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Properties

Label 9.22.a.b.1.1
Level $9$
Weight $22$
Character 9.1
Self dual yes
Analytic conductor $25.153$
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] = [9,22,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, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 22 \)
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: \(25.1529609858\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

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

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.09715e6 q^{4} +1.12398e9 q^{7} +O(q^{10})\) \(q-2.09715e6 q^{4} +1.12398e9 q^{7} -3.70077e11 q^{13} +4.39805e12 q^{16} -3.55406e13 q^{19} -4.76837e14 q^{25} -2.35716e15 q^{28} -9.04007e15 q^{31} -5.77763e16 q^{37} +2.65258e17 q^{43} +7.04792e17 q^{49} +7.76107e17 q^{52} -1.08607e19 q^{61} -9.22337e18 q^{64} +6.94433e18 q^{67} -3.90986e19 q^{73} +7.45341e19 q^{76} +1.68068e20 q^{79} -4.15960e20 q^{91} -1.13207e21 q^{97} +O(q^{100})\)

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
\(4\) −2.09715e6 −1.00000
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 1.12398e9 1.50394 0.751970 − 0.659198i \(-0.229104\pi\)
0.751970 + 0.659198i \(0.229104\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −3.70077e11 −0.744538 −0.372269 − 0.928125i \(-0.621420\pi\)
−0.372269 + 0.928125i \(0.621420\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.39805e12 1.00000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −3.55406e13 −1.32988 −0.664940 − 0.746897i \(-0.731543\pi\)
−0.664940 + 0.746897i \(0.731543\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −4.76837e14 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) −2.35716e15 −1.50394
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −9.04007e15 −1.98095 −0.990476 − 0.137683i \(-0.956035\pi\)
−0.990476 + 0.137683i \(0.956035\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.77763e16 −1.97529 −0.987647 − 0.156694i \(-0.949916\pi\)
−0.987647 + 0.156694i \(0.949916\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 2.65258e17 1.87176 0.935881 − 0.352317i \(-0.114606\pi\)
0.935881 + 0.352317i \(0.114606\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 7.04792e17 1.26183
\(50\) 0 0
\(51\) 0 0
\(52\) 7.76107e17 0.744538
\(53\) 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
\(61\) −1.08607e19 −1.94937 −0.974685 − 0.223583i \(-0.928225\pi\)
−0.974685 + 0.223583i \(0.928225\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −9.22337e18 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 6.94433e18 0.465420 0.232710 − 0.972546i \(-0.425241\pi\)
0.232710 + 0.972546i \(0.425241\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −3.90986e19 −1.06481 −0.532404 − 0.846490i \(-0.678711\pi\)
−0.532404 + 0.846490i \(0.678711\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 7.45341e19 1.32988
\(77\) 0 0
\(78\) 0 0
\(79\) 1.68068e20 1.99711 0.998553 − 0.0537677i \(-0.0171231\pi\)
0.998553 + 0.0537677i \(0.0171231\pi\)
\(80\) 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
\(91\) −4.15960e20 −1.11974
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.13207e21 −1.55872 −0.779362 − 0.626573i \(-0.784457\pi\)
−0.779362 + 0.626573i \(0.784457\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.00000e21 1.00000
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) −2.44432e21 −1.79212 −0.896060 − 0.443932i \(-0.853583\pi\)
−0.896060 + 0.443932i \(0.853583\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 4.46449e21 1.80632 0.903158 − 0.429309i \(-0.141243\pi\)
0.903158 + 0.429309i \(0.141243\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 4.94333e21 1.50394
\(113\) 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
\(121\) −7.40025e21 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 1.89584e22 1.98095
\(125\) 0 0
\(126\) 0 0
\(127\) −1.97512e22 −1.60566 −0.802832 − 0.596205i \(-0.796674\pi\)
−0.802832 + 0.596205i \(0.796674\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −3.99471e22 −2.00006
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) −2.94518e21 −0.0927804 −0.0463902 − 0.998923i \(-0.514772\pi\)
−0.0463902 + 0.998923i \(0.514772\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 1.21166e23 1.97529
\(149\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 4.35403e22 0.574954 0.287477 − 0.957787i \(-0.407183\pi\)
0.287477 + 0.957787i \(0.407183\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −2.68883e22 −0.235840 −0.117920 − 0.993023i \(-0.537623\pi\)
−0.117920 + 0.993023i \(0.537623\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1.34412e23 0.795188 0.397594 − 0.917561i \(-0.369845\pi\)
0.397594 + 0.917561i \(0.369845\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −1.10108e23 −0.445664
\(170\) 0 0
\(171\) 0 0
\(172\) −5.56287e23 −1.87176
\(173\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) −5.35957e23 −1.50394
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 6.96900e23 1.37260 0.686302 − 0.727317i \(-0.259233\pi\)
0.686302 + 0.727317i \(0.259233\pi\)
\(182\) 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
\(193\) 1.98909e24 1.99665 0.998326 − 0.0578425i \(-0.0184221\pi\)
0.998326 + 0.0578425i \(0.0184221\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −1.47806e24 −1.26183
\(197\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 2.33297e24 1.69806 0.849029 − 0.528346i \(-0.177188\pi\)
0.849029 + 0.528346i \(0.177188\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −1.62762e24 −0.744538
\(209\) 0 0
\(210\) 0 0
\(211\) 1.58903e24 0.625412 0.312706 − 0.949850i \(-0.398764\pi\)
0.312706 + 0.949850i \(0.398764\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −1.01609e25 −2.97923
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −3.45915e24 −0.761672 −0.380836 − 0.924643i \(-0.624364\pi\)
−0.380836 + 0.924643i \(0.624364\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 1.17896e25 1.96438 0.982191 − 0.187884i \(-0.0601630\pi\)
0.982191 + 0.187884i \(0.0601630\pi\)
\(230\) 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
\(241\) −7.96668e24 −0.776423 −0.388212 − 0.921570i \(-0.626907\pi\)
−0.388212 + 0.921570i \(0.626907\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 2.27765e25 1.94937
\(245\) 0 0
\(246\) 0 0
\(247\) 1.31528e25 0.990146
\(248\) 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
\(256\) 1.93428e25 1.00000
\(257\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) −6.49396e25 −2.97072
\(260\) 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
\(268\) −1.45633e25 −0.465420
\(269\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) −4.18902e25 −1.19106 −0.595532 − 0.803332i \(-0.703059\pi\)
−0.595532 + 0.803332i \(0.703059\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −2.19618e24 −0.0496169 −0.0248085 − 0.999692i \(-0.507898\pi\)
−0.0248085 + 0.999692i \(0.507898\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 1.05541e26 1.90398 0.951992 − 0.306122i \(-0.0990315\pi\)
0.951992 + 0.306122i \(0.0990315\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −6.90919e25 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 8.19958e25 1.06481
\(293\) 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
\(301\) 2.98146e26 2.81502
\(302\) 0 0
\(303\) 0 0
\(304\) −1.56309e26 −1.32988
\(305\) 0 0
\(306\) 0 0
\(307\) 1.32287e26 1.01523 0.507614 − 0.861585i \(-0.330528\pi\)
0.507614 + 0.861585i \(0.330528\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −3.18989e26 −1.99784 −0.998919 − 0.0464917i \(-0.985196\pi\)
−0.998919 + 0.0464917i \(0.985196\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −3.52465e26 −1.99711
\(317\) 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
\(325\) 1.76466e26 0.744538
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −3.13448e26 −1.09137 −0.545684 − 0.837991i \(-0.683730\pi\)
−0.545684 + 0.837991i \(0.683730\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1.86905e26 0.538898 0.269449 − 0.963015i \(-0.413158\pi\)
0.269449 + 0.963015i \(0.413158\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.64378e26 0.393782
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 9.79269e26 1.95540 0.977699 − 0.210012i \(-0.0673503\pi\)
0.977699 + 0.210012i \(0.0673503\pi\)
\(350\) 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
\(361\) 5.48927e26 0.768580
\(362\) 0 0
\(363\) 0 0
\(364\) 8.72331e26 1.11974
\(365\) 0 0
\(366\) 0 0
\(367\) −1.37670e27 −1.62124 −0.810619 − 0.585574i \(-0.800869\pi\)
−0.810619 + 0.585574i \(0.800869\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −1.60839e27 −1.59753 −0.798767 − 0.601641i \(-0.794514\pi\)
−0.798767 + 0.601641i \(0.794514\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −2.22265e27 −1.86707 −0.933534 − 0.358488i \(-0.883292\pi\)
−0.933534 + 0.358488i \(0.883292\pi\)
\(380\) 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
\(388\) 2.37412e27 1.55872
\(389\) 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
\(397\) −1.94928e27 −1.00595 −0.502973 − 0.864302i \(-0.667761\pi\)
−0.502973 + 0.864302i \(0.667761\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −2.09715e27 −1.00000
\(401\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 3.34552e27 1.47489
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 2.59171e27 0.978343 0.489171 − 0.872188i \(-0.337299\pi\)
0.489171 + 0.872188i \(0.337299\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 5.12611e27 1.79212
\(413\) 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
\(421\) 4.53662e27 1.26407 0.632034 − 0.774940i \(-0.282220\pi\)
0.632034 + 0.774940i \(0.282220\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −1.22072e28 −2.93173
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 8.16275e27 1.69322 0.846611 − 0.532213i \(-0.178639\pi\)
0.846611 + 0.532213i \(0.178639\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −9.36271e27 −1.80632
\(437\) 0 0
\(438\) 0 0
\(439\) −3.72475e27 −0.668683 −0.334341 − 0.942452i \(-0.608514\pi\)
−0.334341 + 0.942452i \(0.608514\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −1.03669e28 −1.50394
\(449\) 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
\(457\) 1.64801e28 1.94017 0.970083 − 0.242775i \(-0.0780577\pi\)
0.970083 + 0.242775i \(0.0780577\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −1.29245e28 −1.32682 −0.663411 − 0.748255i \(-0.730892\pi\)
−0.663411 + 0.748255i \(0.730892\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 7.80531e27 0.699964
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 1.69471e28 1.32988
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 2.13817e28 1.47068
\(482\) 0 0
\(483\) 0 0
\(484\) 1.55194e28 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) −2.61774e27 −0.158078 −0.0790391 − 0.996872i \(-0.525185\pi\)
−0.0790391 + 0.996872i \(0.525185\pi\)
\(488\) 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
\(496\) −3.97587e28 −1.98095
\(497\) 0 0
\(498\) 0 0
\(499\) −3.72846e28 −1.74371 −0.871855 − 0.489765i \(-0.837083\pi\)
−0.871855 + 0.489765i \(0.837083\pi\)
\(500\) 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
\(508\) 4.14213e28 1.60566
\(509\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) −4.39462e28 −1.60141
\(512\) 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
\(523\) −3.66325e28 −1.04616 −0.523081 − 0.852283i \(-0.675217\pi\)
−0.523081 + 0.852283i \(0.675217\pi\)
\(524\) 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
\(532\) 8.37751e28 2.00006
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 9.99080e28 1.99999 0.999997 − 0.00231282i \(-0.000736194\pi\)
0.999997 + 0.00231282i \(0.000736194\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −4.86895e28 −0.868097 −0.434048 − 0.900890i \(-0.642915\pi\)
−0.434048 + 0.900890i \(0.642915\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 1.88906e29 3.00353
\(554\) 0 0
\(555\) 0 0
\(556\) 6.17649e27 0.0927804
\(557\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) −9.81660e28 −1.39360
\(560\) 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
\(571\) −1.36364e29 −1.54889 −0.774443 − 0.632643i \(-0.781970\pi\)
−0.774443 + 0.632643i \(0.781970\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −1.94611e29 −1.98072 −0.990358 − 0.138529i \(-0.955763\pi\)
−0.990358 + 0.138529i \(0.955763\pi\)
\(578\) 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
\(589\) 3.21290e29 2.63443
\(590\) 0 0
\(591\) 0 0
\(592\) −2.54103e29 −1.97529
\(593\) 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
\(601\) −2.22203e29 −1.47424 −0.737119 − 0.675762i \(-0.763815\pi\)
−0.737119 + 0.675762i \(0.763815\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −9.13106e28 −0.574954
\(605\) 0 0
\(606\) 0 0
\(607\) 2.01911e29 1.20692 0.603461 − 0.797392i \(-0.293788\pi\)
0.603461 + 0.797392i \(0.293788\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 3.35322e29 1.80770 0.903851 − 0.427848i \(-0.140728\pi\)
0.903851 + 0.427848i \(0.140728\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 3.54147e29 1.72358 0.861790 − 0.507264i \(-0.169343\pi\)
0.861790 + 0.507264i \(0.169343\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 2.27374e29 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 5.63888e28 0.235840
\(629\) 0 0
\(630\) 0 0
\(631\) −2.04089e29 −0.811914 −0.405957 − 0.913892i \(-0.633062\pi\)
−0.405957 + 0.913892i \(0.633062\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −2.60827e29 −0.939483
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) −1.41918e29 −0.463259 −0.231629 − 0.972804i \(-0.574406\pi\)
−0.231629 + 0.972804i \(0.574406\pi\)
\(644\) 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
\(652\) −2.81883e29 −0.795188
\(653\) 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
\(661\) −8.04773e29 −1.96588 −0.982942 − 0.183916i \(-0.941123\pi\)
−0.982942 + 0.183916i \(0.941123\pi\)
\(662\) 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
\(673\) 8.12894e29 1.64390 0.821951 − 0.569559i \(-0.192886\pi\)
0.821951 + 0.569559i \(0.192886\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 2.30913e29 0.445664
\(677\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) −1.27243e30 −2.34423
\(680\) 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
\(688\) 1.16662e30 1.87176
\(689\) 0 0
\(690\) 0 0
\(691\) −8.65504e28 −0.132663 −0.0663314 − 0.997798i \(-0.521129\pi\)
−0.0663314 + 0.997798i \(0.521129\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 1.12398e30 1.50394
\(701\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 2.05341e30 2.62690
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −7.91584e29 −0.926214 −0.463107 − 0.886302i \(-0.653265\pi\)
−0.463107 + 0.886302i \(0.653265\pi\)
\(710\) 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
\(721\) −2.74737e30 −2.69524
\(722\) 0 0
\(723\) 0 0
\(724\) −1.46151e30 −1.37260
\(725\) 0 0
\(726\) 0 0
\(727\) 2.06610e30 1.85798 0.928990 − 0.370105i \(-0.120678\pi\)
0.928990 + 0.370105i \(0.120678\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −9.86819e29 −0.814040 −0.407020 − 0.913419i \(-0.633432\pi\)
−0.407020 + 0.913419i \(0.633432\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 2.73020e28 0.0206742 0.0103371 − 0.999947i \(-0.496710\pi\)
0.0103371 + 0.999947i \(0.496710\pi\)
\(740\) 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
\(751\) −2.79025e30 −1.78412 −0.892059 − 0.451918i \(-0.850740\pi\)
−0.892059 + 0.451918i \(0.850740\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −2.68402e30 −1.57862 −0.789311 − 0.613993i \(-0.789562\pi\)
−0.789311 + 0.613993i \(0.789562\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 5.01801e30 2.71659
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 2.75585e30 1.37413 0.687066 − 0.726595i \(-0.258898\pi\)
0.687066 + 0.726595i \(0.258898\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −4.17142e30 −1.99665
\(773\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) 4.31064e30 1.98095
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 3.09971e30 1.26183
\(785\) 0 0
\(786\) 0 0
\(787\) 1.38088e30 0.540035 0.270018 − 0.962855i \(-0.412970\pi\)
0.270018 + 0.962855i \(0.412970\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 4.01929e30 1.45138
\(794\) 0 0
\(795\) 0 0
\(796\) −4.89260e30 −1.69806
\(797\) 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
\(811\) −5.30878e30 −1.51452 −0.757261 − 0.653112i \(-0.773463\pi\)
−0.757261 + 0.653112i \(0.773463\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −9.42745e30 −2.48922
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) 7.90132e30 1.93197 0.965987 − 0.258591i \(-0.0832580\pi\)
0.965987 + 0.258591i \(0.0832580\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) −4.85631e30 −1.10023 −0.550116 − 0.835088i \(-0.685416\pi\)
−0.550116 + 0.835088i \(0.685416\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 3.41336e30 0.744538
\(833\) 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
\(841\) −5.13284e30 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) −3.33244e30 −0.625412
\(845\) 0 0
\(846\) 0 0
\(847\) −8.31776e30 −1.50394
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 9.51956e30 1.59828 0.799139 − 0.601147i \(-0.205289\pi\)
0.799139 + 0.601147i \(0.205289\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 1.28204e31 1.99974 0.999872 − 0.0160193i \(-0.00509933\pi\)
0.999872 + 0.0160193i \(0.00509933\pi\)
\(860\) 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
\(868\) 2.13089e31 2.97923
\(869\) 0 0
\(870\) 0 0
\(871\) −2.56994e30 −0.346523
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −4.31229e30 −0.541019 −0.270509 − 0.962717i \(-0.587192\pi\)
−0.270509 + 0.962717i \(0.587192\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 5.16131e30 0.602799 0.301400 − 0.953498i \(-0.402546\pi\)
0.301400 + 0.953498i \(0.402546\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) −2.22000e31 −2.41482
\(890\) 0 0
\(891\) 0 0
\(892\) 7.25436e30 0.761672
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −9.99735e30 −0.881066 −0.440533 − 0.897736i \(-0.645210\pi\)
−0.440533 + 0.897736i \(0.645210\pi\)
\(908\) 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
\(916\) −2.47246e31 −1.96438
\(917\) 0 0
\(918\) 0 0
\(919\) −8.84453e30 −0.678988 −0.339494 − 0.940608i \(-0.610256\pi\)
−0.339494 + 0.940608i \(0.610256\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 2.75499e31 1.97529
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) −2.50488e31 −1.67809
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 6.77423e30 0.424223 0.212111 − 0.977245i \(-0.431966\pi\)
0.212111 + 0.977245i \(0.431966\pi\)
\(938\) 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
\(949\) 1.44695e31 0.792790
\(950\) 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
\(961\) 6.08974e31 2.92417
\(962\) 0 0
\(963\) 0 0
\(964\) 1.67073e31 0.776423
\(965\) 0 0
\(966\) 0 0
\(967\) 2.48166e31 1.11626 0.558128 − 0.829755i \(-0.311520\pi\)
0.558128 + 0.829755i \(0.311520\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) −3.31033e30 −0.139536
\(974\) 0 0
\(975\) 0 0
\(976\) −4.77658e31 −1.94937
\(977\) 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
\(988\) −2.75833e31 −0.990146
\(989\) 0 0
\(990\) 0 0
\(991\) −2.96550e31 −1.03116 −0.515579 − 0.856842i \(-0.672423\pi\)
−0.515579 + 0.856842i \(0.672423\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1.20454e31 0.393118 0.196559 − 0.980492i \(-0.437023\pi\)
0.196559 + 0.980492i \(0.437023\pi\)
\(998\) 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 9.22.a.b.1.1 ✓ 1
3.2 odd 2 CM 9.22.a.b.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.22.a.b.1.1 ✓ 1 1.1 even 1 trivial
9.22.a.b.1.1 ✓ 1 3.2 odd 2 CM