Properties

Label 225.8.a.f.1.1
Level $225$
Weight $8$
Character 225.1
Self dual yes
Analytic conductor $70.287$
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] = [225,8,Mod(1,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 225.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: \(70.2866307339\)
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 \(\chi\) \(=\) 225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-128.000 q^{4} -1255.00 q^{7} +O(q^{10})\) \(q-128.000 q^{4} -1255.00 q^{7} +12605.0 q^{13} +16384.0 q^{16} +43091.0 q^{19} +160640. q^{28} -331387. q^{31} +279710. q^{37} -409495. q^{43} +751482. q^{49} -1.61344e6 q^{52} +1.99835e6 q^{61} -2.09715e6 q^{64} -4.05845e6 q^{67} -6.27481e6 q^{73} -5.51565e6 q^{76} +8.76304e6 q^{79} -1.58193e7 q^{91} -1.75216e7 q^{97} +O(q^{100})\)

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
\(4\) −128.000 −1.00000
\(5\) 0 0
\(6\) 0 0
\(7\) −1255.00 −1.38293 −0.691466 0.722409i \(-0.743035\pi\)
−0.691466 + 0.722409i \(0.743035\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 12605.0 1.59126 0.795630 0.605783i \(-0.207140\pi\)
0.795630 + 0.605783i \(0.207140\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 16384.0 1.00000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 43091.0 1.44128 0.720641 0.693308i \(-0.243848\pi\)
0.720641 + 0.693308i \(0.243848\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 160640. 1.38293
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −331387. −1.99788 −0.998940 0.0460243i \(-0.985345\pi\)
−0.998940 + 0.0460243i \(0.985345\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 279710. 0.907825 0.453912 0.891046i \(-0.350028\pi\)
0.453912 + 0.891046i \(0.350028\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −409495. −0.785433 −0.392716 0.919660i \(-0.628465\pi\)
−0.392716 + 0.919660i \(0.628465\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 751482. 0.912499
\(50\) 0 0
\(51\) 0 0
\(52\) −1.61344e6 −1.59126
\(53\) 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
\(61\) 1.99835e6 1.12724 0.563620 0.826034i \(-0.309408\pi\)
0.563620 + 0.826034i \(0.309408\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −2.09715e6 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −4.05845e6 −1.64854 −0.824269 0.566198i \(-0.808414\pi\)
−0.824269 + 0.566198i \(0.808414\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −6.27481e6 −1.88786 −0.943932 0.330141i \(-0.892904\pi\)
−0.943932 + 0.330141i \(0.892904\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −5.51565e6 −1.44128
\(77\) 0 0
\(78\) 0 0
\(79\) 8.76304e6 1.99968 0.999839 0.0179303i \(-0.00570769\pi\)
0.999839 + 0.0179303i \(0.00570769\pi\)
\(80\) 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
\(91\) −1.58193e7 −2.20060
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.75216e7 −1.94927 −0.974634 0.223805i \(-0.928152\pi\)
−0.974634 + 0.223805i \(0.928152\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 8.02622e6 0.723737 0.361868 0.932229i \(-0.382139\pi\)
0.361868 + 0.932229i \(0.382139\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 2.67457e7 1.97816 0.989081 0.147370i \(-0.0470809\pi\)
0.989081 + 0.147370i \(0.0470809\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −2.05619e7 −1.38293
\(113\) 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
\(121\) −1.94872e7 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 4.24175e7 1.99788
\(125\) 0 0
\(126\) 0 0
\(127\) −4.51256e7 −1.95484 −0.977418 0.211317i \(-0.932225\pi\)
−0.977418 + 0.211317i \(0.932225\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −5.40792e7 −1.99319
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) −5.53251e7 −1.74731 −0.873656 0.486544i \(-0.838257\pi\)
−0.873656 + 0.486544i \(0.838257\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −3.58029e7 −0.907825
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −6.88275e7 −1.62683 −0.813416 0.581683i \(-0.802394\pi\)
−0.813416 + 0.581683i \(0.802394\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −8.58312e7 −1.77009 −0.885047 0.465501i \(-0.845874\pi\)
−0.885047 + 0.465501i \(0.845874\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1.50255e7 −0.271752 −0.135876 0.990726i \(-0.543385\pi\)
−0.135876 + 0.990726i \(0.543385\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 9.61375e7 1.53211
\(170\) 0 0
\(171\) 0 0
\(172\) 5.24154e7 0.785433
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 1.53710e8 1.92676 0.963378 0.268146i \(-0.0864109\pi\)
0.963378 + 0.268146i \(0.0864109\pi\)
\(182\) 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
\(193\) 1.99710e8 1.99963 0.999814 0.0192904i \(-0.00614071\pi\)
0.999814 + 0.0192904i \(0.00614071\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −9.61897e7 −0.912499
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) −7.37380e7 −0.663293 −0.331646 0.943404i \(-0.607604\pi\)
−0.331646 + 0.943404i \(0.607604\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 2.06520e8 1.59126
\(209\) 0 0
\(210\) 0 0
\(211\) −2.88780e7 −0.211630 −0.105815 0.994386i \(-0.533745\pi\)
−0.105815 + 0.994386i \(0.533745\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 4.15891e8 2.76293
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −3.05909e8 −1.84725 −0.923624 0.383299i \(-0.874788\pi\)
−0.923624 + 0.383299i \(0.874788\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −1.61551e8 −0.888965 −0.444482 0.895788i \(-0.646612\pi\)
−0.444482 + 0.895788i \(0.646612\pi\)
\(230\) 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
\(241\) 3.44262e8 1.58427 0.792136 0.610345i \(-0.208969\pi\)
0.792136 + 0.610345i \(0.208969\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −2.55788e8 −1.12724
\(245\) 0 0
\(246\) 0 0
\(247\) 5.43162e8 2.29345
\(248\) 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
\(256\) 2.68435e8 1.00000
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) −3.51036e8 −1.25546
\(260\) 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
\(268\) 5.19482e8 1.64854
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 4.85556e8 1.48200 0.740998 0.671508i \(-0.234353\pi\)
0.740998 + 0.671508i \(0.234353\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −6.15593e8 −1.74026 −0.870131 0.492821i \(-0.835966\pi\)
−0.870131 + 0.492821i \(0.835966\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) −3.10874e8 −0.815326 −0.407663 0.913132i \(-0.633656\pi\)
−0.407663 + 0.913132i \(0.633656\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −4.10339e8 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 8.03176e8 1.88786
\(293\) 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
\(301\) 5.13916e8 1.08620
\(302\) 0 0
\(303\) 0 0
\(304\) 7.06003e8 1.44128
\(305\) 0 0
\(306\) 0 0
\(307\) 9.53811e8 1.88139 0.940693 0.339258i \(-0.110176\pi\)
0.940693 + 0.339258i \(0.110176\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 5.57008e8 1.02673 0.513365 0.858170i \(-0.328399\pi\)
0.513365 + 0.858170i \(0.328399\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −1.12167e9 −1.99968
\(317\) 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
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1.24784e9 −1.89131 −0.945654 0.325175i \(-0.894577\pi\)
−0.945654 + 0.325175i \(0.894577\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −1.14809e9 −1.63408 −0.817039 0.576582i \(-0.804386\pi\)
−0.817039 + 0.576582i \(0.804386\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 9.04366e7 0.121008
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) −6.95221e8 −0.875455 −0.437728 0.899108i \(-0.644217\pi\)
−0.437728 + 0.899108i \(0.644217\pi\)
\(350\) 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
\(361\) 9.62963e8 1.07729
\(362\) 0 0
\(363\) 0 0
\(364\) 2.02487e9 2.20060
\(365\) 0 0
\(366\) 0 0
\(367\) −1.85289e9 −1.95667 −0.978336 0.207021i \(-0.933623\pi\)
−0.978336 + 0.207021i \(0.933623\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −1.95829e9 −1.95387 −0.976937 0.213528i \(-0.931505\pi\)
−0.976937 + 0.213528i \(0.931505\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 8.28135e8 0.781383 0.390692 0.920522i \(-0.372236\pi\)
0.390692 + 0.920522i \(0.372236\pi\)
\(380\) 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
\(388\) 2.24276e9 1.94927
\(389\) 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
\(397\) −2.34403e9 −1.88017 −0.940084 0.340944i \(-0.889254\pi\)
−0.940084 + 0.340944i \(0.889254\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) −4.17713e9 −3.17915
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −4.69220e8 −0.339113 −0.169557 0.985520i \(-0.554234\pi\)
−0.169557 + 0.985520i \(0.554234\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −1.02736e9 −0.723737
\(413\) 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
\(421\) −2.23703e9 −1.46112 −0.730558 0.682850i \(-0.760740\pi\)
−0.730558 + 0.682850i \(0.760740\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −2.50793e9 −1.55890
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −2.20400e9 −1.30468 −0.652341 0.757925i \(-0.726213\pi\)
−0.652341 + 0.757925i \(0.726213\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −3.42346e9 −1.97816
\(437\) 0 0
\(438\) 0 0
\(439\) 4.02008e8 0.226782 0.113391 0.993550i \(-0.463829\pi\)
0.113391 + 0.993550i \(0.463829\pi\)
\(440\) 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
\(448\) 2.63193e9 1.38293
\(449\) 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
\(457\) 4.06709e9 1.99332 0.996661 0.0816509i \(-0.0260192\pi\)
0.996661 + 0.0816509i \(0.0260192\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 1.02273e9 0.478881 0.239440 0.970911i \(-0.423036\pi\)
0.239440 + 0.970911i \(0.423036\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 5.09336e9 2.27981
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 3.52574e9 1.44458
\(482\) 0 0
\(483\) 0 0
\(484\) 2.49436e9 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) −4.48049e9 −1.75782 −0.878910 0.476988i \(-0.841728\pi\)
−0.878910 + 0.476988i \(0.841728\pi\)
\(488\) 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
\(496\) −5.42944e9 −1.99788
\(497\) 0 0
\(498\) 0 0
\(499\) −5.47054e9 −1.97096 −0.985482 0.169780i \(-0.945694\pi\)
−0.985482 + 0.169780i \(0.945694\pi\)
\(500\) 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
\(508\) 5.77608e9 1.95484
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 7.87489e9 2.61079
\(512\) 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
\(523\) 1.19386e9 0.364919 0.182459 0.983213i \(-0.441594\pi\)
0.182459 + 0.983213i \(0.441594\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −3.40483e9 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 6.92214e9 1.99319
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −3.67799e9 −0.998664 −0.499332 0.866411i \(-0.666421\pi\)
−0.499332 + 0.866411i \(0.666421\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 5.98519e9 1.56359 0.781793 0.623537i \(-0.214305\pi\)
0.781793 + 0.623537i \(0.214305\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −1.09976e10 −2.76542
\(554\) 0 0
\(555\) 0 0
\(556\) 7.08161e9 1.74731
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) −5.16168e9 −1.24983
\(560\) 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
\(571\) 2.58916e9 0.582012 0.291006 0.956721i \(-0.406010\pi\)
0.291006 + 0.956721i \(0.406010\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 4.97957e9 1.07914 0.539568 0.841942i \(-0.318588\pi\)
0.539568 + 0.841942i \(0.318588\pi\)
\(578\) 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
\(589\) −1.42798e10 −2.87951
\(590\) 0 0
\(591\) 0 0
\(592\) 4.58277e9 0.907825
\(593\) 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
\(601\) 2.90322e9 0.545530 0.272765 0.962081i \(-0.412062\pi\)
0.272765 + 0.962081i \(0.412062\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 8.80992e9 1.62683
\(605\) 0 0
\(606\) 0 0
\(607\) −2.36131e9 −0.428541 −0.214270 0.976774i \(-0.568737\pi\)
−0.214270 + 0.976774i \(0.568737\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 1.12826e10 1.97832 0.989161 0.146837i \(-0.0469094\pi\)
0.989161 + 0.146837i \(0.0469094\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) −4.00534e9 −0.678770 −0.339385 0.940648i \(-0.610219\pi\)
−0.339385 + 0.940648i \(0.610219\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 1.09864e10 1.77009
\(629\) 0 0
\(630\) 0 0
\(631\) 9.94859e9 1.57637 0.788186 0.615437i \(-0.211021\pi\)
0.788186 + 0.615437i \(0.211021\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 9.47243e9 1.45202
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 1.04945e9 0.155677 0.0778386 0.996966i \(-0.475198\pi\)
0.0778386 + 0.996966i \(0.475198\pi\)
\(644\) 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
\(652\) 1.92327e9 0.271752
\(653\) 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
\(661\) 6.61860e9 0.891376 0.445688 0.895188i \(-0.352959\pi\)
0.445688 + 0.895188i \(0.352959\pi\)
\(662\) 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
\(673\) −4.99918e9 −0.632187 −0.316094 0.948728i \(-0.602371\pi\)
−0.316094 + 0.948728i \(0.602371\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −1.23056e10 −1.53211
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 2.19896e10 2.69570
\(680\) 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
\(688\) −6.70917e9 −0.785433
\(689\) 0 0
\(690\) 0 0
\(691\) −1.52104e10 −1.75375 −0.876876 0.480717i \(-0.840376\pi\)
−0.876876 + 0.480717i \(0.840376\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 1.20530e10 1.30843
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1.77423e10 −1.86960 −0.934800 0.355173i \(-0.884422\pi\)
−0.934800 + 0.355173i \(0.884422\pi\)
\(710\) 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
\(721\) −1.00729e10 −1.00088
\(722\) 0 0
\(723\) 0 0
\(724\) −1.96749e10 −1.92676
\(725\) 0 0
\(726\) 0 0
\(727\) −1.25393e10 −1.21033 −0.605165 0.796100i \(-0.706893\pi\)
−0.605165 + 0.796100i \(0.706893\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −1.97732e10 −1.85445 −0.927223 0.374511i \(-0.877811\pi\)
−0.927223 + 0.374511i \(0.877811\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 1.90404e10 1.73549 0.867743 0.497013i \(-0.165570\pi\)
0.867743 + 0.497013i \(0.165570\pi\)
\(740\) 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
\(751\) 1.45957e10 1.25744 0.628718 0.777633i \(-0.283580\pi\)
0.628718 + 0.777633i \(0.283580\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −2.32934e10 −1.95163 −0.975817 0.218591i \(-0.929854\pi\)
−0.975817 + 0.218591i \(0.929854\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) −3.35659e10 −2.73566
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 2.43003e10 1.92694 0.963472 0.267808i \(-0.0862994\pi\)
0.963472 + 0.267808i \(0.0862994\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −2.55629e10 −1.99963
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 1.23123e10 0.912499
\(785\) 0 0
\(786\) 0 0
\(787\) −2.48906e9 −0.182022 −0.0910110 0.995850i \(-0.529010\pi\)
−0.0910110 + 0.995850i \(0.529010\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 2.51892e10 1.79373
\(794\) 0 0
\(795\) 0 0
\(796\) 9.43846e9 0.663293
\(797\) 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
\(811\) −2.95302e10 −1.94399 −0.971993 0.235010i \(-0.924488\pi\)
−0.971993 + 0.235010i \(0.924488\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −1.76455e10 −1.13203
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) −1.83431e10 −1.14702 −0.573512 0.819197i \(-0.694419\pi\)
−0.573512 + 0.819197i \(0.694419\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 6.33017e9 0.385900 0.192950 0.981209i \(-0.438195\pi\)
0.192950 + 0.981209i \(0.438195\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −2.64346e10 −1.59126
\(833\) 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
\(841\) −1.72499e10 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 3.69638e9 0.211630
\(845\) 0 0
\(846\) 0 0
\(847\) 2.44564e10 1.38293
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 3.54191e10 1.95396 0.976981 0.213326i \(-0.0684297\pi\)
0.976981 + 0.213326i \(0.0684297\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) −1.87486e10 −1.00923 −0.504617 0.863343i \(-0.668366\pi\)
−0.504617 + 0.863343i \(0.668366\pi\)
\(860\) 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
\(868\) −5.32340e10 −2.76293
\(869\) 0 0
\(870\) 0 0
\(871\) −5.11568e10 −2.62325
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 3.64276e9 0.182361 0.0911805 0.995834i \(-0.470936\pi\)
0.0911805 + 0.995834i \(0.470936\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 3.73341e10 1.82492 0.912458 0.409171i \(-0.134182\pi\)
0.912458 + 0.409171i \(0.134182\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 5.66327e10 2.70340
\(890\) 0 0
\(891\) 0 0
\(892\) 3.91564e10 1.84725
\(893\) 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
\(907\) −4.18760e10 −1.86354 −0.931771 0.363046i \(-0.881737\pi\)
−0.931771 + 0.363046i \(0.881737\pi\)
\(908\) 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
\(916\) 2.06785e10 0.888965
\(917\) 0 0
\(918\) 0 0
\(919\) 5.42124e9 0.230406 0.115203 0.993342i \(-0.463248\pi\)
0.115203 + 0.993342i \(0.463248\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) 3.23821e10 1.31517
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 4.52983e10 1.79884 0.899421 0.437084i \(-0.143989\pi\)
0.899421 + 0.437084i \(0.143989\pi\)
\(938\) 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
\(949\) −7.90940e10 −3.00408
\(950\) 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
\(961\) 8.23047e10 2.99153
\(962\) 0 0
\(963\) 0 0
\(964\) −4.40656e10 −1.58427
\(965\) 0 0
\(966\) 0 0
\(967\) −4.22432e10 −1.50233 −0.751163 0.660117i \(-0.770507\pi\)
−0.751163 + 0.660117i \(0.770507\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 6.94330e10 2.41641
\(974\) 0 0
\(975\) 0 0
\(976\) 3.27409e10 1.12724
\(977\) 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
\(988\) −6.95247e10 −2.29345
\(989\) 0 0
\(990\) 0 0
\(991\) −5.77055e10 −1.88347 −0.941737 0.336349i \(-0.890808\pi\)
−0.941737 + 0.336349i \(0.890808\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 5.61437e10 1.79419 0.897093 0.441841i \(-0.145674\pi\)
0.897093 + 0.441841i \(0.145674\pi\)
\(998\) 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 225.8.a.f.1.1 1
3.2 odd 2 CM 225.8.a.f.1.1 1
5.2 odd 4 225.8.b.i.199.1 2
5.3 odd 4 225.8.b.i.199.2 2
5.4 even 2 225.8.a.g.1.1 yes 1
15.2 even 4 225.8.b.i.199.1 2
15.8 even 4 225.8.b.i.199.2 2
15.14 odd 2 225.8.a.g.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
225.8.a.f.1.1 1 1.1 even 1 trivial
225.8.a.f.1.1 1 3.2 odd 2 CM
225.8.a.g.1.1 yes 1 5.4 even 2
225.8.a.g.1.1 yes 1 15.14 odd 2
225.8.b.i.199.1 2 5.2 odd 4
225.8.b.i.199.1 2 15.2 even 4
225.8.b.i.199.2 2 5.3 odd 4
225.8.b.i.199.2 2 15.8 even 4