Properties

Label 588.3.c.c.197.1
Level $588$
Weight $3$
Character 588.197
Self dual yes
Analytic conductor $16.022$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 588.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(16.0218395444\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 12)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 197.1
Character \(\chi\) \(=\) 588.197

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{3} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} +9.00000 q^{9} +22.0000 q^{13} -26.0000 q^{19} +25.0000 q^{25} +27.0000 q^{27} +46.0000 q^{31} +26.0000 q^{37} +66.0000 q^{39} -22.0000 q^{43} -78.0000 q^{57} -74.0000 q^{61} +122.000 q^{67} +46.0000 q^{73} +75.0000 q^{75} -142.000 q^{79} +81.0000 q^{81} +138.000 q^{93} -2.00000 q^{97} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) 3.00000 1.00000
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 22.0000 1.69231 0.846154 0.532939i \(-0.178912\pi\)
0.846154 + 0.532939i \(0.178912\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −26.0000 −1.36842 −0.684211 0.729285i \(-0.739853\pi\)
−0.684211 + 0.729285i \(0.739853\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 25.0000 1.00000
\(26\) 0 0
\(27\) 27.0000 1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 46.0000 1.48387 0.741935 0.670471i \(-0.233908\pi\)
0.741935 + 0.670471i \(0.233908\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 26.0000 0.702703 0.351351 0.936244i \(-0.385722\pi\)
0.351351 + 0.936244i \(0.385722\pi\)
\(38\) 0 0
\(39\) 66.0000 1.69231
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −22.0000 −0.511628 −0.255814 0.966726i \(-0.582343\pi\)
−0.255814 + 0.966726i \(0.582343\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −78.0000 −1.36842
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −74.0000 −1.21311 −0.606557 0.795040i \(-0.707450\pi\)
−0.606557 + 0.795040i \(0.707450\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 122.000 1.82090 0.910448 0.413624i \(-0.135737\pi\)
0.910448 + 0.413624i \(0.135737\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 46.0000 0.630137 0.315068 0.949069i \(-0.397973\pi\)
0.315068 + 0.949069i \(0.397973\pi\)
\(74\) 0 0
\(75\) 75.0000 1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −142.000 −1.79747 −0.898734 0.438494i \(-0.855512\pi\)
−0.898734 + 0.438494i \(0.855512\pi\)
\(80\) 0 0
\(81\) 81.0000 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 138.000 1.48387
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −2.00000 −0.0206186 −0.0103093 0.999947i \(-0.503282\pi\)
−0.0103093 + 0.999947i \(0.503282\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −194.000 −1.88350 −0.941748 0.336321i \(-0.890817\pi\)
−0.941748 + 0.336321i \(0.890817\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) −214.000 −1.96330 −0.981651 0.190684i \(-0.938929\pi\)
−0.981651 + 0.190684i \(0.938929\pi\)
\(110\) 0 0
\(111\) 78.0000 0.702703
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 198.000 1.69231
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 121.000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 146.000 1.14961 0.574803 0.818292i \(-0.305079\pi\)
0.574803 + 0.818292i \(0.305079\pi\)
\(128\) 0 0
\(129\) −66.0000 −0.511628
\(130\) 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
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 22.0000 0.158273 0.0791367 0.996864i \(-0.474784\pi\)
0.0791367 + 0.996864i \(0.474784\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) −286.000 −1.89404 −0.947020 0.321175i \(-0.895922\pi\)
−0.947020 + 0.321175i \(0.895922\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 118.000 0.751592 0.375796 0.926702i \(-0.377369\pi\)
0.375796 + 0.926702i \(0.377369\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −262.000 −1.60736 −0.803681 0.595060i \(-0.797128\pi\)
−0.803681 + 0.595060i \(0.797128\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 315.000 1.86391
\(170\) 0 0
\(171\) −234.000 −1.36842
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 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
\(181\) −314.000 −1.73481 −0.867403 0.497606i \(-0.834213\pi\)
−0.867403 + 0.497606i \(0.834213\pi\)
\(182\) 0 0
\(183\) −222.000 −1.21311
\(184\) 0 0
\(185\) 0 0
\(186\) 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
\(193\) −382.000 −1.97927 −0.989637 0.143590i \(-0.954135\pi\)
−0.989637 + 0.143590i \(0.954135\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −386.000 −1.93970 −0.969849 0.243706i \(-0.921637\pi\)
−0.969849 + 0.243706i \(0.921637\pi\)
\(200\) 0 0
\(201\) 366.000 1.82090
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −166.000 −0.786730 −0.393365 0.919382i \(-0.628689\pi\)
−0.393365 + 0.919382i \(0.628689\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 138.000 0.630137
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −338.000 −1.51570 −0.757848 0.652432i \(-0.773749\pi\)
−0.757848 + 0.652432i \(0.773749\pi\)
\(224\) 0 0
\(225\) 225.000 1.00000
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) −26.0000 −0.113537 −0.0567686 0.998387i \(-0.518080\pi\)
−0.0567686 + 0.998387i \(0.518080\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −426.000 −1.79747
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 286.000 1.18672 0.593361 0.804936i \(-0.297801\pi\)
0.593361 + 0.804936i \(0.297801\pi\)
\(242\) 0 0
\(243\) 243.000 1.00000
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −572.000 −2.31579
\(248\) 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
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 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
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) −242.000 −0.892989 −0.446494 0.894786i \(-0.647328\pi\)
−0.446494 + 0.894786i \(0.647328\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 122.000 0.440433 0.220217 0.975451i \(-0.429324\pi\)
0.220217 + 0.975451i \(0.429324\pi\)
\(278\) 0 0
\(279\) 414.000 1.48387
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −458.000 −1.61837 −0.809187 0.587551i \(-0.800092\pi\)
−0.809187 + 0.587551i \(0.800092\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 289.000 1.00000
\(290\) 0 0
\(291\) −6.00000 −0.0206186
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 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
\(305\) 0 0
\(306\) 0 0
\(307\) 358.000 1.16612 0.583062 0.812428i \(-0.301855\pi\)
0.583062 + 0.812428i \(0.301855\pi\)
\(308\) 0 0
\(309\) −582.000 −1.88350
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 142.000 0.453674 0.226837 0.973933i \(-0.427162\pi\)
0.226837 + 0.973933i \(0.427162\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 550.000 1.69231
\(326\) 0 0
\(327\) −642.000 −1.96330
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 362.000 1.09366 0.546828 0.837245i \(-0.315835\pi\)
0.546828 + 0.837245i \(0.315835\pi\)
\(332\) 0 0
\(333\) 234.000 0.702703
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 482.000 1.43027 0.715134 0.698988i \(-0.246366\pi\)
0.715134 + 0.698988i \(0.246366\pi\)
\(338\) 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
\(349\) 502.000 1.43840 0.719198 0.694805i \(-0.244510\pi\)
0.719198 + 0.694805i \(0.244510\pi\)
\(350\) 0 0
\(351\) 594.000 1.69231
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 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
\(361\) 315.000 0.872576
\(362\) 0 0
\(363\) 363.000 1.00000
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 718.000 1.95640 0.978202 0.207657i \(-0.0665839\pi\)
0.978202 + 0.207657i \(0.0665839\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 698.000 1.87131 0.935657 0.352911i \(-0.114808\pi\)
0.935657 + 0.352911i \(0.114808\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −694.000 −1.83113 −0.915567 0.402165i \(-0.868258\pi\)
−0.915567 + 0.402165i \(0.868258\pi\)
\(380\) 0 0
\(381\) 438.000 1.14961
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −198.000 −0.511628
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −362.000 −0.911839 −0.455919 0.890021i \(-0.650689\pi\)
−0.455919 + 0.890021i \(0.650689\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 1012.00 2.51117
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −626.000 −1.53056 −0.765281 0.643696i \(-0.777400\pi\)
−0.765281 + 0.643696i \(0.777400\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 66.0000 0.158273
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −358.000 −0.850356 −0.425178 0.905110i \(-0.639789\pi\)
−0.425178 + 0.905110i \(0.639789\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 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
\(433\) 862.000 1.99076 0.995381 0.0960028i \(-0.0306058\pi\)
0.995381 + 0.0960028i \(0.0306058\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 94.0000 0.214123 0.107062 0.994252i \(-0.465856\pi\)
0.107062 + 0.994252i \(0.465856\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −858.000 −1.89404
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −814.000 −1.78118 −0.890591 0.454805i \(-0.849709\pi\)
−0.890591 + 0.454805i \(0.849709\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −526.000 −1.13607 −0.568035 0.823005i \(-0.692296\pi\)
−0.568035 + 0.823005i \(0.692296\pi\)
\(464\) 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
\(471\) 354.000 0.751592
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −650.000 −1.36842
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 572.000 1.18919
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 962.000 1.97536 0.987680 0.156489i \(-0.0500176\pi\)
0.987680 + 0.156489i \(0.0500176\pi\)
\(488\) 0 0
\(489\) −786.000 −1.60736
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 26.0000 0.0521042 0.0260521 0.999661i \(-0.491706\pi\)
0.0260521 + 0.999661i \(0.491706\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 945.000 1.86391
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −702.000 −1.36842
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 982.000 1.87763 0.938815 0.344423i \(-0.111925\pi\)
0.938815 + 0.344423i \(0.111925\pi\)
\(524\) 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
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 1034.00 1.91128 0.955638 0.294545i \(-0.0951680\pi\)
0.955638 + 0.294545i \(0.0951680\pi\)
\(542\) 0 0
\(543\) −942.000 −1.73481
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 506.000 0.925046 0.462523 0.886607i \(-0.346944\pi\)
0.462523 + 0.886607i \(0.346944\pi\)
\(548\) 0 0
\(549\) −666.000 −1.21311
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) −484.000 −0.865832
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) −886.000 −1.55166 −0.775832 0.630940i \(-0.782670\pi\)
−0.775832 + 0.630940i \(0.782670\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −962.000 −1.66724 −0.833622 0.552335i \(-0.813737\pi\)
−0.833622 + 0.552335i \(0.813737\pi\)
\(578\) 0 0
\(579\) −1146.00 −1.97927
\(580\) 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
\(589\) −1196.00 −2.03056
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −1158.00 −1.93970
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 526.000 0.875208 0.437604 0.899168i \(-0.355827\pi\)
0.437604 + 0.899168i \(0.355827\pi\)
\(602\) 0 0
\(603\) 1098.00 1.82090
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 814.000 1.34102 0.670511 0.741900i \(-0.266075\pi\)
0.670511 + 0.741900i \(0.266075\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1126.00 −1.83687 −0.918434 0.395574i \(-0.870546\pi\)
−0.918434 + 0.395574i \(0.870546\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 214.000 0.345719 0.172859 0.984947i \(-0.444699\pi\)
0.172859 + 0.984947i \(0.444699\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 625.000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 674.000 1.06815 0.534073 0.845438i \(-0.320661\pi\)
0.534073 + 0.845438i \(0.320661\pi\)
\(632\) 0 0
\(633\) −498.000 −0.786730
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) −314.000 −0.488336 −0.244168 0.969733i \(-0.578515\pi\)
−0.244168 + 0.969733i \(0.578515\pi\)
\(644\) 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
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 414.000 0.630137
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −122.000 −0.184569 −0.0922844 0.995733i \(-0.529417\pi\)
−0.0922844 + 0.995733i \(0.529417\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −1014.00 −1.51570
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1154.00 1.71471 0.857355 0.514725i \(-0.172106\pi\)
0.857355 + 0.514725i \(0.172106\pi\)
\(674\) 0 0
\(675\) 675.000 1.00000
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 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
\(685\) 0 0
\(686\) 0 0
\(687\) −78.0000 −0.113537
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 1318.00 1.90738 0.953690 0.300790i \(-0.0972504\pi\)
0.953690 + 0.300790i \(0.0972504\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) −676.000 −0.961593
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −934.000 −1.31735 −0.658674 0.752428i \(-0.728882\pi\)
−0.658674 + 0.752428i \(0.728882\pi\)
\(710\) 0 0
\(711\) −1278.00 −1.79747
\(712\) 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
\(723\) 858.000 1.18672
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −482.000 −0.662999 −0.331499 0.943455i \(-0.607554\pi\)
−0.331499 + 0.943455i \(0.607554\pi\)
\(728\) 0 0
\(729\) 729.000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −1034.00 −1.41064 −0.705321 0.708888i \(-0.749197\pi\)
−0.705321 + 0.708888i \(0.749197\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −1222.00 −1.65359 −0.826793 0.562506i \(-0.809837\pi\)
−0.826793 + 0.562506i \(0.809837\pi\)
\(740\) 0 0
\(741\) −1716.00 −2.31579
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 1202.00 1.60053 0.800266 0.599645i \(-0.204691\pi\)
0.800266 + 0.599645i \(0.204691\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −838.000 −1.10700 −0.553501 0.832849i \(-0.686708\pi\)
−0.553501 + 0.832849i \(0.686708\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 1534.00 1.99480 0.997399 0.0720749i \(-0.0229621\pi\)
0.997399 + 0.0720749i \(0.0229621\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 1150.00 1.48387
\(776\) 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
\(785\) 0 0
\(786\) 0 0
\(787\) −1562.00 −1.98475 −0.992376 0.123246i \(-0.960669\pi\)
−0.992376 + 0.123246i \(0.960669\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −1628.00 −2.05296
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 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
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) −1514.00 −1.86683 −0.933416 0.358797i \(-0.883187\pi\)
−0.933416 + 0.358797i \(0.883187\pi\)
\(812\) 0 0
\(813\) −726.000 −0.892989
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 572.000 0.700122
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 1058.00 1.28554 0.642770 0.766059i \(-0.277785\pi\)
0.642770 + 0.766059i \(0.277785\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) −458.000 −0.552473 −0.276236 0.961090i \(-0.589087\pi\)
−0.276236 + 0.961090i \(0.589087\pi\)
\(830\) 0 0
\(831\) 366.000 0.440433
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 1242.00 1.48387
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 841.000 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −1374.00 −1.61837
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −1658.00 −1.94373 −0.971864 0.235543i \(-0.924313\pi\)
−0.971864 + 0.235543i \(0.924313\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) −1418.00 −1.65076 −0.825378 0.564580i \(-0.809038\pi\)
−0.825378 + 0.564580i \(0.809038\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 867.000 1.00000
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 2684.00 3.08152
\(872\) 0 0
\(873\) −18.0000 −0.0206186
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −598.000 −0.681870 −0.340935 0.940087i \(-0.610744\pi\)
−0.340935 + 0.940087i \(0.610744\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) −1702.00 −1.92752 −0.963760 0.266771i \(-0.914043\pi\)
−0.963760 + 0.266771i \(0.914043\pi\)
\(884\) 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
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −214.000 −0.235943 −0.117971 0.993017i \(-0.537639\pi\)
−0.117971 + 0.993017i \(0.537639\pi\)
\(908\) 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
\(919\) 866.000 0.942329 0.471164 0.882045i \(-0.343834\pi\)
0.471164 + 0.882045i \(0.343834\pi\)
\(920\) 0 0
\(921\) 1074.00 1.16612
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 650.000 0.702703
\(926\) 0 0
\(927\) −1746.00 −1.88350
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1198.00 1.27855 0.639274 0.768979i \(-0.279235\pi\)
0.639274 + 0.768979i \(0.279235\pi\)
\(938\) 0 0
\(939\) 426.000 0.453674
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 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
\(949\) 1012.00 1.06639
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 1155.00 1.20187
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −1534.00 −1.58635 −0.793175 0.608994i \(-0.791573\pi\)
−0.793175 + 0.608994i \(0.791573\pi\)
\(968\) 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
\(975\) 1650.00 1.69231
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −1926.00 −1.96330
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −46.0000 −0.0464178 −0.0232089 0.999731i \(-0.507388\pi\)
−0.0232089 + 0.999731i \(0.507388\pi\)
\(992\) 0 0
\(993\) 1086.00 1.09366
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1894.00 1.89970 0.949850 0.312707i \(-0.101236\pi\)
0.949850 + 0.312707i \(0.101236\pi\)
\(998\) 0 0
\(999\) 702.000 0.702703
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 588.3.c.c.197.1 1
3.2 odd 2 CM 588.3.c.c.197.1 1
7.2 even 3 588.3.p.b.557.1 2
7.3 odd 6 588.3.p.c.569.1 2
7.4 even 3 588.3.p.b.569.1 2
7.5 odd 6 588.3.p.c.557.1 2
7.6 odd 2 12.3.c.a.5.1 1
21.2 odd 6 588.3.p.b.557.1 2
21.5 even 6 588.3.p.c.557.1 2
21.11 odd 6 588.3.p.b.569.1 2
21.17 even 6 588.3.p.c.569.1 2
21.20 even 2 12.3.c.a.5.1 1
28.27 even 2 48.3.e.a.17.1 1
35.13 even 4 300.3.b.a.149.1 2
35.27 even 4 300.3.b.a.149.2 2
35.34 odd 2 300.3.g.b.101.1 1
56.13 odd 2 192.3.e.b.65.1 1
56.27 even 2 192.3.e.a.65.1 1
63.13 odd 6 324.3.g.b.269.1 2
63.20 even 6 324.3.g.b.53.1 2
63.34 odd 6 324.3.g.b.53.1 2
63.41 even 6 324.3.g.b.269.1 2
77.76 even 2 1452.3.e.b.485.1 1
84.83 odd 2 48.3.e.a.17.1 1
105.62 odd 4 300.3.b.a.149.2 2
105.83 odd 4 300.3.b.a.149.1 2
105.104 even 2 300.3.g.b.101.1 1
112.13 odd 4 768.3.h.a.641.1 2
112.27 even 4 768.3.h.b.641.1 2
112.69 odd 4 768.3.h.a.641.2 2
112.83 even 4 768.3.h.b.641.2 2
140.27 odd 4 1200.3.c.c.449.1 2
140.83 odd 4 1200.3.c.c.449.2 2
140.139 even 2 1200.3.l.b.401.1 1
168.83 odd 2 192.3.e.a.65.1 1
168.125 even 2 192.3.e.b.65.1 1
231.230 odd 2 1452.3.e.b.485.1 1
252.83 odd 6 1296.3.q.b.1025.1 2
252.139 even 6 1296.3.q.b.593.1 2
252.167 odd 6 1296.3.q.b.593.1 2
252.223 even 6 1296.3.q.b.1025.1 2
336.83 odd 4 768.3.h.b.641.2 2
336.125 even 4 768.3.h.a.641.1 2
336.251 odd 4 768.3.h.b.641.1 2
336.293 even 4 768.3.h.a.641.2 2
420.83 even 4 1200.3.c.c.449.2 2
420.167 even 4 1200.3.c.c.449.1 2
420.419 odd 2 1200.3.l.b.401.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.3.c.a.5.1 1 7.6 odd 2
12.3.c.a.5.1 1 21.20 even 2
48.3.e.a.17.1 1 28.27 even 2
48.3.e.a.17.1 1 84.83 odd 2
192.3.e.a.65.1 1 56.27 even 2
192.3.e.a.65.1 1 168.83 odd 2
192.3.e.b.65.1 1 56.13 odd 2
192.3.e.b.65.1 1 168.125 even 2
300.3.b.a.149.1 2 35.13 even 4
300.3.b.a.149.1 2 105.83 odd 4
300.3.b.a.149.2 2 35.27 even 4
300.3.b.a.149.2 2 105.62 odd 4
300.3.g.b.101.1 1 35.34 odd 2
300.3.g.b.101.1 1 105.104 even 2
324.3.g.b.53.1 2 63.20 even 6
324.3.g.b.53.1 2 63.34 odd 6
324.3.g.b.269.1 2 63.13 odd 6
324.3.g.b.269.1 2 63.41 even 6
588.3.c.c.197.1 1 1.1 even 1 trivial
588.3.c.c.197.1 1 3.2 odd 2 CM
588.3.p.b.557.1 2 7.2 even 3
588.3.p.b.557.1 2 21.2 odd 6
588.3.p.b.569.1 2 7.4 even 3
588.3.p.b.569.1 2 21.11 odd 6
588.3.p.c.557.1 2 7.5 odd 6
588.3.p.c.557.1 2 21.5 even 6
588.3.p.c.569.1 2 7.3 odd 6
588.3.p.c.569.1 2 21.17 even 6
768.3.h.a.641.1 2 112.13 odd 4
768.3.h.a.641.1 2 336.125 even 4
768.3.h.a.641.2 2 112.69 odd 4
768.3.h.a.641.2 2 336.293 even 4
768.3.h.b.641.1 2 112.27 even 4
768.3.h.b.641.1 2 336.251 odd 4
768.3.h.b.641.2 2 112.83 even 4
768.3.h.b.641.2 2 336.83 odd 4
1200.3.c.c.449.1 2 140.27 odd 4
1200.3.c.c.449.1 2 420.167 even 4
1200.3.c.c.449.2 2 140.83 odd 4
1200.3.c.c.449.2 2 420.83 even 4
1200.3.l.b.401.1 1 140.139 even 2
1200.3.l.b.401.1 1 420.419 odd 2
1296.3.q.b.593.1 2 252.139 even 6
1296.3.q.b.593.1 2 252.167 odd 6
1296.3.q.b.1025.1 2 252.83 odd 6
1296.3.q.b.1025.1 2 252.223 even 6
1452.3.e.b.485.1 1 77.76 even 2
1452.3.e.b.485.1 1 231.230 odd 2