↑

Properties

Label 2016.3.l.a.433.1
Level $2016$
Weight $3$
Character 2016.433
Self dual yes
Analytic conductor $54.932$
Analytic rank $0$
Dimension $2$
CM discriminant -56
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2016,3,Mod(433,2016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2016, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2016.433");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2016.l (of order \(2\), degree \(1\), not 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: \(54.9320212950\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 433.1
Root \(-2.64575\) of defining polynomial
Character \(\chi\) \(=\) 2016.433

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.29150 q^{5} -7.00000 q^{7} +O(q^{10})\) \(q-5.29150 q^{5} -7.00000 q^{7} +5.29150 q^{13} -37.0405 q^{19} -10.0000 q^{23} +3.00000 q^{25} +37.0405 q^{35} +49.0000 q^{49} -89.9555 q^{59} -121.705 q^{61} -28.0000 q^{65} +110.000 q^{71} +130.000 q^{79} +164.037 q^{83} -37.0405 q^{91} +196.000 q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 14 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 14 q^{7} - 20 q^{23} + 6 q^{25} + 98 q^{49} - 56 q^{65} + 220 q^{71} + 260 q^{79} + 392 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(1765\) \(1793\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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
\(4\) 0 0
\(5\) −5.29150 −1.05830 −0.529150 − 0.848528i \(-0.677489\pi\)
−0.529150 + 0.848528i \(0.677489\pi\)
\(6\) 0 0
\(7\) −7.00000 −1.00000
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 5.29150 0.407039 0.203519 − 0.979071i \(-0.434762\pi\)
0.203519 + 0.979071i \(0.434762\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −37.0405 −1.94950 −0.974750 − 0.223297i \(-0.928318\pi\)
−0.974750 + 0.223297i \(0.928318\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −10.0000 −0.434783 −0.217391 − 0.976085i \(-0.569755\pi\)
−0.217391 + 0.976085i \(0.569755\pi\)
\(24\) 0 0
\(25\) 3.00000 0.120000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 37.0405 1.05830
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 49.0000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −89.9555 −1.52467 −0.762335 − 0.647182i \(-0.775947\pi\)
−0.762335 + 0.647182i \(0.775947\pi\)
\(60\) 0 0
\(61\) −121.705 −1.99516 −0.997578 − 0.0695515i \(-0.977843\pi\)
−0.997578 + 0.0695515i \(0.977843\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −28.0000 −0.430769
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 110.000 1.54930 0.774648 − 0.632393i \(-0.217927\pi\)
0.774648 + 0.632393i \(0.217927\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 130.000 1.64557 0.822785 − 0.568353i \(-0.192419\pi\)
0.822785 + 0.568353i \(0.192419\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 164.037 1.97634 0.988172 − 0.153348i \(-0.0490057\pi\)
0.988172 + 0.153348i \(0.0490057\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) −37.0405 −0.407039
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 196.000 2.06316
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 121.705 1.20500 0.602498 − 0.798121i \(-0.294172\pi\)
0.602498 + 0.798121i \(0.294172\pi\)
\(102\) 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
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −26.0000 −0.230088 −0.115044 − 0.993360i \(-0.536701\pi\)
−0.115044 + 0.993360i \(0.536701\pi\)
\(114\) 0 0
\(115\) 52.9150 0.460131
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 121.000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 116.413 0.931304
\(126\) 0 0
\(127\) 250.000 1.96850 0.984252 − 0.176771i \(-0.0565653\pi\)
0.984252 + 0.176771i \(0.0565653\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −89.9555 −0.686684 −0.343342 − 0.939211i \(-0.611559\pi\)
−0.343342 + 0.939211i \(0.611559\pi\)
\(132\) 0 0
\(133\) 259.284 1.94950
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −50.0000 −0.364964 −0.182482 − 0.983209i \(-0.558413\pi\)
−0.182482 + 0.983209i \(0.558413\pi\)
\(138\) 0 0
\(139\) 89.9555 0.647162 0.323581 − 0.946200i \(-0.395113\pi\)
0.323581 + 0.946200i \(0.395113\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 202.000 1.33775 0.668874 − 0.743376i \(-0.266776\pi\)
0.668874 + 0.743376i \(0.266776\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 5.29150 0.0337038 0.0168519 − 0.999858i \(-0.494636\pi\)
0.0168519 + 0.999858i \(0.494636\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 70.0000 0.434783
\(162\) 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
\(169\) −141.000 −0.834320
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −259.284 −1.49875 −0.749375 − 0.662146i \(-0.769646\pi\)
−0.749375 + 0.662146i \(0.769646\pi\)
\(174\) 0 0
\(175\) −21.0000 −0.120000
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) −248.701 −1.37404 −0.687018 − 0.726640i \(-0.741081\pi\)
−0.687018 + 0.726640i \(0.741081\pi\)
\(182\) 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
\(191\) −130.000 −0.680628 −0.340314 − 0.940312i \(-0.610533\pi\)
−0.340314 + 0.940312i \(0.610533\pi\)
\(192\) 0 0
\(193\) 314.000 1.62694 0.813472 − 0.581605i \(-0.197575\pi\)
0.813472 + 0.581605i \(0.197575\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 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
\(205\) 0 0
\(206\) 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
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −216.952 −0.955734 −0.477867 − 0.878432i \(-0.658590\pi\)
−0.477867 + 0.878432i \(0.658590\pi\)
\(228\) 0 0
\(229\) 386.280 1.68681 0.843405 − 0.537278i \(-0.180547\pi\)
0.843405 + 0.537278i \(0.180547\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 430.000 1.84549 0.922747 − 0.385407i \(-0.125939\pi\)
0.922747 + 0.385407i \(0.125939\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 422.000 1.76569 0.882845 − 0.469664i \(-0.155625\pi\)
0.882845 + 0.469664i \(0.155625\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −259.284 −1.05830
\(246\) 0 0
\(247\) −196.000 −0.793522
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 37.0405 0.147572 0.0737859 − 0.997274i \(-0.476492\pi\)
0.0737859 + 0.997274i \(0.476492\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −274.000 −1.04183 −0.520913 − 0.853610i \(-0.674408\pi\)
−0.520913 + 0.853610i \(0.674408\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −513.276 −1.90809 −0.954044 − 0.299666i \(-0.903125\pi\)
−0.954044 + 0.299666i \(0.903125\pi\)
\(270\) 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
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −338.000 −1.20285 −0.601423 − 0.798930i \(-0.705400\pi\)
−0.601423 + 0.798930i \(0.705400\pi\)
\(282\) 0 0
\(283\) 470.944 1.66411 0.832056 − 0.554692i \(-0.187164\pi\)
0.832056 + 0.554692i \(0.187164\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 289.000 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 375.697 1.28224 0.641121 − 0.767440i \(-0.278470\pi\)
0.641121 + 0.767440i \(0.278470\pi\)
\(294\) 0 0
\(295\) 476.000 1.61356
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −52.9150 −0.176973
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 644.000 2.11148
\(306\) 0 0
\(307\) −418.029 −1.36166 −0.680829 − 0.732443i \(-0.738380\pi\)
−0.680829 + 0.732443i \(0.738380\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 15.8745 0.0488446
\(326\) 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
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 26.0000 0.0771513 0.0385757 − 0.999256i \(-0.487718\pi\)
0.0385757 + 0.999256i \(0.487718\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −343.000 −1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) −121.705 −0.348724 −0.174362 − 0.984682i \(-0.555786\pi\)
−0.174362 + 0.984682i \(0.555786\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) −582.065 −1.63962
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −682.000 −1.89972 −0.949861 − 0.312673i \(-0.898775\pi\)
−0.949861 + 0.312673i \(0.898775\pi\)
\(360\) 0 0
\(361\) 1011.00 2.80055
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 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
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −687.895 −1.74151
\(396\) 0 0
\(397\) −629.689 −1.58612 −0.793059 − 0.609145i \(-0.791513\pi\)
−0.793059 + 0.609145i \(0.791513\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 550.000 1.37157 0.685786 − 0.727804i \(-0.259459\pi\)
0.685786 + 0.727804i \(0.259459\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 629.689 1.52467
\(414\) 0 0
\(415\) −868.000 −2.09157
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 672.021 1.60387 0.801934 − 0.597412i \(-0.203804\pi\)
0.801934 + 0.597412i \(0.203804\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 851.932 1.99516
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −538.000 −1.24826 −0.624130 − 0.781321i \(-0.714546\pi\)
−0.624130 + 0.781321i \(0.714546\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 370.405 0.847609
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −2.00000 −0.00445434 −0.00222717 − 0.999998i \(-0.500709\pi\)
−0.00222717 + 0.999998i \(0.500709\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 196.000 0.430769
\(456\) 0 0
\(457\) −886.000 −1.93873 −0.969365 − 0.245623i \(-0.921007\pi\)
−0.969365 + 0.245623i \(0.921007\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 883.681 1.91688 0.958439 − 0.285297i \(-0.0920922\pi\)
0.958439 + 0.285297i \(0.0920922\pi\)
\(462\) 0 0
\(463\) 226.000 0.488121 0.244060 − 0.969760i \(-0.421520\pi\)
0.244060 + 0.969760i \(0.421520\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 799.017 1.71096 0.855478 − 0.517838i \(-0.173263\pi\)
0.855478 + 0.517838i \(0.173263\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −111.122 −0.233940
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −470.000 −0.965092 −0.482546 − 0.875871i \(-0.660288\pi\)
−0.482546 + 0.875871i \(0.660288\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −770.000 −1.54930
\(498\) 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
\(505\) −644.000 −1.27525
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −386.280 −0.758899 −0.379450 − 0.925212i \(-0.623887\pi\)
−0.379450 + 0.925212i \(0.623887\pi\)
\(510\) 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
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) −418.029 −0.799290 −0.399645 − 0.916670i \(-0.630867\pi\)
−0.399645 + 0.916670i \(0.630867\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −429.000 −0.810964
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −910.000 −1.64557
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 1053.01 1.87035 0.935177 − 0.354181i \(-0.115240\pi\)
0.935177 + 0.354181i \(0.115240\pi\)
\(564\) 0 0
\(565\) 137.579 0.243503
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −1130.00 −1.98594 −0.992970 − 0.118365i \(-0.962235\pi\)
−0.992970 + 0.118365i \(0.962235\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −30.0000 −0.0521739
\(576\) 0 0
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −1148.26 −1.97634
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 164.037 0.279449 0.139725 − 0.990190i \(-0.455378\pi\)
0.139725 + 0.990190i \(0.455378\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 1070.00 1.78631 0.893155 − 0.449748i \(-0.148486\pi\)
0.893155 + 0.449748i \(0.148486\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −640.272 −1.05830
\(606\) 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
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1034.00 −1.67585 −0.837925 − 0.545785i \(-0.816232\pi\)
−0.837925 + 0.545785i \(0.816232\pi\)
\(618\) 0 0
\(619\) −545.025 −0.880492 −0.440246 − 0.897877i \(-0.645109\pi\)
−0.440246 + 0.897877i \(0.645109\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −691.000 −1.10560
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −110.000 −0.174326 −0.0871632 − 0.996194i \(-0.527780\pi\)
−0.0871632 + 0.996194i \(0.527780\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1322.88 −2.08327
\(636\) 0 0
\(637\) 259.284 0.407039
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1030.00 1.60686 0.803432 − 0.595396i \(-0.203005\pi\)
0.803432 + 0.595396i \(0.203005\pi\)
\(642\) 0 0
\(643\) −1053.01 −1.63765 −0.818825 − 0.574043i \(-0.805374\pi\)
−0.818825 + 0.574043i \(0.805374\pi\)
\(644\) 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
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 476.000 0.726718
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 386.280 0.584387 0.292193 − 0.956359i \(-0.405615\pi\)
0.292193 + 0.956359i \(0.405615\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −1372.00 −2.06316
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −670.000 −0.995542 −0.497771 − 0.867308i \(-0.665848\pi\)
−0.497771 + 0.867308i \(0.665848\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1264.67 1.86805 0.934024 − 0.357209i \(-0.116272\pi\)
0.934024 + 0.357209i \(0.116272\pi\)
\(678\) 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
\(685\) 264.575 0.386241
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 1359.92 1.96804 0.984020 − 0.178056i \(-0.0569808\pi\)
0.984020 + 0.178056i \(0.0569808\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −476.000 −0.684892
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −851.932 −1.20500
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 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
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −375.697 −0.512547 −0.256273 − 0.966604i \(-0.582495\pi\)
−0.256273 + 0.966604i \(0.582495\pi\)
\(734\) 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
\(743\) 1430.00 1.92463 0.962315 − 0.271937i \(-0.0876644\pi\)
0.962315 + 0.271937i \(0.0876644\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −998.000 −1.32889 −0.664447 − 0.747335i \(-0.731333\pi\)
−0.664447 + 0.747335i \(0.731333\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −1068.88 −1.41574
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −476.000 −0.620600
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −1275.25 −1.64974 −0.824872 − 0.565320i \(-0.808753\pi\)
−0.824872 + 0.565320i \(0.808753\pi\)
\(774\) 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
\(785\) −28.0000 −0.0356688
\(786\) 0 0
\(787\) 470.944 0.598404 0.299202 − 0.954190i \(-0.403280\pi\)
0.299202 + 0.954190i \(0.403280\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 182.000 0.230088
\(792\) 0 0
\(793\) −644.000 −0.812106
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1529.24 −1.91875 −0.959375 − 0.282133i \(-0.908958\pi\)
−0.959375 + 0.282133i \(0.908958\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −370.405 −0.460131
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −650.000 −0.803461 −0.401731 − 0.915758i \(-0.631591\pi\)
−0.401731 + 0.915758i \(0.631591\pi\)
\(810\) 0 0
\(811\) 724.936 0.893879 0.446939 − 0.894564i \(-0.352514\pi\)
0.446939 + 0.894564i \(0.352514\pi\)
\(812\) 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
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 946.000 1.14945 0.574727 − 0.818345i \(-0.305108\pi\)
0.574727 + 0.818345i \(0.305108\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 1656.24 1.99788 0.998939 − 0.0460600i \(-0.0146666\pi\)
0.998939 + 0.0460600i \(0.0146666\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 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
\(841\) 841.000 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 746.102 0.882961
\(846\) 0 0
\(847\) −847.000 −1.00000
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −1645.66 −1.92926 −0.964629 − 0.263611i \(-0.915087\pi\)
−0.964629 + 0.263611i \(0.915087\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 1232.92 1.43530 0.717648 − 0.696406i \(-0.245218\pi\)
0.717648 + 0.696406i \(0.245218\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −1474.00 −1.70800 −0.853998 − 0.520277i \(-0.825829\pi\)
−0.853998 + 0.520277i \(0.825829\pi\)
\(864\) 0 0
\(865\) 1372.00 1.58613
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −814.891 −0.931304
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 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
\(889\) −1750.00 −1.96850
\(890\) 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
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1316.00 1.45414
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −922.000 −1.01207 −0.506037 − 0.862512i \(-0.668890\pi\)
−0.506037 + 0.862512i \(0.668890\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 629.689 0.686684
\(918\) 0 0
\(919\) −1550.00 −1.68662 −0.843308 − 0.537431i \(-0.819395\pi\)
−0.843308 + 0.537431i \(0.819395\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 582.065 0.630623
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) −1814.99 −1.94950
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1518.66 1.61388 0.806940 − 0.590633i \(-0.201122\pi\)
0.806940 + 0.590633i \(0.201122\pi\)
\(942\) 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
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1010.00 −1.05981 −0.529906 − 0.848057i \(-0.677773\pi\)
−0.529906 + 0.848057i \(0.677773\pi\)
\(954\) 0 0
\(955\) 687.895 0.720309
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 350.000 0.364964
\(960\) 0 0
\(961\) 961.000 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −1661.53 −1.72179
\(966\) 0 0
\(967\) −1430.00 −1.47880 −0.739400 − 0.673266i \(-0.764891\pi\)
−0.739400 + 0.673266i \(0.764891\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1941.98 1.99998 0.999990 − 0.00436935i \(-0.00139081\pi\)
0.999990 + 0.00436935i \(0.00139081\pi\)
\(972\) 0 0
\(973\) −629.689 −0.647162
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1630.00 1.66837 0.834186 − 0.551483i \(-0.185938\pi\)
0.834186 + 0.551483i \(0.185938\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 610.000 0.615540 0.307770 − 0.951461i \(-0.400417\pi\)
0.307770 + 0.951461i \(0.400417\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 259.284 0.260064 0.130032 − 0.991510i \(-0.458492\pi\)
0.130032 + 0.991510i \(0.458492\pi\)
\(998\) 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 2016.3.l.a.433.1 2
3.2 odd 2 224.3.h.b.209.2 2
4.3 odd 2 504.3.l.c.181.1 2
7.6 odd 2 inner 2016.3.l.a.433.2 2
8.3 odd 2 504.3.l.c.181.2 2
8.5 even 2 inner 2016.3.l.a.433.2 2
12.11 even 2 56.3.h.a.13.1 ✓ 2
21.20 even 2 224.3.h.b.209.1 2
24.5 odd 2 224.3.h.b.209.1 2
24.11 even 2 56.3.h.a.13.2 yes 2
28.27 even 2 504.3.l.c.181.2 2
56.13 odd 2 CM 2016.3.l.a.433.1 2
56.27 even 2 504.3.l.c.181.1 2
84.11 even 6 392.3.j.c.117.2 4
84.23 even 6 392.3.j.c.325.2 4
84.47 odd 6 392.3.j.c.325.1 4
84.59 odd 6 392.3.j.c.117.1 4
84.83 odd 2 56.3.h.a.13.2 yes 2
168.11 even 6 392.3.j.c.117.1 4
168.59 odd 6 392.3.j.c.117.2 4
168.83 odd 2 56.3.h.a.13.1 ✓ 2
168.107 even 6 392.3.j.c.325.1 4
168.125 even 2 224.3.h.b.209.2 2
168.131 odd 6 392.3.j.c.325.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.3.h.a.13.1 ✓ 2 12.11 even 2
56.3.h.a.13.1 ✓ 2 168.83 odd 2
56.3.h.a.13.2 yes 2 24.11 even 2
56.3.h.a.13.2 yes 2 84.83 odd 2
224.3.h.b.209.1 2 21.20 even 2
224.3.h.b.209.1 2 24.5 odd 2
224.3.h.b.209.2 2 3.2 odd 2
224.3.h.b.209.2 2 168.125 even 2
392.3.j.c.117.1 4 84.59 odd 6
392.3.j.c.117.1 4 168.11 even 6
392.3.j.c.117.2 4 84.11 even 6
392.3.j.c.117.2 4 168.59 odd 6
392.3.j.c.325.1 4 84.47 odd 6
392.3.j.c.325.1 4 168.107 even 6
392.3.j.c.325.2 4 84.23 even 6
392.3.j.c.325.2 4 168.131 odd 6
504.3.l.c.181.1 2 4.3 odd 2
504.3.l.c.181.1 2 56.27 even 2
504.3.l.c.181.2 2 8.3 odd 2
504.3.l.c.181.2 2 28.27 even 2
2016.3.l.a.433.1 2 1.1 even 1 trivial
2016.3.l.a.433.1 2 56.13 odd 2 CM
2016.3.l.a.433.2 2 7.6 odd 2 inner
2016.3.l.a.433.2 2 8.5 even 2 inner