Properties

Label 688.3.b.a.257.1
Level 688
Weight 3
Character 688.257
Self dual yes
Analytic conductor 18.747
Analytic rank 0
Dimension 1
CM discriminant -43
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 688 = 2^{4} \cdot 43 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 688.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(18.7466421880\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 43)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 257.1
Character \(\chi\) \(=\) 688.257

$q$-expansion

\(f(q)\) \(=\) \(q+9.00000 q^{9} +O(q^{10})\) \(q+9.00000 q^{9} +21.0000 q^{11} -17.0000 q^{13} -9.00000 q^{17} -3.00000 q^{23} +25.0000 q^{25} -19.0000 q^{31} +39.0000 q^{41} +43.0000 q^{43} +78.0000 q^{47} +49.0000 q^{49} +63.0000 q^{53} +54.0000 q^{59} -91.0000 q^{67} +14.0000 q^{79} +81.0000 q^{81} -123.000 q^{83} -193.000 q^{97} +189.000 q^{99} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/688\mathbb{Z}\right)^\times\).

\(n\) \(431\) \(433\) \(517\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 9.00000 1.00000
\(10\) 0 0
\(11\) 21.0000 1.90909 0.954545 0.298065i \(-0.0963413\pi\)
0.954545 + 0.298065i \(0.0963413\pi\)
\(12\) 0 0
\(13\) −17.0000 −1.30769 −0.653846 0.756628i \(-0.726846\pi\)
−0.653846 + 0.756628i \(0.726846\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −9.00000 −0.529412 −0.264706 0.964329i \(-0.585275\pi\)
−0.264706 + 0.964329i \(0.585275\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.00000 −0.130435 −0.0652174 0.997871i \(-0.520774\pi\)
−0.0652174 + 0.997871i \(0.520774\pi\)
\(24\) 0 0
\(25\) 25.0000 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −19.0000 −0.612903 −0.306452 0.951886i \(-0.599142\pi\)
−0.306452 + 0.951886i \(0.599142\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 39.0000 0.951220 0.475610 0.879656i \(-0.342227\pi\)
0.475610 + 0.879656i \(0.342227\pi\)
\(42\) 0 0
\(43\) 43.0000 1.00000
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 78.0000 1.65957 0.829787 0.558080i \(-0.188462\pi\)
0.829787 + 0.558080i \(0.188462\pi\)
\(48\) 0 0
\(49\) 49.0000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 63.0000 1.18868 0.594340 0.804214i \(-0.297414\pi\)
0.594340 + 0.804214i \(0.297414\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 54.0000 0.915254 0.457627 0.889144i \(-0.348699\pi\)
0.457627 + 0.889144i \(0.348699\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −91.0000 −1.35821 −0.679104 0.734042i \(-0.737632\pi\)
−0.679104 + 0.734042i \(0.737632\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 14.0000 0.177215 0.0886076 0.996067i \(-0.471758\pi\)
0.0886076 + 0.996067i \(0.471758\pi\)
\(80\) 0 0
\(81\) 81.0000 1.00000
\(82\) 0 0
\(83\) −123.000 −1.48193 −0.740964 0.671545i \(-0.765631\pi\)
−0.740964 + 0.671545i \(0.765631\pi\)
\(84\) 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
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −193.000 −1.98969 −0.994845 0.101404i \(-0.967667\pi\)
−0.994845 + 0.101404i \(0.967667\pi\)
\(98\) 0 0
\(99\) 189.000 1.90909
\(100\) 0 0
\(101\) 159.000 1.57426 0.787129 0.616789i \(-0.211567\pi\)
0.787129 + 0.616789i \(0.211567\pi\)
\(102\) 0 0
\(103\) 181.000 1.75728 0.878641 0.477483i \(-0.158451\pi\)
0.878641 + 0.477483i \(0.158451\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −42.0000 −0.392523 −0.196262 0.980552i \(-0.562880\pi\)
−0.196262 + 0.980552i \(0.562880\pi\)
\(108\) 0 0
\(109\) −169.000 −1.55046 −0.775229 0.631680i \(-0.782366\pi\)
−0.775229 + 0.631680i \(0.782366\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −153.000 −1.30769
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 320.000 2.64463
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 133.000 1.04724 0.523622 0.851951i \(-0.324580\pi\)
0.523622 + 0.851951i \(0.324580\pi\)
\(128\) 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
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 109.000 0.784173 0.392086 0.919928i \(-0.371753\pi\)
0.392086 + 0.919928i \(0.371753\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −357.000 −2.49650
\(144\) 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
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) −81.0000 −0.529412
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 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
\(167\) −291.000 −1.74251 −0.871257 0.490826i \(-0.836695\pi\)
−0.871257 + 0.490826i \(0.836695\pi\)
\(168\) 0 0
\(169\) 120.000 0.710059
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −342.000 −1.97688 −0.988439 0.151617i \(-0.951552\pi\)
−0.988439 + 0.151617i \(0.951552\pi\)
\(174\) 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
\(181\) −326.000 −1.80110 −0.900552 0.434747i \(-0.856838\pi\)
−0.900552 + 0.434747i \(0.856838\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −189.000 −1.01070
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 343.000 1.77720 0.888601 0.458681i \(-0.151678\pi\)
0.888601 + 0.458681i \(0.151678\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −294.000 −1.49239 −0.746193 0.665730i \(-0.768120\pi\)
−0.746193 + 0.665730i \(0.768120\pi\)
\(198\) 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
\(205\) 0 0
\(206\) 0 0
\(207\) −27.0000 −0.130435
\(208\) 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
\(221\) 153.000 0.692308
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 225.000 1.00000
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 71.0000 0.310044 0.155022 0.987911i \(-0.450455\pi\)
0.155022 + 0.987911i \(0.450455\pi\)
\(230\) 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
\(237\) 0 0
\(238\) 0 0
\(239\) −306.000 −1.28033 −0.640167 0.768235i \(-0.721135\pi\)
−0.640167 + 0.768235i \(0.721135\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −459.000 −1.82869 −0.914343 0.404941i \(-0.867292\pi\)
−0.914343 + 0.404941i \(0.867292\pi\)
\(252\) 0 0
\(253\) −63.0000 −0.249012
\(254\) 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
\(269\) −537.000 −1.99628 −0.998141 0.0609427i \(-0.980589\pi\)
−0.998141 + 0.0609427i \(0.980589\pi\)
\(270\) 0 0
\(271\) 533.000 1.96679 0.983395 0.181479i \(-0.0580884\pi\)
0.983395 + 0.181479i \(0.0580884\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 525.000 1.90909
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) −171.000 −0.612903
\(280\) 0 0
\(281\) −513.000 −1.82562 −0.912811 0.408381i \(-0.866093\pi\)
−0.912811 + 0.408381i \(0.866093\pi\)
\(282\) 0 0
\(283\) −523.000 −1.84806 −0.924028 0.382324i \(-0.875124\pi\)
−0.924028 + 0.382324i \(0.875124\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −208.000 −0.719723
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −102.000 −0.348123 −0.174061 0.984735i \(-0.555689\pi\)
−0.174061 + 0.984735i \(0.555689\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 51.0000 0.170569
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −227.000 −0.739414 −0.369707 0.929148i \(-0.620542\pi\)
−0.369707 + 0.929148i \(0.620542\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 453.000 1.45659 0.728296 0.685263i \(-0.240313\pi\)
0.728296 + 0.685263i \(0.240313\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 591.000 1.86435 0.932177 0.362004i \(-0.117907\pi\)
0.932177 + 0.362004i \(0.117907\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −425.000 −1.30769
\(326\) 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
\(337\) 287.000 0.851632 0.425816 0.904810i \(-0.359987\pi\)
0.425816 + 0.904810i \(0.359987\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −399.000 −1.17009
\(342\) 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
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 663.000 1.87819 0.939093 0.343662i \(-0.111667\pi\)
0.939093 + 0.343662i \(0.111667\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 357.000 0.994429 0.497214 0.867628i \(-0.334356\pi\)
0.497214 + 0.867628i \(0.334356\pi\)
\(360\) 0 0
\(361\) 361.000 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −562.000 −1.53134 −0.765668 0.643236i \(-0.777591\pi\)
−0.765668 + 0.643236i \(0.777591\pi\)
\(368\) 0 0
\(369\) 351.000 0.951220
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 317.000 0.836412 0.418206 0.908352i \(-0.362659\pi\)
0.418206 + 0.908352i \(0.362659\pi\)
\(380\) 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
\(387\) 387.000 1.00000
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 27.0000 0.0690537
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 106.000 0.267003 0.133501 0.991049i \(-0.457378\pi\)
0.133501 + 0.991049i \(0.457378\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −273.000 −0.680798 −0.340399 0.940281i \(-0.610562\pi\)
−0.340399 + 0.940281i \(0.610562\pi\)
\(402\) 0 0
\(403\) 323.000 0.801489
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 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
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 702.000 1.65957
\(424\) 0 0
\(425\) −225.000 −0.529412
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −819.000 −1.90023 −0.950116 0.311897i \(-0.899036\pi\)
−0.950116 + 0.311897i \(0.899036\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −491.000 −1.11845 −0.559226 0.829016i \(-0.688901\pi\)
−0.559226 + 0.829016i \(0.688901\pi\)
\(440\) 0 0
\(441\) 441.000 1.00000
\(442\) 0 0
\(443\) −714.000 −1.61174 −0.805869 0.592094i \(-0.798302\pi\)
−0.805869 + 0.592094i \(0.798302\pi\)
\(444\) 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
\(451\) 819.000 1.81596
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 234.000 0.507592 0.253796 0.967258i \(-0.418321\pi\)
0.253796 + 0.967258i \(0.418321\pi\)
\(462\) 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
\(473\) 903.000 1.90909
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 567.000 1.18868
\(478\) 0 0
\(479\) 117.000 0.244259 0.122129 0.992514i \(-0.461028\pi\)
0.122129 + 0.992514i \(0.461028\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 574.000 1.17864 0.589322 0.807898i \(-0.299395\pi\)
0.589322 + 0.807898i \(0.299395\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 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
\(499\) 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
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −57.0000 −0.111984 −0.0559921 0.998431i \(-0.517832\pi\)
−0.0559921 + 0.998431i \(0.517832\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 1638.00 3.16828
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 171.000 0.324478
\(528\) 0 0
\(529\) −520.000 −0.982987
\(530\) 0 0
\(531\) 486.000 0.915254
\(532\) 0 0
\(533\) −663.000 −1.24390
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1029.00 1.90909
\(540\) 0 0
\(541\) 7.00000 0.0129390 0.00646950 0.999979i \(-0.497941\pi\)
0.00646950 + 0.999979i \(0.497941\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1013.00 1.85192 0.925960 0.377622i \(-0.123258\pi\)
0.925960 + 0.377622i \(0.123258\pi\)
\(548\) 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
\(557\) −993.000 −1.78276 −0.891382 0.453252i \(-0.850264\pi\)
−0.891382 + 0.453252i \(0.850264\pi\)
\(558\) 0 0
\(559\) −731.000 −1.30769
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −1083.00 −1.92362 −0.961812 0.273712i \(-0.911749\pi\)
−0.961812 + 0.273712i \(0.911749\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −969.000 −1.70299 −0.851494 0.524365i \(-0.824303\pi\)
−0.851494 + 0.524365i \(0.824303\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −75.0000 −0.130435
\(576\) 0 0
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 1323.00 2.26930
\(584\) 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
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 909.000 1.51753 0.758765 0.651365i \(-0.225803\pi\)
0.758765 + 0.651365i \(0.225803\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) −819.000 −1.35821
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −1326.00 −2.17021
\(612\) 0 0
\(613\) 538.000 0.877651 0.438825 0.898572i \(-0.355395\pi\)
0.438825 + 0.898572i \(0.355395\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −873.000 −1.41491 −0.707455 0.706758i \(-0.750157\pi\)
−0.707455 + 0.706758i \(0.750157\pi\)
\(618\) 0 0
\(619\) −1066.00 −1.72213 −0.861066 0.508493i \(-0.830203\pi\)
−0.861066 + 0.508493i \(0.830203\pi\)
\(620\) 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
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −833.000 −1.30769
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 262.000 0.407465 0.203733 0.979027i \(-0.434693\pi\)
0.203733 + 0.979027i \(0.434693\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 1134.00 1.74730
\(650\) 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
\(657\) 0 0
\(658\) 0 0
\(659\) 789.000 1.19727 0.598634 0.801022i \(-0.295710\pi\)
0.598634 + 0.801022i \(0.295710\pi\)
\(660\) 0 0
\(661\) 1279.00 1.93495 0.967474 0.252972i \(-0.0814081\pi\)
0.967474 + 0.252972i \(0.0814081\pi\)
\(662\) 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
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 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
\(683\) 741.000 1.08492 0.542460 0.840082i \(-0.317493\pi\)
0.542460 + 0.840082i \(0.317493\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −1071.00 −1.55443
\(690\) 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
\(697\) −351.000 −0.503587
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 714.000 1.01854 0.509272 0.860605i \(-0.329915\pi\)
0.509272 + 0.860605i \(0.329915\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −689.000 −0.971791 −0.485896 0.874017i \(-0.661506\pi\)
−0.485896 + 0.874017i \(0.661506\pi\)
\(710\) 0 0
\(711\) 126.000 0.177215
\(712\) 0 0
\(713\) 57.0000 0.0799439
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −1266.00 −1.76078 −0.880389 0.474251i \(-0.842719\pi\)
−0.880389 + 0.474251i \(0.842719\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 729.000 1.00000
\(730\) 0 0
\(731\) −387.000 −0.529412
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1911.00 −2.59294
\(738\) 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
\(745\) 0 0
\(746\) 0 0
\(747\) −1107.00 −1.48193
\(748\) 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
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 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
\(767\) −918.000 −1.19687
\(768\) 0 0
\(769\) −1214.00 −1.57867 −0.789337 0.613960i \(-0.789575\pi\)
−0.789337 + 0.613960i \(0.789575\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) −475.000 −0.612903
\(776\) 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
\(787\) −26.0000 −0.0330368 −0.0165184 0.999864i \(-0.505258\pi\)
−0.0165184 + 0.999864i \(0.505258\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 906.000 1.13676 0.568381 0.822765i \(-0.307570\pi\)
0.568381 + 0.822765i \(0.307570\pi\)
\(798\) 0 0
\(799\) −702.000 −0.878598
\(800\) 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
\(809\) −1134.00 −1.40173 −0.700865 0.713294i \(-0.747203\pi\)
−0.700865 + 0.713294i \(0.747203\pi\)
\(810\) 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
\(821\) 567.000 0.690621 0.345311 0.938488i \(-0.387774\pi\)
0.345311 + 0.938488i \(0.387774\pi\)
\(822\) 0 0
\(823\) −1603.00 −1.94775 −0.973876 0.227080i \(-0.927082\pi\)
−0.973876 + 0.227080i \(0.927082\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −1482.00 −1.79202 −0.896010 0.444035i \(-0.853547\pi\)
−0.896010 + 0.444035i \(0.853547\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −441.000 −0.529412
\(834\) 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
\(841\) 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
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 1319.00 1.54631 0.773154 0.634219i \(-0.218678\pi\)
0.773154 + 0.634219i \(0.218678\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1038.00 −1.21120 −0.605601 0.795769i \(-0.707067\pi\)
−0.605601 + 0.795769i \(0.707067\pi\)
\(858\) 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
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 294.000 0.338320
\(870\) 0 0
\(871\) 1547.00 1.77612
\(872\) 0 0
\(873\) −1737.00 −1.98969
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −1729.00 −1.97149 −0.985747 0.168235i \(-0.946193\pi\)
−0.985747 + 0.168235i \(0.946193\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1719.00 1.95119 0.975596 0.219574i \(-0.0704666\pi\)
0.975596 + 0.219574i \(0.0704666\pi\)
\(882\) 0 0
\(883\) 1717.00 1.94451 0.972254 0.233929i \(-0.0751583\pi\)
0.972254 + 0.233929i \(0.0751583\pi\)
\(884\) 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
\(891\) 1701.00 1.90909
\(892\) 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
\(901\) −567.000 −0.629301
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 293.000 0.323043 0.161521 0.986869i \(-0.448360\pi\)
0.161521 + 0.986869i \(0.448360\pi\)
\(908\) 0 0
\(909\) 1431.00 1.57426
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) −2583.00 −2.82913
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −763.000 −0.830250 −0.415125 0.909764i \(-0.636262\pi\)
−0.415125 + 0.909764i \(0.636262\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 1629.00 1.75728
\(928\) 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
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1839.00 1.95430 0.977152 0.212542i \(-0.0681742\pi\)
0.977152 + 0.212542i \(0.0681742\pi\)
\(942\) 0 0
\(943\) −117.000 −0.124072
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 213.000 0.224921 0.112460 0.993656i \(-0.464127\pi\)
0.112460 + 0.993656i \(0.464127\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 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
\(961\) −600.000 −0.624350
\(962\) 0 0
\(963\) −378.000 −0.392523
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −1547.00 −1.59979 −0.799897 0.600138i \(-0.795112\pi\)
−0.799897 + 0.600138i \(0.795112\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −867.000 −0.892894 −0.446447 0.894810i \(-0.647311\pi\)
−0.446447 + 0.894810i \(0.647311\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −798.000 −0.816786 −0.408393 0.912806i \(-0.633911\pi\)
−0.408393 + 0.912806i \(0.633911\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −1521.00 −1.55046
\(982\) 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
\(989\) −129.000 −0.130435
\(990\) 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
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 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 688.3.b.a.257.1 1
4.3 odd 2 43.3.b.a.42.1 1
12.11 even 2 387.3.b.a.343.1 1
43.42 odd 2 CM 688.3.b.a.257.1 1
172.171 even 2 43.3.b.a.42.1 1
516.515 odd 2 387.3.b.a.343.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.3.b.a.42.1 1 4.3 odd 2
43.3.b.a.42.1 1 172.171 even 2
387.3.b.a.343.1 1 12.11 even 2
387.3.b.a.343.1 1 516.515 odd 2
688.3.b.a.257.1 1 1.1 even 1 trivial
688.3.b.a.257.1 1 43.42 odd 2 CM