Properties

Label 1521.4.a.y.1.4
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $1$
Dimension $4$
CM discriminant -39
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(89.7419051187\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.8112.1
Defining polynomial: \( x^{4} - 5x^{2} + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 117)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.4
Root \(-0.835000\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.42782 q^{2} +11.6056 q^{4} -20.3925 q^{5} +15.9647 q^{8} +O(q^{10})\) \(q+4.42782 q^{2} +11.6056 q^{4} -20.3925 q^{5} +15.9647 q^{8} -90.2944 q^{10} +70.0332 q^{11} -22.1556 q^{16} -236.667 q^{20} +310.094 q^{22} +290.855 q^{25} -225.819 q^{32} -325.561 q^{40} -486.739 q^{41} -452.000 q^{43} +812.774 q^{44} -71.1653 q^{47} -343.000 q^{49} +1287.85 q^{50} -1428.15 q^{55} -696.914 q^{59} -944.654 q^{61} -822.638 q^{64} -123.807 q^{71} +418.244 q^{79} +451.809 q^{80} -2155.19 q^{82} +1509.37 q^{83} -2001.37 q^{86} +1118.06 q^{88} +155.245 q^{89} -315.107 q^{94} -1518.74 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 32 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 32 q^{4} - 116 q^{10} - 204 q^{16} + 476 q^{22} + 500 q^{25} - 884 q^{40} - 1808 q^{43} - 1372 q^{49} - 3232 q^{55} - 1632 q^{64} - 2924 q^{82} + 2756 q^{88} - 5140 q^{94}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.42782 1.56547 0.782735 0.622356i \(-0.213824\pi\)
0.782735 + 0.622356i \(0.213824\pi\)
\(3\) 0 0
\(4\) 11.6056 1.45069
\(5\) −20.3925 −1.82396 −0.911982 0.410231i \(-0.865448\pi\)
−0.911982 + 0.410231i \(0.865448\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 15.9647 0.705547
\(9\) 0 0
\(10\) −90.2944 −2.85536
\(11\) 70.0332 1.91962 0.959810 0.280652i \(-0.0905506\pi\)
0.959810 + 0.280652i \(0.0905506\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) 0 0
\(16\) −22.1556 −0.346181
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −236.667 −2.64601
\(21\) 0 0
\(22\) 310.094 3.00510
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 290.855 2.32684
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −225.819 −1.24748
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −325.561 −1.28689
\(41\) −486.739 −1.85405 −0.927024 0.375003i \(-0.877642\pi\)
−0.927024 + 0.375003i \(0.877642\pi\)
\(42\) 0 0
\(43\) −452.000 −1.60301 −0.801504 0.597989i \(-0.795967\pi\)
−0.801504 + 0.597989i \(0.795967\pi\)
\(44\) 812.774 2.78478
\(45\) 0 0
\(46\) 0 0
\(47\) −71.1653 −0.220862 −0.110431 0.993884i \(-0.535223\pi\)
−0.110431 + 0.993884i \(0.535223\pi\)
\(48\) 0 0
\(49\) −343.000 −1.00000
\(50\) 1287.85 3.64260
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) −1428.15 −3.50132
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −696.914 −1.53780 −0.768902 0.639367i \(-0.779197\pi\)
−0.768902 + 0.639367i \(0.779197\pi\)
\(60\) 0 0
\(61\) −944.654 −1.98280 −0.991398 0.130879i \(-0.958220\pi\)
−0.991398 + 0.130879i \(0.958220\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −822.638 −1.60672
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −123.807 −0.206947 −0.103474 0.994632i \(-0.532996\pi\)
−0.103474 + 0.994632i \(0.532996\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 418.244 0.595647 0.297824 0.954621i \(-0.403739\pi\)
0.297824 + 0.954621i \(0.403739\pi\)
\(80\) 451.809 0.631422
\(81\) 0 0
\(82\) −2155.19 −2.90245
\(83\) 1509.37 1.99608 0.998040 0.0625815i \(-0.0199333\pi\)
0.998040 + 0.0625815i \(0.0199333\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −2001.37 −2.50946
\(87\) 0 0
\(88\) 1118.06 1.35438
\(89\) 155.245 0.184899 0.0924493 0.995717i \(-0.470530\pi\)
0.0924493 + 0.995717i \(0.470530\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) −315.107 −0.345753
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) −1518.74 −1.56547
\(99\) 0 0
\(100\) 3375.54 3.37554
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) −848.000 −0.811223 −0.405611 0.914046i \(-0.632941\pi\)
−0.405611 + 0.914046i \(0.632941\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) −6323.61 −5.48120
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) −3085.80 −2.40738
\(119\) 0 0
\(120\) 0 0
\(121\) 3573.65 2.68494
\(122\) −4182.76 −3.10401
\(123\) 0 0
\(124\) 0 0
\(125\) −3382.21 −2.42011
\(126\) 0 0
\(127\) −2495.04 −1.74330 −0.871650 0.490129i \(-0.836950\pi\)
−0.871650 + 0.490129i \(0.836950\pi\)
\(128\) −1835.94 −1.26778
\(129\) 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
\(137\) −1082.73 −0.675208 −0.337604 0.941288i \(-0.609616\pi\)
−0.337604 + 0.941288i \(0.609616\pi\)
\(138\) 0 0
\(139\) −340.000 −0.207471 −0.103735 0.994605i \(-0.533079\pi\)
−0.103735 + 0.994605i \(0.533079\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −548.197 −0.323969
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2663.80 1.46461 0.732304 0.680978i \(-0.238445\pi\)
0.732304 + 0.680978i \(0.238445\pi\)
\(150\) 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
\(157\) −934.000 −0.474785 −0.237393 0.971414i \(-0.576293\pi\)
−0.237393 + 0.971414i \(0.576293\pi\)
\(158\) 1851.91 0.932467
\(159\) 0 0
\(160\) 4605.01 2.27536
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) −5648.88 −2.68966
\(165\) 0 0
\(166\) 6683.20 3.12480
\(167\) 3555.90 1.64769 0.823845 0.566815i \(-0.191825\pi\)
0.823845 + 0.566815i \(0.191825\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 0 0
\(172\) −5245.71 −2.32547
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1551.63 −0.664536
\(177\) 0 0
\(178\) 687.398 0.289453
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −4430.00 −1.81922 −0.909611 0.415460i \(-0.863621\pi\)
−0.909611 + 0.415460i \(0.863621\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) −825.912 −0.320403
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −3980.70 −1.45069
\(197\) −3842.18 −1.38956 −0.694782 0.719220i \(-0.744499\pi\)
−0.694782 + 0.719220i \(0.744499\pi\)
\(198\) 0 0
\(199\) −5610.24 −1.99849 −0.999244 0.0388706i \(-0.987624\pi\)
−0.999244 + 0.0388706i \(0.987624\pi\)
\(200\) 4643.42 1.64170
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 9925.85 3.38171
\(206\) −3754.79 −1.26994
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 5740.04 1.87280 0.936399 0.350937i \(-0.114137\pi\)
0.936399 + 0.350937i \(0.114137\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 9217.42 2.92383
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) −16574.5 −5.07934
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1522.31 0.445107 0.222554 0.974920i \(-0.428561\pi\)
0.222554 + 0.974920i \(0.428561\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 1451.24 0.402845
\(236\) −8088.07 −2.23088
\(237\) 0 0
\(238\) 0 0
\(239\) 4191.63 1.13445 0.567226 0.823562i \(-0.308016\pi\)
0.567226 + 0.823562i \(0.308016\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 15823.5 4.20319
\(243\) 0 0
\(244\) −10963.2 −2.87643
\(245\) 6994.64 1.82396
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) −14975.8 −3.78861
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −11047.6 −2.72908
\(255\) 0 0
\(256\) −1548.11 −0.377956
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −4794.11 −1.05702
\(275\) 20369.5 4.46665
\(276\) 0 0
\(277\) −7634.00 −1.65589 −0.827947 0.560806i \(-0.810491\pi\)
−0.827947 + 0.560806i \(0.810491\pi\)
\(278\) −1505.46 −0.324789
\(279\) 0 0
\(280\) 0 0
\(281\) 4539.33 0.963678 0.481839 0.876260i \(-0.339969\pi\)
0.481839 + 0.876260i \(0.339969\pi\)
\(282\) 0 0
\(283\) 490.355 0.102999 0.0514993 0.998673i \(-0.483600\pi\)
0.0514993 + 0.998673i \(0.483600\pi\)
\(284\) −1436.85 −0.300217
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −4913.00 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1396.60 −0.278465 −0.139232 0.990260i \(-0.544463\pi\)
−0.139232 + 0.990260i \(0.544463\pi\)
\(294\) 0 0
\(295\) 14211.8 2.80490
\(296\) 0 0
\(297\) 0 0
\(298\) 11794.8 2.29280
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 19263.9 3.61655
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −6396.25 −1.15507 −0.577536 0.816365i \(-0.695986\pi\)
−0.577536 + 0.816365i \(0.695986\pi\)
\(314\) −4135.58 −0.743262
\(315\) 0 0
\(316\) 4853.95 0.864102
\(317\) −8748.31 −1.55001 −0.775006 0.631954i \(-0.782253\pi\)
−0.775006 + 0.631954i \(0.782253\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 16775.7 2.93059
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) −7770.66 −1.30812
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) 17517.0 2.89570
\(333\) 0 0
\(334\) 15744.9 2.57941
\(335\) 0 0
\(336\) 0 0
\(337\) 10420.0 1.68432 0.842160 0.539228i \(-0.181284\pi\)
0.842160 + 0.539228i \(0.181284\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) −7216.05 −1.13100
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −15814.8 −2.39469
\(353\) −11054.2 −1.66672 −0.833361 0.552728i \(-0.813587\pi\)
−0.833361 + 0.552728i \(0.813587\pi\)
\(354\) 0 0
\(355\) 2524.75 0.377464
\(356\) 1801.71 0.268231
\(357\) 0 0
\(358\) 0 0
\(359\) 7897.23 1.16100 0.580502 0.814259i \(-0.302857\pi\)
0.580502 + 0.814259i \(0.302857\pi\)
\(360\) 0 0
\(361\) −6859.00 −1.00000
\(362\) −19615.2 −2.84794
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −8296.00 −1.17997 −0.589983 0.807416i \(-0.700866\pi\)
−0.589983 + 0.807416i \(0.700866\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −6526.05 −0.905914 −0.452957 0.891532i \(-0.649631\pi\)
−0.452957 + 0.891532i \(0.649631\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −1136.13 −0.155829
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 7961.24 1.06214 0.531071 0.847327i \(-0.321790\pi\)
0.531071 + 0.847327i \(0.321790\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −5475.90 −0.705547
\(393\) 0 0
\(394\) −17012.5 −2.17532
\(395\) −8529.05 −1.08644
\(396\) 0 0
\(397\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(398\) −24841.1 −3.12857
\(399\) 0 0
\(400\) −6444.07 −0.805509
\(401\) −4746.53 −0.591098 −0.295549 0.955328i \(-0.595503\pi\)
−0.295549 + 0.955328i \(0.595503\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 43949.8 5.29397
\(411\) 0 0
\(412\) −9841.51 −1.17684
\(413\) 0 0
\(414\) 0 0
\(415\) −30779.8 −3.64078
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(422\) 25415.8 2.93181
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 40813.1 4.57716
\(431\) 17892.7 1.99968 0.999840 0.0178985i \(-0.00569759\pi\)
0.999840 + 0.0178985i \(0.00569759\pi\)
\(432\) 0 0
\(433\) 36.0555 0.00400166 0.00200083 0.999998i \(-0.499363\pi\)
0.00200083 + 0.999998i \(0.499363\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 11320.0 1.23069 0.615346 0.788257i \(-0.289016\pi\)
0.615346 + 0.788257i \(0.289016\pi\)
\(440\) −22800.1 −2.47034
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) −3165.84 −0.337248
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −19025.4 −1.99969 −0.999846 0.0175361i \(-0.994418\pi\)
−0.999846 + 0.0175361i \(0.994418\pi\)
\(450\) 0 0
\(451\) −34087.9 −3.55906
\(452\) 0 0
\(453\) 0 0
\(454\) 6740.52 0.696802
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 9378.19 0.947475 0.473738 0.880666i \(-0.342905\pi\)
0.473738 + 0.880666i \(0.342905\pi\)
\(462\) 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
\(470\) 6425.82 0.630641
\(471\) 0 0
\(472\) −11126.0 −1.08499
\(473\) −31655.0 −3.07717
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 18559.8 1.77595
\(479\) 20311.6 1.93749 0.968747 0.248050i \(-0.0797898\pi\)
0.968747 + 0.248050i \(0.0797898\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 41474.2 3.89502
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) −15081.1 −1.39896
\(489\) 0 0
\(490\) 30971.0 2.85536
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) −39252.4 −3.51084
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) −28956.3 −2.52900
\(509\) 116.887 0.0101786 0.00508931 0.999987i \(-0.498380\pi\)
0.00508931 + 0.999987i \(0.498380\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 7832.81 0.676102
\(513\) 0 0
\(514\) 0 0
\(515\) 17292.9 1.47964
\(516\) 0 0
\(517\) −4983.93 −0.423971
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) 16148.0 1.35010 0.675050 0.737772i \(-0.264122\pi\)
0.675050 + 0.737772i \(0.264122\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −12167.0 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −24021.4 −1.91962
\(540\) 0 0
\(541\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 23996.0 1.87568 0.937838 0.347073i \(-0.112824\pi\)
0.937838 + 0.347073i \(0.112824\pi\)
\(548\) −12565.6 −0.979520
\(549\) 0 0
\(550\) 90192.6 6.99241
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) −33801.9 −2.59225
\(555\) 0 0
\(556\) −3945.89 −0.300976
\(557\) −13993.2 −1.06447 −0.532237 0.846595i \(-0.678648\pi\)
−0.532237 + 0.846595i \(0.678648\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 20099.3 1.50861
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 2171.20 0.161241
\(567\) 0 0
\(568\) −1976.55 −0.146011
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 25411.9 1.86244 0.931222 0.364451i \(-0.118743\pi\)
0.931222 + 0.364451i \(0.118743\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) −21753.9 −1.56547
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) −6183.88 −0.435928
\(587\) 20858.8 1.46667 0.733335 0.679868i \(-0.237963\pi\)
0.733335 + 0.679868i \(0.237963\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 62927.4 4.39098
\(591\) 0 0
\(592\) 0 0
\(593\) 12660.4 0.876726 0.438363 0.898798i \(-0.355558\pi\)
0.438363 + 0.898798i \(0.355558\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 30914.8 2.12470
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −12626.6 −0.856991 −0.428495 0.903544i \(-0.640956\pi\)
−0.428495 + 0.903544i \(0.640956\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −72875.8 −4.89723
\(606\) 0 0
\(607\) −25504.0 −1.70540 −0.852698 0.522404i \(-0.825035\pi\)
−0.852698 + 0.522404i \(0.825035\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 85297.0 5.66160
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 6923.04 0.451719 0.225860 0.974160i \(-0.427481\pi\)
0.225860 + 0.974160i \(0.427481\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 32614.9 2.08735
\(626\) −28321.4 −1.80823
\(627\) 0 0
\(628\) −10839.6 −0.688768
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 6677.15 0.420257
\(633\) 0 0
\(634\) −38735.9 −2.42650
\(635\) 50880.2 3.17972
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 37439.5 2.31239
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) −48807.1 −2.95200
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 10784.0 0.641836
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 24096.6 1.40833
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 41268.2 2.39029
\(669\) 0 0
\(670\) 0 0
\(671\) −66157.2 −3.80622
\(672\) 0 0
\(673\) 25522.0 1.46181 0.730907 0.682477i \(-0.239097\pi\)
0.730907 + 0.682477i \(0.239097\pi\)
\(674\) 46138.0 2.63675
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 14207.2 0.795936 0.397968 0.917399i \(-0.369715\pi\)
0.397968 + 0.917399i \(0.369715\pi\)
\(684\) 0 0
\(685\) 22079.5 1.23155
\(686\) 0 0
\(687\) 0 0
\(688\) 10014.3 0.554931
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 6933.46 0.378419
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −57612.0 −3.08428
\(705\) 0 0
\(706\) −48945.8 −2.60920
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(710\) 11179.1 0.590908
\(711\) 0 0
\(712\) 2478.45 0.130455
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 34967.5 1.81751
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −30370.4 −1.56547
\(723\) 0 0
\(724\) −51412.6 −2.63914
\(725\) 0 0
\(726\) 0 0
\(727\) −28455.0 −1.45163 −0.725817 0.687888i \(-0.758538\pi\)
−0.725817 + 0.687888i \(0.758538\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) −36733.2 −1.84720
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 35259.5 1.74097 0.870487 0.492191i \(-0.163804\pi\)
0.870487 + 0.492191i \(0.163804\pi\)
\(744\) 0 0
\(745\) −54321.5 −2.67139
\(746\) −28896.1 −1.41818
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −26120.0 −1.26915 −0.634575 0.772861i \(-0.718825\pi\)
−0.634575 + 0.772861i \(0.718825\pi\)
\(752\) 1576.71 0.0764583
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 35009.9 1.68092 0.840460 0.541873i \(-0.182285\pi\)
0.840460 + 0.541873i \(0.182285\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 30422.5 1.44916 0.724582 0.689189i \(-0.242033\pi\)
0.724582 + 0.689189i \(0.242033\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 35250.9 1.66275
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −42841.8 −1.99342 −0.996710 0.0810548i \(-0.974171\pi\)
−0.996710 + 0.0810548i \(0.974171\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −8670.64 −0.397260
\(782\) 0 0
\(783\) 0 0
\(784\) 7599.37 0.346181
\(785\) 19046.6 0.865991
\(786\) 0 0
\(787\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) −44590.6 −2.01583
\(789\) 0 0
\(790\) −37765.1 −1.70079
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) −65109.9 −2.89920
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −65680.5 −2.90270
\(801\) 0 0
\(802\) −21016.8 −0.925346
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 115195. 4.90583
\(821\) −33486.7 −1.42350 −0.711749 0.702433i \(-0.752097\pi\)
−0.711749 + 0.702433i \(0.752097\pi\)
\(822\) 0 0
\(823\) −37352.0 −1.58203 −0.791014 0.611798i \(-0.790446\pi\)
−0.791014 + 0.611798i \(0.790446\pi\)
\(824\) −13538.1 −0.572356
\(825\) 0 0
\(826\) 0 0
\(827\) −35931.1 −1.51082 −0.755409 0.655253i \(-0.772562\pi\)
−0.755409 + 0.655253i \(0.772562\pi\)
\(828\) 0 0
\(829\) −3079.14 −0.129002 −0.0645012 0.997918i \(-0.520546\pi\)
−0.0645012 + 0.997918i \(0.520546\pi\)
\(830\) −136287. −5.69952
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −72513.9 −3.00533
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 12857.9 0.529086 0.264543 0.964374i \(-0.414779\pi\)
0.264543 + 0.964374i \(0.414779\pi\)
\(840\) 0 0
\(841\) −24389.0 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 66616.3 2.71686
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) −43324.3 −1.72085 −0.860423 0.509581i \(-0.829800\pi\)
−0.860423 + 0.509581i \(0.829800\pi\)
\(860\) 106973. 4.24158
\(861\) 0 0
\(862\) 79225.7 3.13044
\(863\) −44880.2 −1.77027 −0.885133 0.465339i \(-0.845932\pi\)
−0.885133 + 0.465339i \(0.845932\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 159.647 0.00626447
\(867\) 0 0
\(868\) 0 0
\(869\) 29291.0 1.14342
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(878\) 50122.9 1.92661
\(879\) 0 0
\(880\) 31641.6 1.21209
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 52410.3 1.99745 0.998724 0.0504988i \(-0.0160811\pi\)
0.998724 + 0.0504988i \(0.0160811\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −14017.8 −0.527952
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −84240.8 −3.13046
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) −150935. −5.57161
\(903\) 0 0
\(904\) 0 0
\(905\) 90338.9 3.31820
\(906\) 0 0
\(907\) 35017.1 1.28195 0.640973 0.767564i \(-0.278531\pi\)
0.640973 + 0.767564i \(0.278531\pi\)
\(908\) 17667.3 0.645715
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 105706. 3.83171
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 36762.2 1.31956 0.659779 0.751460i \(-0.270650\pi\)
0.659779 + 0.751460i \(0.270650\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 41524.9 1.48324
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 56151.6 1.98307 0.991536 0.129832i \(-0.0414439\pi\)
0.991536 + 0.129832i \(0.0414439\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −13275.6 −0.462856 −0.231428 0.972852i \(-0.574340\pi\)
−0.231428 + 0.972852i \(0.574340\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 16842.4 0.584404
\(941\) 55622.6 1.92693 0.963467 0.267826i \(-0.0863052\pi\)
0.963467 + 0.267826i \(0.0863052\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 15440.5 0.532359
\(945\) 0 0
\(946\) −140163. −4.81721
\(947\) −14319.3 −0.491356 −0.245678 0.969352i \(-0.579011\pi\)
−0.245678 + 0.969352i \(0.579011\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 48646.2 1.64574
\(957\) 0 0
\(958\) 89935.9 3.03309
\(959\) 0 0
\(960\) 0 0
\(961\) −29791.0 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 57052.4 1.89435
\(969\) 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
\(976\) 20929.4 0.686407
\(977\) 31890.7 1.04429 0.522146 0.852856i \(-0.325132\pi\)
0.522146 + 0.852856i \(0.325132\pi\)
\(978\) 0 0
\(979\) 10872.3 0.354935
\(980\) 81176.6 2.64601
\(981\) 0 0
\(982\) 0 0
\(983\) −11061.5 −0.358908 −0.179454 0.983766i \(-0.557433\pi\)
−0.179454 + 0.983766i \(0.557433\pi\)
\(984\) 0 0
\(985\) 78351.8 2.53451
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −23272.0 −0.745973 −0.372987 0.927837i \(-0.621666\pi\)
−0.372987 + 0.927837i \(0.621666\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 114407. 3.64517
\(996\) 0 0
\(997\) −21634.0 −0.687217 −0.343609 0.939113i \(-0.611649\pi\)
−0.343609 + 0.939113i \(0.611649\pi\)
\(998\) 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 1521.4.a.y.1.4 4
3.2 odd 2 inner 1521.4.a.y.1.1 4
13.5 odd 4 117.4.b.c.64.1 4
13.8 odd 4 117.4.b.c.64.4 yes 4
13.12 even 2 inner 1521.4.a.y.1.1 4
39.5 even 4 117.4.b.c.64.4 yes 4
39.8 even 4 117.4.b.c.64.1 4
39.38 odd 2 CM 1521.4.a.y.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.4.b.c.64.1 4 13.5 odd 4
117.4.b.c.64.1 4 39.8 even 4
117.4.b.c.64.4 yes 4 13.8 odd 4
117.4.b.c.64.4 yes 4 39.5 even 4
1521.4.a.y.1.1 4 3.2 odd 2 inner
1521.4.a.y.1.1 4 13.12 even 2 inner
1521.4.a.y.1.4 4 1.1 even 1 trivial
1521.4.a.y.1.4 4 39.38 odd 2 CM