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Properties

Label 9.40.a.a.1.1
Level $9$
Weight $40$
Character 9.1
Self dual yes
Analytic conductor $86.706$
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] = [9,40,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, 40, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.1");
 
S:= CuspForms(chi, 40);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 40 \)
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: \(86.7055962508\)
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)
 
\(f(q)\) \(=\) \(q-5.49756e11 q^{4} +5.45957e16 q^{7} +O(q^{10})\) \(q-5.49756e11 q^{4} +5.45957e16 q^{7} +1.05355e22 q^{13} +3.02231e23 q^{16} -1.67975e25 q^{19} -1.81899e27 q^{25} -3.00143e28 q^{28} +2.36867e29 q^{31} -5.15129e29 q^{37} -6.23893e31 q^{43} +2.07115e33 q^{49} -5.79196e33 q^{52} +7.81481e34 q^{61} -1.66153e35 q^{64} +3.61465e35 q^{67} -2.95498e36 q^{73} +9.23454e36 q^{76} +1.09416e37 q^{79} +5.75194e38 q^{91} +6.09196e38 q^{97} +O(q^{100})\)

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
\(4\) −5.49756e11 −1.00000
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 5.45957e16 1.81028 0.905142 − 0.425110i \(-0.139764\pi\)
0.905142 + 0.425110i \(0.139764\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 1.05355e22 1.99876 0.999380 − 0.0352059i \(-0.0112087\pi\)
0.999380 + 0.0352059i \(0.0112087\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 3.02231e23 1.00000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −1.67975e25 −1.94783 −0.973914 − 0.226919i \(-0.927135\pi\)
−0.973914 + 0.226919i \(0.927135\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −1.81899e27 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) −3.00143e28 −1.81028
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 2.36867e29 1.96314 0.981570 − 0.191105i \(-0.0612070\pi\)
0.981570 + 0.191105i \(0.0612070\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.15129e29 −0.135514 −0.0677568 − 0.997702i \(-0.521584\pi\)
−0.0677568 + 0.997702i \(0.521584\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −6.23893e31 −0.875942 −0.437971 − 0.898989i \(-0.644303\pi\)
−0.437971 + 0.898989i \(0.644303\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 2.07115e33 2.27713
\(50\) 0 0
\(51\) 0 0
\(52\) −5.79196e33 −1.99876
\(53\) 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
\(61\) 7.81481e34 1.19946 0.599731 − 0.800202i \(-0.295274\pi\)
0.599731 + 0.800202i \(0.295274\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −1.66153e35 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 3.61465e35 0.890446 0.445223 − 0.895420i \(-0.353124\pi\)
0.445223 + 0.895420i \(0.353124\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −2.95498e36 −1.36697 −0.683484 − 0.729965i \(-0.739536\pi\)
−0.683484 + 0.729965i \(0.739536\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 9.23454e36 1.94783
\(77\) 0 0
\(78\) 0 0
\(79\) 1.09416e37 1.08481 0.542404 − 0.840118i \(-0.317514\pi\)
0.542404 + 0.840118i \(0.317514\pi\)
\(80\) 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
\(91\) 5.75194e38 3.61832
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 6.09196e38 1.10334 0.551669 − 0.834063i \(-0.313991\pi\)
0.551669 + 0.834063i \(0.313991\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.00000e39 1.00000
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 3.27066e39 1.83785 0.918924 − 0.394435i \(-0.129060\pi\)
0.918924 + 0.394435i \(0.129060\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 6.05909e39 1.12873 0.564366 − 0.825525i \(-0.309121\pi\)
0.564366 + 0.825525i \(0.309121\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.65005e40 1.81028
\(113\) 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
\(121\) −4.11448e40 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) −1.30219e41 −1.96314
\(125\) 0 0
\(126\) 0 0
\(127\) 2.08743e41 1.97442 0.987209 − 0.159434i \(-0.0509668\pi\)
0.987209 + 0.159434i \(0.0509668\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −9.17073e41 −3.52612
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) −8.91894e41 −1.45052 −0.725261 − 0.688474i \(-0.758281\pi\)
−0.725261 + 0.688474i \(0.758281\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 2.83195e41 0.135514
\(149\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 5.72363e41 0.185192 0.0925958 − 0.995704i \(-0.470484\pi\)
0.0925958 + 0.995704i \(0.470484\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 1.19344e43 1.80616 0.903082 − 0.429469i \(-0.141299\pi\)
0.903082 + 0.429469i \(0.141299\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 3.78348e42 0.275571 0.137785 − 0.990462i \(-0.456002\pi\)
0.137785 + 0.990462i \(0.456002\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 8.32135e43 2.99504
\(170\) 0 0
\(171\) 0 0
\(172\) 3.42989e43 0.875942
\(173\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) −9.93090e43 −1.81028
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −1.57552e44 −1.48831 −0.744153 − 0.668009i \(-0.767147\pi\)
−0.744153 + 0.668009i \(0.767147\pi\)
\(182\) 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
\(193\) −2.41201e44 −0.651644 −0.325822 − 0.945431i \(-0.605641\pi\)
−0.325822 + 0.945431i \(0.605641\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −1.13862e45 −2.27713
\(197\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) −1.18481e44 −0.176203 −0.0881016 − 0.996111i \(-0.528080\pi\)
−0.0881016 + 0.996111i \(0.528080\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 3.18417e45 1.99876
\(209\) 0 0
\(210\) 0 0
\(211\) −2.88906e45 −1.37167 −0.685834 − 0.727758i \(-0.740563\pi\)
−0.685834 + 0.727758i \(0.740563\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 1.29319e46 3.55384
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −3.38502e45 −0.546533 −0.273266 − 0.961938i \(-0.588104\pi\)
−0.273266 + 0.961938i \(0.588104\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 6.58243e45 0.633278 0.316639 − 0.948546i \(-0.397446\pi\)
0.316639 + 0.948546i \(0.397446\pi\)
\(230\) 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
\(241\) 3.20555e46 1.13912 0.569561 − 0.821949i \(-0.307113\pi\)
0.569561 + 0.821949i \(0.307113\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −4.29624e46 −1.19946
\(245\) 0 0
\(246\) 0 0
\(247\) −1.76971e47 −3.89324
\(248\) 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
\(256\) 9.13439e46 1.00000
\(257\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) −2.81239e46 −0.245318
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −1.98717e47 −0.890446
\(269\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 5.13721e47 1.85279 0.926395 − 0.376554i \(-0.122891\pi\)
0.926395 + 0.376554i \(0.122891\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 3.33109e47 0.783847 0.391923 − 0.919998i \(-0.371810\pi\)
0.391923 + 0.919998i \(0.371810\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 1.22537e48 1.89859 0.949295 − 0.314386i \(-0.101799\pi\)
0.949295 + 0.314386i \(0.101799\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −9.71646e47 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 1.62452e48 1.36697
\(293\) 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
\(301\) −3.40619e48 −1.58570
\(302\) 0 0
\(303\) 0 0
\(304\) −5.07674e48 −1.94783
\(305\) 0 0
\(306\) 0 0
\(307\) −2.20986e48 −0.700110 −0.350055 − 0.936729i \(-0.613837\pi\)
−0.350055 + 0.936729i \(0.613837\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 7.92017e48 1.72036 0.860181 − 0.509989i \(-0.170350\pi\)
0.860181 + 0.509989i \(0.170350\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −6.01521e48 −1.08481
\(317\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −1.91640e49 −1.99876
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 2.40568e49 1.75627 0.878136 − 0.478411i \(-0.158787\pi\)
0.878136 + 0.478411i \(0.158787\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 3.17551e49 1.63317 0.816583 − 0.577228i \(-0.195866\pi\)
0.816583 + 0.577228i \(0.195866\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 6.34185e49 2.31196
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) −5.12513e49 −1.33232 −0.666161 − 0.745808i \(-0.732064\pi\)
−0.666161 + 0.745808i \(0.732064\pi\)
\(350\) 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
\(361\) 2.07789e50 2.79403
\(362\) 0 0
\(363\) 0 0
\(364\) −3.16216e50 −3.61832
\(365\) 0 0
\(366\) 0 0
\(367\) −1.65336e50 −1.61205 −0.806026 − 0.591880i \(-0.798386\pi\)
−0.806026 + 0.591880i \(0.798386\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 2.05894e50 1.46326 0.731630 − 0.681702i \(-0.238760\pi\)
0.731630 + 0.681702i \(0.238760\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −3.75162e50 −1.95323 −0.976617 − 0.214989i \(-0.931029\pi\)
−0.976617 + 0.214989i \(0.931029\pi\)
\(380\) 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
\(388\) −3.34909e50 −1.10334
\(389\) 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
\(397\) 7.87820e50 1.65963 0.829817 − 0.558036i \(-0.188445\pi\)
0.829817 + 0.558036i \(0.188445\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −5.49756e50 −1.00000
\(401\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 2.49552e51 3.92384
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 8.14182e50 0.959659 0.479829 − 0.877362i \(-0.340699\pi\)
0.479829 + 0.877362i \(0.340699\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −1.79806e51 −1.83785
\(413\) 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
\(421\) −2.25787e50 −0.151425 −0.0757126 − 0.997130i \(-0.524123\pi\)
−0.0757126 + 0.997130i \(0.524123\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 4.26655e51 2.17137
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −4.92197e51 −1.90821 −0.954103 − 0.299479i \(-0.903187\pi\)
−0.954103 + 0.299479i \(0.903187\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −3.33102e51 −1.12873
\(437\) 0 0
\(438\) 0 0
\(439\) −1.76366e51 −0.522827 −0.261413 − 0.965227i \(-0.584189\pi\)
−0.261413 + 0.965227i \(0.584189\pi\)
\(440\) 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
\(448\) −9.07127e51 −1.81028
\(449\) 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
\(457\) 3.19135e51 0.432124 0.216062 − 0.976380i \(-0.430679\pi\)
0.216062 + 0.976380i \(0.430679\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −1.36255e49 −0.00143062 −0.000715309 − 1.00000i \(-0.500228\pi\)
−0.000715309 1.00000i \(0.500228\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 1.97344e52 1.61196
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 3.05545e52 1.94783
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −5.42716e51 −0.270859
\(482\) 0 0
\(483\) 0 0
\(484\) 2.26196e52 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) −5.08792e52 −1.99400 −0.997002 − 0.0773819i \(-0.975344\pi\)
−0.997002 + 0.0773819i \(0.975344\pi\)
\(488\) 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
\(496\) 7.15886e52 1.96314
\(497\) 0 0
\(498\) 0 0
\(499\) 2.24594e52 0.547567 0.273784 − 0.961791i \(-0.411725\pi\)
0.273784 + 0.961791i \(0.411725\pi\)
\(500\) 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
\(508\) −1.14758e53 −1.97442
\(509\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) −1.61329e53 −2.47460
\(512\) 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
\(523\) −1.11241e53 −1.08513 −0.542565 − 0.840014i \(-0.682547\pi\)
−0.542565 + 0.840014i \(0.682547\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −1.28052e53 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 5.04166e53 3.52612
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −3.56602e53 −1.79819 −0.899097 − 0.437750i \(-0.855776\pi\)
−0.899097 + 0.437750i \(0.855776\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 3.64176e53 1.48101 0.740506 − 0.672049i \(-0.234586\pi\)
0.740506 + 0.672049i \(0.234586\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 5.97365e53 1.96381
\(554\) 0 0
\(555\) 0 0
\(556\) 4.90324e53 1.45052
\(557\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) −6.57304e53 −1.75080
\(560\) 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
\(571\) 7.72380e53 1.35965 0.679825 − 0.733374i \(-0.262056\pi\)
0.679825 + 0.733374i \(0.262056\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −5.61780e53 −0.806563 −0.403282 − 0.915076i \(-0.632130\pi\)
−0.403282 + 0.915076i \(0.632130\pi\)
\(578\) 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
\(589\) −3.97878e54 −3.82386
\(590\) 0 0
\(591\) 0 0
\(592\) −1.55688e53 −0.135514
\(593\) 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
\(601\) −5.90872e53 −0.383214 −0.191607 − 0.981472i \(-0.561370\pi\)
−0.191607 + 0.981472i \(0.561370\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −3.14660e53 −0.185192
\(605\) 0 0
\(606\) 0 0
\(607\) 2.08886e54 1.11617 0.558084 − 0.829785i \(-0.311537\pi\)
0.558084 + 0.829785i \(0.311537\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 4.52102e54 1.99415 0.997073 − 0.0764609i \(-0.0243620\pi\)
0.997073 + 0.0764609i \(0.0243620\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) −4.70480e54 −1.71622 −0.858112 − 0.513463i \(-0.828362\pi\)
−0.858112 + 0.513463i \(0.828362\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 3.30872e54 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) −6.56100e54 −1.80616
\(629\) 0 0
\(630\) 0 0
\(631\) 5.58618e54 1.40134 0.700668 − 0.713487i \(-0.252885\pi\)
0.700668 + 0.713487i \(0.252885\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 2.18206e55 4.55143
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) −8.89392e54 −1.54518 −0.772590 − 0.634905i \(-0.781039\pi\)
−0.772590 + 0.634905i \(0.781039\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −2.07999e54 −0.275571
\(653\) 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
\(661\) −1.67715e55 −1.70077 −0.850383 − 0.526165i \(-0.823630\pi\)
−0.850383 + 0.526165i \(0.823630\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −1.22384e55 −0.873841 −0.436921 − 0.899500i \(-0.643931\pi\)
−0.436921 + 0.899500i \(0.643931\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −4.57471e55 −2.99504
\(677\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 3.32595e55 1.99735
\(680\) 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
\(688\) −1.88560e55 −0.875942
\(689\) 0 0
\(690\) 0 0
\(691\) 1.49877e55 0.639606 0.319803 − 0.947484i \(-0.396383\pi\)
0.319803 + 0.947484i \(0.396383\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 5.45957e55 1.81028
\(701\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 8.65291e54 0.263957
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −6.06616e55 −1.56788 −0.783939 − 0.620838i \(-0.786793\pi\)
−0.783939 + 0.620838i \(0.786793\pi\)
\(710\) 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
\(721\) 1.78564e56 3.32703
\(722\) 0 0
\(723\) 0 0
\(724\) 8.66152e55 1.48831
\(725\) 0 0
\(726\) 0 0
\(727\) 5.82266e55 0.922997 0.461498 − 0.887141i \(-0.347312\pi\)
0.461498 + 0.887141i \(0.347312\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −1.39449e56 −1.88316 −0.941581 − 0.336788i \(-0.890660\pi\)
−0.941581 + 0.336788i \(0.890660\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −1.37794e56 −1.58731 −0.793656 − 0.608366i \(-0.791825\pi\)
−0.793656 + 0.608366i \(0.791825\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −2.37604e56 −1.99928 −0.999642 − 0.0267535i \(-0.991483\pi\)
−0.999642 + 0.0267535i \(0.991483\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −9.33619e55 −0.672667 −0.336333 − 0.941743i \(-0.609187\pi\)
−0.336333 + 0.941743i \(0.609187\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 3.30801e56 2.04333
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −3.72150e56 −1.97313 −0.986563 − 0.163380i \(-0.947760\pi\)
−0.986563 + 0.163380i \(0.947760\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.32601e56 0.651644
\(773\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) −4.30858e56 −1.96314
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 6.25966e56 2.27713
\(785\) 0 0
\(786\) 0 0
\(787\) −4.40507e56 −1.48746 −0.743730 − 0.668480i \(-0.766945\pi\)
−0.743730 + 0.668480i \(0.766945\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 8.23331e56 2.39744
\(794\) 0 0
\(795\) 0 0
\(796\) 6.51356e55 0.176203
\(797\) 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
\(811\) 6.83722e56 1.28521 0.642604 − 0.766199i \(-0.277854\pi\)
0.642604 + 0.766199i \(0.277854\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 1.04799e57 1.70618
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) 3.66109e56 0.516789 0.258395 − 0.966039i \(-0.416806\pi\)
0.258395 + 0.966039i \(0.416806\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) −1.63114e57 −1.99838 −0.999192 − 0.0401952i \(-0.987202\pi\)
−0.999192 + 0.0401952i \(0.987202\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −1.75051e57 −1.99876
\(833\) 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
\(841\) −1.08024e57 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 1.58828e57 1.37167
\(845\) 0 0
\(846\) 0 0
\(847\) −2.24633e57 −1.81028
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 2.15497e56 0.151334 0.0756669 − 0.997133i \(-0.475891\pi\)
0.0756669 + 0.997133i \(0.475891\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 2.11079e57 1.29294 0.646471 − 0.762939i \(-0.276244\pi\)
0.646471 + 0.762939i \(0.276244\pi\)
\(860\) 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
\(868\) −7.10939e57 −3.55384
\(869\) 0 0
\(870\) 0 0
\(871\) 3.80822e57 1.77979
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −4.69599e57 −1.91971 −0.959855 − 0.280498i \(-0.909500\pi\)
−0.959855 + 0.280498i \(0.909500\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 4.75399e57 1.70147 0.850735 − 0.525595i \(-0.176157\pi\)
0.850735 + 0.525595i \(0.176157\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 1.13965e58 3.57426
\(890\) 0 0
\(891\) 0 0
\(892\) 1.86093e57 0.546533
\(893\) 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
\(907\) −4.74766e56 −0.100725 −0.0503625 − 0.998731i \(-0.516038\pi\)
−0.0503625 + 0.998731i \(0.516038\pi\)
\(908\) 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
\(916\) −3.61873e57 −0.633278
\(917\) 0 0
\(918\) 0 0
\(919\) 7.30639e57 1.19964 0.599819 − 0.800136i \(-0.295239\pi\)
0.599819 + 0.800136i \(0.295239\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 9.37015e56 0.135514
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) −3.47902e58 −4.43545
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1.75168e58 1.97030 0.985152 − 0.171682i \(-0.0549203\pi\)
0.985152 + 0.171682i \(0.0549203\pi\)
\(938\) 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
\(949\) −3.11322e58 −2.73224
\(950\) 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
\(961\) 4.15477e58 2.85392
\(962\) 0 0
\(963\) 0 0
\(964\) −1.76227e58 −1.13912
\(965\) 0 0
\(966\) 0 0
\(967\) −6.59759e57 −0.401392 −0.200696 − 0.979654i \(-0.564320\pi\)
−0.200696 + 0.979654i \(0.564320\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) −4.86936e58 −2.62586
\(974\) 0 0
\(975\) 0 0
\(976\) 2.36188e58 1.19946
\(977\) 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
\(988\) 9.72907e58 3.89324
\(989\) 0 0
\(990\) 0 0
\(991\) 5.01142e58 1.89027 0.945136 − 0.326676i \(-0.105929\pi\)
0.945136 + 0.326676i \(0.105929\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −4.84798e57 −0.162557 −0.0812784 − 0.996691i \(-0.525900\pi\)
−0.0812784 + 0.996691i \(0.525900\pi\)
\(998\) 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 9.40.a.a.1.1 ✓ 1
3.2 odd 2 CM 9.40.a.a.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.40.a.a.1.1 ✓ 1 1.1 even 1 trivial
9.40.a.a.1.1 ✓ 1 3.2 odd 2 CM