Properties

Label 3240.1.bh.d.1349.1
Level $3240$
Weight $1$
Character 3240.1349
Analytic conductor $1.617$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -120
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3240,1,Mod(269,3240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3240, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 1, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3240.269");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3240.bh (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: \(1.61697064093\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1080)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.1080.1
Artin image: $C_6\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

Embedding invariants

Embedding label 1349.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 3240.1349
Dual form 3240.1.bh.d.269.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(0.500000 - 0.866025i) q^{5} -1.00000 q^{8} +O(q^{10})\) \(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(0.500000 - 0.866025i) q^{5} -1.00000 q^{8} +1.00000 q^{10} +(-0.500000 - 0.866025i) q^{11} +(0.500000 - 0.866025i) q^{13} +(-0.500000 - 0.866025i) q^{16} +1.00000 q^{17} +(0.500000 + 0.866025i) q^{20} +(0.500000 - 0.866025i) q^{22} +(-0.500000 + 0.866025i) q^{23} +(-0.500000 - 0.866025i) q^{25} +1.00000 q^{26} +(-0.500000 - 0.866025i) q^{29} +(0.500000 - 0.866025i) q^{31} +(0.500000 - 0.866025i) q^{32} +(0.500000 + 0.866025i) q^{34} +2.00000 q^{37} +(-0.500000 + 0.866025i) q^{40} +(0.500000 + 0.866025i) q^{43} +1.00000 q^{44} -1.00000 q^{46} +(-0.500000 - 0.866025i) q^{47} +(-0.500000 + 0.866025i) q^{49} +(0.500000 - 0.866025i) q^{50} +(0.500000 + 0.866025i) q^{52} -1.00000 q^{55} +(0.500000 - 0.866025i) q^{58} +(1.00000 - 1.73205i) q^{59} +1.00000 q^{62} +1.00000 q^{64} +(-0.500000 - 0.866025i) q^{65} +(-1.00000 + 1.73205i) q^{67} +(-0.500000 + 0.866025i) q^{68} +(1.00000 + 1.73205i) q^{74} +(0.500000 + 0.866025i) q^{79} -1.00000 q^{80} +(0.500000 - 0.866025i) q^{85} +(-0.500000 + 0.866025i) q^{86} +(0.500000 + 0.866025i) q^{88} +(-0.500000 - 0.866025i) q^{92} +(0.500000 - 0.866025i) q^{94} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{4} + q^{5} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - q^{4} + q^{5} - 2 q^{8} + 2 q^{10} - q^{11} + q^{13} - q^{16} + 2 q^{17} + q^{20} + q^{22} - q^{23} - q^{25} + 2 q^{26} - q^{29} + q^{31} + q^{32} + q^{34} + 4 q^{37} - q^{40} + q^{43} + 2 q^{44} - 2 q^{46} - q^{47} - q^{49} + q^{50} + q^{52} - 2 q^{55} + q^{58} + 2 q^{59} + 2 q^{62} + 2 q^{64} - q^{65} - 2 q^{67} - q^{68} + 2 q^{74} + q^{79} - 2 q^{80} + q^{85} - q^{86} + q^{88} - q^{92} + q^{94} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3240\mathbb{Z}\right)^\times\).

\(n\) \(1297\) \(1621\) \(2431\) \(3161\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(3\) 0 0
\(4\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(5\) 0.500000 0.866025i 0.500000 0.866025i
\(6\) 0 0
\(7\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(8\) −1.00000 −1.00000
\(9\) 0 0
\(10\) 1.00000 1.00000
\(11\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.500000 0.866025i −0.500000 0.866025i
\(17\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(21\) 0 0
\(22\) 0.500000 0.866025i 0.500000 0.866025i
\(23\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −0.500000 0.866025i −0.500000 0.866025i
\(26\) 1.00000 1.00000
\(27\) 0 0
\(28\) 0 0
\(29\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(32\) 0.500000 0.866025i 0.500000 0.866025i
\(33\) 0 0
\(34\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(38\) 0 0
\(39\) 0 0
\(40\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(41\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(42\) 0 0
\(43\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(44\) 1.00000 1.00000
\(45\) 0 0
\(46\) −1.00000 −1.00000
\(47\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(50\) 0.500000 0.866025i 0.500000 0.866025i
\(51\) 0 0
\(52\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) −1.00000 −1.00000
\(56\) 0 0
\(57\) 0 0
\(58\) 0.500000 0.866025i 0.500000 0.866025i
\(59\) 1.00000 1.73205i 1.00000 1.73205i 0.500000 0.866025i \(-0.333333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(60\) 0 0
\(61\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(62\) 1.00000 1.00000
\(63\) 0 0
\(64\) 1.00000 1.00000
\(65\) −0.500000 0.866025i −0.500000 0.866025i
\(66\) 0 0
\(67\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(68\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(80\) −1.00000 −1.00000
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(84\) 0 0
\(85\) 0.500000 0.866025i 0.500000 0.866025i
\(86\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(87\) 0 0
\(88\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −0.500000 0.866025i −0.500000 0.866025i
\(93\) 0 0
\(94\) 0.500000 0.866025i 0.500000 0.866025i
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(98\) −1.00000 −1.00000
\(99\) 0 0
\(100\) 1.00000 1.00000
\(101\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(104\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) −0.500000 0.866025i −0.500000 0.866025i
\(111\) 0 0
\(112\) 0 0
\(113\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(116\) 1.00000 1.00000
\(117\) 0 0
\(118\) 2.00000 2.00000
\(119\) 0 0
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 0 0
\(124\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(125\) −1.00000 −1.00000
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(129\) 0 0
\(130\) 0.500000 0.866025i 0.500000 0.866025i
\(131\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −2.00000 −2.00000
\(135\) 0 0
\(136\) −1.00000 −1.00000
\(137\) 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 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −1.00000 −1.00000
\(144\) 0 0
\(145\) −1.00000 −1.00000
\(146\) 0 0
\(147\) 0 0
\(148\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(149\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −0.500000 0.866025i −0.500000 0.866025i
\(156\) 0 0
\(157\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(158\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(159\) 0 0
\(160\) −0.500000 0.866025i −0.500000 0.866025i
\(161\) 0 0
\(162\) 0 0
\(163\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.00000 1.73205i 1.00000 1.73205i 0.500000 0.866025i \(-0.333333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 1.00000 1.00000
\(171\) 0 0
\(172\) −1.00000 −1.00000
\(173\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(177\) 0 0
\(178\) 0 0
\(179\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0.500000 0.866025i 0.500000 0.866025i
\(185\) 1.00000 1.73205i 1.00000 1.73205i
\(186\) 0 0
\(187\) −0.500000 0.866025i −0.500000 0.866025i
\(188\) 1.00000 1.00000
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(192\) 0 0
\(193\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −0.500000 0.866025i −0.500000 0.866025i
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(200\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(201\) 0 0
\(202\) 0.500000 0.866025i 0.500000 0.866025i
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −1.00000 −1.00000
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1.00000 1.00000
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0.500000 0.866025i 0.500000 0.866025i
\(221\) 0.500000 0.866025i 0.500000 0.866025i
\(222\) 0 0
\(223\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −1.00000 −1.00000
\(227\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(228\) 0 0
\(229\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(230\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(231\) 0 0
\(232\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(233\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) −1.00000 −1.00000
\(236\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(240\) 0 0
\(241\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(246\) 0 0
\(247\) 0 0
\(248\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(249\) 0 0
\(250\) −0.500000 0.866025i −0.500000 0.866025i
\(251\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(252\) 0 0
\(253\) 1.00000 1.00000
\(254\) 0 0
\(255\) 0 0
\(256\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(257\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 1.00000 1.00000
\(261\) 0 0
\(262\) −1.00000 −1.00000
\(263\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −1.00000 1.73205i −1.00000 1.73205i
\(269\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(270\) 0 0
\(271\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(272\) −0.500000 0.866025i −0.500000 0.866025i
\(273\) 0 0
\(274\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(275\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(276\) 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 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(282\) 0 0
\(283\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −0.500000 0.866025i −0.500000 0.866025i
\(287\) 0 0
\(288\) 0 0
\(289\) 0 0
\(290\) −0.500000 0.866025i −0.500000 0.866025i
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(294\) 0 0
\(295\) −1.00000 1.73205i −1.00000 1.73205i
\(296\) −2.00000 −2.00000
\(297\) 0 0
\(298\) −1.00000 −1.00000
\(299\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(300\) 0 0
\(301\) 0 0
\(302\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0.500000 0.866025i 0.500000 0.866025i
\(311\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(312\) 0 0
\(313\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(314\) 1.00000 1.00000
\(315\) 0 0
\(316\) −1.00000 −1.00000
\(317\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(318\) 0 0
\(319\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(320\) 0.500000 0.866025i 0.500000 0.866025i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −1.00000 −1.00000
\(326\) −0.500000 0.866025i −0.500000 0.866025i
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 2.00000 2.00000
\(335\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(336\) 0 0
\(337\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(341\) −1.00000 −1.00000
\(342\) 0 0
\(343\) 0 0
\(344\) −0.500000 0.866025i −0.500000 0.866025i
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(348\) 0 0
\(349\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.00000 −1.00000
\(353\) −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 0
\(358\) −1.00000 1.73205i −1.00000 1.73205i
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 1.00000 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(368\) 1.00000 1.00000
\(369\) 0 0
\(370\) 2.00000 2.00000
\(371\) 0 0
\(372\) 0 0
\(373\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(374\) 0.500000 0.866025i 0.500000 0.866025i
\(375\) 0 0
\(376\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(377\) −1.00000 −1.00000
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(392\) 0.500000 0.866025i 0.500000 0.866025i
\(393\) 0 0
\(394\) 0 0
\(395\) 1.00000 1.00000
\(396\) 0 0
\(397\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(398\) −0.500000 0.866025i −0.500000 0.866025i
\(399\) 0 0
\(400\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(401\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(402\) 0 0
\(403\) −0.500000 0.866025i −0.500000 0.866025i
\(404\) 1.00000 1.00000
\(405\) 0 0
\(406\) 0 0
\(407\) −1.00000 1.73205i −1.00000 1.73205i
\(408\) 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 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) −0.500000 0.866025i −0.500000 0.866025i
\(417\) 0 0
\(418\) 0 0
\(419\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −0.500000 0.866025i −0.500000 0.866025i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(440\) 1.00000 1.00000
\(441\) 0 0
\(442\) 1.00000 1.00000
\(443\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −0.500000 0.866025i −0.500000 0.866025i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −1.00000 −1.00000
\(461\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(462\) 0 0
\(463\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(464\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(465\) 0 0
\(466\) −1.00000 1.73205i −1.00000 1.73205i
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −0.500000 0.866025i −0.500000 0.866025i
\(471\) 0 0
\(472\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(473\) 0.500000 0.866025i 0.500000 0.866025i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(480\) 0 0
\(481\) 1.00000 1.73205i 1.00000 1.73205i
\(482\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(491\) 1.00000 1.73205i 1.00000 1.73205i 0.500000 0.866025i \(-0.333333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(492\) 0 0
\(493\) −0.500000 0.866025i −0.500000 0.866025i
\(494\) 0 0
\(495\) 0 0
\(496\) −1.00000 −1.00000
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(500\) 0.500000 0.866025i 0.500000 0.866025i
\(501\) 0 0
\(502\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(503\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(504\) 0 0
\(505\) −1.00000 −1.00000
\(506\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(507\) 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
\(512\) −1.00000 −1.00000
\(513\) 0 0
\(514\) −1.00000 −1.00000
\(515\) 0 0
\(516\) 0 0
\(517\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(518\) 0 0
\(519\) 0 0
\(520\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(524\) −0.500000 0.866025i −0.500000 0.866025i
\(525\) 0 0
\(526\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(527\) 0.500000 0.866025i 0.500000 0.866025i
\(528\) 0 0
\(529\) 0 0
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 1.00000 1.73205i 1.00000 1.73205i
\(537\) 0 0
\(538\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(539\) 1.00000 1.00000
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(543\) 0 0
\(544\) 0.500000 0.866025i 0.500000 0.866025i
\(545\) 0 0
\(546\) 0 0
\(547\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(548\) −2.00000 −2.00000
\(549\) 0 0
\(550\) −1.00000 −1.00000
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 1.00000 1.73205i 1.00000 1.73205i
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 1.00000 1.00000
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(564\) 0 0
\(565\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(566\) −2.00000 −2.00000
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(570\) 0 0
\(571\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(572\) 0.500000 0.866025i 0.500000 0.866025i
\(573\) 0 0
\(574\) 0 0
\(575\) 1.00000 1.00000
\(576\) 0 0
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0.500000 0.866025i 0.500000 0.866025i
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 1.00000 1.73205i 1.00000 1.73205i
\(591\) 0 0
\(592\) −1.00000 1.73205i −1.00000 1.73205i
\(593\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −0.500000 0.866025i −0.500000 0.866025i
\(597\) 0 0
\(598\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(599\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(600\) 0 0
\(601\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −1.00000 −1.00000
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −1.00000 −1.00000
\(612\) 0 0
\(613\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(614\) −0.500000 0.866025i −0.500000 0.866025i
\(615\) 0 0
\(616\) 0 0
\(617\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(620\) 1.00000 1.00000
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(629\) 2.00000 2.00000
\(630\) 0 0
\(631\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(632\) −0.500000 0.866025i −0.500000 0.866025i
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(638\) −1.00000 −1.00000
\(639\) 0 0
\(640\) 1.00000 1.00000
\(641\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(642\) 0 0
\(643\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) −2.00000 −2.00000
\(650\) −0.500000 0.866025i −0.500000 0.866025i
\(651\) 0 0
\(652\) 0.500000 0.866025i 0.500000 0.866025i
\(653\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(654\) 0 0
\(655\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(660\) 0 0
\(661\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1.00000 1.00000
\(668\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(669\) 0 0
\(670\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(681\) 0 0
\(682\) −0.500000 0.866025i −0.500000 0.866025i
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 2.00000 2.00000
\(686\) 0 0
\(687\) 0 0
\(688\) 0.500000 0.866025i 0.500000 0.866025i
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(692\) 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
\(701\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −0.500000 0.866025i −0.500000 0.866025i
\(705\) 0 0
\(706\) 0.500000 0.866025i 0.500000 0.866025i
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(714\) 0 0
\(715\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(716\) 1.00000 1.73205i 1.00000 1.73205i
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(723\) 0 0
\(724\) 0 0
\(725\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(726\) 0 0
\(727\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(732\) 0 0
\(733\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(737\) 2.00000 2.00000
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(741\) 0 0
\(742\) 0 0
\(743\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(746\) 1.00000 1.00000
\(747\) 0 0
\(748\) 1.00000 1.00000
\(749\) 0 0
\(750\) 0 0
\(751\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(752\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(753\) 0 0
\(754\) −0.500000 0.866025i −0.500000 0.866025i
\(755\) 1.00000 1.00000
\(756\) 0 0
\(757\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) −1.00000 −1.00000
\(767\) −1.00000 1.73205i −1.00000 1.73205i
\(768\) 0 0
\(769\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) −1.00000 −1.00000
\(776\) 0 0
\(777\) 0 0
\(778\) 0.500000 0.866025i 0.500000 0.866025i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) −1.00000 −1.00000
\(783\) 0 0
\(784\) 1.00000 1.00000
\(785\) −0.500000 0.866025i −0.500000 0.866025i
\(786\) 0 0
\(787\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) −0.500000 0.866025i −0.500000 0.866025i
\(795\) 0 0
\(796\) 0.500000 0.866025i 0.500000 0.866025i
\(797\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)