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Properties

Label 2304.3.g.a.1279.1
Level $2304$
Weight $3$
Character 2304.1279
Self dual yes
Analytic conductor $62.779$
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] = [2304,3,Mod(1279,2304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2304, 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("2304.1279");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2304.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: \(62.7794529086\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 128)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 1279.1
Character \(\chi\) \(=\) 2304.1279

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.00000 q^{5} +O(q^{10})\) \(q-8.00000 q^{5} -24.0000 q^{13} -30.0000 q^{17} +39.0000 q^{25} -40.0000 q^{29} -24.0000 q^{37} +18.0000 q^{41} +49.0000 q^{49} +56.0000 q^{53} -120.000 q^{61} +192.000 q^{65} -110.000 q^{73} +240.000 q^{85} +78.0000 q^{89} -130.000 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(-0.795167\pi\)
−0.800000 + 0.600000i \(0.795167\pi\)
\(6\) 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
\(13\) −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
\(17\) −30.0000 −1.76471 −0.882353 − 0.470588i \(-0.844042\pi\)
−0.882353 + 0.470588i \(0.844042\pi\)
\(18\) 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
\(37\) −24.0000 −0.648649 −0.324324 − 0.945946i \(-0.605137\pi\)
−0.324324 + 0.945946i \(0.605137\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 18.0000 0.439024 0.219512 − 0.975610i \(-0.429553\pi\)
0.219512 + 0.975610i \(0.429553\pi\)
\(42\) 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
\(61\) −120.000 −1.96721 −0.983607 − 0.180328i \(-0.942284\pi\)
−0.983607 + 0.180328i \(0.942284\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 192.000 2.95385
\(66\) 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 \(-0.771599\pi\)
−0.753425 + 0.657534i \(0.771599\pi\)
\(74\) 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
\(85\) 240.000 2.82353
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 78.0000 0.876404 0.438202 − 0.898876i \(-0.355615\pi\)
0.438202 + 0.898876i \(0.355615\pi\)
\(90\) 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
\(109\) −120.000 −1.10092 −0.550459 − 0.834862i \(-0.685547\pi\)
−0.550459 + 0.834862i \(0.685547\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 30.0000 0.265487 0.132743 − 0.991150i \(-0.457621\pi\)
0.132743 + 0.991150i \(0.457621\pi\)
\(114\) 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
\(137\) 210.000 1.53285 0.766423 − 0.642336i \(-0.222035\pi\)
0.766423 + 0.642336i \(0.222035\pi\)
\(138\) 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 \(-0.111200\pi\)
0.939597 + 0.342282i \(0.111200\pi\)
\(150\) 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
\(157\) 264.000 1.68153 0.840764 − 0.541401i \(-0.182106\pi\)
0.840764 + 0.541401i \(0.182106\pi\)
\(158\) 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
\(169\) 407.000 2.40828
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −104.000 −0.601156 −0.300578 − 0.953757i \(-0.597180\pi\)
−0.300578 + 0.953757i \(0.597180\pi\)
\(174\) 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 \(-0.0334754\pi\)
0.994475 + 0.104972i \(0.0334754\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 192.000 1.03784
\(186\) 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
\(197\) 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
\(205\) −144.000 −0.702439
\(206\) 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
\(221\) 720.000 3.25792
\(222\) 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
\(229\) −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
\(233\) −210.000 −0.901288 −0.450644 − 0.892704i \(-0.648806\pi\)
−0.450644 + 0.892704i \(0.648806\pi\)
\(234\) 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
\(257\) 510.000 1.98444 0.992218 − 0.124514i \(-0.0397370\pi\)
0.992218 + 0.124514i \(0.0397370\pi\)
\(258\) 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
\(277\) −504.000 −1.81949 −0.909747 − 0.415162i \(-0.863725\pi\)
−0.909747 + 0.415162i \(0.863725\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 462.000 1.64413 0.822064 − 0.569395i \(-0.192822\pi\)
0.822064 + 0.569395i \(0.192822\pi\)
\(282\) 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
\(289\) 611.000 2.11419
\(290\) 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
\(305\) 960.000 3.14754
\(306\) 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 \(-0.525451\pi\)
−0.0798722 + 0.996805i \(0.525451\pi\)
\(314\) 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
\(325\) −936.000 −2.88000
\(326\) 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 \(-0.673803\pi\)
−0.519288 + 0.854599i \(0.673803\pi\)
\(338\) 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
\(349\) 360.000 1.03152 0.515759 − 0.856734i \(-0.327510\pi\)
0.515759 + 0.856734i \(0.327510\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −450.000 −1.27479 −0.637394 − 0.770538i \(-0.719988\pi\)
−0.637394 + 0.770538i \(0.719988\pi\)
\(354\) 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
\(373\) −504.000 −1.35121 −0.675603 − 0.737265i \(-0.736117\pi\)
−0.675603 + 0.737265i \(0.736117\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 960.000 2.54642
\(378\) 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 \(-0.838506\pi\)
−0.874036 + 0.485861i \(0.838506\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 456.000 1.14861 0.574307 − 0.818640i \(-0.305271\pi\)
0.574307 + 0.818640i \(0.305271\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −798.000 −1.99002 −0.995012 − 0.0997506i \(-0.968195\pi\)
−0.995012 + 0.0997506i \(0.968195\pi\)
\(402\) 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
\(421\) 840.000 1.99525 0.997625 − 0.0688836i \(-0.0219437\pi\)
0.997625 + 0.0688836i \(0.0219437\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1170.00 −2.75294
\(426\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) −624.000 −1.40225
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −702.000 −1.56347 −0.781737 − 0.623608i \(-0.785666\pi\)
−0.781737 + 0.623608i \(0.785666\pi\)
\(450\) 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 \(-0.191572\pi\)
0.824295 + 0.566161i \(0.191572\pi\)
\(462\) 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
\(481\) 576.000 1.19751
\(482\) 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
\(493\) 1200.00 2.43408
\(494\) 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
\(521\) −558.000 −1.07102 −0.535509 − 0.844530i \(-0.679880\pi\)
−0.535509 + 0.844530i \(0.679880\pi\)
\(522\) 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
\(533\) −432.000 −0.810507
\(534\) 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 \(-0.217074\pi\)
0.776340 + 0.630314i \(0.217074\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 960.000 1.76147
\(546\) 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 \(-0.904271\pi\)
−0.955117 + 0.296230i \(0.904271\pi\)
\(558\) 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
\(565\) −240.000 −0.424779
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −462.000 −0.811951 −0.405975 − 0.913884i \(-0.633068\pi\)
−0.405975 + 0.913884i \(0.633068\pi\)
\(570\) 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
\(593\) −930.000 −1.56830 −0.784148 − 0.620573i \(-0.786900\pi\)
−0.784148 + 0.620573i \(0.786900\pi\)
\(594\) 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
\(613\) 1224.00 1.99674 0.998369 − 0.0570962i \(-0.0181842\pi\)
0.998369 + 0.0570962i \(0.0181842\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −210.000 −0.340357 −0.170178 − 0.985413i \(-0.554434\pi\)
−0.170178 + 0.985413i \(0.554434\pi\)
\(618\) 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
\(629\) 720.000 1.14467
\(630\) 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
\(637\) −1176.00 −1.84615
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1218.00 1.90016 0.950078 − 0.312012i \(-0.101003\pi\)
0.950078 + 0.312012i \(0.101003\pi\)
\(642\) 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 \(-0.160231\pi\)
0.875957 + 0.482389i \(0.160231\pi\)
\(654\) 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
\(661\) −600.000 −0.907716 −0.453858 − 0.891074i \(-0.649953\pi\)
−0.453858 + 0.891074i \(0.649953\pi\)
\(662\) 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 \(-0.693858\pi\)
−0.572065 + 0.820208i \(0.693858\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −104.000 −0.153619 −0.0768095 − 0.997046i \(-0.524473\pi\)
−0.0768095 + 0.997046i \(0.524473\pi\)
\(678\) 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
\(685\) −1680.00 −2.45255
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −1344.00 −1.95065
\(690\) 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
\(697\) −540.000 −0.774749
\(698\) 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
\(709\) 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
\(725\) −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
\(733\) −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 1.00000 \(0\)
−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
\(757\) 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
\(761\) −78.0000 −0.102497 −0.0512484 − 0.998686i \(-0.516320\pi\)
−0.0512484 + 0.998686i \(0.516320\pi\)
\(762\) 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 \(-0.715100\pi\)
−0.625488 + 0.780234i \(0.715100\pi\)
\(770\) 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
\(785\) −2112.00 −2.69045
\(786\) 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
\(793\) 2880.00 3.63178
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1144.00 1.43538 0.717691 − 0.696361i \(-0.245199\pi\)
0.717691 + 0.696361i \(0.245199\pi\)
\(798\) 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
\(809\) −1518.00 −1.87639 −0.938195 − 0.346106i \(-0.887504\pi\)
−0.938195 + 0.346106i \(0.887504\pi\)
\(810\) 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
\(829\) −1080.00 −1.30277 −0.651387 − 0.758745i \(-0.725813\pi\)
−0.651387 + 0.758745i \(0.725813\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −1470.00 −1.76471
\(834\) 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
\(845\) −3256.00 −3.85325
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −1656.00 −1.94138 −0.970692 − 0.240328i \(-0.922745\pi\)
−0.970692 + 0.240328i \(0.922745\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1650.00 1.92532 0.962660 − 0.270712i \(-0.0872590\pi\)
0.962660 + 0.270712i \(0.0872590\pi\)
\(858\) 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
\(877\) −696.000 −0.793615 −0.396807 − 0.917902i \(-0.629882\pi\)
−0.396807 + 0.917902i \(0.629882\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −738.000 −0.837684 −0.418842 − 0.908059i \(-0.637564\pi\)
−0.418842 + 0.908059i \(0.637564\pi\)
\(882\) 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
\(901\) −1680.00 −1.86459
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −2880.00 −3.18232
\(906\) 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
\(925\) −936.000 −1.01189
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 258.000 0.277718 0.138859 − 0.990312i \(-0.455657\pi\)
0.138859 + 0.990312i \(0.455657\pi\)
\(930\) 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 \(-0.711396\pi\)
−0.616366 + 0.787460i \(0.711396\pi\)
\(942\) 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
\(949\) 2640.00 2.78188
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1230.00 −1.29066 −0.645331 − 0.763903i \(-0.723280\pi\)
−0.645331 + 0.763903i \(0.723280\pi\)
\(954\) 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
\(977\) 1890.00 1.93449 0.967247 − 0.253838i \(-0.0816931\pi\)
0.967247 + 0.253838i \(0.0816931\pi\)
\(978\) 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
\(997\) 744.000 0.746239 0.373119 − 0.927783i \(-0.378288\pi\)
0.373119 + 0.927783i \(0.378288\pi\)
\(998\) 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 2304.3.g.a.1279.1 1
3.2 odd 2 256.3.c.b.255.1 1
4.3 odd 2 CM 2304.3.g.a.1279.1 1
8.3 odd 2 2304.3.g.f.1279.1 1
8.5 even 2 2304.3.g.f.1279.1 1
12.11 even 2 256.3.c.b.255.1 1
16.3 odd 4 1152.3.b.a.703.2 2
16.5 even 4 1152.3.b.a.703.1 2
16.11 odd 4 1152.3.b.a.703.1 2
16.13 even 4 1152.3.b.a.703.2 2
24.5 odd 2 256.3.c.a.255.1 1
24.11 even 2 256.3.c.a.255.1 1
48.5 odd 4 128.3.d.a.63.2 yes 2
48.11 even 4 128.3.d.a.63.2 yes 2
48.29 odd 4 128.3.d.a.63.1 ✓ 2
48.35 even 4 128.3.d.a.63.1 ✓ 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.3.d.a.63.1 ✓ 2 48.29 odd 4
128.3.d.a.63.1 ✓ 2 48.35 even 4
128.3.d.a.63.2 yes 2 48.5 odd 4
128.3.d.a.63.2 yes 2 48.11 even 4
256.3.c.a.255.1 1 24.5 odd 2
256.3.c.a.255.1 1 24.11 even 2
256.3.c.b.255.1 1 3.2 odd 2
256.3.c.b.255.1 1 12.11 even 2
1152.3.b.a.703.1 2 16.5 even 4
1152.3.b.a.703.1 2 16.11 odd 4
1152.3.b.a.703.2 2 16.3 odd 4
1152.3.b.a.703.2 2 16.13 even 4
2304.3.g.a.1279.1 1 1.1 even 1 trivial
2304.3.g.a.1279.1 1 4.3 odd 2 CM
2304.3.g.f.1279.1 1 8.3 odd 2
2304.3.g.f.1279.1 1 8.5 even 2