Properties

Label 547.3.b.a.546.1
Level $547$
Weight $3$
Character 547.546
Self dual yes
Analytic conductor $14.905$
Analytic rank $0$
Dimension $3$
CM discriminant -547
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] = [547,3,Mod(546,547)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(547, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("547.546");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 547 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 547.b (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: \(14.9046704605\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.14769.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 33x - 56 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 546.1
Root \(-1.90719\) of defining polynomial
Character \(\chi\) \(=\) 547.546

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.00000 q^{4} +9.00000 q^{9} +O(q^{10})\) \(q+4.00000 q^{4} +9.00000 q^{9} -18.3626 q^{11} +13.9493 q^{13} +16.0000 q^{16} +29.6955 q^{19} +25.0000 q^{25} -38.5223 q^{29} +36.0000 q^{36} -73.4506 q^{44} +8.71650 q^{47} +49.0000 q^{49} +55.7971 q^{52} +105.652 q^{53} +64.0000 q^{64} -127.719 q^{67} -30.7833 q^{73} +118.782 q^{76} +81.0000 q^{81} -48.9878 q^{97} -165.264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 12 q^{4} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 12 q^{4} + 27 q^{9} + 48 q^{16} + 75 q^{25} + 108 q^{36} + 147 q^{49} + 192 q^{64} + 243 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/547\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 4.00000 1.00000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 9.00000 1.00000
\(10\) 0 0
\(11\) −18.3626 −1.66933 −0.834666 0.550757i \(-0.814339\pi\)
−0.834666 + 0.550757i \(0.814339\pi\)
\(12\) 0 0
\(13\) 13.9493 1.07302 0.536511 0.843893i \(-0.319742\pi\)
0.536511 + 0.843893i \(0.319742\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 16.0000 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 29.6955 1.56292 0.781462 0.623953i \(-0.214475\pi\)
0.781462 + 0.623953i \(0.214475\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 25.0000 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −38.5223 −1.32835 −0.664177 0.747575i \(-0.731218\pi\)
−0.664177 + 0.747575i \(0.731218\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 36.0000 1.00000
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) −73.4506 −1.66933
\(45\) 0 0
\(46\) 0 0
\(47\) 8.71650 0.185457 0.0927287 0.995691i \(-0.470441\pi\)
0.0927287 + 0.995691i \(0.470441\pi\)
\(48\) 0 0
\(49\) 49.0000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 55.7971 1.07302
\(53\) 105.652 1.99344 0.996720 0.0809306i \(-0.0257892\pi\)
0.996720 + 0.0809306i \(0.0257892\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 64.0000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −127.719 −1.90626 −0.953128 0.302569i \(-0.902156\pi\)
−0.953128 + 0.302569i \(0.902156\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −30.7833 −0.421689 −0.210845 0.977520i \(-0.567621\pi\)
−0.210845 + 0.977520i \(0.567621\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 118.782 1.56292
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 81.0000 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −48.9878 −0.505029 −0.252515 0.967593i \(-0.581258\pi\)
−0.252515 + 0.967593i \(0.581258\pi\)
\(98\) 0 0
\(99\) −165.264 −1.66933
\(100\) 100.000 1.00000
\(101\) 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
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 209.508 1.85405 0.927025 0.375000i \(-0.122357\pi\)
0.927025 + 0.375000i \(0.122357\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −154.089 −1.32835
\(117\) 125.544 1.07302
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 216.187 1.78667
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −253.641 −1.99717 −0.998587 0.0531329i \(-0.983079\pi\)
−0.998587 + 0.0531329i \(0.983079\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 256.746 1.95990 0.979948 0.199253i \(-0.0638517\pi\)
0.979948 + 0.199253i \(0.0638517\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −273.000 −1.99270 −0.996350 0.0853578i \(-0.972797\pi\)
−0.996350 + 0.0853578i \(0.972797\pi\)
\(138\) 0 0
\(139\) −269.000 −1.93525 −0.967626 0.252389i \(-0.918784\pi\)
−0.967626 + 0.252389i \(0.918784\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −256.146 −1.79123
\(144\) 144.000 1.00000
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −249.000 −1.67114 −0.835570 0.549383i \(-0.814863\pi\)
−0.835570 + 0.549383i \(0.814863\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −233.000 −1.48408 −0.742038 0.670358i \(-0.766141\pi\)
−0.742038 + 0.670358i \(0.766141\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −213.000 −1.27545 −0.637725 0.770264i \(-0.720124\pi\)
−0.637725 + 0.770264i \(0.720124\pi\)
\(168\) 0 0
\(169\) 25.5826 0.151376
\(170\) 0 0
\(171\) 267.260 1.56292
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −293.802 −1.66933
\(177\) 0 0
\(178\) 0 0
\(179\) −189.000 −1.05587 −0.527933 0.849286i \(-0.677033\pi\)
−0.527933 + 0.849286i \(0.677033\pi\)
\(180\) 0 0
\(181\) −314.120 −1.73547 −0.867735 0.497027i \(-0.834425\pi\)
−0.867735 + 0.497027i \(0.834425\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 34.8660 0.185457
\(189\) 0 0
\(190\) 0 0
\(191\) −152.560 −0.798746 −0.399373 0.916789i \(-0.630772\pi\)
−0.399373 + 0.916789i \(0.630772\pi\)
\(192\) 0 0
\(193\) −161.000 −0.834197 −0.417098 0.908861i \(-0.636953\pi\)
−0.417098 + 0.908861i \(0.636953\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 196.000 1.00000
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 341.262 1.71488 0.857441 0.514582i \(-0.172053\pi\)
0.857441 + 0.514582i \(0.172053\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 223.189 1.07302
\(209\) −545.289 −2.60904
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 422.609 1.99344
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 225.000 1.00000
\(226\) 0 0
\(227\) −93.0000 −0.409692 −0.204846 0.978794i \(-0.565669\pi\)
−0.204846 + 0.978794i \(0.565669\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 241.048 1.03454 0.517270 0.855822i \(-0.326948\pi\)
0.517270 + 0.855822i \(0.326948\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 68.3903 0.286152 0.143076 0.989712i \(-0.454301\pi\)
0.143076 + 0.989712i \(0.454301\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 414.232 1.67705
\(248\) 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
\(260\) 0 0
\(261\) −346.700 −1.32835
\(262\) 0 0
\(263\) −274.572 −1.04400 −0.522001 0.852945i \(-0.674814\pi\)
−0.522001 + 0.852945i \(0.674814\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −510.876 −1.90626
\(269\) −9.00000 −0.0334572 −0.0167286 0.999860i \(-0.505325\pi\)
−0.0167286 + 0.999860i \(0.505325\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −459.066 −1.66933
\(276\) 0 0
\(277\) −62.1800 −0.224477 −0.112238 0.993681i \(-0.535802\pi\)
−0.112238 + 0.993681i \(0.535802\pi\)
\(278\) 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
\(287\) 0 0
\(288\) 0 0
\(289\) 289.000 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) −123.133 −0.421689
\(293\) 39.0000 0.133106 0.0665529 0.997783i \(-0.478800\pi\)
0.0665529 + 0.997783i \(0.478800\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 475.129 1.56292
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 500.914 1.61066 0.805328 0.592830i \(-0.201989\pi\)
0.805328 + 0.592830i \(0.201989\pi\)
\(312\) 0 0
\(313\) −615.661 −1.96697 −0.983485 0.180992i \(-0.942069\pi\)
−0.983485 + 0.180992i \(0.942069\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 178.111 0.561864 0.280932 0.959728i \(-0.409356\pi\)
0.280932 + 0.959728i \(0.409356\pi\)
\(318\) 0 0
\(319\) 707.371 2.21746
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 324.000 1.00000
\(325\) 348.732 1.07302
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 147.000 0.423631 0.211816 0.977310i \(-0.432062\pi\)
0.211816 + 0.977310i \(0.432062\pi\)
\(348\) 0 0
\(349\) 695.102 1.99170 0.995848 0.0910333i \(-0.0290170\pi\)
0.995848 + 0.0910333i \(0.0290170\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −577.566 −1.63616 −0.818081 0.575102i \(-0.804962\pi\)
−0.818081 + 0.575102i \(0.804962\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 520.825 1.44273
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −351.382 −0.957444 −0.478722 0.877966i \(-0.658900\pi\)
−0.478722 + 0.877966i \(0.658900\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −537.358 −1.42535
\(378\) 0 0
\(379\) −442.453 −1.16742 −0.583710 0.811962i \(-0.698400\pi\)
−0.583710 + 0.811962i \(0.698400\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 446.300 1.16528 0.582638 0.812732i \(-0.302021\pi\)
0.582638 + 0.812732i \(0.302021\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) −195.951 −0.505029
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) −661.055 −1.66933
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 400.000 1.00000
\(401\) 737.108 1.83817 0.919087 0.394055i \(-0.128928\pi\)
0.919087 + 0.394055i \(0.128928\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 271.000 0.662592 0.331296 0.943527i \(-0.392514\pi\)
0.331296 + 0.943527i \(0.392514\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −65.8074 −0.157058 −0.0785291 0.996912i \(-0.525022\pi\)
−0.0785291 + 0.996912i \(0.525022\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 78.4485 0.185457
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 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
\(437\) 0 0
\(438\) 0 0
\(439\) −851.759 −1.94023 −0.970113 0.242653i \(-0.921982\pi\)
−0.970113 + 0.242653i \(0.921982\pi\)
\(440\) 0 0
\(441\) 441.000 1.00000
\(442\) 0 0
\(443\) 339.000 0.765237 0.382619 0.923906i \(-0.375022\pi\)
0.382619 + 0.923906i \(0.375022\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 688.465 1.53333 0.766665 0.642048i \(-0.221915\pi\)
0.766665 + 0.642048i \(0.221915\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 838.030 1.85405
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) −616.356 −1.32835
\(465\) 0 0
\(466\) 0 0
\(467\) −845.156 −1.80976 −0.904878 0.425671i \(-0.860038\pi\)
−0.904878 + 0.425671i \(0.860038\pi\)
\(468\) 502.174 1.07302
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 742.388 1.56292
\(476\) 0 0
\(477\) 950.871 1.99344
\(478\) 0 0
\(479\) 411.000 0.858038 0.429019 0.903296i \(-0.358859\pi\)
0.429019 + 0.903296i \(0.358859\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 864.747 1.78667
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −328.165 −0.657646 −0.328823 0.944392i \(-0.606652\pi\)
−0.328823 + 0.944392i \(0.606652\pi\)
\(500\) 0 0
\(501\) 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
\(508\) −1014.56 −1.99717
\(509\) 279.158 0.548444 0.274222 0.961666i \(-0.411580\pi\)
0.274222 + 0.961666i \(0.411580\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −160.058 −0.309590
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −366.448 −0.703355 −0.351677 0.936121i \(-0.614389\pi\)
−0.351677 + 0.936121i \(0.614389\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 1026.99 1.95990
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 529.000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −899.770 −1.66933
\(540\) 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
\(547\) −547.000 −1.00000
\(548\) −1092.00 −1.99270
\(549\) 0 0
\(550\) 0 0
\(551\) −1143.94 −2.07611
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −1076.00 −1.93525
\(557\) 567.000 1.01795 0.508977 0.860780i \(-0.330024\pi\)
0.508977 + 0.860780i \(0.330024\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −1062.00 −1.88632 −0.943162 0.332334i \(-0.892164\pi\)
−0.943162 + 0.332334i \(0.892164\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 710.944 1.24509 0.622543 0.782586i \(-0.286100\pi\)
0.622543 + 0.782586i \(0.286100\pi\)
\(572\) −1024.58 −1.79123
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 576.000 1.00000
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −1940.06 −3.32771
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1124.58 −1.91581 −0.957907 0.287079i \(-0.907316\pi\)
−0.957907 + 0.287079i \(0.907316\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1066.38 1.79827 0.899136 0.437670i \(-0.144196\pi\)
0.899136 + 0.437670i \(0.144196\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −996.000 −1.67114
\(597\) 0 0
\(598\) 0 0
\(599\) 651.000 1.08681 0.543406 0.839470i \(-0.317135\pi\)
0.543406 + 0.839470i \(0.317135\pi\)
\(600\) 0 0
\(601\) −627.248 −1.04367 −0.521837 0.853045i \(-0.674753\pi\)
−0.521837 + 0.853045i \(0.674753\pi\)
\(602\) 0 0
\(603\) −1149.47 −1.90626
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 278.468 0.458761 0.229381 0.973337i \(-0.426330\pi\)
0.229381 + 0.973337i \(0.426330\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 121.589 0.199000
\(612\) 0 0
\(613\) 903.915 1.47458 0.737288 0.675579i \(-0.236106\pi\)
0.737288 + 0.675579i \(0.236106\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 625.000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) −932.000 −1.48408
\(629\) 0 0
\(630\) 0 0
\(631\) 1234.54 1.95648 0.978239 0.207480i \(-0.0665261\pi\)
0.978239 + 0.207480i \(0.0665261\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 683.515 1.07302
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 739.000 1.14930 0.574650 0.818399i \(-0.305138\pi\)
0.574650 + 0.818399i \(0.305138\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −894.000 −1.38176 −0.690881 0.722969i \(-0.742777\pi\)
−0.690881 + 0.722969i \(0.742777\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −277.050 −0.421689
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −1321.50 −1.99924 −0.999620 0.0275823i \(-0.991219\pi\)
−0.999620 + 0.0275823i \(0.991219\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −852.000 −1.27545
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1344.76 1.99816 0.999080 0.0428832i \(-0.0136543\pi\)
0.999080 + 0.0428832i \(0.0136543\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 102.330 0.151376
\(677\) −888.389 −1.31224 −0.656122 0.754655i \(-0.727804\pi\)
−0.656122 + 0.754655i \(0.727804\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 426.390 0.624290 0.312145 0.950034i \(-0.398953\pi\)
0.312145 + 0.950034i \(0.398953\pi\)
\(684\) 1069.04 1.56292
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1473.77 2.13900
\(690\) 0 0
\(691\) −806.000 −1.16643 −0.583213 0.812319i \(-0.698205\pi\)
−0.583213 + 0.812319i \(0.698205\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −231.052 −0.329604 −0.164802 0.986327i \(-0.552698\pi\)
−0.164802 + 0.986327i \(0.552698\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −1175.21 −1.66933
\(705\) 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
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −756.000 −1.05587
\(717\) 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
\(724\) −1256.48 −1.73547
\(725\) −963.056 −1.32835
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 729.000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 2345.26 3.18217
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −702.000 −0.944818 −0.472409 0.881379i \(-0.656616\pi\)
−0.472409 + 0.881379i \(0.656616\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 132.540 0.176484 0.0882422 0.996099i \(-0.471875\pi\)
0.0882422 + 0.996099i \(0.471875\pi\)
\(752\) 139.464 0.185457
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −1434.13 −1.89449 −0.947246 0.320506i \(-0.896147\pi\)
−0.947246 + 0.320506i \(0.896147\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1490.51 −1.95862 −0.979312 0.202356i \(-0.935140\pi\)
−0.979312 + 0.202356i \(0.935140\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −610.242 −0.798746
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −644.000 −0.834197
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 784.000 1.00000
\(785\) 0 0
\(786\) 0 0
\(787\) 1027.00 1.30496 0.652478 0.757808i \(-0.273730\pi\)
0.652478 + 0.757808i \(0.273730\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 1365.05 1.71488
\(797\) 1034.98 1.29859 0.649296 0.760536i \(-0.275063\pi\)
0.649296 + 0.760536i \(0.275063\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 565.263 0.703939
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 1546.10 1.90642 0.953208 0.302314i \(-0.0977591\pi\)
0.953208 + 0.302314i \(0.0977591\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 1455.03 1.76796 0.883981 0.467522i \(-0.154853\pi\)
0.883981 + 0.467522i \(0.154853\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 537.879 0.648828 0.324414 0.945915i \(-0.394833\pi\)
0.324414 + 0.945915i \(0.394833\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 892.754 1.07302
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) −2181.15 −2.60904
\(837\) 0 0
\(838\) 0 0
\(839\) 1131.00 1.34803 0.674017 0.738716i \(-0.264568\pi\)
0.674017 + 0.738716i \(0.264568\pi\)
\(840\) 0 0
\(841\) 642.964 0.764524
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 1690.44 1.99344
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −817.617 −0.958519 −0.479260 0.877673i \(-0.659095\pi\)
−0.479260 + 0.877673i \(0.659095\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) −1166.44 −1.35791 −0.678955 0.734180i \(-0.737567\pi\)
−0.678955 + 0.734180i \(0.737567\pi\)
\(860\) 0 0
\(861\) 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
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −1781.59 −2.04545
\(872\) 0 0
\(873\) −440.890 −0.505029
\(874\) 0 0
\(875\) 0 0
\(876\) 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
\(883\) 601.520 0.681224 0.340612 0.940204i \(-0.389366\pi\)
0.340612 + 0.940204i \(0.389366\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −1631.63 −1.83949 −0.919746 0.392513i \(-0.871606\pi\)
−0.919746 + 0.392513i \(0.871606\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −1487.37 −1.66933
\(892\) 0 0
\(893\) 258.841 0.289856
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 900.000 1.00000
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 1798.04 1.98241 0.991204 0.132343i \(-0.0422500\pi\)
0.991204 + 0.132343i \(0.0422500\pi\)
\(908\) −372.000 −0.409692
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −36.2653 −0.0394617 −0.0197309 0.999805i \(-0.506281\pi\)
−0.0197309 + 0.999805i \(0.506281\pi\)
\(920\) 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
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 1455.08 1.56292
\(932\) 964.192 1.03454
\(933\) 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
\(940\) 0 0
\(941\) −1868.28 −1.98542 −0.992709 0.120532i \(-0.961540\pi\)
−0.992709 + 0.120532i \(0.961540\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −294.000 −0.310454 −0.155227 0.987879i \(-0.549611\pi\)
−0.155227 + 0.987879i \(0.549611\pi\)
\(948\) 0 0
\(949\) −429.405 −0.452482
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −565.336 −0.593218 −0.296609 0.954999i \(-0.595856\pi\)
−0.296609 + 0.954999i \(0.595856\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 273.561 0.286152
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 961.000 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 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
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 1656.93 1.67705
\(989\) 0 0
\(990\) 0 0
\(991\) 396.258 0.399857 0.199929 0.979810i \(-0.435929\pi\)
0.199929 + 0.979810i \(0.435929\pi\)
\(992\) 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 547.3.b.a.546.1 3
547.546 odd 2 CM 547.3.b.a.546.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
547.3.b.a.546.1 3 1.1 even 1 trivial
547.3.b.a.546.1 3 547.546 odd 2 CM