Properties

Label 441.6.a.f.1.1
Level $441$
Weight $6$
Character 441.1
Self dual yes
Analytic conductor $70.729$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [441,6,Mod(1,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(70.7292645375\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 441.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-32.0000 q^{4} +O(q^{10})\) \(q-32.0000 q^{4} +427.000 q^{13} +1024.00 q^{16} -3143.00 q^{19} -3125.00 q^{25} -2723.00 q^{31} -6661.00 q^{37} +22475.0 q^{43} -13664.0 q^{52} +38626.0 q^{61} -32768.0 q^{64} -37939.0 q^{67} +78127.0 q^{73} +100576. q^{76} +90857.0 q^{79} +134386. q^{97} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 0 0
\(4\) −32.0000 −1.00000
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 427.000 0.700760 0.350380 0.936608i \(-0.386052\pi\)
0.350380 + 0.936608i \(0.386052\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1024.00 1.00000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −3143.00 −1.99738 −0.998689 0.0511835i \(-0.983701\pi\)
−0.998689 + 0.0511835i \(0.983701\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −3125.00 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −2723.00 −0.508913 −0.254456 0.967084i \(-0.581897\pi\)
−0.254456 + 0.967084i \(0.581897\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6661.00 −0.799899 −0.399949 0.916537i \(-0.630972\pi\)
−0.399949 + 0.916537i \(0.630972\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 22475.0 1.85365 0.926827 0.375489i \(-0.122525\pi\)
0.926827 + 0.375489i \(0.122525\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) −13664.0 −0.700760
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 38626.0 1.32909 0.664546 0.747247i \(-0.268625\pi\)
0.664546 + 0.747247i \(0.268625\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −32768.0 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −37939.0 −1.03252 −0.516260 0.856432i \(-0.672676\pi\)
−0.516260 + 0.856432i \(0.672676\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 78127.0 1.71591 0.857954 0.513727i \(-0.171735\pi\)
0.857954 + 0.513727i \(0.171735\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 100576. 1.99738
\(77\) 0 0
\(78\) 0 0
\(79\) 90857.0 1.63791 0.818956 0.573856i \(-0.194553\pi\)
0.818956 + 0.573856i \(0.194553\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 134386. 1.45019 0.725095 0.688649i \(-0.241796\pi\)
0.725095 + 0.688649i \(0.241796\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 100000. 1.00000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 211477. 1.96413 0.982065 0.188544i \(-0.0603769\pi\)
0.982065 + 0.188544i \(0.0603769\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) −247843. −1.99807 −0.999034 0.0439362i \(-0.986010\pi\)
−0.999034 + 0.0439362i \(0.986010\pi\)
\(110\) 0 0
\(111\) 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
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −161051. −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 87136.0 0.508913
\(125\) 0 0
\(126\) 0 0
\(127\) 347111. 1.90967 0.954837 0.297131i \(-0.0960299\pi\)
0.954837 + 0.297131i \(0.0960299\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) 454657. 1.99594 0.997969 0.0637074i \(-0.0202924\pi\)
0.997969 + 0.0637074i \(0.0202924\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 213152. 0.799899
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −408724. −1.45877 −0.729387 0.684102i \(-0.760194\pi\)
−0.729387 + 0.684102i \(0.760194\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −109214. −0.353614 −0.176807 0.984246i \(-0.556577\pi\)
−0.176807 + 0.984246i \(0.556577\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 678248. 1.99949 0.999746 0.0225538i \(-0.00717969\pi\)
0.999746 + 0.0225538i \(0.00717969\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −188964. −0.508935
\(170\) 0 0
\(171\) 0 0
\(172\) −719200. −1.85365
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 853027. 1.93538 0.967690 0.252142i \(-0.0811351\pi\)
0.967690 + 0.252142i \(0.0811351\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 656375. 1.26841 0.634204 0.773166i \(-0.281328\pi\)
0.634204 + 0.773166i \(0.281328\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 912268. 1.63301 0.816507 0.577336i \(-0.195908\pi\)
0.816507 + 0.577336i \(0.195908\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 437248. 0.700760
\(209\) 0 0
\(210\) 0 0
\(211\) −288976. −0.446844 −0.223422 0.974722i \(-0.571723\pi\)
−0.223422 + 0.974722i \(0.571723\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −304052. −0.409436 −0.204718 0.978821i \(-0.565628\pi\)
−0.204718 + 0.978821i \(0.565628\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 1.45951e6 1.83915 0.919576 0.392913i \(-0.128533\pi\)
0.919576 + 0.392913i \(0.128533\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −1.29697e6 −1.43843 −0.719215 0.694788i \(-0.755498\pi\)
−0.719215 + 0.694788i \(0.755498\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −1.23603e6 −1.32909
\(245\) 0 0
\(246\) 0 0
\(247\) −1.34206e6 −1.39968
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 1.04858e6 1.00000
\(257\) 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
\(268\) 1.21405e6 1.03252
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) −2.25285e6 −1.86341 −0.931707 0.363210i \(-0.881681\pi\)
−0.931707 + 0.363210i \(0.881681\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 738389. 0.578210 0.289105 0.957297i \(-0.406642\pi\)
0.289105 + 0.957297i \(0.406642\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 2.33458e6 1.73277 0.866387 0.499373i \(-0.166436\pi\)
0.866387 + 0.499373i \(0.166436\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1.41986e6 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) −2.50006e6 −1.71591
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −3.21843e6 −1.99738
\(305\) 0 0
\(306\) 0 0
\(307\) −3.20232e6 −1.93919 −0.969593 0.244723i \(-0.921303\pi\)
−0.969593 + 0.244723i \(0.921303\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −2.56708e6 −1.48108 −0.740539 0.672014i \(-0.765430\pi\)
−0.740539 + 0.672014i \(0.765430\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −2.90742e6 −1.63791
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −1.33438e6 −0.700760
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −546151. −0.273995 −0.136998 0.990571i \(-0.543745\pi\)
−0.136998 + 0.990571i \(0.543745\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 2.63172e6 1.26231 0.631155 0.775657i \(-0.282581\pi\)
0.631155 + 0.775657i \(0.282581\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) −4.27561e6 −1.87904 −0.939518 0.342501i \(-0.888726\pi\)
−0.939518 + 0.342501i \(0.888726\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 7.40235e6 2.98952
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 2.58318e6 1.00113 0.500563 0.865700i \(-0.333126\pi\)
0.500563 + 0.865700i \(0.333126\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −5.31727e6 −1.97887 −0.989434 0.144983i \(-0.953687\pi\)
−0.989434 + 0.144983i \(0.953687\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 69893.0 0.0249940 0.0124970 0.999922i \(-0.496022\pi\)
0.0124970 + 0.999922i \(0.496022\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) −4.30035e6 −1.45019
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 5.00342e6 1.59328 0.796638 0.604456i \(-0.206610\pi\)
0.796638 + 0.604456i \(0.206610\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −3.20000e6 −1.00000
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) −1.16272e6 −0.356626
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 5.17021e6 1.52827 0.764134 0.645057i \(-0.223166\pi\)
0.764134 + 0.645057i \(0.223166\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −6.76726e6 −1.96413
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −4.87580e6 −1.34073 −0.670364 0.742033i \(-0.733862\pi\)
−0.670364 + 0.742033i \(0.733862\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 7.49192e6 1.92032 0.960160 0.279450i \(-0.0901522\pi\)
0.960160 + 0.279450i \(0.0901522\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 7.93098e6 1.99807
\(437\) 0 0
\(438\) 0 0
\(439\) −3.99333e6 −0.988950 −0.494475 0.869192i \(-0.664640\pi\)
−0.494475 + 0.869192i \(0.664640\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.18576e6 0.265587 0.132793 0.991144i \(-0.457605\pi\)
0.132793 + 0.991144i \(0.457605\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 2.92217e6 0.633510 0.316755 0.948507i \(-0.397407\pi\)
0.316755 + 0.948507i \(0.397407\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 9.82188e6 1.99738
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −2.84425e6 −0.560537
\(482\) 0 0
\(483\) 0 0
\(484\) 5.15363e6 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) −1.36487e6 −0.260778 −0.130389 0.991463i \(-0.541623\pi\)
−0.130389 + 0.991463i \(0.541623\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −2.78835e6 −0.508913
\(497\) 0 0
\(498\) 0 0
\(499\) −2.17369e6 −0.390793 −0.195397 0.980724i \(-0.562599\pi\)
−0.195397 + 0.980724i \(0.562599\pi\)
\(500\) 0 0
\(501\) 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
\(508\) −1.11076e7 −1.90967
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) 1.17287e7 1.87497 0.937486 0.348023i \(-0.113147\pi\)
0.937486 + 0.348023i \(0.113147\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −6.43634e6 −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
\(539\) 0 0
\(540\) 0 0
\(541\) −3.02235e6 −0.443968 −0.221984 0.975050i \(-0.571253\pi\)
−0.221984 + 0.975050i \(0.571253\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1.27982e7 1.82886 0.914430 0.404744i \(-0.132639\pi\)
0.914430 + 0.404744i \(0.132639\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −1.45490e7 −1.99594
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 9.59682e6 1.29897
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 5.88780e6 0.755723 0.377862 0.925862i \(-0.376660\pi\)
0.377862 + 0.925862i \(0.376660\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 1.29099e7 1.61430 0.807150 0.590347i \(-0.201009\pi\)
0.807150 + 0.590347i \(0.201009\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 8.55839e6 1.01649
\(590\) 0 0
\(591\) 0 0
\(592\) −6.82086e6 −0.799899
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 1.74342e7 1.96887 0.984435 0.175749i \(-0.0562346\pi\)
0.984435 + 0.175749i \(0.0562346\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 1.30792e7 1.45877
\(605\) 0 0
\(606\) 0 0
\(607\) 1.56805e7 1.72739 0.863693 0.504019i \(-0.168146\pi\)
0.863693 + 0.504019i \(0.168146\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1.58234e7 −1.70079 −0.850394 0.526147i \(-0.823636\pi\)
−0.850394 + 0.526147i \(0.823636\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 1.22260e7 1.28251 0.641253 0.767330i \(-0.278415\pi\)
0.641253 + 0.767330i \(0.278415\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 9.76562e6 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 3.49485e6 0.353614
\(629\) 0 0
\(630\) 0 0
\(631\) −1.62439e7 −1.62411 −0.812057 0.583579i \(-0.801652\pi\)
−0.812057 + 0.583579i \(0.801652\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 1.33848e7 1.27668 0.638342 0.769753i \(-0.279620\pi\)
0.638342 + 0.769753i \(0.279620\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −2.17039e7 −1.99949
\(653\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 690949. 0.0615095 0.0307548 0.999527i \(-0.490209\pi\)
0.0307548 + 0.999527i \(0.490209\pi\)
\(662\) 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
\(673\) 1.73115e7 1.47332 0.736661 0.676262i \(-0.236401\pi\)
0.736661 + 0.676262i \(0.236401\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 6.04685e6 0.508935
\(677\) 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
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 2.30144e7 1.85365
\(689\) 0 0
\(690\) 0 0
\(691\) −7.01920e6 −0.559233 −0.279616 0.960112i \(-0.590207\pi\)
−0.279616 + 0.960112i \(0.590207\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 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
\(703\) 2.09355e7 1.59770
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −2.27014e7 −1.69604 −0.848021 0.529962i \(-0.822206\pi\)
−0.848021 + 0.529962i \(0.822206\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) −2.72969e7 −1.93538
\(725\) 0 0
\(726\) 0 0
\(727\) −1.56893e7 −1.10095 −0.550474 0.834853i \(-0.685553\pi\)
−0.550474 + 0.834853i \(0.685553\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 1.75648e7 1.20749 0.603746 0.797177i \(-0.293674\pi\)
0.603746 + 0.797177i \(0.293674\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 1.28170e7 0.863330 0.431665 0.902034i \(-0.357926\pi\)
0.431665 + 0.902034i \(0.357926\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −2.73219e7 −1.76771 −0.883857 0.467758i \(-0.845062\pi\)
−0.883857 + 0.467758i \(0.845062\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.98526e7 1.25915 0.629575 0.776940i \(-0.283229\pi\)
0.629575 + 0.776940i \(0.283229\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −2.50017e7 −1.52459 −0.762296 0.647228i \(-0.775928\pi\)
−0.762296 + 0.647228i \(0.775928\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −2.10040e7 −1.26841
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) 8.50938e6 0.508913
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −3.31061e7 −1.90534 −0.952668 0.304012i \(-0.901674\pi\)
−0.952668 + 0.304012i \(0.901674\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 1.64933e7 0.931375
\(794\) 0 0
\(795\) 0 0
\(796\) −2.91926e7 −1.63301
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) −2.27708e7 −1.21570 −0.607849 0.794053i \(-0.707967\pi\)
−0.607849 + 0.794053i \(0.707967\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −7.06389e7 −3.70245
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) 2.22923e7 1.14725 0.573623 0.819120i \(-0.305538\pi\)
0.573623 + 0.819120i \(0.305538\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 2.26485e7 1.14460 0.572299 0.820045i \(-0.306052\pi\)
0.572299 + 0.820045i \(0.306052\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −1.39919e7 −0.700760
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −2.05111e7 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 9.24723e6 0.446844
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 3.00842e6 0.141568 0.0707842 0.997492i \(-0.477450\pi\)
0.0707842 + 0.997492i \(0.477450\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) −3.06774e6 −0.141852 −0.0709259 0.997482i \(-0.522595\pi\)
−0.0709259 + 0.997482i \(0.522595\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −1.62000e7 −0.723550
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −3.83931e6 −0.168560 −0.0842800 0.996442i \(-0.526859\pi\)
−0.0842800 + 0.996442i \(0.526859\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) −4.58066e7 −1.97709 −0.988545 0.150925i \(-0.951775\pi\)
−0.988545 + 0.150925i \(0.951775\pi\)
\(884\) 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
\(890\) 0 0
\(891\) 0 0
\(892\) 9.72966e6 0.409436
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 2.62120e7 1.05799 0.528995 0.848625i \(-0.322569\pi\)
0.528995 + 0.848625i \(0.322569\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) −4.67042e7 −1.83915
\(917\) 0 0
\(918\) 0 0
\(919\) −4.21821e6 −0.164755 −0.0823777 0.996601i \(-0.526251\pi\)
−0.0823777 + 0.996601i \(0.526251\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 2.08156e7 0.799899
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −5.29708e6 −0.197100 −0.0985501 0.995132i \(-0.531420\pi\)
−0.0985501 + 0.995132i \(0.531420\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 0 0
\(949\) 3.33602e7 1.20244
\(950\) 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
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −2.12144e7 −0.741008
\(962\) 0 0
\(963\) 0 0
\(964\) 4.15032e7 1.43843
\(965\) 0 0
\(966\) 0 0
\(967\) −3.23453e7 −1.11236 −0.556180 0.831062i \(-0.687733\pi\)
−0.556180 + 0.831062i \(0.687733\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 3.95530e7 1.32909
\(977\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 4.29460e7 1.39968
\(989\) 0 0
\(990\) 0 0
\(991\) 6.05528e7 1.95862 0.979310 0.202365i \(-0.0648626\pi\)
0.979310 + 0.202365i \(0.0648626\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −6.04728e7 −1.92674 −0.963368 0.268183i \(-0.913577\pi\)
−0.963368 + 0.268183i \(0.913577\pi\)
\(998\) 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 441.6.a.f.1.1 1
3.2 odd 2 CM 441.6.a.f.1.1 1
7.3 odd 6 63.6.e.b.37.1 2
7.5 odd 6 63.6.e.b.46.1 yes 2
7.6 odd 2 441.6.a.e.1.1 1
21.5 even 6 63.6.e.b.46.1 yes 2
21.17 even 6 63.6.e.b.37.1 2
21.20 even 2 441.6.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.6.e.b.37.1 2 7.3 odd 6
63.6.e.b.37.1 2 21.17 even 6
63.6.e.b.46.1 yes 2 7.5 odd 6
63.6.e.b.46.1 yes 2 21.5 even 6
441.6.a.e.1.1 1 7.6 odd 2
441.6.a.e.1.1 1 21.20 even 2
441.6.a.f.1.1 1 1.1 even 1 trivial
441.6.a.f.1.1 1 3.2 odd 2 CM