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Properties

Label 9.82.a.a.1.1
Level $9$
Weight $82$
Character 9.1
Self dual yes
Analytic conductor $373.951$
Analytic rank $1$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9,82,Mod(1,9)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 82, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.1");
 
S:= CuspForms(chi, 82);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 82 \)
Character orbit: \([\chi]\) \(=\) 9.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: \(373.951156984\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 9.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.41785e24 q^{4} -1.02065e34 q^{7} +O(q^{10})\) \(q-2.41785e24 q^{4} -1.02065e34 q^{7} -1.81813e45 q^{13} +5.84601e48 q^{16} -1.38831e51 q^{19} -4.13590e56 q^{25} +2.46777e58 q^{28} +5.00779e60 q^{31} -2.42763e63 q^{37} +7.54131e65 q^{43} -1.79582e68 q^{49} +4.39598e69 q^{52} +3.59388e72 q^{61} -1.41348e73 q^{64} +1.50074e72 q^{67} -2.10982e75 q^{73} +3.35674e75 q^{76} +6.41546e76 q^{79} +1.85567e79 q^{91} +5.43516e80 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
\(4\) −2.41785e24 −1.00000
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) −1.02065e34 −0.605905 −0.302953 − 0.953006i \(-0.597972\pi\)
−0.302953 + 0.953006i \(0.597972\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −1.81813e45 −1.39611 −0.698056 − 0.716044i \(-0.745951\pi\)
−0.698056 + 0.716044i \(0.745951\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 5.84601e48 1.00000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −1.38831e51 −0.225407 −0.112703 − 0.993629i \(-0.535951\pi\)
−0.112703 + 0.993629i \(0.535951\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −4.13590e56 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 2.46777e58 0.605905
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 5.00779e60 1.99296 0.996478 − 0.0838573i \(-0.0267240\pi\)
0.996478 + 0.0838573i \(0.0267240\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.42763e63 −0.746472 −0.373236 − 0.927736i \(-0.621752\pi\)
−0.373236 + 0.927736i \(0.621752\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 7.54131e65 0.527199 0.263600 − 0.964632i \(-0.415090\pi\)
0.263600 + 0.964632i \(0.415090\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −1.79582e68 −0.632879
\(50\) 0 0
\(51\) 0 0
\(52\) 4.39598e69 1.39611
\(53\) 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
\(61\) 3.59388e72 1.77707 0.888535 − 0.458808i \(-0.151723\pi\)
0.888535 + 0.458808i \(0.151723\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −1.41348e73 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 1.50074e72 0.0166064 0.00830320 − 0.999966i \(-0.497357\pi\)
0.00830320 + 0.999966i \(0.497357\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −2.10982e75 −0.723886 −0.361943 − 0.932200i \(-0.617887\pi\)
−0.361943 + 0.932200i \(0.617887\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 3.35674e75 0.225407
\(77\) 0 0
\(78\) 0 0
\(79\) 6.41546e76 0.898112 0.449056 − 0.893504i \(-0.351760\pi\)
0.449056 + 0.893504i \(0.351760\pi\)
\(80\) 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
\(91\) 1.85567e79 0.845911
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 5.43516e80 1.86619 0.933097 − 0.359624i \(-0.117095\pi\)
0.933097 + 0.359624i \(0.117095\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.00000e81 1.00000
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 1.60845e81 0.485848 0.242924 − 0.970045i \(-0.421893\pi\)
0.242924 + 0.970045i \(0.421893\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 6.53553e82 1.99300 0.996501 − 0.0835789i \(-0.0266351\pi\)
0.996501 + 0.0835789i \(0.0266351\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −5.96671e82 −0.605905
\(113\) 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
\(121\) −2.25324e84 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) −1.21081e85 −1.99296
\(125\) 0 0
\(126\) 0 0
\(127\) 1.40140e85 0.876015 0.438008 − 0.898971i \(-0.355684\pi\)
0.438008 + 0.898971i \(0.355684\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 1.41698e85 0.136575
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) 8.15782e86 1.31666 0.658332 − 0.752728i \(-0.271262\pi\)
0.658332 + 0.752728i \(0.271262\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 5.86966e87 0.746472
\(149\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −2.21768e88 −1.25122 −0.625609 − 0.780137i \(-0.715150\pi\)
−0.625609 + 0.780137i \(0.715150\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −1.33538e89 −1.55478 −0.777388 − 0.629022i \(-0.783456\pi\)
−0.777388 + 0.629022i \(0.783456\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 4.93060e89 1.25690 0.628452 − 0.777849i \(-0.283689\pi\)
0.628452 + 0.777849i \(0.283689\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 1.60967e90 0.949127
\(170\) 0 0
\(171\) 0 0
\(172\) −1.82338e90 −0.527199
\(173\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 4.22130e90 0.605905
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −3.40421e91 −1.24748 −0.623741 − 0.781631i \(-0.714388\pi\)
−0.623741 + 0.781631i \(0.714388\pi\)
\(182\) 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
\(193\) −3.17994e92 −0.865662 −0.432831 − 0.901475i \(-0.642485\pi\)
−0.432831 + 0.901475i \(0.642485\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 4.34202e92 0.632879
\(197\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 2.16982e93 1.70952 0.854759 − 0.519024i \(-0.173705\pi\)
0.854759 + 0.519024i \(0.173705\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −1.06288e94 −1.39611
\(209\) 0 0
\(210\) 0 0
\(211\) 3.24646e93 0.238765 0.119383 − 0.992848i \(-0.461909\pi\)
0.119383 + 0.992848i \(0.461909\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −5.11118e94 −1.20754
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 2.22653e95 1.74294 0.871468 − 0.490452i \(-0.163168\pi\)
0.871468 + 0.490452i \(0.163168\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −4.17848e94 −0.111604 −0.0558022 − 0.998442i \(-0.517772\pi\)
−0.0558022 + 0.998442i \(0.517772\pi\)
\(230\) 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
\(241\) 1.12727e96 0.380483 0.190242 − 0.981737i \(-0.439073\pi\)
0.190242 + 0.981737i \(0.439073\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −8.68946e96 −1.77707
\(245\) 0 0
\(246\) 0 0
\(247\) 2.52414e96 0.314693
\(248\) 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
\(256\) 3.41758e97 1.00000
\(257\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 2.47776e97 0.452291
\(260\) 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
\(268\) −3.62856e96 −0.0166064
\(269\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) −4.31855e98 −1.25916 −0.629582 − 0.776934i \(-0.716774\pi\)
−0.629582 + 0.776934i \(0.716774\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −5.75831e98 −0.691615 −0.345808 − 0.938305i \(-0.612395\pi\)
−0.345808 + 0.938305i \(0.612395\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 3.78626e99 1.90923 0.954617 − 0.297835i \(-0.0962645\pi\)
0.954617 + 0.297835i \(0.0962645\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −4.63834e99 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 5.10124e99 0.723886
\(293\) 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
\(301\) −7.69702e99 −0.319433
\(302\) 0 0
\(303\) 0 0
\(304\) −8.11610e99 −0.225407
\(305\) 0 0
\(306\) 0 0
\(307\) −6.96386e100 −1.29941 −0.649705 − 0.760187i \(-0.725107\pi\)
−0.649705 + 0.760187i \(0.725107\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 1.89922e101 1.61819 0.809093 − 0.587680i \(-0.199959\pi\)
0.809093 + 0.587680i \(0.199959\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −1.55116e101 −0.898112
\(317\) 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
\(325\) 7.51962e101 1.39611
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1.79133e102 −1.58541 −0.792703 − 0.609607i \(-0.791327\pi\)
−0.792703 + 0.609607i \(0.791327\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −4.66644e102 −1.99515 −0.997573 − 0.0696314i \(-0.977818\pi\)
−0.997573 + 0.0696314i \(0.977818\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 4.72901e102 0.989370
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) −5.81047e102 −0.602257 −0.301128 − 0.953584i \(-0.597363\pi\)
−0.301128 + 0.953584i \(0.597363\pi\)
\(350\) 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
\(361\) −3.60077e103 −0.949192
\(362\) 0 0
\(363\) 0 0
\(364\) −4.48674e103 −0.845911
\(365\) 0 0
\(366\) 0 0
\(367\) −1.20573e104 −1.63033 −0.815163 − 0.579232i \(-0.803352\pi\)
−0.815163 + 0.579232i \(0.803352\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −1.29291e104 −0.906484 −0.453242 − 0.891388i \(-0.649733\pi\)
−0.453242 + 0.891388i \(0.649733\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −4.73353e104 −1.73899 −0.869496 − 0.493940i \(-0.835556\pi\)
−0.869496 + 0.493940i \(0.835556\pi\)
\(380\) 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
\(388\) −1.31414e105 −1.86619
\(389\) 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
\(397\) −3.19107e105 −1.79028 −0.895139 − 0.445787i \(-0.852924\pi\)
−0.895139 + 0.445787i \(0.852924\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −2.41785e105 −1.00000
\(401\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) −9.10483e105 −2.78239
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −1.18938e106 −1.99769 −0.998845 − 0.0480387i \(-0.984703\pi\)
−0.998845 + 0.0480387i \(0.984703\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −3.88900e105 −0.485848
\(413\) 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
\(421\) −3.69338e106 −1.92309 −0.961543 − 0.274655i \(-0.911437\pi\)
−0.961543 + 0.274655i \(0.911437\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −3.66808e106 −1.07674
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −8.19437e106 −1.36696 −0.683480 − 0.729970i \(-0.739534\pi\)
−0.683480 + 0.729970i \(0.739534\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −1.58019e107 −1.99300
\(437\) 0 0
\(438\) 0 0
\(439\) −8.48474e106 −0.810636 −0.405318 − 0.914176i \(-0.632839\pi\)
−0.405318 + 0.914176i \(0.632839\pi\)
\(440\) 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
\(448\) 1.44266e107 0.605905
\(449\) 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
\(457\) −2.86999e107 −0.538611 −0.269305 − 0.963055i \(-0.586794\pi\)
−0.269305 + 0.963055i \(0.586794\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 1.79457e108 1.98577 0.992887 − 0.119062i \(-0.0379886\pi\)
0.992887 + 0.119062i \(0.0379886\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) −1.53172e106 −0.0100619
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 5.74194e107 0.225407
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 4.41376e108 1.04216
\(482\) 0 0
\(483\) 0 0
\(484\) 5.44800e108 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) 1.39487e109 1.99348 0.996739 − 0.0806960i \(-0.0257143\pi\)
0.996739 + 0.0806960i \(0.0257143\pi\)
\(488\) 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
\(496\) 2.92756e109 1.99296
\(497\) 0 0
\(498\) 0 0
\(499\) 3.61490e109 1.92763 0.963816 − 0.266569i \(-0.0858900\pi\)
0.963816 + 0.266569i \(0.0858900\pi\)
\(500\) 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
\(508\) −3.38838e109 −0.876015
\(509\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 2.15338e109 0.438607
\(512\) 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
\(523\) 2.50042e110 1.98927 0.994637 − 0.103429i \(-0.0329814\pi\)
0.994637 + 0.103429i \(0.0329814\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −1.99505e110 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) −3.42604e109 −0.136575
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −8.87855e110 −1.79412 −0.897062 − 0.441904i \(-0.854303\pi\)
−0.897062 + 0.441904i \(0.854303\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −1.52709e111 −1.97414 −0.987069 − 0.160296i \(-0.948755\pi\)
−0.987069 + 0.160296i \(0.948755\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −6.54792e110 −0.544171
\(554\) 0 0
\(555\) 0 0
\(556\) −1.97244e111 −1.31666
\(557\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) −1.37111e111 −0.736029
\(560\) 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
\(571\) −5.13393e111 −1.16597 −0.582986 − 0.812482i \(-0.698116\pi\)
−0.582986 + 0.812482i \(0.698116\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −1.83856e111 −0.273437 −0.136718 − 0.990610i \(-0.543656\pi\)
−0.136718 + 0.990610i \(0.543656\pi\)
\(578\) 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
\(589\) −6.95239e111 −0.449226
\(590\) 0 0
\(591\) 0 0
\(592\) −1.41920e112 −0.746472
\(593\) 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
\(601\) 6.73356e112 1.92233 0.961166 − 0.275970i \(-0.0889989\pi\)
0.961166 + 0.275970i \(0.0889989\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 5.36203e112 1.25122
\(605\) 0 0
\(606\) 0 0
\(607\) 6.77722e112 1.29393 0.646963 − 0.762522i \(-0.276039\pi\)
0.646963 + 0.762522i \(0.276039\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 1.07837e113 1.38235 0.691173 − 0.722689i \(-0.257094\pi\)
0.691173 + 0.722689i \(0.257094\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 7.50628e111 0.0648565 0.0324282 − 0.999474i \(-0.489676\pi\)
0.0324282 + 0.999474i \(0.489676\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.71057e113 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 3.22874e113 1.55478
\(629\) 0 0
\(630\) 0 0
\(631\) −5.03319e113 −1.99827 −0.999135 − 0.0415777i \(-0.986762\pi\)
−0.999135 + 0.0415777i \(0.986762\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 3.26503e113 0.883569
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 1.05424e114 1.95163 0.975814 − 0.218602i \(-0.0701496\pi\)
0.975814 + 0.218602i \(0.0701496\pi\)
\(644\) 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
\(652\) −1.19215e114 −1.25690
\(653\) 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
\(661\) −3.24925e114 −1.96619 −0.983093 − 0.183107i \(-0.941384\pi\)
−0.983093 + 0.183107i \(0.941384\pi\)
\(662\) 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
\(673\) −6.81941e114 −1.99130 −0.995649 − 0.0931786i \(-0.970297\pi\)
−0.995649 + 0.0931786i \(0.970297\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −3.89193e114 −0.949127
\(677\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) −5.54738e114 −1.13074
\(680\) 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
\(688\) 4.40866e114 0.527199
\(689\) 0 0
\(690\) 0 0
\(691\) −3.82136e114 −0.383139 −0.191570 − 0.981479i \(-0.561358\pi\)
−0.191570 + 0.981479i \(0.561358\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −1.02065e115 −0.605905
\(701\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 3.37032e114 0.168260
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −3.50255e114 −0.123940 −0.0619701 − 0.998078i \(-0.519738\pi\)
−0.0619701 + 0.998078i \(0.519738\pi\)
\(710\) 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
\(721\) −1.64166e115 −0.294378
\(722\) 0 0
\(723\) 0 0
\(724\) 8.23088e115 1.24748
\(725\) 0 0
\(726\) 0 0
\(727\) 1.47454e116 1.89022 0.945109 − 0.326756i \(-0.105956\pi\)
0.945109 + 0.326756i \(0.105956\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 1.91530e116 1.76006 0.880028 − 0.474922i \(-0.157524\pi\)
0.880028 + 0.474922i \(0.157524\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 2.44061e116 1.61213 0.806063 − 0.591829i \(-0.201594\pi\)
0.806063 + 0.591829i \(0.201594\pi\)
\(740\) 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
\(751\) −3.56317e116 −1.22580 −0.612899 − 0.790161i \(-0.709997\pi\)
−0.612899 + 0.790161i \(0.709997\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 6.66067e116 1.66010 0.830051 − 0.557687i \(-0.188311\pi\)
0.830051 + 0.557687i \(0.188311\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) −6.67047e116 −1.20757
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 3.31137e116 0.436507 0.218254 − 0.975892i \(-0.429964\pi\)
0.218254 + 0.975892i \(0.429964\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 7.68863e116 0.865662
\(773\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) −2.07117e117 −1.99296
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −1.04983e117 −0.632879
\(785\) 0 0
\(786\) 0 0
\(787\) 1.11410e117 0.575373 0.287686 − 0.957725i \(-0.407114\pi\)
0.287686 + 0.957725i \(0.407114\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −6.53415e117 −2.48099
\(794\) 0 0
\(795\) 0 0
\(796\) −5.24630e117 −1.70952
\(797\) 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
\(811\) −8.67387e117 −1.32699 −0.663496 − 0.748180i \(-0.730928\pi\)
−0.663496 + 0.748180i \(0.730928\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −1.04697e117 −0.118834
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) 1.67422e118 1.41292 0.706461 − 0.707752i \(-0.250290\pi\)
0.706461 + 0.707752i \(0.250290\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 1.37191e118 0.862714 0.431357 − 0.902181i \(-0.358035\pi\)
0.431357 + 0.902181i \(0.358035\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 2.56989e118 1.39611
\(833\) 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
\(841\) −2.84602e118 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) −7.84946e117 −0.238765
\(845\) 0 0
\(846\) 0 0
\(847\) 2.29976e118 0.605905
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 9.89142e118 1.95805 0.979023 − 0.203750i \(-0.0653130\pi\)
0.979023 + 0.203750i \(0.0653130\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 8.99919e118 1.34116 0.670580 − 0.741838i \(-0.266045\pi\)
0.670580 + 0.741838i \(0.266045\pi\)
\(860\) 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
\(868\) 1.23581e119 1.20754
\(869\) 0 0
\(870\) 0 0
\(871\) −2.72854e117 −0.0231844
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 2.09272e119 1.34657 0.673283 − 0.739385i \(-0.264884\pi\)
0.673283 + 0.739385i \(0.264884\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) −4.08964e119 −1.99653 −0.998266 − 0.0588721i \(-0.981250\pi\)
−0.998266 + 0.0588721i \(0.981250\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) −1.43034e119 −0.530782
\(890\) 0 0
\(891\) 0 0
\(892\) −5.38342e119 −1.74294
\(893\) 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
\(907\) −9.21405e119 −1.51829 −0.759146 − 0.650920i \(-0.774383\pi\)
−0.759146 + 0.650920i \(0.774383\pi\)
\(908\) 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
\(916\) 1.01029e119 0.111604
\(917\) 0 0
\(918\) 0 0
\(919\) −2.01015e120 −1.94514 −0.972569 − 0.232613i \(-0.925272\pi\)
−0.972569 + 0.232613i \(0.925272\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 1.00405e120 0.746472
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) 2.49316e119 0.142655
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −3.74207e120 −1.65066 −0.825331 − 0.564649i \(-0.809012\pi\)
−0.825331 + 0.564649i \(0.809012\pi\)
\(938\) 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
\(949\) 3.83594e120 1.01063
\(950\) 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
\(961\) 1.87641e121 2.97187
\(962\) 0 0
\(963\) 0 0
\(964\) −2.72556e120 −0.380483
\(965\) 0 0
\(966\) 0 0
\(967\) 1.59724e121 1.96607 0.983033 − 0.183427i \(-0.0587191\pi\)
0.983033 + 0.183427i \(0.0587191\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) −8.32625e120 −0.797773
\(974\) 0 0
\(975\) 0 0
\(976\) 2.10098e121 1.77707
\(977\) 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
\(988\) −6.10300e120 −0.314693
\(989\) 0 0
\(990\) 0 0
\(991\) −6.75490e120 −0.308060 −0.154030 − 0.988066i \(-0.549225\pi\)
−0.154030 + 0.988066i \(0.549225\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1.73137e121 −0.618352 −0.309176 − 0.951005i \(-0.600053\pi\)
−0.309176 + 0.951005i \(0.600053\pi\)
\(998\) 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 9.82.a.a.1.1 ✓ 1
3.2 odd 2 CM 9.82.a.a.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.82.a.a.1.1 ✓ 1 1.1 even 1 trivial
9.82.a.a.1.1 ✓ 1 3.2 odd 2 CM