Properties

Label 2304.3.b.d.127.1
Level $2304$
Weight $3$
Character 2304.127
Analytic conductor $62.779$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2304.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(62.7794529086\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 127.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2304.127
Dual form 2304.3.b.d.127.2

$q$-expansion

\(f(q)\) \(=\) \(q-8.00000i q^{5} +O(q^{10})\) \(q-8.00000i q^{5} -10.0000i q^{13} -16.0000 q^{17} -39.0000 q^{25} -40.0000i q^{29} +70.0000i q^{37} -80.0000 q^{41} +49.0000 q^{49} -56.0000i q^{53} -22.0000i q^{61} -80.0000 q^{65} -110.000 q^{73} +128.000i q^{85} +160.000 q^{89} -130.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 32q^{17} - 78q^{25} - 160q^{41} + 98q^{49} - 160q^{65} - 220q^{73} + 320q^{89} - 260q^{97} + O(q^{100}) \)

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
\(5\) − 8.00000i − 1.60000i −0.600000 0.800000i \(-0.704833\pi\)
0.600000 0.800000i \(-0.295167\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) − 10.0000i − 0.769231i −0.923077 0.384615i \(-0.874334\pi\)
0.923077 0.384615i \(-0.125666\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −16.0000 −0.941176 −0.470588 0.882353i \(-0.655958\pi\)
−0.470588 + 0.882353i \(0.655958\pi\)
\(18\) 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
\(25\) −39.0000 −1.56000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 40.0000i − 1.37931i −0.724138 0.689655i \(-0.757762\pi\)
0.724138 0.689655i \(-0.242238\pi\)
\(30\) 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
\(37\) 70.0000i 1.89189i 0.324324 + 0.945946i \(0.394863\pi\)
−0.324324 + 0.945946i \(0.605137\pi\)
\(38\) 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 1.00000i \(-0.5\pi\)
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
\(53\) − 56.0000i − 1.05660i −0.849057 0.528302i \(-0.822829\pi\)
0.849057 0.528302i \(-0.177171\pi\)
\(54\) 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
\(61\) − 22.0000i − 0.360656i −0.983607 0.180328i \(-0.942284\pi\)
0.983607 0.180328i \(-0.0577159\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −80.0000 −1.23077
\(66\) 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
\(73\) −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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 128.000i 1.50588i
\(86\) 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
\(101\) 40.0000i 0.396040i 0.980198 + 0.198020i \(0.0634510\pi\)
−0.980198 + 0.198020i \(0.936549\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 182.000i 1.66972i 0.550459 + 0.834862i \(0.314453\pi\)
−0.550459 + 0.834862i \(0.685547\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 224.000 1.98230 0.991150 0.132743i \(-0.0423786\pi\)
0.991150 + 0.132743i \(0.0423786\pi\)
\(114\) 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
\(125\) 112.000i 0.896000i
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −176.000 −1.28467 −0.642336 0.766423i \(-0.722035\pi\)
−0.642336 + 0.766423i \(0.722035\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −320.000 −2.20690
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 280.000i 1.87919i 0.342282 + 0.939597i \(0.388800\pi\)
−0.342282 + 0.939597i \(0.611200\pi\)
\(150\) 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
\(157\) 170.000i 1.08280i 0.840764 + 0.541401i \(0.182106\pi\)
−0.840764 + 0.541401i \(0.817894\pi\)
\(158\) 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
\(169\) 69.0000 0.408284
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 104.000i 0.601156i 0.953757 + 0.300578i \(0.0971796\pi\)
−0.953757 + 0.300578i \(0.902820\pi\)
\(174\) 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
\(181\) − 38.0000i − 0.209945i −0.994475 0.104972i \(-0.966525\pi\)
0.994475 0.104972i \(-0.0334754\pi\)
\(182\) 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
\(197\) − 56.0000i − 0.284264i −0.989848 0.142132i \(-0.954604\pi\)
0.989848 0.142132i \(-0.0453957\pi\)
\(198\) 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
\(205\) 640.000i 3.12195i
\(206\) 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
\(221\) 160.000i 0.723982i
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 442.000i 1.93013i 0.262009 + 0.965066i \(0.415615\pi\)
−0.262009 + 0.965066i \(0.584385\pi\)
\(230\) 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
\(245\) − 392.000i − 1.60000i
\(246\) 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
\(257\) −64.0000 −0.249027 −0.124514 0.992218i \(-0.539737\pi\)
−0.124514 + 0.992218i \(0.539737\pi\)
\(258\) 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
\(269\) − 520.000i − 1.93309i −0.256506 0.966543i \(-0.582571\pi\)
0.256506 0.966543i \(-0.417429\pi\)
\(270\) 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
\(277\) − 230.000i − 0.830325i −0.909747 0.415162i \(-0.863725\pi\)
0.909747 0.415162i \(-0.136275\pi\)
\(278\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −33.0000 −0.114187
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 136.000i 0.464164i 0.972696 + 0.232082i \(0.0745537\pi\)
−0.972696 + 0.232082i \(0.925446\pi\)
\(294\) 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 1.00000i \(-0.5\pi\)
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 \(-0.525451\pi\)
−0.0798722 + 0.996805i \(0.525451\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 616.000i − 1.94322i −0.236593 0.971609i \(-0.576031\pi\)
0.236593 0.971609i \(-0.423969\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 390.000i 1.20000i
\(326\) 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
\(337\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) − 598.000i − 1.71347i −0.515759 0.856734i \(-0.672490\pi\)
0.515759 0.856734i \(-0.327510\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −544.000 −1.54108 −0.770538 0.637394i \(-0.780012\pi\)
−0.770538 + 0.637394i \(0.780012\pi\)
\(354\) 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
\(365\) 880.000i 2.41096i
\(366\) 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
\(373\) 550.000i 1.47453i 0.675603 + 0.737265i \(0.263883\pi\)
−0.675603 + 0.737265i \(0.736117\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −400.000 −1.06101
\(378\) 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
\(389\) − 680.000i − 1.74807i −0.485861 0.874036i \(-0.661494\pi\)
0.485861 0.874036i \(-0.338506\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 650.000i 1.63728i 0.574307 + 0.818640i \(0.305271\pi\)
−0.574307 + 0.818640i \(0.694729\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 80.0000 0.199501 0.0997506 0.995012i \(-0.468195\pi\)
0.0997506 + 0.995012i \(0.468195\pi\)
\(402\) 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 \(-0.905214\pi\)
−0.955990 + 0.293399i \(0.905214\pi\)
\(410\) 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
\(421\) 58.0000i 0.137767i 0.997625 + 0.0688836i \(0.0219437\pi\)
−0.997625 + 0.0688836i \(0.978056\pi\)
\(422\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) − 1280.00i − 2.87640i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 560.000 1.24722 0.623608 0.781737i \(-0.285666\pi\)
0.623608 + 0.781737i \(0.285666\pi\)
\(450\) 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 \(-0.119826\pi\)
0.929978 + 0.367615i \(0.119826\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 760.000i − 1.64859i −0.566161 0.824295i \(-0.691572\pi\)
0.566161 0.824295i \(-0.308428\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
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
\(477\) 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
\(485\) 1040.00i 2.14433i
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 640.000i 1.29817i
\(494\) 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
\(505\) 320.000 0.633663
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 440.000i 0.864440i 0.901768 + 0.432220i \(0.142270\pi\)
−0.901768 + 0.432220i \(0.857730\pi\)
\(510\) 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 1.00000i \(-0.5\pi\)
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
\(533\) 800.000i 1.50094i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) − 682.000i − 1.26063i −0.776340 0.630314i \(-0.782926\pi\)
0.776340 0.630314i \(-0.217074\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1456.00 2.67156
\(546\) 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
\(557\) 1064.00i 1.91023i 0.296230 + 0.955117i \(0.404271\pi\)
−0.296230 + 0.955117i \(0.595729\pi\)
\(558\) 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
\(565\) − 1792.00i − 3.17168i
\(566\) 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 1.00000i \(-0.5\pi\)
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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −736.000 −1.24115 −0.620573 0.784148i \(-0.713100\pi\)
−0.620573 + 0.784148i \(0.713100\pi\)
\(594\) 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
\(605\) 968.000i 1.60000i
\(606\) 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
\(613\) 70.0000i 0.114192i 0.998369 + 0.0570962i \(0.0181842\pi\)
−0.998369 + 0.0570962i \(0.981816\pi\)
\(614\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −79.0000 −0.126400
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 1120.00i − 1.78060i
\(630\) 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
\(637\) − 490.000i − 0.769231i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −400.000 −0.624025 −0.312012 0.950078i \(-0.601003\pi\)
−0.312012 + 0.950078i \(0.601003\pi\)
\(642\) 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
\(653\) − 1144.00i − 1.75191i −0.482389 0.875957i \(-0.660231\pi\)
0.482389 0.875957i \(-0.339769\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) − 1178.00i − 1.78215i −0.453858 0.891074i \(-0.649953\pi\)
0.453858 0.891074i \(-0.350047\pi\)
\(662\) 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
\(677\) − 104.000i − 0.153619i −0.997046 0.0768095i \(-0.975527\pi\)
0.997046 0.0768095i \(-0.0244733\pi\)
\(678\) 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
\(685\) 1408.00i 2.05547i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −560.000 −0.812772
\(690\) 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
\(697\) 1280.00 1.83644
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 520.000i − 0.741797i −0.928673 0.370899i \(-0.879050\pi\)
0.928673 0.370899i \(-0.120950\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) − 518.000i − 0.730606i −0.930889 0.365303i \(-0.880965\pi\)
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
\(725\) 1560.00i 2.15172i
\(726\) 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
\(733\) − 1450.00i − 1.97817i −0.147340 0.989086i \(-0.547071\pi\)
0.147340 0.989086i \(-0.452929\pi\)
\(734\) 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
\(745\) 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
\(757\) − 1190.00i − 1.57199i −0.618230 0.785997i \(-0.712150\pi\)
0.618230 0.785997i \(-0.287850\pi\)
\(758\) 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
\(773\) − 1496.00i − 1.93532i −0.252264 0.967658i \(-0.581175\pi\)
0.252264 0.967658i \(-0.418825\pi\)
\(774\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −220.000 −0.277427
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 1144.00i − 1.43538i −0.696361 0.717691i \(-0.745199\pi\)
0.696361 0.717691i \(-0.254801\pi\)
\(798\) 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 1.00000i \(-0.5\pi\)
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
\(821\) − 1400.00i − 1.70524i −0.522533 0.852619i \(-0.675013\pi\)
0.522533 0.852619i \(-0.324987\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) − 1258.00i − 1.51749i −0.651387 0.758745i \(-0.725813\pi\)
0.651387 0.758745i \(-0.274187\pi\)
\(830\) 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
\(845\) − 552.000i − 0.653254i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) − 410.000i − 0.480657i −0.970692 0.240328i \(-0.922745\pi\)
0.970692 0.240328i \(-0.0772551\pi\)
\(854\) 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 1.00000i \(-0.5\pi\)
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
\(865\) 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
\(877\) 1610.00i 1.83580i 0.396807 + 0.917902i \(0.370118\pi\)
−0.396807 + 0.917902i \(0.629882\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1600.00 −1.81612 −0.908059 0.418842i \(-0.862436\pi\)
−0.908059 + 0.418842i \(0.862436\pi\)
\(882\) 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
\(901\) 896.000i 0.994451i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −304.000 −0.335912
\(906\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) − 2730.00i − 2.95135i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1840.00 −1.98062 −0.990312 0.138859i \(-0.955657\pi\)
−0.990312 + 0.138859i \(0.955657\pi\)
\(930\) 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
\(941\) 1160.00i 1.23273i 0.787460 + 0.616366i \(0.211396\pi\)
−0.787460 + 0.616366i \(0.788604\pi\)
\(942\) 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
\(949\) 1100.00i 1.15911i
\(950\) 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
\(965\) 1520.00i 1.57513i
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −496.000 −0.507677 −0.253838 0.967247i \(-0.581693\pi\)
−0.253838 + 0.967247i \(0.581693\pi\)
\(978\) 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
\(997\) − 1850.00i − 1.85557i −0.373119 0.927783i \(-0.621712\pi\)
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.b.d.127.1 2
3.2 odd 2 2304.3.b.e.127.2 2
4.3 odd 2 CM 2304.3.b.d.127.1 2
8.3 odd 2 inner 2304.3.b.d.127.2 2
8.5 even 2 inner 2304.3.b.d.127.2 2
12.11 even 2 2304.3.b.e.127.2 2
16.3 odd 4 576.3.g.a.127.1 1
16.5 even 4 36.3.d.a.19.1 1
16.11 odd 4 36.3.d.a.19.1 1
16.13 even 4 576.3.g.a.127.1 1
24.5 odd 2 2304.3.b.e.127.1 2
24.11 even 2 2304.3.b.e.127.1 2
48.5 odd 4 36.3.d.b.19.1 yes 1
48.11 even 4 36.3.d.b.19.1 yes 1
48.29 odd 4 576.3.g.c.127.1 1
48.35 even 4 576.3.g.c.127.1 1
80.27 even 4 900.3.f.b.199.1 2
80.37 odd 4 900.3.f.b.199.1 2
80.43 even 4 900.3.f.b.199.2 2
80.53 odd 4 900.3.f.b.199.2 2
80.59 odd 4 900.3.c.c.451.1 1
80.69 even 4 900.3.c.c.451.1 1
144.5 odd 12 324.3.f.e.55.1 2
144.11 even 12 324.3.f.e.271.1 2
144.43 odd 12 324.3.f.f.271.1 2
144.59 even 12 324.3.f.e.55.1 2
144.85 even 12 324.3.f.f.55.1 2
144.101 odd 12 324.3.f.e.271.1 2
144.133 even 12 324.3.f.f.271.1 2
144.139 odd 12 324.3.f.f.55.1 2
240.53 even 4 900.3.f.a.199.1 2
240.59 even 4 900.3.c.b.451.1 1
240.107 odd 4 900.3.f.a.199.2 2
240.149 odd 4 900.3.c.b.451.1 1
240.197 even 4 900.3.f.a.199.2 2
240.203 odd 4 900.3.f.a.199.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
36.3.d.a.19.1 1 16.5 even 4
36.3.d.a.19.1 1 16.11 odd 4
36.3.d.b.19.1 yes 1 48.5 odd 4
36.3.d.b.19.1 yes 1 48.11 even 4
324.3.f.e.55.1 2 144.5 odd 12
324.3.f.e.55.1 2 144.59 even 12
324.3.f.e.271.1 2 144.11 even 12
324.3.f.e.271.1 2 144.101 odd 12
324.3.f.f.55.1 2 144.85 even 12
324.3.f.f.55.1 2 144.139 odd 12
324.3.f.f.271.1 2 144.43 odd 12
324.3.f.f.271.1 2 144.133 even 12
576.3.g.a.127.1 1 16.3 odd 4
576.3.g.a.127.1 1 16.13 even 4
576.3.g.c.127.1 1 48.29 odd 4
576.3.g.c.127.1 1 48.35 even 4
900.3.c.b.451.1 1 240.59 even 4
900.3.c.b.451.1 1 240.149 odd 4
900.3.c.c.451.1 1 80.59 odd 4
900.3.c.c.451.1 1 80.69 even 4
900.3.f.a.199.1 2 240.53 even 4
900.3.f.a.199.1 2 240.203 odd 4
900.3.f.a.199.2 2 240.107 odd 4
900.3.f.a.199.2 2 240.197 even 4
900.3.f.b.199.1 2 80.27 even 4
900.3.f.b.199.1 2 80.37 odd 4
900.3.f.b.199.2 2 80.43 even 4
900.3.f.b.199.2 2 80.53 odd 4
2304.3.b.d.127.1 2 1.1 even 1 trivial
2304.3.b.d.127.1 2 4.3 odd 2 CM
2304.3.b.d.127.2 2 8.3 odd 2 inner
2304.3.b.d.127.2 2 8.5 even 2 inner
2304.3.b.e.127.1 2 24.5 odd 2
2304.3.b.e.127.1 2 24.11 even 2
2304.3.b.e.127.2 2 3.2 odd 2
2304.3.b.e.127.2 2 12.11 even 2