Properties

Label 2448.1.de.a.2095.2
Level $2448$
Weight $1$
Character 2448.2095
Analytic conductor $1.222$
Analytic rank $0$
Dimension $8$
Projective image $S_{4}$
CM/RM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2448,1,Mod(319,2448)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2448, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 0, 4, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2448.319");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2448 = 2^{4} \cdot 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2448.de (of order \(12\), degree \(4\), 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: no
Analytic conductor: \(1.22171115093\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.2.1591812.1

Embedding invariants

Embedding label 2095.2
Root \(-0.258819 + 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 2448.2095
Dual form 2448.1.de.a.319.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.258819 - 0.965926i) q^{3} +(-0.866025 - 0.500000i) q^{9} +O(q^{10})\) \(q+(0.258819 - 0.965926i) q^{3} +(-0.866025 - 0.500000i) q^{9} +(0.258819 + 0.965926i) q^{11} +(-0.500000 - 0.866025i) q^{13} -1.00000i q^{17} +1.41421 q^{19} +(0.965926 + 0.258819i) q^{23} +(-0.866025 - 0.500000i) q^{25} +(-0.707107 + 0.707107i) q^{27} +(-0.366025 - 1.36603i) q^{29} +(0.258819 - 0.965926i) q^{31} +1.00000 q^{33} +(-0.965926 + 0.258819i) q^{39} +(0.707107 - 1.22474i) q^{43} +(-1.22474 - 0.707107i) q^{47} +(0.866025 - 0.500000i) q^{49} +(-0.965926 - 0.258819i) q^{51} +1.00000i q^{53} +(0.366025 - 1.36603i) q^{57} +(0.707107 + 1.22474i) q^{59} +(-1.36603 + 0.366025i) q^{61} +(0.500000 - 0.866025i) q^{69} +(0.707107 + 0.707107i) q^{71} +(-0.707107 + 0.707107i) q^{75} +(-0.258819 - 0.965926i) q^{79} +(0.500000 + 0.866025i) q^{81} +(-0.707107 + 1.22474i) q^{83} -1.41421 q^{87} +1.00000 q^{89} +(-0.866025 - 0.500000i) q^{93} +(0.258819 - 0.965926i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{13} + 4 q^{29} + 8 q^{33} - 4 q^{57} - 4 q^{61} + 4 q^{69} + 4 q^{81} + 8 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2448\mathbb{Z}\right)^\times\).

\(n\) \(613\) \(1361\) \(1873\) \(2143\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{3}{4}\right)\) \(-1\)

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
\(3\) 0.258819 0.965926i 0.258819 0.965926i
\(4\) 0 0
\(5\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(6\) 0 0
\(7\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(8\) 0 0
\(9\) −0.866025 0.500000i −0.866025 0.500000i
\(10\) 0 0
\(11\) 0.258819 + 0.965926i 0.258819 + 0.965926i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(12\) 0 0
\(13\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.00000i 1.00000i
\(18\) 0 0
\(19\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.965926 + 0.258819i 0.965926 + 0.258819i 0.707107 0.707107i \(-0.250000\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(24\) 0 0
\(25\) −0.866025 0.500000i −0.866025 0.500000i
\(26\) 0 0
\(27\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(28\) 0 0
\(29\) −0.366025 1.36603i −0.366025 1.36603i −0.866025 0.500000i \(-0.833333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(30\) 0 0
\(31\) 0.258819 0.965926i 0.258819 0.965926i −0.707107 0.707107i \(-0.750000\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(32\) 0 0
\(33\) 1.00000 1.00000
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(38\) 0 0
\(39\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(40\) 0 0
\(41\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(42\) 0 0
\(43\) 0.707107 1.22474i 0.707107 1.22474i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.22474 0.707107i −1.22474 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(48\) 0 0
\(49\) 0.866025 0.500000i 0.866025 0.500000i
\(50\) 0 0
\(51\) −0.965926 0.258819i −0.965926 0.258819i
\(52\) 0 0
\(53\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0.366025 1.36603i 0.366025 1.36603i
\(58\) 0 0
\(59\) 0.707107 + 1.22474i 0.707107 + 1.22474i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(60\) 0 0
\(61\) −1.36603 + 0.366025i −1.36603 + 0.366025i −0.866025 0.500000i \(-0.833333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(68\) 0 0
\(69\) 0.500000 0.866025i 0.500000 0.866025i
\(70\) 0 0
\(71\) 0.707107 + 0.707107i 0.707107 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(72\) 0 0
\(73\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(74\) 0 0
\(75\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −0.258819 0.965926i −0.258819 0.965926i −0.965926 0.258819i \(-0.916667\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(80\) 0 0
\(81\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(82\) 0 0
\(83\) −0.707107 + 1.22474i −0.707107 + 1.22474i 0.258819 + 0.965926i \(0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −1.41421 −1.41421
\(88\) 0 0
\(89\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −0.866025 0.500000i −0.866025 0.500000i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(98\) 0 0
\(99\) 0.258819 0.965926i 0.258819 0.965926i
\(100\) 0 0
\(101\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −1.22474 + 0.707107i −1.22474 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(108\) 0 0
\(109\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.00000i 1.00000i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) −1.00000 1.00000i −1.00000 1.00000i
\(130\) 0 0
\(131\) −0.258819 + 0.965926i −0.258819 + 0.965926i 0.707107 + 0.707107i \(0.250000\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(138\) 0 0
\(139\) −0.258819 + 0.965926i −0.258819 + 0.965926i 0.707107 + 0.707107i \(0.250000\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(140\) 0 0
\(141\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(142\) 0 0
\(143\) 0.707107 0.707107i 0.707107 0.707107i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −0.258819 0.965926i −0.258819 0.965926i
\(148\) 0 0
\(149\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(150\) 0 0
\(151\) 0.707107 1.22474i 0.707107 1.22474i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(152\) 0 0
\(153\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0.707107 0.707107i 0.707107 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) −1.22474 0.707107i −1.22474 0.707107i
\(172\) 0 0
\(173\) 0.366025 + 1.36603i 0.366025 + 1.36603i 0.866025 + 0.500000i \(0.166667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1.36603 0.366025i 1.36603 0.366025i
\(178\) 0 0
\(179\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(180\) 0 0
\(181\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(182\) 0 0
\(183\) 1.41421i 1.41421i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0.965926 0.258819i 0.965926 0.258819i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1.22474 + 0.707107i 1.22474 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(192\) 0 0
\(193\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −0.707107 0.707107i −0.707107 0.707107i
\(208\) 0 0
\(209\) 0.366025 + 1.36603i 0.366025 + 1.36603i
\(210\) 0 0
\(211\) 1.93185 + 0.517638i 1.93185 + 0.517638i 0.965926 + 0.258819i \(0.0833333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(212\) 0 0
\(213\) 0.866025 0.500000i 0.866025 0.500000i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(222\) 0 0
\(223\) −0.707107 + 1.22474i −0.707107 + 1.22474i 0.258819 + 0.965926i \(0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(224\) 0 0
\(225\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(226\) 0 0
\(227\) 0.258819 + 0.965926i 0.258819 + 0.965926i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(228\) 0 0
\(229\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −1.00000 −1.00000
\(238\) 0 0
\(239\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(240\) 0 0
\(241\) 0.366025 + 1.36603i 0.366025 + 1.36603i 0.866025 + 0.500000i \(0.166667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(242\) 0 0
\(243\) 0.965926 0.258819i 0.965926 0.258819i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −0.707107 1.22474i −0.707107 1.22474i
\(248\) 0 0
\(249\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 1.00000i 1.00000i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −0.366025 + 1.36603i −0.366025 + 1.36603i
\(262\) 0 0
\(263\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0.258819 0.965926i 0.258819 0.965926i
\(268\) 0 0
\(269\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0.258819 0.965926i 0.258819 0.965926i
\(276\) 0 0
\(277\) 0.366025 + 1.36603i 0.366025 + 1.36603i 0.866025 + 0.500000i \(0.166667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(278\) 0 0
\(279\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(280\) 0 0
\(281\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(282\) 0 0
\(283\) −0.965926 0.258819i −0.965926 0.258819i −0.258819 0.965926i \(-0.583333\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1.00000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −0.866025 0.500000i −0.866025 0.500000i
\(298\) 0 0
\(299\) −0.258819 0.965926i −0.258819 0.965926i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0.366025 + 1.36603i 0.366025 + 1.36603i
\(310\) 0 0
\(311\) −0.258819 + 0.965926i −0.258819 + 0.965926i 0.707107 + 0.707107i \(0.250000\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(312\) 0 0
\(313\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.36603 0.366025i 1.36603 0.366025i 0.500000 0.866025i \(-0.333333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(318\) 0 0
\(319\) 1.22474 0.707107i 1.22474 0.707107i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1.41421i 1.41421i
\(324\) 0 0
\(325\) 1.00000i 1.00000i
\(326\) 0 0
\(327\) 1.22474 0.707107i 1.22474 0.707107i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 1.00000 1.00000
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(348\) 0 0
\(349\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(350\) 0 0
\(351\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(352\) 0 0
\(353\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(360\) 0 0
\(361\) 1.00000 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0.258819 + 0.965926i 0.258819 + 0.965926i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(378\) 0 0
\(379\) 0.707107 + 0.707107i 0.707107 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(388\) 0 0
\(389\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0.258819 0.965926i 0.258819 0.965926i
\(392\) 0 0
\(393\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −0.366025 + 1.36603i −0.366025 + 1.36603i 0.500000 + 0.866025i \(0.333333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(402\) 0 0
\(403\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) −0.707107 0.707107i −0.707107 0.707107i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(418\) 0 0
\(419\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(420\) 0 0
\(421\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(424\) 0 0
\(425\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −0.500000 0.866025i −0.500000 0.866025i
\(430\) 0 0
\(431\) −0.707107 + 0.707107i −0.707107 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(432\) 0 0
\(433\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.36603 + 0.366025i 1.36603 + 0.366025i
\(438\) 0 0
\(439\) 0.965926 0.258819i 0.965926 0.258819i 0.258819 0.965926i \(-0.416667\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(440\) 0 0
\(441\) −1.00000 −1.00000
\(442\) 0 0
\(443\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −1.00000 1.00000i −1.00000 1.00000i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(458\) 0 0
\(459\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(460\) 0 0
\(461\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(462\) 0 0
\(463\) −1.22474 + 0.707107i −1.22474 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(472\) 0 0
\(473\) 1.36603 + 0.366025i 1.36603 + 0.366025i
\(474\) 0 0
\(475\) −1.22474 0.707107i −1.22474 0.707107i
\(476\) 0 0
\(477\) 0.500000 0.866025i 0.500000 0.866025i
\(478\) 0 0
\(479\) 1.93185 0.517638i 1.93185 0.517638i 0.965926 0.258819i \(-0.0833333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −0.707107 + 0.707107i −0.707107 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(488\) 0 0
\(489\) −0.500000 0.866025i −0.500000 0.866025i
\(490\) 0 0
\(491\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(492\) 0 0
\(493\) −1.36603 + 0.366025i −1.36603 + 0.366025i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0.707107 0.707107i 0.707107 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0.366025 1.36603i 0.366025 1.36603i
\(518\) 0 0
\(519\) 1.41421 1.41421
\(520\) 0 0
\(521\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −0.965926 0.258819i −0.965926 0.258819i
\(528\) 0 0
\(529\) 0 0
\(530\) 0 0
\(531\) 1.41421i 1.41421i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −0.366025 + 1.36603i −0.366025 + 1.36603i
\(538\) 0 0
\(539\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(540\) 0 0
\(541\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −0.965926 + 0.258819i −0.965926 + 0.258819i −0.707107 0.707107i \(-0.750000\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(548\) 0 0
\(549\) 1.36603 + 0.366025i 1.36603 + 0.366025i
\(550\) 0 0
\(551\) −0.517638 1.93185i −0.517638 1.93185i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(558\) 0 0
\(559\) −1.41421 −1.41421
\(560\) 0 0
\(561\) 1.00000i 1.00000i
\(562\) 0 0
\(563\) −0.707107 1.22474i −0.707107 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(570\) 0 0
\(571\) 0.965926 + 0.258819i 0.965926 + 0.258819i 0.707107 0.707107i \(-0.250000\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(572\) 0 0
\(573\) 1.00000 1.00000i 1.00000 1.00000i
\(574\) 0 0
\(575\) −0.707107 0.707107i −0.707107 0.707107i
\(576\) 0 0
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −0.707107 + 1.22474i −0.707107 + 1.22474i 0.258819 + 0.965926i \(0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(588\) 0 0
\(589\) 0.366025 1.36603i 0.366025 1.36603i
\(590\) 0 0
\(591\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(592\) 0 0
\(593\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(600\) 0 0
\(601\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −0.258819 + 0.965926i −0.258819 + 0.965926i 0.707107 + 0.707107i \(0.250000\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.41421i 1.41421i
\(612\) 0 0
\(613\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1.36603 0.366025i −1.36603 0.366025i −0.500000 0.866025i \(-0.666667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(618\) 0 0
\(619\) 0.258819 + 0.965926i 0.258819 + 0.965926i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(620\) 0 0
\(621\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(626\) 0 0
\(627\) 1.41421 1.41421
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(632\) 0 0
\(633\) 1.00000 1.73205i 1.00000 1.73205i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −0.866025 0.500000i −0.866025 0.500000i
\(638\) 0 0
\(639\) −0.258819 0.965926i −0.258819 0.965926i
\(640\) 0 0
\(641\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(642\) 0 0
\(643\) −0.258819 + 0.965926i −0.258819 + 0.965926i 0.707107 + 0.707107i \(0.250000\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(648\) 0 0
\(649\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −1.22474 0.707107i −1.22474 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(660\) 0 0
\(661\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(662\) 0 0
\(663\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1.41421i 1.41421i
\(668\) 0 0
\(669\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(670\) 0 0
\(671\) −0.707107 1.22474i −0.707107 1.22474i
\(672\) 0 0
\(673\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(674\) 0 0
\(675\) 0.965926 0.258819i 0.965926 0.258819i
\(676\) 0 0
\(677\) −0.366025 1.36603i −0.366025 1.36603i −0.866025 0.500000i \(-0.833333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 1.00000 1.00000
\(682\) 0 0
\(683\) 0.707107 + 0.707107i 0.707107 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −0.258819 0.965926i −0.258819 0.965926i
\(688\) 0 0
\(689\) 0.866025 0.500000i 0.866025 0.500000i
\(690\) 0 0
\(691\) 0.258819 + 0.965926i 0.258819 + 0.965926i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(692\) 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
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −0.366025 1.36603i −0.366025 1.36603i −0.866025 0.500000i \(-0.833333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(710\) 0 0
\(711\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(712\) 0 0
\(713\) 0.500000 0.866025i 0.500000 0.866025i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −0.707107 0.707107i −0.707107 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 1.41421 1.41421
\(724\) 0 0
\(725\) −0.366025 + 1.36603i −0.366025 + 1.36603i
\(726\) 0 0
\(727\) 1.22474 + 0.707107i 1.22474 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(728\) 0 0
\(729\) 1.00000i 1.00000i
\(730\) 0 0
\(731\) −1.22474 0.707107i −1.22474 0.707107i
\(732\) 0 0
\(733\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) 0 0
\(741\) −1.36603 + 0.366025i −1.36603 + 0.366025i
\(742\) 0 0
\(743\) 0.517638 1.93185i 0.517638 1.93185i 0.258819 0.965926i \(-0.416667\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 1.22474 0.707107i 1.22474 0.707107i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −0.258819 + 0.965926i −0.258819 + 0.965926i 0.707107 + 0.707107i \(0.250000\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(758\) 0 0
\(759\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(760\) 0 0
\(761\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0.707107 1.22474i 0.707107 1.22474i
\(768\) 0 0
\(769\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(770\) 0 0
\(771\) −0.258819 0.965926i −0.258819 0.965926i
\(772\) 0 0
\(773\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(774\) 0 0
\(775\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(782\) 0 0
\(783\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(798\) 0 0
\(799\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(800\) 0 0
\(801\) −0.866025 0.500000i −0.866025 0.500000i
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(808\) 0 0
\(809\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 1.00000 1.73205i 1.00000 1.73205i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0.366025 + 1.36603i 0.366025 + 1.36603i 0.866025 + 0.500000i \(0.166667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(822\) 0 0
\(823\) 0.965926 + 0.258819i 0.965926 + 0.258819i 0.707107 0.707107i \(-0.250000\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(824\) 0 0
\(825\) −0.866025 0.500000i −0.866025 0.500000i
\(826\) 0 0
\(827\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(828\) 0 0
\(829\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(830\) 0 0
\(831\) 1.41421 1.41421
\(832\) 0 0
\(833\) −0.500000 0.866025i −0.500000 0.866025i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(838\) 0 0
\(839\) −0.258819 0.965926i −0.258819 0.965926i −0.965926 0.258819i \(-0.916667\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(840\) 0 0
\(841\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(842\) 0 0
\(843\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1.36603 0.366025i 1.36603 0.366025i 0.500000 0.866025i \(-0.333333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(858\) 0 0
\(859\) −0.707107 1.22474i −0.707107 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(868\) 0 0
\(869\) 0.866025 0.500000i 0.866025 0.500000i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0.366025 1.36603i 0.366025 1.36603i −0.500000 0.866025i \(-0.666667\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(878\) 0 0
\(879\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(880\) 0 0
\(881\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(882\) 0 0
\(883\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(892\) 0 0
\(893\) −1.73205 1.00000i −1.73205 1.00000i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −1.00000 −1.00000
\(898\) 0 0
\(899\) −1.41421 −1.41421
\(900\) 0 0
\(901\) 1.00000 1.00000
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(908\) 0 0
\(909\) 0.866025 0.500000i 0.866025 0.500000i
\(910\) 0 0
\(911\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(912\) 0 0
\(913\) −1.36603 0.366025i −1.36603 0.366025i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0.258819 0.965926i 0.258819 0.965926i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 1.41421 1.41421
\(928\) 0 0
\(929\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(930\) 0 0
\(931\) 1.22474 0.707107i 1.22474 0.707107i
\(932\) 0 0
\(933\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1.93185 0.517638i 1.93185 0.517638i 0.965926 0.258819i \(-0.0833333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 1.41421i 1.41421i
\(952\) 0 0
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −0.366025 1.36603i −0.366025 1.36603i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 0 0
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −0.707107 1.22474i −0.707107 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(968\) 0 0
\(969\) −1.36603 0.366025i −1.36603 0.366025i
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(976\) 0 0
\(977\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(978\) 0 0
\(979\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(980\) 0 0
\(981\) −0.366025 1.36603i −0.366025 1.36603i
\(982\) 0 0
\(983\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1.00000 1.00000i 1.00000 1.00000i
\(990\) 0 0
\(991\) −0.707107 0.707107i −0.707107 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(998\) 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 2448.1.de.a.2095.2 yes 8
4.3 odd 2 inner 2448.1.de.a.2095.1 yes 8
9.4 even 3 inner 2448.1.de.a.463.1 yes 8
17.13 even 4 inner 2448.1.de.a.1951.2 yes 8
36.31 odd 6 inner 2448.1.de.a.463.2 yes 8
68.47 odd 4 inner 2448.1.de.a.1951.1 yes 8
153.13 even 12 inner 2448.1.de.a.319.1 8
612.319 odd 12 inner 2448.1.de.a.319.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2448.1.de.a.319.1 8 153.13 even 12 inner
2448.1.de.a.319.2 yes 8 612.319 odd 12 inner
2448.1.de.a.463.1 yes 8 9.4 even 3 inner
2448.1.de.a.463.2 yes 8 36.31 odd 6 inner
2448.1.de.a.1951.1 yes 8 68.47 odd 4 inner
2448.1.de.a.1951.2 yes 8 17.13 even 4 inner
2448.1.de.a.2095.1 yes 8 4.3 odd 2 inner
2448.1.de.a.2095.2 yes 8 1.1 even 1 trivial