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Properties

Label 9.58.a.a.1.1
Level $9$
Weight $58$
Character 9.1
Self dual yes
Analytic conductor $185.190$
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] = [9,58,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, 58, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.1");
 
S:= CuspForms(chi, 58);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 58 \)
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: \(185.189790286\)
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)
 
\(f(q)\) \(=\) \(q-1.44115e17 q^{4} +2.40354e24 q^{7} +O(q^{10})\) \(q-1.44115e17 q^{4} +2.40354e24 q^{7} -3.42127e31 q^{13} +2.07692e34 q^{16} -1.48229e36 q^{19} -6.93889e39 q^{25} -3.46387e41 q^{28} -5.52834e42 q^{31} +9.87817e44 q^{37} -2.14359e46 q^{43} +4.29590e48 q^{49} +4.93057e48 q^{52} +7.43829e50 q^{61} -2.99316e51 q^{64} -2.02303e52 q^{67} +2.23891e53 q^{73} +2.13621e53 q^{76} -8.75203e53 q^{79} -8.22317e55 q^{91} +7.40100e56 q^{97} +O(q^{100})\)

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
\(4\) −1.44115e17 −1.00000
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 2.40354e24 1.97496 0.987479 − 0.157753i \(-0.0504250\pi\)
0.987479 + 0.157753i \(0.0504250\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −3.42127e31 −0.612071 −0.306036 − 0.952020i \(-0.599003\pi\)
−0.306036 + 0.952020i \(0.599003\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 2.07692e34 1.00000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −1.48229e36 −0.532667 −0.266334 − 0.963881i \(-0.585812\pi\)
−0.266334 + 0.963881i \(0.585812\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −6.93889e39 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) −3.46387e41 −1.97496
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −5.52834e42 −1.73295 −0.866476 − 0.499218i \(-0.833621\pi\)
−0.866476 + 0.499218i \(0.833621\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 9.87817e44 1.99953 0.999764 − 0.0217237i \(-0.00691541\pi\)
0.999764 + 0.0217237i \(0.00691541\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −2.14359e46 −0.598811 −0.299406 − 0.954126i \(-0.596788\pi\)
−0.299406 + 0.954126i \(0.596788\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 4.29590e48 2.90046
\(50\) 0 0
\(51\) 0 0
\(52\) 4.93057e48 0.612071
\(53\) 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
\(61\) 7.43829e50 0.976277 0.488138 − 0.872766i \(-0.337676\pi\)
0.488138 + 0.872766i \(0.337676\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −2.99316e51 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −2.02303e52 −1.83177 −0.915886 − 0.401438i \(-0.868510\pi\)
−0.915886 + 0.401438i \(0.868510\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 2.23891e53 1.75928 0.879640 − 0.475640i \(-0.157784\pi\)
0.879640 + 0.475640i \(0.157784\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 2.13621e53 0.532667
\(77\) 0 0
\(78\) 0 0
\(79\) −8.75203e53 −0.723997 −0.361998 − 0.932179i \(-0.617905\pi\)
−0.361998 + 0.932179i \(0.617905\pi\)
\(80\) 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
\(91\) −8.22317e55 −1.20881
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 7.40100e56 1.76318 0.881589 − 0.472018i \(-0.156474\pi\)
0.881589 + 0.472018i \(0.156474\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.00000e57 1.00000
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) −4.36096e57 −1.87811 −0.939055 − 0.343766i \(-0.888297\pi\)
−0.939055 + 0.343766i \(0.888297\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) −1.69719e58 −1.45572 −0.727858 − 0.685728i \(-0.759484\pi\)
−0.727858 + 0.685728i \(0.759484\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 4.99196e58 1.97496
\(113\) 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
\(121\) −2.28762e59 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 7.96718e59 1.73295
\(125\) 0 0
\(126\) 0 0
\(127\) −1.04377e60 −1.14868 −0.574339 − 0.818618i \(-0.694741\pi\)
−0.574339 + 0.818618i \(0.694741\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −3.56275e60 −1.05199
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) 2.37052e61 1.99032 0.995159 − 0.0982791i \(-0.0313338\pi\)
0.995159 + 0.0982791i \(0.0313338\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −1.42359e62 −1.99953
\(149\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −2.67248e61 −0.211871 −0.105935 − 0.994373i \(-0.533784\pi\)
−0.105935 + 0.994373i \(0.533784\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 5.32955e62 1.39172 0.695858 − 0.718180i \(-0.255024\pi\)
0.695858 + 0.718180i \(0.255024\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1.41169e63 −1.26588 −0.632939 − 0.774202i \(-0.718151\pi\)
−0.632939 + 0.774202i \(0.718151\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −1.95392e63 −0.625369
\(170\) 0 0
\(171\) 0 0
\(172\) 3.08923e63 0.598811
\(173\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) −1.66779e64 −1.97496
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −2.86627e64 −1.29861 −0.649304 − 0.760529i \(-0.724940\pi\)
−0.649304 + 0.760529i \(0.724940\pi\)
\(182\) 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
\(193\) −2.26091e65 −1.64399 −0.821997 − 0.569492i \(-0.807140\pi\)
−0.821997 + 0.569492i \(0.807140\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −6.19105e65 −2.90046
\(197\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) −4.89101e65 −1.48623 −0.743115 − 0.669164i \(-0.766652\pi\)
−0.743115 + 0.669164i \(0.766652\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −7.10570e65 −0.612071
\(209\) 0 0
\(210\) 0 0
\(211\) −3.37580e66 −1.93342 −0.966712 − 0.255866i \(-0.917639\pi\)
−0.966712 + 0.255866i \(0.917639\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −1.32876e67 −3.42251
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −2.73936e66 −0.324310 −0.162155 − 0.986765i \(-0.551844\pi\)
−0.162155 + 0.986765i \(0.551844\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −2.42079e67 −1.34476 −0.672381 − 0.740206i \(-0.734728\pi\)
−0.672381 + 0.740206i \(0.734728\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 1.54231e68 1.99841 0.999207 − 0.0398276i \(-0.0126809\pi\)
0.999207 + 0.0398276i \(0.0126809\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −1.07197e68 −0.976277
\(245\) 0 0
\(246\) 0 0
\(247\) 5.07132e67 0.326030
\(248\) 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
\(256\) 4.31359e68 1.00000
\(257\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 2.37426e69 3.94898
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 2.91550e69 1.83177
\(269\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) −4.31540e69 −1.97422 −0.987112 − 0.160031i \(-0.948841\pi\)
−0.987112 + 0.160031i \(0.948841\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 6.02310e69 1.47621 0.738103 − 0.674688i \(-0.235722\pi\)
0.738103 + 0.674688i \(0.235722\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 8.73338e69 1.16217 0.581084 − 0.813843i \(-0.302629\pi\)
0.581084 + 0.813843i \(0.302629\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1.36643e70 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) −3.22661e70 −1.75928
\(293\) 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
\(301\) −5.15220e70 −1.18263
\(302\) 0 0
\(303\) 0 0
\(304\) −3.07860e70 −0.532667
\(305\) 0 0
\(306\) 0 0
\(307\) −1.45009e71 −1.89650 −0.948248 − 0.317530i \(-0.897147\pi\)
−0.948248 + 0.317530i \(0.897147\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −1.38078e71 −1.04019 −0.520094 − 0.854109i \(-0.674103\pi\)
−0.520094 + 0.854109i \(0.674103\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 1.26130e71 0.723997
\(317\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 2.37398e71 0.612071
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1.30654e72 −1.99998 −0.999992 − 0.00409738i \(-0.998696\pi\)
−0.999992 + 0.00409738i \(0.998696\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 2.17519e72 1.99547 0.997735 − 0.0672649i \(-0.0214273\pi\)
0.997735 + 0.0672649i \(0.0214273\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 6.76547e72 3.75332
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 5.83448e71 0.197459 0.0987296 − 0.995114i \(-0.468522\pi\)
0.0987296 + 0.995114i \(0.468522\pi\)
\(350\) 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
\(361\) −5.54663e72 −0.716266
\(362\) 0 0
\(363\) 0 0
\(364\) 1.18508e73 1.20881
\(365\) 0 0
\(366\) 0 0
\(367\) 7.83030e72 0.632114 0.316057 − 0.948740i \(-0.397641\pi\)
0.316057 + 0.948740i \(0.397641\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −2.58208e73 −1.31301 −0.656505 − 0.754322i \(-0.727966\pi\)
−0.656505 + 0.754322i \(0.727966\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −4.31811e73 −1.39340 −0.696701 − 0.717362i \(-0.745349\pi\)
−0.696701 + 0.717362i \(0.745349\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) −1.06660e74 −1.76318
\(389\) 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
\(397\) −2.30289e74 −1.98036 −0.990178 − 0.139813i \(-0.955350\pi\)
−0.990178 + 0.139813i \(0.955350\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −1.44115e74 −1.00000
\(401\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 1.89140e74 1.06069
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 2.55647e74 0.940865 0.470432 − 0.882436i \(-0.344098\pi\)
0.470432 + 0.882436i \(0.344098\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 6.28480e74 1.87811
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −9.19083e74 −1.48359 −0.741797 − 0.670625i \(-0.766026\pi\)
−0.741797 + 0.670625i \(0.766026\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1.78783e75 1.92811
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 1.07814e75 0.781216 0.390608 − 0.920557i \(-0.372265\pi\)
0.390608 + 0.920557i \(0.372265\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 2.44592e75 1.45572
\(437\) 0 0
\(438\) 0 0
\(439\) 3.12252e75 1.52850 0.764251 − 0.644919i \(-0.223109\pi\)
0.764251 + 0.644919i \(0.223109\pi\)
\(440\) 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
\(448\) −7.19418e75 −1.97496
\(449\) 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
\(457\) −9.97946e75 −1.55416 −0.777079 − 0.629403i \(-0.783300\pi\)
−0.777079 + 0.629403i \(0.783300\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 1.23358e76 1.32468 0.662340 − 0.749204i \(-0.269564\pi\)
0.662340 + 0.749204i \(0.269564\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) −4.86245e76 −3.61767
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 1.02855e76 0.532667
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −3.37959e76 −1.22385
\(482\) 0 0
\(483\) 0 0
\(484\) 3.29680e76 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) −1.82358e76 −0.463819 −0.231910 − 0.972737i \(-0.574497\pi\)
−0.231910 + 0.972737i \(0.574497\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −1.14819e77 −1.73295
\(497\) 0 0
\(498\) 0 0
\(499\) 1.57216e77 1.99817 0.999083 − 0.0428161i \(-0.0136330\pi\)
0.999083 + 0.0428161i \(0.0136330\pi\)
\(500\) 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
\(508\) 1.50423e77 1.14868
\(509\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 5.38131e77 3.47450
\(512\) 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
\(523\) −3.37194e77 −1.12352 −0.561761 − 0.827300i \(-0.689876\pi\)
−0.561761 + 0.827300i \(0.689876\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −4.15419e77 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 5.13446e77 1.05199
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 9.73959e77 1.23714 0.618569 − 0.785730i \(-0.287713\pi\)
0.618569 + 0.785730i \(0.287713\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −2.02380e78 −1.87728 −0.938640 − 0.344899i \(-0.887913\pi\)
−0.938640 + 0.344899i \(0.887913\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −2.10359e78 −1.42986
\(554\) 0 0
\(555\) 0 0
\(556\) −3.41628e78 −1.99032
\(557\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 7.33379e77 0.366515
\(560\) 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
\(571\) −4.19492e78 −1.14444 −0.572222 − 0.820098i \(-0.693919\pi\)
−0.572222 + 0.820098i \(0.693919\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 5.58909e78 1.13196 0.565979 − 0.824420i \(-0.308498\pi\)
0.565979 + 0.824420i \(0.308498\pi\)
\(578\) 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
\(589\) 8.19461e78 0.923087
\(590\) 0 0
\(591\) 0 0
\(592\) 2.05162e79 1.99953
\(593\) 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
\(601\) 1.35282e79 0.857665 0.428832 − 0.903384i \(-0.358925\pi\)
0.428832 + 0.903384i \(0.358925\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 3.85145e78 0.211871
\(605\) 0 0
\(606\) 0 0
\(607\) −4.04504e79 −1.93217 −0.966084 − 0.258228i \(-0.916862\pi\)
−0.966084 + 0.258228i \(0.916862\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 4.58890e79 1.65609 0.828047 − 0.560659i \(-0.189452\pi\)
0.828047 + 0.560659i \(0.189452\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 6.24984e79 1.70878 0.854389 − 0.519634i \(-0.173932\pi\)
0.854389 + 0.519634i \(0.173932\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 4.81482e79 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) −7.68069e79 −1.39172
\(629\) 0 0
\(630\) 0 0
\(631\) −1.14598e80 −1.81277 −0.906383 − 0.422457i \(-0.861168\pi\)
−0.906383 + 0.422457i \(0.861168\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −1.46975e80 −1.77529
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) −2.16131e80 −1.99850 −0.999250 − 0.0387184i \(-0.987672\pi\)
−0.999250 + 0.0387184i \(0.987672\pi\)
\(644\) 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
\(652\) 2.03446e80 1.26588
\(653\) 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
\(661\) −1.30223e80 −0.548210 −0.274105 − 0.961700i \(-0.588382\pi\)
−0.274105 + 0.961700i \(0.588382\pi\)
\(662\) 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
\(673\) −7.58286e80 −1.91163 −0.955815 − 0.293968i \(-0.905024\pi\)
−0.955815 + 0.293968i \(0.905024\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 2.81590e80 0.625369
\(677\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 1.77886e81 3.48220
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −4.45205e80 −0.598811
\(689\) 0 0
\(690\) 0 0
\(691\) 1.10666e81 1.31489 0.657445 − 0.753502i \(-0.271637\pi\)
0.657445 + 0.753502i \(0.271637\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 2.40354e81 1.97496
\(701\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) −1.46423e81 −1.06508
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 2.65048e81 1.51323 0.756617 − 0.653858i \(-0.226851\pi\)
0.756617 + 0.653858i \(0.226851\pi\)
\(710\) 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
\(721\) −1.04818e82 −3.70919
\(722\) 0 0
\(723\) 0 0
\(724\) 4.13073e81 1.29861
\(725\) 0 0
\(726\) 0 0
\(727\) −5.98601e81 −1.67266 −0.836328 − 0.548229i \(-0.815302\pi\)
−0.836328 + 0.548229i \(0.815302\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −8.08781e81 −1.78800 −0.894001 − 0.448066i \(-0.852113\pi\)
−0.894001 + 0.448066i \(0.852113\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 1.10511e82 1.93660 0.968301 − 0.249785i \(-0.0803598\pi\)
0.968301 + 0.249785i \(0.0803598\pi\)
\(740\) 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
\(751\) −1.56528e82 −1.73322 −0.866610 − 0.498986i \(-0.833706\pi\)
−0.866610 + 0.498986i \(0.833706\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 5.02569e81 0.443571 0.221785 − 0.975095i \(-0.428812\pi\)
0.221785 + 0.975095i \(0.428812\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) −4.07928e82 −2.87498
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 3.13824e82 1.76924 0.884618 − 0.466316i \(-0.154419\pi\)
0.884618 + 0.466316i \(0.154419\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 3.25832e82 1.64399
\(773\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) 3.83606e82 1.73295
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 8.92224e82 2.90046
\(785\) 0 0
\(786\) 0 0
\(787\) −6.37073e82 −1.85742 −0.928709 − 0.370810i \(-0.879080\pi\)
−0.928709 + 0.370810i \(0.879080\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −2.54484e82 −0.597551
\(794\) 0 0
\(795\) 0 0
\(796\) 7.04869e82 1.48623
\(797\) 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
\(811\) −8.25625e82 −1.02256 −0.511280 − 0.859414i \(-0.670828\pi\)
−0.511280 + 0.859414i \(0.670828\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 3.17742e82 0.318967
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) −2.37107e83 −1.93218 −0.966092 − 0.258199i \(-0.916871\pi\)
−0.966092 + 0.258199i \(0.916871\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) −1.85782e83 −1.23083 −0.615417 − 0.788202i \(-0.711012\pi\)
−0.615417 + 0.788202i \(0.711012\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1.02404e83 0.612071
\(833\) 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
\(841\) −2.27345e83 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 4.86505e83 1.93342
\(845\) 0 0
\(846\) 0 0
\(847\) −5.49838e83 −1.97496
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −5.99661e83 −1.76140 −0.880701 − 0.473673i \(-0.842928\pi\)
−0.880701 + 0.473673i \(0.842928\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) −1.49604e83 −0.359863 −0.179931 − 0.983679i \(-0.557588\pi\)
−0.179931 + 0.983679i \(0.557588\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 1.91495e84 3.42251
\(869\) 0 0
\(870\) 0 0
\(871\) 6.92135e83 1.12118
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 4.36113e83 0.580913 0.290456 − 0.956888i \(-0.406193\pi\)
0.290456 + 0.956888i \(0.406193\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 1.80086e84 1.97515 0.987574 − 0.157154i \(-0.0502318\pi\)
0.987574 + 0.157154i \(0.0502318\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) −2.50874e84 −2.26859
\(890\) 0 0
\(891\) 0 0
\(892\) 3.94783e83 0.324310
\(893\) 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
\(907\) 2.07000e84 1.05722 0.528608 − 0.848866i \(-0.322714\pi\)
0.528608 + 0.848866i \(0.322714\pi\)
\(908\) 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
\(916\) 3.48872e84 1.34476
\(917\) 0 0
\(918\) 0 0
\(919\) −4.59992e84 −1.61532 −0.807659 − 0.589650i \(-0.799266\pi\)
−0.807659 + 0.589650i \(0.799266\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −6.85436e84 −1.99953
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) −6.36778e84 −1.54498
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −1.29644e84 −0.261924 −0.130962 − 0.991387i \(-0.541807\pi\)
−0.130962 + 0.991387i \(0.541807\pi\)
\(938\) 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
\(949\) −7.65992e84 −1.07680
\(950\) 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
\(961\) 2.03856e85 2.00312
\(962\) 0 0
\(963\) 0 0
\(964\) −2.22270e85 −1.99841
\(965\) 0 0
\(966\) 0 0
\(967\) −4.54214e84 −0.373771 −0.186886 − 0.982382i \(-0.559839\pi\)
−0.186886 + 0.982382i \(0.559839\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 5.69765e85 3.93079
\(974\) 0 0
\(975\) 0 0
\(976\) 1.54487e85 0.976277
\(977\) 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
\(988\) −7.30854e84 −0.326030
\(989\) 0 0
\(990\) 0 0
\(991\) 6.04003e84 0.247138 0.123569 − 0.992336i \(-0.460566\pi\)
0.123569 + 0.992336i \(0.460566\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −2.04831e85 −0.705641 −0.352820 − 0.935691i \(-0.614777\pi\)
−0.352820 + 0.935691i \(0.614777\pi\)
\(998\) 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 9.58.a.a.1.1 ✓ 1
3.2 odd 2 CM 9.58.a.a.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.58.a.a.1.1 ✓ 1 1.1 even 1 trivial
9.58.a.a.1.1 ✓ 1 3.2 odd 2 CM