Properties

Label 2304.3.h.d.2177.2
Level $2304$
Weight $3$
Character 2304.2177
Analytic conductor $62.779$
Analytic rank $0$
Dimension $4$
CM discriminant -4
Inner twists $8$

Related objects

Downloads

Learn more

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(62.7794529086\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 288)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 2177.2
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 2304.2177
Dual form 2304.3.h.d.2177.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.41421 q^{5} +O(q^{10})\) \(q-1.41421 q^{5} +24.0000i q^{13} -32.5269i q^{17} -23.0000 q^{25} -1.41421 q^{29} -70.0000i q^{37} +69.2965i q^{41} -49.0000 q^{49} +103.238 q^{53} -22.0000i q^{61} -33.9411i q^{65} +96.0000 q^{73} +46.0000i q^{85} -168.291i q^{89} -144.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 92q^{25} - 196q^{49} + 384q^{73} - 576q^{97} + O(q^{100}) \)

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
\(5\) −1.41421 −0.282843 −0.141421 0.989949i \(-0.545167\pi\)
−0.141421 + 0.989949i \(0.545167\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 24.0000i 1.84615i 0.384615 + 0.923077i \(0.374334\pi\)
−0.384615 + 0.923077i \(0.625666\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 32.5269i − 1.91335i −0.291162 0.956674i \(-0.594042\pi\)
0.291162 0.956674i \(-0.405958\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −23.0000 −0.920000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.41421 −0.0487660 −0.0243830 0.999703i \(-0.507762\pi\)
−0.0243830 + 0.999703i \(0.507762\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 70.0000i − 1.89189i −0.324324 0.945946i \(-0.605137\pi\)
0.324324 0.945946i \(-0.394863\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 69.2965i 1.69016i 0.534642 + 0.845079i \(0.320447\pi\)
−0.534642 + 0.845079i \(0.679553\pi\)
\(42\) 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
\(53\) 103.238 1.94788 0.973940 0.226808i \(-0.0728289\pi\)
0.973940 + 0.226808i \(0.0728289\pi\)
\(54\) 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
\(65\) − 33.9411i − 0.522171i
\(66\) 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
\(73\) 96.0000 1.31507 0.657534 0.753425i \(-0.271599\pi\)
0.657534 + 0.753425i \(0.271599\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 46.0000i 0.541176i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 168.291i − 1.89091i −0.325746 0.945457i \(-0.605615\pi\)
0.325746 0.945457i \(-0.394385\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −144.000 −1.48454 −0.742268 0.670103i \(-0.766250\pi\)
−0.742268 + 0.670103i \(0.766250\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 168.291 1.66625 0.833126 0.553084i \(-0.186549\pi\)
0.833126 + 0.553084i \(0.186549\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) − 120.000i − 1.10092i −0.834862 0.550459i \(-0.814453\pi\)
0.834862 0.550459i \(-0.185547\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 137.179i − 1.21397i −0.794713 0.606985i \(-0.792379\pi\)
0.794713 0.606985i \(-0.207621\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −121.000 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 67.8823 0.543058
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 272.943i − 1.99229i −0.0877432 0.996143i \(-0.527965\pi\)
0.0877432 0.996143i \(-0.472035\pi\)
\(138\) 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
\(145\) 2.00000 0.0137931
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −270.115 −1.81285 −0.906425 0.422366i \(-0.861200\pi\)
−0.906425 + 0.422366i \(0.861200\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 170.000i − 1.08280i −0.840764 0.541401i \(-0.817894\pi\)
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
\(173\) 306.884 1.77390 0.886949 0.461867i \(-0.152820\pi\)
0.886949 + 0.461867i \(0.152820\pi\)
\(174\) 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
\(181\) − 360.000i − 1.98895i −0.104972 0.994475i \(-0.533475\pi\)
0.104972 0.994475i \(-0.466525\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 98.9949i 0.535108i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 190.000 0.984456 0.492228 0.870466i \(-0.336183\pi\)
0.492228 + 0.870466i \(0.336183\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 236.174 1.19885 0.599426 0.800431i \(-0.295396\pi\)
0.599426 + 0.800431i \(0.295396\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) − 98.0000i − 0.478049i
\(206\) 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
\(221\) 780.646 3.53233
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) − 120.000i − 0.524017i −0.965066 0.262009i \(-0.915615\pi\)
0.965066 0.262009i \(-0.0843849\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 442.649i − 1.89978i −0.312584 0.949890i \(-0.601194\pi\)
0.312584 0.949890i \(-0.398806\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 240.000 0.995851 0.497925 0.867220i \(-0.334095\pi\)
0.497925 + 0.867220i \(0.334095\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 69.2965 0.282843
\(246\) 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
\(257\) − 405.879i − 1.57930i −0.613560 0.789648i \(-0.710263\pi\)
0.613560 0.789648i \(-0.289737\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) −146.000 −0.550943
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −270.115 −1.00414 −0.502072 0.864826i \(-0.667429\pi\)
−0.502072 + 0.864826i \(0.667429\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 504.000i 1.81949i 0.415162 + 0.909747i \(0.363725\pi\)
−0.415162 + 0.909747i \(0.636275\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 100.409i − 0.357328i −0.983910 0.178664i \(-0.942822\pi\)
0.983910 0.178664i \(-0.0571775\pi\)
\(282\) 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
\(289\) −769.000 −2.66090
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 306.884 1.04739 0.523693 0.851907i \(-0.324554\pi\)
0.523693 + 0.851907i \(0.324554\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 31.1127i 0.102009i
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−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
\(317\) −541.644 −1.70866 −0.854328 0.519735i \(-0.826031\pi\)
−0.854328 + 0.519735i \(0.826031\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) − 552.000i − 1.69846i
\(326\) 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
\(337\) −576.000 −1.70920 −0.854599 0.519288i \(-0.826197\pi\)
−0.854599 + 0.519288i \(0.826197\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000i \(-0.5\pi\)
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
\(353\) 66.4680i 0.188295i 0.995558 + 0.0941474i \(0.0300125\pi\)
−0.995558 + 0.0941474i \(0.969988\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 361.000 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −135.765 −0.371958
\(366\) 0 0
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 550.000i 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
\(377\) − 33.9411i − 0.0900295i
\(378\) 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
\(389\) 748.119 1.92319 0.961593 0.274481i \(-0.0885061\pi\)
0.961593 + 0.274481i \(0.0885061\pi\)
\(390\) 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
\(401\) − 507.703i − 1.26609i −0.774114 0.633046i \(-0.781805\pi\)
0.774114 0.633046i \(-0.218195\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 240.000 0.586797 0.293399 0.955990i \(-0.405214\pi\)
0.293399 + 0.955990i \(0.405214\pi\)
\(410\) 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
\(421\) − 840.000i − 1.99525i −0.0688836 0.997625i \(-0.521944\pi\)
0.0688836 0.997625i \(-0.478056\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 748.119i 1.76028i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) −290.000 −0.669746 −0.334873 0.942263i \(-0.608693\pi\)
−0.334873 + 0.942263i \(0.608693\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 238.000i 0.534831i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 100.409i 0.223628i 0.993729 + 0.111814i \(0.0356662\pi\)
−0.993729 + 0.111814i \(0.964334\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 336.000 0.735230 0.367615 0.929978i \(-0.380174\pi\)
0.367615 + 0.929978i \(0.380174\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 168.291 0.365057 0.182529 0.983201i \(-0.441572\pi\)
0.182529 + 0.983201i \(0.441572\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
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
\(481\) 1680.00 3.49272
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 203.647 0.419890
\(486\) 0 0
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 46.0000i 0.0933063i
\(494\) 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
\(505\) −238.000 −0.471287
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 337.997 0.664041 0.332021 0.943272i \(-0.392270\pi\)
0.332021 + 0.943272i \(0.392270\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1016.82i 1.95167i 0.218511 + 0.975835i \(0.429880\pi\)
−0.218511 + 0.975835i \(0.570120\pi\)
\(522\) 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
\(533\) −1663.12 −3.12029
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 840.000i 1.55268i 0.630314 + 0.776340i \(0.282926\pi\)
−0.630314 + 0.776340i \(0.717074\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 169.706i 0.311386i
\(546\) 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
\(557\) −985.707 −1.76967 −0.884836 0.465903i \(-0.845729\pi\)
−0.884836 + 0.465903i \(0.845729\pi\)
\(558\) 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
\(565\) 194.000i 0.343363i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 408.708i − 0.718291i −0.933282 0.359146i \(-0.883068\pi\)
0.933282 0.359146i \(-0.116932\pi\)
\(570\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 137.179i 0.231330i 0.993288 + 0.115665i \(0.0368999\pi\)
−0.993288 + 0.115665i \(0.963100\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −1102.00 −1.83361 −0.916805 0.399334i \(-0.869241\pi\)
−0.916805 + 0.399334i \(0.869241\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 171.120 0.282843
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 70.0000i 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
\(617\) − 711.349i − 1.15292i −0.817127 0.576458i \(-0.804434\pi\)
0.817127 0.576458i \(-0.195566\pi\)
\(618\) 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
\(625\) 479.000 0.766400
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −2276.88 −3.61985
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 1176.00i − 1.84615i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 578.413i 0.902361i 0.892433 + 0.451180i \(0.148997\pi\)
−0.892433 + 0.451180i \(0.851003\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−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
\(653\) −1254.41 −1.92099 −0.960496 0.278295i \(-0.910231\pi\)
−0.960496 + 0.278295i \(0.910231\pi\)
\(654\) 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 \(0.350047\pi\)
−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 \(-0.306142\pi\)
0.572065 + 0.820208i \(0.306142\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 881.055 1.30141 0.650705 0.759330i \(-0.274473\pi\)
0.650705 + 0.759330i \(0.274473\pi\)
\(678\) 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
\(685\) 386.000i 0.563504i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 2477.70i 3.59608i
\(690\) 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
\(697\) 2254.00 3.23386
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1288.35 1.83787 0.918936 0.394406i \(-0.129050\pi\)
0.918936 + 0.394406i \(0.129050\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 1320.00i 1.86178i 0.365303 + 0.930889i \(0.380965\pi\)
−0.365303 + 0.930889i \(0.619035\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 32.5269 0.0448647
\(726\) 0 0
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 216.000i 0.294679i 0.989086 + 0.147340i \(0.0470711\pi\)
−0.989086 + 0.147340i \(0.952929\pi\)
\(734\) 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
\(745\) 382.000 0.512752
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 936.000i 1.23646i 0.785997 + 0.618230i \(0.212150\pi\)
−0.785997 + 0.618230i \(0.787850\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1019.65i 1.33988i 0.742416 + 0.669940i \(0.233680\pi\)
−0.742416 + 0.669940i \(0.766320\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 962.000 1.25098 0.625488 0.780234i \(-0.284900\pi\)
0.625488 + 0.780234i \(0.284900\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 782.060 1.01172 0.505860 0.862615i \(-0.331175\pi\)
0.505860 + 0.862615i \(0.331175\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 240.416i 0.306263i
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 528.000 0.665826
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1593.82 −1.99977 −0.999886 0.0150826i \(-0.995199\pi\)
−0.999886 + 0.0150826i \(0.995199\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 677.408i − 0.837340i −0.908138 0.418670i \(-0.862496\pi\)
0.908138 0.418670i \(-0.137504\pi\)
\(810\) 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
\(821\) −1596.65 −1.94476 −0.972379 0.233406i \(-0.925013\pi\)
−0.972379 + 0.233406i \(0.925013\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) − 1080.00i − 1.30277i −0.758745 0.651387i \(-0.774187\pi\)
0.758745 0.651387i \(-0.225813\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1593.82i 1.91335i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) −839.000 −0.997622
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 575.585 0.681166
\(846\) 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 \(0.0772551\pi\)
−0.970692 + 0.240328i \(0.922745\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1494.82i 1.74425i 0.489282 + 0.872126i \(0.337259\pi\)
−0.489282 + 0.872126i \(0.662741\pi\)
\(858\) 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
\(865\) −434.000 −0.501734
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1610.00i 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
\(881\) − 609.526i − 0.691857i −0.938261 0.345929i \(-0.887564\pi\)
0.938261 0.345929i \(-0.112436\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−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
\(901\) − 3358.00i − 3.72697i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 509.117i 0.562560i
\(906\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 1610.00i 1.74054i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 1118.64i − 1.20414i −0.798445 0.602068i \(-0.794343\pi\)
0.798445 0.602068i \(-0.205657\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −430.000 −0.458911 −0.229456 0.973319i \(-0.573695\pi\)
−0.229456 + 0.973319i \(0.573695\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −1868.18 −1.98531 −0.992655 0.120982i \(-0.961396\pi\)
−0.992655 + 0.120982i \(0.961396\pi\)
\(942\) 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
\(949\) 2304.00i 2.42782i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1899.29i 1.99296i 0.0838437 + 0.996479i \(0.473280\pi\)
−0.0838437 + 0.996479i \(0.526720\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −961.000 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −268.701 −0.278446
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 985.707i − 1.00891i −0.863437 0.504456i \(-0.831693\pi\)
0.863437 0.504456i \(-0.168307\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) −334.000 −0.339086
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 1850.00i − 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.h.d.2177.2 4
3.2 odd 2 inner 2304.3.h.d.2177.4 4
4.3 odd 2 CM 2304.3.h.d.2177.2 4
8.3 odd 2 inner 2304.3.h.d.2177.3 4
8.5 even 2 inner 2304.3.h.d.2177.3 4
12.11 even 2 inner 2304.3.h.d.2177.4 4
16.3 odd 4 576.3.e.d.449.2 2
16.5 even 4 288.3.e.c.161.1 2
16.11 odd 4 288.3.e.c.161.1 2
16.13 even 4 576.3.e.d.449.2 2
24.5 odd 2 inner 2304.3.h.d.2177.1 4
24.11 even 2 inner 2304.3.h.d.2177.1 4
48.5 odd 4 288.3.e.c.161.2 yes 2
48.11 even 4 288.3.e.c.161.2 yes 2
48.29 odd 4 576.3.e.d.449.1 2
48.35 even 4 576.3.e.d.449.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
288.3.e.c.161.1 2 16.5 even 4
288.3.e.c.161.1 2 16.11 odd 4
288.3.e.c.161.2 yes 2 48.5 odd 4
288.3.e.c.161.2 yes 2 48.11 even 4
576.3.e.d.449.1 2 48.29 odd 4
576.3.e.d.449.1 2 48.35 even 4
576.3.e.d.449.2 2 16.3 odd 4
576.3.e.d.449.2 2 16.13 even 4
2304.3.h.d.2177.1 4 24.5 odd 2 inner
2304.3.h.d.2177.1 4 24.11 even 2 inner
2304.3.h.d.2177.2 4 1.1 even 1 trivial
2304.3.h.d.2177.2 4 4.3 odd 2 CM
2304.3.h.d.2177.3 4 8.3 odd 2 inner
2304.3.h.d.2177.3 4 8.5 even 2 inner
2304.3.h.d.2177.4 4 3.2 odd 2 inner
2304.3.h.d.2177.4 4 12.11 even 2 inner