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Properties

Label 9.52.a.a.1.1
Level $9$
Weight $52$
Character 9.1
Self dual yes
Analytic conductor $148.258$
Analytic rank $0$
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] = [9,52,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, 52, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.1");
 
S:= CuspForms(chi, 52);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 52 \)
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: \(148.258218073\)
Analytic rank: \(0\)
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)
 
\(f(q)\) \(=\) \(q-2.25180e15 q^{4} -1.90393e21 q^{7} +O(q^{10})\) \(q-2.25180e15 q^{4} -1.90393e21 q^{7} -4.99141e28 q^{13} +5.07060e30 q^{16} -3.30998e32 q^{19} -4.44089e35 q^{25} +4.28727e36 q^{28} -2.37320e37 q^{31} -1.69715e40 q^{37} -8.02025e41 q^{43} -8.96431e42 q^{49} +1.12396e44 q^{52} -8.40865e44 q^{61} -1.14180e46 q^{64} +8.90719e45 q^{67} +2.93917e47 q^{73} +7.45341e47 q^{76} -4.10324e48 q^{79} +9.50328e49 q^{91} -4.59047e50 q^{97} +O(q^{100})\)

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
\(4\) −2.25180e15 −1.00000
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) −1.90393e21 −0.536600 −0.268300 − 0.963335i \(-0.586462\pi\)
−0.268300 + 0.963335i \(0.586462\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −4.99141e28 −1.96186 −0.980929 − 0.194368i \(-0.937734\pi\)
−0.980929 + 0.194368i \(0.937734\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 5.07060e30 1.00000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −3.30998e32 −0.815847 −0.407923 − 0.913016i \(-0.633747\pi\)
−0.407923 + 0.913016i \(0.633747\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −4.44089e35 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 4.28727e36 0.536600
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −2.37320e37 −0.221621 −0.110811 − 0.993842i \(-0.535345\pi\)
−0.110811 + 0.993842i \(0.535345\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.69715e40 −1.74011 −0.870056 − 0.492952i \(-0.835918\pi\)
−0.870056 + 0.492952i \(0.835918\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −8.02025e41 −1.78132 −0.890661 − 0.454668i \(-0.849758\pi\)
−0.890661 + 0.454668i \(0.849758\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −8.96431e42 −0.712061
\(50\) 0 0
\(51\) 0 0
\(52\) 1.12396e44 1.96186
\(53\) 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
\(61\) −8.40865e44 −0.250504 −0.125252 − 0.992125i \(-0.539974\pi\)
−0.125252 + 0.992125i \(0.539974\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −1.14180e46 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 8.90719e45 0.242568 0.121284 − 0.992618i \(-0.461299\pi\)
0.121284 + 0.992618i \(0.461299\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 2.93917e47 0.898445 0.449222 − 0.893420i \(-0.351701\pi\)
0.449222 + 0.893420i \(0.351701\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 7.45341e47 0.815847
\(77\) 0 0
\(78\) 0 0
\(79\) −4.10324e48 −1.67354 −0.836769 − 0.547556i \(-0.815558\pi\)
−0.836769 + 0.547556i \(0.815558\pi\)
\(80\) 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
\(91\) 9.50328e49 1.05273
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −4.59047e50 −0.998108 −0.499054 − 0.866571i \(-0.666319\pi\)
−0.499054 + 0.866571i \(0.666319\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.00000e51 1.00000
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) −3.30969e51 −1.55754 −0.778768 − 0.627312i \(-0.784155\pi\)
−0.778768 + 0.627312i \(0.784155\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 1.77145e52 1.96768 0.983839 − 0.179055i \(-0.0573039\pi\)
0.983839 + 0.179055i \(0.0573039\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −9.65406e51 −0.536600
\(113\) 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
\(121\) −1.29130e53 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 5.34398e52 0.221621
\(125\) 0 0
\(126\) 0 0
\(127\) 6.82724e53 1.53904 0.769521 − 0.638622i \(-0.220495\pi\)
0.769521 + 0.638622i \(0.220495\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 6.30196e53 0.437783
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) 2.92526e54 0.659609 0.329805 − 0.944049i \(-0.393017\pi\)
0.329805 + 0.944049i \(0.393017\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 3.82165e55 1.74011
\(149\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 6.48651e55 1.77051 0.885256 − 0.465104i \(-0.153983\pi\)
0.885256 + 0.465104i \(0.153983\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 1.83338e56 1.85273 0.926364 − 0.376628i \(-0.122917\pi\)
0.926364 + 0.376628i \(0.122917\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 2.66649e56 1.03551 0.517757 − 0.855528i \(-0.326767\pi\)
0.517757 + 0.855528i \(0.326767\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 1.84411e57 2.84888
\(170\) 0 0
\(171\) 0 0
\(172\) 1.80600e57 1.78132
\(173\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 8.45514e56 0.536600
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 5.62148e57 1.51024 0.755122 − 0.655584i \(-0.227578\pi\)
0.755122 + 0.655584i \(0.227578\pi\)
\(182\) 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
\(193\) 1.94288e58 1.01563 0.507814 − 0.861467i \(-0.330454\pi\)
0.507814 + 0.861467i \(0.330454\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 2.01858e58 0.712061
\(197\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) −8.35174e58 −1.99997 −0.999985 − 0.00547108i \(-0.998258\pi\)
−0.999985 + 0.00547108i \(0.998258\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −2.53094e59 −1.96186
\(209\) 0 0
\(210\) 0 0
\(211\) −3.69907e59 −1.99017 −0.995085 − 0.0990239i \(-0.968428\pi\)
−0.995085 + 0.0990239i \(0.968428\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 4.51841e58 0.118922
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −1.14080e60 −1.49774 −0.748868 − 0.662719i \(-0.769402\pi\)
−0.748868 + 0.662719i \(0.769402\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −1.84518e60 −1.23093 −0.615467 − 0.788162i \(-0.711033\pi\)
−0.615467 + 0.788162i \(0.711033\pi\)
\(230\) 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
\(241\) 1.05507e61 1.91358 0.956788 − 0.290787i \(-0.0939171\pi\)
0.956788 + 0.290787i \(0.0939171\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 1.89346e60 0.250504
\(245\) 0 0
\(246\) 0 0
\(247\) 1.65215e61 1.60057
\(248\) 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
\(256\) 2.57110e61 1.00000
\(257\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 3.23126e61 0.933744
\(260\) 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
\(268\) −2.00572e61 −0.242568
\(269\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 2.19607e62 1.99954 0.999768 − 0.0215417i \(-0.00685747\pi\)
0.999768 + 0.0215417i \(0.00685747\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −3.78780e62 −1.97312 −0.986559 − 0.163408i \(-0.947751\pi\)
−0.986559 + 0.163408i \(0.947751\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) −6.32150e62 −1.90663 −0.953314 − 0.301981i \(-0.902352\pi\)
−0.953314 + 0.301981i \(0.902352\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −5.66102e62 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) −6.61842e62 −0.898445
\(293\) 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
\(301\) 1.52700e63 0.955857
\(302\) 0 0
\(303\) 0 0
\(304\) −1.67836e63 −0.815847
\(305\) 0 0
\(306\) 0 0
\(307\) −4.30208e63 −1.62798 −0.813992 − 0.580876i \(-0.802710\pi\)
−0.813992 + 0.580876i \(0.802710\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 2.86563e63 0.661973 0.330986 − 0.943636i \(-0.392619\pi\)
0.330986 + 0.943636i \(0.392619\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 9.23966e63 1.67354
\(317\) 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
\(325\) 2.21663e64 1.96186
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 3.55149e64 1.97150 0.985751 − 0.168210i \(-0.0537985\pi\)
0.985751 + 0.168210i \(0.0537985\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −3.59874e64 −1.26354 −0.631768 − 0.775158i \(-0.717670\pi\)
−0.631768 + 0.775158i \(0.717670\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 4.10364e64 0.918691
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) −9.35603e64 −1.34599 −0.672997 − 0.739645i \(-0.734993\pi\)
−0.672997 + 0.739645i \(0.734993\pi\)
\(350\) 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
\(361\) −5.50418e64 −0.334394
\(362\) 0 0
\(363\) 0 0
\(364\) −2.13995e65 −1.05273
\(365\) 0 0
\(366\) 0 0
\(367\) −4.07427e65 −1.62579 −0.812897 − 0.582408i \(-0.802111\pi\)
−0.812897 + 0.582408i \(0.802111\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −7.38373e65 −1.94850 −0.974251 − 0.225466i \(-0.927609\pi\)
−0.974251 + 0.225466i \(0.927609\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 1.02926e66 1.80811 0.904055 − 0.427415i \(-0.140576\pi\)
0.904055 + 0.427415i \(0.140576\pi\)
\(380\) 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
\(388\) 1.03368e66 0.998108
\(389\) 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
\(397\) 3.56085e66 1.91600 0.957999 − 0.286770i \(-0.0925816\pi\)
0.957999 + 0.286770i \(0.0925816\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −2.25180e66 −1.00000
\(401\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 1.18456e66 0.434790
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 4.17769e66 1.05194 0.525972 − 0.850502i \(-0.323702\pi\)
0.525972 + 0.850502i \(0.323702\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 7.45276e66 1.55754
\(413\) 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
\(421\) −9.13328e66 −1.10010 −0.550051 − 0.835131i \(-0.685392\pi\)
−0.550051 + 0.835131i \(0.685392\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1.60095e66 0.134421
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −2.16352e67 −1.27268 −0.636340 − 0.771408i \(-0.719553\pi\)
−0.636340 + 0.771408i \(0.719553\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −3.98896e67 −1.96768
\(437\) 0 0
\(438\) 0 0
\(439\) −1.78729e67 −0.740199 −0.370100 − 0.928992i \(-0.620676\pi\)
−0.370100 + 0.928992i \(0.620676\pi\)
\(440\) 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
\(448\) 2.17390e67 0.536600
\(449\) 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
\(457\) −6.76084e67 −1.00493 −0.502467 − 0.864596i \(-0.667574\pi\)
−0.502467 + 0.864596i \(0.667574\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 8.70492e67 0.927789 0.463895 − 0.885890i \(-0.346452\pi\)
0.463895 + 0.885890i \(0.346452\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) −1.69587e67 −0.130162
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 1.46993e68 0.815847
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 8.47119e68 3.41385
\(482\) 0 0
\(483\) 0 0
\(484\) 2.90775e68 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) −4.41411e68 −1.29675 −0.648373 − 0.761323i \(-0.724550\pi\)
−0.648373 + 0.761323i \(0.724550\pi\)
\(488\) 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
\(496\) −1.20336e68 −0.221621
\(497\) 0 0
\(498\) 0 0
\(499\) −1.52862e68 −0.241399 −0.120699 − 0.992689i \(-0.538514\pi\)
−0.120699 + 0.992689i \(0.538514\pi\)
\(500\) 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
\(508\) −1.53736e69 −1.53904
\(509\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) −5.59596e68 −0.482105
\(512\) 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
\(523\) −1.85659e69 −0.884956 −0.442478 − 0.896779i \(-0.645901\pi\)
−0.442478 + 0.896779i \(0.645901\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −2.80621e69 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) −1.41908e69 −0.437783
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −2.26716e69 −0.455987 −0.227993 − 0.973663i \(-0.573216\pi\)
−0.227993 + 0.973663i \(0.573216\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1.16825e70 1.77361 0.886805 − 0.462145i \(-0.152920\pi\)
0.886805 + 0.462145i \(0.152920\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 7.81226e69 0.898020
\(554\) 0 0
\(555\) 0 0
\(556\) −6.58709e69 −0.659609
\(557\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 4.00323e70 3.49470
\(560\) 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
\(571\) −3.90366e70 −1.98268 −0.991338 − 0.131338i \(-0.958073\pi\)
−0.991338 + 0.131338i \(0.958073\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 3.63162e70 1.41292 0.706460 − 0.707753i \(-0.250291\pi\)
0.706460 + 0.707753i \(0.250291\pi\)
\(578\) 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
\(589\) 7.85526e69 0.180809
\(590\) 0 0
\(591\) 0 0
\(592\) −8.60560e70 −1.74011
\(593\) 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
\(601\) 3.32998e70 0.458292 0.229146 − 0.973392i \(-0.426407\pi\)
0.229146 + 0.973392i \(0.426407\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −1.46063e71 −1.77051
\(605\) 0 0
\(606\) 0 0
\(607\) −1.17095e71 −1.25091 −0.625455 − 0.780260i \(-0.715087\pi\)
−0.625455 + 0.780260i \(0.715087\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1.95122e71 −1.62205 −0.811023 − 0.585015i \(-0.801089\pi\)
−0.811023 + 0.585015i \(0.801089\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 2.70192e71 1.75211 0.876055 − 0.482212i \(-0.160166\pi\)
0.876055 + 0.482212i \(0.160166\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.97215e71 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) −4.12841e71 −1.85273
\(629\) 0 0
\(630\) 0 0
\(631\) 4.27551e71 1.69918 0.849589 − 0.527445i \(-0.176850\pi\)
0.849589 + 0.527445i \(0.176850\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 4.47445e71 1.39696
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 5.71442e71 1.40473 0.702364 − 0.711818i \(-0.252128\pi\)
0.702364 + 0.711818i \(0.252128\pi\)
\(644\) 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
\(652\) −6.00440e71 −1.03551
\(653\) 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
\(661\) 1.64500e72 2.00000 1.00000 0.000411236i \(-0.000130900\pi\)
1.00000 0.000411236i \(0.000130900\pi\)
\(662\) 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
\(673\) 8.91563e71 0.685122 0.342561 − 0.939496i \(-0.388706\pi\)
0.342561 + 0.939496i \(0.388706\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −4.15256e72 −2.84888
\(677\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 8.73992e71 0.535585
\(680\) 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
\(688\) −4.06675e72 −1.78132
\(689\) 0 0
\(690\) 0 0
\(691\) −5.09158e72 −1.99601 −0.998004 − 0.0631485i \(-0.979886\pi\)
−0.998004 + 0.0631485i \(0.979886\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −1.90393e72 −0.536600
\(701\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 5.61755e72 1.41967
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 2.63831e72 0.536843 0.268421 − 0.963302i \(-0.413498\pi\)
0.268421 + 0.963302i \(0.413498\pi\)
\(710\) 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
\(721\) 6.30141e72 0.835774
\(722\) 0 0
\(723\) 0 0
\(724\) −1.26585e73 −1.51024
\(725\) 0 0
\(726\) 0 0
\(727\) 1.86226e73 1.99947 0.999733 − 0.0231240i \(-0.00736125\pi\)
0.999733 + 0.0231240i \(0.00736125\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 7.40810e72 0.644994 0.322497 − 0.946571i \(-0.395478\pi\)
0.322497 + 0.946571i \(0.395478\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 1.29055e73 0.912727 0.456363 − 0.889794i \(-0.349152\pi\)
0.456363 + 0.889794i \(0.349152\pi\)
\(740\) 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
\(751\) 3.28885e73 1.54250 0.771250 − 0.636532i \(-0.219632\pi\)
0.771250 + 0.636532i \(0.219632\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.81520e72 0.0694991 0.0347495 − 0.999396i \(-0.488937\pi\)
0.0347495 + 0.999396i \(0.488937\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) −3.37272e73 −1.05586
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −7.52108e73 −1.92823 −0.964114 − 0.265488i \(-0.914467\pi\)
−0.964114 + 0.265488i \(0.914467\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −4.37498e73 −1.01563
\(773\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) 1.05391e73 0.221621
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −4.54545e73 −0.712061
\(785\) 0 0
\(786\) 0 0
\(787\) −1.40724e74 −1.99992 −0.999958 − 0.00919927i \(-0.997072\pi\)
−0.999958 + 0.00919927i \(0.997072\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 4.19710e73 0.491454
\(794\) 0 0
\(795\) 0 0
\(796\) 1.88064e74 1.99997
\(797\) 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
\(811\) −3.02059e74 −1.99554 −0.997769 − 0.0667642i \(-0.978732\pi\)
−0.997769 + 0.0667642i \(0.978732\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 2.65469e74 1.45329
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) 4.25209e74 1.93155 0.965777 − 0.259374i \(-0.0835162\pi\)
0.965777 + 0.259374i \(0.0835162\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) −3.61374e73 −0.136401 −0.0682003 − 0.997672i \(-0.521726\pi\)
−0.0682003 + 0.997672i \(0.521726\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 5.69918e74 1.96186
\(833\) 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
\(841\) −3.82206e74 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 8.32957e74 1.99017
\(845\) 0 0
\(846\) 0 0
\(847\) 2.45854e74 0.536600
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −1.07065e75 −1.95185 −0.975925 − 0.218107i \(-0.930012\pi\)
−0.975925 + 0.218107i \(0.930012\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) −1.20310e75 −1.83433 −0.917163 − 0.398511i \(-0.869527\pi\)
−0.917163 + 0.398511i \(0.869527\pi\)
\(860\) 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
\(868\) −1.01746e74 −0.118922
\(869\) 0 0
\(870\) 0 0
\(871\) −4.44594e74 −0.475884
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −5.12582e74 −0.460547 −0.230273 − 0.973126i \(-0.573962\pi\)
−0.230273 + 0.973126i \(0.573962\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) −1.50181e75 −1.13401 −0.567005 − 0.823715i \(-0.691898\pi\)
−0.567005 + 0.823715i \(0.691898\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) −1.29986e75 −0.825850
\(890\) 0 0
\(891\) 0 0
\(892\) 2.56885e75 1.49774
\(893\) 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
\(907\) −3.21405e75 −1.22481 −0.612403 − 0.790546i \(-0.709797\pi\)
−0.612403 + 0.790546i \(0.709797\pi\)
\(908\) 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
\(916\) 4.15499e75 1.23093
\(917\) 0 0
\(918\) 0 0
\(919\) 5.47006e75 1.49089 0.745447 − 0.666565i \(-0.232236\pi\)
0.745447 + 0.666565i \(0.232236\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 7.53688e75 1.74011
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) 2.96717e75 0.580932
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 4.44698e75 0.739104 0.369552 − 0.929210i \(-0.379511\pi\)
0.369552 + 0.929210i \(0.379511\pi\)
\(938\) 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
\(949\) −1.46706e76 −1.76262
\(950\) 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
\(961\) −1.09037e76 −0.950884
\(962\) 0 0
\(963\) 0 0
\(964\) −2.37580e76 −1.91358
\(965\) 0 0
\(966\) 0 0
\(967\) 2.24597e76 1.67120 0.835601 − 0.549336i \(-0.185119\pi\)
0.835601 + 0.549336i \(0.185119\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) −5.56948e75 −0.353946
\(974\) 0 0
\(975\) 0 0
\(976\) −4.26369e75 −0.250504
\(977\) 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
\(988\) −3.72030e76 −1.60057
\(989\) 0 0
\(990\) 0 0
\(991\) −4.92861e76 −1.96267 −0.981334 − 0.192313i \(-0.938401\pi\)
−0.981334 + 0.192313i \(0.938401\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 5.66706e76 1.93478 0.967389 − 0.253297i \(-0.0815150\pi\)
0.967389 + 0.253297i \(0.0815150\pi\)
\(998\) 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 9.52.a.a.1.1 ✓ 1
3.2 odd 2 CM 9.52.a.a.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.52.a.a.1.1 ✓ 1 1.1 even 1 trivial
9.52.a.a.1.1 ✓ 1 3.2 odd 2 CM