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Properties

Label 1568.4.a.i.1.1
Level $1568$
Weight $4$
Character 1568.1
Self dual yes
Analytic conductor $92.515$
Analytic rank $0$
Dimension $1$
CM discriminant -4
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1568,4,Mod(1,1568)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1568, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1568.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1568 = 2^{5} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1568.a (trivial)

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: \(92.5149948890\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1568.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.00000 q^{5} -27.0000 q^{9} +O(q^{10})\) \(q+4.00000 q^{5} -27.0000 q^{9} +92.0000 q^{13} +104.000 q^{17} -109.000 q^{25} +130.000 q^{29} -214.000 q^{37} -472.000 q^{41} -108.000 q^{45} +518.000 q^{53} +468.000 q^{61} +368.000 q^{65} -592.000 q^{73} +729.000 q^{81} +416.000 q^{85} +176.000 q^{89} +1816.00 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 0 0
\(5\) 4.00000 0.357771 0.178885 − 0.983870i \(-0.442751\pi\)
0.178885 + 0.983870i \(0.442751\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −27.0000 −1.00000
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 92.0000 1.96279 0.981393 − 0.192012i \(-0.0615011\pi\)
0.981393 + 0.192012i \(0.0615011\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 104.000 1.48375 0.741874 − 0.670540i \(-0.233937\pi\)
0.741874 + 0.670540i \(0.233937\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −109.000 −0.872000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 130.000 0.832427 0.416214 − 0.909267i \(-0.363357\pi\)
0.416214 + 0.909267i \(0.363357\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −214.000 −0.950848 −0.475424 − 0.879757i \(-0.657705\pi\)
−0.475424 + 0.879757i \(0.657705\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −472.000 −1.79790 −0.898951 − 0.438048i \(-0.855670\pi\)
−0.898951 + 0.438048i \(0.855670\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) −108.000 −0.357771
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 518.000 1.34251 0.671253 − 0.741229i \(-0.265757\pi\)
0.671253 + 0.741229i \(0.265757\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 468.000 0.982316 0.491158 − 0.871071i \(-0.336574\pi\)
0.491158 + 0.871071i \(0.336574\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 368.000 0.702227
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −592.000 −0.949156 −0.474578 − 0.880214i \(-0.657399\pi\)
−0.474578 + 0.880214i \(0.657399\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 729.000 1.00000
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 416.000 0.530842
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 176.000 0.209618 0.104809 − 0.994492i \(-0.466577\pi\)
0.104809 + 0.994492i \(0.466577\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1816.00 1.90090 0.950448 − 0.310884i \(-0.100625\pi\)
0.950448 + 0.310884i \(0.100625\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1940.00 1.91126 0.955630 − 0.294570i \(-0.0951766\pi\)
0.955630 + 0.294570i \(0.0951766\pi\)
\(102\) 0 0
\(103\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 1746.00 1.53428 0.767140 − 0.641480i \(-0.221679\pi\)
0.767140 + 0.641480i \(0.221679\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2002.00 1.66666 0.833329 − 0.552778i \(-0.186432\pi\)
0.833329 + 0.552778i \(0.186432\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −2484.00 −1.96279
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1331.00 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −936.000 −0.669747
\(126\) 0 0
\(127\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1606.00 1.00153 0.500766 − 0.865583i \(-0.333052\pi\)
0.500766 + 0.865583i \(0.333052\pi\)
\(138\) 0 0
\(139\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 520.000 0.297818
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −3514.00 −1.93207 −0.966034 − 0.258415i \(-0.916800\pi\)
−0.966034 + 0.258415i \(0.916800\pi\)
\(150\) 0 0
\(151\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) −2808.00 −1.48375
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −3924.00 −1.99471 −0.997354 − 0.0726920i \(-0.976841\pi\)
−0.997354 + 0.0726920i \(0.976841\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 6267.00 2.85253
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2012.00 0.884217 0.442108 − 0.896962i \(-0.354231\pi\)
0.442108 + 0.896962i \(0.354231\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 2860.00 1.17449 0.587243 − 0.809410i \(-0.300213\pi\)
0.587243 + 0.809410i \(0.300213\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −856.000 −0.340186
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 5362.00 1.99982 0.999910 − 0.0134266i \(-0.00427395\pi\)
0.999910 + 0.0134266i \(0.00427395\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1174.00 0.424589 0.212295 − 0.977206i \(-0.431906\pi\)
0.212295 + 0.977206i \(0.431906\pi\)
\(198\) 0 0
\(199\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −1888.00 −0.643237
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 9568.00 2.91228
\(222\) 0 0
\(223\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 2943.00 0.872000
\(226\) 0 0
\(227\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 2684.00 0.774514 0.387257 − 0.921972i \(-0.373423\pi\)
0.387257 + 0.921972i \(0.373423\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 598.000 0.168139 0.0840693 − 0.996460i \(-0.473208\pi\)
0.0840693 + 0.996460i \(0.473208\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 5272.00 1.40913 0.704563 − 0.709641i \(-0.251143\pi\)
0.704563 + 0.709641i \(0.251143\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −8096.00 −1.96504 −0.982519 − 0.186164i \(-0.940394\pi\)
−0.982519 + 0.186164i \(0.940394\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −3510.00 −0.832427
\(262\) 0 0
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 2072.00 0.480309
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 8140.00 1.84500 0.922499 − 0.385999i \(-0.126143\pi\)
0.922499 + 0.385999i \(0.126143\pi\)
\(270\) 0 0
\(271\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 9126.00 1.97952 0.989762 − 0.142727i \(-0.0455871\pi\)
0.989762 + 0.142727i \(0.0455871\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 7430.00 1.57735 0.788677 − 0.614807i \(-0.210766\pi\)
0.788677 + 0.614807i \(0.210766\pi\)
\(282\) 0 0
\(283\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 5903.00 1.20151
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −3452.00 −0.688287 −0.344143 − 0.938917i \(-0.611831\pi\)
−0.344143 + 0.938917i \(0.611831\pi\)
\(294\) 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
\(304\) 0 0
\(305\) 1872.00 0.351444
\(306\) 0 0
\(307\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 8712.00 1.57326 0.786632 − 0.617423i \(-0.211823\pi\)
0.786632 + 0.617423i \(0.211823\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −10274.0 −1.82033 −0.910166 − 0.414243i \(-0.864046\pi\)
−0.910166 + 0.414243i \(0.864046\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −10028.0 −1.71155
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) 0 0
\(333\) 5778.00 0.950848
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 12366.0 1.99887 0.999435 − 0.0336216i \(-0.0107041\pi\)
0.999435 + 0.0336216i \(0.0107041\pi\)
\(338\) 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
\(347\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) −8964.00 −1.37488 −0.687438 − 0.726243i \(-0.741265\pi\)
−0.687438 + 0.726243i \(0.741265\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −12848.0 −1.93720 −0.968598 − 0.248633i \(-0.920019\pi\)
−0.968598 + 0.248633i \(0.920019\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −6859.00 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −2368.00 −0.339580
\(366\) 0 0
\(367\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) 12744.0 1.79790
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −12922.0 −1.79377 −0.896884 − 0.442265i \(-0.854175\pi\)
−0.896884 + 0.442265i \(0.854175\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 11960.0 1.63388
\(378\) 0 0
\(379\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −374.000 −0.0487469 −0.0243735 − 0.999703i \(-0.507759\pi\)
−0.0243735 + 0.999703i \(0.507759\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −12564.0 −1.58834 −0.794168 − 0.607699i \(-0.792093\pi\)
−0.794168 + 0.607699i \(0.792093\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2398.00 0.298629 0.149315 − 0.988790i \(-0.452293\pi\)
0.149315 + 0.988790i \(0.452293\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 2916.00 0.357771
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 14920.0 1.80378 0.901891 − 0.431964i \(-0.142179\pi\)
0.901891 + 0.431964i \(0.142179\pi\)
\(410\) 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
\(419\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −10890.0 −1.26068 −0.630340 − 0.776319i \(-0.717084\pi\)
−0.630340 + 0.776319i \(0.717084\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −11336.0 −1.29383
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −17352.0 −1.92583 −0.962914 − 0.269807i \(-0.913040\pi\)
−0.962914 + 0.269807i \(0.913040\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 704.000 0.0749951
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 16114.0 1.69369 0.846845 − 0.531840i \(-0.178499\pi\)
0.846845 + 0.531840i \(0.178499\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 16506.0 1.68954 0.844768 − 0.535132i \(-0.179738\pi\)
0.844768 + 0.535132i \(0.179738\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 19660.0 1.98624 0.993121 − 0.117093i \(-0.0373577\pi\)
0.993121 + 0.117093i \(0.0373577\pi\)
\(462\) 0 0
\(463\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −13986.0 −1.34251
\(478\) 0 0
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −19688.0 −1.86631
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 7264.00 0.680085
\(486\) 0 0
\(487\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 13520.0 1.23511
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 7760.00 0.683793
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 17996.0 1.56711 0.783555 − 0.621323i \(-0.213404\pi\)
0.783555 + 0.621323i \(0.213404\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 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
\(521\) −1480.00 −0.124453 −0.0622265 − 0.998062i \(-0.519820\pi\)
−0.0622265 + 0.998062i \(0.519820\pi\)
\(522\) 0 0
\(523\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −12167.0 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −43424.0 −3.52890
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −5922.00 −0.470622 −0.235311 − 0.971920i \(-0.575611\pi\)
−0.235311 + 0.971920i \(0.575611\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 6984.00 0.548921
\(546\) 0 0
\(547\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 0 0
\(549\) −12636.0 −0.982316
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −8626.00 −0.656186 −0.328093 − 0.944646i \(-0.606406\pi\)
−0.328093 + 0.944646i \(0.606406\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 8008.00 0.596282
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 26806.0 1.97498 0.987492 − 0.157669i \(-0.0503978\pi\)
0.987492 + 0.157669i \(0.0503978\pi\)
\(570\) 0 0
\(571\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −27504.0 −1.98441 −0.992207 − 0.124603i \(-0.960234\pi\)
−0.992207 + 0.124603i \(0.960234\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −9936.00 −0.702227
\(586\) 0 0
\(587\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 24368.0 1.68748 0.843738 − 0.536755i \(-0.180350\pi\)
0.843738 + 0.536755i \(0.180350\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −24048.0 −1.63218 −0.816089 − 0.577927i \(-0.803862\pi\)
−0.816089 + 0.577927i \(0.803862\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −5324.00 −0.357771
\(606\) 0 0
\(607\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −23222.0 −1.53006 −0.765031 − 0.643994i \(-0.777276\pi\)
−0.765031 + 0.643994i \(0.777276\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −15466.0 −1.00914 −0.504569 − 0.863372i \(-0.668348\pi\)
−0.504569 + 0.863372i \(0.668348\pi\)
\(618\) 0 0
\(619\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 9881.00 0.632384
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −22256.0 −1.41082
\(630\) 0 0
\(631\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −28850.0 −1.77770 −0.888851 − 0.458197i \(-0.848495\pi\)
−0.888851 + 0.458197i \(0.848495\pi\)
\(642\) 0 0
\(643\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −33358.0 −1.99908 −0.999540 − 0.0303236i \(-0.990346\pi\)
−0.999540 + 0.0303236i \(0.990346\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 15984.0 0.949156
\(658\) 0 0
\(659\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 22068.0 1.29856 0.649278 − 0.760551i \(-0.275071\pi\)
0.649278 + 0.760551i \(0.275071\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 4462.00 0.255568 0.127784 − 0.991802i \(-0.459214\pi\)
0.127784 + 0.991802i \(0.459214\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −34996.0 −1.98671 −0.993357 − 0.115072i \(-0.963290\pi\)
−0.993357 + 0.115072i \(0.963290\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 6424.00 0.358319
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 47656.0 2.63505
\(690\) 0 0
\(691\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −49088.0 −2.66763
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −20030.0 −1.07920 −0.539602 − 0.841920i \(-0.681425\pi\)
−0.539602 + 0.841920i \(0.681425\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 36810.0 1.94983 0.974914 − 0.222580i \(-0.0714479\pi\)
0.974914 + 0.222580i \(0.0714479\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −14170.0 −0.725877
\(726\) 0 0
\(727\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) −19683.0 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −8732.00 −0.440005 −0.220003 − 0.975499i \(-0.570607\pi\)
−0.220003 + 0.975499i \(0.570607\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) −14056.0 −0.691238
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −35046.0 −1.68265 −0.841327 − 0.540527i \(-0.818225\pi\)
−0.841327 + 0.540527i \(0.818225\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 27320.0 1.30138 0.650689 − 0.759344i \(-0.274480\pi\)
0.650689 + 0.759344i \(0.274480\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −11232.0 −0.530842
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 41544.0 1.94813 0.974067 − 0.226260i \(-0.0726499\pi\)
0.974067 + 0.226260i \(0.0726499\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 16852.0 0.784119 0.392060 − 0.919940i \(-0.371763\pi\)
0.392060 + 0.919940i \(0.371763\pi\)
\(774\) 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
\(784\) 0 0
\(785\) −15696.0 −0.713649
\(786\) 0 0
\(787\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 43056.0 1.92807
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −16276.0 −0.723370 −0.361685 − 0.932300i \(-0.617798\pi\)
−0.361685 + 0.932300i \(0.617798\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −4752.00 −0.209618
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −23270.0 −1.01129 −0.505643 − 0.862743i \(-0.668745\pi\)
−0.505643 + 0.862743i \(0.668745\pi\)
\(810\) 0 0
\(811\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 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
\(821\) −1850.00 −0.0786424 −0.0393212 − 0.999227i \(-0.512520\pi\)
−0.0393212 + 0.999227i \(0.512520\pi\)
\(822\) 0 0
\(823\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) −41740.0 −1.74872 −0.874361 − 0.485276i \(-0.838719\pi\)
−0.874361 + 0.485276i \(0.838719\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −7489.00 −0.307065
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 25068.0 1.02055
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 45468.0 1.82508 0.912541 − 0.408986i \(-0.134117\pi\)
0.912541 + 0.408986i \(0.134117\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −20056.0 −0.799416 −0.399708 − 0.916642i \(-0.630889\pi\)
−0.399708 + 0.916642i \(0.630889\pi\)
\(858\) 0 0
\(859\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 8048.00 0.316347
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −49032.0 −1.90090
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 42514.0 1.63694 0.818470 − 0.574550i \(-0.194823\pi\)
0.818470 + 0.574550i \(0.194823\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −51808.0 −1.98122 −0.990611 − 0.136714i \(-0.956346\pi\)
−0.990611 + 0.136714i \(0.956346\pi\)
\(882\) 0 0
\(883\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 53872.0 1.99194
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 11440.0 0.420197
\(906\) 0 0
\(907\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 0 0
\(909\) −52380.0 −1.91126
\(910\) 0 0
\(911\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 23326.0 0.829140
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −47480.0 −1.67682 −0.838411 − 0.545038i \(-0.816515\pi\)
−0.838411 + 0.545038i \(0.816515\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −24336.0 −0.848476 −0.424238 − 0.905551i \(-0.639458\pi\)
−0.424238 + 0.905551i \(0.639458\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −48460.0 −1.67880 −0.839400 − 0.543514i \(-0.817093\pi\)
−0.839400 + 0.543514i \(0.817093\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 0 0
\(949\) −54464.0 −1.86299
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −56758.0 −1.92925 −0.964623 − 0.263632i \(-0.915079\pi\)
−0.964623 + 0.263632i \(0.915079\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −29791.0 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 21448.0 0.715477
\(966\) 0 0
\(967\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 56606.0 1.85362 0.926810 − 0.375531i \(-0.122540\pi\)
0.926810 + 0.375531i \(0.122540\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −47142.0 −1.53428
\(982\) 0 0
\(983\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 4696.00 0.151906
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 34164.0 1.08524 0.542620 − 0.839978i \(-0.317432\pi\)
0.542620 + 0.839978i \(0.317432\pi\)
\(998\) 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 1568.4.a.i.1.1 yes 1
4.3 odd 2 CM 1568.4.a.i.1.1 yes 1
7.6 odd 2 1568.4.a.h.1.1 ✓ 1
28.27 even 2 1568.4.a.h.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1568.4.a.h.1.1 ✓ 1 7.6 odd 2
1568.4.a.h.1.1 ✓ 1 28.27 even 2
1568.4.a.i.1.1 yes 1 1.1 even 1 trivial
1568.4.a.i.1.1 yes 1 4.3 odd 2 CM