Properties

Label 576.3.e.d.449.2
Level $576$
Weight $3$
Character 576.449
Analytic conductor $15.695$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [576,3,Mod(449,576)]
 
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([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("576.449");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 576.e (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: no
Analytic conductor: \(15.6948632272\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 288)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 449.2
Root \(1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 576.449
Dual form 576.3.e.d.449.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.41421i q^{5} +O(q^{10})\) \(q+1.41421i q^{5} -24.0000 q^{13} -32.5269i q^{17} +23.0000 q^{25} -1.41421i q^{29} -70.0000 q^{37} -69.2965i q^{41} -49.0000 q^{49} -103.238i q^{53} +22.0000 q^{61} -33.9411i q^{65} -96.0000 q^{73} +46.0000 q^{85} +168.291i q^{89} -144.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 48 q^{13} + 46 q^{25} - 140 q^{37} - 98 q^{49} + 44 q^{61} - 192 q^{73} + 92 q^{85} - 288 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) 1.41421i 0.282843i 0.989949 + 0.141421i \(0.0451672\pi\)
−0.989949 + 0.141421i \(0.954833\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −24.0000 −1.84615 −0.923077 0.384615i \(-0.874334\pi\)
−0.923077 + 0.384615i \(0.874334\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 32.5269i − 1.91335i −0.291162 0.956674i \(-0.594042\pi\)
0.291162 0.956674i \(-0.405958\pi\)
\(18\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 23.0000 0.920000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 1.41421i − 0.0487660i −0.999703 0.0243830i \(-0.992238\pi\)
0.999703 0.0243830i \(-0.00776212\pi\)
\(30\) 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
\(37\) −70.0000 −1.89189 −0.945946 0.324324i \(-0.894863\pi\)
−0.945946 + 0.324324i \(0.894863\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 69.2965i − 1.69016i −0.534642 0.845079i \(-0.679553\pi\)
0.534642 0.845079i \(-0.320447\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 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
\(53\) − 103.238i − 1.94788i −0.226808 0.973940i \(-0.572829\pi\)
0.226808 0.973940i \(-0.427171\pi\)
\(54\) 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
\(65\) − 33.9411i − 0.522171i
\(66\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −96.0000 −1.31507 −0.657534 0.753425i \(-0.728401\pi\)
−0.657534 + 0.753425i \(0.728401\pi\)
\(74\) 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
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 46.0000 0.541176
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 168.291i 1.89091i 0.325746 + 0.945457i \(0.394385\pi\)
−0.325746 + 0.945457i \(0.605615\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −144.000 −1.48454 −0.742268 0.670103i \(-0.766250\pi\)
−0.742268 + 0.670103i \(0.766250\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 168.291i − 1.66625i −0.553084 0.833126i \(-0.686549\pi\)
0.553084 0.833126i \(-0.313451\pi\)
\(102\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 120.000 1.10092 0.550459 0.834862i \(-0.314453\pi\)
0.550459 + 0.834862i \(0.314453\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 137.179i − 1.21397i −0.794713 0.606985i \(-0.792379\pi\)
0.794713 0.606985i \(-0.207621\pi\)
\(114\) 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
\(125\) 67.8823i 0.543058i
\(126\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 272.943i 1.99229i 0.0877432 + 0.996143i \(0.472035\pi\)
−0.0877432 + 0.996143i \(0.527965\pi\)
\(138\) 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
\(145\) 2.00000 0.0137931
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 270.115i 1.81285i 0.422366 + 0.906425i \(0.361200\pi\)
−0.422366 + 0.906425i \(0.638800\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 170.000 1.08280 0.541401 0.840764i \(-0.317894\pi\)
0.541401 + 0.840764i \(0.317894\pi\)
\(158\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 407.000 2.40828
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 306.884i 1.77390i 0.461867 + 0.886949i \(0.347180\pi\)
−0.461867 + 0.886949i \(0.652820\pi\)
\(174\) 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
\(181\) −360.000 −1.98895 −0.994475 0.104972i \(-0.966525\pi\)
−0.994475 + 0.104972i \(0.966525\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 98.9949i − 0.535108i
\(186\) 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 \(-0.336183\pi\)
0.492228 + 0.870466i \(0.336183\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 236.174i − 1.19885i −0.800431 0.599426i \(-0.795396\pi\)
0.800431 0.599426i \(-0.204604\pi\)
\(198\) 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
\(205\) 98.0000 0.478049
\(206\) 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
\(221\) 780.646i 3.53233i
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) −120.000 −0.524017 −0.262009 0.965066i \(-0.584385\pi\)
−0.262009 + 0.965066i \(0.584385\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 442.649i 1.89978i 0.312584 + 0.949890i \(0.398806\pi\)
−0.312584 + 0.949890i \(0.601194\pi\)
\(234\) 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
\(241\) 240.000 0.995851 0.497925 0.867220i \(-0.334095\pi\)
0.497925 + 0.867220i \(0.334095\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 69.2965i − 0.282843i
\(246\) 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
\(257\) − 405.879i − 1.57930i −0.613560 0.789648i \(-0.710263\pi\)
0.613560 0.789648i \(-0.289737\pi\)
\(258\) 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
\(265\) 146.000 0.550943
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 270.115i − 1.00414i −0.864826 0.502072i \(-0.832571\pi\)
0.864826 0.502072i \(-0.167429\pi\)
\(270\) 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
\(277\) 504.000 1.81949 0.909747 0.415162i \(-0.136275\pi\)
0.909747 + 0.415162i \(0.136275\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 100.409i 0.357328i 0.983910 + 0.178664i \(0.0571775\pi\)
−0.983910 + 0.178664i \(0.942822\pi\)
\(282\) 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
\(289\) −769.000 −2.66090
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 306.884i − 1.04739i −0.851907 0.523693i \(-0.824554\pi\)
0.851907 0.523693i \(-0.175446\pi\)
\(294\) 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
\(305\) 31.1127i 0.102009i
\(306\) 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 1.00000 \(0\)
−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
\(317\) − 541.644i − 1.70866i −0.519735 0.854328i \(-0.673969\pi\)
0.519735 0.854328i \(-0.326031\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −552.000 −1.69846
\(326\) 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
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −576.000 −1.70920 −0.854599 0.519288i \(-0.826197\pi\)
−0.854599 + 0.519288i \(0.826197\pi\)
\(338\) 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
\(353\) 66.4680i 0.188295i 0.995558 + 0.0941474i \(0.0300125\pi\)
−0.995558 + 0.0941474i \(0.969988\pi\)
\(354\) 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
\(365\) − 135.765i − 0.371958i
\(366\) 0 0
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 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
\(377\) 33.9411i 0.0900295i
\(378\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 748.119i − 1.92319i −0.274481 0.961593i \(-0.588506\pi\)
0.274481 0.961593i \(-0.411494\pi\)
\(390\) 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
\(401\) − 507.703i − 1.26609i −0.774114 0.633046i \(-0.781805\pi\)
0.774114 0.633046i \(-0.218195\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −240.000 −0.586797 −0.293399 0.955990i \(-0.594786\pi\)
−0.293399 + 0.955990i \(0.594786\pi\)
\(410\) 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
\(421\) −840.000 −1.99525 −0.997625 0.0688836i \(-0.978056\pi\)
−0.997625 + 0.0688836i \(0.978056\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 748.119i − 1.76028i
\(426\) 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 \(-0.608693\pi\)
−0.334873 + 0.942263i \(0.608693\pi\)
\(434\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) −238.000 −0.534831
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 100.409i 0.223628i 0.993729 + 0.111814i \(0.0356662\pi\)
−0.993729 + 0.111814i \(0.964334\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −336.000 −0.735230 −0.367615 0.929978i \(-0.619826\pi\)
−0.367615 + 0.929978i \(0.619826\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 168.291i 0.365057i 0.983201 + 0.182529i \(0.0584282\pi\)
−0.983201 + 0.182529i \(0.941572\pi\)
\(462\) 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 1.00000 \(0\)
−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
\(481\) 1680.00 3.49272
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 203.647i − 0.419890i
\(486\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) −46.0000 −0.0933063
\(494\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 238.000 0.471287
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 337.997i 0.664041i 0.943272 + 0.332021i \(0.107730\pi\)
−0.943272 + 0.332021i \(0.892270\pi\)
\(510\) 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
\(521\) − 1016.82i − 1.95167i −0.218511 0.975835i \(-0.570120\pi\)
0.218511 0.975835i \(-0.429880\pi\)
\(522\) 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
\(529\) 529.000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 1663.12i 3.12029i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −840.000 −1.55268 −0.776340 0.630314i \(-0.782926\pi\)
−0.776340 + 0.630314i \(0.782926\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 169.706i 0.311386i
\(546\) 0 0
\(547\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 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
\(557\) − 985.707i − 1.76967i −0.465903 0.884836i \(-0.654271\pi\)
0.465903 0.884836i \(-0.345729\pi\)
\(558\) 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
\(565\) 194.000 0.343363
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 408.708i 0.718291i 0.933282 + 0.359146i \(0.116932\pi\)
−0.933282 + 0.359146i \(0.883068\pi\)
\(570\) 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
\(577\) −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
\(593\) 137.179i 0.231330i 0.993288 + 0.115665i \(0.0368999\pi\)
−0.993288 + 0.115665i \(0.963100\pi\)
\(594\) 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 \(-0.130759\pi\)
0.916805 + 0.399334i \(0.130759\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 171.120i 0.282843i
\(606\) 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
\(613\) 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
\(617\) 711.349i 1.15292i 0.817127 + 0.576458i \(0.195566\pi\)
−0.817127 + 0.576458i \(0.804434\pi\)
\(618\) 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
\(625\) 479.000 0.766400
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 2276.88i 3.61985i
\(630\) 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
\(637\) 1176.00 1.84615
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 578.413i 0.902361i 0.892433 + 0.451180i \(0.148997\pi\)
−0.892433 + 0.451180i \(0.851003\pi\)
\(642\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 1254.41i − 1.92099i −0.278295 0.960496i \(-0.589769\pi\)
0.278295 0.960496i \(-0.410231\pi\)
\(654\) 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 \(-0.149953\pi\)
0.891074 + 0.453858i \(0.149953\pi\)
\(662\) 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
\(677\) − 881.055i − 1.30141i −0.759330 0.650705i \(-0.774473\pi\)
0.759330 0.650705i \(-0.225527\pi\)
\(678\) 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
\(685\) −386.000 −0.563504
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 2477.70i 3.59608i
\(690\) 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
\(697\) −2254.00 −3.23386
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1288.35i 1.83787i 0.394406 + 0.918936i \(0.370950\pi\)
−0.394406 + 0.918936i \(0.629050\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 1320.00 1.86178 0.930889 0.365303i \(-0.119035\pi\)
0.930889 + 0.365303i \(0.119035\pi\)
\(710\) 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
\(725\) − 32.5269i − 0.0448647i
\(726\) 0 0
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −216.000 −0.294679 −0.147340 0.989086i \(-0.547071\pi\)
−0.147340 + 0.989086i \(0.547071\pi\)
\(734\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) −382.000 −0.512752
\(746\) 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
\(757\) 936.000 1.23646 0.618230 0.785997i \(-0.287850\pi\)
0.618230 + 0.785997i \(0.287850\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 1019.65i − 1.33988i −0.742416 0.669940i \(-0.766320\pi\)
0.742416 0.669940i \(-0.233680\pi\)
\(762\) 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
\(773\) − 782.060i − 1.01172i −0.862615 0.505860i \(-0.831175\pi\)
0.862615 0.505860i \(-0.168825\pi\)
\(774\) 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
\(785\) 240.416i 0.306263i
\(786\) 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
\(793\) −528.000 −0.665826
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 1593.82i − 1.99977i −0.0150826 0.999886i \(-0.504801\pi\)
0.0150826 0.999886i \(-0.495199\pi\)
\(798\) 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
\(809\) 677.408i 0.837340i 0.908138 + 0.418670i \(0.137504\pi\)
−0.908138 + 0.418670i \(0.862496\pi\)
\(810\) 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
\(821\) 1596.65i 1.94476i 0.233406 + 0.972379i \(0.425013\pi\)
−0.233406 + 0.972379i \(0.574987\pi\)
\(822\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 1080.00 1.30277 0.651387 0.758745i \(-0.274187\pi\)
0.651387 + 0.758745i \(0.274187\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1593.82i 1.91335i
\(834\) 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
\(841\) 839.000 0.997622
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 575.585i 0.681166i
\(846\) 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 \(-0.422745\pi\)
0.240328 + 0.970692i \(0.422745\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 1494.82i − 1.74425i −0.489282 0.872126i \(-0.662741\pi\)
0.489282 0.872126i \(-0.337259\pi\)
\(858\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) −434.000 −0.501734
\(866\) 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
\(881\) − 609.526i − 0.691857i −0.938261 0.345929i \(-0.887564\pi\)
0.938261 0.345929i \(-0.112436\pi\)
\(882\) 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 1.00000 \(0\)
−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
\(901\) −3358.00 −3.72697
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 509.117i − 0.562560i
\(906\) 0 0
\(907\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −1610.00 −1.74054
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 1118.64i − 1.20414i −0.798445 0.602068i \(-0.794343\pi\)
0.798445 0.602068i \(-0.205657\pi\)
\(930\) 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 \(-0.426305\pi\)
0.229456 + 0.973319i \(0.426305\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 1868.18i − 1.98531i −0.120982 0.992655i \(-0.538604\pi\)
0.120982 0.992655i \(-0.461396\pi\)
\(942\) 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
\(949\) 2304.00 2.42782
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 1899.29i − 1.99296i −0.0838437 0.996479i \(-0.526720\pi\)
0.0838437 0.996479i \(-0.473280\pi\)
\(954\) 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
\(965\) 268.701i 0.278446i
\(966\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 985.707i − 1.00891i −0.863437 0.504456i \(-0.831693\pi\)
0.863437 0.504456i \(-0.168307\pi\)
\(978\) 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
\(985\) 334.000 0.339086
\(986\) 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
\(997\) −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.e.d.449.2 2
3.2 odd 2 inner 576.3.e.d.449.1 2
4.3 odd 2 CM 576.3.e.d.449.2 2
8.3 odd 2 288.3.e.c.161.1 2
8.5 even 2 288.3.e.c.161.1 2
12.11 even 2 inner 576.3.e.d.449.1 2
16.3 odd 4 2304.3.h.d.2177.3 4
16.5 even 4 2304.3.h.d.2177.2 4
16.11 odd 4 2304.3.h.d.2177.2 4
16.13 even 4 2304.3.h.d.2177.3 4
24.5 odd 2 288.3.e.c.161.2 yes 2
24.11 even 2 288.3.e.c.161.2 yes 2
48.5 odd 4 2304.3.h.d.2177.4 4
48.11 even 4 2304.3.h.d.2177.4 4
48.29 odd 4 2304.3.h.d.2177.1 4
48.35 even 4 2304.3.h.d.2177.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
288.3.e.c.161.1 2 8.3 odd 2
288.3.e.c.161.1 2 8.5 even 2
288.3.e.c.161.2 yes 2 24.5 odd 2
288.3.e.c.161.2 yes 2 24.11 even 2
576.3.e.d.449.1 2 3.2 odd 2 inner
576.3.e.d.449.1 2 12.11 even 2 inner
576.3.e.d.449.2 2 1.1 even 1 trivial
576.3.e.d.449.2 2 4.3 odd 2 CM
2304.3.h.d.2177.1 4 48.29 odd 4
2304.3.h.d.2177.1 4 48.35 even 4
2304.3.h.d.2177.2 4 16.5 even 4
2304.3.h.d.2177.2 4 16.11 odd 4
2304.3.h.d.2177.3 4 16.3 odd 4
2304.3.h.d.2177.3 4 16.13 even 4
2304.3.h.d.2177.4 4 48.5 odd 4
2304.3.h.d.2177.4 4 48.11 even 4