Properties

Label 476.1.o.a.135.1
Level $476$
Weight $1$
Character 476.135
Analytic conductor $0.238$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -68
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [476,1,Mod(67,476)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(476, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("476.67");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 476 = 2^{2} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 476.o (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: \(0.237554946013\)
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 3.1.3332.1

Embedding invariants

Embedding label 135.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 476.135
Dual form 476.1.o.a.67.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{4} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{4} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +(-0.500000 - 0.866025i) q^{11} +(-0.500000 + 0.866025i) q^{12} -1.00000 q^{13} +(0.500000 - 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(-0.500000 - 0.866025i) q^{17} +(0.500000 + 0.866025i) q^{21} +1.00000 q^{22} +(1.00000 - 1.73205i) q^{23} +(-0.500000 - 0.866025i) q^{24} +(-0.500000 - 0.866025i) q^{25} +(0.500000 - 0.866025i) q^{26} -1.00000 q^{27} +(0.500000 + 0.866025i) q^{28} +(1.00000 + 1.73205i) q^{31} +(-0.500000 - 0.866025i) q^{32} +(-0.500000 + 0.866025i) q^{33} +1.00000 q^{34} +(0.500000 + 0.866025i) q^{39} -1.00000 q^{42} +(-0.500000 + 0.866025i) q^{44} +(1.00000 + 1.73205i) q^{46} +1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +(-0.500000 + 0.866025i) q^{51} +(0.500000 + 0.866025i) q^{52} +(0.500000 + 0.866025i) q^{53} +(0.500000 - 0.866025i) q^{54} -1.00000 q^{56} -2.00000 q^{62} +1.00000 q^{64} +(-0.500000 - 0.866025i) q^{66} +(-0.500000 + 0.866025i) q^{68} -2.00000 q^{69} +1.00000 q^{71} +(-0.500000 + 0.866025i) q^{75} +(0.500000 + 0.866025i) q^{77} -1.00000 q^{78} +(-0.500000 + 0.866025i) q^{79} +(0.500000 + 0.866025i) q^{81} +(0.500000 - 0.866025i) q^{84} +(-0.500000 - 0.866025i) q^{88} +(0.500000 - 0.866025i) q^{89} +1.00000 q^{91} -2.00000 q^{92} +(1.00000 - 1.73205i) q^{93} +(-0.500000 + 0.866025i) q^{96} +(-0.500000 + 0.866025i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{3} - q^{4} + 2 q^{6} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - q^{3} - q^{4} + 2 q^{6} - 2 q^{7} + 2 q^{8} - q^{11} - q^{12} - 2 q^{13} + q^{14} - q^{16} - q^{17} + q^{21} + 2 q^{22} + 2 q^{23} - q^{24} - q^{25} + q^{26} - 2 q^{27} + q^{28} + 2 q^{31} - q^{32} - q^{33} + 2 q^{34} + q^{39} - 2 q^{42} - q^{44} + 2 q^{46} + 2 q^{48} + 2 q^{49} + 2 q^{50} - q^{51} + q^{52} + q^{53} + q^{54} - 2 q^{56} - 4 q^{62} + 2 q^{64} - q^{66} - q^{68} - 4 q^{69} + 2 q^{71} - q^{75} + q^{77} - 2 q^{78} - q^{79} + q^{81} + q^{84} - q^{88} + q^{89} + 2 q^{91} - 4 q^{92} + 2 q^{93} - q^{96} - q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/476\mathbb{Z}\right)^\times\).

\(n\) \(239\) \(309\) \(409\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{1}{3}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(3\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(4\) −0.500000 0.866025i −0.500000 0.866025i
\(5\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(6\) 1.00000 1.00000
\(7\) −1.00000 −1.00000
\(8\) 1.00000 1.00000
\(9\) 0 0
\(10\) 0 0
\(11\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(12\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(13\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(14\) 0.500000 0.866025i 0.500000 0.866025i
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(17\) −0.500000 0.866025i −0.500000 0.866025i
\(18\) 0 0
\(19\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(20\) 0 0
\(21\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(22\) 1.00000 1.00000
\(23\) 1.00000 1.73205i 1.00000 1.73205i 0.500000 0.866025i \(-0.333333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(24\) −0.500000 0.866025i −0.500000 0.866025i
\(25\) −0.500000 0.866025i −0.500000 0.866025i
\(26\) 0.500000 0.866025i 0.500000 0.866025i
\(27\) −1.00000 −1.00000
\(28\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(32\) −0.500000 0.866025i −0.500000 0.866025i
\(33\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(34\) 1.00000 1.00000
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(38\) 0 0
\(39\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) −1.00000 −1.00000
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(45\) 0 0
\(46\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(47\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(48\) 1.00000 1.00000
\(49\) 1.00000 1.00000
\(50\) 1.00000 1.00000
\(51\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(52\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(53\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(54\) 0.500000 0.866025i 0.500000 0.866025i
\(55\) 0 0
\(56\) −1.00000 −1.00000
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(60\) 0 0
\(61\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) −2.00000 −2.00000
\(63\) 0 0
\(64\) 1.00000 1.00000
\(65\) 0 0
\(66\) −0.500000 0.866025i −0.500000 0.866025i
\(67\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(68\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(69\) −2.00000 −2.00000
\(70\) 0 0
\(71\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(72\) 0 0
\(73\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(74\) 0 0
\(75\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(76\) 0 0
\(77\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(78\) −1.00000 −1.00000
\(79\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0.500000 0.866025i 0.500000 0.866025i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) −0.500000 0.866025i −0.500000 0.866025i
\(89\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(90\) 0 0
\(91\) 1.00000 1.00000
\(92\) −2.00000 −2.00000
\(93\) 1.00000 1.73205i 1.00000 1.73205i
\(94\) 0 0
\(95\) 0 0
\(96\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(99\) 0 0
\(100\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(101\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(102\) −0.500000 0.866025i −0.500000 0.866025i
\(103\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(104\) −1.00000 −1.00000
\(105\) 0 0
\(106\) −1.00000 −1.00000
\(107\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(108\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(109\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0.500000 0.866025i 0.500000 0.866025i
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 0 0
\(124\) 1.00000 1.73205i 1.00000 1.73205i
\(125\) 0 0
\(126\) 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 0
\(131\) 1.00000 1.73205i 1.00000 1.73205i 0.500000 0.866025i \(-0.333333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(132\) 1.00000 1.00000
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) −0.500000 0.866025i −0.500000 0.866025i
\(137\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(138\) 1.00000 1.73205i 1.00000 1.73205i
\(139\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(143\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −0.500000 0.866025i −0.500000 0.866025i
\(148\) 0 0
\(149\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(150\) −0.500000 0.866025i −0.500000 0.866025i
\(151\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −1.00000 −1.00000
\(155\) 0 0
\(156\) 0.500000 0.866025i 0.500000 0.866025i
\(157\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−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
\(160\) 0 0
\(161\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(162\) −1.00000 −1.00000
\(163\) 1.00000 1.73205i 1.00000 1.73205i 0.500000 0.866025i \(-0.333333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(168\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(169\) 0 0
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(174\) 0 0
\(175\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(176\) 1.00000 1.00000
\(177\) 0 0
\(178\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(179\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(183\) 0 0
\(184\) 1.00000 1.73205i 1.00000 1.73205i
\(185\) 0 0
\(186\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(187\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(188\) 0 0
\(189\) 1.00000 1.00000
\(190\) 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 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(194\) 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
\(199\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(200\) −0.500000 0.866025i −0.500000 0.866025i
\(201\) 0 0
\(202\) 2.00000 2.00000
\(203\) 0 0
\(204\) 1.00000 1.00000
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0.500000 0.866025i 0.500000 0.866025i
\(209\) 0 0
\(210\) 0 0
\(211\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(212\) 0.500000 0.866025i 0.500000 0.866025i
\(213\) −0.500000 0.866025i −0.500000 0.866025i
\(214\) −0.500000 0.866025i −0.500000 0.866025i
\(215\) 0 0
\(216\) −1.00000 −1.00000
\(217\) −1.00000 1.73205i −1.00000 1.73205i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 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 \(-0.333333\pi\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(230\) 0 0
\(231\) 0.500000 0.866025i 0.500000 0.866025i
\(232\) 0 0
\(233\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1.00000 1.00000
\(238\) −1.00000 −1.00000
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) −2.00000 −2.00000
\(254\) 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 \(-0.666667\pi\)
1.00000 \(0\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(263\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(264\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(265\) 0 0
\(266\) 0 0
\(267\) −1.00000 −1.00000
\(268\) 0 0
\(269\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(270\) 0 0
\(271\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(272\) 1.00000 1.00000
\(273\) −0.500000 0.866025i −0.500000 0.866025i
\(274\) −1.00000 −1.00000
\(275\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(276\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(277\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(278\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(279\) 0 0
\(280\) 0 0
\(281\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(282\) 0 0
\(283\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(284\) −0.500000 0.866025i −0.500000 0.866025i
\(285\) 0 0
\(286\) −1.00000 −1.00000
\(287\) 0 0
\(288\) 0 0
\(289\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(294\) 1.00000 1.00000
\(295\) 0 0
\(296\) 0 0
\(297\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(298\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(299\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(300\) 1.00000 1.00000
\(301\) 0 0
\(302\) 0 0
\(303\) −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 \(\pi\)
\(308\) 0.500000 0.866025i 0.500000 0.866025i
\(309\) 0 0
\(310\) 0 0
\(311\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(312\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(313\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(314\) −1.00000 −1.00000
\(315\) 0 0
\(316\) 1.00000 1.00000
\(317\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(318\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(319\) 0 0
\(320\) 0 0
\(321\) 1.00000 1.00000
\(322\) −1.00000 1.73205i −1.00000 1.73205i
\(323\) 0 0
\(324\) 0.500000 0.866025i 0.500000 0.866025i
\(325\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(326\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(335\) 0 0
\(336\) −1.00000 −1.00000
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 1.00000 1.73205i 1.00000 1.73205i
\(342\) 0 0
\(343\) −1.00000 −1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(348\) 0 0
\(349\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(350\) −1.00000 −1.00000
\(351\) 1.00000 1.00000
\(352\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(353\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −1.00000 −1.00000
\(357\) 0.500000 0.866025i 0.500000 0.866025i
\(358\) 0 0
\(359\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(360\) 0 0
\(361\) −0.500000 0.866025i −0.500000 0.866025i
\(362\) 0 0
\(363\) 0 0
\(364\) −0.500000 0.866025i −0.500000 0.866025i
\(365\) 0 0
\(366\) 0 0
\(367\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(368\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(369\) 0 0
\(370\) 0 0
\(371\) −0.500000 0.866025i −0.500000 0.866025i
\(372\) −2.00000 −2.00000
\(373\) 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 0
\(377\) 0 0
\(378\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(379\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(384\) 1.00000 1.00000
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(390\) 0 0
\(391\) −2.00000 −2.00000
\(392\) 1.00000 1.00000
\(393\) −2.00000 −2.00000
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(398\) 1.00000 1.00000
\(399\) 0 0
\(400\) 1.00000 1.00000
\(401\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(402\) 0 0
\(403\) −1.00000 1.73205i −1.00000 1.73205i
\(404\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(409\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(410\) 0 0
\(411\) 0.500000 0.866025i 0.500000 0.866025i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(417\) −0.500000 0.866025i −0.500000 0.866025i
\(418\) 0 0
\(419\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(420\) 0 0
\(421\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(422\) 1.00000 1.73205i 1.00000 1.73205i
\(423\) 0 0
\(424\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(425\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(426\) 1.00000 1.00000
\(427\) 0 0
\(428\) 1.00000 1.00000
\(429\) 0.500000 0.866025i 0.500000 0.866025i
\(430\) 0 0
\(431\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(432\) 0.500000 0.866025i 0.500000 0.866025i
\(433\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(434\) 2.00000 2.00000
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −1.00000 −1.00000
\(443\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −1.00000 −1.00000
\(448\) −1.00000 −1.00000
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 1.00000 1.00000
\(455\) 0 0
\(456\) 0 0
\(457\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(458\) −1.00000 1.73205i −1.00000 1.73205i
\(459\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(460\) 0 0
\(461\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(462\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0.500000 0.866025i 0.500000 0.866025i
\(472\) 0 0
\(473\) 0 0
\(474\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(475\) 0 0
\(476\) 0.500000 0.866025i 0.500000 0.866025i
\(477\) 0 0
\(478\) 0 0
\(479\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 2.00000 2.00000
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) −2.00000 −2.00000
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −2.00000 −2.00000
\(497\) −1.00000 −1.00000
\(498\) 0 0
\(499\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(500\) 0 0
\(501\) −0.500000 0.866025i −0.500000 0.866025i
\(502\) 0 0
\(503\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 1.00000 1.73205i 1.00000 1.73205i
\(507\) 0 0
\(508\) 0 0
\(509\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 1.00000
\(513\) 0 0
\(514\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(522\) 0 0
\(523\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(524\) −2.00000 −2.00000
\(525\) 0.500000 0.866025i 0.500000 0.866025i
\(526\) 0 0
\(527\) 1.00000 1.73205i 1.00000 1.73205i
\(528\) −0.500000 0.866025i −0.500000 0.866025i
\(529\) −1.50000 2.59808i −1.50000 2.59808i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0.500000 0.866025i 0.500000 0.866025i
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −0.500000 0.866025i −0.500000 0.866025i
\(540\) 0 0
\(541\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(545\) 0 0
\(546\) 1.00000 1.00000
\(547\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(548\) 0.500000 0.866025i 0.500000 0.866025i
\(549\) 0 0
\(550\) −0.500000 0.866025i −0.500000 0.866025i
\(551\) 0 0
\(552\) −2.00000 −2.00000
\(553\) 0.500000 0.866025i 0.500000 0.866025i
\(554\) 0 0
\(555\) 0 0
\(556\) −0.500000 0.866025i −0.500000 0.866025i
\(557\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 1.00000 1.00000
\(562\) 0.500000 0.866025i 0.500000 0.866025i
\(563\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 1.00000 1.00000
\(567\) −0.500000 0.866025i −0.500000 0.866025i
\(568\) 1.00000 1.00000
\(569\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(570\) 0 0
\(571\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(572\) 0.500000 0.866025i 0.500000 0.866025i
\(573\) 0 0
\(574\) 0 0
\(575\) −2.00000 −2.00000
\(576\) 0 0
\(577\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(578\) −0.500000 0.866025i −0.500000 0.866025i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0.500000 0.866025i 0.500000 0.866025i
\(584\) 0 0
\(585\) 0 0
\(586\) 0.500000 0.866025i 0.500000 0.866025i
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(594\) −1.00000 −1.00000
\(595\) 0 0
\(596\) −1.00000 −1.00000
\(597\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(598\) −1.00000 1.73205i −1.00000 1.73205i
\(599\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(600\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) −1.00000 1.73205i −1.00000 1.73205i
\(607\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−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 \(\pi\)
\(618\) 0 0
\(619\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(622\) 1.00000 1.00000
\(623\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(624\) −1.00000 −1.00000
\(625\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 0.500000 0.866025i 0.500000 0.866025i
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(633\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(634\) 0 0
\(635\) 0 0
\(636\) −1.00000 −1.00000
\(637\) −1.00000 −1.00000
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(642\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(643\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(644\) 2.00000 2.00000
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(648\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(649\) 0 0
\(650\) −1.00000 −1.00000
\(651\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(652\) −2.00000 −2.00000
\(653\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(662\) 0 0
\(663\) 0.500000 0.866025i 0.500000 0.866025i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −0.500000 0.866025i −0.500000 0.866025i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0.500000 0.866025i 0.500000 0.866025i
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(676\) 0 0
\(677\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(682\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(683\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0.500000 0.866025i 0.500000 0.866025i
\(687\) 2.00000 2.00000
\(688\) 0 0
\(689\) −0.500000 0.866025i −0.500000 0.866025i
\(690\) 0 0
\(691\) 1.00000 1.73205i 1.00000 1.73205i 0.500000 0.866025i \(-0.333333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −2.00000 −2.00000
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(699\) 0 0
\(700\) 0.500000 0.866025i 0.500000 0.866025i
\(701\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(702\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(703\) 0 0
\(704\) −0.500000 0.866025i −0.500000 0.866025i
\(705\) 0 0
\(706\) −1.00000 −1.00000
\(707\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(708\) 0 0
\(709\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0.500000 0.866025i 0.500000 0.866025i
\(713\) 4.00000 4.00000
\(714\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 1.00000 1.00000
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 1.00000 1.00000
\(729\) 1.00000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(734\) 1.00000 1.00000
\(735\) 0 0
\(736\) −2.00000 −2.00000
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 1.00000 1.00000
\(743\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(744\) 1.00000 1.73205i 1.00000 1.73205i
\(745\) 0 0
\(746\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(747\) 0 0
\(748\) 1.00000 1.00000
\(749\) 0.500000 0.866025i 0.500000 0.866025i
\(750\) 0 0
\(751\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) −0.500000 0.866025i −0.500000 0.866025i
\(757\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(758\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(759\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(760\) 0 0
\(761\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(762\) 0 0
\(763\) 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 \(0.666667\pi\)
\(770\) 0 0
\(771\) −1.00000 −1.00000
\(772\) 0 0
\(773\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(774\) 0 0
\(775\) 1.00000 1.73205i 1.00000 1.73205i
\(776\) 0 0
\(777\) 0 0
\(778\) −1.00000 −1.00000
\(779\) 0 0
\(780\) 0 0
\(781\) −0.500000 0.866025i −0.500000 0.866025i
\(782\) 1.00000 1.73205i 1.00000 1.73205i
\(783\) 0 0
\(784\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(785\) 0 0
\(786\) 1.00000 1.73205i 1.00000 1.73205i
\(787\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(797\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 2.00000 2.00000
\(807\) 0 0
\(808\) −1.00000 1.73205i −1.00000 1.73205i
\(809\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(810\) 0 0
\(811\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) −0.500000 0.866025i −0.500000 0.866025i
\(817\) 0 0
\(818\) −1.00000 −1.00000
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(822\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(823\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 1.00000 1.00000
\(826\) 0 0
\(827\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(828\) 0 0
\(829\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −1.00000 −1.00000
\(833\) −0.500000 0.866025i −0.500000 0.866025i
\(834\) 1.00000 1.00000
\(835\) 0 0
\(836\) 0 0
\(837\) −1.00000 1.73205i −1.00000 1.73205i
\(838\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(839\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 1.00000 1.00000
\(842\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(843\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(844\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) −1.00000 −1.00000
\(849\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(850\) −0.500000 0.866025i −0.500000 0.866025i
\(851\) 0 0
\(852\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(853\) 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 \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(858\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(859\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 1.00000 1.00000
\(863\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(864\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(865\) 0 0
\(866\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(867\) 1.00000 1.00000
\(868\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(869\) 1.00000 1.00000
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(878\) −0.500000 0.866025i −0.500000 0.866025i
\(879\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0.500000 0.866025i 0.500000 0.866025i
\(885\) 0 0
\(886\) 0 0
\(887\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0.500000 0.866025i 0.500000 0.866025i
\(892\) 0 0
\(893\) 0 0
\(894\) 0.500000 0.866025i 0.500000 0.866025i
\(895\) 0 0
\(896\) 0.500000 0.866025i 0.500000 0.866025i
\(897\) 2.00000 2.00000
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0.500000 0.866025i 0.500000 0.866025i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(908\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(909\) 0 0
\(910\) 0 0
\(911\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −1.00000 1.73205i −1.00000 1.73205i
\(915\) 0 0
\(916\) 2.00000 2.00000
\(917\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(918\) −1.00000 −1.00000
\(919\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0.500000 0.866025i 0.500000 0.866025i
\(923\) −1.00000 −1.00000
\(924\) −1.00000 −1.00000
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(942\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(948\) −0.500000 0.866025i −0.500000 0.866025i
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(953\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) −2.00000 −2.00000
\(959\) −0.500000 0.866025i −0.500000 0.866025i
\(960\) 0 0
\(961\) −1.50000 + 2.59808i −1.50000 + 2.59808i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(972\) 0 0
\(973\) −1.00000 −1.00000
\(974\) 1.00000 1.00000
\(975\) 0.500000 0.866025i 0.500000 0.866025i
\(976\) 0 0
\(977\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(978\) 1.00000 1.73205i 1.00000 1.73205i
\(979\) −1.00000 −1.00000
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(992\) 1.00000 1.73205i 1.00000 1.73205i
\(993\) 0 0
\(994\) 0.500000 0.866025i 0.500000 0.866025i
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(998\) −0.500000 0.866025i −0.500000 0.866025i
\(999\) 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 476.1.o.a.135.1 yes 2
4.3 odd 2 476.1.o.b.135.1 yes 2
7.2 even 3 3332.1.g.e.883.1 1
7.3 odd 6 3332.1.o.b.67.1 2
7.4 even 3 inner 476.1.o.a.67.1 2
7.5 odd 6 3332.1.g.c.883.1 1
7.6 odd 2 3332.1.o.b.2039.1 2
17.16 even 2 476.1.o.b.135.1 yes 2
28.3 even 6 3332.1.o.a.67.1 2
28.11 odd 6 476.1.o.b.67.1 yes 2
28.19 even 6 3332.1.g.d.883.1 1
28.23 odd 6 3332.1.g.b.883.1 1
28.27 even 2 3332.1.o.a.2039.1 2
68.67 odd 2 CM 476.1.o.a.135.1 yes 2
119.16 even 6 3332.1.g.b.883.1 1
119.33 odd 6 3332.1.g.d.883.1 1
119.67 even 6 476.1.o.b.67.1 yes 2
119.101 odd 6 3332.1.o.a.67.1 2
119.118 odd 2 3332.1.o.a.2039.1 2
476.67 odd 6 inner 476.1.o.a.67.1 2
476.135 odd 6 3332.1.g.e.883.1 1
476.271 even 6 3332.1.g.c.883.1 1
476.339 even 6 3332.1.o.b.67.1 2
476.475 even 2 3332.1.o.b.2039.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
476.1.o.a.67.1 2 7.4 even 3 inner
476.1.o.a.67.1 2 476.67 odd 6 inner
476.1.o.a.135.1 yes 2 1.1 even 1 trivial
476.1.o.a.135.1 yes 2 68.67 odd 2 CM
476.1.o.b.67.1 yes 2 28.11 odd 6
476.1.o.b.67.1 yes 2 119.67 even 6
476.1.o.b.135.1 yes 2 4.3 odd 2
476.1.o.b.135.1 yes 2 17.16 even 2
3332.1.g.b.883.1 1 28.23 odd 6
3332.1.g.b.883.1 1 119.16 even 6
3332.1.g.c.883.1 1 7.5 odd 6
3332.1.g.c.883.1 1 476.271 even 6
3332.1.g.d.883.1 1 28.19 even 6
3332.1.g.d.883.1 1 119.33 odd 6
3332.1.g.e.883.1 1 7.2 even 3
3332.1.g.e.883.1 1 476.135 odd 6
3332.1.o.a.67.1 2 28.3 even 6
3332.1.o.a.67.1 2 119.101 odd 6
3332.1.o.a.2039.1 2 28.27 even 2
3332.1.o.a.2039.1 2 119.118 odd 2
3332.1.o.b.67.1 2 7.3 odd 6
3332.1.o.b.67.1 2 476.339 even 6
3332.1.o.b.2039.1 2 7.6 odd 2
3332.1.o.b.2039.1 2 476.475 even 2