Properties

Label 547.3.b.a.546.2
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 546.2
Root \(-4.54841\) 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)
 
\(f(q)\) \(=\) \(q+4.00000 q^{4} +9.00000 q^{9} +O(q^{10})\) \(q+4.00000 q^{4} +9.00000 q^{9} -1.31201 q^{11} +12.0270 q^{13} +16.0000 q^{16} -35.3814 q^{19} +25.0000 q^{25} +56.8115 q^{29} +36.0000 q^{36} -5.24803 q^{44} -85.4139 q^{47} +49.0000 q^{49} +48.1081 q^{52} -45.3968 q^{53} +64.0000 q^{64} +98.9719 q^{67} +138.989 q^{73} -141.526 q^{76} +81.0000 q^{81} -138.070 q^{97} -11.8081 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 12 q^{4} + 27 q^{9}+O(q^{10}) \) 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
\(11\) −1.31201 −0.119273 −0.0596367 0.998220i \(-0.518994\pi\)
−0.0596367 + 0.998220i \(0.518994\pi\)
\(12\) 0 0
\(13\) 12.0270 0.925155 0.462578 0.886579i \(-0.346925\pi\)
0.462578 + 0.886579i \(0.346925\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 16.0000 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −35.3814 −1.86218 −0.931090 0.364789i \(-0.881141\pi\)
−0.931090 + 0.364789i \(0.881141\pi\)
\(20\) 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
\(29\) 56.8115 1.95902 0.979508 0.201406i \(-0.0645511\pi\)
0.979508 + 0.201406i \(0.0645511\pi\)
\(30\) 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 \(\pi\)
\(44\) −5.24803 −0.119273
\(45\) 0 0
\(46\) 0 0
\(47\) −85.4139 −1.81732 −0.908658 0.417540i \(-0.862892\pi\)
−0.908658 + 0.417540i \(0.862892\pi\)
\(48\) 0 0
\(49\) 49.0000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 48.1081 0.925155
\(53\) −45.3968 −0.856544 −0.428272 0.903650i \(-0.640877\pi\)
−0.428272 + 0.903650i \(0.640877\pi\)
\(54\) 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
\(67\) 98.9719 1.47719 0.738596 0.674148i \(-0.235489\pi\)
0.738596 + 0.674148i \(0.235489\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 138.989 1.90396 0.951979 0.306163i \(-0.0990453\pi\)
0.951979 + 0.306163i \(0.0990453\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −141.526 −1.86218
\(77\) 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
\(97\) −138.070 −1.42341 −0.711703 0.702481i \(-0.752076\pi\)
−0.711703 + 0.702481i \(0.752076\pi\)
\(98\) 0 0
\(99\) −11.8081 −0.119273
\(100\) 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
\(113\) −31.3582 −0.277506 −0.138753 0.990327i \(-0.544309\pi\)
−0.138753 + 0.990327i \(0.544309\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 227.246 1.95902
\(117\) 108.243 0.925155
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −119.279 −0.985774
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 138.508 1.09062 0.545308 0.838236i \(-0.316413\pi\)
0.545308 + 0.838236i \(0.316413\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −173.584 −1.32507 −0.662533 0.749033i \(-0.730518\pi\)
−0.662533 + 0.749033i \(0.730518\pi\)
\(132\) 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
\(143\) −15.7795 −0.110346
\(144\) 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
\(169\) −24.3509 −0.144088
\(170\) 0 0
\(171\) −318.433 −1.86218
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −20.9921 −0.119273
\(177\) 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
\(181\) 312.879 1.72861 0.864306 0.502967i \(-0.167758\pi\)
0.864306 + 0.502967i \(0.167758\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) −341.656 −1.81732
\(189\) 0 0
\(190\) 0 0
\(191\) 379.574 1.98730 0.993649 0.112527i \(-0.0358946\pi\)
0.993649 + 0.112527i \(0.0358946\pi\)
\(192\) 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
\(199\) 6.73451 0.0338417 0.0169209 0.999857i \(-0.494614\pi\)
0.0169209 + 0.999857i \(0.494614\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 192.432 0.925155
\(209\) 46.4207 0.222108
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) −181.587 −0.856544
\(213\) 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
\(233\) −465.906 −1.99960 −0.999799 0.0200583i \(-0.993615\pi\)
−0.999799 + 0.0200583i \(0.993615\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 375.506 1.57115 0.785577 0.618763i \(-0.212366\pi\)
0.785577 + 0.618763i \(0.212366\pi\)
\(240\) 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
\(247\) −425.533 −1.72281
\(248\) 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
\(261\) 511.303 1.95902
\(262\) 0 0
\(263\) −251.255 −0.955344 −0.477672 0.878538i \(-0.658519\pi\)
−0.477672 + 0.878538i \(0.658519\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 395.887 1.47719
\(269\) −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
\(275\) −32.8002 −0.119273
\(276\) 0 0
\(277\) −445.657 −1.60887 −0.804434 0.594042i \(-0.797531\pi\)
−0.804434 + 0.594042i \(0.797531\pi\)
\(278\) 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
\(292\) 555.956 1.90396
\(293\) 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
\(304\) −566.103 −1.86218
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 68.8812 0.221483 0.110742 0.993849i \(-0.464677\pi\)
0.110742 + 0.993849i \(0.464677\pi\)
\(312\) 0 0
\(313\) 209.709 0.669997 0.334998 0.942219i \(-0.391264\pi\)
0.334998 + 0.942219i \(0.391264\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −616.004 −1.94323 −0.971615 0.236570i \(-0.923977\pi\)
−0.971615 + 0.236570i \(0.923977\pi\)
\(318\) 0 0
\(319\) −74.5370 −0.233658
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 324.000 1.00000
\(325\) 300.675 0.925155
\(326\) 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
\(349\) −402.579 −1.15352 −0.576761 0.816913i \(-0.695684\pi\)
−0.576761 + 0.816913i \(0.695684\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 640.408 1.81419 0.907094 0.420928i \(-0.138296\pi\)
0.907094 + 0.420928i \(0.138296\pi\)
\(354\) 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
\(361\) 890.846 2.46772
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 733.782 1.99940 0.999702 0.0243978i \(-0.00776682\pi\)
0.999702 + 0.0243978i \(0.00776682\pi\)
\(368\) 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
\(377\) 683.272 1.81239
\(378\) 0 0
\(379\) −311.784 −0.822649 −0.411324 0.911489i \(-0.634934\pi\)
−0.411324 + 0.911489i \(0.634934\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −762.297 −1.99033 −0.995165 0.0982129i \(-0.968687\pi\)
−0.995165 + 0.0982129i \(0.968687\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) −552.281 −1.42341
\(389\) 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
\(396\) −47.2322 −0.119273
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 400.000 1.00000
\(401\) −642.245 −1.60161 −0.800805 0.598925i \(-0.795595\pi\)
−0.800805 + 0.598925i \(0.795595\pi\)
\(402\) 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
\(419\) 756.392 1.80523 0.902616 0.430448i \(-0.141644\pi\)
0.902616 + 0.430448i \(0.141644\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) −768.725 −1.81732
\(424\) 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
\(439\) 241.373 0.549826 0.274913 0.961469i \(-0.411351\pi\)
0.274913 + 0.961469i \(0.411351\pi\)
\(440\) 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
\(449\) 155.082 0.345394 0.172697 0.984975i \(-0.444752\pi\)
0.172697 + 0.984975i \(0.444752\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −125.433 −0.277506
\(453\) 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 \(\pi\)
\(464\) 908.983 1.95902
\(465\) 0 0
\(466\) 0 0
\(467\) 766.890 1.64216 0.821081 0.570812i \(-0.193371\pi\)
0.821081 + 0.570812i \(0.193371\pi\)
\(468\) 432.973 0.925155
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −884.536 −1.86218
\(476\) 0 0
\(477\) −408.571 −0.856544
\(478\) 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
\(484\) −477.115 −0.985774
\(485\) 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
\(499\) 980.314 1.96456 0.982279 0.187427i \(-0.0600148\pi\)
0.982279 + 0.187427i \(0.0600148\pi\)
\(500\) 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
\(508\) 554.033 1.09062
\(509\) 708.239 1.39143 0.695716 0.718317i \(-0.255087\pi\)
0.695716 + 0.718317i \(0.255087\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 112.064 0.216757
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −661.530 −1.26973 −0.634866 0.772622i \(-0.718945\pi\)
−0.634866 + 0.772622i \(0.718945\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) −694.334 −1.32507
\(525\) 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
\(539\) −64.2883 −0.119273
\(540\) 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
\(551\) −2010.07 −3.64804
\(552\) 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
\(571\) −1129.45 −1.97802 −0.989011 0.147845i \(-0.952766\pi\)
−0.989011 + 0.147845i \(0.952766\pi\)
\(572\) −63.1181 −0.110346
\(573\) 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
\(583\) 59.5609 0.102163
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 270.414 0.460671 0.230335 0.973111i \(-0.426018\pi\)
0.230335 + 0.973111i \(0.426018\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −982.721 −1.65720 −0.828601 0.559840i \(-0.810863\pi\)
−0.828601 + 0.559840i \(0.810863\pi\)
\(594\) 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
\(601\) 1201.61 1.99935 0.999677 0.0254016i \(-0.00808644\pi\)
0.999677 + 0.0254016i \(0.00808644\pi\)
\(602\) 0 0
\(603\) 890.747 1.47719
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −1162.56 −1.91525 −0.957625 0.288019i \(-0.907003\pi\)
−0.957625 + 0.288019i \(0.907003\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −1027.27 −1.68130
\(612\) 0 0
\(613\) 265.337 0.432850 0.216425 0.976299i \(-0.430560\pi\)
0.216425 + 0.976299i \(0.430560\pi\)
\(614\) 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
\(631\) −390.509 −0.618874 −0.309437 0.950920i \(-0.600141\pi\)
−0.309437 + 0.950920i \(0.600141\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 589.324 0.925155
\(638\) 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
\(657\) 1250.90 1.90396
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 629.170 0.951846 0.475923 0.879487i \(-0.342114\pi\)
0.475923 + 0.879487i \(0.342114\pi\)
\(662\) 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
\(673\) −722.369 −1.07336 −0.536678 0.843787i \(-0.680321\pi\)
−0.536678 + 0.843787i \(0.680321\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −97.4034 −0.144088
\(677\) −440.713 −0.650979 −0.325490 0.945546i \(-0.605529\pi\)
−0.325490 + 0.945546i \(0.605529\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 910.687 1.33336 0.666682 0.745343i \(-0.267714\pi\)
0.666682 + 0.745343i \(0.267714\pi\)
\(684\) −1273.73 −1.86218
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −545.988 −0.792436
\(690\) 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
\(701\) −1082.04 −1.54357 −0.771783 0.635886i \(-0.780635\pi\)
−0.771783 + 0.635886i \(0.780635\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −83.9684 −0.119273
\(705\) 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
\(724\) 1251.51 1.72861
\(725\) 1420.29 1.95902
\(726\) 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
\(737\) −129.852 −0.176190
\(738\) 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
\(751\) 1229.43 1.63705 0.818526 0.574470i \(-0.194792\pi\)
0.818526 + 0.574470i \(0.194792\pi\)
\(752\) −1366.62 −1.81732
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 296.830 0.392114 0.196057 0.980593i \(-0.437186\pi\)
0.196057 + 0.980593i \(0.437186\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1011.98 1.32980 0.664902 0.746931i \(-0.268474\pi\)
0.664902 + 0.746931i \(0.268474\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 1518.30 1.98730
\(765\) 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
\(796\) 26.9380 0.0338417
\(797\) −1567.37 −1.96658 −0.983291 0.182039i \(-0.941730\pi\)
−0.983291 + 0.182039i \(0.941730\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −182.354 −0.227091
\(804\) 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
\(811\) −348.394 −0.429585 −0.214793 0.976660i \(-0.568908\pi\)
−0.214793 + 0.976660i \(0.568908\pi\)
\(812\) 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
\(823\) −1393.96 −1.69375 −0.846877 0.531789i \(-0.821520\pi\)
−0.846877 + 0.531789i \(0.821520\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) −1627.15 −1.96279 −0.981394 0.192007i \(-0.938500\pi\)
−0.981394 + 0.192007i \(0.938500\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 769.729 0.925155
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 185.683 0.222108
\(837\) 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
\(841\) 2386.54 2.83774
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) −726.349 −0.856544
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −887.900 −1.04091 −0.520457 0.853888i \(-0.674239\pi\)
−0.520457 + 0.853888i \(0.674239\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) −509.113 −0.592682 −0.296341 0.955082i \(-0.595766\pi\)
−0.296341 + 0.955082i \(0.595766\pi\)
\(860\) 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
\(871\) 1190.34 1.36663
\(872\) 0 0
\(873\) −1242.63 −1.42341
\(874\) 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
\(883\) 1137.19 1.28787 0.643935 0.765080i \(-0.277301\pi\)
0.643935 + 0.765080i \(0.277301\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1418.84 1.59960 0.799799 0.600267i \(-0.204939\pi\)
0.799799 + 0.600267i \(0.204939\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −106.273 −0.119273
\(892\) 0 0
\(893\) 3022.07 3.38417
\(894\) 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
\(907\) −1106.93 −1.22043 −0.610214 0.792236i \(-0.708917\pi\)
−0.610214 + 0.792236i \(0.708917\pi\)
\(908\) −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
\(919\) −1573.31 −1.71198 −0.855991 0.516990i \(-0.827052\pi\)
−0.855991 + 0.516990i \(0.827052\pi\)
\(920\) 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
\(931\) −1733.69 −1.86218
\(932\) −1863.62 −1.99960
\(933\) 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
\(941\) 1130.59 1.20148 0.600738 0.799446i \(-0.294873\pi\)
0.600738 + 0.799446i \(0.294873\pi\)
\(942\) 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
\(949\) 1671.62 1.76146
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1859.03 1.95072 0.975358 0.220629i \(-0.0708109\pi\)
0.975358 + 0.220629i \(0.0708109\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 1502.02 1.57115
\(957\) 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
\(988\) −1702.13 −1.72281
\(989\) 0 0
\(990\) 0 0
\(991\) −1879.94 −1.89701 −0.948505 0.316762i \(-0.897404\pi\)
−0.948505 + 0.316762i \(0.897404\pi\)
\(992\) 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.546.2 3
547.546 odd 2 CM 547.3.b.a.546.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
547.3.b.a.546.2 3 1.1 even 1 trivial
547.3.b.a.546.2 3 547.546 odd 2 CM