Properties

Label 324.3.d.d.163.1
Level $324$
Weight $3$
Character 324.163
Self dual yes
Analytic conductor $8.828$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 324.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(8.82836056527\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 163.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 324.163

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{2} +4.00000 q^{4} -1.19615 q^{5} +8.00000 q^{8} +O(q^{10})\) \(q+2.00000 q^{2} +4.00000 q^{4} -1.19615 q^{5} +8.00000 q^{8} -2.39230 q^{10} +25.7846 q^{13} +16.0000 q^{16} +17.9808 q^{17} -4.78461 q^{20} -23.5692 q^{25} +51.5692 q^{26} -56.3731 q^{29} +32.0000 q^{32} +35.9615 q^{34} +55.7846 q^{37} -9.56922 q^{40} -80.0000 q^{41} +49.0000 q^{49} -47.1384 q^{50} +103.138 q^{52} -56.0000 q^{53} -112.746 q^{58} -92.9230 q^{61} +64.0000 q^{64} -30.8423 q^{65} +71.9230 q^{68} +28.1384 q^{73} +111.569 q^{74} -19.1384 q^{80} -160.000 q^{82} -21.5077 q^{85} -147.550 q^{89} -130.000 q^{97} +98.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 8 q^{4} + 8 q^{5} + 16 q^{8} + O(q^{10}) \) \( 2 q + 4 q^{2} + 8 q^{4} + 8 q^{5} + 16 q^{8} + 16 q^{10} + 10 q^{13} + 32 q^{16} - 16 q^{17} + 32 q^{20} + 36 q^{25} + 20 q^{26} - 40 q^{29} + 64 q^{32} - 32 q^{34} + 70 q^{37} + 64 q^{40} - 160 q^{41} + 98 q^{49} + 72 q^{50} + 40 q^{52} - 112 q^{53} - 80 q^{58} + 22 q^{61} + 128 q^{64} - 176 q^{65} - 64 q^{68} - 110 q^{73} + 140 q^{74} + 128 q^{80} - 320 q^{82} - 334 q^{85} - 160 q^{89} - 260 q^{97} + 196 q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).

\(n\) \(163\) \(245\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 1.00000
\(3\) 0 0
\(4\) 4.00000 1.00000
\(5\) −1.19615 −0.239230 −0.119615 0.992820i \(-0.538166\pi\)
−0.119615 + 0.992820i \(0.538166\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 8.00000 1.00000
\(9\) 0 0
\(10\) −2.39230 −0.239230
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 25.7846 1.98343 0.991716 0.128452i \(-0.0410008\pi\)
0.991716 + 0.128452i \(0.0410008\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 16.0000 1.00000
\(17\) 17.9808 1.05769 0.528846 0.848718i \(-0.322625\pi\)
0.528846 + 0.848718i \(0.322625\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) −4.78461 −0.239230
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −23.5692 −0.942769
\(26\) 51.5692 1.98343
\(27\) 0 0
\(28\) 0 0
\(29\) −56.3731 −1.94390 −0.971949 0.235190i \(-0.924429\pi\)
−0.971949 + 0.235190i \(0.924429\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 32.0000 1.00000
\(33\) 0 0
\(34\) 35.9615 1.05769
\(35\) 0 0
\(36\) 0 0
\(37\) 55.7846 1.50769 0.753846 0.657051i \(-0.228196\pi\)
0.753846 + 0.657051i \(0.228196\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −9.56922 −0.239230
\(41\) −80.0000 −1.95122 −0.975610 0.219512i \(-0.929553\pi\)
−0.975610 + 0.219512i \(0.929553\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−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
\(50\) −47.1384 −0.942769
\(51\) 0 0
\(52\) 103.138 1.98343
\(53\) −56.0000 −1.05660 −0.528302 0.849057i \(-0.677171\pi\)
−0.528302 + 0.849057i \(0.677171\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −112.746 −1.94390
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −92.9230 −1.52333 −0.761664 0.647972i \(-0.775617\pi\)
−0.761664 + 0.647972i \(0.775617\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 64.0000 1.00000
\(65\) −30.8423 −0.474497
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 71.9230 1.05769
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 28.1384 0.385458 0.192729 0.981252i \(-0.438266\pi\)
0.192729 + 0.981252i \(0.438266\pi\)
\(74\) 111.569 1.50769
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) −19.1384 −0.239230
\(81\) 0 0
\(82\) −160.000 −1.95122
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) −21.5077 −0.253032
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −147.550 −1.65786 −0.828932 0.559349i \(-0.811051\pi\)
−0.828932 + 0.559349i \(0.811051\pi\)
\(90\) 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 \(0.733750\pi\)
\(98\) 98.0000 1.00000
\(99\) 0 0
\(100\) −94.2769 −0.942769
\(101\) 40.0000 0.396040 0.198020 0.980198i \(-0.436549\pi\)
0.198020 + 0.980198i \(0.436549\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 206.277 1.98343
\(105\) 0 0
\(106\) −112.000 −1.05660
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) −194.923 −1.78828 −0.894142 0.447783i \(-0.852214\pi\)
−0.894142 + 0.447783i \(0.852214\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 137.981 1.22107 0.610534 0.791990i \(-0.290955\pi\)
0.610534 + 0.791990i \(0.290955\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −225.492 −1.94390
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 121.000 1.00000
\(122\) −185.846 −1.52333
\(123\) 0 0
\(124\) 0 0
\(125\) 58.0962 0.464770
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 128.000 1.00000
\(129\) 0 0
\(130\) −61.6846 −0.474497
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 143.846 1.05769
\(137\) 269.865 1.96982 0.984910 0.173067i \(-0.0553679\pi\)
0.984910 + 0.173067i \(0.0553679\pi\)
\(138\) 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
\(145\) 67.4308 0.465040
\(146\) 56.2769 0.385458
\(147\) 0 0
\(148\) 223.138 1.50769
\(149\) −51.6654 −0.346748 −0.173374 0.984856i \(-0.555467\pi\)
−0.173374 + 0.984856i \(0.555467\pi\)
\(150\) 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
\(157\) −313.631 −1.99765 −0.998824 0.0484851i \(-0.984561\pi\)
−0.998824 + 0.0484851i \(0.984561\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −38.2769 −0.239230
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) −320.000 −1.95122
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 495.846 2.93400
\(170\) −43.0155 −0.253032
\(171\) 0 0
\(172\) 0 0
\(173\) −233.788 −1.35138 −0.675689 0.737187i \(-0.736154\pi\)
−0.675689 + 0.737187i \(0.736154\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) −295.100 −1.65786
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 38.0000 0.209945 0.104972 0.994475i \(-0.466525\pi\)
0.104972 + 0.994475i \(0.466525\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −66.7269 −0.360686
\(186\) 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
\(193\) −195.985 −1.01546 −0.507732 0.861515i \(-0.669516\pi\)
−0.507732 + 0.861515i \(0.669516\pi\)
\(194\) −260.000 −1.34021
\(195\) 0 0
\(196\) 196.000 1.00000
\(197\) 365.750 1.85660 0.928299 0.371834i \(-0.121271\pi\)
0.928299 + 0.371834i \(0.121271\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −188.554 −0.942769
\(201\) 0 0
\(202\) 80.0000 0.396040
\(203\) 0 0
\(204\) 0 0
\(205\) 95.6922 0.466791
\(206\) 0 0
\(207\) 0 0
\(208\) 412.554 1.98343
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) −224.000 −1.05660
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −389.846 −1.78828
\(219\) 0 0
\(220\) 0 0
\(221\) 463.627 2.09786
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 275.962 1.22107
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 117.077 0.511253 0.255627 0.966776i \(-0.417718\pi\)
0.255627 + 0.966776i \(0.417718\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −450.985 −1.94390
\(233\) 389.865 1.67324 0.836621 0.547782i \(-0.184528\pi\)
0.836621 + 0.547782i \(0.184528\pi\)
\(234\) 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
\(241\) 416.846 1.72965 0.864826 0.502072i \(-0.167429\pi\)
0.864826 + 0.502072i \(0.167429\pi\)
\(242\) 242.000 1.00000
\(243\) 0 0
\(244\) −371.692 −1.52333
\(245\) −58.6115 −0.239230
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 116.192 0.464770
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 256.000 1.00000
\(257\) −473.673 −1.84309 −0.921543 0.388277i \(-0.873070\pi\)
−0.921543 + 0.388277i \(0.873070\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −123.369 −0.474497
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 66.9845 0.252772
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −140.488 −0.522262 −0.261131 0.965303i \(-0.584095\pi\)
−0.261131 + 0.965303i \(0.584095\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 287.692 1.05769
\(273\) 0 0
\(274\) 539.731 1.96982
\(275\) 0 0
\(276\) 0 0
\(277\) 230.000 0.830325 0.415162 0.909747i \(-0.363725\pi\)
0.415162 + 0.909747i \(0.363725\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 560.104 1.99325 0.996626 0.0820785i \(-0.0261558\pi\)
0.996626 + 0.0820785i \(0.0261558\pi\)
\(282\) 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
\(289\) 34.3078 0.118712
\(290\) 134.862 0.465040
\(291\) 0 0
\(292\) 112.554 0.385458
\(293\) 425.634 1.45268 0.726339 0.687337i \(-0.241220\pi\)
0.726339 + 0.687337i \(0.241220\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 446.277 1.50769
\(297\) 0 0
\(298\) −103.331 −0.346748
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 111.150 0.364427
\(306\) 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
\(313\) −565.400 −1.80639 −0.903195 0.429231i \(-0.858785\pi\)
−0.903195 + 0.429231i \(0.858785\pi\)
\(314\) −627.261 −1.99765
\(315\) 0 0
\(316\) 0 0
\(317\) −437.904 −1.38140 −0.690700 0.723141i \(-0.742697\pi\)
−0.690700 + 0.723141i \(0.742697\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −76.5538 −0.239230
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −607.723 −1.86992
\(326\) 0 0
\(327\) 0 0
\(328\) −640.000 −1.95122
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 350.000 1.03858 0.519288 0.854599i \(-0.326197\pi\)
0.519288 + 0.854599i \(0.326197\pi\)
\(338\) 991.692 2.93400
\(339\) 0 0
\(340\) −86.0309 −0.253032
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −467.577 −1.35138
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) −598.000 −1.71347 −0.856734 0.515759i \(-0.827510\pi\)
−0.856734 + 0.515759i \(0.827510\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 544.000 1.54108 0.770538 0.637394i \(-0.219988\pi\)
0.770538 + 0.637394i \(0.219988\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −590.200 −1.65786
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 361.000 1.00000
\(362\) 76.0000 0.209945
\(363\) 0 0
\(364\) 0 0
\(365\) −33.6579 −0.0922133
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) −133.454 −0.360686
\(371\) 0 0
\(372\) 0 0
\(373\) −550.000 −1.47453 −0.737265 0.675603i \(-0.763883\pi\)
−0.737265 + 0.675603i \(0.763883\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −1453.56 −3.85559
\(378\) 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
\(386\) −391.969 −1.01546
\(387\) 0 0
\(388\) −520.000 −1.34021
\(389\) −680.000 −1.74807 −0.874036 0.485861i \(-0.838506\pi\)
−0.874036 + 0.485861i \(0.838506\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 392.000 1.00000
\(393\) 0 0
\(394\) 731.500 1.85660
\(395\) 0 0
\(396\) 0 0
\(397\) 69.9076 0.176090 0.0880448 0.996117i \(-0.471938\pi\)
0.0880448 + 0.996117i \(0.471938\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −377.108 −0.942769
\(401\) −651.088 −1.62366 −0.811831 0.583893i \(-0.801529\pi\)
−0.811831 + 0.583893i \(0.801529\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 160.000 0.396040
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −183.154 −0.447809 −0.223905 0.974611i \(-0.571880\pi\)
−0.223905 + 0.974611i \(0.571880\pi\)
\(410\) 191.384 0.466791
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 825.108 1.98343
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −698.461 −1.65905 −0.829527 0.558467i \(-0.811390\pi\)
−0.829527 + 0.558467i \(0.811390\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) −448.000 −1.05660
\(425\) −423.793 −0.997159
\(426\) 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
\(433\) 561.677 1.29717 0.648587 0.761140i \(-0.275360\pi\)
0.648587 + 0.761140i \(0.275360\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −779.692 −1.78828
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 927.254 2.09786
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 176.492 0.396612
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −560.000 −1.24722 −0.623608 0.781737i \(-0.714334\pi\)
−0.623608 + 0.781737i \(0.714334\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 551.923 1.22107
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 134.015 0.293250 0.146625 0.989192i \(-0.453159\pi\)
0.146625 + 0.989192i \(0.453159\pi\)
\(458\) 234.154 0.511253
\(459\) 0 0
\(460\) 0 0
\(461\) 760.000 1.64859 0.824295 0.566161i \(-0.191572\pi\)
0.824295 + 0.566161i \(0.191572\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) −901.969 −1.94390
\(465\) 0 0
\(466\) 779.731 1.67324
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 1438.38 2.99040
\(482\) 833.692 1.72965
\(483\) 0 0
\(484\) 484.000 1.00000
\(485\) 155.500 0.320618
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) −743.384 −1.52333
\(489\) 0 0
\(490\) −117.223 −0.239230
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) −1013.63 −2.05605
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 232.385 0.464770
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) −47.8461 −0.0947447
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −440.000 −0.864440 −0.432220 0.901768i \(-0.642270\pi\)
−0.432220 + 0.901768i \(0.642270\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 512.000 1.00000
\(513\) 0 0
\(514\) −947.346 −1.84309
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) −246.739 −0.474497
\(521\) 880.000 1.68906 0.844530 0.535509i \(-0.179880\pi\)
0.844530 + 0.535509i \(0.179880\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 529.000 1.00000
\(530\) 133.969 0.252772
\(531\) 0 0
\(532\) 0 0
\(533\) −2062.77 −3.87011
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) −280.977 −0.522262
\(539\) 0 0
\(540\) 0 0
\(541\) −386.461 −0.714346 −0.357173 0.934038i \(-0.616259\pi\)
−0.357173 + 0.934038i \(0.616259\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 575.384 1.05769
\(545\) 233.158 0.427812
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 1079.46 1.96982
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 460.000 0.830325
\(555\) 0 0
\(556\) 0 0
\(557\) 246.212 0.442032 0.221016 0.975270i \(-0.429063\pi\)
0.221016 + 0.975270i \(0.429063\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 1120.21 1.99325
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) −165.046 −0.292117
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 920.104 1.61705 0.808527 0.588459i \(-0.200265\pi\)
0.808527 + 0.588459i \(0.200265\pi\)
\(570\) 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
\(577\) 658.138 1.14062 0.570311 0.821429i \(-0.306823\pi\)
0.570311 + 0.821429i \(0.306823\pi\)
\(578\) 68.6156 0.118712
\(579\) 0 0
\(580\) 269.723 0.465040
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 225.108 0.385458
\(585\) 0 0
\(586\) 851.269 1.45268
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 892.554 1.50769
\(593\) 437.404 0.737612 0.368806 0.929506i \(-0.379767\pi\)
0.368806 + 0.929506i \(0.379767\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −206.662 −0.346748
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 135.308 0.225138 0.112569 0.993644i \(-0.464092\pi\)
0.112569 + 0.993644i \(0.464092\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −144.734 −0.239230
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 222.300 0.364427
\(611\) 0 0
\(612\) 0 0
\(613\) −70.0000 −0.114192 −0.0570962 0.998369i \(-0.518184\pi\)
−0.0570962 + 0.998369i \(0.518184\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −426.135 −0.690656 −0.345328 0.938482i \(-0.612232\pi\)
−0.345328 + 0.938482i \(0.612232\pi\)
\(618\) 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
\(625\) 519.739 0.831582
\(626\) −1130.80 −1.80639
\(627\) 0 0
\(628\) −1254.52 −1.99765
\(629\) 1003.05 1.59467
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −875.808 −1.38140
\(635\) 0 0
\(636\) 0 0
\(637\) 1263.45 1.98343
\(638\) 0 0
\(639\) 0 0
\(640\) −153.108 −0.239230
\(641\) 854.819 1.33357 0.666785 0.745250i \(-0.267670\pi\)
0.666785 + 0.745250i \(0.267670\pi\)
\(642\) 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
\(650\) −1215.45 −1.86992
\(651\) 0 0
\(652\) 0 0
\(653\) 1144.00 1.75191 0.875957 0.482389i \(-0.160231\pi\)
0.875957 + 0.482389i \(0.160231\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −1280.00 −1.95122
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −69.3848 −0.104969 −0.0524847 0.998622i \(-0.516714\pi\)
−0.0524847 + 0.998622i \(0.516714\pi\)
\(662\) 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
\(673\) 571.092 0.848577 0.424288 0.905527i \(-0.360524\pi\)
0.424288 + 0.905527i \(0.360524\pi\)
\(674\) 700.000 1.03858
\(675\) 0 0
\(676\) 1983.38 2.93400
\(677\) −104.000 −0.153619 −0.0768095 0.997046i \(-0.524473\pi\)
−0.0768095 + 0.997046i \(0.524473\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −172.062 −0.253032
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) −322.800 −0.471241
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −1443.94 −2.09570
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) −935.154 −1.35138
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −1438.46 −2.06379
\(698\) −1196.00 −1.71347
\(699\) 0 0
\(700\) 0 0
\(701\) −1387.57 −1.97941 −0.989704 0.143129i \(-0.954284\pi\)
−0.989704 + 0.143129i \(0.954284\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 1088.00 1.54108
\(707\) 0 0
\(708\) 0 0
\(709\) 884.154 1.24704 0.623522 0.781806i \(-0.285701\pi\)
0.623522 + 0.781806i \(0.285701\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −1180.40 −1.65786
\(713\) 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
\(722\) 722.000 1.00000
\(723\) 0 0
\(724\) 152.000 0.209945
\(725\) 1328.67 1.83265
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −67.3157 −0.0922133
\(731\) 0 0
\(732\) 0 0
\(733\) −1450.00 −1.97817 −0.989086 0.147340i \(-0.952929\pi\)
−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 \(\pi\)
\(740\) −266.908 −0.360686
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 61.7997 0.0829526
\(746\) −1100.00 −1.47453
\(747\) 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
\(754\) −2907.12 −3.85559
\(755\) 0 0
\(756\) 0 0
\(757\) 1190.00 1.57199 0.785997 0.618230i \(-0.212150\pi\)
0.785997 + 0.618230i \(0.212150\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 692.450 0.909921 0.454961 0.890512i \(-0.349653\pi\)
0.454961 + 0.890512i \(0.349653\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −1520.23 −1.97689 −0.988446 0.151571i \(-0.951567\pi\)
−0.988446 + 0.151571i \(0.951567\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −783.938 −1.01546
\(773\) 1085.75 1.40459 0.702296 0.711885i \(-0.252158\pi\)
0.702296 + 0.711885i \(0.252158\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −1040.00 −1.34021
\(777\) 0 0
\(778\) −1360.00 −1.74807
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 784.000 1.00000
\(785\) 375.150 0.477898
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 1463.00 1.85660
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −2395.98 −3.02142
\(794\) 139.815 0.176090
\(795\) 0 0
\(796\) 0 0
\(797\) 389.288 0.488442 0.244221 0.969720i \(-0.421468\pi\)
0.244221 + 0.969720i \(0.421468\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −754.215 −0.942769
\(801\) 0 0
\(802\) −1302.18 −1.62366
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 320.000 0.396040
\(809\) −1034.63 −1.27890 −0.639448 0.768835i \(-0.720837\pi\)
−0.639448 + 0.768835i \(0.720837\pi\)
\(810\) 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
\(818\) −366.308 −0.447809
\(819\) 0 0
\(820\) 382.769 0.466791
\(821\) 1443.05 1.75767 0.878837 0.477123i \(-0.158320\pi\)
0.878837 + 0.477123i \(0.158320\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) −1258.00 −1.51749 −0.758745 0.651387i \(-0.774187\pi\)
−0.758745 + 0.651387i \(0.774187\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1650.22 1.98343
\(833\) 881.057 1.05769
\(834\) 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
\(841\) 2336.92 2.77874
\(842\) −1396.92 −1.65905
\(843\) 0 0
\(844\) 0 0
\(845\) −593.108 −0.701902
\(846\) 0 0
\(847\) 0 0
\(848\) −896.000 −1.05660
\(849\) 0 0
\(850\) −847.585 −0.997159
\(851\) 0 0
\(852\) 0 0
\(853\) 410.000 0.480657 0.240328 0.970692i \(-0.422745\pi\)
0.240328 + 0.970692i \(0.422745\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1660.94 1.93809 0.969044 0.246887i \(-0.0794076\pi\)
0.969044 + 0.246887i \(0.0794076\pi\)
\(858\) 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
\(865\) 279.647 0.323291
\(866\) 1123.35 1.29717
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) −1559.38 −1.78828
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −1407.75 −1.60519 −0.802596 0.596523i \(-0.796549\pi\)
−0.802596 + 0.596523i \(0.796549\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1600.00 1.81612 0.908059 0.418842i \(-0.137564\pi\)
0.908059 + 0.418842i \(0.137564\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 1854.51 2.09786
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 352.985 0.396612
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −1120.00 −1.24722
\(899\) 0 0
\(900\) 0 0
\(901\) −1006.92 −1.11756
\(902\) 0 0
\(903\) 0 0
\(904\) 1103.85 1.22107
\(905\) −45.4538 −0.0502252
\(906\) 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
\(914\) 268.031 0.293250
\(915\) 0 0
\(916\) 468.308 0.511253
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 1520.00 1.64859
\(923\) 0 0
\(924\) 0 0
\(925\) −1314.80 −1.42141
\(926\) 0 0
\(927\) 0 0
\(928\) −1803.94 −1.94390
\(929\) −1143.43 −1.23082 −0.615411 0.788206i \(-0.711010\pi\)
−0.615411 + 0.788206i \(0.711010\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 1559.46 1.67324
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1794.63 1.91529 0.957647 0.287945i \(-0.0929721\pi\)
0.957647 + 0.287945i \(0.0929721\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −703.450 −0.747555 −0.373778 0.927518i \(-0.621938\pi\)
−0.373778 + 0.927518i \(0.621938\pi\)
\(942\) 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
\(949\) 725.539 0.764530
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1793.21 −1.88165 −0.940824 0.338895i \(-0.889947\pi\)
−0.940824 + 0.338895i \(0.889947\pi\)
\(954\) 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
\(962\) 2876.77 2.99040
\(963\) 0 0
\(964\) 1667.38 1.72965
\(965\) 234.427 0.242930
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 968.000 1.00000
\(969\) 0 0
\(970\) 311.000 0.320618
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) −1486.77 −1.52333
\(977\) 496.000 0.507677 0.253838 0.967247i \(-0.418307\pi\)
0.253838 + 0.967247i \(0.418307\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −234.446 −0.239230
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) −437.493 −0.444155
\(986\) −2027.26 −2.05605
\(987\) 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
\(997\) −280.677 −0.281522 −0.140761 0.990044i \(-0.544955\pi\)
−0.140761 + 0.990044i \(0.544955\pi\)
\(998\) 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 324.3.d.d.163.1 yes 2
3.2 odd 2 324.3.d.a.163.2 2
4.3 odd 2 CM 324.3.d.d.163.1 yes 2
9.2 odd 6 324.3.f.n.271.1 4
9.4 even 3 324.3.f.k.55.2 4
9.5 odd 6 324.3.f.n.55.1 4
9.7 even 3 324.3.f.k.271.2 4
12.11 even 2 324.3.d.a.163.2 2
36.7 odd 6 324.3.f.k.271.2 4
36.11 even 6 324.3.f.n.271.1 4
36.23 even 6 324.3.f.n.55.1 4
36.31 odd 6 324.3.f.k.55.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
324.3.d.a.163.2 2 3.2 odd 2
324.3.d.a.163.2 2 12.11 even 2
324.3.d.d.163.1 yes 2 1.1 even 1 trivial
324.3.d.d.163.1 yes 2 4.3 odd 2 CM
324.3.f.k.55.2 4 9.4 even 3
324.3.f.k.55.2 4 36.31 odd 6
324.3.f.k.271.2 4 9.7 even 3
324.3.f.k.271.2 4 36.7 odd 6
324.3.f.n.55.1 4 9.5 odd 6
324.3.f.n.55.1 4 36.23 even 6
324.3.f.n.271.1 4 9.2 odd 6
324.3.f.n.271.1 4 36.11 even 6