Properties

Label 799.1.c.c.798.4
Level $799$
Weight $1$
Character 799.798
Self dual yes
Analytic conductor $0.399$
Analytic rank $0$
Dimension $4$
Projective image $D_{8}$
CM discriminant -799
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [799,1,Mod(798,799)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(799, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("799.798");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 799 = 17 \cdot 47 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 799.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: \(0.398752945094\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{16})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.2.8671400783.1

Embedding invariants

Embedding label 798.4
Root \(-0.765367\) of defining polynomial
Character \(\chi\) \(=\) 799.798

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.41421 q^{2} +1.00000 q^{4} +0.765367 q^{5} +1.00000 q^{9} +O(q^{10})\) \(q+1.41421 q^{2} +1.00000 q^{4} +0.765367 q^{5} +1.00000 q^{9} +1.08239 q^{10} -1.84776 q^{11} -1.00000 q^{16} -1.00000 q^{17} +1.41421 q^{18} +0.765367 q^{20} -2.61313 q^{22} -0.765367 q^{23} -0.414214 q^{25} +1.84776 q^{29} +1.84776 q^{31} -1.41421 q^{32} -1.41421 q^{34} +1.00000 q^{36} -0.765367 q^{41} -1.84776 q^{44} +0.765367 q^{45} -1.08239 q^{46} -1.00000 q^{47} +1.00000 q^{49} -0.585786 q^{50} -1.41421 q^{55} +2.61313 q^{58} -1.41421 q^{59} +2.61313 q^{62} -1.00000 q^{64} -1.00000 q^{68} -1.84776 q^{73} -0.765367 q^{80} +1.00000 q^{81} -1.08239 q^{82} -0.765367 q^{85} +1.08239 q^{90} -0.765367 q^{92} -1.41421 q^{94} +1.41421 q^{98} -1.84776 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{4} + 4 q^{9} - 4 q^{16} - 4 q^{17} + 4 q^{25} + 4 q^{36} - 4 q^{47} + 4 q^{49} - 8 q^{50} - 4 q^{64} - 4 q^{68} + 4 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/799\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(377\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 1.00000 1.00000
\(5\) 0.765367 0.765367 0.382683 0.923880i \(-0.375000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 1.00000 1.00000
\(10\) 1.08239 1.08239
\(11\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −1.00000 −1.00000
\(17\) −1.00000 −1.00000
\(18\) 1.41421 1.41421
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0.765367 0.765367
\(21\) 0 0
\(22\) −2.61313 −2.61313
\(23\) −0.765367 −0.765367 −0.382683 0.923880i \(-0.625000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(24\) 0 0
\(25\) −0.414214 −0.414214
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.84776 1.84776 0.923880 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(30\) 0 0
\(31\) 1.84776 1.84776 0.923880 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(32\) −1.41421 −1.41421
\(33\) 0 0
\(34\) −1.41421 −1.41421
\(35\) 0 0
\(36\) 1.00000 1.00000
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −0.765367 −0.765367 −0.382683 0.923880i \(-0.625000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) −1.84776 −1.84776
\(45\) 0.765367 0.765367
\(46\) −1.08239 −1.08239
\(47\) −1.00000 −1.00000
\(48\) 0 0
\(49\) 1.00000 1.00000
\(50\) −0.585786 −0.585786
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) −1.41421 −1.41421
\(56\) 0 0
\(57\) 0 0
\(58\) 2.61313 2.61313
\(59\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 2.61313 2.61313
\(63\) 0 0
\(64\) −1.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) −1.00000 −1.00000
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) −0.765367 −0.765367
\(81\) 1.00000 1.00000
\(82\) −1.08239 −1.08239
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −0.765367 −0.765367
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 1.08239 1.08239
\(91\) 0 0
\(92\) −0.765367 −0.765367
\(93\) 0 0
\(94\) −1.41421 −1.41421
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 1.41421 1.41421
\(99\) −1.84776 −1.84776
\(100\) −0.414214 −0.414214
\(101\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(102\) 0 0
\(103\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.84776 1.84776 0.923880 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(108\) 0 0
\(109\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(110\) −2.00000 −2.00000
\(111\) 0 0
\(112\) 0 0
\(113\) 1.84776 1.84776 0.923880 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(114\) 0 0
\(115\) −0.585786 −0.585786
\(116\) 1.84776 1.84776
\(117\) 0 0
\(118\) −2.00000 −2.00000
\(119\) 0 0
\(120\) 0 0
\(121\) 2.41421 2.41421
\(122\) 0 0
\(123\) 0 0
\(124\) 1.84776 1.84776
\(125\) −1.08239 −1.08239
\(126\) 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
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) −0.765367 −0.765367 −0.382683 0.923880i \(-0.625000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −1.00000 −1.00000
\(145\) 1.41421 1.41421
\(146\) −2.61313 −2.61313
\(147\) 0 0
\(148\) 0 0
\(149\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) −1.00000 −1.00000
\(154\) 0 0
\(155\) 1.41421 1.41421
\(156\) 0 0
\(157\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −1.08239 −1.08239
\(161\) 0 0
\(162\) 1.41421 1.41421
\(163\) 0.765367 0.765367 0.382683 0.923880i \(-0.375000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(164\) −0.765367 −0.765367
\(165\) 0 0
\(166\) 0 0
\(167\) 0.765367 0.765367 0.382683 0.923880i \(-0.375000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(168\) 0 0
\(169\) 1.00000 1.00000
\(170\) −1.08239 −1.08239
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.84776 1.84776
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0.765367 0.765367
\(181\) −0.765367 −0.765367 −0.382683 0.923880i \(-0.625000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 1.84776 1.84776
\(188\) −1.00000 −1.00000
\(189\) 0 0
\(190\) 0 0
\(191\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(192\) 0 0
\(193\) 1.84776 1.84776 0.923880 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 1.00000 1.00000
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) −2.61313 −2.61313
\(199\) 0.765367 0.765367 0.382683 0.923880i \(-0.375000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 2.00000 2.00000
\(203\) 0 0
\(204\) 0 0
\(205\) −0.585786 −0.585786
\(206\) 2.00000 2.00000
\(207\) −0.765367 −0.765367
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0.765367 0.765367 0.382683 0.923880i \(-0.375000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 2.61313 2.61313
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −2.61313 −2.61313
\(219\) 0 0
\(220\) −1.41421 −1.41421
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) −0.414214 −0.414214
\(226\) 2.61313 2.61313
\(227\) −0.765367 −0.765367 −0.382683 0.923880i \(-0.625000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) −0.828427 −0.828427
\(231\) 0 0
\(232\) 0 0
\(233\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(234\) 0 0
\(235\) −0.765367 −0.765367
\(236\) −1.41421 −1.41421
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 3.41421 3.41421
\(243\) 0 0
\(244\) 0 0
\(245\) 0.765367 0.765367
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) −1.53073 −1.53073
\(251\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(252\) 0 0
\(253\) 1.41421 1.41421
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 1.84776 1.84776
\(262\) 0 0
\(263\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(272\) 1.00000 1.00000
\(273\) 0 0
\(274\) 0 0
\(275\) 0.765367 0.765367
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) −1.08239 −1.08239
\(279\) 1.84776 1.84776
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −1.41421 −1.41421
\(289\) 1.00000 1.00000
\(290\) 2.00000 2.00000
\(291\) 0 0
\(292\) −1.84776 −1.84776
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) −1.08239 −1.08239
\(296\) 0 0
\(297\) 0 0
\(298\) −2.00000 −2.00000
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) −1.41421 −1.41421
\(307\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 2.00000 2.00000
\(311\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(312\) 0 0
\(313\) −0.765367 −0.765367 −0.382683 0.923880i \(-0.625000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(314\) −2.00000 −2.00000
\(315\) 0 0
\(316\) 0 0
\(317\) −0.765367 −0.765367 −0.382683 0.923880i \(-0.625000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(318\) 0 0
\(319\) −3.41421 −3.41421
\(320\) −0.765367 −0.765367
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000 1.00000
\(325\) 0 0
\(326\) 1.08239 1.08239
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 1.08239 1.08239
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 1.41421 1.41421
\(339\) 0 0
\(340\) −0.765367 −0.765367
\(341\) −3.41421 −3.41421
\(342\) 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
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2.61313 2.61313
\(353\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(354\) 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
\(361\) 1.00000 1.00000
\(362\) −1.08239 −1.08239
\(363\) 0 0
\(364\) 0 0
\(365\) −1.41421 −1.41421
\(366\) 0 0
\(367\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(368\) 0.765367 0.765367
\(369\) −0.765367 −0.765367
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 2.61313 2.61313
\(375\) 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
\(382\) 2.00000 2.00000
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 2.61313 2.61313
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0.765367 0.765367
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) −1.84776 −1.84776
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 1.08239 1.08239
\(399\) 0 0
\(400\) 0.414214 0.414214
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 1.41421 1.41421
\(405\) 0.765367 0.765367
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) −0.828427 −0.828427
\(411\) 0 0
\(412\) 1.41421 1.41421
\(413\) 0 0
\(414\) −1.08239 −1.08239
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0.765367 0.765367 0.382683 0.923880i \(-0.375000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 1.08239 1.08239
\(423\) −1.00000 −1.00000
\(424\) 0 0
\(425\) 0.414214 0.414214
\(426\) 0 0
\(427\) 0 0
\(428\) 1.84776 1.84776
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −1.84776 −1.84776
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 1.00000 1.00000
\(442\) 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
\(449\) −0.765367 −0.765367 −0.382683 0.923880i \(-0.625000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(450\) −0.585786 −0.585786
\(451\) 1.41421 1.41421
\(452\) 1.84776 1.84776
\(453\) 0 0
\(454\) −1.08239 −1.08239
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −0.585786 −0.585786
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) −1.84776 −1.84776
\(465\) 0 0
\(466\) −2.61313 −2.61313
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −1.08239 −1.08239
\(471\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 2.41421 2.41421
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 1.08239 1.08239
\(491\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) −1.84776 −1.84776
\(494\) 0 0
\(495\) −1.41421 −1.41421
\(496\) −1.84776 −1.84776
\(497\) 0 0
\(498\) 0 0
\(499\) −0.765367 −0.765367 −0.382683 0.923880i \(-0.625000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(500\) −1.08239 −1.08239
\(501\) 0 0
\(502\) −2.00000 −2.00000
\(503\) 1.84776 1.84776 0.923880 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(504\) 0 0
\(505\) 1.08239 1.08239
\(506\) 2.00000 2.00000
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.41421 1.41421
\(513\) 0 0
\(514\) 0 0
\(515\) 1.08239 1.08239
\(516\) 0 0
\(517\) 1.84776 1.84776
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 2.61313 2.61313
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −2.00000 −2.00000
\(527\) −1.84776 −1.84776
\(528\) 0 0
\(529\) −0.414214 −0.414214
\(530\) 0 0
\(531\) −1.41421 −1.41421
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 1.41421 1.41421
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −1.84776 −1.84776
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 2.00000 2.00000
\(543\) 0 0
\(544\) 1.41421 1.41421
\(545\) −1.41421 −1.41421
\(546\) 0 0
\(547\) 0.765367 0.765367 0.382683 0.923880i \(-0.375000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 1.08239 1.08239
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −0.765367 −0.765367
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 2.61313 2.61313
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 1.41421 1.41421
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0.317025 0.317025
\(576\) −1.00000 −1.00000
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 1.41421 1.41421
\(579\) 0 0
\(580\) 1.41421 1.41421
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −1.53073 −1.53073
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1.41421 −1.41421
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.84776 1.84776
\(606\) 0 0
\(607\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −1.00000 −1.00000
\(613\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(614\) −2.00000 −2.00000
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 1.41421 1.41421
\(621\) 0 0
\(622\) −2.61313 −2.61313
\(623\) 0 0
\(624\) 0 0
\(625\) −0.414214 −0.414214
\(626\) −1.08239 −1.08239
\(627\) 0 0
\(628\) −1.41421 −1.41421
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −1.08239 −1.08239
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) −4.82843 −4.82843
\(639\) 0 0
\(640\) 0 0
\(641\) 0.765367 0.765367 0.382683 0.923880i \(-0.375000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(648\) 0 0
\(649\) 2.61313 2.61313
\(650\) 0 0
\(651\) 0 0
\(652\) 0.765367 0.765367
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0.765367 0.765367
\(657\) −1.84776 −1.84776
\(658\) 0 0
\(659\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(662\) 2.00000 2.00000
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −1.41421 −1.41421
\(668\) 0.765367 0.765367
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0.765367 0.765367 0.382683 0.923880i \(-0.375000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 1.00000 1.00000
\(677\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) −4.82843 −4.82843
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 1.84776 1.84776 0.923880 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −0.585786 −0.585786
\(696\) 0 0
\(697\) 0.765367 0.765367
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 1.84776 1.84776
\(705\) 0 0
\(706\) 2.00000 2.00000
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −1.41421 −1.41421
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) −0.765367 −0.765367
\(721\) 0 0
\(722\) 1.41421 1.41421
\(723\) 0 0
\(724\) −0.765367 −0.765367
\(725\) −0.765367 −0.765367
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 1.00000 1.00000
\(730\) −2.00000 −2.00000
\(731\) 0 0
\(732\) 0 0
\(733\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(734\) −2.61313 −2.61313
\(735\) 0 0
\(736\) 1.08239 1.08239
\(737\) 0 0
\(738\) −1.08239 −1.08239
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0.765367 0.765367 0.382683 0.923880i \(-0.375000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(744\) 0 0
\(745\) −1.08239 −1.08239
\(746\) 0 0
\(747\) 0 0
\(748\) 1.84776 1.84776
\(749\) 0 0
\(750\) 0 0
\(751\) −0.765367 −0.765367 −0.382683 0.923880i \(-0.625000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(752\) 1.00000 1.00000
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 1.41421 1.41421
\(765\) −0.765367 −0.765367
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.84776 1.84776
\(773\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(774\) 0 0
\(775\) −0.765367 −0.765367
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 1.08239 1.08239
\(783\) 0 0
\(784\) −1.00000 −1.00000
\(785\) −1.08239 −1.08239
\(786\) 0 0
\(787\) 0.765367 0.765367 0.382683 0.923880i \(-0.375000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0.765367 0.765367
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 1.00000 1.00000
\(800\) 0.585786 0.585786
\(801\) 0 0
\(802\) 0 0
\(803\) 3.41421 3.41421
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(810\) 1.08239 1.08239
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0.585786 0.585786
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) −0.585786 −0.585786
\(821\) 1.84776 1.84776 0.923880 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(822\) 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 \(\pi\)
\(828\) −0.765367 −0.765367
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −1.00000 −1.00000
\(834\) 0 0
\(835\) 0.585786 0.585786
\(836\) 0 0
\(837\) 0 0
\(838\) 1.08239 1.08239
\(839\) 1.84776 1.84776 0.923880 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(840\) 0 0
\(841\) 2.41421 2.41421
\(842\) 0 0
\(843\) 0 0
\(844\) 0.765367 0.765367
\(845\) 0.765367 0.765367
\(846\) −1.41421 −1.41421
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0.585786 0.585786
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0.765367 0.765367 0.382683 0.923880i \(-0.375000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(864\) 0 0
\(865\) 0 0
\(866\) 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
\(875\) 0 0
\(876\) 0 0
\(877\) −0.765367 −0.765367 −0.382683 0.923880i \(-0.625000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 1.41421 1.41421
\(881\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(882\) 1.41421 1.41421
\(883\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −1.84776 −1.84776
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −1.08239 −1.08239
\(899\) 3.41421 3.41421
\(900\) −0.414214 −0.414214
\(901\) 0 0
\(902\) 2.00000 2.00000
\(903\) 0 0
\(904\) 0 0
\(905\) −0.585786 −0.585786
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) −0.765367 −0.765367
\(909\) 1.41421 1.41421
\(910\) 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
\(925\) 0 0
\(926\) 0 0
\(927\) 1.41421 1.41421
\(928\) −2.61313 −2.61313
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −1.84776 −1.84776
\(933\) 0 0
\(934\) 0 0
\(935\) 1.41421 1.41421
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −0.765367 −0.765367
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0.585786 0.585786
\(944\) 1.41421 1.41421
\(945\) 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
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 1.08239 1.08239
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 2.41421 2.41421
\(962\) 0 0
\(963\) 1.84776 1.84776
\(964\) 0 0
\(965\) 1.41421 1.41421
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 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
\(977\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0.765367 0.765367
\(981\) −1.84776 −1.84776
\(982\) −2.82843 −2.82843
\(983\) 1.84776 1.84776 0.923880 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −2.61313 −2.61313
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) −2.00000 −2.00000
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) −2.61313 −2.61313
\(993\) 0 0
\(994\) 0 0
\(995\) 0.585786 0.585786
\(996\) 0 0
\(997\) 1.84776 1.84776 0.923880 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(998\) −1.08239 −1.08239
\(999\) 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 799.1.c.c.798.4 yes 4
17.16 even 2 inner 799.1.c.c.798.3 4
47.46 odd 2 inner 799.1.c.c.798.3 4
799.798 odd 2 CM 799.1.c.c.798.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
799.1.c.c.798.3 4 17.16 even 2 inner
799.1.c.c.798.3 4 47.46 odd 2 inner
799.1.c.c.798.4 yes 4 1.1 even 1 trivial
799.1.c.c.798.4 yes 4 799.798 odd 2 CM