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Properties

Label 819.3.d.b.181.1
Level $819$
Weight $3$
Character 819.181
Self dual yes
Analytic conductor $22.316$
Analytic rank $0$
Dimension $1$
CM discriminant -91
Inner twists $2$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [819,3,Mod(181,819)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(819, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.181");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 819.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(22.3161336511\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 181.1
Character \(\chi\) \(=\) 819.181

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.00000 q^{4} +3.00000 q^{5} +7.00000 q^{7} +O(q^{10})\) \(q+4.00000 q^{4} +3.00000 q^{5} +7.00000 q^{7} -13.0000 q^{13} +16.0000 q^{16} +25.0000 q^{19} +12.0000 q^{20} +45.0000 q^{23} -16.0000 q^{25} +28.0000 q^{28} +33.0000 q^{29} -55.0000 q^{31} +21.0000 q^{35} -30.0000 q^{41} -5.00000 q^{43} -81.0000 q^{47} +49.0000 q^{49} -52.0000 q^{52} -15.0000 q^{53} +90.0000 q^{59} +64.0000 q^{64} -39.0000 q^{65} +29.0000 q^{73} +100.000 q^{76} +67.0000 q^{79} +48.0000 q^{80} +159.000 q^{83} -165.000 q^{89} -91.0000 q^{91} +180.000 q^{92} +75.0000 q^{95} -131.000 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/819\mathbb{Z}\right)^\times\).

\(n\) \(92\) \(379\) \(703\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) 0 0
\(4\) 4.00000 1.00000
\(5\) 3.00000 0.600000 0.300000 − 0.953939i \(-0.403013\pi\)
0.300000 + 0.953939i \(0.403013\pi\)
\(6\) 0 0
\(7\) 7.00000 1.00000
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −13.0000 −1.00000
\(14\) 0 0
\(15\) 0 0
\(16\) 16.0000 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 25.0000 1.31579 0.657895 − 0.753110i \(-0.271447\pi\)
0.657895 + 0.753110i \(0.271447\pi\)
\(20\) 12.0000 0.600000
\(21\) 0 0
\(22\) 0 0
\(23\) 45.0000 1.95652 0.978261 − 0.207378i \(-0.0664930\pi\)
0.978261 + 0.207378i \(0.0664930\pi\)
\(24\) 0 0
\(25\) −16.0000 −0.640000
\(26\) 0 0
\(27\) 0 0
\(28\) 28.0000 1.00000
\(29\) 33.0000 1.13793 0.568966 − 0.822361i \(-0.307344\pi\)
0.568966 + 0.822361i \(0.307344\pi\)
\(30\) 0 0
\(31\) −55.0000 −1.77419 −0.887097 − 0.461583i \(-0.847281\pi\)
−0.887097 + 0.461583i \(0.847281\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 21.0000 0.600000
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −30.0000 −0.731707 −0.365854 − 0.930672i \(-0.619223\pi\)
−0.365854 + 0.930672i \(0.619223\pi\)
\(42\) 0 0
\(43\) −5.00000 −0.116279 −0.0581395 − 0.998308i \(-0.518517\pi\)
−0.0581395 + 0.998308i \(0.518517\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −81.0000 −1.72340 −0.861702 − 0.507414i \(-0.830601\pi\)
−0.861702 + 0.507414i \(0.830601\pi\)
\(48\) 0 0
\(49\) 49.0000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) −52.0000 −1.00000
\(53\) −15.0000 −0.283019 −0.141509 − 0.989937i \(-0.545196\pi\)
−0.141509 + 0.989937i \(0.545196\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 90.0000 1.52542 0.762712 − 0.646738i \(-0.223867\pi\)
0.762712 + 0.646738i \(0.223867\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 64.0000 1.00000
\(65\) −39.0000 −0.600000
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 29.0000 0.397260 0.198630 − 0.980075i \(-0.436351\pi\)
0.198630 + 0.980075i \(0.436351\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 100.000 1.31579
\(77\) 0 0
\(78\) 0 0
\(79\) 67.0000 0.848101 0.424051 − 0.905638i \(-0.360608\pi\)
0.424051 + 0.905638i \(0.360608\pi\)
\(80\) 48.0000 0.600000
\(81\) 0 0
\(82\) 0 0
\(83\) 159.000 1.91566 0.957831 − 0.287331i \(-0.0927680\pi\)
0.957831 + 0.287331i \(0.0927680\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −165.000 −1.85393 −0.926966 − 0.375145i \(-0.877593\pi\)
−0.926966 + 0.375145i \(0.877593\pi\)
\(90\) 0 0
\(91\) −91.0000 −1.00000
\(92\) 180.000 1.95652
\(93\) 0 0
\(94\) 0 0
\(95\) 75.0000 0.789474
\(96\) 0 0
\(97\) −131.000 −1.35052 −0.675258 − 0.737582i \(-0.735968\pi\)
−0.675258 + 0.737582i \(0.735968\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −64.0000 −0.640000
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 150.000 1.40187 0.700935 − 0.713226i \(-0.252766\pi\)
0.700935 + 0.713226i \(0.252766\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 112.000 1.00000
\(113\) −135.000 −1.19469 −0.597345 − 0.801984i \(-0.703778\pi\)
−0.597345 + 0.801984i \(0.703778\pi\)
\(114\) 0 0
\(115\) 135.000 1.17391
\(116\) 132.000 1.13793
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 121.000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) −220.000 −1.77419
\(125\) −123.000 −0.984000
\(126\) 0 0
\(127\) −110.000 −0.866142 −0.433071 − 0.901360i \(-0.642570\pi\)
−0.433071 + 0.901360i \(0.642570\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 175.000 1.31579
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 84.0000 0.600000
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 99.0000 0.682759
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −165.000 −1.06452
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 315.000 1.95652
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) −120.000 −0.731707
\(165\) 0 0
\(166\) 0 0
\(167\) −9.00000 −0.0538922 −0.0269461 − 0.999637i \(-0.508578\pi\)
−0.0269461 + 0.999637i \(0.508578\pi\)
\(168\) 0 0
\(169\) 169.000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) −20.0000 −0.116279
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) −112.000 −0.640000
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −267.000 −1.49162 −0.745810 − 0.666159i \(-0.767937\pi\)
−0.745810 + 0.666159i \(0.767937\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) −324.000 −1.72340
\(189\) 0 0
\(190\) 0 0
\(191\) −18.0000 −0.0942408 −0.0471204 − 0.998889i \(-0.515004\pi\)
−0.0471204 + 0.998889i \(0.515004\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 196.000 1.00000
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 231.000 1.13793
\(204\) 0 0
\(205\) −90.0000 −0.439024
\(206\) 0 0
\(207\) 0 0
\(208\) −208.000 −1.00000
\(209\) 0 0
\(210\) 0 0
\(211\) −397.000 −1.88152 −0.940758 − 0.339078i \(-0.889885\pi\)
−0.940758 + 0.339078i \(0.889885\pi\)
\(212\) −60.0000 −0.283019
\(213\) 0 0
\(214\) 0 0
\(215\) −15.0000 −0.0697674
\(216\) 0 0
\(217\) −385.000 −1.77419
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 121.000 0.542601 0.271300 − 0.962495i \(-0.412546\pi\)
0.271300 + 0.962495i \(0.412546\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −246.000 −1.08370 −0.541850 − 0.840475i \(-0.682276\pi\)
−0.541850 + 0.840475i \(0.682276\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.0436681 −0.0218341 − 0.999762i \(-0.506951\pi\)
−0.0218341 + 0.999762i \(0.506951\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −375.000 −1.60944 −0.804721 − 0.593653i \(-0.797685\pi\)
−0.804721 + 0.593653i \(0.797685\pi\)
\(234\) 0 0
\(235\) −243.000 −1.03404
\(236\) 360.000 1.52542
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 365.000 1.51452 0.757261 − 0.653112i \(-0.226537\pi\)
0.757261 + 0.653112i \(0.226537\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 147.000 0.600000
\(246\) 0 0
\(247\) −325.000 −1.31579
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 256.000 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −156.000 −0.600000
\(261\) 0 0
\(262\) 0 0
\(263\) −435.000 −1.65399 −0.826996 − 0.562208i \(-0.809952\pi\)
−0.826996 + 0.562208i \(0.809952\pi\)
\(264\) 0 0
\(265\) −45.0000 −0.169811
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) −290.000 −1.07011 −0.535055 − 0.844817i \(-0.679709\pi\)
−0.535055 + 0.844817i \(0.679709\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −265.000 −0.956679 −0.478339 − 0.878175i \(-0.658761\pi\)
−0.478339 + 0.878175i \(0.658761\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −210.000 −0.731707
\(288\) 0 0
\(289\) 289.000 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 116.000 0.397260
\(293\) −261.000 −0.890785 −0.445392 − 0.895335i \(-0.646936\pi\)
−0.445392 + 0.895335i \(0.646936\pi\)
\(294\) 0 0
\(295\) 270.000 0.915254
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −585.000 −1.95652
\(300\) 0 0
\(301\) −35.0000 −0.116279
\(302\) 0 0
\(303\) 0 0
\(304\) 400.000 1.31579
\(305\) 0 0
\(306\) 0 0
\(307\) −439.000 −1.42997 −0.714984 − 0.699141i \(-0.753566\pi\)
−0.714984 + 0.699141i \(0.753566\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 268.000 0.848101
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 192.000 0.600000
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 208.000 0.640000
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −567.000 −1.72340
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 636.000 1.91566
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −145.000 −0.430267 −0.215134 − 0.976585i \(-0.569019\pi\)
−0.215134 + 0.976585i \(0.569019\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 343.000 1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −330.000 −0.951009 −0.475504 − 0.879713i \(-0.657734\pi\)
−0.475504 + 0.879713i \(0.657734\pi\)
\(348\) 0 0
\(349\) −355.000 −1.01719 −0.508596 − 0.861005i \(-0.669835\pi\)
−0.508596 + 0.861005i \(0.669835\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 594.000 1.68272 0.841360 − 0.540475i \(-0.181756\pi\)
0.841360 + 0.540475i \(0.181756\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −660.000 −1.85393
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 264.000 0.731302
\(362\) 0 0
\(363\) 0 0
\(364\) −364.000 −1.00000
\(365\) 87.0000 0.238356
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 720.000 1.95652
\(369\) 0 0
\(370\) 0 0
\(371\) −105.000 −0.283019
\(372\) 0 0
\(373\) −710.000 −1.90349 −0.951743 − 0.306897i \(-0.900709\pi\)
−0.951743 + 0.306897i \(0.900709\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −429.000 −1.13793
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 300.000 0.789474
\(381\) 0 0
\(382\) 0 0
\(383\) 66.0000 0.172324 0.0861619 − 0.996281i \(-0.472540\pi\)
0.0861619 + 0.996281i \(0.472540\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) −524.000 −1.35052
\(389\) 678.000 1.74293 0.871465 − 0.490457i \(-0.163170\pi\)
0.871465 + 0.490457i \(0.163170\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 201.000 0.508861
\(396\) 0 0
\(397\) 781.000 1.96725 0.983627 − 0.180215i \(-0.0576794\pi\)
0.983627 + 0.180215i \(0.0576794\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −256.000 −0.640000
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 715.000 1.77419
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −755.000 −1.84597 −0.922983 − 0.384841i \(-0.874256\pi\)
−0.922983 + 0.384841i \(0.874256\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 630.000 1.52542
\(414\) 0 0
\(415\) 477.000 1.14940
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 600.000 1.40187
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1125.00 2.57437
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −795.000 −1.79458 −0.897291 − 0.441439i \(-0.854468\pi\)
−0.897291 + 0.441439i \(0.854468\pi\)
\(444\) 0 0
\(445\) −495.000 −1.11236
\(446\) 0 0
\(447\) 0 0
\(448\) 448.000 1.00000
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −540.000 −1.19469
\(453\) 0 0
\(454\) 0 0
\(455\) −273.000 −0.600000
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 540.000 1.17391
\(461\) −870.000 −1.88720 −0.943601 − 0.331085i \(-0.892585\pi\)
−0.943601 + 0.331085i \(0.892585\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 528.000 1.13793
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −400.000 −0.842105
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 615.000 1.28392 0.641962 − 0.766736i \(-0.278121\pi\)
0.641962 + 0.766736i \(0.278121\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 484.000 1.00000
\(485\) −393.000 −0.810309
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −618.000 −1.25866 −0.629328 − 0.777140i \(-0.716670\pi\)
−0.629328 + 0.777140i \(0.716670\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −880.000 −1.77419
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) −492.000 −0.984000
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) −440.000 −0.866142
\(509\) −1005.00 −1.97446 −0.987230 − 0.159302i \(-0.949076\pi\)
−0.987230 + 0.159302i \(0.949076\pi\)
\(510\) 0 0
\(511\) 203.000 0.397260
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1496.00 2.82798
\(530\) 0 0
\(531\) 0 0
\(532\) 700.000 1.31579
\(533\) 390.000 0.731707
\(534\) 0 0
\(535\) 450.000 0.841121
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 275.000 0.502742 0.251371 − 0.967891i \(-0.419119\pi\)
0.251371 + 0.967891i \(0.419119\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 825.000 1.49728
\(552\) 0 0
\(553\) 469.000 0.848101
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 65.0000 0.116279
\(560\) 336.000 0.600000
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) −405.000 −0.716814
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1137.00 1.99824 0.999121 − 0.0419130i \(-0.0133452\pi\)
0.999121 + 0.0419130i \(0.0133452\pi\)
\(570\) 0 0
\(571\) −1133.00 −1.98424 −0.992119 − 0.125298i \(-0.960011\pi\)
−0.992119 + 0.125298i \(0.960011\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −720.000 −1.25217
\(576\) 0 0
\(577\) −146.000 −0.253033 −0.126516 − 0.991965i \(-0.540380\pi\)
−0.126516 + 0.991965i \(0.540380\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 396.000 0.682759
\(581\) 1113.00 1.91566
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −849.000 −1.44634 −0.723169 − 0.690671i \(-0.757315\pi\)
−0.723169 + 0.690671i \(0.757315\pi\)
\(588\) 0 0
\(589\) −1375.00 −2.33447
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1011.00 1.70489 0.852445 − 0.522817i \(-0.175119\pi\)
0.852445 + 0.522817i \(0.175119\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 1077.00 1.79800 0.898998 − 0.437952i \(-0.144296\pi\)
0.898998 + 0.437952i \(0.144296\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 363.000 0.600000
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1053.00 1.72340
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 1030.00 1.66397 0.831987 − 0.554795i \(-0.187203\pi\)
0.831987 + 0.554795i \(0.187203\pi\)
\(620\) −660.000 −1.06452
\(621\) 0 0
\(622\) 0 0
\(623\) −1155.00 −1.85393
\(624\) 0 0
\(625\) 31.0000 0.0496000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −330.000 −0.519685
\(636\) 0 0
\(637\) −637.000 −1.00000
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 993.000 1.54914 0.774571 − 0.632487i \(-0.217966\pi\)
0.774571 + 0.632487i \(0.217966\pi\)
\(642\) 0 0
\(643\) −586.000 −0.911353 −0.455677 − 0.890145i \(-0.650603\pi\)
−0.455677 + 0.890145i \(0.650603\pi\)
\(644\) 1260.00 1.95652
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 150.000 0.229709 0.114855 − 0.993382i \(-0.463360\pi\)
0.114855 + 0.993382i \(0.463360\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −480.000 −0.731707
\(657\) 0 0
\(658\) 0 0
\(659\) 957.000 1.45220 0.726100 − 0.687589i \(-0.241331\pi\)
0.726100 + 0.687589i \(0.241331\pi\)
\(660\) 0 0
\(661\) 1205.00 1.82300 0.911498 − 0.411305i \(-0.134927\pi\)
0.911498 + 0.411305i \(0.134927\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 525.000 0.789474
\(666\) 0 0
\(667\) 1485.00 2.22639
\(668\) −36.0000 −0.0538922
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1255.00 1.86478 0.932392 − 0.361448i \(-0.117717\pi\)
0.932392 + 0.361448i \(0.117717\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 676.000 1.00000
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) −917.000 −1.35052
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −80.0000 −0.116279
\(689\) 195.000 0.283019
\(690\) 0 0
\(691\) −815.000 −1.17945 −0.589725 − 0.807604i \(-0.700764\pi\)
−0.589725 + 0.807604i \(0.700764\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −448.000 −0.640000
\(701\) 873.000 1.24536 0.622682 − 0.782475i \(-0.286043\pi\)
0.622682 + 0.782475i \(0.286043\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −2475.00 −3.47125
\(714\) 0 0
\(715\) 0 0
\(716\) −1068.00 −1.49162
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −528.000 −0.728276
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −1459.00 −1.99045 −0.995225 − 0.0976063i \(-0.968881\pi\)
−0.995225 + 0.0976063i \(0.968881\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1050.00 1.40187
\(750\) 0 0
\(751\) −773.000 −1.02929 −0.514647 − 0.857402i \(-0.672077\pi\)
−0.514647 + 0.857402i \(0.672077\pi\)
\(752\) −1296.00 −1.72340
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 695.000 0.918098 0.459049 − 0.888411i \(-0.348190\pi\)
0.459049 + 0.888411i \(0.348190\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 675.000 0.886991 0.443495 − 0.896277i \(-0.353738\pi\)
0.443495 + 0.896277i \(0.353738\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −72.0000 −0.0942408
\(765\) 0 0
\(766\) 0 0
\(767\) −1170.00 −1.52542
\(768\) 0 0
\(769\) 485.000 0.630689 0.315345 − 0.948977i \(-0.397880\pi\)
0.315345 + 0.948977i \(0.397880\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −246.000 −0.318241 −0.159120 − 0.987259i \(-0.550866\pi\)
−0.159120 + 0.987259i \(0.550866\pi\)
\(774\) 0 0
\(775\) 880.000 1.13548
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −750.000 −0.962773
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 784.000 1.00000
\(785\) 0 0
\(786\) 0 0
\(787\) 1.00000 0.00127065 0.000635324 − 1.00000i \(-0.499798\pi\)
0.000635324 1.00000i \(0.499798\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −945.000 −1.19469
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 945.000 1.17391
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 657.000 0.812114 0.406057 − 0.913848i \(-0.366903\pi\)
0.406057 + 0.913848i \(0.366903\pi\)
\(810\) 0 0
\(811\) −250.000 −0.308261 −0.154131 − 0.988050i \(-0.549258\pi\)
−0.154131 + 0.988050i \(0.549258\pi\)
\(812\) 924.000 1.13793
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −125.000 −0.152999
\(818\) 0 0
\(819\) 0 0
\(820\) −360.000 −0.439024
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) −1630.00 −1.98056 −0.990279 − 0.139092i \(-0.955582\pi\)
−0.990279 + 0.139092i \(0.955582\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −832.000 −1.00000
\(833\) 0 0
\(834\) 0 0
\(835\) −27.0000 −0.0323353
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1650.00 1.96663 0.983313 − 0.181919i \(-0.0582310\pi\)
0.983313 + 0.181919i \(0.0582310\pi\)
\(840\) 0 0
\(841\) 248.000 0.294887
\(842\) 0 0
\(843\) 0 0
\(844\) −1588.00 −1.88152
\(845\) 507.000 0.600000
\(846\) 0 0
\(847\) 847.000 1.00000
\(848\) −240.000 −0.283019
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 1381.00 1.61899 0.809496 − 0.587126i \(-0.199741\pi\)
0.809496 + 0.587126i \(0.199741\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) −60.0000 −0.0697674
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) −1540.00 −1.77419
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −861.000 −0.984000
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) −1510.00 −1.71008 −0.855040 − 0.518563i \(-0.826467\pi\)
−0.855040 + 0.518563i \(0.826467\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −770.000 −0.866142
\(890\) 0 0
\(891\) 0 0
\(892\) 484.000 0.542601
\(893\) −2025.00 −2.26764
\(894\) 0 0
\(895\) −801.000 −0.894972
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −1815.00 −2.01891
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 995.000 1.09702 0.548512 − 0.836143i \(-0.315195\pi\)
0.548512 + 0.836143i \(0.315195\pi\)
\(908\) −984.000 −1.08370
\(909\) 0 0
\(910\) 0 0
\(911\) 453.000 0.497256 0.248628 − 0.968599i \(-0.420020\pi\)
0.248628 + 0.968599i \(0.420020\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) −40.0000 −0.0436681
\(917\) 0 0
\(918\) 0 0
\(919\) −1438.00 −1.56474 −0.782372 − 0.622811i \(-0.785990\pi\)
−0.782372 + 0.622811i \(0.785990\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1845.00 −1.98601 −0.993003 − 0.118087i \(-0.962324\pi\)
−0.993003 + 0.118087i \(0.962324\pi\)
\(930\) 0 0
\(931\) 1225.00 1.31579
\(932\) −1500.00 −1.60944
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −972.000 −1.03404
\(941\) 1875.00 1.99256 0.996281 − 0.0861688i \(-0.0274624\pi\)
0.996281 + 0.0861688i \(0.0274624\pi\)
\(942\) 0 0
\(943\) −1350.00 −1.43160
\(944\) 1440.00 1.52542
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) −377.000 −0.397260
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1815.00 −1.90451 −0.952256 − 0.305301i \(-0.901243\pi\)
−0.952256 + 0.305301i \(0.901243\pi\)
\(954\) 0 0
\(955\) −54.0000 −0.0565445
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 2064.00 2.14776
\(962\) 0 0
\(963\) 0 0
\(964\) 1460.00 1.51452
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 588.000 0.600000
\(981\) 0 0
\(982\) 0 0
\(983\) 1791.00 1.82197 0.910987 − 0.412436i \(-0.135322\pi\)
0.910987 + 0.412436i \(0.135322\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −1300.00 −1.31579
\(989\) −225.000 −0.227503
\(990\) 0 0
\(991\) 1618.00 1.63269 0.816347 − 0.577562i \(-0.195996\pi\)
0.816347 + 0.577562i \(0.195996\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 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 819.3.d.b.181.1 1
3.2 odd 2 91.3.b.a.90.1 ✓ 1
7.6 odd 2 819.3.d.a.181.1 1
13.12 even 2 819.3.d.a.181.1 1
21.20 even 2 91.3.b.b.90.1 yes 1
39.38 odd 2 91.3.b.b.90.1 yes 1
91.90 odd 2 CM 819.3.d.b.181.1 1
273.272 even 2 91.3.b.a.90.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.3.b.a.90.1 ✓ 1 3.2 odd 2
91.3.b.a.90.1 ✓ 1 273.272 even 2
91.3.b.b.90.1 yes 1 21.20 even 2
91.3.b.b.90.1 yes 1 39.38 odd 2
819.3.d.a.181.1 1 7.6 odd 2
819.3.d.a.181.1 1 13.12 even 2
819.3.d.b.181.1 1 1.1 even 1 trivial
819.3.d.b.181.1 1 91.90 odd 2 CM