Properties

Label 168.1.s.a.53.1
Level $168$
Weight $1$
Character 168.53
Analytic conductor $0.084$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -24
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [168,1,Mod(53,168)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(168, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 3, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("168.53");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 168.s (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: \(0.0838429221223\)
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: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.1176.1
Artin image: $C_3\times S_3$
Artin field: Galois closure of 6.0.677376.1

Embedding invariants

Embedding label 53.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 168.53
Dual form 168.1.s.a.149.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{4} +(0.500000 + 0.866025i) q^{5} +1.00000 q^{6} +(-0.500000 + 0.866025i) q^{7} +1.00000 q^{8} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\) \(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{4} +(0.500000 + 0.866025i) q^{5} +1.00000 q^{6} +(-0.500000 + 0.866025i) q^{7} +1.00000 q^{8} +(-0.500000 - 0.866025i) q^{9} +(0.500000 - 0.866025i) q^{10} +(0.500000 - 0.866025i) q^{11} +(-0.500000 - 0.866025i) q^{12} +1.00000 q^{14} -1.00000 q^{15} +(-0.500000 - 0.866025i) q^{16} +(-0.500000 + 0.866025i) q^{18} -1.00000 q^{20} +(-0.500000 - 0.866025i) q^{21} -1.00000 q^{22} +(-0.500000 + 0.866025i) q^{24} +1.00000 q^{27} +(-0.500000 - 0.866025i) q^{28} -1.00000 q^{29} +(0.500000 + 0.866025i) q^{30} +(0.500000 - 0.866025i) q^{31} +(-0.500000 + 0.866025i) q^{32} +(0.500000 + 0.866025i) q^{33} -1.00000 q^{35} +1.00000 q^{36} +(0.500000 + 0.866025i) q^{40} +(-0.500000 + 0.866025i) q^{42} +(0.500000 + 0.866025i) q^{44} +(0.500000 - 0.866025i) q^{45} +1.00000 q^{48} +(-0.500000 - 0.866025i) q^{49} +(0.500000 - 0.866025i) q^{53} +(-0.500000 - 0.866025i) q^{54} +1.00000 q^{55} +(-0.500000 + 0.866025i) q^{56} +(0.500000 + 0.866025i) q^{58} +(0.500000 - 0.866025i) q^{59} +(0.500000 - 0.866025i) q^{60} -1.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +(0.500000 - 0.866025i) q^{66} +(0.500000 + 0.866025i) q^{70} +(-0.500000 - 0.866025i) q^{72} +(-1.00000 + 1.73205i) q^{73} +(0.500000 + 0.866025i) q^{77} +(0.500000 + 0.866025i) q^{79} +(0.500000 - 0.866025i) q^{80} +(-0.500000 + 0.866025i) q^{81} -1.00000 q^{83} +1.00000 q^{84} +(0.500000 - 0.866025i) q^{87} +(0.500000 - 0.866025i) q^{88} -1.00000 q^{90} +(0.500000 + 0.866025i) q^{93} +(-0.500000 - 0.866025i) q^{96} -1.00000 q^{97} +(-0.500000 + 0.866025i) q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{3} - q^{4} + q^{5} + 2 q^{6} - q^{7} + 2 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - q^{3} - q^{4} + q^{5} + 2 q^{6} - q^{7} + 2 q^{8} - q^{9} + q^{10} + q^{11} - q^{12} + 2 q^{14} - 2 q^{15} - q^{16} - q^{18} - 2 q^{20} - q^{21} - 2 q^{22} - q^{24} + 2 q^{27} - q^{28} - 2 q^{29} + q^{30} + q^{31} - q^{32} + q^{33} - 2 q^{35} + 2 q^{36} + q^{40} - q^{42} + q^{44} + q^{45} + 2 q^{48} - q^{49} + q^{53} - q^{54} + 2 q^{55} - q^{56} + q^{58} + q^{59} + q^{60} - 2 q^{62} + 2 q^{63} + 2 q^{64} + q^{66} + q^{70} - q^{72} - 2 q^{73} + q^{77} + q^{79} + q^{80} - q^{81} - 2 q^{83} + 2 q^{84} + q^{87} + q^{88} - 2 q^{90} + q^{93} - q^{96} - 2 q^{97} - q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/168\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(85\) \(113\) \(127\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.500000 0.866025i −0.500000 0.866025i
\(3\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(4\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(5\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(6\) 1.00000 1.00000
\(7\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(8\) 1.00000 1.00000
\(9\) −0.500000 0.866025i −0.500000 0.866025i
\(10\) 0.500000 0.866025i 0.500000 0.866025i
\(11\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(12\) −0.500000 0.866025i −0.500000 0.866025i
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 1.00000 1.00000
\(15\) −1.00000 −1.00000
\(16\) −0.500000 0.866025i −0.500000 0.866025i
\(17\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(18\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(19\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(20\) −1.00000 −1.00000
\(21\) −0.500000 0.866025i −0.500000 0.866025i
\(22\) −1.00000 −1.00000
\(23\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(24\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000 1.00000
\(28\) −0.500000 0.866025i −0.500000 0.866025i
\(29\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(30\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(31\) 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
\(33\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(34\) 0 0
\(35\) −1.00000 −1.00000
\(36\) 1.00000 1.00000
\(37\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(45\) 0.500000 0.866025i 0.500000 0.866025i
\(46\) 0 0
\(47\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(48\) 1.00000 1.00000
\(49\) −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 \(-0.666667\pi\)
1.00000 \(0\)
\(54\) −0.500000 0.866025i −0.500000 0.866025i
\(55\) 1.00000 1.00000
\(56\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(57\) 0 0
\(58\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(59\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(60\) 0.500000 0.866025i 0.500000 0.866025i
\(61\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(62\) −1.00000 −1.00000
\(63\) 1.00000 1.00000
\(64\) 1.00000 1.00000
\(65\) 0 0
\(66\) 0.500000 0.866025i 0.500000 0.866025i
\(67\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) −0.500000 0.866025i −0.500000 0.866025i
\(73\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(78\) 0 0
\(79\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(80\) 0.500000 0.866025i 0.500000 0.866025i
\(81\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(84\) 1.00000 1.00000
\(85\) 0 0
\(86\) 0 0
\(87\) 0.500000 0.866025i 0.500000 0.866025i
\(88\) 0.500000 0.866025i 0.500000 0.866025i
\(89\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(90\) −1.00000 −1.00000
\(91\) 0 0
\(92\) 0 0
\(93\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(94\) 0 0
\(95\) 0 0
\(96\) −0.500000 0.866025i −0.500000 0.866025i
\(97\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(98\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(99\) −1.00000 −1.00000
\(100\) 0 0
\(101\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(102\) 0 0
\(103\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(104\) 0 0
\(105\) 0.500000 0.866025i 0.500000 0.866025i
\(106\) −1.00000 −1.00000
\(107\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(108\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(109\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(110\) −0.500000 0.866025i −0.500000 0.866025i
\(111\) 0 0
\(112\) 1.00000 1.00000
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0.500000 0.866025i 0.500000 0.866025i
\(117\) 0 0
\(118\) −1.00000 −1.00000
\(119\) 0 0
\(120\) −1.00000 −1.00000
\(121\) 0 0
\(122\) 0 0
\(123\) 0 0
\(124\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(125\) 1.00000 1.00000
\(126\) −0.500000 0.866025i −0.500000 0.866025i
\(127\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(128\) −0.500000 0.866025i −0.500000 0.866025i
\(129\) 0 0
\(130\) 0 0
\(131\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(132\) −1.00000 −1.00000
\(133\) 0 0
\(134\) 0 0
\(135\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(136\) 0 0
\(137\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0.500000 0.866025i 0.500000 0.866025i
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(145\) −0.500000 0.866025i −0.500000 0.866025i
\(146\) 2.00000 2.00000
\(147\) 1.00000 1.00000
\(148\) 0 0
\(149\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(150\) 0 0
\(151\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0.500000 0.866025i 0.500000 0.866025i
\(155\) 1.00000 1.00000
\(156\) 0 0
\(157\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(158\) 0.500000 0.866025i 0.500000 0.866025i
\(159\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(160\) −1.00000 −1.00000
\(161\) 0 0
\(162\) 1.00000 1.00000
\(163\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(164\) 0 0
\(165\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(166\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) −0.500000 0.866025i −0.500000 0.866025i
\(169\) 1.00000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(174\) −1.00000 −1.00000
\(175\) 0 0
\(176\) −1.00000 −1.00000
\(177\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(178\) 0 0
\(179\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(180\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0.500000 0.866025i 0.500000 0.866025i
\(187\) 0 0
\(188\) 0 0
\(189\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(190\) 0 0
\(191\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(192\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(193\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(194\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(195\) 0 0
\(196\) 1.00000 1.00000
\(197\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(198\) 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 \(0.666667\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 2.00000 2.00000
\(203\) 0.500000 0.866025i 0.500000 0.866025i
\(204\) 0 0
\(205\) 0 0
\(206\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) −1.00000 −1.00000
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(213\) 0 0
\(214\) 0.500000 0.866025i 0.500000 0.866025i
\(215\) 0 0
\(216\) 1.00000 1.00000
\(217\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(218\) 0 0
\(219\) −1.00000 1.73205i −1.00000 1.73205i
\(220\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(221\) 0 0
\(222\) 0 0
\(223\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(224\) −0.500000 0.866025i −0.500000 0.866025i
\(225\) 0 0
\(226\) 0 0
\(227\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(228\) 0 0
\(229\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(230\) 0 0
\(231\) −1.00000 −1.00000
\(232\) −1.00000 −1.00000
\(233\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(237\) −1.00000 −1.00000
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(241\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(242\) 0 0
\(243\) −0.500000 0.866025i −0.500000 0.866025i
\(244\) 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
\(249\) 0.500000 0.866025i 0.500000 0.866025i
\(250\) −0.500000 0.866025i −0.500000 0.866025i
\(251\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(252\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(253\) 0 0
\(254\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(255\) 0 0
\(256\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(257\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(262\) 0.500000 0.866025i 0.500000 0.866025i
\(263\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(264\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(265\) 1.00000 1.00000
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(270\) 0.500000 0.866025i 0.500000 0.866025i
\(271\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(278\) 0 0
\(279\) −1.00000 −1.00000
\(280\) −1.00000 −1.00000
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 1.00000 1.00000
\(289\) −0.500000 0.866025i −0.500000 0.866025i
\(290\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(291\) 0.500000 0.866025i 0.500000 0.866025i
\(292\) −1.00000 1.73205i −1.00000 1.73205i
\(293\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(294\) −0.500000 0.866025i −0.500000 0.866025i
\(295\) 1.00000 1.00000
\(296\) 0 0
\(297\) 0.500000 0.866025i 0.500000 0.866025i
\(298\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) −1.00000 −1.00000
\(303\) −1.00000 1.73205i −1.00000 1.73205i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) −1.00000 −1.00000
\(309\) 2.00000 2.00000
\(310\) −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
\(313\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(314\) 0 0
\(315\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(316\) −1.00000 −1.00000
\(317\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(318\) 0.500000 0.866025i 0.500000 0.866025i
\(319\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(320\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(321\) −1.00000 −1.00000
\(322\) 0 0
\(323\) 0 0
\(324\) −0.500000 0.866025i −0.500000 0.866025i
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 1.00000 1.00000
\(331\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(332\) 0.500000 0.866025i 0.500000 0.866025i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(337\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(338\) −0.500000 0.866025i −0.500000 0.866025i
\(339\) 0 0
\(340\) 0 0
\(341\) −0.500000 0.866025i −0.500000 0.866025i
\(342\) 0 0
\(343\) 1.00000 1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(347\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(348\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(353\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(354\) 0.500000 0.866025i 0.500000 0.866025i
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 2.00000 2.00000
\(359\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(360\) 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
\(365\) −2.00000 −2.00000
\(366\) 0 0
\(367\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(372\) −1.00000 −1.00000
\(373\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(374\) 0 0
\(375\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(376\) 0 0
\(377\) 0 0
\(378\) 1.00000 1.00000
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0.500000 0.866025i 0.500000 0.866025i
\(382\) 0 0
\(383\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(384\) 1.00000 1.00000
\(385\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(386\) −1.00000 −1.00000
\(387\) 0 0
\(388\) 0.500000 0.866025i 0.500000 0.866025i
\(389\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −0.500000 0.866025i −0.500000 0.866025i
\(393\) −1.00000 −1.00000
\(394\) −1.00000 1.73205i −1.00000 1.73205i
\(395\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(396\) 0.500000 0.866025i 0.500000 0.866025i
\(397\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(398\) 2.00000 2.00000
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −1.00000 1.73205i −1.00000 1.73205i
\(405\) −1.00000 −1.00000
\(406\) −1.00000 −1.00000
\(407\) 0 0
\(408\) 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
\(412\) 2.00000 2.00000
\(413\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(414\) 0 0
\(415\) −0.500000 0.866025i −0.500000 0.866025i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(420\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0.500000 0.866025i 0.500000 0.866025i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −1.00000 −1.00000
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(432\) −0.500000 0.866025i −0.500000 0.866025i
\(433\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(434\) 0.500000 0.866025i 0.500000 0.866025i
\(435\) 1.00000 1.00000
\(436\) 0 0
\(437\) 0 0
\(438\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(439\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(440\) 1.00000 1.00000
\(441\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(442\) 0 0
\(443\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(447\) 2.00000 2.00000
\(448\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(454\) −1.00000 −1.00000
\(455\) 0 0
\(456\) 0 0
\(457\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(462\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(463\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(464\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(465\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(466\) 0 0
\(467\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0.500000 0.866025i 0.500000 0.866025i
\(473\) 0 0
\(474\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(475\) 0 0
\(476\) 0 0
\(477\) −1.00000 −1.00000
\(478\) 0 0
\(479\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(480\) 0.500000 0.866025i 0.500000 0.866025i
\(481\) 0 0
\(482\) −1.00000 −1.00000
\(483\) 0 0
\(484\) 0 0
\(485\) −0.500000 0.866025i −0.500000 0.866025i
\(486\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(487\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(488\) 0 0
\(489\) 0 0
\(490\) −1.00000 −1.00000
\(491\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −0.500000 0.866025i −0.500000 0.866025i
\(496\) −1.00000 −1.00000
\(497\) 0 0
\(498\) −1.00000 −1.00000
\(499\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(500\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(501\) 0 0
\(502\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 1.00000 1.00000
\(505\) −2.00000 −2.00000
\(506\) 0 0
\(507\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(508\) 0.500000 0.866025i 0.500000 0.866025i
\(509\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(510\) 0 0
\(511\) −1.00000 1.73205i −1.00000 1.73205i
\(512\) 1.00000 1.00000
\(513\) 0 0
\(514\) 0 0
\(515\) 1.00000 1.73205i 1.00000 1.73205i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 2.00000 2.00000
\(520\) 0 0
\(521\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(522\) 0.500000 0.866025i 0.500000 0.866025i
\(523\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(524\) −1.00000 −1.00000
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0.500000 0.866025i 0.500000 0.866025i
\(529\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(530\) −0.500000 0.866025i −0.500000 0.866025i
\(531\) −1.00000 −1.00000
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(536\) 0 0
\(537\) −1.00000 1.73205i −1.00000 1.73205i
\(538\) −1.00000 −1.00000
\(539\) −1.00000 −1.00000
\(540\) −1.00000 −1.00000
\(541\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(542\) 0.500000 0.866025i 0.500000 0.866025i
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −1.00000 −1.00000
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(558\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(559\) 0 0
\(560\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(561\) 0 0
\(562\) 0 0
\(563\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −0.500000 0.866025i −0.500000 0.866025i
\(568\) 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 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −0.500000 0.866025i −0.500000 0.866025i
\(577\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(578\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(579\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(580\) 1.00000 1.00000
\(581\) 0.500000 0.866025i 0.500000 0.866025i
\(582\) −1.00000 −1.00000
\(583\) −0.500000 0.866025i −0.500000 0.866025i
\(584\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(585\) 0 0
\(586\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(587\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(588\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(589\) 0 0
\(590\) −0.500000 0.866025i −0.500000 0.866025i
\(591\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(592\) 0 0
\(593\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(594\) −1.00000 −1.00000
\(595\) 0 0
\(596\) 2.00000 2.00000
\(597\) −1.00000 1.73205i −1.00000 1.73205i
\(598\) 0 0
\(599\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(600\) 0 0
\(601\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(605\) 0 0
\(606\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(607\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(608\) 0 0
\(609\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) −1.00000 1.73205i −1.00000 1.73205i
\(619\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(620\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(626\) 0.500000 0.866025i 0.500000 0.866025i
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0.500000 0.866025i 0.500000 0.866025i
\(631\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(632\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(633\) 0 0
\(634\) 0.500000 0.866025i 0.500000 0.866025i
\(635\) −0.500000 0.866025i −0.500000 0.866025i
\(636\) −1.00000 −1.00000
\(637\) 0 0
\(638\) 1.00000 1.00000
\(639\) 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 \(0.666667\pi\)
\(642\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(648\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(649\) −0.500000 0.866025i −0.500000 0.866025i
\(650\) 0 0
\(651\) −1.00000 −1.00000
\(652\) 0 0
\(653\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(654\) 0 0
\(655\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(656\) 0 0
\(657\) 2.00000 2.00000
\(658\) 0 0
\(659\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(660\) −0.500000 0.866025i −0.500000 0.866025i
\(661\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) −1.00000 −1.00000
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0.500000 0.866025i 0.500000 0.866025i
\(670\) 0 0
\(671\) 0 0
\(672\) 1.00000 1.00000
\(673\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(674\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(675\) 0 0
\(676\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(677\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(678\) 0 0
\(679\) 0.500000 0.866025i 0.500000 0.866025i
\(680\) 0 0
\(681\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(682\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(683\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(684\) 0 0
\(685\) 0 0
\(686\) −0.500000 0.866025i −0.500000 0.866025i
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(692\) 2.00000 2.00000
\(693\) 0.500000 0.866025i 0.500000 0.866025i
\(694\) 2.00000 2.00000
\(695\) 0 0
\(696\) 0.500000 0.866025i 0.500000 0.866025i
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0.500000 0.866025i 0.500000 0.866025i
\(705\) 0 0
\(706\) 0 0
\(707\) −1.00000 1.73205i −1.00000 1.73205i
\(708\) −1.00000 −1.00000
\(709\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(710\) 0 0
\(711\) 0.500000 0.866025i 0.500000 0.866025i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −1.00000 1.73205i −1.00000 1.73205i
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(720\) −1.00000 −1.00000
\(721\) 2.00000 2.00000
\(722\) 1.00000 1.00000
\(723\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(728\) 0 0
\(729\) 1.00000 1.00000
\(730\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(734\) −1.00000 −1.00000
\(735\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0.500000 0.866025i 0.500000 0.866025i
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(745\) 1.00000 1.73205i 1.00000 1.73205i
\(746\) 0 0
\(747\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(748\) 0 0
\(749\) −1.00000 −1.00000
\(750\) 1.00000 1.00000
\(751\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(752\) 0 0
\(753\) 0.500000 0.866025i 0.500000 0.866025i
\(754\) 0 0
\(755\) 1.00000 1.00000
\(756\) −0.500000 0.866025i −0.500000 0.866025i
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(762\) −1.00000 −1.00000
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −0.500000 0.866025i −0.500000 0.866025i
\(769\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(770\) 1.00000 1.00000
\(771\) 0 0
\(772\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(773\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −1.00000 −1.00000
\(777\) 0 0
\(778\) 2.00000 2.00000
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −1.00000 −1.00000
\(784\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(785\) 0 0
\(786\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(787\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(788\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(789\) 0 0
\(790\) 1.00000 1.00000
\(791\) 0 0
\(792\) −1.00000 −1.00000
\(793\) 0 0
\(794\) 0 0
\(795\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(796\) −1.00000 1.73205i −1.00000 1.73205i
\(797\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(808\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(809\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(810\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(813\) −1.00000 −1.00000
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) −1.00000 −1.00000
\(819\) 0 0
\(820\) 0 0
\(821\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(822\) 0 0
\(823\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(824\) −1.00000 1.73205i −1.00000 1.73205i
\(825\) 0 0
\(826\) 0.500000 0.866025i 0.500000 0.866025i
\(827\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(828\) 0 0
\(829\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(830\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0.500000 0.866025i 0.500000 0.866025i
\(838\) −1.00000 1.73205i −1.00000 1.73205i
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0.500000 0.866025i 0.500000 0.866025i
\(841\) 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
\(848\) −1.00000 −1.00000
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(857\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(858\) 0 0
\(859\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(864\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(865\) 1.00000 1.73205i 1.00000 1.73205i
\(866\) −1.00000 1.73205i −1.00000 1.73205i
\(867\) 1.00000 1.00000
\(868\) −1.00000 −1.00000
\(869\) 1.00000 1.00000
\(870\) −0.500000 0.866025i −0.500000 0.866025i
\(871\) 0 0
\(872\) 0 0
\(873\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(874\) 0 0
\(875\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(876\) 2.00000 2.00000
\(877\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(878\) 0.500000 0.866025i 0.500000 0.866025i
\(879\) 0.500000 0.866025i 0.500000 0.866025i
\(880\) −0.500000 0.866025i −0.500000 0.866025i
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 1.00000 1.00000
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(886\) 0.500000 0.866025i 0.500000 0.866025i
\(887\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(888\) 0 0
\(889\) 0.500000 0.866025i 0.500000 0.866025i
\(890\) 0 0
\(891\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(892\) 0.500000 0.866025i 0.500000 0.866025i
\(893\) 0 0
\(894\) −1.00000 1.73205i −1.00000 1.73205i
\(895\) −2.00000 −2.00000
\(896\) 1.00000 1.00000
\(897\) 0 0
\(898\) 0 0
\(899\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0.500000 0.866025i 0.500000 0.866025i
\(907\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(908\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(909\) 2.00000 2.00000
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(914\) 0.500000 0.866025i 0.500000 0.866025i
\(915\) 0 0
\(916\) 0 0
\(917\) −1.00000 −1.00000
\(918\) 0 0
\(919\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −1.00000 1.73205i −1.00000 1.73205i
\(923\) 0 0
\(924\) 0.500000 0.866025i 0.500000 0.866025i
\(925\) 0 0
\(926\) −1.00000 1.73205i −1.00000 1.73205i
\(927\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(928\) 0.500000 0.866025i 0.500000 0.866025i
\(929\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(930\) 1.00000 1.00000
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(935\) 0 0
\(936\) 0 0
\(937\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(938\) 0 0
\(939\) −1.00000 −1.00000
\(940\) 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
\(944\) −1.00000 −1.00000
\(945\) −1.00000 −1.00000
\(946\) 0 0
\(947\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(948\) 0.500000 0.866025i 0.500000 0.866025i
\(949\) 0 0
\(950\) 0 0
\(951\) −1.00000 −1.00000
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(955\) 0 0
\(956\) 0 0
\(957\) −0.500000 0.866025i −0.500000 0.866025i
\(958\) 0 0
\(959\) 0 0
\(960\) −1.00000 −1.00000
\(961\) 0 0
\(962\) 0 0
\(963\) 0.500000 0.866025i 0.500000 0.866025i
\(964\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(965\) 1.00000 1.00000
\(966\) 0 0
\(967\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(971\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(972\) 1.00000 1.00000
\(973\) 0 0
\(974\) −1.00000 −1.00000
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(981\) 0 0
\(982\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(983\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(984\) 0 0
\(985\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(991\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(992\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(993\) 0 0
\(994\) 0 0
\(995\) −2.00000 −2.00000
\(996\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(997\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(998\) 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 168.1.s.a.53.1 2
3.2 odd 2 168.1.s.b.53.1 yes 2
4.3 odd 2 672.1.ba.b.305.1 2
7.2 even 3 inner 168.1.s.a.149.1 yes 2
7.3 odd 6 1176.1.n.c.197.1 1
7.4 even 3 1176.1.n.d.197.1 1
7.5 odd 6 1176.1.s.a.1157.1 2
7.6 odd 2 1176.1.s.a.557.1 2
8.3 odd 2 672.1.ba.a.305.1 2
8.5 even 2 168.1.s.b.53.1 yes 2
12.11 even 2 672.1.ba.a.305.1 2
21.2 odd 6 168.1.s.b.149.1 yes 2
21.5 even 6 1176.1.s.b.1157.1 2
21.11 odd 6 1176.1.n.a.197.1 1
21.17 even 6 1176.1.n.b.197.1 1
21.20 even 2 1176.1.s.b.557.1 2
24.5 odd 2 CM 168.1.s.a.53.1 2
24.11 even 2 672.1.ba.b.305.1 2
28.23 odd 6 672.1.ba.b.401.1 2
56.5 odd 6 1176.1.s.b.1157.1 2
56.13 odd 2 1176.1.s.b.557.1 2
56.37 even 6 168.1.s.b.149.1 yes 2
56.45 odd 6 1176.1.n.b.197.1 1
56.51 odd 6 672.1.ba.a.401.1 2
56.53 even 6 1176.1.n.a.197.1 1
84.23 even 6 672.1.ba.a.401.1 2
168.5 even 6 1176.1.s.a.1157.1 2
168.53 odd 6 1176.1.n.d.197.1 1
168.101 even 6 1176.1.n.c.197.1 1
168.107 even 6 672.1.ba.b.401.1 2
168.125 even 2 1176.1.s.a.557.1 2
168.149 odd 6 inner 168.1.s.a.149.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.1.s.a.53.1 2 1.1 even 1 trivial
168.1.s.a.53.1 2 24.5 odd 2 CM
168.1.s.a.149.1 yes 2 7.2 even 3 inner
168.1.s.a.149.1 yes 2 168.149 odd 6 inner
168.1.s.b.53.1 yes 2 3.2 odd 2
168.1.s.b.53.1 yes 2 8.5 even 2
168.1.s.b.149.1 yes 2 21.2 odd 6
168.1.s.b.149.1 yes 2 56.37 even 6
672.1.ba.a.305.1 2 8.3 odd 2
672.1.ba.a.305.1 2 12.11 even 2
672.1.ba.a.401.1 2 56.51 odd 6
672.1.ba.a.401.1 2 84.23 even 6
672.1.ba.b.305.1 2 4.3 odd 2
672.1.ba.b.305.1 2 24.11 even 2
672.1.ba.b.401.1 2 28.23 odd 6
672.1.ba.b.401.1 2 168.107 even 6
1176.1.n.a.197.1 1 21.11 odd 6
1176.1.n.a.197.1 1 56.53 even 6
1176.1.n.b.197.1 1 21.17 even 6
1176.1.n.b.197.1 1 56.45 odd 6
1176.1.n.c.197.1 1 7.3 odd 6
1176.1.n.c.197.1 1 168.101 even 6
1176.1.n.d.197.1 1 7.4 even 3
1176.1.n.d.197.1 1 168.53 odd 6
1176.1.s.a.557.1 2 7.6 odd 2
1176.1.s.a.557.1 2 168.125 even 2
1176.1.s.a.1157.1 2 7.5 odd 6
1176.1.s.a.1157.1 2 168.5 even 6
1176.1.s.b.557.1 2 21.20 even 2
1176.1.s.b.557.1 2 56.13 odd 2
1176.1.s.b.1157.1 2 21.5 even 6
1176.1.s.b.1157.1 2 56.5 odd 6