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Properties

Label 576.3.g.c.127.1
Level $576$
Weight $3$
Character 576.127
Self dual yes
Analytic conductor $15.695$
Analytic rank $0$
Dimension $1$
CM discriminant -4
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] = [576,3,Mod(127,576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(576, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("576.127");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 576.g (of order \(2\), degree \(1\), not 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: \(15.6948632272\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 127.1
Character \(\chi\) \(=\) 576.127

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+8.00000 q^{5} +10.0000 q^{13} +16.0000 q^{17} +39.0000 q^{25} -40.0000 q^{29} +70.0000 q^{37} -80.0000 q^{41} +49.0000 q^{49} +56.0000 q^{53} +22.0000 q^{61} +80.0000 q^{65} +110.000 q^{73} +128.000 q^{85} +160.000 q^{89} -130.000 q^{97} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 8.00000 1.60000 0.800000 − 0.600000i \(-0.204833\pi\)
0.800000 + 0.600000i \(0.204833\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 10.0000 0.769231 0.384615 − 0.923077i \(-0.374334\pi\)
0.384615 + 0.923077i \(0.374334\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 16.0000 0.941176 0.470588 − 0.882353i \(-0.344042\pi\)
0.470588 + 0.882353i \(0.344042\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 39.0000 1.56000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −40.0000 −1.37931 −0.689655 − 0.724138i \(-0.742238\pi\)
−0.689655 + 0.724138i \(0.742238\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 70.0000 1.89189 0.945946 − 0.324324i \(-0.105137\pi\)
0.945946 + 0.324324i \(0.105137\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −80.0000 −1.95122 −0.975610 − 0.219512i \(-0.929553\pi\)
−0.975610 + 0.219512i \(0.929553\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 49.0000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 56.0000 1.05660 0.528302 − 0.849057i \(-0.322829\pi\)
0.528302 + 0.849057i \(0.322829\pi\)
\(54\) 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
\(61\) 22.0000 0.360656 0.180328 − 0.983607i \(-0.442284\pi\)
0.180328 + 0.983607i \(0.442284\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 80.0000 1.23077
\(66\) 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
\(73\) 110.000 1.50685 0.753425 − 0.657534i \(-0.228401\pi\)
0.753425 + 0.657534i \(0.228401\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 128.000 1.50588
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 160.000 1.79775 0.898876 − 0.438202i \(-0.144385\pi\)
0.898876 + 0.438202i \(0.144385\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −130.000 −1.34021 −0.670103 − 0.742268i \(-0.733750\pi\)
−0.670103 + 0.742268i \(0.733750\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −40.0000 −0.396040 −0.198020 − 0.980198i \(-0.563451\pi\)
−0.198020 + 0.980198i \(0.563451\pi\)
\(102\) 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
\(109\) −182.000 −1.66972 −0.834862 − 0.550459i \(-0.814453\pi\)
−0.834862 + 0.550459i \(0.814453\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −224.000 −1.98230 −0.991150 − 0.132743i \(-0.957621\pi\)
−0.991150 + 0.132743i \(0.957621\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 121.000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 112.000 0.896000
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −176.000 −1.28467 −0.642336 − 0.766423i \(-0.722035\pi\)
−0.642336 + 0.766423i \(0.722035\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −320.000 −2.20690
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −280.000 −1.87919 −0.939597 − 0.342282i \(-0.888800\pi\)
−0.939597 + 0.342282i \(0.888800\pi\)
\(150\) 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
\(157\) −170.000 −1.08280 −0.541401 − 0.840764i \(-0.682106\pi\)
−0.541401 + 0.840764i \(0.682106\pi\)
\(158\) 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
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) −69.0000 −0.408284
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 104.000 0.601156 0.300578 − 0.953757i \(-0.402820\pi\)
0.300578 + 0.953757i \(0.402820\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) −38.0000 −0.209945 −0.104972 − 0.994475i \(-0.533475\pi\)
−0.104972 + 0.994475i \(0.533475\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 560.000 3.02703
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) −190.000 −0.984456 −0.492228 − 0.870466i \(-0.663817\pi\)
−0.492228 + 0.870466i \(0.663817\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 56.0000 0.284264 0.142132 − 0.989848i \(-0.454604\pi\)
0.142132 + 0.989848i \(0.454604\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −640.000 −3.12195
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 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
\(221\) 160.000 0.723982
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 442.000 1.93013 0.965066 − 0.262009i \(-0.0843849\pi\)
0.965066 + 0.262009i \(0.0843849\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −416.000 −1.78541 −0.892704 − 0.450644i \(-0.851194\pi\)
−0.892704 + 0.450644i \(0.851194\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −418.000 −1.73444 −0.867220 − 0.497925i \(-0.834095\pi\)
−0.867220 + 0.497925i \(0.834095\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 392.000 1.60000
\(246\) 0 0
\(247\) 0 0
\(248\) 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
\(256\) 0 0
\(257\) 64.0000 0.249027 0.124514 − 0.992218i \(-0.460263\pi\)
0.124514 + 0.992218i \(0.460263\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 448.000 1.69057
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −520.000 −1.93309 −0.966543 − 0.256506i \(-0.917429\pi\)
−0.966543 + 0.256506i \(0.917429\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −230.000 −0.830325 −0.415162 − 0.909747i \(-0.636275\pi\)
−0.415162 + 0.909747i \(0.636275\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −320.000 −1.13879 −0.569395 − 0.822064i \(-0.692822\pi\)
−0.569395 + 0.822064i \(0.692822\pi\)
\(282\) 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
\(289\) −33.0000 −0.114187
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −136.000 −0.464164 −0.232082 − 0.972696i \(-0.574554\pi\)
−0.232082 + 0.972696i \(0.574554\pi\)
\(294\) 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
\(305\) 176.000 0.577049
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 50.0000 0.159744 0.0798722 − 0.996805i \(-0.474549\pi\)
0.0798722 + 0.996805i \(0.474549\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −616.000 −1.94322 −0.971609 − 0.236593i \(-0.923969\pi\)
−0.971609 + 0.236593i \(0.923969\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 390.000 1.20000
\(326\) 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
\(337\) 350.000 1.03858 0.519288 − 0.854599i \(-0.326197\pi\)
0.519288 + 0.854599i \(0.326197\pi\)
\(338\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 598.000 1.71347 0.856734 − 0.515759i \(-0.172490\pi\)
0.856734 + 0.515759i \(0.172490\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 544.000 1.54108 0.770538 − 0.637394i \(-0.219988\pi\)
0.770538 + 0.637394i \(0.219988\pi\)
\(354\) 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
\(361\) 361.000 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 880.000 2.41096
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 550.000 1.47453 0.737265 − 0.675603i \(-0.236117\pi\)
0.737265 + 0.675603i \(0.236117\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −400.000 −1.06101
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 680.000 1.74807 0.874036 − 0.485861i \(-0.161494\pi\)
0.874036 + 0.485861i \(0.161494\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −650.000 −1.63728 −0.818640 − 0.574307i \(-0.805271\pi\)
−0.818640 + 0.574307i \(0.805271\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −80.0000 −0.199501 −0.0997506 − 0.995012i \(-0.531805\pi\)
−0.0997506 + 0.995012i \(0.531805\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 782.000 1.91198 0.955990 − 0.293399i \(-0.0947863\pi\)
0.955990 + 0.293399i \(0.0947863\pi\)
\(410\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 58.0000 0.137767 0.0688836 − 0.997625i \(-0.478056\pi\)
0.0688836 + 0.997625i \(0.478056\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 624.000 1.46824
\(426\) 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
\(433\) 290.000 0.669746 0.334873 − 0.942263i \(-0.391307\pi\)
0.334873 + 0.942263i \(0.391307\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 1280.00 2.87640
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −560.000 −1.24722 −0.623608 − 0.781737i \(-0.714334\pi\)
−0.623608 + 0.781737i \(0.714334\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −850.000 −1.85996 −0.929978 − 0.367615i \(-0.880174\pi\)
−0.929978 + 0.367615i \(0.880174\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −760.000 −1.64859 −0.824295 − 0.566161i \(-0.808428\pi\)
−0.824295 + 0.566161i \(0.808428\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 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
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 700.000 1.45530
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −1040.00 −2.14433
\(486\) 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
\(493\) −640.000 −1.29817
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) −320.000 −0.633663
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 440.000 0.864440 0.432220 − 0.901768i \(-0.357730\pi\)
0.432220 + 0.901768i \(0.357730\pi\)
\(510\) 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
\(521\) 880.000 1.68906 0.844530 − 0.535509i \(-0.179880\pi\)
0.844530 + 0.535509i \(0.179880\pi\)
\(522\) 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
\(529\) 529.000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −800.000 −1.50094
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 682.000 1.26063 0.630314 − 0.776340i \(-0.282926\pi\)
0.630314 + 0.776340i \(0.282926\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −1456.00 −2.67156
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1064.00 1.91023 0.955117 − 0.296230i \(-0.0957294\pi\)
0.955117 + 0.296230i \(0.0957294\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) −1792.00 −3.17168
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −1040.00 −1.82777 −0.913884 − 0.405975i \(-0.866932\pi\)
−0.913884 + 0.405975i \(0.866932\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −1150.00 −1.99307 −0.996534 − 0.0831889i \(-0.973490\pi\)
−0.996534 + 0.0831889i \(0.973490\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 736.000 1.24115 0.620573 − 0.784148i \(-0.286900\pi\)
0.620573 + 0.784148i \(0.286900\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −1102.00 −1.83361 −0.916805 − 0.399334i \(-0.869241\pi\)
−0.916805 + 0.399334i \(0.869241\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 968.000 1.60000
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 70.0000 0.114192 0.0570962 − 0.998369i \(-0.481816\pi\)
0.0570962 + 0.998369i \(0.481816\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1216.00 1.97083 0.985413 − 0.170178i \(-0.0544344\pi\)
0.985413 + 0.170178i \(0.0544344\pi\)
\(618\) 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
\(625\) −79.0000 −0.126400
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1120.00 1.78060
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 490.000 0.769231
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 400.000 0.624025 0.312012 − 0.950078i \(-0.398997\pi\)
0.312012 + 0.950078i \(0.398997\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 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
\(653\) −1144.00 −1.75191 −0.875957 − 0.482389i \(-0.839769\pi\)
−0.875957 + 0.482389i \(0.839769\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −1178.00 −1.78215 −0.891074 − 0.453858i \(-0.850047\pi\)
−0.891074 + 0.453858i \(0.850047\pi\)
\(662\) 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
\(673\) 770.000 1.14413 0.572065 − 0.820208i \(-0.306142\pi\)
0.572065 + 0.820208i \(0.306142\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 104.000 0.153619 0.0768095 − 0.997046i \(-0.475527\pi\)
0.0768095 + 0.997046i \(0.475527\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) −1408.00 −2.05547
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 560.000 0.812772
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −1280.00 −1.83644
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −520.000 −0.741797 −0.370899 − 0.928673i \(-0.620950\pi\)
−0.370899 + 0.928673i \(0.620950\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −518.000 −0.730606 −0.365303 − 0.930889i \(-0.619035\pi\)
−0.365303 + 0.930889i \(0.619035\pi\)
\(710\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1560.00 −2.15172
\(726\) 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
\(733\) 1450.00 1.97817 0.989086 − 0.147340i \(-0.0470711\pi\)
0.989086 + 0.147340i \(0.0470711\pi\)
\(734\) 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
\(745\) −2240.00 −3.00671
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −1190.00 −1.57199 −0.785997 − 0.618230i \(-0.787850\pi\)
−0.785997 + 0.618230i \(0.787850\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1520.00 −1.99737 −0.998686 − 0.0512484i \(-0.983680\pi\)
−0.998686 + 0.0512484i \(0.983680\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 962.000 1.25098 0.625488 − 0.780234i \(-0.284900\pi\)
0.625488 + 0.780234i \(0.284900\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1496.00 1.93532 0.967658 − 0.252264i \(-0.0811751\pi\)
0.967658 + 0.252264i \(0.0811751\pi\)
\(774\) 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
\(785\) −1360.00 −1.73248
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 220.000 0.277427
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1144.00 −1.43538 −0.717691 − 0.696361i \(-0.754801\pi\)
−0.717691 + 0.696361i \(0.754801\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −560.000 −0.692213 −0.346106 − 0.938195i \(-0.612496\pi\)
−0.346106 + 0.938195i \(0.612496\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 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
\(821\) 1400.00 1.70524 0.852619 − 0.522533i \(-0.175013\pi\)
0.852619 + 0.522533i \(0.175013\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 1258.00 1.51749 0.758745 − 0.651387i \(-0.225813\pi\)
0.758745 + 0.651387i \(0.225813\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 784.000 0.941176
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 759.000 0.902497
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −552.000 −0.653254
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −410.000 −0.480657 −0.240328 − 0.970692i \(-0.577255\pi\)
−0.240328 + 0.970692i \(0.577255\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −464.000 −0.541424 −0.270712 − 0.962660i \(-0.587259\pi\)
−0.270712 + 0.962660i \(0.587259\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 832.000 0.961850
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −1610.00 −1.83580 −0.917902 − 0.396807i \(-0.870118\pi\)
−0.917902 + 0.396807i \(0.870118\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1600.00 1.81612 0.908059 − 0.418842i \(-0.137564\pi\)
0.908059 + 0.418842i \(0.137564\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 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
\(901\) 896.000 0.994451
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −304.000 −0.335912
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) 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
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 2730.00 2.95135
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1840.00 1.98062 0.990312 − 0.138859i \(-0.0443435\pi\)
0.990312 + 0.138859i \(0.0443435\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −430.000 −0.458911 −0.229456 − 0.973319i \(-0.573695\pi\)
−0.229456 + 0.973319i \(0.573695\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1160.00 1.23273 0.616366 − 0.787460i \(-0.288604\pi\)
0.616366 + 0.787460i \(0.288604\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) 1100.00 1.15911
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1456.00 1.52781 0.763903 − 0.645331i \(-0.223280\pi\)
0.763903 + 0.645331i \(0.223280\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 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
\(965\) −1520.00 −1.57513
\(966\) 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
\(977\) 496.000 0.507677 0.253838 − 0.967247i \(-0.418307\pi\)
0.253838 + 0.967247i \(0.418307\pi\)
\(978\) 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
\(985\) 448.000 0.454822
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1850.00 −1.85557 −0.927783 − 0.373119i \(-0.878288\pi\)
−0.927783 + 0.373119i \(0.878288\pi\)
\(998\) 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 576.3.g.c.127.1 1
3.2 odd 2 576.3.g.a.127.1 1
4.3 odd 2 CM 576.3.g.c.127.1 1
8.3 odd 2 36.3.d.b.19.1 yes 1
8.5 even 2 36.3.d.b.19.1 yes 1
12.11 even 2 576.3.g.a.127.1 1
16.3 odd 4 2304.3.b.e.127.1 2
16.5 even 4 2304.3.b.e.127.2 2
16.11 odd 4 2304.3.b.e.127.2 2
16.13 even 4 2304.3.b.e.127.1 2
24.5 odd 2 36.3.d.a.19.1 ✓ 1
24.11 even 2 36.3.d.a.19.1 ✓ 1
40.3 even 4 900.3.f.a.199.1 2
40.13 odd 4 900.3.f.a.199.1 2
40.19 odd 2 900.3.c.b.451.1 1
40.27 even 4 900.3.f.a.199.2 2
40.29 even 2 900.3.c.b.451.1 1
40.37 odd 4 900.3.f.a.199.2 2
48.5 odd 4 2304.3.b.d.127.1 2
48.11 even 4 2304.3.b.d.127.1 2
48.29 odd 4 2304.3.b.d.127.2 2
48.35 even 4 2304.3.b.d.127.2 2
72.5 odd 6 324.3.f.f.55.1 2
72.11 even 6 324.3.f.f.271.1 2
72.13 even 6 324.3.f.e.55.1 2
72.29 odd 6 324.3.f.f.271.1 2
72.43 odd 6 324.3.f.e.271.1 2
72.59 even 6 324.3.f.f.55.1 2
72.61 even 6 324.3.f.e.271.1 2
72.67 odd 6 324.3.f.e.55.1 2
120.29 odd 2 900.3.c.c.451.1 1
120.53 even 4 900.3.f.b.199.2 2
120.59 even 2 900.3.c.c.451.1 1
120.77 even 4 900.3.f.b.199.1 2
120.83 odd 4 900.3.f.b.199.2 2
120.107 odd 4 900.3.f.b.199.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
36.3.d.a.19.1 ✓ 1 24.5 odd 2
36.3.d.a.19.1 ✓ 1 24.11 even 2
36.3.d.b.19.1 yes 1 8.3 odd 2
36.3.d.b.19.1 yes 1 8.5 even 2
324.3.f.e.55.1 2 72.13 even 6
324.3.f.e.55.1 2 72.67 odd 6
324.3.f.e.271.1 2 72.43 odd 6
324.3.f.e.271.1 2 72.61 even 6
324.3.f.f.55.1 2 72.5 odd 6
324.3.f.f.55.1 2 72.59 even 6
324.3.f.f.271.1 2 72.11 even 6
324.3.f.f.271.1 2 72.29 odd 6
576.3.g.a.127.1 1 3.2 odd 2
576.3.g.a.127.1 1 12.11 even 2
576.3.g.c.127.1 1 1.1 even 1 trivial
576.3.g.c.127.1 1 4.3 odd 2 CM
900.3.c.b.451.1 1 40.19 odd 2
900.3.c.b.451.1 1 40.29 even 2
900.3.c.c.451.1 1 120.29 odd 2
900.3.c.c.451.1 1 120.59 even 2
900.3.f.a.199.1 2 40.3 even 4
900.3.f.a.199.1 2 40.13 odd 4
900.3.f.a.199.2 2 40.27 even 4
900.3.f.a.199.2 2 40.37 odd 4
900.3.f.b.199.1 2 120.77 even 4
900.3.f.b.199.1 2 120.107 odd 4
900.3.f.b.199.2 2 120.53 even 4
900.3.f.b.199.2 2 120.83 odd 4
2304.3.b.d.127.1 2 48.5 odd 4
2304.3.b.d.127.1 2 48.11 even 4
2304.3.b.d.127.2 2 48.29 odd 4
2304.3.b.d.127.2 2 48.35 even 4
2304.3.b.e.127.1 2 16.3 odd 4
2304.3.b.e.127.1 2 16.13 even 4
2304.3.b.e.127.2 2 16.5 even 4
2304.3.b.e.127.2 2 16.11 odd 4