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Properties

Label 16.11.c.a.15.1
Level $16$
Weight $11$
Character 16.15
Self dual yes
Analytic conductor $10.166$
Analytic rank $0$
Dimension $1$
CM discriminant -4
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [16,11,Mod(15,16)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(16, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("16.15");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 16.c (of order \(2\), degree \(1\), minimal)

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: \(10.1657160428\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 15.1
Character \(\chi\) \(=\) 16.15

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+474.000 q^{5} +59049.0 q^{9} +O(q^{10})\) \(q+474.000 q^{5} +59049.0 q^{9} +683050. q^{13} +2.18685e6 q^{17} -9.54095e6 q^{25} -3.23198e7 q^{29} +1.11786e7 q^{37} +2.07190e8 q^{41} +2.79892e7 q^{45} +2.82475e8 q^{49} -7.83189e8 q^{53} -1.33009e9 q^{61} +3.23766e8 q^{65} +3.74036e9 q^{73} +3.48678e9 q^{81} +1.03657e9 q^{85} -8.56202e9 q^{89} -8.68453e9 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/16\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(15\)
\(\chi(n)\) \(1\) \(-1\)

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
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 0 0
\(5\) 474.000 0.151680 0.0758400 − 0.997120i \(-0.475836\pi\)
0.0758400 + 0.997120i \(0.475836\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 59049.0 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 683050. 1.83965 0.919826 − 0.392326i \(-0.128330\pi\)
0.919826 + 0.392326i \(0.128330\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.18685e6 1.54019 0.770095 − 0.637929i \(-0.220209\pi\)
0.770095 + 0.637929i \(0.220209\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −9.54095e6 −0.976993
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.23198e7 −1.57572 −0.787859 − 0.615855i \(-0.788811\pi\)
−0.787859 + 0.615855i \(0.788811\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.11786e7 0.161206 0.0806029 − 0.996746i \(-0.474315\pi\)
0.0806029 + 0.996746i \(0.474315\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.07190e8 1.78834 0.894169 − 0.447729i \(-0.147767\pi\)
0.894169 + 0.447729i \(0.147767\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 2.79892e7 0.151680
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 2.82475e8 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −7.83189e8 −1.87278 −0.936390 − 0.350960i \(-0.885855\pi\)
−0.936390 + 0.350960i \(0.885855\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −1.33009e9 −1.57482 −0.787412 − 0.616427i \(-0.788579\pi\)
−0.787412 + 0.616427i \(0.788579\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.23766e8 0.279038
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 3.74036e9 1.80426 0.902131 − 0.431463i \(-0.142003\pi\)
0.902131 + 0.431463i \(0.142003\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 3.48678e9 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 1.03657e9 0.233616
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −8.56202e9 −1.53330 −0.766648 − 0.642068i \(-0.778077\pi\)
−0.766648 + 0.642068i \(0.778077\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −8.68453e9 −1.01132 −0.505659 − 0.862733i \(-0.668751\pi\)
−0.505659 + 0.862733i \(0.668751\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.14158e10 −1.08618 −0.543088 − 0.839676i \(-0.682745\pi\)
−0.543088 + 0.839676i \(0.682745\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 2.99829e10 1.94868 0.974340 − 0.225080i \(-0.0722642\pi\)
0.974340 + 0.225080i \(0.0722642\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −2.27576e10 −1.23519 −0.617595 − 0.786496i \(-0.711893\pi\)
−0.617595 + 0.786496i \(0.711893\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 4.03334e10 1.83965
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 2.59374e10 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −9.15132e9 −0.299870
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −9.07993e10 −1.88139 −0.940697 − 0.339248i \(-0.889827\pi\)
−0.940697 + 0.339248i \(0.889827\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −1.53196e10 −0.239005
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1.44612e11 −1.96913 −0.984565 − 0.175020i \(-0.944001\pi\)
−0.984565 + 0.175020i \(0.944001\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 1.29131e11 1.54019
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 5.29202e10 0.554783 0.277392 − 0.960757i \(-0.410530\pi\)
0.277392 + 0.960757i \(0.410530\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 3.28699e11 2.38432
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.37263e10 0.0885771 0.0442886 − 0.999019i \(-0.485898\pi\)
0.0442886 + 0.999019i \(0.485898\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) −1.95015e11 −1.00386 −0.501932 − 0.864907i \(-0.667377\pi\)
−0.501932 + 0.864907i \(0.667377\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 5.29868e9 0.0244517
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) −2.88269e11 −1.07649 −0.538247 − 0.842787i \(-0.680913\pi\)
−0.538247 + 0.842787i \(0.680913\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −4.48835e11 −1.51271 −0.756354 − 0.654162i \(-0.773021\pi\)
−0.756354 + 0.654162i \(0.773021\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 9.82081e10 0.271255
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.49373e12 2.83341
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) −5.63383e11 −0.976993
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 3.05850e11 0.485658 0.242829 − 0.970069i \(-0.421925\pi\)
0.242829 + 0.970069i \(0.421925\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 9.89218e11 1.44050 0.720249 − 0.693716i \(-0.244028\pi\)
0.720249 + 0.693716i \(0.244028\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −1.39831e12 −1.71996 −0.859979 − 0.510330i \(-0.829523\pi\)
−0.859979 + 0.510330i \(0.829523\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.33893e11 0.151680
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −1.81950e12 −1.62288 −0.811438 − 0.584439i \(-0.801315\pi\)
−0.811438 + 0.584439i \(0.801315\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −1.90845e12 −1.57572
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) −3.71231e11 −0.284063
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 2.71212e12 1.92552 0.962758 − 0.270364i \(-0.0871440\pi\)
0.962758 + 0.270364i \(0.0871440\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −2.74627e12 −1.68401 −0.842003 − 0.539473i \(-0.818624\pi\)
−0.842003 + 0.539473i \(0.818624\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 3.48173e12 1.98730 0.993651 − 0.112505i \(-0.0358875\pi\)
0.993651 + 0.112505i \(0.0358875\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2.76632e12 1.37219
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.68068e12 0.778300 0.389150 − 0.921174i \(-0.372769\pi\)
0.389150 + 0.921174i \(0.372769\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −6.30463e11 −0.238869
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 2.33856e12 0.778444 0.389222 − 0.921144i \(-0.372744\pi\)
0.389222 + 0.921144i \(0.372744\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −5.95371e12 −1.85991 −0.929955 − 0.367674i \(-0.880154\pi\)
−0.929955 + 0.367674i \(0.880154\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −6.51695e12 −1.79733
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 6.60088e11 0.161206
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −3.47719e12 −0.799979 −0.399990 − 0.916520i \(-0.630986\pi\)
−0.399990 + 0.916520i \(0.630986\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 9.40330e12 1.81616 0.908078 − 0.418801i \(-0.137549\pi\)
0.908078 + 0.418801i \(0.137549\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −3.38551e12 −0.617661 −0.308830 − 0.951117i \(-0.599938\pi\)
−0.308830 + 0.951117i \(0.599938\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 6.13107e12 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1.77293e12 0.273670
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 1.22344e13 1.78834
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 1.21781e13 1.68669 0.843344 − 0.537374i \(-0.180584\pi\)
0.843344 + 0.537374i \(0.180584\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −2.20760e13 −2.89877
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 1.01301e13 1.13728 0.568640 − 0.822586i \(-0.307470\pi\)
0.568640 + 0.822586i \(0.307470\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −1.96559e13 −1.99315 −0.996574 − 0.0827093i \(-0.973643\pi\)
−0.996574 + 0.0827093i \(0.973643\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −1.82028e13 −1.75556 −0.877782 − 0.479060i \(-0.840977\pi\)
−0.877782 + 0.479060i \(0.840977\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 1.65274e12 0.151680
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −1.87253e12 −0.163611 −0.0818053 − 0.996648i \(-0.526069\pi\)
−0.0818053 + 0.996648i \(0.526069\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 8.93792e12 0.675812 0.337906 − 0.941180i \(-0.390281\pi\)
0.337906 + 0.941180i \(0.390281\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −2.08646e13 −1.50476
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 3.01584e13 1.98138 0.990691 − 0.136132i \(-0.0434671\pi\)
0.990691 + 0.136132i \(0.0434671\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 1.66799e13 1.00000
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) −4.05840e12 −0.232570
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 3.55762e13 1.94952 0.974759 − 0.223258i \(-0.0716692\pi\)
0.974759 + 0.223258i \(0.0716692\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.22156e13 −0.612819 −0.306409 − 0.951900i \(-0.599128\pi\)
−0.306409 + 0.951900i \(0.599128\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 5.49695e12 0.264008 0.132004 − 0.991249i \(-0.457859\pi\)
0.132004 + 0.991249i \(0.457859\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −4.62465e13 −1.87278
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 7.63558e12 0.296563
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −4.11647e12 −0.153397
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) −7.06786e13 −2.42691
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) −5.41109e12 −0.164751
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 4.21090e13 1.23250 0.616248 − 0.787552i \(-0.288652\pi\)
0.616248 + 0.787552i \(0.288652\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 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
\(521\) −2.38622e13 −0.621616 −0.310808 − 0.950473i \(-0.600600\pi\)
−0.310808 + 0.950473i \(0.600600\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 4.14265e13 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 1.41521e14 3.28992
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −2.45613e13 −0.529986 −0.264993 − 0.964250i \(-0.585370\pi\)
−0.264993 + 0.964250i \(0.585370\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1.42119e13 0.295576
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) −7.85405e13 −1.57482
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.06986e14 1.99550 0.997751 − 0.0670317i \(-0.0213529\pi\)
0.997751 + 0.0670317i \(0.0213529\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) −1.07871e13 −0.187354
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −1.03553e14 −1.73620 −0.868101 − 0.496388i \(-0.834659\pi\)
−0.868101 + 0.496388i \(0.834659\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −1.16980e14 −1.82908 −0.914541 − 0.404494i \(-0.867448\pi\)
−0.914541 + 0.404494i \(0.867448\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 1.91180e13 0.279038
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1.43563e14 −1.95781 −0.978903 − 0.204327i \(-0.934500\pi\)
−0.978903 + 0.204327i \(0.934500\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 7.28552e13 0.929155 0.464578 − 0.885532i \(-0.346206\pi\)
0.464578 + 0.885532i \(0.346206\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.22943e13 0.151680
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −4.87781e13 −0.563537 −0.281768 − 0.959482i \(-0.590921\pi\)
−0.281768 + 0.959482i \(0.590921\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.34951e14 1.50921 0.754606 − 0.656179i \(-0.227828\pi\)
0.754606 + 0.656179i \(0.227828\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 8.88356e13 0.931509
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 2.44460e13 0.248288
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 1.92945e14 1.83965
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −3.41001e12 −0.0315113 −0.0157556 − 0.999876i \(-0.505015\pi\)
−0.0157556 + 0.999876i \(0.505015\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1.38880e14 −1.16970 −0.584848 − 0.811143i \(-0.698846\pi\)
−0.584848 + 0.811143i \(0.698846\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 2.20865e14 1.80426
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −1.78319e14 −1.41316 −0.706580 − 0.707633i \(-0.749763\pi\)
−0.706580 + 0.707633i \(0.749763\pi\)
\(662\) 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
\(673\) 2.66003e13 0.192668 0.0963342 − 0.995349i \(-0.469288\pi\)
0.0963342 + 0.995349i \(0.469288\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −2.63669e14 −1.85403 −0.927014 − 0.375027i \(-0.877633\pi\)
−0.927014 + 0.375027i \(0.877633\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) −4.30389e13 −0.285370
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −5.34957e14 −3.44527
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 4.53094e14 2.75438
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1.09411e14 0.646357 0.323179 − 0.946338i \(-0.395248\pi\)
0.323179 + 0.946338i \(0.395248\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −3.42414e14 −1.91127 −0.955633 − 0.294561i \(-0.904826\pi\)
−0.955633 + 0.294561i \(0.904826\pi\)
\(710\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 3.08362e14 1.53947
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 2.05891e14 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 3.12697e14 1.47776 0.738879 − 0.673838i \(-0.235355\pi\)
0.738879 + 0.673838i \(0.235355\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) −6.85463e13 −0.298678
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 4.88148e14 1.96369 0.981843 − 0.189694i \(-0.0607496\pi\)
0.981843 + 0.189694i \(0.0607496\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.29427e14 −0.507111 −0.253555 − 0.967321i \(-0.581600\pi\)
−0.253555 + 0.967321i \(0.581600\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 6.12082e13 0.233616
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −1.26380e14 −0.469943 −0.234971 − 0.972002i \(-0.575500\pi\)
−0.234971 + 0.972002i \(0.575500\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −5.28028e14 −1.91320 −0.956598 − 0.291410i \(-0.905876\pi\)
−0.956598 + 0.291410i \(0.905876\pi\)
\(774\) 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
\(784\) 0 0
\(785\) 2.50842e13 0.0841496
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −9.08518e14 −2.89713
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 4.19227e14 1.30364 0.651820 − 0.758373i \(-0.274006\pi\)
0.651820 + 0.758373i \(0.274006\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −5.05579e14 −1.53330
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.35171e14 0.390068 0.195034 − 0.980796i \(-0.437518\pi\)
0.195034 + 0.980796i \(0.437518\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 2.85345e14 0.764988 0.382494 − 0.923958i \(-0.375065\pi\)
0.382494 + 0.923958i \(0.375065\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) −7.19582e14 −1.83784 −0.918920 − 0.394445i \(-0.870937\pi\)
−0.918920 + 0.394445i \(0.870937\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 6.17731e14 1.54019
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 6.23862e14 1.48289
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.55803e14 0.361654
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 8.46148e14 1.87371 0.936853 − 0.349724i \(-0.113725\pi\)
0.936853 + 0.349724i \(0.113725\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1.83803e14 0.397602 0.198801 − 0.980040i \(-0.436295\pi\)
0.198801 + 0.980040i \(0.436295\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 6.50624e12 0.0134354
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −5.12813e14 −1.01132
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −4.69342e14 −0.904673 −0.452337 − 0.891847i \(-0.649409\pi\)
−0.452337 + 0.891847i \(0.649409\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 8.81991e14 1.66182 0.830911 − 0.556405i \(-0.187820\pi\)
0.830911 + 0.556405i \(0.187820\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −1.71272e15 −2.88444
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −9.24370e13 −0.152266
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) −6.74092e14 −1.08618
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −1.06655e14 −0.157497
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 8.87879e14 1.28314 0.641572 − 0.767063i \(-0.278283\pi\)
0.641572 + 0.767063i \(0.278283\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −1.32296e15 −1.83168 −0.915839 − 0.401546i \(-0.868473\pi\)
−0.915839 + 0.401546i \(0.868473\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −1.45205e15 −1.96804 −0.984022 − 0.178048i \(-0.943022\pi\)
−0.984022 + 0.178048i \(0.943022\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) 2.55485e15 3.31921
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 5.62187e14 0.715182 0.357591 − 0.933878i \(-0.383598\pi\)
0.357591 + 0.933878i \(0.383598\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 8.19628e14 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −1.36640e14 −0.163283
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 5.04935e14 0.567234 0.283617 − 0.958938i \(-0.408466\pi\)
0.283617 + 0.958938i \(0.408466\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 1.77046e15 1.94868
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) −2.12748e14 −0.229448
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −6.58976e14 −0.668950 −0.334475 − 0.942405i \(-0.608559\pi\)
−0.334475 + 0.942405i \(0.608559\pi\)
\(998\) 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 16.11.c.a.15.1 ✓ 1
3.2 odd 2 144.11.g.a.127.1 1
4.3 odd 2 CM 16.11.c.a.15.1 ✓ 1
8.3 odd 2 64.11.c.a.63.1 1
8.5 even 2 64.11.c.a.63.1 1
12.11 even 2 144.11.g.a.127.1 1
16.3 odd 4 256.11.d.a.127.1 2
16.5 even 4 256.11.d.a.127.2 2
16.11 odd 4 256.11.d.a.127.2 2
16.13 even 4 256.11.d.a.127.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.11.c.a.15.1 ✓ 1 1.1 even 1 trivial
16.11.c.a.15.1 ✓ 1 4.3 odd 2 CM
64.11.c.a.63.1 1 8.3 odd 2
64.11.c.a.63.1 1 8.5 even 2
144.11.g.a.127.1 1 3.2 odd 2
144.11.g.a.127.1 1 12.11 even 2
256.11.d.a.127.1 2 16.3 odd 4
256.11.d.a.127.1 2 16.13 even 4
256.11.d.a.127.2 2 16.5 even 4
256.11.d.a.127.2 2 16.11 odd 4