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Properties

Label 9.46.a.a.1.1
Level $9$
Weight $46$
Character 9.1
Self dual yes
Analytic conductor $115.430$
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] = [9,46,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, 46, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.1");
 
S:= CuspForms(chi, 46);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 46 \)
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: \(115.430153467\)
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)
 
\(f(q)\) \(=\) \(q-3.51844e13 q^{4} -1.58014e19 q^{7} +O(q^{10})\) \(q-3.51844e13 q^{4} -1.58014e19 q^{7} +1.85551e24 q^{13} +1.23794e27 q^{16} +1.05959e29 q^{19} -2.84217e31 q^{25} +5.55962e32 q^{28} +5.28564e33 q^{31} +2.05067e35 q^{37} -1.07411e37 q^{43} +1.42677e38 q^{49} -6.52850e37 q^{52} -7.56907e39 q^{61} -4.35561e40 q^{64} +2.40665e41 q^{67} -2.40214e41 q^{73} -3.72808e42 q^{76} +7.05431e42 q^{79} -2.93197e43 q^{91} -8.57398e44 q^{97} +O(q^{100})\)

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
\(4\) −3.51844e13 −1.00000
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) −1.58014e19 −1.52753 −0.763765 − 0.645494i \(-0.776651\pi\)
−0.763765 + 0.645494i \(0.776651\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 1.85551e24 0.160228 0.0801140 − 0.996786i \(-0.474472\pi\)
0.0801140 + 0.996786i \(0.474472\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.23794e27 1.00000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 1.05959e29 1.79135 0.895674 − 0.444712i \(-0.146694\pi\)
0.895674 + 0.444712i \(0.146694\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −2.84217e31 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 5.55962e32 1.52753
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 5.28564e33 1.47048 0.735241 − 0.677806i \(-0.237069\pi\)
0.735241 + 0.677806i \(0.237069\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.05067e35 1.06502 0.532510 − 0.846424i \(-0.321249\pi\)
0.532510 + 0.846424i \(0.321249\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −1.07411e37 −1.89675 −0.948374 − 0.317155i \(-0.897272\pi\)
−0.948374 + 0.317155i \(0.897272\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 1.42677e38 1.33335
\(50\) 0 0
\(51\) 0 0
\(52\) −6.52850e37 −0.160228
\(53\) 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
\(61\) −7.56907e39 −0.511825 −0.255912 − 0.966700i \(-0.582376\pi\)
−0.255912 + 0.966700i \(0.582376\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −4.35561e40 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 2.40665e41 1.97120 0.985598 − 0.169103i \(-0.0540872\pi\)
0.985598 + 0.169103i \(0.0540872\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −2.40214e41 −0.285650 −0.142825 − 0.989748i \(-0.545619\pi\)
−0.142825 + 0.989748i \(0.545619\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −3.72808e42 −1.79135
\(77\) 0 0
\(78\) 0 0
\(79\) 7.05431e42 1.41855 0.709276 − 0.704931i \(-0.249022\pi\)
0.709276 + 0.704931i \(0.249022\pi\)
\(80\) 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
\(91\) −2.93197e43 −0.244753
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −8.57398e44 −1.70145 −0.850724 − 0.525613i \(-0.823836\pi\)
−0.850724 + 0.525613i \(0.823836\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.00000e45 1.00000
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 5.28551e44 0.271800 0.135900 − 0.990723i \(-0.456607\pi\)
0.135900 + 0.990723i \(0.456607\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) −1.28300e46 −1.84557 −0.922784 − 0.385318i \(-0.874092\pi\)
−0.922784 + 0.385318i \(0.874092\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1.95612e46 −1.52753
\(113\) 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
\(121\) −7.28905e46 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) −1.85972e47 −1.47048
\(125\) 0 0
\(126\) 0 0
\(127\) −3.94342e47 −1.82091 −0.910456 − 0.413606i \(-0.864269\pi\)
−0.910456 + 0.413606i \(0.864269\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −1.67429e48 −2.73634
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) −2.77379e48 −1.67974 −0.839869 − 0.542790i \(-0.817368\pi\)
−0.839869 + 0.542790i \(0.817368\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −7.21516e48 −1.06502
\(149\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −1.87056e49 −1.75788 −0.878942 − 0.476929i \(-0.841750\pi\)
−0.878942 + 0.476929i \(0.841750\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 5.11200e49 1.99917 0.999583 − 0.0288712i \(-0.00919126\pi\)
0.999583 + 0.0288712i \(0.00919126\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1.16477e50 1.95893 0.979466 − 0.201610i \(-0.0646173\pi\)
0.979466 + 0.201610i \(0.0646173\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −1.30664e50 −0.974327
\(170\) 0 0
\(171\) 0 0
\(172\) 3.77920e50 1.89675
\(173\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 4.49103e50 1.52753
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 8.30879e50 1.32364 0.661820 − 0.749663i \(-0.269784\pi\)
0.661820 + 0.749663i \(0.269784\pi\)
\(182\) 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
\(193\) −5.33329e50 −0.200427 −0.100213 − 0.994966i \(-0.531953\pi\)
−0.100213 + 0.994966i \(0.531953\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −5.02001e51 −1.33335
\(197\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) −7.79746e51 −1.47150 −0.735748 − 0.677255i \(-0.763169\pi\)
−0.735748 + 0.677255i \(0.763169\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 2.29701e51 0.160228
\(209\) 0 0
\(210\) 0 0
\(211\) −3.55089e52 −1.79466 −0.897329 − 0.441363i \(-0.854495\pi\)
−0.897329 + 0.441363i \(0.854495\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −8.35206e52 −2.24621
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 1.31836e53 1.91945 0.959723 − 0.280949i \(-0.0906491\pi\)
0.959723 + 0.280949i \(0.0906491\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 2.19969e53 1.76223 0.881115 − 0.472902i \(-0.156794\pi\)
0.881115 + 0.472902i \(0.156794\pi\)
\(230\) 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
\(241\) 6.36590e53 1.61613 0.808065 − 0.589094i \(-0.200515\pi\)
0.808065 + 0.589094i \(0.200515\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 2.66313e53 0.511825
\(245\) 0 0
\(246\) 0 0
\(247\) 1.96607e53 0.287024
\(248\) 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
\(256\) 1.53250e54 1.00000
\(257\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) −3.24035e54 −1.62685
\(260\) 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
\(268\) −8.46765e54 −1.97120
\(269\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) −1.08083e55 −1.95861 −0.979303 − 0.202400i \(-0.935126\pi\)
−0.979303 + 0.202400i \(0.935126\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 8.69170e54 0.962299 0.481149 − 0.876639i \(-0.340219\pi\)
0.481149 + 0.876639i \(0.340219\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 1.90743e53 0.0130393 0.00651963 − 0.999979i \(-0.497925\pi\)
0.00651963 + 0.999979i \(0.497925\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −2.34532e55 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 8.45178e54 0.285650
\(293\) 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
\(301\) 1.69725e56 2.89734
\(302\) 0 0
\(303\) 0 0
\(304\) 1.31170e56 1.79135
\(305\) 0 0
\(306\) 0 0
\(307\) −1.11169e56 −1.21723 −0.608613 − 0.793468i \(-0.708274\pi\)
−0.608613 + 0.793468i \(0.708274\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 2.65309e56 1.87934 0.939668 − 0.342088i \(-0.111134\pi\)
0.939668 + 0.342088i \(0.111134\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −2.48202e56 −1.41855
\(317\) 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
\(325\) −5.27368e55 −0.160228
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −9.38604e56 −1.88952 −0.944762 − 0.327757i \(-0.893707\pi\)
−0.944762 + 0.327757i \(0.893707\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −2.01522e56 −0.270801 −0.135400 − 0.990791i \(-0.543232\pi\)
−0.135400 + 0.990791i \(0.543232\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −5.63644e56 −0.509198
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) −3.27037e57 −1.99998 −0.999989 − 0.00460188i \(-0.998535\pi\)
−0.999989 + 0.00460188i \(0.998535\pi\)
\(350\) 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
\(361\) 7.72846e57 2.20893
\(362\) 0 0
\(363\) 0 0
\(364\) 1.03160e57 0.244753
\(365\) 0 0
\(366\) 0 0
\(367\) 1.01393e58 1.99997 0.999983 − 0.00585107i \(-0.00186246\pi\)
0.999983 + 0.00585107i \(0.00186246\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −3.77327e57 −0.516736 −0.258368 − 0.966047i \(-0.583185\pi\)
−0.258368 + 0.966047i \(0.583185\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 2.34723e57 0.224478 0.112239 − 0.993681i \(-0.464198\pi\)
0.112239 + 0.993681i \(0.464198\pi\)
\(380\) 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
\(388\) 3.01670e58 1.70145
\(389\) 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
\(397\) −5.37096e58 −1.80828 −0.904139 − 0.427237i \(-0.859487\pi\)
−0.904139 + 0.427237i \(0.859487\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −3.51844e58 −1.00000
\(401\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 9.80758e57 0.235612
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −1.16051e59 −1.99929 −0.999644 − 0.0266953i \(-0.991502\pi\)
−0.999644 + 0.0266953i \(0.991502\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −1.85967e58 −0.271800
\(413\) 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
\(421\) −7.19190e58 −0.646393 −0.323197 − 0.946332i \(-0.604757\pi\)
−0.323197 + 0.946332i \(0.604757\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1.19602e59 0.781828
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 3.47392e59 1.65898 0.829492 − 0.558519i \(-0.188630\pi\)
0.829492 + 0.558519i \(0.188630\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 4.51415e59 1.84557
\(437\) 0 0
\(438\) 0 0
\(439\) −5.69860e59 −1.99672 −0.998358 − 0.0572837i \(-0.981756\pi\)
−0.998358 + 0.0572837i \(0.981756\pi\)
\(440\) 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
\(448\) 6.88248e59 1.52753
\(449\) 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
\(457\) 1.31327e60 1.86312 0.931561 − 0.363586i \(-0.118448\pi\)
0.931561 + 0.363586i \(0.118448\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 4.50938e59 0.477028 0.238514 − 0.971139i \(-0.423340\pi\)
0.238514 + 0.971139i \(0.423340\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) −3.80284e60 −3.01106
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −3.01152e60 −1.79135
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 3.80505e59 0.170646
\(482\) 0 0
\(483\) 0 0
\(484\) 2.56461e60 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) −5.44280e60 −1.84680 −0.923402 − 0.383835i \(-0.874603\pi\)
−0.923402 + 0.383835i \(0.874603\pi\)
\(488\) 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
\(496\) 6.54331e60 1.47048
\(497\) 0 0
\(498\) 0 0
\(499\) −9.99129e60 −1.96047 −0.980233 − 0.197846i \(-0.936606\pi\)
−0.980233 + 0.197846i \(0.936606\pi\)
\(500\) 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
\(508\) 1.38747e61 1.82091
\(509\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 3.79572e60 0.436339
\(512\) 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
\(523\) 2.92818e61 1.99669 0.998344 − 0.0575316i \(-0.0183230\pi\)
0.998344 + 0.0575316i \(0.0183230\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −1.89563e61 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 5.89089e61 2.73634
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −5.67163e61 −1.80622 −0.903108 − 0.429413i \(-0.858720\pi\)
−0.903108 + 0.429413i \(0.858720\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −6.60341e61 −1.64079 −0.820396 − 0.571795i \(-0.806247\pi\)
−0.820396 + 0.571795i \(0.806247\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −1.11468e62 −2.16688
\(554\) 0 0
\(555\) 0 0
\(556\) 9.75942e61 1.67974
\(557\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) −1.99303e61 −0.303912
\(560\) 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
\(571\) −7.16885e61 −0.677856 −0.338928 − 0.940812i \(-0.610064\pi\)
−0.338928 + 0.940812i \(0.610064\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −2.20891e62 −1.65091 −0.825453 − 0.564471i \(-0.809080\pi\)
−0.825453 + 0.564471i \(0.809080\pi\)
\(578\) 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
\(589\) 5.60059e62 2.63414
\(590\) 0 0
\(591\) 0 0
\(592\) 2.53861e62 1.06502
\(593\) 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
\(601\) 1.28527e61 0.0383988 0.0191994 − 0.999816i \(-0.493888\pi\)
0.0191994 + 0.999816i \(0.493888\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 6.58144e62 1.75788
\(605\) 0 0
\(606\) 0 0
\(607\) −3.49972e61 −0.0836156 −0.0418078 − 0.999126i \(-0.513312\pi\)
−0.0418078 + 0.999126i \(0.513312\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −2.82796e62 −0.541518 −0.270759 − 0.962647i \(-0.587275\pi\)
−0.270759 + 0.962647i \(0.587275\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 2.34301e61 0.0360358 0.0180179 − 0.999838i \(-0.494264\pi\)
0.0180179 + 0.999838i \(0.494264\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 8.07794e62 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) −1.79863e63 −1.99917
\(629\) 0 0
\(630\) 0 0
\(631\) −1.56376e63 −1.56138 −0.780691 − 0.624918i \(-0.785132\pi\)
−0.780691 + 0.624918i \(0.785132\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 2.64740e62 0.213640
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 1.59370e62 0.104150 0.0520751 − 0.998643i \(-0.483416\pi\)
0.0520751 + 0.998643i \(0.483416\pi\)
\(644\) 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
\(652\) −4.09817e63 −1.95893
\(653\) 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
\(661\) −1.55677e63 −0.546628 −0.273314 − 0.961925i \(-0.588120\pi\)
−0.273314 + 0.961925i \(0.588120\pi\)
\(662\) 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
\(673\) 7.86123e63 1.84141 0.920707 − 0.390254i \(-0.127613\pi\)
0.920707 + 0.390254i \(0.127613\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 4.59733e63 0.974327
\(677\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 1.35481e64 2.59901
\(680\) 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
\(688\) −1.32969e64 −1.89675
\(689\) 0 0
\(690\) 0 0
\(691\) 8.61624e63 1.11445 0.557227 − 0.830360i \(-0.311865\pi\)
0.557227 + 0.830360i \(0.311865\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −1.58014e64 −1.52753
\(701\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 2.17286e64 1.90782
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −8.53419e63 −0.618903 −0.309451 − 0.950915i \(-0.600145\pi\)
−0.309451 + 0.950915i \(0.600145\pi\)
\(710\) 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
\(721\) −8.35184e63 −0.415182
\(722\) 0 0
\(723\) 0 0
\(724\) −2.92339e64 −1.32364
\(725\) 0 0
\(726\) 0 0
\(727\) −4.17299e64 −1.72156 −0.860781 − 0.508975i \(-0.830025\pi\)
−0.860781 + 0.508975i \(0.830025\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 2.53420e64 0.868965 0.434482 − 0.900680i \(-0.356931\pi\)
0.434482 + 0.900680i \(0.356931\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −5.38015e64 −1.53566 −0.767830 − 0.640654i \(-0.778663\pi\)
−0.767830 + 0.640654i \(0.778663\pi\)
\(740\) 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
\(751\) 3.27777e64 0.651147 0.325574 − 0.945517i \(-0.394443\pi\)
0.325574 + 0.945517i \(0.394443\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −1.84867e64 −0.307045 −0.153522 − 0.988145i \(-0.549062\pi\)
−0.153522 + 0.988145i \(0.549062\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 2.02732e65 2.81916
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 1.71308e65 1.99727 0.998633 − 0.0522732i \(-0.0166467\pi\)
0.998633 + 0.0522732i \(0.0166467\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.87649e64 0.200427
\(773\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) −1.50227e65 −1.47048
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 1.76626e65 1.33335
\(785\) 0 0
\(786\) 0 0
\(787\) −2.70052e65 −1.87075 −0.935374 − 0.353661i \(-0.884937\pi\)
−0.935374 + 0.353661i \(0.884937\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −1.40445e64 −0.0820087
\(794\) 0 0
\(795\) 0 0
\(796\) 2.74349e65 1.47150
\(797\) 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
\(811\) −2.22603e65 −0.784443 −0.392222 − 0.919871i \(-0.628293\pi\)
−0.392222 + 0.919871i \(0.628293\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −1.13811e66 −3.39773
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) −5.58202e65 −1.41350 −0.706748 − 0.707465i \(-0.749839\pi\)
−0.706748 + 0.707465i \(0.749839\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 6.66803e65 1.43391 0.716953 − 0.697121i \(-0.245536\pi\)
0.716953 + 0.697121i \(0.245536\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −8.08190e64 −0.160228
\(833\) 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
\(841\) −6.42554e65 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 1.24936e66 1.79466
\(845\) 0 0
\(846\) 0 0
\(847\) 1.15177e66 1.52753
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −1.05218e66 −1.19052 −0.595262 − 0.803532i \(-0.702952\pi\)
−0.595262 + 0.803532i \(0.702952\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 1.23405e66 1.19258 0.596289 − 0.802770i \(-0.296641\pi\)
0.596289 + 0.802770i \(0.296641\pi\)
\(860\) 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
\(868\) 2.93862e66 2.24621
\(869\) 0 0
\(870\) 0 0
\(871\) 4.46557e65 0.315841
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −2.27209e66 −1.37700 −0.688502 − 0.725235i \(-0.741731\pi\)
−0.688502 + 0.725235i \(0.741731\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) −7.91671e65 −0.411556 −0.205778 − 0.978599i \(-0.565972\pi\)
−0.205778 + 0.978599i \(0.565972\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 6.23115e66 2.78150
\(890\) 0 0
\(891\) 0 0
\(892\) −4.63858e66 −1.91945
\(893\) 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
\(907\) −6.71059e66 −1.90808 −0.954040 − 0.299679i \(-0.903120\pi\)
−0.954040 + 0.299679i \(0.903120\pi\)
\(908\) 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
\(916\) −7.73946e66 −1.76223
\(917\) 0 0
\(918\) 0 0
\(919\) −6.39054e66 −1.35188 −0.675941 − 0.736956i \(-0.736262\pi\)
−0.675941 + 0.736956i \(0.736262\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −5.82836e66 −1.06502
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) 1.51179e67 2.38849
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1.13518e67 1.55212 0.776059 − 0.630660i \(-0.217216\pi\)
0.776059 + 0.630660i \(0.217216\pi\)
\(938\) 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
\(949\) −4.45720e65 −0.0457691
\(950\) 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
\(961\) 1.50176e67 1.16232
\(962\) 0 0
\(963\) 0 0
\(964\) −2.23980e67 −1.61613
\(965\) 0 0
\(966\) 0 0
\(967\) 2.46061e67 1.65557 0.827785 − 0.561045i \(-0.189600\pi\)
0.827785 + 0.561045i \(0.189600\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 4.38298e67 2.56585
\(974\) 0 0
\(975\) 0 0
\(976\) −9.37006e66 −0.511825
\(977\) 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
\(988\) −6.91751e66 −0.287024
\(989\) 0 0
\(990\) 0 0
\(991\) −1.34236e67 −0.520250 −0.260125 − 0.965575i \(-0.583764\pi\)
−0.260125 + 0.965575i \(0.583764\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −4.52855e67 −1.53221 −0.766104 − 0.642716i \(-0.777807\pi\)
−0.766104 + 0.642716i \(0.777807\pi\)
\(998\) 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 9.46.a.a.1.1 ✓ 1
3.2 odd 2 CM 9.46.a.a.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.46.a.a.1.1 ✓ 1 1.1 even 1 trivial
9.46.a.a.1.1 ✓ 1 3.2 odd 2 CM