Properties

Label 1560.1.dk.a.1349.1
Level $1560$
Weight $1$
Character 1560.1349
Analytic conductor $0.779$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -120
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1560,1,Mod(1109,1560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1560, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 3, 3, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1560.1109");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1560.dk (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: \(0.778541419707\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.16039857600.1

Embedding invariants

Embedding label 1349.1
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1560.1349
Dual form 1560.1.dk.a.1109.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.866025 + 0.500000i) q^{2} +(-0.500000 - 0.866025i) q^{3} +(0.500000 - 0.866025i) q^{4} +1.00000i q^{5} +(0.866025 + 0.500000i) q^{6} +1.00000i q^{8} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\) \(q+(-0.866025 + 0.500000i) q^{2} +(-0.500000 - 0.866025i) q^{3} +(0.500000 - 0.866025i) q^{4} +1.00000i q^{5} +(0.866025 + 0.500000i) q^{6} +1.00000i q^{8} +(-0.500000 + 0.866025i) q^{9} +(-0.500000 - 0.866025i) q^{10} +(-0.866025 + 0.500000i) q^{11} -1.00000 q^{12} -1.00000 q^{13} +(0.866025 - 0.500000i) q^{15} +(-0.500000 - 0.866025i) q^{16} -1.00000i q^{18} +(0.866025 + 0.500000i) q^{20} +(0.500000 - 0.866025i) q^{22} +(-0.866025 - 1.50000i) q^{23} +(0.866025 - 0.500000i) q^{24} -1.00000 q^{25} +(0.866025 - 0.500000i) q^{26} +1.00000 q^{27} +(-0.866025 - 1.50000i) q^{29} +(-0.500000 + 0.866025i) q^{30} -1.73205i q^{31} +(0.866025 + 0.500000i) q^{32} +(0.866025 + 0.500000i) q^{33} +(0.500000 + 0.866025i) q^{36} +(-1.50000 + 0.866025i) q^{37} +(0.500000 + 0.866025i) q^{39} -1.00000 q^{40} +(0.500000 - 0.866025i) q^{43} +1.00000i q^{44} +(-0.866025 - 0.500000i) q^{45} +(1.50000 + 0.866025i) q^{46} +1.00000i q^{47} +(-0.500000 + 0.866025i) q^{48} +(0.500000 + 0.866025i) q^{49} +(0.866025 - 0.500000i) q^{50} +(-0.500000 + 0.866025i) q^{52} +(-0.866025 + 0.500000i) q^{54} +(-0.500000 - 0.866025i) q^{55} +(1.50000 + 0.866025i) q^{58} +(-0.866025 - 0.500000i) q^{59} -1.00000i q^{60} +(0.866025 + 1.50000i) q^{62} -1.00000 q^{64} -1.00000i q^{65} -1.00000 q^{66} +(-0.866025 + 1.50000i) q^{69} +(-0.866025 - 0.500000i) q^{72} +(0.866025 - 1.50000i) q^{74} +(0.500000 + 0.866025i) q^{75} +(-0.866025 - 0.500000i) q^{78} -1.00000 q^{79} +(0.866025 - 0.500000i) q^{80} +(-0.500000 - 0.866025i) q^{81} +1.00000i q^{86} +(-0.866025 + 1.50000i) q^{87} +(-0.500000 - 0.866025i) q^{88} +1.00000 q^{90} -1.73205 q^{92} +(-1.50000 + 0.866025i) q^{93} +(-0.500000 - 0.866025i) q^{94} -1.00000i q^{96} +(-0.866025 - 0.500000i) q^{98} -1.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 2 q^{4} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} + 2 q^{4} - 2 q^{9} - 2 q^{10} - 4 q^{12} - 4 q^{13} - 2 q^{16} + 2 q^{22} - 4 q^{25} + 4 q^{27} - 2 q^{30} + 2 q^{36} - 6 q^{37} + 2 q^{39} - 4 q^{40} + 2 q^{43} + 6 q^{46} - 2 q^{48} + 2 q^{49} - 2 q^{52} - 2 q^{55} + 6 q^{58} - 4 q^{64} - 4 q^{66} + 2 q^{75} - 4 q^{79} - 2 q^{81} - 2 q^{88} + 4 q^{90} - 6 q^{93} - 2 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1560\mathbb{Z}\right)^\times\).

\(n\) \(391\) \(521\) \(781\) \(937\) \(1081\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(3\) −0.500000 0.866025i −0.500000 0.866025i
\(4\) 0.500000 0.866025i 0.500000 0.866025i
\(5\) 1.00000i 1.00000i
\(6\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(7\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(8\) 1.00000i 1.00000i
\(9\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(10\) −0.500000 0.866025i −0.500000 0.866025i
\(11\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(12\) −1.00000 −1.00000
\(13\) −1.00000 −1.00000
\(14\) 0 0
\(15\) 0.866025 0.500000i 0.866025 0.500000i
\(16\) −0.500000 0.866025i −0.500000 0.866025i
\(17\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(18\) 1.00000i 1.00000i
\(19\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(20\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(21\) 0 0
\(22\) 0.500000 0.866025i 0.500000 0.866025i
\(23\) −0.866025 1.50000i −0.866025 1.50000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(-0.5\pi\)
\(24\) 0.866025 0.500000i 0.866025 0.500000i
\(25\) −1.00000 −1.00000
\(26\) 0.866025 0.500000i 0.866025 0.500000i
\(27\) 1.00000 1.00000
\(28\) 0 0
\(29\) −0.866025 1.50000i −0.866025 1.50000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(-0.5\pi\)
\(30\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(31\) 1.73205i 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(32\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(33\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(34\) 0 0
\(35\) 0 0
\(36\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(37\) −1.50000 + 0.866025i −1.50000 + 0.866025i −0.500000 + 0.866025i \(0.666667\pi\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(40\) −1.00000 −1.00000
\(41\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(42\) 0 0
\(43\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(44\) 1.00000i 1.00000i
\(45\) −0.866025 0.500000i −0.866025 0.500000i
\(46\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(47\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(49\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(50\) 0.866025 0.500000i 0.866025 0.500000i
\(51\) 0 0
\(52\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(55\) −0.500000 0.866025i −0.500000 0.866025i
\(56\) 0 0
\(57\) 0 0
\(58\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(59\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(60\) 1.00000i 1.00000i
\(61\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) 0.866025 + 1.50000i 0.866025 + 1.50000i
\(63\) 0 0
\(64\) −1.00000 −1.00000
\(65\) 1.00000i 1.00000i
\(66\) −1.00000 −1.00000
\(67\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(68\) 0 0
\(69\) −0.866025 + 1.50000i −0.866025 + 1.50000i
\(70\) 0 0
\(71\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(72\) −0.866025 0.500000i −0.866025 0.500000i
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0.866025 1.50000i 0.866025 1.50000i
\(75\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(76\) 0 0
\(77\) 0 0
\(78\) −0.866025 0.500000i −0.866025 0.500000i
\(79\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(80\) 0.866025 0.500000i 0.866025 0.500000i
\(81\) −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
\(85\) 0 0
\(86\) 1.00000i 1.00000i
\(87\) −0.866025 + 1.50000i −0.866025 + 1.50000i
\(88\) −0.500000 0.866025i −0.500000 0.866025i
\(89\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(90\) 1.00000 1.00000
\(91\) 0 0
\(92\) −1.73205 −1.73205
\(93\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(94\) −0.500000 0.866025i −0.500000 0.866025i
\(95\) 0 0
\(96\) 1.00000i 1.00000i
\(97\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(98\) −0.866025 0.500000i −0.866025 0.500000i
\(99\) 1.00000i 1.00000i
\(100\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(101\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 1.00000i 1.00000i
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(108\) 0.500000 0.866025i 0.500000 0.866025i
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(111\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(112\) 0 0
\(113\) −0.866025 + 1.50000i −0.866025 + 1.50000i 1.00000i \(0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(114\) 0 0
\(115\) 1.50000 0.866025i 1.50000 0.866025i
\(116\) −1.73205 −1.73205
\(117\) 0.500000 0.866025i 0.500000 0.866025i
\(118\) 1.00000 1.00000
\(119\) 0 0
\(120\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(121\) 0 0
\(122\) 0 0
\(123\) 0 0
\(124\) −1.50000 0.866025i −1.50000 0.866025i
\(125\) 1.00000i 1.00000i
\(126\) 0 0
\(127\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(128\) 0.866025 0.500000i 0.866025 0.500000i
\(129\) −1.00000 −1.00000
\(130\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(131\) −1.73205 −1.73205 −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(132\) 0.866025 0.500000i 0.866025 0.500000i
\(133\) 0 0
\(134\) 0 0
\(135\) 1.00000i 1.00000i
\(136\) 0 0
\(137\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(138\) 1.73205i 1.73205i
\(139\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(140\) 0 0
\(141\) 0.866025 0.500000i 0.866025 0.500000i
\(142\) 0 0
\(143\) 0.866025 0.500000i 0.866025 0.500000i
\(144\) 1.00000 1.00000
\(145\) 1.50000 0.866025i 1.50000 0.866025i
\(146\) 0 0
\(147\) 0.500000 0.866025i 0.500000 0.866025i
\(148\) 1.73205i 1.73205i
\(149\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(150\) −0.866025 0.500000i −0.866025 0.500000i
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1.73205 1.73205
\(156\) 1.00000 1.00000
\(157\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(158\) 0.866025 0.500000i 0.866025 0.500000i
\(159\) 0 0
\(160\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(161\) 0 0
\(162\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(163\) 1.50000 + 0.866025i 1.50000 + 0.866025i 1.00000 \(0\)
0.500000 + 0.866025i \(0.333333\pi\)
\(164\) 0 0
\(165\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(166\) 0 0
\(167\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 1.00000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) −0.500000 0.866025i −0.500000 0.866025i
\(173\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(174\) 1.73205i 1.73205i
\(175\) 0 0
\(176\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(177\) 1.00000i 1.00000i
\(178\) 0 0
\(179\) 0.866025 + 1.50000i 0.866025 + 1.50000i 0.866025 + 0.500000i \(0.166667\pi\)
1.00000i \(0.5\pi\)
\(180\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 1.50000 0.866025i 1.50000 0.866025i
\(185\) −0.866025 1.50000i −0.866025 1.50000i
\(186\) 0.866025 1.50000i 0.866025 1.50000i
\(187\) 0 0
\(188\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(192\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(193\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(194\) 0 0
\(195\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(196\) 1.00000 1.00000
\(197\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(198\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(199\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(200\) 1.00000i 1.00000i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1.73205 1.73205
\(208\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(216\) 1.00000i 1.00000i
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) −1.00000 −1.00000
\(221\) 0 0
\(222\) −1.73205 −1.73205
\(223\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(224\) 0 0
\(225\) 0.500000 0.866025i 0.500000 0.866025i
\(226\) 1.73205i 1.73205i
\(227\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) −0.866025 + 1.50000i −0.866025 + 1.50000i
\(231\) 0 0
\(232\) 1.50000 0.866025i 1.50000 0.866025i
\(233\) 1.73205 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(234\) 1.00000i 1.00000i
\(235\) −1.00000 −1.00000
\(236\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(237\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) −0.866025 0.500000i −0.866025 0.500000i
\(241\) −1.50000 0.866025i −1.50000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(244\) 0 0
\(245\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(246\) 0 0
\(247\) 0 0
\(248\) 1.73205 1.73205
\(249\) 0 0
\(250\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(251\) 0.866025 1.50000i 0.866025 1.50000i 1.00000i \(-0.5\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(252\) 0 0
\(253\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(254\) 0 0
\(255\) 0 0
\(256\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(257\) −0.866025 1.50000i −0.866025 1.50000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(-0.5\pi\)
\(258\) 0.866025 0.500000i 0.866025 0.500000i
\(259\) 0 0
\(260\) −0.866025 0.500000i −0.866025 0.500000i
\(261\) 1.73205 1.73205
\(262\) 1.50000 0.866025i 1.50000 0.866025i
\(263\) −0.866025 1.50000i −0.866025 1.50000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(-0.5\pi\)
\(264\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(270\) −0.500000 0.866025i −0.500000 0.866025i
\(271\) −1.50000 + 0.866025i −1.50000 + 0.866025i −0.500000 + 0.866025i \(0.666667\pi\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 1.00000 1.00000
\(275\) 0.866025 0.500000i 0.866025 0.500000i
\(276\) 0.866025 + 1.50000i 0.866025 + 1.50000i
\(277\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(283\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(287\) 0 0
\(288\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(289\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(290\) −0.866025 + 1.50000i −0.866025 + 1.50000i
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(294\) 1.00000i 1.00000i
\(295\) 0.500000 0.866025i 0.500000 0.866025i
\(296\) −0.866025 1.50000i −0.866025 1.50000i
\(297\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(298\) −1.00000 −1.00000
\(299\) 0.866025 + 1.50000i 0.866025 + 1.50000i
\(300\) 1.00000 1.00000
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(315\) 0 0
\(316\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(320\) 1.00000i 1.00000i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −1.00000 −1.00000
\(325\) 1.00000 1.00000
\(326\) −1.73205 −1.73205
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 1.00000i 1.00000i
\(331\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(332\) 0 0
\(333\) 1.73205i 1.73205i
\(334\) 0.500000 0.866025i 0.500000 0.866025i
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(339\) 1.73205 1.73205
\(340\) 0 0
\(341\) 0.866025 + 1.50000i 0.866025 + 1.50000i
\(342\) 0 0
\(343\) 0 0
\(344\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(345\) −1.50000 0.866025i −1.50000 0.866025i
\(346\) 0 0
\(347\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(348\) 0.866025 + 1.50000i 0.866025 + 1.50000i
\(349\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(350\) 0 0
\(351\) −1.00000 −1.00000
\(352\) −1.00000 −1.00000
\(353\) 1.73205 1.00000i 1.73205 1.00000i 0.866025 0.500000i \(-0.166667\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(354\) −0.500000 0.866025i −0.500000 0.866025i
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) −1.50000 0.866025i −1.50000 0.866025i
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0.500000 0.866025i 0.500000 0.866025i
\(361\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(362\) 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 \(0.333333\pi\)
\(368\) −0.866025 + 1.50000i −0.866025 + 1.50000i
\(369\) 0 0
\(370\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(371\) 0 0
\(372\) 1.73205i 1.73205i
\(373\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(374\) 0 0
\(375\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(376\) −1.00000 −1.00000
\(377\) 0.866025 + 1.50000i 0.866025 + 1.50000i
\(378\) 0 0
\(379\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(384\) −0.866025 0.500000i −0.866025 0.500000i
\(385\) 0 0
\(386\) 0 0
\(387\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(388\) 0 0
\(389\) 1.73205 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(390\) 0.500000 0.866025i 0.500000 0.866025i
\(391\) 0 0
\(392\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(393\) 0.866025 + 1.50000i 0.866025 + 1.50000i
\(394\) 0 0
\(395\) 1.00000i 1.00000i
\(396\) −0.866025 0.500000i −0.866025 0.500000i
\(397\) −1.50000 0.866025i −1.50000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
−1.00000 \(\pi\)
\(398\) 2.00000i 2.00000i
\(399\) 0 0
\(400\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(401\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(402\) 0 0
\(403\) 1.73205i 1.73205i
\(404\) 0 0
\(405\) 0.866025 0.500000i 0.866025 0.500000i
\(406\) 0 0
\(407\) 0.866025 1.50000i 0.866025 1.50000i
\(408\) 0 0
\(409\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(410\) 0 0
\(411\) 1.00000i 1.00000i
\(412\) 0 0
\(413\) 0 0
\(414\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(415\) 0 0
\(416\) −0.866025 0.500000i −0.866025 0.500000i
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(422\) 0 0
\(423\) −0.866025 0.500000i −0.866025 0.500000i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −0.866025 0.500000i −0.866025 0.500000i
\(430\) −1.00000 −1.00000
\(431\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(432\) −0.500000 0.866025i −0.500000 0.866025i
\(433\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(434\) 0 0
\(435\) −1.50000 0.866025i −1.50000 0.866025i
\(436\) 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 \(-0.666667\pi\)
\(440\) 0.866025 0.500000i 0.866025 0.500000i
\(441\) −1.00000 −1.00000
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 1.50000 0.866025i 1.50000 0.866025i
\(445\) 0 0
\(446\) 0 0
\(447\) 1.00000i 1.00000i
\(448\) 0 0
\(449\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(450\) 1.00000i 1.00000i
\(451\) 0 0
\(452\) 0.866025 + 1.50000i 0.866025 + 1.50000i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 1.73205i 1.73205i
\(461\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) −0.866025 + 1.50000i −0.866025 + 1.50000i
\(465\) −0.866025 1.50000i −0.866025 1.50000i
\(466\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) −0.500000 0.866025i −0.500000 0.866025i
\(469\) 0 0
\(470\) 0.866025 0.500000i 0.866025 0.500000i
\(471\) −0.500000 0.866025i −0.500000 0.866025i
\(472\) 0.500000 0.866025i 0.500000 0.866025i
\(473\) 1.00000i 1.00000i
\(474\) −0.866025 0.500000i −0.866025 0.500000i
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(480\) 1.00000 1.00000
\(481\) 1.50000 0.866025i 1.50000 0.866025i
\(482\) 1.73205 1.73205
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 1.00000i 1.00000i
\(487\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(488\) 0 0
\(489\) 1.73205i 1.73205i
\(490\) 0.500000 0.866025i 0.500000 0.866025i
\(491\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 1.00000 1.00000
\(496\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) −0.866025 0.500000i −0.866025 0.500000i
\(501\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(502\) 1.73205i 1.73205i
\(503\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −1.73205 −1.73205
\(507\) −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 0
\(512\) 1.00000i 1.00000i
\(513\) 0 0
\(514\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(515\) 0 0
\(516\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(517\) −0.500000 0.866025i −0.500000 0.866025i
\(518\) 0 0
\(519\) 0 0
\(520\) 1.00000 1.00000
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(523\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(524\) −0.866025 + 1.50000i −0.866025 + 1.50000i
\(525\) 0 0
\(526\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(527\) 0 0
\(528\) 1.00000i 1.00000i
\(529\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(530\) 0 0
\(531\) 0.866025 0.500000i 0.866025 0.500000i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0.866025 1.50000i 0.866025 1.50000i
\(538\) 0 0
\(539\) −0.866025 0.500000i −0.866025 0.500000i
\(540\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(541\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(542\) 0.866025 1.50000i 0.866025 1.50000i
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(548\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(549\) 0 0
\(550\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(551\) 0 0
\(552\) −1.50000 0.866025i −1.50000 0.866025i
\(553\) 0 0
\(554\) 1.00000i 1.00000i
\(555\) −0.866025 + 1.50000i −0.866025 + 1.50000i
\(556\) 0 0
\(557\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(558\) −1.73205 −1.73205
\(559\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(564\) 1.00000i 1.00000i
\(565\) −1.50000 0.866025i −1.50000 0.866025i
\(566\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(567\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 1.00000i 1.00000i
\(573\) 0 0
\(574\) 0 0
\(575\) 0.866025 + 1.50000i 0.866025 + 1.50000i
\(576\) 0.500000 0.866025i 0.500000 0.866025i
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) −0.866025 0.500000i −0.866025 0.500000i
\(579\) 0 0
\(580\) 1.73205i 1.73205i
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(586\) 0 0
\(587\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(588\) −0.500000 0.866025i −0.500000 0.866025i
\(589\) 0 0
\(590\) 1.00000i 1.00000i
\(591\) 0 0
\(592\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(593\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(594\) 0.500000 0.866025i 0.500000 0.866025i
\(595\) 0 0
\(596\) 0.866025 0.500000i 0.866025 0.500000i
\(597\) 2.00000 2.00000
\(598\) −1.50000 0.866025i −1.50000 0.866025i
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(601\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 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
\(611\) 1.00000i 1.00000i
\(612\) 0 0
\(613\) 1.50000 0.866025i 1.50000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
1.00000 \(0\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0.866025 1.50000i 0.866025 1.50000i
\(621\) −0.866025 1.50000i −0.866025 1.50000i
\(622\) 0 0
\(623\) 0 0
\(624\) 0.500000 0.866025i 0.500000 0.866025i
\(625\) 1.00000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0.500000 0.866025i 0.500000 0.866025i
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(632\) 1.00000i 1.00000i
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −0.500000 0.866025i −0.500000 0.866025i
\(638\) −1.73205 −1.73205
\(639\) 0 0
\(640\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(641\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(642\) 0 0
\(643\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(644\) 0 0
\(645\) 1.00000i 1.00000i
\(646\) 0 0
\(647\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(648\) 0.866025 0.500000i 0.866025 0.500000i
\(649\) 1.00000 1.00000
\(650\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(651\) 0 0
\(652\) 1.50000 0.866025i 1.50000 0.866025i
\(653\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(654\) 0 0
\(655\) 1.73205i 1.73205i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0.866025 1.50000i 0.866025 1.50000i 1.00000i \(-0.5\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(660\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(661\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0.866025 + 1.50000i 0.866025 + 1.50000i
\(667\) −1.50000 + 2.59808i −1.50000 + 2.59808i
\(668\) 1.00000i 1.00000i
\(669\) 0 0
\(670\) 0 0
\(671\) 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
\(675\) −1.00000 −1.00000
\(676\) 0.500000 0.866025i 0.500000 0.866025i
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) −1.50000 0.866025i −1.50000 0.866025i
\(683\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(684\) 0 0
\(685\) 0.500000 0.866025i 0.500000 0.866025i
\(686\) 0 0
\(687\) 0 0
\(688\) −1.00000 −1.00000
\(689\) 0 0
\(690\) 1.73205 1.73205
\(691\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) −1.50000 0.866025i −1.50000 0.866025i
\(697\) 0 0
\(698\) 0 0
\(699\) −0.866025 1.50000i −0.866025 1.50000i
\(700\) 0 0
\(701\) −1.73205 −1.73205 −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(702\) 0.866025 0.500000i 0.866025 0.500000i
\(703\) 0 0
\(704\) 0.866025 0.500000i 0.866025 0.500000i
\(705\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(706\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(707\) 0 0
\(708\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(709\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(710\) 0 0
\(711\) 0.500000 0.866025i 0.500000 0.866025i
\(712\) 0 0
\(713\) −2.59808 + 1.50000i −2.59808 + 1.50000i
\(714\) 0 0
\(715\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(716\) 1.73205 1.73205
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(720\) 1.00000i 1.00000i
\(721\) 0 0
\(722\) −0.866025 0.500000i −0.866025 0.500000i
\(723\) 1.73205i 1.73205i
\(724\) 0 0
\(725\) 0.866025 + 1.50000i 0.866025 + 1.50000i
\(726\) 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 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(736\) 1.73205i 1.73205i
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(740\) −1.73205 −1.73205
\(741\) 0 0
\(742\) 0 0
\(743\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(744\) −0.866025 1.50000i −0.866025 1.50000i
\(745\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(746\) 1.00000i 1.00000i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0.500000 0.866025i 0.500000 0.866025i
\(751\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(752\) 0.866025 0.500000i 0.866025 0.500000i
\(753\) −1.73205 −1.73205
\(754\) −1.50000 0.866025i −1.50000 0.866025i
\(755\) 0 0
\(756\) 0 0
\(757\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(758\) 0 0
\(759\) 1.73205i 1.73205i
\(760\) 0 0
\(761\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 1.00000 1.00000
\(767\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(768\) 1.00000 1.00000
\(769\) −1.50000 + 0.866025i −1.50000 + 0.866025i −0.500000 + 0.866025i \(0.666667\pi\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) −0.866025 + 1.50000i −0.866025 + 1.50000i
\(772\) 0 0
\(773\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(774\) −0.866025 0.500000i −0.866025 0.500000i
\(775\) 1.73205i 1.73205i
\(776\) 0 0
\(777\) 0 0
\(778\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(779\) 0 0
\(780\) 1.00000i 1.00000i
\(781\) 0 0
\(782\) 0 0
\(783\) −0.866025 1.50000i −0.866025 1.50000i
\(784\) 0.500000 0.866025i 0.500000 0.866025i
\(785\) 1.00000i 1.00000i
\(786\) −1.50000 0.866025i −1.50000 0.866025i
\(787\) −1.50000 0.866025i −1.50000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
−1.00000 \(\pi\)
\(788\) 0 0
\(789\) −0.866025 + 1.50000i −0.866025 + 1.50000i
\(790\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(791\) 0 0
\(792\) 1.00000 1.00000
\(793\) 0 0
\(794\) 1.73205 1.73205
\(795\) 0 0
\(796\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(797\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −0.866025 0.500000i −0.866025 0.500000i
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) −0.866025 1.50000i −0.866025 1.50000i
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(810\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(814\) 1.73205i 1.73205i
\(815\) −0.866025 + 1.50000i −0.866025 + 1.50000i
\(816\) 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 \(0.5\pi\)
\(822\) −0.500000 0.866025i −0.500000 0.866025i
\(823\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(824\) 0 0
\(825\) −0.866025 0.500000i −0.866025 0.500000i
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0.866025 1.50000i 0.866025 1.50000i
\(829\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(830\) 0 0
\(831\) 1.00000 1.00000
\(832\) 1.00000 1.00000
\(833\) 0 0
\(834\) 0 0
\(835\) −0.500000 0.866025i −0.500000 0.866025i
\(836\) 0 0
\(837\) 1.73205i 1.73205i
\(838\) 0 0
\(839\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(840\) 0 0
\(841\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.00000i 1.00000i
\(846\) 1.00000 1.00000
\(847\) 0 0
\(848\) 0 0
\(849\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(850\) 0 0
\(851\) 2.59808 + 1.50000i 2.59808 + 1.50000i
\(852\) 0 0
\(853\) 1.73205i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1.73205 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(858\) 1.00000 1.00000
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0.866025 0.500000i 0.866025 0.500000i
\(861\) 0 0
\(862\) 0 0
\(863\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(864\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(865\) 0 0
\(866\) 0 0
\(867\) 0.500000 0.866025i 0.500000 0.866025i
\(868\) 0 0
\(869\) 0.866025 0.500000i 0.866025 0.500000i
\(870\) 1.73205 1.73205
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1.50000 + 0.866025i 1.50000 + 0.866025i 1.00000 \(0\)
0.500000 + 0.866025i \(0.333333\pi\)
\(878\) 1.73205 + 1.00000i 1.73205 + 1.00000i
\(879\) 0 0
\(880\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(881\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(882\) 0.866025 0.500000i 0.866025 0.500000i
\(883\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(884\) 0 0
\(885\) −1.00000 −1.00000
\(886\) 0 0
\(887\) 0.866025 + 1.50000i 0.866025 + 1.50000i 0.866025 + 0.500000i \(0.166667\pi\)
1.00000i \(0.5\pi\)
\(888\) −0.866025 + 1.50000i −0.866025 + 1.50000i
\(889\) 0 0
\(890\) 0 0
\(891\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(892\) 0 0
\(893\) 0 0
\(894\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(895\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(896\) 0 0
\(897\) 0.866025 1.50000i 0.866025 1.50000i
\(898\) 0 0
\(899\) −2.59808 + 1.50000i −2.59808 + 1.50000i
\(900\) −0.500000 0.866025i −0.500000 0.866025i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) −1.50000 0.866025i −1.50000 0.866025i
\(905\) 0 0
\(906\) 0 0
\(907\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(908\) 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 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 1.00000 1.73205i 1.00000 1.73205i 0.500000 0.866025i \(-0.333333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(920\) 0.866025 + 1.50000i 0.866025 + 1.50000i
\(921\) 0 0
\(922\) 1.00000 1.00000
\(923\) 0 0
\(924\) 0 0
\(925\) 1.50000 0.866025i 1.50000 0.866025i
\(926\) 0 0
\(927\) 0 0
\(928\) 1.73205i 1.73205i
\(929\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(930\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(931\) 0 0
\(932\) 0.866025 1.50000i 0.866025 1.50000i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(941\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(942\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(943\) 0 0
\(944\) 1.00000i 1.00000i
\(945\) 0 0
\(946\) −0.500000 0.866025i −0.500000 0.866025i
\(947\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(948\) 1.00000 1.00000
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0.866025 1.50000i 0.866025 1.50000i 1.00000i \(-0.5\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 1.73205i 1.73205i
\(958\) 0 0
\(959\) 0 0
\(960\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(961\) −2.00000 −2.00000
\(962\) −0.866025 + 1.50000i −0.866025 + 1.50000i
\(963\) 0 0
\(964\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(972\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(973\) 0 0
\(974\) 0 0
\(975\) −0.500000 0.866025i −0.500000 0.866025i
\(976\) 0 0
\(977\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(978\) 0.866025 + 1.50000i 0.866025 + 1.50000i
\(979\) 0 0
\(980\) 1.00000i 1.00000i
\(981\) 0 0
\(982\) 0 0
\(983\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −1.73205 −1.73205
\(990\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(991\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(992\) 0.866025 1.50000i 0.866025 1.50000i
\(993\) 0 0
\(994\) 0 0
\(995\) −1.73205 1.00000i −1.73205 1.00000i
\(996\) 0 0
\(997\) 1.00000 1.73205i 1.00000 1.73205i 0.500000 0.866025i \(-0.333333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(998\) 0 0
\(999\) −1.50000 + 0.866025i −1.50000 + 0.866025i
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 1560.1.dk.a.1349.1 yes 4
3.2 odd 2 inner 1560.1.dk.a.1349.2 yes 4
5.4 even 2 1560.1.dk.b.1349.2 yes 4
8.5 even 2 1560.1.dk.b.1349.1 yes 4
13.4 even 6 inner 1560.1.dk.a.1109.1 4
15.14 odd 2 1560.1.dk.b.1349.1 yes 4
24.5 odd 2 1560.1.dk.b.1349.2 yes 4
39.17 odd 6 inner 1560.1.dk.a.1109.2 yes 4
40.29 even 2 inner 1560.1.dk.a.1349.2 yes 4
65.4 even 6 1560.1.dk.b.1109.2 yes 4
104.69 even 6 1560.1.dk.b.1109.1 yes 4
120.29 odd 2 CM 1560.1.dk.a.1349.1 yes 4
195.134 odd 6 1560.1.dk.b.1109.1 yes 4
312.173 odd 6 1560.1.dk.b.1109.2 yes 4
520.69 even 6 inner 1560.1.dk.a.1109.2 yes 4
1560.1109 odd 6 inner 1560.1.dk.a.1109.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1560.1.dk.a.1109.1 4 13.4 even 6 inner
1560.1.dk.a.1109.1 4 1560.1109 odd 6 inner
1560.1.dk.a.1109.2 yes 4 39.17 odd 6 inner
1560.1.dk.a.1109.2 yes 4 520.69 even 6 inner
1560.1.dk.a.1349.1 yes 4 1.1 even 1 trivial
1560.1.dk.a.1349.1 yes 4 120.29 odd 2 CM
1560.1.dk.a.1349.2 yes 4 3.2 odd 2 inner
1560.1.dk.a.1349.2 yes 4 40.29 even 2 inner
1560.1.dk.b.1109.1 yes 4 104.69 even 6
1560.1.dk.b.1109.1 yes 4 195.134 odd 6
1560.1.dk.b.1109.2 yes 4 65.4 even 6
1560.1.dk.b.1109.2 yes 4 312.173 odd 6
1560.1.dk.b.1349.1 yes 4 8.5 even 2
1560.1.dk.b.1349.1 yes 4 15.14 odd 2
1560.1.dk.b.1349.2 yes 4 5.4 even 2
1560.1.dk.b.1349.2 yes 4 24.5 odd 2