Properties

Label 1792.1.o.a.1663.1
Level $1792$
Weight $1$
Character 1792.1663
Analytic conductor $0.894$
Analytic rank $0$
Dimension $4$
Projective image $A_{4}$
CM/RM no
Inner twists $4$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1792,1,Mod(639,1792)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1792, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1792.639");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1792.o (of order \(6\), degree \(2\), 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: no
Analytic conductor: \(0.894324502638\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 224)
Projective image: \(A_{4}\)
Projective field: Galois closure of 4.0.3136.1

Embedding invariants

Embedding label 1663.1
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1792.1663
Dual form 1792.1.o.a.639.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.500000 + 0.866025i) q^{3} +(-0.866025 + 0.500000i) q^{5} -1.00000i q^{7} +O(q^{10})\) \(q+(-0.500000 + 0.866025i) q^{3} +(-0.866025 + 0.500000i) q^{5} -1.00000i q^{7} +(-0.500000 + 0.866025i) q^{11} -1.00000i q^{15} +(-0.500000 + 0.866025i) q^{17} +(-0.500000 - 0.866025i) q^{19} +(0.866025 + 0.500000i) q^{21} +(-0.866025 + 0.500000i) q^{23} -1.00000 q^{27} +(-0.866025 - 0.500000i) q^{31} +(-0.500000 - 0.866025i) q^{33} +(0.500000 + 0.866025i) q^{35} +(0.866025 - 0.500000i) q^{37} +(-0.866025 + 0.500000i) q^{47} -1.00000 q^{49} +(-0.500000 - 0.866025i) q^{51} +(-0.866025 - 0.500000i) q^{53} -1.00000i q^{55} +1.00000 q^{57} +(0.500000 - 0.866025i) q^{59} +(-0.866025 + 0.500000i) q^{61} +(-0.500000 + 0.866025i) q^{67} -1.00000i q^{69} +(-0.500000 + 0.866025i) q^{73} +(0.866025 + 0.500000i) q^{77} +(0.866025 - 0.500000i) q^{79} +(0.500000 - 0.866025i) q^{81} -1.00000i q^{85} +(-0.500000 - 0.866025i) q^{89} +(0.866025 - 0.500000i) q^{93} +(0.866025 + 0.500000i) q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} - 2 q^{11} - 2 q^{17} - 2 q^{19} - 4 q^{27} - 2 q^{33} + 2 q^{35} - 4 q^{49} - 2 q^{51} + 4 q^{57} + 2 q^{59} - 2 q^{67} - 2 q^{73} + 2 q^{81} - 2 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(-1\)

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
\(3\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(4\) 0 0
\(5\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 1.00000i 1.00000i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 1.00000i 1.00000i
\(16\) 0 0
\(17\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(22\) 0 0
\(23\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.00000 −1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(32\) 0 0
\(33\) −0.500000 0.866025i −0.500000 0.866025i
\(34\) 0 0
\(35\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(36\) 0 0
\(37\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −1.00000 −1.00000
\(50\) 0 0
\(51\) −0.500000 0.866025i −0.500000 0.866025i
\(52\) 0 0
\(53\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(54\) 0 0
\(55\) 1.00000i 1.00000i
\(56\) 0 0
\(57\) 1.00000 1.00000
\(58\) 0 0
\(59\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(60\) 0 0
\(61\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 1.00000i 1.00000i
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(78\) 0 0
\(79\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(80\) 0 0
\(81\) 0.500000 0.866025i 0.500000 0.866025i
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 1.00000i 1.00000i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0.866025 0.500000i 0.866025 0.500000i
\(94\) 0 0
\(95\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(96\) 0 0
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) −1.00000 −1.00000
\(106\) 0 0
\(107\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(108\) 0 0
\(109\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(110\) 0 0
\(111\) 1.00000i 1.00000i
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 0.500000 0.866025i 0.500000 0.866025i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000i 1.00000i
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(132\) 0 0
\(133\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(134\) 0 0
\(135\) 0.866025 0.500000i 0.866025 0.500000i
\(136\) 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 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 1.00000i 1.00000i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0.500000 0.866025i 0.500000 0.866025i
\(148\) 0 0
\(149\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1.00000 1.00000
\(156\) 0 0
\(157\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(158\) 0 0
\(159\) 0.866025 0.500000i 0.866025 0.500000i
\(160\) 0 0
\(161\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(162\) 0 0
\(163\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(164\) 0 0
\(165\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(166\) 0 0
\(167\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 1.00000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(178\) 0 0
\(179\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(180\) 0 0
\(181\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 1.00000i 1.00000i
\(184\) 0 0
\(185\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(186\) 0 0
\(187\) −0.500000 0.866025i −0.500000 0.866025i
\(188\) 0 0
\(189\) 1.00000i 1.00000i
\(190\) 0 0
\(191\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) −0.500000 0.866025i −0.500000 0.866025i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1.00000 1.00000
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(218\) 0 0
\(219\) −0.500000 0.866025i −0.500000 0.866025i
\(220\) 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
\(227\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(228\) 0 0
\(229\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(232\) 0 0
\(233\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(234\) 0 0
\(235\) 0.500000 0.866025i 0.500000 0.866025i
\(236\) 0 0
\(237\) 1.00000i 1.00000i
\(238\) 0 0
\(239\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0.866025 0.500000i 0.866025 0.500000i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(252\) 0 0
\(253\) 1.00000i 1.00000i
\(254\) 0 0
\(255\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(256\) 0 0
\(257\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) −0.500000 0.866025i −0.500000 0.866025i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(264\) 0 0
\(265\) 1.00000 1.00000
\(266\) 0 0
\(267\) 1.00000 1.00000
\(268\) 0 0
\(269\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 0 0
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(294\) 0 0
\(295\) 1.00000i 1.00000i
\(296\) 0 0
\(297\) 0.500000 0.866025i 0.500000 0.866025i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(304\) 0 0
\(305\) 0.500000 0.866025i 0.500000 0.866025i
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 1.00000i 1.00000i
\(310\) 0 0
\(311\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(312\) 0 0
\(313\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −1.00000 −1.00000
\(322\) 0 0
\(323\) 1.00000 1.00000
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0.866025 0.500000i 0.866025 0.500000i
\(328\) 0 0
\(329\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(330\) 0 0
\(331\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1.00000i 1.00000i
\(336\) 0 0
\(337\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0.866025 0.500000i 0.866025 0.500000i
\(342\) 0 0
\(343\) 1.00000i 1.00000i
\(344\) 0 0
\(345\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(346\) 0 0
\(347\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(358\) 0 0
\(359\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1.00000i 1.00000i
\(366\) 0 0
\(367\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(372\) 0 0
\(373\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(374\) 0 0
\(375\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) −1.00000 −1.00000
\(386\) 0 0
\(387\) 0 0
\(388\) 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
\(391\) 1.00000i 1.00000i
\(392\) 0 0
\(393\) −1.00000 −1.00000
\(394\) 0 0
\(395\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(396\) 0 0
\(397\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(398\) 0 0
\(399\) 1.00000i 1.00000i
\(400\) 0 0
\(401\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 1.00000i 1.00000i
\(406\) 0 0
\(407\) 1.00000i 1.00000i
\(408\) 0 0
\(409\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(410\) 0 0
\(411\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(412\) 0 0
\(413\) −0.866025 0.500000i −0.866025 0.500000i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(438\) 0 0
\(439\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(444\) 0 0
\(445\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(446\) 0 0
\(447\) 1.00000i 1.00000i
\(448\) 0 0
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0.500000 0.866025i 0.500000 0.866025i
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(466\) 0 0
\(467\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(468\) 0 0
\(469\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(470\) 0 0
\(471\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −1.00000 −1.00000
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(488\) 0 0
\(489\) −1.00000 −1.00000
\(490\) 0 0
\(491\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(500\) 0 0
\(501\) −1.73205 1.00000i −1.73205 1.00000i
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) −1.00000 −1.00000
\(506\) 0 0
\(507\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(508\) 0 0
\(509\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(510\) 0 0
\(511\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(512\) 0 0
\(513\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(514\) 0 0
\(515\) 0.500000 0.866025i 0.500000 0.866025i
\(516\) 0 0
\(517\) 1.00000i 1.00000i
\(518\) 0 0
\(519\) 1.00000i 1.00000i
\(520\) 0 0
\(521\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0.866025 0.500000i 0.866025 0.500000i
\(528\) 0 0
\(529\) 0 0
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −0.866025 0.500000i −0.866025 0.500000i
\(536\) 0 0
\(537\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(538\) 0 0
\(539\) 0.500000 0.866025i 0.500000 0.866025i
\(540\) 0 0
\(541\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(542\) 0 0
\(543\) −1.73205 1.00000i −1.73205 1.00000i
\(544\) 0 0
\(545\) 1.00000 1.00000
\(546\) 0 0
\(547\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −0.500000 0.866025i −0.500000 0.866025i
\(554\) 0 0
\(555\) −0.500000 0.866025i −0.500000 0.866025i
\(556\) 0 0
\(557\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 1.00000 1.00000
\(562\) 0 0
\(563\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −0.866025 0.500000i −0.866025 0.500000i
\(568\) 0 0
\(569\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 1.00000i 1.00000i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(578\) 0 0
\(579\) −0.500000 0.866025i −0.500000 0.866025i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0.866025 0.500000i 0.866025 0.500000i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 1.00000i 1.00000i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) −1.00000 −1.00000
\(596\) 0 0
\(597\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(598\) 0 0
\(599\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(620\) 0 0
\(621\) 0.866025 0.500000i 0.866025 0.500000i
\(622\) 0 0
\(623\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(624\) 0 0
\(625\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(626\) 0 0
\(627\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(628\) 0 0
\(629\) 1.00000i 1.00000i
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(642\) 0 0
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(648\) 0 0
\(649\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(650\) 0 0
\(651\) −0.500000 0.866025i −0.500000 0.866025i
\(652\) 0 0
\(653\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) −0.866025 0.500000i −0.866025 0.500000i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0.500000 0.866025i 0.500000 0.866025i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1.00000i 1.00000i
\(672\) 0 0
\(673\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(682\) 0 0
\(683\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 1.00000i 1.00000i
\(686\) 0 0
\(687\) 1.00000i 1.00000i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −1.00000 −1.00000
\(700\) 0 0
\(701\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) −0.866025 0.500000i −0.866025 0.500000i
\(704\) 0 0
\(705\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(706\) 0 0
\(707\) 0.500000 0.866025i 0.500000 0.866025i
\(708\) 0 0
\(709\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1.00000 1.00000
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −1.73205 1.00000i −1.73205 1.00000i
\(718\) 0 0
\(719\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(720\) 0 0
\(721\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(722\) 0 0
\(723\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 1.00000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 1.00000i 1.00000i
\(736\) 0 0
\(737\) −0.500000 0.866025i −0.500000 0.866025i
\(738\) 0 0
\(739\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0.500000 0.866025i 0.500000 0.866025i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0.866025 0.500000i 0.866025 0.500000i
\(750\) 0 0
\(751\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(754\) 0 0
\(755\) −1.00000 −1.00000
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(760\) 0 0
\(761\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(762\) 0 0
\(763\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) 1.00000 1.00000
\(772\) 0 0
\(773\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 1.00000 1.00000
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −1.00000 −1.00000
\(786\) 0 0
\(787\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(788\) 0 0
\(789\) 0.866025 0.500000i 0.866025 0.500000i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 1.00000i 1.00000i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −0.500000 0.866025i −0.500000 0.866025i
\(804\) 0 0
\(805\) −0.866025 0.500000i −0.866025 0.500000i
\(806\) 0 0
\(807\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(808\) 0 0
\(809\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 1.00000i 1.00000i
\(814\) 0 0
\(815\) −0.866025 0.500000i −0.866025 0.500000i
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(830\) 0 0
\(831\) 0.866025 0.500000i 0.866025 0.500000i
\(832\) 0 0
\(833\) 0.500000 0.866025i 0.500000 0.866025i
\(834\) 0 0
\(835\) −1.00000 1.73205i −1.00000 1.73205i
\(836\) 0 0
\(837\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 1.00000 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −0.500000 0.866025i −0.500000 0.866025i
\(850\) 0 0
\(851\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(864\) 0 0
\(865\) 0.500000 0.866025i 0.500000 0.866025i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1.00000i 1.00000i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −1.00000 −1.00000
\(876\) 0 0
\(877\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(878\) 0 0
\(879\) 1.73205 + 1.00000i 1.73205 + 1.00000i
\(880\) 0 0
\(881\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) 0 0
\(885\) −0.866025 0.500000i −0.866025 0.500000i
\(886\) 0 0
\(887\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(892\) 0 0
\(893\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(894\) 0 0
\(895\) 1.00000i 1.00000i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0.866025 0.500000i 0.866025 0.500000i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1.00000 1.73205i −1.00000 1.73205i
\(906\) 0 0
\(907\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(916\) 0 0
\(917\) 0.866025 0.500000i 0.866025 0.500000i
\(918\) 0 0
\(919\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(930\) 0 0
\(931\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(932\) 0 0
\(933\) 0.866025 0.500000i 0.866025 0.500000i
\(934\) 0 0
\(935\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(936\) 0 0
\(937\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 1.00000 1.00000
\(940\) 0 0
\(941\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −0.500000 0.866025i −0.500000 0.866025i
\(946\) 0 0
\(947\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 1.00000i 1.00000i
\(952\) 0 0
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) 0.500000 0.866025i 0.500000 0.866025i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −0.866025 0.500000i −0.866025 0.500000i
\(960\) 0 0
\(961\) 0 0
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 1.00000i 1.00000i
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(970\) 0 0
\(971\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 1.00000 1.00000
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −1.00000 −1.00000
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) 1.00000 1.00000
\(994\) 0 0
\(995\) −1.00000 −1.00000
\(996\) 0 0
\(997\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(998\) 0 0
\(999\) −0.866025 + 0.500000i −0.866025 + 0.500000i
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 1792.1.o.a.1663.1 4
4.3 odd 2 1792.1.o.b.1663.1 4
7.2 even 3 inner 1792.1.o.a.639.2 4
8.3 odd 2 inner 1792.1.o.a.1663.2 4
8.5 even 2 1792.1.o.b.1663.2 4
16.3 odd 4 448.1.r.a.319.2 4
16.5 even 4 224.1.r.a.95.2 yes 4
16.11 odd 4 224.1.r.a.95.1 4
16.13 even 4 448.1.r.a.319.1 4
28.23 odd 6 1792.1.o.b.639.2 4
48.5 odd 4 2016.1.cd.a.991.2 4
48.11 even 4 2016.1.cd.a.991.1 4
56.37 even 6 1792.1.o.b.639.1 4
56.51 odd 6 inner 1792.1.o.a.639.1 4
112.3 even 12 3136.1.d.d.1471.2 2
112.5 odd 12 1568.1.r.a.863.2 4
112.11 odd 12 1568.1.d.b.1471.2 2
112.13 odd 4 3136.1.r.b.3007.2 4
112.19 even 12 3136.1.r.b.2431.2 4
112.27 even 4 1568.1.r.a.1439.2 4
112.37 even 12 224.1.r.a.191.1 yes 4
112.45 odd 12 3136.1.d.d.1471.1 2
112.51 odd 12 448.1.r.a.191.1 4
112.53 even 12 1568.1.d.b.1471.1 2
112.59 even 12 1568.1.d.a.1471.1 2
112.61 odd 12 3136.1.r.b.2431.1 4
112.67 odd 12 3136.1.d.b.1471.1 2
112.69 odd 4 1568.1.r.a.1439.1 4
112.75 even 12 1568.1.r.a.863.1 4
112.83 even 4 3136.1.r.b.3007.1 4
112.93 even 12 448.1.r.a.191.2 4
112.101 odd 12 1568.1.d.a.1471.2 2
112.107 odd 12 224.1.r.a.191.2 yes 4
112.109 even 12 3136.1.d.b.1471.2 2
336.107 even 12 2016.1.cd.a.415.1 4
336.149 odd 12 2016.1.cd.a.415.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
224.1.r.a.95.1 4 16.11 odd 4
224.1.r.a.95.2 yes 4 16.5 even 4
224.1.r.a.191.1 yes 4 112.37 even 12
224.1.r.a.191.2 yes 4 112.107 odd 12
448.1.r.a.191.1 4 112.51 odd 12
448.1.r.a.191.2 4 112.93 even 12
448.1.r.a.319.1 4 16.13 even 4
448.1.r.a.319.2 4 16.3 odd 4
1568.1.d.a.1471.1 2 112.59 even 12
1568.1.d.a.1471.2 2 112.101 odd 12
1568.1.d.b.1471.1 2 112.53 even 12
1568.1.d.b.1471.2 2 112.11 odd 12
1568.1.r.a.863.1 4 112.75 even 12
1568.1.r.a.863.2 4 112.5 odd 12
1568.1.r.a.1439.1 4 112.69 odd 4
1568.1.r.a.1439.2 4 112.27 even 4
1792.1.o.a.639.1 4 56.51 odd 6 inner
1792.1.o.a.639.2 4 7.2 even 3 inner
1792.1.o.a.1663.1 4 1.1 even 1 trivial
1792.1.o.a.1663.2 4 8.3 odd 2 inner
1792.1.o.b.639.1 4 56.37 even 6
1792.1.o.b.639.2 4 28.23 odd 6
1792.1.o.b.1663.1 4 4.3 odd 2
1792.1.o.b.1663.2 4 8.5 even 2
2016.1.cd.a.415.1 4 336.107 even 12
2016.1.cd.a.415.2 4 336.149 odd 12
2016.1.cd.a.991.1 4 48.11 even 4
2016.1.cd.a.991.2 4 48.5 odd 4
3136.1.d.b.1471.1 2 112.67 odd 12
3136.1.d.b.1471.2 2 112.109 even 12
3136.1.d.d.1471.1 2 112.45 odd 12
3136.1.d.d.1471.2 2 112.3 even 12
3136.1.r.b.2431.1 4 112.61 odd 12
3136.1.r.b.2431.2 4 112.19 even 12
3136.1.r.b.3007.1 4 112.83 even 4
3136.1.r.b.3007.2 4 112.13 odd 4