pi\)
0.772818 + 0.634627i \(0.218846\pi\)
\(384\) 0 0
\(385\) 1.26795 0.0646207
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −20.5359 −1.04121 −0.520606 0.853797i \(-0.674294\pi\)
−0.520606 + 0.853797i \(0.674294\pi\)
\(390\) 0 0
\(391\) 4.39230 0.222128
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 6.39230 0.321632
\(396\) 0 0
\(397\) 2.67949 0.134480 0.0672399 0.997737i \(-0.478581\pi\)
0.0672399 + 0.997737i \(0.478581\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −26.7846 −1.33756 −0.668780 0.743461i \(-0.733183\pi\)
−0.668780 + 0.743461i \(0.733183\pi\)
\(402\) 0 0
\(403\) 11.6077 0.578220
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 7.26795 0.360259
\(408\) 0 0
\(409\) 9.85641 0.487368 0.243684 0.969855i \(-0.421644\pi\)
0.243684 + 0.969855i \(0.421644\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 3.12436 0.153739
\(414\) 0 0
\(415\) −9.12436 −0.447897
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −14.5359 −0.710125 −0.355063 0.934843i \(-0.615540\pi\)
−0.355063 + 0.934843i \(0.615540\pi\)
\(420\) 0 0
\(421\) −27.7846 −1.35414 −0.677070 0.735919i \(-0.736750\pi\)
−0.677070 + 0.735919i \(0.736750\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.26795 0.0615046
\(426\) 0 0
\(427\) −2.92820 −0.141706
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 2.41154 0.116160 0.0580800 0.998312i \(-0.481502\pi\)
0.0580800 + 0.998312i \(0.481502\pi\)
\(432\) 0 0
\(433\) 4.53590 0.217981 0.108991 0.994043i \(-0.465238\pi\)
0.108991 + 0.994043i \(0.465238\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 8.53590 0.408327
\(438\) 0 0
\(439\) 27.3923 1.30736 0.653682 0.756770i \(-0.273224\pi\)
0.653682 + 0.756770i \(0.273224\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −40.7321 −1.93524 −0.967619 0.252415i \(-0.918775\pi\)
−0.967619 + 0.252415i \(0.918775\pi\)
\(444\) 0 0
\(445\) 5.19615 0.246321
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0.124356 0.00586871 0.00293435 0.999996i \(-0.499066\pi\)
0.00293435 + 0.999996i \(0.499066\pi\)
\(450\) 0 0
\(451\) 1.39230 0.0655611
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −1.07180 −0.0502466
\(456\) 0 0
\(457\) 11.8038 0.552161 0.276080 0.961135i \(-0.410964\pi\)
0.276080 + 0.961135i \(0.410964\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 21.5885 1.00547 0.502737 0.864439i \(-0.332326\pi\)
0.502737 + 0.864439i \(0.332326\pi\)
\(462\) 0 0
\(463\) 18.3923 0.854763 0.427381 0.904071i \(-0.359436\pi\)
0.427381 + 0.904071i \(0.359436\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −15.8038 −0.731315 −0.365657 0.930750i \(-0.619156\pi\)
−0.365657 + 0.930750i \(0.619156\pi\)
\(468\) 0 0
\(469\) −10.5359 −0.486503
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 11.6603 0.536139
\(474\) 0 0
\(475\) 2.46410 0.113061
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 20.6603 0.943991 0.471996 0.881601i \(-0.343534\pi\)
0.471996 + 0.881601i \(0.343534\pi\)
\(480\) 0 0
\(481\) −6.14359 −0.280124
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −2.73205 −0.124056
\(486\) 0 0
\(487\) −37.4641 −1.69766 −0.848830 0.528666i \(-0.822693\pi\)
−0.848830 + 0.528666i \(0.822693\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 7.73205 0.348943 0.174471 0.984662i \(-0.444178\pi\)
0.174471 + 0.984662i \(0.444178\pi\)
\(492\) 0 0
\(493\) 5.41154 0.243724
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0.588457 0.0263959
\(498\) 0 0
\(499\) 6.60770 0.295801 0.147901 0.989002i \(-0.452748\pi\)
0.147901 + 0.989002i \(0.452748\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0.679492 0.0302970 0.0151485 0.999885i \(-0.495178\pi\)
0.0151485 + 0.999885i \(0.495178\pi\)
\(504\) 0 0
\(505\) 18.1244 0.806523
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −14.7846 −0.655316 −0.327658 0.944796i \(-0.606259\pi\)
−0.327658 + 0.944796i \(0.606259\pi\)
\(510\) 0 0
\(511\) 7.46410 0.330192
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −2.39230 −0.105418
\(516\) 0 0
\(517\) 8.19615 0.360466
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −10.3923 −0.455295 −0.227648 0.973744i \(-0.573103\pi\)
−0.227648 + 0.973744i \(0.573103\pi\)
\(522\) 0 0
\(523\) 24.3923 1.06660 0.533301 0.845926i \(-0.320952\pi\)
0.533301 + 0.845926i \(0.320952\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −10.0526 −0.437896
\(528\) 0 0
\(529\) −11.0000 −0.478261
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −1.17691 −0.0509778
\(534\) 0 0
\(535\) 3.46410 0.149766
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −11.1962 −0.482252
\(540\) 0 0
\(541\) −4.46410 −0.191927 −0.0959634 0.995385i \(-0.530593\pi\)
−0.0959634 + 0.995385i \(0.530593\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 5.92820 0.253936
\(546\) 0 0
\(547\) −36.7846 −1.57280 −0.786398 0.617720i \(-0.788057\pi\)
−0.786398 + 0.617720i \(0.788057\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 10.5167 0.448025
\(552\) 0 0
\(553\) 4.67949 0.198992
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −16.3923 −0.694564 −0.347282 0.937761i \(-0.612895\pi\)
−0.347282 + 0.937761i \(0.612895\pi\)
\(558\) 0 0
\(559\) −9.85641 −0.416882
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −19.2679 −0.812047 −0.406024 0.913863i \(-0.633085\pi\)
−0.406024 + 0.913863i \(0.633085\pi\)
\(564\) 0 0
\(565\) 17.6603 0.742972
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 4.94744 0.207408 0.103704 0.994608i \(-0.466931\pi\)
0.103704 + 0.994608i \(0.466931\pi\)
\(570\) 0 0
\(571\) −26.8564 −1.12391 −0.561953 0.827169i \(-0.689950\pi\)
−0.561953 + 0.827169i \(0.689950\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 3.46410 0.144463
\(576\) 0 0
\(577\) 22.1962 0.924038 0.462019 0.886870i \(-0.347125\pi\)
0.462019 + 0.886870i \(0.347125\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −6.67949 −0.277112
\(582\) 0 0
\(583\) 18.5885 0.769855
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −14.1962 −0.585938 −0.292969 0.956122i \(-0.594643\pi\)
−0.292969 + 0.956122i \(0.594643\pi\)
\(588\) 0 0
\(589\) −19.5359 −0.804963
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 36.9282 1.51646 0.758230 0.651987i \(-0.226065\pi\)
0.758230 + 0.651987i \(0.226065\pi\)
\(594\) 0 0
\(595\) 0.928203 0.0380526
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 45.5885 1.86269 0.931347 0.364133i \(-0.118635\pi\)
0.931347 + 0.364133i \(0.118635\pi\)
\(600\) 0 0
\(601\) 10.3205 0.420982 0.210491 0.977596i \(-0.432494\pi\)
0.210491 + 0.977596i \(0.432494\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −8.00000 −0.325246
\(606\) 0 0
\(607\) −28.5885 −1.16037 −0.580185 0.814485i \(-0.697020\pi\)
−0.580185 + 0.814485i \(0.697020\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −6.92820 −0.280285
\(612\) 0 0
\(613\) 3.60770 0.145713 0.0728567 0.997342i \(-0.476788\pi\)
0.0728567 + 0.997342i \(0.476788\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 12.0000 0.483102 0.241551 0.970388i \(-0.422344\pi\)
0.241551 + 0.970388i \(0.422344\pi\)
\(618\) 0 0
\(619\) −10.0000 −0.401934 −0.200967 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 3.80385 0.152398
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 5.32051 0.212143
\(630\) 0 0
\(631\) −13.9282 −0.554473 −0.277237 0.960802i \(-0.589419\pi\)
−0.277237 + 0.960802i \(0.589419\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −6.19615 −0.245887
\(636\) 0 0
\(637\) 9.46410 0.374981
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 23.1962 0.916193 0.458096 0.888902i \(-0.348531\pi\)
0.458096 + 0.888902i \(0.348531\pi\)
\(642\) 0 0
\(643\) −16.5885 −0.654185 −0.327092 0.944992i \(-0.606069\pi\)
−0.327092 + 0.944992i \(0.606069\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 48.2487 1.89685 0.948426 0.316998i \(-0.102675\pi\)
0.948426 + 0.316998i \(0.102675\pi\)
\(648\) 0 0
\(649\) 7.39230 0.290173
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 9.46410 0.370359 0.185179 0.982705i \(-0.440713\pi\)
0.185179 + 0.982705i \(0.440713\pi\)
\(654\) 0 0
\(655\) −13.7321 −0.536556
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 9.46410 0.368669 0.184335 0.982864i \(-0.440987\pi\)
0.184335 + 0.982864i \(0.440987\pi\)
\(660\) 0 0
\(661\) −5.39230 −0.209736 −0.104868 0.994486i \(-0.533442\pi\)
−0.104868 + 0.994486i \(0.533442\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.80385 0.0699502
\(666\) 0 0
\(667\) 14.7846 0.572462
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −6.92820 −0.267460
\(672\) 0 0
\(673\) −17.6077 −0.678727 −0.339363 0.940655i \(-0.610212\pi\)
−0.339363 + 0.940655i \(0.610212\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 28.6410 1.10076 0.550382 0.834913i \(-0.314482\pi\)
0.550382 + 0.834913i \(0.314482\pi\)
\(678\) 0 0
\(679\) −2.00000 −0.0767530
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −14.5359 −0.556201 −0.278100 0.960552i \(-0.589705\pi\)
−0.278100 + 0.960552i \(0.589705\pi\)
\(684\) 0 0
\(685\) 16.3923 0.626318
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −15.7128 −0.598610
\(690\) 0 0
\(691\) −10.0000 −0.380418 −0.190209 0.981744i \(-0.560917\pi\)
−0.190209 + 0.981744i \(0.560917\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −5.39230 −0.204542
\(696\) 0 0
\(697\) 1.01924 0.0386064
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 42.1244 1.59101 0.795507 0.605944i \(-0.207204\pi\)
0.795507 + 0.605944i \(0.207204\pi\)
\(702\) 0 0
\(703\) 10.3397 0.389971
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 13.2679 0.498993
\(708\) 0 0
\(709\) 17.4641 0.655878 0.327939 0.944699i \(-0.393646\pi\)
0.327939 + 0.944699i \(0.393646\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −27.4641 −1.02854
\(714\) 0 0
\(715\) −2.53590 −0.0948372
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 6.80385 0.253741 0.126870 0.991919i \(-0.459507\pi\)
0.126870 + 0.991919i \(0.459507\pi\)
\(720\) 0 0
\(721\) −1.75129 −0.0652214
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 4.26795 0.158508
\(726\) 0 0
\(727\) 33.1769 1.23046 0.615232 0.788346i \(-0.289062\pi\)
0.615232 + 0.788346i \(0.289062\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 8.53590 0.315712
\(732\) 0 0
\(733\) −39.5692 −1.46152 −0.730761 0.682633i \(-0.760835\pi\)
−0.730761 + 0.682633i \(0.760835\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −24.9282 −0.918242
\(738\) 0 0
\(739\) 36.1769 1.33079 0.665395 0.746492i \(-0.268263\pi\)
0.665395 + 0.746492i \(0.268263\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 36.5885 1.34230 0.671150 0.741321i \(-0.265801\pi\)
0.671150 + 0.741321i \(0.265801\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 2.53590 0.0926597
\(750\) 0 0
\(751\) −0.784610 −0.0286308 −0.0143154 0.999898i \(-0.504557\pi\)
−0.0143154 + 0.999898i \(0.504557\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 3.39230 0.123459
\(756\) 0 0
\(757\) 0.392305 0.0142586 0.00712928 0.999975i \(-0.497731\pi\)
0.00712928 + 0.999975i \(0.497731\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 5.87564 0.212992 0.106496 0.994313i \(-0.466037\pi\)
0.106496 + 0.994313i \(0.466037\pi\)
\(762\) 0 0
\(763\) 4.33975 0.157109
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −6.24871 −0.225628
\(768\) 0 0
\(769\) −13.2487 −0.477761 −0.238880 0.971049i \(-0.576780\pi\)
−0.238880 + 0.971049i \(0.576780\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −0.339746 −0.0122198 −0.00610991 0.999981i \(-0.501945\pi\)
−0.00610991 + 0.999981i \(0.501945\pi\)
\(774\) 0 0
\(775\) −7.92820 −0.284789
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.98076 0.0709682
\(780\) 0 0
\(781\) 1.39230 0.0498206
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 6.73205 0.240277
\(786\) 0 0
\(787\) −25.8038 −0.919808 −0.459904 0.887969i \(-0.652116\pi\)
−0.459904 + 0.887969i \(0.652116\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 12.9282 0.459674
\(792\) 0 0
\(793\) 5.85641 0.207967
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 11.3205 0.400993 0.200496 0.979694i \(-0.435744\pi\)
0.200496 + 0.979694i \(0.435744\pi\)
\(798\) 0 0
\(799\) 6.00000 0.212265
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 17.6603 0.623217
\(804\) 0 0
\(805\) 2.53590 0.0893787
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 39.5885 1.39186 0.695928 0.718112i \(-0.254993\pi\)
0.695928 + 0.718112i \(0.254993\pi\)
\(810\) 0 0
\(811\) 23.2487 0.816373 0.408186 0.912899i \(-0.366161\pi\)
0.408186 + 0.912899i \(0.366161\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 15.2679 0.534813
\(816\) 0 0
\(817\) 16.5885 0.580357
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −20.6603 −0.721048 −0.360524 0.932750i \(-0.617402\pi\)
−0.360524 + 0.932750i \(0.617402\pi\)
\(822\) 0 0
\(823\) −3.07180 −0.107076 −0.0535381 0.998566i \(-0.517050\pi\)
−0.0535381 + 0.998566i \(0.517050\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 38.5359 1.34002 0.670012 0.742350i \(-0.266289\pi\)
0.670012 + 0.742350i \(0.266289\pi\)
\(828\) 0 0
\(829\) −10.2154 −0.354795 −0.177398 0.984139i \(-0.556768\pi\)
−0.177398 + 0.984139i \(0.556768\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −8.19615 −0.283980
\(834\) 0 0
\(835\) −3.12436 −0.108123
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 39.5885 1.36675 0.683373 0.730070i \(-0.260512\pi\)
0.683373 + 0.730070i \(0.260512\pi\)
\(840\) 0 0
\(841\) −10.7846 −0.371883
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −10.8564 −0.373472
\(846\) 0 0
\(847\) −5.85641 −0.201229
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 14.5359 0.498284
\(852\) 0 0
\(853\) 10.1962 0.349110 0.174555 0.984647i \(-0.444151\pi\)
0.174555 + 0.984647i \(0.444151\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 42.9282 1.46640 0.733200 0.680013i \(-0.238026\pi\)
0.733200 + 0.680013i \(0.238026\pi\)
\(858\) 0 0
\(859\) −33.7846 −1.15272 −0.576358 0.817197i \(-0.695527\pi\)
−0.576358 + 0.817197i \(0.695527\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 4.48334 0.152615 0.0763073 0.997084i \(-0.475687\pi\)
0.0763073 + 0.997084i \(0.475687\pi\)
\(864\) 0 0
\(865\) −24.2487 −0.824481
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 11.0718 0.375585
\(870\) 0 0
\(871\) 21.0718 0.713991
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0.732051 0.0247478
\(876\) 0 0
\(877\) −34.2487 −1.15650 −0.578248 0.815861i \(-0.696264\pi\)
−0.578248 + 0.815861i \(0.696264\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −17.1962 −0.579353 −0.289677 0.957125i \(-0.593548\pi\)
−0.289677 + 0.957125i \(0.593548\pi\)
\(882\) 0 0
\(883\) 32.8372 1.10506 0.552529 0.833493i \(-0.313663\pi\)
0.552529 + 0.833493i \(0.313663\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −16.7321 −0.561807 −0.280904 0.959736i \(-0.590634\pi\)
−0.280904 + 0.959736i \(0.590634\pi\)
\(888\) 0 0
\(889\) −4.53590 −0.152129
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 11.6603 0.390196
\(894\) 0 0
\(895\) −12.1244 −0.405273
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −33.8372 −1.12853
\(900\) 0 0
\(901\) 13.6077 0.453338
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −9.53590 −0.316984
\(906\) 0 0
\(907\) −42.7846 −1.42064 −0.710320 0.703879i \(-0.751450\pi\)
−0.710320 + 0.703879i \(0.751450\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 17.1962 0.569734 0.284867 0.958567i \(-0.408051\pi\)
0.284867 + 0.958567i \(0.408051\pi\)
\(912\) 0 0
\(913\) −15.8038 −0.523031
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −10.0526 −0.331965
\(918\) 0 0
\(919\) 30.6077 1.00965 0.504827 0.863220i \(-0.331556\pi\)
0.504827 + 0.863220i \(0.331556\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −1.17691 −0.0387386
\(924\) 0 0
\(925\) 4.19615 0.137969
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −46.5167 −1.52616 −0.763081 0.646303i \(-0.776314\pi\)
−0.763081 + 0.646303i \(0.776314\pi\)
\(930\) 0 0
\(931\) −15.9282 −0.522026
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 2.19615 0.0718219
\(936\) 0 0
\(937\) 32.9282 1.07572 0.537859 0.843035i \(-0.319233\pi\)
0.537859 + 0.843035i \(0.319233\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 15.4641 0.504115 0.252058 0.967712i \(-0.418893\pi\)
0.252058 + 0.967712i \(0.418893\pi\)
\(942\) 0 0
\(943\) 2.78461 0.0906794
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −16.6410 −0.540760 −0.270380 0.962754i \(-0.587149\pi\)
−0.270380 + 0.962754i \(0.587149\pi\)
\(948\) 0 0
\(949\) −14.9282 −0.484590
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 9.46410 0.306572 0.153286 0.988182i \(-0.451014\pi\)
0.153286 + 0.988182i \(0.451014\pi\)
\(954\) 0 0
\(955\) 19.0526 0.616526
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 12.0000 0.387500
\(960\) 0 0
\(961\) 31.8564 1.02763
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 20.5885 0.662766
\(966\) 0 0
\(967\) 38.5885 1.24092 0.620461 0.784238i \(-0.286946\pi\)
0.620461 + 0.784238i \(0.286946\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −25.7321 −0.825781 −0.412890 0.910781i \(-0.635481\pi\)
−0.412890 + 0.910781i \(0.635481\pi\)
\(972\) 0 0
\(973\) −3.94744 −0.126549
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 16.3923 0.524436 0.262218 0.965009i \(-0.415546\pi\)
0.262218 + 0.965009i \(0.415546\pi\)
\(978\) 0 0
\(979\) 9.00000 0.287641
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 49.2679 1.57140 0.785702 0.618605i \(-0.212302\pi\)
0.785702 + 0.618605i \(0.212302\pi\)
\(984\) 0 0
\(985\) −13.8564 −0.441502
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 23.3205 0.741549
\(990\) 0 0
\(991\) 14.2154 0.451567 0.225783 0.974178i \(-0.427506\pi\)
0.225783 + 0.974178i \(0.427506\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −11.8564 −0.375873
\(996\) 0 0
\(997\) 41.8038 1.32394 0.661971 0.749530i \(-0.269720\pi\)
0.661971 + 0.749530i \(0.269720\pi\)
\(998\) 0 0
\(999\) 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 1620.2.a.h.1.2 yes 2
3.2 odd 2 1620.2.a.g.1.2 2
4.3 odd 2 6480.2.a.bp.1.1 2
5.2 odd 4 8100.2.d.l.649.3 4
5.3 odd 4 8100.2.d.l.649.2 4
5.4 even 2 8100.2.a.s.1.1 2
9.2 odd 6 1620.2.i.n.1081.1 4
9.4 even 3 1620.2.i.m.541.1 4
9.5 odd 6 1620.2.i.n.541.1 4
9.7 even 3 1620.2.i.m.1081.1 4
12.11 even 2 6480.2.a.bh.1.1 2
15.2 even 4 8100.2.d.m.649.3 4
15.8 even 4 8100.2.d.m.649.2 4
15.14 odd 2 8100.2.a.t.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1620.2.a.g.1.2 2 3.2 odd 2
1620.2.a.h.1.2 yes 2 1.1 even 1 trivial
1620.2.i.m.541.1 4 9.4 even 3
1620.2.i.m.1081.1 4 9.7 even 3
1620.2.i.n.541.1 4 9.5 odd 6
1620.2.i.n.1081.1 4 9.2 odd 6
6480.2.a.bh.1.1 2 12.11 even 2
6480.2.a.bp.1.1 2 4.3 odd 2
8100.2.a.s.1.1 2 5.4 even 2
8100.2.a.t.1.1 2 15.14 odd 2
8100.2.d.l.649.2 4 5.3 odd 4
8100.2.d.l.649.3 4 5.2 odd 4
8100.2.d.m.649.2 4 15.8 even 4
8100.2.d.m.649.3 4 15.2 even 4