Properties

Label 324.3.d.d.163.2
Level $324$
Weight $3$
Character 324.163
Self dual yes
Analytic conductor $8.828$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $2$

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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 163.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 324.163

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{2} +4.00000 q^{4} +9.19615 q^{5} +8.00000 q^{8} +O(q^{10})\) \(q+2.00000 q^{2} +4.00000 q^{4} +9.19615 q^{5} +8.00000 q^{8} +18.3923 q^{10} -15.7846 q^{13} +16.0000 q^{16} -33.9808 q^{17} +36.7846 q^{20} +59.5692 q^{25} -31.5692 q^{26} +16.3731 q^{29} +32.0000 q^{32} -67.9615 q^{34} +14.2154 q^{37} +73.5692 q^{40} -80.0000 q^{41} +49.0000 q^{49} +119.138 q^{50} -63.1384 q^{52} -56.0000 q^{53} +32.7461 q^{58} +114.923 q^{61} +64.0000 q^{64} -145.158 q^{65} -135.923 q^{68} -138.138 q^{73} +28.4308 q^{74} +147.138 q^{80} -160.000 q^{82} -312.492 q^{85} -12.4500 q^{89} -130.000 q^{97} +98.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 8 q^{4} + 8 q^{5} + 16 q^{8} + O(q^{10}) \) \( 2 q + 4 q^{2} + 8 q^{4} + 8 q^{5} + 16 q^{8} + 16 q^{10} + 10 q^{13} + 32 q^{16} - 16 q^{17} + 32 q^{20} + 36 q^{25} + 20 q^{26} - 40 q^{29} + 64 q^{32} - 32 q^{34} + 70 q^{37} + 64 q^{40} - 160 q^{41} + 98 q^{49} + 72 q^{50} + 40 q^{52} - 112 q^{53} - 80 q^{58} + 22 q^{61} + 128 q^{64} - 176 q^{65} - 64 q^{68} - 110 q^{73} + 140 q^{74} + 128 q^{80} - 320 q^{82} - 334 q^{85} - 160 q^{89} - 260 q^{97} + 196 q^{98} + O(q^{100}) \)

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
\(5\) 9.19615 1.83923 0.919615 0.392820i \(-0.128501\pi\)
0.919615 + 0.392820i \(0.128501\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 8.00000 1.00000
\(9\) 0 0
\(10\) 18.3923 1.83923
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −15.7846 −1.21420 −0.607100 0.794625i \(-0.707667\pi\)
−0.607100 + 0.794625i \(0.707667\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 16.0000 1.00000
\(17\) −33.9808 −1.99887 −0.999434 0.0336351i \(-0.989292\pi\)
−0.999434 + 0.0336351i \(0.989292\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 36.7846 1.83923
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 59.5692 2.38277
\(26\) −31.5692 −1.21420
\(27\) 0 0
\(28\) 0 0
\(29\) 16.3731 0.564589 0.282294 0.959328i \(-0.408905\pi\)
0.282294 + 0.959328i \(0.408905\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 32.0000 1.00000
\(33\) 0 0
\(34\) −67.9615 −1.99887
\(35\) 0 0
\(36\) 0 0
\(37\) 14.2154 0.384200 0.192100 0.981375i \(-0.438470\pi\)
0.192100 + 0.981375i \(0.438470\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 73.5692 1.83923
\(41\) −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
\(50\) 119.138 2.38277
\(51\) 0 0
\(52\) −63.1384 −1.21420
\(53\) −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
\(58\) 32.7461 0.564589
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 114.923 1.88398 0.941992 0.335635i \(-0.108951\pi\)
0.941992 + 0.335635i \(0.108951\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 64.0000 1.00000
\(65\) −145.158 −2.23320
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) −135.923 −1.99887
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −138.138 −1.89231 −0.946154 0.323718i \(-0.895067\pi\)
−0.946154 + 0.323718i \(0.895067\pi\)
\(74\) 28.4308 0.384200
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 147.138 1.83923
\(81\) 0 0
\(82\) −160.000 −1.95122
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) −312.492 −3.67638
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −12.4500 −0.139888 −0.0699439 0.997551i \(-0.522282\pi\)
−0.0699439 + 0.997551i \(0.522282\pi\)
\(90\) 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
\(100\) 238.277 2.38277
\(101\) 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 \(\pi\)
\(104\) −126.277 −1.21420
\(105\) 0 0
\(106\) −112.000 −1.05660
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 12.9230 0.118560 0.0592800 0.998241i \(-0.481120\pi\)
0.0592800 + 0.998241i \(0.481120\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 86.0192 0.761232 0.380616 0.924733i \(-0.375712\pi\)
0.380616 + 0.924733i \(0.375712\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 65.4923 0.564589
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 121.000 1.00000
\(122\) 229.846 1.88398
\(123\) 0 0
\(124\) 0 0
\(125\) 317.904 2.54323
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 128.000 1.00000
\(129\) 0 0
\(130\) −290.315 −2.23320
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) −271.846 −1.99887
\(137\) −93.8653 −0.685148 −0.342574 0.939491i \(-0.611299\pi\)
−0.342574 + 0.939491i \(0.611299\pi\)
\(138\) 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
\(145\) 150.569 1.03841
\(146\) −276.277 −1.89231
\(147\) 0 0
\(148\) 56.8616 0.384200
\(149\) −228.335 −1.53245 −0.766223 0.642574i \(-0.777866\pi\)
−0.766223 + 0.642574i \(0.777866\pi\)
\(150\) 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
\(157\) 143.631 0.914845 0.457423 0.889249i \(-0.348773\pi\)
0.457423 + 0.889249i \(0.348773\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 294.277 1.83923
\(161\) 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
\(169\) 80.1539 0.474283
\(170\) −624.985 −3.67638
\(171\) 0 0
\(172\) 0 0
\(173\) 337.788 1.95253 0.976267 0.216570i \(-0.0694871\pi\)
0.976267 + 0.216570i \(0.0694871\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) −24.9000 −0.139888
\(179\) 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
\(185\) 130.727 0.706632
\(186\) 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
\(193\) 385.985 1.99992 0.999960 0.00895123i \(-0.00284930\pi\)
0.999960 + 0.00895123i \(0.00284930\pi\)
\(194\) −260.000 −1.34021
\(195\) 0 0
\(196\) 196.000 1.00000
\(197\) −309.750 −1.57233 −0.786167 0.618014i \(-0.787938\pi\)
−0.786167 + 0.618014i \(0.787938\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 476.554 2.38277
\(201\) 0 0
\(202\) 80.0000 0.396040
\(203\) 0 0
\(204\) 0 0
\(205\) −735.692 −3.58874
\(206\) 0 0
\(207\) 0 0
\(208\) −252.554 −1.21420
\(209\) 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
\(218\) 25.8461 0.118560
\(219\) 0 0
\(220\) 0 0
\(221\) 536.373 2.42703
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 172.038 0.761232
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 324.923 1.41888 0.709439 0.704767i \(-0.248948\pi\)
0.709439 + 0.704767i \(0.248948\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 130.985 0.564589
\(233\) 26.1347 0.112166 0.0560830 0.998426i \(-0.482139\pi\)
0.0560830 + 0.998426i \(0.482139\pi\)
\(234\) 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
\(241\) 1.15390 0.00478798 0.00239399 0.999997i \(-0.499238\pi\)
0.00239399 + 0.999997i \(0.499238\pi\)
\(242\) 242.000 1.00000
\(243\) 0 0
\(244\) 459.692 1.88398
\(245\) 450.611 1.83923
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 635.808 2.54323
\(251\) 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
\(257\) 409.673 1.59406 0.797029 0.603941i \(-0.206404\pi\)
0.797029 + 0.603941i \(0.206404\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −580.631 −2.23320
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) −514.985 −1.94334
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −379.512 −1.41082 −0.705412 0.708798i \(-0.749238\pi\)
−0.705412 + 0.708798i \(0.749238\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) −543.692 −1.99887
\(273\) 0 0
\(274\) −187.731 −0.685148
\(275\) 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
\(281\) −240.104 −0.854462 −0.427231 0.904143i \(-0.640511\pi\)
−0.427231 + 0.904143i \(0.640511\pi\)
\(282\) 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
\(289\) 865.692 2.99547
\(290\) 301.138 1.03841
\(291\) 0 0
\(292\) −552.554 −1.89231
\(293\) −561.634 −1.91684 −0.958421 0.285359i \(-0.907887\pi\)
−0.958421 + 0.285359i \(0.907887\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 113.723 0.384200
\(297\) 0 0
\(298\) −456.669 −1.53245
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1056.85 3.46508
\(306\) 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
\(313\) 515.400 1.64664 0.823322 0.567574i \(-0.192118\pi\)
0.823322 + 0.567574i \(0.192118\pi\)
\(314\) 287.261 0.914845
\(315\) 0 0
\(316\) 0 0
\(317\) −178.096 −0.561818 −0.280909 0.959734i \(-0.590636\pi\)
−0.280909 + 0.959734i \(0.590636\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 588.554 1.83923
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −940.277 −2.89316
\(326\) 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 \(0.326197\pi\)
\(338\) 160.308 0.474283
\(339\) 0 0
\(340\) −1249.97 −3.67638
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 675.577 1.95253
\(347\) 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
\(356\) −49.8001 −0.139888
\(357\) 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
\(365\) −1270.34 −3.48039
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 261.454 0.706632
\(371\) 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
\(377\) −258.442 −0.685524
\(378\) 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
\(386\) 771.969 1.99992
\(387\) 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
\(394\) −619.500 −1.57233
\(395\) 0 0
\(396\) 0 0
\(397\) −719.908 −1.81337 −0.906685 0.421809i \(-0.861395\pi\)
−0.906685 + 0.421809i \(0.861395\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 953.108 2.38277
\(401\) 731.088 1.82316 0.911581 0.411120i \(-0.134862\pi\)
0.911581 + 0.411120i \(0.134862\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 160.000 0.396040
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −598.846 −1.46417 −0.732086 0.681213i \(-0.761453\pi\)
−0.732086 + 0.681213i \(0.761453\pi\)
\(410\) −1471.38 −3.58874
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) −505.108 −1.21420
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 756.461 1.79682 0.898410 0.439157i \(-0.144723\pi\)
0.898410 + 0.439157i \(0.144723\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) −448.000 −1.05660
\(425\) −2024.21 −4.76284
\(426\) 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
\(433\) −851.677 −1.96692 −0.983460 0.181123i \(-0.942027\pi\)
−0.983460 + 0.181123i \(0.942027\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 51.6922 0.118560
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 1072.75 2.42703
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) −114.492 −0.257286
\(446\) 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
\(452\) 344.077 0.761232
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 715.985 1.56671 0.783353 0.621577i \(-0.213508\pi\)
0.783353 + 0.621577i \(0.213508\pi\)
\(458\) 649.846 1.41888
\(459\) 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 \(\pi\)
\(464\) 261.969 0.564589
\(465\) 0 0
\(466\) 52.2693 0.112166
\(467\) 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
\(481\) −224.384 −0.466496
\(482\) 2.30781 0.00478798
\(483\) 0 0
\(484\) 484.000 1.00000
\(485\) −1195.50 −2.46495
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 919.384 1.88398
\(489\) 0 0
\(490\) 901.223 1.83923
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) −556.369 −1.12854
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 1271.62 2.54323
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 367.846 0.728408
\(506\) 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
\(514\) 819.346 1.59406
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) −1161.26 −2.23320
\(521\) 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
\(530\) −1029.97 −1.94334
\(531\) 0 0
\(532\) 0 0
\(533\) 1262.77 2.36917
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) −759.023 −1.41082
\(539\) 0 0
\(540\) 0 0
\(541\) 1068.46 1.97497 0.987487 0.157698i \(-0.0504073\pi\)
0.987487 + 0.157698i \(0.0504073\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −1087.38 −1.99887
\(545\) 118.842 0.218059
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) −375.461 −0.685148
\(549\) 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
\(557\) 817.788 1.46820 0.734101 0.679040i \(-0.237604\pi\)
0.734101 + 0.679040i \(0.237604\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −480.207 −0.854462
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 791.046 1.40008
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 119.896 0.210714 0.105357 0.994434i \(-0.466401\pi\)
0.105357 + 0.994434i \(0.466401\pi\)
\(570\) 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
\(577\) 491.862 0.852446 0.426223 0.904618i \(-0.359844\pi\)
0.426223 + 0.904618i \(0.359844\pi\)
\(578\) 1731.38 2.99547
\(579\) 0 0
\(580\) 602.277 1.03841
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) −1105.11 −1.89231
\(585\) 0 0
\(586\) −1123.27 −1.91684
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 227.446 0.384200
\(593\) −1173.40 −1.97876 −0.989379 0.145358i \(-0.953567\pi\)
−0.989379 + 0.145358i \(0.953567\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −913.338 −1.53245
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 966.692 1.60847 0.804236 0.594309i \(-0.202575\pi\)
0.804236 + 0.594309i \(0.202575\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1112.73 1.83923
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 2113.70 3.46508
\(611\) 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
\(617\) −789.865 −1.28017 −0.640085 0.768304i \(-0.721101\pi\)
−0.640085 + 0.768304i \(0.721101\pi\)
\(618\) 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
\(625\) 1434.26 2.29482
\(626\) 1030.80 1.64664
\(627\) 0 0
\(628\) 574.523 0.914845
\(629\) −483.050 −0.767965
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −356.192 −0.561818
\(635\) 0 0
\(636\) 0 0
\(637\) −773.446 −1.21420
\(638\) 0 0
\(639\) 0 0
\(640\) 1177.11 1.83923
\(641\) −1254.82 −1.95760 −0.978798 0.204828i \(-0.934336\pi\)
−0.978798 + 0.204828i \(0.934336\pi\)
\(642\) 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
\(650\) −1880.55 −2.89316
\(651\) 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
\(661\) −1108.62 −1.67718 −0.838589 0.544764i \(-0.816619\pi\)
−0.838589 + 0.544764i \(0.816619\pi\)
\(662\) 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
\(673\) −1341.09 −1.99271 −0.996354 0.0853191i \(-0.972809\pi\)
−0.996354 + 0.0853191i \(0.972809\pi\)
\(674\) 700.000 1.03858
\(675\) 0 0
\(676\) 320.616 0.474283
\(677\) −104.000 −0.153619 −0.0768095 0.997046i \(-0.524473\pi\)
−0.0768095 + 0.997046i \(0.524473\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −2499.94 −3.67638
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) −863.200 −1.26015
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 883.938 1.28293
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 1351.15 1.95253
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 2718.46 3.90023
\(698\) −1196.00 −1.71347
\(699\) 0 0
\(700\) 0 0
\(701\) 867.565 1.23761 0.618805 0.785544i \(-0.287617\pi\)
0.618805 + 0.785544i \(0.287617\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 1088.00 1.54108
\(707\) 0 0
\(708\) 0 0
\(709\) −1402.15 −1.97765 −0.988825 0.149082i \(-0.952368\pi\)
−0.988825 + 0.149082i \(0.952368\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −99.6001 −0.139888
\(713\) 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
\(725\) 975.331 1.34528
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −2540.68 −3.48039
\(731\) 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 \(\pi\)
\(740\) 522.908 0.706632
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) −2099.80 −2.81852
\(746\) −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
\(754\) −516.885 −0.685524
\(755\) 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
\(761\) 827.550 1.08745 0.543725 0.839263i \(-0.317013\pi\)
0.543725 + 0.839263i \(0.317013\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 558.230 0.725917 0.362959 0.931805i \(-0.381767\pi\)
0.362959 + 0.931805i \(0.381767\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1543.94 1.99992
\(773\) 410.250 0.530725 0.265362 0.964149i \(-0.414508\pi\)
0.265362 + 0.964149i \(0.414508\pi\)
\(774\) 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
\(785\) 1320.85 1.68261
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) −1239.00 −1.57233
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −1814.02 −2.28754
\(794\) −1439.82 −1.81337
\(795\) 0 0
\(796\) 0 0
\(797\) −1533.29 −1.92382 −0.961912 0.273358i \(-0.911866\pi\)
−0.961912 + 0.273358i \(0.911866\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1906.22 2.38277
\(801\) 0 0
\(802\) 1462.18 1.82316
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 320.000 0.396040
\(809\) 1594.63 1.97111 0.985554 0.169361i \(-0.0541703\pi\)
0.985554 + 0.169361i \(0.0541703\pi\)
\(810\) 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
\(818\) −1197.69 −1.46417
\(819\) 0 0
\(820\) −2942.77 −3.58874
\(821\) −43.0498 −0.0524358 −0.0262179 0.999656i \(-0.508346\pi\)
−0.0262179 + 0.999656i \(0.508346\pi\)
\(822\) 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
\(832\) −1010.22 −1.21420
\(833\) −1665.06 −1.99887
\(834\) 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
\(841\) −572.923 −0.681240
\(842\) 1512.92 1.79682
\(843\) 0 0
\(844\) 0 0
\(845\) 737.108 0.872317
\(846\) 0 0
\(847\) 0 0
\(848\) −896.000 −1.05660
\(849\) 0 0
\(850\) −4048.41 −4.76284
\(851\) 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
\(857\) −1196.94 −1.39667 −0.698333 0.715774i \(-0.746074\pi\)
−0.698333 + 0.715774i \(0.746074\pi\)
\(858\) 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
\(865\) 3106.35 3.59116
\(866\) −1703.35 −1.96692
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 103.384 0.118560
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −202.246 −0.230612 −0.115306 0.993330i \(-0.536785\pi\)
−0.115306 + 0.993330i \(0.536785\pi\)
\(878\) 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 \(\pi\)
\(884\) 2145.49 2.42703
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −228.985 −0.257286
\(891\) 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
\(901\) 1902.92 2.11201
\(902\) 0 0
\(903\) 0 0
\(904\) 688.154 0.761232
\(905\) 349.454 0.386137
\(906\) 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
\(914\) 1431.97 1.56671
\(915\) 0 0
\(916\) 1299.69 1.41888
\(917\) 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
\(925\) 846.800 0.915459
\(926\) 0 0
\(927\) 0 0
\(928\) 523.938 0.564589
\(929\) −696.565 −0.749801 −0.374901 0.927065i \(-0.622323\pi\)
−0.374901 + 0.927065i \(0.622323\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 104.539 0.112166
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −1364.63 −1.45638 −0.728191 0.685374i \(-0.759639\pi\)
−0.728191 + 0.685374i \(0.759639\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1863.45 1.98029 0.990143 0.140058i \(-0.0447290\pi\)
0.990143 + 0.140058i \(0.0447290\pi\)
\(942\) 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
\(949\) 2180.46 2.29764
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 337.211 0.353842 0.176921 0.984225i \(-0.443386\pi\)
0.176921 + 0.984225i \(0.443386\pi\)
\(954\) 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
\(962\) −448.769 −0.466496
\(963\) 0 0
\(964\) 4.61561 0.00478798
\(965\) 3549.57 3.67831
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 968.000 1.00000
\(969\) 0 0
\(970\) −2391.00 −2.46495
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 1838.77 1.88398
\(977\) 496.000 0.507677 0.253838 0.967247i \(-0.418307\pi\)
0.253838 + 0.967247i \(0.418307\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 1802.45 1.83923
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) −2848.51 −2.89189
\(986\) −1112.74 −1.12854
\(987\) 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
\(997\) −1569.32 −1.57405 −0.787023 0.616924i \(-0.788378\pi\)
−0.787023 + 0.616924i \(0.788378\pi\)
\(998\) 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.163.2 yes 2
3.2 odd 2 324.3.d.a.163.1 2
4.3 odd 2 CM 324.3.d.d.163.2 yes 2
9.2 odd 6 324.3.f.n.271.2 4
9.4 even 3 324.3.f.k.55.1 4
9.5 odd 6 324.3.f.n.55.2 4
9.7 even 3 324.3.f.k.271.1 4
12.11 even 2 324.3.d.a.163.1 2
36.7 odd 6 324.3.f.k.271.1 4
36.11 even 6 324.3.f.n.271.2 4
36.23 even 6 324.3.f.n.55.2 4
36.31 odd 6 324.3.f.k.55.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
324.3.d.a.163.1 2 3.2 odd 2
324.3.d.a.163.1 2 12.11 even 2
324.3.d.d.163.2 yes 2 1.1 even 1 trivial
324.3.d.d.163.2 yes 2 4.3 odd 2 CM
324.3.f.k.55.1 4 9.4 even 3
324.3.f.k.55.1 4 36.31 odd 6
324.3.f.k.271.1 4 9.7 even 3
324.3.f.k.271.1 4 36.7 odd 6
324.3.f.n.55.2 4 9.5 odd 6
324.3.f.n.55.2 4 36.23 even 6
324.3.f.n.271.2 4 9.2 odd 6
324.3.f.n.271.2 4 36.11 even 6