Properties

Label 1521.4.a.y.1.2
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $1$
Dimension $4$
CM discriminant -39
Inner twists $4$

Related objects

Downloads

Learn more

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 1.2
Root \(2.07431\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.52058 q^{2} +4.39445 q^{4} -9.17304 q^{5} +12.6936 q^{8} +O(q^{10})\) \(q-3.52058 q^{2} +4.39445 q^{4} -9.17304 q^{5} +12.6936 q^{8} +32.2944 q^{10} +20.4780 q^{11} -79.8444 q^{16} -40.3104 q^{20} -72.0942 q^{22} -40.8554 q^{25} +179.549 q^{32} -116.439 q^{40} -196.898 q^{41} -452.000 q^{43} +89.9894 q^{44} +640.490 q^{47} -343.000 q^{49} +143.834 q^{50} -187.845 q^{55} +579.506 q^{59} +944.654 q^{61} +6.63840 q^{64} +1190.09 q^{71} -418.244 q^{79} +732.416 q^{80} +693.193 q^{82} +94.6440 q^{83} +1591.30 q^{86} +259.939 q^{88} -1672.06 q^{89} -2254.89 q^{94} +1207.56 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 32 q^{4}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.52058 −1.24471 −0.622356 0.782735i \(-0.713824\pi\)
−0.622356 + 0.782735i \(0.713824\pi\)
\(3\) 0 0
\(4\) 4.39445 0.549306
\(5\) −9.17304 −0.820462 −0.410231 0.911982i \(-0.634552\pi\)
−0.410231 + 0.911982i \(0.634552\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 12.6936 0.560984
\(9\) 0 0
\(10\) 32.2944 1.02124
\(11\) 20.4780 0.561304 0.280652 0.959810i \(-0.409449\pi\)
0.280652 + 0.959810i \(0.409449\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) 0 0
\(16\) −79.8444 −1.24757
\(17\) 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 \(0.5\pi\)
\(20\) −40.3104 −0.450685
\(21\) 0 0
\(22\) −72.0942 −0.698661
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −40.8554 −0.326843
\(26\) 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 \(0.5\pi\)
\(32\) 179.549 0.991879
\(33\) 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
\(40\) −116.439 −0.460266
\(41\) −196.898 −0.750006 −0.375003 0.927024i \(-0.622358\pi\)
−0.375003 + 0.927024i \(0.622358\pi\)
\(42\) 0 0
\(43\) −452.000 −1.60301 −0.801504 0.597989i \(-0.795967\pi\)
−0.801504 + 0.597989i \(0.795967\pi\)
\(44\) 89.9894 0.308327
\(45\) 0 0
\(46\) 0 0
\(47\) 640.490 1.98777 0.993884 0.110431i \(-0.0352231\pi\)
0.993884 + 0.110431i \(0.0352231\pi\)
\(48\) 0 0
\(49\) −343.000 −1.00000
\(50\) 143.834 0.406825
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) −187.845 −0.460528
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 579.506 1.27873 0.639367 0.768902i \(-0.279197\pi\)
0.639367 + 0.768902i \(0.279197\pi\)
\(60\) 0 0
\(61\) 944.654 1.98280 0.991398 0.130879i \(-0.0417798\pi\)
0.991398 + 0.130879i \(0.0417798\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 6.63840 0.0129656
\(65\) 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
\(71\) 1190.09 1.98926 0.994632 0.103474i \(-0.0329957\pi\)
0.994632 + 0.103474i \(0.0329957\pi\)
\(72\) 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 \(-0.596261\pi\)
−0.297824 + 0.954621i \(0.596261\pi\)
\(80\) 732.416 1.02358
\(81\) 0 0
\(82\) 693.193 0.933541
\(83\) 94.6440 0.125163 0.0625815 0.998040i \(-0.480067\pi\)
0.0625815 + 0.998040i \(0.480067\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1591.30 1.99528
\(87\) 0 0
\(88\) 259.939 0.314882
\(89\) −1672.06 −1.99143 −0.995717 0.0924493i \(-0.970530\pi\)
−0.995717 + 0.0924493i \(0.970530\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) −2254.89 −2.47420
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 1207.56 1.24471
\(99\) 0 0
\(100\) −179.537 −0.179537
\(101\) 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 \(0.5\pi\)
\(110\) 661.323 0.573224
\(111\) 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
\(118\) −2040.20 −1.59165
\(119\) 0 0
\(120\) 0 0
\(121\) −911.653 −0.684938
\(122\) −3325.73 −2.46801
\(123\) 0 0
\(124\) 0 0
\(125\) 1521.40 1.08862
\(126\) 0 0
\(127\) 2495.04 1.74330 0.871650 0.490129i \(-0.163050\pi\)
0.871650 + 0.490129i \(0.163050\pi\)
\(128\) −1459.77 −1.00802
\(129\) 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
\(137\) 3018.79 1.88258 0.941288 0.337604i \(-0.109616\pi\)
0.941288 + 0.337604i \(0.109616\pi\)
\(138\) 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
\(142\) −4189.80 −2.47606
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2477.09 1.36196 0.680978 0.732304i \(-0.261555\pi\)
0.680978 + 0.732304i \(0.261555\pi\)
\(150\) 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 \(0.576293\pi\)
\(158\) 1472.46 0.741409
\(159\) 0 0
\(160\) −1647.01 −0.813799
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) −865.256 −0.411983
\(165\) 0 0
\(166\) −333.201 −0.155792
\(167\) −2446.51 −1.13363 −0.566815 0.823845i \(-0.691825\pi\)
−0.566815 + 0.823845i \(0.691825\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 0 0
\(172\) −1986.29 −0.880542
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1635.05 −0.700265
\(177\) 0 0
\(178\) 5886.60 2.47876
\(179\) 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
\(188\) 2814.60 1.09189
\(189\) 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
\(196\) −1507.30 −0.549306
\(197\) −3977.33 −1.43844 −0.719220 0.694782i \(-0.755501\pi\)
−0.719220 + 0.694782i \(0.755501\pi\)
\(198\) 0 0
\(199\) 5610.24 1.99849 0.999244 0.0388706i \(-0.0123760\pi\)
0.999244 + 0.0388706i \(0.0123760\pi\)
\(200\) −518.602 −0.183354
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 1806.15 0.615351
\(206\) 2985.45 1.00974
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −5740.04 −1.87280 −0.936399 0.350937i \(-0.885863\pi\)
−0.936399 + 0.350937i \(0.885863\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 4146.21 1.31521
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) −825.476 −0.252971
\(221\) 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
\(227\) −6668.65 −1.94984 −0.974920 0.222554i \(-0.928561\pi\)
−0.974920 + 0.222554i \(0.928561\pi\)
\(228\) 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
\(235\) −5875.24 −1.63089
\(236\) 2546.61 0.702416
\(237\) 0 0
\(238\) 0 0
\(239\) 6085.88 1.64712 0.823562 0.567226i \(-0.191984\pi\)
0.823562 + 0.567226i \(0.191984\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 3209.54 0.852550
\(243\) 0 0
\(244\) 4151.24 1.08916
\(245\) 3146.35 0.820462
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) −5356.19 −1.35502
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −8783.98 −2.16991
\(255\) 0 0
\(256\) 5086.11 1.24173
\(257\) 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
\(274\) −10627.9 −2.34326
\(275\) −836.635 −0.183458
\(276\) 0 0
\(277\) −7634.00 −1.65589 −0.827947 0.560806i \(-0.810491\pi\)
−0.827947 + 0.560806i \(0.810491\pi\)
\(278\) 1197.00 0.258241
\(279\) 0 0
\(280\) 0 0
\(281\) −8255.10 −1.75252 −0.876260 0.481839i \(-0.839969\pi\)
−0.876260 + 0.481839i \(0.839969\pi\)
\(282\) 0 0
\(283\) −490.355 −0.102999 −0.0514993 0.998673i \(-0.516400\pi\)
−0.0514993 + 0.998673i \(0.516400\pi\)
\(284\) 5229.79 1.09272
\(285\) 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
\(293\) −9933.00 −1.98052 −0.990260 0.139232i \(-0.955537\pi\)
−0.990260 + 0.139232i \(0.955537\pi\)
\(294\) 0 0
\(295\) −5315.83 −1.04915
\(296\) 0 0
\(297\) 0 0
\(298\) −8720.79 −1.69524
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −8665.35 −1.62681
\(306\) 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 \(-0.304014\pi\)
0.577536 + 0.816365i \(0.304014\pi\)
\(314\) 3288.22 0.590971
\(315\) 0 0
\(316\) −1837.95 −0.327193
\(317\) 7133.53 1.26391 0.631954 0.775006i \(-0.282253\pi\)
0.631954 + 0.775006i \(0.282253\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −60.8943 −0.0106378
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) −2499.34 −0.420741
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) 415.908 0.0687528
\(333\) 0 0
\(334\) 8613.11 1.41104
\(335\) 0 0
\(336\) 0 0
\(337\) −10420.0 −1.68432 −0.842160 0.539228i \(-0.818716\pi\)
−0.842160 + 0.539228i \(0.818716\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) −5737.51 −0.899262
\(345\) 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
\(352\) 3676.81 0.556745
\(353\) −7331.69 −1.10546 −0.552728 0.833361i \(-0.686413\pi\)
−0.552728 + 0.833361i \(0.686413\pi\)
\(354\) 0 0
\(355\) −10916.7 −1.63211
\(356\) −7347.77 −1.09391
\(357\) 0 0
\(358\) 0 0
\(359\) −11077.3 −1.62852 −0.814259 0.580502i \(-0.802857\pi\)
−0.814259 + 0.580502i \(0.802857\pi\)
\(360\) 0 0
\(361\) −6859.00 −1.00000
\(362\) 15596.1 2.26441
\(363\) 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 \(-0.350369\pi\)
0.452957 + 0.891532i \(0.350369\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 8130.13 1.11511
\(377\) 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
\(383\) −12702.2 −1.69465 −0.847327 0.531071i \(-0.821790\pi\)
−0.847327 + 0.531071i \(0.821790\pi\)
\(384\) 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
\(392\) −4353.91 −0.560984
\(393\) 0 0
\(394\) 14002.5 1.79044
\(395\) 3836.57 0.488706
\(396\) 0 0
\(397\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(398\) −19751.3 −2.48754
\(399\) 0 0
\(400\) 3262.07 0.407759
\(401\) −15342.6 −1.91066 −0.955328 0.295549i \(-0.904497\pi\)
−0.955328 + 0.295549i \(0.904497\pi\)
\(402\) 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 \(0.5\pi\)
\(410\) −6358.68 −0.765934
\(411\) 0 0
\(412\) −3726.49 −0.445609
\(413\) 0 0
\(414\) 0 0
\(415\) −868.173 −0.102691
\(416\) 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 \(0.5\pi\)
\(422\) 20208.2 2.33109
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) −14597.1 −1.63705
\(431\) 320.305 0.0357971 0.0178985 0.999840i \(-0.494302\pi\)
0.0178985 + 0.999840i \(0.494302\pi\)
\(432\) 0 0
\(433\) −36.0555 −0.00400166 −0.00200083 0.999998i \(-0.500637\pi\)
−0.00200083 + 0.999998i \(0.500637\pi\)
\(434\) 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 \(0.289016\pi\)
\(440\) −2384.43 −0.258349
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 15337.8 1.63390
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −333.683 −0.0350723 −0.0175361 0.999846i \(-0.505582\pi\)
−0.0175361 + 0.999846i \(0.505582\pi\)
\(450\) 0 0
\(451\) −4032.06 −0.420981
\(452\) 0 0
\(453\) 0 0
\(454\) 23477.5 2.42699
\(455\) 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
\(461\) −17433.8 −1.76133 −0.880666 0.473738i \(-0.842905\pi\)
−0.880666 + 0.473738i \(0.842905\pi\)
\(462\) 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
\(470\) 20684.2 2.02998
\(471\) 0 0
\(472\) 7356.03 0.717349
\(473\) −9256.04 −0.899774
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) −21425.8 −2.05019
\(479\) 5200.84 0.496101 0.248050 0.968747i \(-0.420210\pi\)
0.248050 + 0.968747i \(0.420210\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −4006.21 −0.376241
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 11991.1 1.11232
\(489\) 0 0
\(490\) −11077.0 −1.02124
\(491\) 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 \(0.5\pi\)
\(500\) 6685.70 0.597988
\(501\) 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
\(508\) 10964.3 0.957605
\(509\) −22966.8 −1.99997 −0.999987 0.00508931i \(-0.998380\pi\)
−0.999987 + 0.00508931i \(0.998380\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −6227.90 −0.537572
\(513\) 0 0
\(514\) 0 0
\(515\) 7778.74 0.665577
\(516\) 0 0
\(517\) 13115.9 1.11574
\(518\) 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
\(539\) −7023.94 −0.561304
\(540\) 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 \(0.112824\pi\)
\(548\) 13265.9 1.03411
\(549\) 0 0
\(550\) 2945.44 0.228352
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 26876.1 2.06111
\(555\) 0 0
\(556\) −1494.11 −0.113965
\(557\) −22258.1 −1.69319 −0.846595 0.532237i \(-0.821352\pi\)
−0.846595 + 0.532237i \(0.821352\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 29062.7 2.18138
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 1726.33 0.128203
\(567\) 0 0
\(568\) 15106.6 1.11595
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) −25411.9 −1.86244 −0.931222 0.364451i \(-0.881257\pi\)
−0.931222 + 0.364451i \(0.881257\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) 17296.6 1.24471
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 34969.9 2.46517
\(587\) 19338.0 1.35974 0.679868 0.733335i \(-0.262037\pi\)
0.679868 + 0.733335i \(0.262037\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 18714.8 1.30589
\(591\) 0 0
\(592\) 0 0
\(593\) 25958.2 1.79760 0.898798 0.438363i \(-0.144442\pi\)
0.898798 + 0.438363i \(0.144442\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 10885.5 0.748131
\(597\) 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 \(-0.359044\pi\)
0.428495 + 0.903544i \(0.359044\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 8362.63 0.561966
\(606\) 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
\(610\) 30507.0 2.02491
\(611\) 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
\(617\) 29859.9 1.94832 0.974160 0.225860i \(-0.0725191\pi\)
0.974160 + 0.225860i \(0.0725191\pi\)
\(618\) 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
\(625\) −8848.92 −0.566331
\(626\) −22518.5 −1.43773
\(627\) 0 0
\(628\) −4104.42 −0.260803
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) −5309.03 −0.334148
\(633\) 0 0
\(634\) −25114.1 −1.57320
\(635\) −22887.1 −1.43031
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 13390.5 0.827040
\(641\) 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
\(649\) 11867.1 0.717758
\(650\) 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
\(656\) 15721.2 0.935684
\(657\) 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
\(664\) 1201.37 0.0702144
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −10751.0 −0.622710
\(669\) 0 0
\(670\) 0 0
\(671\) 19344.6 1.11295
\(672\) 0 0
\(673\) 25522.0 1.46181 0.730907 0.682477i \(-0.239097\pi\)
0.730907 + 0.682477i \(0.239097\pi\)
\(674\) 36684.5 2.09649
\(675\) 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
\(683\) 32750.6 1.83480 0.917399 0.397968i \(-0.130285\pi\)
0.917399 + 0.397968i \(0.130285\pi\)
\(684\) 0 0
\(685\) −27691.5 −1.54458
\(686\) 0 0
\(687\) 0 0
\(688\) 36089.7 1.99986
\(689\) 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
\(695\) 3118.83 0.170222
\(696\) 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
\(704\) 135.941 0.00727765
\(705\) 0 0
\(706\) 25811.8 1.37597
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(710\) 38433.2 2.03151
\(711\) 0 0
\(712\) −21224.4 −1.11716
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 38998.5 2.02704
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 24147.6 1.24471
\(723\) 0 0
\(724\) −19467.4 −0.999310
\(725\) 0 0
\(726\) 0 0
\(727\) 28455.0 1.45163 0.725817 0.687888i \(-0.241462\pi\)
0.725817 + 0.687888i \(0.241462\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 29206.7 1.46872
\(735\) 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
\(743\) −19936.4 −0.984382 −0.492191 0.870487i \(-0.663804\pi\)
−0.492191 + 0.870487i \(0.663804\pi\)
\(744\) 0 0
\(745\) −22722.5 −1.11743
\(746\) −22975.4 −1.12760
\(747\) 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 \(0.718825\pi\)
\(752\) −51139.6 −2.47988
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −35009.9 −1.68092 −0.840460 0.541873i \(-0.817715\pi\)
−0.840460 + 0.541873i \(0.817715\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −28936.4 −1.37838 −0.689189 0.724582i \(-0.742033\pi\)
−0.689189 + 0.724582i \(0.742033\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 44719.1 2.10936
\(767\) 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
\(773\) −3484.00 −0.162110 −0.0810548 0.996710i \(-0.525829\pi\)
−0.0810548 + 0.996710i \(0.525829\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 24370.6 1.11658
\(782\) 0 0
\(783\) 0 0
\(784\) 27386.6 1.24757
\(785\) 8567.62 0.389543
\(786\) 0 0
\(787\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) −17478.2 −0.790144
\(789\) 0 0
\(790\) −13506.9 −0.608297
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 24653.9 1.09778
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −7335.55 −0.324189
\(801\) 0 0
\(802\) 54014.8 2.37821
\(803\) 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
\(820\) 7937.03 0.338016
\(821\) 33048.4 1.40487 0.702433 0.711749i \(-0.252097\pi\)
0.702433 + 0.711749i \(0.252097\pi\)
\(822\) 0 0
\(823\) −37352.0 −1.58203 −0.791014 0.611798i \(-0.790446\pi\)
−0.791014 + 0.611798i \(0.790446\pi\)
\(824\) −10764.2 −0.455083
\(825\) 0 0
\(826\) 0 0
\(827\) −31167.2 −1.31051 −0.655253 0.755409i \(-0.727438\pi\)
−0.655253 + 0.755409i \(0.727438\pi\)
\(828\) 0 0
\(829\) 3079.14 0.129002 0.0645012 0.997918i \(-0.479454\pi\)
0.0645012 + 0.997918i \(0.479454\pi\)
\(830\) 3056.47 0.127821
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 22441.9 0.930101
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 46872.5 1.92875 0.964374 0.264543i \(-0.0852210\pi\)
0.964374 + 0.264543i \(0.0852210\pi\)
\(840\) 0 0
\(841\) −24389.0 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) −25224.3 −1.02874
\(845\) 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 \(-0.170200\pi\)
0.860423 + 0.509581i \(0.170200\pi\)
\(860\) 18220.3 0.722451
\(861\) 0 0
\(862\) −1127.66 −0.0445571
\(863\) −23594.8 −0.930678 −0.465339 0.885133i \(-0.654068\pi\)
−0.465339 + 0.885133i \(0.654068\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 126.936 0.00498091
\(867\) 0 0
\(868\) 0 0
\(869\) −8564.79 −0.334339
\(870\) 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 \(0.5\pi\)
\(878\) −39852.9 −1.53186
\(879\) 0 0
\(880\) 14998.4 0.574540
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) −52410.3 −1.99745 −0.998724 0.0504988i \(-0.983919\pi\)
−0.998724 + 0.0504988i \(0.983919\pi\)
\(884\) 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
\(890\) −53998.0 −2.03373
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 1174.75 0.0436549
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 14195.2 0.524000
\(903\) 0 0
\(904\) 0 0
\(905\) 40636.6 1.49260
\(906\) 0 0
\(907\) −35017.1 −1.28195 −0.640973 0.767564i \(-0.721469\pi\)
−0.640973 + 0.767564i \(0.721469\pi\)
\(908\) −29305.0 −1.07106
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 1938.12 0.0702545
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −36762.2 −1.31956 −0.659779 0.751460i \(-0.729350\pi\)
−0.659779 + 0.751460i \(0.729350\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 61377.1 2.19235
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −7352.53 −0.259665 −0.129832 0.991536i \(-0.541444\pi\)
−0.129832 + 0.991536i \(0.541444\pi\)
\(930\) 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 \(-0.425660\pi\)
0.231428 + 0.972852i \(0.425660\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −25818.4 −0.895856
\(941\) −15462.1 −0.535652 −0.267826 0.963467i \(-0.586305\pi\)
−0.267826 + 0.963467i \(0.586305\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −46270.3 −1.59531
\(945\) 0 0
\(946\) 32586.6 1.11996
\(947\) 56498.4 1.93870 0.969352 0.245678i \(-0.0790105\pi\)
0.969352 + 0.245678i \(0.0790105\pi\)
\(948\) 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
\(956\) 26744.1 0.904775
\(957\) 0 0
\(958\) −18309.9 −0.617502
\(959\) 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 \(0.5\pi\)
\(968\) −11572.2 −0.384239
\(969\) 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
\(976\) −75425.4 −2.47368
\(977\) −52089.2 −1.70571 −0.852856 0.522146i \(-0.825132\pi\)
−0.852856 + 0.522146i \(0.825132\pi\)
\(978\) 0 0
\(979\) −34240.3 −1.11780
\(980\) 13826.5 0.450685
\(981\) 0 0
\(982\) 0 0
\(983\) −60639.0 −1.96753 −0.983766 0.179454i \(-0.942567\pi\)
−0.983766 + 0.179454i \(0.942567\pi\)
\(984\) 0 0
\(985\) 36484.2 1.18019
\(986\) 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
\(995\) −51462.9 −1.63968
\(996\) 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.1.2 4
3.2 odd 2 inner 1521.4.a.y.1.3 4
13.5 odd 4 117.4.b.c.64.3 yes 4
13.8 odd 4 117.4.b.c.64.2 4
13.12 even 2 inner 1521.4.a.y.1.3 4
39.5 even 4 117.4.b.c.64.2 4
39.8 even 4 117.4.b.c.64.3 yes 4
39.38 odd 2 CM 1521.4.a.y.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.4.b.c.64.2 4 13.8 odd 4
117.4.b.c.64.2 4 39.5 even 4
117.4.b.c.64.3 yes 4 13.5 odd 4
117.4.b.c.64.3 yes 4 39.8 even 4
1521.4.a.y.1.2 4 1.1 even 1 trivial
1521.4.a.y.1.2 4 39.38 odd 2 CM
1521.4.a.y.1.3 4 3.2 odd 2 inner
1521.4.a.y.1.3 4 13.12 even 2 inner