Properties

Label 1456.1.bs.a.991.2
Level $1456$
Weight $1$
Character 1456.991
Analytic conductor $0.727$
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] = [1456,1,Mod(191,1456)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1456, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 2, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1456.191");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1456.bs (of order \(6\), degree \(2\), 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: \(0.726638658394\)
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: yes
Projective image: \(A_{4}\)
Projective field: Galois closure of 4.0.132496.1

Embedding invariants

Embedding label 991.2
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1456.991
Dual form 1456.1.bs.a.191.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{3} +(0.500000 + 0.866025i) q^{5} +(-0.866025 + 0.500000i) q^{7} +O(q^{10})\) \(q+1.00000i q^{3} +(0.500000 + 0.866025i) q^{5} +(-0.866025 + 0.500000i) q^{7} +1.00000i q^{11} -1.00000 q^{13} +(-0.866025 + 0.500000i) q^{15} -1.00000i q^{19} +(-0.500000 - 0.866025i) q^{21} +1.00000i q^{27} +(0.500000 + 0.866025i) q^{29} +(-0.866025 - 0.500000i) q^{31} -1.00000 q^{33} +(-0.866025 - 0.500000i) q^{35} -1.00000i q^{39} +(-0.500000 - 0.866025i) q^{41} +(0.866025 + 0.500000i) q^{43} +(0.866025 - 0.500000i) q^{47} +(0.500000 - 0.866025i) q^{49} +(-0.500000 + 0.866025i) q^{53} +(-0.866025 + 0.500000i) q^{55} +1.00000 q^{57} +1.00000 q^{61} +(-0.500000 - 0.866025i) q^{65} +1.00000i q^{67} +(0.866025 + 0.500000i) q^{71} +(-0.500000 + 0.866025i) q^{73} +(-0.500000 - 0.866025i) q^{77} +(-0.866025 + 0.500000i) q^{79} -1.00000 q^{81} -2.00000i q^{83} +(-0.866025 + 0.500000i) q^{87} +(0.866025 - 0.500000i) q^{91} +(0.500000 - 0.866025i) q^{93} +(0.866025 - 0.500000i) q^{95} +(-0.500000 + 0.866025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{5} - 4 q^{13} - 2 q^{21} + 2 q^{29} - 4 q^{33} - 2 q^{41} + 2 q^{49} - 2 q^{53} + 4 q^{57} + 4 q^{61} - 2 q^{65} - 2 q^{73} - 2 q^{77} - 4 q^{81} + 2 q^{93} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1456\mathbb{Z}\right)^\times\).

\(n\) \(561\) \(911\) \(1093\) \(1249\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) 0 0
\(5\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(6\) 0 0
\(7\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(12\) 0 0
\(13\) −1.00000 −1.00000
\(14\) 0 0
\(15\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(16\) 0 0
\(17\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 0 0
\(19\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(20\) 0 0
\(21\) −0.500000 0.866025i −0.500000 0.866025i
\(22\) 0 0
\(23\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000i 1.00000i
\(28\) 0 0
\(29\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(30\) 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
\(33\) −1.00000 −1.00000
\(34\) 0 0
\(35\) −0.866025 0.500000i −0.866025 0.500000i
\(36\) 0 0
\(37\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(38\) 0 0
\(39\) 1.00000i 1.00000i
\(40\) 0 0
\(41\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 0 0
\(49\) 0.500000 0.866025i 0.500000 0.866025i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(56\) 0 0
\(57\) 1.00000 1.00000
\(58\) 0 0
\(59\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(60\) 0 0
\(61\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.500000 0.866025i −0.500000 0.866025i
\(66\) 0 0
\(67\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(72\) 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
\(77\) −0.500000 0.866025i −0.500000 0.866025i
\(78\) 0 0
\(79\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −1.00000 −1.00000
\(82\) 0 0
\(83\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(88\) 0 0
\(89\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(90\) 0 0
\(91\) 0.866025 0.500000i 0.866025 0.500000i
\(92\) 0 0
\(93\) 0.500000 0.866025i 0.500000 0.866025i
\(94\) 0 0
\(95\) 0.866025 0.500000i 0.866025 0.500000i
\(96\) 0 0
\(97\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(102\) 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
\(105\) 0.500000 0.866025i 0.500000 0.866025i
\(106\) 0 0
\(107\) 1.73205 + 1.00000i 1.73205 + 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(108\) 0 0
\(109\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 0.866025 0.500000i 0.866025 0.500000i
\(124\) 0 0
\(125\) 1.00000 1.00000
\(126\) 0 0
\(127\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(128\) 0 0
\(129\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(130\) 0 0
\(131\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(132\) 0 0
\(133\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(134\) 0 0
\(135\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(136\) 0 0
\(137\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(138\) 0 0
\(139\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(142\) 0 0
\(143\) 1.00000i 1.00000i
\(144\) 0 0
\(145\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(146\) 0 0
\(147\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(148\) 0 0
\(149\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(150\) 0 0
\(151\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1.00000i 1.00000i
\(156\) 0 0
\(157\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) −0.866025 0.500000i −0.866025 0.500000i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(164\) 0 0
\(165\) −0.500000 0.866025i −0.500000 0.866025i
\(166\) 0 0
\(167\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(168\) 0 0
\(169\) 1.00000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 1.00000i 1.00000i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −0.500000 0.866025i −0.500000 0.866025i
\(190\) 0 0
\(191\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(192\) 0 0
\(193\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(194\) 0 0
\(195\) 0.866025 0.500000i 0.866025 0.500000i
\(196\) 0 0
\(197\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(198\) 0 0
\(199\) 1.73205 1.00000i 1.73205 1.00000i 0.866025 0.500000i \(-0.166667\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(200\) 0 0
\(201\) −1.00000 −1.00000
\(202\) 0 0
\(203\) −0.866025 0.500000i −0.866025 0.500000i
\(204\) 0 0
\(205\) 0.500000 0.866025i 0.500000 0.866025i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1.00000 1.00000
\(210\) 0 0
\(211\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(212\) 0 0
\(213\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(214\) 0 0
\(215\) 1.00000i 1.00000i
\(216\) 0 0
\(217\) 1.00000 1.00000
\(218\) 0 0
\(219\) −0.866025 0.500000i −0.866025 0.500000i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(228\) 0 0
\(229\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(230\) 0 0
\(231\) 0.866025 0.500000i 0.866025 0.500000i
\(232\) 0 0
\(233\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(236\) 0 0
\(237\) −0.500000 0.866025i −0.500000 0.866025i
\(238\) 0 0
\(239\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(240\) 0 0
\(241\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.00000 1.00000
\(246\) 0 0
\(247\) 1.00000i 1.00000i
\(248\) 0 0
\(249\) 2.00000 2.00000
\(250\) 0 0
\(251\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(264\) 0 0
\(265\) −1.00000 −1.00000
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(270\) 0 0
\(271\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(272\) 0 0
\(273\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(284\) 0 0
\(285\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(286\) 0 0
\(287\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(288\) 0 0
\(289\) 0.500000 0.866025i 0.500000 0.866025i
\(290\) 0 0
\(291\) −0.866025 0.500000i −0.866025 0.500000i
\(292\) 0 0
\(293\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −1.00000 −1.00000
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −1.00000 −1.00000
\(302\) 0 0
\(303\) 1.00000i 1.00000i
\(304\) 0 0
\(305\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) −0.500000 0.866025i −0.500000 0.866025i
\(310\) 0 0
\(311\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(312\) 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
\(317\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(320\) 0 0
\(321\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(322\) 0 0
\(323\) 0 0
\(324\) 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
\(331\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(336\) 0 0
\(337\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(338\) 0 0
\(339\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(340\) 0 0
\(341\) 0.500000 0.866025i 0.500000 0.866025i
\(342\) 0 0
\(343\) 1.00000i 1.00000i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −1.73205 + 1.00000i −1.73205 + 1.00000i −0.866025 + 0.500000i \(0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(348\) 0 0
\(349\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(350\) 0 0
\(351\) 1.00000i 1.00000i
\(352\) 0 0
\(353\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(354\) 0 0
\(355\) 1.00000i 1.00000i
\(356\) 0 0
\(357\) 0 0
\(358\) 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
\(365\) −1.00000 −1.00000
\(366\) 0 0
\(367\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 1.00000i 1.00000i
\(372\) 0 0
\(373\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(374\) 0 0
\(375\) 1.00000i 1.00000i
\(376\) 0 0
\(377\) −0.500000 0.866025i −0.500000 0.866025i
\(378\) 0 0
\(379\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(382\) 0 0
\(383\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(384\) 0 0
\(385\) 0.500000 0.866025i 0.500000 0.866025i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(394\) 0 0
\(395\) −0.866025 0.500000i −0.866025 0.500000i
\(396\) 0 0
\(397\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(398\) 0 0
\(399\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(400\) 0 0
\(401\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(402\) 0 0
\(403\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(404\) 0 0
\(405\) −0.500000 0.866025i −0.500000 0.866025i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(410\) 0 0
\(411\) −1.73205 + 1.00000i −1.73205 + 1.00000i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 1.73205 1.00000i 1.73205 1.00000i
\(416\) 0 0
\(417\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(418\) 0 0
\(419\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(428\) 0 0
\(429\) 1.00000 1.00000
\(430\) 0 0
\(431\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(432\) 0 0
\(433\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) −0.866025 0.500000i −0.866025 0.500000i
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 1.00000i 1.00000i
\(448\) 0 0
\(449\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(450\) 0 0
\(451\) 0.866025 0.500000i 0.866025 0.500000i
\(452\) 0 0
\(453\) 0.500000 0.866025i 0.500000 0.866025i
\(454\) 0 0
\(455\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(456\) 0 0
\(457\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 1.00000 1.00000
\(466\) 0 0
\(467\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) −0.500000 0.866025i −0.500000 0.866025i
\(470\) 0 0
\(471\) −0.866025 0.500000i −0.866025 0.500000i
\(472\) 0 0
\(473\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −1.00000 −1.00000
\(486\) 0 0
\(487\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(488\) 0 0
\(489\) 1.00000 1.00000
\(490\) 0 0
\(491\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1.00000 −1.00000
\(498\) 0 0
\(499\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(500\) 0 0
\(501\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(502\) 0 0
\(503\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(504\) 0 0
\(505\) −0.500000 0.866025i −0.500000 0.866025i
\(506\) 0 0
\(507\) 1.00000i 1.00000i
\(508\) 0 0
\(509\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(510\) 0 0
\(511\) 1.00000i 1.00000i
\(512\) 0 0
\(513\) 1.00000 1.00000
\(514\) 0 0
\(515\) −0.866025 0.500000i −0.866025 0.500000i
\(516\) 0 0
\(517\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(518\) 0 0
\(519\) 1.00000i 1.00000i
\(520\) 0 0
\(521\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(522\) 0 0
\(523\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −0.500000 0.866025i −0.500000 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(534\) 0 0
\(535\) 2.00000i 2.00000i
\(536\) 0 0
\(537\) 1.00000 1.00000
\(538\) 0 0
\(539\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(540\) 0 0
\(541\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1.00000 1.00000
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0.866025 0.500000i 0.866025 0.500000i
\(552\) 0 0
\(553\) 0.500000 0.866025i 0.500000 0.866025i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(558\) 0 0
\(559\) −0.866025 0.500000i −0.866025 0.500000i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(564\) 0 0
\(565\) 1.00000 1.00000
\(566\) 0 0
\(567\) 0.866025 0.500000i 0.866025 0.500000i
\(568\) 0 0
\(569\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(570\) 0 0
\(571\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(572\) 0 0
\(573\) 1.00000 1.00000
\(574\) 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
\(579\) 1.00000i 1.00000i
\(580\) 0 0
\(581\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(582\) 0 0
\(583\) −0.866025 0.500000i −0.866025 0.500000i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(590\) 0 0
\(591\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(592\) 0 0
\(593\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(598\) 0 0
\(599\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(608\) 0 0
\(609\) 0.500000 0.866025i 0.500000 0.866025i
\(610\) 0 0
\(611\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(612\) 0 0
\(613\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(614\) 0 0
\(615\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(616\) 0 0
\(617\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(618\) 0 0
\(619\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(626\) 0 0
\(627\) 1.00000i 1.00000i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(632\) 0 0
\(633\) −0.500000 0.866025i −0.500000 0.866025i
\(634\) 0 0
\(635\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(636\) 0 0
\(637\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(642\) 0 0
\(643\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(644\) 0 0
\(645\) −1.00000 −1.00000
\(646\) 0 0
\(647\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 1.00000i 1.00000i
\(652\) 0 0
\(653\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(654\) 0 0
\(655\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(660\) 0 0
\(661\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(662\) 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
\(669\) −0.500000 0.866025i −0.500000 0.866025i
\(670\) 0 0
\(671\) 1.00000i 1.00000i
\(672\) 0 0
\(673\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 1.00000i 1.00000i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(684\) 0 0
\(685\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(686\) 0 0
\(687\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(688\) 0 0
\(689\) 0.500000 0.866025i 0.500000 0.866025i
\(690\) 0 0
\(691\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1.00000i 1.00000i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0.866025 0.500000i 0.866025 0.500000i
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(706\) 0 0
\(707\) 0.866025 0.500000i 0.866025 0.500000i
\(708\) 0 0
\(709\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0.866025 0.500000i 0.866025 0.500000i
\(716\) 0 0
\(717\) 2.00000 2.00000
\(718\) 0 0
\(719\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(720\) 0 0
\(721\) 0.500000 0.866025i 0.500000 0.866025i
\(722\) 0 0
\(723\) 0 0
\(724\) 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
\(733\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 1.00000i 1.00000i
\(736\) 0 0
\(737\) −1.00000 −1.00000
\(738\) 0 0
\(739\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(740\) 0 0
\(741\) −1.00000 −1.00000
\(742\) 0 0
\(743\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(744\) 0 0
\(745\) −0.500000 0.866025i −0.500000 0.866025i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −2.00000 −2.00000
\(750\) 0 0
\(751\) 1.73205 + 1.00000i 1.73205 + 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(752\) 0 0
\(753\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(754\) 0 0
\(755\) 1.00000i 1.00000i
\(756\) 0 0
\(757\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(762\) 0 0
\(763\) 1.00000i 1.00000i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(780\) 0 0
\(781\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(782\) 0 0
\(783\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(784\) 0 0
\(785\) −1.00000 −1.00000
\(786\) 0 0
\(787\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(788\) 0 0
\(789\) 1.00000 1.00000
\(790\) 0 0
\(791\) 1.00000i 1.00000i
\(792\) 0 0
\(793\) −1.00000 −1.00000
\(794\) 0 0
\(795\) 1.00000i 1.00000i
\(796\) 0 0
\(797\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −0.866025 0.500000i −0.866025 0.500000i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0.866025 0.500000i 0.866025 0.500000i
\(816\) 0 0
\(817\) 0.500000 0.866025i 0.500000 0.866025i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(822\) 0 0
\(823\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(830\) 0 0
\(831\) 1.73205 1.00000i 1.73205 1.00000i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(836\) 0 0
\(837\) 0.500000 0.866025i 0.500000 0.866025i
\(838\) 0 0
\(839\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 0 0
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 1.00000 1.00000
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(858\) 0 0
\(859\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(860\) 0 0
\(861\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(862\) 0 0
\(863\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(866\) 0 0
\(867\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(868\) 0 0
\(869\) −0.500000 0.866025i −0.500000 0.866025i
\(870\) 0 0
\(871\) 1.00000i 1.00000i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(876\) 0 0
\(877\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(878\) 0 0
\(879\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(880\) 0 0
\(881\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(888\) 0 0
\(889\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(890\) 0 0
\(891\) 1.00000i 1.00000i
\(892\) 0 0
\(893\) −0.500000 0.866025i −0.500000 0.866025i
\(894\) 0 0
\(895\) 0.866025 0.500000i 0.866025 0.500000i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1.00000i 1.00000i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 1.00000i 1.00000i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 2.00000 2.00000
\(914\) 0 0
\(915\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(916\) 0 0
\(917\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(918\) 0 0
\(919\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −0.866025 0.500000i −0.866025 0.500000i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(930\) 0 0
\(931\) −0.866025 0.500000i −0.866025 0.500000i
\(932\) 0 0
\(933\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) 0 0
\(939\) 0.866025 0.500000i 0.866025 0.500000i
\(940\) 0 0
\(941\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0.500000 0.866025i 0.500000 0.866025i
\(946\) 0 0
\(947\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(948\) 0 0
\(949\) 0.500000 0.866025i 0.500000 0.866025i
\(950\) 0 0
\(951\) 0.866025 0.500000i 0.866025 0.500000i
\(952\) 0 0
\(953\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(954\) 0 0
\(955\) 0.866025 0.500000i 0.866025 0.500000i
\(956\) 0 0
\(957\) −0.500000 0.866025i −0.500000 0.866025i
\(958\) 0 0
\(959\) −1.73205 1.00000i −1.73205 1.00000i
\(960\) 0 0
\(961\) 0 0
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −0.500000 0.866025i −0.500000 0.866025i
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(972\) 0 0
\(973\) −1.00000 −1.00000
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 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
\(985\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(986\) 0 0
\(987\) −0.866025 0.500000i −0.866025 0.500000i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(992\) 0 0
\(993\) −1.00000 −1.00000
\(994\) 0 0
\(995\) 1.73205 + 1.00000i 1.73205 + 1.00000i
\(996\) 0 0
\(997\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(998\) 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 1456.1.bs.a.991.2 yes 4
4.3 odd 2 inner 1456.1.bs.a.991.1 yes 4
7.2 even 3 1456.1.dg.a.1199.1 yes 4
13.9 even 3 1456.1.dg.a.1439.2 yes 4
28.23 odd 6 1456.1.dg.a.1199.2 yes 4
52.35 odd 6 1456.1.dg.a.1439.1 yes 4
91.9 even 3 inner 1456.1.bs.a.191.2 yes 4
364.191 odd 6 inner 1456.1.bs.a.191.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1456.1.bs.a.191.1 4 364.191 odd 6 inner
1456.1.bs.a.191.2 yes 4 91.9 even 3 inner
1456.1.bs.a.991.1 yes 4 4.3 odd 2 inner
1456.1.bs.a.991.2 yes 4 1.1 even 1 trivial
1456.1.dg.a.1199.1 yes 4 7.2 even 3
1456.1.dg.a.1199.2 yes 4 28.23 odd 6
1456.1.dg.a.1439.1 yes 4 52.35 odd 6
1456.1.dg.a.1439.2 yes 4 13.9 even 3