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Properties

Label 64.22.a.d.1.1
Level $64$
Weight $22$
Character 64.1
Self dual yes
Analytic conductor $178.866$
Analytic rank $0$
Dimension $1$
CM discriminant -4
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [64,22,Mod(1,64)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(64, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("64.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 64.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: \(178.865500344\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 32)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 64.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.33986e7 q^{5} -1.04604e10 q^{9} +O(q^{10})\) \(q-1.33986e7 q^{5} -1.04604e10 q^{9} -9.70515e11 q^{13} -1.50959e13 q^{17} -2.97314e14 q^{25} -6.17267e14 q^{29} -1.87640e16 q^{37} -5.56716e15 q^{41} +1.40154e17 q^{45} -5.58546e17 q^{49} -2.30506e18 q^{53} +1.00221e19 q^{61} +1.30036e19 q^{65} -6.97145e19 q^{73} +1.09419e20 q^{81} +2.02265e20 q^{85} -4.36441e20 q^{89} -1.16329e21 q^{97} +O(q^{100})\)

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
\(3\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 0 0
\(5\) −1.33986e7 −0.613586 −0.306793 − 0.951776i \(-0.599256\pi\)
−0.306793 + 0.951776i \(0.599256\pi\)
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) −1.04604e10 −1.00000
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −9.70515e11 −1.95253 −0.976264 − 0.216586i \(-0.930508\pi\)
−0.976264 + 0.216586i \(0.930508\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.50959e13 −1.81612 −0.908062 − 0.418836i \(-0.862438\pi\)
−0.908062 + 0.418836i \(0.862438\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −2.97314e14 −0.623512
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.17267e14 −0.272455 −0.136227 − 0.990678i \(-0.543498\pi\)
−0.136227 + 0.990678i \(0.543498\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.87640e16 −0.641516 −0.320758 − 0.947161i \(-0.603938\pi\)
−0.320758 + 0.947161i \(0.603938\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −5.56716e15 −0.0647744 −0.0323872 − 0.999475i \(-0.510311\pi\)
−0.0323872 + 0.999475i \(0.510311\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 1.40154e17 0.613586
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −5.58546e17 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.30506e18 −1.81045 −0.905223 − 0.424937i \(-0.860296\pi\)
−0.905223 + 0.424937i \(0.860296\pi\)
\(54\) 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
\(61\) 1.00221e19 1.79886 0.899428 − 0.437069i \(-0.143983\pi\)
0.899428 + 0.437069i \(0.143983\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.30036e19 1.19804
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −6.97145e19 −1.89860 −0.949299 − 0.314374i \(-0.898205\pi\)
−0.949299 + 0.314374i \(0.898205\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.09419e20 1.00000
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 2.02265e20 1.11435
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −4.36441e20 −1.48365 −0.741823 − 0.670596i \(-0.766039\pi\)
−0.741823 + 0.670596i \(0.766039\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.16329e21 −1.60172 −0.800860 − 0.598852i \(-0.795624\pi\)
−0.800860 + 0.598852i \(0.795624\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1.92430e21 1.73340 0.866701 − 0.498827i \(-0.166236\pi\)
0.866701 + 0.498827i \(0.166236\pi\)
\(102\) 0 0
\(103\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 8.00191e20 0.323754 0.161877 − 0.986811i \(-0.448245\pi\)
0.161877 + 0.986811i \(0.448245\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −4.14933e21 −1.14988 −0.574942 − 0.818194i \(-0.694975\pi\)
−0.574942 + 0.818194i \(0.694975\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.01519e22 1.95253
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −7.40025e21 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.03726e22 0.996165
\(126\) 0 0
\(127\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 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
\(137\) −2.75544e22 −1.01071 −0.505353 − 0.862913i \(-0.668638\pi\)
−0.505353 + 0.862913i \(0.668638\pi\)
\(138\) 0 0
\(139\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 8.27054e21 0.167174
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −3.37065e22 −0.511986 −0.255993 − 0.966679i \(-0.582402\pi\)
−0.255993 + 0.966679i \(0.582402\pi\)
\(150\) 0 0
\(151\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 1.57908e23 1.81612
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 1.11174e23 0.975120 0.487560 − 0.873089i \(-0.337887\pi\)
0.487560 + 0.873089i \(0.337887\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 6.94835e23 2.81236
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −6.30410e23 −1.99590 −0.997950 − 0.0639995i \(-0.979614\pi\)
−0.997950 + 0.0639995i \(0.979614\pi\)
\(174\) 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
\(181\) 3.18312e23 0.626942 0.313471 − 0.949598i \(-0.398508\pi\)
0.313471 + 0.949598i \(0.398508\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 2.51412e23 0.393625
\(186\) 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
\(193\) 1.98363e24 1.99117 0.995586 − 0.0938508i \(-0.0299177\pi\)
0.995586 + 0.0938508i \(0.0299177\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.46466e24 1.99462 0.997312 − 0.0732738i \(-0.0233447\pi\)
0.997312 + 0.0732738i \(0.0233447\pi\)
\(198\) 0 0
\(199\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 7.45924e22 0.0397447
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 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
\(221\) 1.46508e25 3.54603
\(222\) 0 0
\(223\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 3.11001e24 0.623512
\(226\) 0 0
\(227\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 1.12423e25 1.87319 0.936596 − 0.350412i \(-0.113958\pi\)
0.936596 + 0.350412i \(0.113958\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 8.00003e24 1.11136 0.555680 − 0.831396i \(-0.312458\pi\)
0.555680 + 0.831396i \(0.312458\pi\)
\(234\) 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
\(241\) −1.41416e25 −1.37822 −0.689111 − 0.724656i \(-0.741999\pi\)
−0.689111 + 0.724656i \(0.741999\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 7.48375e24 0.613586
\(246\) 0 0
\(247\) 0 0
\(248\) 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
\(256\) 0 0
\(257\) 3.89475e25 1.93278 0.966390 − 0.257082i \(-0.0827610\pi\)
0.966390 + 0.257082i \(0.0827610\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 6.45684e24 0.272455
\(262\) 0 0
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 3.08847e25 1.11086
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 2.33847e25 0.718677 0.359338 − 0.933207i \(-0.383002\pi\)
0.359338 + 0.933207i \(0.383002\pi\)
\(270\) 0 0
\(271\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −4.76431e25 −1.07637 −0.538186 − 0.842826i \(-0.680890\pi\)
−0.538186 + 0.842826i \(0.680890\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 7.95676e24 0.154639 0.0773197 − 0.997006i \(-0.475364\pi\)
0.0773197 + 0.997006i \(0.475364\pi\)
\(282\) 0 0
\(283\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1.58794e26 2.29831
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1.23899e26 −1.55224 −0.776119 − 0.630587i \(-0.782814\pi\)
−0.776119 + 0.630587i \(0.782814\pi\)
\(294\) 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
\(304\) 0 0
\(305\) −1.34283e26 −1.10375
\(306\) 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
\(313\) −3.18866e26 −1.99707 −0.998534 − 0.0541278i \(-0.982762\pi\)
−0.998534 + 0.0541278i \(0.982762\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −3.60685e26 −1.97700 −0.988499 − 0.151230i \(-0.951676\pi\)
−0.988499 + 0.151230i \(0.951676\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 2.88547e26 1.21742
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) 0 0
\(333\) 1.96278e26 0.641516
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −6.74526e26 −1.94484 −0.972422 − 0.233228i \(-0.925071\pi\)
−0.972422 + 0.233228i \(0.925071\pi\)
\(338\) 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
\(349\) −5.59999e26 −1.11820 −0.559101 − 0.829099i \(-0.688854\pi\)
−0.559101 + 0.829099i \(0.688854\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1.10901e27 −1.96473 −0.982364 − 0.186977i \(-0.940131\pi\)
−0.982364 + 0.186977i \(0.940131\pi\)
\(354\) 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
\(361\) −7.14209e26 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 9.34079e26 1.16495
\(366\) 0 0
\(367\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) 5.82345e25 0.0647744
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −2.00931e27 −1.99574 −0.997871 − 0.0652182i \(-0.979226\pi\)
−0.997871 + 0.0652182i \(0.979226\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 5.99067e26 0.531975
\(378\) 0 0
\(379\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 5.31434e26 0.339609 0.169804 − 0.985478i \(-0.445686\pi\)
0.169804 + 0.985478i \(0.445686\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −3.83746e27 −1.98036 −0.990178 − 0.139811i \(-0.955350\pi\)
−0.990178 + 0.139811i \(0.955350\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 3.73301e27 1.73398 0.866988 − 0.498329i \(-0.166053\pi\)
0.866988 + 0.498329i \(0.166053\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −1.46607e27 −0.613586
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −7.89595e25 −0.0298064 −0.0149032 − 0.999889i \(-0.504744\pi\)
−0.0149032 + 0.999889i \(0.504744\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 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
\(421\) 7.16422e27 1.99621 0.998107 − 0.0615082i \(-0.0195910\pi\)
0.998107 + 0.0615082i \(0.0195910\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 4.48822e27 1.13237
\(426\) 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
\(433\) 9.08474e27 1.88447 0.942236 − 0.334949i \(-0.108719\pi\)
0.942236 + 0.334949i \(0.108719\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 5.84259e27 1.00000
\(442\) 0 0
\(443\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 5.84771e27 0.910345
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −9.99517e27 −1.41646 −0.708229 − 0.705983i \(-0.750505\pi\)
−0.708229 + 0.705983i \(0.750505\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.17004e28 −1.37746 −0.688731 − 0.725017i \(-0.741832\pi\)
−0.688731 + 0.725017i \(0.741832\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 1.36306e28 1.46439 0.732194 − 0.681096i \(-0.238496\pi\)
0.732194 + 0.681096i \(0.238496\pi\)
\(462\) 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
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 2.41117e28 1.81045
\(478\) 0 0
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 1.82107e28 1.25258
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1.55866e28 0.982793
\(486\) 0 0
\(487\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 9.31821e27 0.494812
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) −2.57831e28 −1.06359
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −5.26605e28 −1.99962 −0.999812 − 0.0193791i \(-0.993831\pi\)
−0.999812 + 0.0193791i \(0.993831\pi\)
\(510\) 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
\(521\) 6.09812e28 1.81301 0.906504 − 0.422196i \(-0.138741\pi\)
0.906504 + 0.422196i \(0.138741\pi\)
\(522\) 0 0
\(523\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −3.94716e28 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 5.40301e27 0.126474
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −9.94860e28 −1.99155 −0.995773 − 0.0918449i \(-0.970724\pi\)
−0.995773 + 0.0918449i \(0.970724\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −1.07215e28 −0.198651
\(546\) 0 0
\(547\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 0 0
\(549\) −1.04835e29 −1.79886
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −9.74811e28 −1.43695 −0.718474 − 0.695554i \(-0.755159\pi\)
−0.718474 + 0.695554i \(0.755159\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 5.55953e28 0.705553
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −7.57212e28 −0.892358 −0.446179 − 0.894944i \(-0.647215\pi\)
−0.446179 + 0.894944i \(0.647215\pi\)
\(570\) 0 0
\(571\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 1.50763e29 1.53444 0.767219 − 0.641385i \(-0.221640\pi\)
0.767219 + 0.641385i \(0.221640\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −1.36022e29 −1.19804
\(586\) 0 0
\(587\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1.92277e29 −1.46843 −0.734214 − 0.678918i \(-0.762449\pi\)
−0.734214 + 0.678918i \(0.762449\pi\)
\(594\) 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
\(601\) −2.77759e29 −1.84283 −0.921417 − 0.388576i \(-0.872967\pi\)
−0.921417 + 0.388576i \(0.872967\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 9.91533e28 0.613586
\(606\) 0 0
\(607\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −6.84475e28 −0.368997 −0.184498 − 0.982833i \(-0.559066\pi\)
−0.184498 + 0.982833i \(0.559066\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 2.11191e29 1.06336 0.531680 − 0.846945i \(-0.321561\pi\)
0.531680 + 0.846945i \(0.321561\pi\)
\(618\) 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
\(625\) 2.79194e27 0.0122791
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 2.83260e29 1.16507
\(630\) 0 0
\(631\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 5.42077e29 1.95253
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −5.82289e29 −1.96395 −0.981973 − 0.189024i \(-0.939468\pi\)
−0.981973 + 0.189024i \(0.939468\pi\)
\(642\) 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
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −7.04299e29 −1.95510 −0.977549 − 0.210707i \(-0.932424\pi\)
−0.977549 + 0.210707i \(0.932424\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 7.29238e29 1.89860
\(658\) 0 0
\(659\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 1.89886e29 0.463849 0.231924 − 0.972734i \(-0.425498\pi\)
0.231924 + 0.972734i \(0.425498\pi\)
\(662\) 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
\(673\) 7.72812e29 1.56284 0.781422 − 0.624003i \(-0.214495\pi\)
0.781422 + 0.624003i \(0.214495\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 7.60327e29 1.44484 0.722419 − 0.691456i \(-0.243030\pi\)
0.722419 + 0.691456i \(0.243030\pi\)
\(678\) 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
\(685\) 3.69191e29 0.620155
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 2.23709e30 3.53494
\(690\) 0 0
\(691\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 8.40413e28 0.117638
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −1.13807e30 −1.50013 −0.750066 − 0.661363i \(-0.769978\pi\)
−0.750066 + 0.661363i \(0.769978\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −7.28411e26 −0.000852297 0 −0.000426149 − 1.00000i \(-0.500136\pi\)
−0.000426149 1.00000i \(0.500136\pi\)
\(710\) 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
\(724\) 0 0
\(725\) 1.83522e29 0.169879
\(726\) 0 0
\(727\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) −1.14456e30 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −4.38241e28 −0.0361511 −0.0180756 − 0.999837i \(-0.505754\pi\)
−0.0180756 + 0.999837i \(0.505754\pi\)
\(734\) 0 0
\(735\) 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
\(743\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 4.51621e29 0.314147
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 2.22925e30 1.31115 0.655576 − 0.755129i \(-0.272426\pi\)
0.655576 + 0.755129i \(0.272426\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 8.78891e29 0.489098 0.244549 − 0.969637i \(-0.421360\pi\)
0.244549 + 0.969637i \(0.421360\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −2.11576e30 −1.11435
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −4.00959e30 −1.99928 −0.999639 − 0.0268553i \(-0.991451\pi\)
−0.999639 + 0.0268553i \(0.991451\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 4.01857e30 1.89752 0.948761 − 0.315996i \(-0.102339\pi\)
0.948761 + 0.315996i \(0.102339\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 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
\(785\) −1.48958e30 −0.598320
\(786\) 0 0
\(787\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −9.72662e30 −3.51232
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −4.97466e30 −1.70393 −0.851964 − 0.523601i \(-0.824588\pi\)
−0.851964 + 0.523601i \(0.824588\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 4.56533e30 1.48365
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −3.68221e30 −1.07807 −0.539037 − 0.842282i \(-0.681212\pi\)
−0.539037 + 0.842282i \(0.681212\pi\)
\(810\) 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
\(820\) 0 0
\(821\) −2.16755e30 −0.543709 −0.271854 − 0.962338i \(-0.587637\pi\)
−0.271854 + 0.962338i \(0.587637\pi\)
\(822\) 0 0
\(823\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) −3.47964e30 −0.788337 −0.394169 − 0.919038i \(-0.628967\pi\)
−0.394169 + 0.919038i \(0.628967\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 8.43176e30 1.81612
\(834\) 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
\(841\) −4.75182e30 −0.925768
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −9.30984e30 −1.72563
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −2.27544e30 −0.382033 −0.191016 − 0.981587i \(-0.561178\pi\)
−0.191016 + 0.981587i \(0.561178\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1.20812e31 −1.93112 −0.965562 − 0.260175i \(-0.916220\pi\)
−0.965562 + 0.260175i \(0.916220\pi\)
\(858\) 0 0
\(859\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 8.44663e30 1.22466
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 1.21685e31 1.60172
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −6.61666e30 −0.830125 −0.415062 − 0.909793i \(-0.636240\pi\)
−0.415062 + 0.909793i \(0.636240\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1.36982e31 1.63839 0.819193 − 0.573518i \(-0.194422\pi\)
0.819193 + 0.573518i \(0.194422\pi\)
\(882\) 0 0
\(883\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) 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
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 3.47970e31 3.28799
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −4.26494e30 −0.384683
\(906\) 0 0
\(907\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 0 0
\(909\) −2.01289e31 −1.73340
\(910\) 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
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 5.57879e30 0.399993
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 2.27439e31 1.55848 0.779238 − 0.626729i \(-0.215607\pi\)
0.779238 + 0.626729i \(0.215607\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −3.18479e31 −1.99442 −0.997208 − 0.0746795i \(-0.976207\pi\)
−0.997208 + 0.0746795i \(0.976207\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 2.57568e31 1.54241 0.771206 − 0.636586i \(-0.219654\pi\)
0.771206 + 0.636586i \(0.219654\pi\)
\(942\) 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
\(949\) 6.76590e31 3.70706
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1.11542e31 0.584739 0.292369 − 0.956305i \(-0.405556\pi\)
0.292369 + 0.956305i \(0.405556\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −2.08255e31 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −2.65779e31 −1.22176
\(966\) 0 0
\(967\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 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
\(976\) 0 0
\(977\) 4.46734e31 1.80366 0.901832 − 0.432086i \(-0.142222\pi\)
0.901832 + 0.432086i \(0.142222\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −8.37028e30 −0.323754
\(982\) 0 0
\(983\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) −3.30230e31 −1.22387
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 3.93620e31 1.28463 0.642315 − 0.766440i \(-0.277974\pi\)
0.642315 + 0.766440i \(0.277974\pi\)
\(998\) 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 64.22.a.d.1.1 1
4.3 odd 2 CM 64.22.a.d.1.1 1
8.3 odd 2 32.22.a.a.1.1 ✓ 1
8.5 even 2 32.22.a.a.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.22.a.a.1.1 ✓ 1 8.3 odd 2
32.22.a.a.1.1 ✓ 1 8.5 even 2
64.22.a.d.1.1 1 1.1 even 1 trivial
64.22.a.d.1.1 1 4.3 odd 2 CM